Monetary Theory and Policy - Carl E. Walsh (2017)

Monetary Theory and Policy Monetary Theory and Policy Fourth Edition Carl E. Walsh The MIT Press Cambridge, Massachusetts London, England © 20 1 7 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set in Times Roman by diacriTech, Chennai. and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Names: Walsh, Carl E. Title: Monetary theory and policy I Carl E .Walsh. Description: Fourth edition. I Cambridge, MA : MIT Press, 20 1 7 . I Includes bibliographical references and index. Identifiers: LCCN 20 1 6034549 I ISBN 97802620358 1 1 (hardcover : alk. paper) Subjects: LCSH: Monetary policy. I Money. Classification: LCC HG230.3 .W35 20 1 7 I DDC 332.4/6-dc23 LC record available at https://lccn.loc.gov/20 1 6034549 10 9 8 7 6 5 4 3 2 Contents Preface Introduction 1 Evidence on Money, Prices, and Output 1 .1 1 .2 1 .3 1 .4 1 .5 2 Introduction Some Basic Correlations Estimating the Effect of Monetary Policy on Output and Inflation 1 .3 . 1 The Evidence of Friedman and Schwartz 1 .3.2 Granger Causality 1 .3.3 Policy Uses 1 .3.4 The VAR Approach 1 .3.5 Structural Econometric Models 1 .3.6 Alternative Approaches Monetary Policy at Very Low Interest Rates 1 .4.1 Measuring Policy at the Effective Lower Bound (ELB) 1 .4.2 The Effects of Quantitative Easing (QE) Policies Summary Money-in-the-Utility Function 2.1 2.2 2.3 2.4 Introduction The Basic MIU Model 2.2. 1 Steady-State Equilibrium 2.2.2 Multiple Equilibria in Monetary Models 2.2.3 The Interest Elasticity of Money Demand 2.2.4 Limitations The Welfare Cost of Inflation Extensions Xlll XVll 1 1 1 8 9 13 14 17 25 27 31 32 33 40 41 41 43 48 55 57 61 61 66 vi Contents 2.5 2.6 2.7 2.8 3 Money and Transactions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4 2.4. 1 Interest on Money 2.4.2 Nonsuperneutrality Dynamics in an MIU Model 2.5 . 1 The Decision Problem 2.5.2 The Steady State 2.5.3 The Linear Approximation 2.5.4 Calibration 2.5.5 Simulation Results Summary Appendix: Solving the MIU Model 2.7 . 1 The Linear Approximation 2.7.2 Collecting all Equations 2.7.3 Solving Linear Rational-Expectations Models with Forward-Looking Variables Problems Introduction Resource Costs of Transacting 3.2. 1 Shopping-Time Models 3.2.2 Real Resource Costs Cash-in-Advance (CIA) Models 3 .3 . 1 The Certainty Case 3.3.2 A Stochastic CIA Model Search 3.4. 1 Centralized and Decentralized Markets 3 .4.2 The Welfare Costs of Inflation Summary Appendix: The CIA Approximation 3.6. 1 The Steady State 3.6.2 The Linear Approximation Problems Money and Public Finance 4.1 4.2 4.3 4.4 Introduction Budget Accounting 4.2. 1 Intertemporal Budget Balance Money and Fiscal Policy Frameworks Deficits and Inflation 66 67 69 70 73 74 78 80 83 83 85 90 91 94 97 97 98 98 102 103 104 113 120 122 127 129 1 30 1 30 1 30 132 1 37 1 37 138 143 144 145 vii Contents 4.5 4.6 4.7 4.8 5 Informational and Portfolio Rigidities 5.1 5.2 5.3 5.4 5.5 5.6 6 4.4. 1 Ricardian and (Traditional) Non-Ricardian Fiscal Policies 4.4.2 The Government Budget Constraint and the Nominal Rate of lnterest 4.4.3 Equilibrium Seigniorage 4.4.4 Cagan's Model 4.4.5 Rational Hyperinflation The Fiscal Theory of the Price Level 4.5 . 1 Multiple Equilibria 4.5.2 The Basic Idea of the Fiscal Theory 4.5.3 Empirical Evidence on the Fiscal Theory Optimal Taxation and Seigniorage 4.6. 1 A Partial Equilibrium Model 4.6.2 Optimal Seigniorage and Temporary Shocks 4.6.3 Friedman's Rule Revisited 4.6.4 Nonindexed Tax Systems Summary Problems Introduction Informational Frictions 5 .2. 1 Imperfect Information 5.2.2 The Lucas Model 5.2.3 Sticky Information 5.2.4 Learning Limited Participation and Liquidity Effects 5.3 . 1 A Basic Limited-Participation Model 5.3.2 Endogenous Market Segmentation 5.3.3 Assessment Summary Appendix: An Imperfect-Information Model Problems 148 151 153 157 159 162 163 164 168 169 170 173 174 1 86 188 1 89 193 193 194 194 195 200 204 206 208 212 214 215 215 219 Discretionary Policy and Time Inconsistency 221 Introduction Inflation under Discretionary Policy 6.2. 1 Policy Objectives 6.2.2 The Economy 6.2.3 Equilibrium Inflation 221 223 223 225 227 6.1 6.2 viii Contents Solutions to the Inflation Bias 6.3 . 1 Reputation 6.3.2 Preferences 6.3.3 Contracts 6.3.4 Institutions 6.3.5 Targeting Rules Is the Inflation Bias Important? Summary Problems 234 235 247 25 1 256 259 264 27 1 27 1 Nominal Price and Wage Rigidities 277 6.3 6.4 6.5 6.6 7 7.1 7.2 7.3 7.4 7.5 7.6 8 Introduction Sticky Prices and Wages 7.2. 1 An Example of Nominal Rigidities in General Equilibrium 7.2.2 Early Models of Intertemporal Nominal Adjustment 7.2.3 Imperfect Competition 7.2.4 Time-Dependent Pricing (TDP) Models 7.2.5 State-Dependent Pricing (SDP) Models 7.2.6 Frictions in the Timing of Price Adjustment or in the Adjustment of Prices? Assessing Alternatives 7.3 . 1 Microeconomic Evidence 7.3.2 Evidence on the New Keynesian Phillips Curve 7.3.3 Sticky Prices versus Sticky Information Summary Appendix: A Sticky-Wage MIU Model Problems New Keynesian Monetary Economics 8.1 8.2 8.3 Introduction The Basic Model 8.2. 1 Households 8.2.2 Firms 8.2.3 Market Clearing A Linearized New Keynesian Model 8.3 . 1 The Linearized Phillips Curve 8.3.2 The Linearized IS Curve 8.3.3 Local Uniqueness of the Equilibrium 8.3.4 The Monetary Transmission Mechanism 8.3.5 Adding Economic Disturbances 277 277 278 282 285 288 295 300 301 302 304 312 313 3 14 316 319 319 320 321 323 325 327 327 33 1 332 336 339 ix Contents 8.4 8.5 8.6 8.7 8.8 9 Monetary Policy Analysis in New Keynesian Models 8.4. 1 Policy Objectives 8.4.2 Policy Trade-offs 8.4.3 Optimal Commitment and Discretion 8.4.4 Commitment to a Rule 8.4.5 Endogenous Persistence 8.4.6 Targeting Regimes and Instrument Rules 8.4.7 Model Uncertainty Labor Market Frictions and Unemployment 8.5 . 1 Sticky Wages and Prices 8.5.2 Unemployment Summary Appendix 8.7 . 1 The New Keynesian Phillips Curve 8.7.2 Approximating Utility Problems Monetary Policy in the Open Economy 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Introduction A Two-Country Open-Economy Model 9.2. 1 Households 9.2.2 International Consumption Risk Sharing 9.2.3 Firms 9.2.4 Equilibrium 9.2.5 Optimal Policy A Model of the Small Open Economy 9.3 . 1 Households 9.3.2 International Risk Sharing and Uncovered Interest Parity 9.3.3 Domestic Firms 9.3.4 Equilibrium Conditions 9.3.5 Monetary Policy in the Linear Model Additional Sources of Nominal Distortions 9.4. 1 Imperfect Pass-Through 9.4.2 Local Currency Pricing 9.4.3 Sticky Tradeable and Nontradeable Goods Prices Currency Unions Summary Appendix Problems 341 342 344 346 353 354 358 363 365 366 369 377 378 378 381 388 397 397 398 398 401 402 404 414 424 424 428 429 430 433 437 437 440 442 442 447 448 449 Contents X 10 Financial Markets and Monetary Policy Introduction Interest Rates and Monetary Policy 10.2. 1 Interest Rate Rules and the Price Level 10.2.2 Interest Rate Policies in General Equilibrium The Term Structure of Interest Rates 10.3. 1 The Basic Expectations Theory 10.3.2 Expected Inflation and the Term Structure Macrofinance 1 0.4. 1 Affine Models of the Term Structure 10.4.2 A Preferred Habitat Term Structure Model Policy and the Term Structure 10.5 . 1 A Simple Example 10.5.2 An Affine Example Financial Frictions in Credit Markets 10.6. 1 Adverse Selection 10.6.2 Moral Hazard 10.6.3 Monitoring Costs 10.6.4 Agency Costs 10.6.5 Intermediary-to-Intermediary Credit Flows Macroeconomic Implications 10.7 . 1 General Equilibrium Models 10.7.2 Agency Costs and General Equilibrium 10.7.3 Agency Costs and Sticky Prices Summary Problems 455 456 456 459 462 463 465 467 467 469 473 473 476 478 480 482 484 488 491 495 496 501 504 505 505 The Effective Lower Bound and Balance Sheet Policies 509 Introduction The Effective Lower Bound Liquidity Traps Conventional Policies at the ELB 1 1 .4.1 Equilibria at the ELB 1 1 .4.2 Analytics at the ELB 1 1 .4.3 Commitment and Forward Guidance 1 1 .4.4 Summary on the ELB Balance Sheet Policies 1 1 .5.1 Asset Pricing Wedges 1 1 .5.2 Market Segmentation and Transaction Costs 509 510 512 515 516 518 522 53 1 532 534 538 1 0.1 1 0.2 1 0.3 1 0.4 1 0.5 1 0.6 1 0.7 1 0.8 1 0.9 11 455 1 1 .1 1 1 .2 1 1 .3 1 1 .4 1 1 .5 xi Contents 1 1 .6 1 1 .7 12 1 1 .5.3 Costly Intermediation 1 1 .5.4 Moral Hazard in Banking 1 1 .5.5 Resaleability Constraints 1 1 .5.6 Summary on Balance Sheet Policies Appendix: Derivation of the Asset Pricing Wedges Problems Monetary Policy Operating Procedures 1 2.1 1 2.2 1 2.3 1 2.4 1 2.5 1 2.6 1 2.7 1 2.8 1 2.9 Introduction From Instruments to Goals The Instrument Choice Problem 12.3 . 1 Poole's Analysis 12.3.2 Policy Rules and Information 12.3.3 Intermediate Targets 12.3.4 Real Effects of Operating Procedures Operating Procedures and Policy Measures 12.4. 1 Money Multipliers 12.4.2 The Reserve Market Interest on Reserves in a Channel System A Brief History of Fed Operating Procedures 12.6. 1 1972-1979 12.6.2 1979-1982 12.6.3 1982-1988 12.6.4 1988-2008 12.6.5 2009-2016 Other Countries Summary Problems 543 547 552 556 556 557 561 561 562 563 563 568 570 578 579 579 581 590 595 595 596 599 599 600 603 605 605 References 609 Name Index 645 Subject Index 65 1 Preface This book covers the most important topics in monetary economics and models that economists have employed to understand the interactions between real and monetary fac­ tors. It deals with topics in both monetary theory and monetary policy and is designed for second-year graduate students specializing in monetary economics, for researchers in monetary economics wishing to have a systematic summary of recent developments, and for economists working in policy institutions such as central banks. It can also be used as a supplement for first-year graduate courses in macroeconomics, because it provides a more in-depth treatment of inflation and monetary policy topics than is customary in graduate macroeconomics textbooks. The chapters on monetary policy may be useful for advanced undergraduate courses. For the fourth edition of Monetary Theory and Policy, every chapter has been revised to improve the exposition and to incorporate recent research contributions. When the first edition appeared in 1998, the use of models based on dynamic optimization and nominal rigidities in consistent general equilibrium frameworks was still relatively new. By the time of the second edition, these models had become the common workhorse for monetary pol­ icy analysis. They have continued to provide the theoretical framework for most monetary policy analysis, and they also provide the foundation for empirical models that have been estimated for a number of countries, with many central banks now employing or developing dynamic stochastic general equilibrium (DSGE) models that build on the new Keynesian model. The third edition incorporated expanded material on money in search equilibria, sticky information, adaptive learning, state-contingent pricing models, and channel sys­ tems of implementing monetary policy, among other topics. The fourth edition includes an entirely new chapter on the effective lower bound on nominal interest rates, forward guid­ ance policies, and quantitative and credit-easing policies that have been at the forefront of monetary policy discussions since the global financial crisis of 2008-2009. In addition, the material on the basic new Keynesian model has been reorganized into a single chapter to provide a comprehensive analysis of this model and its policy implications. The chapter on the open economy has been completely rewritten to reflect the dominance of the new Keynesian approach. xiv Preface In the introduction to the first edition, I cited three innovations of the book: the use of calibration and simulation techniques to evaluate the quantitative significance of the chan­ nels through which monetary policy and inflation affect the economy; a stress on the need to understand the incentives facing central banks and to model the strategic interactions between the central bank and the private sector; and a focus on interest rates in the dis­ cussion of monetary policy. All three aspects remain in the current edition, but each is now commonplace in monetary research. For example, it is rare today to see research that treats monetary policy in terms of money supply control, yet this was common well into the 1990s. Monetary economics is a large field, and one must decide whether to provide broad coverage, giving students a brief introduction to many topics, or to focus more narrowly and in more depth. I have chosen to focus on particular models, models that monetary economists have employed to address topics in theory and policy. I discuss the major topics in monetary economics in order to provide sufficiently broad coverage of the field, but the focus within each topic is often on a small number of papers or models that I have found useful for gaining insight into a particular issue. As an aid to students, derivations of basic results are often quite detailed, but deeper technical issues of existence, multiple equilibria, and stability receive somewhat less attention. This choice was not made because the latter are unimportant. Instead, the relative emphasis reflects an assessment that to do these topics justice, while still providing enough emphasis on the core insights offered by monetary economics, would have required a much longer book. By reducing the dimensionality of problems and not treating them in full generality, I sought to achieve the right balance of insight, accessibility, and rigor. The many references guide students to the extensive treatments in the literature of all the topics touched on in this book. 1 The organization of chapters 1-5 is similar to that of previous editions, although the appendix of chapter 2 gives more detail on Blanchard-Kahn conditions. The simulation results from all chapters are now done only using Dynare; the programs are available at http://people.ucsc.edu/�walshc/mtp4e/. Chapters 6-1 2 have seen major revisions in content and organization. Chapter 6 now deals with issues of time inconsistency and the average inflation bias under discretion. The workhorse model employed in that literature can be motivated by the models based on informational frictions covered in chapter 5, so it seems natural to cover this model immediately after chapter 5. Chapter 7 then discusses nominal price rigidities and provides the background for the material on the new Keynesian model that begins in chapter 8. Chapter 8 on the new Keynesian model includes a new section on labor market rigidities that contains new material on search and matching labor market frictions and unemploy­ ment. The derivation in the chapter appendix of the quadratic approximate to the welfare of the representative household has been rewritten to more closely parallel the model used 1 . A BibTex file containing all references cited in the fourth edition is available at http://people.ucsc.edu /�walshc/mtp4e/. Preface XV in the chapter. Chapter 9 covers open-economy models and has been completely rewritten. Previous editions began with the Obstfeld-Rogoff model. This material has been cut but is available online. The chapter now begins with a two-country new Keynesian model, then moves to a model of a small open economy. These models assume sticky prices are the only nominal frictions. The implications of imperfect pass-through, local currency pricing, and sticky tradeable and nontradeable goods prices as additional sources of nominal fric­ tions are discussed. The chapter ends with a model of a currency union. Chapter 10 has been expanded to discuss the role of intermediary-to-intermediary financial frictions based on moral hazard. An important addition is a new chapter 1 1 that focuses on the effective lower bound (ELB) on nominal interest rates and on balance sheet policies. The chapter opens with a discussion of why standard models imply the nominal interest rate cannot be negative in equilibrium and discusses one modification to the basic model that could account for negative rates. The discussion then moves to the existence of a liquidity trap when policy follows a Taylor rule, material that was in chapter 10 of the third edition but which fits better now in a chapter devoted to ELB issues. Optimal interest rate policy at the ELB is discussed, as is forward guidance about the future path of interest rates. The existence of multiple equilibria at the ELB is emphasized, and it is shown that the economy may suffer deflation and depressed output while at the ELB or it may experience a boom, depending on the central bank's commitment to inflation in the post-ELB period. The chapter ends with a review of several models of balance sheet policies. Finally, chapter 12 provides a discussion of operating procedures and channel systems. It is not possible to discuss here all the areas of monetary economics in which economists are pursuing active research, or to give adequate credit to all the interesting work that has been done. The topics covered and the space devoted to them reflect my own biases toward research motivated by policy questions or influential in affecting the conduct of monetary policy. The field has simply exploded with new and interesting research, much of it motivated by the financial crisis of 2008-2009, the Great Recession, the limits on policy due to the ELB, and the active use of balance sheet policies in ways not seen during the 40 years prior to 2008. At best, this edition, like the earlier ones, can only scratch the surface of many topics. To those whose research has been slighted, I offer my apologies. I am grateful to all those who have read and commented on drafts of the various editions, and some deserve special mention. Lars Svensson and Berthold Herrendorf each made extensive comments on complete drafts of the first edition. Henning Bohn, Betty Daniel, Jordi Galf, Eric Leeper, Tim Fuerst, Ed Nelson, Federico Ravenna, and Kevin Salyer pro­ vided very helpful comments on early draft versions of some of the chapters of the second edition. Henrik Jensen provided a host of useful suggestions that helped improve the third edition in terms of substance and clarity. Federico Ravenna and Chris Limnios commented on material new to the fourth edition. Addressing the issues raised greatly improved each edition. xvi Preface I have received many useful comments from users that have guided this and previous revisions. My thanks go to Jonathan Benchimol, Luigi Buttiglione, Julia Chiriaeva, Vasco Curdia, David Coble Fernandez, Oliver Fries, William Gatt, Federico Guerrero, Basil Halperin, Marco Hoeberichts, Stefan Homburg, Michael Hutchison, Nancy Jianakoplos, Beka Lamazoshvili, Sendor Lczel, Jaewoo Lee, Haroan Lei, Francesco Lippi, Carlo Migliardo, Stephen Miller, Rasim Mutlu, Jim Nason, Mario Nigrinis, Doug Pearce, Xingyun Peng (who translated the third edition into Chinese), Gustavo Piga, Alvaro Pina, Glenn Rudebusch, Bo Sandemann, Stephen Sauer, Claudio Shikida, Teresa Simoes, Paul Soderlind, Ulf Soderstrom, Robert Tchaidze, Oreste Tristani, Willem Verhagen, Yuichiro Waki, Chris Waller, Ken West, and Jizhong Zhou (who translated the second edition into Chinese). My apologies to anyone I have failed to mention. Numerous graduate students at the University of California, Santa Cruz, have offered helpful comments and assistance on the various editions of this book. They include Alina Carare, Cesar Carrera, Wei Chen, David Florian-Hoyle, Peter Kriz, Sergio Lago Alves, Jamus Lim, Chris Limnios, Jerry Mcintyre, David Munro, Akatsuki Sukeda, and Ethel Wang. Jules Leichter and Conglin Xu deserve special mention for providing excellent research assistance during the process of preparing earlier editions. The first edition was based on lecture notes developed when I taught in the first-year macroeconomics sequence at Stanford; feedback from students in that course and, in particular, Fabiano Schivardi, was most helpful. Many of the changes in the book are the result of comments and suggestion from students and participants at intensive courses and lectures in monetary economics I have taught at the IMF Institute, the Bank of England, the Bank of Korea, the Bank of Portugal, the Bank of Spain, the Central Bank of Brazil, the Federal Reserve Bank of Philadelphia, the Finnish Post-Graduate Program in Economics, the Hong Kong Institute for Monetary Research, the Norges Bank Training Program for Economists, the Swiss National Bank Studienzentrum Gerzensee, the University of Oslo, the University of Rome "Tor Vergata," and the ZEI Summer School. As always, remaining errors are my own. I would also like to thank Jane MacDonald and Emily Taber, who have been my editors at MIT Press for the third and fourth editions; Nancy Lombardi, the production editor for the first and second editions; Deborah Cantor-Adams, production editor for the third edition; Virginia Crossman, production editor for the fourth edition; and Alice Cheyer, copy editor on the third and fourth editions, for their excellent assistance on the manuscript. Needless to say, all the remaining weaknesses and errors are my own responsibility. Terry Vaughan, my original editor at MIT Press, was instrumental in ensuring this project got off the ground initially, and Elizabeth Murry served ably as editor for the second edition. I owe an enormous debt to my wife, Judy Walsh, for all her support, encouragement, and assistance on this fourth edition. Judy carefully read every chapter, editing my writing and improving the exposition. Introduction Monetary economics investigates the relationship between real economic variables at the aggregate level (such as real output, real rates of interest, employment, and real exchange rates) and nominal variables (such as the inflation rate, nominal interest rates, nominal exchange rates, and the supply of money). So defined, monetary economics overlaps con­ siderably with macroeconomics more generally, and these two fields have to a large degree shared a common history over most of the past 50 years. This statement was particularly true during the 1970s after the monetarist/Keynesian debates led to a reintegration of mone­ tary economics with macroeconomics. The seminal work of Lucas (1972) provided theoret­ ical foundations for models of economic fluctuations in which money was the fundamental driving factor behind movements in real output. The rise of real business cycle models during the 1980s and early 1990s, building on the contribution of Kydland and Prescott (1982) and focusing explicitly on nonmonetary factors as the driving forces behind busi­ ness cycles, tended to separate monetary economics from macroeconomics. More recently, the real business cycle approach to aggregate modeling has been used to incorporate mon­ etary factors into dynamic general equilibrium models. Today, macroeconomics and mon­ etary economics share the common tools associated with dynamic stochastic approaches to modeling the aggregate economy. Despite these close connections, a book on monetary economics is not a book on macroeconomics. The focus in monetary economics is distinct, emphasizing price level determination, inflation, and the role of monetary policy. Monetary economics is currently dominated by three alternative modeling strategies. The first two, representative agent models and overlapping-generations models, share a common methodological approach in building equilibrium relationships explicitly on the foundations of optimizing behavior by individual agents. The third approach is based on sets of equilibrium relationships that are often not derived directly from any decision problem. Instead, they are described as ad hoc by critics and as convenient approximations by proponents. The latter characterization is generally more appropriate, and these models have demonstrated great value in help­ ing economists understand issues in monetary economics. This book deals with models in the representative agent class and with ad hoc models of the type more common in policy analysis. xviii Introduction There are several reasons for ignoring the overlapping-generations (OLG) approach. First, systematic expositions of monetary economics from the perspective of overlapping generations are already available. For example, Sargent (1987) and Champ, Freeman, and Haslag (2016) cover many topics in monetary economics using OLG models. Second, many of the issues studied in monetary economics require understanding the time series behavior of macroeconomic variables such as inflation or the relationship between money and business cycles. It is helpful if the theoretical framework can be mapped directly into implications for behavior that can be compared with actual data. This mapping is more easily done with infinite-horizon representative agent models than with OLG models. This advantage, in fact, is one reason for the popularity of real business cycle models that employ the representative agent approach, and so a third reason for limiting the coverage to representative agent models is that they provide a close link between monetary economics and other popular frameworks for studying business cycle phenomena. Fourth, monetary policy issues are generally related to the dynamic behavior of the economy over time peri­ ods associated with business cycle frequencies, and here again the OLG framework seems less directly applicable. Finally, OLG models emphasize the store-of-value role of money at the expense of the medium-of-exchange role that money plays in facilitating transac­ tions. McCallum ( 1983b) has argued that some of the implications of OLG models that contrast most sharply with the implications of other approaches (the tenuousness of mone­ tary equilibria, for example) are directly related to the lack of a medium-of-exchange role for money. This book is about monetary theory and the theory of monetary policy. There are some references to empirical results, but no attempt is made to provide a systematic survey of the vast body of empirical research in monetary economics. Most of the debates in monetary economics, however, have at their root issues of fact that can only be resolved by empir­ ical evidence. Empirical evidence is needed to choose among theoretical approaches, but theory is also needed to interpret empirical evidence. How one links the quantities in the theoretical model to measurable data is critical, for example, in developing measures of monetary policy actions that can be used to estimate the impact of policy on the economy. Because empirical evidence aids in discriminating between alternative theories, it is helpful to begin with a brief overview of some basic facts. Chapter 1 does so, focusing primarily on the estimated impact of monetary policy actions on real output. Here, as in the chapters that deal with institutional details of monetary policy, the evidence comes primarily from research on U.S. data. However, an attempt is made to cite cross-country studies and to focus on empirical regularities that seem to characterize most industrialized economies. Chapters 2-4 emphasize the role of inflation as a tax, using models that provide the basic microeconomic foundations of monetary economics. These chapters cover topics of funda­ mental importance for understanding how monetary phenomena affect the general equilib­ rium behavior of the economy and how nominal prices, inflation, money, and interest rates are linked. Because the models studied in these chapters assume that prices are perfectly Introduction xix flexible, they are most useful for understanding longer-run correlations between inflation, money, and output and cross-country differences in average inflation. However, they do have implications for short-run dynamics as real and nominal variables adjust in response to aggregate productivity disturbances and random shocks to money growth. These dynam­ ics are examined by employing simulations based on linear approximations around the steady-state equilibrium. Chapters 2 and 3 employ a neoclassical growth framework to study monetary phenom­ ena. The neoclassical model is one in which growth is exogenous, and money either has no effect on the real economy's long-run steady state or has effects that are likely to be small empirically. However, because these models allow one to calculate the welfare implications of exogenous changes in the economic environment, they provide a natural framework for examining the welfare costs of alternative steady-state rates of inflation. Stochastic versions of the basic models are calibrated, and simulations are used to illustrate how monetary fac­ tors affect the behavior of the economy. Such simulations aid in assessing the ability of the models to capture correlations observed in actual data. Since policy can be expressed in terms of both exogenous shocks and endogenous feedbacks from real shocks, the models can be used to study how economic fluctuations depend on monetary policy. In chapter 4 the focus turns to public finance issues associated with money, inflation, and monetary policy. The ability to create money provides governments with a means of generating revenue. As a source of revenue, money creation, along with the inflation that results, can be analyzed from the perspective of public finance as one among many tax tools available to governments. The link between the dynamic general equilibrium models of chapters 2-4 and the mod­ els employed for short-run and policy analysis is developed in two stages. In the first stage, chapter 5 reviews some attempts to understand the short-run effects of monetary policy shocks while still maintaining the assumption of flexible prices. Lucas's misperceptions model provides an important example of one such attempt. Models of sticky information with flexible prices due to the work of Mankiw and Reis provide a modern approach that can be thought of as building on Lucas's original insight that imperfect information is important for understanding the short-run effects of monetary shocks. Chapter 6 turns to the analysis of monetary policy using a model due to Barro and Gordon ( 1983a) that can be motivated by the information frictions discussed in chapter 5. The focus in chapter 6 is on monetary policy objectives and the ability of policy authorities to achieve these objectives. Understanding monetary policy requires an understanding of how policy actions affect macroeconomic variables, but it also requires models of policy behavior to understand why particular policies are undertaken. A large body of research over the past three decades has used game-theoretic concepts to model the monetary pol­ icymaker as a strategic agent. These models have provided new insights into the rules­ versus-discretion debate, positive theories of inflation, and justification for many of the actual reforms of central banking legislation that have been implemented in recent years. XX Introduction Despite the growing research on sticky information and on models with portfolio rigidi­ ties (see chapter 5), it remains the case that the most research in monetary economics in recent years has assumed that prices and/or wages adjust sluggishly in response to eco­ nomic disturbances. Chapter 7 discusses some important models of price and inflation adjustment, and reviews some of the new microeconomic evidence on price adjustment by firms. This evidence is helping to guide research on nominal rigidities and has renewed interest in models of state-contingent pricing. Models of sticky prices in dynamic stochastic general equilibrium form the foundation of the new Keynesian models that over the past two decades have become the standard models for monetary policy analysis. These models build on the joint foundations of optimizing behavior by economic agents and nominal rigidities, and they form the core material of chapter 8. The basic new Keynesian model is covered, and some of its policy implications are explored. Chapter 9 extends the analysis to the open economy by focusing on two questions. First, what additional channels from monetary policy actions to the real economy are present in the open economy that were absent in the closed-economy analysis? Second, how do conclusions about monetary policy obtained in the context of a closed economy need to be modified when open-economy considerations are included? There is a long tradition of treating the money stock or even the inflation rate as the direct instrument of monetary policy. In fact, major central banks have employed interest rates as their operational policy instrument, so chapter lO emphasizes explicitly the role of the interest rate as the instrument of monetary policy and the term structure that link policy rates to long-term interest rates. While the channels of monetary policy emphasized in traditional models operate primarily through interest rates and exchange rates, an alter­ native view is that credit markets play an independent role in affecting the transmission of monetary policy actions to the real economy. The nature of credit markets and their role in the transmission process are affected by market imperfections arising from imperfect information, so chapter lO also examines theories that stress the role of credit and credit market imperfections in the presence of moral hazard, adverse selection, and costly moni­ toring. Much of the literature has focused on financial frictions between firms (borrowing to finance capital projects) and lenders (financial intermediaries). A model of credit fric­ tions that affect the flow of funds among financial intermediaries is also discussed. Models that assume the central bank implements policy through its control over a short­ term interest rate are useful in normal times when the policymaker can raise or lower the policy rate. However, when short-term rates are constrained by a lower bound, and rates have hit that bound and can no longer be reduced, new issues arise. These issues are the subject of chapter 1 1 . Among the topics covered are liquidity traps, the role of future com­ mitments when the policy rate is at its effective lower bound, and the importance of wedges in the standard asset-pricing formula if the composition of the central bank's balance sheet is to matter. A number of recent models of balance sheet policies are discussed. These Introduction xxi models are based on frictions due to transaction costs, moral hazard, and limitations on asset sales. Chapter 12 focuses on monetary policy implementation. Here the discussion deals with the monetary instrument choice problem and monetary policy operating procedures. A long tradition in monetary economics has debated the usefulness of monetary aggregates versus interest rates in the design and implementation of monetary policy, and chapter 12 reviews the approach economists have used to address this issue. A simple model of the market for bank reserves is used to stress how the observed responses of short-term interest rates and reserve aggregates depend on the operating procedures used in the conduct of policy. New material on channel systems for interest rate control is added in this edition. A basic understanding of policy implementation is important for empirical studies that attempt to measure changes in monetary policy. Monetary Theory and Policy 1 1.1 Evidence on Money, Prices, and Output Introduction This chapter reviews some of the basic empirical evidence on money, nominal interest rates, inflation, and output. This review serves two purposes. First, these basic results about long-run and short-run relationships are benchmarks for judging theoretical models. Sec­ ond, reviewing the empirical evidence provides an opportunity to discuss the approaches monetary economists have taken to estimate the effects of money and monetary policy on real economic activity. The discussion focuses heavily on evidence from vector autore­ gressions (VARs) because these have served as a primary tool for uncovering the impact of monetary phenomena on the real economy. The findings obtained from VARs have been criticized, and these criticisms as well as other methods that have been used to investigate the money-output relationship are also discussed. 1.2 Some Basic Correlations What are the basic empirical regularities that monetary economics must explain? Monetary economics focuses on the behavior of prices, monetary aggregates, nominal and real inter­ est rates, and output, so a useful starting point is to summarize briefly what macroeconomic data tell us about the relationships among these variables. McCandless Jr. and Weber (1995) provided a summary of the long-run relationships based on data for inflation, the output gap, and the growth rate of various measures of money covering a 30-year period from 1 10 countries using several definitions of money. By examining data from many time periods and countries, they provided evidence on rela­ tionships that are unlikely to be dependent on unique, country-specific events (such as the particular means employed to implement monetary policy) that might influence the actual evolution of money, prices, and output in a particular country. The first of two primary conclusions that emerged from their analysis was that the correlation between inflation and the growth rate of the money supply is almost 1 , varying between 0.92 and 0.96, depend­ ing on the definition of the money supply used. This strong positive relationship between 2 Chapter 1 inflation and money growth is consistent with many other studies based on smaller sam­ ples of countries and different time periods. 1 This correlation is normally taken to support one of the basic tenets of the quantity theory of money: a change in the growth rate of money induces "an equal change in the rate of price inflation" (Lucas 1980b, 1005). Using U.S. data from 1955 to 1975, Lucas plotted annual inflation against the annual growth rate of money. While the scatter plot suggests only a loose but positive relationship between inflation and money growth, a much stronger relationship emerged when Lucas filtered the data to remove short-run volatility. Berentsen, Menzio, and Wright (201 1) repeated Lucas's exercise using data from 1955 to 2005, and like Lucas, they found a strong corre­ lation between inflation and money growth as they removed more and more of the short-run fluctuations in the two variables. 2 This high correlation between inflation and money growth does not, however, have any implication for causality. If countries followed policies under which money supply growth rates were exogenously determined, then the correlation could be taken as evidence that money growth causes inflation, with an almost one-to-one relationship between the two. An alternative possibility, equally consistent with the high correlation, is that other factors generate inflation, and central banks allow the growth rate of money to adjust. Most of the models examined in this book are consistent with a one-to-one long-run relationship between money growth and inflation. 3 Money growth is not an exogenous variable; it depends on the actions of the central bank as well as the actions of the private sector. Inflation is also not exogenous. Because both money growth and inflation are endogenous variables, the correlation between the two depends on the types of disturbances affecting the economy as well as on changes in policy. Sargent and Surico (20 1 1 ) emphasized that the strong relationship between money growth and inflation that Lucas found in the filtered data for 1955-1975 does not characterize other periods of U.S. history. They found that the regression of inflation on money growth yielded a coefficient of 1 .01 for the 1960-1983 sample period when they first filtered out short-run volatility in the data. However, for 1 984-2005, the regression coefficient was essentially equal to 0 (the point estimate was -0.03). They attributed this changing rela­ tionship to shifts in U.S. monetary policy. Their interpretation is that the close association of money growth and inflation found by Lucas is likely to occur during periods in which the monetary authority has allowed persistent movements in money growth and fails to respond sufficiently to offset movements in inflation. The association breaks down when the monetary authority responds more strongly to inflation, leading to more stable inflation. 1 . Examples include Lucas ( 1 980b), Geweke ( 1 986), and Rolnick and Weber ( 1 997), among others. 2. Berentsen, Menzio, and Wright (20 1 1 ) employed an HP filter and progressively increased the smoothing parameter from 0 to 1 60,000. 3. Haldane ( 1 997) found, however, that the money growth rate-inflation correlation is much less than 1 among low-inflation countries. Evidence on Money, Prices, and Output 3 15 c 0 � 'E 10 5 0 -5 0 0 0 <0 10 II 5 c... I c 0 � 'E 0 0 -5 5 0 20 15 10 M2 growth 25 0 -5 1 2 3 4 5 M2 growth : 6 HP = 7 16000 8 9 10 Figure 1.1 Upper: Quarterly inflation (GDP deflator) versus M2 quarterly growth rate (both at annual rate). Lower: Filtered inflation and M2 growth using HP smoothing parameter of 16,000 .45 ° line adjusted by mean annual rate of real GDP growth of 3.0 2percent for 1 960 :1- 20 1 5 :4 . Dotted line in lower panel is fitted regression line. The relationship between money growth and inflation in the United States for the period 1960: l-20 15:4 is illustrated in figure 1 . 1 . The upper panel is a scatter plot of money growth on the horizontal axis, measured by the quarterly growth rate of the M2 measure of the money stock (expressed at an annual rate) and the quarterly inflation rate (at annual rates) measured by the GDP price deflator. The models investigated in chapters 2 and 3 imply inflation should equal the rate of money growth minus the rate of growth of real output. That is, on average the inflation points should lie along a 45 degree line with negative intercept equal to the average growth rate of real output (solid line in figure 1 . 1). The lower panel of the figure plots the same two variables after they have been filtered to remove much of the short-run volatility from each series. This is done using a Hodrick-Prescott (HP) filter with a smoothing weight of 1 6,000. The upper panel shows a weak relationship between money growth and inflation; the contemporaneous correlation between the two is 0.2 1 . The lower panel, however, reveals a very positive relationship for the filtered data; the contemporaneous correlation is 0.66. The slope of the regression line in the lower panel is close to the adjusted 45 degree line, suggesting a one-to-one relationship between growth rate of money and inflation. Figure 1 .2 presents the same variables for the 1985: 1-2006:4 period, an era sometimes referred to as the Great Moderation because macroeconomic volatility was much lower than it had been earlier (notice the differences in the inflation scales in figures 1 . 1 and 1 .2). Chapter 1 4 6 0 O L-----�-----L--�--L--� -2 6 "iii � 0 2 4 6 8 M2 growth 10 12 0 L_--�_____L_____L____�----�----L---�L---� - -2 2 1 8 3 4 5 6 7 9 M2 growth : HP = 16000 Figure 1.2 Upper: Quarterly inflation (GDP deflator) versus M2 quarterly growth rate (both at annual rate). Lower: Filtered inflation and M2 growth using HP smoothing parameter of 16,000. 45 ° line adjusted by mean annual rate of real GOP growth of 2 .5 9 percent for 1985 : 1-2006:4 .Dotted line in lower panel is fitted regression line . The correlation between the data in the upper panel of figure 1 .2 is actually negative ( -0. 15); money growth varied significantly over the period while inflation remained within a narrow band. The lower panel shows the filtered data. There is little relationship to infla­ tion. In fact, the correlation between money growth and inflation in the smoothed data is -0.05. Later chapters discuss the conduct of monetary policy. Changes in the conduct of policy over the two periods shown in these figures are important in accounting for the changing relationship between money growth and inflation, the point made by Sargent and Surico (201 1). Models in monetary economics imply that nominal interest rates and inflation should tend to move together one-to-one. Figure 1 .3 presents the data on the federal funds interest rate and inflation for 1960: 1-2015:4. The funds rate plays an important role in monetary policy in the United States. A strong positive relationship between this interest rate and inflation shows up in both the quarterly and the smoothed data. The solid line in the lower panel is the 45° degree line with an intercept equal to 2 percent to adjust for the average real interest rate (see chapter 2). The regression line is essentially on top of this 45° degree line. Figure 1 .4 shows the funds rate and inflation data for 1985: 1-2015:4, a period that includes the Great Moderation between 1985 and 2007 and the Great Recession of 2008-2009. The upper panel shows the correlation between the funds rate and inflation is much weaker than that seen in figure 1 .3. In addition, the regression line in the lower Evidence on Money, Prices, and Output 20 � (f) "0 c :::> LL. 10 II c... I . . § Q) (f) "0 c :::> LL. 0 5 0 0 0 0 <D oo oo 15 5 -2 2 0 4 6 Inflation ooo 0 0 0 0 0 8 10 12 14 5 6 7 8 15 10 5 0 -5 2 0 3 Inflation : 4 HP = 16000 Figure 1.3 Upper: Federal funds interest rate versus inflation (GOP deflator) .Lower: Filtered funds rate and inflation using HP smoothing parameter of 1 6,000 . 45 ° line adjusted by 2 .0 percent as an estimate of the real interest rate for 1 960: 1- 20 1 5 :4. Dotted line in lower panel is fitted regression line. 10 2 � (f) "0 c :::> LL. 5 0 0 0 Oo o 0 o o 0 0@ 0 0 0 0 ��oC:So ocfJO o 0 ooo c9 0 0 o8 0 0 0 000 00 o0o oo 0 <§> II c... I " § Q) (f) "0 c :::> LL. 0 co 0 0.5 0 0 0 <D c§J Cb o0o 0 0 0 00 2 1 .5 2.5 Inflation 3.5 3 4.5 4 5 0 10 8 6 4 2 1 .4 Figure 1.4 1 .6 1 .8 2 2.2 Inflation 2.4 : HP = 2.6 16000 2.8 3 3.2 3.4 Upper: Federal funds interest rate versus inflation (GOP deflator) .Lower: Filtered funds rate and inflation using HP smoothing parameter of 1 6,000 . 45 ° line adjusted by 2 .0 percent as an estimate of the real interest rate for 1 985: 1-2006:4 .Dotted line in lower panel is fitted regression line . 6 Chapter 1 panel now has a slope that is greater than one. If regression line is interpreted as reflecting the reaction of policy to inflation, a slope that is greater than one suggests the Fed increased the funds rate more than one-for-one in response to changes in inflation. The appropriate interpretation of money-inflation correlations, both in terms of causality and in terms of tests of long-run relationships, also depends on the statistical properties of the underlying series. As Fischer and Seater (1993) noted, one cannot ask how a perma­ nent change in the growth rate of money affects inflation unless actual money growth has exhibited permanent shifts. They showed how the order of integration of money and prices influences the testing of hypotheses about the long-run relationship between money growth and inflation. In a similar vein, McCallum ( 1984b) demonstrated that regression-based tests of long-run relationships in monetary economics may be misleading when expectational relationships are involved. The second general conclusion that emerged from McCandless and Weber's (1995) work was that there is no correlation between either inflation or money growth and the growth rate of real output. Thus, there are countries with low output growth and low money growth and inflation, countries with low output growth and high money growth and inflation, and countries with every other combination as well. Figure 1 .5 illustrates the lack of correlation between inflation and real GDP growth for the United States over the 1960: 1-2015:4 period. This conclusion is not as robust as the money growth-inflation one; McCandless 20 � .r: 10 e (!) I a.. (!) 0 0 0 -10 0 0 g 0 0 0 -2 v0 00 oo 0 0 0 0 0 2 4 0 6 o o 0 '60 0 0 0 Inflation 0 0 8 10 12 14 5 6 7 8 6 a.. I 4 � .r: 2 (!) I a.. o § 0 Figure 1.5 2 3 I nflation : 4 HP = 16000 Real GDP growth rate versus quarterly inflation rate (GDP deflator) (both at annual rate) for 1 960: 1-20 1 5 :4. Lower: Filtered real GDP growth rate and inflation using HP smoothing parameter of 1 6,000. Dot­ ted line in lower panel is fitted regression line. Upper: Evidence on Money, Prices, and Output 7 and Weber reported a positive correlation between real growth and money growth, but not inflation, for a subsample of OECD countries. Kormendi and Meguire (1984) for a sample of almost 50 countries and Geweke (1986) for the United States argued that the data reveal no long-run effect of money growth on real output growth. Barro (1995; 1996) reported a negative correlation between inflation and growth in a cross-country sample. Bullard and Keating (1995) examined post-World War II data from 58 countries, concluding for the sample as a whole that the evidence that permanent shifts in inflation produce permanent effects on the level of output is weak, with some evidence of positive effects of inflation on output among low-inflation countries and zero or negative effects for higher-inflation coun­ tries. Similarly, Boschen and Mills (1995a) concluded that permanent monetary shocks in the United States made no contribution to permanent shifts in GDP, a result consistent with the findings of King and Watson (1997). Bullard (1999) surveyed much of the existing empirical work on the long-run relation­ ship between money growth and real output, discussing both methodological issues asso­ ciated with testing for such a relationship and the results of a large literature. Specifically, while shocks to the level of the money supply do not appear to have long-run effects on real output, this is not the case with respect to shocks to money growth. For example, the evidence based on postwar U.S. data reported in King and Watson (1997) is consistent with an effect of money growth on real output. Bullard and Keating (1995) did not find any real effects of permanent inflation shocks with a cross-country analysis, but Berentsen, Menzio, and Wright (201 1), using the same filtering approach described earlier, argued that inflation and unemployment are positively related in the long run. A positive correlation between inflation and unemployment characterizes the Great Moderation period 1985: 1-2006:4, as seen in figure 1 .6. However, despite this diversity of empirical findings concerning the long-run relation­ ship between inflation and real growth, and other measures of real economic activity such as unemployment, the general consensus is summarized by the proposition, "about which there is now little disagreement . . . that there is no long-run trade-off between the rate of inflation and the rate of unemployment" (Taylor 1996, 1 86). Monetary economics is also concerned with the relationship between interest rates, infla­ tion, and money. According to the Fisher equation, the nominal interest rate equals the real return plus the expected rate of inflation. If real returns are independent of inflation, then nominal interest rates should be positively related to expected inflation. This relationship is an implication of the theoretical models discussed throughout this book. In terms of long-run correlations, it suggests that the level of nominal interest rates should be pos­ itively correlated with average rates of inflation. Because average rates of inflation are positively correlated with average money growth rates, nominal interest rates and money growth rates should also be positively correlated. Monnet and Weber (2001) examined annual average interest rates and money growth rates over the period 1961-1998 for a sam­ ple of 3 1 countries. They found a correlation of 0.87 between money growth and long-term 8 Chapter 1 8 "§ 6 0 0 6b 0 Q) c. E � 4 ::::> o a. I Qj � _: c. Inflation 7 o ooo 6 5 oo o 0 0 o� o 0 o � ... .. .. . . .. . o0 0 0 oo o o o o ......... ... .. . . . ..� �;; o o o oo o?:ac ···· ····· ··· e· o . ..J.._ ___/. _J_ _ ._ ._ _, __J___...._ _ _ __ ___.___.J..._ __,_____, � 4 c_ 1 .4 E 0 2 L----L--�L----�---L--� 4.5 2 5 3 4 2.5 1 .5 3.5 0.5 0 0 0 8 CD II 0 ::::> 1 .6 1 .8 2 2.2 Inflation 2.4 : HP = 2.6 1 6000 2.8 3 3.2 3.4 Figure 1.6 Upper: Unemployment rate versus quarterly inflation rate (GDP deflator) for 1 985 : 1-2006:4. Lower: Filtered real GDP growth rate and inflation using HP smoothing parameter of 1 6,000. Dotted line in lower panel is fitted regression line . interest rates. For the developed countries, the correlation is somewhat smaller (0.70); for the developing countries, it is 0.84, although this falls to 0.66 when Venezuela is excluded. 4 This evidence is consistent with the Fisher equation. 5 1.3 Estimating the Effect of Monetary Policy on Output and Inflation While long-run effects of money may fall entirely, or almost entirely, on prices and have little impact on real variables, most economists believe that monetary disturbances can, in the short run, have important effects on real variables such as output. 6 As Lucas (1996) put it in his Nobel lecture, "This tension between two incompatible ideas-that changes in money are neutral unit changes and that they induce movements in employment and production in the same direction-has been at the center of monetary theory at least since 4. Venezuela's money growth rate averaged over 28 percent, the highest among the countries in Monnet and Weber's sample. 5. Consistent evidence on the strong positive long-run relationship between inflation and interest rates was reported by Berentsen, Menzio, and Wright (20 1 1). 6. For an exposition of the view that monetary factors have not played an important role in U.S. business cycles, see Kydland and Prescott ( 1 982). Evidence on Money, Prices, and Output 9 Hume wrote" (664). 7 The evidence of the short-run effects of money on real output comes from a variety of approaches. The tools that have been employed to estimate the impact of monetary policy have evolved over time as the result of developments in time series econometrics and changes in the specific questions posed by theoretical models. This section reviews some of the empirical evidence on the relationship between monetary policy and U.S. macroeconomic behavior. One objective of this literature has been to determine whether monetary policy disturbances actually have played an important role in U.S. economic fluctuations. Equally important, the empirical evidence is useful in judging whether the predictions of differ­ ent theories about the effects of monetary policy are consistent with the evidence. Among the excellent discussions of these issues are Leeper, Sims, and Zha (1996) and Christiano, Eichenbaum, and Evans (1999), where the focus is on the role of identified VARs in esti­ mating the effects of monetary policy; King and Watson (1996), where the focus is on using empirical evidence to distinguish among competing business cycle models; and Boivin, Kiley, and Mishkin (2010), where the focus is on the channels through which monetary shocks affect the economy. Much of the empirical literature has focused on estimating the impact of a monetary shock such as an unpredicted change in policy on other macroeco­ nomic variables. A discussion of the literature on estimating the effects of such shocks is provided by Ramey (2016). 1.3.1 The Evidence of Friedman and Schwartz M. Friedman and Schwartz's (1963) classic study of the relationship between money and business cycles still represents probably the most influential empirical evidence that money does matter for business cycle fluctuations. Their evidence, based on almost 100 years of U.S. data, relies heavily on patterns of timing; systematic evidence that money growth rate changes lead changes in real economic activity is taken to support a causal interpretation in which money causes output fluctuations. Friedman and Schwartz concluded that the data "decisively support treating the rate of change series [of the money supply] as conforming to the reference cycle positively with a long lead" (36). That is, faster money growth tends to be followed by increases in output above trend, and slowdowns in money growth tend to be followed by declines in output. The inference Friedman and Schwartz drew was that variations in money growth rates cause, with long (and variable) lags, variations in real economic activity. The nature of this evidence for the United States is apparent in figure 1 .7, which shows the logs of the M2 measure of the money supply and real GDP. Both variables are detrended using a Hodrick-Prescott filter. The sample is quarterly and spans 1960: 1 to 2015 : 1 , so this figure starts after the Friedman and Schwartz study ends. The figure reveals 7. The reference is to David Hume's 1 752 essays Of Money and Of Interest. Chapter 1 10 3 2 0 -1 -2 i f I i i i �. .fi I -3 i Reai G ii •j I -4 0 1 -60 0 1 -70 0 1 -80 0 1 -90 0 1 -00 01-10 Figure 1.7 Detrended log M2 and real GDP. Shaded regions are NBER recession dates. slowdowns in money leading most business cycle downturns through the early 1980s. How­ ever, the pattern is not so apparent after 1982. B. Friedman and Kuttner (1992) documented the seeming breakdown in the relationship between monetary aggregates and real output; this changing relationship between money and output has affected the manner in which monetary policy has been conducted, at least in the United States (see chapter 12). While suggestive, evidence based on timing patterns and simple correlations may not indicate the true causal role of money. Since the Federal Reserve and the banking sector respond to economic developments, movements in the monetary aggregates are not exoge­ nous, and the correlation patterns need not reflect any causal effect of monetary policy on economic activity. If, for example, the central bank is implementing monetary policy by controlling the value of some short-term market interest rate, the nominal stock of money will be affected both by policy actions that change interest rates and by developments in the economy that are not related to policy actions. An economic expansion may lead banks to expand lending in ways that produce an increase in the stock of money, even if the central bank has not changed its policy. If the money stock is used to measure monetary policy, the relationship observed in the data between money and output may reflect the impact of output on money, not the impact of money and monetary policy on output. Tobin (1970) was the first to model formally the idea that the positive correlation between money and output, the correlation that M. Friedman and Schwartz interpreted as providing evidence that money caused output movements, could in fact reflect just the opposite-output might be causing money. This reverse causation argument was inves­ tigated by King and Plosser ( 1984). They showed that inside money-the component of Evidence on Money, Prices, and Output 11 a monetary aggregate such as M 1 that represents the liabilities of the banking sector-is more highly correlated with output movements in the United States than is outside money, the liabilities of the Federal Reserve. King and Plosser interpreted this finding as evidence that much of the correlation between broad aggregates such as Ml or M2 and output arises from the endogenous response of the banking sector to economic disturbances that are not the result of monetary policy actions. Coleman (1996), in an estimated equilibrium model with endogenous money, found that the implied behavior of money in the model cannot match the lead-lag relationship in the data. Specifically, a money supply measure such as M2 leads output, whereas Coleman found that his model implied money should be more highly correlated with lagged output than with future output. 8 The endogeneity problem is likely to be particularly severe if the monetary authority has employed a short-term interest rate as its main policy instrument, and this has gener­ ally been the case in the United States. Changes in the money stock are then endogenous and cannot be interpreted as representing policy actions. Figure 1 .8 shows the behavior of the federal funds rate and the 5-year U.S. government bond rate, together with detrended real GDP. The figure provides some support for the notion that monetary policy actions have contributed to U.S. business cycles. Interest rates have typically increased prior to economic downturns. But whether this is evidence that monetary policy has caused or 16 14 12 10 8 6 2 0 ·2 ·4 i. i \;t� . If 01 -60 �rJ � I 0 1 -70 Ii .j._• ' I � ,I 0 1 -80 ; ; •j I i Real GO 01 -90 0 1 -00 01-1 0 Figure 1.8 Nominal federal funds rate, 5-year U.S. bond rate, Baa corporate bond rate, and detrended real GDP, 1 960: 1-20 1 5 :4. 8. Lacker ( 1 988) showed how the correlations between inside money and future output could also arise if move­ ments in inside money reflected new information about future monetary policy. Chapter 1 12 contributed to cyclical fluctuations cannot be inferred from the figure; the movements in interest rates may simply reflect the Federal Reserve's response to the state of the economy. Simple plots and correlations are suggestive, but they cannot be decisive. Other factors may be the cause of the joint movements of output, monetary aggregates, and interest rates. The comparison with business cycle reference points also ignores much of the information about the time series behavior of money, output, and interest rates that could be used to determine what impact, if any, monetary policy has on output. And the appropriate variable to use as a measure of monetary policy depends on how policy has been implemented. One of the earliest time series econometric attempts to estimate the impact of money was due to M. Friedman and Meiselman (1963). Their objective was to test whether monetary or fiscal policy was more important for the determination of nominal income. To address this issue, they estimated the following equation: 9 (1.1) i=O i=O i=O where yn denotes the log of nominal income, equal to the sum of the logs of output and the price level, A is a measure of autonomous expenditures, and m is a monetary aggregate; z can be thought of as a vector of other variables relevant for explaining nominal income fluctuations. Friedman and Meiselman reported finding a much more stable and statistically significant relationship between output and money than between output and their measure of autonomous expenditures. In general, they could not reject the hypothesis that the a ; coefficients were zero, while the b ; coefficients were always statistically significant. The use of equations such as ( 1 . 1 ) for policy analysis was promoted by a number of economists at the Federal Reserve Bank of St. Louis, so regressions of nominal income on money are often called St. Louis equations (see Andersen and Jordan 1968; B. Friedman 1977a; Carlson 1978). Because the dependent variable is nominal income, the St. Louis approach does not address directly the question of how a money-induced change in nominal spending is split between a change in real output and a change in the price level. The impact of money on nominal income was estimated to be quite strong, and Andersen and Jordan (1968, 22) concluded that this finding suggested monetary policy should be used to promote economic stabilization. 10 The original Friedman-Meiselman result generated responses by Modigliani and Ando (1976) and De Prano and Mayer (1965), among others. This debate emphasized that an 9. This is not exactly correct; because M. Friedman and Meiselman included autonomous expenditures as an explanatory variable, they also used consumption as the dependent variable (basically, output minus autonomous expenditures). They also reported results for real variables as well as nominal ones. Following modern practice, ( 1 . 1 ) is expressed in terms of logs; Friedman and Meiselman estimated their equation in levels. 1 0 . B. Friedman ( 1 977a) argued that updated estimates of the St. Louis equation did yield a role for fiscal policy, although the statistical reliability of this finding was questioned by Carlson ( 1 978). Carlson also provided a bibliography listing many of the papers on the St. Louis equation (see his footnote 2, p. 1 3). Evidence on Money, Prices, and Output 13 equation such as ( 1 . 1 ) is misspecified if m is endogenous. To illustrate the point with an extreme example, suppose that the central bank is able to manipulate the money supply to offset almost perfectly shocks that would otherwise generate fluctuations in nominal income. In this case, yn would simply reflect the random control errors the central bank had failed to offset. As a result, m and yn might be completely uncorrelated, and a regression of n y on m would not reveal that money actually played an important role in affecting nominal income. If policy is able to respond to the factors generating the error term u1, then m1, and u1 will be correlated, ordinary least squares estimates of ( 1 . 1) will be inconsistent, and the resulting estimates will depend on the manner in which policy has induced a correlation between u and m. Changes in policy that altered this correlation will also alter the least squares regression estimates one would obtain in estimating ( 1 . 1). Belongia and Ireland (2016) updated Friedman and Schwartz's evidence on money­ output correlations by examining the 1967-201 3 period and by employing a measure of the money supply that differentially weighs the various components added together in a standard measure such as M2. For example, M2 consists of the sum of currency, checkable deposits, savings accounts, small time deposits, and retail money market mutual funds. If these components are not perfect substitutes, then it is incorrect to simply add them together dollar-for-dollar, as is done to obtain M2. Instead, Barnett (1980) advocated the use of Divisia measures of money that construct weighted averages rather than simply sums, with the weights a function of the user cost of each component of the monetary aggregate. I 1 Belongia and Ireland (2016) found large and positive correlations between Divisia monetary aggregates and GDP, with money leading output, and they argued that the U.S. data since 1967 are consistent with the findings of M. Friedman and Schwartz for the prior 100 years. Belongia and Ireland found that the exact correlations and the lead of money over output varied over different subsamples, with some evidence that the lead time between changes in money and subsequent changes in real output has lengthened. 1 .3.2 Granger Causality The St. Louis equation related nominal output to the past behavior of money. Simi­ lar regressions employing real output have also been used to investigate the connection between real economic activity and money. In an important contribution, Sims (1972) intro­ duced the notion of Granger causality into the debate over the real effects of money. A vari­ able X is said to Granger-cause Y if and only if lagged values of X have marginal predictive content in a forecasting equation for Y. In practice, testing whether money Granger-causes output involves testing whether the a ; coefficients equal zero in a regression of the form Yr = Yo + L a ;mt -i + L b ;Yt-i + L C iZt-i + et, i=l i=l i=l l l . See Barnett et al. (20 1 3). (1 .2) Chapter 1 14 where key issues involve the treatment of trends in output and money, the choice of lag lengths, and the set of other variables (represented by z) that are included in the equation. Sims's original work used log levels of U.S. nominal GNP and money (both Ml and the monetary base). He found evidence that money Granger-caused GNP. That is, the past behavior of money helped to predict future GNP. However, using the index of industrial production to measure real output, Sims (1980) found that the fraction of output variation explained by money was greatly reduced when a nominal interest rate was added to the equation (so that z consisted of the log price level and an interest rate). Thus, the conclusion seemed sensitive to the specification of z. Eichenbaum and Singleton (1986) found that money appeared to be less important if the regressions were specified in log first difference form rather than in log levels with a time trend. Stock and Watson ( 1989) provided a systematic treatment of the trend specification in testing whether money Granger-causes real output. They concluded that money does help to predict future output (they actually use industrial production) even when prices and an interest rate are included. A large literature has examined the value of monetary indicators in forecasting output. One interpretation of Sims's finding was that including an interest rate reduced the apparent role of money because, at least in the United States, a short-term interest rate, rather than the money supply, provided a better measure of monetary policy actions (see chapter 1 2). B. Friedman and Kuttner (1992) and Bernanke and Blinder (1992), among others, looked at the role of alternative interest rate measures in forecasting real output. Friedman and Kuttner examined the effects of alternative definitions of money and different sample peri­ ods, concluding that the relationship in the United States is unstable and deteriorated in the 1990s. Bernanke and Blinder found that the federal funds rate "dominates both money and the bill and bond rates in forecasting real variables." Regressions of real output on money were also popularized by Barro (1977; 1978; 1979b) as a way of testing whether only unanticipated money matters for real output. By dividing money into anticipated and unanticipated components, Barro obtained results suggesting that only the unanticipated part affected real variables (see also Barro and Rush 1980 and the critical comment by Small l979). Subsequent work by Mishkin (1982) found a role for anticipated money as well. Cover (1992) employs a similar approach and finds differences in the impacts of positive and negative monetary shocks. Negative shocks were estimated to have significant effects on output, while the effect of positive shocks was usually small and statistically insignificant. 1.3.3 Policy Uses Before reviewing other evidence on the effects of money on output, it is useful to ask whether equations such as (1 .2) can be used for policy purposes. That is, can a regression of this form be used to design a policy rule for setting the central bank's policy instrument? If it can, then the discussions of theoretical models that form the bulk of this book would be unnecessary, at least from the perspective of conducting monetary policy. Evidence on Money, Prices, and Output 15 Suppose that the estimated relationship between output and money takes the form (1 .3) According to (1 .3), systematic variations in the money supply affect output. Consider the problem of adjusting the money supply to reduce fluctuations in real output. If this objec­ tive is interpreted to mean that the money supply should be manipulated to minimize the variance of y1 around yo , then m1 should be set equal to cz a1 mr = - - m r- 1 - - zr - 1 + Vt ao ao (1 .4) = 7T1 m r- 1 + 7T2 Zt -1 + Vr, where for simplicity it is assumed the monetary authority's forecast of z1 is equal to zero. The term v1 represents the control error experienced by the monetary authority in setting the money supply. Equation (1 .4) represents a feedback rule for the money supply whose parameters are themselves determined by the estimated coefficients in the equation for y. A key assumption is that the coefficients in (1 .3) are independent of the choice of the policy rule for m. Substituting (1 .4) into (1 .3), output under the policy rule given in (1 .4) would be equal to Yr = Yo + c 1 zr + u r + a o vr . Notice that a policy rule was derived using only knowledge of the policy objective (min­ imizing the expected variance of output) and knowledge of the estimated coefficients in ( 1 .3). No theory of how monetary policy actually affects the economy was required. Sar­ gent (1976) showed, however, that the use of ( 1 .3) to derive a policy feedback rule may be inappropriate. To see why, suppose that real output actually depends only on unpredicted movements in the money supply; only surprises matter, with predicted changes in money simply being reflected in price-level movements with no impact on output. l2 From (1 .4), the unpredicted movement in m1 is just v1, so let the true model for output be (1 .5) Now from (1 .4), v1 = m1 - (rr 1 mr -1 + rrzzr- 1 ), so output can be expressed equivalently as Yt = Yo + do [mr - (n 1 m r- 1 + nzzr- 1 ) ] + d 1 zr + dzzr - 1 + ur = Yo + domr - don 1 mr- 1 + d 1Zr + (dz - donz) Zr- 1 + Ur. (1 .6) which has exactly the same form as (1 .3). Equation (1 .3), which was initially interpreted as consistent with a situation in which systematic feedback rules for monetary policy could affect output, is observationally equivalent to (1 .6), which was derived under the assump­ tion that systematic policy had no effect and only money surprises mattered. The two are observationally equivalent because the error term in both (1 .3) and (1 .6) is just u1; both equations fit the data equally well. 1 2. The influential model of Lucas ( 1 972) has this implication. See chapter 5. 16 Chapter 1 A comparison of (1 .3) and (1 .6) reveals another important conclusion. The coefficients of ( 1 .6) are functions of the parameters in the policy rule ( 1 .4 ). Thus, changes in the con­ duct of policy, interpreted to mean changes in the feedback rule parameters, change the parameters estimated in an equation such as (1 .6) (or in a St. Louis-type regression). This is an example of the Lucas (1976) critique: empirical relationships are unlikely to be invari­ ant to changes in policy regimes. Of course, as Sargent stressed, it may be that (1 .3) is the true structure that remains invariant as policy changes. In this case, ( 1 .5) will not be invariant to changes in policy. To demonstrate this point, note that (1 .4) implies mr = ( 1 - IT JL) - 1 (7T2Zt- l + Vr ) , where L is the lag operator. 13 Hence, one can write ( 1 .3) as Yt = Yo + a omr + atmr - 1 + C!Zt + C2Zt - ! + u r = Yo + a o (1 - rr 1 L) - 1 (rr2Zt - 1 + vr ) + a 1 ( 1 - 7T 1 L) - l (7T2Zt- 2 + Vt - 1 ) + C1Zt + C2Zt - 1 + Ut = ( 1 - 7T ! ) Yo + 7T 1Yt-1 + a o vr + G! Vt - 1 + C!Zt + (C2 + G07T2 - C1 7T 1 ) Zt -1 + (a 1 7T2 - C2 7Tl ) Zt -2 + Ut - 7T 1 Ut -1 • ( 1 .7 ) where output is now expressed as a function of lagged output, the z variable, and money surprises (the v realizations). If this were interpreted as a policy-invariant expression, one would conclude that output was independent of any predictable or systematic feedback rule for monetary policy; only unpredicted money appears to matter. Yet, under the hypothesis that ( 1 .3 ) is the true invariant structure, changes in the policy rule (the rr 1 coefficients) cause the coefficients in ( 1 .7 ) to change. Note that starting with ( 1 .5 ) and ( 1 .4) , one derives an expression for output that is obser­ vationally equivalent to ( 1 .3 ) . But starting with ( 1 .3 ) and ( 1 .4 ) , an expression for output is obtained that was not equivalent to ( 1 .5 ) ; ( 1 .7 ) contains lagged values of output, v, and u, and two lags of z, while ( 1 .5 ) contains only the contemporaneous values of v and u and one lag of z. These differences would allow one to distinguish between the two, but they arise only because this example placed a priori restrictions on the lag lengths in ( 1 .3 ) and ( 1 .5 ) . In general, one would not have the type of a priori information that would allow this. The lesson from this simple example is that policy cannot be designed without a theory of how money affects the economy. A theory should identify whether the coefficients in a specification of the form (1 .3) or the form ( 1 .5 ) will remain invariant as policy changes. While output equations estimated over a single policy regime may not allow the true struc­ ture to be identified, information from several policy regimes might succeed in doing so. Evidence on Money, Prices, and Output 17 If a policy regime change means that the coefficients in the policy rule (1 .4) have changed, this would identify whether an expression of the form ( 1 .3) or the form ( 1 .5) was policy­ invariant. 1 .3.4 The VAR Approach Much of our understanding of the empirical effects of monetary policy on real economic activity has come from the use of vector autoregression (VAR) frameworks. The use of VARs to estimate the impact of money on the economy was pioneered by Sims (1972; 1980). The development of the approach as it has moved from bivariate to trivariate to larger and larger systems, as well as the empirical findings the literature has produced, were summarized by Leeper, Sims, and Zha (1996). Christiano, Eichenbaum, and Evans ( 1999) provided a thorough discussion of the use of VARs to estimate the impact of money, and they provide an extensive list of references to work in this area. 14 More recent surveys on the effects of monetary policy that utilize VAR or related frameworks are Boivin, Kiley, and Mishkin (2010) and Ramey (2016) and the references they cite. Suppose there is a bivariate system in which Yt is the natural log of real output at time t, and Xt is a candidate measure of monetary policy such as a measure of the money stock or a short-term market rate of interest. 15 The VAR system can be written as [] [ ] [ ] Yt = A(L) Yt-1 + Uyt , (1 .8) Uxt Xt Xt -1 where A (L) is a 2 2 matrix polynomial in the lag operator L, and U if is a time t serially independent innovation to the ith variable. These innovations can be thought of as linear combinations of independently distributed shocks to output (eyt ) and to policy (ext ): x [ Uyt = [ ¢eyt�++()ex�t = [ ¢1 ()1 [ �eyt ] = B [ �eyt ] . J J J The one-period-ahead error made in forecasting the policy variable � (1 .9) Xt is equal to Uxt. and since from (1 .9) Uxt = ¢eyt + ext. these errors are caused by the exogenous output and policy disturbances eyt and ext · The random variable ext represents the exogenous shock to policy. To determine the role of policy in causing movements in output or other macroeconomic variables, one needs to estimate the effect of ex on these variables. If ¢ # 0, the innovation to the observed 14. Two references on the econometrics of VARs are Hamilton ( 1 994) and Maddala ( 1 992). 15. How one measures monetary policy is a critical issue in the empirical literature and is a topic of ongoing debate. See, for example, Romer and Romer ( 1 989), Bernanke and Blinder ( 1 992), Gordon and Leeper ( 1 994), Christiano, Eichenbaum, and Evans ( 1 996; 1 999), Bernanke and Mihov ( 1 998), Rudebusch and Svensson ( 1 999), Leeper, Sims, and Zha ( 1 996), and Leeper ( 1 997)). Zha ( 1 997) and Ramey (20 16) provided useful discussions of the general identification issues that arise in attempting to measure the impact of monetary policy. This issue is discussed further in chapter 12. 18 Chapter 1 policy variable x1 depends both on the shock to policy exr and on the nonpolicy shock ey1; obtaining an estimate of Uxr does not provide a measure of the policy shock unless ¢ = 0. To make the example even more explicit, suppose the VAR system is [] [ ][ J [] Yt = Yt-1 + Uyt , ( 1 . 10) 0 Uxt Xt - 1 Xt O with 0 < a 1 < 1 . Then x1 = Uxr and y1 = a 1Yr-1 + Uyt + a 2 Uxr- J , and one can write y1 in moving average form as a1 az 00 00 Yr = L a 'l Uyt-i + L a � a 2 Uxr-i- 1 . i=O i=O Estimating (1 .8) yields estimates of A(L) and b u , and from these the effects of Uxr on {Yr. Yt+ 1 , ... } can be calculated. If one interpreted Uxr as an exogenous policy disturbance, then the implied response of y1, Yt+ 1 , . . . to a policy shock would be 1 6 0, To estimate the impact of a policy shock on output, however, one needs to calculate the effect on {y1, Yr+l , ... } of a realization of the policy shock ext · In terms of the true underlying structural disturbances ey1 and ex1, ( 1 .9) implies 00 00 Yt = L a i (eyr-i + 8exr-i ) + L a i a 2 (exr-i- 1 + </Jeyr-i- 1 ) i=O i=O 00 00 ( 1 . 1 1) = eyr + 2:: a i ca 1 + a 2 ¢) eyr-i- 1 + eexr + 2:: a i ca1e + a 2 ) exr-i-1 · i=O s o that the impulse response function giving the true response of y to the exogenous policy shock ex is e, This response involves the elements of A(L) and the elements of B. And while A(L) can be estimated from (1 .8), B and b e are not identified without further restrictions. 17 Letting b e denote the 2 2 diagonal variance matrix of the e ;r , and b u denote the variance-covariance matrix of the VAR residues u ;1, then b u = Bb e B'. From estimates of the VAR, an esti­ mate of the three elements of bu can be obtained. But these are functions of six unknown x 1 6. This represents the response to an nonorthogonalized innovation. The basic point, however, is that if (! and 4> are nonzero, the underlying shocks are not identified, so the estimated response to ux or to the component of ux that is orthogonal to uy will not identify the response to the policy shock ex. 17. I n this example, the three elements o f L u , the two variances and the covariance term, are functions o f the four unknown parameters: 1/>, IJ, and the variances of ey and ex. Evidence on Money, Prices, and Output 19 parameters (two elements of :E e and four elements of B). In this example, the diagonal ele­ ments of B were normalized to equal 1 , so the three elements of :Eu (the two variances and the covariance term) are functions of four unknown parameters: ¢, e, and the variances of ey and ex. Several approaches have been taken to solving this identification problem. One approach imposes additional restrictions on the matrix B that links the observable VAR residuals to the underlying structural disturbances (see 1 .9). This approach was used by Sims (1972; 1988), Bernanke ( 1986), Walsh (1987), Bernanke and Blinder (1992), Gordon and Leeper (1994), and Bernanke and Mihov (1998), among others. Sims (1972) treated the nominal money supply (Ml ) as the measure of monetary policy (the x variable) and identified policy shocks by assuming that ¢ = 0. This approach corresponds to the assumption that the money supply is predetermined and that policy innovations are exogenous with respect to the nonpolicy innovations (see 1 .9). Alternatively, if policy shocks affect output with a lag, for example, the restriction that e = 0 would allow the other parameters of the model to be identified. This type of restriction was imposed by Bernanke and Blinder (1992) and Bernanke and Mihov (1998). A second approach achieves identification by imposing restrictions on the long-run effects of the disturbances on observed variables. For example, the assumption of long­ run neutrality of money would imply that a monetary policy shock (ex) has no long-run permanent effect on output. In terms of the example that led to ( 1 . 1 1 ), long-run neutral­ ity of the policy shock would imply that e + (a l e + a 2 ) L a i = 0 or e = -a 2 . Examples of this approach include Blanchard and Watson ( 1986), Blanchard ( 1989), Blanchard and Quah (1989), Judd and Trehan (1989), Hutchison and Walsh (1992), and Galf (1992). The use of long-run restrictions is criticized by Faust and Leeper (1997). A third approach relies on sign restrictions. For example, Uhlig (2005) identifies a mone­ tary shock by imposing restrictions on how such a shock should affect some of the variables in the VAR. He left unrestricted the effects of the shock on real GDP because estimating the effect of monetary policy shocks on GDP is his primary interest. Specifically, he assumes a contractionary monetary shock does not decrease the federal funds rate, increase prices, or increase nonborrowed reserves. He found that the contractionary policy shocks he iden­ tified had ambiguous effects on output. Monetary policy shocks might be identified by using information from outside the VAR. For example, Romer and Romer (1989) developed a narrative measure of policy. Rather than identify a variable of interest like a monetary policy shock as directly observable, an alternative approach is to distinguish between the variables of interest, which might be unobservable, and the variables that are actually observed. In the factor-augmented VAR (FAVAR) approach of Bernanke (2005), the model structure takes the following form: Xr = AFt + er , Fr = A(L)Fr- 1 + Ur , 20 Chapter 1 where Xr is a vector of observable indicators that are a function of a vector Fr of potentially unobserved variables. The vector er consists of measurement errors that are variable spe­ cific. Fr follows a VAR, with innovations ur that are functions of the underlying structural shocks. Monetary policy shocks are one of the structural disturbances that affect elements of ur. 18 If the variables Fr that govern the evolution of Xr are identified with observable variables, then Xr = Fr and the system reduces to a standard VAR. However, the FAVAR framework is more general and allows one to deal with situations in which there may be data available on many macroeconomic variables that are functions of a smaller number of fundamental factors Fr. That is, Xr may contain many more variables than appear in a stan­ dard VAR. For example, many interest rates and monetary aggregates may be affected by monetary policy, but none of these variables is an exact measure of monetary policy. These variables can all be included in Xr, and their comovements help identify the evolution of monetary policy. For example, Boivin, Kiley, and Mishkin (2010) included almost 200 variables in Xr and assumed they were functions of five factors in Fr. To identify monetary policy shocks, they assumed monetary policy responds contemporaneously to real GDP, prices, and the unemployment rate, while there is at least a one-month lag in the response of these variables to monetary policy. Money and Output Sims (1992) provided a useful summary of the VAR evidence on money and output from France, Germany, Japan, the United Kingdom, and the United States. He estimated separate VARs for each country, using a common specification that includes industrial production, consumer prices, a short-term interest rate as the measure of monetary policy, a measure of the money supply, an exchange rate index, and an index of commodity prices. Sims ordered the interest rate variable first. This corresponds to the assumption that ¢ = 0; innovations to the interest rate variable potentially affect the other variables contemporaneously (Sims used monthly data), while the interest rate is not affected contemporaneously by innova­ tions in any of the other variables. 19 The response of real output to an interest rate innovation was similar for all five of the countries Sims examined. In all cases, monetary shocks led to an output response that is usually described as following a hump-shaped pattern. The negative output effects of a contractionary shock, for example, build to a peak after several months and then gradually die out. Eichenbaum ( 1992) compared the estimated effects of monetary policy in the United States using alternative measures of policy shocks and discussed how different choices can produce puzzling results, or at least puzzling relative to certain theoretical expectations. 18. 19. See Boivin and Giannoni (2006) and Boivin, Kiley, and Mishkin (2010). Sims noted that the correlations among the VAR residuals, the u ; � o are small, so the ordering has little impact on his results (i.e., sample estimates of ¢ and e are small). Evidence on Money, Prices, and Output 21 He based his discussion on the results obtained from a VAR containing four variables: the price level and output (these correspond to the elements of y in (1 .8)), M1 as a measure of the money supply, and the federal funds rate as a measure of short-term interest rates (these correspond to the elements of x). He considered interpreting shocks to M1 as policy shocks versus the alternative of interpreting funds rate shocks as policy shocks. He found that a positive innovation to M1 is followed by an increase in the federal funds rate and a decline in output. This result is puzzling if M1 shocks are interpreted as measuring the impact of monetary policy. An expansionary monetary policy shock would be expected to lead to increases in both Ml and output. The interest rate was also found to rise after a positive M1 shock, also a potentially puzzling result; a standard model in which money demand varies inversely with the nominal interest rate would suggest that an increase in the money supply would require a decline in the nominal rate to restore money market equilibrium. Gordon and Leeper (1994) showed that a similar puzzle emerges when total reserves are used to measure monetary policy shocks. Positive reserve innovations are found to be associated with increases in short-term interest rates and unemployment increases. The suggestion that a rise in reserves or the money supply might raise, not lower, market interest rates generated a large literature that attempted to search for a liquidity effect of changes in the money supply (e.g., Reichenstein 1987; Christiano and Eichenbaum 1992b; Leeper and Gordon 1992; Strongin 1995; Hamilton 1996). When Eichenbaum used innovations in the short-term interest rate as a measure of mon­ etary policy actions, a positive shock to the funds rate represented a contractionary policy shock. No output puzzle was found in this case; a positive interest rate shock was fol­ lowed by a decline in the output measure. Instead, what has been called the price puzzle emerges: a contractionary policy shock is followed by a rise in the price level. The effect is small and temporary (and barely statistically significant) but still puzzling. The most com­ monly accepted explanation for the price puzzle is that it reflects the fact that the variables included in the VAR do not span the full information set available to the Federal Reserve. Suppose the Fed tends to raise the funds rate whenever it forecasts that inflation might rise in the future. To the extent that the Fed is unable to offset the factors that led it to forecast higher inflation, or it acts too late to prevent inflation from rising, the increase in the funds rate will be followed by a rise in prices. This interpretation would be consistent with the price puzzle. One solution is to include commodity prices or other asset prices in the VAR. Since these prices tend to be sensitive to changing forecasts of future inflation, they are a proxy for some of the Fed's additional information (Sims 1992; Chari, Christiano, and Eichenbaum 1995; Bernanke and Mihov 1998). Sims 1992 showed that the price puzzle is not confined to U.S. studies. He reported VAR estimates of monetary policy effects for France, Germany, Japan, and the United Kingdom as well as for the United States, and in all cases, a positive shock to the interest rate leads to a positive price response. These price responses tend to become smaller but do not in all cases disappear when a commodity price index and a nominal exchange rate are included in the VAR. In fact, Hansen (2004) 22 Chapter 1 failed to find much relationship between an indicator's ability to forecast future prices and its ability to reduce the size of the price puzzle. An alternative interpretation of the price puzzle was provided by Barth and Ramey (2002). They argued that contractionary monetary policy operates on aggregate supply as well as aggregate demand. For example, an increase in interest rates raises the cost of holding inventories and thus acts as a positive cost shock. This negative supply effect raises prices and lowers output. Such an effect is called the cost channel of monetary policy. In this interpretation, the price puzzle is simply evidence of the cost channel rather than evi­ dence that the VAR is misspecified. Barth and Ramey combined industry-level data with aggregate data in a VAR and reported evidence supporting of the cost channel interpreta­ tion of the price puzzle (see also Ravenna and Walsh 2006 and Gaiotti and Secchi 2004). One difficulty in measuring the impact of monetary policy shocks arises when operating procedures change over time. The best measure of policy during one period may no longer accurately reflect policy in another period if the implementation of policy has changed. Many of the earlier VAR papers employed measures of monetary aggregates as measures of monetary policy. However, during most of the past 50 years, the federal funds interest rate has been the key policy instrument in the United States, suggesting that unforecasted changes in this interest rate may provide good estimates of policy shocks. Bemanke and Blinder (1992) and Bernanke and Mihov (1998) argued for using the federal funds rate as the measure of monetary policy. While the Fed's operating procedures have varied over time, the funds rate is likely to be the best indicator of policy in the United States dur­ ing the pre- 1979 and post- 1982-2008 periods. Policy during the period 1979-1982 is less adequately characterized by the funds rate. 20 The Fed's funds rate target remained fixed at 25 basis points between December 2008 and December 2015, while the Fed used other instruments to influence the economy. Boivin, Kiley, and Mishkin (2010) summarized evidence on the impact of monetary pol­ icy on real GDP and the GDP price deflator. They found that the impact of monetary policy on real GDP was smaller in the 1984-2008 period than before 1980, evidence consistent with the findings of Boivin and Giannoni (2006) but not with those of Canova and Gam­ betti (2009) or Primiceri (2006), who used VAR approaches with time-varying coefficients. Thus, the issue of whether the effects of a monetary policy shock have changed over time is an open empirical issue. While researchers disagree on the best means of identifying policy shocks, there is a surprising consensus on the general nature of the economic responses to monetary policy shocks. A variety of VARs estimated for a number of countries all indicate that in response 20. During this period, nonborrowed reserves were set to achieve a level of interest rates consistent with the desired monetary growth targets. In this case, the funds rate may still provide a satisfactory policy indicator. Cook ( 1 989) found that most changes in the funds rate during the 1 979-1 982 period reflected policy actions. See chapter 12 for a discussion of operating procedures and the reserve market. Evidence on Money, Prices, and Output 23 to a policy shock, output follows a hump-shaped pattern in which the peak impact occurs several quarters after the initial shock. Monetary policy actions appear to be taken in antic­ ipation of inflation, so a price puzzle emerges if forward-looking variables such as com­ modity prices are not included in the VAR. If monetary policy shocks cause output movements, how important have these shocks been in accounting for actual business cycle fluctuations? Leeper, Sims, and Zha ( 1996) concluded that monetary policy shocks have been relatively unimportant. However, their assessment is based on monthly data for the period from the beginning of 1960 until early 1996. This sample contains several distinct periods, characterized by differences in how the Fed implemented monetary policy and differing contributions of monetary shocks over various subperiods. Christiano, Eichenbaum, and Evans (1999) concluded that estimates of the importance of monetary policy shocks for output fluctuations are sensitive to the way monetary policy is measured. When they used a funds rate measure of monetary policy, policy shocks accounted for 21 percent of the four-quarter-ahead forecast error variance for quarterly real GDP. This figure rose to 38 percent of the 12-quarter-ahead forecast error variance. Smaller effects were found using policy measures based on monetary aggregates. Christiano, Eichenbaum, and Evans found that very little of the forecast error variance for the price level could be attributed to monetary policy shocks. Romer and Romer (2004) found a larger role for monetary policy using their measure of policy shocks. Criticisms of the VAR Approach Measures of monetary policy based on the estimation of VARs have been criticized on sev­ eral grounds. 21 First, some of the impulse responses do not accord with most economists' priors. In particular, the price puzzle-the finding that a contractionary policy shock, as measured by a funds rate shock, tends to be followed by a rise in the price level-is trouble­ some. As noted earlier, the price puzzle can be solved by including oil prices or commodity prices in the VAR system, and the generally accepted interpretation is that lacking these inflation-sensitive prices, a standard VAR misses important information that is available to policymakers. A related but more general point is that many of the VAR models used to assess monetary policy fail to incorporate forward-looking variables. Central banks look at a lot of information in setting policy. Because policy is likely to respond to forecasts of future economic conditions, VARs may attribute the subsequent movements in output and inflation to the policy action. However, the argument that puzzling results indicate a mis­ specification implicitly imposes a prior belief about what the correct effects of monetary shocks should look like. Eichenbaum (1992), in fact, argued that short-term interest rate innovations have been used to represent policy shocks in VARs because they produce the types of impulse response functions for output that economists expect. 2 1 . These criticisms are detailed in Rudebusch ( 1 998). 24 Chapter 1 In addition, the residuals from the VAR regressions that are used to represent exoge­ nous policy shocks often bear little resemblance to standard interpretations of the histor­ ical record of past policy actions and periods of contractionary and expansionary policy (Sheffrin 1995; Rudebusch and Svensson 1999). They also differ considerably depending on the particular specification of the VAR. Rudebusch reported low correlations between the residual policy shocks he obtained based on funds rate futures and those obtained from a VAR by Bernanke and Mihov. How important this finding is depends on the question of interest. If the objective is to determine whether a particular recession was caused by a policy shock, then it is important to know if and when the policy shock occurred. If alter­ native specifications provide differing and possibly inconsistent estimates of when policy shocks occurred, then their usefulness as a tool of economic history would be limited. If, however, the question of interest is how the economy responds when a policy shock occurs, then the discrepancies among the VAR residual estimates may be less important. Sims ( 1998a) argued that in a simple supply-demand model, different authors using dif­ ferent supply curve shifters may obtain quite similar estimates of the demand curve slope (since they all obtain consistent estimators of the true slope). At the same time, they may obtain quite different residuals for the estimated supply curve. If the true interest is in the parameters of the demand curve, the variations in the estimates of the supply shocks may not be important. Thus, the type of historical analysis based on a VAR, as in Walsh ( 1 993), is likely to be more problematic than the use of a VAR to determine the way the economy responds to exogenous policy shocks. While VARs focus on residuals that are interpreted as policy shocks, the systematic part of the estimated VAR equation for a variable such as the funds rate can be interpreted as a policy reaction function; it provides a description of how the policy instrument has been adjusted in response to lagged values of the other variables included in the VAR system. Rudebusch and Svensson ( 1999) argued that the implied policy reaction functions look quite different than results obtained from more direct attempts to estimate reaction functions or to model actual policy behavior. 22 A related point is that VARs are typically estimated using final, revised data and therefore do not capture accurately the historical behavior of the monetary policymaker who is reacting to preliminary and incomplete data. Woolley (1995) showed how the perception of the stance of monetary policy in the United States in 1972, and President Richard Nixon's attempts to pressure Fed Chairman Arthur F. Burns into adopting a more expansionary policy were based on initial data on the money supply that were subsequently very significantly revised. At best the VAR approach identifies only the effects of monetary policy shocks, shifts in policy unrelated to the endogenous response of policy to developments in the economy. 22 . For example, Taylor ( 1 993a) employed a simple interest rate rule that closely matches the actual behavior of the federal funds rate in recent years. Such a rule is now the standard way to model Fed behavior. Yet as Khoury ( 1 990) noted in an earlier survey of many studies of the Fed's reaction function, few systematic conclusions had emerged from this empirical literature prior to Taylor's work. Evidence on Money, Prices, and Output 25 Yet most, if not all, of what one thinks of in terms of policy and policy design represents the endogenous response of policy to the economy, and "most variation in monetary policy instruments is accounted for by responses of policy to the state of the economy, not by ran­ dom disturbances to policy" (Sims 1998a, 933). This is also a major conclusion of Leeper, Sims, and Zha ( 1996). So it is unfortunate that VAR analysis, a primary empirical tool used to assess the impact of monetary policy, is uninformative about the role played by policy rules. If policy is completely characterized as a feedback rule on the economy, so that there are no exogenous policy shocks, then the VAR methodology would conclude that mone­ tary policy doesn't matter. Yet while monetary policy is not causing output movements in this example, it does not follow that policy is unimportant; the response of the economy to nonpolicy shocks may depend importantly on the way monetary policy endogenously adjusts. Cochrane (1998) made a similar point that is related to the issues discussed in section 1 .3.3. In that section, it was noted that one must know whether it is anticipated money with real effects (as in (1 .3)) or unanticipated money (as in (1 .5)) that matters. Cochrane argued that most of the VAR literature has focused on issues of lag length, detrending, ordering, and variable selection, and has largely ignored another fundamental identification issue: is it anticipated or unanticipated monetary policy that matters? If only unanticipated policy matters, then the subsequent systematic behavior of money after a policy shock is irrelevant. This means that the long hump-shaped response of real variables to a policy shock must be due to inherent lags of adjustment and the propagation mechanisms that characterize the structure of the economy. If anticipated policy matters, then subsequent systematic behavior of money after a policy shock is relevant. This means that the long hump-shaped response of real variables to a policy shock may only be present because policy shocks are followed by persistent, systematic policy actions. If this is the case, the direct impact of a policy shock, if it were not followed by persistent policy moves, would be small. Attempts have been made to use VAR frameworks to assess the systematic effects of monetary policy. Sims (1998b), for example, estimated a VAR for the interwar years and used it to simulate the behavior of the economy if policy had been determined according to the feedback rule obtained from a VAR estimated using postwar data. 1 .3.5 Structural Econometric Models The empirical assessment of the effects of alternative feedback rules for monetary pol­ icy has traditionally been carried out using structural macroeconometric models. During the 1960s and early 1970s, the specification, estimation, use, and evaluation of large­ scale econometric models for forecasting and policy analysis represented a major research agenda in macroeconomics. Important contributions to our understanding of investment, consumption, the term structure, and other aspects of the macroeconomy grew out of the need to develop structural equations for various sectors of the economy. An equation 26 Chapter 1 describing the behavior of a policy instrument such as the federal funds rate was incorpo­ rated into these structural models, allowing model simulations of alternative policy rules to be conducted. These simulations would provide an estimate of the impact on the economy's dynamic behavior of changes in the way policy was conducted. For example, a policy under which the funds rate was adjusted rapidly in response to unemployment movements could be contrasted with one in which the response was more muted. A key maintained hypothesis, one necessary to justify this type of analysis, was that the estimated parameters of the model would be invariant to the specification of the policy rule. If this were not the case, then one could no longer treat the model's parameters as unchanged when altering the monetary policy rule (as the example in section 1 .3.3 shows). In a devastating critique of this assumption, Lucas (1976) argued that economic theory predicts that the decision rules for investment, consumption, and expectations formation will not be invariant to shifts in the systematic behavior of policy. The Lucas critique emphasized the problems inherent in the assumption, common in the structural economet­ ric models of the time, that expectations adjust mechanically to past outcomes. While large-scale econometric models of aggregate economies continued to play an important role in discussions of monetary policy, they fell out of favor among academic economists during the 1970s, in large part as a result of Lucas's critique, the increasing emphasis on the role of expectations in theoretical models, and the dissatisfaction with the empirical treatment of expectations in existing large-scale models. 23 The academic lit­ erature witnessed a continued interest in small-scale rational-expectations models, both single and multicountry versions (e.g., the work of Taylor 1993b) as well as the develop­ ment of larger-scale models (Fair 1984), all of which incorporated rational expectations into some or all aspects of the model's behavioral relationships. However, recent empir­ ical work investigating the impact of monetary policy has relied on estimated dynamic stochastic general equilibrium (DSGE) models. These models combine rational expecta­ tions with a microeconomic foundation in which households and firms are assumed to behave optimally, given their objectives (utility maximization, profit maximization) and the constraints they face. In general, these models are built on the theoretical foundations of the new Keynesian model. As discussed in chapter 8, this model is based on the assump­ tion that prices and wages display rigidities and that this nominal stickiness accounts for the real effects of monetary policy. Early examples include the work of Yun (1996), Ire­ land (1997a), and Rotemberg and Woodford (1997). Among more recent examples are the DSGE models of Christiano, Eichenbaum, and Evans (2005), who estimated their model by matching VAR impulse responses, and Smets and Wouters (2003), who estimated their model using Bayesian techniques. The use of Bayesian estimation is now common; early examples include work by Smets and Wouters (2003; 2007); Levin et al. (2006), and Lubik 23. For an example of a small-scale model in which expectations play no explicit role, see Rudebusch and Svensson ( 1 999). Evidence on Money, Prices, and Output 27 and Schorfueide (2005). Many central banks have built and estimated DSGE models to use for policy analysis, and many more central banks are in the process of doing so. A major advantage of these structural models is that they can be used to evaluate the effects of alter­ native, systematic rules for monetary policy rather than just the effects of policy shocks on macroeconomic variables. The basic structure of these models can be expressed as ( 1 . 12) where Y1 is a vector of endogenous variables, X1 is a vector of exogenous variables, i1 is the policy instrument, and u1 is an i.i.d. vector of mean zero shocks. The endogenous variables depend on expectations of their future values, on policy, and on the exogenous variables and shocks. The assumption in structural models is that the parameters in the coefficient matrices A 1 , A 2 , and B are invariant to the particular policy rule followed by the central bank. Suppose the rule is where v1 is an i.i.d. policy shock. Finally, assume the exogenous variables evolve according to X1 = r Xt- 1 + e1, where r is also independent of the parameters of the policy rule. Assuming rational expectations, the solution to this model takes the form Y1 = MX1 + N (Bv1 + u1), ( 1 . 1 3) where M satisfies and The key implication is that the M and N matrices in ( 1 . 1 3) depend on the coefficients C 1 and C2 in the policy rule. However, if the structural model ( 1 . 12) can be estimated, then one can investigate how M and N and the behavior of Y changes as the policy rule coefficients C 1 and C2 are changed, because A 1 , A 2 , and B remain constant as C 1 and C2 are varied. 1 .3.6 Alternative Approaches The VAR approach is the most commonly used empirical methodology, and the accompa­ nying results provide a fairly consistent view of the impact of monetary policy shocks. But other approaches have also influenced views on the role of policy. Two such approaches, one based on deriving policy directly from a reading of policy statements, the other based on case studies of disinflations, have influenced academic discussions of monetary policy. 28 Chapter 1 Announcement Effects Monetary policy meetings of the Fed's Federal Open Market Committee (FOMC) are fol­ lowed by announcements. The FOMC releases a statement describing any change in the target for the federal funds rate and guidance about the direction of future policy. Using data on asset prices from immediately before a policy announcement and immediately after the announcement can provide evidence on the impact of policy actions on financial markets and on how information about future policy affects those markets. Measuring the impact of policy expectations by examining the reaction of financial mar­ kets to the release of new information has a long history. In the early 1980s, for example, attention focused on the weekly release of new data on the money supply. Because the Federal Reserve had established targets for money growth, if actual money growth was faster than expected, markets interpreted this as a sign that the Fed would tighten future policy to bring money growth back to target. Roley and Walsh (1985) described empiri­ cal work to investigate the impact of weekly money surprises on interest rates. Cook and Hahn (1989) focused on the effects of announced changes in the funds rate target on asset prices. Kuttner (2001 ) used data on Fed funds futures to distinguish between anticipated and unanticipated changes in the funds rate target and found significant effects on Treasury yields of the latter but not the former. Rigobon and Sack (2004) and Bernanke and Kuttner (2005) examined the stock market reaction to monetary policy. Giirkaynak, Sack, and Swanson (2005) distinguished between the effects of announce­ ments about policy changes and announcements providing information about future policy. For example, they pointed out that long-term interest rates jumped in response to a January 2004 post-policy meeting announcement even though there was no change in actual pol­ icy. Instead, the Fed changed its language about future conditions, which led market partici­ pants to anticipate a future rise in the policy rate. Using information on Fed announcements over a 15-year period, they showed that policy announcements affect asset prices through two factors: surprise changes in the actual funds rate target and surprise changes in the expected future path of the funds rate. To assess the effects of Fed guidance about future policy on inflation and the real econ­ omy, Campbell et al. (2012) estimated the effects of policy surprises on professional fore­ casts of future inflation and unemployment. For the 1996-2007 period, they argued, the Fed was able to signal future policy actions that moved private sector forecasts in ways consistent with policy intentions. 24 Event studies that estimate the effects of Federal Reserve policy announcements on asset prices have been used extensively to investigate the impact of Fed balance sheet policies undertaken between early 2009 and late 2015, when the Fed funds rate target was fixed at 0-25 basis points. These are discussed in section 1 .4. 24. Kool and Thornton (20 1 2) provided a more skeptical assessment of the effectiveness of forward guidance by examining the experiences of the central banks of New Zealand, Norway, Sweden, and the United States. Evidence on Money, Prices, and Output 29 Narrative Measures of Monetary Policy An alternative to the VAR statistical approach is to develop a measure of the stance of monetary policy from a direct examination of the policy record. This approach was taken by Romer and Romer (1989; 2004) and Boschen and Mills (199 1 ), among others. 25 Boschen and Mills developed an index of policy stance that takes on integer values from -2 (strong emphasis on inflation reduction) to + 2 (strong emphasis on "promoting real growth"). Their monthly index is based on a reading of the FOMC policy directives and the records of the FOMC meetings. Boschen and Mills showed that innovations in their index corresponding to expansionary policy shifts are followed by subsequent increases in monetary aggregates and declines in the federal funds rate. They also concluded that all the narrative indices they examined yielded relatively similar conclusions about the impact of policy on monetary aggregates and the funds rates. And in support of the approach described in section 1 .3.4, Boschen and Mills concluded that the funds rate is a good indi­ cator of monetary policy. These findings were extended in Boschen and Mills (1995b), which compared several narrative-based measures of monetary policy, finding them to be associated with permanent changes in the level of M2 and the monetary base and temporary changes in the funds rate. Romer and Romer (1989) used the Fed's "Record of Policy Actions" and, prior to 1976 when they were discontinued, minutes of FOMC meetings to identify episodes in which policy shifts occurred that were designed to reduce inflation. They found six different months during the postwar period that saw such contractionary shifts in Fed policy: Octo­ ber 1947, September 1955, December 1968, April 1974, August 1978, and October 1979. Leeper (1993) argued that the Romer-Romer index is equivalent to a dummy variable that picks up large interest rate innovations. Hoover and Perez ( 1994) provided a critical assess­ ment of the Romers' narrative approach, noting that the Romer dates are associated with oil price shocks, while Leeper ( 1997) found that the exogenous component of the Romers' policy variable does not produce dynamic effects on output and prices that accord with general beliefs about the effects of monetary policy. Romer and Romer (2004) used a narrative approach to identify changes in the Fed's tar­ get for the federal funds rate and then took the component that was orthogonal to the Fed's forecasts of macroeconomic variables. Using this measure of policy shocks, they found a much larger role for policy shocks both in affecting inflation and output and in accounting for historical fluctuations. Coibion (201 2) reconciled these results with the smaller effects found in most VAR analyses by showing that the lag length structure assumed can play an important role, as does the treatment of the 1979-1982 period, during which Fed policy was better characterized as a nonborrowed reserve aggregates procedure (see chapter 12), implying movements of the funds rate were not the appropriate measures of policy. 25. Boschen and Mills ( 1 99 1 ) provided a discussion and comparison of some other indices of policy. 30 Chapter 1 Case Studies of Disinflations Case studies of specific episodes of disinflation provide, in principle, an alternative means of assessing the real impact of monetary policy. Romer and Romer's approach to dating periods of contractionary monetary policy is one form of case study. However, the most influential example of this approach is that of Sargent (1986), who examined the ends of several hyperinflations. As discussed more fully in chapter 5, the distinction between anticipated and unanticipated changes in monetary policy played an important role in the 1980s in academic discussions of monetary policy, and a key hypothesis is that anticipated changes should affect prices and inflation, with little or no effect on real economic activity. This implies that a credible policy to reduce inflation should succeed in actually reducing inflation without causing a recession. This implication contrasts sharply with the view that any policy designed to reduce inflation would succeed only by inducing an economic slowdown and temporarily higher unemployment. Sargent tested these competing hypotheses by examining the ends of the post-World War I hyperinflations in Austria, German, Hungary, and Poland. In each case, he found that the hyperinflations ended abruptly. In Austria, for example, prices rose by over a factor of 20 from December 1921 to August 1922, an annual inflation rate of over 8, 800 percent. Prices then stopped rising in September 1922, actually declining by more than 10 percent during the remainder of 1922. While unemployment did rise during the price stabilizations, Sargent concluded that the output cost per percentage point reduction in inflation was much smaller than what some economists had estimated would be the costs of reducing U.S. inflation. Sargent's interpretation of the experiences in Germany, Poland, and Hungary is similar. In each case, the hyperinflation was ended by a regime shift that involved a credible change in monetary and fiscal policy designed to reduce government reliance on inflationary finance. Because the end of inflation reduced the opportunity cost of holding money, money demand grew and the actual stock of money continued to grow rapidly after prices had stabilized. Sargent's conclusion that the output costs of these disinflations were small has been questioned, as have the lessons he drew for the moderate inflations experienced by the industrialized economies in the 1970s and early 1980s. As Sargent noted, the ends of the hyperinflations "were not isolated restrictive actions within a given set of rules of the game" but represented changes in the rules of the game, most importantly in the ability of the fiscal authority to finance expenditures by creating money. In contrast, the empirical evidence from VARs of the type discussed earlier in this chapter reflects the impact of policy changes within a given set of rules. Schelde-Andersen (1992) and Ball (1993) provided other examples of the case study approach. In both cases, the authors examined disinflationary episodes in order to estimate the real output costs associated with reducing inflation. 26 Their cases, all involving OECD countries, represent evidence on the costs of ending moderate inflations. Ball calculated 26. See also Gordon ( 1 982) and Gordon and King ( 1 982). Evidence on Money, Prices, and Output 31 the deviation of output from trend during a period of disinflation and expressed this as a ratio to the change in trend inflation over the same period. The 65 disinflation periods he identifies in annual data yield an average sacrifice ratio of 0.77 percent; each percentage point reduction in inflation was associated with a 0.77 percent loss of output relative to trend. The estimate for the United States was among the largest, averaging 2.3 percent based on annual data. The sacrifice ratios are negatively related to nominal wage flexibility; countries with greater wage flexibility tend to have smaller sacrifice ratios. The costs of a disinflation also appear to be larger when inflation is brought down more gradually over a longer period of time. 27 The case study approach can provide interesting evidence on the real effects of mone­ tary policy. Unfortunately, as with the VAR and other approaches, the issue of identifica­ tion needs to be addressed. To what extent have disinflations been exogenous, so that any resulting output or unemployment movements can be attributed to the decision to reduce inflation? If policy actions depend on whether they are anticipated or not, then estimates of the cost of disinflating obtained by averaging over episodes--episodes that are likely to have differed considerably in terms of whether the policy actions were expected or, if announced, credible-may yield little information about the costs of ending any specific inflation. 1.4 Monetary Policy at Very Low Interest Rates In December 2008 the Federal Reserve cut its federal funds rate target range to 0-25 basis points, and it remained there until December 2015. Bemanke and Reinhart (2004) argued that this was the effective lower bound for Fed's target. Historically, zero was treated as the lower bound on nominal interest rates, but subsequently several central banks set negative interest rates (e.g., the central banks of Denmark, Japan, Sweden, and Switzerland as well as the European Central Bank). 28 How negative rates can go is uncertain, but it is certain that zero is not the lower bound for nominal interest rates. Thus, this book generally refers to the minimum possible level of the nominal interest rate as the effective lower bound (ELB) rather than the more common zero lower bound (ZLB). Regardless of what the value of the ELB is, with the funds rate effectively fixed for seven years in the United States, standard empirical strategies that used unforecastable movements in the funds rate to measure monetary policy shocks were no longer useful. While the funds rate target did not change, the Federal Reserve engaged in policy actions that expanded its balance sheet 27. Brayton and Tinsley ( 1 996) showed how the costs of disinflation can be estimated under alternative assump­ tions about expectations and credibility using the FRB/US structural model. Their estimates of the sacrifice ratio, expressed in terms of the cumulative annual unemployment rate increase per percentage point decrease in the inflation rate, range from 2.6 under imperfect credibility and VAR expectations to 1 .3 under perfect credibility and VAR expectations. Under full-model expectations, the sacrifice ratio is 2.3 with imperfect credibility and 1 . 7 with full credibility. 28 . Why zero was viewed as the lower bound is discussed in chapter 1 1 . Chapter 1 32 from around $850 billion in 2008 to $4.5 trillion by 2015. It also altered the composition of the assets on its balance sheet by selling holdings of short-term government securities and purchasing long-term government securities and large quantities of mortgage-backed securities. These actions raise two questions. How can the stance of monetary policy be measured when the policy interest rate is at its lower bound? What have been the effects of balance sheet policies on financial markets and the macroeconomy? 1.4.1 Measuring Policy at the Effective Lower Bound (ELB) If the central bank is at its effective lower bound but is making announcements about the future path of the policy rate (forward guidance), expanding its balance sheet, and altering its asset holdings, it can be difficult to develop a summary measure of monetary policy. One approach is to employ data on various policy instruments and treat them as observable indicators of the unobservable policy stance. This is the approach, for example, in the FAVAR strategy see section 1 .3.4. An alternative approach was developed by Wu and Xia (2016) to estimate an effective short-term rate when the policy rate is fixed at zero. They used a theory of the relationship between interest rates on government bonds of different maturities to estimate the value of the short-term rate that is consistent with the observed behavior of long-term rates. When the actual short-term rate is positive, their estimate corresponds to the actual short-term rate. When the actual short-term rate is fixed at its lower bound, they obtain an estimated shadow short-term rate. If nonstandard policies are effective at reducing long-term interest rates, even though the actual policy rate has not changed, the shadow rate will be below the policy rate. Its level can proxy for the impact of the nonstandard policies. Figure 1 .9 shows the Fed's target for the funds rate and the Wu-Xia shadow rate. The data are monthly from January 2006 to November 2015. The shadow rate has been negative since July 2009, suggesting that the unconventional balance sheet policies of the Fed suc­ ceeded in lowering long-term rates even though the funds rate target remained unchanged. The models examined in this book generally imply both the current value of the short­ term rate and its expected future path are important for households and firms making con­ sumption and investment decisions. Long-term rates that affect spending decisions should respond directly to changes in expectations about future short-term rates. When the short­ term rate is at the ELB, central bank announcements designed to affect expectations of future short-term rates may allow the central bank to influence economic activity, and a large literature has investigated the impact of central bank announcements on expecta­ tions and on long-term rates. 29 However, the effects of announcements can be difficult to interpret. Suppose the central bank announces it will keep interest rates lower for longer than it previously planned. If this is interpreted as indicating a more expansionary future 29. See Kiley (20 14) for evidence that aggregate demand is affected by both short-term and long-term interest rates. Evidence on Money, Prices, and Output 33 6 .-------,--,---,,---� 4 3 Fed funds rate target 2 1 \ 0 �------��==== � \ ' ... ,' ' .. - -1 -2 ' ,,,, ' ... ... ... .... ' - , ... ... _ ... .. Wu-Xia shadow funds rate " ' ', I ' \ \ I ' , I -3 Jan06 JanOS Jan 1 0 Jan 1 2 Jan 1 4 Jan 1 6 Figure 1.9 The Fed's target for the funds rate and the Wu-Xia shadow rate. policy stance, the effect should be expansionary. Alternatively, if the public interprets the announcement of lower future rates as a signal that the central bank is pessimistic about future economic activity, the effect can be contractionary. Campbell et al. (2012) called the first effect Odyssian--the central bank is committing itself to keeping rates low in the future--and the second Delphic--the central bank is signaling a change in its outlook for the economy. 1 .4.2 The Effects of Quantitative Easing (QE) Policies Between 2008 and 2015, the Fed employed balance sheet policies in an attempt to stimulate economic activity. These policies involved asset purchases and were collectively referred to as large-scale asset purchases (LSAP) programs or simply as quantitative easing (QE) policies. Many authors have described in detail the specific nature and timing of each of the Fed's QE policies. For example, see Gagnon et al. (201 1 ), Krishnamurthy and Vissing­ Jorgensen (2013), and D'Amico et al. (201 2) for the United States and Joyce et al. (20 1 1 ) for the United Kingdom. 3 0 The net effects of these policies were to expand the Fed's bal­ ance sheet from $850 billion to $4.5 trillion and to extend the maturity and riskiness of the assets held by the Fed. 30 . See also http://projects.marketwatch.com/short-history-of-qe-and-the-market-timeline/#O. Chapter 1 34 6 X 10 5 ��,--------.---------,---------.---------� 4.5 4 - Traditional Security holdings - LOLR c=:J Mortgage Backed Securities c=:J L-T Treasuries 3.5 f/7 0 � � :0 3 2.5 2 1 .5 0.5 0 JanOS Jan 1 0 Jan 1 2 Jan 1 4 Jan 1 6 Figure 1.10 Asset holdings of the Federal Reserve. Among the traditional tools a central bank has at its disposal in a financial crisis is the ability to provide short-term loans to solvent institutions. Providing liquidity in a crisis is part of the lender of last resort (LOLR) function of a central bank. Figure 1 . 10 shows the balance sheet assets of the Federal Reserve. The component labeled LOLR shows the rapid increase in the Federal Reserve's provision of loans to financial institutions and liquidity to key markets during the 2008-2009 financial crisis. These actions expanded the balance sheet from its precrisis level of roughly $850 billion to a peak in December 2008 of just under $ 1 .9 trillion. As one would expect of LOLR activities, the crisis created a temporary expansion of the balance sheet, but the effects of these actions were quickly reversed as financial markets returned to more normal conditions. As figure 1 . 10 clearly illustrates, however, the overall size of the Fed's balance sheet did not return to precrisis levels. The Fed also undertook QE policies. This section focuses on models designed to understand how expansions of the balance sheet and changes in the composition of the assets held by a central bank may affect asset prices and economic activity. The figure shows the consequences for the Fed's balance sheet of these QE poli­ cies, which involved the purchase of mortgage-backed securities and long-term Treasury securities. The balance sheet continued to grow after the end of the financial crisis, reaching $4.5 trillion in early 2014, where it remained as of early 2016. The first large expansion, under QE1 from November 2008 to March 2010, resulted in the purchase of $300 billion in U.S. Treasuries, $ 1 .25 trillion of agency mortgage-backed Evidence on Money, Prices, and Output 35 securities, and $170 billion of agency debt. After a pause, the balance sheet expanded again under QE2, which began in November 2010 and lasted until June 201 1 . QE2 involved the purchases of long-term U.S. Treasuries. From July 201 1 to December 2012, the total bal­ ance sheet remained relatively constant at about $2.8-$2.9 trillion. However, this period saw the Fed alter the composition of its balance sheet by purchasing long-term Treasuries financed by selling short-term Treasuries. This modern-day Operation Twist, or matu­ rity extension program (MEP), began in September 201 1 and continued through 2012. 31 September 201 2 saw the start of QE3, under which the Fed shifted from announcing a fixed amount of purchases and instead committed to purchasing $45 billion of U.S. Treasuries and $40 billion of mortgage-backed securities per month with no end date. QE3 continued until late 2013, when the monthly amount purchased was reduced by $10 billion in December 201 3 . A gradual tapering of purchases continued until QE3 ended in October 2014. LSAP programs were designed to reduce long-term interest rates to stimulate spending. Normally, the Fed would lower its policy rate, a very short-term rate, if it wanted to lower longer-term rates. But if the policy rate is at its ELB, this option is not available. By pur­ chasing long-term assets, the Fed reduced the supply of these assets available to the private sector. For example, when the Fed purchases long-term bonds, fewer are available for the private sector to hold, and this may increase their price, reducing long-term interest rates. Any assessment of balance sheet policies must address two separate questions. First, are such policies effective in altering yields? Second, if the answer to the first question is yes, are these changes effective in influencing real economic activity? Most of the empirical work on balance sheet policies has focused on the first question, but obtaining the answer to the second is clearly essential. On Yields and Asset Prices The bulk of the empirical work designed to estimate the impact of balance sheet policies has focused on the effects of the announcements of balance sheet policies on asset prices and bond yields. In this context, an important issue is determining which asset prices and interest rates are most important for affecting the real economy. Consider a very simple economy with short-term and long-term government debt and a private security such as a corporate bond. There are three interest rates: the rate on short-term government debt, the rate on long-term government debt, and the rate on risky private debt. When the short-term rate is the policy instrument, increases or decreases in this rate are assumed to affect the other two rates. However, if the short-term rate is at its lower bound, are balance sheet policies more effective if they work by lowering long-term rates relative to the short-term rate or by lowering risk premiums so that the rate on private debt falls relative to riskless government debt? 3 1 . See Swanson (20 1 1 ) for a comparison of MEP with the Operation Twist of the 1 960s. 36 Chapter 1 One way to address this question is to ask whether future real activity is better fore­ cast by spreads between long-term and short-term rates on government securities or by credit spreads that reflect risk premiums. This forecasting exercise was conducted by Rude­ busch, Sack, and Swanson (2007), who also summarized the earlier literature. Gilchrist and Zakrajsek (2012) undertook it, also using a number of new alternative measures of credit spreads. Rudebusch, Sack, and Swanson (2007) found that a rise in the long-rate relative to the short-rate predicts higher future real activity, a finding confirmed with more recent data by Walsh (2014). Rudebusch argued that if changes in spreads rather than levels are used, a rise in the long-term rate predicts slower future growth. However, Walsh found this was the case for industrial production but not for unemployment. Results for risk premiums as measured by spreads between the Aaa corporate bond rate and the 10-year government bond rate, or between Aaa and Baa bonds, were more robust, consistent with the findings of Gilchrist and Zakrajsek (201 2) based on corporate credit risks. Increases in these risk spreads predicted weaker future industrial production and higher unemployment. Gilchrist and Zakrajsek (2013) found that the LSAP programs lowered overall credit risk as mea­ sured by the cost of default risk insurance outside the financial sector. One of the first and most influential analyses of the Fed's LSAP policies is the work of Gagnon et al. (201 1 ). They concluded "LSAPs cause economically meaningful and long­ lasting reductions in longer-term interest rates on a range of securities, including securities that were not included in the purchase programs"(S) (italics added). They also concluded that the policies reduced risk premiums rather than expectations of future short-term rates. This suggested a low degree of substitutability between reserves and assets purchased (long-term Treasuries and mortgage-backed securities) and a high degree of substitutability between the assets purchased and corporate debt. Krishnamurthy and Vissing-Jorgensen (2013) found, somewhat in contrast to Gagnon et al. (201 1), that QE policies primarily affect the prices of the assets that the Fed purchases rather than broadly all long-term bonds. 32 This is an important finding, since it suggests that effects depend on particular assets and that QE policies are not good substitutes for general changes in the level of interest rates when the policy rate itself can be used. It may also suggest that the level of segmentation in financial markets is particularly high, limiting the arbitrage across broad categories that is implicitly assumed by arguments that lowering long-term rates on Treasuries have effects on a wide range of asset prices. 33 A number of authors have used term structure factor models to investigate the effects of bond supply on interest rates. See, for example, Li and Wei (201 3), Greenwood, Hanson, and Vayanos (2015), Hamilton and Wu (2012b), D'Amico et al. (2012), and 32. See also Krishnamurthy and Vissing-Jorgensen (20 l l). Krishnamurthy and Vissing-Jorgensen (20 1 3 , table 1 , p. 10), provided a summary of their findings for LSAP programs. 33. Attention is restricted to studies of the Fed's QE policies. Papers that focus on the Bank of England's policies include Joyce et al. (20 1 1 ) and Kapetanios and Mumtaz (20 1 2). As noted previously, Christensen and Rudebusch (20 12) also estimated the effects of QE policies in the United Kindom. Evidence on Money, Prices, and Output 37 Swanson and Williams (2013). If financial assets are imperfect substitutes in investors' portfolios, then changes in the outstanding stocks of these assets should cause relative rates of return to adjust. Hamilton and Wu (20 1 2b) echoed the earlier work by Bernanke, Rein­ hart, and Sack (2004) in stating, "Our conclusion is that although it appears to be possible for the Fed to influence the slope of the yield curve in normal times . . . very large operations are necessary to have an appreciable immediate impact. If there is no concern about a ZLB constraint, this potential tool should clearly be secondary to the traditional focus of open­ market operations on the short end of the yield curve" (24). D'Amico et al. (2012) reached a more positive conclusion in arguing that changes in debt stocks affect yields independent of any signaling effects and that their results argued for the effectiveness of LSAPs as a useful tool of monetary policy. Even in the absence of portfolio balance effects arising from investor heterogeneity or segmented markets, long-term yields could be affected by QE policies if these policies provide new information about the future path of short-term rates. This signaling channel is the only channel that operates in pure expectations models of the term structure. Chris­ tensen and Rudebusch (201 2) and Bauer and Rudebusch (20 13) argued that the commonly used Kim-Wright estimate of the term premium, the estimate used in several studies of QE policies, is based on a model in which the short-term rate's speed of reversion is over­ stated. Hence, they argued, work using the Kim-Wright model of the term premium, such as Gagnon et al. (201 1), tend to overattribute movements in the long-term rate due to QE to movements in the term premium rather than to persistent movements in expected future short-term rates. Bauer and Rudebusch (201 3) argued that the effects of QE policies on long-term rates in the United States and United Kingdom were similar but worked through the signaling channel in the United States and because of declines in term premium in the United Kingdom. They attributed these differences to a greater focus on providing forward guidance in the communications of the Fed. While much of literature has focused on the effects of LSAPs on Treasury yields, Gilchrist and Zakrajsek (2013) focused on the default risk channel by looking at effects on measures of corporate credit risk. If LSAP programs help stimulate the economy, expected defaults should fall, reducing the default risk premium and increasing investor appetite for risk. They argued that event study estimates of LSAP policies are biased downward because of endogeneity of interest rate and credit risk responses to common shocks. They identified the credit risk response to QE policies using shifts in the variance of monetary policy shocks on announcement dates, based on the premise that a larger share of news is associated with monetary policy on these dates. Gilchrist and Zakrajsek (2013) concluded that declines in risk-free rates due to LSAP programs did succeed in reducing measures of risk for the corporate sector but not for the financial intermediary sector. Giirkaynak, Sack, and Swanson (2005) identified two factors associated with the effects of Fed announcements on asset prices, with one factor associated with changes in the target for the funds rate and the other associated with information about the future path 38 Chapter 1 of the target. Swanson (2015) applied this approach to the 2009-2015 period and found that the two factors can be identified as reflecting forward guidance and LSAP programs. He looked at the impact on 2-year, 5-year, and 10-year yields and found that LSAP policies significantly lowered long-term interest rates, with effects increasing as the maturity of the security increased. Chen, Curdia, and Ferrero (2012, table 1) summarized the results of several papers that provided estimates of the impact of LSAP policies on the U.S. 10-year Treasury yield. A $100 billion QE policy is estimated to reduce the 10-year rate from about 3 basis points (e.g., Hamilton and Wu 2012b) to 1 5 basis points (D'Amico et a!. (2012)). On the Macroeconomy While most studies of QE have focused on financial markets, understanding their impact on asset prices and yields provides at best a partial answer to the question of whether these policies have supported economic growth. Estimating such effects is inherently much more difficult than estimating the effects of announcements on asset yields. Several authors have utilized DSGE models to simulate the effects of QE policies. For example, Chen, Curdia, and Ferrero (201 2), building on the work of Andres, L6pez-Salido, and Nelson (2004), simulated the effects of a QE program in an estimated DSGE model with segmented financial markets and a transaction cost that limits arbitrage. 34 This trans­ action cost appears as a wedge between one-period returns on the short-term and long­ term government bonds, and this wedge is assumed to depend on the maturity structure of publicly held government debt. Central bank balance sheet policies that alter the ratio of long-term to short-term debt held by the public affect the wedge between long-term rates and short-term rates. The resulting interest rate adjustments affect consumption behavior and real economic activity. The simulation results of Chen, Curdia, and Ferrero (201 2) seem consistent with earlier findings that very large QE policies are necessary to move interest rate premiums significantly. 35 They conclude, "Asset purchase programmes are in principle effective at stimulating the economy because of limits to arbitrage and market segmentation between short-term and long-term government bonds. The data, however, provide little support for these frictions to be pervasive" (F3 13). Another example of a DSGE model developed to investigate QE policies is that of Carlstrom, Fuerst, and Paustian (2014). Their model incorporated market segmentation and because of moral hazard issues, the net worth of financial intermediaries limits the ability of these institutions to arbitrage away the spread between long-term rates and deposit rates. 3 4 . The models o f Andres, L6pez-Salido, and Nelson (2004) and Chen, Cordia, and Ferrero (20 12) are discussed in chapter 1 1 . 35. For example, they estimated that a commitment to keep the short-term rate at zero for four quarters combined with an LSAP of $600 billion raises GDP growth by 0 . 1 3 percent at an annual rate and increases inflation by 3 basis points. The effects of LSAP are similar to a 25 basis point cut in the short-term rate, but (see their Figure 5, p. 3 1 3) it is interesting that the interest rate cut has a large impact on GDP growth but only a tiny impact on the 1 0-year rate, raising questions about the transmission channel of monetary policy in the model. Evidence on Money, Prices, and Output 39 They also assumed that new investment is financed with long-term nominal debt, arguing that this leads to larger effects of QE policies because investment is more interest sensi­ tive than is the consumption spending that is the focus of the segmented market's model of Chen, Curdia, and Ferrero (2012). 36 Financial intermediaries are the sole purchasers of long-term government bonds and investment bonds, but these are perfect substitutes from the perspective of the intermediaries, so they carry the same yield. Thus, QE policies that lower long-term rates on government debt automatically lower interest rates on private debt used to finance investment. DelNegro et al. (201 6) also developed a DSGE model to assess the Fed's policies. They found that the Fed's provision of liquidity during the financial crisis of 2008-2009 helped avert another Great Depression. An alternative approach to specifying a DSGE model is provided by Baumeister and Benati (2013). They used a time-varying VAR that allows for stochastic volatility to esti­ mate the impact of a decline in the long-term interest rate relative to the short-term policy rate. They then used estimates of the impact of QE policies in the United States and the United Kingdom in reducing long-term interest rates to obtain an estimate of the effects of these policies on inflation and output. Baumeister and Benati argued that for both countries the QE policies significantly reduce the risks of a major contraction. Earlier, the shadow interest rate that Wu and Xia (2016) constructed from a term struc­ ture model as a measure of monetary policy was discussed. Wu and Xia found that the impact of their shadow interest rate on macroeconomic variables was similar to the esti­ mated impact of the funds rate target in the prior zero interest rate period. They used their shadow rate term structure model in three ways to estimate the impact of uncon­ ventional monetary policies on the real economy. First, using their shadow rate in a VAR, they identified the estimated monetary shocks as reflecting unconventional policies. Setting the shocks to zero, they found the shadow rate would have been 0.4 percent higher during 201 1-2013. They attributed this to unconventional policy generating expansionary shocks, leading the actual shadow rate to be below the counterfactual, no-shock path. However, the effects on the real economy were small. Without these shocks, unemployment in December 201 3 would have been 6.83 percent rather than the actual 6.70 percent. The index of indus­ trial production would have been 101.0 rather than the actual 101 .8. Housing starts would have been 988,000 rather than the actual 999,000. They concluded that unconventional policy succeeded in stimulating the economy, but the effects seemed small. Second, they considered a counterfactual exercise in which the shadow rate never falls below a lower bound. In this case, they concluded the unemployment rate would have been 1 percentage point higher. Third, they estimated the impact of forward guidance by simulating expected lift-off dates (dates when the shadow rate is expected to exceed a lower bound). They found that a one-year increase in the expected time until lift-off leads to a 0.25 percent decrease in 36. Since debt is issued in nominal terms, inflation has real effects even with flexible prices. Chapter 1 40 the unemployment rate (but the impulse response function was not statistically significant at the 10 percent level). Overall, they concluded this has roughly the same effect as a 35 basis point decline in the policy rate. 1.5 Summary The consensus from the empirical literature on the long-run relationship between money, prices, and output is clear. Money growth and inflation essentially display a correlation of 1 ; the correlation between money growth or inflation and real output growth is probably close to 0, although it may be slightly positive at low inflation rates and negative at high rates. The consensus from the empirical literature on the short-run effects of money is that exogenous monetary policy shocks produce hump-shaped movements in real economic activity. The peak effects occur after a lag of several quarters (as much as two or three years in some of the estimates) and then die out. The exact manner in which policy is measured makes a difference, and using an incorrect measure of monetary policy can significantly affect the empirical estimates obtained. There is less consensus, however, on the role played by the systematic feedback responses of monetary policy. Structural econometric models have the potential to fill this gap, and they are widely used in policymaking settings. Disagreements over the "true" structure and the potential dependence of estimated relationships on the policy regime have, however, posed problems for the structural modeling approach. A major theme of the next 1 1 chapters is that the endogenous response of monetary policy to economic devel­ opments can have important implications for the empirical relationships observed among macroeconomic variables. Finally, balance sheet policies that many central banks implemented during and after the 2008-2009 financial crisis appear to have been effective in lowering long-term interest rates. There is more uncertainty about the exact channels through which these policies affect the general level of economic activity. In addition, the conclusions of Bernanke, Reinhart, and Sack (2004) appear to have been supported by more recent work: large-scale balance sheet policies are required to have even modest effects on the real economy. 2 2.1 Money -in-the-Utility Function Introduction The neoclassical growth model due to Ramsey (1928) and Solow (1956) provides the basic framework for much of modem macroeconomics. Solow's growth model has just three key ingredients: a production function allowing for smooth substitutability between labor and capital in the production of output; a capital accumulation process in which a fixed fraction of output is devoted to investment each period; and a labor supply process in which the quantity of labor input grows at an exogenously given rate. Solow showed that such an economy would converge to a steady-state growth path along which output, the capital stock, and the effective supply of labor all grew at the same rate. When the assumption of a fixed savings rate is replaced by a model of forward-looking households choosing savings and labor supply to maximize lifetime utility, the Solow model becomes the foundation for dynamic stochastic general equilibrium (DSGE) mod­ els of the business cycle. Productivity shocks or other real disturbances affect output and savings behavior, with the resultant effect on capital accumulation propagating the effects of the original shock over time in ways that can mimic some features of actual business cycles (see Cooley 1995). The neoclassical growth model is a model of a nonmonetary economy, and while goods are exchanged and transactions must be taking place, there is no medium of exchange­ that is, no "money"-used to facilitate these transactions. Nor is there an asset like money that has a zero nominal rate of return and is therefore dominated in rate of return by other interest-bearing assets. To employ the neoclassical framework to analyze monetary issues, a role for money must be specified so that the agents will wish to hold positive quantities of money. A positive demand for money is necessary if, in equilibrium, money is to have positive value. 1 1 . This is just another way of saying that we would like the money price of goods to be bounded. If the price of goods in terms of money is denoted by P, then 1 unit of money will purchase 1 / P units of goods. If money has positive value, l jP > 0, and P is bounded (0 < P < oo). Bewley ( 1 983) referred to the issue of why money has positive value as the Hahn problem (Hahn 1 965). 42 Chapter 2 Fundamental questions in monetary economics are the following: How should the demand for money be modeled? How do real economies differ from Arrow-Debreu economies in ways that give rise to a positive value for money? Three general approaches to incorporating money into general equilibrium models have been followed: (1) assume that money yields direct utility by incorporating money balances into the utility functions of the agents of the model (Sidrauski 1967); (2) impose transaction costs of some form that give rise to a demand for money, by making asset exchanges costly (Baumol 1952; Tobin 1956), requiring that money must be used for certain types of transactions (Clower 1967; Lagos and Wright 2005), assuming that time and money can be combined to pro­ duce transaction services that are necessary for obtaining consumption goods (Brock 1974; McCallum and Goodfriend 1987; Croushore 1993), or assuming that direct barter of com­ modities is costly (Kiyotaki and Wright 1989); or (3) treat money like any other asset used to transfer resources intertemporally (Samuelson 1 958; Sims 201 3). All three approaches involve shortcuts; some aspects of the economic environment are simply specified exogenously to introduce a role for money. This can be a useful device, allowing one to focus on questions of primary interest without being unduly distracted by secondary issues. But confidence in the ability of a model to answer the questions brought to it is reduced if exogenously specified aspects appear to be critical to the primary issue. An important consideration in evaluating different approaches is to determine whether con­ clusions generalize beyond the specific model or depend on the exact manner in which a role for money has been introduced. Subsequent examples include results that are robust, such as the connection between money growth and inflation, and others that are sensitive to the specification of money's role, such as the impact of inflation on the steady-state capital stock. This chapter develops the first of the three approaches by incorporating into the basic neoclassical model agents whose utility depends directly on their consumption of goods and their holdings of money. 2 Given suitable restrictions on the utility function, such an approach can guarantee that, in equilibrium, agents choose to hold positive amounts of money and money is positively valued. The money-in-the-utility function (MIU) model developed in this chapter is originally due to Sidrauski (1967), and it has been used widely. 3 It can be employed to examine some of the important issues in monetary economics: the relationship between money and prices, the effects of inflation on equilibrium, and the optimal rate of inflation. To better understand the role of money in such a framework, a linear approximation to the model is presented. This approximation can be used to derive 2. The second approach, focusing on the transaction role of money, is discussed in chapter 3. The third approach has been developed primarily within the context of overlapping generation models ; see Sargent ( 1 987) or Champ, Freeman, and Haslag (20 1 6). 3 . See Patinkin ( 1 965, ch. 5) for an earlier discussion of an MIU model, although he did not integrate capital accumulation into his model. However, the first-order condition for optimal money holdings that he presented (see his eq. I, p. 89) is equivalent to the one derived in the next section. Money-in-the-Utility Function 43 some analytical implications and to study numerically the MIU model's implications for macrodynamics. 2.2 The Basic MIU Model To develop the basic MIU approach, uncertainty and any labor-leisure choice are initially ignored to focus on the implications of the model for money demand, the value of money, and the costs of inflation. Suppose the utility function of the representative household takes the form where z1 is the flow of services yielded by money holdings and c1 is time t per capita consumption. Utility is assumed to be increasing in both arguments, strictly concave, and continuously differentiable. The demand for monetary services is always positive if one assumes that limz--> 0 Uz(C , z) = for all c, where Uz = au(c, z) I az. What constitutes z1? To maintain the assumption of rational economic agents, what enters the utility function cannot just be the number of dollars (or euro or yen) that the individual holds. What should matter is the command over goods that are represented by those dollar holdings, or some measure of the transaction services, expressed in terms of goods, that money yields. In other words, z should be related to something like the number of dollars, M, times their price, 1 I P, in terms of goods: M (1 I P) = M I P. If the service flow is proportional to the real value of the stock of money and N1 is the population, then z can be set equal to real per capita money holdings: Mt Zt = P N = m t . I t To ensure that a monetary equilibrium exists, it is often assumed that for all c, there exists a finite m > 0 such that um ( c, m) ::::: 0 for all m > m. This means that the marginal utility of money eventually becomes nonpositive for sufficiently high money balances. The role of this assumption is made clear later, when the existence of a steady state is discussed. It is, however, not necessary for the existence of equilibrium, and some common functional forms employed for the utility function (which are used later in this chapter) do not satisfy this condition. 4 The assumption that money enters the utility function is often criticized on the grounds that money itself is intrinsically useless (e.g., paper currency) and that it is only through its use in facilitating transactions that it yields valued services. Approaches that emphasize the transaction role of money are discussed in chapter 3, but models in which money helps 00 -- 4. For example, u(c, m) = log e + b log m does not exhibit this property, since Um = bjm > 0 for all finite m. Chapter 2 44 to reduce the time needed to purchase consumption goods can be represented by the MIU approach adopted in this chapter. 5 The representative household is viewed as choosing time paths for consumption and real money balances subject to budget constraints specified later, with total utility given by 00 00 L f3 1 Ur L f3 1 u(ct, mr ), (2. 1) t=O t=O where 0 < f3 < 1 is a subjective rate of discount. Equation (2. 1 ) implies a much stronger notion of the utility provided by holding money than simply that the household would prefer having more money to less money. If the marginal utility of money is positive, then (2. 1 ) implies that, holding constant the path of real consumption for all t, the individual's utility is increased by an increase in money holdings. That is, even though the money holdings are never used to purchase consumption, they yield utility. This should seem strange; one usually thinks the demand for money is instrumental in that money is held to engage in transactions leading to the purchase of the goods and services that actually yield utility. All this is just a reminder that the money-in­ the-utility function may be a useful shortcut for ensuring that there is a demand for money, but it is just a shortcut. To complete the specification of the model, assume that households can hold money, bonds that pay a nominal interest rate it. and physical capital. Physical capital produces output according to a standard neoclassical production function. Given its current income, its assets, and any net transfers received from the government ( r1 ), the household allocates its resources among consumption, gross investment in physical capital, and gross accumu­ lation of real money balances and bonds. If the rate of depreciation of physical capital is 8, the aggregate economywide budget constraint of the household sector takes the form (1 + it- l )Bt -1 Mt- 1 Mt Bt = Ct + Kt + - + - , (2.2) Yt + TtNt + (1 - 8)Kt -1 + + Pr Pt Pt Pt where Y1 is aggregate output, Kr -1 is the aggregate stock of capital at the start of period t, and r1N1 is the aggregate real value of any lump-sum transfers or taxes. The timing implicit in this specification of the MIU model assumes that it is the house­ hold's real money holdings at the end of the period, Mt! Pt. after having purchased con­ sumption goods, that yield utility. Carlstrom and Fuerst (2001) criticized this timing assumption, arguing that the appropriate way to model the utility from money is to assume it is money balances available before the purchase of consumption goods that yield utility. As they demonstrated, alternative timing assumptions can affect the correct definition of the opportunity cost of holding money and whether multiple real equilibria can be ruled W = = -- 5. Brock ( 1 974), for example, developed two simple transaction stories that can be represented by putting money directly in the utility function. See also Feenstra ( 1 986). Money-in-the-Utility Function 45 out. Because it is standard in the MIU approach to assume that it is end-of-period money holdings that yield utility, this assumption is maintained in the development of the model. 6 The aggregate production function relates output Y1 to the available capital stock K1_ 1 and employment N1: Y1 = F(K1- t , N1) . 7 Assuming this production function is linear homo­ geneous with constant returns to scale, output per capita y1 is a function of the per capita capital stock kt -1 :8 kt- 1 (2.3) Yt = f 1 n ' + where n is the population growth rate (assumed to be constant). Note that output is pro­ duced in period t using capital carried over from period t - 1 . The production function is assumed to be continuously differentiable and to satisfy the usual Inada conditions Uk ::: 0, !kk ::S O, limk---> 0 fk(k) = oo, limk---> oo fk (k) = 0). Dividing both sides of the budget constraint (2.2) by population Nt. the per capita version becomes ( 1 it t)bt t + m l- 1 1-8 kt k1 - 1 + + - w1 f - l + r1 + 1 +n 1 +n ( 1 + 1Tt ) ( 1 + n) (2.4) = Ct + kt + mt + bt, where n1 is the rate of inflation, b1 = BtlP1Nt. and m1 = MtfP1N1• The representative household's problem is to choose paths for c1, kt. b1, and m1 to max­ imize (2. 1) subject to (2.4). This is a problem in dynamic optimization, and it is conve­ nient to formulate the problem in terms of a value function. The value function gives the maximized present discounted value of utility that the household can achieve by optimally choosing consumption, capital holdings, bond holdings, and money balances, given its cur­ rent state. 9 The state variable for the problem is the household's initial level of resources w1, and the value function is defined by (2.5) V(w1) = max {u(c1, m1) + ,B V(w1+1 )} , = ( ) ( ) ( ) c 1 , kr , b r , m r where the maximization is subject to the budget constraint (2.4) and 1 8 (1 + i1)bt + mt kt rt+t + - k1 + wt+1 = f + l +n l +n (1 + nl+ t ) (1 + n) · ( ) ( ) 6. Problems 1 and 2 at the end of this chapter ask you to derive the first-order conditions for money holdings under an alternative timing assumption. 7. Since any labor-leisure choice is ignored in this section, N1 is used interchangeably for population and employment. 8. That is, if Y1 = F(K1_ 1 , Nt ), where Y is output, K is the capital stock, and N is labor input, and F(I.. K , I..N ) = I.. F (K,N) = I.. Y, we can write YtfN1 =. y1 = F(K1_ , , N1)/N1 = F(K1_ J INt. l ) =.f(k1_J I(l + n)), where n = (N1 - N1_ , ) jN1_ 1 is the constant labor force growth rate. In general, a lowercase letter denotes the per capita value of the corresponding uppercase variable. 9. Introductions to dynamic optimization designed for economists can be found in Sargent ( 1 987), Lucas and Stokey ( 1989), Dixit ( 1 990), Chiang ( 1 992), Obstfeld and Rogoff ( 1 996), Ljungquist and Sargent (2000), Wickens (2008), and Miao (20 1 4). Chapter 2 46 Using (2.4) to express kt as Wt - Ct - mt - bt and making use of the definition of Wt+I , (2.5) can be written as { u(cr , mt) V(wt ) = max Ct , ht , mt Wt - Ct - mt - bl 1-o +,B V ! + Tt+l + l n (wt - Ct - mt - bt) + l+n (1 + it)bt + mt + (l + nt+J ) (l + n) ' with the maximization problem now an unconstrained one over Ct, bt, and mt. The first­ order necessary conditions for this problem are (( )} ) ( ) (2.6) 1 + it 1 ! kt [ ) ( + l - o ] = 0, --k ----n (l ( 1 + nt+d + n) 1 + n 1+ l ] ,B Vw (Wt+l ) Um (Ct, m l ) - __ [fk ( � ) + 1 - 0 - l l +n 1 +n + nt+l (2.7) = 0, together with the transversality conditions lim ,8 1 A tXt = 0, for = k, b, m, 1-+ 00 x (2.8) (2.9) where A t is the marginal utility of period t consumption. The envelope theorem implies ,8 [fk (�) ] + 1 - o Vw (Wt+L ), _ Vw (Wt) = _ l +n l +n which together with (2.6) yields (2. 10) The first-order conditions have straightforward interpretations. Since initial resources w1 must be divided between consumption, capital, bonds, and money balances, each use must yield the same marginal benefit at an optimum allocation J 0 Using (2.6) and (2. 10), (2.8) can be written as ,B uc ( Ct+l , mt+I ) = (2. 1 1) Uc (Ct, mt), Um ( Ct, mt) + ( 1 + ni+I ) (l + n) which states that the marginal benefit of adding to money holdings at time t must equal the marginal utility of consumption at time t. The marginal benefit of additional money holdings has two components. First, money directly yields utility Um . Second, real money 10. For a general equilibrium analysis of asset prices in an MIU framework, see LeRoy ( 1 984a; 1 984b). Money-in-the-Utility Function 47 balances at time t add l I ( l + 1l"t+ 1 ) ( 1 + n) to real per capita resources at time t + 1 ; this addition to Wt+ 1 is worth Vw ( Wt+ 1 ) at t + 1 , or f3 Vw ( Wt+ 1 ) at time t. Thus, the total marginal benefit of money at time t is um (ct. mt)+ f3 Vw (Wt+d f(1 + 1l"t+J ) ( 1 + n) . Equation (2. 1 1) is then obtained by noting that Vw (Wt+l ) = uc ( Ct+l , mt+l ) . From (2.6), (2.7), and (2. 1 1), um (Ct. mt) f3 Uc (Ct+l , mt+l ) =1( 1 + nt+d ( 1 + n) Uc (Ct. mt) Uc (Ct. mt) 1 =1(1 + rt) ( 1 + 1l"t+1 ) it ( 2. 1 2 ) - 1 + it Y where 1 + rt = fk (ktf 1 + n) + 1 - 8 is the real return on capital, and (2.6) implies f3uc ( Ct+l , mt+1 )/uc (ct. mt) = ( 1 + n)/ ( 1 + rt) . Equation (2. 12) also makes use of (2.7), which links the nominal return on bonds, inflation, and the real return on capital. This latter equation can be written as [ - 1 + it = -----­ - -- = ] [, �k C � n ) + 1 - 8 ] ( 1 + 1l"t+l ) = ( 1 + rt) ( l + 1l"t+l ) . (2. 1 3) This relationship between real and nominal rates of interest is called the Fisher relationship after Irving Fisher (1 896). It expresses the gross nominal rate of interest as equal to the gross real return on capital times 1 plus the expected rate of inflation. Note that ( 1 + x) ( 1 + y) 1 + x + y when x and y are small, so (2. 13) is often written as � To interpret (2. 12), consider a very simple choice problem in which the agent must pick and z to maximize u(x , z) subject to a budget constraint of the form x + pz = y, where p is the relative price of z. The first-order conditions imply uz lux = p; in words, the marginal rate of substitution between z and x equals the relative price of z in terms of x. Comparing this to (2. 12) shows that Y can be interpreted as the relative price of real money balances in terms of the consumption good. The marginal rate of substitution between money and consumption is set equal to the price, or opportunity cost, of holding money. The opportu­ nity cost of holding money is directly related to the nominal rate of interest. The household could hold one unit less of money, purchasing instead a bond yielding a nominal return of i; the real value of this payment is i/ ( 1 + n ), and since it is received in period t + 1 , its present value is i/[(1 + r) (1 + n)] = i/ (1 + i) . l l Since money is assumed to pay no inter­ est, the opportunity cost of holding money is affected both by the real return on capital and x I I . Suppose households gain utility from the real money balances they have at the start of period t rather than the balances they hold at the end of the period, as has been assumed. Then the marginal rate of substitution between money and consumption will be set equal to i1 (see Lucas 1 982; Carlstrom and Fuerst 200 1 ) . See also problem 1 at the end of this chapter. Chapter 2 48 the rate of inflation. If the price level is constant (so n = 0), then the forgone earnings from holding money rather than capital are determined by the real return to capital. If the price level is rising (n > 0), the real value of money in terms of consumption declines, and this adds to the opportunity cost of holding money. In deriving the first-order conditions for the household's problem, it could have been equivalently assumed that the household rented its capital to firms, receiving a rental rate of rk. and sold its labor services at a wage rate of Household income would then be rkk + (expressed on a per capita basis and ignoring population growth). With competitive firms hiring capital and labor in perfectly competitive factor markets under constant returns to scale, rk = f' (k) and = f(k) - kf' (k) , so household income would be rkk + = fk(k)k + [f(k) - kfk(k)] = f(k), as in ( 2.4) 12 While this system could be used to study analytically the dynamic behavior of the econ­ omy (e.g., Sidrauski 1967; Fischer 1979; Blanchard and Fischer 1989), the properties of the steady-state equilibrium are the initial focus. And because the main focus here is not on the exogenous growth generated by population growth, it provides some slight simplification to set = 0 in the following. After examining the steady state, we study the dynamic prop­ erties implied by a stochastic version of the model, a version that also includes uncertainty, a labor-leisure choice, and variable employment. w w. w _ w n 2.2.1 Steady-State Equilibrium Consider the properties of this economy when it is in a steady-state equilibrium with = 0 and the nominal supply of money growing at the rate e . Let the superscript ss denote values evaluated at the steady state. The steady-state values of consumption, the capital stock, real money balances, inflation, and the nominal interest rate must satisfy the first­ order necessary conditions for the household's decision problem given by (2.6)-(2.8), the economywide budget constraint, and the specification of the exogenous growth rate of M. Note that with real money balances constant in the steady state, it must be that the prices are growing at the same rate as the nominal stock of money, or n ss = e . 13 Using (2. 1 0) to eliminate V ( w ss ), the equilibrium conditions can be written as n w (2. 14) (2. 15) 12. This follows from Euler's theorem: i f the aggregate constant-returns-to-scale production function i s F(N, K) , then F(N, K) = FNN + FKK. In per capita terms, this becomes f(k) = FN + FKk = w + rk if labor and capital are paid their marginal products. 1 3 . If the population is growing at the rate n, then I + rr ss = (l + 8 ) / ( 1 + n). Money-in-the-Utility Function 49 (2. 16) J(kss ) + r ss + (1 ss 8 ) kss + m (2. 17) c ss + kss + m ss ' 1+8 where w ss J(Ps ) + r ss + (1 - 8 ) ps + m ss /(1 + n). In (2. 14)-(2. 17) use has been made of the fact that in this representative agent model, borrowing and lending must equal zero in equilibrium, b 0. Equation (2. 15) is the steady-state form of the Fisher relationship linking real and nominal interest rates. This can be seen by noting that the real return on capital (net of depreciation) is rss fk (kss ) - 8, so (2. 15) can be written as _ __ = = = = (2. 1 8) Notice that in (2. 14)-(2. 17) money appears only in the form of real money balances. Thus, any change in the nominal quantity of money that is matched by a proportional change in the price level, leaving m ss unchanged, has no effect on the economy's real equilibrium. This is described by saying that the model exhibits neutrality of money. One­ time changes in the level of the nominal quantity of money affect only the level of prices. If prices do not adjust immediately in response to a change in M, then a model might display non-neutrality in the short run but still exhibit monetary neutrality in the long run, once all prices have adjusted. In fact, this is the case with the models used in chapters 5-12 to examine issues related to short-run monetary policy. Dividing (2. 14) by Uc ( C ss , m ss ) yields 1 - f3 [fkCkss ) + 1 - 8 ] 0, or 1 (2. 19) fk(kss ) � - 1 + 8. = = This equation defines the steady-state capital-labor ratio ps as a function of f3 and 8 . If the production function i s Cobb-Douglas, say, f (k) ka for 0 < a .:::= 1 , then fk (k) aka - 1 and = kss _ [ 1 + {3a(/38 - 1) ] l�a = (2.20) What is particularly relevant for our purposes is the implication from (2. 19) that the steady­ state capital-labor ratio is independent of (1) all parameters of the utility function other than the subjective discount rate f3, and (2) the steady-state rate of inflation n ss . In fact, kss depends only on the production function, the depreciation rate, and the discount rate. It is independent of the rate of inflation and the growth rate of money. Because changes in the nominal quantity of money are engineered in this model by mak­ ing lump-sum transfers to the public, the real value of these transfers must equal (M1 M1_ J )fP1 = 8M1_ J / P1 = ti m1 - 1 / ( 1 + n1) . Hence, steady-state transfers are given by Chapter 2 so r55 = em ss I (1 + JT55) = em ss I (1 + 8), and the budget constraint (2. 17) reduces to the economy's resource constraint (2.21) The steady-state level of consumption per capita is equal to output minus replacement investment and is completely determined once the level of steady-state capital is known. Assumingf(k) = k01 , k55 is given by (2.20) and css = [ 1 + f3af3(8 - 1 ) ] 1«a - 8 [ 1 + f3af3(8 - 1) ] I�a Steady-state consumption per capita depends on the parameters of the production function (a), the rate of depreciation (8), and the subjective rate of time discount ({3). The Sidrauski MIU model exhibits a property called the superneutrality of money; the steady-state values of the capital stock, consumption, and output are all independent of the rate of growth of the nominal money stock. That is, not only is money neutral, so that proportional changes in the level of nominal money balances and prices have no real effects, but changes in the rate of growth of nominal money also have no effect on the steady-state capital stock or therefore on output or per capita consumption. Because the real rate of interest is equal to the marginal product of capital, it also is invariant across steady states that differ only in their rates of money growth. Thus, the Sidrauski MIU model possesses the properties of both neutrality and superneutrality. To understand why superneutrality holds, note from (2. 10), Uc = V,.v(w1), so using (2.6), Uc (Ct. mt ) = f3 [fk(kr) + 1 - 8] Uc ( Ct+l , m t+I ) , or Uc (Ct+l , m t+l ) Uc (Ct . m r) 1 1{3 fk(kr) + 1 - 8 (2.22) Recall from (2. 19) that the right side of this expression is equal to 1 in the steady state. If k < k55 so that fk(k) > fk(k55), then the right side is smaller than 1 , and the marginal utility of consumption declines over time. It is optimal to postpone consumption to accu­ mulate capital and have consumption grow over time (so Uc declines over time). As long as fk + 1 - 8 > 1 1 {3, this process continues, but as the capital stock grows, the marginal prod­ uct of capital declines until eventually fk(k) + 1 - 8 = 1 1 f3 . The converse holds if k > k55• Consumption remains constant only when fk + 1 - 8 = llf3. If an increase in the rate of money growth (and therefore an increase in the rate of inflation) were to induce households to accumulate more capital, this would lower the marginal product of capital, leading to a situation in which fk + 1 - 8 < 1 I f3 . Households would then want their consumption path to decline over time, so they would immediately attempt to increase current consumption and reduce their holdings of capital. The value of k55 consistent with a steady state is inde­ pendent of the rate of inflation. Money-in-the-Utility Function 51 What is affected by the rate of inflation? One thing to expect is that the interest rate on any asset that pays off in units of money at some future date will be affected; the real value of those future units of money will be affected by inflation, and this will be reflected in the interest rate required to induce individuals to hold the asset, as shown by (2. 1 3). To understand this equation, consider the nominal interest rate that an asset must yield if it is to give a real return of r1 in terms of the consumption good. That is, consider an asset that costs 1 unit of consumption in period t and yields (1 + r1) units of consumption at t + 1 . In units of money, this asset costs P1 units of money at time t. Because the cost of each unit of consumption at t + 1 is P1+ 1 in terms of money, the asset must pay an amount equal to (1 + r)P1+ 1 . Thus, the nominal return is [ (1 + r1)Pt+ 1 - P1 J I P1 = ( 1 + r1) (1 + rr1+ 1 ) - 1 i1• In the steady state, 1 + rss = 1 1 {3 and rr ss = 8 , so the steady­ state nominal rate of interest is given by [ ( 1 + ())I f3] - 1 and varies (approximately) one­ to-one with inflation. 14 = Existence of the Steady State To ensure a steady-state monetary equilibrium, there must exist a positive but finite level of real money balances m55 that satisfies (2. 1 2), evaluated at the steady-state level of consump­ tion. If utility is separable in consumption and money balances, say, u(c, m) = v(c) + ¢ (m) , this condition can be written as ¢m (m ss ) = y ss vc (css ) . The right side of this expression is a non-negative constant; the left side approaches oo as m -+ 0. If ¢m (m) :S 0 for all m greater than some finite level, a steady-state equilibrium with positive real money balances is guaranteed to exist. This was the role of the earlier assumption that the marginal util­ ity of money eventually becomes negative. Note that this assumption is not necessary; ¢ (m) = log m yields a positive solution to (2. 1 2) as long as 155vc (c55) > 0. 15 When util­ ity is not separable, one can still write (2. 1 2) as Um (c55 , m ss ) = y ss uc ( c55 , m s5 ) . If U cm < 0, so that the marginal utility of consumption decreases with increased holdings of money, both Urn and U c decrease with m and the solution to (2. 1 2) may not be unique; multiple steady-state equilibria may exist. 16 However, it may be more plausible to assume money and consumption are complements in utility, an assumption that would imply Ucm ::::_ 0. When u(c, m) = v(c) + ¢ (m), the dynamics of real balances around the steady state can be described easily by multiplying both sides of (2. 1 1) by M1 and noting that M1+ 1 = ( 1 + ())Mt : (2.23) 14. Outside of the steady state, the nominal rate can still be written as the sum of the expected real rate plus the expected rate of inflation, but there is no longer any presumption that short-run variations in inflation will leave the real rate unaffected. 1 5 . I ss > 0 requires that iss > 0. 16. For more on the conditions necessary for the existence of monetary equilibria, see Brock ( 1 974; 1 975) and Bewley ( 1 983). Chapter 2 52 which gives a difference equation in m. The properties of this equation have been examined by Brock (1974) and Obstfeld and Rogoff (1983; 1986). A steady-state value for m satisfies B(m ss ) = A(m ss ) . The functions B(m) and A(m) are illustrated in figure 2. 1 . B(m) is a straight line with slope f3vc (css )/( l + 8 ) . A(m) has slope (vc - c/Jm - c/Jmmm) . For the case drawn, limm-.o ¢mm = 0, so there are two steady-state solutions to (2.23), one at m* and one at 0. Only one of these involves positive real money balances (and a positive value for money). If limm-.o ¢mm = m > 0, then limm-.oA(m) < 0 and there is only one solution. Paths for m1 originating to the right of m* involve mr+s -+ oo as s -+ oo. When e :::: 0 (non-negative money growth), such explosive paths for m, involving a price level going to zero, violate the transversality condition that the discounted value of asset holdings must go to zero. 17 More recently, Benhabib, Schmitt-Grohe, and Uribe (2001b; 2001a; 2002) noted that the existence of an effective lower bound on the nominal interest rate may not allow ruling out paths that begin to the right of m* . Suppose the effective lower bound is at zero. As the rate of deflation rises along these deflationary paths, the nominal interest . ·· • ·· ·· · ·· · ·· · ·· · · • • · ·· ·· · ·· · • • ·· · ·· · ·· · · · · • • • • · ·· ·· · •·· . ... ·· ··· · · ··· ·· B(m) • • • · ·· ··· ·· · · • • • ··· ··· • ··· ·· · · ·· ··· • ··· ·· · · ··· ·· • ··· · ··· m' m(t) Figure 2.1 Steady-state real balances (separable utility). 1 7 . Obstfeld and Rogoff ( 1 986) showed that any such equilibrium path with an implosive price level violates the transversality condition unless lim111 _. 00 ¢ (m ) = oo. This condition is implausible because it would require that the utility yielded by money be unbounded. See also Obstfeld and Rogoff ( 1983). Money-in-the-Utility Function 53 rate must fall. Once it reaches zero, the process cannot continue, so the economy may find itself in a zero interest rate equilibrium that does not violate any transversality condition. 18 When limm---> O A(m) < 0, paths originating to the left of m* converge to m < 0; but this is clearly not possible because real balances cannot be negative. For the case drawn in figure 2. 1 , however, some paths originating to the left of m* converge to zero without ever involving negative real balances. For example, a path that reaches m" at which A(m") = 0 then jumps to m = 0. Along such an equilibrium path, the price level is growing faster than the nominal money supply (so that m declines). Even if e = 0, so that the nominal money supply is constant, the equilibrium path would involve a speculative hyperinflation with the price level going to infinity. 19 Unfortunately, Obstfeld and Rogoff showed that the conditions needed to ensure limm---> 0 ¢mm = fh > 0, so that speculative hyperinftations can be ruled out, are restrictive. They showed that limm---> O ¢mm > 0 implies limm---> o ¢ (m) = oo ; essentially, money must be so necessary that the utility of the representative agent goes to minus infinity if real balances fall to zero. 20 When paths originating to the left of m* cannot be ruled out, the model exhibits multiple equilibria. For example, suppose the nominal stock of money is held constant, with M1 = Mo for all t > 0. Then there is a rational-expectations equilibrium path for the price level and real money balances starting at any price level Po as long as Mo/ Po < m* . Chapter 4 examines an approach called the .fiscal theory of the price level, which argues that the initial price level may be determined by fiscal policy. - Steady States with a Time-Varying Money Stock The previous section considered the steady state associated with a constant growth rate of the nominal supply of money. Often, particularly when the focus is on the relation­ ship between money and prices, one might be more interested in a steady state in which real quantities such as consumption and the capital stock are constant but the growth rate of money varies over time. Assume that c1 = c * and k1 = k* for all t. Setting popu­ lation growth to zero and using (2. 10), the equilibrium conditions (2.6) and (2.7) can be written as n (2.24) 1 + 1'1 -= [fkC k * ) + 1 - o ] , (1 + lTt+I ) 18. 19. (2.25) See the discussion of the Taylor principle in chapter 8 and of liquidity traps in chapter 1 1 . The hyperinflation is labeled speculative, since it is not driven by fundamentals such as the growth rate of the nominal supply of money. 20. Speculative hyperinflations are shown by Obstfeld and Rogoff ( 1986) to be ruled out if the government holds real resources to back a fraction of the outstanding currency. This ensures a positive value below which the real value of money cannot fall. Chapter 2 54 while (2. 12) implies Um ( c* , m ) Uc ( c* , m t ) and the economy's resource constraint becomes r c* = f(k* ) - ok* (2.26) 0 The evolution of the real stock of money is given by 1 + et (2.27) m mt = 1 + 7Tt t -1 · If e is constant, one has the situation previously studied. There is a steady state with inflation equal to the rate of growth of money (n = 8 ) , and real money balances are constant. With m constant, (2.24) uniquely determines the capital stock such that f3 [fk(kss) + 1 - o ] = 1 . The economy's resource constraint then detennines c* . There may also be steady-state equilibria in the real variables in which m is changing over time. Reis (2007) investigated how monetary policies that allow the money stock to be time-varying can alter the steady-state values of consumption and capital. To understand intuitively how c* and k* could be affected by monetary policy, consider (2.24) for k* > kss . 21 Because of diminishing marginal productivity, f3 [fk (k*) + 1 - o] < 1 , so for (2.24) to hold requires the marginal utility of consumption to rise over time such that U c ( c *, m + ! ) 1 ------- > 1 . (2.28) Uc ( c* , m ) f3 [fk(k*) + 1 - 8] For example, suppose Ucm > 0, so that higher levels of real money balances increase the marginal utility of consumption. Then (2.28) can be satisfied if real money balances grow over time. For real money balances to grow over time, (2. 12) implies that the nom­ inal interest rate must be decreasing, reducing the opportunity cost of holding money. Of course, a steady state that satisfies (2.28) may not be feasible. If the marginal util­ ity of money goes to zero for some m > 0, then such a steady state does not exist. Note also that if utility is separable in consumption and real money balances, (2.24) becomes U c ( c *) = f3 [fk (k*) + 1 - o] U c (c*), which implies k* = kss, and the steady state is inde­ pendent of real money balances. If, following Fischer (1979), the utility function takes the form l( c l- Y m Y ) � (2.29) u(c, m) = , 1 - 1] ( ) r r 2 1 . Recall kss is such that f3 [fkCkss ) + I - 8] = I . Money-in-the-Utility Function with 17 < 1 and according to ={ y E 55 (0, 1), then (2.28) requires that real money balances evolve } r<l-�l I 1 (2.30) --;;;;f3 [fk (k*) + 1 - 8] Rather than characterize the steady state in terms of the growth rate of the nominal stock of money, Reis (2007) examined the behavior of the nominal interest rate directly, since central banks today generally employ a nominal interest rate and not a nominal quantity as their policy instrument. The equilibrium condition (2.26) implicitly defines a money demand function of the form mt = </> ( it. c* ) , mt+I so (2.30) implies the path of the nominal rate must satisfy { } y(l-•1l I <!> C it+J , c * ) 1 <f> (it. c*) f3 [fk(k*) + 1 - 8] With k constant, (2.25) implies the real interest rate, given by ( 1 + i1)/(1 + n1+ t ), is con­ stant, so the required path for the nominal rate also pins down the path followed by the inflation rate. Advancing (2.27) one period then determines the growth rate of the nominal money stock consistent with the specified equilibrium path. Reis discussed how the mone­ tary authority could, through a policy of declining nominal interest rates, sustain a steady state in which consumption and output remain above the levels that would be reached under a constant growth rate of money policy. 22 2.2.2 Multiple Equilibria in Monetary Models Section 2.2. 1 considered the stationary, steady-state equilibrium of the MIU model, in which real money balances were constant. With Mt f P1 constant in such an equilibrium, inflation was pinned down by the growth rate of the nominal money supply (perhaps adjusted for income growth) and one-time permanent changes to the level of M1 would produce proportional changes in the price level. These conclusions are typically associated with the quantity theory of money. The discussion of figure 2. 1 suggested the existence of a unique steady state with a constant level of real money balances could not be taken for granted. This section focuses on dynamic paths for the price level and examines whether, given a path for the nominal money supply, there exists a unique equilibrium path for the price level. Or, can there be multiple values of P1 all of which are consistent with the model's equilibrium conditions? 22. Of course, an effective lower bound on the nominal interest rate (conventionally assumed to be zero) would halt the decline in the nominal interest rate when rates reached the effective lower bound. See chapter I I . Chapter 2 56 1. It is convenient to restrict attention to the case of separable utility in which supemeutral­ ity holds and focus on the case in which real consumption and the gross real interest rate are constant, with the latter equal to its steady-state value { r The analysis can also be simplified by assuming the nominal money supply is fixed and equal to Mo . In this case, the key equilibrium condition in the MIU model, (2. 12), can be written as Um ( ��) [ 1 - � ( /�1 ) ] Uc(c). = (2.3 1) This i s a forward difference equation in the price level; does it uniquely determine P1? One solution to (2.3 1) is Pr+i = P* for all i ::': 0, where Um ( �� ) = ( 1 - �) Uc(c), or P* = Mo /u; / ((1 - �) uc(c)). In this equilibrium, the quantity theory holds, and the price level is proportional to the money supply. However, this may not be the only equilib­ rium for the price level. Rewriting (2.3 1) as uc(c) ­ makes explicit that it defines a difference equation in the price level. Because ( ��) > 0, one solution' is characterized by a constant price level P* = ¢ (P*) . 23 Since ::=: 0, it follows that ¢ (P1) > 0. In figure 2.2, ¢ (P1) is shown as an increasing function of P1• Also shown in the figure is the 45° line. Using the fact that P* = ¢ (P*) implies um Umm [ Uc(C)�-Uc(UmC) ��) ] ( = l ' [ Uc(C) - UmUc(c)(�) -UmUm:0()�) (�) ] the slope of ¢ (P1), evaluated at P*, is cp' (P * ) = - ( > 1. P* Thus, ¢ cuts the 45° line from below at P* . Any price path starting at Po > P* is consistent with (2.3 1) and involves a positive rate of inflation. As the figure illustrates, P ---+ oo, but the equilibrium condition (2.3 1 ) is satisfied along this path. As the price level explodes, real money balances go to zero. But this is consistent with private agents' demand for money because inflation and therefore nominal interest rates are rising, lowering the real , 23. From (2.3 1 ) Uc - Urn ( ��) = f3 ( /� ) 1 Uc > 0. Money-in-the-Utility Function . .. . . . .. . .. .. . . .. . .. . . . ·· .. .. ·· · · ·· · ·· · ·· ·· ·· · · ·· · ·· 57 . ·· . P* Figure 2.2 Equilibrium inflationary path with a fixed nominal money supply. demand for money. Any price level to the right of P* is a valid equilibrium. These equilibria all involve speculative hyperinftations. Equilibria originating to the left of P* eventually violate a transversality condition because M/P is exploding as P ---+ 0. 24 By itself, (2.3 1 ) i s not sufficient to uniquely determine the equilibrium value of the initial price level, even though the nominal quantity of money is fixed. Monetary models typically focus on stationary equilibria. In this case, P* is the unique stationary equilibrium for the price level, and the focus is on the properties of this equilib­ rium. 2.2.3 The Interest Elasticity of Money Demand Equation (2. 12) characterizes the demand for real money balances as a function of the nom­ inal rate of interest and real consumption. For example, suppose that the utility function in consumption and real balances is of the constant elasticity of substitution (CES) form: ( ) [ u Ct. m 1 = ac1 -b l + ] ( 1 - a ) m11 -b i=b , I (2.32) 24. As P falls toward zero, the nominal interest rate will eventually be driven to zero, an issue ignored here but explored in chapter l l . Chapter 2 58 with 0 < a < 1 and b > 0, b =/= 1 . Then Um Uc = ) (�) b ' (� a mt and (2. 12) can be written25 as i -i 1 a t (2.33) mt - -Ct . 1 + 1. a In terms of the more common log specification used to model empirical money demand equations, 1 a M i log t = � log (2.34) + log c � log _-. , a b 1+1 PtNt b which gives the real demand for money as a negative function of the nominal rate of interest and a positive function of consumption. 26 The consumption (income) elasticity of money demand is equal to 1 in this specification. The elasticity of money demand with respect to the opportunity cost variable It = ir / ( 1 + it ) is 1 /b. For simplicity, this is often referred to as the interest elasticity of money demand. 27 As b approaches 1 in the limit, the CES specification yields a Cobb-Douglas utility function u(c1, m1) = c�mf -a . Note from (2.34) that in this case the consumption (income) elasticity of money demand and the elasticity with respect to the opportunity cost measure It are both equal to 1 . While the parameter b governs the interest elasticity of demand, the steady-state level of money holdings depends on the value of a. From (2.33), the ratio of real money balances to consumption in the steady state is 28 - ( - ) ( ) -- ( - ) )t (1 + (� a 1+ - l[ ss b ---+ _ n ss ) {3 - i 25. In the limit, as oo, (2.33) implies that m = c . This is then equivalent to the cash-in-advance models examined in chapter 3. 26. The standard specification of money demand would use income in place of consumption; but see Mankiw and Summers ( 1 986). 27. The elasticity of money demand with respect to the nominal interest rate is amt it 1 1 a it tn t b l + it Empirical work often estimates money demand equations in which the log of real money balances is a function of log income and the level of the nominal interest rate. The coefficient on the nominal interest rate is then equal to the semielasticity of money demand with respect to the nominal interest rate (m - 1 3m f 3i), which for (2.34) is 1 /bi( l + i). Note that an increase in the nominal interest rate reduces money demand, but the elasticity is expressed as a positive value. 28. This makes use of the fact that l + iss = (l + r ss ) ( l + n ss ) = (l + n ss ) / f3 in the steady state. Money-in-the-Utility Function 59 The ratio of m ss to css is decreasing in a; an increase in a reduces the weight given to real money balances in the utility function and results in smaller steady-state holdings of money (relative to consumption). Increases in inflation also reduce the ratio of money holdings to consumption by increasing the opportunity cost of holding money. Empirical Evidence on the Interest Elasticity of Money Demand The empirical literature on money demand is vast. See, for example, the references in Judd and Scadding (1982), Laidler (1985), or Goldfeld and Sichel (1990) for earlier surveys. More recent contributions include Lucas (1988), Hoffman and Rasche ( 199 1), Stock and Watson (1993), Ball (2001), Knell and Stix (2005), Teles and Zhou (2005), Bae and De Jong (2007), and Ireland (2009). Ball argued that in postwar samples ending prior to the late 1980s, the high degree of collinearity between output and interest rates made it difficult to obtain precise estimates of the income and interest elasticities of money demand. Based on data from 1946 to 1996, he found the income elasticity of the demand for the M 1 mon­ etary aggregate to be about 0.5 and the interest semielasticity to be about 0.5. An income elasticity less than 1 (the value implied by equation 2.34) is consistent with the findings of Knell and Stix. Teles and Zhou argued that M1 is not the relevant measure of money after 1980 because of the widespread changes in financial regulations. They focused on a monetary aggregate constructed by the Federal Reserve Bank of St. Louis, called money zero maturity (MZM), which measures balances available immediately for transactions at zero cost. Teles and Zhou also assumed an income elasticity of 1 and estimated the interest elasticity of money demand to be 0.24. Holman (1998) directly estimated the parameters of the utility function under various alternative specifications of its functional form, including (2.32), using annual U.S. data from 1 889 to 199 1 . 29 She obtained estimates of b of about 0. 1 and a of about 0.95. This value of b implies an elasticity of money demand equal to I 0. However, in shorter samples, the data fail to reject b = 1 , the case of Cobb-Douglas preferences, indicating that the interest elasticity of money demand is estimated very imprecisely. Using annual data, Lucas (2000) obtained an estimate of 0.5 for the interest elasticity of M1 demand. Chari, Kehoe, and McGrattan (2000) estimated (2.34) using quarterly U.S. data and the M1 definition of money. They obtained an estimate for a of about 0.94 and an estimate of the interest elasticity of money demand of 0.39, implying a value of b on the order of 1 /0.39 2.6. Christiano, Eichenbaum, and Evans (2005) reported an interest semielasticity of 0.96 (the partial of log real money holdings with respect to the gross nominal interest rate), obtained as part of the estimation of a dynamic stochastic general equilibrium (DSGE) model of the United States. � 29. Holman considered a variety of specifications for the utility function, including Cobb-Douglas (b = I ) and nested CES functions of the form given in section 2.5. Chapter 2 60 Hoffman, Rasche, and Tieslau ( 1995) conducted a cross-country study of money demand and found a value of about 0.5 for the U.S. and Canadian money demand interest elastic­ ities, with somewhat higher values for the United Kingdom and lower values for Japan and Germany. An elasticity of 0.5 implies a value of 2 for b. Ireland (2001) estimated the interest elasticity as part of a general equilibrium model and obtained a value of 0. 19 for the pre- 1979 period and 0. 12 for the post- 1979 period. These translate into values for b of 5.26 and 8.33, respectively. The log-log specification for money demand given by (2.33) is consistent with the spec­ ification adopted by Lucas (2000) and is also used by Bae and De Jong (2007). Ireland (2009) focused on what recent data on interest rates and Ml reveal about the appropriate functional form for the money demand equation. He contrasted two alternative functions. The first is a standard log-log specification, in which the log of real money balances rel­ ative to income is related to the log of the nominal interest rate. The second is a semilog specification linking the log of real money balances relative to income to the level of the nominal interest rate: log Mt = ao + log e - � i. PtNt -- Estimated elasticities for the log-log form were in the range of 0.05 to 0.09, corresponding to a value of b in (2.34) ranging from 1 1 to 20. The semilog form yielded a coefficient in the range of 1 .5 to 1 .9 on the level of the interest rate. Ireland found that the semilog specification fits the post- 1980 data for the United States much better than the log-log spec­ ification. The form of the money demand equation and the sensitivity of money demand to the opportunity cost of holding money are important for assessing the welfare costs of inflation (see section 2.3). Reynard (2004) found that an increase in financial market participation had increased the interest elasticity of U.S. money demand. He reported the interest rate elasticity rose from 0.065 for the 1949-1969 period to 0. 1 34 for 1977-1999. Obtaining estimates of the money demand equation is important when monetary policy is implemented through control of a monetary aggregate. The extent to which interest rates adjust in response to a change in the money supply, for example, depends on the interest elasticity of money demand. As many central banks switched during the 1990s to poli­ cies that focused directly on using a short-term market interest rate as the instrument of monetary policy, the money demand equation became less relevant for monetary policy, and interest in estimating money demand equations declined. However, as Ireland (2009) showed, estimates of the welfare cost of inflation can depend importantly on the value of the interest elasticity of demand that is used. Most empirical estimates of the interest elasticity of money demand employ aggregate time series data. At the household level, many U.S. households hold no interest-earning assets, so the normal substitution between money and interest-earning assets as the nominal Money-in-the-Utility Function 61 interest rate changes is absent. As nominal interest rates rise, more households find it worthwhile to hold interest-earning assets. Changes in the nominal interest rate then affect both the extensive margin (the decision whether to hold interest-earning assets) and the intensive margin (the decision of how much to hold in interest-earning assets, given that the household already holds some wealth in this form). Mulligan and Sala-i-Martin (2000) focused on these two margins and used cross-sectional evidence on household holdings of financial assets to estimate the interest elasticity of money demand. They found that the elasticity increases with the level of nominal interest rates and is low at low nominal rates of interest. 2.2.4 Limitations Before moving on to use the MIU framework to analyze the welfare cost of inflation, one needs to consider the limitations of the money-in-the-utility approach. In the MIU model, there is a clearly defined reason for individuals to hold money: it provides utility. However, this essentially solves the problem of generating a positive demand for money by assumption; it doesn't address the reasons that money, particularly money in the form of unbacked pieces of paper, might yield utility. The money-in-the-utility function approach should be thought of as a shortcut for a fully specified model of the transaction technology faced by households that gives rise to a positive demand for a medium of exchange. Shortcuts are often extremely useful. But one problem with such a shortcut is that it does not provide any real understanding of, or possible restrictions on, partial derivatives such as Um or Ucm that play a role in determining equilibrium and the outcome of comparative static exercises. One possible scenario that can generate a rationale for money to appear in the utility function is based on the idea that money can reduce the time needed to purchase consumption goods. This shopping-time model is discussed in chapter 3. 2.3 The Welfare Cost of Inflation Because money holdings yield direct utility and higher inflation reduces real money bal­ ances, inflation generates a welfare loss. This raises two questions: Is there an optimal rate of inflation that maximizes the steady-state welfare of the representative household? How large is the welfare cost of inflation? Some important results on these questions are illus­ trated here, and chapters 4 and 8 provide more discussion on the optimal rate of inflation. The optimal rate of inflation was originally addressed by Bailey (1956) and M. Friedman (1969). Their basic intuition was that the private opportunity cost of hold­ ing money depends on the nominal rate of interest (see 2. 12). The social marginal cost of producing money, that is, running the printing presses, is essentially zero. The wedge that arises between the private marginal cost and the social marginal cost when the nominal rate of interest is positive generates an inefficiency. This inefficiency would be eliminated if the private opportunity cost were also equal to zero, and this is the case if the nominal Chapter 2 62 rate of interest equals zero. But i = 0 requires that n = -r/(1 + r) -r. So the optimal rate of inflation is a rate of deflation approximately equal to the real return on capita1. 30 In the steady state, real money balances are directly related to the inflation rate, so the optimal rate of inflation is also frequently discussed under the heading of the optimal quan­ tity of money (M. Friedman 1969). With utility depending directly on m, one can think of the government choosing its policy instrument e (and therefore n ) to achieve the steady­ state optimal value of m. Steady-state utility is maximized when u(c55, m55) is maximized subject to the constraint that c55 = f(k55) - 8k55 • But because c55 is independent of e , the first-order condition for the optimal 8 is just Um (amj a e) = 0, or Um = 0, and from (2. 12), this occurs when i = 0. 31 The major criticism of this result is due to Phelps (1973), who pointed out that money growth generates revenue for the government-the inflation tax. The implicit assumption so far has been that variations in money growth are engineered via lump-sum transfers. Any effects on government revenue can be offset by a suitable adjustment in these lump-sum transfers (taxes). But if governments only have distortionary taxes available for financ­ ing expenditures, then reducing inflation tax revenue to achieve the Friedman rule of a zero nominal interest rate requires that the lost revenue be replaced through increases in other distortionary taxes. Reducing the nominal rate of interest to zero would increase the inefficiencies generated by the higher level of other taxes needed to replace the lost infla­ tion tax revenue. To minimize the total distortions associated with raising a given amount of revenue, it may be optimal to rely on the inflation tax to some degree. A number of authors have reexamined these results. See, for example, Chari, Christiano, and Kehoe (199 1 ; 1996), Correia and Teles (1996; 1999), and Mulligan and Sala-i-Martin (1997)). The revenue implications of inflation and optimal inflation are major themes of chapter 4. Now let's return to the question, what is the welfare cost of inflation? Beginning with Bailey (1956), this welfare cost has been calculated from the area under the money demand curve (showing money demand as a function of the nominal rate of interest) because this provides a measure of the consumer surplus lost as a result of having a positive nominal rate of interest. Figure 2.3 is based on a money demand function given by ln(m ) = B - � i1• At a nominal interest rate of i* , agents hold real money balances m(i * ), and the shaded area mea­ sures the loss in consumer surplus relative to zero nominal interest rate. The darker shaded area represents the inflation tax revenue the government gains when the nominal interest rate is positive, so only the light shaded area represents a deadweight loss. 32 Consumer surplus is maximized when i = 0. � 30. Since ( l + i) = ( l + r) ( L + n), i = 0 implies n = - rj ( l + r) "" - r. 3 1 . Note that the earlier assumption that the marginal utility of money goes to zero at some finite level of real balances ensures that Um = 0 has a solution with m < oo . The focus here is on the steady state, but a more appropriate perspective for addressing the optimal inflation question would not restrict attention solely to the steady state. The more general case is considered in chapter 4. 32. See chapter 4. Money-in-the-Utility Function 63 c (ij E 0 z 2 m(i') Real money balances Figure 2.3 Welfare costs of inflation as measured by area under money demand curve. Nominal interest rates reflect expected inflation, so calculating the area under the money demand curve provides a measure of the costs of anticipated inflation and is therefore appropriate for evaluating the costs of alternative constant rates of inflation. However, not all the shaded area in the figure represents a deadweight loss to society. The rectangle equal to i * times m(i* ) equals the seigniorage revenue the inflation tax generates for the govern­ ment (see chapter 4). In addition to the loss of consumer surplus when agents economize on their holdings of money when the nominal interest rate is positive, there are costs of inflation associated with tax distortions and with variability in the rate of inflation; these are discussed in the survey on the costs of inflation by Driffill, Mizon, and Ulph ( 1990). In the presence of multiple economic distortions, it may not be optimal to completely elimi­ nate the distortion generated by inflation; doing so may worsen the other distortions. The interactions of inflation with other distortions is discussed in connection with search mod­ els of money demand (chapter 3), the inflation tax when integrated into a model of optimal taxation (chapter 4 ), and the role of inflation in generating relative price distortions when prices are sticky (chapter 8). Lucas (2000) provided estimates of the welfare costs of inflation, starting with the fol­ lowing specification of the instantaneous utility function: (2.35) Chapter 2 64 With this utility function, (2. 1 2) becomes q/ (x) i Um (2.36) - 1' Uc - rp (x) - xrp' (x) - 1 + i where x mjc . 33 Normalize so that steady-state consumption equals 1 ; then u(1 , m) is maximized when Y = 0, implying that the optimal x is defined by rp ' (m*) = 0. Lucas pro­ posed to measure the costs of inflation by the percentage increase in steady-state consump­ tion necessary to make the household indifferent between a nominal interest rate of i and a nominal rate of 0. If this cost is denoted w(Y), it is defined by - = u (1 + w(Y), m(Y)) = u(l , m * ), (2.37) where m(Y) denotes the solution of (2.36) for real money balances evaluated at steady­ state consumption c = 1 . 1 Suppose, following Lucas, that rp (m) = ( 1 + Bm - 1 f , where B is a positive constant. Solving (2.36), one obtains m(i) = B- 5 1 - · 5 . 34 Note that rp ' = 0 requires that m * = oo. But rp (oo) = 1 , and u(l, oo) = 0, so w(Y) is the solution to u ( 1 + w(Y),B- 5 Y - · 5 ) = u( 1 , oo) = 0. Using the definition of the utility function, one obtains 1 + w(Y) = 1 + .JBY, or w ( Y ) = �. (2.38) Based on U.S. annual data from 1900 to 1985, Lucas reported an estimate of 0.0018 for B. Hence, the welfare loss arising from a nominal interest rate of I 0 percent would be J(0.0018) (0. 1 / 1 . 1 ) = 0.013, or just over 1 percent of aggregate consumption. Since U.S. government bond yields were about 10 percent in 1979 and 1980, one can use 1980 aggregate personal consumption expenditures of $2,447. 1 billion to get a rough esti­ mate of the dollar welfare loss (although consumption expenditures include purchases of durables). In this example, 1 .3 percent of $2,447. 1 billion is about $32 billion. Because this is the annual cost in terms of steady-state consumption, one needs the present discounted value of $32 billion. Using a real rate of return of 2 percent, one obtains to $32(1 .02)/ 0.02 $ 1 .632 billion; at 4 percent, the cost would be $832 billion. An annual welfare cost of $32 billion seems a small number, especially when compared to the estimated costs of reducing inflation. For example, Ball (1993) reported a "sacrifice ratio" of 2.4 percent of output per 1 percentage point inflation reduction for the United States. Inflation was reduced from about 10 percent to about 3 percent in the early 1980s, so Ball's estimate would put the cost of this disinflation at approximately 17 percent of GDP (2.4 percent times an inflation reduction of 7 percentage points). Based on a 1980 GDP of = 33. In Lucas's framework, the relevant expression is um f u c = i; problem I at the end of this chapter provides an example of the timing assumptions Lucas employed. 34. Lucas actually started with the assumption that money demand is equal to m = Ai - · 5 for A equal to a constant. He then derived rp (m) as the utility function necessary to generate such a demand function, where B = A 2 . Money-in-the-Utility Function 65 $3,776.3 billion (1987 prices), this would be $642 billion. This looks large when compared to the $32 billion annual welfare cost, but the trade-off starts looking more worthwhile if the costs of reducing inflation are compared to the present discounted value of the annual welfare cost; see also Feldstein (1979). Gillman ( 1995) provided a useful survey of different estimates of the welfare cost of inflation. The estimates differ widely. One important reason for these differences arises from the choice of the base inflation rate. Some estimates compare the area under the money demand curve between an inflation rate of zero and, say, 10 percent. This is incorrect in that a zero rate of inflation still results in a positive nominal rate (equal to the real rate of return) and therefore a positive opportunity cost associated with holding money. Gillman concluded, based on surveying the empirical estimates, that a reasonable value of the welfare cost of inflation for the United States is in the range of 0.85 percent to 3 percent of real GNP per percentage rise in the nominal interest rate above zero, a loss in 2008 dollars of $120 billion to $426 billion per year. 35 It should be clear from figure 2.3 that the size of the area under the demand curve depends importantly on both the shape and the position of the demand curve. For example, if money demand exhibits a constant elasticity with respect to the nominal interest rates, than at low levels of interest rates, further declines in the interest rate generate larger and larger increases in the absolute level of money demand, as illustrated in the figure. The area under the demand curve, and thus the welfare costs of inflation, will correspondingly be large. Lucas (2000) calculated the welfare costs of inflation for two alternative specifications of money demand. The first takes the form ln(m) = ln(A) - TJ ln(i) ; (2 . 39 ) the second takes the form ln(m) = ln(B) - � i. (2.40) Based on annual U.S. data from the period 1900-1994, Lucas obtained estimates of 0.5 for TJ and 7 for �. Ireland (2009) illustrated how these two functional forms have very dif­ ferent curvatures at low nominal interest rates. Real money demand becomes very large as i approaches zero under the log-log specification but approaches the finite limit ln(B) with the semilog version. Equation (2.40) implies that a fall of interest rates from 3 per­ cent to 2 percent produces the same increase in money demand as a fall from 10 percent to 9 percent, unlike the functional form in figure 2.3. If the welfare costs of positive nom­ inal interest rates are measured from the area under the money demand function, these costs appear much larger when using (2 . 39 ) rather than (2.40). For example, at a real inter­ est rate of 3 percent, an average inflation rate of 2 percent carries a welfare cost of just 35. These estimates apply to the United States, which has experienced relatively low rates of inflation. They may not be relevant for high-inflation countries. Chapter 2 66 over 1 percent of income if (2.39) is the correct specification of money demand, but only 0.25 percent if (2.40) is correct. Ireland (2009) argued that the support for the log-log specification comes primarily from two historical periods. The first is the late 1940s, when interest rates were very low and money demand very high (relative to income). The second is the period of the disinflation beginning in 1979 through the early 1980s, when interest rates were very high and money demand was unexpectedly low (often referred to as the period of missing money; see Gold­ feld 1976). Ireland found, using a measure of the money stock that accounts for some of the changes due to financial market deregulation, that the data since 1980 provide much more support for the semilog specification with a small value of � . Rather than the value of 7 estimated by Lucas, Ireland found values below 2. His estimates imply the welfare cost of 2 percent inflation is less than 0.04 percent of income. The Sidrauski model provides a convenient framework for calculating the steady-state welfare costs of inflation, both because the lower level of real money holdings that result at higher rates of inflation has a direct effect on welfare when money enters the utility function and because the supemeutrality property of the model means that the other argument in the utility function, real consumption, is invariant across different rates of inflation. This latter property simplifies the calculation because it is not necessary to account for both variations in money holdings and variations in consumption when making the welfare cost calcula­ tion. However, the area under the demand curve is a partial equilibrium measure of the welfare costs of inflation if supemeutrality does not hold, because steady-state consump­ tion is no longer independent of the inflation rate. Gomme (1993) and Dotsey and Ireland (1996) examined the effects of inflation in general equilibrium frameworks that allow for the supply of labor and the average rate of economic growth to be affected (in models that do not display superneutrality; see section 2.4.2). Gomme found that even though inflation reduces the supply of labor and economic growth, the welfare costs are small because of the increased consumption of leisure that households enjoy. 36 Dotsey and Ireland found much larger welfare costs of inflation in a model that generates an interest elasticity of money demand that matches estimates for the United States. See also De Gregorio (1993) and Imrohorolu and Prescott ( 199 1 ). 2.4 2.4.1 Extensions Interest on Money If the welfare costs of inflation are related to the positive private opportunity costs of holding money, paying explicit interest on money would be an alternative to deflation as 36. The effect of money (and inflation) on labor supply is discussed in section 2.4.2. Money-in-the-Utility Function 67 a means of eliminating these costs. 37 There are obvious technical difficulties in paying interest on cash, but ignoring these, assume that the government pays a nominal interest rate of im on money balances. Assume further that these interest payments are financed by lump-sum taxes. The household's budget constraint, (2.4), now becomes (setting = 0) l + imt- l f(kt- l ) + Tt + ( 1 - 8)kt -l + ( 1 + rt- t)bt- l + mt = Ct + kt + mt + bt. (2.41) 1 + lft -l where Tt represents transfers net of taxes. The first-order condition (2.8) becomes ,8 ( 1 + i;") Vw (Wt+l ) = 0, (2.42) -uc ( Ct, mt) + Um (Ct. mt ) + ( 1 + lft+l ) while (2. 12) is now Um (Ct. mt) it - i;" 1 + it Uc(Ct, mt ) The opportunity cost of money is related to the interest rate gap i - im , which represents the difference between the nominal return on bonds and the nominal return on money. Thus, the optimal quantity of money can be achieved as long as i - im = 0, regardless of the rate of inflation. The optimal quantity of money is obtained with a positive nominal interest rate as long as iss im rss 0. The assumption that interest payments are financed by revenue from lump-sum taxes is critical for this result. Problem 7 at the end of this chapter considers what happens if the government simply finances the interest payments on money by printing more money. n = 2.4.2 = > Nonsuperneutrality Calculations of the steady-state welfare costs of inflation in the Sidrauski model are greatly simplified by the fact that the model exhibits superneutrality. But how robust is the result that money is superneutral? The empirical evidence of Barro (1995) suggests that inflation has a negative effect on growth, a finding inconsistent with superneutrality. 38 Berentsen, Menzio, and Wright (201 1) also argued that there is evidence of a long-run positive rela­ tionship between inflation and unemployment. One channel through which inflation can have real effects in the steady state is introduced if households face a labor supply choice. That is, suppose utility depends on consumption, real money holdings, and leisure: u = u(c, m, l) . (2.43) 37. Since 2008 the Federal Reserve has been paying interest on reserves. The implications this has for the imple­ mentation of monetary policy are discussed in chapter 12. 38. Of course, the empirical relationship may not be causal; both growth and inflation may be reacting to common factors. As noted in chapter I, McCandless Jr. and Weber ( 1 995) found no relationship between inflation and average real growth. Chapter 2 68 The economy's production function becomes (2.44) y = f(k, n), where n is employment. If the total supply of time is normalized to equal l , then n = 1 - l. The additional first-order condition implied by the optimal choice of leisure is U f( C , m, l) (2.45) = f (k, 1 - l) . Uc (c, m, l) n Now, both steady-state labor supply and consumption may be affected by variations in the rate of inflation. Specifically, an increase in the rate of inflation reduces holdings of real money balances. If this affects the marginal rate of substitution between leisure and consumption uf/uc, then (2.45) implies the supply of labor will be affected, leading to a change in the steady-state per capita stock of capital, output, and consumption. But why would changes in money holdings affect U f Uc ? Because money has simply been assumed to yield utility, with real no explanation for the reason, it is difficult to answer this question. Chapter 3 examines a model in which money helps to reduce the time spent in carrying out the transactions necessary to purchase consumption goods; in this case, a rise in inflation would lead to more time spent engaged in transactions, and this would raise the marginal utility of leisure time. But one might expect that this channel is unlikely to be important empirically, so superneutrality may remain a reasonable first approximation to the effects of inflation on steady-state real magnitudes. Equation (2.45) suggests that if uf/ Uc were independent of m, then supemeutrality would hold. This is the case because the steady-state values of k, c, and l could then be found from !!!_ = fn (kss , 1 - zss ), Uc 1 + 8, fk(e, 1 - 155 ) = *- css = f(kss , 1 - zss ) - okss Superneutrality reemerges when the utility function takes the general form u(c, m, l) = v(c, l)g(m) ; in this case uf/uc = vf/vc is independent of m. Variations in inflation affect the agent's holdings of money, but the consumption-leisure choice is not directly affected. As McCallum ( 1990a) noted, Cobb-Douglas specifications of utility, which are quite com­ mon in the literature, satisfy this condition. So with Cobb-Douglas utility, the ratio of the marginal utility of leisure to the marginal utility of consumption is independent of the level of real money balances, and superneutrality holds. Superneutrality also holds if utility is separable in money holdings. Another channel through which inflation can affect the steady-state stock of capital is if money enters directly into the production function (Fischer 1974). Since steady states with different rates of inflation have different equilibrium levels of real money balances, they also lead to different marginal products of capital for given levels of the capital-labor ratio. 0 Money-in-the-Utility Function 69 With the steady-state marginal product of capital determined by 1 I f3 - 1 + o (see 2. 19), the two steady states can have the same marginal product of capital only if their capital­ labor ratios differ. If a MPKj a m > 0 (so that money and capital are complements), higher inflation, by leading to lower real money balances, also leads to a lower steady-state capital stock. 39 This is the opposite of the Tobin effect; Tobin (1965) argued that higher inflation would induce a portfolio substitution toward capital that would increase the steady-state capital-labor ratio (see also Stein 1969; Fischer 1972). For higher inflation to be associated with a higher steady-state capital-labor ratio requires that a MPKj a m < 0 (that is, higher money balances reduce the marginal product of capital; money and capital are substitutes in production). This discussion has, by ignoring taxes, excluded what is probably the most important reason that superneutrality may fail in actual economies. Taxes generally are not fully indexed to inflation and are levied on nominal capital gains instead of real capital gains. Effective tax rates depend on the inflation rate, generating real effects on capital accumula­ tion and consumption as inflation varies (e.g., see Feldstein 1978; 1998; Summers 198 1 ). 2.5 Dynamics in an MIU Model The analysis of the MIU approach has, up to this point, focused on steady-state properties. It is also important to understand the model's implications for the dynamic behavior of the economy as it adjusts to exogenous disturbances. Even the basic Sidrauski model can exhibit nonsuperneutralities during the transition to the steady state. For example, Fischer (1979) showed that for the constant relative risk aversion class of utility functions, the rate of capital accumulation is positively related to the rate of money growth except for the case of log separable utility. Section 2.2. 1 discussed how the steady state can be affected when money growth varies over time (Reis 2007).40 One way to study the model's dynamics is to employ numerical methods to carry out simulations using the model. The results can then be compared to actual data generated by real economies. This approach was popularized by the real business cycle literature (see Cooley 1995). Since the parameters of theoretical models can be varied in ways the characteristics of real economies cannot, simulation methods allow one to answer a vari­ ety of "what if' questions. For example, how does the dynamic response to a temporary 39. That is, in the steady state, fk (kss , m ss ) = {3 - 1 I + 8 , where f(k, m) is the production function and fi denotes the partial derivative with respect to argument i. It follows that dkss jdm ss = -fkm lfkk, so with fkk � 0, sign(dkss jdm ss ) signifkm l · 40. Superneutrality holds during the transition if u (c, m) = ln(c) + b ln(m) . The general class of utility functions Fischer considered is of the form u (c, m) = l �<l> (ca m b ) l - <1> ; log utility obtains when <I> = l . See also Asako ( 1 983), who showed that faster money growth can lead to slower capital accumulation under certain conditions if c and m are perfect complements. These effects of inflation on capital accumulation apply during the transition from one steady-state equilibrium to another; they differ therefore from the Tobin ( 1 965) effect of inflation on the steady-state capital-labor ratio. - = Chapter 2 70 change in the growth rate of the money supply depend on the degree of intertemporal substitution characterizing individual preferences or the persistence of money growth rate disturbances? Numerical solutions allow one to investigate whether simulation results are sensitive to parameter values. In addition, easily adaptable programs for solving lin­ ear dynamic stochastic general equilibrium rational-expectations models are now freely available.41 This section develops a linearized version of an MIU model that also incorporates a labor-leisure choice. This introduces a labor supply decision into the analysis, an important and necessary extension for studying business cycle fluctuations because employment vari­ ation is an important characteristic of cycles. It is also important to allow for uncertainty by adding exogenous shocks that disturb the system from its steady-state equilibrium. The two types of shocks considered are productivity shocks, the driving force in real business cycle models, and shocks to the growth rate of the nominal stock of money. 2.5.1 The Decision Problem The household's decision problem is conveniently expressed using the value function. In studying a similar problem without a labor-leisure choice (see section 2.2), the state could be summarized by the resource variable w1, which included current income. When the household chooses how much labor to supply, current income is no longer predetermined from the perspective of the household. Consequently, income (output) y1 cannot be part of the state vector for period t. Instead, let + it- 1 m r - 1 + Tr a1 = b r- 1 + + lrt + lrt be the household's real financial wealth plus net transfer at the start of period t. If n1 denotes the fraction of time the household devotes to market employment (so that n1 = - lr. where !1 is the fraction of time spent in leisure activities), output per household y1 is given by ( 11 ) ( -1 1- ) 1 where z1 is a stochastic productivity disturbance. Define the value function V(a1, k1_ I ) as the maximum present value of utility the house­ hold can achieve if the current state is (a1, k1- J ) . The value function for the household's decision problem satisfies (2.46) 4 1 . For example, MATLAB programs provided by Harald Uhlig can be obtained from https://www.wiwi.hu­ berlin.de/de/professurenlvwllwipo/research/MATLAB_Toolkit, and Paul Siiderlind's Gauss and MATLAB pro­ grams are available at https://sites.google.com/site/paulsoderlindecon/home/software. Dynare for MATLAB is available at http://www.dynare.org/. Money-in-the-Utility Function 71 where the maximization is over (cr. mt, bt , kt , n t) and is subject to a t+1 = Tt+1 + (2.47) ( l +1 +nt+1it ) bt + l +mntt+l . (2.48) Note that the presence of uncertainty arising from the stochastic productivity and money growth rate shocks means that the expected value of V(a t+1 , kt ) appears in the value func­ tion (2.46). The treatment of a t as a state variable assumes that the money growth rate is known at the time the household decides on Ct , kt, bt, and mt because it determines the current value of the transfer Tt. Assume also that the productivity disturbance Zt is known at the start of period t. Equation (2.47) always holds with equality (as long as Uc > 0); it can be used to elim­ inate kr. and (2.48) can be used to substitute for a t+ 1 , allowing the value function to be rewritten42 as { V(at. kt- d = maxm u (cr, mr, 1 - nt ) ( 1 +l +7Tt+1ir ) br + 1 +mr7Tt+l , f(kr- t , nr, Zt) + ( 1 - o)kt-1 + a t - Ct - br - mr ) } • ( Ct,nt. b t. r + .BErV Tt+t + where this is now an unconstrained maximization problem. The first-order necessary con­ ditions with respect to cr , nr, br, and m1 are (2.49) -UJ(Ct, mr. 1 - nr) + .BEtVk(ar+l , kt ) fn (kr -l , nr. Zt ) = 0, (2.50) .BEt (2.5 1) ( 1 +1 +7Tt+lir ) Va (at+1 , kr) - .BErVk(at+1 , kr) 0, Va (at+l , kr) ] - .BEtVk(a t+l , kt) Um (Cr, mt , 1 - n t ) + .BEt [ 1 + 7Tt+l = = 0, (2.52) and the envelope theorem yields (2.53) (2.54) 42. Rather than introduce firms explicitly, households are assumed to directly operate the production technology. Chapter 2 72 Updating (2.54) one period and using (2.53), one obtains Vk(at+ ] , kt ) = Et [fk(kt , nt+I , Zt+I ) + 1 - o ] Va (at+ ] , kt ) . Now using this to substitute for Vk(a1+I , k1) in (2.49) yields Uc (Ct. mt, 1 - nt) - f3Et [fk(kt , n t+I , Zt+d + 1 - 8 ] Va (a t+I , kt ) = 0. (2.55) When it is recognized that u c ( Ct. m1, 1 - n1) = f3E1Vk(a1+I , k1), (2.52), (2.55), and (2.53) take the same form as (2.8), (2.6), and (2. 10), the first-order conditions for the basic Sidrauski model that did not include a labor-leisure choice. The only new condition is (2.50), which can be written, using (2.49), as u,(ct, mt. 1 - nt) = fn (kt- l , nt. Zt ) . ---Uc (Ct. mt. 1 - n1) -This states that at an optimum, the marginal rate of substitution between leisure and con­ sumption must equal the marginal product of labor. Note that ( 2.49 ) , (2.5 1 ) , and (2.53) imply that uc (Ct+l , m t+l , 1 - n t+l ) . Uc ( Ct. mt, 1 _ nt ) - f3 ( 1 + lt. )Et 1 + 7Tt+] Using this relationship and ( 2.49 ) , one can now write ( 2.50) , ( 2.52) , and ( 2.55 ) as _ ] [ ( ) i1 Um ( Ct , mt. 1 - n1) = uc (c1, mt. 1 - n1) --. , 1 + lt uc (Ct. mt, 1 - nt ) = f3Et( l + rt) Uc ( Ct+I , mt+I , 1 - nt+I ), where in (2.58 ) ( 2.56 ) ( 2.57 ) ( 2.58 ) ( 2.59 ) is the marginal product of capital net of depreciation. In addition, the economy's aggregate resource constraint, expressed in per capita terms, requires that kt = ( 1 - o)kt- 1 + Yt - c1, while the production function is ( 2.60) ( 2.61 ) Finally, real money balances evolve according to 1 + ()1 mt = m I, 1 + 7Tt t where 81 is the stochastic growth rate of the nominal stock of money. ( ) ( 2.62 ) Money-in-the-Utility Function 73 Once processes for the exogenous disturbances Zt and et have been specified, equations (2.56)-(2.62) constitute a nonlinear system of equations to determine the equilibrium val­ ues of the model's seven endogenous variables: y1, cr , k1, m1, n1, rr, n1•43 2.5.2 The Steady State Consider a steady-state equilibrium of this model in which all real variables (including m) are constant and shocks are set to zero. It follows immediately from (2.58) that 1 + r55 = {3 - 1 and from (2.59) that (2.63) Thus, the marginal product of capital is a function only of f3 and 8. If the production function exhibits constant returns to scale,fk depends only on the capital-labor ratio psI n55 • In this case, (2.63) uniquely determines kin. That is, the capital-labor ratio is independent of inflation or the real quantity of money. With constant returns to scale, ¢ (kin) = fIn can be defined as the intensive production function. Then, from the economy's resource constraint, [ ( �:: ) 8 ( �:: ) ] nss = ifynss , css = f(kss , n ss , O) 8kss = ¢ _ _ where ¢ ¢ (P5 ln55) - 8 (k55 ln55) does not depend on anything related to money. Now, (2.56) implies that , m-ss , 1 - n ss ) U f(Css---- = fn (kss , n ss , Q) . SS Uc (c , m SS , � - nSS ) In the constant returns to scale case, Jn depends only on k55 I n55, which is a function of f3 and 8, so using the definition of ¢, one can rewrite this last equation as = (2.64) This relationship provides the basic insight into how money can affect the real equilibrium. Suppose the utility function is separable in money so that neither the marginal utility of leisure nor the marginal utility of consumption depends on the household's holdings of real money balances. Then (2.64) becomes u1 (ify n55, 1 - n ss ) cp kss kss ' kss = ¢ , S nss nss nss s) n Uc (c/Jnss, ] which determines the steady-state supply of labor. Steady-state consumption is then given by ¢n55• Thus, separable preferences imply superneutrality. Changes in the steady-state rate of inflation alter nominal interest rates and the demand for real money balances (see 2.57), ( ) ( ) ( ) _ 43. Since all households are identical, b1 = 0 in equilibrium. Chapter 2 74 but different inflation rates have no effect on the steady-state values of the capital stock, labor supply, or consumption. If utility is nonseparable, so that either U t or Uc (or both) depend on m ss , then money is not superneutral. Variations in average inflation that affect the opportunity cost of holding money affect m ss . Different levels of m ss change the value of n ss that satisfies (2 . 64) . Since 1 + iss = ( 1 + rss ) ( 1 + rr ss ) = (3 - 1 (1 + e ss ), one can rewrite (2.57) as 1 + fJ ss _ f3 iss Um (fP n ss , mss , 1 - n ss ) = = 1 + fJ SS · 1 + iss Uc(¢nss, mss, 1 - nss) This equation, together with (2 . 64) must be jointly solved for m ss and nss . Even in this case, however, the ratios of output, consumption, and capital to labor are independent of the rate of money growth. The steady-state levels of the capital stock, output, and consumption depend on the money growth rate through the effects of inflation on labor supply, with inflation-induced changes in n ss affecting is , css , and ps equiproportionally. The effect of faster money growth depends on how Uc and U t are affected by m. For example, suppose money holdings do not affect the marginal utility of leisure (utm = 0), but money and consumption are Edgeworth complements; higher inflation that reduces real money balances decreases the marginal utility of consumption (ucm > 0). In this case, faster money growth reduces m ss and the marginal utility of consumption. Households substitute away from labor and toward leisure. Steady-state employment, output, and con­ sumption fall. These effects go in the opposite direction if consumption and money are Edgeworth substitutes (ucm < 0). ( 2.5.3 ) The Linear Approximation To further explore the effects of money outside the steady state, it is useful to approx­ imate the model's equilibrium conditions around the steady state. The steps involved in obtaining the linear approximation around the steady state follow the approach of Camp­ bell (1994) and Uhlig (1999). Details on the approach used to linearize (2.56)-(2.62) are discussed in the chapter appendix. With the exception of interest rates and inflation, vari­ ables are expressed as percentage deviations around the steady state. Percentage deviations of a variable q1 around its steady-state value are denoted by q1, where q1 q ss ( l + q1). For interest rates and inflation, rr , lr , and frr denote rr - rss , it - iss , and ITt - 7T ss respectively. 44 In what follows, uppercase letters denote economywide variables, lowercase letters denote random disturbances and variables expressed in per capita terms, and the superscript ss indicates the steady-state value of a variable. However, m, m ss , and m refer to real money balances per capita, whereas M represents the aggregate nominal stock of money. = 44. That is, if the interest rate is 0.0 1 25 at a quarterly rate (i.e., 5 percent at an annual rate) and the steady­ state value of the interest rate is 0.0 1 , then r1 = 0.0 1 25 - 0.01 = 0.0025, i.e., 25 basis points, not (0.0 1 25 0.0 1 ) /0.01 = 0.25, a 25 percent deviation. Money-in-the-Utility Function 75 As is standard, the production function is taken to be Cobb-Douglas with constant returns to scale, so 1Y t = ez'kta- 1 n t a , (2.65) with 0 < a < 1. Note the timing convention in (2.65): the capital carried over from period t - 1 , kt -1 , is available for use in producing output during period t. For the utility function, assume act1-b + (1 - a ) m t1-b (1 - nt ) 1- 'l (2.66) + \II 1 - ry 1 - <l> King, Plosser, and Rebelo (1988) demonstrated that with the exception of the log case, utility must be multiplicatively separable in labor to be consistent with steady-state growth, in which the share of time devoted to work remains constant as real wages rise. Equation (2.66) does not have this property. However, abstracting from growth factors and assuming linear separability in leisure is common in the literature on business cycles. The problems at the end of this chapter present an example using a utility function consistent with growth. The resulting linearized system consists of the exogenous processes for the productivity shock and the money growth rate plus the eight additional equilibrium conditions: the production function, the goods market clearing condition, the definition of the real return on capital, the Euler equation for optimal intertemporal consumption allocation, the first­ order conditions for labor supply and money holdings, the Fisher equation linking nominal and real interest rates, and the money market equilibrium condition. These can be solved for the capital stock, money holdings, output, consumption, employment, the real rate of interest, the nominal interest rate, and the inflation rate. To this system of eight endogenous variables, it is convenient to add investment, Xt , given by [ ] 1 - <1> T=b Xt = kt - ( 1 - 8) kt - 1 , and to define A t as the marginal utility of consumption. The linearized expression for � t is (2.67) where Q 1 = [(b - <l>) y - b] , Q 2 = (b - <l>) (l - y) , y = a(css ) 1 -b / a(css ) 1 -b + (1 - a) (m ss ) 1 -b . [ ] Then, in linearized form, the equilibrium conditions are (see the chapter appendix): (2 . 68) Chapter 2 76 (2.69) (2.70) (2.7 1) (2.72) (2.73) , , mt - Ct = - ( b1 ) ( � 1 iss ) It · - , (2.74) (2.75) A process for the nominal stock of money needs to be specified. In previous sections, e denoted the growth rate of the nominal money supply. Assume that the average growth rate is e ss , and let Ut 8t - e ss be the deviation in period t of the growth rate from its unconditional average value. Then = (2.76) and Ut will be treated as a stochastic process given by (2.77) Ut = PuUt-1 + </J Zt-L + (/Jt , where CfJt is a white noise process and I Pu l < 1 . This formulation allows the growth rate of the money stock to display persistence (if Pu > 0), respond to the real productivity shock z (if ¢ '1- 0), and be subject to random disturbances through the realizations of CfJt · Consistent with the real business cycle literature, a stochastic disturbance to total factor productivity that follows an AR(l ) process is incorporated: (2.78) Zt = Pz Zt-1 + C t . Assume et is a serially uncorrelated mean zero process and I Pz l < 1 . Equation (2.68) is the economy's production function in which output deviations from the steady state are a linear function of the percentage deviations of the capital stock and labor supply from the steady state plus the productivity shock. Equation (2.69) is the resource constraint derived from the condition that output equals consumption plus invest­ ment. Deviations of the marginal product of capital are tied to deviations of the real return by (2.7 1). Equations (2.72)-(2.75) are derived from the representative household's first­ order conditions for consumption, leisure, and money holdings. Changes in the deviation from steady state of real money balances are related by (2.76) to the inflation rate and Money-in-the-Utility Function 77 the growth of the nominal money stock. Finally, the exogenous disturbances for nominal money growth and productivity are given by (2.78) and (2.77). One conclusion follows immediately from inspecting this system. If <I> = h, then Qz = 0 and money no longer affects the marginal utility of consumption. Thus, money drops out of both (2.72) and (2.73) and (2.68)-(2.73) can be solved for y, c, r, k, and ii independently of the money supply process and inflation. This implies that superneutrality characterizes dynamics around the steady state as well as the steady state itself.45 Separability allows the real equilibrium to be solved independently of money and infla­ tion, but the assumption is more commonly used in monetary economics to allow the study of inflation and money growth to be conducted independently of the real equilibrium. When <I> = h, (2.75) and (2.76) constitute a two-equation system in inflation and real money bal­ ances, with u representing an exogenous random disturbance and c and r determined by (2.68)-(2.73) and exogenous to the determination of inflation and real money balances. Using (2.74), equation (2.75) can then be written as hiss hiss ( Mt - Pt ) + Xt , Et nt+l EtPt+l - Pt = - --. ss m t + Xt = - --. ss 1 11 1= A A ( _ ) A ( _ ) A A where M1 represents the nominal money stock (so m1 = M1 - p 1 ) . This is an expectational difference equation that can be solved for the equilibrium path of p for a given process for the nominal money supply and the exogenous variable Xt [ (hiss I (1 - iss )) c1 - r1J . Models of this type have been widely employed in monetary economics (see chapter 4). A second conclusion revealed by the dynamic system is that when money does matter (i.e., when h # <I>), it is only anticipated changes in money growth that matter. To see this, suppose Pu = ¢ = 0, so that u1 = q;1 is a purely unanticipated change in the growth rate of money that has no effect on anticipated future values of money growth. Now consider a positive realization of q;1 (nominal money growth is faster than average). This increases the nominal stock of money. If Pu = ¢ = 0, future money growth rates are unaffected by the value of q;1• This means that future expected inflation, E1rr1+ 1 , is also unaffected. Therefore, a permanent jump in the price level that is proportional to the unexpected rise in the nom­ inal money stock leaving m1 unaffected also leaves (2.68)-(2.75) unaffected. From (2.76), for q;1 to have no effect on m1 requires that rr1 = q;1• So an unanticipated money growth rate disturbance has no real effects and simply leads to a one-period change in the inflation rate (and a permanent change in the price level). Unanticipated money doesn't matter. 46 Now consider what happens when ¢ = 0 but Pu differs from zero. In the United States, money growth displays positive serial correlation, so assume that Pu > 0. A positive shock to money growth (q;1 > 0) now has implications for the future growth rate of money. With = 45 . This result, for the preferences given by (2.66), generalizes the findings of Brock ( 1 974) and Fischer ( 1 979). 46. During the 1 970s macroeconomics was heavily influenced by a model developed by Lucas ( 1 972) in which only unanticipated changes in the money supply had real effects. See chapter 5 . Chapter 2 78 Pu > 0, future money growth will be above average, so expectations of future inflation will rise. From (2.75), however, for real consumption and the expected real interest rate to remain unchanged in response to a rise in expected future inflation, current real money balances must fall. This means that p1 would need to rise more than in proportion to the rise in the nominal money stock. But when S1 2 =/:. 0, the decline in fn1 affects the first-order conditions given by (2.73) and (2.75), so the real equilibrium does not remain unchanged. Monetary disturbances have real effects by affecting the expected rate of inflation. A positive monetary shock increases the nominal rate of interest. Monetary policy actions that increase the growth rate of money are usually thought to reduce nominal inter­ est rates, at least initially. The negative effect of money on nominal interest rates is usually called the liquidity effect, and it arises if an increase in the nominal quantity of money also increases the real quantity of money because nominal interest rates would need to fall to ensure that real money demand also increased. However, in the MIU model, prices have been assumed to be perfectly flexible; the main effect of money growth rate shocks when Pu > 0 is to increase expected inflation and raise the nominal interest rate. Because prices are perfectly flexible, the monetary shock generates a jump in the price level immediately. The real quantity of money actually falls, consistent with the decline in real money demand that occurs as a result of the increase in the nominal interest rate. To actually determine how the equilibrium responds to money growth rate shocks and how the response depends quantitatively on Pu and ¢, one must calibrate the parameters of the model and numerically solve for the rational-expectations equilibrium. 2.5.4 Calibration Thirteen parameters appear in the equations that characterize behavior around the steady state: a, 8, pz , i {3, a, b, TJ, <t> , g ss , Pu, ¢, a; . Some of these parameters are common to stan­ dard real business cycle models; for example, Cooley and Prescott (1995) report values of, in our notation, a (the share of capital income in total income), 8 (the rate of depreci­ ation of physical capital), Pz (the autoregressive coefficient in the productivity process), CJe (the standard deviation of productivity innovations), and f3 (the subjective rate of time discount in the utility function). These values are based on a time period equal to three months (one quarter). Cooley and Prescott's values are adopted except for the depreciation rate 8 ; Cooley and Prescott calibrate 8 = 0.012 based on a model that explicitly incorpo­ rates growth. Here the somewhat higher value of 0.019 given in Cooley and Hansen (1995) is used. The value of ae is set to match the standard deviation of quarterly HP-filtered log U.S. real GDP for the 1985: 1-2014:4 period of 1 . 10 percent. Over this same period, U.S. money growth as measured by Ml averaged 5.54 percent. An annual rate of 5.54 percent would imply a quarterly value of 1 .38 for 1 + g ss , so we set 1 + g ss = 1 .0138 to match M I . Estimating an AR(1) process for M1 growth rate (expressed at quarterly rates to be consistent with the timing of the model) yields Pu = 0.69 for 1985: 1-2014:4 and an estimated standard error of the residual of 1 . 17 percent. Various alternative values for a , Money-in-the-Utility Function 79 the autoregression coefficient for money growth, Pu , and the coefficient on the productiv­ ity shock, ¢, are considered to see how the implications of the model are affected by the manner in which money growth evolves. The remaining parameters are those in the utility function. The value of \II can be chosen so that the steady-state value of nss is equal to one-third, as in Cooley and Prescott. Ireland (2009) estimated a money demand equation for M1 that is of the same form as (2.75), except that he uses GDP rather than consumption. For the 1980-2008 period, he finds the coefficient on the level of the interest rate to be around 1 .85. The coefficient on !1 in (2.75) is b - 1 (1 - iss) /iss, implying b = (1 - iss) / (1 .85 iss) to match Ireland's estimate. The average for the 3-month Treasury bill rate over this period was 5.64. Taking this value for iss, b = ( 1 - 0.0056) / ( 1 .85 0.056) � 9. 47 The chapter appendix shows that the steady-state value of real money balances relative to consumption is equal to [ayss;( l - a)r ' 1b , where yss = ( 1 + ess - ,8) /(1 + ess) = iss I (1 + iss). For real Ml, this ratio in the data is just under 0.2 when consumption is expressed at annual rates, or about 0.78 at quarterly rates. If b = 9, this would imply a = 0.997. The inverse of the intertemporal elasticity of substitution, <t> , is set equal to 2 in the benchmark simulations. With b = 9, this means b - <t> > 0 and faster expected money growth decreases employment and output. Finally, 17 is set equal to 1 . With nss = 1 /3, a value of 11 = 1 yields a labor supply elasticity of [ 17 nss 1 ( 1 - nss) r 1 2. These parameter values are summarized in table 2. 1 . Using the information in this table, the steady-state values for the variables can be evaluated. These are given in table 2.2. The effect of money growth on the steady-state level of employment can be derived using (2.80). The elasticity of the steady-state labor supply with respect to the growth rate of x X = Table 2.1 Baseline Parameter Values 0.36 8 fJ 0.0 1 9 0.989 1) 2 a b I + e ss Pz 0.997 9 1 .0 1 4 0.95 Pu 0.34 0.69 0.85 Table 2.2 Steady-State Values at Baseline Parameter Values 1 + r" 1 .0 1 1 111 s s 0.084 0.065 FS 0.05 1 0.021 b 47. To match the estimates of Ireland (2009), this value of is larger than was used in the third edition of this book. Chapter 2 80 the nominal money supply depends on the sign of Ucm ; this, in tum, depends on the sign of b - <1> . For the benchmark parameter values, this is positive. With <I> less than b, the marginal utility of consumption is increasing in real money balances. Hence, higher infla­ tion decreases the marginal utility of consumption, increases the demand for leisure, and decreases the supply of labor (see 2.45). If b - <I> is negative, higher inflation leads to a rise in labor supply and output. The dependence of the elasticity of labor with respect to infla­ tion on the partial derivatives of the utility function in a general MIU model is discussed more fully by Wang and Yip (1992). 2.5.5 Simulation Results Figure 2.4 shows the effect of a one standard deviation monetary shock on output, employ­ ment, real and nominal interest rates, inflation, and the real stock of money. 48 Because b > <1>, a positive money growth rate shock reduces employment and output. 49 Notice that a positive monetary shock increases the nominal rate of interest. In the MIU model, prices are assumed to be perfectly flexible; when Pu > 0, money growth rate shocks increase o.o2 o -0.02 -o.o4 .--�-��-0"-'u::.:Ctp"-'u'-'-t�-�-.-----, --- ,....- 1 -p � 1 0 L�-��-�-�=====-.J - - - p = 0 .9 = .67 10 15 20 25 30 35 40 10 15 20 25 30 35 10 15 20 25 30 35 ::[-' ' '�:��- - - - - - - -- -0.06 5 10 15 20 25 30 35 40 40 10 15 20 25 30 35 40 40 10 15 20 25 30 35 40 Figure 2.4 Responses to a positive money growth rate shock in the MIU model; Pu = 0.67, Pu = 0.9. 48. Simulation results are obtained using Dynare. See the chapter appendix and the programs available at http://people.ucsc.edu/�walshc/mtp4e/ for details. 49. Recall that the transitional dynamics exhibit superneutrality when <I> = b. In this case, neither output nor employment would be affected by the monetary shock. Money-in-the-Utility Function 81 expected inflation and raise the nominal interest rate. The price level jumps immediately and the real quantity of money actually falls, consistent with the decline in real money demand that occurs as a result of the increase in the nominal interest rate. The effects clearly depend on the degree of persistence in the money growth process. Higher values of Pu generate much larger effects on labor input and output. 5 ° The value of b relative to <I> is critical for determining the real effects of a money growth rate shock. The results in figure 2.4 are for the baseline calibration in which b = 9 > <1> . The effects when b = 3 (i.e., smaller than the baseline value but still greater than <I>) and b = 1 < <I> are shown in figure 2.5. 51 When b < <I> , higher expected inflation (and therefore lower real money balances) raise the marginal utility of consumption and lead to a decrease in leisure demand; labor supply and output rise in this case, as shown in the figure. In all cases, inflation jumps immediately and then quickly returns to its steady-state value. How do the properties of the model vary if money growth responds to productivity shocks. Figure 2.6 illustrates the effects of varying ¢, the response of money growth to 10 15 20 25 30 35 40 : b; :�· : 10 :� I : : : 1 : f: -3 5 10 15 20 25 30 35 40 5 10 '"'"'"" '� 15 20 25 30 35 40 -0.1 5 10 5 10 15 20 25 �-'"" '"-.. 15 20 25 '�' """'""P'" 15 20 25 30 35 40 I ::I 30 35 40 30 35 40 Figure 2.5 Responses to a positive money growth rate shock; b = 9 > <t>, b = 3 > <t>, b = 1 < <t> . 5 0 . Effects would also b e larger i f the model were calibrated to match a broader monetary aggregate b y reducing a, increasing b, and increasing CJrp . 5 1 . The value b = 3 was used as the baseline in the third edition of this book. Chapter 2 82 :� r:;;:� __ : 1 :r::��: : , • • :� : : I : � ,_, 25 30 35 40 35 40 35 40 _ -0.01 5 10 15 ·� � 10 15 20 ·· · 25 30 35 40 30 35 40 -0.4 5 10 15 20 25 30 5 10 15 20 25 30 � · " 20 25 � - Figure 2.6 Responses in the MIU model for different values of <P in the money growth process (2.77). the productivity shock. 5 2 The major effect of ¢ is on the behavior of inflation and the nom­ inal rate of interest. When money growth does not respond to a productivity shock or when it decreases in response (i.e., when ¢ .:=: 0), output and inflation are negatively correlated, as the positive shock to productivity increases output and reduces prices. When ¢ < 0, a positive technology shock leads to lower expected money growth and inflation. Lower expected inflation raises real money balances, increases the marginal utility of consump­ tion, and increases the labor supply when, as in the case here, b > <t> . Hence, employment and output are slightly higher after a technology shock when ¢ < 0 than when ¢ 0. Con­ versely, when ¢ > 0, a positive technology shock leads to higher expected inflation, and the output-inflation correlation becomes positive. Employment and output respond less than in the base case. Changes in the money growth process have their main effect on the behavior of the nominal interest rate and inflation. Both the sign and the magnitude of the correlation between these variables and output depend on the money growth process. Consistent with the earlier discussion, the monetary shock cp1 affects the labor-leisure choice only when the nominal money growth rate process exhibits serial correlation ( Pu =/:. 0) or responds to the technology shock (¢ =/:. 0). = 52. When <P f= 0, the variance of the innovation to u is adjusted to keep the standard deviation of nominal money growth equal to its value in the baseline case. Money-in-the-Utility Function 2.6 83 Summary Assuming that holdings of real money balances yield direct utility is a means of ensuring a positive demand for money so that, in equilibrium, money is held and has value. This assumption is clearly a shortcut; it does not address the issue of why money yields utility or why certain pieces of paper that we call money yield utility but other pieces of paper presumably do not. The Sidrauski model, because it assumes that agents act systematically to maximize util­ ity, allows the welfare effects of alternative inflation rates to be assessed. The model illus­ trates the logic behind Friedman's conclusion that the optimal inflation rate is the rate that produces a zero nominal rate of interest, a result that also appears in the models discussed in chapters 3 and 4. Finally, by developing a linear approximation to the basic money in the utility function model (augmented to include a labor supply choice), it was shown how the effects of variations in the growth rate of the money supply on the short-run dynamic adjustment of the economy depend on effects of money holdings on the marginal utility of consumption and leisure. 2.7 Appendix: Solving the MIU Model The basic MIU model is linearized around the nonstochastic steady state, so the first task is to derive the steady-state equilibrium. Setting all shocks to zero and all endogenous variables equal to constants, and using the functional forms assumed for production and utility, the Euler condition, the definition of the real return, the production function, the capital accumulation equation, and the goods market clearing condition imply r ss ( ) ( ) ( ) ( -fJ1 - 1 + 8) ' s 1 ls - 8 =? l = =a SS SS k k a ( ) ( _!. ) [_!. - ] s css 1 + (1 - a)8 . css = yss - xss :::} SS = lSS - 8 = k k a fJ These five equations pin down the steady-state values of the real return as well as the steady-state ratios of output, employment, investment, and consumption to the capital stock. Chapter 2 84 In the text, the intensive production function was defined as is lnss = ¢ (kss lnss ), css = ySS - okss = [¢n - 8 (kss lnss ) ]ssnssnss= {jmSS 0 Section 20 5 0 2 y = f = rjJ ,Jn = ¢ - (k I )¢'0 mss = ( 11 ++ rre ssss ) mss ' one obtains rr ss = e ss , and this means 1 + iss 1 + e ss ' 1 + rss = --� 1 + rr 0 1 + iss = (1 + rss )(1 + rr ss ) = --or iss = (1 + e ss - �) I� 0 The first-order condition for money holdings then becomes Then also made use of the fact that because From :::} a ) ( 1 + e ss _ � )] - i ( mcssss ) = [( 1 - a 1 + e ss (2079) From the first-order condition for the household's choice of hours and the definition of the marginal utility of consumption, Ut(Css ,m;,ss , 1 _ nss ) ss where Q [a(css ) l- b + (1 - a)(mss ) l- b ] . This can be rewritten as is ) - 1¥( 1 - nss1-) ---,-� _,<l>,-- = (1 - a) ( nss a [a + ( l - a) ( 7:: ) b ] (css ) - <1> Rearranging, and using the earlier results, nss satisfies (1 - nss ) - � (nss ) <l> = : , = ------- b 1-::_b - 0 mss ) l-b ] �-::_b ( -css ) - <1> ( -kss ) l- <1> H = ( l - a) ( -iksss ) a [a + ( l - a) ( kss nss css 1] �-::_b ss - <1> ss ) <l>l - a ss b ( �kSS = (1 - a)a a + ( l - a) ( � ) � SS SS ) k C ( [ where (20 80) Money-in-the-Utility Function 85 I ( �:: ) ( �:: ) = ( �) ( i - 1 o ) , �;; = ( � ) [ i - 1 (1 )o ] , and (2.79), (.!. - 1 + o ) [a + (1 - a) ( ---a 1 e ss (3)- 1 "b" ] ---,---'a (l 1 - a 1 ess X [ *-' r Using �;: = H = a) + -r=a , fJ a + .( ' - • l ' i.::-� -a + + + _ �-=_'!; · Only H depends on the rate of money growth (and thus on the steady-state rate of infla­ tion), and if b = <t>, then H, too, is independent of 1 + e ss . In this case, n ss and all other real variables (except m ss ) are independent of the rate of money growth. The next step is to obtain the linear approximation for each equilibrium condition of the model so that the dynamic behavior as the economy fluctuates around the steady state can be studied. 2.7.1 The Linear Approximation Three basic rules are employed in deriving the linear approximations (see Uhlig 1999). First, for two variables u and w, (2.81) That is, assume that product terms like uw are approximately equal to zero. Second, (2.82) ua = (uss ) a (l + u) a (uss ) a ( l + au), � which can be obtained as a repeated application of the first rule. Furthermore, ln u = ln u ss ( l + u) = ln u ss + ln(l + u) ln u ss + it. (2.83) Finally, because variables such as interest rates and inflation rates are already expressed in percentages, it is natural to write them as absolute deviations from steady state. So, for example, r1 r1 - rss . Assuming interest rates and inflation rates are small, (2.83) per­ mits approximating the log deviation of (1 + r1) around the steady state by ln( l + r1) ln( l + rss ) r1 - rss = ft. and similarly for i1 and n1• This also means (1 + r1)/(1 + rss ) is approximately equal to 1 - r1 - rss = 1 + r1. 5 3 By applying these rules, one obtains a system of linear equations that characterizes the dynamic behavior of the MIU model for small deviations around its steady state. � = � 53. This requires that terms such as r1 be small. Otherwise, one should use the exact Taylor series expansion. For example, in the case of (l + r1)j(l + r55), this would be 1 I + rt "" ss - ( -- ) ( -- ) ! r I + l rss ( r1 - r ) - l + + l + rss t · With the calibration employed, rss = 0.0 1 1 , so l/( l + rss ) = fJ = 0.989. l + rss Chapter 2 86 The Production Function First, rewrite the production relationship (2.65) by replacing each variable with its steady­ state value times one plus the percent deviation of its time t value from the steady state, noting that ezr can be approximated by 1 + z1 for small z1: 1 + kl - 1 ( 1 + iit) l - a ( 1 + Yl) = (1 + Zl ) (ksst ( r (nss) l-a ls Because ls (ksst (nss) 1-a , both sides can be divided by is to obtain (1 + .Y�) c 1 + z1) ( 1 + k1 - 1 f (1 + n t) 1 - "' 0 = = 1 + akt - 1 + (1 - a)n1 + z1, or Y1 ak1- 1 + (1 - a)n1 + z1. � = Goods Market Clearing y1 (2. 84) Goods market clearing requires that c1 + x�> where x1 is investment. Write this as + X1) . + c1) + + Yl ) Because + it follows that = css ( l Xss ( l is css xss , Dividing both sides by kss and noting that xss / kss ( y-ssSS ) Yt - ( -cssSS ) cI + 8xI · k k /s ( l = = A A = 8 gives A (2. 85) Capital Accumulation The capital stock evolves according to k1 1 + kt = (1 - 8)kss 1 + kt- 1 + kss ( ) which implies k1 = (1 - 8)k1- 1 + xss /kss ( ( XssSS ) X1, k = (1 - 8)k1- t + x�> or ) Xss ( l + X1), = 8, so but k1 (1 - 8)k1- 1 + 8x1. = Labor Hours The first-order condition for the choice of labor hours is (2. 86) Money-in-the-Utility Function 87 where A t is the marginal utility of consumption. Using the production and utility functions, this becomes IJ! (l - nr ) - '1 wt; " Uf = = (l - a) Yt . = nr At At At Written in terms of deviation, this is - " ( l + lr ) - '1 IJ! (lss l + .Yr, yss -----' ) --,----:- - = ( 1 - a) . n 55 l + n1 Ass (l + Ar) But in the steady state, ( ( )( ) -- ) so 1 + �� ( 1 + lr! - " , or = 1 + nr ( 1 + A1) ) ( ( 1 - ryl1) ( 1 - � �) � 1 - ry lt - � � � 1 + Yt - Ytr . From 11 = 1 - nr , ( ) ss zss (l + lr ) = 1 - n 55 ( 1 + nr) :::} lr = - n nr. zss Hence, which can be written as [ 1 + ry ( �:: ) ] n1 = .Yr + ��. (2.87) The Marginal Utility o f Consumption The marginal utility of consumption is -a [actl -b + (1 - a)mt1 -b] T=b ct-b . Define Qr = ac� -b + ( 1 - a)m� -b · Then At = a Q1T=b c1- b , or b - <l> At - 1 b - <l> Chapter 2 88 A ss (I + � t ) = a ( Qss ) �-::_b ( css r b 1 + [ ( �-=_:) qt] ( 1 _ bet) . Since A ss = a ( Q ss ) �-::_b ( css r b , the right side of the previous equation can be approximated by so ( ) b - <t> A' t = qt - bct . 1-b To obtain an expression for qt, note that from the definition of Q t , 1-b a (css ) 1-b 1 -b + ( 1 - a) (m ss ) (1 + m t ) 1 -b l (I + qt + c ) C t ')= Q SS Q SS = y [1 + ( 1 - b)ct] + ( 1 - y ) [1 + ( 1 - b) mt] , where a (css ) 1 -b y= -- Hence, qt = y ( l - b)ct + (1 - y) ( l - b)mt. Combining these results, � t = Q 1 ct + Q 2 mr. (2.88) where Qt = b (y - 1) - y <t> and Q 2 = (b - <P) ( 1 - y) . Note that if b = <t> , � t = -bet. The Euler Condition The Euler condition is A t = f3Et( l + rt )A t+L , which, because f3 = (1 + rss ) - 1 , can be written as A ss ( 1 + A' t ) = f3 A ss ( 1 + rt ) Et ( 1 + A' t+ I ) = A ss ( l1 ++rssrt ) Et ( 1 + A' t+ ) I . Dividing both sides by A ss , recalling that rt = rt - rss , and using (2.8 1 ), ( 1 + � t) ( 1 + rt + Et� t+l ) ; � then � t = ft + Et � t+l · Money-in-the-Utility Function 89 Marginal Product, Real Return Condition Start with I + r1 = 1 - 8 + aE1 c��� ) . Using the same general approach as applied to the other equations, s 8 a i . r � kss Since rss a (yss /kss ) - 8, ( 1+ r 1- + = r1 = r1 - rss = a A - kA r) ) Er ( 1 + Yt+l ( �:: ) E1 (Yr+l - kr) . (2.89) Money Holdings The first-order condition for money holdings is i Um (cr , mr , l - n r ) Uc(Cr, mt, nr ) From the specification of the utility function, the left side can be approximated as m ss b Um ( �m-; b � css Uc c1 b 1- = = = ( 1 +r ir ) · 1 - a ) ( ) - (1 - bmr + bcr) 1 -a ( a a (� 1 + iss ) (1 - bmr + bcr) · Therefore, s � iss Multiplying both sides by i1 and approximating ( 1 � ) ( 1 - bmr + her) ( 1 � it ) . 1+ t ( 1 + tr - t ) ( 1 - bmA t + bCtA ) � tr , or (tr - bmA r + bcA r) tr . ·SS •SS 1 • . ·SS •SS = - 1 • SS • - 1 (1 + i1) / (1 + iss ) by 1 + i1 - iss yields . Therefore, the money demand equation is given by 1 - -;ss- ) (tt - t ) . mA t CtA - ( b1 ) ( ---iSS = Real Money Growth Because e ss • •SS = rr ss , one can approximate (2.90) Chapter 2 90 by mt = or ( l1 ++ rr()1t ) ( 1l ++ rressss ) mt- 1 � ( l1 ++ fi:{)1t ) mt-1 , mss ( 1 + m t ) � ( 1 + (jl - ii:t ) m ss ( 1 + m t -1 ) , where e1 = 81 - e ss . Dividing both sides by mss and using (2.81) yields The Fisher Equation The relationship between the nominal interest rate, the real interest rate, and expected inflation is 1 + r1 = E1 1 + it , or ( 1 + lft+1 ) Subtracting steady state values from both sides, 2.7.2 Collecting all Equations The linearized model consists of twelve equations to determine the exogenous disturbances Zt and Ut and the ten endogenous variables Yt, k(, n(, Xt , Ct, �(, r(, l(, fi:(, m(. These twelve equations are Zt = Pz Zt-1 + et , ( yss ) Yt = ( css ) Ct + OXt , A kSS A kSS A kt = (1 - o)kt- 1 + oxt, 1 + ( ;:: ) n 1 = Yt + ��, r1 = a G:: ) (EtYt+l - kt) , �� = r1 + Et� t+l , [ � ] Money-in-the-Utility Function 91 m1 = Ut - irt + m 1- 1 , it = r-t + Et irt+ t , U t = PuUt- 1 + c/J Zt- 1 + rpt, where S1t = b (y - 1) - y <l> and S1 2 = (b - <I>) (1 - y ) . Note that if b = <I> so that S1 2 = 0, the first eight equations can be solved for the behavior of the real variables z1, y1, k1, n1, Xt. Ct , A t , and r(, while the last four then determine Ut , lt , irt , and mt . 2.7.3 Solving Linear Rational-Expectations Models with Forward-Looking Variables This section provides a brief overview of the approach used to solve linear rational expec­ tations models numerically. The basic reference is Blanchard and Kahn (1980). General discussions can be found in Wickens (2008), DeJong and Dave (201 1 ), and Miao (2014). Following the solutions methods of Blanchard and Kahn ( 1980), a linear rational expec­ tations model can be written in the form where X are predetermined variables (n 1 in number) and x are non-predetermined (forward­ looking) variables (nz in number). Predetermined means that X1 is known at time t and not jointly determined with Xt. while x1 is endogenously determined at time t. G1 consists of a vector of exogenous variables. Premultiplying both sides by A 1 inverse, we obtain 1/J Xt+t X = A t + BGt + A ! ! t+l (2.9 1) 0 Xt EtXt+1 where A = A 1 1 A z and B = A 1 1 A 3 . King and Watson (1998) consider the case in which A 1 is singular. Blanchard and Kahn showed that the number of eigenvalues of A that are outside the unit circle must equal the number of forward-looking variables. Decompose A as Q - 1 A Q, where A is a diagonal matrix of the eigenvalues of A, and Q is the corresponding matrix of eigenvectors. Next, order A so that A. t is the smallest and A. n is the largest eigenvalue, where n n 1 + n z . Then, the Blanchard and Kahn conditions require that the first n t eigenvalues must be inside the unit circle and the last nz must be outside the unit circle if the system is to have a unique stationary rational-expectations equilibrium. If fewer than nz eigenvalues are outside the unit circle, multiple equilibria exist and the system is said to be characterized by indeterminacy. If too many eigenvalues are outside the unit circle, no solution exists. To understand the role these conditions play, it is convenient to write (2.91) as (2.92) ] [] [ = [ ] ' Chapter 2 92 Z1 [X1 where xtl' is a n 1 vector of predetermined and non-predetermined variables and G1 is a stationary (possibly stochastic) vector of exogenous variables. Under the assumption 0. Writing A of rational expectations, Q, both sides of (2.92) can be multiplied by Q, yielding = x = Q- 1 A E1 (Z1+ 1 -E1Z1+ 1 ) = (2.93) z1 Z1 81 ¢1+ 1 = Q(Z1+ t -E1ZZ1i+,l+1).l = A.;z;,A1 z; 1 , 8i,to A.; A z1 = -ii1 A. 21+ t = [ PY11++ tl ] = [ A0t A02 l [ PY1]1 + [ 82,8t,11 ] + [ ¢2,¢t,1l++ 1t ] · Y1 consists of the first iit elements of Zto i.e., those corresponding to the eigenvalues in A less than 1 in absolute value and contained in the diagonal matrix A t , and P1 consists of the ii2 remaining elements of z1 associated with the eigenvalues equal to or greater than 1 in absolute value and contained in A 2 . Writing this system out explicitly, one has Y1+ t = A tYI + 8U + ¢t,l+ t• Pt+ t = A 2Pt + 82,1 + ¢2,t+ t· This second set of equations is explosive (the elements of A 2 are outside the unit circle). Hence, because E1 ¢;, 1+ t = 0, it must be the case in any nonexplosive equilibrium that PI = A 2 1 (EIPI+ I - 82,1) = - Li=O A 2i- l 82,1+i· This uniquely determines P1 as the only value of P1 consistent with a stationary equilibrium. Any other value of P1 leads to explosive behavior. For example, for P1 scalar and 82,1 = P82,l- t + where is white noise and less than 1 in absolute value, where Q , Since is a diagonal matrix, QBGt. and (2.93) consists of n independent equations of the form + g ;,t. where is the ith element of and similarly for while is the ith diagonal element of Suppose has ii] elements within the unit circle and ih n on or outside the unit circle. The system can be written as = = 00 e1 , e A. A.ii2t -equations0 because I 2I of the form Y1+ t = A tYI + 8t,1 where The p > p 0:: 1 and IPI < l. Money-in-the-Utility Function 93 can be solved backward because IA. i l recursive substitution leads to < 1 for all i _::: i'it. That is, for these i'i t equations, 00 = l:)�8i,t-j· j=O Recall that Y1 and P1 were linear combinations of the variables of interest X1 and x1• Having obtained unique stationary solutions for Y1 and Pt. when can unique solutions for the variables of interest be obtained? Let W Q -1 . Then, since z1 = [Y1 Ptl' = Q[X1 x1]', = one can write where W has been partitioned to conform with the dimensions of the different vectors. This system yields which gives n 1 equations. Since X1 is predetermined, let Xo denote the initial conditions on the system. A unique value for Po was obtained earlier. Thus, Yo must satisfy Xo = Wll Yo + W1 2Po. (2.94) If the number of predetermined variables n 1 > n 1 , then (2.94) consists of n 1 equations in the n 1 < n 1 unknown elements of Yo. This imposes n 1 > n 1 conditions on Yo, and so there will generally be no solution. If n 1 < then there are n 1 initial conditions Xo but > n 1 unknowns in Yo, a situation of too few equations, and generally multiple solution will exist. Thus, the Blanchard-Kahn condition for a unique stationary rational-expectations equilibrium is n 1 = or, as it is more commonly expressed, n2 = n - n 1 = n - = n 2 , that is, the number of forward-looking variables n2 must equal the number of eigenvalues outside the unit circle n2 . Assuming this condition is met, the unique solution of the original model (2.9 1) takes the form i'i t , i'it, i'i t i'it Xt = CXt . The MATLAB code used to solve the MIU model is available at http://people.ucsc.edu J�walshc/mtp4e/. The programs use Dynare, available at http://www.dynare.org/. Chapter 2 94 2.8 Problems 1 . The MIU model of section 2.2 implied that the marginal rate of substitution between money and consumption was set equal to / 1 (see (2. 12)). That model assumed it ( + i1) w1 that agents entered period t with resources and used those to purchase capital, consumption, nominal bonds, and money. The real value of these money holdings yielded utility in period t. Assume instead that money holdings chosen in period t do not yield utility until period t 1 . Utility is L fJ ' as before, but the budget constraint takes the form U(c1+; , M1+JPt+i ) + Wt = Ct + MtPt+ L + ht + kt , and the household chooses c�> k�, b1, and Mt+ 1 in period t. The household's real wealth w1 is given by -- Derive the first-order condition for the household's choice of Um (Ct+t ,mt+d = I.t· Uc(Ct+t , mt+ t ) Mt+ L and show that (Suggested by Kevin Salyer.) 2. Carlstrom and Fuerst (2001). Assume that the representative household's utility depends on consumption and the level of real money balances available for spending on consumption. Let be the real stock of money that enters the utility function. If capital is ignored, the household's objective is to maximize L fJ ' subject to the budget constraint At !P1 U(cr+i ,At+JPr+i ) Yr + MrPt-1 + rr + (1 + irPt- t)Br-1 = Cr + -MrPt + -BrPt , where income Y1 is treated as an exogenous process. Assume that the stock of money -- that yields utility is the real value of money holdings after bonds have been purchased but before income has been received or consumption goods have been purchased: At- = Mt-1 + Tt + (1 + it- t)Bt- 1 - -Br . Pr Pr Pr Pr a. Derive the first-order conditions for B1 and for A1. -- b. How do these conditions differ from those obtained in the text? 3. Assume the representative household's utility function is given by (2.29). Show that (2.24) implies (2.30). Now suppose = (1 - y ) In c +y In Show that if consumption is constant in the steady state, there is a unique steady-state capital stock u(c, m) m. Money-in-the-Utility Function 95 - k55 such that f3 [fk (k55) + 1 8 ] = 1 . Explain why variations in the growth rate of the money supply do not affect the steady-state k in this case but do when the utility func­ tion is (2.29). 4. Suppose W = L (in + ) , y > 0, and f3 = 0.95. Assume that the produc­ tion function is f(k ) = k/ and 8 = 0.02. What rate of inflation maximizes steady­ state welfare? How do real money balances at the welfare-maximizing rate of inflation depend on y ? 5. Suppose that the utility function (2.66) is replaced by {3 1 c1 m1e- y m, 1 a. Derive the first-order conditions for the household's optimal money holdings. b. Show how (2.72) and (2.73) are altered with this specification of the utility function. 6. Suppose the utility function (2.66) is replaced by u (cr,m1, 1 - nr) - [a Ctl -b + (1 - a) mt1-b ] \-=_';; [ ( l - nr) l- '1 1 - <1> 1 - ry l · a. Derive the first-order conditions for the household's optimal money holdings. b. Show how (2.72) and (2.73) are altered with this specification of the utility function. 7. Suppose a nominal interest rate of is paid on money balances. These payments are financed by a combination of lump-sum taxes and printing money. Let be the fraction financed by lump-sum taxes. The government's budget identity is + = with Using Sidrauski's model, do the following: and = = a. Show that the ratio of the marginal utility of money to the marginal utility of con­ sumption equals r + n Explain why. = b. Show how is affected by the method used to finance the interest payments on money. Explain the economics behind your result. 8. Suppose agents do not treat as a lump-sum transfer but instead assume their transfers will be proportional to their own holdings of money (because in equilibrium, = Solve for an agent's demand for money. What is the welfare cost of inflation? 9. Suppose money is a productive input into production, so that the aggregate produc­ tion function becomes y = f(k, Incorporate this modification into the model of section 2.2. Is money still supemeutral? Explain. im Tt aimmr v emr. i - im a r1 v1 immr , - im i - im . r1 r ()m). m). Chapter 2 96 10. Consider the following two alternative specifications for the demand for money given by (2.39) and (2.40). a. Using (2.39), calculate the welfare cost as a function of rJ . b . Using (2.40), calculate the welfare cost as a function of � . 1 1 . In Sidrauski ' s MIU model augmented to include a variable labor supply, money is superneutral if the representative agent's preferences are given by but not if they are given by " L f3 i (ct+i + km t+ i ) b ldt+ i· L f3 i u(ct+ i , m t+ i , lt+ i ) = " Discuss. (Assume output depends on capital and labor, and the aggregate production function is Cobb-Douglas.) 12. Suppose preferences over consumption, money holdings, and leisure are given be = a In + (1 - a) In + } I (1 - ry) ). Fischer (1979) showed that the transi­ tion paths are independent of the money supply in this case because the marginal rate of substitution between leisure and consumption is independent of real money balances. Write the equilibrium conditions for this case, and show that the model dichotomizes into a real sector that determines output, consumption, and investment, and a monetary sector that determines the price level and the nominal interest rate. 13. For the model of section 2.5, is the response of output and employment to a money growth rate shock increasing or decreasing in b? Explain. (See http://people.ucsc .edu/�walshc/mtp4e/ for programs to answer this question.) 14. For the model of section 2.5, is the response of output and employment to a money growth rate shock increasing or decreasing in a? Explain. (See http://people.ucsc .edu/�walshc/MTP4e/ for programs to answer this question.) (u c1 m1 \ll l - 17 3 3.1 Money and Transactions Introduction The previous chapter introduced a role for money by assuming that individuals derive utility directly from holding real money balances. Therefore, real money balances appeared in the utility function along side consumption and leisure. Yet one usually thinks of money as yielding utility indirectly through use; it is valued because it is useful in facilitating transactions to obtain the consumption goods that do directly provide utility. As described by Clower (1967), goods buy money, and money buys goods, but goods don't buy goods. And because goods don't buy goods, a monetary medium of exchange that aids the process of transacting will have value. A medium of exchange that facilitates transactions yields utility indirectly by allow­ ing certain transactions to be made that would not otherwise occur or by reducing the costs associated with transactions. The demand for money is then determined by the nature of the economy's transaction technology. The first formal models of money demand that emphasized the role of transaction costs are due to Baumol (1952) and Tobin (1956). 1 Niehans (1978) developed a systematic treatment of the theory of money in which transac­ tion costs play a critical role. These models are partial equilibrium models, focusing on the demand for money as a function of the nominal interest rate and income. In keeping with the approach used in examining money-in-the-utility function (MIU) models, the focus in this chapter is on general equilibrium models in which the demand for money arises from its use in carrying out transactions. In the first models examined in this chapter, real resources and money are used to pro­ duce transaction services, which are required to purchase consumption goods. These real resources can take the form of either time or goods. Most of this chapter, however, is devoted to the study of models that impose a rigid restriction on the nature of transactions. Rather than allowing substitutability between time and money in carrying out transactions, l. Jovanovic ( 1 982) and Romer ( 1 986) embedded the Baumol-Tobin model in general equilibrium frameworks. Chapter 3 98 cash-in-advance models (CIA) simply require that money balances be held to finance cer­ tain types of purchases; without money, these purchases cannot be made. CIA models, like the MIU models of chapter 2, assume that money is special; unlike other financial assets, it either yields direct utility and therefore belongs in the utility function, or it has unique properties that allow it to be used to facilitate transactions. This chapter concludes with a look at some recent work based on search theory to explain how the nature of transactions gives rise to money. 3.2 Resource Costs of Transacting A direct approach to modeling the role of money in facilitating transactions is to assume that the purchase of goods requires the input of transaction services. First a model is con­ sidered in which these services are produced using inputs of money and time. Then an alternative approach is studied in which there are real resource costs in terms of goods that are incurred in purchasing consumption goods. Larger holdings of money allow the household to reduce the resource costs of producing transaction services. 3.2.1 Shopping-Time Models When transaction services are produced by time and money, the consumer must balance the opportunity cost of holding money against the value of leisure in deciding how to combine time and money to purchase consumption goods. The production technology used to pro­ duce transaction services determines how much time must be spent "shopping" for given levels of consumption and money holdings. Higher levels of money holdings reduce the time needed for shopping, thereby increasing the individual agent's leisure. When leisure enters the utility function of the representative agent, shopping-time models provide a link between the MIU approach of chapter 2 and models of money that focus more explicitly on transaction services and money as a medium of exchange. 2 Suppose that purchasing consumption requires transaction services 1J; , with units chosen so that consumption of requires transaction services 1J; = These transaction services are produced with inputs of real cash balances M /P and shopping time c m c. = ns : (3. 1 ) where o/m :=::: 0, o/ns :=::: 0 , and o/mm ::::0 0 , o/nsns ::::0 0 . This specification assumes that it i s the agent's holdings of money balances that produce transaction services; a change in the price level requires a proportional change in nominal money holdings to generate the same real 2. See Brock ( 1 974) for an earlier use of a shopping-time model to motivate an MIU approach. The use of a shopping-time approach to the study of the demand for money is presented in McCallum and Goodfriend ( 1 987) and Croushore ( 1 993). Money and Transactions 99 level of real consumption purchases, holding shopping time n5 constant. Rewriting (3. 1 ) in terms of the shopping time required for given levels of consumption and money holdings gives n5 = g (c, m ) ; gc > 0, gm ::S 0. Household utility is assumed to depend on consumption and leisure: v(c, l) . Leisure is equal to l = 1 - n - n5 , where n is time spent in market employment and n5 is time spent shopping. Total time available is normalized to equal 1 . With shopping time n5 an increasing function of consumption and a decreasing function of real money holdings, time available for leisure is 1 - n - g (c, m) . Now define a function u (c, m , n) = v [ c, 1 - n - g (c, m ) ] that gives utility as a function of consumption, labor supply, and money holdings. Thus, a simple shopping-time model can motivate the appearance of an MIU function and, more important, can help determine the properties of the partial derivatives of the function u with respect to m. By placing restrictions on the partial derivatives of the shopping-time produc­ tion function g (c, m), one can potentially determine what restrictions might be placed on the utility function u (c, m, n) . For example, if the marginal productivity of money goes to zero for some finite level of real money balances m, that is, limm->m gm = 0, then this property will carry over to Um . In the MIU model, higher expected inflation lowers money holdings, but the effect on leisure and consumption depends on the signs of UJm and Ucm ·3 The shopping time model implies that Um = - v1gm :=:: 0, so (3.2) The sign of Ucm depends on such factors as the effect of variations in leisure time on the marginal utility of consumption (vel) and the effect of variations in consumption on the marginal productivity of money in reducing shopping time (gem) . In the benchmark MIU model of chapter 2, Ucm was taken to be positive. 4 Relating Ucm to the partials of the under­ lying utility function v and the transaction production function g can suggest whether this assumption was reasonable. From (3.2), the assumption of diminishing marginal utility of leisure (vu ::::: 0) and gm ::::: 0 implies that vugc gm :=:: 0. If greater consumption raises the marginal productivity of money in reducing shopping time (gem ::S 0), then -v1gcm :=:: 0 as well. Wang and Yip ( 1992) characterized the situation in which these two dominate, so that Ucm :=:: 0, as the transaction services version of the MIU model. In this case, the MIU model implies that a rise in expected inflation would lower m and Uc, and this would lower 3. This is a statement about the partial equilibrium effect of inflation on the representative agent's decision. In general equilibrium, consumption and leisure are independent of inflation in models that display superneutrality. 4. This corresponded to > <P in the benchmark utility function used in chapter 2. b Chapter 3 100 consumption, labor supply, and output (see section 2.5.2). The reduction in labor supply is reinforced by the fact that = < 0, so the reduction in raises the marginal 5 utility of leisure. If consumption and leisure are strong substitutes so that Vel ::::: 0, then could be negative, a situation Wang and Yip describe as corresponding to an asset sub­ stitution model. With < 0, a monetary injection that raises expected inflation increases consumption, labor supply, and output. The household's intertemporal problem analyzed in chapter 2 for the MIU model can be easily modified to incorporate a shopping-time role for money. The household's objective is to maximize m Utm -vugm Ucm Ucm L f3 i v [Ct+i• 1 - nt+i - g(cr+i• mt+ ; ) ] , 0 00 i=O < f3 < 1, subject to f(kr-1 , nr) + Tt + ( 1 - o )kr-1 + (1 + it- J)1 +ht-1nr + mt-1 Ct + kr + br + mr, (3 .3) wheref is a standard neoclassical production function, k is the capital stock, 8 is the depre­ ciation rate, b and m are real bond and money holdings, and is a real lump-sum transfer from the government. 6 Defining a1 T1 + [ (1 + ir - I)br- I + m1_ !] /0 + n1), the house­ hold's decision problem can be written in terms of the value function V (a1, kr- I ): = r = where the maximization is subject to the constraints f(kr , n1) + ( 1 - 8)k1 - I + a1 c1 + kt + br + mr and a1+ 1 Tt+ 1 + [ (1 + it)br + mr ] / (1 + Trr-1+ 1 ). Proceeding as in chapter 2 by using these two constraints to eliminate k1 and a1+ 1 from the expression for the value function, the necessary first-order conditions for consumption, real money holdings, real = = bond holdings, and labor supply are Vtgc - f3Vk(ar+I,kt) 0, -Vtgm + f3 Va1 (a+t+Inr+1,kr) - f3Vk(ar+I,kr) Vc - = (3.4) = 0, (3.5) (3.6) (3.7) 5. I thank Henrik Jensen for pointing this out. 6. It is assumed that transaction services are needed only for the purchase of consumption goods, not for the purchase of capital goods. In the next section, alternative treatments of investment and the transaction technology are shown to have implications for the steady state. Money and Transactions 101 and the envelope theorem yields (3.8) Va (ar,kr-1 ) = f3Vk(ar+1 ,kr), (3.9) Vk(ar,kr- 1 ) = f3Vk (ar+ l ,kr) [{k(kr-l , nr) + 1 - 8] . Letting w1 denote the marginal product of labor, that is, w1 = fn(kr - 1 , n 1), (3.6) and (3.8) yield v1 = w1 Va (ar,k1_ t), This implies that (3.4) can be written as (3.10) The marginal utility of consumption is set equal to the marginal utility of wealth, Va (ar,kr- 1 ), plus the cost, in utility units, of the marginal time needed to purchase con­ sumption. Thus, the total cost of consumption includes the value of the shopping time involved. A marginal increase in consumption requires an additional 8c in shopping time. The value of this time in terms of goods is obtained by multiplying 8c by the real wage w, and its value in terms of utility is Va (a,k)wgc. With 8m 0, v1gm Va wgm is the value in utility terms of the shopping time savings that result from additional holdings of real money balances. Equations (3.5) and (3.8) imply = :S that money will be held to the point where the marginal net benefit, equal to the value of shopping time savings plus the discounted value of money's wealth value in the next period, or ), just equals the net marginal utility of wealth. + + The first-order condition for optimal money holdings, together with (3. 7) and (3.8), implies -v1gm f3Va (at+ 1 , k1) / (1 n1+ 1 Va (ar+l ,k-r) -vlgm = f3Vk(ar+J,kr) - /3 --;__ 1 + JTt+1 V (a ,kr) / Va (ar,kr-1 ) ] = Va (ar,kr-l ) [ l - f3 a r+ l 1 + JTt+ 1 ] = Va (ar ,kr-1 ) [ 1 C - � iJ = Va (ar ,kr- 1 ) (�) , (3.11) 1 + lt where i1 is the nominal rate of interest and, using (3.7) and (3.8), ( Va (a r+ 1 ,kr) / Va (ar ,kr- d) (1 + JTr+d /(1 + i1).7 Further insight can be gained by using (3.6) and (3.8) to note that (3.11) can also be = written as (3.12) + 7. Note that (3 . 1 1 ) implies - vlgm / Va = ij ( l i) . The left side is the value of the shopping time savings from holding additional real money balances relative to the marginal utility of income. The right side is the opportunity cost of holding money. This expression can be compared to the result from the MIU model, which showed that the marginal utility of real balances relative to the marginal utility of income would equal ij ( 1 i) . In the MIU model, however, the marginal utility of income and the marginal utility of consumption were equal. + Chapter 3 102 The left side of this equation is the value of the transaction time saved by holding additional real money balances. At the optimal level of money holdings, this is just equal to the opportunity cost of holding money, i/ ( + i) . Since no social cost of producing money has been introduced, optimality would require that the private marginal product of money, gm , be driven to zero. Equation (3. 2 implies that gm = if and only if i = one thus obtains the standard result for the optimal rate of inflation, as seen earlier in the MIU model. The chief advantage of the shopping time approach as a means of motivating the pres­ ence of money in the utility function is its use in tying the partials of the utility function with respect to money to the specification of the production function relating money, shop­ ping time, and consumption. But this representation of the medium-of-exchange role of money is also clearly a shortcut. The transaction services production function n5) is simply postulated; this approach does not help to determine what constitutes money. Why, for example, do certain types of green paper facilitate transactions (at least in the United States), while yellow pieces of paper don't? Section 3.4 reviews models based on search theory that attempt to derive money demand from a more primitive specification of the transaction process. 1 0 1) 0; 1/J (m, 3.2.2 Real Resource Costs An alternative approach to the CIA and shopping-time models is to assume that transaction costs take the form of real resources that are used up in the process of exchange (Brock An increase in the volume of goods exchanged leads to a rise in transaction costs, while higher average real money balances for a given volume of transactions lower costs. In a shopping time model, these costs are time costs and so enter the utility function indirectly by affecting the time available for leisure. If goods must be used up in transacting, the household's budget constraint must be mod­ ified, for example, by adding a transaction costs term that depends on the volume of transactions (represented by and the level of money holdings. The budget constraint (3.3) then becomes 1974; 1990). I (c, m) c) f(kr- 1 ) + (l - o)kt- 1 + Tt + (l + (1986) rr-1 )bt-l + 1mr+-1rr1 :::: Ct + mt + bt + kt + l(cr,mr). -- Feenstra considered a variety of transaction cost formulations and showed that they all lead to the presence of a function involving and appearing on the right side of the budget constraint. He also showed that transaction costs satisfy the following condition for all is twice continuously differentiable and = :::; :=:: e :::; and is quasi -convex, with expansion paths + m mm m having a non-negative slope. These conditions all have intuitive meaning: = means the consumer bears no transaction costs if consumption is zero. The sign restrictions on the partial derivatives reflect the assumptions that transaction costs rise at an increasing c,m :::: 0: I I 0; Icc. I 0; I 0; c I (c, m) c m I :::: 0; I(O,m) 0; lc :::: 0; 1(0, m) 0 Money and Transactions 103 rate as consumption increases and money has positive but diminishing marginal productiv­ ity in reducing transaction costs. The assumption em _:::: 0 means the marginal transaction costs of additional consumption do not increase with money holdings. Expansion paths with non-negative slopes imply increases with income. Positive money holdings can be ensured by the additional assumption that limm--> 0 m ( , m ) = - oo ; that is, money is essential. Now consider how the MIU approach compares to a transaction cost approach. Suppose a function W(x, m) has the following properties: for all x, m :::: 0, W is twice continuously differentiable and satisfies W :::: 0; W(O, m) = 0; W(x, m) --+ oo as x --+ oo for fixed m; Wm :=:: 0; 0 _:::: Wx _:::: 1 ; Wxx _:::: 0; Wmm _:::: 0; Wxm :=:: 0; W is quasi-concave with Engel curves with a non-negative slope. Now simplify by dropping capital and consider the following two static problems repre­ senting simple transaction cost and MIU approaches: max U(c) subject to ( , m) (3 . 1 3) +m=y (3. 14) max V(x, m) subject to x m = y, I c+I l c c+l c +b +b+ where V(x, m) U [W(x, m)]. These two problems are equivalent if (c* ,b * ,m * ) solves (3. 1 3) if and only if (x* , b * , m * ) solves (3. 14) with x* = c * + l(c * , m * ) . Feenstra (1986) showed that equivalence holds if the functions I (c, m) and W (x, m) satisfy the stated = conditions. This "functional equivalence" (Wang and Yip 1992) between the transaction cost and MIU approaches suggests that conclusions derived within one framework also hold under the alternative approach. However, this equivalence is obtained by redefining variables. So, for example, the consumption variable x in the utility function is equal to consumption of transaction costs (x = + m)) and is therefore not independent of money holdings. At the very least, the appropriate definition of the consumption variable needs to be considered if one attempts to use either framework to draw implications for actual macroeconomic time series. 8 inclusive 3.3 c l(c, Cash-in-Advance (CIA) Models A direct approach to generating a role for money, proposed by Clower (1967) and devel­ oped formally by Grandmont and Younes (1972) and Lucas (1980a), captures the role of money as a medium of exchange by requiring explicitly that money be used to purchase goods. Such a requirement can also be viewed as replacing the substitution possibil­ ities between time and money highlighted in the shopping-time model with a trans­ action technology in which shopping time is zero if M /P :::: and infinite otherwise c 8. When distortionary taxes are introduced, Mulligan and Sala-i-Martin ( 1 997) showed the functional equiva­ lence between the two approaches can depend on whether money is required to pay taxes. Chapter 3 104 (McCallum 1990a). This specification can be represented by assuming that the individ­ ual faces, in addition to a standard budget constraint, a cash-in-advance (CIA) constraint. 9 The exact form of the CIA constraint depends on which transactions or purchases are subject to the CIA requirements. For example, both consumption goods and investment goods might be subject to the requirement. Or only consumption might be subject to the constraint. Or only a subset of all consumption goods might require cash for their purchase. The constraint will also depend on what constitutes cash. Can bank deposits that earn interest, for example, also be used to carry out transactions? The exact specification of the transactions subject to the CIA constraint can be important. Timing assumptions also are important in CIA models. Lucas (1982) allows agents to allocate their portfolio between cash and other assets at the start of each period, after observing any current shocks but prior to purchasing goods. This timing is often described by saying that the asset market opens first and then the goods market opens. If there is a positive opportunity cost of holding money and the asset market opens first, agents will only hold an amount of money that is just sufficient to finance their desired level of con­ sumption. Svensson (1985) has the goods market open first. This implies that agents have available for spending only the cash carried over from the previous period, and so cash bal­ ances must be chosen before agents know how much spending they will wish to undertake. For example, if uncertainty is resolved after money balances are chosen, agents may find they are holding cash balances that are too low to finance their desired spending level. Or they may be left with more cash than they need, thereby forgoing interest income. To elucidate the structure of CIA models, section 3.3. 1 reviews a simplified version of a model due to Svensson (1985). The simplification involves eliminating uncertainty. Once the basic framework has been reviewed, a stochastic CIA model is considered as a means of studying the role of money in a stochastic dynamic general equilibrium model (DSGE) in which business cycles are generated by both real productivity shocks and shocks to the growth rate of money. Developing a linearized version of the model illustrates how the CIA approach differs from the MIU approach. 3.3.1 The Certainty Case This section develops a simple cash-in-advance model. Issues arising in the presence of uncertainty or the presence of labor-leisure choices are postponed. The timing of transac­ tions and markets follows Svensson (1985), although the alternative timing used by Lucas (1982) is also discussed. After the model and its equilibrium conditions are set out, the steady state is examined and the welfare costs of inflation in a CIA model are discussed. 9. Boianovsky (2002) discussed the early use in the 1 960s of a CIA constraint by the Brazilian economist Mario Simonsen. Money and Transactions 105 The Model Consider the following representative agent model. The agent's objective is to choose a path for consumption and asset holdings to maximize (3. 1 5) for 0 < f3 < 1 , where u(. ) is bounded, continuously differentiable, strictly increasing, and strictly concave, and the maximization is subject to a sequence of CIA and budget con­ straints. The agent enters the period with money holdings and receives a lump-sum transfer (in nominal terms). If the goods market opens first, the CIA constraint takes the form M1- 1 T1 c P is the aggregate price level, and T represents lump-sum (3. 1 6) Ct MtPt-1 + PtTt 1m+t-1nt + Tf> where m1 -1 M1_ l !P1_ I , n1 (Pt/P1 - I ) - 1 is the inflation rate, and 1 Tt /P1• Note the timing: Mt-1 refers to nominal money balances chosen by the agent in period t - 1 and carried into period t. The real value of these balances is determined by the period t price level P1• Since certainty is assumed, the agent knows P1 at the time Mt -1 is chosen. This specification of the CIA constraint assumes that income from production during period t is not be available for consumption purchases until period t + 1 . The budget constraint, in nominal terms, is Ptwt PJ(kt-1 ) + (1 - o )Ptkt-1 + Mt-1 + Tt + (1 + it- I) Bt-l ::;: Ptct + Ptkt + Mt + B�> (3. 1 7) where w1 is the agent's time t real resources, consisting of income generated during period t, f(k1- I ); the undepreciated capital stock, (1 - 8)k1- I ; money holdings, m; the transfer from the government, and gross nominal interest earnings on the agent's t - 1 holdings of nominal one-period bonds, (1 + i1 - I )B1 -] . Physical capital depreciates at the rate 8. where is real consumption, transfers. In real terms, :S -- - = -- = = r = = r; These resources are used to purchase consumption, capital, bonds, and nominal money holdings, which are then carried into period t + 1 . Dividing through by the time t price level, the budget constraint can be rewritten in real terms as Wt =f(kt- 1 ) + ( 1 - o )kt-1 + Tt + mt-1 + (11 ++nit-I)bt-1 ::;: Ct + ml + bt + kt , t (3. 1 8) Chapter 3 106 m b +1 + it) bt . (3.19) Wt+l f(kt) + (1 - o )kt + Tt+l + mt +1 +(17Tt+1 The period t gross nominal interest rate 1 + it divided by 1 + ITt+ 1 is the gross real rate of return from period to t + 1 and can be denoted by 1 + rt (1 + it )/(1 + 7Tt+ d · With this notation, (3 .19) can be written as Wt+l f(kt) + (1 - o )kt + Tt+l + (1 + rt )at - ( 1 + i7Ttt + 1 ) mt. where a t m t + bt is the agent's holding of nominal financial assets (money and bonds). This form highlights that there is a cost to holding money when the nominal interest rate is positive. This cost is it f(l + 7Tt+J). Since this is the cost in terms of period + 1 real resources, the discounted cost at time t of holding an additional unit of money is it / (1 + rt)(1 + 7Tt+ 1 ) it /(1 + it). This is the same expression for the opportunity cost of money obtained in chapter 2 in an MIU model. Equation (3 .16) is based on the timing convention that the goods market opens before the asset market. The model of Lucas (1982) assumed the reverse, and individuals can engage where and are real cash and bond holdings. Note that real resources available to the representative agent in period t are given by = t = = = t = in asset transactions at the start of each period before the goods market has opened. In the present model, this would mean that the agent enters period t with financial wealth that can be used to purchase nominal bonds or carried as cash into the goods market to purchase consumption goods. The CIA constraint would then take the form Bt mt-1 Ct -1 + 7Tt + Tt - bt. (3. 20) :S In this case, the household is able to adjust its portfolio between money and bonds before entering the goods market to purchase consumption goods. To understand the implications of this alternative timing, suppose there is a positive opportunity cost of holding money. Then, if the asset market opens first, the agent will only hold an amount of money that is just sufficient to finance the desired level of consumption. Since the opportunity cost of holding is positive whenever the nominal interest rate is greater than zero, (3.20) will always hold with equality as long as the nominal rate of interest is positive. When uncertainty is introduced, the CIA constraint may not bind when is used and the goods market opens before the asset market. For example, if period t's income is uncertain and is realized after has been chosen, a bad income realization may cause the agent to reduce consumption to a point where the CIA constraint is no longer binding. Or a disturbance that causes an unexpected price decline might, by increasing the real value of the agent's money holdings, result in a nonbinding constraint. 10 Since m (3.16) Mt-1 10. While uncertainty may cause the CIA constraint not to bind, it does not follow that the nominal interest rate will be zero. If money is held, the constraint must be binding in some states of nature. The nominal interest rate will equal the discounted expected value of money; see problem 4 at the end of this chapter. Money and Transactions 107 a nonstochastic environment holds in this section, the CIA constraint binds under either timing assumption if the opportunity cost of holding money is positive. For a complete discussion and comparison of alternative assumptions about the timing of the asset and goods markets, see Salyer (1991). The remainder of this chapter follows Svensson (1985) in using (3. 1 6) and assuming that consumption in period t is limited by the cash carried over from period t 1 plus any net transfer. The choice variables at time t are and An individual agent's state at time t can be characterized by resources and real cash holdings m t- 1 ; both are relevant since consumption choice is constrained by the agent's resources and by cash holdings. To ana­ lyze the agent's decision problem, one can define the value function - Wt V(wt,mt- 1 ) = max c 1 , k1 , b1 , m 1 Ct , mr, bt , kt . {u(ct) + ,B V(wt+t,mt) } , (3.21) where the maximization is subject to the budget constraint (from 3 . 1 8) (3.22) Wt+ l the CIA constraint (3. 1 6), and the definition of given by (3. 19). Using this expression for in (3 .21 ) and letting denote the Lagrangian multiplier associated with the budget constraint (the CIA constraint), the first-order necessary conditions for the agent's choice of consumption, capital, bond, and money holdings take the form 1 1 Wt+ l At (M t) [fkCkt) + 1 - 8 ] Vw(Wt+ t,mt) - At = 0, ,8 (1 + rt )Vw(Wt+l ,mt ) - A t = 0, it ] Vw(Wt+ L,mt) + .BVm (Wt+ [,mt) - At = 0. ,B [ 1 + rt - 1 + 1Tt+1 ,8 (3.23) (3.24) (3.25) (3.26) From the envelope theorem, (3.27) (3.28 Vm (Wt,mt- d = ( -1 +1-1rt ) f.it· From (3.27), A t is equal to the marginal utility of wealth. According to (3.23), the marginal utility of consumption exceeds the marginal utility of wealth by the value of liquidity services, f.it· The individual must hold money in order to purchase consumption, ) so the "cost," to which the marginal utility of consumption is set equal, is the marginal utility of wealth plus the cost of the liquidity services needed to finance the transaction. 12 ll. The first-order necessary conditions also include the transversality conditions. 12. Equation (3 .23) can be compared to (3. 1 0) from the shopping-time model. Chapter 3 108 In terms of A., (3.25) becomes (3.29) which is a standard asset pricing equation and is a familiar condition from problems involv­ ing intertemporal optimization. Along the optimal path, the marginal cost (in terms of today's utility) from reducing wealth slightly, must equal the utility value of carrying that wealth forward one period, earning a gross real return l + where tomorrow's utility is discounted back to today at the rate {3 ; that is, fJ ( l + along the optimal path. Using (3.27) and (3.28), the first-order condition (3.26) can be expressed as + f3 (3.30) l+ Equation (3.30) can also be interpreted as an asset pricing equation for money. The price of a unit of money in terms of goods is just 1 at time t; its value in utility terms is By dividing (3.30) through by it can be rewritten as + Solving this equation forward implies that At, At = rt. rt )At+ 1 At = ( A t+ I 7Tt+1tLt+l ) . /P1 Pt.1 3 A.;/P1• A. ;/P1 = f3(A.1+ J /Pt+l tLt+ I !Pt+J ). (3.3 1) tLt+dPt+i (w1+; , mt+i- d /Pt+i- 1· From (3.28), This last expression, though, is equal to V is just the partial of the value function with respect to time t + i - 1 nominal money balances: m av(wt+i ,mt+i- 1 ) = Vm (Wt+i ,mt+i- 1 ) ( amt+i- 1 ) aMt+t.- 1 aMt+t.- 1 ,m (Wt ) Vm +i t+i- 1 Pr+i- 1 = ( �;:; ) 0 This means one can rewrite (3.3 1 ) as In other words, the current value of money in terms of utility is equal to the present value of the marginal utility of money in all future periods. Equation (3.3 1) is an interesting result; it says that money is just like any other asset in the sense that its value (i.e., its price today) is equal to the present discounted value of the stream of returns generated by the asset. In 1 3 . For references on solving difference equations forward in the context of rational-expectations models, see Blanchard and Kahn ( 1 980) or McCallum ( 1 989). Money and Transactions 109 the case of money, these returns take the form of liquidity services. 14 If the CIA constraint were not binding, these liquidity services would not have value = Vm = 0) and neither would money. But if the constraint is binding, then money has value because it yields valued liquidity services. 15 The result that the value of money, satisfies an asset pricing relationship is not unique to the CIA approach. For example, a similar relationship is implied by the MIU approach. The model employed in analyzing the MIU approach (see chapter 2) implied that (JL A./P, At {3 ( At+ l ) + Um (Cc,mc) , Pc Pc+l Pc -= - which can be solved forward to yield um (Ct+i ,mt+i) ] . i f f3 [ Pc i = O Pc+r � = Here, the marginal utility of money Um plays a role exactly analogous to that played by the Lagrangian on the CIA constraint The one difference is that in the MIU approach, yields utility at time t, whereas in the CIA approach, the value of money accumulated at time t is measured by since the cash cannot be used to purchase consumption goods until period t + An expression for the nominal rate of interest can be obtained by using (3.29) and (3.30) = or + to get = = ), the nominal interest rate is given by ) . Since = + + 1 1 1 + (3.32) = = fL. m1 11. 6 /Lt+l , A. c {3(1 + rc )A.c+l f3 (A.c+l + fL t+l ) /(1 + ITt+ I ), (1 rc)(1 + 7Tt+I)A.c+ l + 1 + i1 ( 1 r1) ( 1 ITt+ (A.t+ fL t+ ic ( A.t+ lA. t+ !Ll t+l ) 1 MA.ct++ll . Thus, the nominal rate of interest is positive if and only if money yields liquidity services (/Lt+l 0). In particular, if the nominal interest rate is positive, the CIA constraint is bind­ ing (JL 0). _ > > One can use the relationship between the nominal rate of interest and the Lagrangian multipliers to rewrite the expression for the marginal utility of consumption, given in (3.23), as Uc = A.( l + JL/A.) A.(1 + i) A.. = (3.33) ::::_ 14. The parallel expression for the shopping-time model can b e obtained from (3.5) and (3.8). See problem 2 at the end of this chapter. 1 5 . Bohn (1991 b) analyzed the asset pricing implications of a CIA model. See also Salyer ( 1 99 1 ) . 16. Carlstrom and Fuerst (200 1 ) argued that utility a t time t should depend o n money balances available for spending during period t, or M1 _ 1 f P1 • This would make the timing more consistent with CIA models. With this i timing, m 1 is chosen at time t but yields utility at t 1 . In this case, A t /P1 2:: � 1 {J [u111 (cr+i, m t+i)/ Pr+i] , and the timing is the same as in the CIA model. + = Chapter 3 110 Since A. represents the marginal value of income, the marginal utility of consumption exceeds that of income whenever the nominal interest rate is positive. Even though the economy's technology allows output to be directly transformed into consumption, the "price" of consumption is not equal to it is i, since the household must hold money to finance consumption. Thus, in this CIA model, a positive nominal interest rate acts as a tax on consumption; it raises the price of consumption above its production cost. 17 The CIA constraint holds with equality when the nominal rate of interest is positive, so Since the lump-sum monetary transfer is equal to = this implies that = = Consequently, the consumption velocity of money is identically equal to (velocity = Since actual velocity varies over time, = CIA models have been modified in ways that break this tight link between and One way to avoid this is to introduce uncertainty (see Svensson If money balances have to be chosen prior to the resolution of uncertainty, it may turn out after the realization of shocks that the desired level of consumption is less than the amount of real money balances being held. In this case, some money balances will be unspent, and velocity can be less than Velocity may also vary if the CIA constraint only applies to a subset of consumption goods. Then variations in the rate of inflation can lead to substitution between goods whose purchase requires cash and those whose purchase does not (see problem 6 at the end of this chapter). 1; 1 + c1 M1-1 /P1 + r1• c1 Mt fP1 m1• 1 P1ct/M1 1). r1 1985). (M1 - M1- L) /P1, c m. 1. The Steady State (1 + r88 ) 1/ If consideration is restricted to the steady state, (3.29) implies that = {3 , and i = In addition, (3.24) gives the steady-state capital stock as the solution to (1 + n 88 )//3 - 1 l //3 - 1 + n 88 • R:! So this CIA model, like the Sidrauski MIU model, exhibits superneutrality. The steady­ state capital stock depends only on the time preference parameter {3, the rate of depreciation 8, and the production function. It is independent of the rate of inflation. Since steady-state consumption is equal to f (kss ) - 8P , it, too, is independent of the rate of inflation. 18 It has been shown that the marginal utility of consumption could be written as the marginal utility of wealth (A.) times plus the nominal rate of interest, reflecting the oppor­ tunity cost of holding the money required to purchase goods for consumption. Using (3.32), the ratio of the liquidity value of money, measured by the Lagrangian multiplier J.,L , to the 8 1 1 7 . In the shopping-time model, consumption is also taxed. See problem 3 at the end of this chapter. 1 8 . The expression for steady-state consumption can be obtained from (3 . 1 8) by noting that m1 = r1 + m1_ J ! n 1 and, with all households identical, = 0 in equilibrium. Then (3. 1 8) reduces to c ss + kss = f (kss ) + ( 1 - 8 )kss , or c ss = f (kss ) - 8kss . b Money and Transactions 111 marginal utility of consumption is A.(l + i) 1 + i This expression is exactly parallel to the result in the MIU framework, where the ratio of the marginal utility of money to the marginal utility of consumption was equal to the nominal interest rate divided by 1 plus the nominal rate, that is, the relative price of money in terms of consumption. With the CIA constraint binding, real consumption is equal to real money balances. In the steady state, constant consumption implies that the stock of nominal money balances and the price level must be changing at the same rate. Define e as the growth rate of the nominal quantity of money (so that T1 = 8M1 - 1 ); then Uc The steady-state inflation rate is, as usual, determined by the rate of growth of the nominal money stock. One difference between the CIA model and the MIU model is that with c ss independent of inflation and the cash-in-advance constraint binding, the fact that css = m ss in the CIA model implies that steady-state money holdings are also independent of inflation. The Welfare Costs of Inflation The CIA model, because it is based explicitly on behavioral relationships consistent with utility maximization, can be used to assess the welfare costs of inflation and to determine the optimal rate of inflation. The MIU approach had very strong implications for the opti­ mal inflation rate. Steady-state utility of the representative household was maximized when the nominal rate of interest equaled zero. It has already been suggested that this conclusion continues to hold when money produces transaction services. In the basic CIA model, however, there is no optimal rate of inflation that maximizes the steady-state welfare of the representative household. The reason follows directly from the specification of utility as a function only of consumption and the result that consumption is independent of the rate of inflation (superneutrality). Steady-state welfare is equal to oo L ,B t u(css ) = ( ss ) 1 - .B !!....!:.___ 1= 0 and is invariant to the inflation rate. Comparing across steady states, any inflation rate is as good as any other. 1 9 This finding is not robust to modifications in the basic CIA model. In particular, once the model is extended to incorporate a labor-leisure choice, consumption will no longer be 19. By contrast, the optimal rate of inflation was well defined even in the basic Sidrauski model that exhibited superneutrality, since real money balances vary with inflation and directly affect utility in an MIU model. 112 Chapter 3 independent of the inflation rate, and there will be a well-defined optimal rate of inflation. Because leisure can be "purchased" without the use of money (i.e., leisure is not subject to the CIA constraint), variations in the rate of inflation affect the marginal rate of substitution between consumption and leisure (see section 3.3.2). With different inflation rates leading to different levels of steady-state consumption and leisure, steady-state utility is a function of inflation. This type of substitution plays an important role in the model of Cooley and Hansen (1989) (see section 3.3.2). In their model, inflation leads to an increased demand for leisure and a reduction in labor supply. But before including a labor-leisure choice, it is useful to review briefly some other modifications of the basic CIA model, modifications that will, in general, generate a unique optimal rate of inflation. Lucas and Stokey (1983; 1987) introduced the idea that the CIA constraint may only apply to a subset of consumption goods. They modeled this by assuming that the representative agent's utility function is defined over consumption of two types of goods: cash goods and credit goods. In this case, paralleling (3.23), the marginal utility of cash goods is equated to A. + f.1, ::= A., while the marginal utility of credit goods is equated to A.. Hence, the CIA requirement for cash goods drives a wedge between the marginal utilities of the two types of goods. It is exactly as if the consumer faces a tax of f.l,/A. = i on purchases of cash goods. Higher inflation, by raising the opportunity cost of holding cash, raises the tax on cash goods and generates a substitution away from the cash good and toward the credit good (see also Hartley 1988). The obvious difficulty with this approach is that the classifications of goods into cash and credit goods is exogenous. And it is common to assume a one-good technology so that the goods are not differentiated by any technological considerations. The advantage of these models is that they can produce time variation in velocity. Recall that in the basic CIA model, any equilibrium with a positive nominal rate of interest is characterized by a binding CIA constraint, and this means that c = m. With both cash and credit goods, m will equal the consumption of cash goods, allowing the ratio of total consumption to money holdings to vary with expected inflation. 20 Cash and Credit Goods A second modification to the basic model involves extend­ ing the CIA constraint to cover investment goods. In this case, the inflation tax applies to both consumption and investment goods. Higher rates of inflation tend to discourage capital accumulation, and Stockman (1981) showed that higher inflation would lower the steady-state capital-labor ratio (see also Abel 1985 and problem 9 at the end of this chapter). 21 CIA and Investment Goods 20. Woodford ( 1 998) studied a model with a continuum of goods indexed by i E [0, 1 ] . A fraction s, 0 � s � l, are cash goods. He then approximated a cashless economy by letting s ---+ 0. 21. Abel ( 1 985) studied the dynamics of adjustment in a model in which the CIA constraint applies to both consumption and investment. Money and Transactions 113 In CIA models, inflation acts as a tax on goods or activities whose purchase requires cash. This tax then introduces a distortion by creating a wedge between the marginal rates of transformation implied by the economy's technology and the marginal rates of substitution faced by consumers. Since the CIA model, like the MIU model, offers no reason for such a distortion to be introduced (there is no inefficiency that calls for Pigovian taxes or subsidies on particular activities, and the government's revenue needs can be met through lump-sum taxation), optimality calls for setting the inflation tax equal to zero. The inflation tax is directly related to the nominal rate of interest; a zero inflation tax is achieved when the nominal rate of interest is equal to zero. Implications for Optimal Inflation 3.3.2 A Stochastic CIA Model While the models of Lucas (1982), Svensson (1985), and Lucas and Stokey (1987) provide theoretical frameworks for assessing the role of inflation, they do not provide any guide to the empirical magnitude of inflation effects or to the welfare costs of inflation. What one would like is a dynamic equilibrium model that could be simulated under alternative monetary policies-for example, for alternative steady-state rates of inflation or alternative policy responses to shocks-in order to assess quantitatively the effects of inflation and monetary policy. Such an exercise was conducted by Cooley and Hansen (1989; 1991), who were the first to add money and a cash-in-advance constraint to a calibrated real business cycle model. They followed the basic framework of Lucas and Stokey ( 1987). However, important aspects of their specification include ( 1 ) introduction of capital, and consequently an investment decision; (2) the introduction of a labor-leisure choice; and (3) the identification of consumption as the cash good and investment and leisure as credit goods. Inflation represents a tax on the purchases of the cash good, and therefore higher rates of inflation shift household demand away from the cash good and toward the credit good. In Cooley and Hansen's formulation, this implies that higher inflation increases the demand for leisure. One effect of higher inflation, then, is to reduce the supply of labor. This then reduces output, consumption, investment, and the steady-state capital stock. Cooley and Hansen expressed welfare losses across steady states in terms of the con­ sumption increase (as a percentage of output) required to yield the same utility as would arise if the CIA constraint were nonbinding. 22 For a 10 percent inflation rate, they reported a welfare cost of inflation of 0.387 percent of output if the CIA constraint is assumed to apply at a quarterly time interval. Not surprisingly, if the constraint binds only at a monthly time interval, the cost falls to 0. 1 12 percent of output. These costs are small. For much higher rates of inflation, they start to look significant. For example, with a monthly time 22. Refer to Cooley and Hansen ( 1 989, sec. II) or Hansen and Prescott ( 1 994) for discussions of the computa­ tional aspects of this exercise. Chapter 3 114 period for the CIA constraint, a 400 percent annual rate of inflation generates a welfare loss equal to 2. 1 37 percent of output. The welfare costs of inflation are discussed further in section 3.4.2 and in chapter 4. The Basic Model To model the behavior of the representative agent faced with uncertainty and a CIA con­ straint, assume the agent's objective is to maximize (3.34) c1 n1 with 0 < f3 < 1 . Here is real consumption, and is labor supplied to market activities, expressed as a fraction of the total time available, so that 1 is equal to leisure time. The parameters <t> , W , and T} are restricted to be positive. Households supply labor and rent capital to firms that produce goods. The household enters each period with nominal money balances and receives a nominal lump-sum transfer equal to In the aggregate, this transfer is related to the growth rate of the nom­ inal supply of money. Letting the stochastic variable denote the rate of money growth At the start of period t, is ), the per capita transfer equals = (1 + known to all households. Households purchase bonds and their remaining cash is avail­ able for purchasing consumption goods. Thus, the timing has asset markets opening first, and the CIA constraint, which is taken to apply only to the purchase of consumption goods, takes the form T1• 81)M1-1 (M1 23 - n1 M1_ 1 81 81Mt- 1· B1, 81 P1 (3.35) Ct 1m+t-llft + Tt - bt. Here l + n1 is equal to l plus the rate of inflation. The CIA constraint will always be where is the time t price level. Note that time t transfers are available to be spent in period t. In real terms, the CIA constraint becomes _:::: --- binding if the nominal interest rate is positive. In addition to the CIA constraint, the household faces a flow budget constraint in nominal terms of the form b 23. In order to allow for comparison between the MIU model developed earlier and a CIA model, the preference function used earlier, (2.66) in chapter 2, is modified by setting a = I and = 0 so that real balances do not yield direct utility. The resulting utility function given in (3.34) differs from Cooley and Hansen's specification; they assume that the preferences of the identical (ex ante) households are log separable in consumption and leisure, a case obtained when <I> = 1J = I . Money and Transactions 115 In real terms, this becomes m . (3.36) m t = Yt + (1 - o)kt- 1 + lt ht - kt + t - 1 + it - Ct, 1 + lft where 0 ::=: o ::=: 1 is the depreciation rate. The household is assumed to own the economy's technology, given by a Cobb-Douglas constant returns to scale production function, which can be expressed in per capita terms as -- (3.37) where 0 ::=: a process: ::=: 1. The exogenous productivity shock Zt is assumed to follow an AR(l) Zt = Pz Zt-1 + et. with 0 ::=: Pz ::=: 1 . The innovation et has mean zero and variance cr'j. The individual's decision problem can be characterized by the value function where the maximization is subject to the constraints (3.35) and (3.36). If A t is the Lagrangian multiplier on the budget constraint and /h t is the multiplier on the cash-in-advance constraint, these first-order conditions take the form (3.38) IJ! ( l - n) - " = (1 - a) (�:) At. (3.39) (3.40) A t = .BEt (l + rt )A t+ 1 , (3.41) it A t - !ht = 0, A + /h t+1 (3.42) , A t = ,BEt t+1 1 + lft+l where r1 = a (Yt+ t !kt ) - o . Finally, let Ut = et - ess be the deviation of money growth from its steady-state average rate and assume [ ] Ut = PuUt- 1 + 4>Zt- l + cpt. where cp1 is a white noise innovation with variance a�. This is the same process for the nominal growth rate of money that was used in chapter 2. Chapter 3 116 The Steady State With the same parameter calibrations as those reported in section 2.5.4 for the MIU model, the steady-state values of the ratios that were reported for the MIU model are also the steady-state values for the CIA model (see the chapter appendix). The Euler condition ensures 1 + = 1 {J, which then implies and, with investment in = + the steady state equal to = Even though the method used to gen­ erate a demand for money has changed in moving from the MIU model to the CIA model, the steady-state values of the output-capital and consumption-capital ratios are unchanged. Note that none of these steady-state ratios depends on the growth rate of the nominal money supply. The level of real money balances in the steady state is then determined by the cash­ in-advance constraint, which is binding as long as the nominal rate of interest is positive. Hence, = so = + + = The steady-state labor supply depends on the money growth rate and therefore on the rate of inflation. The chapter appendix shows that satisfies rss / ls /kss (rss 8) /a 8kss , css /kss Cls /kss ) - 8. css mss /(1 rr ss ) r ss mss , mss ;ps css /Ps . nss ( 1 - a ) ( fJ SS ) ( lSSs ) i�: ( cssSS ) - <1> ' 1 n ss n ss (3.43) k k 1+e where e is the steady-state rate of money growth. Since the left side of this expression is increasing in n ss , a rise in e ss , which implies a rise in the inflation rate, lowers the steady­ _ ( ) - ry ( ) <!> _ - __ llJ ___ _ _ state labor supply. Higher inflation taxes consumption and causes households to substitute toward more leisure. This is the source of the welfare cost of inflation in this CIA model. The elasticity of labor supply with respect to the growth rate of money is negative. It is useful to note the similarity between the expression for steady-state labor supply in the CIA model and the corresponding expression (see (2.80) in chapter 2) that was obtained in the MIU model. With the MIU specification, faster money growth had an ambiguous effect on the supply of labor. With the calibrated values of the parameters of the utility function used in chapter 2, money and consumption were complements, so higher inflation, by reducing real money holdings, lowered the marginal utility of consumption and also reduced the supply of labor. Dynamics The dynamic implications of the CIA model can be explored by obtaining a first-order linear approximation around the steady state of the model's equilibrium conditions. The derivation of the approximation is contained in the chapter appendix. As in chapter 2, a variable x denotes the percentage deviation of around the steady state. 24 The CIA model can be approximated around the steady state by the following ten linear equations: x (3.44) 24. The exceptions again being that r and i are expressed in percentage terms (e.g., r1 = r1 - r55 ). Money and Transactions 117 (3.45) 'kt = c 1 - o ) kt-1 + ox(, r1 = a (�:: ) (EtYt+ l - kt) , At = ft + EtAt+l , (3.46) (3.47) (3.48) (3.49) - <t> c1 = A t + i�> At = - <t>Etct+l - Edrt+J , (3.50) (3.5 1) (3.52) (3.53) The first six equations (production function, resource constraint, capital accumulation equation, marginal product of capital equation, Euler condition, and labor-leisure condi­ tion) are identical to those found with the MIU approach. The critical differences between the two approaches appear in a comparison of (3.50), (3.5 1), and (3.52) with (2.74) and (2.75) of chapter 2. In the MIU model, utility depended directly on money holdings, so In the CIA (2.75) expressed the marginal utility of consumption in terms of and model, the marginal utility of income can differ from the marginal utility of consumption; (3.5 1) reflects the fact that an extra dollar of income received in period t cannot be spent on consumption until t l . Equation (3.42) gives I ) /(1 this becomes Since the marginal utility of consumption c;-ct> is equated to Linearizing this result produces (3.5 1). Equa­ + + tion (2.75) was the MIU money demand condition derived from the first-order condition for the household's holdings of real money balances. In the CIA model, (3.50) and (3.52) reflect the presence of the nominal interest rate as a tax on consumption and the binding cash-in-advance constraint in the CIA model. Finally, note that (3.48), (3.50), and (3.5 1) can b e combined to yield the Fisher equation: Ct mt . + f3Etc�"; !O 7rt+d = f3Etm�"; !O 7rt+J). Calibration and Simulations At = f3Et (A t+ l + It t+ At + It t > + 7rt+d .25 At = rt = Et (it+l - frt+l ) . To assess the effects of money in this CIA model, values must be assigned to the specific parameters; that is, the model must be calibrated. The steady state depends on the values of fJ, IJ!, and The baseline values reported in section 2.5.4 for the MIU model can be employed for the CIA model as well. a,f3,8, <t> . 25. Equation (3.30) is the corresponding equation for the nonstochastic CIA model of section 3.3. 1 . Chapter 3 118 Recall that the MIU model displayed short-run dynamics in which the real variables such as output, consumption, the capital stock, and employment were independent of the nominal money supply process when utility was log-linear in consumption and money balances. 26 While does not directly enter the utility function in the CIA model, note that in the case of log utility in consumption (that is, when <1> = 1), the short-run real dynamics in the CIA model are not independent of the process followed by as they were in the MIU model. Equations (3.49), (3.5 1), and (3.53) imply, when <1> = 1 , that m m, �� = -Er (mr+l + nr+l) = - (mr + Er ur+l ) = ( 1 + TJ �:: ) nr - Yr· Thus, variations in the expected future growth rate of money, E1ur+ J , force adjustment to y, c or (or all three). In particular, for given output and consumption, higher expected money growth (and therefore higher expected inflation) produces a fall in This is the effect, discussed earlier, by which higher inflation reduces labor supply and output. The current growth rate of the nominal money stock, and the current rate of inflation, (see 3.53). Hence, as seen in the MIU model, unantic­ only appear in the form ipated monetary shocks affect only current inflation and have no real effects unless they alter expectations of future money growth (i.e., unless 1 is affected). The responses of output, employment, and other variables to a positive money growth rate shock are illustrated in figure 3 . 1 . As in the MIU model under the baseline calibration, a positive money growth rate shock reduces output and employment, and the impact is larger the more highly positively serially correlated the shock is. The rise in money growth immediately raises expected inflation when Pu > 0 and the nominal interest rate. Greater persistence of the money growth rate process leads to larger movements in expected infla­ tion in response to a monetary shock. By raising the expected rate of inflation and thereby increasing the inflation tax on consumption, the money growth rate increase induces a sub­ stitution toward leisure that lowers labor supply and output. These effects are larger the more persistent the rise in expected inflation. 27 The economy's response to a productivity shock depends on the money growth rate process when ¢ differs from zero. This is illustrated in figure 3.2. For example, when ¢ is negative, a positive productivity shock implies that money growth will decline in the future. Consequently, expected inflation also declines. The resulting reduction in the nominal interest rate lowers the effective inflation tax on consumption and increases labor supply. In contrast, when ¢ is positive, a positive productivity shock increases expected inflation and reduces labor supply. This tends to partially offset the effect of the productivity shock on (m), n n1• u1 - n1 lfr , u1, E1u1+ b 26. This was the case in which <I> = = 1 . 27. Comparing figure 3 . 1 with figure 2.4 reveals that a money growth rate shock has a larger real impact in the CIA model than in the MIU model of chapter 2; this difference would be larger if a smaller value of the money demand parameter had been used in the MIU model. b Money and Transactions 119 o.o5 .-��-�0"-u::.tcc.p.::.ut'-.-----, 0 ·0.05 /� 10 15 20 -- p , = 0. 67 -- p u = 0. 9 25 30 35 1 .�r=5==1 � ·0.1 40 -0 . 1 5 L_��-��-��-�---' 10 15 20 25 30 35 40 ; E?J o : E:"'� I 5 10 15 20 25 30 35 40 10 15 20 25 30 35 40 Figure 3.1 5 10 15 20 25 30 35 40 10 15 20 25 30 35 40 Responses to a positive money growth rate shock in the CIA model; Pu = 0.67, Pu = 0.9. : ��· . : , I : �;:=: : 1l : : � . I "l)? I l5?�... I : p;·'"" � . 0 5 10 15 20 25 30 35 40 -0.5 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Figure 3.2 5 10 15 20 25 30 35 40 35 40 Nominal Interest Rate Real Interest Rate -0.01 -0.2 -0.5 5 10 15 20 25 30 -0.2 L_��-��-��-�_j 5 10 15 20 25 30 35 40 Effects of tP on responses to a productivity shock in the CIA modeL Chapter 3 120 output. Thus, output variability is less when ¢ is positive than when it is zero or negative. However, the effects are small; as ¢ goes from -0.5 to 0 to 0.5, the standard deviation of output falls from 1 . 3 1 to 1 . 1 6 to 1 .02. 3.4 Search Both the MIU and the CIA approaches are useful alternatives for introducing money into a general equilibrium framework. However, neither approach is very specific about the exact role played by money. MIU models assume the direct utility yielded by money proxies for the services money produces in facilitating transactions. However, the nature of these transactions and, more important, the resource costs they might involve, and how these costs might be reduced by holding money, are not specified. Use of the CIA model is motivated by appealing to the idea that some form of nominal asset is required to facili­ tate transactions. Yet the constraint used is extreme, implying that there are no alternative means of carrying out certain transactions. The CIA constraint is meant to capture the essential role of money as a medium of exchange, but in this case one might wish to start from a specification of the transaction technology to understand why some commodities and assets serve as money and others do not. A number of papers have employed search theory to motivate the development of media of exchange; this has been one of the most active areas of monetary theory. Examples include Jones (1976), Diamond ( 1984), Kiyotaki and Wright (1989; 1993), Oh (1989), Trejos and Wright (1993 ; 1995), Ritter (1995), Shi (1995), Rupert, Schindler, and Wright (2001), Lagos and Wright (2005), Rocheteau and Wright (2005), and the papers in the May 2005 issue of the Williamson and Wright (201 1) and Nosal and Rocheteau (201 1) provided excellent surveys of the literature. In these models, individual agents must exchange the goods they produce (or with which they are endowed) for the goods they consume. During each period, individuals randomly meet other agents; exchange takes place if it is mutually beneficial. In a barter economy, exchange is possible only if an agent holding good and wishing to consume goodj (call this an agent) meets an individual holding goodj who wishes to consume good ( aj agent). This requirement is known as the and limits the feasibility of direct barter exchange when production is highly specialized. Trade could occur if agent meets a agent for k =!= j as long as exchange of goods is costless and the probability of meeting a jk agent is the same as meeting a j agent. In this case, agent ij would be willing to exchange for k (thereby becoming an kj agent). In the basic Kiyotaki-Wright model, direct exchange of commodities is assumed to be costly, but there exists a fiat money that can be traded costlessly for commodities. The assumption that there exists money with certain exchange properties (costless trade with commodities) serves a role similar to that of putting money directly into the utility function International Economic Review. i double coincident of wants i ij i i ij ki i Money and Transactions 121 in the MIU approach or specifying that money must be used in certain types of transac­ tions in the CIA approach. 28 More recent work on search and exchange assumes trading is anonymous, so credit is precluded; one would not accept an IOU from a trading partner if one were unable to identify or locate that person when wanting to collect. 29 However, whether an agent will accept money in exchange for goods will depend on the probability the agent places on being able later to exchange money for a consumption good. Suppose agents are endowed with a new good according to a Poisson process with arrival rate a. 3 0 Trading opportunities arrive at rate b. A successful trade can occur if there is a double coincidence of wants. If x is the probability that another agent chosen at random is willing to accept the trader's commodity, the probability of a double coincidence of wants is x2 . A successful trade can also take place if there is a single coincidence of wants (i.e., one of the agents has a good the other wants), if one agent has money and the other agent is willing to accept it. That is, a trade can take place when an ij agent meets a jk agent if the ij agent has money and the jk agent is willing to accept it. In this simple framework, agents can be in one of three states: an agent can be waiting for a new endowment to arrive (state 0), can have a good to trade and be waiting to find a trading partner (state 1), or have money and be waiting for a trading opportunity (state m). Three equilibria are possible. Suppose the probability of making a trade holding money is less than the probability of making a trade holding a commodity. In this case, individ­ uals will prefer to hold on to their good when they meet another trader (absent a double coincidence) rather than trade for money. With no one willing to trade for money, money will be valueless in equilibrium. A second equilibrium arises when holding money makes a successful trade more likely than continuing to hold a commodity. So every agent will be willing to hold money, and in equilibrium all agents will be willing to accept money in exchange for goods. A mixed monetary equilibrium can also exist: agents accept money with some probability as long as they believe other agents will accept it with the same probability. The Kiyotaki-Wright model emphasized the exchange process and the possibility for an intrinsically valueless money to be accepted in trade. It does so, however, by assuming a fixed rate of exchange-one unit of money is exchanged for one unit of goods whenever a trade takes place. The value of money in terms of goods is either 0 (in a nonmonetary 28. In an early analysis, Alchian ( 1 977) attempted to explain why there might exist a commodity with the types of exchange properties assumed in the search literature. He stressed the role of information and the costs of assessing quality. Any commodity whose quality can be assessed at low cost can facilitate the acquisition of information about other goods by serving as a medium of exchange. Models assuming an absence of a technology for record keeping rule out credit. For an analysis of "money as memory," see Kocherlakota ( 1 998). 29. Anonymity is treated as given, and the role of third parties, such as credit card companies and banks, that solve this problem in monetary economies is precluded by assumption. 30. Kiyotaki and Wright ( 1 993) interpreted this as a production technology. Chapter 3 122 equilibrium) or 1 . In the subsequent literature, however, the goods price of money is deter­ mined endogenously as part of the equilibrium. For example, Trejos and Wright (1995) made price the outcome of a bargaining process between buyers and sellers who meet through a process similar to that in the Kiyotaki-Wright model. However, Trejos and Wright assumed money is indivisible, while goods are infinitely divisible (i.e., all trades involve one dollar, but the quantity of goods exchanged for that dollar may vary). Shi (1997) extended the Kiyotaki-Wright search model to include divisible goods and divis­ ible money, and Shi (1999) also analyzed inflation and its effects on growth in a search model. 3.4.1 Centralized and Decentralized Markets Lagos and Wright (2005) provided the core example of a monetary search model and the insights about the costs of inflation that this literature has provided. Money is assumed to be perfectly divisible and is the only storable good available to agents. Each period is divided into subperiods, called day and night. Agents consume and supply labor (produce) in both subperiods. The subperiods differ in terms of their market structure. Night markets are centralized and competitive; day markets are decentralized, and prices (and quantities) are set via bargaining between individual agents in bilateral meetings. The preferences of agents are identical and given by U = U(x,h,X,H) = u(x) - c(h) + U(X) - H, where x (X) is consumption during the day (night), and h (H) labor supply during the day (night). The utility functions c, and U have standard properties, and it is assumed that there exist q * and X* such that (q * ) = c' (q * ) and U' (X* ) = 1 . Utility is linear in night labor supply H. The technology allows one unit of H to be transformed into one unit of X. u, u ' Hence X* is the quantity of the night good such that marginal utility equals marginal cost. During the night, trading takes place in a centralized Walrasian market. Consider the decision problem of an agent who enters the night market with nominal money balances Let denote the price of money in terms of goods (i.e., the price level, the price of goods in terms of money, is 1 I Let be the value function for an agent at the start of the night market with money holdings and let V1+ 1 be the value function for the agent entering the day market with money holdings (described later). Then is defined as max [ .B Vt+l ] m. ¢1 W1(m) = ¢1). W1(m) m, X,H,m' U(X) - H + m' (m') W1(m) (m1) , where the maximization is subject to a budget constraint of the form ¢1m + H = X + ¢1m' . The left side represents the agent's real money holdings on entering the night market plus income generated from production. The right side is consumption plus real balances carried Money and Transactions 123 into the next day market. Using the budget constraint to eliminate H, the problem can be rewritten as W1(m) = max [U(X) + ¢1m - X + ¢1m ' + X, m' .BVr+ l (m') J . (3.54) The first-order conditions for an interior solution take the form31 U(X) = 1 ::::? X = X* , (m' ) = 0. + (3.55) (3.56) <Pr .sv;+ t Equations (3.55) and (3.56) imply that X and m' are independent of m. This is a con­ sequence of the assumption that utility is linear in H. Intuitively, the marginal value of accumulating an extra dollar in the centralized market is (m') . The marginal cost of acquiring an extra dollar is ¢1 times the utility cost of the extra labor needed to produce and sell more output. But the marginal disutility of work is a constant (equal to 1). So the marginal cost of acquiring an extra dollar is just which is the same for all agents. But if all agents exit the night market holding the same level of money balances, that is, the same m', the distribution of money holdings across agents at the start of each day will be degen­ erate. This is extremely useful in dealing with a model in which agents may have different market experiences, as they will in Lagos and Wright's day market, while still preserving the idea of a representative agent. Shi (1999) adopted the notion of a large family whose individual members may have different experiences during each period but who reunite into a representative family at the end of each period. This approach, originally introduced by Lucas ( 1990), is used in chapter 5 when discussing models that impose restrictions on access by some agents to credit markets. A final useful result from (3.54) is that W can be written as (m1 ) J , W1(m) = ¢1m + max [U(X) - X + ¢1m' + .sv;+ l </Jr . X. m' ,B Vr+ l showing that W is linear in m. The subperiods differ in the nature of the trading process that occurs in each. The day good comes in different varieties, and agents each consume a different variety than the one they produce. Hence, there is a motive for trade. As in the night market, one unit of labor can be converted into one unit of the good. In the day market, agents search for trading partners. With probability a , they meet another agent. One of three possible outcomes can occur as a result of this meeting. First, each consumes what the other produces. This corresponds to a double coincidence of wants; no money or credit is necessary for a trade to occur. Assume the probability of a double coincidence of wants is 8 . Second, there could be a single coincidence of wants; one agent consumes what the other produces, but not vice x 3 1 . Because of the linearity of utility in H, Lagos and Wright (2005) needed to verify that H < fl in equililbrium, where fl is the maximum labor time an agent has available. Chapter 3 124 versa. Assume the probability of this occurring is 2CJ . 32 Finally, neither agent consumes what the other produces, an event that occurs with probability 1 - 8 - 2CJ . Recall that V1(m) is the value function for an agent with money holdings m who is entering the decentralized day market, and W1(m) is the value function when entering the centralized night market. Let F1(ih) be the fraction of agents at the beginning of day t with m ::S ih. Then f B1(m, ih)dF1(ih) + aCJ J {u [qt(m, ih )] + Wt [m - d1(m, ih)]} dF1(ih) + aCJ J { - c [qt (ih, m)] + Wt [m + d1(ih, m)] } dF1(ih) Vt(m) = ao (3.57) + (1 - ao - 2a CJ ) W1(m), where B1 (m, ih) is the payoff to an agent holding m who meets an agent holding ih when there is a double coincidence of wants. The four terms in V1(m) are (1) the probability of a double coincidence times the expected payoff; (2) the probability the agent meets another agent with ih, there is a single coincidence of wants, and d1(m, ih) is exchanged for q1(m, ih) of the consumption good; (3) the probability of a single coincidence meeting in which the agent produces q1(fh, m) and receives d1(fh, m) ; and (4) the probability that no meeting (or trade) occurs and the agent enters the night market with m. Because the day meetings each involve just two agents, the search literature has gener­ ally assumed the price and quantity exchanged, q1 and d1, are determined by Nash bargain­ ing between the agents. When a double coincidence of wants occurs, the joint surplus is maximized when q * is exchanged, where u ' (q * ) = c' (q * ) . Hence, B1(m, ih) = u(q * ) - c(q * ) + W1(m) . When a single coincidence occurs, bargaining is more complicated. Let the buyer's share of the joint surplus from a bargain be 8 E [0 1]. The threat point of a buyer is W1(m) ; that of the seller is W1(ih), where m and ih are the buyer's and the seller's initial money holdings. The exchange of q for d units of money maximizes (3.58) [u(q) + Wr(m - d) - W1(m)]11 [ -c(q) + W1(m + d) - W1(m)] l -ll , subject to d ::::_ 0, q ::::_ 0. Recall that W1(m) is linear in m. Hence, (3.58) can be rewritten as (3.59) [u(q) - cf>td]11 [ -c(q) + ¢1d] 1- 11 • 32. For agents i and ), the probability i consumes what } produces but not vice versa is a ; the probability j con­ sumes what i produces but not vice versa is also a . Thus, the probability a meeting satisfies a single coincidence of wants is 2a . Money and Transactions 125 d m, d q -B cf>t [u(q) - cf>tdr 1 + (1 - B) cf>t [ -c(q) + cf>tdr 1 Bu' (q) [u(q) - ¢>1dr 1 - ( 1 - B)c' (q) [ - c(q) + ¢>1dr 1 0, or u' (q) c' (q) qt q* , cf>td* B c(q* ) + (1 - B)u(q* ). The monetary cost of q , d* , i s a weighted average of the cost of producing it and the value of consuming it, with weights reflecting the bargaining power of the buyer and seller. If d* m, then the buyer does not have the cash necessary to purchase q * ; in effect, the cash-in-advance constraint is binding. In this case, Lagos and Wright (2005 ) showed that If _::: money holdings are not a binding constraint, and the first-order conditions with respect to and yield = = :::} = o, = = > the seller receives all the buyer's money, so (3.60) q1 where is the solution to a constrained Nash bargaining problem. 33 The quantity trans­ acted and the price depend on the buyer's money holdings but do not depend on the seller's. This quantity can be expressed as a function of = Lagos and Wright showed that the amount of money agents carry out of the night market, is less than whenever the inflation rate, 1 , exceeds fJ 1 . Recall I) that an inflation rate of fJ 1 corresponds to the Friedman rule of a zero nominal interest rate. So, just as in the earlier CIA models, the cash-in-advance constraint is binding when the nominal rate of interest is positive. Of course, the constraint only binds for agents who find themselves as buyers in single coincidence of wants meetings. Sellers, or those in a double coincidence of wants meeting, or in no meeting, exit the period with unchanged money holdings. Now consider the value to an agent of entering the day market with money holdings This value arises from the effects of on price and quantity when the agent is the buyer in a single coincidence meeting. Since the probability this occurs is a a , it can be expressed, using (3.57), as d* m', - m Vt (m) = aa I {u [q1(m)] - ¢>1d1(m) } dF1(fh). 33. q(m) solves ec(q)u' (q) + ( I - 8)u(q)c1 (q) eu'(q) + ( I - 8)c '(q ) ----'�:-'-'-'---:-:--' ---:� :'-- ,---c:_ ---:-: = <Pt m . See Lagos and Wright (2005) for details. m: q1 q1(m). (cf>t !¢>1+ - - m. Chapter 3 126 The value of money is then given by the pricing equation 3 .6 1 = fJ + where is the aggregate nominal quantity of money. Because = 1 (an increase in the quantity of money increases the number of dollars needed to purchase goods by the same amount), c/Jt [v;+l (M) c/Jt+d , M d;+ l (M) ( ) v;+l (M) = [u' [qt+1 (M) ] q;+l (M) - c/Jt+l] . Using this in ( 3 . 6 1 ) , c/Jt fJaCJu' [qt+l] q;+l (M) + fJ (1 - aCJ) c/Jt+l· cw = (3.62) The value of money is determined by the marginal utility of the goods the agent is able to consume when faced with a single coincidence of wants trading opportunity. If such meetings are uncommon (aCJ is small), money will be less useful and therefore less valu­ able. This implication of search models of money emphasizes the importance of the trading environment for determining the value of money. Equation (3.62) can be rewritten34 using (3.60) as c/Jt = fJ [ aCJ u'z, ((qt+d qt+l ) + (1 - aCJ) ] c/Jt+l · Now consider a steady state in which the money stock grows at the rate r . The inflation rate will also equal r : (c/Jt ! ¢1+ 1 ) - 1 = r . Thus, (qt+l) + (1 - aCJ) ] c/Jt+l ::::} 1 = fJ [aCJ u'(q) + (1 - aCJ) ] ( -1- ) c/Jt = fJ [aCJ u'z'(qt+J) 1+r z'(q) using (3.60). Solving for u' / z', u'(q) = l + r - fJ ( l - aCJ) = 1 + l + r - fJ . -{JaCJ {JaCJ z' (q) The left side of this equation, u' / z', is the marginal utility of consumption divided by the marginal cost of the good. The right side is 1 plus a term that can be written as fJ - 1 ( 1 + r) - 1 divided by aCJ. But since fJ - 1 is the gross real interest rate and r is the inflation rate, fJ - 1 ( 1 + r) - 1 is the nominal rate of interest, so u' (q) 1 + -i . (3.63) z'(q) aCJ - = This looks very similar to earlier results from a CIA model (see (3.33)). A positive nominal interest rate acts as a tax on consumption. But this tax now also depends on the nature of trading. An increase in the frequency of single coincidence meetings, by raising the usefulness of money, reduces the net cost of holding money. 34. Equation (3.60) implies q' = ¢!z' . Money and Transactions 3.4.2 127 The Welfare Costs of Inflation While it is clear from (3.63) that the inflation tax is zero if the Friedman rule of a zero nominal interest rate is followed, Lagos and Wright showed that the equilibrium with i = 0 is still not fully efficient because of the trading frictions associated with bargaining in the decentralized market. Efficiency requires that all the surplus go to the buyer (e = 1). 35 In standard models such as the MIU model in chapter 2 or the CIA model in section 3.3, full efficiency is attained with i = 0. Then, since i = 0 maximizes welfare, small deviations have small effects on welfare (basically an application of the envelope theorem). But if e < 1 , the equilibrium with i = 0 in the search model does not fully maximize utility. Hence, small deviations from the Friedman rule can have first-order effects on welfare. By calibrating their model, Lagos and Wright found much larger welfare costs of positive nominal interest rates than other authors had found. The importance of the trading environment in determining the costs of inflation is further explored by Rocheteau and Wright (2005). They compared welfare costs in three settings: a search model similar to Lagos and Wright (2005), a competitive market model, and a search model with posted prices (rather than the bilateral bargaining of the basic search model). By allowing for endogenous determination of the number of market participants, Rocheteau and Wright introduced an extensive margin (the effects on the value of money as the number of traders varies) as well as an intensive margin (the effects for a given number of traders as individual agents' money holdings vary). The Friedman rule always ensures efficiency along the intensive margin, but the extensive margin may still generate a source of inefficiency. Interestingly, if the market makers in the competitive search version of the model internalize the effects of the prices they post on the number of traders they attract, the model endogenously ensures that the Hosios condition is satisfied, as shown by Moen (1997), and the equilibrium is fully efficient when the nominal rate of interest is zero. Lagos and Rocheteau (2005) explored the interactions of the pricing mechanism (bilateral bargaining versus posted pricing) and found that with directed search, inflation can increase search intensities when inflation is low but reduce them when inflation is high. Thus, at low inflation rates, an increase in inflation can raise output, but they showed that this actually reduces welfare, and the Friedman rule supports the efficient equilibrium. Craig and Rocheteau (2008) demonstrated the search approach can account for the esti­ mates of the welfare costs of inflation obtained from examining the area under the money demand curve, as discussed in chapter 2. The Lagos and Wright model has only one nominal asset money. If an interest-bearing nominal asset such as a bond were introduced into the analysis, it would dominate money whenever the nominal interest rate is positive. To explain the simultaneous existence of l. 35. This is essentially the Hosios ( 1 990) condition for this model; since the quantity transacted is independent of the seller's money holdings, all the surplus is due to the buyer, so efficiency would require e = Chapter 3 128 interest-bearing nominal bonds and non-interest-bearing money, Shi (2005) employed a model with a decentralized goods market and a centralized bond market but in which there are assumed to be barriers to trading across markets. Households can use either bonds or money in the goods market, but only money can be used to purchase bonds. At the start of each period, households must allocate their money holdings between the two markets. Assume a fraction a is sent to the goods market and 1 - a to the bond market. Let w'J' denote the value of money at the end of period t. Then Shi showed that w'J' = f3 aa CJ )... ':t, 1 + f3 w':t, 1 , where f3 is the discount factor, a and CJ are the probability of meeting a potential trading partner and the probability there is single coincidence of wants, and ).., m is the Lagrangian multiplier on the constraint that the money payment from buyer to seller in the goods market must be less than the buyer's money holdings. Thus, aaCJ )... ':t, 1 is the service value of money in facilitating a goods purchase. The current value of money is equal to this service value plus the discounted future value of money. Money the household sends to the bond market cannot be used to purchase current goods, nor can the newly purchased bonds be used to exchange for goods. While bonds can, in future periods, be used to purchase goods, purchasing bonds initially entails a one­ period loss of liquidity. Therefore, bonds must sell at a discount relative to money; if S is the money price of a bond, S < 1 . Shi demonstrates that the nominal interest rate, ( 1 - S)/S, is given by 1 - S aaCJ A m (3.64) wm S which is positive if ).., m is positive. This expression for the nominal interest rate can be compared to (3.32), obtained in a basic cash-in-advance model. Similar to the result in other models in the search literature, (3.64) reveals how the nature of transactions in the decentralized market as reflected in the parameters a and CJ affects the value of money and the nominal interest rate. In Shi's basic model, old bonds and money can both circulate in the goods market and be used in purchasing goods. Suppose, however, that the government also engages as a seller in the goods market, and assume the government only accepts money in payment for goods. Since there is a chance a household will encounter a government seller in the decentralized market, and frictions are assumed to prevent the household from locating another seller, there is a smaller probability of a successful trade if the household carries only bonds into the goods market than if it carries money. This difference drives bonds out of the goods market, and Shi showed that only money circulates as a means of payment. The search-theoretic approach to monetary economics provides a natural framework for addressing a number of issues. Ritter (1995) used it to examine the conditions necessary Money and Transactions 129 for fiat money to arise, linking it to the credibility of the issuer. Governments lacking credibility would be expected to overissue the currency to gain seigniorage. In this case, agents would be unwilling to hold the fiat money. Soller and Waller (2000) used a search­ theoretic approach to study the coexistence of legal and illegal currencies. By stressing the role of money in facilitating exchange, the search-theoretic approach emphasizes the role of money as a medium of exchange. The approach also emphasizes the social aspect of valued money; agents are willing to accept fiat money only in environments in which they expect others to accept such money. 36 3.5 Summary The models studied in this chapter are among the basic frameworks monetary economists have found useful for studying the effects of inflation and the welfare implications of alter­ native rates of inflation. These models, and those examined in chapter 2, assume prices are perfectly flexible, adjusting to ensure that market equilibrium is continuously maintained. The MIU, CIA, shopping-time, and search models all represent means of introducing val­ ued money into a general equilibrium framework. Each approach captures some aspects of the role that money plays in facilitating transactions. Despite the different approaches, several conclusions are common to all. First, because the price level is completely flexible, the value of money, equal to 1 over the price of goods, behaves like an asset price. 37 The return money yields, however, differs in the various approaches. In the MIU model, the marginal utility of money is the direct return, while in the CIA model, this return is measured by the Lagrangian multiplier on the CIA constraint. In the shopping-time model, the return arises from the time savings provided by money in carrying out transactions, and the value of this time savings depends on the real wage. In search models, it depends on the probability of trading opportunities. All these models have similar implications for the optimal rate of inflation. An effi­ cient equilibrium is characterized by equality between social and private costs. Because the social cost of producing money is taken to be zero, the private opportunity cost of holding money must be zero in order to achieve optimality. The private opportunity cost is measured by the nominal interest rate, so the optimal rate of inflation in the steady state is the rate that achieves a zero nominal rate of interest. While this result is quite general, two important considerations have been ignored: the effects of inflation on government revenue and the interaction of inflation with other taxes in a nonindexed tax system. These are among the topics of chapter 4. 36. Samuelson ( 1 958) provided one of the earliest modern treatments of money as a social construct. 37. Of course, this is clearly not the case in the search models that assume fixed prices. Chapter 3 130 Appendix: The CIA Approximation 3.6 The method used to obtain a linear approximation around the steady state for the CIA model is discussed here. Since the approach is similar to the one followed for the MIU model, some details are skipped. The basic equations of the model are given by (3.44)­ (3.53). 3.6.1 T h e Steady State I With a binding CIA constraint, but in a steady state with I constant, Thus, and I From the first-order condition for the household's choice of css r ss ss ss m r ss + mss (1 + rr ss ) = mss .= +cssm= m(1ss+, rr ),mss css = l . n, (3.65) (1 nss ) - ry = (1 a) ( �:: ) A ss , and since y5s kss takes on the same values as in the MIU model (because the production technology and the discount factor are identical), it only remains to determine the marginal utility of income A ss . From (3.38) and ( 3.4 1) , (c ss ) - <1> = A ss + /t ss = A ss ( l + iss ). Using this relationship in ( 3.42 ) yields Ass = [ Ass1(+l +e ssiss ) ] l + iss = l +f3e ss , where e ss = rr ss . This is the steady-state version of the Fisher equation, and it means one can write Ass = (cl ss+) -iSS<1> = fJ(cl +ss e) -ss<1> Combining this with (3.65) and multiplying and dividing appropriately by kss and n ss , \11 _ _ I f3 :::} _ _ _ nss lkss = (yss lkss ) so one obtains - a ) ( f3 ) ( yss ) f.::-: ( css ) - <1> (1 - nss ) - ry (nss ) <l> = ( -1--qJ l + e ss kss kss It is useful to note that the expressions for is kss , c ss kss , rss , and n ss kss are identical to those obtained in the MIU model. Only the equation determining nss differs from the one I 1 -a , The production function implies that I I I found in chapter 2. 3.6.2 The Linear Approximation Expressions linear in the percentage deviations around the steady state can be obtained for the economy's production function, and resource constraint, the definition of the marginal Money and Transactions 131 product of capital, and the first-order conditions for consumption, money holdings, and labor supply, just as was done for the MIU model of chapter 2. The economy's production function, and resource constraint, the definition of the marginal product of capital, and the labor-leisure first-order condition are identical to those of the MIU modei, 38 so they are simply stated here: (3.66) (3.67) (3.68) (3.69) The Euler condition linking the marginal utility of income to its expected future value and the real return on capital, (3.40), becomes (3.70) Equations (3.38) and (3.41) imply = At ( 1 +it) . When linearized, this yields -<f>ct = �t + It · c;-ct> (3.7 1) From (3.38) and (3.42), At = f3Et [ At1+ +l +7Tt/h+tl+ l ] = f3Et [ 1 +c 7Tt�+ l l · � When this is linearized around the steady state, one obtains �t -<f>Etct+ l - Etrrt+ l · = (3.72) From the CIA constraint, (3.73) in an equilibrium with a positive nominal rate of interest. Finally, define as the percentage deviation of investment around the steady state: Xt 38. See the chapter 2 appendix. (3.74) Chapter 3 132 Collecting All Equations To summarize, the linearized model consists of equations (3.66), (3.67), (3.68), (3.69), (3.70), (3.7 1), (3.72), (3.73), and (3.74), together with the processes for the two exogenous shocks and the equation governing the evolution of real money balances. The resulting twelve equations solve for Zt, Ut, Yt, kt, fzt, � �, Ct, Xt, mt, rl, lt, and lrt. Collecting all the equilibrium conditions together, they are Zt = PzZt - 1 U t = PuU t - 1 + et , + </>Zt- 1 + C(Jt, and (3.44)-(3.53). Additional details on the derivation of the linearized CIA model and the MATLAB pro­ gram used to simulate it are available at http://people.ucsc.edu/�walshc/mtp4e/. 3. 7 Problems 1 . Suppose the production function for shopping takes the form 1jl = c = ex (n, f mb , where a and b are both positive but less than 1 , and x is a productivity factor. The agent's utility is given by v(c, l) = c 1 - <t> 1 (1 - <I>) + 1 ( 1 - 17 ), where l = 1 - n ­ ns , and n is time spent in market employment. a. Derive the transaction time function g(c, m ) = ns . b. Derive the money-in-the-utility function specification implied by the shopping pro­ duction function. How does the marginal utility of money depend on the parame­ ters a and b? How does it depend on x? c. Is the marginal utility of consumption increasing or decreasing in m? 2. Using (3.5) and (3.8), show that z l- ry Interpret this equation. How does it compare to (3.3 1)? 3. Show that, for the shopping-time model (section 3.2. 1), the tax on consumption is given by - c �iJ (::) 0 (Recall that money reduced shopping time, so gm tation for this expression. .:::: 0.) Provide an intuitive interpre­ Money and Transactions 133 4. In the model of section 3.3.2, suppose the current CIA constraint is not binding. This implies f.-L t = 0. Use (3.41) and (3.42) to show that money still has value at time t (that is, the price level at time t is finite) as long as the CIA constraint is expected to bind in the future. 5. MIU and CIA models are alternative approaches to constructing models in which money has positive value in equilibrium. a. What strengths and weaknesses do you see in each of these approaches? b. Suppose you wanted to study the effects of the growth of credit cards on money demand. Which approach would you adopt? Why? 6. Modify the basic model of section 3.3.1 by assuming utility depends on the consump­ tion of two goods, C';' and Cf . Purchases of c;n are subject to a cash-in-advance con­ straint; purchases of Cf are not. The two goods are produced by the same technology: C';' + c�· = Yt = f(kt ) . a . Write the household's decision problem. b. Write the first-order conditions for the household's optimal choices for C';' and Cf . How are these affected by the cash-in-advance constraint? c. Show that the nominal rate of interest acts as a tax on the consumption of c;n . 7. Assume the model of section 3.3 . 1 is modified so that only a fraction 1/f of consump­ tion must be purchased using cash. In this case, the cash-in-advance constraint takes the form m t-1 1/f ct ::::: 1 + Trt + r1, 0 < 1/f ::::: 1 . a. Write the household's decision problem. b. Write the household's first-order conditions. How are these affected by 1/f ? c . If 1jf were a choice variable of the household, would it ever choose 1jf > 0? 8. Modify the model of section 3.3.2 so that only a fraction 1/f of consumption is subject to the cash-in-advance constraint. How is the impact of a serially correlated shock to the money growth rate on real output affected by 1/f ? (Use the programs available at http://people.ucsc.edu/�walshc/mtp4e/ to answer this question, and compare the impulse response of output for 1/f = 0.25, 0.5, 0.75, and 1 .) 9. Consider the model of section 3.3 . 1 . Suppose that money is required to purchase both consumption and investment goods. The CIA constraint then becomes c1 + x1 ::::: m1- 1 /0 + n1) + r1, where x is investment. Assume that the aggregate production function takes the form y1 = ez' k�_ 1 nf - a . Show that the steady-state capital-labor ratio is affected by the rate of inflation. Does a rise in inflation raise or lower the steady-state capital-labor ratio? Explain. 134 Chapter 3 10. Consider the following model. Preferences are given by i [1nct+i + () 1n dt+i] , ,B i= O 00 Et L and the budget and CIA constraints take the form mt 1 (3.75) Ct + dt + mt + kt = Akt- 1 + (1 - o)kt-1 + Tt + 1 - , + Trt mt 1 (3.76) Ct .:S Tt + 1 - , + Trt where m denotes real money balances, and rr1 is the inflation rate from period t 1 to period t. The two consumption goods, and d, represent cash and credit (d) goods. The net transfer r is viewed as a lump-sum payment (or tax) by the household. a. Does this model exhibit superneutrality? Explain. b. What is the rate of inflation that maximizes steady-state utility? 1 1 . Consider the following specification for the representative household. Preferences are given by a -- -- c - (c) 00 Et L {3 ; lnc ln dt , i=O [ t+i + +i] and the budget constraint is Ct + dt + mt + kt = Ak1_ 1 + Tt + a m t-1 + (1 - o)kt - J , 1 + Trt -- where m denotes real money balances, and rr1 is the inflation rate from period t 1 to period t. Utility depends on the consumption of two types of good: must be purchased with cash, while d can be purchased using either cash or credit. The net transfer r is viewed as a lump-sum payment (or tax) by the household. If a fraction () of d is purchased using cash, then the household also faces a CIA constraint that takes the form mt 1 Ct + () dt _:::: 1 - + Tt. + Trt c -- What is the relationship between the nominal rate of interest and whether the CIA constraint is binding? Explain. Will the household ever use cash to purchase d (i.e., will the optimal () ever be greater than zero)? 12. Suppose the representative household enters period t with nominal money balances M1 _ 1 and receives a lump-sum transfer T1• During period t, the bond market opens first, and the household receives interest payments and purchases nominal bonds in the Money and Transactions 135 amount B1• With its remaining money (Mt -1 + T1 + (1 + i1 - I )Bt- l - B1), the house­ hold enters the goods market and purchases consumption goods subject to The household receives income at the end of the period and ends period t with nominal money holdings M1, given by M1 = P1ez'K�_ 1 N11 - a + (1 - 8)P1Kt -1 - PtKt + Mt- 1 + Tt + (1 + it - I )Bt- 1 - B t - Ptct. [ If the household's objective is to maximize 00 00 1 -<1> ct+i (1 - Nt+ i ) 1 Eo L f3 ; u(ct+i· 1 - Nt+ i ) = Eo L f3 ; -+ \II 1 - <1> 1 - 1] �0 �0 � ], do the equilibrium conditions differ from (3.38)-(3.42)? 1 3 . Trejos and Wright (1993) found that if no search is allowed while bargaining takes place, output tends to be too low (the marginal utility of output exceeds the marginal production costs). Show that output is also too low in a basic CIA model. (For sim­ plicity, assume that only labor is needed to produce output according to the production function y = ) Does the same hold true for an MIU model? 14. For the bargaining problem of section 3.4. 1 , the buyer and seller exchange q for d, where these two values maximize (3.58). Verify that when money holdings are not a constraint, <Ptd* = ec(q * ) + (1 - 8)u(q * ) . n. 15. Equation (3.63) shows how the nominal interest rate acts as a positive tax on consump­ tion. Discuss how this condition compares to (3.33) from the basic CIA model. If the CIA model is interpreted as one in which trading takes place with certainty and always involves a single coincidence of wants, can the CIA model be viewed as a special case of the search model? 16. This question deals with the Lagos and Wright (2005) model. a. Lagos and Wright divide each period into a decentralized market and a central­ ized market. What aspects of the model ensure that all agents leave the centralized market with the same money holdings, even though different agents enter the cen­ tralized market with different money holdings? b. In the double coincidence of wants case analyzed by Lagos and Wright, show that the joint surplus is maximized without any money changing hands. c. Show that m .::=: [Bc(q* ) + (1 - 8)u(q *) J /¢, where q* maximizes u(q) - c(q) (so that u' (q*) = c' (q *) ), when the gross inflation rate is greater than or equal to {3 . Chapter 3 136 17. In the bilateral bagaining problem of Lagos and Wright (2005), the quantity transacted and the money exchanged d solve the following problem: q u(q) + W1(m - d) - W1(m) ]0 [- c(q) + W1(m + d) - W1(m) ] 1-e + A. (m - d), where A. is the Lagrangian multiplier on the constraint d m. Show that when the constraint binds, q solves Bu'(q)c(q) + (1 - B)c'(q)u(q) ¢rm = Bu' (q) + ( 1 - B)c' (q) . max [ q.d _::: 18. Lagos and Wright (2005) showed that the solution to the bilateral bargaining problem when the cash constraint binds implies zMt(qt) = f3 zMt(qt++ l ) [ az�u'(q(qtt++Jl)) + (1 a ) ] , l t+ l where the quantity traded q satisfies z (q1 ) = ¢1M1• Suppose the money stock grows at the rate so that Mt+ 1 = ( 1 + )M1• Show that in a steady-state with the real variables constant, u' (q) = 1 + -, _.1!._ = 1 + and ai z' (q) ¢t+ l _ r a r r -- a where i is the nominal rate of interest. 19. Rocheteau and Wright (2005) consider three different market structures. For which ones does the Friedman rule deliver the first-best equilibrium? Explain. For which ones doesn't it? Explain. 4 4.1 Money and Public Finance Introduction Inflation is a tax. And as a tax, it both generates revenue for the government and distorts private sector behavior. Chapters 2 and 3 focused on these distortions. In the Sidrauski model, inflation distorts the demand for money, thereby generating welfare effects because real money holdings directly yield utility. In the cash-in-advance model, inflation serves as an implicit tax on consumption, so a higher inflation rate generates a substitution toward leisure, leading to lower labor supply, output, and consumption. In the analysis of these distortions, the revenue side of the inflation tax was ignored except to note that the Friedman rule for the optimal rate of inflation may need to be mod­ ified if the government does not have lump-sum sources of revenue available. Any change in inflation that affects the revenue from the inflation tax will have budgetary implications for the government. If higher inflation allows other forms of distortionary taxation to be reduced, this fact must be incorporated into any assessment of the costs of the inflation tax. This chapter introduces the government sector's budget constraint and examines the revenue implications of inflation. This allows a more explicit focus on the role of inflation in a theory of public finance and draws on the literature on optimal taxation to analyze the effects of inflation. A public finance approach yields several insights. Among the most important is the recognition that fiscal and monetary policies are linked through the government sector's budget constraint. Variations in the inflation rate can have implications for the fiscal author­ ity's decisions about expenditures and taxes, and, conversely, decisions by the fiscal author­ ity can have implications for money growth and inflation. When inflation is viewed as a distortionary revenue-generating tax, the degree to which it should be relied upon depends on the set of alternative taxes available to the government and on the reasons individuals hold money. Whether the most appropriate strategy is to think of money as entering the utility function as a final good or as an intermediate input into the production of transac­ tion services can have implications for whether money should be taxed. The optimal tax perspective also has empirical implications for inflation. Chapter 138 4 In the next section, the consolidated government's budget identity is set out, and some of the revenue implications of inflation are examined. Section 4.3 introduces various assumptions that can be made about the relationship between monetary and fiscal policies. Section 4.4 discusses situations of fiscal dominance in which a fixed amount of revenue must be raised from the inflation tax. It then discusses the equilibrium relationship between money and the price level. Section 4.5 turns to recent theories that emphasize what has come to be called the fiscal theory of the price level. In section 4.6, inflation revenue (seigniorage) and other taxes are brought together to analyze the joint determination of the government's tax instruments. This theme is developed first in a partial equilibrium model, and then Friedman's rule for the optimal inflation rate is revisited. The implica­ tions of optimal Ramsey taxation for inflation are discussed. Finally, section 4.6.4 contains a brief discussion of some additional effects that arise when the tax system is not fully indexed. 4.2 Budget Accounting To obtain goods and services, governments in market economies need to generate revenue. One way they can obtain goods and services is by printing money, which is then used to purchase resources from the private sector. However, to understand the revenue implica­ tions of inflation (and the inflation implications of the government's revenue needs), one must start with the government's budget constraint. 1 Consider the following identity for the fiscal branch of a government: (4. 1) where all variables are in nominal terms. The left side consists of government expendi­ tures on goods, services, and transfers Gr. plus interest payments on the outstanding debt it 1 1 (the superscript T denoting total debt, assumed to be one period in maturity, where debt issued in period t 1 earns the nominal interest rate it - 1 ), and the right side consists of tax revenue Tt , plus new issues of interest-bearing debt plus any direct receipts from the central bank RCB t . As an example of RCB, the U.S. Federal Reserve turns over to the Treasury almost all the interest earnings on its portfolio of government debt. 2 Equation (4. 1) represents the Treasury 's budget constraint. The monetary authority, or central bank, also has a budget identity that links changes in its assets and liabilities. If the central bank's assets consist of government debt, its budget - B'{_' - Bf - B'{'_ l ' l . Bohn ( 1 992) provided a general discussion of government deficits and accounting. 2. In 20 14 the Federal Reserve banks turned over $96.9 billion to the Treasury (JOist Annual Report of the Federal Reserve System 20 14, 1 1 3). In contrast, the payment to the Treasury in 2007, before the huge expansion of the Federal Reserve's balance sheet during the financial crisis of 2008-2009 and the Great Recession, was only $34.6 billion (93rd Annual Report of the Federal Reserve System 2007, 1 6 1 ) . Klein and Neumann ( 1 990) showed how the revenue generated by seigniorage and the revenue received by the fiscal branch may differ. Money and Public Finance 139 identity takes the form3 (4.2) (B'(l - B� 1 ) + RCB t = it - tB� 1 + (Ht - Ht- t), where B'(l - B� 1 is equal to the central bank's purchases of government debt, it- tB� 1 is the central bank's receipt of interest payments from the Treasury, and Ht - Ht - 1 is the change in the central bank's own liabilities. These liabilities are called high-powered money, or sometimes the monetary base, because they form the stock of currency held by the nonbank public plus bank reserves, and they represent the reserves private banks can use to back deposits. Changes in the stock of high-powered money lead to changes in broader measures of the money supply, measures that normally include various types of bank deposits as well as currency held by the public (see chapter 12). By letting B = B T - BM be the stock of government interest-bearing debt held by the public, the budget identities of the Treasury and the central bank can be combined to pro­ duce the consolidated government sector budget identity: (4.3) From the perspective of the consolidated government sector, only debt held by the public (i.e., outside the government sector) represents an interest-bearing liability. According to (4.3), the dollar value of government purchases Gr. plus its payment of interest on outstanding privately held liabilities i1 - tBt - 1 , must be funded by revenue that can be obtained from one of three alternative sources. First, Tt represents revenue generated by taxes (other than inflation). Second, the government can obtain funds by borrowing from the private sector. This borrowing is equal to the change in the debt held by the private sector, B1 - B t- l · Finally, the government can print currency to pay for its expenditures, and this is represented by the change in the outstanding stock of non-interest-bearing debt, Ht - Ht -1 · 4 Dividing (4.3) by the price level P1 yields T B -B Gt Ht - Ht -1 Bt- l . = t + t t- l + + it -l P1 P1 P1 Pt Pt ( ) 3. The Federal Reserve has paid interest on reserves since 2008 , a factor ignored in (4.2). Accounting for it would add i� l H1 _ 1 to the left side of ( 4.2) if, for simplicity, one assumed the rate { is paid on reserves plus currency. From 1 9 85 to 2007 currency averaged just over 80 percent of high-powered money; since 2008 , currency has fallen to less than half of high-powered money because of the tremendous increase in bank reserves. Also ignored here is any income to the central bank from interest charged on borrowed reserves. Chapter 1 2 discusses the implications of interest on reserves for monetary policy implementation. See Hall and Reis (20 1 3). 4. If the central bank holds private sector assets on its balance sheet and pays interest on its liabilities, (4.3) becomes Gt + it - I Bt - 1 + i�- I Ht - 1 + At - A t - I = i:_ I A t - 1 + Tt + (Bt - Bt- l ) + (Ht - Ht - 1 ), where the left side now includes the cost of interest payments on the central bank's liabilities and net purchases of private sector assets At - A1_ 1 . On the right side the income from the private assets, denoted here by t;_ 1 A t - 1 , now appears as a revenue source. Chapter 4 140 Note that terms like Br - 1 1 P1 can be multiplied and divided by Pr - 1 , yielding B�� � (!:=�) ( p�� � ) = = br- 1 c � JrJ , 5 where hr - 1 = Br- 1 I Pr- 1 represents real debt and n1 is the inflation rate. Employing the convention that lowercase letters denote variables deflated by the price level, the govern­ ment's budget identity is 1 (4.4) gt + rr -1 ht- 1 = tr + (br - br- 1 ) + h r - -- h r -1 , 1 + Jrr where 1-r- 1 = [ (1 + ir- 1 ) I ( 1 + n1) J - 1 is the ex post real return from t - 1 to t. To highlight the respective roles of anticipated and unanticipated inflation, let r1 be the ex ante real rate of return, and let n1 be the expected rate of inflation; then 1 + ir -1 = (1 + rr_I ) (l + nn . Adding (rr- 1 - rr- d br- 1 = (nr - nn ( l + rr- l )br - 1 /(1 + nr) to both sides of (4.4) and rearranging, the budget constraint becomes ( gt + rr - l ht- 1 = ) ( ) [ ( ) ] 1 n1 tr + (br - br- d + Jrr hr . (1 + rr- l )br - 1 + h r 1 + Jrr 1 + Irt - 1 (4.5) -- The third term on the right side of this expression, involving (n1 - nnbr - J , represents the revenue generated when unanticipated inflation reduces the real value of the government's outstanding interest-bearing nominal debt. To the extent that inflation is anticipated, this term will be zero; n1 will be reflected in the nominal interest rate that the government must pay. Inflation by itself does not reduce the burden of the government's interest-bearing debt; only unexpected inflation has such an effect. The last bracketed term in (4.5) represents seigniorage, the revenue from money cre­ ation. Seigniorage can be written as Hr - Hr- l = (hr - h r -1 ) + ____!!__!___ h r -1 · s1 (4.6) 1 + Irt Pr Seigniorage arises from two sources. First, h1 - h r- L is equal to the change in real high­ powered money holdings. Since the government is the monopoly issuer of high-powered money, an increase in the amount of high-powered money that the private sector is willing to hold allows the government to obtain real resources in return. In a steady-state equi­ librium, h is constant, so this source of seigniorage then equals zero. The second term in (4.6) is normally the focus of analyses of seigniorage because it can be nonzero even in the steady state. To maintain a constant level of real money holdings, the private sector needs = ( ) 5. If one is dealing with a growing economy, it is appropriate to deflate nominal variables by the price level and the level of output, i.e., by P1 Y1 • If the growth rate of output is J-ir, then Bt- I fP1Y1 = bt - l [ 1 / ( 1 + n1 ) ( 1 + i-ir)] . Money and Public Finance 141 to increase its nominal holdings of money at the rate n (approximately) to offset the effects of inflation on real holdings. By supplying money to meet this demand, the government is able to obtain goods and services or reduce other taxes. Denote the growth rate of the nominal monetary base H by e; the growth rate of h is then ce - n) I (1 + n) e - n . 6 In a steady state, h is constant, implying that n = e . In this case, (4.6) shows that seigniorage equals n e -- h -- h · (4.7) 1 + rr 1+B For small values of the rate of inflation, n I (1 + n) is approximately equal to n , so s can be thought of as the product of a tax rate of n, the rate of inflation, and a tax base of h, the real stock of base money. Since base money does not pay interest, its real value is depreciated by inflation whether or not inflation is anticipated. The definition of s would appear to imply that the government receives no revenue if inflation is zero. But this inference neglects the real interest savings to the government of issuing h, which is non-interest-bearing debt, as opposed to b, which is interest-bearing debt. That is, for a given level of the government's total real liabilities d = b + h, interest costs are a decreasing function of the fraction of this total that consists of h. A shift from interest-bearing to non-interest-bearing debt would allow the government to reduce total tax revenue or increase transfers or purchases. This observation suggests that one should consider the government's budget constraint expressed in terms of the total liabilities of the government. Using (4.5) and (4.6), the budget constraint can be rewritten7 as ir-1 n, - n( h (4.8) ( 1 + r, _ I )dr- 1 + -g , + rr -1 dr - 1 = tr + (dr - dr - 1 ) + 1 + n, r- 1 · 1 + n, Seigniorage, defined as the last term in (4.8), becomes i s = t -1 h r -1 · (4.9) l + nr This shows that the relevant tax rate on high-powered money depends directly on the nom­ inal rate of interest. Thus, under the Friedman rule for the optimal rate of inflation, which calls for setting the nominal rate of interest equal to zero (see chapters 2 and 3), the govern­ ment collects no revenue from seigniorage. The budget constraint also illustrates that any change in seigniorage requires an offsetting adjustment in the other components of (4.8). Reducing the nominal interest rate to zero implies that the lost revenue must be replaced by � ( ) ( ) ( - ( ) ( ) ) 6. Problem 2 at this end of this chapter deals with the case in which there is population growth and real per capita income growth. 7. To obtain this, add r1_ , h1_ , to both sides of (4.5). 142 Chapter 4 an increase in other taxes, real borrowing that increases the government's net indebtedness, or reductions in expenditures. The various forms of the government's budget identity suggest at least three alternative measures of the revenue from money creation. First, the measure that might be viewed as appropriate from the perspective of the Treasury is simple RCB, total transfers from the central bank to the Treasury (see 4. 1). Under this definition, shifts in the ownership of government debt between the private sector and the central bank affect the measure of seigniorage even if high-powered money remains constant. That is, from (4.2), if the central bank used interest receipts to purchase debt, EM would rise, RCB would fall, and the Trea­ sury would, from (4. 1), need to raise other taxes, reduce expenditures, or issue more debt. But this last option means that the Treasury could simply issue debt equal to the increase in the central bank's debt holdings, leaving private debt holdings, government expenditures, and other taxes unaffected. Thus, changes in RCB do not represent real changes in the Treasury's finances and are therefore not the appropriate measure of seigniorage. A second possible measure of seigniorage is given by ( 4.6), the real value of the change in high-powered money. 8 This measure of seigniorage equals the revenue from money creation for a given path of interest-bearing government debt. That is, s equals the total expenditures that could be funded, holding constant other tax revenue and the total private sector holdings of interest-bearing government debt. Finally, (4.9) provides a third defini­ tion of seigniorage as the nominal interest savings from issuing non-interest-bearing rather than interest-bearing debt. 9 This third definition equals the revenue from money creation for a given path of total (interest-bearing and non-interest-bearing) government debt; it equals the total expenditures that could be funded, holding constant other tax revenue and the total private sector holdings of real government liabilities. The difference between s and s arises from alternative definitions of fiscal policy. To understand the effects of monetary policy, one normally wants to consider changes in mon­ etary policy while holding fiscal policy constant. Suppose tax revenue t is simply treated as lump-sum taxes. Then one definition of fiscal policy would be in terms of a time series for government purchases and interest-bearing debt: (g1+;, b1+d �o · Changes in s, together with the changes in t necessary to maintain (g1+;, b1+d �o unchanged, would constitute monetary policy. Under this definition, monetary policy would change the total liabilities of the government (i.e., b + h). An open market purchase by the central bank would, ceteris paribus, lower the stock of interest-bearing debt held by the public. The Treasury would then need to issue additional interest-bearing debt to keep the bt+i sequence unchanged. Total government liabilities would rise. Alternatively, under the definition s, fiscal policy 8. Since the Fed began paying interest on reserves in 2008, the formulas for seigniorage need to be adjusted to reflect the payment of interest (which reduces seigniorage) and the income from Fed holdings of non-Treasury debt such as mortgage-backed securities. 9. These are not the only three possible definitions. See King and Plosser ( 1 985) for an additional three. Money and Public Finance 143 sets the path {g t+i , dt+ d �o and monetary policy determines the division of d between interest-bearing and non-interest-bearing debt but not its total. 4.2.1 Intertemporal Budget Balance The budget relationships derived in the previous section link the government's choices concerning expenditures, taxes, debt, and seigniorage at each point in time. However, unless there are restrictions on the government's ability to borrow or to raise revenue from seigniorage, (4.8) places no direct constraint on expenditure or tax choices. If governments, like individuals, are constrained in their ability to borrow, then this constraint limits the government's choices. To see exactly how it does so requires focusing on the intertemporal budget constraint of the government. Ignoring the effect of surprise inflation, the single-period budget identity of the govern­ ment given by (4.5) can be written as gt + rt -1 bt- 1 = ft + (bt - ht- 1 ) + St . Assuming the interest factor r is a constant (and is positive), this equation can be solved forward to obtain 00 00 00 tt+ t. s t+ t. b . gt+ t. = '"" . + lim t+ t . . '"" (4. 1 0) ( 1 + r)bt - I + '"" + . . L L ( 1 + r) ' L ( l + r) ' ' i ( 1 r) -+oo ( 1 + + r) ' t=O t=O t=O The government's expenditure and tax plans are said to satisfy the requirement of intertem­ poral budget balance (the no Ponzi condition) if the last term in (4. 10) equals zero: b (4. 1 1) lim t+i . = 0. i-+oo (1 + r) ' In this case, the right side of (4. 1 0) becomes the present discounted value of all current and future tax and seigniorage revenue, and this is equal to the left side, which is the present discounted value of all current and future expenditures plus current outstanding debt (prin­ cipal plus interest). In other words, the government must plan to raise sufficient revenue, in present value terms, to repay its existing debt and finance its planned expenditures. Defining the primary deficit as /'::,. = g - t - s, intertemporal budget balance implies, from (4. 10), that ( 1 + r)bt - I 00 t:,. t+i . = - '"" L ( 1 + r) '. t=O (4. 12) Thus, if the government has outstanding debt (bt - 1 > 0), the present value of future primary deficits must be negative (i.e., the government must run a primary surplus in present value). This surplus can be generated through adjustments in expenditures, taxes, or seigniorage. Chapter 4 144 Is (4. 12) a constraint on the government? Must the government (the combined monetary and fiscal authorities) pick expenditures, taxes, and seigniorage to ensure that (4. 12) holds for all possible values of the initial price level and interest rates? Or is it an equilibrium condition that need only hold at the equilibrium price level and interest rate? Buiter (2002) argued strongly that the intertemporal budget balance condition represents a constraint on government behavior, and this is the perspective generally adopted here. However, Sims (1994), Woodford (1995; 2001a), and Cochrane (1999) argued that (4. 12) is an equilibrium condition; this alternative perspective is taken up in section 4.5. 4.3 Money and Fiscal Policy Frameworks Most analyses of monetary phenomena and monetary policy assume, usually without state­ ment, that variations in the stock of money matter but that how that variation occurs does not. The nominal money supply could change because of a shift from tax-financed govern­ ment expenditures to seigniorage-financed expenditures. Or it could change as the result of an open-market operation in which the central bank purchases interest-bearing debt, financing the purchase by an increase in non-interest-bearing debt, holding other taxes constant (see 4.2). Because these two means of increasing the money stock have differing implications for taxes and the stock of interest-bearing government debt, they may lead to different effects on prices and/or interest rates. The government sector's budget constraint links monetary and fiscal policies in ways that can matter for determining how a change in the money stock affects the equilibrium price levei. 10 The budget link also means that one needs to be precise about defining mon­ etary policy as distinct from fiscal policy. An open-market purchase increases the stock of money, but by reducing the interest-bearing government debt held by the public, it has implications for the future stream of taxes needed to finance the interest cost of the gov­ ernment's debt. So an open-market operation potentially has a fiscal side to it, and this fact can lead to ambiguity in defining what one means by a change in monetary policy, holding fiscal policy constant. The literature in monetary economics has analyzed several alternative assumptions about the relationship between monetary and fiscal policies. In most traditional analyses, fiscal policy is assumed to adjust to ensure that the government's intertemporal budget is always in balance, while monetary policy is free to set the nominal money stock or the nominal rate of interest. This situation is described as one of monetary dominance (Sargent 1982) or one in which fiscal policy is passive and monetary policy is active (Leeper 1991). The models of chapters 2 and 3 implicitly fall into this category in that fiscal policy was ignored and monetary policy determined the price level. Traditional quantity theory relationships were 10. See, for example, Sargent and Wallace ( 1 9 8 1 ) and Wallace ( 1 9 8 1 ). The importance of the budget constraint for the analysis of monetary topics is clearly illustrated in Sargent ( 1 987). Money and Public Finance 145 obtained, with one-time proportional changes in the nominal quantity of money leading to equal proportional changes in the price level. If fiscal policy affects the real rate of interest, then the price level is not independent of fiscal policy, even under regimes of monetary dominance. A balanced budget increase in expenditures that raises the real interest rate raises the nominal interest rate and lowers the real demand for money. Given an exogenous path for the nominal money supply, the price level must jump to reduce the real supply of money. A second policy regime is one in which the fiscal authority sets its expenditures and taxes without regard to any requirement of intertemporal budget balance. If the present discounted value of these taxes is not sufficient to finance expenditures (in present value terms), seigniorage must adjust to ensure that the government's intertemporal budget con­ straint is satisfied. This regime is one of fiscal dominance (or active fiscal policy) and passive monetary policy, as monetary policy must adjust to deliver the level of seigniorage required to balance the government's budget. Prices and inflation are affected by changes in fiscal policy because these fiscal changes, if they require a change in seigniorage, alter the current and/or future money supply. Any regime in which either taxes or seigniorage always adjust to ensure that the government's intertemporal budget constraint is satisfied is called a Ricardian regime (Sargent 1982). Regimes of fiscal dominance are analyzed in section 4.4. A final regime leads to what has become known as the fiscal theory of the price level (Sims 1994; Woodford 1995; 2001a; Cochrane 1999). In this regime, the government's intertemporal budget constraint may not be satisfied for arbitrary price levels. Following Woodford (1995), these regimes are described as non-Ricardian. The discussion of non­ Ricardian regimes is in section 4.5. 4.4 Deficits and Inflation The intertemporal budget constraint implies that any government with a current outstand­ ing debt must run, in present value terms, future surpluses. One way to generate a surplus is to increase revenue from seigniorage, and for that reason, economists have been inter­ ested in the implications of budget deficits for future money growth. Two questions have formed the focus of studies of deficits and inflation. First, do fiscal deficits necessarily imply that inflation will eventually occur? Second, if inflation is not a necessary conse­ quence of deficits, is it in fact a historical consequence? The literature on the first question has focused on the implications for inflation if the monetary authority must act to ensure that the government's intertemporal budget is bal­ anced. This interpretation views fiscal policy as set independently, so that the monetary authority is forced to generate enough seigniorage to satisfy the intertemporal budget bal­ ance condition. Leeper (1991) describes this as a situation with an active fiscal policy and a passive monetary policy. It is also described as a situation offiscal dominance. Chapter 4 146 From (4. 12), the government's intertemporal budget constraint takes the form br - 1 00 = -R - 1 L R -i (gr+ i - tr+i - St+i ) , i=O where R = 1 + r is the gross real interest rate, g1 - t1 - s1 is the primary deficit, and s1 is real seigniorage revenue. Let J; t1 - g1 be the primary fiscal surplus (i.e., tax revenue minus expenditures but excluding interest payments and seigniorage revenue). Then the government's budget constraint can be written as = 00 00 " R -i /t+i + R - 1 L " R -i St+i · (4. 1 3) bt - 1 = R - 1 L i=O i=O The current real liabilities of the government must be financed, in present value terms, by either a fiscal primary surplus or seigniorage. Given the real value of the government's liabilities br- 1 , (4. 1 3) illustrates what Sargent and Wallace (1981) described as "unpleasant monetarist arithmetic" in a regime of fiscal dominance. If the present value of the fiscal primary surplus is reduced, the present value of seigniorage must rise to maintain (4. 13). Or, for a given present value of sf, an attempt by the monetary authority to reduce inflation and seigniorage today must lead to higher inflation and seigniorage in the future, because the present discounted value of seignior­ age cannot be altered. The mechanism is straightforward; if current inflation tax revenues are lowered, the deficit grows and the stock of debt rises. This implies an increase in the present discounted value of future tax revenue, including revenue from seigniorage. If the fiscal authority does not adjust, the monetary authority will be forced eventually to produce higher inflation. 1 1 The literature on the second question-has inflation been a consequence of deficits historically?-has focused on estimating empirically the effects of deficits on money growth. Joines (1985) found money growth in the United States to be positively related to major war spending but not to nonwar deficits. Grier and Neiman (1987) summarized a number of earlier studies of the relationship between deficits and money growth (and other measures of monetary policy) in the United States. That the results are generally inconclu­ sive is perhaps not surprising, as the studies they reviewed were all based on postwar but pre- 1980 data. Thus, the samples covered periods in which there was relatively little deficit variation and in which much of the existing variation arose from the endogenous response of deficits to the business cycle as tax revenue varied procyclically. 12 Grier and Neiman did find that the structural (high-employment) deficit is a determinant of money growth. l l . In a regime of monetary dominance, the monetary authority can determine inflation and seigniorage; the fiscal authority must then adjust either taxes or spending to ensure that (4. 1 3 ) is satisfied. 12. For that reason, some of the studies cited by Grier and Neiman employed a measure of the high-employment surplus (i.e., the surplus estimated to occur if the economy had been at full employment). Grier and Neiman concluded, "The high employment deficit (surplus) seems to have a better 'batting average' " (204). Money and Public Finance 147 This finding is consistent with that of King and Plosser ( 1985), who reported that the fiscal deficit did help to predict future seigniorage for the United States. They interpreted this as mixed evidence for fiscal dominance. Demopoulos, Katsimbris, and Miller (1987) provided evidence on debt accommodation for eight OECD countries. These authors estimated a variety of central bank reaction func­ tions (regression equations with alternative policy instruments on the left side) in which the government deficit is included as an explanatory variable. For the post-Bretton Woods period, they found a range of outcomes, from no accommodation by the Federal Reserve and the Bundesbank to significant accommodation by the Bank of Italy and the Neder­ landse Bank. One objection to this empirical literature is that simple regressions of money growth on deficits, or unrestricted VAR used to assess Granger causality (i.e., whether deficits con­ tain any predictive information about future money growth), ignore information about the long-run behavior of taxes, debt, and seigniorage that is implied by intertemporal budget balance. lntertemporal budget balance implies a cointegrating relationship between the pri­ mary deficit and the stock of debt. This link between the components of the deficit and the stock of debt restricts the time series behavior of expenditures, taxes, and seigniorage, and this fact in turn implies that empirical modeling of their behavior should be carried out within the framework of a vector error correction model (VECM). 13 Suppose X1 (g1 T1 br- t ), where T t + s is defined as total government receipts from taxes and seigniorage. If the elements of X are nonstationary, intertemporal bud­ get balance implies that the deficit inclusive of interest, or ( 1 - 1 r)X1 fJ'X1 g1 - T1 + rb1 - t , is stationary. Hence, fJ ' (1 - 1 r) is a cointegrating vector for X. The appropriate specification of the time series process is then a VECM of the form = = = = = C(L) IiX1 = -afJ 'Xt + et . (4.14) The presence of the deficit inclusive of interest, fJ'Xt. ensures that the elements of X can­ not drift too far apart; doing so would violate intertemporal budget balance. A number of authors have tests for cointegration to examine the sustainability of budget policies (see Trehan and Walsh 1988; 199 1 for one approach). However, Bohn (2007) argued that time series based on cointegration relationships are not capable of rejecting intertemporal bud­ get balance. Bohn (199 1 a) estimated a model of the form (4. 14) using U.S. data from 1 800 to 1988. Unfortunately for our purposes, Bohn did not treat seigniorage separately, and thus his results are not directly relevant for determining the effects of spending or tax shocks on the adjustment of seigniorage. He did find, however, that one-half to two-thirds of deficits initiated by a tax revenue shock were eventually eliminated by spending adjustments, while about one-third of spending shocks were essentially permanent and resulted in tax changes. 13. See Engle and Granger ( 1 987). Chapter 4 148 4.4.1 Ricardian and (Traditional) Non-Ricardian Fiscal Policies Changes in the nominal quantity of money engineered through lump-sum taxes and trans­ fers (as in chapters 2 and 3) may have different effects than changes introduced through open-market operations in which non-interest-bearing government debt is exchanged for interest-bearing debt. In an early contribution, Metzler (195 1 ) argued that an open-market purchase, that is, an increase in the nominal quantity of money held by the public and an offsetting reduction in the nominal stock of interest-bearing debt held by the public, would raise the price level less than proportionally to the increase in M. An open-market opera­ tion would therefore affect the real stock of money and lead to a change in the equilibrium rate of interest. Metzler assumed that households' desired portfolio holdings of bonds and money depended on the expected return on bonds. An open-market operation, by altering the ratio of bonds to money, requires a change in the rate of interest to induce private agents to hold the new portfolio composition of bonds and money. A price-level change propor­ tional to the change in the nominal money supply would not restore equilibrium, because it would not restore the original ratio of nominal bonds to nominal money. An important limitation of Metzler's analysis was its dependence on portfolio behavior that was not derived directly from the decision problem facing the agents of the model. The analysis was also limited in that it ignored the consequence for future taxes of shifts in the composition of the government's debt, a point made by Patinkin (1965). The government's intertemporal budget constraint requires the government to run surpluses in present value terms equal to its current outstanding interest-bearing debt. An open-market purchase by the monetary authority reduces the stock of interest-bearing debt held by the public, and this reduction has consequences for future expected taxes. Sargent and Wallace (1981) showed that the backing for government debt, whether it is ultimately paid for by taxes or by printing money, is important in determining the effects of debt issuance and open-market operations. This finding can be illustrated following the analysis of Aiyagari and Gertler (1985). They used a two-period overlapping-generations model that allows debt policy to affect the real intergenerational distribution of wealth. This effect is absent from the representative agent model used here, but the representative agent framework can still be used to show how the specification of fiscal policy has important implications for conclusions about the link between the money supply and the price level. 14 In order to focus on debt, taxes, and seigniorage, set government purchases equal to zero and ignore population and real income growth, in which case the government's budget constraint takes the simplified form (4. 15) with s1 denoting seigniorage. 14. See also Woodford ( 1 995; 200 1 a) and section 4.5.2. Money and Public Finance 149 In addition to the government's budget constraint, one needs to specify the budget con­ straint of the representative agent. Assume that this agent receives an exogenous endow­ ment y in each period and pays (lump-sum) taxes in period The agent also receives interest payments on any government debt held at the start of the period; these payments, in real terms, equal where is the nominal interest rate in period is the number of bonds held at the start of period and is the period price level. This can be written equivalently as where i )I = is the ex post real rate of interest. Finally, the agent has real money balances equal to that are carried into period from period I = The agent allo­ cates these resources to consumption, real money holdings, and real bond purchases, sub­ ject to t1 (I + i1-1 )B1-l /Pt, Bt-l Mt- 1 Pt (I + 7Tt) -l m1- 1 t. it-1 t - 1, t, Pt t (I + rt-1 )bt-1 , rt-1 (1 + t-1 (I + 7Tt) - 1 t t - 1. mt-1 - ft. Ct + mt + bt + (1 + rt- d ht-1 + -1 + 7Tt (4. 16) =y Aiyagari and Gertler (1985) asked whether the price level will depend only on the stock of money or whether debt policy and the behavior of the stock of debt might also be rel­ evant for price level determination. They assumed that the government sets taxes to back a fraction 1f; of its interest-bearing debt liabilities, with 0 .::: 1f; .::: If 1f; = government interest-bearing debt is completely backed by taxes in the sense that the government com­ mits to maintaining the present discounted value of current and future tax receipts equal to its outstanding debt liabilities. Such a fiscal policy was called Ricardian by Sargent ( 1982). l 5 If 1f; < Aiyagari and Gertler characterized fiscal policy as non-Ricardian. To avoid confusion with the more recent interpretations of non-Ricardian regimes (see section 4.5.2), regimes where 1f; < are referred to here as traditional non-Ricardian regimes. In such regimes, seigniorage must adjust to maintain the present value of taxes plus seignior­ age equal to the government's outstanding debt. Let now denote the present discounted value of taxes. Under the assumed debt pol­ icy, the government ensures that = 1/1 because is the net liability of the government (including its current interest payment). Because is a present value, one can also write 1. 1, 1, 1 Tt Tt (I + rt- dbt- L (I + rt- t)bt- L Tt Tt+l ) = tt + Et [ lj;(1 + rt)bt ] Tt = lt + Et ( -( 1 + rt) 1 + rt or Tt = t1 + 1/l bt. Now because Tt = lj;( l + rt- d h t-1 , it follows that (4. 1 7) 1 5 . It is more common for Ricardo's name to be linked with debt in the form of the Ricardian equivalence theorem, under which shifts between debt and tax financing of a given expenditure stream have no real effects. See Barro ( 1 974) or Romer (20 1 2). Ricardian equivalence holds in the representative agent framework used here; the issue is whether debt policy, as characterized by 1/J , matters for price level determination. Chapter 150 4 R 1+ s1 (1 - 1/!) (R1-1 br-l - b1). 1jf 1 - 1jf, (4.17), (4.16) mr-1 Cr + mr + (1 - 1/!)br = Y + (1 - 1/!)Rr-l bt-1 + -1 + n1 . In the Ricardian case (1/! = 1), all terms involving the government's debt drop out; only the stock of money matters. If 1jf 1, however, debt does not drop out. One can then rewrite the budget constraint as y + Rt - 1 w1 - 1 = c1 + w1 + ir- 1 m1 - 1 / (1 + n1), where w = m + ( 1 - 1jf )b, showing that the relevant measure of household income is y + Rr- 1 wr- 1 and this is then used to purchase consumption, financial assets, or money balances (where the opportunity cost of money is i/(1 + n)). With asset demand depending on 1jf through Wr-1 , the equilibrium price level and nominal rate of interest generally depend on 1jf . 16 where = With taxes adjusting to ensure r. Similarly, = that the fraction of the government's debt liabilities is backed by taxes, the remaining fraction, represents the portion backed by seigniorage. Using the household's budget constraint becomes < Having derived the representative agent's budget constraint and shown how it is affected by the means the government uses to back its debt, to actually determine the effects on the equilibrium price level and nominal interest rate, one must determine the agent's demand for money and bonds and then equate these demands to the (exogenous) sup­ plies. To illustrate the role of debt policy, assume log separable utility, ln 8 ln and consider a perfect-foresight equilibrium. From chapter 2, the marginal rate of sub­ stitution between money and consumption is set equal to With log utility, this implies = The Euler condition for the optimal consumption path yields = r c • Using these in the agent's budget constraint, c1 + m1, ir /(1 + i1). m1 o c1(1 + i1)/i1. cr+l ,8(1 + 1) 1 c1 = y, so this becomes Rt- 1 w1-1 = (8/,B )y + w1• In the steady state, w1 = Wr-1 wss oy/ ,B (R - 1). But w [M + (1 - 1/!)B]/P, so the equilibrium steady-state price level is equal to ( ,B;; ) [M + ( l - 1/!)B] . (4.18) If government debt is entirely backed by taxes ( 1jf = 1 ) , one gets the standard result: the price level is proportional to the nominal stock of money. The stock of debt has no effect on the price level. With 0 1jf 1, however, both the nominal money supply and the nominal stock of debt play a role in price level determination. Proportional changes in M and B In equilibrium, = p ss = = = s < < produce proportional changes in the price level. 1 6. In this example, c = y in equilibrium, since there is no capital good that would allow the endowment to be transferred over time. Money and Public Finance 151 In a steady state, all nominal quantities and the price level must change at the same rate because real values are constant. Thus, if M grows, then B must also grow at the same rate. The real issue is whether the composition of the government's liabilities matters for the price level. To focus more clearly on that issue, let A. = MI (M + B) be the fraction of gov­ ernment liabilities that consists of non-interest-bearing debt. Since open-market operations affect the relative proportions of money and bonds in government liabilities, open-market operations determine A.. Equation (4. 1 8) can then be written as p ss = ( ,B;; ) [1 - 1jr ( 1 - A.)] (M + B) . s Open market purchases (an increase in A.) that substitute money for bonds but leave M + B unchanged raise pss when 1jr > 0. The rise in pss is not proportional to the increase in M. Shifting the composition of its liabilities away from interest-bearing debt reduces the present discounted value of the private sector's tax liabilities by less than the fall in debt holdings; a rise in the price level proportional to the rise in M would leave households' real wealth lower (their bond holdings are reduced in real value, but the decline in the real value of their tax liabilities is only 1jr < 1 times as large). Leeper (1991) argued that even if 1jr = 1 on average (that is, all debt is backed by taxes), the means used to finance shocks to the government's budget have important implica­ tions. He distinguished between active and passive policies; with an active monetary pol­ icy and a passive fiscal policy, monetary policy acts to target nominal interest rates and does not respond to the government's debt, while fiscal policy must then adjust taxes to ensure intertemporal budget balance. Conversely, with an active fiscal policy and a pas­ sive monetary policy, the monetary authority must adjust seigniorage revenue to ensure intertemporal budget balance, while fiscal policy does not respond to shocks to debt. Leeper showed that the inflation and debt processes are unstable if both policy authori­ ties follow active policies, and there is price level indeterminacy if both follow passive policies. 4.4.2 The Government Budget Constraint and the Nominal Rate of Interest Earlier, Sargent and Wallace's "unpleasant monetarist" arithmetic was examined using (4. 1 3). Given the government's real liabilities, the monetary authority would be forced to finance any difference between these real liabilities and the present discounted value of the government's fiscal surpluses. Fiscal considerations determine the money supply, but the traditional quantity theory holds and the price level is proportional to the nominal quantity of money. Suppose, however, that the initial nominal stock of money is set exoge­ nously by the monetary authority. Does this mean that the price level is determined solely by monetary policy, with no effect of fiscal policy? The following example shows that the answer is no; fiscal policy can affect the initial equilibrium price level even when the initial nominal quantity of money is given and the government's intertemporal budget constraint must be satisfied at all price levels. Chapter 152 4 Consider a perfect-foresight equilibrium. In such an equilibrium, the government's bud­ get constraint must be satisfied and the real demand for money must equal the real sup­ ply of money. The money-in-the-utility function (MIU) model of chapter 2 can be used, for example, to derive the real demand for money. That model implied that agents would equate the marginal rate of substitution between money and consumption to the cost of holding money, where this cost depended on the nominal rate of interest: it Um (Ct,mt) Uc(Ct, mt) 1 + it 17 [ ( 1 +it it ) ( 1 ) ] t Ct . With the utility function employed in chapter 2, this condition implies that mt = Mt ?; = a -a Evaluated at the economy's steady state, this can be written as Mt p; = f(Rm ,t ), (4. 19) i f(Rm) = [ ( R��: 1 ) c : ) r t Ct. where Rm = 1 + is the gross nominal rate of interest and a Given the nominal interest rate, (4. 19) implies a proportional relationship between the nominal quantity of money and the equilibrium price level. If the initial money stock is Mo, then the initial price level is Po = Mo/f(Rm). The government's budget constraint must also be satisfied. In a perfect-foresight equi­ librium, there are no inflation surprises, so the government's budget constraint given by (4.5) can be written as 8t + rbt-1 = lt + (bt - bt- d + mt - ( -I +1-nt ) mt-1 · (4.20) Now consider a stationary equilibrium in which government expenditures and taxes are constant, as are the real stocks of government interest-bearing debt and money. In such a stationary equilibrium, the budget constraint becomes (1 )b f3 g+ - - 1 =t ( lrt ) m -- + 1+� 17. In chapter 2 it was assumed that [ac1 u (cr , mr ) = -=- 1 -b + (l - a) m11 - b ] T=li - 1 _ <1> 1 - <1> -=---- =t ( ) R + f3 {3m - 1 f(Rm), � (4.21) Money and Public Finance 153 which uses the steady-state results that the gross real interest rate is 11 {3, Rm ( l + n1) I {3 , and real money balances must be consistent with the demand given by (4. 19). Suppose the fiscal authority sets g, t, and b. Then (4.21) determines the nominal inter­ est rate Rm . With g, t, and b given, the government needs to raise g + ( I I f3 - 1) b - t in seigniorage. The nominal interest rate is determined by the requirement that this level of seigniorage be raised. 18 Because the nominal interest rate is equal to ( 1 + n ) I f3, one can alternatively say that fiscal policy determines the inflation rate. Once the nominal interest rate is determined, the initial price level is given by (4.19) as Po = Molf(Rm), where Mo is the initial stock of money. In subsequent periods, the price level is equal to P1 = Po (f3Rm) 1 , where f3 Rm = ( 1 + n) is the gross inflation rate. The nominal stock of money in each future period is endogenously determined by M1 = PJ(Rm). In this case, even though the mone­ tary authority has set Mo exogenously, the initial price level is determined by the need for fiscal solvency because the fiscal authority's budget requirement (4.21) determines Rm and therefore the real demand for money. The initial price level is proportional to the initial money stock, but the factor of proportionality, 11f(Rm), is determined by fiscal policy, and both the rate of inflation and the path of the future nominal money supply are determined by the fiscal requirement that seigniorage equal g + ( 1 I f3 - 1) b - t. If the fiscal authority raises expenditures, holding b and t constant, then seigniorage must rise. The equilibrium nominal interest rate rises to generate this additional seigniorage. 19 With a higher Rm, the real demand for money falls, and this increases the equilibrium value of the initial price level Po, even though the initial nominal quantity of money is unchanged. = 4.4.3 Equilibrium Seigniorage Suppose that given its expenditures and other tax sources, the government has a fiscal deficit of t:,.f that must be financed by money creation. When will it be feasible to raise t:,.f in a steady-state equilibrium? And what will be the equilibrium rate of inflation? The answers to these questions would be straightforward if there were a one-to-one relationship between the revenue generated by the inflation tax and the inflation rate. If this were the case, the inflation rate would be uniquely determined by the amount of revenue that must be raised. But the inflation rate affects the base against which the tax is levied. For a given base, a higher inflation rate raises seigniorage, but a higher inflation rate raises the opportunity cost of holding money and reduces the demand for money, thereby lowering the base against which the tax is levied. This raises the possibility that a given amount of b- 1 8 . The nominal interest rate that raises seigniorage equal to g + ( I J f3 - I) t may not be unique. A rise in Rm increases the tax rate on money, but it also erodes the tax base by reducing the real demand for money. A given amount of seigniorage may be raised with a low tax rate and a high base or a high tax rate and a low base. 19. This assumes that the economy is on the positively sloped portion of the Laffer curve so that raising the tax rate increases revenue; see section 4.4.3. Chapter 4 154 revenue can be raised by more than one rate of inflation. For example, the nominal rate of interest that satisfies (4.21) may not be unique. It will be helpful to impose additional structure so that one can say more about the demand for money. The standard approach used in most analyses of seigniorage is to spec­ ify directly a functional form for the demand for money as a function of the nominal rate of interest. An early example of this approach, and one of the most influential, is that of Cagan (1956). This approach is discussed in section 4.4.4, but here Calvo and Leiderman (1992) are followed in using a variant of the Sidrauski model of chapter 2 to motivate a demand for money. That is, suppose the economy consists of identical individuals, and the utility of the representative agent is given by Rm L {3 1 u(ct,mt), 00 t=O (4.22) 1, m where 0 < f3 < c is per capita consumption, is per capita real money holdings, and the function is strictly concave and twice continuously differentiable. The representative agent chooses consumption, money balances, and holdings of interest-earning bonds to maximize the expected value of (4.22), subject to the following budget constraint: u(. ) Cr + br + mr = Yr - Tt + ( 1 + r)br-1 + mrITt- 1 , where b is the agent's holdings of bonds, y is real income, is equal to the net taxes of the agent, r is the real rate of interest, assumed constant for simplicity, and ITt = Pt!Pt -1 = 1 + n:r , where n:1 is the inflation rate. Thus, the last term in the budget constraint, m1_ J / ITt, is equal to the period t real value of money balances carried into period t, that is, M1 -1 I Pt, where M represents nominal money holdings. In what follows, attention is restricted to perfect-foresight equilibria. If w1 is the agent's real wealth in period t, Wt = b1 + m1, and let Rt = 1 + rt, then the budget constraint can be rewritten as c1 + w1 y1 - Tr +Rr-1 Wt- 1 - ( Rr-1 ITrITt - 1 ) mr-1 i = Yt - Tt + Rt - 1 Wr -1 - ( r-1 ) mt -1 ITt by using the fact that R IT 1 + i, where i is the nominal rate of interest. Writing the bud­ get constraint in this way, it is clear that the cost of holding wealth in the form of money rather than interest-earning bonds is i/ IT. 20 The first-order condition for optimal money -- r = = 20. Recall from the derivation of (4.8) that the term for the government's revenue from seigniorage was Ur- ! J n1 )hr- ! · Comparing this to the household's budget constraint (with h1_ , = m1_ , ) shows that the cost of holding money is exactly equal to the revenue obtained by the government. Money and Public Finance 155 holdings sets the marginal utility of money equal to the cost of holding money times the marginal utility of wealth. Since the interest forgone by holding money in period t is a cost that is incurred in period t + 1, this cost must be discounted back to period t using the discount factor to compare with the marginal utility of money in period t. Thus, But the standard Euler condition for optimal con­ = 1) 1, sumption implies that Combining these first-order condi­ = tions yields i i (4.23) = --_ = -_-. 1 + lt Now suppose the utility function takes the form Using = + the functional form in (4.23), one obtains f3 um (er, mt) f3 (it! fi t+ uc(et+ mt+J). uc(et,mt) f3Rtuc(et+ l ,mt+ t). Um (er ,mt) ( Rt fitt+l ) Uc(er,mt) ( t ) Uc(er,mt). u(er,m1) lne1 m1(B - Dlnm1). (4.24) where A = e ( � - 1 ) and w = i/( 1 + i) . Equation (4.24) provides a convenient functional representation for the demand for money. Since the time of Cagan's seminal contribution to the study of seigniorage and hyper­ inflations (Cagan 1956, 158-161), many economists have followed him in specifying a money demand function of the form = Ke -a rre ; (4.24) shows how something similar can be derived from an underlying utility function. As Calvo and Leiderman (1992) pointed out, the advantage is that one sees how the parameters K and a depend on more primitive parameters of the representative agent's preferences and how they may actually be time­ dependent. For example, a depends on and therefore will be time-dependent unless K varies appropriately or e itself is constant. The reason for deriving the demand for money as a function of the rate of inflation is that, having done so, one can express seigniorage as a function of the rate of inflation. Recall from (4.9) that seigniorage was equal to i m/( 1 + n) = (1 + r)i m/(1 + i) . Using the expression for the demand for money, steady-state seigniorage is equal to i i s = (1 + r) -. A exp + t). 1+t If supemeutrality is assumed to characterize the model, then is constant in the steady state and independent of the rate of inflation. The same is true of the real rate of interest. To determine how seigniorage varies with the rate of inflation, think of choos­ ing w = i/(1 + i) through the choice of n . Then s = (1 + r)wAe - w fDc , and as jan = (as;aw) (aw;ai) (ai;an ) = (as;aw) (1 + r)/(1 + i) 2 , so the sign of as; an is determined by the sign of (as;aw) . Since as w s w - = (1 + r)Ae - w fDc 1 - = - 1 - - , aw w m et x ( ) [ - De( l ] . [ - De J [ De J x e Chapter 156 4 Q) Ol � 0 ·c: Ol ·a; (f) Inflation rate Figure 4.1 Seigniorage as a function of inflation. the sign of 'dsj'dw depends on the sign of 1 - (wjDe). As illustrated in figure 4. 1, seignior­ age increases with inflation initially but eventually begins to decline with further increases in n as the demand for real balances shrinks. 21 To determine the inflation rate that maximizes seigniorage, note that 'ds/'drr = 0 if and only if w = 1 i i = De , or + -- m rr ax 1 - I+r _ ( ) ( I -1 De ) __ ___ _ 1. For inflation rates less than rr max , the government's revenue is increasing in the inflation rate. The effect of an increase in the tax rate dominates the effect of higher inflation in reducing the real demand for money. As inflation increases above rr max , the tax base shrinks sufficiently that revenue from seigniorage declines. Consequently, governments face a seigniorage Laffer curve; raising inflation beyond a certain point results in lower real tax revenue. 2 1 . Whether a Laffer curve exists for seigniorage depends on the specification of utility. For example. in chapter 2 I it was noted that with a CES utility function the demand for money was given by m1 = A [i/ ( 1 + i)] - b Ct. where I A is a constant. Hence. seigniorage is A [ij ( l + i)] 1 - li c1• which is monotonic in i. Money and Public Finance 4.4.4 157 Cagan's Model Since 1970 the consumer price index for the United States has risen just over sixfold; that's inflation. 22 In Hungary, the index of wholesale prices was 38,500 in January 1923 and 1 ,026,000 in January 1924, one year later, a 27-fold increase; that's hyperinflation (Sargent 1986). Cagan ( 1956) provided one of the earliest studies of the dynamics of money and prices during hyperinflation. The discussion here follows Cagan in using continuous time. Sup­ pose the real per capita fiscal deficit that needs to be financed is exogenously given and is equal to /").! . This means that HH /").! = HP Y = 8h ' where h has been expressed as real balances relative to income to allow for real economic growth. The demand for real balances depends on the nominal interest rate and therefore the expected rate of inflation. Treating real variables such as the real rate of interest and real output as constant (which is appropriate in a steady state characterized by superneutrality and is usually taken as reasonable during hyperinflations because all the action involves money and prices), write the demand for the real monetary base as h = exp( -an: e ) . Then the government's revenue requirement implies that (4.25) For h to be constant in equilibrium requires that n: = e - f.J.,, where fJ., is the growth rate of real income. And in a steady-state equilibrium, n: e = n: , so (4.25) becomes (4.26) the solution( s) of which give the rates of money growth that are consistent with raising the amount /").! through seigniorage. The right side of (4.26) equals zero when money growth is equal to zero, rises to a maximum at e = ( 1 /a), and then declines. 23 That is, for rates of money growth above ( 1 /a) (and therefore inflation rates above ( 1 /a) - f.J.,), higher inflation actually leads to lower revenues because the tax base falls sufficiently to offset the rise in inflation. Thus, any deficit less than /"). * = ( 1 /a) exp(af.J., - 1) can be financed by either a low rate of inflation or a high rate of inflation. Figure 4.2, based on Bruno and Fischer (1990), illustrates the two inflation rates consis­ tent with seigniorage revenue of /').f . The curve SR is derived from (4.25) and shows, for 22. The CPI was equal to 37.9 in January 1 970 and reached 238.0 in December 20 1 5 . 2 3 . More generally, with h a function o f the nominal interest rate and r a constant, seigniorage can b e written as s Eih(IJ ) . This is maximized at the point where the elasticity of real money demand with respect to (! is equal to - I : Eih' (Ei)/h -1. = = Chapter 158 4 Figure 4.2 Money growth and seigniorage revenue. each rate of money growth, the expected rate of inflation needed to generate the required seigniorage revenue. 24 The 45° line gives the steady-state inflation rate as a function of the money growth rate: rr e = rr = e f..t. The two points of intersection labeled A and D are the two solutions to (4.26). What determines whether, for a given deficit, the economy ends up at the high inflation equilibrium or the low inflation equilibrium? Which equilibrium emerges depends on the stability properties of the economy. Determining this, in turn, requires a more complete specification of the dynamics of the model. Recall that the demand for money depends on expected inflation through the nominal rate of interest, while the inflation tax rate depends on actual inflation. In considering the effects of variations in the inflation rate, one needs to determine how expectations will adjust. Cagan (1956) addressed this by assuming that expectations adjust adaptively to actual inflation: - a rr e Bt = n e = TJ (Tl - rr e ) , (4.27) where T) captures the "speed of adjustment" of expectations. A low T) implies that expec­ tations respond slowly to inflation forecast errors. Since h = exp( - a rr e) , differentiate this expression with respect to time, obtaining h - = e f.L Tl = -arr e . h - . - = 24 . That is. SR plots ne (ln e - In t:/)ja . A reduction in e continues to yield and this would require a fall in expected inflation. t:/ only if money holdings rise, Money and Public Finance 159 Solving for rr using (4.27) yields rr = e - JL + ane = e - JL + arJ (n - ne), or rr = (8 - fL - a rJne) I (1 - a rJ) . Substituting this back into the expectations adjustment equa­ tion gives . 1] (8 - fL - ne) (4.28) 7T e 1 - arJ which implies that the low inflation equilibrium will be stable as long as a17 < 1. This requires that expectations adjust sufficiently slowly (17 < 1 /a). If expectations adjust adaptively and sufficiently slowly, what happens when the deficit is increased? Since the demand for real money balances depends on expected inflation, and because the adjustment process does not allow the expected inflation rate to jump imme­ diately, the higher deficit can be financed by an increase in the rate of inflation (assuming the new deficit is still below the maximum that can be financed, t:. *). Since actual inflation now exceeds expected inflation, n-e > 0, and ne begins to rise. The economy converges into a new equilibrium at a higher rate of inflation. In terms of figure 4.2, an increase in the deficit shifts the SR curve to the right to SR" (for a given expected rate of inflation, money growth must rise in order to generate more revenue). Assume that initially the economy is at point A, the low inflation equilibrium. Budget balance requires that the economy be on the SR" line, so e jumps to the rate asso­ ciated with point B. But now, at point B, inflation has risen and ne < rr = e - fL . Expected inflation rises (as long as a 17 < 1 ; see (4.28)), and the economy converges to C. The high inflation equilibrium, in contrast, is unstable. Adaptive expectations of the sort Cagan assumed disappeared from the literature under the onslaught of Lucas and Sargent's rational-expectations revolution of the early 1970s. If agents are systematically attempting to forecast inflation, then their forecasts will depend on the actual process governing the evolution of inflation; rarely will this imply an adjustment process such as (4.27). Stability in the Cagan model also requires that expec­ tations not adjust too quickly (17 < 1 /a), and this requirement conflicts with the rational­ expectations notion that expectations adjust quickly in response to new information. Bruno and Fischer (1990) showed that, to some degree, assuming agents adjust their holdings of real money balances slowly plays a role under rational expectations similar to the role played by the slow adjustment of expectations in Cagan's model in ensuring stability under adaptive expectations. == 4.4.5 ' Rational Hyperinflation Why do countries find themselves in situations of hyperinflation? Most explanations of hyperinflation point to fiscal policy as the chief culprit. Governments that are forced to print money to finance real government expenditures often end up generating hyperinflations. In that sense, rapid money growth does lead to hyperinflation, consistent with the relationship between money growth and inflation implied by the models examined so far, but money Chapter 160 4 growth is no longer exogenous. Instead, it is endogenously determined by the need to finance a fiscal deficit. Two explanations for the development of hyperinflation suggest themselves. In the Cagan model with adaptive expectations, suppose that a ry < 1 , so that the low inflation equilibrium is stable. Now suppose that a shock pushes the inflation rate above the high inflation equilibrium (above point D in figure 4.2). If that equilibrium is unstable, the econ­ omy continues to diverge, moving to higher and higher rates of inflation. So one explana­ tion for hyperinflations is that they represent situations in which exogenous shocks push the economy into an unstable region. Alternatively, suppose the deficit that needs to be financed with seigniorage grows. If it rises above /"),. * , the maximum that can be financed by money creation, the government finds itself unable to obtain enough revenue, so it runs the printing presses faster, further reducing the real revenue it obtains and forcing it to print money even faster. Most hyperin­ flations have occurred after wars (and on the losing side). Such countries face an economy devastated by war and a tax system that no longer functions effectively. At the same time, there are enormous demands on the government for expenditures to provide the basics of food and shelter and to rebuild the economy. Revenue needs outpace the government's ability to raise tax revenue. The ends of such hyperinflations usually involve a fiscal reform that allows the government to reduce its reliance on seigniorage (see Sargent 1 986). When expected inflation falls in response to the reforms, the opportunity cost of holding money is reduced and the demand for real money balances rises. Thus, the growth rate of the nominal money supply normally continues temporarily at a very high rate after a hyper­ inflation has ended. A similar, if smaller-scale, phenomenon occurred in the United States in the mid- 1980s. The money supply, as measured by Ml, grew very rapidly. At the time, there were concerns that this growth would lead to a return of higher rates of inflation. Instead, it seemed to reflect the increased demand for money resulting from the decline in inflation from its peak levels in 1979-1980. The need for real money balances to grow as inflation is reduced often causes problems for establishing and maintaining the credi­ bility of policies designed to reduce inflation. If a disinflation is credible, so that expected inflation falls, it may be necessary to increase the growth rate of the nominal money sup­ ply temporarily. But when inflation and rapid money growth are so closely related, letting money growth rise may be misinterpreted as a signal that the central bank has given up on its disinflation policy. Fiscal theories of seigniorage, inflation, and hyperinflations are based on fundamentals-there really is a deficit that needs to be financed, and that is what leads to money creation. An alternative view of hyperinflations is that they are simply bubbles, similar to bubbles in financial markets. Such phenomena are based on the possibility of multiple equilibria in which expectations can be self-fulfilling. 25 25. A recent modern example of such a fiscally driven hyperinflation was provided by Zimbabwe. Money and Public Finance 161 To illustrate this possibility, suppose the real demand for money is given by, in log terms, mt - Pt = -a (EtPt+I - Pt ), where E1p1+ 1 denotes the expectation formed at time t of time t + 1 prices and a > 0. This money demand function is the log version of Cagan's demand function. One can rearrange this equation to express the current price level as 1 a (4.29) mt + - EtPt+l · Pt = l +a l +a Suppose that the growth rate of the nominal money supply process is given by m1 = eo + (1 - y WI t + y mt - I · Since m is the log money supply, the growth rate of the money supply is m1 - m1 - I = ( 1 - y WI + y (m1 - I - m1- 2 ), and the trend (average) growth rate is 8 1 . Given this process, and the assumption that agents make use of it and the equilibrium condition (4.29) in forming their expectations, one solution for the price level is given by a ( 1 - y )e l 1 a [Bo + (1 - y)e! ( l + a)] t+ mt + Pt = 1 + a (l - y) 1 + a (1 - y ) 1 + a(1 - y) = A o + A 1 t + A 2 m1• That this is a solution can be verified by noting that it implies E1Pt+I = A o + A 1 (t + 1) + A 2 Etmt+ I = A o + A I (t + 1) + A 2 [Bo + (1 - y )e 1 (t + 1) + ymt] ; substituting this into (4.29) yields the proposed solution. Under this solution, the inflation rate p1 - Pt-1 converges to 8 1 , the average growth rate of the nominal supply of money. 26 Consider, now, an alternative solution: ( ) ( ) [ ] [ ] (4.30) where 81 is time-varying. Does there exist a B1 process consistent with (4.29)? Substituting the new proposed solution into the equilibrium condition for the price level yields a [A o + A 1 (t + 1) + A 2 Etmt+l + EtBt+I ] m1 A o + A 1 t + A 2 m t + B t = -, + 1+a 1 +a which, to hold for all realizations of the nominal money supply, requires that, as before, A o = a [Bo + (1 - Y WI ( 1 + a)] / [1 + a ( l - y)], A 1 = a ( 1 - Y Wt / [1 + a ( 1 - y)], and A 2 = 1 / [1 + a ( 1 - y )]. This then implies that the 81 process must satisfy a B1 = -- E1Bt+! , 1 +a which holds if B follows the explosive process ( ) (4.3 1) 26. This follows, since P t - P r - 1 = A 1 + A2 (m1 - m1- 1 ) converges to A 1 + A2 81 = 81 . Chapter 162 4 for k = ( 1 + a)ja > 1 . In other words, (4.30) is an equilibrium solution for any process B1 satisfying (4.3 1). Since B grows at the rate k - 1 = 1 /a, and since a, the elasticity of money demand with respect to expected inflation, is normally thought to be small, its inverse would be large. The actual inflation rate along a bubble solution path could greatly exceed the rate of money growth. As discussed in section 2.2.2, speculative hyperinflation in unbacked fiat money systems cannot generally be ruled out. Equilibrium paths may exist along which real money balances eventually converge to zero as the price level goes to +oo. (See also section 4.5 . 1 .) The methods developed to test for bubbles are similar to those that have been employed to test for intertemporal budget balance. For example, if the nominal money stock is non­ stationary, then the absence of bubbles implies that the price level will be nonstationary but cointegrated with the money supply. This is a testable implication of the no-bubble assump­ tion. Equation (4.3 1 ) gives the simplest example of a bubble process. Evans (199 1 ) showed how the cointegration tests can fail to detect bubbles that follow periodically collapsing processes. For more on asset prices and bubbles, see Shiller (1981), Mattey and Meese (1986), West (1987; 1988), Diba and Grossman (1988a; 1988b), and Barlevy (2007). 4.5 The Fiscal Theory of the Price Level A number of researchers have examined models in which fiscal factors replace the money supply as the key determinant of the price level. See, for example, Leeper ( 1 99 1 ), Sims (1994), Woodford (1995; 1998; 2001 a), Bohn (1999), Cochrane (1999; 2001), Kocherlakota and Phelen (1999), Daniel (200 I), the excellent discussions by Carlstrom and Fuerst (2000), Christiano and Fitzgerald (2000), Canzoneri, Cumby, and Diba (201 1) and the references they list, and the criticisms of the approach by McCallum (200 1 ), Buiter (2002), and McCallum and Nelson (2005). The fiscal theory of the price level raises some important issues for both monetary theory and monetary policy. 27 There are two ways fiscal policy might matter for the price level. First, equilibrium requires that the real quantity of money equal the real demand for money. If fiscal variables affect the real demand for money, the equilibrium price level will also depend on fiscal factors (see section 4.4.2). This, however, is not the channel emphasized in fiscal theories of the price level. Instead, these theories focus on a second aspect of monetary models: there may be multiple price levels consistent with a given nominal quantity of money and equality between money supply and money demand. The possibility of multiple equilibria was discussed in section 2.2.2 in the context of the MIU model, but the same possibility arises in other models of the demand for money. Fundamentally, the real demand for money depends on the nominal interest rate, which in turn depends on the expected future price 27. For an analysis of the 2008 financial crisis and its implications for monetary and fiscal policy from the perspective of the fiscal theory, see Cochrane (20 I I b). Money and Public Finance 163 level. There may be multiple paths for the price level at each point on which the real demand for money is equal to the real supply of money. When standard monetary models are consistent with multiple equilibrium values for the price level, fiscal policy may then determine which of these is the equilibrium price level. And in some cases, the equilibrium price level picked out by fiscal factors may be independent of the nominal supply of money. In contrast to the standard monetary theories of the price level, the fiscal theory assumes that the government's intertemporal budget equation represents an equilibrium condition rather than a constraint that must hold for all price levels. At some price levels, the intertem­ poral budget constraint would be violated. Such price levels are not consistent with equi­ librium. Given the stock of nominal debt, the equilibrium price level must ensure that the government's intertemporal budget is balanced. In that way, fiscal considerations may pin down the equilibrium price level. 4.5.1 Multiple Equilibria The traditional quantity theory of money highlights the role the nominal stock of money plays in determining the equilibrium price level. Using the demand for money given by (4. 19), a proportional relationship is obtained between the nominal quantity of money and the equilibrium price level that depends on the nominal rate of interest. However, the nom­ inal interest rate is also an endogenous variable, so (4. 19) by itself may not be sufficient to determine the equilibrium price level. Because the nominal interest rate depends on the rate of inflation, (4. 19) can be written as M t f (Rt Pt+l Pt Pt ) , where R is the gross real rate of interest. This forward difference equation in the price level - = may be insufficient to determine a unique equilibrium path for the price level. Consider a perfect-foresight equilibrium with a constant nominal supply of money, Mo. Suppose the real rate of return is equal to its steady-state value of 1 I f3, and the demand for real money balances is given by (4. 19). One can then write the equilibrium between the real supply of money and the real demand for money as Mo Pt+l ) , g' g ( Pt Pt = < 0. Under suitable regularity conditions on g(), this condition can be rewritten as (4.32) Pt+l Ptg - 1 ( ��) ¢ (Pt) . Equation (4.32) defines a difference equation in the price level. One solution is Pt+ i P* for all i 0, where P* Mo I g ( 1 ). In this equilibrium, the quantity theory holds, and the = = ::=: = price level is proportional to the money supply. = Chapter 4 164 This constant price level equilibrium is not, however, the only possible equilibrium. As noted in sections 2.2.2 and 4.4.5, there may be equilibrium price paths starting from that are fully consistent with the equilibrium condition (4.32). This possibility was illustrated in figure 2.2. Thus, standard models in which equilibrium depends on forward­ looking expectations of the price level, a property of the models discussed in chapters 2 and 3, generally have multiple equilibria. An additional equilibrium condition may be needed to uniquely determine the price level. The fiscal theory of the price level focuses on situations in which the government's intertemporal budget constraint may supply that additional condition. Po =!= P* 4.5.2 The Basic Idea of the Fiscal Theory The fiscal theory can be illustrated in the context of a model with a representative house­ hold and a government, but with no capital. The implications of the fiscal theory are easiest to see if attention is restricted to perfect-foresight equilibria. The representative household chooses its consumption and asset holdings optimally, sub­ ject to an intertemporal budget constraint. Suppose the period t budget constraint of the representative household takes the form i 0:: B = l Dr +PrYt - Tr Prcr +Mf + f Prcr + ( -+t lt. ) Mf + ( -1 +1-lt. ) D�+l ' where D1 is the household's beginning-of-period financial wealth and D�+l = (1 + i1)Bf + Mf. The superscripts denote that Md and Bd are the household's demand for money and _- interest-bearing debt. In real terms, this budget constraint becomes i dr + Yr - ir Ct + mf + bf = Cr + ( 1 +r lt ) mf + (-1 +1-) rr d�+ l ' where it = Tr /Pr, mf = Mf /Pr , 1 + rr = (1 + ir ) ( 1 + 1rt+t ), and dr = Dr/Pr. Let Ar.t+i jn= L ( 1 +1rr+j ) be the discount factor, with A r,t = 1. Under standard assumptions, the household intertem­ poral budget constraint takes the form (4.33) At,t+i (Yr+i - it+i ) f A r,t+i [cr+i + ( 1 ��it+l. ) m�+i ] . dr + f i=O i=O -_. 0:: = = Household choices must satisfy this intertemporal budget constraint. The left side is the present discounted value of the household's initial real financial wealth and after-tax income. The right side is the present discounted value of consumption spending plus the real cost of holding money. This condition holds with equality because any path of Money and Public Finance 165 consumption and money holdings for which the left side exceeded the right side would not be optimal; the household could increase its consumption at time t without reducing consumption or money holdings at any other date. As long as the household is unable to accumulate debts that exceed the present value of its resources, the right side cannot exceed the left side. The budget constraint for the government sector, in nominal terms, takes the form (4.34) Dividing by P�. this can be written as i = 1 8t + dt Tt + ( 1 +t lt ) mt + ( 1 + -rt ) dt+l· Recursively substituting for future values of dt+ i · this budget constraint implies that (4.35) dt + L A t,t+i [8t+i - Tt+ i - St+i] Tlim A t,t+ TdT , i=O where St = itmt f( l + it ) is the government's real seigniorage revenue. In previous sec­ -_ . 00 = ---+ 00 tions, it was assumed that the expenditures, taxes, and seigniorage choices of the con­ solidated government (the combined monetary and fiscal authorities) were assumed to be constrained by the requirement that limT---+ oo T T = 0 for all price levels Pt . Policy paths for such that At,t+ d (gt+i · Tt+i , St+; , dt+i) ;:o:o dt + L At,t+i [8t+i - Tt+i - St+i] = Tlim A t,t+TdT = 0 i=O for all price paths Pt+ i• i 0, are called Ricardian policies. Policy paths for (8t+i• Tt+i• St+i• dt+i) ;>o for2 which limT At,t+ TdT may not equal zero for all price paths 00 ---+ 00 � ---+ oo are called non-Ricardian. 8 Now consider a perfect-foresight equilibrium. Regardless of whether the government follows a Ricardian or a non-Ricardian policy, equilibrium in the goods market in this simple economy with no capital requires that = The demand for money must also equal the supply of money: 1 = Substituting for and for 1 in (4.33) and rearranging yields Ct + 8t· m mt. Yt Yt - 8t Ct mt m dt + fi=O A t,t+i [8t+i - Tt+i - ( � 1 + lt+t ) mt+i] = 0. (4.36) 28. Notice that this usage differs somewhat from the way Sargent ( 1 982) and Aiyagari and Gertler ( 1 985) employed the terms. In these earlier papers, a Ricardian policy was one in which the fiscal authority fully adjusted taxes to ensure intertemporal budget balance for all price paths. A non-Ricardian policy was a policy in which the monetary authority was required to adjust seigniorage to ensure intertemporal budget balance for all price paths. Both these policies would be labeled Ricardian under the current section's use of the term. Chapter 166 4 Thus, an implication of the representative household's optimization problem and mar­ ket equilibrium is that (4.36) must hold in equilibrium. Under Ricardian policies, (4.36) does not impose any additional restrictions on equilibrium because the policy variables are always adjusted to ensure that this condition holds. Under a non-Ricardian policy, however, it does impose an additional condition that must be satisfied in equilibrium. To see what this condition involves, one can use the definition of d1 and seigniorage to write (4.36) as oo DI = P1 L (4.37) Mt- = f(l + lt).. Pt (4.38) A i=O t,t+i [ Tt+i + St+i - 8t+i ] . At time t, the government's outstanding nominal liabilities D1 are predetermined by past policies. Given the present discounted value of the government's future surpluses (the right side of 4.37), the only endogenous variable is the current price level P1• The price level must adjust to ensure that (4.37) is satisfied. Equation (4.37) is an equilibrium condition under non-Ricardian policies, but it is not the only equilibrium condition. It is still the case that real money demand and real money supply must be equal. Suppose the real demand for money is given by (4. 19) , rewritten here as Equations (4.37) and (4.38) must both be satisfied in equilibrium. However, which two variables are determined jointly by these two equations depends on the assumptions that are made about fiscal and monetary policies. For example, suppose the fiscal authority and for all :=:: 0, and the monetary authority pegs the nominal rate of determines interest = I for all :=:: 0. Seigniorage is equal to If (I + l) / (1 + /) and so is fixed by monetary policy. With this specification of monetary and fiscal policies, the right side of (4.37) is given. Since D1 is predetermined at date t, (4.37) can be solved for the equilibrium price level PJ given by 8t+ Tt+ it+i i ii i (4.39) The current nominal money supply is then determined by (4.38): Mt = PJf( l + !) . One property of this equilibrium is that changes in fiscal policy (g or r) directly alter the equilibrium price level, even though seigniorage as measured by L � o is unaf­ fected. 29 The finding that the price level is uniquely determined by (4.39) contrasts with A.t,t+iSt+i 29. A change in g or r causes the price level to jump, and this transfers resources between the private sector and the government. This transfer can also be viewed as a form of seigniorage. Money and Public Finance 167 a standard conclusion that the price level is indeterminate under a nominal interest rate peg. This conclusion is obtained from (4.38): with i pegged, the right side of (4. 38 ) is fixed, but this only determines the real supply of money. Any price level is consistent with equilibrium, as M then adjusts to ensure that (4. 38 ) holds. Critical to the fiscal theory is the assumption that (4. 3 7), the government's intertemporal budget constraint, is an equilibrium condition that holds at the equilibrium price level and not a condition that must hold at all price levels. This means that at price levels not equal to P7, the government is planning to run surpluses (including seigniorage) whose real value, in present discounted terms, is not equal to the government's outstanding real liabilities. Similarly, it means that the government could cut current taxes, leaving current and future government expenditures and seigniorage unchanged, and not simultaneously plan to raise future taxes. 3 0 When (4.37) is interpreted as a budget constraint that must be satisfied for all price levels, that is, under Ricardian policies, any decision to cut taxes today (and so lower the right side of (4. 3 7)) must be accompanied by planned future tax increases to leave the right side unchanged. In standard infinite-horizon representative agent models, a tax cut (current and future government expenditures unchanged) has no effect on equilibrium (i.e., Ricardian equiva­ lence holds) because the tax reduction does not have a real wealth effect on private agents. Agents recognize that in a Ricardian regime, future taxes have risen in present value terms by an amount exactly equal to the reduction in current taxes. Alternatively expressed, the government cannot engineer a permanent tax cut unless government expenditures are also cut (in present value terms). Because the fiscal theory of the price level assumes that (4. 3 7) holds only when evaluated at the equilibrium price level, the government can plan a per­ manent tax cut. If it does, the price level must rise to ensure that the new, lower value of discounted surpluses is again equal to the real value of government debt. For (4. 3 9) to define an equilibrium price level, it must hold that D1 i- 0. Niepelt (2004) has argued that the fiscal theory cannot hold if there is no initial outstanding stock of nom­ inal government debt. However, Daniel (2007) showed that one can define non-Ricardian policies in a consistent manner when the initial stock of debt is zero. Her argument is most clearly seen in a two-period example. If the monetary authority pegs the nominal rate of interest, then any initial value of the price level is consistent with equilibrium, a standard result under interest rate pegs (see chapter 10). The nominal interest rate peg does pin down the expected inflation rate, or equivalently, the expected price level in the second period. However, this policy does not pin down the actual price level in period 2. Under a Ricardian fiscal policy, any realization of the price level in period 2, consistent with the value expected, is an equilibrium. If the realized price level were to result in the govern­ ment's budget constraint not balancing, then the Ricardian nature of policy means that 30. However, as Bassetto (2002) emphasized, the ability of the government to run a deficit in any period under a non-Ricardian policy regime is constrained by the willingness of the public to lend to the government. Chapter 168 4 taxes and/or spending must adjust to ensure intertemporal budget balance at the realized price level. Under a non-Ricardian fiscal policy, only realizations of the price level that sat­ isfy intertemporal budget balance can be consistent with an equilibrium. Thus, whatever quantity of nominal debt the government issued in the first period, the realized price level must ensure the real value of this debt in period 2 balances with the real value the gov­ ernment chooses for its primary surplus (including seigniorage). Under rational expecta­ tions, however, a non-Ricardian government cannot systematically employ price surprises in period 2 to finance spending because the monetary authority's interest peg has deter­ mined the expected value of the period 2 price level. Equilibrium must be consistent with those expectations. An interest rate peg is just one possible specification for monetary policy. As an alter­ native, suppose as before that the fiscal authority sets the paths for and but now suppose that the government adjusts tax revenue to offset any variations in seigniorage. In this case, becomes an exogenous process. Then (4.37) can be solved for the + equilibrium price level, independent of the nominal money stock. Equation (4.38) must still hold in equilibrium. If the monetary authority sets this equation determines the nom­ inal interest rate that ensures that the real demand for money is equal to the real supply. If the monetary authority sets the nominal rate of interest, (4.38) determines the nominal money supply. The extreme implication of the fiscal theory (relative to traditional quantity theory results) is perhaps most stark when the monetary authority fixes the nominal sup­ M for all i ::': 0. Then, under a fiscal policy that makes ply of money: + an exogenous process, the price level is proportional to and, for a given level of is independent of the value chosen for M . 8t+i r1+ ; , Tt+ i St+i Mt. Mr+i 4.5.3 = D1 Tr+i :Sr+i Dt. Empirical Evidence on the Fiscal Theory Under the fiscal theory of the price level, (4.37) holds at the equilibrium value of the price level. Under traditional theories of the price level, (4.37) holds for all values of the price level. If one only observes equilibrium outcomes, it is impossible empirically to distinguish between the two theories. As Sims (1994) put it, "Determinacy of the price level under any policy depends on the public's beliefs about what the policy authority would do under conditions that are never observed in equilibrium" (38 1). Canzoneri, Cumby, and Diba (201 1) discussed the identification issues that arise in attempting to test whether fiscal policy is Ricardian or non-Ricardian. Canzoneri, Cumby, and Diba (2001) examined VAR evidence on the response of U.S. liabilities to a positive innovation to the primary surplus. Under a non-Ricardian policy, a positive innovation to + (see 4.37) unless it also signals should increase future reductions in the surplus, that is, unless + is negatively serially correlated. The authors argued that in a Ricardian regime, a positive innovation to the current primary r1 s1 - g1 D r / P1 r1 s1 - g1 Money and Public Finance 169 surplus will reduce real liabilities. This can be seen by writing the budget constraint (4.34) in real terms as (4.40) Examining U.S. data, the authors found the responses were inconsistent with a non­ Ricardian regime. Increases in the surplus were associated with declines in current and future real liabilities, and the surplus did not display negative serial correlation. Cochrane (1999) pointed out the fundamental problem with this test: both (4.40) and (4.37) must hold in equilibrium, so it can be difficult to develop testable restrictions that can distinguish between the two regimes. The two regimes have different implications only if one can observe nonequilibrium values of the price level. Cochrane (20 1 1 b) used the fiscal theory to analyze the role of monetary and fiscal policy during the Great Recession. Bohn (1999) examined the U.S. deficit and debt processes and concluded that the pri­ mary surplus responds positively to the debt to GDP ratio. In other words, a rise in the debt to GDP ratio leads to an increase in the primary surplus. Thus, the surplus does adjust, and Bohn found that it responds enough to ensure that the intertemporal budget constraint is satisfied. This is evidence that the fiscal authority seems to act in a Ricardian fashion. Finally, an older literature (see section 4.4) attempted to estimate whether fiscal deficits tended to lead to faster money growth. Such evidence might be interpreted to imply a Ricardian regime of fiscal dominance. 4.6 Optimal Taxation and Seigniorage If the government can raise revenue by printing money, how much should it raise from this source? Suppose only distortionary revenue sources are available. To raise a given amount of revenue while causing the minimum deadweight loss from tax-induced distortions, the government should generally set its tax instruments so that the marginal distortionary cost per dollar of revenue raised is equalized across all taxes. As first noted by Phelps (1973), this suggests that an optimal tax package should include some seigniorage. This prescrip­ tion links the optimal inflation tax to a more general problem of determining the optimal levels of all tax instruments. If governments are actually attempting to minimize the distor­ tionary costs of raising revenue, then the optimal tax literature provides a positive theory of inflation. This basic idea, which is developed in section 4.6. 1 was originally used by Mankiw ( 1987) to explain nominal interest rate setting by the Federal Reserve. However, the implications of this approach are rejected for the industrialized economies (Poterba and Rotemberg 1990; Trehan and Walsh 1990), although this may not be too surprising because seigniorage plays a fairly small role as a revenue source for these countries. Calvo and Chapter 170 4 Leiderman (1992) used the optimal tax approach to examine the experiences of some Latin American economies, with more promising results. A survey of optimal seigniorage that links the topic with the issues of time inconsistency (see chapter 6) can be found in Herrendorf (1997). Section 4.6.2 considers the role inflation might play as an optimal response to the need to finance temporary expenditure shocks. Section 4.6.3 revisits Fried­ man's rule for the optimal rate of inflation in an explicit general equilibrium framework. 4.6.1 A Partial Equilibrium Model Assume a Ricardian regime in which the government has two revenue sources available to it. The government can also borrow. It needs to finance a constant, exogenous level of real expenditures g, plus interest on any borrowing. To simplify the analysis, the real rate of interest is assumed to be constant, and ad hoc descriptions of both money demand and the distortions associated with the two tax instruments are specified. With these assumptions, the basic real budget identity of the government can be obtained by dividing (4.3) by the time t price level to obtain bt = Rbt-1 + Tt - St , where R i s the gross interest factor (i.e., 1 plus the rate of interest), revenue, and s is seigniorage revenue. Seigniorage is given by mt-1 . St = Mt - Mt-1 = mt - -1 + 1ft (4.41) g- r is nonseigniorage tax (4.42) Taking expectations of (4.41 ) conditional on time t information and recursively solving forward yields the intertemporal budget constraint of the government: R ) - 1(Tt+i + st+i) = Rbt-l + ( R-(4.43) Et LR -l i=O Note that, given bt - 1 , (4.43) imposes a constraint on the government, because Et limi-+ oo R-ibt+i has been set equal to zero. Absent this constraint, the problem of choos­ ing the optimal time path for taxes and seigniorage becomes trivial. Just set both equal to 00 . g. zero and borrow continually to finance expenditures plus interest because debt never needs to be repaid. The government is assumed to set and the inflation rate as well as planned paths for their future values to minimize the present discounted value of the distortions gener­ ated by these taxes, taking as given the inherited real debt the path of expenditures, and the financing constraint (4.43). The assumption that the government can commit to a planned path for future taxes and inflation is an important one. Much of chapter 6 deals with outcomes when governments cannot precommit to future policies. In order to understand the key implications of the joint determination of inflation and taxes, assume that the distortions arising from income taxes are quadratic in the Tt Trt bt- 1 , Money and Public Finance (r1 ¢1) 2 171 4> tax rate: + /2, where is a stochastic term that allows the marginal costs of taxes to vary randornly. 31 Similarly, costs associated with seigniorage are taken to equal /2, where is a stochastic shift in the cost function. Thus, the present discounted + value of tax distortions is given by (s1 c1) 2 c (4.44) The government's objective is to choose paths for the tax rate and inflation to minimize (4.44) subject to (4.43). Letting A represent the Lagrangian multiplier associated with the intertemporal budget constraint, the necessary first-order conditions for the government's setting of T and s take the form Et(Tt+i 4>t+i) A Et(St+i Ct+i ) A, + + = = , These conditions simply state that the government will arrange its tax collections to equalize the marginal distortionary costs across tax instruments, that is, + = + + + for each i ::': 0, and across time, that is, = and = + + for all i andj. For i = 0, the first-order condition implies that + = + = A; this represents an infratemporal optimality condition. Since the value of depends on the total revenue needs of the government, increases in Rg I (R - l) + Rb 1 cause the government to increase the revenue raised from both tax sources. Thus, one would expect to observe and moving in similar directions (given and lntertemporal optimality requires that marginal costs be equated across time periods for each tax instrument: E1(s1+i c1+; ) E1(s1+ i c1+ ;) E1(s1+j l':t+j) E1(Tt+ 4>t+ ) E1(Tt+i 4>t+;) E1(Tti +j 4>it+j) T1 ¢1 s1 c1 A t T1 s1 - ¢1 c1). EtTt+ l = Tt - Et4>t+ l + 4>t. Etst+ 1 = St - Etct+ l + ct. (4.45) (4.46) These intertemporal conditions lead to standard tax-smoothing conclusions; for each tax instrument, the government will equate the expected marginal distortionary costs in differ­ ent time periods. If the random shocks to tax distortions follow /(1) processes such that = = 0, these intertemporal optimality conditions imply that both and follow Martingale processes, an implication of the tax-smoothing model originally developed by Barro (1979a). If = 0, (4.46) implies that changes in seigniorage revenue should be unpredictable based on information available at time t. E14>t+ 1 - ¢1 E1ct+ J - c1 T s E1ct+ l - c1 3 1 . This approach follows that of Poterba and Rotemberg ( 1 990), who specified tax costs directly, as done here, although they assumed a more general functional form for which the quadratic specification is a special case. See also Trehan and Walsh ( 1 988). Chapter 172 4 Changes in revenue sources might be predictable and still be consistent with this model of optimal taxation if the expected t 1 values of ¢> and/or E:, conditional on period t information, are nonzero. For example, if E: > 0, that is, if the distortionary cost of seigniorage revenue were expected to rise, it would be optimal to plan to reduce future seigniorage. Using a form of (4.46), Mankiw ( 1987) argued that the near random walk behavior of inflation (actually nominal interest rates) is consistent with U.S. monetary policy having been conducted in a manner consistent with optimal finance considerations. Poterba and Rotemberg ( 1990) provided some cross-country evidence on the joint movements of infla­ tion and other tax revenue. In general, this evidence was not favorable to the hypothesis that inflation (or seigniorage) has been set on the basis of optimal finance considerations. While Poterba and Rotemberg found the predicted positive relationship between tax rates and inflation for the United States and Japan, there was a negative relationship for France, Germany, and the United Kingdom. The implications of the optimal finance view of seigniorage are, however, much stronger than simply that seigniorage and other tax revenue should be positively correlated. Since the unit root behavior of both s and r arises from the same source (their dependence on 1) through A), the optimizing model of tax setting has the joint implica­ tion that both tax rates and inflation should contain unit roots (they respond to permanent shifts in government revenue needs) and that they should be cointegrated. 32 Trehan and Walsh ( 1990) showed that this implication is rejected for U.S. data. The optimal finance view of seigniorage fails for the United States because seignior­ age appears to behave more like the stock of debt than like general tax revenue. Under a tax-smoothing model, temporary variations in government expenditures should be met through debt financing. Variations in seigniorage should reflect changes in expected per­ manent government expenditures or, from (4.46), stochastic shifts in the distortions associ­ ated with raising seigniorage (because of the E: realizations). In contrast, debt should rise in response to a temporary revenue need (such as a war) and then gradually decline over time. However, the behavior of seigniorage in the United States, particularly during the World War II period, mimics that of the deficit much more than it does that of other tax revenue (Trehan and Walsh 1988). One drawback of this analysis is that the specification of the government's objective function is ad hoc; the tax distortions were not related in any way to the underlying sources of the distortions in terms of the allocative effects of taxes or the welfare costs of inflation. These costs depend on the demand for money; therefore, the specification of the distortions should be consistent with the particular approach used to motivate the demand for money. Calvo and Leiderman ( 1992) provided an analysis of optimal intertemporal inflation taxation using a money demand specification that is consistent with utility maximization. + Erft+l - r Rg/(R - +Rbr-1 32. That is, if ¢ and E: are /(0) processes, then r and s are /( 1 ) , but r - s = E: - ¢ is /(0) . Money and Public Finance 173 They showed that the government's optimality condition requires that the nominal rate of interest vary with the expected growth of the marginal utility of consumption. Optimal tax considerations call for high taxes when the marginal utility of consumption is low and low taxes when the marginal utility of consumption is high. Thus, models of inflation in an optimal finance setting generally imply restrictions on the joint behavior of inflation and the marginal utility of consumption, not just on inflation alone. Calvo and Leiderman estimated their model using data from three countries that have experienced periods of high inflation: Argentina, Brazil, and Israel. While the overidentifying restrictions implied by their model are not rejected for the first two countries, they are for Israel. 4.6.2 Optimal Seigniorage and Temporary Shocks The prescription to smooth marginal distortionary costs over time implies that tax levels are set on the basis of some estimate of permanent expenditure needs. Allowing tax rates to fluctuate in response to temporary and unanticipated fluctuations in expenditures would result in a higher total efficiency loss in present value terms because of the distortions induced by non-lump-sum taxes. As extended to seigniorage by Mankiw (1987), the same argument implies that seigniorage should be set on the basis of permanent expenditure needs, not adjusted in response to unanticipated temporary events. The allocative distortions induced by the inflation tax, however, were shown in chapters 2 and 3 to be based on anticipated inflation. Consumption, labor supply, and money-holding decisions are made by households on the basis of expected inflation, and for this reason, variations in expected inflation generate distortions. In contrast, unantici­ pated inflation has wealth effects but no substitution effects. It therefore serves as a form of lump-sum tax. Given real money holdings, which are based on the public's expectations about inflation, a government interested in minimizing distortionary tax costs should engi­ neer a surprise inflation. If sufficient revenue could be generated in this way, socially costly distortionary taxes could be avoided. Unfortunately, private agents are likely to anticipate that the government will have an incentive to attempt a surprise inflation; the outcome in such a situation is the major focus of chapter 6. But suppose the government can commit itself to, on average, only inflating at a rate consistent with its revenue needs based on average expenditures. That is, aver­ age inflation is set according to permanent expenditures, as implied by the tax-smoothing model. But if there are unanticipated fluctuations in expenditures, these should be met through socially costless unanticipated inflation. Calvo and Guidotti (1993) made this argument rigorous. They showed that when the government can commit to a path for anticipated inflation, it is optimal for unanticipated 33 33. Auernheimer ( 1 974) provided a guide to seigniorage for an "honest" government, one that does not generate revenue by allowing the price level to jump unexpectedly, even though this would represent an efficient lump­ sum tax. Chapter 4 174 inflation to respond flexibly to unexpected disturbances. 34 This implication is consistent with the behavior of seigniorage in the United States, which for most of the twentieth cen­ tury followed a pattern that appeared to be more similar to that of the federal government deficit than to a measure of the average tax rate. During war periods, when most of the rise in expenditures could be viewed as temporary, taxes were not raised sufficiently to fund the war effort. Instead, the U.S. government borrowed heavily, just as the Barro tax-smoothing model implies. But the United States did raise the inflation tax; seigniorage revenue rose during the war, falling back to lower levels at the war's conclusion. This behavior is much closer to that implied by Calvo and Guidotti's theory than to the basic implications of Mankiw's. 35 Rockoff (2015) examined the U.S. evidence from the revolutionary war to the Iraq war, finding a common theme for when money creation was relied on to finance wars. He concluded that the United States used borrowing and taxes to finance wars against minor powers. In major wars, however, it resorted to inflationary finance in the face of per­ ceived limits on further tax increases or when further borrowing would push interest rates to levels considered too high. 4.6.3 Friedman 's Rule Revisited The preceding analysis has gone partway toward integrating the choice of inflation with the general public finance choice of tax rates. The discussion was motivated by Phelps's conclusion that if only distortionary tax sources are available, some revenue should be raised from the inflation tax. However, this conclusion has been questioned by Kimbrough ( 1986a; 1986b), Faig (1988), Chari, Christiano, and Kehoe (199 1 ; 1996), and Correia and Teles (1996; 1999). 36 They showed that there are conditions under which Friedman's rule for the optimal inflation rate-a zero nominal rate of interest-continues to be optimal even in the absence of lump-sum taxes. Mulligan and Sala-i-Martin (1997) provided a general discussion of the conditions necessary for taxing (or not taxing) money. This literature integrates the question of the optimal inflation tax into the general prob­ lem of optimal taxation. By doing so, the analysis can build on findings in the optimal tax literature that identify situations in which the structure of optimal indirect taxes calls for different final goods to be taxed at the same rate or for the tax rate on goods that serve as intermediate inputs to be zero (see Diamond and Mirrlees 197 1 ; Atkinson and Stiglitz 1972). Using an MIU approach, for example, treats money as a final good; in contrast, a shopping-time model, or a more general model in which money produces transaction services, treats money as an intermediate input. Thus, it is important to examine the impli­ cations of these alternative assumptions about the role of money have for the optimal tax 34. 35. 36. See also Benigno and Woodford (2004) and Angeletos (2004). Chapter 8 revisits the optimal choice of taxes and inflation in a new Keynesian model. An early example of the use of optimal tax models to study the optimal inflation rate issue is Drazen ( 1 979). See also Walsh ( 1 984) . Chari and Kehoe ( 1 999) provided a survey. Money and Public Finance 175 approach to inflation determination, and how optimal inflation tax results might depend on particular restrictions on preferences or on the technology for producing transaction services. The Basic Ramsey Problem The problem of determining the optimal structure of taxes to finance a given level of expen­ ditures is called the Ramsey problem, after the classic treatment by Ramsey (1928). In the representative agent model studied here, the Ramsey problem involves setting taxes to maximize the utility of the representative agent, subject to the government's revenue requirement. The following static Ramsey problem, based on Mulligan and Sala-i-Martin (1997), can be used to highlight the key issues. 37 The utility of the representative agent depends on consumption, real money balances, and leisure: u = u(c, m, l). Agents maximize utility subject to the following budget constraint: (4.47) f(n) 2: (1 + r)c + Tm m, where f(n) is a standard production function, n = 1 - l is the supply of labor, c is con­ sumption, r is the consumption tax, Tm = i/(1 + i) is the tax on money, and m is the household's holdings of real money balances. The representative agent picks consump­ tion, money holdings, and leisure to maximize utility, taking the tax rates as given. Letting A be the Lagrangian multiplier on the budget constraint, the first-order conditions from the agent's maximization problem are Uc = A ( l + r), Ut = Aj' , (4.48) (4.49) (4.50) (4.5 1) f ( l - l) - (1 + r)c - Tmm = 0. From these first-order conditions and the budget constraint, c, m, and l can be expressed as functions of the two tax rates: c(r, Tm), m(r, Tm ) , and /(r, Tm). The government's problem i s to set r and Tm to maximize the representative agent's utility, subject to three types of constraints. First, the government must satisfy its budget constraint; tax revenue must be sufficient to finance expenditures. This constraint takes the form (4.52) 37. I thank Bo Sandemann for pointing out an error in my derivation of the model in earlier editions and for suggesting the approach taken in this edition. Fortunately, the key equation, (4.55) in the third edition, is not affected. Chapter 176 4 where g is real government expenditures. These expenditures are taken to be exogenous. Second, the government is constrained by the fact that consumption, labor supply, and real money must be consistent with the choices of private agents. That means that (4.48)-(4.5 1) represent constraints on the government's choices. Finally, the government is constrained by the economy's resource constraint: f( l l) :::: c + g. (4.53) However, (4.5 1 ) and (4.52) imply (4.53) is redundant. There are two approaches to solving this problem. The first approach, often called the dual approach, employs the indirect utility function to express utility as a function of taxes. These tax rates are treated as the government's control variables, and the optimal values of the tax rates are found by solving the first-order conditions from the government's opti­ mization problem. The second approach, called the primal approach, treats quantities as the government's controls. The tax rates are found from the representative agent's first­ order conditions to ensure that private agents choose the quantities that solve the govern­ ment's maximization problem. The dual approach is presented first. The primal approach is employed later. Substituting the solutions to the representative agent's decision problem into the utility function yields the indirect utility function: (4.54) v (T, Tm) u(c(T, Tm), m(T, Tm), l(T, Tm)). From Roy's identity, 38 - = C = - Vr A -· Vrm m = --. A The government's problem is to pick r and Tm to maximize (4.54), subject to the gov­ ernment's budget constraint (4.52). Thus, the government's problem can be written as ft [Tmm(T, Tm) + rc(r, Tm) - g] , max T,Tm {v(r, Tm) + 38. To see this, differentiate the indirect utility function with respect to r and use (4.48)-(4.50) to obtain Vr = UcCr + Umlnr + U[lr = A [0 + r) Ct + Tmlnr + f' lr ] . From (4.5 1), -f' lr = c + ( 1 + r) ct + Tm m r . Combining these two expressions implies vr = - A.c, which when rearranged yields the desired result. A simi