Labor Market Search and Monetary Shocks

Labor Market Search and Monetary Shocks Carl E. Walsh∗ First draft: February 2002 This draft: August 2002 Abstract A number of recent papers have focused on the role of real factors in accounting for the persistent effects of monetary shocks. These factors include habit persistence in consumption, variable capital utilization, and produced inputs. In this chapter, I focus on labor market rigidities that prevent unemployed workers from Þnding new jobs immediately and Þrms with job vacancies from Þlling them immediately. I employ dynamic simulations to investigate the role the job matching process plays in affecting the economy’s adjustment to real and monetary shocks. By combining labor market rigidities with sticky prices, it is possible to study the relative roles of real and nominal rigidities in accounting for the impact of monetary shocks. JEL: E52, E58 ∗ Department of Economics, SS1, University of California, Santa Cruz, CA 95064, [email protected]. Pre- pared for Elements of Dynamic Macroeconomic Analysis, S. Altuĝ, J. Chadha, and C. Nolan (eds). I would like to thank Ryota Kojima for research assistance, and the referees, seminar participants at UC Davis and UCSB and participants in the Brown Bag workshop at UCSC for helpful comments on this research. 1 1 Introduction In recent years, dynamic stochastic general equilibrium (DSGE) models of monetary economies have focused on the role of nominal rigidities in affecting the economy’s adjustment to monetary policy and nonpolicy disturbances. While these rigidities appear important for understanding the impact nominal shocks have on such real variables as output and employment, models with only nominal rigidities have been unable to match the responses to monetary disturbances that have been estimated in the data. Typically, empirical studies have concluded that monetary shocks generate large and persistent real responses that display a hump shape. After a positive money shock, for example, output rises over several quarters and then declines. Christiano, Eichenbaum, and Evans (1999) document this effect and provide an extensive discussion of the empirical evidence on the effects of monetary shocks. Sims (1992) Þnds large, hump shaped responses of real output to monetary shocks in several OECD countries. Inßation also displays a hump shaped response, although inßation is usually found to respond more slowly than output to monetary shocks. The “stylized facts” emphasized by Christiano, Eichenbaum, and Evans, by Sims, and by others are illustrated in Figure 1, which shows estimated impulse responses of output and inßation following a shock to the growth rate of money. These responses were obtained from a three-variable VAR (output, inßation, and money growth) estimated using U.S. quarterly data for 1965-2001. Output is real GDP , inßation is measured by the Consumer Price Index, and M2 is the aggregate used to measure of money. The real persistence and inßation inertia seen in Figure 1 has been hard for models based on nominal rigidities to match.1 As Dotsey and King (2001) have recently 1 Chari, Kehoe, and McGratten (2000) Þnd little persistence in their model of staggered price adjustment, and Nelson (1998) demonstrates that several optimizing models of price stickiness are unable to match the time series properties of inßation. 2 expressed it, “...modern optimizing sticky price models have displayed a chronic inability to generate large and persistent real responses to monetary shock...” In order to capture at least some of the real persistence seen in empirical studies, models based on nominal rigidity generally must assume a high degree of price stickiness. For example, it is common to assume individual prices remain Þxed on average for as much as nine months. Micro data on individual prices, however, suggests that prices typically change more frequently than this. Consequently, a number of researchers have recently argued that simply adding nominal rigidities to an otherwise standard DSGE model is not sufficient to match the persistence observed in the data. Instead, the real side of the economy must also be modiÞed to capture additional factors affecting both aggregate production and aggregate spending decisions. Christiano, Eichenbaum and Evans (2001), in particular, have argued that models with nominal rigidities are capable of matching macro evidence but only if the real side of the model is properly speciÞed. Despite the recent interest on the real side of monetary models, almost all existing DSGE models of monetary economies continued to assume labor can be costlessly and instantaneously reallocated across Þrms. Yet the complex process through which workers seeking jobs and Þrms with open vacancies are matched is likely to be important in inßuencing the way economic disturbances are propagated over time. The work of Mortensen and Pissarides (1994, 1999) and Pissarides (2000) has emphasized the costly and time consuming process of matching workers seeking employment with Þrms seeking to Þll job vacancies. How quickly unemployment returns to its steady-state level after an adverse shock, whether it is a real or nominal shock, is likely to be inßuenced by how efficiently the labor market is able to generate new matches between Þrms and unemployment workers. In this chapter, the dynamic implications of labor market search and sticky prices are analyzed. The model that is developed is the Þrst to combine a labor market structure based on 3 a Mortensen-Pissarides aggregate matching function with an optimizing model of price rigidity. The introduction of price stickiness allows the interactions between labor market rigidities and nominal price rigidities to be investigated. The next section reviews some of the related literature that has focused on the labor market and other real aspects of the economy that affect the economy’s response to nominal shocks. Section 3 develops the basic model. In section 4, the dynamic adjustment of the economy to nominal money growth rate shocks is examined under ßexible prices and under sticky prices. Conclusions and some suggestions for further research are discussed in section 5. 2 Related literature This paper brings together two previously unrelated strands of the literature — models such as those of Merz (1995), Andolfatto (1996) and Cooley and Quadrini (1999, 2000) that study the implications of matching models of the labor market for DSGE models of the business cycle, and models such as those of Rotemberg and Woodford (1997), Yun (1996), Goodfriend and King (1997), Chari, Kehoe, and McGratten (2000), among others, that introduce price stickiness in DSGE models of monopolistic competition. Cole and Rogerson (1999) argue that aggregate labor market matching models based on the work of Mortensen and Pissarides (1994, 1999) can replicate important aspects of business cycles, but only if the models are calibrate to imply average durations of unemployment spells that are much longer than data on actual duration suggest. Similarly, models of price stickiness are typically calibrated to imply individual prices are Þxed for durations that are also much longer than suggested by evidence on individual prices (Bils and Klenow 2002). By incorporating both a labor market matching model and price stickiness within a single model, one can investigate whether the interactions of these two allow the model to match important business cycle facts with more plausible calibrations of unemployment duration and 4 price rigidity. In fact, it has been the failure of models based solely on nominal price stickiness to account for output persistence and inertia in inßation that has led some researchers to explore the role of real factors that might interact with nominal rigidities to account for the dynamic responses seen in the data. In an early contribution, Ball and Romer (1990) argued that real rigidities act to amplify the effects of nominal rigidities. More recently, Dotsey and King (2001) argue that “real ßexibilities” — variable capital utilization and produced inputs — are critical for generating the persistence displayed in the data. These real ßexibilities allow output to vary with relatively small effects on marginal cost, reducing the elasticity of marginal costs with respect to output. A nominal shock to aggregate demand and output has only a small impact on marginal cost, and therefore on inßation, when this elasticity is small.2 Similarly, Christiano, Eichenbaum, and Evans (2001) emphasize that the interaction of real and nominal rigidities seems to be critical in matching empirical evidence. Christiano, Eichenbaum, and Evans (hereafter CEE) allow for habit persistence in consumption, variable capital utilization, and investment adjustment costs (real rigidities) as well as both price and nominal wage stickiness. They conclude that nominal wage rigidity (as opposed to price rigidity) is critical in matching aggregate data for the U.S. However, King and Goodfriend (2001) take the position that nominal wage rigidity may not be important for business cycle phenomena, arguing that “The labor market is characterized by long-term relationships where there is opportunity and reason for Þrms and workers to neutralize the allocative effects of temporarily sticky nominal wages.” (p. 4). The real rigidities introduced by long-term relationships between Þrms and workers has, to date, not been incorporated into models with nominal rigidities. This neglect is perhaps surpris2 See Burnside and Eichenbaum (1996) for an earlier analysis of variable capital utilization in a real business cycle model. 5 ing. Jeanne (1998) combined a Calvo-type model of price stickiness with an ad hoc speciÞcation of the equilibrium real wage. This allowed him to exogenously vary the response of the real wage to output movements. He showed that an increase in labor market real wage rigidity reduced the degree of price stickiness that was needed to match the response of output to a monetary shock. This work suggested that the speciÞcation of the labor market might play an important role in explaining the persistent output response to nominal shocks, but his model provided no underlying theory to explain the source of the labor market rigidity. Merz (1995) and Andolfatto (1996) have shown that a real business cycle model that incorporates a Mortensen—Pissarides aggregate matching function (Mortensen and Pissarides 1994, 1999, Pissarides 2000) to represent the search process in the labor market is able to provide a better match with evidence on employment and wages than do models based on a traditional Walrasian labor market. Cole and Rogerson (1999) note, however, that one must assume workers face a small probability each period of Þnding a match if aggregate matching models are to capture important labor market behavior. den Haan, Ramey and Watson (2000) show how a search model of the labor market can serve to amplify and propagate productivity shocks. These models capture the long-term nature of employment relationships that King and Goodfriend emphasize alter the allocative implications of observed rigidity in nominal wages. The only examples to date of monetary models with a matching model of the labor force are due to Cooley and Quadrini (1999, 2000). They introduced money into a DSGE model with a matching model of the labor market and show that monetary shocks have highly persistent impacts on inßation and the real economy. However, they assume prices are completely ßexible, and persistent real effects arise, in part, because they assume nominal portfolios adjust slowly over time. This portfolio rigidity generates a liquidity effect (a fall in the nominal interest rate after an increase in money growth).3 By assuming portfolio readjustments take several periods 3 This form of limited participation model (Fuerst 1992) assumes households cannot immediately reallocate their 6 to occur, the real impact of money shocks is propagated over time in the Cooley-Quadrini model. Thus, their dynamics reßect both the speciÞcation of the labor market and the assumption of sticky portfolio adjustment. While the work by Cooley and Quadrini has helped to highlight the role of the matching process in a monetary economy, their assumption of ßexible prices meant they were unable to study the possible interactions between the dynamic adjustment of employment through the matching process and the dynamic adjustment of inßation when prices are sticky. 3 The model economy To study the interaction of labor search and price stickiness, I employ a model that distinguishes between wholesale and retail sectors. Goods are produced in the wholesale sector and then sold by retail Þrms to households. The production of wholesale goods requires that a Þrm and a worker be matched. Unmatched workers and Þrms with vacancies are matched through a process characterized by an aggregate matching function. Because some workers and Þrms will be matched, while others will not be, distributional issues arise. To avoid these issues, I follow Merz (1995), Andolfatto (1996), and den Haan, Ramey, and Watson (2000) in assuming households pool consumption, both market purchased consumption and home consumption produced by workers who do not have an employment match in period t. The output produced by worker-Þrm matches is sold in a competitive goods market to retail Þrms. These Þrms costlessly transform the wholesale good into retail goods that are sold to households in markets characterized by monopolistic competition. Prices at the retail level are sticky, with only a fraction of retail Þrms optimally adjusting their price each period. This separation between wholesale Þrms participating in the labor market matching process and retail Þrms with sticky prices follows the approach bond and money holdings after a monetary shock. 7 of Bernanke, Gertler and Gilchrist (1999) in their study of credit market imperfections. This separation simpliÞes the structure of the model. Money is introduced through a cash-in-advance constraint.4 Income generated in period t is not available for consumption until period t + 1, so the nominal interest rate affects the present discounted value of current production. This generates a channel through which nominal interest rate changes affect output and employment. An increase in the nominal interest rate reduces the present value of production and leads to an increase in job destruction, a fall in employment, and a decline in output in the wholesale sector. The role played by the labor market speciÞcation can be highlighted by considering the effects of an unanticipated increase in money growth in period t that persists for periods t and t + 1, after which money growth returns to its steady-state value. When prices are ßexible, inßation rises immediately and remains above its steady-state value during periods t and t + 1. In period t, expected inßation also rises, causing the nominal rate of interest to rise. In period t + 1, the nominal rate returns to its steady-state value. In a model with an inelastic labor supply and no capital but with a standard Walrasian labor market, this money growth shock has no impact on employment or output. If labor supply is elastic, however, the rise in the nominal interest rate causes a substitution towards leisure (a “cash” good), and employment and output fall, but only for one period.5 In contrast, in the present model, the rise in the nominal interest rate leads to a rise in job destruction; employment and output fall. The economy enters period t + 1 with fewer employment matches. This propagates the output and employment decline into period t + 1. The matching process causes the return to steady-state to be spread over several periods. Thus, the labor market dynamics contribute to the persistence displayed by the economy is response to a 4 For a general discussion of cash-in-advance models, see Walsh (1998, Chapter 3). 5 The effect can be spread over several periods if capital is introduced into the model. See, for example, Walsh (1998, Chapter 3). 8 money growth rate shock. The adverse output effects of a rise in the nominal interest rate also has implications for inßation. The decline in wholesale output raises wholesale prices relative to retail prices. The fall in the retail price markup acts to raise retail price inßation. A positive impact on inßation of an increase in the nominal interest rate is variously called the cost channel of monetary policy or the Wright Patman effect, after the late, popularist Texas Congressman. Barth and Ramey (2001) have used industry-level data to examine this cost channel. They argue that the cost channel can account for the price puzzle — the Þnding in empirical VARs that inßation initially rises after an interest rate increase. Christiano, Eichenbaum, and Evans (2001) also incorporate a cost channel into their model, but they do not provide information on its contribution to matching the dynamic responses of output and inßation to a monetary shock. In the remainder of this section, the details of the model are developed. 3.1 Households The representative household consists of a worker and a shopper. Each of these actors engage in different activities during the period before reuniting at the end of each period (Lucas and Stokey 1983, 1987). There are assumed to be a continuum of such households on the interval [0, 1]. Shoppers carry cash balances to the goods market to purchase market consumption goods; these purchases are subject to a cash-in-advance constraint. For simplicity, households are assumed to supply their unit of labor inelasticly. Households are also the owners of all Þrms in the economy. Households maximize the expected present discounted value of utility: Et ∞ X β i [u(Ct+i ) + (1 − χt )h − χt A] , (1) i=0 where Ct is a composite consumption good consisting of the differentiate products produced by monopolisticly competitive retail Þrms. There are a continuum of such Þrms of measure 1. Ct is 9 deÞned as Ct = ·Z 1 0 θ−1 θ cjt dj θ ¸ θ−1 θ > 1. (2) The variable χt is an indicator variable, equal to 1 if the household’s worker is employed and 0 otherwise. The disutility of work is A, and h is the utility of home production when unemployed. Households maximize expected utility subject to two constraints. First, they face a cash-inadvance constraint that takes the form h Pt Ct ≤ Mt−1 + Tt − Bt , (3) where Mth (Bt ) is the household’s nominal holdings of money (one-period nominal bonds), Pt is the retail price index, and Tt is a lump-sum transfer received from the government. In the aggregate, this transfer is equal to Mt − Mt−1 = (Gt − 1)Mt−1 where M (without the superscript h) is the aggregate nominal money stock. Note that current income is unavailable for purchasing current market consumption. This timing assumes Þnancial asset markets open before the goods market. Bonds purchased at the start of period t, Bt , pay a gross nominal interest rate of Rt , These interest payments are received when the asset market reopens in period t + 1. Thus, the budget constraint households face can be written as h + Tt − Bt − Pt Ct , Mth = Pt Ytl + Dt + Rt Bt + Mt−1 (4) where Ytl is the household’s real labor income, and Dt is their share of aggregate proÞts from wholesale and retail Þrms. Given prices pjt for the Þnal goods, this preference speciÞcation implies the household’s demand for good j is µ cjt = pjt Pt ¶−θ Ct (5) where the aggregate retail price index Pt is deÞned as Pt = ·Z 0 1 1−θ dj pjt 10 1 ¸ 1−θ . (6) The following two conditions, obtained from the household’s Þrst order conditions and the cash-in-advance constraint, must hold in equilibrium: u0t = βRt Et Pt Ct = µ u0t+1 Pt+1 ¶ (7) Mt Pt (8) where u0 denotes the marginal utility of consumption. It will be convenient to deÞne the oneperiod discount factor δt ≡ βEt µ Pt u0t+1 Pt+1 u0t ¶ , (9) so that equation (7) can be written as δ t Rt = 1. 3.2 The labor and goods markets The production side of the model, and the labor market speciÞcation, is similar to that used by den Haan, Ramey, and Watson (2000). Their focus is on the role of aggregate productivity shocks, and because their model does not incorporate money, they do not study the role of price stickiness. In order to simplify the non-monetary aspects of the model, I ignore the capital stock dynamics that den Haan, Ramey, and Watson include. Production takes place in the wholesale sector, where Þrms and workers are paired through a matching process. 3.2.1 The wholesale sector At the beginning of the period, there are Nt matched workers and Þrms; Ut = 1 − Nt workers who are unmatched. If a worker is part of an existing match at the start of period t, she travels to her place of employment. At that point, there is an exogenous probability 0 ≤ ρx < 1 the match is terminated. For the (1 − ρx )Nt surviving matches, the worker and Þrm jointly observe the current realization of productivity and decide whether to continue the match. If the realization of productivity is low enough, it will be unproÞtable for the match to continue. If the match does 11 continue, production occurs. The output of a matched worker/Þrm pair i in period t that does produce is yit = ait zt (10) where ait is a serially uncorrelated, match-speciÞc productivity disturbance and zt is a common, aggregate productivity disturbance. The means of both productivity disturbances are equal to 1, and both are bounded below by zero. Wholesale Þrms sell their output in a competitive market at the price Ptw . Firms seeking workers must incur a cost of posting a vacancy, and workers seeking jobs must engage in a search process that takes time. As a consequence, existing matches may earn an economic surplus, and both the Þrm and the worker will wish to maintain a match with a positive expected surplus. The expected surplus an existing match generates depends, in part, on the value of the current output the match produces. Because of the cash-in-advance constraint, proceeds from output produced in period t are only available for consumption in period t + 1. Thus, the time t value of the revenues obtained from production in period t is δ t Ptw ait zt /Pt , where δ t is the discount rate given by equation (9). In addition, there is a continuation value of being part of an existing match that survives into period t + 1. Therefore, the expected value of a match that produces in period t is δt µ ait zt µt ¶ − A + git , where µt = Pt /Ptw is the markup of retail over wholesale prices, and git is the expected present value of a match that continues into period t + 1. To simplify, assume that the share of the surplus from a match received by each participant is Þxed.6 The surplus is the difference between (δ t ait zt /µt )−A+git and the alternative opportunities available to the Þrm and the worker. If the Þrm has no alternative opportunities, the match’s 6 The assumption of Þxed shares is common and would arise under risk neutrality in a Nash bargaining solution. 12 opportunity cost, wtu , is equal to the value of home consumption an unmatched worker can produce plus the present value of future worker opportunities if unmatched in period t. DeÞne qt ≡ gt − wtu as the expected excess value of a match that continues into period t + 1. Since all matches are identical, the subscript i has been suppressed. A match continues as long as (δ t ait zt /µt )−A+qt ≥ 0. Matches endogenously separate if the match speciÞc productivity shock is less than ãt , where this critical value is deÞned using the deÞnition of δ t in equation (9) as ãt = µt Rt (A − qt ) . zt (11) If A − qt < 0, then matches would never endogenously end since the support of a is strictly positive. When A − qt > 0, matches do endogenously breakup. In this case, a higher realization of the aggregate productivity shock zt will, ceteris paribus, lower ãt , making it more likely that existing matches produce. A higher zt realization directly increases the production of all matched worker/Þrms (see equation 10). It also leads more matches to produce because fewer endogenously separate (see equation 11). Thus, the role of zt in affecting ãt tends to amplify the impact of the aggregate productivity shock on output, an effect emphasized by den Haan, Ramey, and Watson (2000). A decrease in the nominal interest rate also leads more matches to produce. Because income earned in period t is only available for the household to consume in period t + 1, a rise in the value of future income (as would be caused by a fall in Rt ) makes current production more valuable and decreases the probability that a match will be dissolved.7 A rise in the markup of retail over wholesale prices reduces the proÞtability of wholesale production and increases ãt . 7 The nonneutrality this creates is similar to other cash-in-advance models that, for example, require Þrms to pay wages prior to the receipt of revenues from production as in Carlstrom and Fuerst (1995), Cooley and Quadrini (1999), or Christiano, Eichenbaum, and Evans (2001). For evidence on the existence of such a “cost channel” of nominal interest rates, see Barth and Ramey (2001). 13 These results are only partial equilibrium effects, since changes in aggregate productivity or the nominal interest rate also affect wtu , the present discounted value of unemployment, and gt , the present discounted value of a match. Let ρnt be the aggregate fraction of matches that endogenously separate, and let F denote the cumulative distribution function of the match speciÞc productivity shock. Then the probability match i endogenously separates is F (ãt ) and, because all matches are identical, the aggregate endogenous separation rate is the probability that at ≤ ãt : ρnt = Pr [at ≤ ãt ] = F (ãt ) (12) The aggregate total separation rate ρt is equal to ρt = ρx + (1 − ρx )ρnt (13) while the survival rate, ϕt ≡ (1 − ρt ) = (1 − ρx ) [1 − F (ãt )], is decreasing in ã. DeÞne the joint surplus of a worker-Þrm pair who are matched at the start of t + 1 and do not separate as sit+1 = δt+1 µ ait+1 zt+1 µt+1 ¶ − A + qt+1 (14) Note that this is expressed in terms of the present value as of the beginning of period t + 1. Let η denote the share of this surplus received by the worker; the Þrm receives 1− η of the joint surplus. If an unmatched worker in period t succeeds in making a match that produces in period t + 1, she u . The u receives her opportunity utility wt+1 plus the fraction η of the joint surplus, or ηsit+1 +wt+1 probability of this occurring is ktw (1 − ρt+1 ), where ktw is the period t probability an unmatched worker Þnds a job and 1 − ρt+1 is the probability that the match actually produces in period t + 1. With probability 1 − ktw (1 − ρt+1 ) the worker either fails to make a match or makes a match that fails to survive to produce in t + 1. In either case, the worker is unmatched in t + 1 and receives u . Therefore, the expected discounted value to an unmatched worker in the labor matching wt+1 14 market is8 wtu = h + βEt µ u0t+1 u0t ¶" Z w x kt (1 − ρ ) ā ãt+1 u ηsit+1 f(ai )dai + wt+1 # (15) since an unmatched worker is assumed to enjoy utility h while unmatched. For a worker and Þrm who are already matched, the joint discounted value of an existing match is gt = βEt µ u0t+1 u0t ¶" Z x (1 − ρ ) ā ãt+1 # u sit+1 f(ai )dai + wt+1 . (16) Hence, subtracting (15) from (16), qt ≡ gt − wtu = (1 − ρx )(1 − ηktw )βEt µ u0t+1 u0t ¶ "Z # ā st+1 f(ai )dai − h. ãt+1 (17) Unmatched Þrms, or Þrms whose matches terminated, may choose to enter the labor matching market and post vacancies. Posting a vacancy costs γ per period. If an unmatched Þrm does post a vacancy and succeeds in making a match that produces in period t+1, it receives (1−η)st+1 −γ. Otherwise (i.e., if no match is made or if the match separates before production), the Þrm receives nothing. If ktf is the probability a vacancy is Þlled, free entry ensures that Þrms post vacancies until βEt µ u0t+1 u0t ¶" Z f x kt (1 − ρ ) # ā (1 − η)st+1 f (a)da − γ = 0. ãt+1 (18) Combining (17) and (18), qt = γ(1 − ηktw ) (1 − η)ktf − h. (19) Increases in either kw or kf reduce the value of continuing an existing match by making it easier to Þnd a new match. 8 The expected surplus is ktw (1 − ρt+1 ) Z ā ηst+1 ãt+1 f (a) da = ktw (1 − ρx ) 1 − F (ãt+1 ) 15 Z ā ηst+1 f (a)da. ãt+1 A total of ρt Nt matches dissolve prior to engaging in production during period t. If the worker is not part of an existing match, or if her current match ends, she travels to the labor matching market. Thus, a total of ut ≡ Ut + ρt Nt = 1 − (1 − ρt )Nt (20) workers will not produce market goods during the period and will be searching for a new match. Based on an aggregate matching function, a fraction of workers and Þrms in the labor market establish new matches. These, plus the worker-Þrm matches that produced during the period, constitute the stock of matches that enter period t+1. The number of matches is equal to m(ut , Vt ) where Vt is the number of posted vacancies and m( ) is the aggregate matching function. The probability an unemployed worker makes a match, ktw , is equal to ktw = m(ut , Vt ) . ut (21) Similarly, the probability a Þrm with a posted vacancy Þnds a match, ktf , is ktf = m(ut , Vt ) . Vt (22) The total number of matches evolves according to Nt+1 = (1 − ρt )Nt + m(ut , Vt ). (23) The aggregate output of the wholesale sector is obtained by aggregating over all matches that actually produce: Qt = (1 − ρt )Nt zt ·Z ∞ at ãt 3.2.2 µ ¶ ¸ f (a) da . 1 − F (ãt ) (24) The retail section Firms in the retail sector purchase output from wholesale producers at the price Ptw and sell directly to households. For simplicity, assume retail Þrms have no other inputs. Given the 16 structure of demand facing each retail Þrm (see equation 5), all retail Þrms would charge the same price in a ßexible price equilibrium. This price would be a constant market up over wholesale prices, with the markup equal to θ/(θ − 1). Rather than assume ßexible prices, I assume prices at the retail level are sticky. To model this price stickiness, I adopt the approach due originally to Calvo (1983) and now widely used in macroeconomics (see, for example, Erceg, Henderson, and Levin 2000, Woodford 1999, 2000, or Christiano, Eichenbaum, and Evans 2001). The Calvo model is based on the assumption that each period a randomly chosen fraction of all Þrms are allowed to adjust their price. Let the probability a Þrm adjusts its price each period be given by 1 − ω. If Þrm j sets its price at time t, it will do so to maximize expected proÞts, subject to the demand curve it faces. The price of retail Þrm j is pjt . Let p∗t be the price chosen by all Þrms who set prices in period t; all retail Þrms setting prices in period t will choose the same price. Each retail Þrm’s nominal marginal cost is just Ptw . Real marginal cost is Ptw /Pt = µ−1 t . Using equation (5), the Þrm’s decision problem when it adjusts its price involves picking p∗t to maximize Et ∞ X i=0 ω i ∆i,t+1 "µ p∗t Pt+i ¶1−θ − µ−1 t+i µ p∗t Pt+i ¶−θ # Ct+i , where the i-period discount factor ∆i,t+1 is given by β i (u0t+i /u0t+1 ). The Þrst order condition implies9 ¸ · ´ ³ Pt+i θ −1 µ ∗¶ µ ¶ Et Ct+i µt+i Pt θ pt ¸ . · = ´ ³ P∞ i Pt θ−1 Pt+i θ−1 Ct+i Et i=0 ω ∆i,t+1 Pt (25) 1−θ . Pt1−θ = (1 − ω)(p∗t )1−θ + ωPt−1 (26) P∞ i i=0 ω ∆i,t+1 The aggregate retail price index is Equations (25) and (26) jointly determine p∗t and Pt . 9 See, for example, Sbordone (2002) for a more complete derivation. 17 3.3 The monetary authority To close the model, it is necessary to specify the behavior of the monetary authority. Most central banks implement monetary policy by controlling a short-term nominal rate of interest. Thus, one way to analyze the impact of monetary policy is to specify a rule for setting the nominal interest rate. For example, Taylor rules, in which the nominal rate responds to output and inßation, are commonly used to describe monetary policy. However, one cannot simply specify arbitrary, exogenous (stationary) rules for the nominal interest rate and compare the adjustments to policy shocks in a ßexible-price version of the model with a sticky-price version. The reason is that the nominal rate must react endogenously to inßation in the sticky price model to ensure the existence of a unique, stationary, rational expectations equilibrium (Svensson and Woodford 1999). The model may have multiple stationary equilibria if the nominal interest rate is a function solely of exogenous disturbances. It is common in recent sticky price models to require that the policy rule satisfy the “Taylor Principle” under which the nominal interest rate responds more than one-forone to changes in either actual or expected inßation, yet such rules would seem less relevant when prices are perfectly ßexible. Thus, to provide the most transparent comparisons between the ßexible-price and sticky-price versions of the model, I assume monetary policy can be represented by an exogenous process for the growth rate of the money supply.10 SpeciÞcally, let Θt denote the growth rate of the nominal money supply. It is assumed that Θt = (1 − ρm ) Θ̄ + ρm Θt−1 + φt Θ̄ (27) where φt is a serially uncorrelated, mean zero stochastic process. The steady-state gross inßation rate is equal to the average growth rate of money Θ̄. A similar stochastic process for the growth rate of the nominal money supply is employed by Cooley and Quadrini (1999) and Christiano, 10 Cooley and Quadrini (1999), Dostey and King (2001), and Christiano, Eichenbaum, and Evans (2001) are other recent examples of papers using money growth rate rules to evaluate the effects of monetary shocks. 18 Eichenbaum, and Evans (2001). 3.4 Equilibrium and the steady-state The Þnal equilibrium condition in the model requires that consumption equal aggregate household income which, in turn, is equal to production net of vacancy posting costs: x Ct = Yt = Qt − γVt = zt (1 − ρ )Nt Z ∞ ait f(a)da − γVt . (28) ãt An equilibrium in the model consists of an initial value of the nominal money stock Mt , the initial number of matchers Nt , and sequences for ãt , ρnt , ρt , qt , ut , Vt , ktw , ktf , Nt+1 , Yt , Ct , Pt , p∗t , µt , and Rt that satisfy equations (4), (7), (8), (11)—(13), (17), (19)—(26), and (28), and the central bank’s policy rule governing the evolution of the nominal money stock Mt given by (27). If prices are ßexible, the model consists of (4), (7), (8), (11)—(13), (17), (19)—(24), (27), p∗t = Pt and µt = θ/(θ − 1). The steady-state is the same for both the ßexible price and sticky price versions of the model. ¡ ¢ In a zero-inßation steady-state, (7) implies R = Θ̄/β. Using this in (11), q = A − β/Θ̄µ ã. Equation (17) can then be written as µ ¶ ¸ µ ¶ ·Z ā β β x w ã + (1 − ρ )(1 − ηk )β (a − ã) f(ai )dai = (A + h) Θ̄. µ µ ã (29) Rearranging this condition yields the following steady-state condition for ã: [1 − (1 − ρx )(1 − ηkw )β (1 − F (ã))] ã = G(ã), where G(ã) ≡ µ (A + h) Θ̄/β − (1 − ρx )(1 − ηkw )β hR ā ã (30) i af(ai )dai . Note that this condition also depends on the endogenous kw . Both the sides of equation (30) are continuous and increasing in ã. The support of a is (0, ā]. For a given kw , G(ā) = µ (A + h) Θ̄/β = µ (A + h) R, while the left side of (30) is equal to ā. Evaluated at zero, the left side of (30) is zero and the right side is equal 19 to G(0) = µ (A + h) R − (1 − ρx )(1 − ηkw )β as the expected value of a is equal to 1. A unique solution ã(k w ) as a function of kw exists as long as G(0) > 0 and G(ā) < ā, or (1 − ρx )(1 − ηkw )β < µ (A + h) R < ā. Assume A (the disutility of work) and ā are such that this holds. Then ρ = ρx + (1 − ρx )F [ã(k w )] and the steady-state values of N, u, V , kf , kw , and C are given by the solution to u = 1 − (1 − ρ)N ρN = m(u, V ) kf = m(u, V ) V m(u, V ) u ¶ µ γ(1 − ηkw ) β ã(k w ) =A+h− (1 − η)k f Θ̄µ # ¶ "Z ∞ µ 1 C = (1 − ρ)N af(a)da − γV . 1 − F [ã(kw )] ã(kw ) kw = The steady-state markup is equal to θ/(θ−1), while real money balances are equal to consumption. 4 Simulations The recent literature in monetary economics has used simulations extensively to study the dynamic properties of stochastic general equilibrium models. To cite just a few recent examples, Dotsey, King and Wolman (1999), Dotsey and King (2001), Fuhrer (2000), McCallum and Nelson (1999), Jensen (2001), Walsh (2002). In keeping with that literature, the model of the previous section is expressed in terms of percentage deviations and linearized around the steady state. The basic approach is described in Uhlig (1999), and the model solution and its properties are obtained using the “toolkit” of programs written by Harald Uhlig.11 11 Uhlig’s programs are available at http://cwis.kub.nl/˜few5/center/STAFF/uhlig/toolkit.dir/toolkit.htm. 20 In solving the model, functional forms of the utility function and the aggregate matching function need to be speciÞed. The utility function for the composite consumption good is assumed to be of isoelastic form: u(Ct ) = Ct1−σ ; 1−σ σ > 0, where σ is the coefficient of relative risk aversion. The matching function is taken to be m(ut , Vt ) = µuat Vtξ , 0 < a < 1, 0 < ξ < 1. (31) A Cobb-Douglas speciÞcation of the matching function is common, and is the form used by Cooley and Quadrini (1999). With constant returns to scale, a + ξ = 1.12 Equation (31) does allow for the matching function to display increasing or decreasing returns to scale if a + ξ 6= 1.13 Let ẑt denote the log deviation from steady-state of the aggregate productivity disturbance. 12 Blanchard and Diamond (1989, 1990) provide evidence that the aggregate matching function displays constant returns to scale. 13 den Haan, et. al (2000) assume the matching function displays constant returns to scale and is of the form m(ut , Vt ) = ut Vt [ust + Vts ]1/s . As den Haan, et. al., note, this functional form ensures the probabilities k w and kf are bounded between 0 and 1. This speciÞcations and the Cobb-Douglas lead to similar equilibrium conditions when the model is linearized around the steady-state. For example, the speciÞcation of den Haan, et. al. implies ktw = V / [ust + Vts ]1/s which is approximated by ³ ´ ³ ´s k̂tw = [1 − (kw )s ] V̂t − k f ût = [1 − (kw )s ] V̂t − ût , where x̂ denotes the percentage deviation around the steady state. With constant returns to scale, (31) implies Vt1−a which leads to ktw = µua−1 t ´ ³ k̂tw = (1 − a) V̂t − ût . When a = (kw )s , the two speciÞcations produce identical dynamic simulations. In the calibration used by den Haan, et. al., s = 1.27 and k w = 0.45, implying that (kw )s = 0.3627. Cooley and Quadrini set a = 0.4 in their base calibration, implying the two speciÞcations are essentially identical. 21 As is standard in the literature, ẑt is assumed to follow an AR(1) process with innovation εt : ẑt = ρz ẑt−1 + εt . 4.1 (32) The linearized model Given the assumed functional forms, the model is linearized around the steady-state.14 Let x̂t denote the percentage deviation of a variable Xt around its steady-state value. The linearized model consists of equation (32) for the aggregate productivity disturbance and the following thirteen equations: • The policy rule for nominal money growth, equation (27): Θ̂t = ρm Θ̂t−1 + φ̂t ; (33) • The cash-in-advance constraint (in Þrst difference form), equation (8): Θ̂t = ŷt − ŷt−1 + π̂t ; (34) • The evolution of the number of matches, equation (23): n̂t+1 = ϕϕ̂t + ϕn̂t + µ vkf N ¶ v̂t + µ vkf N ¶ k̂tf ; (35) • The endogenous job destruction margin, equation (11): ât = r̂t + µ̂t − µ µRq ã ¶ q̂t − ẑt ; (36) • The survival rate ϕt = 1 − ρt , using equation (12): ϕ̂t = − 14 µ ρn 1 − ρn ¶ eF,a ât ; Details are in an appendix available at http://econ.ucsc.edu/˜walshc/. 22 (37) • The number of unemployed job seekers, equation (20): µ ϕN ût = − u ¶ n̂t − µ ϕN u ¶ ϕ̂t ; (38) • The probability a vacancy is Þlled, equation (22): k̂tf = aût − (1 − ξ)v̂t ; (39) • The equality of Þrms Þlling vacancies and workers Þnding matches: v̂t + k̂tf = ût + k̂tw ; (40) • The job posting condition, equation (19): k̂tf µ ηkw =− 1 − ηkw ¶ k̂tw − µ ¶ q q̂t ; q+h (41) • The output equation (28): ŷt = µ Q Y ¶ (eH,a ât + n̂t + zt ) − µ γV Y ¶ v̂t ; (42) • The Euler condition from the household’s optimization problem equation (7): µ ¶ µ ¶ 1 1 0 = Et ŷt+1 − ŷt − r̂t + Et π̂t+1 ; σ σ (43) • The inßation equation from the retail Þrms’ pricing decisions, obtained from equations (25) and (26): 0 = βEt π̂t+1 − π̂t − κµ̂t (44) • The present value condition for matches, equation (17): q̂t ¸ · ¡ ¢ (1 − ηkw )βϕ(q − A) Et ϕ̂t+1 = AB eH,a Et ât+1 − Et µ̂t+1 − Et r̂t+1 + Et zt+1 + q ¶ µ µ µ ¶ ¶ q+h q+h ηkw − (r̂t − Et π̂t+1 ) − k̂tw + (1 − ηk w )βϕEt q̂t+1 (45) w q 1 − ηk q 23 In these conditions, eF,a is the elasticity of the cumulative density function of a, H(ã) ≡ Et (a | a ≥ ã), eH,a is the elasticity of H(ã) with respect to ã, evaluated at the steady-state, and AB = (1 − ηkw )βH(ã) . µRq Note that, while the distribution of the idiosyncratic shock a appears in the form of the function H(ã) and the elasticities eF,a and eH,a , the actual realizations of the ait shocks average out across matches, so they do not appear in the equilibrium conditions. In the ßexible-price version of the model, the mark-up µ is constant, equal to θ/(θ − 1), and equation (44) is dropped. Let xt = (r̂t , n̂t , ϕ̂t , ŷt , ât , v̂t , k̂tf , k̂tw , q̂, ût , µ̂t , π̂t )0 be the vector of endogenous variables, and let ψt = (Θ̂t , zt )0 be the vector of exogenous aggregate disturbances. Equations (34) - (45) can be written as AEt xt+1 + Bxt + CEt ψt+1 + Dψt = 0, (46) where and ψt+1 = Nψ t + χt+1 , where χt = (φt , εt )0 is the innovation vector. If an equilibrium solution to this system of equations exists, it takes the form of stable laws of motion given by xt = P xt−1 + Qψt . Uhlig (1999) provides a complete discussion of the methods used to solve systems such as (46). 4.2 Calibration The model is characterized by Þve sets of parameters — those describing household preferences, those describing the aggregate matching function, those characterizing the degree of price rigidity 24 at the retail level, those specifying the behavior of the growth rate of money, and those characterizing the stochastic distribution of the exogenous shocks. Parameter values are chosen to be largely consistent with those shown to match U. S. data in nonmonetary models. The baseline parameters values are shown in Table 1 and discussed in this section. Preferences Household preferences are characterized by the parameters β, σ, θ, h, and A. The discount rate and the coefficient of relative risk aversion appear in standard DSGE models. Choosing the time period to correspond to a calendar quarter, β is set equal to 0.989, implying a steady-state real annual return of 4.5%. A value σ = 2 is chosen for the coefficient of relative risk aversion, implying greater risk aversion than log utility. The parameter θ determines the elasticity of demand for the differentiated retail goods. This elasticity in turn determines the markup µ. This markup is set equal to 1.1, corresponding to a value of θ = 11. The value of home production while unemployed, h, is set equal to zero. Finally, A is determined by the steady-state condition (11) once q is found from the labor market calibrations. Matching and the labor market den Haan, et. al. (2000) set the steady-state separation rate ρs equal to 0.1. This is based on Hall’s conclusion that “around 8 or 10 percent of workers separate from their employer each quarter” (Hall, 1995, p.235) and the Davis, Haltiwanger, and Schuh (1996) Þnding of about an 11 percent quarterly separation rate. This is higher that the 0.07 value adopted by Merz (1995), but lower than the 0.15 used by Andolfatto (1996). Given a value of 10 percent for ρs , den Haan, et. al. use evidence on permanent job destruction to calibrate the exogenous separation probability ρx as 0.068. I use this value for the baseline simulations. These values for ρx and ρs imply an endogenous separation probability ρn of 0.0343. From this value, and the assumed distribution function for the match speciÞc productivity shock, the steady-state value of the cut-off productivity realization ã can be derived. I assume ã is log 25 normally distributed with standard deviation 0.15; this is somewhat higher than the value used by den Haan, et. al. (they set this standard deviation equal to 0.1). For the Cobb-Douglas matching function (31), I follow Cooley and Quadrini and set a = 0.4 and ξ = 0.6 based on the estimates of Blanchard and Diamond (1989). Both Cooley and Quadrini and den Haan, et. al. Þx kf = 0.7. Cooley and Quadrini cite Cole and Rogerson (1996) to set the average duration of unemployment at 1.67 quarters, which implies kw = 0.6. I set N = 0.94, implying a steady-state unemployment rate of 0.06 and a value of 0.154 for u, the steady-state number of workers searching each period. The steady-state value of V is 0.132.15 I follow den Haan, et. al. and set the share of the match surplus that the worker receives, η, equal to 0.5. Finally, the steady-state value of a match q, is obtained from (19). Finally, (17) is used to calibrate γ. Price rigidity The degree of nominal rigidity is determined by ω, the fraction of Þrms each period that do not adjust their price. Empirical estimates of forward-looking price setting models of the type employed here suggest prices are Þxed on the order of 9 months. This would imply a value of 0.67 for ω (Sborbone 2001) and this is taken as the baseline value. However, evidence based on BLS data on price changes suggestions a median time between changes of 6 months (Bils and Klenow 2002); this would imply a value of 0.5 for ω. In the simulations, various values for ω ranging from ω = 0 (price ßexibility) to ω = 0.9 are used to explore the impact of nominal rigidity on the dynamic response of the economy. 15 den Haan, et. al (2000) chose a value of 1.27 for the parameter s that appears in their matching function (see footnote 9). This, together with the steady-state value of u implies a value of 0.0993 for V and 0.4515 for kw . This last value is close to the value of 0.45 used by den Haan, et. al. and is signiÞcantly lower than the 0.6 employed by Cooley and Quadrini. 26 Policy The process for the growth rate of M is calibrated by estimating an AR(1) process for U.S. M2 using quarterly data over the period 1965:3 - 2001:4. This yielded ρm = 0.73 and the standard deviation of φ is set to 0.00624.16 . Shocks In addition to the money growth rate shock, there are two other exogenous disturbances in the model: the match speciÞc shock and the aggregate productivity shock. The speciÞcation of the distribution of the match speciÞc shock has already been discussed. The log aggregate productivity shock was given in equation (32). Standard calibrations for the productivity disturbance process in the RBC literature are ρz = 0.95 and σε = 0.007 (see Cooley and Prescott 1995). This calibration is, however, based on models in which the productivity disturbance is the sole source of ßuctuations. In the present model there are nominal money growth shocks in addition to the aggregate productivity shocks. Thus, a smaller value of σε is appropriate for matching output ßuctuations. Cooley and Quadrini choose σε so the model’s prediction for the standard deviation of output matches the standard deviation of U. S. real GDP . For their baseline parameter values, this implies σ ε = 0.0033 when prices are assumed to be ßexible. I employ this as the baseline value for σε , together with ρz = 0.95. 4.3 4.3.1 Results Flexible prices Column (1) of Table 2 presents standard deviations based on U. S. data for 1959:1 - 1996:4; column (2) expresses these relative to the standard deviation of output. These values are taken from den Haan, et. al. and Cooley and Quadrini. Columns (3) and (4) report the results from the ßexible price model. The model based on productivity and money growth rate shocks implies 16 Cooley and Quadrini (1999) set the standard deviation of φ is set to 0.00623 and ρm = 0.49. Christiano, Eichenbaum, and Evans (2001) set ρm = 0.5. 27 less output variability than is observed in the data. The model implies slightly less employment volatility relative to the standard deviation of output than found in the data (0.56 compared to 0.62), and it also reverses the relative volatility of job creation and destruction. In the data, destruction exhibits more volatility, while in the ßexible price model job creation has a larger standard deviation. Greater volatility in job creation than destruction is also a property of the real model studied by den Haan, Ramey and Watson (2000). Perhaps not surprisingly, the ßexible price model implies greater inßation volatility than is evident in the U. S. data. Figure 2 illustrates the impact of a money growth rate shock when prices are ßexible. Nominal money growth rate ßuctuations have very small real effects in a basic CIA model with a neoclassical speciÞcation of the labor market and ßexible prices (for example, see Walsh 1998, Chapter 3). In those models, higher inßation taxes consumption and leads households to reduce their labor supply. In the present model, labor supply is completely inelastic, so this channel is absence. Instead, ßuctuations in the nominal interest rate alter the job destruction margin and have real effects even when prices are ßexible. Figure 2 shows that a 1 percentage point money growth rate shock (which raises the nominal interest rate — see panel b) causes a fall in employment and output (panel a). For a given level of employment at the beginning of each period, fewer worker-Þrm matches actually remain together to produce when the nominal interest rate increases. The contraction in production at the wholesale level increases wholesale prices. While a Þxed markup of retail over wholesale prices, inßation spikes as the retail price level jumps (panel b). Job creation and job destruction initially move in opposite directions, as the number of endogenous separations rise and fewer jobs are created (panel c). However, job creation rebounds after one period. The job Þnding probability falls for workers and rises for Þrms (panel d), reßecting the rise in the number of searching workers relative to vacancies (panel e). The results in Figure 2 are qualitatively similar to those reported by Cooley and Quadrini 28 (1999) for a contractionary money shock. The key difference is the presence of a liquidity effect in the Cooley-Quadrini model and the lack of one in the present model. Thus, a negative money shock raises the nominal interest rate in their model while a positive money shock does so in the present model. 4.3.2 Sticky prices When prices are sticky, monetary disturbances have traditional demand effects as well as the supply-side effects. Columns (5) — (7) of Table 2 show the impact of increasing price rigidity on the relative variability of key variables. As expected, the relative standard deviation of inßation falls as the degree of price stickiness increases. In other sticky price models, ω = 0.67 is a common parameterization. It implies prices are Þxed, on average, for 9 months. This degree of price stickiness does provide a better match between the model’s predictions for the standard deviation of inßation relative to that of output. When ω = 0.67, the standard deviation of job destruction is increased relative to that of job creation, moving these statistics closer to the values found in U. S. data. Figure 3 shows how important price stickiness is in affecting the model’s predictions for the response to a money supply disturbance. In contrast to Figure 2, a positive money shock now increases real output and employment. There is a jump in job creation and a signiÞcant drop in job destruction. As a consequence, the number of workers searching for matches falls and then gradually returns to steady state. In this CIA model, there is no liquidity effect — the nominal interest rate rises immediately in reaction to the positive shock to money growth. As expected, the inßation impact of the money growth shock is smaller when prices are sticky. Two effects operate on inßation. First, the expansion in output induced by the rise in demand increases wholesale prices and leads to a rise in retail inßation. Second, the associated rise in the nominal interest 29 rate (reßecting the rise in expected inßation) reduces wholesale output and also contributes to an increase in wholesale prices and retail inßation. If the cost channel is eliminated, the standard deviation of inßation relative to output falls from 0.84 to 0.54, a 35% decline. For the baseline parameter values used for Figure 3, the immediate impact effect on output of a 1 percentage point rise in the money growth rate is 0.20. This increases to a peak effect of 0.44, with this peak occurring three periods after the initial shock. The total multiplier is 5.49 and the mean lag is 8.37 periods. There are several elements of the model that account for the dynamic pattern displayed by this output response. First, the degree of nominal price stickiness inßuences the model dynamics. Second, the money growth rate process itself is serially correlated, and this accounts for some of the persistence in the output effects seen in the Figure 3. Third, the nominal interest rate affects job separation and creation, and fourth, the labor market matching process affects the evolution of employment after a shock. Insight into how the dynamic response of output is affected by the various aspects of the model speciÞcation is provided by Table 3. Each row of that table represents a different set of parameter values. Row (1) reports results for the baseline parameter values. Row (2) has a lower degree of price rigidity than in the baseline parameter set (ω = 0.50 versus 0.67). In addition, ρm is set equal to 0 so that the growth rate of money is serially uncorrelated, and the direct cost channel of nominal interest rates is eliminated. Finally, both kf and k w are increased to 0.95 to capture a matching process in which workers and Þrms are able to Þnd new matches much more quickly. As a result of these changes, the impact effect of a money growth shock on output is actually increased to 0.39 but this impact effect is also the peak effect as output starts declining in the period immediately following the shock. The response no longer displays the typical hump-shape seen in estimated VARs. The total multiplier falls dramatically to 1.03 from 5.49 for the baseline parameter set (row 1), and the mean lag falls to 1.60 periods. Row (3) of the table adds back in 30 the supply channel of the nominal interest rate. This increases the impact multiplier on output and the total impact, but it has only minor effects on the dynamic pattern of output’s response. The maximum effect still occurs in the period of the shock, with output declining thereafter. Row (4) increase the degree of price rigidity by raising ω from 0.5 to 0.67. This, like the supply channel, increases the impact of a money shock but has little impact on the shape of the response. The maximum effect occurs immediately, although the mean lag and median lags are both increased. Row (5) returns kf and kw to their baseline values, so row (5) serves to illustrate the impact of more sluggish labor market adjustment. While the impact effect of a money shock is not signiÞcantly changed, the median lag is doubled to 4 periods, while the mean lag rises from 3.45 to 5.60 periods. Labor market search stretches out the response, adding to the overall persistence due to a monetary shock, but it does not induce the hump-shape response seen in VARs. Row (5) differs from the baseline parameter set only in setting ρm = 0. If ρm is increased to its baseline value of 0.73, the impact is quite dramatic. To facilitate the comparison, row (6) repeats row (1), the outcomes for the baseline parameters. Comparing rows (5) and (6) shows that serially correlated money shocks reduce the impact effects of a money growth shock on output. While the impact effect is smaller, the peak effect is now both larger and delayed, occurring 3 periods after the shock. The total effect also rises, and the mean lag increases to over 7 periods. Thus, the persistence in the money supply growth process appears to have an important effect on the response to a monetary shock, and it is serial correlation in the money shock that generates the hump-shaped response of output. Rows (1) - (6) employed a value of 0.5 for the share parameter η. This is the value used by den Haan, Ramey and Watson (2000) and is common in other applications of the MortensenPissarides framework. In contrast, Cooley and Quadrini (1999) set η equal to 0.01 and 0.1 in 31 their alternative model economies. Row (7) of Table 4 sets η equal to 0.1. This has a major impact on the model’s response to a money growth rate shock. The impact effect rises from 0.20 in row (1) to 0.27 in row (7), the peak effect increases from 0.44 to 0.75, and the total impact rises from 5.49 to 15.20. The peak impact now occurs Þve periods after the shock, and the mean lag rises to over 10 periods. The results reported in Table 3 help to identify the key parameters affecting both the magnitude and the persistence of the response of output. The magnitude of the total impact depends importantly on the values of ω, ρm , ktw , ktf , and η. The persistence displayed by output in response to a nominal money growth shock, as measured by the median lag, is affected by the degree of price rigidity, the speed of labor market adjustment, and the degree of persistence in the money growth process itself. The role of ω, the degree of price rigidity, is not surprising. Nominal rigidities have long been viewed as the key explanation for sizable real effects of nominal disturbances. The ease with which employment matches are formed, as reßected in the values of kf and kw , also have a major effect on the size of the output effect of a money growth shock. The values of ω, ktw , and ktf affect mainly the total impact multiplier but not the shape of the output response. The hump-shape response of output is determined by the degree of serial correlation in the money growth rate (ρm ) and η (the labor share parameter). The role of η is unexpected, in part because this share parameter plays no role in traditional dynamic general equilibrium models. Cooley and Quadrini Þnd that the volatility of output and employment increases with η in their ßexible-price model. This is no longer the case when prices are sticky, and increasing the share of the surplus going to the Þrm (a reduction in η) increases overall volatility.17 As η goes to zero, equation 19 implies the present value of a match becomes less sensitive to kw and more sensitive to kf . As a consequence, a contractionary monetary 17 For example, decreasing η from 0.5 to 0.1 increases the standard deviation of output from 0.73 to 1.64. De- creasing it further to 0.01 raises the standard deviation to 2.15. 32 policy shock that reduces employment and increases the number of searching workers, increases kf and reduces the continuation value of a match. Becuase Þrms are able to Þnd new matches more easily, the value of continuing in an existing match falls. This raises the probability of endogenous separations, and, as a result, the economy takes longer to return to its steady-state level of employment. Thus, a decline in η leads to greater persistence in the real effects of a monetary policy shock. Table 3 serves to separate the inßuence on output dynamics of various aspects of the model. To assess the “match” between the model and the data, it is useful to compare impulse responses directly. Figure 4 shows the response of output to a money growth shock estimated from US data and the response obtained from the model simulations.18 Under the baseline parameter values, the peak impact of a money shock on output is too large (0.44 — see row 1 of Table 3 — versus 0.23 from the estimated VAR). A better match is obtained with ω = 0.62, implying slightly more price ßexibility than with the baseline parameter values, and this value of ω is used to generate Figure 4. The model captures the basic hump-shaped response, but there are several aspects of the estimated impulse response that are not captured by the model. The estimated output dynamics are much more complex than those exhibited by the model. The model implies output reaches a peak and then declines smoothly to the steady-state; the estimate VAR shows output reaching its peak and then falling below steady-state. The impulse response function from the estimated VAR also shows output peaking Þve periods after the shock versus three periods in the case of the model simulation.19 18 The estimated response for output is the same as in Figure 1. 19 However, it should be noted that the VAR impulse responses are obtain using a Choleski decomposition with money ordered last. This means that the VAR restricts money growth shocks to have no impact on output until at least one period after the shock. 33 5 Conclusions Dynamic stochastic general equilibrium models are well suited for studying the interactions of real and nominal stickiness, and this chapter has examine the role of the labor market matching function and price stickiness in affecting the way the economy responds to money growth shocks. The model incorporated both an aggregate labor market matching function and price stickiness by incorporating a wholesale production sector, in which matched Þrms and workers produce output, and a retail sector characterized by monopolistic competition and sticky prices. Money growth shocks led to employment and output responses that were hump shaped, just as the empirical evidence suggests. Replacing the Walrasian labor market with a simple model of labor market search appears to be a promising avenue to pursue in understanding the dynamic adjustment of the economy to monetary policy shocks. In the sticky price version of the model, monetary shocks had both demand and supply effects. Increases in money growth lead to increases in consumption and the nominal interest rate (via the expected inßation channel). The rise in the nominal rate reduced production among wholesale Þrms by altering the job destruction margin. By pushing up wholesale prices relative to retail prices, positive nominal interest rate movements also affected retail price inßation. This supply, or cost channel, effect reinforced the inßationary impact of a money growth rate increase. As in tradition models, the response of output to monetary shocks depended on the degree of nominal price stickiness. It also depended on the degree of persistence displayed by the money process itself. Perhaps more interestingly, the dynamic behavior of the model economy was sensitive to the parameter that determined how a match surplus was divided between the worker and the Þrm. An interesting direction for future research will be to explore further the implications of this share parameter for economic dynamics. Hosios (1990) has shown that the relationship between the share parameter and the elasticity of the vacancy matching probability with respect 34 to labor market tightness (measured by the ratio of vacancies to searchers) is critical for determining the efficiency of the steady-state unemployment rate (Pissarides, 2000, Chapter 8). While Friedman argued that the optimal rate of inßation is the rate that produces a zero nominal interest rate, Cooley and Quadrini (2000) argue that the optimal level of the nominal interest rate may be positive if steady-state unemployment is inefficiently low.20 The positive nominal rate increases job destruction and raises the average unemployment rate. Monetary policy has been represented by a process for the growth rate of the nominal money supply. It is common in much of the recent monetary literature to represent policy by a rule for the nominal interest rate. An extension of the present model would be to replace the money growth rate rule with a nominal interest rate rule. A policy shock that raised the nominal interest rate would reduce output through a traditional demand channel and through its effect on job destruction. These channels, however, would have countervailing effects on inßation. The negative impact on wholesale output would raise wholesale prices relative to retail prices and lead to an increase in retail price inßation. The demand side reduction in consumption would put downward pressure on inßation. These opposing effects could account for the small initial net impact on inßation of a nominal interest rate shock that is observed in the data. 20 This occurs in the model of this chapter when η < α as in the calibration of Cooley and Quadrini (1999). 35 References [1] Andolfatto, D., “Business Cycles and Labor-Market Search,” American Economic Review, 86 (1), March 1996, 112-132. [2] Ball, L. and D. Romer, “Real Rigidities and the Nonneutrality of Money,” Review of Economic Studies, 57, April 1990, 183-203. [3] Barth, M. J. III and V. A. Ramey, “The Cost Channel of Monetary Transmission,” NBER Macroeconomic Annual 2001, Cambridge, MA: MIT Press, 199-239. [4] Bernanke, B., M. Gertler, and S. 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[48] Yun, “Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles,” Journal of Monetary Economics, 37 (2), April 1996, 345-370. 40 Table 1 Calibrated Parameters Preferences β σ θ h 0.989 2 11 0 Labor Market ρs ρx a ξ kf kw N η 0.10 0.068 0.4 0.6 0.7 0.6 0.94 0.5 Price rigidity ω 0.67 Policy ρm 0.73 σφ Θ̄ 0.00624 1.01 Productivity σε ρz 0.0033 0.95 41 σa 0.15 Table 2 Business cycle properties U. S. (1) (2) Model outcomes (3) (4) (5) (6) (7) ω=0 ω=0 ω = 0.33 ω = 0.67 ω = 0.9 σi σi /σy σi σi /σy σi /σy σ i /σy σi /σy output 1.60 1.00 1.18 1.00 1.00 1.00 1.00 employment 0.99 0.62 0.66 0.56 0.53 0.57 0.78 job creation rate 4.62 2.89 4.77 4.05 3.93 4.93 4.26 job destruction rate 6.81 4.26 4.05 3.44 3.34 5.97 7.10 inßation 0.56 0.35 1.05 0.89 0.96 0.84 0.14 nominal interest rate 1.27 0.79 0.41 0.35 0.48 0.87 0.62 Variable 42 Table 3 Effects of parameter variation Response of output cost Multipliers Lags ω kf kw ρm η channel Impact Peak Total Peak Median Mean 1. 0.67 0.70 0.60 0.73 0.5 Yes 0.20 0.44 5.49 3 7 8.37 2. 0.50 0.95 0.95 0 0.5 No 0.39 0.39 1.03 0 1 1.60 3. 0.50 0.95 0.95 0 0.5 Yes 0.43 0.43 1.43 0 1 2.31 4. 0.67 0.95 0.95 0 0.5 Yes 0.63 0.63 2.83 0 2 3.45 5. 0.67 0.70 0.60 0 0.5 Yes 0.68 0.68 4.55 0 4 5.60 6. 0.67 0.70 0.60 0.73 0.5 Yes 0.20 0.44 5.49 3 7 8.37 7. 0.67 0.70 0.60 0.73 0.1 Yes 0.27 0.75 13.05 5 13 15.20 43 0.8 0.7 0.6 0.5 0.4 0.3 In fla t io n 0.2 O utput 0.1 0 -0 . 1 -0 . 2 0 2 4 Ye 6a rs 8 10 12 Figure 1: Impulse responses to a money growth rate shock (estimated VAR) 0 1.5 1 employment % deviation % deviation -0.2 -0.4 -0.6 -0.8 -1 output 0 1 2 3 4 (a) Years after shock 5 nominal rate 0 -0.5 -1 6 4 1 % deviation 2 % deviation 0 1 2 3 4 (b) Years after shock 5 6 0 1 2 3 4 (d) Years after shock 5 6 2 destruction 0 -2 -4 -1 π 0.5 creation 0 1 0 -1 -2 2 3 4 (c) Years after shock 5 6 5 6 kf -3 -1 kw 6 % deviation 4 searchers 2 0 vacancies -2 -1 0 1 2 3 4 (e) Years after shock Figure 2: Effects of a money growth shock with ßexible prices 44 0.5 1 % deviation % deviation output 0.4 0.3 0.2 employment 0.5 π 0 nominal rate 0.1 0 -1 0 1 2 3 4 5 -0.5 -1 6 0 1 (a) Years after shock 1 3 4 5 6 4 5 6 2 0 % deviation -1 -2 -3 0 w k f 0 -1 destruction -4 -1 k 1 creation % deviation 2 (b) Years after shock 1 2 3 4 5 -2 -1 6 0 (c) Years after shock 1 2 3 (d) Years after shock 1 vacancies % deviation 0 -1 -2 searchers -3 -1 0 1 2 3 4 5 6 (e) Years after shock Figure 3: Effects of a money growth rate shock with sticky prices (ω = 0.67) 0.3 0.25 0.2 0.15 % deviation 0.1 M odel sim ulation 0.05 Estim ated VA R 0 -0.05 -0.1 -0.15 -0.2 0 2 4 6 Years after shock 8 10 12 Figure 4: Output responses: VAR and model simulations 45