Essays on Empirical Dynamic Games and Imperfect Information

E SSAYS ON E MPIRICAL D YNAMIC G AMES AND I MPERFECT I NFORMATION by Arvind Magesan A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Economics University of Toronto c Copyright by Arvind Nathan Magesan (2011) Essays on Empirical Dynamic Games and Imperfect Information Arvind Magesan Doctor of Philosophy Department of Economics University of Toronto 2011 Abstract This thesis collects three papers that study applied problems in economics dealing with dynamic strategic behavior and imperfect information. In the first chapter I study the relationship between participation in United Nations Human Rights Treaties (HRT), foreign aid receipts and domestic human rights institutions. I provide empirical evidence that countries with relatively high HRT participation rates receive more foreign aid. Further, countries with high quality institutions are more likely to participate in HRTs, but that high levels of HRT participation leads to a decline in the quality of domestic human rights institutions. Based on these findings, I propose and estimate a dynamic game of HRT ratification. The estimates show that economic factors play an important role in HRT ratification and that the ratification costs countries incur vary significantly across treaties and country regime types. I use the estimated model to evaluate the effects of counterfactual policies on HRT ratification decisions, human rights behavior, and the distribution of foreign aid. The second chapter considers environmental regulation under imperfect information and political constraints. We compare the value of two types of information to a regulator: the cost of pollution and the profitability of firms in the economy. We find that in environments where small increases in the losses to regulated firms greatly affect the regulator’s ability to implement the ii policy, it is most valuable to learn the types of firms, while it is most valuable to learn the cost of pollution when small increases in losses are relatively ineffectual. The third chapter deals with the identification and estimation of dynamic games when players maximize expected payoffs given beliefs about other players’ actions, but their beliefs may not be in equilibrium. First, we derive conditions for point-identification of structural parameters and players’ beliefs, and propose a simple two-step estimation method and sequential generalization of the method that improves its asymptotic and finite sample properties. We also present a procedure for testing the null hypothesis of equilibrium beliefs. Finally, we illustrate our model and methods with an application of a dynamic game of store location by retail chains. iii Acknowledgement I thank the three members of my supervisory committee, Victor Aguirregabiria, Matthew Turner and Carlos Serrano, without whom these pages would be completely empty. I thank Victor for his persistent support and optimism at every step of the way, and most importantly for teaching me everything I know about conducting and presenting empirical research in economics. I am forever in debt. I thank Matthew for his encouragement and for teaching me how to properly write a theory paper in economics. Finally, I thank Carlos for his many insightful comments and criticisms which helped to shape the first chapter. I also thank the rest of my thesis examination committee, Jordi Mondria, Junichi Suzuki and especially Matt Shum, for useful comments. I thank those who taught and supported me during my time at the University of Toronto, especially Li, Hao and Martin Osborne, from whom I have learned a very great deal. I thank Ben Amsel, Branko Boskovic, David Byrne, Sacha Kapoor and Ali Emre Konukoǧlu for their help, support and friendship and I also gratefully acknowledge financial support from the University of Toronto and The Royal Bank Graduate Fellowship in Public and Economic Policy. In addition to those mentioned above, each chapter has benefitted from the comments of several individuals. “Human Rights Treaty Ratification of Aid Receiving Countries" has benefitted from the comments of Gustavo Bobonis, Patrick Francois, Ken Jackson, Eik Swee, Wang, Hui, and participants of the 2010 EARIE Conference in Istanbul, the 2009 Meeting of the Canadian Economics Association as well as seminar audiences at the Universities of Calgary, Houston, New South Wales, Sydney, Toronto and Western Ontario. “The Value of Information In Regulation" has benefitted from the comments of Patrick Francois, Sumon Majumdar as well as participants of the 2008 Canadian Public Economics Group Meetings and the 2009 Camp Resources Meetings. “Identification and Estimation of Dynamic Games when Players’ Beliefs Are Not in Equilibrium" has benefitted from the comments of Dan Bernhardt, Sridhar Moorthy, Katsumi Shimotsu, and seminar participants at Carlos III-Madrid, New York University, UI Urbana-Champaign, the Canadian Econometric Study Group in Ottawa, and the Econometric Society World Congress in Shanghai. iv I thank my parents, Murugesapillai Magesan and Sugi Magesan for a lifetime of support and encouragement, and for always pushing me to expect more from myself. Lastly I thank Deniz Çorbacı. Deniz, hem bütün kötü günlerin hem de bütün güzel günlerin her saniyesi benim yanimdaydin, sensiz bunu yapamazdim. Verdiǧin sonsuz destek ve yardim için sana nasil teşekkur edeceǧimi bilemiyorum. v Contents List of Tables ix List of Figures x 1 Human Rights Treaty Ratification of Aid Receiving Countries 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Aid, Treaty Ratification and Human Rights . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 1.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5 Estimation of the Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.6 Counterfactual Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.8.1 Treaty-Specific Cost Heterogeneity and Observable Treaty Characteristics . . 58 1.8.2 Representation of Choice Probabilities . . . . . . . . . . . . . . . . . . . . . . . 63 The Value of Information in Regulation 66 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4 Planner’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.5 Optimal taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 vi 3 2.6 The Value of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Identification and Estimation of Dynamic Games When Players’ Beliefs are not in Equilibrium 100 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.1 Basic Framework and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.4 Estimation and Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5 3.6 3.4.1 Estimation with nonparametric payoff function . . . . . . . . . . . . . . . . . 118 3.4.2 Estimation with parametric payoff function . . . . . . . . . . . . . . . . . . . . 120 3.4.3 Test of Equilibrium Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.5.1 Data and descriptive evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.5.3 Estimation of the structural model . . . . . . . . . . . . . . . . . . . . . . . . . 135 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References 141 vii List of Tables 1.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Sample Quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Foreign Aid Receipts: Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Human Rights-Democracy Episodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 Number of Different Regimes Visited in Country History . . . . . . . . . . . . . . . . 21 1.6 Human Rights: Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Treaty Capital: Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8 Treaty Decision: Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.9 Structural Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.10 Structural Parameter Estimates : Treaty Specific Costs . . . . . . . . . . . . . . . . . . 48 1.11 Structural Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.12 Human Rights Treaties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.1 Distribution of probability over states . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 Information partition Ωθ induced by information about firms . . . . . . . . . . . . . 75 2.3 Information partition Ωc induced by cost information . . . . . . . . . . . . . . . . . . 77 2.4 Optimal taxes in each possible state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.5 Optimal taxes when the planner knows firm types only . . . . . . . . . . . . . . . . . 80 2.6 Optimal taxes when social cost is known . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.7 Optimal taxes in each possible state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.8 Optimal taxes when planner knows who has political power . . . . . . . . . . . . . . 98 2.9 Optimal taxes when planner knows social cost of pollution . . . . . . . . . . . . . . . 98 2.10 Optimal taxes when planner knows firm types . . . . . . . . . . . . . . . . . . . . . . 99 viii 2.11 Optimal taxes in each possible state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.12 Optimal taxes when profit is known . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.13 Optimal taxes when the planner knows firm cost types only . . . . . . . . . . . . . . 99 3.1 Descriptive Statistics on Local Markets (Year 1991) . . . . . . . . . . . . . . . . . . . . . 125 3.2 Evolution of the Number of Stores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3 Transition Probability Matrix for Market Structure . . . . . . . . . . . . . . . . . . . . . 127 3.4 Reduced Form Probits for the Decision to Open a Store . . . . . . . . . . . . . . . . . . 127 3.5 Estimates from Myopic Game of Entry for McDonalds and Burger King Under the Assumption of Equilibrium Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.6 Estimates from Dynamic Game of Entry for McDonalds and Burger King Under the Assumption of Equilibrium Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.7 Estimated Bias in BK Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 ix List of Figures 1.1 Argentina and Chile Treaty Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Kenya and Chad Treaty Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Uganda and Chad Treaty Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Ratification Cost as a Function of Human Rights Institutions . . . . . . . . . . . . . . 45 1.5 Frequency Distribution of Country Specific Treaty Costs . . . . . . . . . . . . . . . . 51 1.6 Number of Ratified Treaties Under Factual and Counterfactual Equilibria . . . . . . 54 1.7 K Under Factual and Counterfactual Equilibria 1.8 Number of Ratified Treaties Under Counterfactual Equilibria with No Heterogeneity 55 1.9 Argentina and Somalia Treaty Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . . 54 1.10 Argentina and Somalia ln(Aid) Receipts . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.11 Argentina Treaty Capital and ln(Aid) Receipts . . . . . . . . . . . . . . . . . . . . . . 59 1.12 Somalia Treaty Capital and ln(Aid) Receipts . . . . . . . . . . . . . . . . . . . . . . . . 59 1.13 Reservations/Objections per Ratifier and Treaty Costs . . . . . . . . . . . . . . . . . . 60 1.14 Treaty Costs and Words per Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 x Chapter 1 Human Rights Treaty Ratification of Aid Receiving Countries 1.1 Introduction With the horrors of two catastrophic world wars fresh in its collective memory, the United Nations General Assembly adopted the Universal Declaration of Human Rights (UDHR) on December 10, 1948. In the years since, the UDHR has formed the basis of international human rights norms and their formal legal embodiment in the United Nations Human Rights Treaties (HRT). The ratification and in many cases subsequent integration of HRT’s into domestic law by countries from all over the political and cultural spectrum is considered one of the great achievements of the international community in the post Second World War era. At the same time, it is natural to ask why countries ratify HRT’s at all. First, formal enforcement mechanisms generally do not exist. Countries party to a treaty can in practice violate its terms without formally specified punishment. This stands in contrast to treaties in trade and the environment, where monitoring is possible and non-compliant parties can be punished. Second, the benefits to ratifying a human rights treaty are not immediately clear. Treaties in trade and the environment solve a collective action problem by clearly stipulating and enforcing the rules of the game, and the mutual benefit associated with treaty participation is clear and tangible. When a country ratifies an HRT, on the other hand, it agrees not to take actions that affect its own citizens. It is not obvious what one country has to gain by agreeing to limit domestic behaviour while receiving unenforceable promises from other parties to do the same.1 1 To the best of our knowledge this is the first study in the economics literature to address the question of why countries commit to human rights institutions. However a small but growing literature in political science and international law, triggered by Hathaway (2003) and Hathaway (2007), have studied the question of why autocratic countries often participate in HRTs more frequently than democratic countries. Von Stein (2005) studies the extent to which HRTs actually act as a constraint on treaty participant behaviour, and considers the possibility that HRTs simply select countries that are ex-ante compliant as participants and thus do not constrain behaviour at all. 1 Despite the seemingly inconsequential nature of HRT’s, we observe ratification throughout the2 history of the UN, and perhaps more importantly, substantial and persistent variation in the timing and frequency of ratification across countries and treaties. Exploiting a panel of 83 foreign aid receiving countries over the years 1973-2001, we investigate who ratifies when and why. Specifically, we examine the role of economic, political and strategic incentives in the decision to ratify HRT’s at the UN. Acknowledging that global human rights standards are not constant and evolve over time, our measure of a country’s treaty participation is always relative to the treaty participation of other countries at that point in time. We approach the problem in two stages. In the first stage we utilize dynamic panel data methods to study the relationship between aid, human rights institutions, and human rights treaty participation. In the first set of results we find that, conditional on domestic human rights institutions, high levels of relative treaty participation has a significant and positive effect on foreign aid receipts. This result is robust to the inclusion of many political and economic control variables, and to the presence of unobserved country and time heterogeneity. By contrast, the quality of domestic human rights institutions themselves do not have a significant effect on aid receipts once we control for other political and economic variables. These findings are a contribution to the literature on the determinants of foreign aid, at the forefront of which is Alesina and Dollar (2000) and Alesina and Weder (2002), as well as the literature on the determinants of HRT participation (i.e., Hathaway (2007)). We rationalize this set of results by arguing that aid donors use HRT participation as a criteria to distribute foreign aid among recipients, and actually prefer this “nominal" measure instead of a “real" measure of human rights behavior. 2 Further, we find that countries with high quality human rights institutions tend to participate more frequently in HRT’s relative to other countries, but that the converse is not true; countries 2 There are several possible reasons aid donors prefer to use HRT participation as the criteria for making aid allocation decisions. For example, the actual recipient of aid, usually the current government of the receiving country, is often constrained by other domestic political forces (i.e., the military) in affecting immediate change in actual human rights institutions. Ratifying a treaty is a task that is not only simpler, but often under the government’s direct control. Additionally, there is often disagreement among outside actors about the quality of a country’s human rights institutions, and further, about how to interpret any given measure and compare it across countries. For this reason, when a donor needs to justify to its domestic polity its decision of who to allocate aid to, HRT participation may be a more favorable statistic to point to. 3 with a higher propensity to ratify HRT’s actually experience a subsequent decline in their human rights institutions. The first finding is intuitive. The costs of complying with the terms of a treaty will be lower for a country that is already compliant at the time of ratification than for a noncompliant country. Compliant countries have a higher net benefit to ratifying, and ratify more frequently. The second result, that HRT participation causes a decline in domestic human rights institutions, is somewhat surprising. Previous studies in political science (i.e., Hathaway (2007)) find a similar result, and argue that countries under scrutiny by the international community for their practices may use HRT participation to take the “spotlight" off their real behavior. Once the spotlight is off, so too is the pressure for real improvements to domestic human rights institutions, and HRT participation may be followed by a decline in real human rights institutions. Finally, we show that high rates of relative participation in previous years has a strong negative effect on the country’s current ratification decision. Or, viewed in another light, the probability a country ratifies a treaty increases as countries in the rest of the world increase their relative treaty participation. This finding is consistent with the hypothesis that countries are strategic and forward-looking in their decision to ratify an HRT. Countries are strategic in the sense that other countries’ treaty participation decisions affect their own decision, and countries are forward looking in the sense that their own past decisions influence current decisions. These empirical results, while providing new insight regarding the determinants of foreign aid and the relationship between foreign aid, international human rights treaties, and domestic human rights institutions, raise three interesting questions that we study here. First, motivated by the empirical relationship between relative treaty participation and aid receipts, we investigate how much of the patterns of behavior we observe are explained by the economic benefits to ratification. In other words, is increased foreign aid receipts the motivation behind ratification? Second, finding that treaty participation is rewarded by aid raises the question, why don’t all countries ratify all available treaties immediately? Treaty ratification must be costly in some dimension. We explore the sources of the costs of treaty ratification, as understanding where these costs come from and the role they play in the participation decisions of countries is of interest 4 from a policy perspective. Third, it is clear also that there is significant heterogeneity in participation rates across countries, suggesting that there is heterogeneity in the net benefit to participation across countries. One other question of interest is, what specifically is driving this heterogeneity in net benefit? Absent a theoretical model of treaty ratification, these issues are difficult to investigate. As such, to further understand how economic and strategic factors drive country behavior, in the second part of the paper we develop and estimate a structural dynamic game of treaty ratification. The model we consider is analogous to a model of quality competition in an oligopolistic industry (eg., Pakes and McGuire (1994)). In our model countries compete for foreign aid by ratifying costly HRTs in the same way firms compete for consumer demand by investing in costly product quality improvement. Donor countries make resource allocation decisions based on the relative treaty participation of recipient countries. Recipient countries “invest" in the quality they offer donor countries by ratifying treaties at a cost. These investment costs are allowed to vary by country, treaty and by the type of domestic institutions in place. We estimate the investment costs as well as the structural parameters of the ratification benefit function. Our first finding is that the economic benefit to ratification that we found evidence for at least partly explains the ratification decision. Second, there are significant permanent differences in ratification costs across countries, and moreover countries with good human rights institutions and democratic institutions in place find it less costly to participate, all else equal. Finally, we find significant variation in costs across treaties. We relate this cost variation to variation in institutional characteristics of the treaties, in particular the verifiability of the treaty terms, and discuss the implications of this finding for institutional design. We then use the estimated model to consider several counterfactual experiments. In the first set of experiments we quantify the relative importance of foreign aid receipts and human rights as motives in the treaty ratification decision. We find that the average rate of participation drops by 33% when we decrease the aid motive by 10%, confirming that the economic returns to ratification we find evidence of in the first part of the paper are a significant part of the reason countries ratify 5 in the first place. In the second experiment we investigate the importance of regime heterogeneity in explaining the observed heterogeneity in country ratification behaviour. Existing studies on HRT ratification emphasize variation across regimes in the “cost of commitment" (Hathaway (2007)) to treaty terms as the primary source of heterogeneity in ratification decisions. As we allow for heterogeneity in both the economic benefits and cost of treaty ratification, estimating a structural model lets us compare the relative importance of these two sources of heterogeneity in explaining ratification behavior. While both sources of heterogeneity are important, we find that heterogeneity in the benefits to ratification explain a more significant proportion of the variation in behavior. As well as contributing to the literatures on the determinants of foreign aid and human rights treaty participation, this paper contributes to the nascent literature on the estimation of dynamic structural models in political economy. Several recent key developments in the estimation of dynamic games, in particular Aguirregabiria and Mira (2007), Bajari, Benkard and Levin (2007) and Pakes, Ostrovsky and Berry (2007), have allowed applied researchers to address questions that, for mainly computational reasons, were previously unanswerable. But with the exception of the pioneering work of Merlo (1997) and Diermeier et al (2003) who develop and structurally estimate a dynamic bargaining model of government formation, and, more relevant to the present study, Wagner (2008), who estimates a dynamic game of environmental treaty ratification, the tools of structural dynamic game estimation, and especially the most recently developed tools, have yet to be applied to questions in political economy. We aim to contribute by bridging the gap between the most recent methodological developments in empirical industrial organization and questions of interest in the field of political economy. 1.2 Data As our study is targeted at the developing world, we restrict ourselves to the countries of Latin America, Africa, Asia and the Middle East. We consider every country that existed on these con- 6 tinents at any point during the period 1973-2000, was a member of the United Nations, and has a population of at least 500,000.3 We exclude countries that are major oil exporters throughout the time horizon of the sample, as these countries were largely non-aid recipients during this time. Altogether this sample contains 83 countries and 2043 country-year observations. Treaty ratification dates come from the United Nations Treaty Collection.4 We restrict consideration to treaties that opened after the founding of the United Nations in 1945 (See table 1.12 in the data appendix for the full list of treaties). 5 From these data we construct all variables related to ratification. There are dozens of treaties in the collection, many pertaining to human rights related issues.6 While chapter IV of the treaty collection is titled “Human Rights," many other treaties in other chapters have an important (often predominantly) human rights dimension, and we thus do not restrict our attention only to treaties from this chapter. The set of treaties we select is broad enough that basic human rights (torture, political killings etc.), property rights, civil rights (religious and political freedoms), and emancipatory rights (worker rights, discrimination) are each considered explicitly by at least one of the treaties in the data set. We consider seven of the eight “core" human rights treaties,7 as well as eight other treaties considered human rights treaties by the Encyclopedia of Human Rights (1996). The data on country level democracy over time comes from the Polity IV data set (Marshall and Jaggers, 2004). Each of the democracy and autocracy indexes in the Polity data set, which range from 0-10 (0 being the lowest level of democracy (autocracy) and 10 being the highest level of democracy (autocracy)) are composites of other political variables. First, democracy is conceived 3 Much of the data we use, most importantly the Polity IV data set, does not provide values for countries with a population below 500,000. The human rights data starts in 1973. 4 Available at http://treaties.un.org/Pages/Home.aspx?lang=en 5 While treaties dealing with substantive human rights issues did exist in the pre-world war II era, the concept of international human rights law is widely considered to have been a product of the founding of the UN . 6 There are other international governmental organizations (IGO’s) that have as part of their raison d’etre a well established treaty regime that is valid under international law. A prominent example is the International Labour Organization (ILO). While we could have included treaties from these organizations in our sample, doing so would have raised additional problems. For example, there are countries that are members of the United Nations and not the ILO. Thus there are countries that are “players" in the ratification game associated with UN treaties, but not ILO treaties. Further, treaty rules vary across organizations. In sum, the nature of the treaty ratification game differs across IGO’s, and we wish to minimize these differences. 7 We are unable to consider the last of these treaties, the Convention on the Rights of Persons with Disabilities (in force 3 May 2008), because it opened too late with respect to the scope of our data set. 7 as the composite of three things: the degree to which citizens can freely express preferences over political leaders and policies, the constraints on the exercise of power by the executive, and the guarantee of civil liberties to citizens. Autocracy on the other hand is determined by how sharply political participation and competition is restricted, and how freely the executive, once selected, exercises power. We follow the literature and use the difference between these two scores (the Polity Composite Index) as our measure of a country’s level of democracy. Our measure of political stability comes from the “durable" variable in the Polity IV data set. This variable simply measures the number of years since a major political regime change in the country. Any missing data from the polity data set is imputed using the suggestions of the authors. The measure of trade openness we use is the Sachs and Warner (1996) trade openness indicator (updated by Wacziarg and Welch (2003)). This is a binary variable which designates a country as “closed" if one of five policy criteria are met: the country has an average tariff rate greater than 40%, non-tariff barriers cover more than 40% of imports, there is a state monopoly over major exports, the country has a socialist economic system, or there is a black market exchange rate premium greater than 20%. Country GDP data is measured in thousands of international Geary-Khamis dollars, and is taken from Maddison (2003). Population data is also taken from Maddison (2003). The data on foreign aid comes from the OECD Overseas Development Assistance (net ODA). We consider both Total Net ODA data itself, as well as (as a robustness check) the Net Aid Transfers (NAT) data set constructed by David Roodman (2005). While this data is constructed from the same underlying OECD Overseas Development Assistance (net ODA) data, the NAT data corrects for two sources of concern with the original net ODA data among practitioners. First, Net ODA does not account for interest payments paid on past loans by recipients to donors, it is only net of principle payments received on past ODA loans. NAT is net of both principle and interest payments. Second, NAT does not account for cancelation of old non-ODA loans, while the ODA does. The data is in billions of constant 2005 US dollars. While DAC members (aid donors) change over time, the countries that are members at some point during the time horizon of our sample are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, 8 Japan, Luxembourg, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, United Kingdom, United States, and the Commission of the European Communities. Lastly, the human rights data comes from Freedom House’s Comparative Survey of Freedom. We specifically consider the “Civil Liberties" variable, which rates countries on a scale from 17 according to a checklist (see above) of civil freedoms and rights. The measure is essentially a grade given to each country each year based on the number of positive answers to a checklist of fourteen questions. These include (Gastil, 1990): Free media, Open Public Discussion, Freedom of Assembly and Demonstration, Freedom of Political Organization, Nondiscriminatory Rule of Law in Politically Relevant Cases, Freedom from Political Terror and Imprisonment, Free Trade Unions and Peasant Organizations, Free Businesses/cooperatives, Free Professional/Private Organizations, Free Religious Institutions, Personal Social Rights, Socioeconomic Rights, Freedom From Gross Socioeconomic Inequality, Freedom From Gross Government Indifference or Corruption. Crucially, there is significant overlap between the realms of human rights behavior measured by the civil liberties variable we use, and the subjects covered by the human rights treaties that comprise the treaty capital variable (see the appendix for a list of the treaties used in this paper). Summary statistics for the key variables are provided in table 1.1. We include a variable measuring the rate of treaty participation in the table. This variable measures, for each country at each point in time, the ratio of the number of treaties it is a member of to the number of treaties open for ratification. This variable will be key in the analysis below and we discuss it in much more detail in the next section. 1.3 Aid, Treaty Ratification and Human Rights In this section we use dynamic panel data techniques to examine the relationship between foreign aid receipts, human rights treaty ratification and domestic human rights institutions. First, we establish that aid receiving countries that ratify HRTs more frequently than other aid recipients experience an increase in foreign aid receipts, but that the quality of domestic human rights in- 9 Table 1.1: Summary Statistics Variable Aid per capita Obs 2043 Mean 51.83335 Std. Dev. 52.73943 Min -19.45861 Max 459.4547 Measurement Units 2006 US dollars Human Rights 2043 3.399902 1.5162 1 7 Ratification Rate 2043 0 2.561963 -8.246575 6.5 Discrete Index Ranging from 1 to 7 Continuous Index per capita 2043 2.406353 2.109192 .218 12.34 Political Stability 2043 13.8605 14.36674 0 81 Democracy 2043 -1.181596 6.695256 -10 10 Openness 2043 .3220754 .4673861 0 1 ln(Population) 2043 16.25582 1.42699 13.18815 20.95647 GDP Thousands of “Geary-Khamis" International Dollars Number of Years Discrete Index Ranging from -10 to 10 Binary Indicator Natural Log of Number of People stitutions themselves are inconsequential for aid receipts. We then discuss these findings in the context of the foreign aid literature. Second, we find that countries with relatively good human rights institutions in place participate in HRT’s more frequently but that the converse is not true. Countries with high levels of predetermined treaty participation experience a decline in human rights institutions. Finally we provide some evidence that countries are strategic and forward looking in their decisions to participate in HRT’s, and in tying all of these results together, motivate the structural model of treaty ratification that we develop in the following section. It is useful to conceptualize human rights treaty ratification as a lumpy investment decision. At any point in time there are several treaties open for ratification by a country, as new treaties appear throughout the history of our sample (see data section below). Countries rarely ratify many treaties at once, and instead decisions are spaced over time. As with any other investment decision, there is a “capital stock" associated with ratification, and we imagine that as countries ratify treaties they accumulate capital. We now describe the measure of treaty capital that we use in our analysis. 10 Indexing countries by i, the year by t, and treaties (in chronological order) by r, the “stock" of treaties ratified by country i before year t is given by: Tit = ∑ xirt (1.1) r where xirt is a binary variable taking the value 1 if country i has ratified treaty r at any year before t and 0 otherwise. While Tit clearly measures a country’s treaty participation, it does not suit our purposes for one important reason. We would like to understand how treaty participation effects foreign aid receipts. A “high level" of participation is a moving target, however. New treaties appear regularly, and more importantly, global human rights standards change over time, and what may have been perceived as a high level of treaty participation in 1970 is not likely to be regarded as such in 1995. As such, we define: Kit = Tit − ∑ j6=i Tjt Ct−1 − 1 (1.2) where Ct is the number of aid receiving countries in the world at time t. Kit represents the cumulative number of human rights treaties ratified by country i relative to the average cumulative number of ratified treaties by other aid receiving countries. As a country remains idle and other countries ratify, the country’s stock depreciates. The measure Kit is particularly effective in that it allows for depreciation of the capital stock, while also making transparent in the context of our model how recipient countries compete with one another for economic attention from the more developed world. It is useful to illustrate some ways in which the path of treaty capital can differ across countries, and the consequent issues for estimation. Figure 1.1 displays the capital paths over the duration of the sample for Argentina and Chile, figure 1.2 displays the paths for Kenya and Chad, and figure 1.3 displays the paths for Uganda and Chad. We purposefully select countries that are within close geographical proximity to one another to “control" for regional differences. In figure 11 1.1 it is clear that each country has a period of heavy investment (Argentina in the mid 1980’s, Chile in the Early 1970’s), and each has long periods characterized by stagnation and decline. But most interesting is that each country’s heavy investment comes at a time when the other’s treaty capital is well into a period of decline. The striking feature in figure 1.2 is that, while both country’s investment dynamics are similar, the treaty capital stock of Kenya is persistently higher than the stock of Chad up until the last few years of our sample. This suggests there is important permanent country level heterogeneity in ratification patterns as well. Finally in figure 1.3 we see that the investment dynamics of Uganda and Chad track one another fairly closely for many years, but most importantly, in 1995 both countries experience a significant upward spike in investment. This suggests the possibility of region and time specific heterogeneity in behavior. As we discuss below, these pictures are very useful both in terms of understanding what is required for identification of the effects of interest, as well as for interpreting results. Regression Analysis In this section we establish and analyze the empirical relationships between foreign aid, human rights practices and HRT participation. Our two equations of primary interest are given by: ait = γaa ait−1 + γha hit−1 + γka Kit + β a zit−1 + uita (1.3) hit = γah ait−1 + γhh hit−1 + γkh Kit + βh zit−1 + uith (1.4) where: ait is foreign aid receipts per capita8 for country i at time t, hit is the quality of human 8 While in principle one could consider total foreign aid receipts instead of foreign aid receipts per capita as the explained variable in equation 1.3, we prefer per-capita aid receipts for two reasons. First, we are interested in discussing our results in the context of the existing literature on the determinants of foreign aid, in particular the landmark work of Alesina and Dollar (2000). As the literature has almost exclusively considered per capita foreign aid (or the log of per capita foreign aid), to facilitate comparison we do so as well. Second, as there are several country-year observations where foreign aid receipts are zero or negative (aid receipts are measured as negative if the amount of principal a recipient repaid on earlier loans exceeded the amount of aid it received during the year, or if there was a recovery on grants or other unused funds previously reported as disbursed) the distribution of total aid receipts is much less “wellbehaved" (from an econometric point of view) than the distribution of per-capita foreign aid receipts. Total aid receipts are typically either zero or a very large number, where as per-capita foreign aid has a more smooth distribution. In principle we could use a log to compress the total foreign aid receipts variable, but will involve some form of artificial transformation of the data. For example, we would have to add some constant positive number (greater than 1, since aid can be negative) to each observation before a log transformation to avoid having to take the log of a number less than or equal to zero. 12 Figure 1.1: Argentina and Chile Treaty Capital Figure 1.2: Kenya and Chad Treaty Capital Figure 1.3: Uganda and Chad Treaty Capital 13 rights institutions in country i at time t, and Kit is the treaty capital of country i at time t. zit is a j vector of covariates and uit is the unobserved error term. We assume the following structure for j uit , j ∈ { a, h}: j j j j j uit = ωi + δt + νr(i),t + ũit j (1.5) j j where ωi and δt are country and time fixed effects respectively, and νr(i),t is a region-time fixed j effect (the function r (i ) simply selects the region of country i). We thus explicitly allow uit to be correlated across time and across countries but maintain the usual assumption  j  E ũit | ait−1 , hit−1 , Kit , zit−1 = 0. (1.6) We consider the hypothesis that treaty capital Kit Granger causes foreign aid receipts ait and human rights practices hit . Formally, HRT Granger causes aid receipts and human rights practices if: E( ait |Kit , Jt−1 ) 6= E( ait | Jt−1 ) (1.7) E(hit |Kit , Jt−1 ) 6= E(hit | Jt−1 ) (1.8) where Jt−1 contains information on past aid, human rights institutions and all other explanatory variables except Kit (Wooldridge, 2008). In words, these equations say that past treaty behavior of a country is still informative for current foreign aid and human rights institutions even after conditioning on past aid, human rights institutions and the other explanatory variables. While we readily admit that establishing Granger causality is a less ambitious task than establishing strict causality, we favor identifying the effect of past HRT participation for two reasons. First, identifying a strict causal relationship between variables with complicated contemporaneous feedback effects in the absence of a valid instrument is not a realistic objective. Second, the timing of political events, particularly with respect to foreign aid giving, is possibly better captured through a 14 model with lagged explanatory variables rather than contemporaneous ones. The OECD aid data that we consider here measures actual disbursements of aid. Seeing as how most OECD countries make donation decisions in the fiscal year prior to the year of actual disbursement, aid donation decisions themselves are lagged by one year (Kuziemko and Werker, 2006). Then if we are trying to determine how treaty capital investment influences donor aid decisions, it is more correct to consider current aid disbursement as a function of the previous year’s treaty capital stock. For completeness, in the regression analysis we also consider the equation: Kit+1 = γak ait−1 + γhk hit−1 + γkk Kit + βk zit−1 + uitk (1.9) While understanding the dynamic relationship between aid receipts, human rights behavior and HRT participation is a stated goal of this study, and thus considering equation 1.9 together with equations 1.3 and 1.4 is logical from the point of view of completeness, we should be cautious in interpreting the estimates of equation 1.9. Recall that Kit measures the HRT participation of country i against all other countries in the world at year t, not simply the decision of country i at year t. Clearly, Kit will largely be determined by variables other than political and economic variables of country i. This is part of the motivation for the structural model of treaty ratification decisions that we develop and estimate later, which fully incorporates the strategic and forward looking behavior of countries. However, considering equation 1.9 now does help guide the structural model because it allows us to get a better idea of the possible mechanism underlying the data generating process. Further, to try and address the shortcomings associated with estimating equation 1.9, we also consider a linear probability model of country i’s ratification decision itself below. To aid in the interpretation of the results below, we include in table 1.2 the quartiles of the three endogenous variables, aid per capita, quality of human rights institutions and HRT participation, as well as the country-year identity corresponding to each quartile. Foreign Aid Receipts 15 Table 1.2: Sample Quartiles Minimum Aid Receipts/Capita -19.46 (Malaysia - 1996) Human Rights Institutions 1.0 (Angola - 1978) Participation -8.25 (South Africa - 1994) 25% 14.47 (Indonesia - 1981) 2.0 (Albania - 1995) -1.69 (Thailand - 1974) 50% 37.34 (The Gambia - 1996) 3.0 (Guinea- 1992) 0.20 (Zimbabwe - 1997) 75% 68.58 (Rwanda - 1981) 5.0 (El Salvador -1993) 1.79 (Venezuela - 1988) 459.50 (Mauritania - 1976) 7.0 (Uruguay - 2001) 6.50 (Philippines - 1996) Maximum HRT We interpret equation (3) as an aid supply function that describes the decision process of aid donor countries. Our primary interest here is in uncovering how the supply of aid responds to human rights treaty participation and human rights behavior in recipient countries, holding fixed other variables that may effect HRT participation, human rights institutions and aid receipts. We are not the first to try to disentangle the effects of different economic and political variables on aid receipts. The studies by Alesina and Dollar (2000) and Alesina and Weder (2002) are particularly relevant to ours, and we use their findings to guide our analysis. Alesina and Dollar (2000) find that aid flows are as much dictated by strategic and political interest as they are by the economic conditions and performance in recipient countries. While we do not directly control for strategic importance and political importance of the recipient to the donor(s), the inclusion of country specific fixed effects is useful in this regard. If strategic importance is relatively time invariant, at least over the time horizon of our sample, country specific fixed effects allow us to control for the average strategic value of the recipient across the donors (OECD members). Alesina and Dollar also find that aid donors reward democratization. As many episodes of democratization are accompanied by increased participation in HRTs as well as improvements in human rights institutions, democracy is an important control variable in our analysis. Alesina and Weder (2002) examine the relationship between foreign aid and domestic corruption. They find no evidence that less corrupt governments receive more aid (per capita) than more corrupt governments. Similar to 16 Alesina and Weder (2002) and in contrast to Alesina and Dollar (2000), we focus specifically on a relatively narrow set of institutions, and try to discern whether, after controlling for other determinants of aid flows, countries with one type of institutions are more likely to receive aid than countries without them. Table 1.3 displays the parameter estimates of equation (3). Column 2 .6737234* (.0362061) -.3595292 (.5269335) 1.235768* (.3781004) YES NO NO .8272 2042 Column 3 .6354525* (.0375123) .9460418 (.5122686) 1.490251* (.3797569) YES YES NO .8364 2042 Column 4 .6295216* (.0377596) 1.241887* (.5183479) 1.438282* (.3830961) YES YES YES .8392 2042 .6116126* (.1664197) YES YES YES .8402 2042 Column 5 .6207469* (.0382736) -.3084337 (.6649017) 1.380379* (.3811949) Column 6 .615328* (.0380997) -.1835161 (.6735464) 1.311195* (.3787861) -2.145677* (1.092854) .0182174 (.0755765) .6137618* (.169889) -4.218408* (1.806804) .2193509 (12.22033 ) YES YES YES .8412 2042 Notes: Standard errors estimated using Newey-West (allows for autocorrelation and heteroskedasticity) Country FE Time FE Region*Time FE R2 N popit−1 openit−1 democit−1 Stabilityit−1 gdpit−1 Kit hit−1 ait−1 Column 1 .8938801* (.0163584) -.2783102 (.3047436) -.0533879 (.1508015) NO NO NO .8041 2042 Table 1.3: Foreign Aid Receipts: Parameter Estimates Column 7 .8187365* (.0215647) .303654 (.4906844) .1241684 (.1527717) -1.294062* (.2988315) .0365486 (.0323007) .0858431 ( .1120373) -2.520494* (.9867822) -3.913687* (.496932) NO NO NO .8127 2042 17 18 We include estimates of the regression model with and without fixed effects for illustrative purposes. The importance of accounting for permanent and unobserved country level heterogeneity in cross country studies is made clear by Acemoglu et al’s (2008) study of the effect of income on democracy. Permanent and unobserved country level factors simultaneously cause many country level outcomes of interest, and since pooled OLS only yields consistent estimates of the parameters of interest if there is no correlation between time invariant country characteristics and the explanatory variables, not accounting for permanent unobserved heterogeneity can result in significantly biased estimates. As an example in our context, Argentina is one of the few countries to have ratified every treaty in our sample, and throughout the history of our sample has a larger stock of treaty capital than Somalia (see figure 1.9 in appendix). However, Somalia annually receives more aid per capita than nearly every country in our sample, and certainly much more than Argentina at any point in time ( figure 1.10 in the appendix). By comparing across the two countries we may conclude that past human rights treaty ratification results in a decrease in foreign aid. But few would dispute that there is something intrinsically different and time-invariant (at least over the duration of our sample) between these countries that potentially explain both aid receipts and human rights treaty participation. One should look at the relationship between aid and human rights within a country over time to get a clear picture of how aid responds to treaty participation and human rights behavior (figures 1.11 and 1.12 in the appendix). Controlling for country specific heterogeneity in the regression model allows us to do precisely this. For the purposes of this paper, the key result from this section is that, once we control for country-specific unobserved heterogeneity HRT participation has an economically and statistically significant positive effect on per capita foreign aid receipts. Looking at the first two columns of table 1.3, the estimates confirm the message in figures 1.9 - 1.12. In column 1 we regress current aid receipts on lagged aid, human rights institutions and treaty capital, and in the second column we add only country level fixed effects to the regression in column 1. In column 1, the only statistically significant variable is lagged foreign aid receipts. If there is any relationship at all, treaty capital negatively effects aid receipts. Once we allow for the presence of permanent country 19 level heterogeneity however, we see that the estimated effect of treaty capital becomes positive and statistically significant. To quantify the economic significance of the effect, the estimate in column 2 suggests that if country i were to ratify one more treaty, holding the behavior of all other countries in the world constant, in the short-run country i would receive 1.24 dollars more of aid per capita. As the model is dynamic, it makes sense to also consider the long run effect of ratifying an additional treaty. By recursive substitution of equation 1.3, it is straightforward to see that entering year t having ratified one additional treaty in the previous year implies an additional total stream of γka ∑sT=1 (γaa )s dollars of aid T years in the future. Thus, far enough into the future, one additional treaty implies an additional γka 1−γaa ∼ 3.80 dollars of aid per-capita. By similar calculations, a one-standard deviation increase in treaty participation implies a short-run increase of about 3.17 dollars and a long-run increase of 9.62 dollars of aid per-capita. At first glance the effect of HRT participation on aid seems small given the mean (51.8) and standard deviation (52.7) of aid per capita, but we note that the most important determinant of differences in aid receipts across countries is actually permanent unobserved country specific heterogeneity, and not any of the covariates we consider in the analysis. This is actually in line with Alesina and Dollar (2000) who find that the colonial heritage and (permanent) strategic political importance of a country are the most important determinants of aid receipts. Among time-varying covariates, HRT participation is a relatively important determinant of aid receipts, as we discuss in more detail below. The difference in the estimates between columns 1 and 2 suggests that unobserved permanent country heterogeneity is negatively correlated with treaty participation and positively correlated with foreign aid receipts. One potential hypothesis is that there are some countries, such as Somalia, with permanent, poor social, political and economic institutions. These countries tend to be economically poor and rely on foreign aid and moreover, their poor institutions are also inimical to participation in international human rights initiatives. On the other hand, relatively wealthy countries like Argentina do not rely on foreign aid, and possess more sound domestic institutions which are conducive to participation in international institutions. While this argument has intu- 20 itive appeal, we provide some evidence below that the “omitted variable" of column 1 is not likely to be domestic institutions. The results in columns 1 and 2 suggest that the quality of real human rights institutions has no statistically significant effect on aid receipts. The measure of human rights institutions we use here is the “civil freedoms" variable from the Freedom House Comparative Survey of Freedom (see Gastil(1990) for details).9 In columns 3 and 4 we add time fixed effects and region-time effects respectively. While the reasons for including time fixed effects here may be obvious, the choice to add region-time interactions is motivated by what we observed in figure 1.3 above: regional political/economic initiatives which simultaneously cause aid and participation in international treaties are potentially important. As we move to these columns we see that lagged human rights becomes economically and statistically significant. Before interpreting this as a positive result, we should be sure we are not simply obtaining an upwards bias in our estimate of the effect of human rights institutions on foreign aid receipts. The quality of domestic human rights institutions is potentially correlated with other factors that explain foreign aid receipts, such as political institutions. It has been established in the foreign aid literature (Alesina and Dollar, 2000) that countries with more democratic political institutions are rewarded with more aid. It is also well known that democratization episodes are often accompanied by improvements in human rights institutions (Hafner-Burton et al. (2008)). Moreover, Alesina and Dollar present some evidence that, controlling for democracy, human rights institutions have no effect on aid receipts. These findings motivate column 5 of table 1.3. When we control for domestic political institutions (democracy score) in column 5, the effect of human rights institutions on HRT participation becomes statistically insignificant, and is negative if anything, while the estimate of the effect of democracy is economically and statistically significant, perfectly consistent with the results in the existing aid literature. From an identification perspective one may be concerned about a potentially high level 9 The Comparative Survey of Freedom contains two variables, a “political freedoms" variable, generally interpreted as a measure of how democratic a country is, and the civil freedoms variable, which has been generally interpreted in the political science literature as a measure of human rights (i.e., Hathaway, 2007). The political freedoms variable has been used extensively in the economics literature, but the civil freedoms variable has not, perhaps owing to the fact that there are relatively few studies on human rights in economics. One important exception is Alesina and Dollar (2000), who use both variables. 21 Table 1.4: Human Rights-Democracy Episodes Democracy low high Total Human low 65 28 93 Rights high 44 51 95 Total 109 79 Table 1.5: Number of Different Regimes Visited in Country History Number of Regimes 1 2 3 4 Total Number of Countries 20 29 26 8 83 of correlation between democracy and human rights institutions. While we discuss these two measures in more detail below, for now as a check, we dichotomize each into “high" and “low" for each country-time observation 10 and for each of the four possible democracy - human rights combinations, count the number of countries that have an episode at that combination at any point in the history of our sample. The statistics are presented in table 1.4. As there are 83 countries in our sample, it is clear that several countries necessarily visited multiple democracy-human rights pairs in the duration of our sample. To further illustrate this point, in table 1.5 we provide, for each number of possible regime pairs a country can visit in its history, a count of the number of countries that visited that many pairs. Table 1.4 suggests there are many episodes of low democracy-high human rights and high democracy-low human rights. Moreover, table 1.5 suggests the presence of the most important type of variation in institutions from an identification perspective, within country, over-time variation. While they are bound to be highly correlated, our democracy and human rights variables 10 For example democracy ranges from −10 to +10, so any country-time observation with greater than or equal to zero democracy score is labeled “high." We do the analogous relabeling for the human rights variable. 22 are not measuring the exactly the same institutional features. In column 6 of table 1.3 we include several variables that have been considered in the foreign aid literature. Not surprisingly, countries with higher GDP per capita receive less aid. Political stability has no significant effect on foreign aid receipts. “Openness" of a country, a general measure of economic policy created by Sachs and Werner (1995) is associated with less foreign aid. This is possibly explained by the fact that countries closed to economic activity are more reliant on foreign aid than are more open countries. This finding is also generally consistent with the findings of Alesina and Dollar (2000). In the final column of table 1.3, we consider the same model as in column 6, but without allowing for controlling for unobserved heterogeneity of any form. We see that the coefficient on treaty capital is economically insignificant and far from statistically significant. This provides some evidence against the hypothesis that the “omitted variable" in column 1 is simply permanent differences in the levels of political institutions and wealth across countries. Even when we control for these factors, unless we allow for country specific heterogeneity, we get an estimate of the effect of treaty capital on aid that is not statistically different from zero. The question then is, why do aid receipts respond to treaty participation? On possible explanation is that treaty participation is a costly signal of a country’s preferences and intentions. As an example: In Estonia, 28 treaties...were ratified in one session, a month after independence, without even having been translated into Estonian. Estonia “wanted to send a strong signal that it would respect human rights and was not a part of the Soviet Union anymore." (Heyns and Viloen, 2002) That a government of a country would agree to abide by the terms of 28 complicated documents potentially not even knowing the language they were written in lends credence to the possibility that countries with a dark history signal to the Western world that they are now “open for business." The link between treaty participation and aid receipts may be even more direct. For example, the EU’s Generalised System of Preferences Plus (GSP Plus) offers large tariff reductions as 23 well as other benefits to developing countries in exchange for the ratification of key international treaties, 8 of which are UN human rights treaties that we consider here.11 While this particular example is not one of foreign aid giving per se, it does indicate that HRT participation is used as a criteria when the developed world decides who to give a “carrot" to. (b) Human Rights Behavior To the best of our knowledge this is the first study to consider the economic and political determinants of human rights behavior in the economics literature. In table 1.6 we present estimates of equation 1.4. 11 See:www.ec.europa.eu/trade/wider-agenda/development/generalised-system-of-preferences/ http://trade.ec.europa.eu/doclib/docs/2005/june/tradoc1 23861.pd f and Column 2 .0001513 (.0003201) .8058864 * (.0160688 ) -.0157269 (.009243) YES NO NO .8802 2042 Column 3 .0005233 (.0003461 ) .7900121* (.0168019 ) -.0167405 (.0090792) YES YES NO .8867 2042 Column 4 .0006184 (.0003521 ) .7852115* (.0171473) -.015276 ( .0093057) YES YES YES .8911 2042 YES YES YES .8925 2042 .021051* (.0048281) Column 5 .0003164 (.0003482) .7318513* (.0207996) -.0172689 (.0092487) Column 6 .000252 (.0003504) .7282466* (.0209071) -.0191326* (.0092234) -.0173823 (.016706) -.0032785 (.0019546) .0188709* (.0052418) .1065236* ( .0458571 ) -.4198933 (.2395068 ) YES YES YES .8931 2042 Column 7 .0002227 (.0002521) .8624241* (.0149096) .0073752 (.0048028) .019666* (.0067289) -.0004692 (.0007621) .01263* (.0032013) .0607946* (.0275029) -.0070408 (.0105671) NO NO NO .8732 2042 Notes: Standard errors estimated using Newey-West (allows for autocorrelation and heteroskedasticity). The null hypothesis of unit root in hrit is rejected in a standard Dickey Fuller test (the test statistic is almost -10). Country FE Time FE Region*Time FE R2 N popit−1 openit−1 democit−1 Stabilityit−1 gdpit−1 Kit hit−1 ait−1 Column 1 .0000956 (.0002048) .9248508* (.0082207) .0060268 (.004748) NO NO NO .8703 2042 Table 1.6: Human Rights: Parameter Estimates 24 25 We consider the same set of specifications as in the case of foreign aid. In table 1.6 we see that in columns 1-4 only lagged human rights institutions is statistically significant. Treaty capital has an economically (but not statistically) significant negative effect. However, once we control for democracy in column 5, treaty capital becomes statistically significant at the 10% level, and negative. Not surprisingly the democracy score has a positive estimated effect and is very statistically significant. Once we add other covariates in column 6 we also see that openness has a positive and statistically significant estimated effect on human rights behavior, but none of the estimates from column 5 are affected very much. The most surprising result here is that HRT participation has a negative effect on human rights behavior. This would suggest that the institutions designed with the intention of preventing governments from violating the fundamental rights of their own citizenry actually have the opposite effect. HRT participation makes governments more likely to commit human rights abuses. We are not the first to find results suggestive of a negative relationship between HRT participation and human rights behavior. Hathaway (2007) also finds a relative decline in human rights practices among countries with a higher rate of treaty participation. In rationalizing this result she argues that HRT participation often relieves “pressure for real change in (human rights) performance in countries that ratify the treaty." In other words, aid recipients may take the international spotlight off their real practices by ratifying unenforceable human rights treaties. While this “smokescreen" argument does have some intuitive appeal, we have reason to be cautious. Goodman and Jinks (2003) point out an alternative interpretation of Hathaway’s result, in that “greater compliance with one obligation... can show up as lower compliance with another." Countries may ratify and comply with the terms of a treaty concerned with one human right, and “substitute" into abuse of another human right. Thus HRT ratification may be associated with future declines in human rights behavior. In the case of our study this is not as much of a concern. Both our ratification variable Kit and our human rights institution variable are summary measures of treaty participation and quality of domestic human rights institutions, and so what we are capturing in equation 1.4 is the effect of an increase in relative treaty participation on human rights institutions in general. 26 The other concern Goodman and Jinks have with Hathaway’s study which is relevant in our case is the potential “measurement error" in the human rights behavior variable. They argue that the measurement error is not exogenous, and is instead highly correlated with treaty ratification in the sense that ratification is associated with improvements in the reporting of abuses. However the Freedom House “civil liberties" variable is not a violation count variable, and we should be confident that our study is not subject to this criticism (see Gastil (1990) for details). The other result of interest in this regression is the positive and significant effect of trade openness on human rights behavior. This result is not terribly surprising considering that many trade deals between developed and less-developed countries have human rights related strings attached, including the EU’s GSP plus program discussed above. Countries that are open to trade are then more likely to have better human rights practices, because the return to good human rights practices in the form of trade benefits are higher for these countries. Treaty Capital We present estimates of equation 1.9 in table 1.7. Column 1 -.0004006 ( .0002662 ) .0283617* ( .0110861) .9545335* (.008096) NO NO NO .9230 2042 Column 2 .0006869 ( .0004728 ) .0455078* ( .018455 ) .819478 * ( .0188985) YES NO NO .9323 2042 Column 3 .0000689 (.0005001) .0493893 * ( .0187422) .8230187* (.0187433) YES YES NO .9343 2042 Column 4 .0001304 (.0005024) .0448022 * ( .0193904) .8173725* ( .0189641) YES YES YES .9354 2042 YES YES YES .9354 2042 -9.41e-06 (.0056218) Column 5 .0001305 (.0005122) .0448261 (.0247881) .8173734* (.0189968) Notes: Standard errors estimated using Newey-West (allows for autocorrelation and heteroskedasticity) Country FE Time FE Region*Time FE R2 N popit−1 openit−1 democit−1 Stabilityit−1 gdpit−1 Kit hit−1 ait−1 Table 1.7: Treaty Capital: Parameter Estimates Column 6 .0001309 (.0005052 ) .0422902 ( .0247236) .8116114* (.0196132 ) -.0174141 ( .0212753 ) -.0006018 ( .0022304 ) .0003837 (.0058654) .0288315 (.0570071) .6424783* (.3290319 ) YES YES YES .9356 2042 Column 7 -.0004831 (.000326) .0600241* (.0169391) .9547278* (.0078307) .004651 ( .0090556) -.0013737 (.000896) -.0090468* (.0034549) -.0471121 (.0390001) .0003138 (.0129301) NO NO NO .9251 2042 27 28 As we discussed above, these estimates should be interpreted with caution. A significant amount of the variation in country i’s Treaty capital Kit is due to aggregate behavior of countries in the rest of the world. We shouldn’t expect to be able to explain this variation with political and economic variables of country i only. Not surprisingly, the only variable other than lagged treaty participation that is significant or close to significant throughout the columns of table 5 is lagged human rights. The effect of human rights institutions on treaty capital is positive. To get a better understanding of how the key variables of interest affect treaty participation decisions, we also estimate the following equation: dirt = γad ait−1 + γhd hit−1 + γkd Kit−1 + βd zit−1 + ωid + δtd + κrd + εdirt (1.10) where dirt is country i’s ratification decision (binary) in treaty r at time t, and κrd is a treaty specific fixed effect. In table 6 we present the estimates of the full specification (including all controls and fixed effects) of equation 1.10. The two key results here are the economic and statistical significance of treaty capital and human rights. Countries with a large stock of treaty capital are less likely to ratify a treaty than a country with a low stock. Dynamic strategic effects are important, as countries are more compelled to ratify a given treaty if the number of treaties they have already ratified is low relative to the number of treaties that other countries in the world have ratified. Countries with good human rights institutions are more likely to ratify than countries with poor institutions. Summary of Main Results The results of the previous section, taken together, are suggestive of the following short run relationships between foreign aid, human rights and human rights treaty participation. 1. Aid donors use relative HRT participation to decide who to donate foreign aid to. In contrast, 29 Table 1.8: Treaty Decision: Parameter Estimates ait−1 hit−1 Kit gdpit−1 Stabilityit−1 democit−1 openit−1 popit−1 R2 N 0.000034 (0.000059) 0.007456* (0.002546) -0.016046* (0.001364) -0.002122 (0.002770) -0.000021 (0.000261) -0.000055 (0.000637) 0.005442 (0.006720) 0.089781* (0.034281) .094 14112 the quality of a country’s domestic human rights institutions are inconsequential for aid receipts. While rewarding a nominal measure of human rights and ignoring the real measure seems like strange behavior on the part of donor countries, this behavior is possibly rationalized by the fact that the actual aid recipients themselves often do not have direct control over actual human rights institutions in the short run, but do have control over the decision to participate in an HRT. Moreover, from the point of view of donors, HRT participation is an easier object to quantify than human rights institutions. 2. Countries with good human rights institutions are more likely to participate in HRTs. Surprisingly, countries with a high level of predetermined HRT participation experience a decline in the quality of domestic human rights institutions. Countries with good institutions are by definition ex-ante compliant, and so ratification is less costly for these countries. Further, in light of point (1), if donors used real human rights institutions instead of the nominal HRT participation as the criteria for distributing aid, countries with good domestic human rights institutions would have no reason to ratify HRTs. It is then intuitive that these coun- 30 tries are more willing to participate. While it is surprising that countries that participate frequently experience a decline in domestic human rights institutions, it is possible that HRT participation is used as something of a “smokescreen." Countries under scrutiny for their practices participate in order to take the spotlight off their real behavior. 3. Countries with a low level of HRT participation are more likely to ratify a given treaty than countries with a high level of HRT participation. This suggests that dynamic strategic effects are important. The past ratifications of other countries increase the likelihood of ratification for a country. These results raise further questions. We have established that predetermined HRT participation has an economically and statistically significant effect on aid receipts, but is increased foreign aid receipts the motivation behind ratification? And if so, why don’t all countries ratify all available treaties immediately? There must be some cost to ratification. A study of the decision to ratify HRTs is not complete without understanding where these costs come from and the role they play in the participation decision. It is clear also that there is significant heterogeneity in participation rates across countries, suggesting that there is heterogeneity in the net benefit to participation across countries. One other question of interest is, what specifically is driving this heterogeneity in net benefit? To address these questions we develop and estimate a structural model that allows for the dynamic strategic interaction we have found evidence for. We then use the estimated model to consider counterfactual experiments. 1.4 1.4.1 Model Discussion The structural model we propose here has as an analogue in the empirical IO literature to a dynamic game of oligopoly competition where firms invest in quality (eg.,Pakes and McGuire (1994)). Before formally laying out the model it is worthwhile to make explicit the relation between a dynamic game of quality competition and the problem we study here. In a model of qual- 31 ity competition, each firm in an industry composed of several firms produces a product which is indexed by quality. Consumers derive utility only from the quality (net of price) of the good they choose to purchase, and so demand for a given product in the industry is determined fully by its quality relative to the quality on offer from the other firms. Firms also have the option to increase the demand for their products by making a costly investment in quality. Firm profits thus depend on quality through both the revenue and cost channel: a higher quality means more demand and the ability to command a higher price, but comes at an economic cost. Firms who do not invest in quality may see their demand decrease if other firms in the market continue to invest. In the model we consider here, poorer countries in the world compete with each other to increase aid receipts from OECD countries. We also allow for the possibility that countries have an intrinsic preference for human rights. Donor countries have a finite amount of aid to distribute among recipient countries, and rely on K to make allocation decisions. Recipient countries have the option to increase the aid they receive by making a costly ratification of a human rights treaty. This is a costly investment because ratifying a treaty commits a country to the terms of the treaty, at least in principle. Different treaties are allowed to have different ratification costs for a country, and different countries can have different costs of ratification of a given treaty. For example, countries with different human rights institutions or different political institutions may have different costs of participation. As in the quality investment model country payoffs depends on quality through both the benefit and cost channel: a larger treaty capital stock means more aid and economic growth, but comes at a cost. Further, countries that do not invest in quality may see the economic resources they receive decline if other countries continue to increase their participation in human rights treaties. 1.4.2 Formal Model At year t, the international community is configured by Ct recipient countries and Nt treaties. Countries and treaties are given exogenously in our model. Let xirt ∈ {0, 1} indicate country i’s status in treaty r at year t. If xirt = 1, we say country i has ratified treaty r at some time τ < t. 32 We can represent country i’s membership status in the set of Rt treaties at time t by the vector xit = { xirt : r = 1, 2, ..., Nt }, and we can represent the ratification status of the entire international community as the vector xt = {xit : i = 1, 2, ..., Ct }. Ratification is irreversible: once a country has ratified a treaty it may not exit (erase its name) from the treaty, an assumption clearly validated by the data. We represent the ratification decisions at period t as dit ≡ {dirt : r = 1, 2, ..., Nt }. Country payoffs in year t are the difference between per-period economic payoff Ri and a ratification “investment" cost ECi : Πi (xt , zt , dt , ε it ) = Ri (xt , zt ) − ECi (xt , zt , dit , ε it ) (1.11) zt is a vector of exogenous political and economic variables and ε it is a vector of private information shocks of country i. We specify the benefit and ratification cost functions in turn. Economic Payoffs We make the following assumption on the economic payoff function Ri (xt , zt ): • Assumption (E1) The function Ri (xt , zt ) depends on the vector xt , zt only through its effect on foreign aid receipts and human rights. Specifically, let ait = ai (xt , zt ) and hit = hi (xt , zt ) represent country i’s foreign aid received and human rights score respectively at year t. Then: Ri ( x t , z t ) = α a ai ( x t , z t ) + α h hi ( x t , z t ) ai (xt , zt ) and hi (xt , zt ) are functions that make explicit the fact that these variables depend on treaty status xt as well as other political and economic variables zt . α a and αh are parameters to be estimated, representing the relative weights of aid and human rights in country payoffs. What remains then is to specify the functions ait = ai (xt , zt ) and hit = hi (xt , zt ). While it is natural here to use the results of the first stage estimation, we considered several versions of the equations aid and human rights equations 1.3 and 1.4, and we must settle on a specification. The trade-off here 33 is that a richer specification provides a more accurate depiction of the true relationship among the variables, but comes at the conceptual cost of clarity and computational cost of a larger state space. We would like a specification rich enough so as not to sacrifice reality but concise enough so that we can narrowly focus on answering our primary questions of interest. Note that in the empirical results above, once we condition on all three forms of unobserved heterogeneity as well as democracy, adding other covariates does not significantly alter the results. With this in mind, the specification we consider is given by: ait = γaa ait−1 + γha hit−1 + γka Kit + β a demit−1 + ũita (1.12) hit = γah ait−1 + γhh hit−1 + γkh Kit + βh demit−1 + ũith (1.13) where: j j j j j uit = ωi + δt + νr(i),t + ũit (1.14) We allow for all three forms of unobserved heterogeneity in the transition of the endogenous variables: country-specific across time, time-specific across country, and time-specific within region across country. This is the specification that we considered in column 5 of tables 1.3 and 1.6 above. Implicit in this specification of Ri (xt , zt ) is the following assumption: • Assumption (E2) The vector of ratification statuses xt of the international community enter the payoff of country i in the dynamic game only through Kit . Treaties are interconnected in the sense that we allow ratification decisions in one treaty to affect the payoff to ratifying any other treaty. Country i0 s status, and the status of all other countries in all other treaties, influences country i’s decision in any given treaty. This is a departure from the traditional “isolated markets" assumption typically made in the literature in empirical industrial 34 organization. As we discuss below in the estimation of the dynamic model, this specification of the payoff function plays an important practical role in alleviating the computational burden associated with the solution and estimation of the dynamic game. Investment Costs The ratification cost function ECi (xt , zt , ε it ) is directly analogous to the concept of irreversible investment cost in the IO literature. It should be interpreted as the sum of a one time cost paid upon ratification of (investment in) the treaty plus the discounted present value of a sequence of fixed costs of remaining in the treaty. We specify the following investment cost function for ratifying treaty r in country i at time t: ECirt = γi + ξ r + γde demit + γh hrit + γdh demit hrit + ε irt where γi is the country specific component of the entry cost, ξ r is a treaty specific cost, and demit and hrit are the levels of democracy and human rights institutions in country i at time t. γde and γhr represent the affect on entry cost to having democratic institutions and to having good human rights institutions respectively. ε irt is a random shock with distribution G, which is private information of country i. We discuss the private information shock in more detail below. It is important to note here that we can not estimate a fixed cost of being a party to a treaty. To separately identify fixed costs from our investment costs we would need to observe both entry and exit into treaties. Since ratification decisions are irreversible, we do not observe exit. The choice to include democracy and human rights in the cost function is not arbitrary. Hathaway (2007) finds that for a given level of democracy, countries with worse human rights practices have lower probability of ratifying HRT’s, while for a given level of human rights, more democratic countries have a lower rate of ratification. Hathaway theoretically links state decisions to ratify a human rights treaty to the domestic enforceability of the treaty by arguing that ratification is only costly for those countries that a) are not compliant with the treaty’s terms ex-ante of ratification and b) 35 are to the terms of the treaty by some domestic enforcement mechanism post-ratification. While the cost of abiding by the terms of a treaty is in principle the same for any ratifier, only countries that need to change their behavior post ratification actually pay the cost. Our specification of ratification cost allows us to test Hathaway’s hypothesis in the context of our model. We also allow for treaty and country specific heterogeneity in ratification cost. There is substantial and persistent variation in the rate of ratification across countries and treaties. As an example nearly every country in the world has ratified the Convention on the Rights of the Child, but a comparatively small fraction have ratified the Convention on the Protection of the Rights of all Migrant Workers, even though the latter opened 5 years prior. The country specific differences are often stark as well. Armenia and Kazakhstan, two countries with similar recent political histories follow very different HRT ratification paths post independence in 1991. Of the treaties in our data set that both countries have ratified, Armenia ratified all but one well before Kazakhstan. These suggest that ignoring the potentially permanent differences across countries and treaties in the estimation of the model may lead to spurious results. Further, by allowing costs to vary across treaty type for a given regime we are also able to study the possible relationships between observable institutional features of the treaties and the cost of ratification. Implicit in our definition of the payoff function Πi is a time to build assumption: • Assumption (E3): Ratification costs are paid at the time period of ratification, but the ratification decision is not effective until the following time period. As we discussed above, countries do not see the benefits of a ratification made in period t until period t + 1. This is especially justified in the case of aid, as we are considering aid disbursement data. This means that the aid received in period t + 1 was decided by donors in period t. A treaty ratification in period t does not affect aid receipts in period t. Country Strategies and Equilibrium We assume that countries are forward looking and maximize intertemporal payoffs, and take 36 into account the direct effect of their actions on their own future payoffs as well as the indirect effect through the expected reaction of other countries. Further, we assume that strategies depend only on payoff relevant variables. That is, we restrict players to use Markov strategies. Given the above discussion, country i’s payoffs at time t depends on the vector of state variables  ait , hrit , Kit , demit , ε it . The endogenous state variables are ait , hrit , Kit , while the exogenous state variables are demit , ε it . The model constitutes a dynamic game because the evolution of the variable Kit depends on the actions of all the players.  Keeping in mind that the payoff-relevant variables for player i at time t are ait , hrit , Kit , demit , ε it , for notational simplicity, we continue to represent the full state of the game at time t for player i by the vector {xt , zt , ε it }. Let σi (xt , zt , ε it ) be a strategy function for country i. Given this strategy function we can define a conditional choice probability (CCP) function as
Pi dit |xt , zt ≡ Z  I σi (xt , zt , ε it ) = dit dG (ε) In words, this is the probability with which country i takes action dit at time period t given the ratification status xt and state zt .
We define country i’s value function given equilibrium entry probabilities Pi d−i |xt , zt as V Pi (xt , zt ,ε it ), and we say that a strategy function σ is a Markov Perfect Equilibrium if for any possible state (xt , zt ,ε it ): n  o σi (xt , zt ,ε it ) = arg max Πi (xt , zt , ε it ) + δE V Pi (xt+1 , zt+1 , ε it+1 )|xt , zt dit We discuss the dynamic model and further assumptions in more detail in the estimation section below. 37 1.5 Estimation of the Dynamic Model The dynamic model we have described above leads to three interrelated dimensionality problems that render estimation of the model in its current form impossible: 1. (P1) In the model section above we have formally defined the Markov states of our game to be xt , zt , and thus player strategies depend on the full vector (xt , zt ) of treaty statuses and exogenous variables. Ignoring zt , the dimension of xt alone is 2 Nt ∗Ct , which, given the data we consider, can be as large as 21245 . Solving the value functions associated with a game defined on this space is computationally infeasible. 2. (P2) Player i’s action space at time t, Dit , is the set of treaties yet to be ratified by player i. This can be as large as 215 . 3. (P3) Player payoffs depend on the behaviour of the other players in the game. Thus the expectation of future payoffs depend on expected behaviour of all (83) players, and the transition of the state variables has a very high dimension. We describe in detail how we deal with each of these dimensionality problems so as to facilitate estimation of the model. As described above, a key benefit of the modeling approach we have taken is that the economic payoffs of players in the game depends on the vector (xt ) only through the functions ai (xt , zt ) and hi (xt , zt ). This effectively solves problem (P1) above. Since potentially many values of xt , zt yield the same value of ait , hit , 12 the space over which player strategies are defined is considerably reduced. To deal with problem (P2) we adopt the method proposed by Aguirregabiria and Ho (2009). We assume first that at each period t, each country appoints a committee to each treaty it has not ratified yet. This committee observes some private information about the treaty that neither any other country, nor any other committee in its own country observes. Based on this information, the state of the game, and beliefs about the strategies of other committees within the country 12 This of course depends on the coarseness of the grid ait and hit are discretized on, which we discuss later. 38 and committees in other countries, the committee makes a recommendation to the government to either ratify or not. The government then takes the decision that was recommended. We imagine that the government finds it too costly to research the implications of ratification of all the treaties open to it and delegates this task to the committee. Committees within a country can not share all their private information with each other, and thus do not fully co-ordinate. In this sense we are moving away from the “state as monolithic decision maker" assumption that has been used in similar problems (i.e., Wagner (2008)). What does this assumption buy us? It allows us to treat each country as a “different player" in each treaty, while still relaxing the “isolated markets" assumption. The country payoff to ratifying each treaty is still affected by the decisions made in other treaties through the K variable. 13 More formally, we make the following assumptions: • (D1) Committee r in country i at time t makes recommendation dirt ∈ {0, 1} to maximize the expected discounted value of the stream of country-treaty (committee) payoffs:
s Et ∑∞ s=1 β Πir,t+s , where: Πirt ≡ xirt Ri (1, x−irt , zt ) + (1 − xirt ) Ri (0, x−irt , zt ) − dirt (1 − xirt ) ECirt ( xt , zt , ε irt ) • (D2) The shocks {ε irt } are private information of committee r in country i at time t. These shocks are unknown to the other committees in country i and unknown to all other countries. Assumption D1 explicitly says that committees that enter period t having ratified the treaty for which they are responsible (xirt = 1) obtain Ri (1, x−irt , zt ), and committees that have yet to ratify earn Ri (0, x−irt , zt ). Further, committees that have yet to ratify and recommend ratification in year t, i.e., (dirt = 1, xirt = 0), pay the cost ECirt ( xt , zt , ε irt ). Assumption D2 says that there is statistical independence across treaties within a country. Note that committees within a country have the 13 This assumption is motivated partly by the literature on “formal" and “real" authority in organizations, spawned by Aghion and Tirole (1997). In this literature “formal" authority is defined as the right to decide, while “real" authority is defined as effective control over a decision. The difference between the two types of authority is generated by private information. Aghion and Tirole derive conditions under which a principal (here the leader of a country) may allocate authority over a decision to an agent (here the treaty committees) who possesses private information. Importantly, they show that when a principal can “trust" an agent (i.e., aligned incentives), authority will be delegated. Here our agents’ payoffs are closely aligned with those of the country. 39 same general objective, as the variables that enter the committee economic payoff are country level aid and human rights. While committees in the same country are playing “against" one another in the treaty ratification game, the decisions of committees within the same country and the decisions of committees in other countries enter the payoffs of a committee differently. Technically speaking, the portfolio of treaties chosen by the country (the set selected by the committees) will be the optimal portfolio for the country up to a deviation in one treaty holding decisions in all other treaties fixed. 14 While he have solved the computational issues associated with P1 and P2, we are still left with P3, the problems associated with each player conditioning his strategy on the state variables of all players in the game. Note now that for any treaty committee r in country i, current payoff is fully determined by it’s own status xirt , three endogenous variables, treaty capital Kit and the payoff variables ait , hit and the exogenous variable demit Then define the reduced vector of variables wirt : wirt ≡ { xirt , Kit , ait , hit , demit }. (1.15) To alleviate the computational burden associated with P3, we make the following further assumption: Assumption (D3): The strategy function of treaty office (i,r) is given by σir (wirt , ε irt ) that maps from W × R into {0, 1} As we did above for the full state space, we can now define the vector of conditional choice probabilities (CCPs) associated with the strategy functions σ as: Pir (wirt ) ≡ 14 The Z I {σir (wirt , ε irt ) = 1}dGε (ε irt ). (1.16) inability of committees within a country to perfectly co-ordinate and select the best overall portfolio of treaties for the country to hold at a given time can be alternatively viewed as a bounded rationality assumption. Information about treaties is costly to aggregate and process for countries, and so each decision is made in some isolation. 40 With these assumptions in hand we now write the value function of committee r in country i at time t:   VirP (1, wirt ) = Ri (1, wirt ) + δE VirP (1, wirt+1 )|1, wirt      VirP (0, wirt ) = Ri (0, wirt ) + max − ECir (wirt , ε irt ) + δE VirP (1, wirt+1 )|1, wirt , δE VirP (0, wirt+1 )|0, wirt While the payoffs (and thus strategies) of each player still depend on the treaty statuses of all other players through K, an issue we deal with in the following subsection, the assumption D3 has considerably reduced the dimension of the vector of variables that each player conditions his behavior on. Markov Perfect Equilibrium under Assumptions D1 - D3 We now define a Markov Perfect equilibrium of the treaty ratification game given assumptions D1 − D3. Note that by the time to build assumption, we can write the flow economic payoff to choosing action d for treaty office r in country i at time t as Πirt (d) = α a air + αh hir − d ∗ ECirt . Formally, let œ ≡ {σir (wirt , ε irt ) : i = 1, 2, ..., C; r = 1, 2, ..., R; } be a set of strategy functions, one for each treaty office in each country, such that σir maps from the space W × R into {0, 1}. Then σ is an MPE if for every treaty office in every country (i, r ) that has yet to ratify treaty r (i.e., xirt = 0) and every possible state (wirt , ε irt ): {σir (wirt , ε irt ) = 1} ⇐⇒      ε irt ≤ − ECirt + δE VirP (1, wirt+1 )|1, wirt − δE VirP (0, wirt+1 )|0, wirt , and {σir (xirt , ε irt ) = 1} for every treaty office in every country (i, r ) that has ratified treaty r (xirt = 1). In words, each treaty office that has yet to ratify ratifies the treaty if and only if doing so maximizes the value of the country given the state and the decisions of all other treaty offices in all countries. By the assumption of irreversibility treaty offices that have already ratified in a past period do not make a decision in the current period. 41 Then, a Markov Perfect Equilibrium of our dynamic game can be expressed as a vector P = { Pir (w)} of conditional choice probabilities (CCPs) such that for every (i, r, wirt ) such that xirt = 0:   
 Pir (wirt ) = Gε − ECirt + δE VirP (1, wirt+1 )|1, wirt − δE VirP (0, wirt+1 )|0, wirt (1.17) and Pir (wirt ) = 1 otherwise. More explicitly:  
Pir (wirt ) = Gε − ECirt + δ ∑ VirP (w0 ) f irw,P (w0 |1, wirt ) − f irw,P (w0 |0, wirt ) (1.18) w0 where f irw,P (wirt+1 |dirt , wirt ) is the transition probability of the vector of payoff relevant state variables w given equilibrium probabilities P. Finally, before moving to the estimation of the structural parameters, we need to address further the computational burden caused by the number of players in the game. Here the issue is estimation of the transition probabilities f irw,P (wirt+1 |dirt , wirt ) of the state vector wit . In more standard applications the transition of the exogenous payoff relevant state variables is estimated separately from the conditional choice probabilities using simple maximum likelihood methods, while the transition of the endogenous payoff relevant state variables which depend on the choices made by the players in the game is estimated jointly with the parameters of the model. In the current application, the large number of players in the game renders the standard method computationally infeasible. The transition probability of the payoff variables ait = ai (xt , zt ) and hit = hi (xt , zt ) depends on the actions of all players in the game. To minimize the computational burden, we estimate the transition of wit separately from the choice probabilities P, in a manner analogous to Hendel and Nevo (2006). Specifically, we consider a VAR process for the vector wi , allowing for country-specific transitions. Then using the estimated parameters of the VAR model, the transition probability vectors f irw,P (wirt+1 |dirt , wirt ) are calculated following Tauchen (1996). 42 Now we have the elements we need to estimate the full dynamic model. Given the model we have described above, the vector of structural parameters θ of interest is given by: θ = {α a , αh , {γi }iN=1 , {ξ r }rR=−11 , γde , γh , γdh } that is, the weights on foreign aid and human rights in the countries per-period payoff, the country specific entry cost parameter, the treaty specific entry cost parameters, and institutional cost parameters. We assume that the country-treaty private information ε irt is normally distributed with variance σε2 . Following Aguirregabiria and Mira (2007), we express the entry thresholds and conditional choice probabilities in a form that is more convenient for the purposes of estimation (see the appendix for a detailed derivation):   P P θ + ẽirt Pir (wirt ) = Φ z̃irt σε Estimator For notational simplicity, let us redefine θ = θ σε . For some arbitrary value of the parameters and entry probabilities θ, P, define the log likelihood function: T
L θ, P = Nt Ct ∑ ∑ ∑ dirt ln Φ


P P P P z̃irt θ + ẽirt + (1 − dirt ) ln 1 − Φ z̃irt θ + ẽirt (1.19) t =1 r =1 i =1 We estimate the model using the Nested Pseudo-Likelihood Estimator (NPL). As a way of providing a simple description of the estimator and explaining our reasons for choosing it, consider
first the two-step Pseudo-Maximum Likelihood (PML) Estimator. Specifically, let θ0 , P0 represent the true parameter vector and CCP vector in the population. The two-step PML Estimator of
the above likelihood function is a pair θ̂, P̂ such that P̂ is a consistent non-parametric estimator
of P0 , and θ̂ maximizes the Pseudo-Likelihood function L θ, P̂ . In many applications, in partic- 43 ular those involving permanent unobserved heterogeneity, the implementation of this estimator is problematic, because obtaining an unbiased estimator of the choice probabilities P̂ is not feasible, and thus the PML estimator is not an attractive option. Given the very likely importance of country and treaty specific permanent unobserved heterogeneity (the rough analogue of firm and market level permanent unobserved heterogeneity in more standard IO applications), we face such a problem in our model. The NPL estimator has a clear advantage in such cases, as a consistent estimator of choice probabilities is not required. One can obtain the NPL estimator in the following way. Given any (consistent or not) initial estimate of the true choice probabilities P0 , say
P1 , one may obtain the vector θ 1 that maximizes the pseudo-likelihood L θ, P1 . This estimate allows us to obtain an updated estimate of the choice probabilities P2 using the mapping described above: 1 1 P 1 P P2 = Φ z̃irt θ + ẽirt
With these estimates of the choice probabilities in hand, we now find the parameters θ that
maximize L θ, P2 , and again obtain a new estimate of the choice probabilities using these parameter estimates. We continue iterating in this fashion until the sequence of probability estimates
converges to the limit P∗ . The vector θ ∗ that maximizes L θ, P∗ is the NPL estimator.1516 Estimation Results For expositional purposes, we divide the presentation and discussion of the structural parameter estimates into three subsections, key payoff parameters, treaty specific costs, and country specific 15 See Aguirregabiria and Mira (2002,2007) for details. alternative method for estimating the model we have presented here is that of Benkhard, Bajari and Levin (BBL) (2007). There are two potential reasons a practitioner may prefer to use BBL over NPL. First, when using BBL one never has to invert a large matrix to solve value functions, whereas in using NPL we must solve the value functions once for each NPL iteration. Second, BBL allows for continuous variables in the state space, while NPL does not. Since many applications are naturally modeled with continuous state variables, this is an important consideration. However, we prefer the NPL estimator for several reasons. First, implementation of BBL requires a consistent estimate of the conditional choice probabilities, which as we argued above, is typically not available for applications with permanent unobserved heterogeneity. Second, NPL generally delivers more efficient estimates of the structural parameters. Finally, while it is true that not having to incur the computational cost of solving value functions is an important virtue, we will be using the estimated parameters to perform counterfactuals. In order to perform counterfactuals, regardless of the estimation procedure, one must solve the value functions. Given that we must endure this cost at the counterfactual stage in any case, the benefit of avoiding it at the estimation stage is not as large. 16 One Table 1.9: Structural Parameter Estimates 44 γ̄i 5.097 Mean Country Specific Ratification Cost γde 0.0610 ( 0.1611) 0.0509 (0.0394) -0.1032 (0.0380) 0.0945 ( 0.0167 ) 1.4390 ( 0.6355) 0.1268 14077 Contribution to Cost from Democracy Status γh γdh αa αh Pseudo-R2 N Contribution to Cost from Human Rights Status Democracy-Human Rights Interaction Weight on Aid in Payoff Function Weight on Human Rights in Payoff Function Notes: Standard errors should be interpreted with caution. We are not accounting for the variability of the first stage estimates in our calculation of the standard errors. costs. (a) Payoff Parameters The estimates of the weights α a , αh in the revenue function and the γ vector of cost parameters are presented in table 1.9 We find that countries have a positive preference both for aid and for human rights institutions. That countries have a positive preference for foreign aid in our game confirms that foreign aid benefits are not only associated with HRT participation, but they motivate a country’s decision to participate in an HRT. To quantify the relative importance of foreign aid and human rights in a country’s flow payoff, we can use the estimated parameters to calculate how much foreign aid a country at the 25th percentile of human rights institutions would be willing to forego in order to jump to the 75th percentile, and vice versa. We find that a country would be willing to pay $45.71 per capita in foreign aid to improve human rights institutions form the 25th to the 75th percentile (an increase of 3 on the human rights institutions score), and that a country would be willing to accept a drop of 3.55 on the human rights institutions score to jump from the 25th to 75th percentile in foreign aid (an increase of $54 per capita of aid). While these numbers are informative in that they help interpret the estimated weights in the payoff function, we should note that these calculations do not take into account the fact that changing domestic human rights institutions 45 Figure 1.4: Ratification Cost as a Function of Human Rights Institutions changes the cost of treaty participation and thus the cost of obtaining aid in the future, and moreover that this effect is heterogeneous across countries depending on political institutions. Further, the calculations do not take into account that changing human rights institutions, by changing a country’s payoff to ratification directly through cost, also effects the decisions of other countries which enter into country payoffs through foreign aid. Generally speaking, the calculations ignore the dynamic and strategic effects. Below we use counterfactual analysis to more fully quantify the relative importance of foreign aid and human rights institutions in explaining the ratification decision. Our regime cost estimates allow us to directly consider the hypothesis of Hathaway (2007) that democracies may ratify less frequently if domestic human rights institutions are such that the country is ex ante non-compliant with the treaty terms. We find that in the context of our model, this is not the case. Our results suggest that, for two otherwise identical countries, one democratic and one autocratic, regardless of the human rights record of the countries, the democratic country always pays a lower cost of ratification than the autocratic country and this difference increases as the quality of domestic human rights institutions improve. To illustrate this result in figure 1.4 we plot the estimated contribution to ratification cost from being a democracy and from being an autocracy as a function of human rights behavior, holding all else constant. 46 This result stands in contrast to the findings of Hathaway (2007), who argues that democracies with poor human rights records find it more costly to ratify than autocracies with poor human rights records, because democracies actually realize the cost of ratification through their domestic institutions while autocracies do not. We in fact find that, for a democracy, the cost of ratification decreases in the quality of human rights institutions, but that the opposite is true for autocracies. Ratification for an autocracy with good human rights practices is costly relative to an autocracy with good human rights practices. We acknowledge that the source of the difference between our findings and Hathaway’s may lie simply in the fact that we are looking at different sets of countries (we consider only recipients of OECD aid, while Hathaway considers both aid donors and recipients), or in the fact that she considers specific treaties and human rights individually while we are studying more general measures of human rights behaviour and treaty participation. However, we must also consider the possibility that the difference in our results is being driven by differences in our methodological approaches. First, in our analysis we are controlling for a variable that Hathaway “omits" from hers: the benefits of ratification. Second, we are controlling for country specific heterogeneity both in costs and benefits to ratification. 17 Ignoring permanent unobserved heterogeneity in ratifica- tion decisions can result in biased estimates of ratification costs if the unobserved heterogeneity is also correlated with the democracy and human rights measures in the cost functions. But supposing the cost estimates presented here are correct, we face the burden of rationalizing what we see in figure 1.4. The costs of HRT ratification in Hathaway’s theory are largely domestic costs which depend on the ability of actors within the country to hold the government to the terms of the treaty. It is plausible that the cost of ratification actually has a significant non-domestic dimension in the sense that the international community monitors and either implicitly or explicitly punishes non-compliance. As we discussed above, we envision the cost of ratification itself being the sum of two costs, the cost of complying (given that the ratifier complies) and the cost of not 17 We are sympathetic to the fact that Hathaway(2007) employs hazard analysis, with which it is difficult to allow for permanent unobserved heterogeneity. 47 complying. To take an extreme case, suppose that democratic countries comply and autocratic countries do not. As long as the costs from non-compliance are severe enough, autocracies will always have a larger cost to participation. While the source of the cost of complying is relatively clear, this is not the case for the cost of not complying. One possibility is reputation (Goodman and Jinks, 2004). Countries who ratify a treaty and later flout its terms risk their international reputation and thus their opportunities in other realms of global interaction. However we still need to explain the widening gap in costs as domestic human rights institutions improve. That the cost of ratification for a democracy decreases as the quality of domestic human rights institutions improves is intuitive. Democracies are likely to comply with the terms of the treaty, and compliance is cheaper if the democracy is ex-ante compliant. Then why does the cost of ratification for an autocracy increase in the quality of its human rights institutions? We have argued that an autocracy’s ratification cost is likely to be comprised primarily by the cost of not complying, which would be imposed by the international community. Suppose further that non-compliance is not perfectly observed by the international community. The international community relies on media and NGO presence, for example, for information about violations of treaty commitment. The human rights institutions variable measures, among other things, freedom of media, freedom to demonstrate and open public discussion. A country with high quality human rights institutions, according to our measure, is thus more likely to have any violations of HRT commitments become known to the international community, and is thus more likely to realize the cost of non-compliance. Thus, an autocracy may actually have a higher cost of ratification if it has high quality human rights institutions. Of course without more detailed data on human rights institutions we can not verify this hypothesis, but it does present one way to rationalize our findings. (b) Treaty Specific Costs The estimates of the treaty specific costs are in table 1.10. The most striking pattern in the treaty specific cost estimates is the substantial variance in the costs across treaties. For example, the cost of ratifying the Convention on the Rights of the Child 48 Table 1.10: Structural Parameter Estimates : Treaty Specific Costs ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ 10 ξ 11 ξ 12 ξ 13 ξ 14 ξ 15 N -0.0213 ( 0.1322) -0.1811 ( 0.1638 ) 0.3811 ( 0.1607 ) -1.5183 ( 0.4390) 0.8208 (0.2079 ) 1.2245 ( 0.2416 ) 0.0385 ( 0.1373 ) -0.5286 (0.1787) 0.4105 (0.1469 ) 0.5587 (0.1605) 0.7794 (0.1809) 0.3923 (0.1630 ) 0.1282 (0.1648 ) -0.0039 (0.1362) 14077 Economic, Social and Cultural Rights Civil and Political Rights Elimination of All Forms of Racial Discrimination Prevention and Punishment of the Crime of Genocide Rights of Child Protection of the Rights of All Migrant Workers and Members of their Families Non-applicability of statutory limitations to war crimes and crimes against Humanity Suppression and Punishment of the Crime of Apartheid Elimination of All Forms of Discrimination against Women Against the Taking of Hostages Prohibitions or Restrictions on the Use of Certain Conventional Weapons Protection of Performers, Producers of Phonograms and Broadcasting Organisations Political Rights of Women Status of Refugees Torture and Other Cruel, Inhuman or Degrading Treatment or Punishment Notes: Standard errors should be interpreted with caution. We are not accounting for the variability of the first stage estimates in our calculation of the standard errors. 49 is nearly half the cost of ratifying the treaty on The Non-Applicability of Statutory Limitations to War Crimes and Crimes Against Humanity. To understand this result, let us consider closely two treaties, the Convention on the Elimination of all Forms of Discrimination against Women (CEDAW) (1979) , and the Convention on the Political Rights of Women (CPRW) (1952). We choose these treaties to illustrate our point because they address a similar issue, but have very different estimated costs of ratification: CPRW is much more costly to ratify than CEDAW is. This is somewhat puzzling. Ex-ante, if one treaty was more costly than another, we would naturally expect CEDAW to be more costly. Discrimination against women in the political sphere is merely one form of discrimination against women. If a country is willing to ratify a treaty concerned with all forms of discrimination against women (CEDAW), it should be willing to ratify a treaty concerned with just one type of discrimination against women (CPRW). Consider articles 1-2 of CPRW: Article 1 : Women shall be entitled to vote in all elections on equal terms with men, without any discrimination. Article 2 : Women shall be eligible for election to all publicly elected bodies, established by national law, on equal terms with men, without any discrimination. By contrast, consider now articles 2 and 3 of CEDAW (article 1 simply defines “discrimination against women"): Article 2 : States Parties condemn discrimination against women in all its forms, agree to pursue by all appropriate means and without delay a policy of eliminating discrimination against women, and to this end undertake: – To embody the principle of the equality of men and women in their national constitutions or other appropriate legislation... – To adopt appropriate legislative and other measures, including sanctions where appropriate, prohibiting all discrimination against women... 50 Article 3 : States Parties shall take in all fields, in particular in the political, social, economic and cultural fields, all appropriate measures, including legislation, to ensure the full development and advancement of women... on a basis of equality with men. In particular, note the ease with which articles 1-3 of CPRW can be verified. Election monitoring by reputable organizations deployed by the OECD and the EU occur regularly in countries all over the world.18 Further, the monitoring is generally done over a long period prior to the election itself. Any failure to comply with any of articles 1-2 of CPRW would not go unnoticed, and at the very minimum would be brought to the attention of the international community. On the other hand, what is demanded by CEDAW is to a great extent open to interpretation. Verification of compliance ex-post of ratification would require consensus in the international community on whether a violation has occurred. In a world where strategic alliances are important but fluid, the set of countries who agree that another’s actions violate the terms of a treaty can be small and uncertain. One hypothesis then is that the more open to interpretation, or more difficult a violation is to verify, the cheaper the ratification cost should be. We discuss this hypothesis and its possible implications in more detail in the appendix. (c) Country Specific Costs There are significant permanent differences across countries in the cost of ratification. To illustrate this variation, in table 1.11 we display the country-specific cost estimates for ten of the eighty-three countries in our sample, and the variance and coefficient of variation of the costs. In figure 1.5 we plot a histogram of country specific costs. Nearly all the country-specific costs are statistically significant. The range over which country specific costs vary is larger and more economically significant that the range over which treaty specific costs vary. Mauritania has the smallest estimated cost, and Malaysia has the largest. We purposely include in the table cost estimates for countries that were discussed above. In particular, figures 1.1, 1.2, and 1.3 comparing the HRT participation of Argentina and Chile, Kenya and Chad 18 See for example the Handbook for European Union Election Observation (2008) 51 Table 1.11: Structural Parameter Estimates Mauritania El Salvador Chile Argentina Uganda Kenya Somalia Chad Turkey Malaysia Variance Coefficient of Variation 0.4173 (1.227) 4.260 (0.6075) 4.2903 (0.6023) 4.3504 (0.6041) 4.7963 (0.7281) 5.383 (0.8005) 5.5389 (0.745) 6.1012 (1.003) 7.3958 (1.0141) 11.9950 (1.729) 3.0414 0.3422 Notes: Standard errors should be interpreted with caution. We are not accounting for the variability of the first stage estimates in our calculation of the standard errors. Figure 1.5: Frequency Distribution of Country Specific Treaty Costs 52 and Uganda and Chad respectively as well as figure 1.9 in the appendix comparing Argentina and Somalia all are consistent with these cost estimates. 1.6 Counterfactual Experiments In this section we use the estimated model to consider counterfactual policy experiments. The two goals of counterfactual analysis here are first to evaluate the relative importance of potential competing theories of HRT participation, and second to evaluate equilibrium effects of alternative policies. Performing counterfactual analysis in dynamic games is often complicated by the potential multiplicity of equilibria. How behaviour responds to a change in the structural parameters of interest directly depends on the equilibrium played under the counterfactual parameters. If the model has multiple equilibria, we can not know which equilibrium is played in the counterfactual scenario, and thus we can not make counterfactual predictions.19 Many applications in the dynamic game literature simply get around this problem by assuming a unique equilibrium, or assuming that the counterfactual equilibrium is the same as the equilibrium in the data. These assumptions are strong and in many cases not realistic. Here we adopt the approach of Aguirregabiria (2009). Let (θ̂, Pˆ0 ) represent the estimated parameters and equilibrium choice probabilities
(i.e., P0 = Φ θ̂, P0 ). Then under the assumption that the equilibrium selection mechanism is a smooth function of the structural parameters, by taking a Taylor expansion around the estimated structural parameters θ̂ one can obtain an approximation to the counterfactual equilibrium associated with the counterfactual parameters θ ∗ , say P∗app . 20 As we discussed above, it is difficult to assess the relative importance of purported key variables such as foreign aid receipts in the treaty ratification decision simply by looking at the estimated parameters. One of the key benefits of estimating a structural model which allows for 19 If we had a method to solve for every equilibrium in the model we could at least say what all the potential counterfactual scenarios would be. Note however that if the number of equilibria is very large model has little ability to predict behaviour under counterfactual scenarios. 20 The approximation error can be quite large if the counterfactual is large (i.e., if θ ∗ is very different from θ̂). Supposing the error is small enough so that P∗app lies in the dominion of attraction of the counterfactual equilibrium P∗ , by Pk Pk
starting from P∗app and iterating in the mapping Pk+1 = Φ z̃irt θ + ẽirt we will reach the counterfactual equilibrium P∗ . 53 several competing theories is that we can “shut down" one theory while allowing for the other(s) and observe how much the original patterns in the behavioral responses we were interested in changed. This is precisely what we do in the first set of counterfactual experiments. To get a sense of the importance of foreign aid receipts in the treaty participation decision, we consider a counterfactual world where the weight on foreign aid receipts in the payoff function is 10% lower than in the factual world. In the terms discussed above, in this counterfactual world a country would be willing to forego $50.79 in foreign aid (as opposed to $45.71) for a jump from a 2.0 score on human rights institutions to 5.0. In figure 1.6 we present a plot of the number of treaties ratified for each country in the factual and counterfactual equilibrium scenarios. To obtain the number of treaties ratified we create a dynamic sequence of decisions using the equilibrium choice probabilities in both the factual and counterfactual scenarios, starting from the same state in each case. The average number of treaties ratified across countries drops from 6.78 under the factual equilibrium to 4.5 in the counterfactual scenario, a drop of about 33%. The standard deviation of the number of treaties ratified increases from 2.56 in the factual scenario to 2.89 in the counterfactual scenario. We also plot treaty capital K in the two scenarios in figure 1.7. While we can not learn anything by comparing the means across the two scenarios (the mean across countries of K is always 0 by definition), it is worthwhile to note the difference in standard deviations across the two equilibria. In the factual case the standard deviation is 2.52 while in the counterfactual case it is 3.42. Though aid receipts are an important factor in explaining HRT ratification behavior, the same is not true for human rights. We consider a counterfactual analogous to the one we considered above, and reduce the weight on human rights in the payoff function by 10%. In this counterfactual world a country is now willing to drop almost 4 points in human rights institutional quality for an increase from the 25th to 75th percentile in aid receipts. In this case, the mean number of treaties ratified in the factual case is 6.78, and in the counterfactual case 6.80. In the context of our model, aid is by far the most significant factor in the treaty ratification decision, at least on the benefit side. In the second set of counterfactuals, we try to determine whether country-specific benefit het- 54 Figure 1.6: Number of Ratified Treaties Under Factual and Counterfactual Equilibria Figure 1.7: K Under Factual and Counterfactual Equilibria 55 Figure 1.8: Number of Ratified Treaties Under Counterfactual Equilibria with No Heterogeneity erogeneity or cost heterogeneity is more important in explaining the heterogeneity we observe in ratification decisions. Recall that in the structural model we allow for both permanent differences in the flow of foreign aid into a country (equation 1.13), and permanent differences in the ratification costs across countries. The experiment we consider here then is to shut down each source of heterogeneity in turn, and compare ratification behavior in each counterfactual scenario. By comparing the variances in treaty participation across these two counterfactual cases we get an idea of whether heterogeneity on the cost side or benefit side explains more of the participation decision; if the variance across countries is larger in the case where there is no cost side heterogeneity we can conclude that more of the heterogeneity in behaviour is explained by benefit side heterogeneity. Similarly, if the variance is larger in the case where there is no benefit side heterogeneity, we would conclude the cost side differences are more important. In figure 1.8 we plot the number of treaties ratified in each case. The key result is that the standard deviation of the number of treaties ratified in the counterfactual case of no benefit heterogeneity (2.89) is smaller than the standard deviation of the number of treaties ratified in the case of no cost heterogeneity (3.74). This tells us that benefit heterogeneity is more significant in explaining heterogeneity in behavior. This is result is particularly important because benefit heterogeneity has been overlooked as a factor in explaining heterogeneity in behavior in past studies of HRT participation. 56 1.7 Conclusion We model the decision of aid receiving countries to participate in human rights treaties at the United Nations. Using dynamic panel data techniques, we first establish that there are significant economic returns to treaty ratification. Countries that participate in HRT’s receive more foreign aid than those that do not. We further establish that countries with good human rights practices are more likely to participate than countries with bad practices, but that the converse is not true. Countries with relatively large levels of predetermined treaty participation experience a decline in human rights practices. Motivated by these findings, we develop and estimate a dynamic game of treaty ratification. The dynamic model we consider is analogous to a dynamic game of oligopoly competition where firms invest in quality (eg.,Pakes and McGuire (1994)). Here, aid receiving countries compete to attract economic resources from the developed world by ratifying costly human rights treaties. We estimate the costs and benefits of ratification, allowing for heterogeneity across treaties and regimes. We find that the attendant economic returns to ratification induce countries to ratify. This is a contribution of the paper, as the literature on treaty ratification has generally focused on the cost of ratification as opposed to the possible benefits in rationalizing observed behaviour. We also find that ratification costs vary significantly across regimes and across treaties. Specifically, autocracies with poor human rights behavior have a larger cost of ratification than democracies with poor human rights behavior, but this difference shrinks as human rights behavior improves. We are the first to examine how costs may vary across treaties. We find significant variance in the cost of ratification, and discuss how this variance may be explained by observable institutional details of the treaties, in particular the verifiability of treaty terms. In future work we hope to address this issue in more detail, as this result can have interesting policy implications for the design of international treaties from a welfare perspective. We then use the estimated model to consider several counterfactual experiments. We evaluate the relative importance of aid and domestic human rights as factors for explaining ratification 57 behavior, and we find that aid is by far the more significant factor. Decreasing the aid motive by 10% decreases ratification by 33% while a 10% decrease in the human rights motive hardly changes ratification behavior at all. We also evaluate the relative importance of benefit side and cost side heterogeneity in explaining heterogeneity in behavior and find that benefit heterogeneity is more important. 58 Figure 1.9: Argentina and Somalia Treaty Capital Figure 1.10: Argentina and Somalia ln(Aid) Receipts 1.8 1.8.1 Appendix Treaty-Specific Cost Heterogeneity and Observable Treaty Characteristics There are several candidate characteristics that vary across treaties that measure how “open to interpretation" a treaty is. For example each treaty contains a section on reservations ratifiers may have with respect to the contents of the treaty, and objections existing parties may have to the reservations. Objections and reservations are easier to make (and perhaps more necessary) when the treaty is less interpretable. Interpreting the treaty specific cost as the choice of the treaty designer (x-axis) and the rate of reservation/objection as the outcome (y-axis), in figure 1.13 we 59 Figure 1.11: Argentina Treaty Capital and ln(Aid) Receipts Figure 1.12: Somalia Treaty Capital and ln(Aid) Receipts 60 Figure 1.13: Reservations/Objections per Ratifier and Treaty Costs Figure 1.14: Treaty Costs and Words per Article plot the rate of reservation/objection per treaty participant against the estimated treaty specific entry costs. There appears to be a negative relationship here; the larger the treaty specific cost, the fewer reservations and objections made per treaty ratifier. This then begs the question, what exogenous characteristic of HRT’s determines the costliness of ratification? In light of our example above, in figure 1.14 we plot the estimated treaty specific entry costs against the number of words per article for each treaty. Consistent with the example in the text, treaties with more words per article are cheaper to ratify. This raises potentially interesting optimal institutional design questions. An optimal treaty design balances verifiability and enforceability of the treaty’s terms with the 61 number of participants to the treaty. While some of these questions could be addressed in the context of our structural model, we leave this for future work. Treaty International Covenant on Economic, Social and Cultural Rights International Covenant on Civil and Political Rights International Convention on the Elimination of All Forms of Racial Discrimination Convention on the Prevention and Punishment of the Crime of Genocide Convention on the Rights of child International Convention on the Protection of the Rights of All Migrant Workers and Members of their Families Convention on the non-applicability of statutory limitations to war crimes and crimes against humanity International Convention on the Suppression and Punishment of the Crime of Apartheid Convention on the Elimination of All Forms of Discrimination against Women International Convention Against the Taking of Hostages Convention on Prohibitions or Restrictions on the Use of Certain Conventional Weapons which may be deemed to be Excessively Injurious or to have Indiscriminate Effects International Convention for the Protection of Performers, Producers of Phonograms and Broadcasting Organisations Convention on the Political Rights of Women Convention relating to the Status of Refugees Convention against Torture and Other Cruel, Inhuman or Degrading Treatment or Punishment Table 1.12: Human Rights Treaties 10/4/1981 26/10/1961 20/12/1952 28/7/1951 10/12/1984 Date Opened 19/12/1966 19/12/1966 7/3/1966 9/12/1948 20/11/1989 18/12/1990 26/11/1968 30/11/1973 1/3/1980 18/12/1979 62 63 1.8.2 Representation of Choice Probabilities   P θ + ẽP We show how we arrived at the expression Pir (wirt ) = Φ z̃irt irt explicitly. For notational σε simplicity, let us redefine θ = θ 21 σε . Recalling that wirt contains all payoff relevant information for treaty office r in country i at time t, for the sake of clarity where necessary we will write separately the ratification status and all other payoff relevant variables in wirt as ( xirt , wirt ), with it being P (1, w ), vP (0, w ) represent the choice-specific understood that xirt is included in wirt . Let virt irt irt irt values to ratification and non-ratification of treaty r in state wirt . The entry threshold for treaty office r in country i is given by the difference in the choice specific values: P P ∆ir = virt (1, wirt ) − virt (0, wirt ) Treaty r is ratified in country i when the private information shock ε irt is no larger than this difference, i.e., when P P virt (1, wirt ) − virt (0, wirt ) ≥ ε Then we have the ex-ante probability of ratification in any ( xirt , wirt ) : P P Pir (0, wirt ) = Φ (virt (1, wirt ) − virt (0, wirt ))/σε
Pir (1, wirt ) = 1 (1.20) (1.21) P (1, w ) − vP (0, w ) = z̃P θ + ẽP . Define the integrated What we need to show now is that virt irt irt irt irt irt value function of treaty office r in country i as:    P P VirP (w) = Eε max virt (1, wirt ) − ε irt , virt (0, wirt ) 21 Note however that given the structure of the model we are able to identify σε separately. 64 The model we have described above implies that: P virt (1, wirt ) = Rirt (wirt ; θ ) − (1 − xirt ) ECirt (wirt ; θ ) + δFirw,P (1, wirt )0 Vir P virt (0, wirt ) = Rirt (wirt ; θ ) + δFirw,P (0, wirt )0 Vir where Vir is the vector of values (as many value functions as there are values of wirt ) and Firw,P (0, wirt ) and Firw,P (1, wirt ) are vectors with transition probabilities. We are able to express the discounted expected continuation values as δFirw,P ( xirt , wirt )0 Vir by our assumption that the private information shock ε irt is independent from the state variables wirt . Note that we can express the choice specific values in this way even though we assume that entry is irreversible, as long as we have Pir (wirt ) = 1 for any vector wirt such that xirt = 1. Then, by the definition of the integrated value function, we have that:
P P P VirP (wirt ) = 1 − Pir (wirt ) virt (0, wirt ) + Pir (wirt )virt (1, wirt ) + eirt (1.22)   P P P eirt = Pir (wirt ) E ε irt |virt (1, wirt ) + ε irt > virt (0, wirt ) + (1 − Pir (wirt ))0 (1.23) where
P = σ φ Φ−1 ( P ( w )) (see AguirIn the case of normally distributed error, we have that eirt ε ir irt regabiria and Mira (2007) for details). Now, by substituting the expressions for the choice-specific values into the above expression for the integrated value function, we get:   VirP (wirt ) = Rirt (wirt ; θ ) + Pir (wirt ) − (1 − xirt )Cirt (wirt ; θ ) + δFirw,P (1, wirt )0 VirP +
  P 1 − Pir (wirt ) δFirw,P (0, wirt )0 VirP + eirt 65 By stacking the value functions by state w, we can express this more compactly as: VirP = Rir (θ ) + Pir × [−(1 − xir )Cir (θ ) + δFirw,P (1)VirP 
  1 − Pir × δFirw,P (0)VirP + eirP + where × denotes the element-wise product. Further, define: Firw,P ≡ (1 − Pir ) × Firw,P (0) + Pir × Firw,P (1) A ≡  I − δFirw,P −1 Then we have:
VirP = A ∗ Rir (θ ) − Pir × [(1 − xir )Cir (θ )] + A ∗ eirP Now recalling the definition of the entry threshold and the expressions for the choice-specific values, we write the vector of entry thresholds as: ∆irP = virP (1) − virP (0)
= −(1 − x)Cir (θ ) + δ Firw,P (1) − Firw,P (0) Vir   P = −(1 − x)Cir (θ ) + δFw,P A ∗ R ( θ ) − P × [( 1 − x ) C ( θ )] + δFw,P ir ir ir D D A ∗ eir
w,P w,P P = δFw,P D A ∗ Pir − 1 × (1 − x ) Cir ( θ ) + δF D A ∗ Rir ( θ ) + δF D A ∗ eir w,P w,P where Fw,P D ≡ Fi,r (1) − Fi,r (0). Note first that the parameters of the model θ enter into the thresholds in the first two terms of the expression only. Further, we have assumed that the functions Rirt (wirt , θ ) and ECirt (wirt , θ ) are linear in the parameters θ. We can therefore express the vector of entry thresholds as ∆irP = Z̃Pir θ + ẽPir Chapter 2 The Value of Information in Regulation 2.1 Introduction We consider the choice of environmental policy by an imperfectly informed regulator whose choice of regulation is constrained by the ability of regulated firms to block regulation. We compare the value to the regulator of two distinct types of information, information about the social cost of pollution and information about the profitability of firms. Determining when the regulator would prefer learning the social cost of pollution to learning about firm profitability is important for three reasons. First, cost-benefit analysis has long held the status as the fundamental yardstick in policymaking, though perhaps by default. As Richard Posner argues, “My own justification for using cost-benefit analysis... [is based on] ...what I claim to be the inability of judges to get better results using any alternative approach" (Posner, 2000). The Clinton administration institutionalized the use of cost-benefit analysis in public decision making with Executive Order 12866, mandating that agencies “propose or adopt a regulation only upon a reasoned determination that the benefits of the intended regulation justify its costs" (W.J. Clinton, 1993). Consequently, public policy decisions worth billions of dollars are made on the basis of cost-benefit analysis each year. Given this, it is important to assess the extent to which public decisions can be improved by better information. Further, in environments with several sources of uncertainty, determining what type of information to learn is of practical use. Second, cost-benefit analyses are notoriously inaccurate. One particularly relevant example is the uncertainty surrounding the cost of meeting the Kyoto Protocol targets. In a study comparing the cost estimates from eight different sources, the Energy Information Administration1 finds 1 Information available at the EIA Kyoto website: http://www.eia.doe.gov/oiaf/kyoto/cost.html 66 67 significant variance in estimated costs along many dimensions. For example, estimates of annual GDP loss in the United States from meeting the targets range from 91 to 311 billion dollars, while estimated carbon prices range from 147 to 360 1996 dollars per metric ton. Thus, at least seven out of the eight projections are inaccurate to some degree. This wide range for the estimates indicates that there is considerable “noise" present in cost-benefit analysis. By characterizing the value of information we address the question of how much the noise matters, and which type of noise is most worth reducing. Finally, executive order 12866 actually requires cost benefit analyses only for regulatory actions “likely to result in a rule that [has] an annual effect on the economy of 100 million or more," (W.J. Clinton, 1993). That is, US policy makers have a binding research budget, and this budget is currently being directed to large regulatory decisions. Our goal is to provide a basis for refining the way this research budget is allocated. In particular, we determine conditions where equally costly research capacity is best directed at learning the social cost of pollution and, alternatively, where this research capacity is best directed at learning the cost structure of regulated firms. To fix ideas, we consider the specific case of environmental tax policy. Environmental policy is necessarily chosen under uncertainty about both the costs (lost profit to firms) and benefits (reduced pollution) of any particular regulation. Moreover, environmental regulators are often constrained by industry influence on the legislative process, making the probability of implementation a central consideration in the choice of tax. Thus, environmental regulation seems a useful example of our more general problem. We find that the value of learning costs relative to learning benefits hinges on how politically powerful firms are, where we say firms are “politically powerful" if a marginal increase in tax has a large effects on the regulator’s ability to implement the policy. Our primary finding is that when firms are not politically powerful, information about how costly pollution is to consumer welfare (benefits of regulation) is more valuable than information about the types of firms in the economy (costs of regulation). On the other hand, when firms are politically powerful, it pays more to learn precisely what types of firms are in the economy. 68 The intuition behind this result is best seen by considering an extreme case. If firms possess no lobbying ability and take any policy as given, all a regulator needs to know to set a first-best tax is how costly pollution is to consumers. With powerful firms, by contrast, the regulator faces a trade-off: choosing a tax high enough so that if it passes the legislative process only efficient firms remain in production, or choosing a regulation that is more likely to pass but that leaves some inefficient firms in the market. Information about firm types helps the regulator make this trade off optimally. 2.2 Literature Review The political economy of regulation has its roots in the work of Olson (1964), Stigler (1971) and Becker (1985). Broadly speaking, the contribution of this literature is to offer a convincing theory of “who gets regulated and why". While Olson (1964) frames lobbying by regulated parties as a collective choice problem and examines how the regulated group’s power to influence regulation depends on collective action, it is Stigler (1971) who first developed a theory of the demand for regulation. He argues that the state, which has the unique ability to “prohibit or compel, to take or give money," is largely at the service of well organized groups. Becker (1985) formalizes the ideas developed in the earlier papers into a model of strategic competition among pressure (interest) groups for political influence. This canonical model of regulation largely neglects the role of information and uncertainty. Lewis (1996), the foundation of the present work, remedies this by introducing regulatory uncertainty into the Becker model. Lewis’ model incorporates the political constraints of Olson and Becker in the sense that the probability of implementing a policy decreases in the aggregate losses incurred by firms. Lewis also assumes the regulator is unable to observe the type (profitability) of any particular firm in the economy, and shows that in this context a welfare maximizing planner chooses an inefficient policy. Lewis’ planner would ideally shut down all socially inefficient firms, but a tax which accomplishes this objective is costly both for firms remaining in the industry and 69 for firms which are forced to shut down. A lower tax, which places a smaller burden on efficient firms, is more likely to be implemented. On the other hand, with a lower tax, some inefficient firms remain in the industry. The optimal (second best) policy is one that does not regulate almost efficient firms but induces very inefficient firms to exit. The optimal policy is inefficient in the sense that, even if the policy survives the legislative process, firms whose profit is smaller than the externality cost they create go unregulated. To determine what type of information is most valuable and when, our model departs from Lewis’s in that we assume the planner is not only uninformed about firm types, but also about the social cost of pollution. In Lewis’s model the planner knows the exact cost of pollution. Our purposes are also different. The punchline of Lewis’s paper is that, although the first best is achieved by setting tax equal to cost, the regulator chooses a lower tax when he doesn’t know firm types and must be concerned with lobbying. Lewis is able to characterize the second best tax in this instance. We start from Lewis’s model knowing that a second-best tax will be chosen, and show how the second best tax and welfare varies with information. There is a literature, e.g., Laffont and Tirole (1993), which seeks to characterize an optimal regulatory mechanism under imperfect information. While this literature is broadly related to the current paper, it seeks to answer a fundamentally different question. Instead of characterizing an optimal regulatory strategy under imperfect information, the current exercise takes as exogenous a stylized description of prevailing regulatory institutions and examines the value of information in the context of these institutions. Informally, the existing literature seeks to inform a more ambitious policy reform than we do. The existing literature asks how we can best constitute our regulatory agencies. We ask, more simply, how should existing regulatory agencies allocate their research efforts. That is, we take as given the simpler observed mechanism and ask, given this mechanism, which type of information is worth more to a welfare maximizing regulator. Value of Information The statistical measure of the value of information is the expected value of an informed decision less the expected value of the uninformed decision. To illustrate this idea,2 70 consider a risk-neutral gambler who is offered the following gamble: A fair coin will be tossed once. If the gambler predicts the side of the coin that faces up correctly, he wins a dollar, while if the prediction is incorrect, he gets nothing. The gambler can expect to earn 50 cents from the gamble, and so will pay at most fifty cents to play. Now suppose the gambler knows that before making his “prediction", the true outcome of the coin toss will be revealed to him. In this case the gambler expects to earn a dollar from the gamble. Whatever the outcome of the coin toss, he will be able to predict it correctly. In this simple example, the value of information is fifty cents: the expected value of a fully informed decision minus the expected value of an uninformed decision. In the context of our problem, the “gambler" is a welfare maximizing regulator with the option to learn about either the profitability of the firms present in the economy, or the cost they impose on society. We perform calculations analogous to the one above in order to determine when each type of information is more valuable. 2.3 Model We consider the choice of tax by a regulator who is initially uncertain about firms’ profits and about the social cost of pollution. The regulator chooses a tax to induce the exit of firms whose profit does not cover the social cost of production. The implementation of the tax is less likely as the losses to firms increase. After deriving the optimal taxes under different information structures we determine what information is most valuable to the regulator in different circumstances. To be more precise, we consider an economy where heterogeneous firms make a binary decision to either produce or not. Firms that produce earn some profit, and firms that do not earn zero profit. We assume that the firms face perfectly elastic demand.3 Through productive activity, firms 2 See Arrow and Fisher (1973) for an early application of the value of information in economics. A more rigorous exposition of the value of information, particularly in the context of irreversibility, is provided in Freixas and Laffont (1984) 3 While the assumption of perfectly elastic demand is not crucial, it does simplify the value of information calculations considerably. In the absence of this assumption, when the regulator sets a tax he must take into account that the number of firms in the economy will typically decrease when the tax is higher, and the market price will depend on the tax. 71 impose a social cost (pollution) on consumers. We assume that, whatever this cost is, it is the same for all firms. The regulator’s proposed policy may be blocked by firms, and the probability of the policy being blocked increases in the losses the tax imposes on firms. We obtain optimal taxes and (expected) social welfare for each possible information set, or ‘type’ of planner, and calculate the statistical values of different types of information. Formally, the set of firm types in the population is Θ = {θ L , θ H }. The population frequency of low types θ L is α ∈ [0, 1]. The θi ’s describe firm profit conditional on entry, with a firm of type θi earning profit θi gross of any tax. For expositional clarity we suppose that at most two firms may operate in the economy. Let i ∈ {1, 2} index the realized set of firms. The sample of realized firms, (θ1 , θ2 ), is binomially distributed, with parameters (2, α), with α ∈ [0, 1]. That is,   Prob (θ1 , θ2 ) = (θ L , θ L ) = α2 ,     Prob (θ1 , θ2 ) = (θ L , θ H ) = Prob (θ1 , θ2 ) = (θ H , θ L ) = α(1 − α),   Prob (θ1 , θ2 ) = (θ H , θ H ) = (1 − α)2 . The social cost of pollution, c, takes one of two values, {c L , c H }, where the population frequency of low costs is q ∈ [0, 1]. Thus there are six possible states of the world, with representative element s ∈ S. The distribution of probability µ(s) over the set of states is given in table 2.3. A planner who learns c will be said to “learn the social cost of pollution", while a planner who learns (θ1 , θ2 ) will be said to “learn firm types". This model is among the simplest in which we can sensibly talk about more than one type of information. Under any proposed policy, firms either pay the tax and remain in production, or exit and pay nothing. The losses firms incur are the sum of profit forgone by firms that exit (those with profit less than the size of the tax), and the taxes paid by firms who remain in production. The firm’s stay/exit decision is completely determined by whether profit net of the tax is positive or negative. We assume that indifferent firms exit. Alternatively, we can imagine that the regulator faces a single firm which produces 0, 1, or 2 72 Table 2.1: Distribution of probability over states State s {θ L , θ L , c L } {θ L , θ H , c L } {θ H , θ H , c L } {θ L , θ L , c H } {θ L , θ H , c H } {θ H , θ H , c H } Prob(s) ≡ µ(s) α2 q 2α(1 − α)q (1 − α )2 q α2 (1 − q ) 2α(1 − α)(1 − q) (1 − α )2 (1 − q ) units of output. θi denotes the firm’s profit from producing unit i. The firm’s production decision hinges on whether the profit from a marginal unit is sufficiently high to cover the tax. This is precisely the logic behind the Pigouvian tax. With this alternative interpretation noted, we adhere to our original interpretation for the remainder of the paper to preserve the parallel with Lewis (1996), the motivation of the present work. To capture the idea that lobbying makes implementation uncertain we follow Lewis (1996), and suppose that if a tax τ imposes aggregate losses L on firms, the probability of the policy being implemented is P(L), where P0 ( L) < 0 and P00 ( L) < 0.4 In words, as aggregate losses increase the chances of the policy passing decrease at an increasing rate. From the planner’s perspective, proposed policy is not implemented with certainty, so that choosing a tax is a gamble. The political power of firms depends on the shape of the P(·) function. If P(·) is steep, small changes in the losses the policy imposes on firms greatly decrease the probability of the policy being implemented. In other words, marginal changes in L are important, and in such a case we say that firms are politically powerful.5 We also assume: Ec > θ H > c L ≥ θ L , 4 We are assuming here that while firms have the ability to organize and (probabilistically) affect regulation outcomes, consumers do not. One could imagine a more general model where consumer and industry interest groups compete over the regulation outcome, as in Grossman and Helpman (2001). 5 One could imagine allowing firms to make a lobbying decision where there is some effort cost to lobbying intensity. In this case strategic interactions both between firms and between firms and the regulator become important. In some instances firms may find it profitable to free-ride on the lobbying effort of other firms, which the regulator must consider when choosing a tax. Further, the regulator must also take into account the fact that lobbying intensity will depend on the tax he selects. While these issues are potentially interesting in their own right, these extensions render our model intractable. 73 where Ec denotes the expected cost of pollution, qc L + (1 − q)c H . θ H > c L means that when the planner draws the sample of firm types and the social cost of pollution, there is strictly positive probability that at least one firm is efficient. c L ≥ θ L means that, for whatever social cost the planner has drawn, there is strictly positive probability that the planner has drawn an inefficient firm. We discuss the role and restrictiveness of each of the three inequalities in turn. First consider the assumption Ec > θ H . This assumption does not effect the main results and is made in order to keep the scope for regulation large. Notice that without this assumption, a planner informed about firm types who realizes {θ H , θ H } would not want to set a tax at all. Avoiding such cases keeps the analysis more tractable (see section 5 below). The assumption θ H > c L is made to keep the regulator’s problem interesting. If instead θ H < c L there are no efficient firms. Finally consider the assumption c L ≥ θ L . If instead c L < θ L , if the planner chooses to learn social cost first and realizes c = c L , he does not wish to set any tax; any firm draw is efficient. As in the case of Ec > θ H , this assumption is made to keep the analysis more tractable. We argue that this assumption is not restrictive, however. To see this, suppose that instead of what was presented above, our planner draws the firms from a continuum Θ = [θ, θ ] and social cost from a continuum C = [c, c]. In this alternative framework, as long as Θ ∩ C 6= ∅ and c ≥ θ, this assumption holds. This alternative setup would also ensure that there is an efficient firm in the population, i.e., θ H > c L . 2.4 Planner’s Problem The planner’s objective is to choose a tax to induce the exit of firms whose profit is no larger than the externality they create. The planner chooses the tax taking into consideration the information he has, and that policies are not implemented with certainty. The planner’s welfare criterion does not directly include the loss the tax imposes on firms since the tax is simply a transfer. 74 Let the function 1`i indicate that the profit of firm i is greater than the tax τ. That is, 1 `i =    1 if θi > τ   0 otherwise. Then for any given realization {θ1 , θ2 , c} ∈ S, the payoff to a planner who chooses tax τ is given by:     G (τ, s) = P( L(τ, s)) 1θ1 (θ1 − c) + 1θ2 (θ2 − c) + (1 − P( L(τ, s))) ∑(θi − c) i where: L(τ, s) = ∑  1 θi τ + (1 − 1 θi ) θ i  i At any state s, a fully informed planner or government chooses a tax that solves:      max P( L(τ, s)) 1θ1 (θ1 − c) + 1θ2 (θ2 − c) + (1 − P( L(τ, s))) ∑(θi − c) τ ∈< (2.1) i Let τF∗ (s) denote the optimal full information tax in state s, and G (τF∗ (s), s) be the planner’s payoff. We can write the expected full information payoff as: EGF = ∑ G(τF∗ (s), s)µ(s) s∈S In contrast, an uninformed planner chooses a tax independent of the true state of the world. In particular, he chooses a tax to solve:    max ∑ P( L(τ, s)) 1θ1 (s) (θ1 (s) − c(s)) + 1θ2 (s) (θ2 (s) − c(s)) τ ∈< s∈S +(1 − P( L(τ, s)))  ∑(θi (s) − c(s)) i  µ(s) (2.2) 75 Table 2.2: Information partition Ωθ induced by information about firms Event ωθ {{θ L , θ L , c L }, {θ L , θ L , c H }} {{θ L , θ H , c L }, {θ L , θ H , c H }} {{θ H , θ H , c L }, {θ H , θ H , c H }} Probability µ(ωθ ) α2 2α(1 − α) (1 − α )2 Let τU∗ denote the optimal uninformed tax, and G (τU∗ , s) be the planner’s payoff to choosing this tax when the true state is s. Note that here the planner may choose only one tax regardless of what the true state is, as information about the state will not be revealed to him. We can thus write the expected uninformed payoff as: EGU = ∑ G(τU∗ , s)µ(s) s∈S To calculate the value of learning the social cost of pollution or firm type, we must state the partially informed planner’s problem in each case. To describe how new data changes the planner’s beliefs, it is convenient to use the concept of information “partitions." Formally, a partition of the sample space S is a collection of subsets, or events, A1 , ..., An such that ∪in=1 Ai = S and ∩in=1 Ai = ∅. A partition groups realizations {θ1 , θ2 , c} together, and allows a planner to distinguish across, but not within groups of realizations. Letting Ωθ represent the partition induced by information on firm types, ωθ an event in the partition, and µ(ωθ ) the probability measure defined on the partition, the elements of the partition and their population frequencies are given in table 2.4. Information about firm types partitions the set of states into three disjoint sets, which the planner can distinguish from one another. Each of these sets contains two elements, between which the planner is unable to distinguish. In words, when the planner learns firm types he learns whether he faces two low type firms, two high types, or one of each type. However, he does not learn whether social cost of pollution is high or low. A planner who learns the types of firms but is uncertain of the social cost has expected payoff, max ∑ τ ∈< s∈ω θ      P( L) 1`1 (θ1 − c(s)) + 1`2 (θ2 − c(s)) + (1 − P( L)) ∑(θi − c(s)) µ(s|s ∈ ωθ ). (2.3) i 76 Here the planner does not know the realization {θ1 , θ2 , c} exactly, but he knows that the realization falls in the set ωθ . As before, for any state s and tax τ the payoff G (τ, s) is obtained. The planner thus chooses τ to maximize the expected value of G (τ, s), given that s ∈ ωθ . Let τΩ∗ θ (ωθ ) be the solution to this problem, the optimal tax under partial information about firm types. This tax balances the benefit of shutting down socially inefficient firms against the cost of reducing the probability of implementation. We can write the expected partial information payoff from learning firm types as, EGΩθ = ∑ ∑ ωθ ∈ Ω θ s ∈ ωθ G (τΩ∗ θ (ωθ ), s)µ(s|s ∈ ωθ )µ(ωθ ). In words, a planner anticipating information about firm types forms an expectation over the information he could receive (the elements ωθ of the partition Ωθ ), knowing that once he obtains this information he will choose the tax that maximizes expected social welfare conditional on his information. Note that this partially informed problem is intermediate between the ones the planner faces when fully informed and uninformed. In (3) the planner can choose one of three taxes, one for each element of the partition. This contrasts with the fully informed and uninformed planner’s problem, where the planner chooses six taxes and one tax respectively. A planner who chooses to be informed about the social cost of pollution learns whether the cost of pollution is high or low, but not the types of firms that are present. Formally, information about social cost of pollution c partitions the set of states of the world into two disjoint sets between which the planner can distinguish. Each set contains three elements, between which the planner cannot distinguish. Letting Ωc represent this partition and ωc an event in Ωc , table 2.4 gives the elements of the partition and their population frequencies. We can now write the partial information payoffs when the planner learns the social cost of pollution but not firm types, max ∑ τ ∈< s∈ω c      P( L(τ, s)) 1`1 (s) (θ1 (s) − c) + 1`2 (s) (θ2 (s) − c) + (1 − P( L(τ, s))) ∑(θi (s) − c) µ(s|s ∈ ωc ). i 77 Table 2.3: Information partition Ωc induced by cost information Event ωc {{θ L , θ L , c L }, {θ L , θ H , c L }, {θ H , θ H , c L }} {{θ L , θ L , c H }, {θ L , θ H , c H }, {θ H , θ H , c H }} Probability µ(ωc ) q 1−q Letting τc∗ (ωc ) denote the optimal tax under partial information about social cost given the information ωc , the expected partial information payoff is: EGΩc = ∑ ∑ ωc ∈ Ω c s ∈ ωc G (τc∗ (ωc ), s)µ(s|s ∈ ωc )µ(ωc ). The model we have presented is extremely simple. In particular, we have greatly simplified our task by assuming that all firms impose the same pollution cost on society. In an extension we consider the possibility that firms have heterogeneous costs of pollution (see section 7). Finally, we note that we have also assumed the cost of obtaining information to be constant across the two types of information. One could easily allow for the costs to be different across information types, and introduce a parameter measuring this difference into the problem. The substantive results presented below will not change; all comparisons between values of information will be made relative to this parameter. 2.5 Optimal taxes To calculate the value of the three types of information, we need to obtain the tax that maximizes social welfare for a planner with each possible type of information: a fully informed planner, either type of partially informed planner, and an uninformed planner. We can then characterize and compare the values of the different types of information. We first observe that a welfare-maximizing planner may restrict himself to choosing a tax from {θ L , θ H } without loss. That is, a planner with any information can ignore taxes outside the set {θ L , θ H } when choosing the optimal tax. To see this, note first that a planner is indifferent between 78 Table 2.4: Optimal taxes in each possible state State s {θ L , θ L , c L } {θ L , θ H , c L } {θ H , θ H , c L } {θ L , θ L , c H } ( {θ L , θ H , c H } {θ H , θ H , c H } θL Optimal Tax τF∗ (s) {θ L , θ H } θL θL {θ L , θ H } (θ H −c H )+(θ L −c H ) if ≤ P(Pθ(2θ+Lθ) ) (θ −c ) θ H if L H (θ H −c H )+(θ L −c H ) (θ L −c H ) H > L P(2θ L ) P(θ H +θ L ) θH taxes τ = θ H and τ ∈ (θ H , ∞). The former leaves a high type firm with zero profit, and we have assumed that firms exit when they earn zero profit, while the latter set of taxes clearly induces shut-down. Both types of firm exit for any τ ∈ [θ H , ∞), and aggregate losses are constant over [θ H , ∞). Notice as well that, as we have assumed c L ≥ θ L , a tax of τ = θ L is strictly preferred by the planner to any tax τ < θ L . Under the latter, no firms exit when the policy is passed, while under the former low-types exit, and a planner would always like to induce low firms to exit. Together these imply that, without any loss in generality, we need only consider taxes τ ∗ ∈ [θ L , θ H ]. It only remains to show that the optimal tax may not lie in the open interval (θ L , θ H ). Suppose to the contrary that τ ∗ ∈ (θ L , θ H ). By decreasing the tax by ε ∈ (0, τ ∗ − θ L ), the planner reduces the loss directly incurred by firms while leaving firm exit decisions unchanged. But if this is true, then the planner has increased the probability of implementing the tax policy without changing expected welfare, contradicting the optimality of τ ∗ . Therefore the only taxes we need consider are in the set {θ L , θ H }. We can now calculate the optimal tax in each of the six possible states in S. Letting τF∗ (s) denote the optimal full information tax in state s, the optimal taxes are given in table 2.5. Given the realization {θ L , θ L , c L }, the planner is indifferent between any taxes that induce low types to exit the market. Any tax at least as large as θ L induces low type firms to exit, and losses incurred by the low type firms do not increase once tax exceeds θ L . As we can always restrict ourselves to the set {θ L , θ H } when looking for the optimal tax, this completes the argument. The 79 same argument applies to the realization {θ L , θ L , c H }. Now consider {θ L , θ H , c L }. In this state the planner optimally shuts down the low-type and leaves the high type unregulated. A tax of θ L uniquely solves this problem. In the case of {θ H , θ H , c L } the planner would prefer to leave both firms unregulated. Under a tax of θ L neither firm exits.6 For the realization {θ H , θ H , c H } the planner optimally shuts down both firms, which any tax at least as large as θ H does. Finally, consider {θ L , θ H , c H }. In this case the planner faces a trade-off. On the one hand, both of the firms are inefficient, and the planner would ideally shut down both of them, which requires a high tax, τ = θ H . However, while a high tax may shut down both firms, such a tax imposes larger losses on firms and is less likely to be enacted than a low tax that shuts down only the most inefficient firm θ L . The optimal tax can be θ L or θ H depending on the importance of lobbying and the relative efficiencies of the firms. To see this more clearly, note that when the P(·) function decreases quickly, as is the case when lobbying is important, the ratio P(2θ L ) P(θ H +θ L ) is very large. The planner would choose a low tax in this case, as it is preferable to shut down the more inefficient low type with higher probability than risk not shutting down either firm by setting a high tax and regulating both firms. Note as well that when θ H ∼ θ L we have that (θ H −c H )+(θ L −c H ) (θ L −c H ) >1∼ P(2θ L ) P(θ H +θ L ) and the planner regulates both firms by setting a high tax. The firms are almost equally efficient and the planner does not benefit by treating them differently. On the other hand, when θ L < θ H ∼ c H so that the firms are different and the high type is almost efficient we get that (θ H −c H )+(θ L −c H ) (θ L −c H ) ∼1< P(2θ L ) , P(θ H +θ L ) and the planner prefers a low tax, and thus treats the two firms differently. We now turn attention to the calculation of optimal taxes when the planner learns firm types but not the social cost of pollution. Optimal taxes for each element of the relevant partition are given in table 2.5. When {{θ L , θ L , c L }, {θ L , θ L , c H }} is realized, whatever the social cost of pollution, both firms 6 Note enacted. that for the social welfare criterion we use, in this case the planner does not care whether or not regulation is 80 Table 2.5: Optimal taxes when the planner knows firm types only Optimal Tax τθ∗ (ωθ ) {θ L , θ H } (θ H − Ec)+(θ L − Ec) if ≤ P(Pθ(2θ+Lθ) ) (θ − Ec) Event ωθ {{θ L , θ L , c L }, {θ L , θ L , c H }} ( {{θ L , θ H , c L }, {θ L , θ H , c H }} θL L θ H if (θ H − Ec)+(θ L − Ec) (θ L − Ec) H > L P(2θ L ) P(θ H +θ L ) {{θ H , θ H , c L }, {θ H , θ H , c H }} θH Table 2.6: Optimal taxes when social cost is known Event ωc {{θ L , θ L , c L }, {θ L , θ H , c L }, {θ H , θ H , c L }} ( {{θ L , θ L , c H }, {θ L , θ H , c H }, {θ H , θ H , c H }} θL Optimal tax τc∗ (ωc ) θL P(2θ L )(θ L −c H )− P(θ H +θ L )[(θ H −c H )+(θ L −c H )] 1− α > α if P(2θ )[θ −c ] θ H if H H H P(2θ L )(θ L −c H )− P(θ H +θ L )[(θ H −c H )+(θ L −c H )] P(2θ H )[θ H −c H ] ≤ 1− α α are inefficient. Any tax at least as large as θ L induces the firms to exit, and all such taxes impose equal losses on the firms. Thus the optimal tax is θ L or θ H . When {{θ L , θ H , c L }, {θ L , θ H , c H }}, is realized the planner faces a trade-off similar to that when he is fully informed and the state of the world is given by {θ L , θ H , c H }}. We have assumed that Ec > θ H so both firms are inefficient in expectation. From the point of view of expected efficiency a high tax is strictly optimal. However, high taxes are less likely to be implemented than low taxes, and thus the preferred tax hinges on the relative inefficiencies of the two types of firms. The more inefficient the low type is relative to the high type, or equivalently, the faster the probability of implementation decreases in losses, the greater the planner’s inclination to choose a low tax. Last, when the realization is {{θ H , θ H , c L }, {θ H , θ H , c H }}, both firms are inefficient in expectation. Given the planner’s information it is optimal to regulate both firms and thus a high tax is chosen. We now calculate optimal taxes under the assumption that the planner knows the social cost but not firm types. The optimal taxes under the partition induced by this information are given in table 2.5. When costs are low, i.e., {{θ L , θ L , c L }, {θ L , θ H , c L }, {θ H , θ H , c L }} is realized, only the low type is 81 inefficient. A low tax is uniquely optimal since a high tax shuts down efficient firms. When costs are high, {{θ L , θ L , c H }, {θ L , θ H , c H }, {θ H , θ H , c H }} is realized. If the planner knew that the true state was an element of {{θ L , θ L , c H }, {θ H , θ H , c H }} the optimal policy would clearly be a high tax. That the planner cannot distinguish between these states and {θ L , θ H , c H } complicates matters, because the fact that {θ L , θ H , c H } is possible forces the planner to weigh the net benefit of a low tax against the net benefit of a high tax. The planner must choose between a higher probability of implementation at the cost of efficiency, and more efficiency at the cost of lower probability of implementation. If, conditional on the true state being an element of {{θ L , θ L , c H }, {θ H , θ H , c H }}, the likelihood of drawing the realization {θ L , θ H , c H } is sufficiently high, and the probability of implementation decreases quickly enough as losses increase, the planner will prefer a low tax to a high one. Finally, we derive the uninformed planner’s optimal tax. This tax is given by: τU∗ =    θL if P(2θ L )(θ L − Ec)− P(θ H +θ L )[(θ H − Ec)+(θ L − Ec)] P(2θ H )(θ H − Ec) > 1− α α   θ H otherwise. Recalling that an uninformed planner must trade-off probability of implementation with efficiency, this condition says that when marginal changes in political resistance are important, an uninformed planner chooses a low tax, and when marginal changes are not so important he chooses a high tax. Of course the relative frequencies of firm types and social cost play a role as well. For example if the probability of drawing a high type firm is sufficiently large relative to the probability of drawing a low type firm (i.e., α → 0) it is easy to see from the above expression that a high tax is optimal. 2.6 The Value of Information In this section we use the optimal taxes derived above to determine which information is more valuable to the planner, and when. 82 By definition, the value of learning firm types is given by, VΩθ = EGΩθ − EGU , while the value of learning social cost of pollution is, VΩc = EGΩc − EGU . With this notation in place, we can state our first proposition. Proposition 1 1. The value of learning the types of firms, VΩθ , is zero if firms’ political power is sufficiently small. Formally: If P(2θ L )(θ L − Ec) ≥ P(θ H + θ L )[(θ L − Ec) + (θ H − Ec)] then VΩθ = 0. 2. The value of learning social cost of pollution, VΩc , is zero if the firms’ political power is sufficiently large. Formally: If P(2θ L )(θ L − c H ) ≤ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] + 1−α P(2θ H )(θ H − c H ) α then VΩc = 0. Beginning with proposition 1.1, consider an uninformed planner who optimally taxes τU∗ . Suppose that once the planner learns the firm types but not social cost of pollution, the tax τU∗ remains optimal for any possible pair of firm types. That is τU∗ = τθ∗ (ωθ ) for all ωθ ∈ Ωθ . Then, in an exante sense the planner receives the same expected payoff whether he is informed about firm types or has no information at all. For parameter values such that this is true, information about firm types cannot have value. Proposition 1 provides conditions under which the planner’s optimal tax with and without information are the same. When this is the case, information has no value. 83 To understand intuitively when the value of information about firm types can be zero, let us first determine when it is strictly positive. Suppose first that the planner only regulates the low type firm when uninformed: τU∗ = θ L . Looking at table 2.5, if ωθ = ({θ H , θ H , c L }, {θ H , θ H , c H }) is realized, that is, the planner learns that the sample contains two high types, the optimal tax decision is τ ∗ (ωθ ) = θ H . Thus, with probability (1 − α)2 information about firm types changes the optimal tax decision when the optimal uninformed tax is low, and in this case learning the sample must be of value. Now suppose an uninformed planner optimally chooses: τU∗ = θ H . Then, again looking at table 2.5, if either ωθ = ({θ L , θ L , c L }, {θ L , θ L , c H }), or ωθ = ({θ H , θ H , c L }, {θ H , θ H , c H }) is realized, then τU∗ = θ H remains optimal. In the first case the planner is indifferent over all taxes that induce low types to exit, and in the second case, since Ec > θ H the planner prefers a high tax. However, if the realization is ({θ L , θ H , c L }, {θ L , θ H , c H }), the optimal tax may be high or low, depending on how quickly P(·) decreases. If P(·) decreases quickly, then going from a low tax to a high tax can drastically reduce the probability of implementing the policy, while if P(·) does not decrease quickly the optimal tax under this realization is high. If the latter is true, then for any possible realization of firm types the planner prefers a high tax. If he also optimally chooses a high tax when he has no information, then information about firm types cannot have value, because it does not change the optimal tax. The above argument is exactly summarized by the inequality in proposition 1.1, which simply says that a planner who realizes {{θ H , θ L }, c} prefers a high tax. Example: Consider the parameterization, P( L) = 1 − 1 2 L , B where B ∈ [(2θ H )2 , ∞) so that 84 P( L) is a proper probability distribution. This choice for the function P( L) provides a convenient way to consider comparative statics with respect to political power. Note that P0 ( L) = − 2L , B P00 ( L) = − 2 B As we noted above, the political power of firms as we have defined it here, depends on the shape of the P(·) function. If marginal changes in L are important we say that firms are politically powerful. Changing the parameter B allows us to change how important marginal changes in L are, holding all else fixed. The way we have set up this example, small values of B correspond to high political power, while larger values of B correspond to low political power. Now, under this parameterization, proposition 1.1 says that the value of learning firm types is zero if:  1−     1 1 (2θ L )2 (θ L − Ec) ≥ 1 − (θ L + θ H )2 (θ L − Ec) + (θ H − Ec) B B which, after rearranging yields:   1 − B1 (2θ L )2 1− 1 B (θ L  + θ H )2  ≤ 1+ θ H − Ec . θ L − Ec Note that the right hand side of this expression is greater than 1 by our assumption Ec > θ H . Then, as B gets large,   1 − B1 (2θ L )2  1 − B1 (θ L + θ H )2  → 1 < 1+ θ H − Ec θ L − Ec In words, as the slope of P( L) gets very flat so that marginal changes in L leave the probability of implementation unaffected, the inequality holds with certainty and the value of learning firm types is zero. On the other hand, it is straightforward to find parameter values for which the inequality does not hold. If θ L ≈ 0 and θ H ≈ Ec, the right hand side becomes 1 while one can find appropriate values for B and Ec so that the left hand side is larger than 1.  85 To understand proposition 1.2, note that information about social cost can only have zero value if the planner chooses the same tax for any value of social cost he draws. Further, recall that information about social cost has value if τU∗ = θ H . This is true because when social cost is low the optimal tax is τ ∗ = θ L for any possible sample. Thus information about social cost changes the planner’s tax choice with positive probability (probability q). So for information about social cost to have no value, the planner must optimally regulate only the low type for any realization of social cost. In other words, the planner must tax low even in the event c = c H . For this to be true, lobbying must be very important. Indeed, the inequality in proposition 1.2 says that when the planner does not know firm types but knows that social cost is high, he prefers a low tax. Example cont’d Continuing with the parameterization given above, proposition 1.2 says that the value of learning the social cost of pollution is zero if:  1−      1 − α 1 1 1 (2θ L )2 (θ L − c H ) ≤ 1 − (θ L + θ H )2 (θ L − c H ) + (θ H − ch ) + 1 − (2θ H )2 (θ H − c H ) B B α B First, letting B get large we have:  1 (2θ L )2 (θ L − c H ) → θ L − c H B    1 1 − ( θ L + θ H )2 ( θ L − c H ) + ( θ H − c h ) → ( θ L − c H ) + ( θ H − c H ) B  1 − α 1 1−α 1 − (2θ H )2 (θ H − c H ) → (θ H − c H ) α B α  1− and the inequality thus approaches: 0 ≤ θH − cH and as we have assumed θ H < c H , the inequality does not hold. This is indeed consistent with proposition 1.2, as it shows that when firm political power becomes arbitrarily small the value of learning social cost of pollution cannot be zero. Now, letting B approach its lower bound, (2θ H )2 86 we have:   1 (2θ L )2  (θ L − c H ) (2θ L )2 (θ L − c H ) → 1 − B (2θ H )2      1 ( θ L + θ H )2   ( θ − c ) + ( θ − c ) 1 − ( θ L + θ H )2 ( θ L − c H ) + ( θ H − c h ) → 1− L H H H B (2θ H )2   1−α 1 1 − (2θ H )2 (θ H − c H ) → 0 α B  1− In this case, the inequality reduces to  1−   (2θ L )2  ( θ L + θ H )2   ( θ − c ) ≤ 1 − (θ L − c H ) + (θ H − c H ) L H 2 2 (2θ H ) (2θ H ) This inequality holds for certain parameter values. One example of where it does hold is when θ L ≈ 0 and θ H ≈ c H .  Proposition 2 1. Learning the social cost of pollution yields the expected full information payoff if firms’ political power is sufficiently small. Formally: If P(2θ L )(θ L − c H ) ≥ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] then VΩc = VF . 2. Learning only the types of firms never yields the expected full information payoff. That is, VΩθ < VF . For information about social cost of pollution to yield the full information payoff, learning firm types once social cost is known must not change the planner’s tax choice in any state of the world. Similarly, for information about firm types to yield the full information payoff, learning social cost once firm types are known must not change the planner’s tax choice in any state of the world. 87 Let us first establish that learning firm types can never yield the full information payoff. To see this, note that if the planner learns that both firms are high types, i.e., {(θ H , θ H , c L ), (θ H , θ H , c H )} is realized, the planner would strictly prefer to leave the firms unregulated if the true social cost of pollution is c L , but would optimally regulate them if true social cost of pollution is c H . Thus we have found a possible realization of firm types where information on social costs would strictly change the planner’s tax choice. θ does not yield the full information payoff. The planner knows that there is positive probability of drawing a sample of firm types such that it will still pay to learn social cost. Learning social cost of pollution can yield the full information payoff, however. To see this, note first that the optimal tax does not vary across the states {θ L , θ L , c L }, {θ L , θ H , c L }, {θ H , θ H , c L }; once the planner observes that social cost of pollution is low, regardless of what the types of firms are, a low tax is optimal. Now suppose that c = c H . The three possible states of the world are then: { θ L , θ L , c H }, { θ L , θ H , c H }, { θ H , θ H , c H } In the first and last case a high tax is optimal. However, for a mixed sample the level of the tax depends on how quickly P(·) decreases, or how likely the policy is to pass. If lobbying is not important, the planner would choose to regulate both firms, and set a high tax. If this is true, then there is a unique optimal tax across all possible firm pairs for each possible level of social cost, and learning c yields the full information payoff. Example cont’d After some re-arranging, under the parameterization above proposition 2.1 says that learning social cost yields the full information payoff when:   1 − B1 (2θ L )2  1 − B1 (θ L + θ H )2  ≤ 1+ θH − cH . θL − cH Clearly, as B becomes arbitrarily large the inequality holds. In words, as the function P( L) becomes arbitrarily flat and marginal changes in L do not affect the probability of implementation, learning 88 the social cost of pollution yields the full information payoff.  Corollary: Marginal value of information 1. Learning firm types after social cost is known is of no value if firm political power is sufficiently small. 2. Information about social cost always has value when firm types are known. The “marginal" values of information, (i.e., what the planner would pay to learn social cost of pollution after information about firm types has been obtained, and vice versa) also depends on how important lobbying is. When firms possess significant political power, both marginal values are strictly positive. This is the same as saying that when lobbying is very important, learning only one type of information is not enough for a first-best solution. When lobbying is not important, it is always worthwhile to learn c after learning firm types, but it may be of no value to learn the sample after learning c. Whether or not the marginal value to either type of information is positive follows immediately from our discussion about whether partial information can yield the expected full information payoff. We noted that once the types of firms have been observed the planner is always willing to pay for information about the social cost of pollution. Thus the marginal value of information is always positive in this case. We also noted that if lobbying is not important, information about social cost of pollution yields the expected full information payoff, and thus the marginal value of information is zero. However if lobbying is important, partial information about social cost can’t yield the expected full information payoff, and the marginal value of information is positive. 2.7 Extensions In this section we discuss potential extensions to the model we have presented above. 7.1 Firm Political Power 89 We now consider the possibility that firms differ in their political power. Specifically, we assume that with some probability γ the losses of firm i do not enter into the probability of implementation function P( L). Formally we define two indicator functions, 1 Ri , i ∈ {1, 2} such that Pr (1 Ri = 0) = γ and now write the loss function associated with tax τ in state s as L(τ, s) = ∑ 1 Ri Li (τ, s) (2.4) i =1,2 where, as before, Li (τ, s) = 1θi τ + (1 − 1θi τ )θi . (2.5) In this framework there are three possible types of information a planner can acquire, information about profit types {θ1 , θ2 } , information about social cost of pollution c, and information about firm political power {1 R1 , 1 R2 }. Tables 2.7-2.10 in the appendix describe the optimal taxes for a fully informed planner, a planner informed about political power, a planner informed about social costs, and a planner informed about firm types respectively. We find that learning which firms have political power never yields the full information payoff, and has no value at all if (holding all else constant) either α is very small, θ H ∼ θ L or P(·) is not steep (lobbying is not important). In such cases an uninformed planner chooses a high tax, and a planner who learns political power, regardless of what he learns, also chooses a high tax. Further, for γ very small (so that both firms are likely to have power), the propositions in the paper continue to hold. This is straightforward: As γ → 0 we approach the model in the paper. On the other hand as γ approaches 1 and neither firm has power, information about social cost yields the full information payoff. In this case when cost is low the planner only wants to regulate low type firms, and when the cost is high he wants to regulate both. 90 Notice that a planner who does not directly observe firm profit types or the social cost of production does learn something about the economic environment simply by observing the expenditures/effort undertaken by firms in opposition to a proposed tax policy. In a further extension to the analysis we have presented one could examine the value that accrues to simply observing firm expenditures as a response to a tax. Considering such an extension would require further assumptions and modifications to the model we have presented here. Firstly we do not model explicitly how firm expenditures in opposition to a policy affect the probability of a policy’s implementation; we simply allow the probability of implementation to be an unspecified function of the firm types. One would need to explicitly model this relationship to consider the value of observing expenditures. Second, the timing implicit in the model we have presented is such that one would need to consider a dynamic version of the model to answer the question at all. Specifically, a planner chooses a tax and then observes expenditures made by firms. In the context of our model, it is too late to learn from firm expenditures once they occur. One would need to consider the problem in the context of a dynamic game. 7.2 Heterogeneity in Pollution Costs Now suppose that the firms earn identical profit θ ∈ {θ L , θ H }, but differ in the costs they impose on society ci ∈ {c L , c H }, i = 1, 2. We maintain all other parametric assumptions made in the paper. Tables 2.11-2.13 in the appendix describe the optimal taxes for each possible type of planner, a fully informed planner, a planner who knows profit θ but not the costs of pollution {c1 , c2 }, a planner who knows the costs of pollution {c1 , c2 } but not profit θ, and a completely uninformed planner. The key result in this extension is that the value of learning the common profit level θ is zero, and learning the cost types of the firms {c1 , c2 } yields the full information payoff. While firms are inefficient in expectation by assumption, what is important is that firms differ in terms of how inefficient they are. This is true both in the model we present in the paper and in the extension here. In the model presented in the paper, the difference in efficiency across firms is driven by 91 differences in profit type θ. The cost of pollution is common across the firms. By contrast, in the model we present in the extension profit is common across the two firms and the difference in efficiency depends on the cost of pollution. Notice that the probability of implementation always depends only on firm profit types. Thus, in the model in the paper, the planner faces the following trade-off: set a lower tax and shut down only the most inefficient firm with higher probability (a lower tax is more likely to be implemented) while leaving the more efficient firm in the market, or set a higher tax and try to shut down both firms, but with less probability. The probability of implementation decreases as the planner includes more efficient firms in his regulation. In the model in the extension however, the nature of the trade-off is quite different. Here firms have the same profit, and their response to the tax depends only on their profits. Thus it is not possible to regulate the less efficient firm and not regulate the more efficient firm; the planner cannot regulate firms according to their efficiency as he can in the original model. Since both firms are inefficient in expectation, the uninformed planner regulates both by setting a high tax. Notice that a planner who learns the firm profit type, regardless of whether he learns that the type is low or high, can do no better than set a high tax. If he learns the common profit type is low, both firms are inefficient with certainty and he wants to shut down both firms. A tax of either θ L or θ H does this. If he learns that the profit type is θ H , the firms are still inefficient in expectation and he shuts down both firms. Thus there is no value to learning the firm profit type; learning this information does not change the planner’s tax choice. It immediately follows that learning cost types must yield the full information payoff 2.8 Conclusion We model an economy where a regulator chooses optimal environmental regulation in the presence of political constraints and hidden information. We use the model to compare the value of information about the social cost of pollution with the value of information about the types of firms present in the economy. We find that in environments where firms are politically powerful, 92 it is most valuable to learn the types of firms. On the other hand, when firms are not politically powerful, it is most valuable to learn the social cost of pollution. While we present our results using a deliberately simple model, this intuition is general. Indeed, the main results appear to generalize to an environment with many firm types and levels of social costs. The relative values of different types of information hinge on the political power of firms. When the probability of implementing a policy is not sensitive to the losses that firms incur as a consequence of the policy, choosing the optimal regulation amounts to setting the tax to equal the externality (pollution) cost of production. This is the first best tax, and so information about social cost of pollution is much more valuable than is information about firm types. On the other hand, when a high tax is much less likely to be implemented than a low tax, a welfare-maximizing planner would treat a sample of high profit, inefficient firms differently from a mixed sample of high and low profit inefficient firms. In the latter case there is more surplus at stake from not passing the policy, and information about firm types is more valuable. To say this another way, information about firm types allows the regulator to choose regulation that affects only firms that are less able to influence the political process. In an environment where marginal changes in political resistance are important, this means that information about firm type can be very valuable. If marginal changes in political resistance are not important, information that leads to the first best, i.e., information about social cost, is more valuable. We acknowledge that the assumption of a binary production decision is a strong one. Relaxing the assumption of binary production choice and allowing firms to choose quantity in the context of the model we present here comes at great cost in terms of tractability and clarity, and generates small benefit in intuition over and above the propositions we have discussed here. Finding the correct way to extend the model in this direction without losing the simplicity inherent to the binary setup remains a subject for future research. Given the results we have obtained here, the conclusions of Olson (1982), Becker (1985) and Stigler (1971) suggest that when a regulator is facing a concentrated industry he would do well to learn firm types as opposed to the true social cost of pollution, as concentrated industries are 93 conducive to firm mobilization. On the other hand, if the regulator finds himself in an industry with many small firms that cannot organize effectively, learning the social cost of pollution is preferable, because the benefit from getting the tax closer to the actual cost of pollution outweighs the cost from potentially not implementing the tax. 94 2.9 Appendix Proof of Proposition 1 Proposition 1 asserts that under some conditions on the parameters the value to either type of information is zero. To prove this result, we need only show that when these conditions hold, the uninformed planner’s optimal tax choice is the same as the tax choice of a planner that has the information in question. We prove each of proposition 1.1 and 1.2 in turn. First we show that: P(2θ L )(θ L − Ec) ≥ P(θ H + θ L )[(θ L − Ec) + (θ H − Ec)] ⇒ VΩθ = 0. Note first that if the parameters satisfy: P(2θ L )(θ L − Ec) ≥ P(θ H + θ L )[(θ L − Ec) + (θ H − Ec)], then looking at Table 2.5, for the realization {{θ L , θ H , c L }, {θ L , θ H , c H }} the planner prefers a high tax. This then implies that a high tax is optimal for any of the three samples that are possible when the planner gains information about firm types. Further, P(2θ L )(θ L − Ec) ≥ P(θ H + θ L )[(θ L − Ec) + (θ H − Ec)] ⇒ P(2θ L )(θ L − Ec) ≥ P(θ H + θ L )[(θ H − Ec) + (θ L − Ec)] + 1−α P(2θ H )(θ H − Ec), α which implies that the planner prefers a high tax with no information. Altogether, the planner prefers a high tax with no information, and chooses a high tax for any of the three possible realizations when he gains information about firm types. There cannot be a value to learning firm types. 95 Now we show that: P(2θ L )(θ L − c H ) ≤ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] + 1−α P(2θ H )(θ H − c H ) ⇒ VΩc = 0 α First, from table 6, if P(2θ L )(θ L − c H ) ≤ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] + 1−α P(2θ H )(θ H − c H ) α then the planner chooses a low tax even when the social cost is high. Then, for either realization of social cost, the optimal tax is low. Further, 1−α P(2θ H )(θ H − c H ) α 1−α ⇒ p(2θ L )(θ L − Ec) ≤ P(θ H + θ L )[(θ H − Ec) + (θ L − Ec)] + P(2θ H )(θ H − Ec) α P(2θ L )(θ L − c H ) ≤ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] + and thus an uninformed planner optimally taxes low. Altogether, a planner taxes low for either realization when information about social cost is gained, but taxes low when he has no information anyways. The value of learning social cost can’t be positive.  Proof of Proposition 2 Proposition 2.1 establishes conditions under which learning only the social cost of pollution leaves the planner as well off as if he would be with full information. To prove this we need to show that, for each possible level of social cost, once information about social cost is obtained, learning the types of firms would not change the planner’s choice of tax. Now suppose P(2θ L )(θ L − c H ) ≥ P(θ H + θ L )[(θ L − c H ) + (θ H − c H )] 96 Note that (see table 4) if the parameter values satisfy this condition a fully informed planner regulates both types for the realization {θ L , θ H , c H }. This implies that a high tax is always optimal when c = c H , regardless of firm types. Further, we know that if c = c L a low tax is always optimal. Thus once the planner knows the social cost, learning firm types does not change his choice of tax regardless of the types, and thus does not change his ex-post payoff. Information about social cost yields the full information payoff. We now prove proposition 2.1, that information about firm types can never yield the full information payoff: Vθ 6= VF To prove this, we need find only one possible realization of firm types such that learning social cost in addition to information about firm types changes the planner’s tax choice. If we can find such a case, information about firm types can’t yield full information payoff, because the planner’s ex-post payoff can change with more information. So consider the realization {{θ H , θ H , c L }, {θ H , θ H , c H }} In this case, without further information the planner chooses a high tax, as both firms are inefficient in expectation. However, if the planner were to learn that the true social cost is c L , a low tax would be strictly preferable, as both firms are efficient. Thus information about firm types alone can’t provide full information payoff. 97 Table 2.7: Optimal taxes in each possible state State s {θ L , θ L , c L , 0, 0} {θ L , θ L , c L , 1, 0} {θ L , θ L , c L , 0, 1} {θ L , θ L , c L , 1, 1} {θ L , θ H , c L , 0, 0} {θ L , θ H , c L , 1, 0} {θ L , θ H , c L , 0, 1} {θ L , θ H , c L , 1, 1} {θ H , θ L , c L , 0, 0} {θ H , θ L , c L , 1, 0} {θ H , θ L , c L , 0, 1} {θ H , θ L , c L , 1, 1} {θ H , θ H , c L , 0, 0} {θ H , θ H , c L , 1, 0} {θ H , θ H , c L , 0, 1} {θ H , θ H , c L , 1, 1} {θ L , θ L , c H , 0, 0} {θ L , θ L , c H , 1, 0} {θ L , θ L , c H , 0, 1} {θ L , θ L , c H , 1, 1} Optimal Tax τF∗ (s) {θ L , θ H } {θ L , θ H } {θ L , θ H } {θ L , θ H } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L } {θ L , θ H } {θ L , θ H } {θ L , θ H } {θ L , θ H } 98 Table 2.7 (cont’d) Optimal Tax τF∗ (s) {θ H } {θ H } State s {θ L , θ H , c H , 0, 0} {θ L , θ H , c H , 1, 0} ( {θ L , θ H , c H , 0, 1} ( {θ L , θ H , c H , 1, 1} (θ −c H )+(θ L −c H ) (θ L −c H ) θ L if H θ H o.w. (θ −c H )+(θ L −c H ) (θ L −c H ) θ L if H θ H o.w. ≤ ≤ P(2θ L ) P(θ H +θ L ) {θ H , θ L , c H , 0, 0} ( {θ H , θ L , c H , 1, 0} (θ H −c H )+(θ L −c H ) (θ L −c H ) θ L if θ H o.w. {θ H , θ L , c H , 0, 1} ( {θ H , θ L , c H , 1, 1} (θ H −c H )+(θ L −c H ) (θ L −c H ) θ L if θ H o.w. ≤ P(θ L ) P(θ H ) ≤ {θ H } P(θ L ) P(θ H ) {θ H } P(2θ L ) P(θ H +θ L ) {θ H , θ H , c H , 0, 0} {θ H , θ H , c H , 1, 0} {θ H , θ H , c H , 0, 1} {θ H , θ H , c H , 1, 1} {θ H } {θ H } {θ H } {θ H } Table 2.8: Optimal taxes when planner knows who has political power State s Neither firm has power ( θ L if P(Lθ )+αP(θ )H < L H ( θ H o.w. P(θ )+ 1−α P(θ ) θ L if P(Lθ )+αP(θ )H < L H θ H o.w. Firm 1 has power Firm 2 has power ( Both firms have power P(θ )+ 1−α P(θ ) Optimal Tax τR∗ (s) θH (θ H − Ec)+(θ L − Ec) (θ H − Ec)+(θ H − Ec) (θ H − Ec)+(θ L − Ec) (θ H − Ec)+(θ H − Ec) P(2θ L )(θ L − Ec)− P(θ H +θ L )[(θ H − Ec)+(θ L − Ec)] P(2θ H )(θ H − Ec) θ L if θ H o.w. > 1− α α Table 2.9: Optimal taxes when planner knows social cost of pollution State s c = cL c = cH   θL  θ H Optimal Tax τω∗ c (s) θL   2 (1−γ) 2α γ(1− P(θ L ))+α(1−α)[γ( P(θ H )− P(θ L ))+(1−γ)( P(θ H +θ L )− P(θ L ))] −c H if γ2 (1−α)+γ(1−γ)(1−α) P(θ )+(1−γ)2 (α(1−α) P(θ +θ )+(1−α)2 P(θ )) > θθHL − cH H L H H o.w. 99 Table 2.10: Optimal taxes when planner knows firm types State s { θ1 , θ2 } = { θ L , θ L } { θ1 , θ2 } = { θ L , θ H }   θL  θ H if Optimal Tax τω∗ c (s) {θ L , θ H } γ(1−γ)[ P(θ L )− P(θ H )]−(1−γ)2 [ P(θ L +θ H )− P(2θ L )]
1−(1−γ) γ(1− P(θ L )+1− P(θ H ))+(1−γ)(1− P(θ L +θ H )) o.w. { θ1 , θ2 } = { θ H , θ L } { θ1 , θ2 } = { θ H , θ H } − θH Table 2.11: Optimal taxes in each possible state State s {θ L , c L , c L } {θ L , c L , c H } {θ L , c H , c H } {θ H , c L , c L }  {θ H , c L , c H } {θ H , c H , c H } θL θH Optimal Tax τF∗ (s) {θ L , θ H } {θ L , θ H } {θ L , θ H } θL if [(θ H − c H ) + (θ H − c L )] > 0 o.w. θH Table 2.12: Optimal taxes when profit is known Event ωθ Optimal tax τθ∗ (ωθ ) {{θ L , c L , c L }, {θ L , c L , c H }, {θ L , c H , c H }}  {θ L , θ H } θ L if q(θ H − c L ) + (1 − q)(θ H − c H ) > 0 {{θ H , c L , c L }, {θ H , c L , c H }, {θ H , c H , c H }} θ H o.w. Table 2.13: Optimal taxes when the planner knows firm cost types only Event ωc {{θ L , c L , c L }, {θ H , c L , c L }}  {{θ L , c L , c H }, {θ H , c L , c H }} {{θ L , c H , c H }, {θ H , c H , c H }} θL θH Optimal Tax τc∗ (ωc ) {θ L } if θ H − c L + θ H − c H > 0 o.w. θH Optimal tax when the planner is uninformed τU∗  = θ L if q(θ H − c L ) + (1 − q)(θ H − c H ) > 0 θ H o.w. Chapter 3 Identification and Estimation of Dynamic Games When Players’ Beliefs are not in Equilibrium 3.1 Introduction The principle of revealed preference (Samuelson, 1938) is a cornerstone in the structural empirical analysis of decision models, either static or dynamic, single-agent decision problems or games. Under the principle of revealed preference, agents maximize expected payoffs and their actions reveal information on the structure of payoff functions. This simple but powerful concept has allowed econometricians to use data on agents’ decisions to identify important structural parameters for which there is very limited information from other sources. Agents’ degree of risk aversion, intertemporal rates of substitution, market entry costs, adjustment costs and switching costs, consumer willingness to pay, preference for a political party, or the cost of a merger, are just some examples of the type of structural parameters that have been estimated under the principle of revealed preference. In the context of empirical games, a player’s expected payoff depends on his beliefs about the behavior of other players. Most empirical applications of games have combined the principle of revealed preference with the assumption that players’ beliefs are in equilibrium. There are multiple reasons why the assumption of equilibrium beliefs is useful in the estimation of games. Equilibrium restrictions have identification power even in models with multiple equilib- 100 ria (Tamer, 2003, and Aradillas-Lopez and Tamer, 2008).1 101 Imposing these restrictions contributes to improve asymptotic and finite sample properties of estimators (Kasahara and Shimotsu, 2008). Furthermore, structural models where agents’ beliefs are endogenously determined in equilibrium are attractive for the evaluation of counterfactual policy experiments because they take into account how agents’ beliefs may change in the counterfactual scenario. Despite the important and attractive implications of the assumption of equilibrium beliefs, there are empirical applications of games where the assumption is not realistic and it is of interest to relax it. To motivate the contributions of this paper, we start by presenting three examples. First, competition in actual oligopoly industries is often characterized by strategic uncertainty (Besanko et al., 2010). Firm managers are very secretive about their own strategies and face significant uncertainty about the strategies of their competitors. In fact, it is often the case that firms have incentives to misrepresent their own strategies (e.g., entry deterrence). The second example deals with the empirical evaluation of the effect of a policy change in a strategic environment. Consider a game of competition in an oligopoly industry where firms choose whether or not to adopt a new technology. In this game, firms’ adoption decisions are strategic complements and the game has multiple stable equilibria, e.g., an equilibrium with low probabilities of adoption, and an equilibrium with high probabilities of adoption. We have panel data of firms in this industry over several periods of time. Suppose that an important policy change occurred in the middle of our sample period, e.g., the introduction of a new government subsidy that tries to encourage firms’ adoption of the new technology. It seems reasonable to think that it will take time for firms to learn about the new strategies of competitors after the policy change, and for some time firms’ strategies will be out of equilibrium. Imposing the restriction of equilibrium strategies may bias our estimates of the effects of the new policy. Finally, there is significant empirical evidence in 1 In games with multiple equilibria, the assumption of equilibrium beliefs is key to have point identification of struc- tural parameters and beliefs, and for the implementation of relatively simple methods of estimation. That is the case in games of incomplete information under the assumption that the observations in the data come from the same equilibrium (Aguirregabiria and Mira, 2007). Under this assumption, it is possible to estimate players’ beliefs consistently using a nonparametric estimator of the distribution of players’ actions. This nonparametric estimator of beliefs can be used to construct players’ expected payoffs and to obtain an estimator of structural parameters that optimizes a sample criterion function based on players’ best responses to the estimated beliefs from the data. This simple two-step approach for identification and estimation cannot be applied when players beliefs are not in equilibrium. 102 experimental economics showing large heterogeneity in agents’ elicited beliefs (Camerer, 2003). In many laboratory experiments, agents’ heterogeneity in beliefs is one of the most important factors to explain heterogeneity in behavior. This evidence contrasts dramatically with the standard approach in empirical applications of games using non-experimental data. Most of these applications impose the assumption of equilibrium beliefs, and this implies that beliefs play no role in explaining heterogeneous behavior.2 In this paper we study identification, estimation, and inference in dynamic discrete games of incomplete information when we relax the assumption of equilibrium beliefs and the researcher does not have data on elicited beliefs.3 First, we present new results on the point-identification of structural parameters and beliefs. If players are rational, in the sense that each player maximizes expected payoff given some arbitrary beliefs, an exclusion restriction and a large-support condition on one of the explanatory variables are sufficient for point-identification of structural parameters and players’ beliefs. While the type of exclusion restriction that we need is quite plausible in dynamic games of oligopoly competition, the large support condition is not satisfied in most applications. However, we show that this condition can be replaced with a "normalization" restriction on beliefs. Second, we propose a simple two-step estimation method of structural parameters and beliefs, and an extension of this method that provides a sequence of estimators with asymptotic variances and finite sample bias that decline monotonically. We also present a procedure for testing the null hypothesis of equilibrium beliefs. Finally, we illustrate our model and methods with an empirical application of a dynamic game of store location by retail chains. In the class of econometric models that we consider, players’ beliefs are probability distributions over the set of other players’ actions. These distributions are nonparametrically specified and they are treated as incidental parameters that, together with the structural parameters of the game, determine the stochastic process followed by players’ actions. When players beliefs are not in equilibrium they are different from the actual distribution of players’ actions in the population. 2 An exception is the recent paper by Goldfarb and Xiao (2009) that studies entry decisions in the US local telephone industry and finds significant heterogeneity in firms’ beliefs about other firms’ strategic behavior. 3 Data on elicited beliefs of firm managers is very rare and typically of low quality. 103 Therefore, without other restrictions, beliefs cannot be identified and estimated by simply using a nonparametric estimator of the distribution of players’ actions. We show that an exclusion restriction in the one-period payoff function and a large-support condition on one of the state variables are sufficient conditions for point-identification of structural parameters and players’ beliefs. The exclusion restriction consists in the assumption that for each player there is an observable state variable that does not enter in the player’s payoff function but that enters in the payoff function of the other players. The large-support condition establishes that the ’special’ state variable of the exclusion restriction has unbounded support on the real line. To illustrate our model and methods, we consider an empirical application where players are firms. Most empirical studies on bounded rationality have concentrated on individual behavior, and there is very little empirical work on bounded rationality of firms. However, there are good reasons to consider that, in some contexts, firms’ do not have ’correct’ (equilibrium) beliefs about their competitors’ behavior. For instance, firms tend to be very secretive about their business strategies and this fact can make it difficult for them to construct correct beliefs about the behavior of competitors. The hypothesis of social learning, that seems plausible in the context of individual decision making, is much less convincing in strategic settings involving firms. We consider a dynamic game of store location between McDonalds (MD) and Burger King (BK). Using panel data of 422 local markets (districts) in United Kingdom during the period 19911995, we present empirical evidence that documents a puzzling regularity in the entry decisions of MD and BK. Reduced form estimates show that the probability that a firm opens a new store in a local market depends significantly and negatively on the own (predetermined) number of stores in that market, but it does not respond (or even it responds positively) to the number of stores of the competing firm. This firm behavior cannot be explained by a standard static model of store location by firms that sell substitute products. After presenting robust empirical evidence showing that unobserved market heterogeneity alone cannot alone explain this puzzle, we propose and estimate a structural dynamic game that provides three possible explanations: spillover effects; forward looking behavior; and biased beliefs about the behavior of the competitor 104 This paper is related to several topics on the econometrics of empirical games. It builds and extends the literature on estimation of dynamic games of incomplete information, with recent methodological contributions by Aguirregabiria and Mira (2007), Bajari, Benkard and Levin (2007), Pakes, Ostrovsky and Berry (2007), Pesendorfer and Schmidt-Dengler (2008), and Kasahara and Shimotsu (2009), and empirical applications by Ryan (2008), Collard-Wexler (2008), Sweeting (2007), Dunne et al (2009), Xu (2008), or Aguirregabiria and Mira (2009), among others. All the papers in this literature have assumed that the data come from a Markov Perfect Equilibrium. We relax that assumption. Our paper is related to that of Aradillas-Lopez and Tamer (2008) who study the identification power of the assumption of equilibrium beliefs in static games. There are important differences between our model and econometric approach and those in Aradillas-Lopez and Tamer. First, we study identification and estimation of dynamic games. There are different aspects in which the assumption of equilibrium beliefs and identification issues are substantially different in dynamic games than in static ones (Magnac and Thesmar, 2002, Aguirregabiria, 2010). In particular, the characterization and derivation of bounds on choice probabilities is significantly more complicated in dynamic than in static games. Some important results in Aradillas-Lopez and Tamer either do not extend to dynamic games or they are impractical. That is the main reason why we incorporate additional restrictions that provide point identification of parameters and beliefs. Second, Aradillas-Lopez and Tamer concentrate on identification, while we propose and implement new estimators and tests. The rest of the paper includes the following sections. Section 2 presents the model and basic assumptions. In section 3, we derive our characterization of the set of players’ rational strategies in our class of dynamic games. Section 4 presents our identification results. Section 5 describes estimation methods and testing procedures. The empirical application is described in section 6. We summarize and conclude in section 7. 105 3.2 3.2.1 Model Basic Framework and Assumptions This section presents a dynamic game of incomplete information where two players make binary choices over T periods.4 The time horizon T can be either finite or infinite. We use the indexes i ∈ {1, 2} and j ∈ {1, 2} to represent a player and his opponent, respectively. Time is discrete and indexed by t ∈ {1, 2, ..., T }. Every period t, players choose simultaneously and non-cooperatively between alternatives 0 and 1. Let Yit ∈ {0, 1} represent the choice of player i at period t. Each player makes this decision to maximize his expected and discounted payoff, Et (∑sT=0 βsi Πi,t+s ), where β i ∈ (0, 1) is player i’s discount factor, and Πit is his payoff at period t. The one-period payoff function has the following structure:5 Πit =     πit (Yjt , Xt ) − ε it if Yit = 1          0 (3.1) if Yit = 0 Yjt represents the current action of the other player; Xt is a vector of state variables which are common knowledge for both players; variable ε it is private information of firm i at period t; and πit (.) is a real valued function. We distinguish several components in the vector of common knowledge state variables: Xt = (Wt , Xit , X jt ). Wt ∈ W is a vector of state variables that evolve exogenously according to a Markov process with transition probability function f W (Wt+1 |Wt ). The variable Xit is an endogenous state variable for player i with discrete support {0, 1, 2, ..., K } and it evolves 4 The results in the paper can be generalized to models with more than two players or choice alternatives. A key result in our paper is the characterization of rational beliefs in section 3. That characterization is used in the identification results and in the estimation methods in sections 4 and 5. It is possible to extend that representation to multinomial choice models and games with more than two players. 5 For the nonparametric identification of the payoff function we need to impose some "normalization" assumption. We normalize to zero the one-period payoff of choice alternative 0. In contrast with static decision problems, this normalization is not innocuous in dynamic decision models and it has implications for some counterfactual experiments using the model (see Aguirregabiria, 2010). However, the results in this paper do not depend on the specific choice of normalization of the payoff function, and they apply both to models with nonparametric and with parametric specification of the payoff function. over time according to a transition probability function f i ( Xit+1 |Yit , Xit ).6 106 We will refer to Xit as the stock or the "capacity" of player i. The private information shocks ε 1t and ε 2t are independent of Wt , independent of each other, and independently and identically distributed over time. Their distribution functions, Λ1 and Λ2 , are absolutely continuous and strictly increasing with respect to the Lebesgue measure on R. EXAMPLE: Dynamic game of market entry and exit. Consider two firms competing in a market. Each firm sells a differentiated product. Every period, firms decide whether to be active in the market or not. Then, incumbent firms compete in prices. Let Yit ∈ {0, 1} represent the decision of firm i to be active in the market at period t. Let Xit be a state variable that represents the incumbent status of firm i at previous period, i.e., Xit = Yit−1 . And let Wt be a vector of exogenous market characteristics, e.g., market size. The profit function of firm i is: Πit = 
   πi Yjt , Xit , Wt − ε it if Yit = 1          0 (3.2) if Yit = 0 In this example, we have that πi (0, 0, Wt ) (alternatively πi (0, 1, Wt )) is the profit of firm i when it is a monopolist and a new entrant (an incumbent last period). Similarly, πi (1, 0, Wt ) (alternatively πi (1, 1, Wt )) is the profit of firm i when it is a duopolist and a new entrant (an incumbent last period). The variable ε it is a private information shock in the fixed operating cost, and it is i.i.d. normally distributed. Without further restrictions, this model considers a nonparametric specification of the profit function πi . We study identification and estimation in this context. 6 For instance, this transition probability could be based on the transition rule: Xit+1 =   max 0 , min K , Xit + κ Yit − δi,t+1 . K and κ are parameters that belong to the natural numbers. κ represents the increment in the stock associated with the investment decision Yit = 1, and K represents the maximum possible level of the stock. δi,t+1 ∈ {0, 1} is a random variable that captures the exogenous depreciation in the stock, and it is i.i.d. with a multinomial distribution with discrete and finite support. This specification includes as particular case a model where the evolution of the state variable Xit is deterministic. i.e., the probability distribution of the depreciation variable δi,t+1 has all the probability mass at a single point. 107 Alternatively, the model could be parametric. For instance, the profit function could be: πi Yjt , Xit , Wt

= Wt (1 − Yjt ) θiM + Yjt θiD − θiFC − (1 − Xit ) θiEC (3.3) where Wt represents market size, and θiM , θiD , θiFC , and θiEC are structural parameters. θiM and θiD represent the per capita variable profit of firm i when the firm is a monopolist and when it is a duopolist, respectively. θiEC and θiFC are parameters that represent the fixed operating cost and the entry cost, respectively. We also study the estimation of parametric models. Most previous literature on estimation of dynamic discrete games assumes that the data comes from a Markov Perfect Equilibrium (MPE). This equilibrium concept incorporates three main assumptions. ASSUMPTION 1 (Payoff relevant state variables): Players’ strategy functions depend only on payoff relevant state variables: Xt and ε it . ASSUMPTION 2 (Rational beliefs on own future behavior): Players are forward looking, maximize expected intertemporal payoffs, and have rational expectations on their own behavior in the future. ASSUMPTION ’EQUIL’: (Rational or equilibrium beliefs on other players’ actions): Strategy functions are common knowledge, and players’ have rational expectations on the current and future behavior of other players. That is, players beliefs about other players’ behavior are consistent with the actual behavior of other players. Suppose that we impose only Assumption 1. The payoff-relevant information set of player i is {Xt , ε it }. The space of Xt is X ≡ W × {0, 1, ..., K } × {0, 1, ..., K }. At period t, players observe Xt and choose their respective actions. Let σit (Xt , ε it ) be a strategy function for player i at period t.7 This is a function from the support of (Xt , ε it ) into the binary set {0, 1}, i.e., σit : X × R → {0, 1}. Given any strategy function σit , we can define a choice probability function Pit (Xt ) that represents 7 Strategy functions may change over time. This is the case for the equilibrium strategies in a model with finite time horizon T. More generally, when we do not impose the assumption of Markov Perfect equilibrium, we allow strategies to vary over time. 108 the probability of Yit = 1 conditional on Xt given that player i follows strategy σit . That is, Pit (Xt ) ≡ 1 {σit (Xt , ε it ) = 1} dΛi (ε it ) (3.4) where 1{.} is the binary indicator function. It is convenient to represent players’ behavior and beliefs using these Conditional Choice Probability (CCP) functions. Note that, given that the variables in Xt have a discrete support we can represent the CCP function Pit (.) using a finite dimension vector Pit ≡ { Pi (Xt ) : Xt ∈ X }. Throughout the paper we use either the function Pit (.) or the vector Pit to represent the actual behavior of player i at period t. Without imposing Assumption ’Equil’ (’Equilibrium Beliefs’), a player’s beliefs about the behavior of other players do not necessarily represent the actual behavior of the other player. Therefore, we need functions other than σjt (.) and Pjt (.) to represent players i’ beliefs about the strat
egy of player j. Let b jt Xt , ε jt be player i’s belief about the strategy function of player j at
period t. And let Bjt (Xt ) be the choice probability associated with b jt Xt , ε jt , i.e., Bjt (Xt ) ≡

1{b jt Xt , ε jt = 1} dΛ j ε jt . We can represent function Bjt (.) using a finite-dimensional vector B jt ≡ { Bjt (Xt ) : Xt ∈ X }, and B̃ j represents {B jt : t = 1, 2, ..., T }. Now, suppose that we impose Assumptions 1 and 2, but not Assumption ’Equil’. We say that a strategy function σit (.) (and the associated CCP function Pit ) is rational if for every possible value of (Xt , ε it ) ∈ X × R the action σit (Xt , ε it ) maximizes player i’s expected and discounted value given his beliefs on the opponent’s strategy. Given beliefs B̃ j , player i’s best response is the optimal solution of a single-agent dynamic programming (DP) problem. A DP problem can be described in terms of a discount factor, a payoff function, and a transition probability function for the state variables. In this case, the discount factor is β i , the expected one-period payoff function is Yit (πitB (Xt ) − ε it ), where: πitB (Xt ) = (1 − Bjt (Xt )) πit (0, Xt ) + Bjt (Xt ) πit (1, Xt ) (3.5) 109 And the transition probability of the state variables is: f itB (Xt+1 |Yit , Xt ) =   f i ( Xit+1 |Yit , Xt ) (1 − Bjt (Xt )) f j ( X jt+1 |0, Xt ) + Bjt (Xt ) f j ( X jt+1 |1, Xt ) (3.6) Let VitB (Xt , ε it ) be the value function for player i’s dynamic programming problem given beliefs B̃ j . By Bellman’s principle, the sequence of value functions {VitB : t = 1, 2, ..., T } can be obtained recursively using backwards induction in the following Bellman equation: VitB (Xt ) = max Yit (πitB (Xt ) − ε it ) + β i VitB+1 (Xt+1 , ε it+1 ) dΛi (ε it+1 ) d f itB (Xt+1 |Yit , Xt ) (3.7) Yit ∈{0,1} The best response function of player i at period t given beliefs B̃ j is the optimal decision rule of this DP problem. Define the threshold value function: vitB (Xt ) ≡ πitB (Xt ) + β i ∑   f itB (Xt+1 |1, Xt ) − f itB (Xt+1 |0, Xt ) V̄itB+1 (Xt+1 ) (3.8) X t +1 where V̄itB (Xt ) is the integrated value function R VitB (Xt , ε it )dΛi (ε it ). We denote vitB a threshold value function because it represents the threshold value of ε it that makes player i indifferent between the choice of alternatives 0 and 1. The optimal response function is: {Yit = 1} iff  ε it ≤ vitB (Xt ) (3.9) The best response probability function is Λi (vitB (Xt )). Therefore, under Assumptions 1 and 2 the actual behavior of player i, represented by the CCP function Pit (.), satisfies the following condition: Pit (Xt ) = Λi vitB (Xt )
(3.10) This equation summarizes all the restrictions that Assumptions 1 and 2 impose on players’ choice probabilities. The right hand side of equation (3.10) is the best response function of a rational 110 player. We use Λi (vitB ) to represent the vector-value function {Λi vitB (Xt )
: X t ∈ X }. The concept of Markov Perfect Equilibrium (MPE) is completed with Assumption ’Equil’ (’Equilibrium Beliefs’). Under this assumption, players’ beliefs are in equilibrium, i.e., Bjt (Xt ) = Pjt (Xt ) for any period t and state Xt . A MPE can be described as a sequence of CCP pairs, {Pit , P jt : t = 1, 2, ..., T } such that for every player i and any time period t, we have that  P Pit = Λi vitj (3.11) For the rest of the paper, we maintain Assumptions 1 and 2 but we relax Assumption ’Equil’ on ’Equilibrium Beliefs’. Our approach is agnostic about the formation of players’ beliefs. Assumption 2 establishes that every player is rational, in the sense that his strategy maximizes his expected and discounted payoff given his beliefs on other players’ behavior. For some of our results, we need a stronger concept of rationality that is defined as level 2 rationalizability: every player is rational and knows that his opponents are rational as well, such that players’ beliefs are consistent with other players’ having rational strategies. 3.3 Identification Suppose that the researcher has a random sample with realizations of the game over multiple locations and time periods. Using the terminology in empirical applications of games in Industrial Organization, we employ the term local market to refer to a location. We use the letter m to index local markets. The researcher observes a random sample of M local markets with information on {Yimt , Ximt , Wmt } for every player i ∈ {1, 2} and every period t ∈ {1, 2, ..., T }. For the moment, we consider that the dynamic game has a finite horizon T. The number of local markets, M, is large and for the identification results in this section we consider that M is infinite. The unobservable variables {ε imt } are assumed to be independently and identically distributed across markets and over time.8 8 This framework can be extended to incorporate unobservable state variables for the econometrician which are common knowledge to players and have a distribution with finite support (see Kasahara and Shimotsu, 2008b). 111 We want to use this sample to estimate the models structural parameters (functions) {πi , f i , f W , Λi , β i : i = 1, 2}. We assume that the distribution of the unobservables, Λi , is known to the researcher up to a scale parameter. We consider identification of the payoff function πi up to scale, but for notational convenience we omit the scale parameter. Following the standard approach in dynamic decision models, we assume that the discount factors, β i , is known to the researcher. Note that the transition probability functions f i and f W are nonparametrically identified by using 0 0 M sample frequency estimators, i.e., fˆi ( Xi |Yi , Xi ) = [∑m =1 1{ Ximt+1 = Xi , Yimt+1 = Yi , Ximt = Xi }] 0 0 M M ˆ / [∑m =1 1{Yimt+1 = Yi , Ximt = Xi }], and f W ( W | W ) = [ ∑m=1 1{ Wmt+1 = W , Wmt = W }] / M [∑m =1 1{ Wmt = W }]. Therefore, we concentrate on the identification of the payoff functions πi and π j and consider that { f i , f W , Λi , β i : i = 1, 2} are known. 0 ( X ) be the true (population) conditional probability function Pr(Y Let Pimt imt = 1| Xmt = X ) that represents the actual behavior of player i in market m at period t. Let B0jmt (X) be the probability function with player i’s ’true’ beliefs in market m at period t. And let π 0 ≡ {πi0 , π 0j } be the true payoff functions in the population. Assumption 3 summarizes our conditions on the Data Generating Process. 0 is the best response of player i given his beliefs B0 and the ASSUMPTION 3: (A) For every player i, Pimt jm payoff functions π 0 , i.e., P0imt = Λi (B0im ; π 0 ). (B) Players’ beliefs may vary over time (in an unrestricted way) and over markets with the observable characteristics X, but a player has the same beliefs in markets with the same observable characteristics X at the same time period, i.e., for every market m with Xmt = X, 0 ( X ) = B0 ( X ). Bimt it Assumption 3(A) simply establishes that players are rational in the sense that their actual behavior is the best response given their beliefs. Assumption 3(B) establishes that a player should have the same beliefs in two markets with the same observable characteristics and at the same period of time. Note that beliefs can vary across markets according to the observable variables in Xmt .9 This assumption plays a similar role for identification as the assumption of "no multiple equi9 Our sampling design with large M and small T is the standard case in applications of empirical games in Industrial Organization. Alternatively, if the sampling design is such that the number of periods T is large and the number of markets M is small, then we can allow beliefs to vary over markets but being constant over time. 112 libria in the data" in dynamic games under the condition that beliefs are in equilibrium.In dynamic games where beliefs are in equilibrium, the no multiple equilibria in the data assumption effectively allows the researcher to identify player beliefs. This is because conditional choice probabilities are non-parametrically identified under this assumption (Aguirregabiria and Mira (2007)), and if beliefs are in equilibrium, a player’s belief about his opponents probability of taking any given action is that probability itself. When beliefs are not in equilibrium we of course can not identify beliefs in this way, but assumption 3B does allow us to identify CCPS from the data because it implies that the probability of taking any given action across two markets with the same observable characteristics is the same. This in turn, as we will show, is important for the identification of beliefs themselves.10 Assumption 3 implies that choice probabilities describing players’ actual behavior (CCPs) should be the same in two markets with the same observable characteristics, i.e., for any market 0 ( X ) = P0 ( X ). It also implies that, given a large sample of local m with Xmt = X, we have that , Pimt it markets, we can identify these probabilities nonparametrically from the data. For any player i, any period t, and any value of X ∈X , we have that Pit0 (X) = E(Yimt |Xmt = X), and this conditional expectation can be estimated consistently using data on Yimt and Xmt . For instance, given that Xmt M is a vector of discrete random variables the frequency or ’cell’ estimator ∑m =1 Yimt 1{ Xmt = X } / M 0 ∑m =1 1{ Xmt = X } is a consistent estimator of Pit ( X ). For notational simplicity, we omit the market subindex m for the rest of this section. The model  0  restrictions are summarized in the best response conditions Pit0 (Xt ) = Λi vitB (Xt , π 0 ) . Given these conditions, we want to identify payoffs π 0 and beliefs B0j . It is simple to verify that, without further restriction, the order condition for identification is not satisfied. For a given time period and player, the number of restrictions is |X |, the number of parameters in the payoff function is 2|X |, and the number of parameters in beliefs is |X |, i.e., |X | < 3|X |. 10 See Assumptions 5(A) and 5’(A) in Aguirregabiria and Mira (2007) and the discussion of these assumptions in pages 14-15 and 25-26 of that paper. Other recent papers on the estimation of dynamic games, such as Bajari, Benkard, and Levin (2007), Pakes, Ostrovsky, and Berry (2007), and Pesendorfer ans Schmidt-Dengler (2008) consider similar assumptions. Aguirregabiria and Mira (2009) relax that assumption to allow for multiple equilibria in the data. 113 The following is our key identification assumption. ASSUMPTION 4 (Exclusion Restriction): The one-period payoff function of player i depends on the actions of both players, Yit and Yjt , the common state variables Wt , and the own stock variable, Xit , but it does not depend on the stock variable of the other player, X jt . πit (Yjt , Xt ) = πit (Yjt , Wt , Xit ) This type of exclusion restriction appears naturally in some dynamic games. For instance, this assumption holds in the dynamic game of market entry and exit of our Example in section 2 above. A firm’s profits depend on whether the competitor is currently in the market or not but, given that, it does not depend on whether the competitor was in the market or not at previous period. Similarly, in a dynamic game of Cournot competition with capacity constraints (e.g., Besanko and Doraszelski, 2004) a firm’s profit depend on the current capacity of competing firms but not on the capacity of these firms at previous periods. A similar structure appears in the dynamic game of retail store competition that we present and estimate in section 5. Though exclusion restriction in Assumption 4 reduces the number of parameters in the payoff function, it is not enough for the identification of the model. The order condition of identification is not satisfied. The number of restrictions is still the same, |X | = |W |(K + 1)2 , the number of parameters in the payoff function is now 2|W |(K + 1), and the number of parameters in beliefs is still |X | = |W |(K + 1)2 . ASSUMPTION 5 (Monotonic Beliefs and Large support condition): (A) The beliefs function B0jt (Xt ) is strictly monotonically decreasing (or increasing) in X jt : i.e., ∂B0jt (Xt )/∂X jt < 0. (B) Conditional on (Wmt , Ximt ) the variable X jmt has variability over the whole real line. ASSUMPTION 6 (Normalization of beliefs): There are two values in the support of X j , say X jlow and high Xj , such that for every value of ( Xi , W ) we have that beliefs are in equilibrium, i.e., B0jt ( Xi , X jlow , W ) = high Pjt0 ( Xi , X jlow , W ), and B0jt ( Xi , X j high , W ) = Pjt0 ( Xi , X j , W ). 114 We use Assumptions 5 and 6 as two alternative conditions that, together with the exclusion restriction, can provide nonparametric identification of payoff functions and beliefs. These two assumptions are related. Assumption 6 may be interpreted as a generalization of Assumption 5. The intuition behind it is that when the opponent’s stock variable is ’bad’ enough (or ’good’ enough) strategic uncertainty disappears such that the player knows perfectly the strategy of his opponent. Strategic uncertainty may disappear because the opponent’s choice becomes certain (i.e., the choice probability gets arbitrarily close to zero or one), as under Assumption 5. But the conditions for the absence of strategic uncertainty may be more general, as under Assumption 6. The following Proposition summarizes our main identification result. PROPOSITION: Under Assumptions 1-4 and 5 (and under Assumptions 1-4 and 6) the payoff functions {πit0 for any i, t} and the beliefs functions { Bit0 for any i, t} are nonparametrically identified. The proof of this Proposition follows a backwards induction approach. First, we show identification of beliefs and payoff functions at period T. Then, given identification at period T, we apply backwards induction to prove identification at any period t < T. Through this proof, we assume that the time horizon T is finite and that the researcher has panel data that covers all the periods until T. At the end of this section we discuss identification when we relax this conditions. Consider the identification at the last period T. For notational simplicity we omit the time subindex. Under Assumptions 1 to 3, the CCP functions Pi0 and Pj0 are identified everywhere in the support of X. At period T, the threshold function viB (X, π ) is equal to πiB (X) = (1 − Bj (X)) πi (0, X) + Bj (X) πi (1, X). Therefore, the best response condition is: Pi0 (X) = Λi ((1 − B0j (X))πi0 (0, X) + B0j (X)πi0 (1, X)). Define the function q0i (X) ≡ Λi−1 ( Pi0 (X)). Given that the distribution function Λi is invertible and it is known (up to scale) to the researcher, the function q0i (.) is identified everywhere in the support of X. We have that:   q0i (X) = πi0 (0, X) + πi0 (1, X) − πi0 (0, X) B0j (X) (3.12) Let X ja , X jb , X jc ,and X jd be four arbitrary values in the support of X j such that X ja 6= X jb and X jc 6= X jd . Under Assumption 4 (Exclusion Restriction), we have that πi0 (0, X a ) = πi0 (0, Xb ) = 115 = πi0 (0, Xc ) πi0 (0, Xd ), and     {πi0 (0, X a ) + πi0 (1, X a ) − πi0 (0, X a ) Bj (X a )} − {πi0 (0, Xb ) + πi0 (1, Xb ) − πi0 (0, Xb ) Bj (Xb )}     {πi0 (0, Xc ) + πi0 (1, Xc ) − πi0 (0, Xc ) Bj (Xc )} − {πi0 (0, Xd ) + πi0 (1, Xd ) − πi0 (0, Xd ) Bj (Xd )} = Bj (X a ) − Bj (Xb ) Bj (Xc ) − Bj (Xd ) Therefore, we have that: Bj (X a ) − Bj (Xb ) q0i (X a ) − q0i (Xb ) = Bj (Xc ) − Bj (Xd ) q0i (Xc ) − q0i (Xd ) (3.13) This expression shows that, under assumptions 1 to 4, there is a function of beliefs that is identified, without having to impose the assumption of equilibrium beliefs. Note that this result relies on the exclusion restriction but it does not require the monotonicity of beliefs (assumption 5a), the large support condition (assumption 5b), or the restriction on beliefs in assumption 6. This result also provides a nonparametric test for the null hypothesis of equilibrium beliefs. Define the function: ( δi (X a , Xb , Xc , Xd ) ≡ q0i (X a ) − q0i (Xb ) q0i (Xc ) − q0i (Xd ) ) ( − Pj0 (X a ) − Pj0 (Xb ) Pj0 (Xc ) − Pj0 (Xd ) ) (3.14) It is clear that we can nonparametrically identify δi . If beliefs are in equilibrium, δi is equal to zero for any value of (X a , Xb , Xc , Xd ). Let δ̂i be a root-M consistent and asymptotically nonparametric estimator of δi . The hypothesis of equilibrium beliefs implies δi = 0, and we can test equilibrium beliefs using a simple LM test of the null hypothesis of δi = 0. We describe this test in more detail in section 4. Let’s consider the identification of the payoff function πi0 from equation (3.12). First, we present the proof using Assumption 5 (monotonic beliefs and large support condition). By the monotonicity of the beliefs function Bj , as X j → +∞ we have that Bj (X) → 0. Therefore, for arbitrarily large values of X j we have that πi0 (0, Xi , W) = q0i (X) and variables ( Xi , W) have variability over all their support. Therefore, πi0 (0, .) is identified. Similarly, as X j → −∞ we have that Bj (X) → 1, and πi0 (1, Xi , W) = q0i (X) such that πi0 (1, .) is identified over the whole sup- πi0 port of ( Xi , W).Given that 116 is identified, then we can obtain the beliefs function as Bj (X) = [q0i (X) − πi0 (0, X)]/[πi0 (1, X) − πi0 (0, X)] for any value of X with πi0 (1, X) − πi0 (0, X) 6= 0. We consider now identification of the model when we replace Assumption 5 by Assumption 6. First, note that the order condition of identification is satisfied: the number of restrictions is |X | = |W |(K + 1)2 , the number of parameters in the payoff function is now 2|W |(K + 1), and the number of parameters in beliefs is |W |(K + 1)2 − 2|W |(K + 1). The number of unknown parameters is equal to the number of restrictions. Solving the restrictions imposed by Assumption 6 into equation (3.12), we have that for any value of ( Xi , W): q0i ( Xi , X jlow , W) =   πi0 (0, Xi , W) + πi0 (1, Xi , W) − πi0 (0, Xi , W) Pj0 ( Xi , X jlow , W) (3.15) high q0i ( Xi , X j   high , W) = πi0 (0, Xi , W) + πi0 (1, Xi , W) − πi0 (0, Xi , W) Pj0 ( Xi , X j , W) high Combining these two equations, we obtain [πi0 (1, Xi , W) − πi0 (0, Xi , W)] = [q0i ( Xi , X j high q0i ( Xi , X jlow , W)] / [ Pj0 ( Xi , X j , W) − , W) − Pj0 ( Xi , X jlow , W)]. And solving this expression into the equa- tions in (3.15), we identify the payoff function as: πi0 (0, X) = q0i (Xlow ) − Pj0 (Xlow )  q0i (Xhigh )−q0i (Xlow ) Pj0 (Xhigh )− Pj0 (Xlow )  (3.16) h πi0 (1, X) = q0i (Xlow ) + 1 − Pj0 (Xlow ) high where Xlow ≡ ( Xi , X jlow , W) and Xhigh ≡ ( Xi , X j i q0i (Xhigh )−q0i (Xlow ) Pj0 (Xhigh )− Pj0 (Xlow )  , W). Again, given the identification of the payoff function, we can obtain beliefs as Bj (X) = [q0i (X) − πi0 (0, X)] / [πi0 (1, X) − πi0 (0, X)]. The equations in (3.16) deserves further comments. First, these equations contain as a particular case the model with the large support condition in Assumption 5. That is, the conditions of Assumption 5 can be seen as a particular case of the conditions in Assumption 6. In particular, suphigh pose that X j high is so large that Pj0 ( Xi , X j , W) = 0, and X jlow is so small that Pj0 ( Xi , X jlow , W) = 1. high Then, the equations in (3.16) become πi0 (0, X) = q0i ( Xi , X jlow , W) and πi0 (1, X) = q0i ( Xi , X j , W ), 117 which are also the expressions that identify the payoff function under the large support conditions in assumption 6a. Second, the equations in (3.16) relate also to the model with equilibrium beliefs. When we impose the assumption of equilibrium beliefs, the equations in (3.16) are satisfied not high only for the pair of values ( X jlow , X j ) but for every pair of values of X j , say X ja and X jb , such that Pj0 ( Xi , X ja , W) − Pj0 ( Xi , X jb , W) 6= 0. Therefore, the payoff function is over-identified, and we can test these over identifying restrictions, as we have mentioned above. Given beliefs and payoffs at period T, we we use backwards induction to prove the identification of the payoff function and beliefs at any period t < T. Suppose that the functions πit0 +s and B0jt+s are known for every s ≥ 1. Given this information, we want to identify func0 tions πit0 and B0jt . Player i’s best response implies that Pit0 (Xt ) = Λi (vitB (Xt )), where vitB (Xt ) = (1 − Bjt (Xt )) πi (0, Xt ) + Bjt (Xt ) πi (1, Xt )+ β i ∑ [ f itB (Xt+1 |1, Xt )− f itB (Xt+1 |0, Xt )] V̄itB+1 (Xt+1 ), and X t +1 R B B V̄it+1 is the integrated values function Vit+1 (Xt+1 , ε it+1 )dΛi (ε it+1 ). Given { Pit0+s , πit0 +s , B0jt+s : s = 1, 2, ..., T − t}, the integrated value function V̄itB+1 is known. Define the function: q0it (X) ≡ Λi−1 ( Pit0 (X)) − β i ∑ [ fitB (Xt+1 |1, Xt ) − fitB (Xt+1 |0, Xt )] V̄itB+1 (Xt+1 ) 0 0 0 (3.17) X t +1 This function is identified everywhere in the support of X. Furthermore, by the best response condition and the definition of q0it , we have that:   q0it (X) = πit0 (0, X) + πit0 (1, X) − πit0 (0, X) B0jt (X) (3.18) This equation has the same structure as the one at the last period T. Therefore, we can use exactly the same arguments to show the identification of functions πit0 and B0jt . So far, we have assumed that the time horizon T is finite and that the researcher has panel data that covers all the periods until T. What if the time horizon of the decision problem is infinite? For the sake of saving notation, we continue using T to represent the number of periods in the sample (but not the time horizon that now is infinite). To obtain identification in this case, we make the 118 following assumption: from period T and forever in the future, agents have rational beliefs on the behavior of the opponent, i.e., Bit0 (X) = Pit0 (X) for any player i, any state X, and any period t ≥ T. Aguirregabiria (2005) and Tan (2008) have studied identification in this case. 3.4 Estimation and Inference The proof of identification in the previous section implicitly suggests a method for the estimation of the just-identified nonparametric model. We start with a detailed description of that estimation method. In most empirical applications, the specification of the payoff function involves parametric restrictions. Therefore, we extend the estimation method to deal with parametric models in a very simple way. Finally, for parametric models, we describe how the two-step method can be extended recursively to generate a sequence of estimators with better statistical properties. 3.4.1 Estimation with nonparametric payoff function Step 1: Nonparametric estimation of CCPs, P̂it0 , for every player, time period, and state X, and (if needed) of the transition probabilities f i and f W . 0 ( X )), the estimated Step 2: Estimation of preferences and beliefs. At the last period T, q̂0iT (X) = Λi−1 ( P̂iT payoff function is: 0 (0, X ) = q̂0 ( Xlow ) − P̂0 ( X high ) π̂iT iT jT  q̂0iT (Xhigh )−q̂0iT (Xlow ) 0 ( X high )− P̂0 ( Xlow ) P̂jT jT  (3.19) h i B̂0jT (X) = 0 (0,X ) q̂0iT (X)−π̂iT 0 0 (0,X ) π̂iT (1,X)−π̂iT 0 (1, X ) = q̂0 ( Xlow ) + 1 − P̂0 ( X high ) π̂iT iT jT q̂0iT (Xhigh )−q̂0iT (Xlow ) 0 ( X high )− P̂0 ( Xlow ) P̂jT jT  and the estimated beliefs function is: (3.20) We also construct an estimate of the integrated value function at period T, that we will use later 119 for the estimation of payoffs and beliefs at periods earlier than T. For instance, if the distribution Λi is logistic:  n 0 o B0 B V̂iT (X) = ln 1 + exp v̂iT (X) (3.21) 0 B ( X ) ≡ (1 − B̂0 ( X )) π̂ 0 (0, X ) + B̂0 ( X ) π̂ 0 (1, X ). where v̂iT t jT iT jT iT At any period t < T, we construct function: ∑ [ fitB (Xt+1 |1, X) − fitB (Xt+1 |0, X)] V̂itB+1 (Xt+1 ) 0 q̂0it (X) = Λi−1 ( P̂it0 (X)) − β i 0 0 (3.22) X t +1 Given q̂0it , we can estimate payoffs and beliefs using the same type of expression as for period T: π̂it0 (0, X) = q̂0it (Xlow ) − P̂jt0 (Xhigh )  q̂0it (Xhigh )−q̂0it (Xlow ) P̂jt0 (Xhigh )− P̂jt0 (Xlow )  (3.23) h i B̂0jT (X) = 0 (0,X ) q̂0iT (X)−π̂iT 0 0 (0,X ) π̂iT (1,X)−π̂iT π̂it0 (1, X) = q̂0it (Xlow ) + 1 − P̂jt0 (Xhigh ) q̂0it (Xhigh )−q̂0it (Xlow ) P̂jt0 (Xhigh )− P̂jt0 (Xlow )  and (3.24) And the integrated value function at period t is: 0 V̂itB (X) = β i ∑ 0 0 f itB (Xt+1 |0, X)] V̂itB+1 (Xt+1 ) − ln 1 − P̂it0 (X)
(3.25) X t +1 It is straightforward to show that this estimator is consistent and asymptotically normal. The derivation of the asymptotic variance is cumbersome. In our empirical application we use the bootstrap method to obtain standard errors and confidence intervals for the estimates. 120 3.4.2 Estimation with parametric payoff function In most applications, we consider a parametric specification of the payoff function. A very common class of parametric specifications is the linear in parameters model: πit (Yjt , Xt ) = zit (Yjt , Xt ) θ (3.26) where zit (Yjt , Xt ) is a 1 × K vector of known functions, and θ is a K × 1 vector of unknown structural parameters. Note that, given the subindexes i and t in the vector of functions zit (.), this specification is very general and it includes models with player-specific and time-specific parameters. Given this specification, the model implies the following relationship: 0 q0it (X) = zitB (X) θ 0 (3.27) 0 where zitB (X) ≡ (1 − B0jt (X)) zit (0, X) + B0jt (X) zit (1, X), and θ 0 is the true vector of parameters in the population. Expression (3.27) is a key equation in the estimation of θ 0 . To estimate θ 0 we consider a simple three steps method. The first two-steps are the same as for the nonparametric model. B = Step 3: Given the estimates from step 2, we construct the variables: q̂imt = q̂0it (Xmt ) and ẑimt (1 − B̂0jt (Xmt )) zit (0, Xmt ) + B̂0jt (Xmt ) zit (1, Xmt ), for every sample observation (i, m, t). Then, we B . The estimator is root-M consistent and estimate θ 0 by running an OLS regression of q̂imt on ẑimt asymptotically normal. Again, the derivation of the asymptotic variance of this estimator is cumbersome, and we use the bootstrap method to obtain standard errors and confidence intervals for the estimates. It is possible to apply step 3 recursively to obtain a sequence of estimators of θ 0 and of belief functions with better asymptotic and finite sample properties. To describe this recursive method, it is important to take into account the model under Assumption 6 implies the following expression 121 for beliefs at any value of X:11 h B0j (X) = Pj0 (Xlow ) + Pj0 (Xhigh ) − Pj0 (Xlow ) i " q0i (X) − q0i (Xlow ) q0i (Xhigh ) − q0i (Xlow ) # (3.28) Let’s denote steps 1 to 3 as the first stage of the recursive procedure. Let θ̂ (1) be the estimator of `0 that results from this first stage. Given θ̂ (1) and using backwards induction we can obtain new estimates of choice probabilities, q’s, beliefs, and value functions. (1) (1) (1) At period T, the functions q̂iT (.), P̂iT (.), and B̂jT (.) for both players are the result of per(1) forming R iterations in the following iterative procedure. We initialize B̂jT (.) = B̂0jT (.).At each iteration, we perform tasks (a) to (d): h i (1) (1) (1) (a) q̂iT (X) = (1 − B̂jT (X)) ziT (0, X) + B̂jT (X) ziT (1, X) θ̂ (1)   (1) (1) (b) P̂iT (X) = Λi q̂iT (X) (c) Updating beliefs: (1) B̂jT (X) = (1) P̂jT (Xlow ) + h (1) P̂jT (Xhigh ) − (1) P̂jT (Xlow ) i (1) " (1) q̂iT (X) − q̂iT (Xlow ) (1) # (1) q̂iT (Xhigh ) − q̂iT (Xlow ) (d) Repeat (a)-(c) using the new beliefs functions. (1) Given P̂iT (.), we update the integrated value function:   (1) (1) V̂iT (X) = − ln 1 − P̂iT (X) (1) (1) (3.29) (1) At period t < T, the functions q̂it (.), P̂it (.), and B̂jt (.) for both players are the result of (1) performing R iterations in the following iterative procedure. We initialize B̂jt (.) = B̂0jt (.). At each iteration, we perform tasks (a) to (d): 11 As B0 (X a )− B0 (Xb ) q0i (X a )−q0i (Xb ) = B0j (Xc )− B0j (Xd ) . q0i (Xc )−q0i (Xd ) j j Xhigh , and Xd = Xlow , we have that shown in equation (3.13) above, for any vectors X a , Xb , Xc , and Xd , we have that If we particularize this expression at the vectors X a = X, Xb = Xlow , Xc = q0i (X)−q0i (Xlow ) q0i (Xhigh )−q0i (Xlow ) B0j (X)− B0j (Xlow ) 0 Bj (Xhigh )− B0j (Xlow ) . By Assumption 6, B0j (Xlow ) = Pj0 (Xlow ) and B0j (Xhigh ) = Pj0 (Xhigh ). Thus, solving h ih 0 i q (X)−q0 (Xlow ) for B0j (X) we obtain B0j (X) = Pj0 (Xlow ) + Pj0 (Xhigh ) − Pj0 (Xlow ) q0 (iXhigh )−i q0 (Xlow ) . = i i 122 h (a) (1) q̂it (X) (b) (1) P̂iT (X) = (1 − (1) B̂jt (X)) zit (0, X) + (1) B̂jt (X) zit (1, X) i θ̂ (1) ! (1) q̂it (X) + = Λi B0 B0 β i ∑ [ f it (Xt+1 |1, X) − f it (Xt+1 |0, X)] X t +1 (1) V̂it+1 (Xt+1 ) (c) Updating beliefs: (1) B̂jt (X) = (1) P̂jt (Xlow ) + h (1) P̂jt (Xhigh ) − (1) P̂jt (Xlow ) i " (1) (1) q̂it (X) − q̂it (Xlow ) (1) # (1) q̂it (Xhigh ) − q̂it (Xlow ) (d) Repeat (a)-(c) using the new beliefs functions. (1) Given P̂it (.), we update the integrated value function:   0 (1) (1) (1) V̂it (X) = β i ∑ f itB (Xt+1 |0, X)] V̂it+1 (Xt+1 ) − ln 1 − P̂it (X) X t +1 (1) (1) (1) Finally, given the new estimated functions q̂it (.) and B̂jt (.), we construct the variables: q̂imt = (1) (1) (1) (1) q̂it (Xmt ) and ẑimt = (1 − B̂jt (Xmt )) zit (0, Xmt ) + B̂jt (Xmt ) zit (1, Xmt ), for every sample observa(1) (1) tion (i, m, t). Then, we estimate θ 0 by running an OLS regression of q̂imt on ẑimt . Let θ̂ (2) be the estimator of θ 0 that results from this first stage. This same procedure can be repeated to obtain a sequence of estimators {θ̂ (K) , B̂(K) : K ≥ 1} with better statistical properties. 3.4.3 Test of Equilibrium Beliefs In this subsection, we present a simple test of the null hypothesis of equilibrium beliefs. In principle, we could considered a standard Lagrange Multiplier (LM) or Score test based on the constrained maximum likelihood estimation (MLE) of structural parameters and beliefs. Define the log-likelihood function: M l (θ, P) ≡ T 2 ∑ ∑ ∑ Yit m =1 t =1 i =1 log Λi (vitB (Xmt , θ )) + (1 − Yit ) log(1 − Λi (vitB (Xmt , θ ))) (3.30) 123 The constrained MLE is defined as a vector (θ̂ MLE , P̂ MLE ) such that: (θ̂ MLE , P̂ MLE ) = arg max l (θ, P) (θ,P) (3.31) subject to: P = Λ(vP (θ )) We want to test the null hypothesis P = Λ(vP (θ )), that consists of 2|X | constrains on (θ, P). The standard LM statistic for testing this null hypothesis is: LM( MLE) ∂l (θ̂ MLE , P̂ MLE ) = ∂(θ, P)0 " ∂2 l (θ̂ MLE , P̂ MLE ) ∂(θ, P)∂(θ, P)0 # −1 ∂l (θ̂ MLE , P̂ MLE ) ∂(θ, P) (3.32) Under the null hypothesis, this statistic is asymptotically distributed as a chi-square with 2|X | degrees of freedom. The test that we propose is the following. Let X ja , X jb , X jc ,and X jd be four arbitrary values in the support of X j such that X ja 6= X jb and X jc 6= X jd . And define the function δi0 : ( δi0 (X a , Xb , Xc , Xd ) ≡ q0i (X a ) − q0i (Xb ) q0i (Xc ) − q0i (Xd ) ) ( − Pj0 (X a ) − Pj0 (Xb ) ) (3.33) Pj0 (Xc ) − Pj0 (Xd ) Given our model under Assumptions 1-4, we have that if player i has rational beliefs then δi0 (X a , Xb , Xc , Xd ) = 0 for every value of (X a , Xb , Xc , Xd ). Therefore, testing δi0 (X a , Xb , Xc , Xd ) = 0 implies testing the null hypothesis of rational beliefs. More precisely, we test the following null hypothesis. Let H be the number of all possible combinations of four different values of X j . We index (h) these values by h. Let Xmt the quadruplet h when the values of Xi and W are the ones in ob(h) (h,a) (h,b) (h,c) (h,d) (h,a) servation (m, t): i.e., Xmt = (Xmt , Xmt , Xmt , Xmt ) = ([ Ximt , X j (h,c) [ Ximt , X j (h,d) , Wmt ], [ Ximt , X j (h,b) , Wmt ], [ Ximt , X j , Wmt ], , Wmt ]). Under the hypothesis of equilibrium beliefs, we have that (h) E(δi0 (Xmt )) = 0 for every h. This is exactly the null hypothesis that we test:    (h) H0 : E δi0 Xmt = 0 for every quadruple h (3.34) 124 Let be the estimator of that we obtain when we replace and by nonparametric esti  (h) (h) mates of these CCP functions. Define the statistic dˆi that is simply the sample mean of δ̂i0 Xmt ,   (h) M T 0 X(h) . Then, define the statistic: i.e., dˆi = ( MT )−1 ∑m δ̂ ∑ =1 t =1 i mt δ̂i0 δi0 Pi0 Ŝ = ∑ H h =1 (h) dˆi (h) se(dˆ ) Pj0 !2 (3.35) i (h) (h) where se(dˆi ) is the standard error of dˆi , that we can obtains using nonparametric bootstrap. Under the null hypothesis, Ŝ is asymptotically distributed as a Chi-square with H degrees of freedom. 3.5 Empirical Application We illustrate our model and method with an application of a dynamic game of store location. There has been significant interest in recent years in the estimation of game theoretic models of entry in local markets and store location. Most studies have considered static games: see Mazzeo (2002), Seim (2006), Jia (2008), Nishida (2008), and Zhu and Singh (2009), among others. Recent applications estimate dynamic games of store location: Holmes (2010), Beresteanu and Ellickson (2005), Aguirregabiria, Mira, and Roman (2006), Walrath (2008), and Suzuki (2010). We study store location of McDonalds (MD) and Burger King (BK) using data for United Kingdom during the period 1991-1995. We divide the UK in local markets (districts) and consider these companies’ decisions of how many stores, if any, to operate in each local market. The profits of a store in a market depend on local demand and cost conditions and on the degree of competition from other firms’ stores and from stores of the same chain. There are sunk costs associated with opening a new store, and therefore this decision has implications on future profits. Firms are forward-looking and maximize the value of expected and discounted profits. Each firm has uncertainty about future demand and cost conditions in local markets. Firms also have uncertainty about the current and future behavior of the competitor. In this context, the standard assump- 125 tion is that firms have rational expectations about other firms’ strategies, and that these strategies constitute a Markov Perfect Equilibrium. Here we relax this assumption. The main question that we want to analyze in this empirical application is whether the beliefs of each of these companies about the store location strategy of the competitor are consistent with the actual behavior of the competitor. 3.5.1 Data and descriptive evidence The dataset was collected by Otto Toivanen and Michael Waterson, who have used it in their paper Toivanen and Waterson (2005).12 Our working sample is a five year panel that tracks 422 local authority districts (local markets), including the information on the stock and flow of MD and BK stores into each district. It also contains socio-economic variables at the district level such as population, density, age distribution, average rent, income per capita, local retail taxes, and distance to the UK headquarters of each of the firms. The local authority district is the smallest unit of local government in the UK, and generally consists of a city or a town sometimes with a surrounding rural area. There are almost 500 local authority districts in Great Britain. Our working sample of 422 does not included the districts that belong to Greater London.13 The median district in our sample has an area of 300 square kilometers and a population of 95,000 people.14 Table 1 presents descriptive statistics for socio-economic and geographic characteristics of our sample of local authority districts. Table 2 presents descriptive statistics on the evolution of the number of stores for the two firms. Toivanen and Waterson present a detailed discussion of why the retail chain fast food hamburger industry in the UK during this period can be considered as a duopoly of BK and MD. In 1990, MD had more than three times the number of stores of BK, and it was active in more than twice 12 We want to thank Otto Toivanen and Michael Waterson for generously sharing their data with us. See also Yang (2010) for other empirical application with these data. 13 The reason to exclude from our sample the districts in Greater London is that they do not satisfy the standard criteria of isolated geographic markets. 14 As a definition of geographic market for the fast food retail industry, the district is perhaps a bit wide. However, an advantage of using district as definition of local market is that most of the markets in our sample are geographically isolated. Most districts contain a single urban area. And, in contrast to North America where many fast food restaurants are in transit locations, in UK these restaurants are mainly located in the centers of urban areas. 126 Table 3.1 Descriptive Statistics on Local Markets (Year 1991) 422 local authority districts (excluding Greater London districts) Variable Median Std. Dev. Pctile 5% Pctile 95% Area (thousand square km) 0.30 0.73 0.03 1.67 Population (thousands) 94.85 93.04 37.10 280.50 Children: Age 5-14 (%) 12.43 1.00 10.74 14.07 Young: 15-29 (%) 21.24 2.46 17.80 25.17 Pensioners: 65-74 (%) 9.01 1.50 6.89 11.82 GDP per capita (thousand £) 92.00 12.14 74.40 112.70 Claimants of UB / Population ratio (%) 2.75 1.27 1.24 5.11 Avg. Weekly Rent per dwelling (£) 25.31 10.61 19.11 35.07 Council tax (thousand £) 0.24 0.05 0.11 0.31 Number of BK stores 0.00 0.62 0.00 1.00 Number of MD stores 1.00 1.16 0.00 3.00 127 the number of local markets than BK. Conditional on being active in a local market, MD had also significantly more stores per market than BK. These differences between MD and BK have not declined significantly over the period 1991-1995. While BK has entered in more new local markets than MD (69 new markets for BK and 48 new markets for MD), MD has open more stores (143 new stores for BK and 166 new stores for MD). Table 3 presents the annual transition probability of market structure in local markets as described by the number of stores of the two firms. According to this transition matrix, opening a new store is an irreversible decision, i.e., no store closings are observed during this sample period. In Britain during our sample period, the fast food hamburger industry was still young and expanding, as shown by the large proportion of observations/local markets without stores (41.6%). Although there is significant persistence in every state, the less persistent market structures are those where BK is the leader. For instance, if the state is "BK=1 & MD=0", there is a 20% probability that the next year MD opens at least one store in the market. Similarly, when the state is "BK=2 & MD=1", the chances that MD opens one more store the next year are 31%. 128 Table 3.2 Evolution of the Number of Stores 422 local authority districts (excluding Greater London districts) 1990 #Markets with Stores Change in #Markets with Stores # of Stores Change in # of Stores Mean #Stores per Market (Conditional on#Stores > 0) #Markets with Stores 98 104 118 131 150 - 17 6 14 13 19 79 115 128 153 181 222 - 36 13 25 28 41 1.11 1.17 1.23 1.30 1.38 1.48 1990 1991 McDonalds 1992 1993 1994 1995 206 213 220 237 248 254 7 7 17 11 6 316 344 382 421 447 35 28 38 39 26 1.49 1.56 1.61 1.70 1.76 281 Change in # of Stores Mean #Stores per Market (Conditional on#Stores > 0) 1995 71 Change in #Markets with Stores # of Stores Burger King 1991 1992 1993 1994 1.36 129 Table 3.3 Transition Probability Matrix for Market Structure Annual Transitions. Market structure: BK=x & MD=y, where x and y are number of stores % Market Structure at t+1 2 BK≥ MD=0 MD=1 MD≥ - - 0.1 - 7.4 1.0 - - 1.4 - - 15.8 - - 1.4 - 76.0 18.0 2.0 4.0 - - - - - 87.1 8.1 - 3.3 1.4 - - - - - 86.5 - - 13.5 2 &MD=0 - - - - - - 84.6 15.4 - 2 &MD=1 - - - - - - - 69.0 31.0 2 &MD≥ 2 - - - - - - - - 100.0 41.6 23.3 6.6 2.2 10.9 8.8 0.6 1.4 4.5 BK=0 BK=0 MD=0 MD=1 BK=0 &MD=0 95.1 3.6 BK=0 &MD=1 - BK=1 BK=1 MD=0 MD=1 0.2 1.0 - 87.2 4.2 - 2 - - 82.7 BK=1 &MD=0 - - BK=1 &MD=1 - 2 BK≥ BK≥ BK=0 &MD≥ BK=1 &MD≥ BK≥ Frequency BK=0 MD≥ 2 BK≥ BK=1 MD≥ 2 2 BK≥ 2 2 130 Table 3.4 Reduced Form Probits for the Decision to Open a Store Explanatory Variable Estimated Marginal Effects1 (∆P( x ) when dummy goes from 0 to 1) Burger King McDonalds No FE County FE District FE No FE County FE District FE Own number of stores at t-1 Dummy: Own #stores = 1 Dummy: Own #stores = 2 Dummy: Own #stores ≥ 3 -0.021∗∗ (0.005) -0.023∗∗ (0.004) -0.019∗∗ (0.005) -0.036∗∗ (0.007) -0.030∗∗ (0.005) -0.027∗∗ (0.005) -0.885∗∗ (0.063) -0.210∗ (0.085) -0.056 (0.036) -0.035∗∗ (0.010) -0.047∗∗ (0.006) -0.043∗∗ (0.006) -0.045∗∗ (0.012) -0.060∗ (0.008) -0.053∗∗ (0.008) -0.550∗∗ (0.056) -0.757∗∗ (0.041) -0.816∗∗ (0.038) 0.032∗∗ (0.011) 0.045∗ (0.023) 0.089∗ (0.048) 0.037∗ (0.014) 0.052∗ (0.029) 0.101∗ (0.059) -0.025 (0.055) -0.017 (0.031) 0.011 (0.084) 0.020 (0.013) 0.041 (0.029) -0.041∗∗ (0.007) 0.032∗ (0.018) 0.076 (0.046) -0.050∗∗ (0.009) 0.052∗∗ (0.073) -0.007∗∗ (0.093) -0.104∗∗ (0.020) 0.024 0.027 0.014 0.045 0.054 0.085 YES YES NO NO 2110 422 -371.89 0.229 YES YES YES NO 1715 343 -340.26 0.252 YES YES NO YES 535 107 -110.54 0.624 YES YES NO NO 2110 422 -467.46 0.159 YES YES YES NO 1855 371 -449.02 0.161 YES YES NO YES 640 128 -198.50 0.441 Competitor’s number of stores at t-1 Dummy: Comp.’s #stores = 1 Dummy: Comp.’s #stores = 2 Dummy: Comp.’s #stores ≥ 3 Pred. Prob. Y=1 at mean X Time dummies Control variables2 County Fixed Effects District Fixed Effects Number of Observations3 Number of Local Districts3 log likelihood Pseudo R-square Note 1: Estimated Marginal Effects are evaluated at the mean value of the rest of the explanatory variables. Note 2: Every estimation includes as control variables log of population, log of GDP per capita, log of population density, proportion population 5-14, proportion population 15-29, average rent, and proportion of claimants of unemployment benefits. Note 3: Fixed effects estimations do not include districts for which the dependent variable does not have enough time variation. 131 Table 4 presents estimates of reduced form Probit models for the decision to open a new store. We obtain separate estimates for MD and BK. Our main interest is in the estimation of the effect of previous year number of stores (own stores and competitor’s stores) on the probability of opening a new store. We include as control variables population, GDP per capita, population density, proportion of population 5-14, proportion population 15-29, average rent, and proportion of claimants of unemployment benefits. To control for unobserved local market heterogeneity we also present two fixed effects estimations, one with county fixed effects and other with local district fixed effects. We report only estimates of the marginal effects associated to the dummy variables that represent previous year number of stores. The main empirical result from Table 4 is that, regardless the set of control variables that we use, the own number of stores has a strong negative effect on the probability of opening a new store but the effect of the competitor’s number of stores is either negligible or even positive. This result is very robust to different specification of the reduced form model is along the lines of the empirical results from the reduced forms in Toivanen and Waterson’s paper. Controlling for unobserved heterogeneity using fixed effects reveals that the estimation without fixed effects suffers of a very important upward bias in the marginal effect of the number of own stores. However, the estimated marginal effect of the number of competitor’s stores barely changes. The estimates show also a certain asymmetry between the two firms: the absence of response to the competitor’s number of stores is more clear for BK than for MD. In particular, when BK has three stores in the market there is significant reduction on MD’s probability of opening a new store. That negative effect does not appear in the reduced form probit for BK. This empirical evidence cannot be explained by a standard static model of store location by firms that sell substitute products. Here we explore three, non-mutually exclusive, explanations: (a) spillover effects; (b) forward looking behavior (dynamic game); and (c) biased beliefs about the behavior of the competitor. (a) Spillover effects. The competitor’s number of stores may have a positive spillover effect on the profit of a firm. There are different possible sources of this spillover effect, e.g., informational, 132 product expansion through advertising, costs reductions etc. Since we do not have price and quantity data at the level of local markets, we do not try to identify the source of the spillover effect. We allow for this effect in our specification of demand such that the natural interpretation in the context of our model is a product expansion coming from the advertising effect of retail stores. However, this should be interpreted as a shortcut or ’reduced form’ specification of different possible spillover effects. (b) Forward looking behavior. Opening a store is a partly irreversible decision that involves significant sunk costs. Therefore, it is reasonable to assume that firms are forward looking when they make this decision and that they play a dynamic game. Forward looking behavior might explain the apparent lack of competitive effects when we look at these decisions from the point of view of a static model of entry. Suppose that firms anticipate, with some uncertainty, the total number of hamburger stores that a local market can sustain in the long-run given the size and the socioeconomic characteristics of the market. For simplicity, suppose that this number of "available slots" does not depend on the ownership of the stores because the products sold by the two firms are very close substitutes. In this context, firms play a ’racing’ game to fill as many ’slots’ as possible with their own stores. Diseconomies of scale and scope may generate a negative effect of the own number of stores on the decision of opening new stores. However, in this model, during most of the period of expansion the number of slots of the competitor has zero effect on the decision of opening a new store. Only when the market is filled or close to filled do the competitor’s stores have a significant effect on entry decisions. (c) Biased beliefs. As mentioned in the Introduction, competition in actual oligopoly industries is often characterized by strategic uncertainty. Firm face significant uncertainty about the strategies of their competitors, and they try to deceive competitors about their own strategies. In the context of our application, it may be the case that MD’sand/or BK’s beliefs overestimate the negative effect of the competitor’s stores on the competitor’s entry decisions. For instance, if MD has one store in a local market, BK may believe that the probability that MD opens a second store is close 133 to zero. This over-optimistic belief about the competitor’s behavior may generate an apparent lack of response of BK’s entry decisions to the number of MD’s stores. 3.5.2 Model Consider two retail chains competing in a local market. Each firm sells a differentiated product. The demand for product i is qit = (St /2)(1 + (bit − b jt ) − ( pit − p jt )) where: qit is the quantity sold by firm i; St is market size that is exogenous; bit is the ’quality’ of product i at period t;15 and pit is the ’price’ of product i at period t. Production costs are linear in the quantity produced, i.e., Cit = ci qit , where ci is firm i’s constant marginal cost. The variable profit of firm i is VPit = ( pit − ci )qit . Given market size and qualities at period t, firms compete in prices ala Bertrand to maximize current variable profit. It is simple to show that in the Bertrand equilibrium prices are pit = 1 + c j + bit − b jt , and variable profits are:16 VPit = St 2 1− 1 3
ci − c j + 1 3 bit − b jt

2 (3.36) A firm’s quality increases with the number of stores that the firm has in the market. There are at least two ways in which the number of stores in the market affects the willingness to pay of the average consumer. First, an increase in the number of stores implies a reduction in consumer transportation cost to visit a store of the chain. Second, stores are like ’advertisements’ in the sense that they increase the awareness of local consumers about the retail chain. We assume the following specification: (0) bit = bi (1) + bi Yit (3.37) where Yit ∈ {0, 1, ..., K } represents the firm’s choice of number of stores in the local market at (0) period t. bi ≥ 0 is a parameter that represents the exogenous quality of firm i in every local (1) market. And bi 15 This ≥ 0 is a parameter that measures the effect of the number of stores in the local ’quality’ is the just the willingness to pay for the product of the average consumer in the market. that this is the total variable profit of the firm in the local market, not the profit of a single store. 16 Note market on the ’quality’ of the firm in that 134 Firm i is active in the market at period t if Yit is market.17 strictly positive. Given the specification of firm quality and the previous formula for equilibrium variable profits, we have the following expression for variable profits in terms of the state variables and the decision variables of the dynamic game:  VPit = 1{Yit > 0} St (3.38) VP + θ VP Y + θ VP Y 2 + θ VP Y + θ VP Y 2 + θ VP Y Y θ0i it jt jt jt it it 5i 4i 3i 2i 1i VP : k = 0, 1, ..., 5} are structural parameters that are known 1{.} is the indicator function. {θki (0) (0) (1) (1) functions of the ’deep’ parameters bi , b j , bi , b j , ci , and c j .18 It is simple to verify that given VP : k = 0, 1, ..., 5; i = 1, 2} we can (over) identify the ’deep’ a value of the vector of parameters {θki structural parameters di , d j , and ([bi − ci ] − [b j − c j ]). In order to distinguish decision and state variables, we use the variable Xit to represent the number of stores at period t − 1, i.e., Xit ≡ Yit−1 . Every period, the two firms know the current ’stocks’ of stores in the market at previous period and choose simultaneously the new number of stores. The firm’s total profit function is: Πit = VPit − θiEC 1{Yit > 0 and Xit = 0} (3.39)   − 1{Yit > 0} θ0iFC − θ1iFC Yit − θ2iFC Yit 2 − 1{Yit > Xit } ε it FC , θ FC and θ FC are parameters in the cost function. θ EC is an entry cost that is paid where θiEC , θ0i 1i 2i 0i FC is a lump-sum cost associated the first time that the firm opens a store in the local market. θ0i FC Y + θ FC Y 2 takes into with having any positive number of stores in the market. The function θ1i it 2i it 17 It is important to note that in the static version of the game, the number of stores of the two firms are strategic substitutes, i.e., ∂2 VPit /∂Yit ∂Yjt = −di d j St /9 < 0. 18 In (1) (0) VP ≡ (1 + ( b particular, θ0i i (1) (0) (1) (0) VP ≡ (2/3) b (1 + ( b − b j )/3 − (ci − c j )/3)2 , θ1i i i (0) VP ≡ −(2/3) b (1 + ( b [bi /3]2 , θ3i j i (0) (1) (0) VP ≡ − b j )/3 − (ci − c j )/3), θ2i (1) (1) VP ≡ [ b /3]2 , and θ VP ≡ −2[ b /3][ b /3]2 . − b j )/3 − (ci − c j )/3), θ4i 5i j i j 135 account that operating costs increase with the number of stores in a linear or quadratic form. The variable ε it is a private information shock in the cost of opening a new store, and it is i.i.d. normally distributed. In this example, the vector of structural parameters for firm i is VP VP VP VP VP VP FC FC FC 0 θi ≡ (θ0i , θ1i , θ2i , θ3i , θ4i , θ5i , θiEC , θ0i , θ1i , θ2i ) (3.40) And observable part of the payoff function, πit , is: πit ≡ 1{Yit > 0}  St , St Yit , St Yit 2 , (3.41) St Yjt , St Yjt 2 , St Yit Yjt , −1{ Xit = 0}, − 1, − Yit , − Yit 2 θi Note that our model implies the exclusion restriction that, given Yjt , the profit of firm i does not depend on X jt = Yjt−1 . That is, a firm’s current profit depends on his own and his opponents current number of stores in the market, but given these variables it does not depend on the number of stores of the competitors at period t − 1. Of course, a firm’s beleifs about the probability distribution of Yjt depends on X jt . 3.5.3 Estimation of the structural model As described in section 5.1 above, we do not observe store closings in our sample. Furthermore, for almost all the observations with store openings the number of new stores is one. Therefore, we consider that Yit ∈ { Xit , Xit + 1} or equivalently, Yit − Xit ∈ {0, 1}. The stock variable Ximt that represents the number of installed stores of firm i in market m at the beginning of the year. The maximum value of Ximt in the sample is 13, and we consider that the set of possible values of Ximt is {0, 1, ..., 15}. Therefore, the state space X is {0, 1, ..., 15} × {0, 1, ..., 15} that has 256 grid points. Yimt − Ximt is the binary indicator of the event "firm i opens a new store in market m at year t". We 136 consider that market characteristics are constant over time. The measure of market size Sm is total population in the district. For some specifications, we allow the cost of investment to depend on market characteristics such as average rent, retail taxes, population density, or average income. Therefore, each market has its own vector of players’ CCPs. The dimension of the vectors Pi in this model is equal to 108, 032, i.e., 422 markets times 256 states X. Tables 5 and 6 present estimates of the structural model under the assumption that firms are myopic, β = 0, and under the assumption that firms are forward looking, β = 0.95, respectively. We report two different sets of point estimates: estimates using a simple two-step method Pseudo Maximum Likelihood method where the estimator of (equilibrium) players’ beliefs in the first step is a nonparametric frequency estimator; and estimates using the Nested Pseudo Likelihood (NPL) method proposed in Aguirregabiria and Mira (2007). The NPL method imposes the equilibrium restrictions in the sample (i.e., the estimated beliefs should be equal to the estimated best response probabilities), while the two-step method only satisfies the equilibrium restrictions asymptotically. The NPL estimator has smaller asymptotic variance and finite sample bias than the two-step method. There are very substantial differences between two models, particularly in the estimates of the parameters that capture cannibalization and competition effects. While these effects have the ’wrong’ sign in the myopic model, the signs are the expected ones in the forward looking model. All the parameter estimates in the forward looking model have the expected signs and have reasonable magnitudes. Therefore, it seems that forward looking behavior explains part of the puzzle in the reduced form estimates. 137 Table 3.5 Myopic Game of Entry for McDonalds and Burger King Under the Assumption that Players’ Beliefs are in Equilibrium Data: 422 markets, 2 firms, 5 years = 4,220 observations β = 0.00 (not estimated) Two Step Estimates NPL Estimates Burger King McDonalds Burger King McDonalds Variable Profits: θ0VP 4.904 (1.070)∗ 7.909 (2.289)∗ 4.864 (1.081)∗ 7.898 (2.287)∗ θ1VP cannibalization 2.005 (0.869)∗ 3.510 (0.659)∗ 2.035 (0.831)∗ 3.466 (0.647)∗ θ2VP competition 0.014 (0.046) 0.032 (0.051) 0.016 (0.044) 0.037 (0.053) θ0FC fixed 0.378 (0.212)∗ 0.806 (0.248)∗ 0.374 (0.212)∗ 0.808 (0.247)∗ θ1FC linear 3.099 (0.436)∗ 2.662 (0.405)∗ 3.103 (0.436)∗ 2.659 (0.405)∗ θ2FC quadratic -0.054 (0.064) 0.085 (0.041) -0.052 (0.063) 0.087 (0.041) Fixed Costs: Pseudo R-square Log-Likelihood Distance||PK −PK− || # NPL iterations 0.154 0.154 -895.5 -895.4 0.00 1 5 138 Table 3.6 Dynamic Game of Entry for McDonalds and Burger King Under the Assumption that Players’ Beliefs are in Equilibrium Data: 422 markets, 2 firms, 5 years = 4,220 observations β = 0.95 (not estimated) Two Step Estimates NPL Estimates Burger King McDonalds Burger King McDonalds Variable Profits: θ0VP 0.5849 (0.1077)∗ 0.8303 (0.2968)∗ 1.098 (0.2169)∗ 0.9737 (0.3091)∗ θ1VP cannibalization -0.2096 (0.0552)∗ -0.0024 (0.0392) -0.0765 (0.0725) 0.2874 (0.0986)∗ θ2VP competition -0.0110 (0.0029)∗ 0.0008 (0.0027) -0.0129 (0.0065)∗ -0.0074 (0.0073) Fixed Costs: θ0FC fixed 0.0784 (0.0213)∗ 0.0822 (0.0332)∗ 0.0788 (0.0307)∗ 0.0773 (0.0261)∗ θ1FC linear 0.0790 (0.0420)∗ 0.1076 (0.0400)∗ 0.1509 (0.0282)∗ 0.1302 (0.0185)∗ θ2FC quadratic -0.0078 (0.0059) -0.0034 (0.0023) -0.0054 (0.0026)∗ 0.0001 (0.016) Pseudo R-square Log-Likelihood Distance||PK −PK− || # NPL iterations 0.323 0.146 -655.7 -893.4 4831.26 0.00 1 31 Table 7 presents results of our test of equilibrium beliefs. We implement separate tests for MD and BK. We impose the restrictions that beliefs for X jt = 0 and X jt = 3 are unbiased. 3.6 Conclusion This paper considers a class of dynamic games of incomplete information where players’ beliefs about the other players’ actions may not be in equilibrium. We present new results on identification, estimation, and inference of structural parameters and beliefs in this class of games when the researcher does not have data on elicited beliefs. Specifically, we derive sufficient conditions 139 Table 3.7 Estimated Bias in BK Beliefs Difference Between BMD and PMD Stores of BK 0 1 Stores of MD 1 -0.17 (0.04) -0.10 (0.06) 2 -0.08 (0.07) -0.06 (0.10) Estimated Bias in MD Beliefs Difference Between BBK and PBK Stores of MD 0 1 Stores of BK 1 -0.03 (0.05) 0.02 (0.04) 2 0.03 (0.10) 0.04 (0.12) 140 under which payoffs and beliefs are point identified. These conditions then lead naturally to a two-step estimator of payoffs and beliefs, which we show can be extended to provide a sequence of estimators with asymptotic variances and finite sample biases that decline monotonically. We also present a procedure for testing the null hypothesis that beliefs are in equilibrium. We illustrate our model and methods with an empirical application of a dynamic game of store location by McDonalds and Burger King. Our main interest in this application is to explain a puzzling empirical regularity, that the probability a firm opens a new store in a local market depends negatively on the number of stores it currently has open in the location, and does not depend (or may even positively depend) on the number of stores its competitor currently has open in the location. In the context of our model we explore three alternative (but not mutually exclusive) explanations for these: cross-firm spillovers, forward looking behavior, and out of equilibrium (i.e., biased) beliefs. We find empirical evidence for the hypotheses of forward looking behavior and biased beliefs. References 1. Acemoglu, Daron, Simon Johnson, James A. Robinson and Pierre Yared (2008) “Income and Democracy" American Economic Review Vol. 98, Issue 3: 808:842. 2. Aguirregabiria, Victor (2009) “A Method for Implementing Counterfactual Experiments in Models with Multiple Equilibria." Working Paper. 3. Aguirregabiria, Victor (2010) "Another Look at the Identification of Dynamic Discrete Decision Processes: An Application to Retirement Behavior,” Journal of Business and Economic Statistics, 28(2), 201–218. 4. Aguirregabiria, Victor and C-Y. Ho (2009): "A Dynamic Oligopoly Game of the US Airline Industry: Estimation and Policy Experiments," Manuscript. University of Toronto. 5. 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