CONTENTS EDITORIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ANDREAS PARK AND HAMID SABOURIAN: Herding and Contrarian Behavior in Financial
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Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Risk Sharing in Private Information Models With Asset Accumulation: Explaining the Excess Smoothness of Consumption . . . ANA CECÍLIA FIELER: Nonhomotheticity and Bilateral Trade: Evidence and a Quantitative Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ALESSANDRA FOGLI AND LAURA VELDKAMP: Nature or Nurture? Learning and the Geography of Female Labor Force Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIRGILIU MIDRIGAN: Menu Costs, Multiproduct Firms, and Aggregate Fluctuations . . . . P. GAGLIARDINI, C. GOURIEROUX, AND E. RENAULT: Efficient Derivative Pricing by the Extended Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NOAH WILLIAMS: Persistent Private Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . JOHANNES HÖRNER, TAKUO SUGAYA, SATORU TAKAHASHI, AND NICOLAS VIEILLE: Recursive Methods in Discounted Stochastic Games: An Algorithm for δ → 1 and a Folk Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ANNOUNCEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 FORTHCOMING PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323 SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES . . . . . . . . . . 1325
VOL. 79, NO. 4 — July, 2011
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Econometrica, Vol. 79, No. 4 (July, 2011), 971–972
VOLUME 79
JULY, 2011
NUMBER 4
EDITORIAL In Econometrica’s first issue, published in January 1933, Ragnar Frisch declared that promoting “studies that aim at a unification of the theoreticalquantitative and the empirical-quantitative approach to economic problems and that are penetrated by constructive and rigorous thinking similar to that which has come to dominate in the natural sciences” would be the new journal’s main objective. The Econometric Society and Econometrica have been at the forefront of this endeavor over the last 80 years. Econometrica has not only been a prime outlet for some of the most important methodological contributions in economic theory and econometrics, which have formed the intellectual foundations of our discipline, but has also played a central role in forging the quantitative and rigorous nature of many sub-areas of economics. Economics has advanced in many dimensions since 1933. Among these developments, two stand out. First, economics has become much more, and much more richly, empirical—both in terms of the growth in high-quality empirical work and in the role that empirical motivation plays in shaping theoretical work. Second, the set of questions that economics and economists focus on is now much broader, covering a rich array of empirical questions as well as issues with significant organizational, political, social and psychological dimensions. These developments have brought new challenges for “theoretical-quantitative” and “empirical-quantitative” approaches. There is likely no single answer to questions such as: How should theory and new empirical strategies be combined? How should we evaluate research branching into areas where our standard assumptions must be relaxed or even abandoned? How should new methodological contributions make contact with more applied questions? And perhaps most importantly, how can we ensure that research on new topics meets the highest standards without being excessively constrained by rigid methodological straitjackets? The Editorial Board must confront these questions with an open mind. We must also ask whether Econometrica has changed sufficiently in response to these developments. I believe that we are only part way through a process of fundamental changes within economics, both in terms of the emergence of new applied theoretical questions, and the development and application of new empirical methods and approaches. Econometrica needs to be at the forefront of today’s developments in the same way it has pioneered other major transformations in economic research. But change, particularly measured and gradual change, is hard for well-established organizations. Econometrica has built © 2011 The Econometric Society
DOI: 10.3982/ECTA794ED
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a stellar brand name as the prime outlet for methodological contributions in many areas upon which the rest of the profession continues to rely. It would be a mistake for Econometrica to abandon this great tradition and strength. However, it would be an equally costly mistake for Econometrica to just play to its strengths and not aspire to be the leading journal in all areas of economics. Our intent therefore is to be more receptive and willing to take risks in new lines of research both in theory and empirical work without compromising our strengths and our reputation as the prime outlet for methodological contributions in econometrics, economic theory, and structural applied work. The way to achieve this two-pronged objective is to expand Econometrica not only intellectually but also in terms of number of papers published. In line with this objective, we are targeting a 20% increase in the number of papers Econometrica publishes within the next several years. Our hope is to contribute to a healthy and measured transformation of research in the economics profession through the high quality and intellectual vision of the papers we publish. Ultimately, it is the tremendous intellectual energy, enthusiasm and open-mindedness of young economists that makes us optimistic that quantitative research in economics, and thus Econometrica, will live up to the high standards that Ragnar Frisch set out in 1933. THE EDITOR
Econometrica, Vol. 79, No. 4 (July, 2011), 973–1026
HERDING AND CONTRARIAN BEHAVIOR IN FINANCIAL MARKETS BY ANDREAS PARK AND HAMID SABOURIAN1 Rational herd behavior and informationally efficient security prices have long been considered to be mutually exclusive but for exceptional cases. In this paper we describe the conditions on the underlying information structure that are necessary and sufficient for informational herding and contrarianism. In a standard sequential security trading model, subject to sufficient noise trading, people herd if and only if, loosely, their information is sufficiently dispersed so that they consider extreme outcomes more likely than moderate ones. Likewise, people act as contrarians if and only if their information leads them to concentrate on middle values. Both herding and contrarianism generate more volatile prices, and they lower liquidity. They are also resilient phenomena, although by themselves herding trades are self-enforcing whereas contrarian trades are self-defeating. We complete the characterization by providing conditions for the absence of herding and contrarianism. KEYWORDS: Social learning, herding, contrarians, asset price volatility, market transparency.
1. INTRODUCTION IN TIMES OF GREAT ECONOMIC UNCERTAINTY, financial markets often appear to behave frantically, displaying substantial price spikes as well as drops. Such extreme price fluctuations are possible only if there are dramatic changes in behavior with investors switching from buying to selling or the reverse. This pattern of behavior and the resulting price volatility is often claimed to be inconsistent with rational traders and informationally efficient asset prices, and is attributed to investors’ animal instincts. We argue in this paper, however, that such behavior can be the result of fully rational social learning where agents change their beliefs and behavior as a result of observing the actions of others. 1
This manuscript is a substantially revised and generalized version of our earlier paper “Herd Behavior in Efficient Financial Markets.” Financial support from the ESRC (Grants RES-15625-0023 and R00429934339) and the TMR Marie Curie Fellowship Program is gratefully acknowledged. Andreas thanks the University of Copenhagen for its hospitality while some of this research was developed. We thank three anonymous referees for detailed comments, and the co-editor for very helpful suggestions and advice. We are also grateful to Markus Brunnermeier, Christophe Chamley, Tony Doblas-Madrid, Scott Joslin, Rob McMillan, Peter Sorensen, and Wei Xiong for helpful discussions. Finally, we thank seminar participants at the following conferences, workshops, and departments for useful comments: St. Andrews ESRC Macro, Gerzensee Economic Theory, SAET Vigo Spain, ESRC World Economy Birkbeck, ESRC British Academy Joint Public Policy Seminar, Cambridge–Princeton Workshop, Cambridge Social Learning Workshop, Copenhagen Informational Herding Workshop, CEA, NFA, Said Oxford, St. Andrews, Leicester, Nottingham, Queen Mary London, Cambridge, LSE Finance, Naples, EUI Florence, Paris School of Economics, Yonsei Korea, Singapore National University, Singapore Management University, Queens, Rotman Toronto, Penn State, and Wharton. © 2011 The Econometric Society
DOI: 10.3982/ECTA8602
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One example of social learning is herd behavior in which agents switch behavior (from buying to selling or the reverse) following the crowd. So-called rational herding can occur in situations with information externalities, when agents’ private information is swamped by the information derived from observing others’ actions. Such “herders” rationally act against their private information and follow the crowd.2 It is not clear, however, that such herd behavior can occur in informationally efficient markets, where prices reflect all public information. For example, consider an investor with unfavorable private information about a stock. Suppose that a crowd of people buys the stock frantically. Such an investor will update his information and upon observing many buys, his expectation of the value of the stock will rise. At the same time, prices also adjust upward. Then it is not clear that the investor buys—to him the security may still be overvalued. So for herding, private expectations and prices must diverge. In models with only two states of the world, such divergence is impossible, as prices always adjust so that there is no herding.3 Yet two-state models are rather special and herding can emerge once there are at least three states. In this paper, we characterize the possibility of herding in the context of a simple, informationally efficient financial market. Moreover, we show that (i) during herding, prices can move substantially and (ii) herding can induce lower liquidity and higher price volatility than if there were no herding. Herd behavior in our setup is defined as any history-switching behavior in the direction of the crowd (a kind of momentum trading).4 Social learning can also arise as a result of traders switching behavior by acting against the crowd. Such contrarian behavior is the natural counterpart of herding, and we also characterize conditions for such behavior. Contrary to received wisdom that contrarian behavior is stabilizing, we also show that contrarian behavior leads to higher volatility and lower liquidity, just as herd behavior does. The key insight of our characterization result is that social learning in financial markets occurs if and only if investors receive information that satisfies a compelling and intuitive property. Loosely, herding arises if and only if private information satisfies a property that we call U-shaped. An investor who receives such information believes that extreme states are more likely to have generated the information than more moderate ones. Therefore, when forming his posterior belief, the recipient of such a signal will shift weight away from the center to the extremes so that the posterior distribution of the trader 2 See Banerjee (1992) or Bikhchandani, Hirshleifer, and Welch (1992) for early work on herding. 3 With two states, the price will adjust so that it is always below the expectation of traders with favorable information and above the expectation of those with unfavorable information irrespective of what is observed. As we show, with more than two states, this strict separation no longer applies. 4 We are concerned with short-term behavior. In the literature, there are also other definitions of herding; see Section 4 for a discussion.
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is “fat-tailed.” The recipient of a U-shaped signal thus discounts the possibility of the intermediate value and as a consequence, will update the probabilities of extreme values faster than an agent who receives only the public information. Contrarianism occurs if and only if the investor’s signal indicates that moderate states are more likely to have generated the signal than extreme states. We describe such signals as being “hill-shaped.” The recipient of a hill-shaped signal always puts more weight on middle outcomes relative to the market so that this trader’s posterior distribution becomes “thin-tailed.” He thus discounts the possibility of extreme states and, therefore, updates extreme outcomes slower than the market maker. We follow the microstructure literature and establish our results in the context of a stylized trading model in the tradition of Glosten and Milgrom (1985). In such models, the bid and ask prices are set by a competitive market maker. Investors trade with the market maker either because they receive private information about the asset’s fundamental value or because they are “noise traders” and trade for reasons outside of the model. The simplest possible Glosten–Milgrom trading model that allows herding or contrarianism is one with at least three states. For this case, we show that (i) a U-shaped (hill-shaped) signal is necessary for herding (contrarianism) and (ii) herding (contrarianism) occurs with positive probability if there exists at least one U-shaped (hill-shaped) signal and there is a sufficient amount of noise trading. The latter assumption on the minimum level of noise trading is not required in all cases and is made as otherwise the bid and ask spread may be too large to induce appropriate trading. In Section 9, we show that the intuition for our three-state characterization carries over to a setup with an arbitrary number of states. We obtain our characterization results without restrictions on the signal structure. In the literature on asymmetric information, it is often assumed that information structures satisfy the monotone likelihood ratio property (MLRP). Such information structures are “well behaved” because, for example, investors’ expectations are ordered. It may appear that such a strong monotonicity requirement would prohibit herding or contrarianism. Yet MLRP not only admits the possibility of U-shaped signals (and thus herding) or hill-shaped signals (and thus contrarianism), but also the trading histories that generate herding and contrarianism are significantly simpler to describe with than without MLRP signals. Our second set of results concerns the impact of social learning on prices. We first show that the range of price movements can be very large during both contrarianism and herding. We then compare price movements in our setup where agents observe one another with those in a hypothetical economy that is otherwise identical to our setup except that the informed traders do not switch behavior. We refer to the former as the transparent economy and to the latter as the opaque economy. In contrast to the transparent economy, in the opaque economy there is no social learning by assumption. We show, for the
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case of MLRP, that once herding or contrarianism begins, prices respond more to individual trades relative to the situation without social learning so that price rises and price drops are greater in the transparent setup than in the opaque one.5 As a corollary, liquidity, measured by the inverse of the bid–ask spread, is lower with social learning than without. The price volatility and liquidity results may have important implications for the discussion on the merits of “market transparency.” The price path in the opaque economy can be interpreted as the outcome of a trading mechanism in which people submit orders without knowing the behavior of others or the market price. Our results indicate that in the less transparent setup, price movements are less pronounced and liquidity is higher. While the results on price ranges, volatility, and liquidity indicate similarities between herding and contrarianism, there is also a stark difference. Contrarian trades are self-defeating because a large number of such trades will cause prices to move against the crowd, thus ending contrarianism. During herding, on the other hand, investors continue to herd when trades are in the direction of the crowd, so herding is self-enforcing. Examples of Situations That Generate U- and Hill-Shaped Signals First, a U-shaped signal may be interpreted as a “volatility signal.”6 Very informally, an example is a signal that generates a mean preserving spread of a symmetric prior distribution. Conversely, a mean preserving signal that decreases the variance is hill-shaped. Second, U-shaped and hill-shaped signals may also be good descriptions of situations with a potential upcoming event that has an uncertain impact. For example, consider the case of a company or institution that contemplates appointing a new leader who is an uncompromising “reformer.” If this person takes power, then either the necessary reforms take place or there will be strife with calamitous outcomes. Thus the institution will not be the same, as the new leader will be either very good or disastrous. Any private information signifying that the person is likely to be appointed exemplifies a U-shaped signal and any information revealing that this person is unlikely to be appointed (and thus the institution will carry on as before) represents a hill-shaped signal.7 5
The increase in price volatility associated with herding is only relative to a hypothetical scenario. Even when herding is possible, in the long-run, volatility settles down and prices react less to individual trades. It is well known that the variance of martingale price processes such as ours is bounded by model primitives. 6 We thank both Markus Brunnermeier and an anonymous referee for this interpretation. 7 Other examples include an upcoming merger or takeover with uncertain merits, the possibility of a government stepping down, announcements of Food and Drug Administration (FDA) drug approvals, and outcomes of lawsuits. Degenerate examples for such signals were first discussed by Easley and O’Hara (1992) and referred to as “event uncertainty.”
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Third, consider a financial institution (FI) that is a competitor to a bank that has recently failed. Suppose there are three possible scenarios: (i) the FI will also fail because it has deals with the failed bank that will not be honored and/or that the business model of the FI is as bad as that of the failed bank; (ii) the FI’s situation is entirely unrelated to the bank and the latter’s collapse will not affect the FI; (iii) the FI may benefit greatly from the bank’s collapse, as it is able to attract the failed bank’s customers and most capable employees. Cases (i) and (iii) resemble extreme outcomes and case (ii) resembles a middle outcome. In this environment, some investor’s information might have implied that the most likely outcome is either that the FI will also go down as well or that it will benefit greatly from the failed banks’ demise. Such information is an example of a U-shaped signal. Alternatively, some investors’ assessments might have implied that the most likely outcome is that the FI is unaffected. Such information is an example of a hill-shaped signal. It is conceivable that in the Fall of 2008 (after the collapse of Lehman) and early 2009, many investors believed that for individual financial institutions the two extreme states (collapse or thrive) were the most likely outcomes. Then our theory predicts the potential for herd behavior, with investors changing behavior in the direction of the crowd, causing strong short-term price fluctuations. Hill-shaped private signals, signifying that the institutions were likely to be unaffected, may also have occurred, inducing contrarianism and changes of behavior against the crowd. The Mechanism That Induces Herding and Contrarianism Consider the above banking example and assume that all scenarios are equally likely. Let the value of the stock of the FI in each of the three scenarios (i), (ii), and (iii) be V1 < V2 < V3 , respectively. We are interested in the behavior of an investor, who has a private signal S, after different public announcements. Specifically, consider a good public announcement G that rules out the worst state, Pr(V1 |G) = 0, and a bad public announcement B that rules out the best state, Pr(V3 |B) = 0. Assume that the price of the stock is equal to the expected value of the asset conditional on the public information and that the investor buys (sells) if his expectation exceeds (is less than) the price. Note that the price will be higher after G and lower after B, compared to the ex ante situation when all outcomes are equally likely. Both G and B eliminate one state, so that after each such announcement, there are only two states left. In two-state models, an investor has a higher (lower) expectation than the market if and only if his private information is more (less) favorable toward the better state than toward the worse state. Thus, in the cases of G and B, E[V |G] ≶ E[V |S G] is equivalent to Pr(S|V2 ) ≶ Pr(S|V3 ) and E[V |S B] ≶ E[V |B] is equivalent to Pr(S|V2 ) ≶ Pr(S|V1 ). Hence, for example, after good news G, an investors buys (sells) if he thinks, relative
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to the market, that it is more (less) likely that the FI will thrive than being unaffected. It follows from the above discussion that the investor buys after G and sells after B if and only if Pr(S|V3 ) > Pr(S|V2 ) and Pr(S|V1 ) > Pr(S|V2 ). Such an investor, loosely, herds in the sense that he acts like a momentum trader, buying with rising and selling with falling prices. The private information (conditional probabilities) that is both necessary and sufficient for such behavior thus has a U shape. Conversely, the investor sells after G and buys after B if and only if Pr(S|V3 ) < Pr(S|V2 ) and Pr(S|V1 ) < Pr(S|V2 ). Such an investor, loosely, trades contrary to the general movement of prices. The private information that is both necessary and sufficient to generate such a behavior thus has a hill shape.8 There are several points to note about this example. First, the public announcements G and B are degenerate as they each exclude one of the extreme states. Yet the same kind of reasoning holds if we replace G by an announcement that attaches arbitrarily small probability to the worst outcome, V1 , and if we replace B by an announcement that attaches arbitrarily small probability to the best outcome, V3 . Second, in the above illustration, G and B are exogenous public signals. In the security model described in this paper, on the other hand, public announcements or, more generally, public information, are created endogenously by the history of publicly observable transactions. Yet the intuition behind our characterization results is similar to the above illustration. The analysis in the paper involves describing public histories of trades that allow investors to almost rule out some extreme outcome, either V1 or V3 .9 Such histories are equivalent to public announcements G and B (or, to be more precise, to perturbations of G and B), and demonstrating their existence is crucial for demonstrating the existence of herding and contrarianism. Overview The next section discusses some of the related literature. Section 3 outlines the setup. Section 4 defines herding and contrarian behavior. Section 5 discusses the necessary and sufficient conditions that ensure herding and contrarianism. Section 6 discusses the special case of MLRP signals. Section 7 considers the resiliency, and fragility of herding and contrarianism and describes the range of prices for which there may be herding and contrarianism. Section 8 discusses the impact of social learning on prices with respect to volatility and liquidity. Section 9 extends the result to a setting with an arbitrary number of 8 In our formal definition of herding and contrarianism, we benchmark behavior against the decision that the trader would take at the initial history, but the switching mechanism is akin to what we describe here. 9 In the asset market described in the paper, every state has a positive probability at all finite trading histories because of the existence of noise traders. Therefore, we describe public histories at which the probabilities of extreme states are arbitrarily small, but not zero.
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states. Section 10 discusses the relation of our findings to an earlier important paper on financial market herding. Section 11 concludes. Proofs that are not given in the text are either in the Appendix or in the Supplemental Material (Park and Sabourian (2011)). 2. RELATED LITERATURE Extensive literature surveys on herding in financial markets are given in Brunnermeier (2001), Chamley (2004), and Vives (2008). Our work relates to the part of the literature that focuses on short-run herding. The work closest to ours is Avery and Zemsky (1998) (henceforth AZ), who were the first to present an intuitively appealing example of informational herding in financial markets. They argued that herd behavior with informationally efficient asset prices is not possible unless signals are “nonmonotonic” and they attributed the herding result in their example to “multidimensional uncertainty” (investors have a finer information structure than the market). In their main example, however, prices hardly move under herding. To generate extreme price movements (bubbles) with herding, AZ expanded their example to a second level of information asymmetry that leads to an even finer information partition. Yet even with these further informational asymmetries, the likelihood of large price movements during herding is extremely small (see Chamley (2004)). The profession, for instance, Brunnermeier (2001), Bikhchandani and Sunil (2000), and Chamley (2004), has derived three messages from AZ’s paper. First, with “unidimensional” or “monotonic” signal structures, herding is impossible. Second, the information structure needed to induce herding is very special. Third, herding does not involve violent price movements except in the most unlikely environments. AZ’s examples are special cases of our framework. Our paper demonstrates that the conclusions derived from AZ’s examples should be reconsidered. First, we show that it is U-shaped signals, and not multidimensionality or nonmonotonicity of the information structure, that is both necessary and sufficient for herding. Second, while AZ’s examples are intuitively appealing, due to their extreme nature (with several degenerate features), it may be argued that they are very special and, therefore, have limited economic relevance. Our results show instead that herding may apply in a much more general fashion and, therefore, there may be a great deal more rational informational herding than is currently expected in the literature. Third, we show that extreme price movements with herding are possible under not so unlikely situations, even with MLRP signals and without “further dimensions of uncertainty.” In Section 10, we discuss these concepts in detail by comparing and contrasting the work and conclusions of AZ with ours. A related literature on informational learning explains how certain facets of market organization or incentives can lead to conformism and informational cascades. In Lee (1998), fixed transaction costs temporarily keep traders out
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of the market. When they enter suddenly and en masse, the market maker absorbs their trades at a fixed price, leading to large price jumps after this “avalanche.” In Cipriani and Guarino (2008), traders have private benefits from trading in addition to the fundamental value payoff. As the private and public expectations converge, private benefits gain importance to the point when they overwhelm the informational rents. Then learning breaks down and an informational cascade arises. In Dasgupta and Prat (2008), an informational cascade is triggered by traders’ reputation concerns, which eventually outweigh the possible benefit from trading on information. Chari and Kehoe (2004) also studied a financial market with efficient prices; herding in their model arises with respect to a capital investment that is made outside of the financial market. Our work also relates to the literature that shows how public signals can have a larger influence on stock price fluctuations than warranted by their information content. Beginning with He and Wang (1995), who described the relation between public information and nonrandom trading volume patterns caused by the dynamic trading activities of long-lived traders, this literature has identified how traders who care for future prices (as opposed to fundamentals) rely excessively on public expectations (see also Allen, Morris, and Shin (2006), Bacchetta and Wincoop (2006, 2008), Ozdenoren and Yuan (2007), and Goldstein, Ozdenoren, and Yuan (2010)). All of the above contributions highlight important aspects, facets, and mechanisms that can trigger conformism in financial markets. Our findings complement the literature in that the effects that we identify may be combined with many of the above studies and they may amplify the effects described therein. 3. THE MODEL We model financial market sequential trading in the tradition of Glosten and Milgrom (1985). Security: There is a single risky asset with a value V from a set of three potential values V = {V1 V2 V3 } with V1 < V2 < V3 . Value V is the liquidation or true value when the game has ended and all uncertainty has been resolved. States V1 and V3 are the extreme states; state V2 is the moderate state. The prior distribution over V is denoted by Pr(·). To simplify the computations, we assume that {V1 V2 V3 } = {0 V 2V }, V > 0, and that the prior distribution is symmetric around V2 ; thus Pr(V1 ) = Pr(V3 ).10 Traders: There is a pool of traders that consists of two kinds of agents: noise traders and informed traders. At each discrete date t, one trader arrives at the market in an exogenous and random sequence. Each trader can only trade once at the point in time at which he arrives. We assume that at each date, the 10
The ideas of this paper remain valid without these symmetry assumptions.
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entering trader is an informed agent with probability μ > 0 and a noise trader with probability 1 − μ > 0. The informed agents are risk neutral and rational. Each receives a private, conditionally independent and identically distributed (i.i.d.) signal about the true value of the asset V . The set of possible signals or types of informed agents is denoted by S and consists of three elements S1 , S2 , and S3 . The signal structure of the informed can, therefore, be described by a 3-by-3 matrix I = {Pr(Si |Vj )}ij=123 , where Pr(Si |Vj ) is the probability of signal Si if the true value of the asset is Vj . Noise traders have no information and trade randomly. These traders are not necessarily irrational, but they trade for reasons not included in this model, such as liquidity.11 Market Maker: Trade in the market is organized by a market maker who has no private information. He is subject to competition and thus makes zero expected profits. In every period t, prior to the arrival of a trader, he posts a bid price bidt at which he is willing to buy the security and an ask price askt at which he is willing to sell the security. Consequently, he sets prices in the interval [V1 V3 ]. Traders’ Actions: Each trader can buy or sell one unit of the security at prices posted by the market maker, or he can be inactive. So the set of possible actions for any trader is {buy hold sell}. We denote the action taken in period t by the trader who arrives at that date by at . We assume that noise traders trade with equal probability. Therefore, in any period, a noise-trader buy, hold, or sale occurs with probability γ = (1 − μ)/3 each. Public History: The structure of the model is common knowledge among all market participants. The identity of a trader and his signal are private information, but everyone can observe past trades and transaction prices. The history (public information) at any date t > 1, the sequence of the traders’ past actions together with the realized past transaction prices, is denoted by H t = ((a1 p1 ) (at−1 pt−1 )) for t > 1, where aτ and pτ are traders’ actions and realized transaction prices at any date τ < t, respectively. Also, H 1 refers to the initial history before any trade takes place. Public Belief and Public Expectation: For any date t and any history H t , denote the public belief/probability that the true liquidation value of the asset is Vi by qit = Pr(Vi |H t ) for each i = 1 2 3. The public expectation, which we sometimes also refer to asthe market expectation, of the liquidation value at H t is given by E[V |H t ] = qit Vi . Also, we shall, respectively, denote the probability of a buy and the probability of a sale at any history H t , when the true value of the asset is Vi , by βti = Pr(buy|H t Vi ) and σit = Pr(sell|H t Vi ). For instance, suppose that at history H t , the only informed type that buys is Sj . Then 11 As is common in the microstructure literature with asymmetric information, we assume that noise traders have positive weight (μ < 1) to prevent “no-trade” outcomes à la Milgrom and Stokey (1982).
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when the true value is Vi , the probability of a buy is given by the probability that there is a noise trader who buys plus the probability that there is an informed trader with signal Sj : βti = (1 − μ)/3 + μ Pr(Sj |Vi ). The Informed Trader’s Optimal Choice: The game played by the informed agents is one of incomplete information; therefore, the optimal strategies correspond to a perfect Bayesian equilibrium. Here, the equilibrium strategy for each trader simply involves comparing the quoted prices with his expected value, taking into account both the public history and his own private information. For simplicity, we restrict ourselves to equilibria in which each agent trades only if he is strictly better off (in the case of indifference, the agents do not trade). Therefore, the equilibrium strategy of an informed trader who enters the market in period t, receives signal S t , and observes history H t is (i) to buy if E[V |H t S t ] > askt , (ii) to sell if bidt > E[V |H t S t ], and (iii) to hold in all other cases. The Market Maker’s Price-Setting: To ensure that the market maker receives zero expected profits, the bid and ask prices must satisfy the following criteria at any date t and any history H t : askt = E[V |at = buy at askt H t ] and bidt = E[V |at = sell at bidt H t ]. Thus if there is a trade at H t , the public expectation E[V |H t+1 ] coincides with the transaction price at time t (askt for a buy, bidt for a sale). If the market maker always sets prices equal to the public expectation, E[V |H t ], he makes an expected loss on trades with an informed agent. However, if the market maker sets an ask price and a bid price, respectively, above and below the public expectation, he gains on noise traders, as their trades have no information value. Thus, in equilibrium, the market maker must make a profit on trades with noise traders to compensate for any losses against informed types. This implies that if, at any history H t , there is a possibility that the market maker trades with an informed trader, then there is a spread between the bid–ask prices at H t and the public expectation E[V |H t ] that satisfies askt > E[V |H t ] > bidt . Trading by the Informed Types and No Cascade Condition: At any history H t , either informed types do not trade and every trade is by a noise trader or there is an informed type that would trade at the quoted prices. The game played by the informed types in the former case is trivial, as there will be no trade by the informed from H t onward and an informational cascade occurs. The reason is that if there were no trades by the informed at H t , no information will be revealed, and the expectations and prices remain unchanged; hence, by induction, we would have no trading by the informed and no information revelation at any date after H t . In this paper, we thus consider only the case in which at every history, there is an informed type that would trade at the quoted prices. Informative Private Signals: The private signals of the informed traders are informative at history H t if (1)
there exists S ∈ S
such that
E[V |H t S] = E[V |H t ]
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First note that (1) implies that at H t there is an informed type that buys and an informed type that sells. To see this, observe that by (1) there must exist two signals S and S such that E[V |H t S ] < E[V |H t ] < E[V |H t S ]. If no informed type buys at H t , then there is no informational content in a buy and askt = E[V |H t ]. Then, by E[V |H t ] < E[V |H t S ], type S must be buying at H t ; a contradiction. Similarly, if no informed type sells at H t , then bidt = E[V |H t ]. Then, by E[V |H t ] > E[V |H t S ], type S must be selling at H t ; a contradiction. Second, it is also the case that if there is an informed type that trades at H t , then (1) must hold. Otherwise, for every signal S ∈ S, E[V |H t S] = E[V |H t ] = askt = bidt and the informed types that not trade at H t . It follows from the above discussion that (1) is both necessary and sufficient for trading by an informed type at H t . Since we are interested in the case when the informed types trade, we therefore assume throughout this paper that (1) holds at every history H t .12 One important consequence of condition (1) is that past behavior can be inferred from past transaction prices alone: since the bid and ask prices always differ by condition (1), one can infer behavior iteratively, starting from date 1. Therefore, all the results of this paper are valid if traders observe only past transaction prices and no-trades. Long-Run Behavior of the Model: Since price formation in our model is standard, (1) also ensures that standard asymptotic results on efficient prices hold. More specifically, by standard arguments as in Glosten and Milgrom (1985), we have that transaction prices form a martingale process. Since by (1), buys and sales have some information content (at every date there is an informed type that buys and one that sells), it also follows that beliefs and prices converge to the truth in the long run (see, for instance, Proposition 4 in AZ). However, here we are solely interested in short-run behavior and fluctuations. Conditional Signal Distributions: As we outlined in the Introduction, the possibility of herding or contrarian behavior for any informed agent with signal S ∈ S depends critically on the shape of the conditional signal distribution of S. Henceforth, we refer to the conditional signal distribution of the signal as the c.s.d. Furthermore, we will also employ the following terminology to describe four different types of c.s.d.s: increasing:
Pr(S|V1 ) ≤ Pr(S|V2 ) ≤ Pr(S|V3 )
decreasing:
Pr(S|V1 ) ≥ Pr(S|V2 ) ≥ Pr(S|V3 )
U-shaped:
Pr(S|Vi ) > Pr(S|V2 )
for i = 1 3
hill-shaped:
Pr(S|Vi ) < Pr(S|V2 )
for i = 1 3
12 A sufficient condition for (1) to hold at every H t is that all minors of order 2 of matrix I are nonzero.
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An increasing c.s.d. is strictly increasing if all three conditional probabilities for the signal are distinct; a strictly decreasing c.s.d. is similarly defined. For the results in our paper, it is also important whether the likelihood of a signal is higher in one of the extreme states V1 or V3 relative to the other extreme state. We thus define the bias of a signal S as Pr(S|V3 ) − Pr(S|V1 ). A U-shaped c.s.d. with a negative bias, Pr(S|V3 ) − Pr(S|V1 ) < 0, will be labeled as an nU-shaped c.s.d. and a U-shaped c.s.d. with a positive bias, Pr(S|V3 ) − Pr(S|V1 ) > 0, will be labeled as a pU-shaped c.s.d. Similarly, we use nhill (phill) to describe a hill-shaped c.s.d. with a negative (positive) bias. In describing the above properties of a type of c.s.d. for a signal, we shall henceforth drop the reference to the c.s.d. and attribute the property to the signal itself, when the meaning is clear. Similarly, when describing the behavior of a signal recipient, we attribute the behavior to the signal itself. 4. DEFINITIONS OF HERDING AND CONTRARIAN BEHAVIOR In the literature, there are several definitions of herding. Some require “action convergence” or even complete informational cascades where all types take the same action in each state, irrespective of their private information; see Brunnermeier (2001), Chamley (2004), and Vives (2008). The key feature of this early literature was that herding can induce, after some date, the loss of all private information and wrong or inefficient decisions thereafter. A situation like an informational cascade in which all informed types act alike may not, however, be very interesting in an informationally efficient financial market setting. In such a framework, prices account for the information contained in the traders’ actions. If all informed types act alike, then their actions would be uninformative and, as result, price would not move. Therefore, such uniformity of behavior cannot explain price movements, which are a key feature of financial markets. Moreover, if the uniform action involves trading, then a large imbalance of trades would accumulate without affecting prices—contrary to common empirical findings.13 Furthermore, as we have explained in the previous section, in our “standard” microstructure trading model, at any history, uniform behavior by the informed types is possible if and only if all private signals are uninformative, in the sense that the private expectations of all informed types are equal that of the market expectation (condition (1) is violated). The case of such uninformative private signals is trivial and uninteresting, as it implies that no further information is revealed, that all informed types have the same expectation, and that, because we assume that informed agents trade only if trading makes them strictly better off, a no-trade cascade results. In this paper we thus focus on the social learning (learning from others) aspect of behavior for individual traders that is implied by the notion of herding from the earlier literature. Specifically, we follow Brunnermeier’s (2001, 13
See, for instance, Chordia, Roll, and Subrahmanyam (2002).
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Chap. 5) description of herding as a situation in which “an agent imitates the decision of his predecessor even though his own signal might advise him to take a different action,” and we consider the behavior of a particular signal type by looking at how the history of past trading can induce a trader to change behavior and trade against his private signal.14 Imitative behavior is not the only type of behavior that learning from others’ past trading activities may generate: a trader may also switch behavior and go against what most have done in the past. We call such social learning contrarianism and differentiate it from herding by describing the latter as a history-induced switch of opinion in the direction of the crowd and the former as a history-induced switch against the direction of the crowd. The direction of the crowd here is defined by the “recent” price movement. Thus, there is a symmetry in our definitions, making herding the intuitive counterpart to contrarianism. DEFINITION: Herding. A trader with signal S buy herds in period t at history H t if and only if (i) E[V |S] < bid1 , (ii) E[V |S H t ] > askt , or (iii-h) E[V |H t ] > E[V ]. Sell herding at history H t is defined analogously with the required conditions E[V |S] > ask1 , E[V |S H t ] < bidt , and E[V |H t ] < E[V ]. Type S herds if he either buy herds or sell herds at some history. Contrarianism. A trader with signal S engages in buy contrarianism in period t at history H t if and only if (i) E[V |S] < bid1 , (ii) E[V |S H t ] > askt , or (iii-c) E[V |H t ] < E[V ]. Sell contrarianism at history H t is defined analogously with the required conditions E[V |S] > ask1 , E[V |S H t ] < bidt , and E[V |H t ] > E[V ]. Type S engages in contrarianism if he engages in either buy contrarianism or sell contrarianism at some history. Both with buy herding and buy contrarianism, type S prefers to sell at the initial history before observing other traders’ actions (condition (i)), but prefers to buy after observing the history H t (condition (ii)). The key differences between buy herding and buy contrarianism are conditions (iii-h) and (iii-c). The former requires the public expectation, which is the last transaction price and an average of the bid and ask prices, to rise at history H t so that a change of action from selling to buying at H t is with the general movement of the prices (crowd), whereas the latter condition requires the public expectation to have dropped so that a trader who buys at H t acts against the movement of prices. Henceforth, we refer to a buy herding history as one at which some type, were they to trade, would buy herd at that history; similarly for a buy contrarianism history. 14
Vives (2008, Chap. 6) adopted a similar view of herding, defining it as a situation in which agents put “too little” weight on their private signals with respect to a well defined welfare benchmark.
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Our definition of herding is identical to that in Avery and Zemsky (1998),15 and it has also been used in other work on social learning in financial markets (see, for instance, Cipriani and Guarino (2005) or Drehmann, Oechssler, and Roider (2005)). It describes histories at which a trader acts against his signal (judgement) and follows the trend, where “against his signal” is defined by comparing the herding action to the benchmark without public information. “The trend” is identified by price movements based on the idea that prices rise (fall) when there are more (less) buys than sales. The contrarianism definition, on the other hand, captures the contra-trend action that is also against one’s signal.16 Our definitions also capture well documented financial market trading behavior. In particular, our herding definition is a formalization of the idea of rational momentum trading. It also captures the situation in which traders behave as if their demand functions are increasing. Contrarianism has a meanreversion flavor. Both momentum and contrarian trading have been analyzed extensively in the empirical literature and have been found to generate abnormal returns over some time horizon.17 Our analysis thus provides a characterization for momentum and mean-reversion behavior, and shows that it can be rational. As we have argued above, in our setup we assume that trades are informative to avoid the uninteresting case of a no-trade cascade. Thus, we cannot have a situation in which all informed types act alike.18 Nevertheless, our setup allows for the possibility that a very large portion of informed traders are involved in herding or contrarianism. The precise proportion of such informed traders is, in fact, determined exogenously by the information structure through the like15 Avery and Zemsky’s (1998) definition of contrarianism is stronger than ours (they also imposed an additional bound on price movements that reflects the expectations that would obtain if the traders received an infinite number of draws of the same signal). We adopt the definition of contrarianism above because, as we explained before, it is a natural and simple counterpart to the definition of herd behavior. 16 Herding and contrarianism here refer to extreme switches of behavior from selling to buying or the reverse. One could expand the definition to switches from holding to buying or to selling (or the reverse). For consistency with the earlier literature, we use the extreme cases where switches do not include holding. 17 In the empirical literature, contrarian behavior is found to be profitable in the very short run (1 week and 1 month, Jegadeesh (1990) and Lehmann (1990)) and in the very long run (3–5 years, de Bondt and Thaler (1985)). Momentum trading is found to be profitable over the medium term (3–12 months, Jegadeesh and Titman (1993)) and exceptionally unprofitable beyond that (the 24 months following the first 12, Jegadeesh and Titman (2001)). Sadka (2006) studied systematic liquidity risk and linked a component of it, which is caused by informed trading, to returns on momentum trading. 18 Note that even though with no cascades (the market expects that), the different types do not take the same actions, in degenerate settings when a particular type is not present in certain states, it is possible that in these particular states all informed types that occur with positive probability take the same action. An example of such a degenerate situation is the information structure in Avery and Zemsky (1998) that we discuss in Section 10.
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lihood that a trader receives the relevant signal. We discuss this point further in Section 11 and show that this proportion can be arbitrarily large. Although the informative trade assumption ensures that asymptotically the true value of the security is learned, our definition of herding and contrarianism admits the possibility of switches of trades “in the wrong direction” in the short run. For instance, traders may buy herd even though the true value of the asset is V1 (the lowest). Finally, social learning can have efficiency consequences. In any setting where agents learn from the actions of others, an informational externality is inevitable as future agents are affected when earlier agents take actions that reveal private information. For example, in the early herding models of Bikhchandani, Hirshleifer, and Welch (1992) and Banerjee (1992), once a cascade begins, no further information is revealed. However, in this paper, we do not address the efficiency element, because it requires a welfare benchmark, which “is generally lacking in asymmetric information models of asset markets” (see Avery and Zemsky (1998, p. 728)). Instead, our definition is concerned with observable behavior and outcomes that are sensitive to the details of the trading history. Such sensitivities may dramatically affect prices in financial markets. 5. CHARACTERIZATION RESULTS: THE GENERAL CASE The main characterization result for herding and contrarianism is as follows: THEOREM 1: (a) HERDING. (i) Necessity: If type S herds, then S is U-shaped with a nonzero bias. (ii) Sufficiency: If there is a U-shaped type with a nonzero bias, there exists μh ∈ (0 1] such that some informed type herds when μ < μh . (b) CONTRARIANISM. (i) Necessity: If type S acts as a contrarian, then S is hill-shaped with a nonzero bias. (ii) Sufficiency: If there is a hill-shaped type with a nonzero bias, there exists μc ∈ (0 1] such that some informed type acts as a contrarian when μ < μc . Theorem 1 does not specify when we have buy or sell herding or when we have buy or sell contrarianism. In what follows, we first consider the necessary and then the sufficient conditions for each of these cases. The proof of Theorem 1 then follows from these characterization results at the end of this section. We begin by stating three useful lemmas. The first lemma provides a characterization for the difference between private and public expectations. LEMMA 1: For any S, time t, and history H t , E[V |S H t ] − E[V |H t ] has the same sign as (2)
q1t q2t [Pr(S|V2 ) − Pr(S|V1 )] + q2t q3t [Pr(S|V3 ) − Pr(S|V2 )] + 2q1t q3t [Pr(S|V3 ) − Pr(S|V1 )]
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Second, as the prior on the liquidation values is symmetric, it follows that the expectation of the informed is less (greater) than the public expectation at the initial history if and only if his signal is negatively (positively) biased. LEMMA 2: For any signal S, E[V |S] is less than E[V ] if and only if S has a negative bias, and E[V |S] is greater than E[V ] if and only if S has a positive bias. An immediate implication of this lemma is that someone sells at the initial history only if this type’s signal is negatively biased and buys only if the signal is positively biased. Third, note that herding and contrarianism involve switches in behavior after changes in public expectations relative to the initial period. A useful way to characterize these changes is the following lemma. LEMMA 3: If E[V |H t ] > E[V ], then q3t > q1t , and if E[V ] > E[V |H t ], then q1t > q. t 3
Thus the public expectation rises (falls) if and only if the public belief attaches a lower (higher) probability to the lowest value, V1 , than to the highest value, V3 . 5.1. Necessary Conditions Herding and contrarianism by a given signal type involve buying at some history and selling at another. Our first result establishes that this cannot happen if the signal is decreasing or increasing. Consider a decreasing type S. Since the ask price always exceeds the public expectation, it follows that, at any H t , type S does not buy if the expectation of S, E[V |S H t ], is no more than the public expectation, E[V |H t ]. The latter must indeed hold because, for any two valuations V and Vh such that V < Vh , the likelihood that a decreasing type S attaches to V relative Vh at any history H t cannot exceed that of the market: Pr(Vl |S H t ) Pr(Vh |S H t )
=
Pr(Vl |H t ) Pr(S|Vl ) Pr(Vh |H t ) Pr(S|Vh )
≤
Pr(Vl |H t ) Pr(Vh |H t )
Formally, every term in (2) is nonpositive when S is decreasing; therefore, it follows immediately from Lemma 1 that E[V |S H t ] ≤ E[V |H t ]. An analogous set of arguments demonstrates that an increasing type does not sell at any history. Therefore, we can state the following proposition. PROPOSITION 1: If S is decreasing, then type S does not buy at any history. If S is increasing, then type S does not sell at any history. Thus recipients of such signals cannot herd or behave as contrarians.
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Proposition 1 demonstrates that any herding and contrarian type must be either U- or hill-shaped. We next refine these necessary conditions further and state the main result of this subsection as follows. PROPOSITION 2: (a) Type S buy herds only if S is nU-shaped and sell herds only if S is pU-shaped. (b) Type S acts as a buy contrarian only if S is nhill-shaped and acts as a sell contrarian only if S is phill-shaped. A sketch of the proof of Proposition 2 for buy herding and buy contrarianism is as follows. Suppose that S buy herds or acts as a buy contrarian. Then it must be that at H 1 , type S sells and thus his expectation is below the public expectation. By Lemma 2, this implies that S is negatively biased.19 Thus, by Proposition 1, S is either nU- or nhill-shaped. The proof is completed by showing that buy herding is inconsistent with an nhill-shaped c.s.d. and that buy contrarianism is inconsistent with an nUshaped c.s.d. To see the intuition, for example, for the case of buy herding, note that in forming his belief, a trader with an nhill-shaped c.s.d. puts less weight on the tails of his belief (and thus more on the center) relative to the public belief; furthermore, the shift from the tail towards the center is more for value V3 than for V1 because of the negative bias. When buy herding occurs, the public belief must have risen and thus, by Lemma 3, the public belief attaches more weight to V3 relative to V1 (i.e., q1t < q3t ). Such a redistribution of probability mass ensures that S’s expectation is less than that of the public. Hence an nhill-shaped S cannot be buying. The arguments for sell herding and sell contrarianism are analogous except that here the bias has to be positive to ensure that the informed type buys at the initial history. 5.2. Sufficient Conditions The above necessary conditions—U shape for herding and hill shape for contrarianism—turn out to be almost sufficient as well. Before stating the sufficiency results, it will be useful to discuss two of the ideas that provide insight into the analysis. (I) With a U-shaped signal S, the difference between the private and the public expectation, E[V |S H t ] − E[V |H t ], is positive at histories at which the public probability of state V1 is sufficiently small relative to the probability of the other states (both q1t /q2t and q1t /q3t are close to zero) and is negative at histories at which the public probability of state V3 is sufficiently small relative to the probability of the other states (both q3t /q1t and q3t /q1t are close to zero). For a hill-shaped signal S the sign of this difference is the reverse. 19
The bias is only required because we assume that priors are symmetric.
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The intuition for this statement relates to the example from the Introduction. Consider first any history H t at which both q1t /q2t and q1t /q3t are close to zero. Then there are effectively two states V2 and V3 at H t . This means that at such a history, the difference between the expectations of an informed type S and of the public (which has no private information), E[V |S H t ] − E[V |H t ], has the same sign as Pr(S|V3 ) − Pr(S|V2 ).20 The latter is positive for a Ushaped S and negative for a hill-shaped S. Thus, at such history, E[V |S H t ] − E[V |H t ] is positive if S is U-shaped and is negative if S is hill-shaped. By a similar reasoning, the opposite happens at any history H t at which both q3t /q1t and q3t /q2t are close to zero. At such a history, there are effectively two states, V1 and V2 ; therefore, E[V |S H t ] − E[V |H t ] has the same sign as Pr(S|V2 ) − Pr(S|V1 ).21 Since the latter is negative for a U-shaped S and positive for a hill-shaped S, it follows that at such a history, E[V |S H t ] − E[V |H t ] is negative if S is U-shaped and is positive if S is hill-shaped. (II) The probability of noise trading may have to be sufficiently large to ensure that the bid–ask spread is not too wide both at the initial history and later at the history at which the herding or contrarian candidate changes behavior. In (I), we have compared the private expectation of the informed trader with that of the public. To establish the existence of herding or contrarian behavior, however, we must compare the private expectations with the bid and ask prices. The difference is that bid and ask prices form a spread around the public expectation. To ensure the possibility of herding or contrarianism, this spread must be sufficiently “tight.” Tightness of the spread, in turn, depends on the extent of noise trading: the more noise there is (the smaller the likelihood of the informed types, μ), the tighter is the spread. More specifically, to ensure buy herding or buy contrarianism (the other cases are similar) by an informed type, a minimal amount of noise trading may be necessary, so that the informed type (i) sells initially and (ii) switches to buying after some history H t . Below, we formalize these minimal noise trading restrictions by introducing two bounds, one to ensure a tight spread for the initial sell and the second to ensure a tight spread for the subsequent switch to buying, and we require the likelihood of informed trading μ to be less than both these bounds. Appealing to the above ideas, we can now state a critical lemma. To save space, we state the result only for the case of buy herding and buy contrarianism. 20 Formally, at any H t at which both q1t /q2t and q1t /q3t are close to zero, the first and the third terms in (2) are arbitrarily small; moreover, the second term in (2) has the same sign as Pr(S|V3 ) − Pr(S|V2 ); therefore, it follows from Lemma 1 that E[V |S Ht ] − E[V |Ht ] has the same sign as Pr(S|V3 ) − Pr(S|V2 ). 21 This claim also follows from Lemma 1: at such history, the second and the third terms in (2) are arbitrarily small and the second term in (2) has the same sign as Pr(S|V3 ) − Pr(S|V2 ).
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LEMMA 4: (i) Suppose that signal S is nU-shaped. Then there exist μi and μ ∈ (0 1] such that S buy herds if μ < μbh ≡ min{μi μsbh } and if the following statement holds: s bh
(3)
For any ε > 0 there exists a history H t such that q1t /qlt < ε for all l = 2 3
(ii) Suppose signal S is nhill-shaped. Then there exist μi and μsbc ∈ (0 1] such that S acts as a buy contrarian if μ < μbc ≡ min{μi μsbc } and if the following statement holds: (4)
For anyε > 0 there exists a history H t such that q3t /qlt < ε for all l = 1 2
Conditions (3) and (4) above ensure that there are histories at which the probability of an extreme state, V1 or V3 , can be made arbitrarily small relative to the other states. Value μbh is the minimum of the two bounds μi and μsbh , mentioned in (II), that respectively ensure that spreads are small enough at the initial history and at the time of the switch of behavior by a buy herding type. Similarly, μbc is the minimum of the two bounds μi and μsbc that respectively ensure that spreads are small enough at the initial history and at the time of the switch of behavior by a buy contrarian type S. Below we will discuss how restrictive these bounds are and whether they are necessary for the results. A sketch of the proof for part (i) of Lemma 4 is as follows. First, by Lemma 2, S having a negative bias implies that E[V |S] < E[V ]. Then it follows from the arguments outlined in (II) above that one can find an upper bound μi > 0 on the size of the informed trading such that if μ < μi , then at H 1 the bid price bid1 is close enough to the public expectation E[V ] so that E[V |S] is also below bid1 , that is, S sells at H 1 . Second, since S is U-shaped, it follows from the arguments outlined in (I) that at any history H t at which q1t /q2t and q1t /q3t are sufficiently small, E[V |S H t ] > E[V |H t ] (by condition (3) such history H t exists). Then, by the arguments outlined in (II) above, one can find an upper bound μsbh > 0 such that if μ < μsbh , then at H t the ask price askt is sufficiently close to E[V |H t ] so that E[V |S H t ] also exceeds askt , that is, S switches to buying. Finally, since q1t /q3t is small at H t , by Lemma 3, E[V |H t ] > E[V ]. Thus, the switch to buying at H t by S is in the direction of the crowd. The argument for contrarianism in part (ii) is analogous except that to ensure E[V |S H t ] exceeds E[V |H t ] at some history H t for a hill-shaped S, by (I), H t must be such that q3t /q1t and q3t /q2t are sufficiently small (condition (4) ensures that such history H t exists). Then by (II) there exists μsbc > 0 such that if μ < μsbc , askt is sufficiently close to E[V |H t ] so that E[V |S H t ] > askt . Furthermore, at such a history H t , S will be buying against the crowd because when q3t /q1t is small, which is the case at H t , by Lemma 3, E[V |H t ] < E[V ].
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A set of results similar to Lemma 4 can be obtained for the cases of sell herding, and sell contrarianism except that to ensure sell herding, the appropriate assumptions are that S is pU and (4) holds, and to ensure sell contrarian, we need that S is phill and (3) holds.22 The sufficiency results in Lemma 4 are nonvacuous provided that conditions (3) and (4) are satisfied. As these conditions depict properties of endogenous variables, to complete the analysis, we need to show the existence of the histories assumed by conditions (3) and (4). In some cases, this is a straightforward task. The easiest case arises when at any t there is a trade that reduces both q1t /q2t and q1t /q3t or both q3t /q1t and q3t /q2t uniformly (independent of time). For example, suppose that the probability of a buy is uniformly increasing in the liquidation value at any date and history: (5)
for some ε > 0 βtj > βti + ε for any j > i and any t.
Since qit+1 /qjt+1 = (qit βti )/(qjt βtj ) when there is a buy at date t, it follows that in this case a buy reduces both q1t /q2t and q1t /q3t uniformly. Thus if (5) holds, a sufficiently large number of buys induces the histories described in (3). Similarly, with a sale, qit+1 /qjt+1 = (qit σit )/(qjt σjt ). Thus, if the probability of a sale is uniformly decreasing in the liquidation value at any H t , (6)
for some ε > 0 σit > σjt + ε for any j > i and any t
then a sale reduces both q3t /q1t and q3t /q2t uniformly. Thus, if (6) holds, a sufficiently large number of sales induces the histories described in (4).23 Demonstrating conditions (3) and (4) generally, however, requires a substantially more complex construction. In particular, in some cases there are no paths that result in both q1t /q2t and q1t /q3t decreasing at every t or in both q3t /q1t and q3t /q2t decreasing at every t24 For these cases, we construct outcome paths consisting of two different stages. For example, to ensure (3), the path is constructed so that in the first stage, q1t /q2t becomes small while ensuring that q1t /q3t does not increase by too much. Then in the second stage, once q1t /q2t is 22 These differences arise because with sell herding or sell contrarianism the informed needs to buy initially and switch to selling later. To ensure the former, by Lemma 2, we need to assume a positive bias and, to ensure the latter, the appropriate “extreme” histories at which the switches happen are the opposite of the buy herding and buy contrarian case. Also, the values for the bounds on the size of informed trading might be different for sell herding and sell contrarianism from those for buy herding and buy contrarianism. 23 These monotonicity properties of the probability of a buy and the probability of a sale as defined in (5) and (6) are satisfied, for instance, by MLRP information structures; see the next section. 24 For instance, if for every action at (= buy, sell, or hold), the probability of at in state V1 is no less than the probability of at in the other two states, there will not be a situation that reduces both q1t /q2t and q1t /q3t .
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sufficiently small, the continuation path makes q1t /q3t small while ensuring that q1t /q2t does not increase by too much. A similar construction is used for (4). Such constructions work for most signal distributions. The exceptions are cases with two U-shaped signals with opposite biases or two hill-shaped signals with opposite biases. In these cases, we can show, depending on the bias of the third signal, that either (3) or (4) holds, but we cannot show both. For example, if one of the three signals is nU and another is pU, then we can show that (3) holds if the third signal has a nonnegative bias and (4) holds if the third signal has a nonpositive bias. This implies, by Lemma 4(i), that in the former case, the nU type buy herds, and in the latter case, by an analogous argument, the pU type sell herds (similarly for the contrarian situation). The next proposition is our main sufficiency result. It follows from the discussion above, concerning (3) and (4), and Lemma 4 (for completeness, we state the result for buy and sell herding and for buy and sell contrarianism). PROPOSITION 3: (a) Let S be nU-shaped. If another signal is pU-shaped, assume the third signal has a non negative bias. Then there exists μbh ∈ (0 1] such that S buy herds if μ < μbh . (b) Let S be pU-shaped. If another signal is nU-shaped, assume the third signal has a nonpositive bias. Then there exists μsh ∈ (0 1] such that S sell herds if μ < μsh . (c) Let S be nhill-shaped. If another signal is phill-shaped assume the third signal has a nonpositive bias. Then there exists μbc ∈ (0 1] such that S is a buy contrarian if μ < μbc . (d) Let S be phill-shaped. If another signal is nhill-shaped, assume the third signal has a nonnegative bias. Then there exists μsc ∈ (0 1] such that S is a sell contrarian if μ < μsc . Discussion of the Noise Restriction For each of our sufficiency results above (Lemma 4 and Proposition 3), we assume that μ is less than some upper bound. We will now discuss whether these bounds are necessary for our results, and whether they are restrictive. Consider the case of buy herding by an nU-shaped type S as in Lemma 4(i). For this sufficiency result, we assume two restrictions: μ < μi and μ < μsbh . These conditions respectively ensure that the spread is small enough at the initial history and at history H t at which there is a switch in behavior. In the Appendix, we show that there exists a unique value for the first bound, μi ∈ (0 1], such that μ < μi is also necessary for buy herding. In general, we cannot find a unique upper bound for the second value, μsbh , such that μ < μsbh is also necessary for buy herding. The reason is that the upper bound that ensures that the spread is not too large at the history H t at which S switches to buying may depend on what the types other than S do at H t . Since there may be more than one history at which S switches to buying, this upper bound may not be unique.
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The above difficulty with respect to the necessity of the second noise condition μ < μsbh does not arise, for instance, when types other than S always take the same action.25 More generally, we show in the Supplemental Material (Proposition 3a), that μ < μsbh is also necessary for buy herding provided there is at most one U-shaped signal. A similar argument applies to contrarian behavior. The noise restriction with respect to the initial trade is necessary, whereas the noise restriction with respect to the switch of behavior is necessary as long as there is at most one hill-shaped type. Finally, note that the bounds on μ in our sufficiency results do not always constitute a restriction. Consider again the case of buy herding for an nU type S. If at H 1 , type S has the lowest expectation among all informed types, then his expectation must be less than the bid price at H 1 and, thus, there is no need for any restriction on the size of the informed at H 1 (μi can be set to 1). Likewise, if at H t at which buy herding occurs, type S has the highest expectation (for example, this happens if no type other than S buys at H t ), then his expectation must be greater than the ask price at H t and, therefore, no restriction on the size of the informed is needed (μsbh can be set to 1). Consequently, to obtain our sufficiency results, there are restrictions on the value of μ only if at H 1 or at the switch history H t the expectation of the herding candidate is in between those of the other signal types.26 5.3. Proof of Theorem 1 The necessity part in Theorem 1 follows immediately from Proposition 2. The proof of the sufficiency part for case (a) of herding is as follows. Let μh = min{μsh μbh }, where μbh and μsh are the bounds for herding given in Proposition 3. Also, assume that μ < μh and that there exists a U-shaped signal as in part (a) of Theorem 1. Then there are two possibilities: either there is another U-shaped signal with the opposite bias or there is not. If there is no other U-shaped signal with the opposite bias, then by part (a) of Proposition 3, the U-shaped type buy herds if it has a negative bias, and by part (b) of Proposition 3, the U-shaped type sell herds if it has a positive bias. If there is another U-shaped signal with the opposite bias, then by parts (a) and (b) of Proposition 3, one of the U-shaped signals must herd. This is because if the third signal is weakly positive, then the U-shaped signal with a negative bias buy herds, and if the third signal is weakly negative, then the U-shaped signal 25
For example, this happens when the information structure satisfies MLRP; see the next section. 26 Romano and Sabourian (2008) extended the present model to the case where traders can trade a continuum of quantities. In this setting, they showed that no restrictions on μ are needed because at each history, each quantity is traded by a specific signal type.
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with a positive bias sell herds. The reasoning for sufficiency in part (b) of Theorem 1 is analogous and is obtained by setting μc = min{μsc μbc }, where μsc and μbc are the bounds for contrarianism from Proposition 3.27 Q.E.D. 6. SOCIAL LEARNING WITH MLRP INFORMATION STRUCTURE The literature on asymmetric information often assumes that the information structure is monotonic in the sense that it satisfies the monotone likelihood ratio property (MLRP). Here this means that for any signals Sl Sh ∈ S and values Vl Vh ∈ V such that Sl < Sh and Vl < Vh , Pr(Sh |Vl ) Pr(Sl |Vh ) < Pr(Sh |Vh ) Pr(Sl |Vl ). MLRP is very restrictive (it is stronger than first order stochastic dominance) and at first might seem to be too strong to allow herding or contrarianism. This turns out to be false. Not only does MLRP not exclude such possibilities, it actually enables one to derive a sharper set of results for the existence of herding and contrarianism. Moreover, with MLRP the histories that can generate herding or contrarianism can be easily identified. MLRP is a set of restrictions on the conditional probabilities for the entire signal structure and is equivalent to assuming that all minors of order 2 of the information matrix I are positive. Herding or contrarianism, on the other hand, relates to the c.s.d. of a signal being U- or hill-shaped, that is, to the individual row in matrix I that corresponds to the signal. Therefore, to analyze the possibility of herding or contrarianism with MLRP, we need to consider the c.s.d. of the different signals under MLRP. In the next lemma, we describe some useful implications of MLRP. LEMMA 5: Assume S1 < S2 < S3 and that the information structure satisfies MLRP. Then the following statements hold: (i) E[V |S1 H t ] < E[V |S2 H t ] < E[V |S3 H t ] at any t and any H t . (ii) In any equilibrium, S1 types always sell and S3 types always buy. (iii) S1 is strictly decreasing and S3 is strictly increasing. (iv) The probability of a buy is uniformly increasing in the liquidation value as specified in (5) and the probability of a sale is uniformly decreasing as specified in (6). Part (i) states that MLRP imposes a natural order on the signals in terms of their conditional expectations after any history. Part (iv) implies that with MLRP, the probability of a buy is uniformly increasing and the probability of a sell is uniformly decreasing in the liquidation values. Parts (ii) and (iii) state that MLRP restricts the behavior and the shape of the lowest and the highest signals S1 and S3 . In particular, these two types do not change behavior and they are decreasing and increasing, respectively. 27
The bounds μbh , μsh , μsc , and μbc in the proof of Proposition 3 are such that μh = μc .
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Lemma 5, however, does not impose any restrictions on the behavior or the shape of the middle signal S2 . In fact, MLRP is consistent with a middle signal S2 that is decreasing, increasing, hill-shaped, or U-shaped with a negative or a positive bias: Table I describes a robust example of all these possibilities. Thus, with MLRP, S2 is the only type that can display herding or contrarian behavior. We can then state the following characterization result for MLRP information structures (again we omit the analogous sell herding and sell contrarian cases). THEOREM 2: Assume S1 < S2 < S3 and the signal structure satisfies MLRP. (a) If S2 is nU, there exists μbh ∈ (0 1] such that S2 buy herds if and only if μ < μbh . (b) If S2 is nhill, there exists μbc ∈ (0 1] such that S2 acts as a buy contrarian if and only if μ < μbc . PROOF: Fix the critical levels μbh ≡ min{μi μsbh } for buy herding and μbc = min{μi μsbc } for buy contrarianism, where μi is defined in Lemma 8, and μsbh and μsbc are, respectively, defined in (12) and (13) in the Appendix. The “if” part follows from Lemma 4: By Lemma 5(iv), conditions (5) and (6) hold. As outlined in the last section, both q1t /q2t and q1t /q3t can be made arbitrarily close to zero by considering histories that involve a sufficiently large number of buys, and both q3t /q1t and q3t /q2t can be made arbitrarily close to zero by considering histories with a sufficiently large number of sales. Then conditions (3) and (4) hold, and by Lemma 4, an nU-shaped S2 buy herds and an nhill-shaped S2 acts as a buy contrarian. The “only if” part follows from Proposition 3a in the Supplemental MateQ.E.D. rial.28 The “if” part of the above proof demonstrates that with MLRP, it is strikingly easy to describe histories that induce herding by a U-shaped type or contrarianism by a hill-shaped type: an nU-shaped S2 buy herds after a sufficient number of buys and an nhill-shaped S2 acts as a buy contrarian after a sufficient number of sales.29 28 The “only if” part of the theorem also follows from the discussion in the previous section on noise trading: if types other than S2 always take the same action, then there is a unique upper bound on the size of the informed trading that ensures that the spreads are sufficiently tight for S2 to buy herd (or to act as a buy contrarian). By Lemma 5, S3 always buys and S1 always sells. Therefore, with MLRP, the upper bound μbh is unique. 29 Note that the bounds on the size of the informed μbh and μbc in Theorem 2 must be strictly below 1. To see this, recall that by part (i) of Lemma 5, the expectation of the herding or contrarian candidate type S2 is always between those of the other two types at every history. Also, by part (ii) of Lemma 5, S1 always sells and S3 always buys. Therefore, if μ is arbitrarily close to 1, then, to ensure zero profits for the market maker, E[V |S1 H t ] < bidt < E[V |S2 H t ] and E[V |S2 H t ] < askt < E[V |S3 H t ] for every H t . This implies that S2 does not trade; a contradiction.
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HERDING AND CONTRARIAN BEHAVIOR TABLE I AN EXAMPLE OF AN MLRP SIGNAL DISTRIBUTIONa Pr(S|V )
S1 S2 S3
V1
V2
V3
δ(1 − β)/(β + δ(1 − β)) β/(β + δ(1 − β)) 0
δ(1 − α) α (1 − δ)(1 − α)
0 β/(β + (1 − δ)(1 − β)) (1 − β)(1 − δ)/(β + (1 − δ)(1 − β))
a For α δ ∈ (0 1) and β ∈ (0 α), the above distribution satisfies MLRP. Moreover, S is nU-shaped if β ∈ (α(1 − 2 δ)/(1 − αδ) α) and δ < 1/2, is pU-shaped if β ∈ (αδ/(1 − α(1 − δ)) α) and δ > 1/2, is nhill-shaped if β ∈ (0 αδ/(1 − α(1 − δ))) and δ < 1/2, is phill-shaped if β ∈ (0 α(1 − δ)/(1 − αδ)) and δ > 1/2, is decreasing if β ∈ (αδ/(1 − α(1 − δ)) α(1 − δ)/(1 − αδ)) and δ < 1/2, and is increasing if β ∈ (α(1 − δ)/(1 − αδ) αδ/(1 − α(1 − δ))) and δ > 1/2.
7. RESILIENCE, FRAGILITY, AND LARGE PRICE MOVEMENTS We now consider the robustness of herding and contrarianism, and describe the range of prices for which herding and contrarianism can occur. Throughout this section, we assume that signals satisfy the well behaved case of MLRP (we will return to this later), and perform the analysis for buy herding and buy contrarianism; the other cases are analogous. We first show that buy herding persists if and only if the number of sales during an episode of buy herding is not too large. This implies, in particular, that buy herding behavior persists if the buy herding episode consists only of buys. We also show that during a buy herding episode, as the number of buys increases, it takes more sales to break the herd. For buy contrarianism, the impact of buys and sales work in reverse: in particular, buy contrarianism persists if and only if the number of buys during an episode of buy contrarianism is not too large. This means that buy contrarianism does not end if the buy contrarianism episode consists only of sales. We also show that during a buy contrarianism episode, as the number of sales increases, it takes more buys to end buy contrarianism. PROPOSITION 4: Assume MLRP. Consider any history H r = (a1 ar−1 ) and suppose that H r is followed by b ≥ 0 buys and s ≥ 0 sales in some order; denote this history by H t = (a1 ar+b+s−1 ).30 (a) If there is buy herding by S at H r , then there exists an increasing function s¯(·) > 1 such that S continues to buy herd at H t if and only if s < s¯(b). (b) If there is buy contrarianism by S at H r , then there exists an increasing ¯ > 1 such that S continues to act as a buy contrarian at H t if function b(·) ¯ and only if b < b(s). One implication of the above result is that herding is resilient and contrarianism is self-defeating. The reason is that when buy herding or buy contrarian30
We will henceforth omit past prices from the history H t to simplify the exposition.
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ism begins, buys become more likely relative to a situation where the herding or contrarian type does not switch. Thus, in both buy herding and buy contrarianism there is a general bias toward buying (relative to the case of no social learning). By Proposition 4, buy herding behavior persists if there are not too many sales and buy contrarian ends if there is a sufficiently large number of buys. Thus herding is more likely to persist, whereas contrarianism is more likely to end. To see the intuition for Proposition 4, consider the case of buy herding in part (a). At any history, the difference between the expectation of the herding type S and that of the market is determined by the relative likelihood that they attach to each of the three states. Since the herding type S must have an nUshaped c.s.d., it follows that in comparing the expectation of the herding type S with that of the market there are two effects: first, S attaches more weight to V3 relative to V2 than the market and, second, S attaches more weight to V1 relative to both V2 and V3 than the market. Since V1 < V2 < V3 and at any history H r with buy herding, the expectation of the herding type S exceeds that of the market, it then follows that at H r the first effect must dominate the second one, that is, q1r /q2r and q1r /q3r are sufficiently small so that the first effect dominates. Also, by Lemma 5(iv), when MLRP holds, buys reduce the probability of V1 relative to the other states. Therefore, further buys after H r make the second effect more insignificant and thereby ensure that the expectation of the herding type S remains above the ask price. On the other hand, by Lemma 5(iv), when MLRP holds, sales reduce the probability of V3 relative to the other states; thus sales after H r make the first effect less significant. Therefore, with sufficiently many sales, the expectation of the herding type S will move below the ask price so that type S will no longer buy. This ends herding. The intuition for the buy contrarian case is analogous except that the effect of further buys and further sales work in the opposite direction. Next, we show that with MLRP, large price movements are consistent with both herding and contrarianism. In fact, the range of price movements in both cases can include (almost) the entire set of feasible prices. Specifically, for buy herding, the range of feasible prices is [V2 V3 ], and for buy contrarianism, the range is [V1 V2 ].31 As argued above, with MLRP, buys increase prices and sales decrease prices. Furthermore, by Proposition 4, buy herding persists when there are only buys and buy contrarianism persists when there are only sales. Thus, once buy herding starts, a large number of buys can induce prices to rise to levels arbitrarily close to V3 without ending buy herding, and once buy contrarianism starts, large numbers of sales can induce prices to fall to levels arbitrarily close to V1 without ending buy contrarianism. 31 The reason is, by Lemma 3, at any date t, buy herding implies q3t > q1t , and buy contrarian implies q1t > q3t .
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We complete the analysis by showing that there exists a set of priors on V such that herding and contrarianism can start when prices are close to the middle value, V2 . Together with the arguments in the last paragraph, we have that herding and contrarian prices can span almost the entire range of feasible prices. Formally, we have the following proposition. PROPOSITION 5: Let signals obey MLRP. (a) Consider any history H r = (a1 ar−1 ) at which there is buy herding (contrarianism). Then for any ε > 0, there exists history H t = (a1 at−1 ) following H r such that there is buy herding (contrarianism) at every H τ = (a1 aτ−1 ), r ≤ τ ≤ t, and E[V |H r+τ ] exceeds V3 − ε (is less than V1 + ε). (b) Suppose the assumptions in Theorem 2 that ensure buy herding (contrarianism) hold. Then for every ε > 0, there exists a δ > 0 such that if Pr(V2 ) > 1 − δ, there is a history H t = (a1 at−1 ) and a date r < t such that (i) there is buy herding (contrarianism) at every H τ = (a1 aτ−1 ), r ≤ τ ≤ t, (ii) E[V |H r ] < V2 + ε (E[V |H r ] > V2 − ε), and (iii) E[V |H t ] > V3 − ε (E[V |H t ] > V1 + ε). The results of this section (and those in the next section on volatility) assume that the information structure satisfies the well behaved case of MLRP. This ensures that the probabilities of buys and sales are uniformly increasing and decreasing in V (see Lemma 5(iv)). As a result, we have that the relative probability q1t /qt falls with buys and rises with sales for all = 2 3, and the opposite holds for q3t /qt for all = 1 2. This monotonicity in the relative probabilities of the extreme states is the feature that allows us to establish our persistence and fragility results. If MLRP were not to hold, then the probability of buys and sales may not be monotonic in V , and the results of this section may not hold.32 An example of this is the herding example in Avery and Zemsky (1998); also see Section 10. 8. THE IMPACT OF SOCIAL LEARNING ON VOLATILITY AND LIQUIDITY In this section, we are concerned with the impact of social learning on price movements. In particular, we ask the following questions: Do buys move prices more with than without social learning? Will sales move prices more with than without social learning? To address these questions, we compare price movements in our setup with those in a hypothetical benchmark economy in which informed traders do not switch behavior. This economy is identical to our setup except that each informed type always takes the same action as the one he chooses at the initial 32
The monotonicity of the probability of buys and sales in V can hold under weaker conditions than MLRP. For example, we could assume the existence of a strictly increasing and a strictly decreasing signal. All the results of this paper with MLRP also hold with this weaker assumption.
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history (before receiving any public information). Consequently, in the hypothetical benchmark economy, informed traders act as if they do not observe prices and past actions of others. We thus refer to this world as the opaque market and discuss examples for such situations at the end of the section. In contrast, in the standard setting, traders observe and learn from the actions of their predecessors. To highlight the difference, in this section we refer to the standard case as the transparent market. In both the transparent and the opaque economies, the market maker correctly accounts for traders’ behavior when setting prices. Volatility We show that with MLRP signals, at any histories at which either herding or contrarianism occurs, trades move prices more in the transparent market than in the opaque one. We found it interesting that larger price movements in the transparent market occur both after buys and after sales. Moreover, the result holds for MLRP information structures which, taken at face value, are “well behaved.” We present the result for the case of buy herding and buy contrarianism; the results for sell herding and sell contrarianism are identical and will thus be omitted. Specifically, fix any history H r at which buy herding starts and consider the difference between the most recent transaction price in the transparent market with that in the opaque market at any buy herding history that follows H r . Assuming MLRP signals, we show (a) that the difference between the two prices is positive if the history since H r consists of only buys, (b) that the difference is negative if the history since H r consists of only sales and the number of sales is not too large,33 and (c) that the difference is positive if the history following H r is such that the number of buys is arbitrarily large relative to the number of sales. We also show an analogous result for buy contrarianism. t t βtio , and σio be, respectively, Formally, for any history H t , let Eo [V |H t ] qio the market expectation, the probability of Vi , the probability of a buy in state Vi , and the probability of a sale in state Vi in the opaque market at H t . Then we can show the following situations. PROPOSITION 6: Assume MLRP. Consider any finite history H r = (a1 r for i = 1 2 3. a ) at which the priors in the two markets coincide: qir = qio r Suppose that H is followed by b ≥ 0 buys and s ≥ 0 sales in some order; denote this history by H t = (a1 ar+b+s−1 ). I. Assume that there is buy herding at H τ for every τ = r r + b + s. (a) Suppose s = 0. Then E[V |H t ] > Eo [V |H t ] for any b > 0. r−1
33 Note that, by Proposition 5, buy herding cannot persist with an arbitrarily large number of sales.
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(b) Suppose b = 0. Then there exists s ≥ 1 such that E[V |H t ] < Eo [V |H t ] for any s ≤ s. (c) For any s, there exists b such that E[V |H t ] > Eo [V |H t ] for any b > b. II. Assume that there is buy contrarianism at H τ for every τ = r r + b + s. (a) Suppose b = 0. Then E[V |H t ] < Eo [V |H t ] for any s > 0. (b) Suppose s = 0. Then there exists b ≥ 1 such that E[V |H t ] > Eo [V |H t ] for any b ≤ b. (c) For any b, there exists s such that E[V |H t ] < Eo [V |H t ] for any s > s. The critical element in demonstrating the result is the U-shaped nature of the herding candidate’s signal and the hill-shaped nature of the contrarian candidate’s signal in combination with the public belief once herding/contrarianism starts. To see this, consider any buy herding history H t = (a1 ar+b+s−1 ) that satisfies the above proposition for the case described in part (I); the arguments for a buy contrarian history described in part (II) are analogous. Then the prices in the transparent and opaque markets differ because at any buy herding history in the transparent market, the market maker assumes that the buy herding candidate S buys, whereas in the opaque market, the market maker assumes that S sells.34 Since the buy herding type must have a U-shaped signal, we also have Pr(S|V3 ) > Pr(S|V2 ). Then the following must hold: (i) the market maker upon observing a buy increases his belief about the likelihood of V3 relative to that of V2 faster in the transparent market (where S is a buyer) than in the opaque market (where S is a seller) and (ii) the market maker upon observing a sale decreases his belief about the likelihood of V3 relative to V2 faster in the transparent market than in the opaque market. Now if it is also the case that the likelihood of V1 is small relative to that of V3 in both worlds, then it follows from (i) and (ii), respectively, that the market expectation (which is the most recent transaction price) in the transparent market exceeds that in the opaque market after a buy and it is less after a sale. r ); At H r in both markets the likelihoods of each state coincide (qir = qio moreover, the likelihood of V1 in both markets is small relative to that of V3 (to ensure buy herding). Then the following two conclusions follow from the discussion in the previous paragraph: First, if H t involves only a single buy after H r (i.e., if s = 0 and b = 1), then E[V |H t ] > Eo [V |H t ]. Second, if H t involves only a single sale after H r (i.e., if b = 0 and s = 1), then E[V |H t ] < Eo [V |H t ]. Part I(b) follows from the latter. To complete the intuition for I(a) and I(c), note that further buys after H r reduce the probabilities of V1 relative to V3 in both markets (see Lemma 5(iv)). Thus if either the history after H r involves no sales (as in part I(a)) or the number of buys is large relative to the number of sales (as in part I(c)), then the first conclusion is reinforced and E[V |H t ] remains above Eo [V |H t ] after any such histories. 34 If we assume S1 < S2 < S3 , then with MLRP, the buy herding (buy contrarian) candidate must be S2 .
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Notice that with MLRP, any sale beyond H r increases the probability of V1 relative to V3 (and relative to V2 ) both in the transparent and in the opaque market. Furthermore, the increase may be larger in the latter than in the former. As a result, for the buy herding case, we cannot show that, in general, prices in the transparent market fall more than in the opaque market after any arbitrary number of sales. However, if the relative likelihood of a sale in state V1 to V3 in the transparent market is no less than that in the opaque market (i.e., (σ1 /σ3 ) ≥ (σ1o /σ3o )), then we can extend the conclusion in part I(b) to show that the price in the transparent market falls more than in the opaque market after any arbitrary number of sales (the proof is given in the Supplemental Material).35 Proposition 6 of course does not address the likelihood of a buy or a sale after herding or contrarianism begins. It is important to note, however, that once buy herding or buy contrarianism starts, there will also be more buys in the transparent market compared to the opaque market because the herding type buys at such histories. Thus, given the conclusions of Proposition 6, price paths must have a stronger upward bias in the transparent market than in the opaque market. Finally, it is often claimed that herding generates excess volatility, whereas contrarianism tends to stabilize markets because the contrarian types act against the crowd. The conclusions of this section are consistent with the former claim but contradict the latter. Both herding and contrarianism increase price movements compared to the opaque market and they do so for similar reasons—namely because of the U-shaped nature of the herding type’s c.s.d. and the hill-shaped nature of the contrarian type’s c.s.d. Liquidity In sequential trading models in the tradition of Glosten and Milgrom (1985), liquidity is measured by the bid–ask spread, as a larger spread implies higher adverse selection costs and thus lower liquidity. Since, at any date, market expectations after a buy and market expectations after a sale respectively coincide with the ask and the bid price at the previous date, the next corollary to Proposition 6 follows: COROLLARY: At any history H t at which type S engages in buy herding or buy contrarianism, two situations occur: (a) The ask price when the buy herding or buy contrarian candidate S rationally buys exceeds the ask price when he chooses not to buy. (b) The bid price when the buy herding or buy contrarian candidate S rationally buys is lower than the bid price when he chooses to sell. 35 The condition (σ1 /σ3 ) ≥ (σ1o /σ3o ) is satisfied if, for example, the bias of the herding candidate is close to zero.
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Part (a) of the result follows from Proposition 6.I(a) and II(b) when b = 1 and part (b) follows from Proposition 6.I(b) and II(a) when s = 1.36 The above corollary implies that in equilibrium, liquidity (as measured by the bid– ask spread) is lower when some informed types herd or act as a contrarian than when they do not. Interpretation of the Opaque Market and the Volatility Result One can think of the traders in the opaque market as automata that always buy or sell depending on their signals. One justification for such naive behavior is that traders do not observe or remember the public history of actions and prices (including current prices). Alternatively, the nonchanging behavior may represent actions of rational traders in a trading mechanism where traders submit their orders through a market maker some time before the orders get executed. The market maker would receive these orders in some sequence and he would execute them sequentially at prices which reflect all the information contained in the orders received so far. The actions of other traders and the prices are unknown at the time of the order submission and, thus, as in the opaque market, the order of each trader is independent of these variables.37 As traders effectively commit to a particular trade before any information is revealed, the price sequence in this alternative model would coincide with the price sequence in the opaque market that we depict above. Therefore, Proposition 6 can also be used to claim that volatility is greater in the transparent market than in this alternative setup in which all orders are submitted before any execution. A slightly more transparent market than the opaque one is one where each trader with herding/contrarian signal S compares his prior expectations, E[V |S], with the current price and buys if E[V |S] exceeds the ask price, sells if E[V |S] is less than the bid price, and does not trade otherwise. In this “almost opaque” market, there is a different kind of nontransparency in that at each period, the traders do not observe or recall past actions and prices, but they know the bid and ask prices at that period; furthermore, they act semirationally by comparing their private expectation with current prices without learning about the liquidation value from the current price (e.g., for cognitive reasons). For the case of herding, the same excess volatility result as in part (I) of Proposition 6 holds if we compare the transparent market with the above almost opaque market. To see this, note that at the initial history H 1 , every buy 36 We show in the Appendix that the proof of Proposition 6 in these cases does not require MLRP. Thus, the corollary is stated without assuming MLRP information structure; see footnotes 46 and 47. 37 A possible example of such mechanism is a market after a “circuit breaker” is introduced. The latter triggers a trading halt after “large” movements in stock prices. Before trading recommences, traders submit their orders without knowing others’ actions. We thank Markus Brunnermeier for this interpretation.
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herding type sells. Also, at every buy herding history, the prices are higher than at H 1 ; therefore, in an almost opaque market, the herding type must also sell at every buy herding history.38 Since Proposition 6 compares price volatility only at histories at which buy herding occurs, it follows that the same excess volatility result holds if we compare the transparent with the almost opaque market. 9. HERDING AND CONTRARIANISM WITH MANY STATES Our results intuitively extend to cases with more signals and more values. In fact, with three states and an arbitrary number of signals, our characterization results, in terms of U-shaped signals for herding and hill-shaped signals for contrarianism, and all our conclusions in the previous two sections with respect to fragility, persistence, large price movements, liquidity, and price volatility remain unchanged.39 With more than three states, U shape and Hill shape are no longer the only possible signal structures that can lead to herding and contrarianism. The intuition for our results with many states does, however, remain the same: the herding type must distribute probability weight to the tails and the contrarian types must distribute weight to the middle. Assume there are N > 2 states and N signals. Denote the value of the asset in state j by Vj and assume that V1 ≤ V2 ≤ · · · ≤ VN . Signal S is said to have an increasing c.s.d. if Pr(S|Vi ) ≤ Pr(S|Vi+1 ) for all i = 1 N −1 and a decreasing c.s.d. if Pr(S|Vi ) ≥ Pr(S|Vi+1 ) for all i = 1 N − 1. By the same reasoning as in Lemma 1, one can show that E[V |S H t ] − E[V |H t ] has the same sign as (7)
N−1 N−j (Vi+j − Vi ) · qi qi+j [Pr(S|Vi+j ) − Pr(S|Vi )] j=1 i=1
For an increasing c.s.d., (7) is always nonnegative since Pr(S|Vi+j ) − Pr(S|Vi ) is nonnegative for all i j; similarly, for decreasing c.s.d.s, (7) is always nonpositive since Pr(S|Vi+j ) − Pr(S|Vi ) is nonpositive for all i j. Therefore, an increasing or decreasing type cannot switch behavior and we have the following necessity result which is analogous to Proposition 1. LEMMA 6: An increasing or decreasing S never switches from buying to selling or vice versa. 38 Since at a buy contrarian history, prices are lower than at the initial history, the same claim cannot be made for the contrarian case. 39 With three states, hill and U shape are still well defined, irrespective of the number of signals; even with a continuum of signals, these concepts can be defined in terms of conditional densities.
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Next, we describe two sufficient conditions that yield herding and contrarianism that have a similar flavor as our sufficiency results in Section 5. We will focus only on buy herding and buy contrarianism; sell herding and sell contrarianism are analogous. In line with the previous analysis, we assume for the remainder of this section that the values of the asset are distinct in each state and that they are on an equal grid. Moreover, we assume that the prior probability distribution is symmetric. Thus we set {V1 V2 VN } = {0 V 2V (N − 1)V } and Pr(Vi ) = Pr(VN+1−i ) for all i. We begin with the analysis of the decision problem of selling at H 1 and generalize the concept of a negative bias as follows. Signal S is said to have a negative bias if for any pair of values that are equally far from the middle value, the signal happens more frequently when the true value is the smaller one than when it is the larger one: Pr(S|Vi ) > Pr(S|VN+1−i ) for all i < (N + 1)/2. In the Supplemental Material, we show that this property ensures that E[V |S] < E[V ]. This means, by a similar argument as with the three value case, that a negatively biased S must be selling at H 1 if μ is sufficiently small. Next we generalize the sufficient conditions for switching to buying at some history. Recall that in the three value case, we considered histories at which the probability of one extreme value was small to the point where it can be effectively ignored. Then the expectation of the informed exceeds that of the market if the informed puts more weight on the larger remaining value than on the smaller remaining one. The sufficient conditions that we describe for the switches in the general case have a similar intuition and are very simple, as we impose restrictions only on the most extreme values. Specifically, to ensure buy herding, we assume Pr(S|VN−1 ) < Pr(S|VN ) and consider histories at which the probabilities of all values are small relative to the two largest values VN−1 and VN . Since at such histories all but the two largest values can be ignored, it must be that the price must have risen and the expectation of type S must exceed that of the public expectation if Pr(S|VN−1 ) < Pr(S|VN ). If, in addition, the bid–ask spread is not too large (enough noise trading), the expectation of S will also exceed the ask price at H t and type S switches from selling to buying after a price rise. Similarly, to ensure buy contrarianism, we assume Pr(S|V1 ) < Pr(S|V2 ) and consider histories at which the probabilities of all values are small relative to the two smallest ones V1 and V2 . Since at such histories all but the two smallest values can be ignored and the price must have fallen, the expectation of type S must exceed that of the public expectation if Pr(S|V1 ) < Pr(S|V2 ). If, in addition, the bid–ask spread is not too large at such histories, then S switches from selling to buying after a price fall. Formally, we can show the following analogous result to Lemma 4.
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LEMMA 7: (i) Suppose S is negatively biased and satisfies Pr(S|VN−1 ) < Pr(S|VN ). Then there exists a μbh ∈ (0 1] such that S buy herds if μ < μbh and if (8)
for all ε > 0 there exists H t such that qit /qlt < ε for all l = N − 1 N and i < N − 1
(ii) Suppose S is negatively biased and satisfies Pr(S|V1 ) < Pr(S|V2 ). Then there exists a μbc ∈ (0 1] such that S is a buy contrarian if μ < μbc and if (9)
for all ε > 0 there exists H t such that qit /qlt < ε for all l = 1 2 and i > 2
The simplest way to ensure the existence of histories that satisfy (8) and (9) is to assume MLRP. Then, as in Lemma 5(iv) for the three state case, the probability of a buy is increasing and the probability of a sale is decreasing in V . As a result, with MLRP we can always ensure (8) by considering histories that contain a sufficiently large number of buys and ensure (9) by considering histories that contain a sufficiently large number of sales. Hence, Lemma 7 yields the following situations for buy herding and buy contrarianism.40 THEOREM 3: Assume MLRP and suppose signal S is negatively biased. (a) If Pr(S|VN−1 ) < Pr(S|VN ), then there exists μbh ∈ (0 1] such that S buy herds if μ < μbh . (b) If Pr(S|V1 ) < Pr(S|V2 ), then there exists μbh ∈ (0 1] such that S is a buy contrarian if μ < μbh . The description in this section has assumed that each state is associated with a unique liquidation value of the underlying security. There can be, however, other uncertainties that do not affect the liquidation value but that do have an impact on the price. One example is a situation in which some agents may have superior information about the distribution of information in the economy (e.g., as in Avery and Zemsky’s (1998) “composition uncertainty”; see the next section). In the Supplemental Material, we prove all the sufficiency results from this section for such a generalized setup. 10. AVERY AND ZEMSKY (1998) As mentioned in the literature review, Avery and Zemsky (1998) (AZ) argue that herd behavior with informationally efficient asset prices is not possible 40 Conditions that ensure sell herding and sell contrarian are defined analogously. In particular, to ensure the initial buy, we need to assume a positive bias Pr(S|Vi ) < Pr(S|VN+1−i ) for all i < (N + 1)/2. For the switches, we reverse the two conditions that ensure switching for buy herding and buy contrarian: for sell herding, we need Pr(S|V1 ) > Pr(S|V2 ); for sell contrarian, we need Pr(S|VN−1 ) > Pr(S|VN ).
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unless signals are “nonmonotonic” and uncertainty is “multidimensional.” AZ reached their conclusions by (i) showing that herding is not possible when the information structure satisfies their definition of monotonicity and (ii) providing an example of herding that has “multidimensional uncertainty.” In this section, we explain why our conclusions differ from theirs. We will also discuss the issue of price movements (or lack thereof) in their examples. AZ’s conclusion with respect to monotonicity arises because their adopted definition is nonstandard and excludes herding almost by assumption. Specifically, they define a monotonic information structure as one that satisfies (10)
∀S ∃w s.t. ∀H t E(V |H t S) is weakly between w and E(V |H t )
This definition neither implies nor is implied by the standard MLRP definition of monotonicity. Also, in contrast to MLRP, it is not a condition on the primitives of the signal distribution. Instead, it is a requirement on endogenous variables that must hold for all trading histories.41 Furthermore, it precludes herding almost by definition.42 AZ’s example of herding uses event uncertainty, a concept first introduced by Easley and O’Hara (1992). Specifically, in their example, the value of the asset and the signals can take three values {0 12 1} and the information structure can be described by Pr(S|V )
S1 = 0 S2 = 12 S3 = 1
V1 = 0
V2 = 12
V3 = 1
p 0 1−p
0 1 0
1−p 0 p
for some p > 1/2. The idea behind the notion of event uncertainty as used by AZ is that, first, informed agents know if something (an event) has happened (they know whether V = V2 or V ∈ {V1 V3 }). Second, they receive noisy information with precision p about how this event has influenced the asset’s liquidation value. This two-stage information structure makes the uncertainty multidimensional. Thus multidimensionality is equivalent to informed traders having a finer information partition than the market maker. AZ attributed herding to this feature of their example. Multidimensionality is, however, neither necessary nor sufficient for herding and it is relevant to herding only to the extent that it may generate U41 Condition (10) does not imply that each signal has a increasing or decreasing c.s.d.; however, one can show that if every signal has either a strictly increasing or a strictly decreasing c.s.d., then the information structure satisfies (10). 42 For example, for buy herding by type S to occur at some history H t , we must have E[V |S] < E[V ] and a subsequent price rise E[V ] < E[V |H t ]; but then (10) implies immediately that w < E[V |H t S] < E[V |H t ] and hence buy herding by S at H t is not possible.
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shaped signals. First, since AZ’s example has three states and three signals, it is a special case of our main setup and our characterization results apply. Specifically, the two herding types in AZ’s example are S1 and S3 . In addition to having finer partitions of the state space than the market maker, both types are also U-shaped and so our Proposition 3 explains the possibility of buy herding by S1 and sell herding by S3 .43 Second, our Proposition 3 demonstrates that there would also be herding if the AZ example is changed in such a way that all signals occur with positive probability in all states, while maintaining the U-shaped nature of signals S1 and S3 .44 Such an information structure is no longer multidimensional (the informed trader’s partition would be the same as the market maker’s). Third, consider an information structure for which signals S1 and S3 are such that informed traders know whether or not V = 0 has occurred (and the market did not). Such signals are multidimensional (they generate a finer partition), but they are not U-shaped and thus do not admit herding. Our general analysis with three states also provides us with an appropriate framework to understand the nature of histories that generate herding and contrarianism. For example, as we explained in Section 7, to induce buy herding, the trading history must be such that the probabilities of the lowest state V1 is sufficiently small relative to the two other states. With MLRP, such beliefs arise after very simple histories consisting of a sufficiently large number of buys. In AZ’s example, one also needs to generate such beliefs, but the trading histories that generate them are more complicated and involve a large number of holds followed by a large number of buys.45 Turning to price movements, in AZ’s event uncertainty example, herding has limited capacity to explain price volatility, as herding is fragile and price movements during herding are strictly limited. To allow for price movements during herding, AZ introduced a further level of informational uncertainty to their event uncertainty example. Specifically, they assumed additionally that 43
In AZ’s example, not all signals arise in all states (in states V1 and V3 only signals S1 and S3 arise; in state V2 only signal S2 arises). Thus when there is herding (by types S1 or S3 ), all informed types that occur with positive probability act alike. Nevertheless, it is important to note that in AZ’s example, herding does not constitute an informational cascade since at any history, not all types take the same action. The reason is that it is never common knowledge that there is herding. Instead, at any finite history, the market maker attaches nonzero probability to all three states and thus he always attaches nonzero probability to the state in which some trades are made by type S2 . Moreover, the market expects that when type S1 buy herds, type S2 sells, and when type S3 sell herds, type S2 buys. See also our discussion in footnote 18. 44 For example, take Pr(Si |Vi ) = p(1 − ε) and Pr(S2 |Vi ) = Pr(Si |V2 ) = ε for all i = 1 3 for 0 < ε < (1 − p)/2. 45 The holds bring down the probability of V1 relative to V2 and the number of buys is chosen to bring down the probability of V1 relative to V3 sufficiently while not increasing the probability of V1 relative to V2 by too much. Formally, their construction is similar to Subcase D2 in the proof of our Proposition 3.
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for each signal there are high and low quality informed traders. They also assumed that there is uncertainty about the proportion of each type of informed trader. AZ claimed that this additional level of uncertainty, which they label composition uncertainty, complicates learning and allows for large price movements during a buy herding phase (they do this by simulation). A state of the world in AZ’s example with composition uncertainty refers to both the liquidation value of the asset and the proportion of different types of informed traders in the market (the latter influences the prices). Thus, there is more than one state associated with a given value V of the asset. This example is, therefore, formally a special case of the multistate version of our model, and the possibility of herding follows from Lemma IV in the Supplemental Material. More specifically, our result establishes that to ensure herding in AZ’s example with composition uncertainty, we need the analogue of U-shaped signals with the property that the probability of a signal in each state with V = 1/2 is less than the probability of the signal in each of the states with V = 0 or with V = 1. This is indeed the case in the example with composition uncertainty. Therefore, herding there is also due to U-shaped signals. (See the discussion in Section F of the Supplemental Material.) To understand the differences in price movements and persistence, recall our discussion of fragility in Section 7 with regard to buy herding by type S at some history H t . The probability of the lowest state V1 relative to those of the other two states, q1t /q2t and q1t /q3t , must be sufficiently small to start herding, and these relative probabilities need to remain small for herding to persist beyond H t . With MLRP, since Pr(buy|V ) is increasing in V , further buys reduce both q1t /q2t and q1t /q3t ; at the same time, further buys increase prices. Thus herding can persist and prices can move significantly. In AZ’s example without composition uncertainty, while buys result in price increases, during any buy herding phase we have that Pr(buy|V2 ) < Pr(buy|V1 ) = Pr(buy|V3 ). Thus, once buy herding begins, further buys cannot ensure that the relative probabilities of V1 remain low, as buys increase q1t /q2t , while leaving q1t /q3t unaffected. Hence, buys during buy herding in AZ’s example without composition uncertainty are self-defeating. In AZ’s example with composition uncertainty, we have that Pr(buy|V2 ) < Pr(buy|V1 ) < Pr(buy|V3 ) once herding starts. Further buys during herding thus reduce q1t /q3t while increasing q1t /q2t . As the former offsets somewhat the effect of the latter, buy herding may persist (and allow price increases) for longer than without composition uncertainty. In conclusion, what makes herding less fragile and more consistent with significant price movements are the relative probabilities of a specific trade in the different states of the world and not so much the addition of extra dimensions of uncertainty. In fact, when the probability of a buy is increasing in the value of the asset, buy herding is least fragile and most consistent with large price movements. As we have shown, this can happen with only three states, without different dimensions of uncertainty and with “well behaved” information structures satisfying MLRP.
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11. EXTENSIONS, DISCUSSION AND CONCLUSION Herding and contrarian behavior are examples of history-dependent behavior that may manifest itself in real market data as momentum or mean reversion. Understanding the causes for the behavior that underlies the data can thus help interpret these important nonstationarities. In the first part of this paper, we characterized specific circumstances under which herding and contrarian behavior can and cannot occur in markets with efficient prices. In the second part, we showed that both herding and contrarianism can be consistent with large movement in prices, and that they both can reduce liquidity and increase volatility relative to situations where these kinds of social learning are absent. In the early literature on informational social learning (e.g., Banerjee (1992)), herding was almost a generic outcome and would arise, loosely, with any kinds of signals. Herding as defined in our setup does not arise under all circumstances, but only under those that we specify here. Namely, the underlying information generating process must be such that there are some signals that people receive under extreme outcomes more frequently than under moderate outcomes. Therefore, to deter herding, mixed messages that predict extreme outcomes (U-shaped signal) should be avoided. It is important to note that, depending on the information structure, the prevalence of types who herd (or act as contrarians) can vary and in some cases they can be very substantial. For example, consider the MLRP information structure in Table I when S2 is U-shaped. Then, in any state, the likelihood that an informed trader is a herding type S2 has lower bound α. This bound can range between 0 and 1. Thus, when α is sufficiently close to 1, the likelihood that an informed trader is a herding type is arbitrarily close to 1 and the impact of herding switches can then be very significant. In this paper, we have presented the results for which we were able to obtain clear-cut analytical results. In the Supplemental Material, we also explore other implications with numerical simulations. First, an important implication of our analysis for applied research is that when social learning arises according to our definition, simple summary statistics such as the number of buys and sales are not sufficient statistics for trading behavior. Instead, as some types of traders change their trading modes during herding or contrarianism, prices become history-dependent. Thus as the entry order of traders is permutated, prices with the same population of traders can be strikingly different, as we illustrate with numerical examples. Second, herding results in price paths that are very sensitive to changes in some key parameters. Specifically, in the case with MLRP, comparing the situation where the proportion of informed agents is just below the critical levels described in Theorem 2 with that where the proportion is just above that threshold (so there is no herding), prices deviate substantially in the two cases. Third, herding slows down the convergence to the true value if the herd moves away from that true value, but it accelerates convergence if the herd moves in the direction of the true value.
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APPENDIX A: OMITTED PROOFS A.1. Proof of Proposition 2 To save space, we shall prove the result for the case of buy herding and buy contrarian; the proof for the sell cases is analogous. Thus suppose that S buy herds or acts as a buy contrarian at some H t . The proof proceeds in several steps. Step 1—S has a negative bias. Buy herding and buy contrarianism imply E[V |S] < bid1 . Since bid1 < E[V ], we must have E[V |S] < E[V ]. Then by Lemma 2, S has a negative bias. Step 2—(Pr(S|V1 ) − Pr(S|V2 ))(q3t − q1t ) > 0. It follows from the definition of buy herding and buy contrarianism that E[V |S H t ] > askt . Since E[V |H t ] < askt , we must have E[V |S H t ] > E[V |H t ] By Lemma 1, this implies that (2) is positive at H t . Also, by the negative bias (Step 1), the third term in (2) is negative. Therefore, the sum of the first two terms in (2) is positive: q3t (Pr(S|V3 ) − Pr(S|V2 )) + q1t (Pr(S|V2 ) − Pr(S|V1 )) > 0. But this means, by negative bias, that (Pr(S|V1 ) − Pr(S|V2 ))(q3t − q1t ) > 0. Step 3a—If S buy herds at H t , then S is nU-shaped. It follows from the definition of buy herding that E[V |H t ] > E[V ]. By Lemma 3, this implies that q3t > q1t . Then it follows from Step 2 that Pr(S|V1 ) > Pr(S|V2 ). Also, since S buy herds, by Lemma 1, S cannot have a decreasing c.s.d. and we must have Pr(S|V2 ) < Pr(S|V3 ). Thus, S is nU-shaped. Step 3b—If S acts as a buy contrarian at H t , then S is nhill-shaped. It follows from the definition of buy contrarianism that E[V |H t ] < E[V ]. By Lemma 3, this implies that q3t < q1t . But then it follows from Step 2 that Pr(S|V1 ) < Pr(S|V2 ). Since, by Step 1, S has a negative bias, we have Pr(S|V2 ) > Pr(S|V1 ) > Pr(S|V3 ). Thus S is nhill-shaped. A.2. Proof of Lemma 4 First consider the decision problem of type S at H 1 . If S has a negative bias, by Lemma 2, E[V |S] < E[V ]. Also, E[V ] − bid1 > 0 and limμ→0 E[V ] − bid1 = 0. We can thus establish the following lemma. LEMMA 8: If S has a negative bias, then there exists μi ∈ (0 1] such that E[V |S] − bid1 < 0 if and only if μ < μi . Simple calculations (see the Supplemental Material for details) establish the following useful characterization of the buying decision of type S at any H t . LEMMA 9: E[V |S H t ] − askt has the same sign as (11)
q1t q2t [βt1 Pr(S|V2 ) − βt2 Pr(S|V1 )] + q2t q3t [βt2 Pr(S|V3 ) − βt3 Pr(S|V2 )] + 2q1t q3t [βt1 Pr(S|V3 ) − βt3 Pr(S|V1 )]
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To establish buy herding or buy contrarianism, we need to show that (11) is positive at some history H t . To analyze the sign of the expression in (11), we first show that the signs of the first and the second terms in (11) are, respectively, determined by the signs of the expressions Pr(S|V2 ) − Pr(S|V1 ) and Pr(S|V3 ) − Pr(S|V2 ) if and only if μ is sufficiently small. To establish this, let, for any i = 1 2 and any signal type S , mi ≡ Pr(S|Vi+1 ) − Pr(S|Vi ), M i (S ) ≡ Pr(S |Vi ) Pr(S|Vi+1 ) − Pr(S |Vi+1 ) Pr(S|Vi ), and ⎧ mi ⎪ ⎨ if mi and M i (S ) are nonzero μi (S ) ≡ mi − 3M i (S ) and have opposite signs, ⎪ ⎩ 1 otherwise. Clearly, μi (S ) ∈ (0 1]. The next lemma shows that for some S , μ1 (S ) and μ2 (S ) are, respectively, the critical bounds on the value of μ that characterize the signs of the first and the second terms in (11). LEMMA 10: In any equilibrium the following cases hold: (i) Suppose that Pr(S|V3 ) > Pr(S|V2 ). Then at any H t at which S buys and S = S S does not, the second term in (11) is positive if and only if μ < μ2 (S ). (ii) Suppose that Pr(S|V2 ) > Pr(S|V1 ). Then at any H t at which S buys and S = S S does not, the first term in (11) is positive if and only if μ < μ1 (S ). PROOF: First we establish (i). By simple computation, it follows that the second term in (11) equals γm2 + μM 2 (S ). Also, since in this case Pr(S|V3 ) > Pr(S|V2 ), we have from the definition of m2 and M 2 (S ) that γm2 +μM 2 (S ) > 0 if and only if μ < μ2 (S ). This completes the proof of (i). The proofs of (ii) is analogous: By simple computation, it follows that the first term in (11) equals γm1 + μM 1 (S ). Also, since in this case Pr(S|V1 ) < Pr(S|V2 ), we have from the definition of m1 and M 1 (S ) that γm1 +μM 1 (S ) > 0 if and only if μ < μ1 (S ). Q.E.D. Each of the two cases in Lemma 10 provides a set of conditions that determine the sign of one of the terms in (11). If the other terms in (11) are sufficiently small, then these conditions also determine whether S is a buyer. Specifically, if q1t is arbitrarily small relative to q2t and q3t , then the first and the last terms are close to zero (as they are multiplied by q1t and the second term is not) and can be ignored; thus, at such a history, type S buys if the second term in (11) is positive. Also, if q3t is arbitrarily small relative to q1t and q2t , then the last two terms are close to zero (as they are multiplied by q3t and the first term is not) and can be ignored; thus, at such history, type S buys if the first term in (11) is positive. We can now prove Lemma 4 by appealing to Lemmas 3, 8, and 10 as follows:
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OF
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LEMMA 4: Consider case (i). Let μi be as defined in Lemma 8
μ2 (S ) μsbh ≡ min S
Assume also that μ < μbh ≡ min{μi μsbh }. Since by assumption S has a negative bias and μ < μi , it follows from Lemma 8 that S sells at the initial history. Also, since S is U-shaped, we have Pr(S|V3 ) > Pr(S|V2 ). Therefore, by μ < μsbh and Lemma 10(i), there exists some η > 0 such that the second term in (11) always exceeds η. qt 2qt By condition (3), there exists a history H t such that q1t /q3t < 1 and q1t + qt1 < 3 2 η. Then by the former inequality and Lemma 3, we have E[V |H t ] > E[V ]. Also, qt since the sum of the first and the third terms in (11) is greater than −q2 q3 ( q1t + 2q1t
qt
2qt
3
), it follows from q1t + qt1 < η that the sum must also be greater than −η. 3 2 This, together with the second term in (11) exceeding η, implies that (11) is greater than zero and hence S must be buying at H t . The proof of (ii) is analogous. Let q2t
(13)
μ1 (S ) μsbc ≡ min S
and assume that μ < μbc ≡ min{μi μsbc }. Then by the same reasoning as above, S sells at H 1 . Also, since S has a hill shape, we have Pr(S|V1 ) > Pr(S|V1 ). Therefore, by μ < μsbc and Lemma 10(ii), there exists some η > 0 such that the first term in (11) always exceeds η. qt 2qt By condition (4), there exists a history H t such that q3t /q1t < 1 and q3t + qt3 < 1 1 η. Then by the former inequality and Lemma 3, we have E[V |H t ] < E[V ]. Also, since the sum of the second and the third terms in (11) is greater than qt 2qt qt 2qt −q1 q2 ( q3t + qt3 ), it follows from q3t + qt3 < η that the sum must also be greater 1 2 1 1 than −η. Since the first term in (11) exceeds η, this implies that (11) is greater Q.E.D. than zero and hence S must be buying at H t . A.3. Proof of Proposition 3 Below we provide a proof for part (a) of the proposition; the arguments for the other parts are analogous and, therefore, are omitted. The proof of part (a) is by contradiction. Suppose that S is nU-shaped and that all the other assumptions in part (a) of the proposition hold. Also assume, contrary to the claim in part (a), that S does not buy herd. Then by Lemma 4(i), we have a contradiction if it can be shown that (3) holds. This is indeed what we establish in the rest of the proof. First note that the no buy herding supposition implies that S does not buy at any history H t . Otherwise, since S has a negative bias, by Step 2 in the proof
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of Proposition 2, (Pr(S|V1 ) − Pr(S|V2 ))(q3t − q1t ) > 0. Because S is U-shaped this implies that q3t > q1t ; but then since by assumption μ < μi , it follows from Lemma 8 that S buy herds—a contradiction. Next, we describe conditions that ensure that q1t /qlt is decreasing in t for any l = 2 3. Denote an infinite path of actions by H ∞ = {a1 a2 }. For any date t and any finite history H t = {a1 at−1 }, let atk be the action that would be taken by type Sk ∈ S\S at H t ; thus if the informed trader at date t receives a signal Sk ∈ S\S, then at , the actual action taken at H t , equals atk . Also denote the action taken by S at H t by at (S). Then we have the following lemma. LEMMA 11: Fix any infinite path H ∞ = {a1 a2 } and any signal Sk ∈ S\S Let Sk ∈ S\S be such that Sk = Sk Suppose that at = atk . Then for any date t and l = 2 3, we have the following cases: (A) If atk = atk then q1t /qlt is decreasing. (B) If atk = at (S) and the inequality Pr(Sk |Vl ) ≤ Pr(Sk |V1 ) holds, then q1t /qlt is nonincreasing. Furthermore, if the inequality is strict, then q1t /qlt is decreasing. (C) If atk = atk and atk = at (S) and the inequality Pr(Sk |Vl ) ≥ Pr(Sk |V1 ) holds, then q1t /qlt is nonincreasing. Furthermore, if the inequality is strict, then q1t /qlt is decreasing. PROOF: Fix any l = 2 3 Since
q1t+1 qlt+1
=
q1t Pr(at |H t V1 ) qlt Pr(at |H t Vl ) t t l
to establish that q1t /qlt
is decreasing, it suffices to show that Pr(a |H V ) is (greater) no less than Pr(at |H t V1 ). Now consider each of the following three cases: (A) Since signal S is nU-shaped, the combination of Sk and Sk is phillshaped. This together with at = atk = atk implies that Pr(at |H t Vl ) exceeds Pr(at |H t V1 ). (B) If Pr(Sk |Vl ) ≤ Pr(Sk |V1 ), we have Pr(Sk |Vl ) + Pr(S|Vl ) ≥ Pr(Sk |V1 ) + Pr(S|V1 ) This, together with at = atk = at (S) implies that Pr(at |H t Vl ) ≥ Pr(at |H t V1 ). Furthermore, the latter inequality must be strict if Pr(Sk |Vl ) were less than Pr(Sk |V1 ). (C) If Pr(Sk |Vl ) ≥ Pr(Sk |V1 ), atk = atk , and atk = at (S), we have immediately that Pr(at |H t Vl ) ≥ Pr(at |H t V1 ). Furthermore, the latter inequality is strict if Pr(Sk |Vl ) were less than Pr(Sk |V1 ). This concludes the proof of Lemma 11. Q.E.D. Now we show that (3) holds and thereby obtain the required contradiction. This will be done for each feasible c.s.d. combination of signals. Case A—Either There Exists a Signal That Is Decreasing or There Are Two Hill-Shaped Signals Each With a Nonnegative Bias: Consider an infinite path of actions consisting of an infinite number of buys. We demonstrate (3) by showing that along this infinite history, at any date t, both q1t /q2t and q1t /q3t are decreasing and hence converge to zero (note that there are a finite number of states and signals). We show this in several steps.
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Step 1—If more than one informed type buys at t, then q1t /q2t and q1t /q3t are both decreasing at any t. Since S does not buy at any t, this follows immediately from Lemma 11(A). Step 2—If exactly one informed type buys at period t, then (i) q1t /q2t is decreasing and (ii) q1t /q3t is decreasing if the informed type that buys has a nonzero bias and is nonincreasing otherwise. Let Si be the only type that buys at t. This implies that Si cannot be decreasing; therefore, by assumption, Si must be phill-shaped and the step follows from Lemma 11(C). Step 3—If a type has a zero bias, he cannot be a buyer at any date t. Suppose not. Then there exist a type Si with a zero bias such that E[V |H t Si ] − E[V |H t ] > 0. By Lemma 1, we then have (14)
[Pr(Si |V3 ) − Pr(Si |V2 )](q3t − q1t ) > 0
Also, by Steps 1 and 2, q1t /q3t is nonincreasing at every t. Moreover, by assumption, q11 /q31 = 1. Therefore, q1t /q3t ≤ 1. Since Si buys at t Si must be hill-shaped, contradicting (14). Step 4—q1t /q2t and q1t /q3t are both decreasing at any t. This follows Steps 1–3. Case B—There Exists an Increasing Si Such That Pr(Si |Vk ) = Pr(Si |Vk ) for Some k and k : Let Sj be the third signal other than S and Si . Now we obtain (3) in two steps. Step 1—If Pr(Si |V1 ) = Pr(Si |V2 ), then for any ε > 0, there exists a finite history H τ = {a1 aτ−1 } such that q1τ /q2τ < ε. Consider an infinite path H ∞ = {a1 a2 } such that at = atj (recall that atj is the action taken by Sj at history H t = (a1 at−1 )). Note that S is nU-shaped and Pr(Si |V1 ) = Pr(Si |V2 ) < Pr(Si |V3 ). Therefore, Pr(Sj |V2 ) > max{Pr(Sj |V1 ) Pr(Sj |V3 )}. Then it follows from Lemma 11 that q1t /q2t is decreasing if at = at (S) and it is constant if at = at (S). To establish the claim, it suffices to show that at = at (S) infinitely often. Suppose not. Then there exists T such that for all t > T atj = at (S). Since type S does not buy at any date and there cannot be more than one informed type holding at any date (there is always a buyer or a seller), we must have Sj (and S) selling at every t > T . Then, by Lemma 1, we have (15)
q3t [Pr(Sj |V3 ) − Pr(Sj |V2 )] + [Pr(Sj |V2 ) − Pr(Sj |V1 )] q1t +2
q3t [Pr(Sj |V3 ) − Pr(Sj |V1 )] < 0 q2t
for all t > T . Also, by Pr(Si |V1 ) = Pr(Si |V2 ) < Pr(Si |V3 ), we have Pr(Sj |Vl ) + Pr(S|Vl ) > Pr(Sj |V3 ) + Pr(S|V3 ) for l = 1 2. Therefore,
q3t
→ 0 as t → ∞ for any l = 1 2. This, together with Pr(Sj |V2 ) > Pr(Sj |V1 ) contradicts (15). Step 2—For any ε > 0, there exists a history H t such that q1t /qlt < ε for any l = 2 3. Fix any ε > 0. Let H τ be such that q1τ /q2τ < ε if Pr(Si |V1 ) = Pr(Si |V2 ) (by qlt
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the previous step such a history exists) and be the empty history H 1 otherwise. Consider any infinite path H ∞ = {H τ aτ aτ+1 }, where for any t ≥ τ, at is the action that type Si takes at history H t = {H τ aτ at−1 }; that is, we first have the history H τ and then we look at a subsequent history that consists only of the actions that type Si takes. Since Si is increasing, it follows from Proposition 1 that at any history, Si does not sell. Also by the supposition, S does not buy at any history. Therefore, Si and S always differ at every history H t with t ≥ τ (there cannot be more than one type holding). But since at is the action that type Si takes at history H t , Si is increasing and Pr(Si |Vk ) = Pr(Si |Vk ) for some k and k , it then follows qt from parts (A) and (C) of Lemma 11 that for every t ≥ τ, (i) q1t is decreasing qt
3
and (ii) q1t is nonincreasing. This, together with q1τ /q2τ < ε when Pr(Si |V1 ) = 2 Pr(Si |V2 ), establishes that there exists t such that q1t /qlt < ε for any l = 2 3. Case C: There Are Two Hill-Shaped Signals and One Has a Negative Bias: Let Si be the hill-shaped signal with the negative bias. Also, let Sj be the other hill-shaped signal. Since both S and Si have negative biases, Sj must have a positive bias. Next fix any ε > 0, and define y and ϕlm for any l m = 1 2 3, as (16)
[Pr(Si |V2 ) − Pr(Si |V1 )] > 0 2[Pr(Si |V1 ) − Pr(Si |V3 )] γ + μ Pr(Si |Vl ) γ + μ(1 − Pr(S|Vl )) ϕlm := max γ + μ Pr(Si |Vm ) γ + μ(1 − Pr(S|Vm )) γ + μPr(Sj |Vl ) γ + μ Pr(Sj |Vm )
y :=
Since both Si and Sj are hill-shaped, we have ϕ12 < 1 This implies that there exists an integer M > 0 and δ ∈ (0 ε) such that y(ϕ12 )M < ε and δ(ϕ13 )M < ε. Consider the infinite path H ∞ = {a1 a2 }, where at = atj at every t Then we make the following claims. Claim 1—q1t /q3t is decreasing at every t. As Si and Sj have a negative and a positive bias, respectively, by Lemma 11, q1t /q3t is decreasing at every t. Claim 2—q1t /q2t converge to zero if there exists T such that ati = atj for all t > T . Since Sj is hill-shaped, this follows immediately from parts (A) and (C) of Lemma 11. Claim 3—There exists a history H τ such that q1τ /q3τ < δ and q1τ /q2τ < y. Suppose not. Then by Claims 1 and 2, there exists a date τ such that q1τ /q3τ < δ and aτi = aτj . Since S does not buy at any history, it follows that Si and Sj must be buying at τ (there is always at least one buyer and seller; thus Si and Sj cannot both be holding at τ). Then E[V |Si H t ] − E[V |H t ] > 0. By Proposition 2, this
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implies [Pr(Si |V3 ) − Pr(Si |V2 )] + +
q1τ [Pr(Si |V2 ) − Pr(Si |V1 )] q3τ
2q1τ [Pr(Si |V3 ) − Pr(Si |V1 )] > 0 q2τ
Since Si is nhill-shaped, it follows from the last inequality that
(17)
q1τ < q2τ
q1τ [Pr(Si |V2 ) − Pr(Si |V1 )] q3τ 2[Pr(Si |V1 ) − Pr(Si |V3 )]
[Pr(Si |V3 ) − Pr(Si |V2 )] +
q1τ [Pr(Si |V2 ) − Pr(Si |V1 )] q3τ < 2[Pr(Si |V1 ) − Pr(Si |V3 )] As q1τ /q3τ < δ and δ < 1, we have q1τ /q2τ < y. This contradicts the supposition. To complete the proof for this case, fix any τ and H τ such that q1τ /q3τ < δ and t q1τ /q2τ < y (by Claim 3, such a history exists). Consider a history H that consists of path H τ = (a1 aτ−1 ) followed by M periods of buys. Thus t = τ + M and H t = {H τ a1 aM }, where for any m ≤ M, am = buy. Since a buy must be either from Sj or Si or both, it then follows from the definitions of ϕ13 M, and δ and from q1τ /q3τ < δ that (18)
q1t /q3t ≤ (ϕ13 )M (q1τ /q3τ ) < (ϕ13 )M δ < ε
Also, since q1τ /q2τ < y, we have (19)
q1t /q2t < (ϕ12 )M (q1τ /q2τ ) < (ϕ12 )M y < ε
Since the initial choice of ε was arbitrary, (3) follows immediately from (18) and (19). Case D—There Exists a U-Shaped Signal Si ∈ S\S: Since both S and Si are U-shaped it follows that the third signal Sj is hill-shaped. Moreover, by assumption, Sj must have a nonnegative bias. To establish (3), fix any ε > 0 and consider the two possible subcases that may arise. Subcase D1—Si Has a Zero Bias: We establish the result using two claims. Claim 1—There exists a history H τ such that q1τ /q3τ < ε Consider the infinite path H ∞ = {a1 a2 } such that at = buy for each t. Since a buy must be from either Sj or Si or both, and Si has a zero bias, it follows from parts (A) and (C) of Lemma 11 that q1t /q3t is nonincreasing at every t Furthermore, q1t /q3t is decreasing if at = atj . Therefore, the claim follows if Sj buys infinitely often along the path H ∞ . To show that the latter is true, suppose it is not. Then
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there exists T such that for all t ≥ T , atj = ati = buy. Then for all t > T , by Lemma 9, (20)
q2t t q2t t t [β Pr (S |V ) − β Pr (S |V )] + [β Pr(Sj |V2 ) − βt2 Pr(Sj |V1 )] j 3 j 2 3 q1t 2 q3t 1 + [βt1 Pr(Sj |V3 ) − βt3 Pr(Sj |V1 )] < 0 qt
qt
Also, since Si is U-shaped, both q2t and q2t must be decreasing at every t > 1 3 T . But this is a contradiction because at every t > T , the last term in (20) is positive: βt1 Pr(Sj |V3 ) − βt3 Pr(Sj |V1 ) = γ(Pr(Sj |V3 ) − Pr(Sj |V1 )) > 0 (the equality follows from Si ’s zero bias). Claim 2—There exists a history H t such that q1t /qlt < ε for all l = 2 3. By the previous claim, there exists a history H τ such that q1τ /q3τ < ε. Next consider a history H ∞ = {H τ aτ aτ+1 } that consists of path H τ followed by a sequence of actions {aτ aτ+1 } such that at = atj at every history H t = {H τ aτ at−1 }. Since Si has a zero bias, it follows from Lemma 11 that at every t > τ, q1t /q3t is nonincreasing. Also, we have q1τ /q3τ < ε; therefore we have that at every t > τ, q1t /q3t < ε. Furthermore, since S and Si are U-shaped, and Sj is hill-shaped, by Lemma 11, q1t /q2t is decreasing at every t > τ; hence there must exist t > τ such that q1t /q2t < ε Since the initial choice of ε was arbitrary, (3) follows from Claim 2. Subcase D2—Both Si And Sj Have Nonzero Bias: Consider first the infinite path H ∞ = {a1 a2 } such that at = atj at every history H t = {a1 at−1 }. Then the following claims must hold. Claim 1—q1t /q2t is decreasing at every t. Since Sj and Si are, respectively, hill-shaped and U-shaped, it follows from Lemma 11 that q1t /q2t is decreasing. Claim 2—If Si has a negative bias, then q1t /q3t is decreasing at every t. Since Sj has a positive bias and Si has a negative bias, by Lemma 11, q1t /q3t is decreasing at every t. Claim 3—If Sj has a positive bias and there exists a period T such that for all t > T atj = buy, then q1t /q3t is decreasing at every t > T . Since Sj has a positive bias and S does not buy at any date, by Lemma 11, q1t /q3t must be decreasing at every t > T . Before stating the next claim, consider ϕml defined in (16). If both Si and Sj have positive biases, ϕ13 < 1. Thus, if Si has a positive bias, there exists an integer M such that (21)
(φml )M < ε if Sj has a positive bias and M
γ + μ Pr(Si |Vl ) (ii) < ε γ + μ Pr(Si |Vm ) (i)
Fix any such M. Then there also exists δ ∈ (0 ε) such that (22)
δ(ϕ12 )M < ε
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HERDING AND CONTRARIAN BEHAVIOR
Claim 4—If Sj has a zero bias, then there exists a history H τ such that q /q2τ < δ and q1τ /q3τ = 1. Since q11 /q31 = 1, it follows that at date 1, Sj holds. By recursion, it follows that at every history H t = {a1 at−1 }, we have q1t /q3t = 1 and the claim follows from Claim 1. Claim 5—If both Si and Sj have positive biases, then there exists a history H τ such that q1τ /q2τ < δ and q1τ /q3τ < x, where δ satisfies (21) and τ 1
x≡
[Pr(Si |V3 ) − Pr(Si |V2 )] + 2ε[Pr(Si |V3 ) − Pr(Si |V1 )] [Pr(Si |V1 ) − Pr(Si |V2 )]
Suppose not. Then by Claims 1 and 3, there exists date τ such that q1τ /q2τ < δ and aτj = buy. Since S also does not buy at H τ , it follows that only Si buys at τ. Then E[V |Si H τ ] > E[V |H τ ]. By Proposition 2, this implies [Pr(Si |V3 ) − Pr(Si |V2 )] + +
q1τ [Pr(Si |V2 ) − Pr(Si |V1 )] q3τ
2q1τ [Pr(Si |V3 ) − Pr(Si |V1 )] > 0 q2τ
Since q1τ /q2τ < δ < ε and Si is pU-shaped, we can rearrange the above expression to show that q1τ ≤ q3τ <
2q1τ [Pr(Si |V3 ) − Pr(Si |V1 )] q2τ Pr(Si |V1 ) − Pr(Si |V2 )
[Pr(Si |V3 ) − Pr(Si |V2 )] +
[Pr(Si |V3 ) − Pr(Si |V2 )] + 2ε[Pr(Si |V3 ) − Pr(Si |V1 )] = x [Pr(Si |V1 ) − Pr(Si |V2 )]
Claim 6—If Si has a positive bias, then there exists a history H t such that q1t /qlt < ε for any l = 2 3 Fix any history H τ = (a1 aτ−1 ) such that q1τ /q2τ < δ and q1τ /q3τ = 1 if Sj has a zero bias, and q1τ /q2τ < δ and q1τ /q3τ < x if Sj has a positive bias (by Claims 4 and 5 such histories exist). Next, consider a history H t that consists of path H τ followed by M periods of buys. Thus t = τ + M and H t = {hτ a1 aM }, where for any m ≤ M, am = buy. Since a buy must be either from Sj or Si or both, it then follows from the definitions of qt
qτ
ϕ12 in (16), from (22), and from q1τ /q2τ < δ that q1t ≤ q1τ (ϕ12 )M < δ(ϕ12 )M < ε. 2 2 To show that q1τ /q3τ < ε, consider the two cases of Sj having a zero bias and Sj having a positive bias separately. In the latter case, we have q1τ /q3τ < x. Then, by (i) in (21), we have
q1t q3t
<
q1τ q3τ
(ϕ13 )M < x(ϕ13 )M < ε In the former
= aτ+m . Hence, case, since q1τ /q3τ = 1, it must be that am = at+1 i j
q1t q3t
< 1. Re-
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A. PARK AND H. SABOURIAN
cursively, it then follows that am = at+1 = aτ+m for all m ≤ m. Thus, by (ii) j i in (21),
q1t q3t
<
q1τ q3τ
γ+μ Pr(Si |Vl ) M ( γ+μ ) < ε Pr(Si |Vm )
Since the initial choice of ε was arbitrary, (3) follows from Claims 1, 2, and 5. A.4. Proof of Proposition 4 (a) At H t , buy herding occurs if and only if E[V |S H t ] − askt > 0 and E[V |H t ] − E[V ] > 0. Thus, to demonstrate the existence of the function s, we need to characterize the expressions E[V |S H t ] − askt and E[V |H t ] − E[V ] for different values of b and s. Let βi = Pr(buy|Vi ) and σi = Pr(sale|V3 ) at every buy herding history (these probabilities are always the same at every history at which S buy herds). Note that, by Lemma 9, E[V |S H t ] − askt has the same sign as
(23)
β1 β3
b
σ1 σ3
s q2r q1r [β1 Pr(S|V2 ) − β2 Pr(S|V1 )]
+ q3r q2r [β2 Pr(S|V3 ) − β3 Pr(S|V2 )]
b s β1 σ1 +2 q3r q1r [β1 Pr(S|V3 ) − β3 Pr(S|V1 )] β2 σ2 Also, by MLRP and Lemma 5(iv), we have (24)
β1 < β2 < β3
and
σ3 < σ 2 < σ 1
Since by Proposition 2, S must have an nU-shaped c.s.d., it then follows that (25)
β1 Pr(S|V2 ) − β2 Pr(S|V1 ) < 0
β1 Pr(S|V3 ) − β3 Pr(S|V1 ) < 0
β2 Pr(S|V3 ) − β3 Pr(S|V2 ) > 0
(The last inequality in (25) follows from the first two and from (11) being positive at H r .) Thus, the first and the third terms in (23) are negative, and the second is positive. Hence it follows from (24) that the expression in (23) satisfies the following three properties: (i) it increases in b, (ii) it decreases in s, and (iii) for any b, it is negative for sufficiently large s. By (24), the expression E(V |H t )− E(V ) must also satisfy (i)–(iii) (note that q3t /q1t is increasing in b and decreasing in s). Since (23) and E(V |H t ) − E(V ) are both increasing in b and since, by assumption, there is buy herding at H r , it must be that for any b, both (23) and E(V |H t ) − E(V ) are positive when s = 0. Thus, it follows from (ii) and (iii) that for any b, there exists an integer s > 1 such that both (23) and E(V |H t ) − E(V ) are positive for any integer s < s and either (23) or E(V |H t ) − E(V ) is nonpositive for any integer s ≥ s.
HERDING AND CONTRARIAN BEHAVIOR
1021
To complete the proof of this part, we need to show that s is increasing in b. To show this, suppose otherwise; then there exist b and b such that b < b and s > s , where s and s are, respectively, the critical values of sales corresponding to b and b described in the previous paragraph. Now since s > s , it follows that both (23) and E(V |H t ) − E(V ) are positive if b = b and s = s . But since both (23) and E(V |H t ) − E(V ) are increasing in b, we must then have that both (23) and E(V |H t ) − E(V ) are positive if b = b and s = s . By the definition of s , this is a contradiction. (b) At H t , buy contrarianism occurs if and only if E[V |S H t ] − askt > 0 and E[V ] − E[V |H t ] > 0. By Lemma 9 E[V |S H t ] − askt has the same sign as (26)
q2r q1r [β1 Pr(S|V2 ) − β2 Pr(S|V1 )]
b s β3 σ3 + q3r q2r [β2 Pr(S|V3 ) − β3 Pr(S|V2 )] β1 σ1
b s σ3 β3 q3r q1r [β1 Pr(S|V3 ) − β3 Pr(S|V1 )] +2 β2 σ2
Also, with buy contrarianism, S must have an nhill-shaped c.s.d. and, therefore, β1 Pr(S|V2 ) − β2 Pr(S|V1 ) > 0 β1 Pr(S|V3 ) − β3 Pr(S|V1 ) < 0, and β2 Pr(S|V3 ) − β3 Pr(S|V2 ) < 0 Thus, the second and the third terms in (26) are negative, and the first is positive. Hence, by (24), the expression in (26) satisfies the following properties: (i) it increases in s, (ii) it decreases in b, and (iii) for each s, it is negative for sufficiently large b. The expression E(V ) − E(V |H t ) also satisfies the same three properties. The existence of the function b is now analogous to that for part (a), with reversed roles for buys and sales. A.5. Proof of Proposition 5 We show the proof for buy herding; the proof for buy contrarianism is analogous. (a) In the proof of Proposition 4, we have shown for the case of buy herding that if the history following H r consists only of buys, then type S herds at any point during that history. What remains to be shown is that for an arbitrary number of buys after herding has started, the price will approach V3 . Ob qt serve that E[V |H t ] = i Vi qit = q3t ( q2t V2 + V3 ). Also, q2t /q3t is arbitrarily small 3 at any history H t that includes a sufficiently large number of buys as outlined following conditions (5) and (6). Consequently, for every ε > 0, there exists a history H t consisting of H r followed by sufficiently many buys such that E[V |H t ] > V3 − ε
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A. PARK AND H. SABOURIAN
(b) Since the assumptions of Theorem 2 that ensure buy herding hold, S2 must be selling initially and also μ < μsbh ≤ μ2 (S3 ). The latter implies that there exists η > 0 such that (27)
[βt2 Pr(S2 |V3 ) − βt3 Pr(S2 |V2 )] > η for every t
By MLRP, type S1 does not buy at any history. Therefore, for any history Pr(S3 |V1 ) H r consisting only of r − 1 buys, it must be that q1r /q3r ≤ (max{ γ+μ , γ+μ Pr(S |V ) 3
γ+μ(1−Pr(S1 |V1 )) r−1 γ+μ(1−Pr(S1 |V3 ))
3
) . Also, by MLRP, S3 is strictly increasing and S1 is strictly decreasing. Thus, there must exist r > 1 such that q1r /q3r < η/2. Fix one such r. Then it follows from (27) that (28)
[βr2 Pr(S2 |V3 ) − βr3 Pr(S2 |V2 )] +
q1r r [β Pr(S2 |V2 ) − βr2 Pr(S2 |V1 )] > η/2 q3r 1
Next, fix any ε > 0. Note that there exists δ > 0 such that if q21 > 1 − δ, then askr = E[V |H r buy] = q2r V2 + q3r V3 ∈ (V2 V2 + ε) and (29)
2
q1r r [β Pr(S|V1 ) − βr1 Pr(S|V3 )] < η/2 q2r 3
Fix any such δ. Then it follows from (28) and (29) that q2r q3r [βr2 Pr(S2 |V3 ) − βr3 Pr(S2 |V2 )] + q1r q2r [βr1 Pr(S2 |V2 ) − βr2 Pr(S2 |V1 )] + 2q1r q3r [βr1 Pr(S2 |V3 ) − βr3 Pr(S2 |V1 )] > 0 Since S2 sells initially and q1r /q3r < η/2 < 1, it follows from the last inequality that S2 is buy herding at H r at an ask price that belongs to the interval (V2 V2 + ε). Next, as shown in part (a), there must also exist a history H t with t = r + b following H r such that there is buy herding at any history H τ , r ≤ τ ≤ t, and E[V |H t ] > V3 − ε. A.6. Proof of Proposition 6 We shall prove the two results for the case of buy herding; the proof for the buy contrarian case is analogous and will be omitted. PROOF OF PART I(A) OF PROPOSITION 6: Let βi and σi be, respectively, the probability of a buy and the probability of a sale in the transparent world at any date τ = r r + b + s. Also, let βio and σio be the analogous probabilities
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HERDING AND CONTRARIAN BEHAVIOR
in the opaque world. Then t t E[V |H t ] − Eo [V |H t ] = V {(q2t − q2o ) + 2(q3t − q3o )}
=V q
r 2
s βb2o σ2o βb σ s − 2 2 s qir βbi σis qir βbio σio
+ 2q3r
i
i
s βb3o σ3o βb σ s − 3 3 s qir βbi σis qir βbio σio i
i
Therefore, E[V |H t ] − Eo [V |H t ] has the same sign as (30)
q2r q1r [(β2 β1o )b (σ2 σ1o )s − (β2o β1 )b (σ2o σ1 )s ] + q3r q2r [(β3 β2o )b (σ3 σ2n )s − (β3o β2 )b (σ3o σ2 )s ] + 2q3r q1r [(β3 β1o )b (σ3 σ1o )s − (β3o β1 )b (σ3o σ1 )s ]
Suppose that S buy herds at H r . Then, by Lemma 9, we have (31)
q2r q1r [β1 Pr(S|V2 ) − β2 Pr(S|V1 )] + q3r q2r [β2 Pr(S|V3 ) − β3 Pr(S|V2 )] + 2 q3r q1r [β1 Pr(S|V3 ) − β3 Pr(S|V1 )] > 0
By simple computation we also have (32)
β2 β1o − β2o β1 = μ[β1 Pr(S|V2 ) − β2 Pr(S|V1 )] β3 β1o − β3o β1 = μ[β1 Pr(S|V3 ) − β3 Pr(S|V1 )] β3 β2o − β3o β2 = μ[β2 Pr(S|V3 ) − β3 Pr(S|V2 )]
Therefore, it follows from (31) that (33)
q2r q1r [β2 β1o − β2o β1 ] + q3r q2r [β3 β2o − β3o β2 ] + 2q3r q1r [β3 β1o − β3o β1 ] > 0
To prove I(a) in Proposition 6, suppose that s = 0 (thus t = b). Then by expanding (30), it must be that E[V |H t ] − Eo [V |H t ] has the same sign as b−1 r r b−1−τ τ (34) q2 q1 (β2 β1o − β2o β1 ) (β2 β1o ) (β2o β1 ) τ=0
+ q3r q2r [(β3 β2o ) − (β3o β2 )]
b−1 (β3 β2o )b−1−τ (β3o β2 )τ τ=0
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A. PARK AND H. SABOURIAN
+ 2q q
r r 3 1
b−1 b−1−τ τ (β3 β1o − β3o β1 ) (β3 β1o ) (β3o β1 ) τ=0
Also, by MLRP, β3 > β2 > β1 and β3o > β2o > β1o . Therefore, (35)
b−1 b−1 (β3 β2o )b−1−τ (β3o β2 )τ > (β2 β1o )b−1−τ (β2o β1 )τ τ=0
(36)
τ=0
b−1 b−1 (β3 β2o )b−1−τ (β3o β2 )τ > (β3 β1o )b−1−τ (β3o β1 )τ τ=0
τ=0
Also, by (25) and (32), the first and the third terms in (34) are negative and the second is positive. Therefore, by (33), (35), and (36), E[V |H t ] − Eo [V |H t ] > 0 Q.E.D. for s = 0 This completes the proof of part I(a) of Proposition 6.46 PROOF OF PART I(B) OF PROPOSITION 6: Suppose that b = 0 and s = 1 (t = r + 1). Since S buys in the transparent world, E[V |S H r ] − bidr > 0. Simple computations analogous to the proof of Lemma 9 show that this is equivalent to (37)
q2r q1r [σ1 Pr(S|V2 ) − σ2 Pr(S|V1 )] + q3r q2r [σ2 Pr(S|V3 ) − σ3 Pr(S|V2 )] + 2q3r q1r [σ1 Pr(S|V3 ) − σ3 Pr(S|V1 )] > 0
Also, by the definition of σi and σi , we have (38)
σ3 σ2o − σ3o σ2 = −μ[σ2 Pr(S|V3 ) − σ3 Pr(S|V2 )] σ3 σ1o − σ3o σ1 = −μ[σ1 Pr(S|V3 ) − σ3 Pr(S|V1 )] σ2 σ1o − σ2o σ1 = −μ[σ1 Pr(S|V2 ) − σ2 Pr(S|V1 )]
Therefore, (37) is equivalent to (39)
q2r q1r [σ2 σ1o − σ2o σ1 ] + q3r q2r [σ3 σ2o − σ3o σ2 ] + 2q3r q1r [σ3 σ1o − σ3o σ1 ] < 0
Since the left hand side of (39) is the same as the expression in (30) when b = 0 and s = 1, it follows that in this case, E[V |H t ] − Eo [V |H t ] < 0.47 This completes the proof of this part. Q.E.D. 46 Note that MLRP is assumed in the above proof so as to establish conditions (35), and (36). When b = 1 and s = 0, these conditions, and hence MLRP, are not needed, as E[V |H t ] − Eo [V |H t ] > 0 follows immediately from (30) and (33). 47 This claim also does not require the assumption of MLRP.
HERDING AND CONTRARIAN BEHAVIOR
1025
PROOF OF I(C) IN PROPOSITION 6: First, note that (30) can be written as (40)
(β2o β1 )b (β2 β1o )b s s (σ σ ) − (σ σ ) 2 1o 2o 1 (β3 β2o )b (β2o β1 )b (β3o β2 )b r r s s (σ3o σ2 ) + q3 q2 (σ3 σ2n ) − (β3 β2o )b b (β3 β1o )b r r (β3o β1 ) s s (σ3 σ1o ) − (σ3o σ1 ) + 2q3 q1 (β3 β2o )b (β3o β1 )b
q2r q1r
Fix s and let b → ∞. Then since by (25) and (32), β3 β2o > β3o β2 , we have that the second term in (40) converges to q3r q2r (σ3 σ2n )s as b → ∞. Also, since β3 > β2 > β1 , it follows that β2o β1 < β2o β3 and β3o β2 > β30 β1 . The former, together with (25) and (32), implies that the first term in (40) vanishes as b → ∞. The latter, together with (25) and (32), implies that β3 β2o > β3o β1 ; therefore, using (25) and (32) again, the last term in (40) also vanishes. Consequently, as b → ∞, the expression in (40) converges to q3r q2r (σ3 σ2n )s . Since (σ3 σ2n )s > 0 and E[V |H t ] − Eo [V |H t ] has the same sign as the expression in (40), the claim in I(c) of the proposition is established. Q.E.D. REFERENCES ALLEN, F., S. MORRIS, AND H. S. SHIN (2006): “Beauty Contests and Iterated Expectations in Asset Markets,” Review of Financial Studies, 19, 719–752. [980] AVERY, C., AND P. ZEMSKY (1998): “Multi-Dimensional Uncertainty and Herd Behavior in Financial Markets,” American Economic Review, 88, 724–748. [979,986,987,999,1006] BACCHETTA, P., AND E. V. WINCOOP (2006): “Can Information Heterogeneity Explain the Exchange Rate Determination Puzzle?” American Economic Review, 96, 552–576. [980] (2008): “Higher Order Expectations in Asset Pricing,” Journal of Money, Credit and Banking, 40, 837–866. [980] BANERJEE, A. (1992): “A Simple Model of Herd Behavior,” Quarterly Journal of Economics, 107, 797–817. [974,987,1010] BIKHCHANDANI, S., AND S. SUNIL (2000): “Herd Behavior in Financial Markets: A Review,” Staff Paper WP/00/48, IMF. [979] BIKHCHANDANI, S., D. HIRSHLEIFER, AND I. WELCH (1992): “Theory of Fads, Fashion, Custom, and Structural Change as Informational Cascades,” Journal of Political Economy, 100, 992–1026. [974,987] BRUNNERMEIER, M. K. (2001): Asset Pricing Under Asymmetric Information—Bubbles, Crashes, Technical Analysis and Herding. Oxford, England: Oxford University Press. [979,984] CHAMLEY, C. (2004): Rational Herds. Cambridge, United Kingdom: Cambridge University Press. [979,984] CHARI, V., AND P. J. KEHOE (2004): “Financial Crises as Herds: Overturning the Critiques,” Journal of Economic Theory, 119, 128–150. [980] CHORDIA, T., R. ROLL, AND A. SUBRAHMANYAM (2002): “Order Imbalance, Liquidity, and Market Returns,” Journal of Financial Economics, 65, 111–130. [984] CIPRIANI, M., AND A. GUARINO (2005): “Herd Behavior in a Laboratory Financial Market,” American Economic Review, 95, 1427–1443. [986]
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(2008): “Herd Behavior and Contagion in Financial Markets,” The B.E. Journal of Theoretical Economics (Contributions), 8, Article 24. [980] DASGUPTA, A., AND A. PRAT (2008): “Information Aggregation in Financial Markets With Career Concerns,” Journal of Economic Theory, 143, 83–113. [980] DE BONDT, W. F. M., AND R. THALER (1985): “Does the Stock Market Overreact?” The Journal of Finance, 40, 793–805. [986] DREHMANN, M., J. OECHSSLER, AND A. ROIDER (2005): “Herding and Contrarian Behavior in Financial Markets: An Internet Experiment,” American Economic Review, 95, 1403–1426. [986] EASLEY, D., AND M. O’HARA (1992): “Time and the Process of Security Price Adjustment,” Journal of Finance, 47, 577–605. [976,1007] GLOSTEN, L., AND P. MILGROM (1985): “Bid, Ask and Transaction Prices in a Specialist Market With Heterogenously Informed Traders,” Journal of Financial Economics, 14, 71–100. [975,980, 983,1002] GOLDSTEIN, I., E. OZDENOREN, AND K. YUAN (2010): “Trading Frenzy and Its Impact on Real Investment,” Working Paper, The Wharton School, University of Pennsylvania. [980] HE, H., AND J. WANG (1995): “Differential Information and Dynamic Behavior of Stock Trading Volume,” The Review of Financial Studies, 8, 919–972. [980] JEGADEESH, N. (1990): “Evidence of Predictable Behavior of Security Returns,” The Journal of Finance, 45, 881–898. [986] JEGADEESH, N., AND S. TITMAN (1993): “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” The Journal of Finance, 48, 65–91. [986] (2001): “Profitability of Momentum Strategies: An Evaluation of Alternative Explanations,” The Journal of Finance, 56, 699–720. [986] LEE, I. H. (1998): “Market Crashes and Informational Avalanches,” Review of Economic Studies, 65, 741–759. [979] LEHMANN, B. N. (1990): “Fads, Martingales, and Market Efficiency,” The Quarterly Journal of Economics, 105, 1–28. [986] MILGROM, P., AND N. STOKEY (1982): “Information, Trade and Common Knowledge,” Journal of Economic Theory, 26, 17–27. [981] OZDENOREN, E., AND K. YUAN (2007): “Feedback Effects and Asset Prices,” Journal of Finance, 6, 1939–1975. [980] PARK, A., AND H. SABOURIAN (2011): “Supplement to ‘Herding and Contrarian Behavior in Financial Markets’,” Econometrica Supplemental Material, 79, http://www.econometricsociety. org/ecta/Supmat/8602_proofs.pdf. [979] ROMANO, M. G., AND H. SABOURIAN (2008): “Herd Behaviour in Financial Markets With Continuous Investment Decisions,” Unpublished Manuscript, University of Cambridge. [994] SADKA, R. (2006): “Momentum and Post-Earnings-Announcement Drift Anomalies: The Role of Liquidity Risk,” Journal of Financial Economics, 80, 309–349. [986] VIVES, X. (2008): Information and Learning in Markets. Princeton, NJ: Princeton University Press. [979,984,985]
Dept. of Economics, University of Toronto, Toronto, Ontario M5S 3G7, Canada;
[email protected] and Faculty of Economics, Cambridge University, Sidgwick Avenue, Cambridge CB3 9DD, United Kingdom;
[email protected]. Manuscript received May, 2009; final revision received October, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1027–1068
RISK SHARING IN PRIVATE INFORMATION MODELS WITH ASSET ACCUMULATION: EXPLAINING THE EXCESS SMOOTHNESS OF CONSUMPTION BY ORAZIO P. ATTANASIO AND NICOLA PAVONI1 We study testable implications for the dynamics of consumption and income of models in which first-best allocations are not achieved because of a moral hazard problem with hidden saving. We show that in this environment, agents typically achieve more insurance than that obtained under self-insurance with a single asset. Consumption allocations exhibit “excess smoothness,” as found and defined by Campbell and Deaton (1989). We argue that excess smoothness, in this context, is equivalent to a violation of the intertemporal budget constraint considered in a Bewley economy (with a single asset). We also show parameterizations of our model in which we can obtain a closedform solution for the efficient insurance contract and where the excess smoothness parameter has a structural interpretation in terms of the severity of the moral hazard problem. We present tests of excess smoothness, applied to U.K. microdata and constructed using techniques proposed by Hansen, Roberds, and Sargent (1991) to test the intertemporal budget constraint. Our theoretical model leads us to interpret them as tests of the market structure faced by economic agents. We also construct a test based on the dynamics of the cross-sectional variances of consumption and income that is, in a precise sense, complementary to that based on Hansen, Roberds, and Sargent (1991) and that allows us to estimate the same structural parameter. The results we report are consistent with the implications of the model and are internally coherent. KEYWORDS: Consumption, private information, hidden savings, excess smoothness.
1. INTRODUCTION IN THIS PAPER, we study the intertemporal allocation of consumption in a model with moral hazard and hidden saving. We characterize some of the empirical implications of such a model and show how it can explain some well known puzzles in the consumption literature. The ability to smooth out income shocks and avoid large changes in consumption is an important determinant of individual households’ welfare. Resources can be moved over time and across states of the world using a variety of instruments, ranging from saving and borrowing to formal and informal insurance arrangements. The study of the intertemporal allocation of consumption is equivalent to establishing what instruments are available to individual households (that is, what are the components of the intertemporal budget constraint) and how they are used. 1
We are grateful for comments received from Marco Bassetto, Richard Blundell, Harald Uhlig, three anonymous referees, participants at the 2004 SED annual meeting, and several seminar audiences. Attanasio’s research was financed by the ESRC Professorial Fellowship Grant RES-051-27-0135. Nicola Pavoni thanks the Spanish Ministry of Science and Technology for Grant BEC2001-1653. © 2011 The Econometric Society
DOI: 10.3982/ECTA7063
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When thinking of the smoothing of individual shocks, two mechanisms come immediately to mind. On one hand, households can use a private technology to move resources over time, for instance, saving in “good times” and dis-saving in “bad times.” On the other, they can enter implicit or explicit insurance arrangements against idiosyncratic shocks. There might be links between these two mechanisms: with endogenously incomplete markets, the availability of specific instruments to move resources over time is likely to affect the type of insurance arrangements that can be sustained in equilibrium. We study consumption smoothing within a unifying framework and consider explicitly the links between private (anonymous) savings and insurance in a moral hazard model. We therefore contribute to and bring together two strands of the literature: that on consumption smoothing and that on endogenously incomplete markets. In particular, we show how some empirical results in the consumption literature can be interpreted, with the help of the model we construct, as providing evidence on partial insurance and on the relevance of specific imperfections. The permanent income/life-cycle (PILC) model has been, for a long time, the workhorse to study individual consumption smoothing and the intertemporal allocation of resources. During the 1980s and 1990s, a number of papers, starting with Campbell and Deaton (1989), pointed to the fact that consumption seems “too smooth” to be consistent with the model’s predictions, in that consumption does not react sufficiently to innovation to the permanent component of income. This evidence, derived from aggregate data, was interpreted as a failure of the PILC model. In such a model, allocations are determined by a sequence of Euler equations and an intertemporal budget constraint (IBC), often characterized by the presence of a single asset with a fixed rate of return. We refer to the latter as the risk-free IBC. Campbell and Deaton (1989), as other papers in this literature (including West (1988) and Gali (1991)), took the risk-free IBC (and in particular, the instruments available to an individual to move resources over time) as given and therefore interpreted the evidence of “excess smoothness” as a failure of some of the Euler equations. Indeed, Campbell and Deaton (1989, p. 372) concluded their paper by saying, “Whatever it is that causes changes in consumption to be correlated with lagged changes in income whether [. . . ] the marginal utility of consumption depends on other variables besides consumption, or that consumers are liquidity constrained or that consumers adjust slowly through inertia or habit formation, the same failure is responsible for the smoothness of consumption . . . .” All these explanations, however, maintain a risk-free IBC in which individuals move resources over time through a single asset with a fixed interest rate. We propose a model in which excess smoothness is related to a violation of the risk-free IBC which arises because the standard model neglects trades in state-contingent assets (or statecontingent transfers) that give individuals more insurance than they would be
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able to get by self-insurance and saving. In our model, the amount of risk sharing enjoyed by individuals is determined endogenously and depends on the severity of the moral hazard problem. As we hint above, an IBC defines the structure of the market to which individuals have access. Therefore, a violation of a specific IBC can be interpreted as a violation of a given market structure. Given that our approach focuses on violation of a risk-free IBC, we use a test of an IBC proposed by Hansen, Roberds, and Sargent (1991; henceforth HRS). The HRS test is particularly appropriate in our context as it can be explicitly interpreted as a test of a market structure. Moreover, because of the way it is formulated, not only does it encompass other tests (such as the Campbell and Deaton (1989) or West (1988)), but it is also robust to informational advantages of the agents relative to the econometrician. The cost of applying the HRS test is that it requires the Euler equation to hold. However, this is consistent with the model we construct and, more importantly, is an empirically testable proposition. When framed within the context of endogenously incomplete markets, it seems intuitive that a model that predicts some additional insurance relative to the amount of (self-) insurance one observes in a Bewley model would predict that consumption reacts less to shocks than is predicted by the Bewley model. To make such a statement precise, however, is not a trivial exercise. In what follows, in addition to such a formal statement, we also obtain, for some versions of our model, closed-form solutions for the optimal intertemporal allocations, and we map the coefficient of excess smoothness of the consumption literature and of the HRS test into a structural parameter of our model that reflects the severity of the moral hazard problem. This paper, therefore, makes two main contributions. The first is of a theoretical nature: we show how a model with moral hazard and hidden saving, by providing additional insurance relative to a Bewley economy, generates what in the consumption literature has been called the excess smoothness of consumption. In addition, we construct specific examples in which we can derive a closed-form solution and relate directly the excess-smoothness coefficients to a structural parameter of our model. While the examples are surely special, we stress that, in equilibrium, they deliver an income process that coincides with the process typically used in the permanent income hypothesis (PIH) literature. Moreover, the examples give an intuition that can be generalized to a larger class of models. Our theoretical result allows us to interpret in a specific fashion the empirical findings we obtain. The second contribution is empirical: we adapt the HRS test to microdata and apply it on a time series of cross sections from the United Kingdom. This evidence constitutes one of the first examples of an excess smoothness test performed on microdata. We also complement the HRS test with a test based on movements in the cross-sectional variance of consumption. Because of the nature of our data, we are forced to use the HRS test on time series of cross sections and, therefore, focus on risk sharing across cohorts. The variance test complements the HRS test because it focuses on insurance within cohorts.
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Our empirical results, obtained from U.K. time series of cross sections, are remarkably in line with the predictions of the model we construct. The rest of the paper is organized as follows. In Section 2, we present the building blocks of our model, provide the equilibrium definition, and discuss alternative market structures. In Section 3, we characterize the equilibrium allocation for our model and present some examples that yield useful closed-form solutions. In Section 4, we discuss the empirical implications of the equilibria we considered in Section 3, and present our tests and the results we obtain. Section 5 concludes. The Appendices contain the proofs of the results stated in the text: Appendices B and C comprise the Supplemental Material (Attanasio and Pavoni (2011)). 2. MODEL 2.1. Tastes and Technology Consider an economy consisting of a large number of agents that are ex ante identical. Each agent lives T ≤ ∞ periods and is endowed with a private stochastic production technology which takes the form (neglecting individual indices for notational ease) yt = f (θt et ) That is, the individual income yt ∈ Y ⊂ is determined by the agent’s effort level et ∈ E ⊂ and the shock θt ∈ Θ ⊂ . The history of income up to period t is denoted by y t = (y1 yt ), while the history of shocks is θt = (θ1 θt ). Let Φ(θt+1 |θt ) be the conditional probability of θt+1 conditioned on θt ∈ Θt . The component θt can be interpreted as the agent’s skill level at date t. At this stage, we do not impose any specific structure on the time series properties of θt , but we assume that θt are independent and identically distributed (i.i.d.) across individuals. In each period, the effort et is chosen after observing θt . The function f : Θ × E → Y is assumed to be continuous and increasing in both arguments. Both the effort e and the shocks θ are assumed to be private information, giving rise to moral hazard problems, while yt is publicly observable. Agents are born with no wealth, have von Neumann–Morgenstern preferences, and rank deterministic sequences according to T
δt−1 u(ct et )
t=1
with ct ∈ C and δ ∈ (0 1). We assume uto be real valued, continuous, strictly concave, smooth, strictly increasing in ct , and strictly decreasing in et . Notice that, given a plan for effort levels, there is a deterministic and one-to-one mapping between histories of the private shocks θt and income y t . Therefore, we
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can consider θt alone. Let μt be the probability measure on Θt and assume that the law of large numbers applies, so that μt (A) is also the fraction of agents with histories θt ∈ A at time t. Since θt are unobservable, we make use of the revelation principle and define a reporting strategy σ = {σ t }Tt=1 as a sequence of θt -measurable functions such that σ t : Θt → Θ and σ t (θt ) = θˆ t for some θˆ t ∈ Θ. A truthful reporting strategy σ ∗ is such that σ ∗t (θt ) = θt almost surely (a.s.) for all θt . Let Σ be the set of all possible reporting strategies. A reporting strategy essentially generates publicly observable histories according to ht = σ(θt ) = (σ1 (θ1 ) σt (θt )), with ht = θt when σ = σ ∗ . An allocation (e c y) consists of a triplet {et ct yt }Tt=1 of θt -measurable functions for effort, consumption, and income growths (production) such that they are “technically” attainable: Ω = (e c y) : ∀t ≥ 1 θt et (θt ) ∈ E ct (θt ) ∈ C and yt (θt ) = f (θt et (θt )) The idea behind this notation is that incentive compatibility will guarantee that the agent announces truthfully his endowments (i.e., uses σ ∗ ) so that, in equilibrium, private histories are public information. For simplicity, we disregard aggregate shocks and we do not allow for productive assets such as capital. In the baseline model, we assume the availability of a constant return technology that allows q ∈ (0 1) units of consumption at time t to be transformed into one unit of consumption at time t + 1 or vice versa. Equivalently, we are assuming a small open economy or a village economy where a money lender has access to an external market with an exoge1 can be interpreted as the time nously given interest rate r. The number q = 1+r constant price of a one-period bond in the credit market. In this case, absent aggregate shocks, thanks to the law of large numbers, the feasibility condition is constituted to be a unique inequality: (1)
T Θt
t=1
qt−1 ct (θt ) dμt (θt ) ≤
T Θt
qt−1 yt (θt ) dμt (θt )
t=1
Although we present our results under the assumption of a small open economy with a constant interest rate, in the Appendices we show that the same results can be derived with a time-varying interest rate and even for the case of a closed economy.2 2 For a similar model in a small open economy, see Abraham and Pavoni (2004, 2005, 2008). For another similar analysis in a closed economy with capital, see Golosov and Tsyvinski (2007).
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2.2. Equilibrium and Market Arrangements Having described agents’ tastes and the technological environment they face, to characterize intertemporal allocations we need to specify the market arrangements in which they operate and define the relevant equilibrium concepts. Our economy is characterized by private information, as we assume that effort and skill level are not observable. In addition to the moral hazard problem, consumption is also not observable (and/or contractable) and agents have hidden access to an anonymous credit market where they trade risk-free bonds at price q. The agents do not have private access to any other anonymous asset markets.3 The market structure we adopt is an open economy version of that proposed in Golosov and Tsyvinski (2007). In addition to the anonymous (secondary) market for assets, there is a (primary) insurance market where a continuum of identical firms offer exclusive contracts to agents. All insurance firms are owned equally by all agents. At the beginning of period 1, each firm signs a contract (e c y) with a continuum of agents in the economy that is binding for both parties. Firms operate in a competitive market and agents sign a contract with the firm that promises the highest ex ante expected discounted utility. After the contract is signed, each agent chooses a reporting strategy σ, supplies effort levels so as to generate, at each node, the agreed income level y t (σ(θt )), and receives ct (σ(θt )) units of consumption. Firms can borrow and lend at the ongoing interest rate, which they take as given. We define expected utility from reporting strategy σ ∈ Σ, given the allocation (e c y) ∈ Ω as T t−1 δ u(ct et )(e c y) σ E t=1
:=
T t=1
u ct (σ(θt )) g θt yt (σ(θt )) dμt (θt )
δt−1 Θt
where g(y θ) represents the effort level needed to generate y when the shock is θ, that is, g is the inverse of f with respect to e keeping θ fixed. Since y is observable, the misreporting agent must adjust his/her effort level so that the lie is not detected. Let b := {bt+1 }Tt=1 be a plan of risk-free asset holding, where bt is a θt−1 measurable function with b1 = 0 and limt→T qt−1 bt+1 (θt ) ≥ 0. Given the price 3
The assumption that in the anonymous asset market only the risk-free asset is traded is done without loss of generality in our environment, as no other Arrow security would be traded (see Golosov and Tsyvinski (2007)).
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of the bond q, the typical firm, in equilibrium, solves the problem (2)
(3)
max E
b(ecy)∈Ω
T t=1
δt−1 u(ct et )(e c y) σ ∗
subject to E
T
q
t−1
∗ (ct − yt )(e c y) σ ≤ 0
t=1
with the incentive compatibility constraint (4)
T t−1 ∗ E δ u(ct et )(e c y) σ t=1
≥E
T
δ
t−1
u(cˆ eˆ )(e c y) σ σ t
σ t
for all σ ∈ Σ
t=1
where the deviation for consumption cˆσ must be such that the new path of consumption can be replicated by the use of a risk-free bond. More precisely, given (e c y b), for each σ, a deviation cˆσ is admissible if there is a plan of bond holdings bˆ σ such that for all t and a.s. for all histories θt , cˆ σt (σ t (θt )) + qbˆ σt+1 (σ t (θt )) − bˆ σt (σ t−1 (θt−1 )) = ct (σ t (θt )) + qbt+1 (σ t (θt )) − bt (σ t−1 (θt−1 )) and limt→T qt−1 bˆ σt+1 (σ t (θt )) ≥ 0. DEFINITION 1: Given q, an equilibrium for the economy is an allocation (e∗ c ∗ y ∗ ) and bond trades b∗ such that the following statements hold: (i) Firms choose (e∗ c ∗ y ∗ b∗ ), solving problem (2)–(4) (taking q as given). (ii) Agents choose their reporting strategy, effort levels, consumption, and asset trades optimally as described above, given the contract (and q). (iii) The intertemporal aggregate feasibility constraint (1) holds. It is straightforward to see that the incentive constraint (4) (considered at σ ∗ ) implies that the equilibrium allocation must satisfy the Euler equation (5)
u c (ct∗ e∗t ) =
δ ∗ Et [u c (ct+1 e∗t+1 )] q
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where the marginal utilities are evaluated at the equilibrium values dictated by (e∗ c ∗ y ∗ ) and Et [·] is the conditional expectation operator on future histories given θt .4 2.3. Two Useful Extreme Cases With an eye to our empirical strategy, it is useful to compare the allocations in our imperfect information model with those that obtain in two well known alternative and extreme cases: a full information setting and a situation where intertemporal trades are exogenously restricted to the risk-free bond. Both situations can be obtained, under certain parameter settings, as special cases of our model. Moreover, both can be characterized by a different intertemporal budget constraint. Full Information In the model where there is no private information problem, in equilibrium, the typical firm offers exclusive contracts solving problem (2) and (3) alone. That is, the firm will solve the same problem as before with the crucial exception of condition (4). Since in this case we are in a complete market setting, we can apply the well known welfare theorems which imply that the equilibrium allocation solves the problem of a planner aiming to maximize the representative agent’s utility subject to the feasibility constraint, condition (1). Note that, as in our setup, with full information the equilibrium allocation satisfies the Euler equation (5). In fact, the equality between present and future discounted marginal utility is true state by state, u c (ct et ) = qδ u c (ct+1 et+1 ), and hence also in expected terms. The Bewley Economy (Self-Insurance) We define a Bewley economy (or self-insurance) as an economy where agents can move resources over time only by participating in a simple credit market with a constant one-period bond with price q. Recall that an asset holding plan b is made of a set of θt−1 -measurable function bt , t = 1 T , with b1 = 0. In this economy, a typical agent solves T δt−1 u(ct et ) max E b(ecy)∈Ω
t=1
subject to (6)
ct (θt ) + qbt+1 (θt ) ≤ bt (θt−1 ) + yt (θt )
4 This condition is the first-order equivalent of the incentive constraint that ensures that the agent is not willing to deviate in assets decisions alone, while contemplating telling the truth about shock histories θt .
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where b0 = 0. As usual, we rule out Ponzi games by requiring that limt→T qt−1 × bt (θt ) ≥ 0. Condition (6) is the budget constraint typically used in permanent income models when the agent has access only to a risk-free bond market. This problem can be seen as an extension of the permanent-income model studied by Bewley (1977), which allows for endogenous labor supply and nonstationary income. The main implication of the self-insurance model is the well known Euler equation (7)
u c (ct et ) =
δ Et [u c (ct+1 et+1 )] q
Another important necessary condition that individual intertemporal allocations have to satisfy in this model can be derived by repeatedly applying the budget constraint (6). Starting from any node θ¯ t−1 , t ≥ 1, with asset holding level bt (θ¯ t−1 ), the following net present value condition (NPVC) must be satisfied: (8)
T
qn−t (cn (θn ) − yn (θn )) ≤ bt (θ¯ t−1 )
n=t
a.s. for all histories θT emanating from node θ¯ t−1 . Given the income process and the price for the bond q, conditions (7) and (8) define consumption (even when a closed-form solution does not exist). It is interesting to compare NPVC (8) for t = 1 with the corresponding equation for the full information case, equation (1). In the latter case, the agent has available a wide array of state-contingent securities that are linked in an individual budget constraint that sums over time and across histories, as all trades can be made at time 1. In the Bewley economy, instead, the agent can only trade in a single asset. This restriction on trade requires that the net present value of consumption minus income equals zero a.s. for all histories θT . Notice that there is no expectation operator involved in condition (8): the intertemporal transfer technology implied by a risk-free asset does not allow for cross-subsidizations of consumption across income histories. The lesson we should retain from this section is that all allocations considered here satisfy the Euler equation: clearly such an equation is consistent with many stochastic processes for consumption. The key feature that distinguishes the three different equilibrium allocations we have considered is the intertemporal budget constraint, which defines the relevant market structure. For instance, the equilibrium allocation in our moral hazard model typically violates the NPVC (8) based on a single asset which is relevant for the permanent income model because it ignores the state-contingent payments the agents might receive from the insurance firm in our economy.
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3. CHARACTERIZING THE EQUILIBRIUM ALLOCATION In this section, we analyze the properties of the endogenously incomplete markets model we presented above. It is easy to show that the equilibrium allocation (e∗ c ∗ y ∗ ) and b∗ can be replicated by an incentive compatible plan of lump sum transfers τ∗ = {τ ∗t (θt )}Tt=1 that solves the firm’s problem T t−1 max E δ u(ct et ) τ(ecy)∈Ω
t=1
subject to (9)
ct (θt ) = yt (θt ) + τ t (θt )
the incentive constraint (4), and the intertemporal budget constraint T t−1 E q τt ≤ 0 t=1
It is clear from condition (9) that we study equilibrium allocations where agents do not trade intertemporally (b∗t ≡ 0). This is done without loss of generality since the firms and the agents face the same interest rate.5 Alternatively, τ could be chosen so that the transfer τt = τ t (θt ) represents the net trade on state-contingent assets the agent implements at each date t and node θt . In this case, the lump sum transfer solves E[τt ] = 0 and all the intertemporal transfer of resources are, in effect, made by the agents. In Appendix A, we show that, under some conditions, the equilibrium allocation (e∗ c ∗ y ∗ ) and b∗t ≡ 0 can be implemented using a transfer scheme τ∗ which is a function of income histories y t alone.6 This simplifies the analysis and allows us to describe the consumption allocation in terms of observables. At history y t , an agent with asset level bt faces the following budget constraint for all y t : ct + qbt+1 = yt + τ ∗t (y t ) + bt 3.1. Excess Smoothness and the Bewley Economy Allen (1985) and Cole and Kocherlakota (2001) presented asymmetric information models that give rise to intertemporal allocations that coincide with those that prevail in the Bewley economy we described in Section 2.3. In this section, we first present a specific version of our model where the Allen and 5 6
For example, see Fudenberg, Holmstrom, and Milgrom (1990). In particular, we assume that the optimal plan of consumption c is y t -measurable.
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Cole–Kocherlakota (ACK) result obtains. As stressed in Abraham and Pavoni (2004), to obtain the self-insurance result, a crucial restriction on the way effort is converted into output is needed. We then move on to relax this restriction. Within the more general case, we consider a specific parameterization of the income process that allows us to obtain a closed-form solution for the equilibrium transfers. While this example is useful because it gives very sharp predictions, some of the properties of the allocations we discuss generalize to the more general case and inform our empirical specification. Self-Insurance Let us assume agents’ preferences and production satisfy (10)
u(c e) = u(c − e) and
f (θ e) = θ + e
with Θ = (θmin θmax ) and E = (emin emax ). Obviously, in this environment the optimal plan of effort levels is indeterminate. As long as 0 ∈ E, we can hence set, without any loss of generality, e∗t ≡ 0. This normalization has two advantages. First, since et does not change with θt while f (θt et ) is strictly increasing in θt , all variations in θt will induce variations in yt , automatically guaranteeing the y t -measurability of c (see Appendix A). Second, with constant effort we can focus on the risk sharing dimension of the equilibrium allocation. This last argument also motivates the modelling choice for our closed-form solution below. PROPOSITION 1: Assume T < ∞, and that the utility function u and the production function f are as in (10). Then the equilibrium allocation coincides with self-insurance. The proof of the proposition is reported in Appendix A, where we show that incentive compatibility fully characterizes the equilibrium allocation. The main intuition of the result can be easily seen in a two-period model, which we solve backward. Consider the last period of the program. Using the budget constraint, given any history y T −1 and θT (and last period wealth bT ), the incentive constraint (for e∗T = 0) is u(cT∗ − e∗T ) = u(θT + τ ∗T (y T −1 θT ) + bT ) ≥ u(θT + τ ∗T (y T −1 θT + eˆ T ) + bT )
for all eˆ T ∈ E
Clearly, to be incentive compatible, the transfer scheme must be constant across yT s. Now consider the problem in period T − 1. For ease of exposition, let us assume that the transfer scheme is differentiable7 and write the 7
The formal proof does not assume differentiability. See Appendix A.
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agent’s first-order conditions with respect to eT −1 (evaluated at the optimum b∗T = b∗T −1 = e∗T = e∗T −1 = 0). A necessary condition for incentive compatibility is8 ∗ ∂τ ∗T −1 (y T −1 ) u (c ) ∂τ ∗T (y T ) = 0 +δ ET −1 ∗ T ∂yT −1 ∂yT −1 u (cT −1 ) The Euler equation—that is, the (local) incentive constraint for bond u (c ∗ ) holding—reads as ET −1 [ u (c∗ T ) ] = qδ , which implies T −1
(11)
∂τ ∗T −1 (y T −1 ) ∂τ ∗ (y T ) +q T = 0 ∂yT −1 ∂yT −1
Equation (11) states that the net present value of transfers must be constant across income histories. The intuition for the fact that the firm is unable to provide any consumption insurance on top of self-insurance is relatively simple in this case. First, at each t with equilibrium income yt∗ = θt , the agent can always deviate, locally, and generate any income level such that yt∗ − ε ≤ yˆt ≤ yt∗ + ε. Second, the perfect substitutability between consumption and effort in the utility function on one side, and between income and effort in production on the other side imply that such deviation has zero direct cost to the agent. Since this is true for all income levels, in a static setting, these two observations together imply that the firm will never be able to provide a transfer scheme that induces anything different from constant payments over income levels. In our model, this simple intuition extends to a general dynamic setting. Roughly speaking, free access to the credit market implies that the agent only cares about the present value of transfers (i.e., he/she does not care about the exact timing of deterministic transfer payments). Hence, a simple extension of the previous argument implies that the present value of transfers must be constant across income histories, as otherwise the agent would find it profitable to perform some local deviation on effort and engage in an appropriate bond plan. Now recall that the self-insurance allocation has two defining properties: first, it must satisfy the Euler equation; second, it must satisfy the intertemporal budget constraint with one bond, that is, the period zero net present value must be zero for all y T . Since the Euler equation is always satisfied here, the only way to obtain a different allocation is that the transfers scheme τ∗ permits violation of the agent’s period zero self-insurance intertemporal budget constraint for some history y T . The previous argument demonstrates that this cannot be the case; hence the only incentive compatible allocation coincides with self-insurance. This implies that the “relaxed-optimal” contract obtained 8
∂τ ∗ (y T )
Note that in the expression below, ∂yTT −1 has been taken out from the expectation operator as we saw that τ T is constant in yT shocks.
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by using the local (first-order-condition) version of the incentive constraint corresponds to the self-insurance allocation. Since this allocation is obviously globally incentive compatible, it must be the optimal one. The proof of Proposition 1, in Appendix A, never uses the time-series properties of θt , which can, therefore, be very general. Likewise, we do not require a constant q.9 Proposition 1 is, therefore, in some dimensions, a general result. Indeed, as we prove in Appendix B, it holds even in a more general model that allows for two types of income shocks, with different degrees of persistence. Proposition 1 and its proof, however, also make it clear that the coincidence between the equilibrium allocation and self-insurance is, in other dimensions, very fragile. Below, we present a full class of models with slightly more general u and f functions, where the equilibrium allocation coincides with selfinsurance only for a zero measure set of parameters.10 Excess Smoothness Consider now the generalization of agent preferences u(c e) = u(c − v(e)) with v increasing and convex. Since consumption and effort cost enter the utility function in a linear fashion, we eliminate the wealth effects. This simplifies the analysis and allows for closed-form solutions. Moreover, it is crucial for the self-insurance result we derived above. While we only consider interior solutions for e, we leave the function f unspecified. Again, the main intuition regarding the model can be gained by studying the last two periods. Let us start with the final period (T ). The first-order condition of the incentive constraint in this last period is (12)
1+
v (e∗ ) ∂τ ∗T (y T −1 yT ) = ∗ T ∂yT fe (eT θT )
Recall that in the ACK model we have v (e) = fe (e θ) = 1 for all e θ. Since risk sharing requires
∂τ ∗T (y T −1 yT ) ∂yT v (eT ) fe (eT θT )
< 0, no insurance is possible in that environ-
ment. However, in general, might be less than 1. In fact, it is easy to see that under some regularity conditions on f and v, for all θT , the optimal 9 Both Allen (1985) and Cole and Kocherlakota (2001) assumed i.i.d. shocks and constant q. We generalize Allen’s result, while our model does not nest, strictly speaking, Cole and Kocherlakota’s, as we do not impose exogenous and binding liquidity constraints. 10 Abraham and Pavoni (2004) also obtained similar results. They considered a more general class of models but did not derive sufficient conditions for the self-insurance result and did not obtain closed-form solutions.
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T) contract implements effort levels such that fe v(e(eT θ ≤ 1 (as long as this is techT) 11 nologically feasible). A strict inequality is compatible with some risk sharing. What is the intuition for this fact? If the firm’s aim is to make agents share risk, the key margin for an optimal contract is to guarantee that the agent does not shirk or, equivalently, that she does not reduce effort. When the agent shirks so that output is reduced by one unit, the gain she gets from this re duction in effort is equivalent to fve units of consumption. This is the right-hand side of (12). The left-hand side is the net consumption loss. When the marginal tax/transfer is negative, this loss will be less than 1 as the direct reduction of one unit of consumption is mitigated by the increase in net transfers. A small v reduces shirking returns, making it easier for the firm to satisfy the incentive fe compatibility and hence to provide insurance. The same intuition carries over to a multiperiod setting, where the generalized version of (11) takes the form
∂τ ∗T −1 (y T −1 ) v (e∗T −1 ) ∂τ ∗ (y T ) − 1 +q T = ∂yT −1 ∂yT −1 fe (θT −1 e∗T −1 ) 3.2. Closed Forms We now assume that income yt depends on exogenous shocks θt and effort et as (13)
yt = f (θt et ) = θt + a min{et 0} + b max{et 0}
with a ≥ 1 ≥ b. Notice that when a = b = 1, one obtains the linear specification used to obtain the self-insurance result. Preferences are as in the previous section12 : u(ct et ) = u(ct − et ) Moreover, we assume that θt follows an ARIMA(p) process (14)
θt − θt−1 = β(L)vt
where β(L) is a polynomial of order p in the lag operator L, invertible, and the innovation vt is i.i.d. Linearity of the autoregressive integrated moving average 11
Assume there is at least one e¯ ∈ E such that
v (eT )
fe (eT θT )
¯ v (e) ¯ fe (eθ)
≤ 1. Implementing an effort level where
> 1 is dominated by a lower effort, since by marginally reducing eT , for such θT , there is f v
d [ fθe ] ≥ 0 (i.e., the Spence–Mirrlees a production gain. In the static case, it can be shown that if de condition holds for the static version of our problem), the firm can provide the same ex post welfare to the agent by saving net transfer costs as it also improves on insurance. Details on the formal derivation are available upon request. 12 Equivalently, we could have assumed a fully linear production function, f (e θ) = e + θ and a kinked effort cost function, v(e) = a1 min{et 0} + b1 max{et 0}.
RISK SHARING IN PRIVATE INFORMATION MODELS
1041
(ARIMA) process helps us to find the closed-form solution. Since effort will be time constant, in equilibrium, θt = yt , and hence the income process will display the standard representation often used in the consumption literature.13 In Proposition 3, stated and proved in Appendix B, we show that if u is exponential (constant absolute risk aversion (CARA)), u(c − e) = − ρ1 exp{−ρ(c − e)}, and the shocks vt are normally distributed with zero mean and variance σv2 , we get the expression for consumption growth T −t ln(δ/q) ρ 2 1 (1 − q) ∗ ∗ n ∗ + σc + ct − ct−1 = (Et − Et−1 ) q yt ρ 2 a 1 − qT −t−1 n=0 where σc2 indicates the variance of consumption growth. For t ≤ T − p, the previous expression reduces to (see Appendix B for details) (15)
ct∗ =
ρ ln(δ/q) 1 + 2 [β(q)]2 σv2 + β(q)vt ρ 2a a
For a = 1, we are back to the self-insurance case where innovations to permanent income are fully reflected in consumption changes. For a > 1, we get some more risk sharing over and above self-insurance, with full insurance obtainable as a limit case for a → ∞. Proposition 5 in Appendix B, shows that a very similar (simpler) closed form can be obtained assuming quadratic u, δ = q, and differentiability of the transfer scheme.14 With CARA utility (as opposed to the model where u is quadratic), the presence of a precautionary saving motive implies that the equilibrium allocation displays increasing consumption. Notice, however, that in our economy, a > 1 permits both reduction of the cross-sectional dispersion of consumption and mitigation of the precautionary saving motive, hence the steepness of consumption (i.e., intertemporal dispersion). The model implies a very tight relationship between these two moments. 3.3. The Case With Isoelastic Utility We now discuss the case in which agents have isoelastic preferences. Full details are to be found in Appendix B. We assume the production function yt = θt et This model corresponds to a simple dynamic variant of the most standard version of the well known model of Mirrlees (1971), where y is labor income, θ 13 See, for example, Abowd and Card (1989), Meghir and Pistaferri (2004), and Blundell, Pistaferri, and Preston (2008). 14 In this case, no parametric assumptions are needed for the income process and consumption does not grow because a precautionary saving motive is absent. See Appendix B for details.
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O. P. ATTANASIO AND N. PAVONI
represents worker productivity, and e is hours of work. We assume that preferences take the CRRA (or isoelastic) form15 u(ct et ) =
)1−γ (ct · e−1/a t 1−γ
for et ≤ 1 and
)1−γ (ct · e−1/b t u(ct et ) = 1−γ 1 ln et a 1 u(ct et ) = ln ct − ln et b u(ct et ) = ln ct −
for et ≥ 1 for et ≤ 1 for
for γ > 1; and
et ≥ 1
for γ = 1
where a ≥ 1 ≥ b. As before, therefore, we assume that the marginal cost of effort changes discontinuously above a threshold level, here et = 1. Moreover, we assume that ln θt − ln θt−1 = β(L) ln vt where (with a small abuse in notation) we posit that ln vt is normally distributed with zero mean and variance σv2 . In this context, we can obtain a closed form for discounted net transfers that leads, for t ≤ T − p, to the permanent income formulation (16)
ln ct∗ =
ln(δ/q) γ 2 1 + σc + β(qλ) ln vt γ 2 a
where λ = exp{ ln γδ/q + 1−γ σc2 } and σc2 = a12 [β(qλ)]2 σv2 .16 Note that whenever u is 2 logarithmic (γ = 1), then qλ = δ (see Appendix B for details). When a = 1 and income shocks are a random walk, β(qλ) = 1 and one obtains the same expression as in the self-insurance model with zero wealth (e.g., Constantinides and Duffie (1996)). With nonzero wealth, the comparison with the Bewley model is more delicate. First, the coefficient that relates innovations to permanent income to consumption needs to be corrected for the wealth–income ratio, as (c ·e−1 )1−γ
t Again, we could have assumed, equivalently, u(ct et ) = t 1−γ and a modified Cobb– a Douglas production function of the form yt = θt et for et ≤ 1 and yt = θt ebt for et ≥ 1 with a ≥ 1 ≥ b. 16 More precisely, given a γ σv and the coefficients βj , j = 0 1 2 p, defining the persistence of the income processthe conditional variance of consumptionsolves the system of equations
15
p
ln δ/q 1 − γ 2 λ = exp + σc γ 2
and
σc =
βj qj λj σv
j=0
a
RISK SHARING IN PRIVATE INFORMATION MODELS
1043
pointed out by Banks, Blundell, and Brugiavini (2001) and Blundell, Pistaferri, and Preston (2008). Second, equation (16) does not necessarily hold and log consumption is not necessarily a martingale, and this might have implications for our empirical approach, which we discuss below. “Pure” Moral Hazard Finally, we can use equation (16) to draw attention to an important difference between our model and a model with observable assets. It is well known that in that case, when preferences are additive separable, the inverse of the marginal utility follows a martingale and, therefore, the Euler equation is violated (see Rogerson (1985) and Ligon (1998)). In that framework, it is easy to show that with isoelastic utility, consumption growth follows17 (17)
Et−1 ln ct∗ =
ln(δ/q) γ 2 − σc γ 2
The key difference between this expression and (16) is that, according to (17), consumption growth decreases with an increase in the variance of consumption and, therefore, income shocks. This prediction is inconsistent with the evidence in, for instance, Banks, Blundell, and Brugiavini (2001). 4. EMPIRICAL IMPLICATIONS OF THE MODEL The model we developed in the previous section has some important empirical implications. In it, while the Euler equation holds, the risk-free IBC is violated. This violation gives rise to what the literature has labelled the excess smoothness of consumption: consumption does not react enough to innovations in permanent income. We provide one of the first excess-smoothness tests performed on microdata and we give a structural interpretation to the excess-smoothness coefficient as reflecting the severity of moral hazard.18 In this section, we pursue these ideas by developing two different and complementary tests of the implications of our model on microdata. The first is an excess-smoothness test derived from a time series model of consumption and income, and based on the test of the risk-free IBC proposed by HRS. As mentioned above, testing the validity of an IBC is equivalent, in our context, to testing a market structure. The second test is based on the dynamics of consumption and income inequality within groups, as measured by variances of log consumption. As we discuss below, the two tests complement each other. As we study the extent to which income shocks are insured or are reflected in consumption, we need to use individual-level data. Unfortunately, however, 17
With CARA utility, the same expression holds true for consumption in levels. Nalewaik (2006) adapted the HRS test to U.S. microdata, but did not stress the relationship with excess smoothness or provides an excess sensitivity test. 18
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O. P. ATTANASIO AND N. PAVONI
longitudinal surveys that follow individuals over time and contain complete information on consumption are extremely rare. One of the most commonly used panels from the United States, the Panel Study of Income Dynamics (PSID), for instance, only contains information on food consumption. Other data sets, which contain complete information on consumption, have a very short longitudinal dimension (such as the Consumer Expenditure Survey (CEX) in the United States) or lack it completely (such as the Family Expenditure Survey (FES) in the United Kingdom). As a long time period is crucial to identify the time series properties of the variables of interest, this lack of longitudinal data is a problem for us. To overcome this difficulty, we use synthetic cohort data or pseudopanels, along the lines proposed by Deaton (1985) and Browning, Deaton, and Irish (1985). Our main data source is the U.K. FES from 1974 to 2002. The FES is a time series of repeated cross sections which is collected for the main purpose of computing the weights for the Consumer Price Index. Each survey consists of about 7,000 families contacted over 2-week periods throughout the year. As households are interviewed every week throughout the year, the FES data are used to construct quarterly time series. This allows us to exploit a relatively long time series horizon. We use data on households headed by individuals born in the 1930s, 1940s, 1950s, and 1960s to form pseudopanels for four yearof-birth cohorts. As we truncate the samples so as to have individuals aged between 25 and 60, the four cohorts form an unbalanced sample. The 1930s cohort is observed over later periods of its life cycle and exits before the end of our sample, while the opposite is true for the 1960s cohort. Apart from the year of birth, the other selection criterion we used for this study is marital status. As we want to study relatively homogeneous groups, we excluded from our sample unmarried individuals. We also excluded the self-employed. This data set, which has been used in many studies of consumption (see, for instance, Attanasio and Weber (1993)), probably constitutes the best quality microdata set with consumption information and that covering the longest time period. It contains detailed information on consumption, income, and various demographic and economic variables. We report results obtained using two different definitions of consumption. The first takes as consumption expenditure on nondurable items and services, in real terms and divided by the number of adult equivalents in the household (where for the latter we use the McClements definition of adult equivalents). The second also includes expenditure on durables. Within our model, the risk-free IBC is violated because it considers disposable income from labor and a single asset, and ignores state-contingent transfers. The latter can be interpreted as the returns to other assets with statecontingent returns or other income sources, such as government transfers. This implies that if one were to modify the definition of income to include all these state-contingent transfers, one would go back to a version of the PIH and to its implications. Therefore, one should not be able to detect excess smoothness
RISK SHARING IN PRIVATE INFORMATION MODELS
1045
with this changed definition of income. Intuitively, this suggests changing the definition of income in our empirical exercise: if one considers income definitions that include transfers that are conceived to insure individual shocks, one should observe less excess smoothness when applying our tests. This, however, is not a formal statement, as our results are silent about the decentralization of the constrained efficient allocations and, empirically, we do not observe any individual or cohort of individuals for the entire life cycle. However, in what follows, we apply our tests using different definitions of income: first gross earnings, then gross earnings plus public transfers, and finally net earnings and transfers. 4.1. The HRS Approach In constructing our first test, we follow HRS. They considered an income process yt which is one element of the information structure available to the consumer and assumed that it admits the representation (18)
(1 − L)yt := yt = β(L)wt
where wt is an n-dimensional vector of orthogonal covariance stationary random variables that represent the information available to the consumer and β(L) is a 1 × n vector of polynomials in the lag operator L. Notice that this specification for the income process encompasses both models considered in Section 3. Here we adopt only in part the HRS notation and adapt it to ours. In particular, without loss of generality, we start from a representation for the income process that has already been rotated so that its first component represents the innovation for the process that generates consumption. It is useful to decompose the right-hand side of equation (18) into its first component and the remaining ones: (19)
yt = β1 (L)w1t + β2 (L)w2t
The first result that HRS proved in their paper is that an intertemporal budget constraint has empirical content, in that it imposes testable restrictions on the time series behavior of income and consumption only if one has restrictions on the time series property of consumption, for example, that it follows a martingale. When this is not true, one can always find a time-series representation for consumption and disposable income that satisfies an IBC. In our model, a Euler equation is always satisfied because of the availability of the hidden asset, so that the HRS test has empirical content. Moreover, whether consumption satisfies a Euler equation can be checked empirically, which we do.19 19 A short digression is in order here. Evidence of the “excess sensitivity” of consumption to income, as reported by Campbell and Mankiw (1989) and others, is usually interpreted as a vio-
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O. P. ATTANASIO AND N. PAVONI
The martingale restriction on consumption implies that it can be represented by (20)
ct = πw1t
where deterministic trends have been removed from consumption and π is a scalar different from zero whenever insurance markets are not complete. The coefficient π represents the extent to which income news is reflected in consumption. Notice that equation (20) does not include lags. HRS showed that, given this structure, the net present value (NPV) condition implies some restrictions on the coefficients of equations (20) and (19). In particular, the intertemporal budget constraint implies that (21)
q) π = β1 (
(22)
β2 ( q) = 0
where q = q in the case in levels, while q = λq for the case in logs.20 HRS showed that restriction (21) is testable, while restriction (22) is not, in that there exist other representations for income that are observationally equivalent to (19) for which the restriction holds by construction. The alternative q) is equivalent to what Campbell and Deaton (1989) hypothesis that π < β1 ( and West (1988) defined as excess smoothness of consumption. The theoretical structure we have illustrated in the previous section provides a structural interpretation of the excess smoothness and HRS tests. Notice the similarity of equations (14) and (15) to equations (19) and (20).21 The null considered by HRS corresponds to the Bewley model we considered in Section 2.3, which also corresponds to a special case of our model (see Section 3.1). The moral hazard model we constructed generates a specific deviation from the q)/π = a. In our context, the extent of excess smoothness null: it implies β1 ( represents the severity of the incentive problem. lation of the orthogonality conditions implied by the Euler equation for consumption. Such evidence would constitute a problem for the use of the HRS test we are proposing. However, there is evidence that many of the violations reported in the literature are based on aggregate time series data and are explained by aggregation problems and/or the failure to control for the evolution of needs over the life cycle. See Attanasio and Weber (1993, 1995). Attanasio (2000) presented a critical discussion of these issues. Another reason that can explain income and consumption tracking each other is the nonseparability of consumption and leisure in the utility function, a point made early on by Heckman (1974), and on which Attanasio and Weber (1993, 1995) and Blundell, Browing, and Meghir (1994) provided some evidence. 20 HRS worked with quadratic utility and q = δ. As we saw above, similar relations can be obtained for CARA utility and for CRRA when working in logs. In these last two cases, we allow for q = δ. 21 Simply set π = a1 , w1t = vt , β1 (L) = β(L), and β2 (L) = 0.
RISK SHARING IN PRIVATE INFORMATION MODELS
1047
An important feature of the HRS approach is that the test of the NPV restriction does not require the econometrician to identify all information available to the consumer. Intuitively, the test uses two facts. First, under the null, the intertemporal budget constraint with a single asset must hold whatever is the information set available to the agent. Second, under the assumption that the agent has no coarser information than the econometrician, the validity of the Euler equation implies that consumption innovations reveal part of the information available to the agent. By following the HRS strategy, we test the intertemporal budget constraint along the dimensions of information identified via the Euler equation (which is not zero as long as insurance markets are not complete). The representation in equations (20) and (19) is derived under the assumption that preferences are time separable. A more general formulation of equation (20) is (23)
ct = πψ(L)w1t
where c in equation (23) represents total consumption expenditure, which enters the budget constraint, while utility is defined over the consumption services, which, in turn, are a function of current and, possibly, past expenditure. The polynomial in the lag operator ψ(L) reflects these nonseparabilities or other complications, such as i.i.d. taste shocks to the instantaneous utility ˜ = 1, where function. The nonseparabilities considered in HRS imply that ψ(δ) ˜δ = δ = q if utility is quadratic, while δ = 1 with CARA and constant relative ˜ = 1 imposes restrictions on risk aversion (CRRA) utility. The condition ψ(δ) the way lagged shocks enter the equation for consumption growth which imply q), as in equathat the NPV restriction, in this case, takes the form π = β1 ( tion (21).22 We will need this extension to interpret some of our results. We follow HRS and consider a time series representation of consumption and income as a function of two (unobservable) factors. As in HRS, this gives rise to an moving average (MA) representation of the form (24)
cht = αcc (L)vh1t yht = αyc (L)vh1t + αyy (L)vh2t
where we have added the subscript h to denote households, and we stress that the application will be on microdata and assume that the two vectors v are 22 Attanasio (2000, Section 3.6) discussed the case with preference shocks which leads to extended equations such as (23). HRS (in Section 4) and Alessie and Lusardi (1997) consided habit formation. HRS only consided the case with quadratic utility, while Alessie and Lusardi, within q) is ima simpler model, allowed for both quadratic and CARA utilities. The condition π = β1 ( mediate from their equations (3) and (9) for the quadratic and CARA utilities respectively. The CRRA utility case is similar as long as habit is multiplicative. Details are available upon request.
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O. P. ATTANASIO AND N. PAVONI
independent of each other and over time (we allow, however, correlation between vhit and vkit i = 1 2, h = k, to take into account aggregate shocks). The system (24), which is identified by a standard triangular assumption that consumption is only affected by the first factor, can be estimated by maximum likelihood, making some assumptions about the distribution of the relevant variables. As in HRS, it is straightforward to show that in the case of the intertemporal nonseparability implied by equation (23), the intertemporal budget constraint with a single asset can still be tested as a hypothesis on the coefficients of the system (24), provided that the process for consumption satisfies some restrictions we discuss below. In particular, if the IBC holds (and therefore there is no excess smoothness), the relevant restrictions to be tested are ˜ = αyc (q), ˜ ≤ αyc (q), ˜ while excess smoothness will imply αcc (δ) ˜ and once αcc (δ) cc ˜ yc ˜ = 1/a reflects the severity of the moral hazard problem in again, α (δ)/α (q) our model. The presence of intertemporal nonseparabilities implies an equation like (23) for consumption changes and, in particular, the presence of lagged values of w1 . Consumption is not a martingale anymore. However, as we mention above, the relevant Euler equation still provides enough restrictions to make the IBC testable. Moreover, some of these restrictions are testable in a system like (24), as they translate into restrictions on the α coefficients. While ˜ = 1 cannot be tested because the coefficients of ψ are the hypothesis that ψ(δ) ˜ is proportional to ψ(q)), ˜ we can identify coefficients on not identified (αcc (δ) lagged values of v2 in the first equation of the system (24) and test the hypothesis that they are zero. The alternative that they are different from zero corresponds to the standard excess sensitivity tests, which have been applied many times in the literature since Hall (1978). Intuitively, a coefficient on lagged values of v2 in the consumption equation would imply that consumption is excessively sensitive to expected changes in income to be consistent with the PIH. We implement these tests on our data below. In our context, these tests are important because, as stressed by HRS, if consumption does not satisfy the restrictions implied by the Euler equation, we cannot meaningfully test the intertemporal budget constraint. To implement the estimation of system (24) on our data, some important modifications of the standard procedure used by HRS are necessary. In particular, we need to take into account that we are using household-level data and that we do not have longitudinal data. To address these two problems, we use the approach recently developed by Attanasio and Borella (2006). Because our data do not have a longitudinal dimension, we do not observe the quantities on the left-hand side of equations (24). As mentioned above, we overcome this problem by using synthetic cohort techniques (see Deaton (1985)). In particular, given groups of fixed membership, indexed by g, and an h , we can state, without loss of generality, individual variable zgt h = z gt + ηhgt zgt
1049
RISK SHARING IN PRIVATE INFORMATION MODELS
where the first term on the right-hand side defines the population group mean. zct of it from We do not observe z gt , but we can obtain a consistent estimate our sample. This differs from the population mean by an error whose variance can be consistently estimated given the within-cell variability and cell size (see Deaton (1985)). The presence of this measurement error in the levels induces an additional MA(1) component in the time series behavior of the changes in the variables of interest. The variability of this component has to be taken into account when estimating the parameters of the model. We do so by assuming that the information on within-cell variability provides an exact measurement of the variance of this component. Given the sample sizes involved, this assumption is not very strong. Given the known values for the variance– covariance matrix of the sampling error component, the likelihood function of the MA system in (24) can be computed using the Kalman filter (for details see Attanasio and Borella (2006)). Given these considerations, aggregating the household-level equations (24) at the group level and assuming that the degree of the polynomials in the lag operator is 2, the system that we estimate can be written as (25)
cc cgt = v1gt + αcc 1 v1gt−1 + α2 v1gt−2 yc
yc
yc
yy
yy
ygt = α0 v1gt + α1 v1gt−1 + α2 v1gt−2 + v2gt + α1 v2gt−1 + α2 v2gt−2 where we have normalized the coefficient of the contemporaneous first factor in the consumption equation and of the second factor in the income equation to be 1. In addition to estimating the coefficients in the above system, we also test the hypothesis that coefficients on the lagged values of v2gt do not enter the consumption equation. 4.1.1. Results in Levels In Tables I and II, we report the estimates we obtain estimating the MA system (25) by maximum likelihood. As we allow the variance–covariance matrix in the system to be cohort specific, we limit the estimation to cohorts that are observed over a long time period. This means using balanced pseudopanels with two cohorts: that born in the 1940s and that born in the 1950s. We experimented with several specifications that differed in terms of the number of lags considered in the system. The most general specification included up to eight lags in both the consumption and income equation. However, no coefficient beyond lag 2 was either individually or jointly significant. In the tables, therefore, we focus on the specification with two lags. Table I uses as a definition of consumption the expenditure on nondurables and services. We use three different definitions of income. The first is gross earnings, the second is gross earnings plus all public transfers (the most important of which are unemployment insurance and housing benefits—benefits awarded to the needy to cover housing expenses), and the third is net earnings
1050
TABLE I NONDURABLE CONSUMPTIONa
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
Income Equation Own Shock ayy(t)
ayy(t − 2) Income/Consumption Shockb ayc(t) ayc(t − 1) ayc(t − 2)
1 – – – – –
1 – 0.333 (25.683) −0.748 (18.183)
1 – – – – –
1 – 0.376 (18.240) −0.507 (10.247)
1 – – – – –
1.159 (0.346) −1.140 (0.454) 0.992 (0.370)
1.171 (0.317) −1.148 (0.415) 0.995 (0.331)
1.073 (0.329) −0.887 (0.492) 0.827 (0.356)
1.064 (0.317) −0.894 (0.459) 0.844 (0.311)
0.771 (0.253) −0.602 (0.342) 0.619 (0.257)
0.793 (0.243) −0.587 (0.323) 0.568 (0.227)
Consumption Equation Income/Consumption Shockb acc(t)
log L
1 – −0.577 (0.196) 0.084 (0.201) −773.3
1 – −0.493 (0.114) – – −773.9
1 – −0.604 (0.193) 0.116 (0.191) −759.5
1 – −0.491 (0.116) – – −760.6
1 – −0.612 (0.191) 0.118 (0.193) −685.0
1 – −0.499 (0.111) – – −685.5
Excess smoothnessc SE
−0.491 (0.171)
−0.499 (0.165)
−0.489 (0.160)
−0.493 (0.153)
−0.273 (0.132)
−0.263 (0.128)
acc(t − 1) acc(t − 2)
(Continues)
O. P. ATTANASIO AND N. PAVONI
ayy(t − 1)
1 – 0.505 (20.735) −0.672 (12.252)
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
Excess sensitivityd log L unrestricted model LR p-Value
−771.87 2.9 0.235
−773.4 0.94 0.625
−759.45 0.1 0.951
−758.9 3.38 0.185
−684.64 0.68 0.712
−684.4 2.22 0.330
Comparison with 4 lags model log L4 lags model LR p-Value
−771.5 3.7 0.717
−773.6 0.54 0.763
−756.3 6.34 0.386
−760.4 0.42 0.811
−681.7 6.52 0.368
−685.3 0.34 0.844
a All data are in (first diff of) levels. Standard Errors (SE) are given in parentheses. b Income/consumption shock is the shock that enters both the income and the consumption equation. c Excess smoothness tests are computed as sum(acc(t − L)) − sum(ayy(t − L)) = 0, with L = 0 4; interest rate = 001. d Excess sensitivity test computed as LR test (restrictions: acy(t − 1) = acy(t − 2) = 0).
RISK SHARING IN PRIVATE INFORMATION MODELS
TABLE I—Continued
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1052
TABLE II TOTAL CONSUMPTION EXPENDITUREa
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
Income Equation Own Shock ayy(t)
ayy(t − 2) Income/Consumption Shock ayc(t) ayc(t − 1) ayc(t − 2)
1 – – – – –
1 – 0.400 (9.124) −0.968 (7.222)
1 – – – – –
1 – 0.321 (28.027) −0.765 (20.287)
1 – – – – –
0.596 (0.427) −0.341 (0.638) 0.572 (0.451)
0.586 (0.351) −0.673 (0.473) 0.981 (0.467)
0.730 (0.383) −0.472 (0.572) 0.561 (0.430)
0.497 (0.346) −0.533 (0.477) 0.910 (0.451)
0.683 (0.236) −0.277 (0.358) 0.330 (0.280)
0.662 (0.205) −0.273 (0.304) 0.363 (0.238)
Consumption Equation Income/Consumption Shock acc(t)
log L
1 – −0.349 (0.406) −0.069 (0.358) −885.2
1 – −0.346 (0.208) – – −886.0
1 – −0.386 (0.366) −0.022 (0.330) −869.4
1 – −0.345 (0.202) – – −870.5
1 – −0.372 (0.367) −0.031 (0.319) −788.0
1 – −0.395 (0.172) – – −788.9
Excess smoothnessb SE
−0.233 (0.128)
−0.224 (0.144)
−0.217 (0.116)
−0.203 (0.113)
−0.131 (0.099)
−0.174 (0.128)
acc(t − 1) acc(t − 2)
(Continues)
O. P. ATTANASIO AND N. PAVONI
ayy(t − 1)
1 – 51.615 (1535.290) −36.884 (1072.450)
Excess sensitivityc log L unrestricted model LR p-Value Comparison with 4 lags model log L 4 lags model LR
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
−882.9 4.64 0.098
−885.3 1.36 0.507
−869.25 0.36 0.835
−868.7 3.64 0.162
−786.8 2.4 0.301
−788.8 0.1 0.951
−882.3 5.78 0.448
−885.4 1.24 0.538
−867.8 3.26 0.776
−869.6 1.82 0.403
−786.2 3.54 0.739
−788.3 1.1 0.577
a All data are in (first difference of) levels. Standard errors (SE) are given in parentheses. b Excess smoothness test computed as sum(axx(t − L)) − sum(ayx(t − L)) = 0, with L = 0 4; interest rate = 001. c Excess sensitivity test computed as LR test (restrictions: acy(t − 1) = acy(t − 2) = 0).
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TABLE II—Continued
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O. P. ATTANASIO AND N. PAVONI
plus benefits. For each of the three definitions, we report two specifications: one with two lags in each of the two equations and one where the insignificant coefficients are restricted to zero. Several interesting elements come out of the table. First, the dynamics of income are richer than those of consumption. However, and perhaps surprisingly, the coefficients on the lags of v2gt are not statistically significant and are constrained to zero in the third, fifth, and seventh columns. In the consumption equation, the coefficient on the first lag of v1gt is consistently significant and attracts a negative sign. As discussed above, this could be a sign of intertemporal nonseparability of preference, maybe induced by some elements of nondurable consumption having some durability at the quarterly frequency. To implement the HRS test of the IBC, it is necessary that the marginal utility of consumption satisfies a martingale property. In Table I, such an assumption is imposed, in that lagged income shocks do not enter the consumption equation. This hypothesis, however, can be tested. If we do not impose the restriction, the estimated coefficients are small in size and never significantly different from zero, either individually or jointly. In the table, for each specification, we report the value of the likelihood ratio (LR) test that the coefficients on the first and second lag are jointly zero and its corresponding p-value. This is an important result as it corresponds to a nonviolation of the excess sensitivity test. The test of the intertemporal budget constraint, which is parameterized as ˜ − β1 (q), ˜ clearly shows the presence of excess smoothness. Interestingly, πψ(δ) such evidence is stronger for gross earnings. The value of the test does not change much when we add benefits to gross earnings (as in the fourth and fifth columns). However, when we consider net earnings plus benefits, the value of the test is greatly reduced in absolute value (moving from −0.49 to −0.26), although still statistically different from zero. Therefore, when we use a definition of income that includes an important smoothing mechanism, we find much less evidence of consumption excess smoothness relative to that income definition. Table II mirrors the content of Table I, with the difference that the definition of consumption we use now includes expenditure on durables. The results we obtain are, in many ways, similar to those of Table I. Perhaps surprisingly, the coefficient on lagged v1gt in the consumption equation is smaller in absolute value than in Table I and for two of the three income definitions is not statistically different from zero. The test of excess sensitivity, as in Table I, does not reject the null that the coefficients on lagged income shocks are zero in the consumption equation (although the p-value for the test corresponding to the second column is 0.098). The most interesting piece of evidence, however, is that the coefficient that measures excess smoothness is now considerably lower in absolute value, indicating less consumption smoothing relative to the null of the Bewley model. This is suggestive of the fact that durables might be playing an important role in the absorption of shocks, as speculated, for instance, by
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Browning and Crossley (2009). However, when we consider different income definitions, the evidence is consistent with that reported in Table I, in that consumption exhibits much less excess smoothness relative to net earnings than to gross earnings. 4.1.2. Results in Logs When reestimating the system using the specification in logs, we try to use the same sample used in the specification in levels. However, as we aggregate the nonlinear relationship (i.e., we take the group average of logs), we are forced to drop observations that have zero or negative income. Apart from this, the sample is the same. We report our estimates in Table III. Given the evidence on the dynamics of consumption discussed above, we only report the results for total consumption, which includes expenditure on durables. Results for nondurables and services are available upon request. Our estimates of a are obtained under the assumption that λ = 1, which corresponds to log utility, and δ = q. If q > δ and γ > 1, λ would be less than unity. We have explored values for λ below 1, obtaining similar results. As with Tables I and II, we test the hypothesis that lagged coefficients of the income shock do not appear significantly in the consumption equation. In the case of the log specification, we stress, as mentioned in Section 3.3, that one could get a rejection either because of binding liquidity constraints or because the second moments in equation (16) might depend on lagged variables. While this is possible, empirically, this excess sensitivity test does not reject the null at standard significance values.23 As with the results in levels, we do find evidence of excess smoothness. The drop in the size of the excess-smoothness parameter when we move to definitions of income that include some smoothing mechanisms is even more dramatic than in Table II. In the last column, corresponding to net earnings plus benefits, the excess-smoothness parameter, while still negative, is not significantly different from zero. This result suggests that most of the insurance available over and above self-insurance comes from public transfers and taxation. 4.2. The Evolution of Cross-Sectional Variances The empirical implications of the model we have stressed so far focus on the means of consumption and income. Because of the lack of longitudinal data at the individual level, we were forced to use synthetic panel data. This implies that purely idiosyncratic shocks are averaged away and the excess-smoothness test identifies risk sharing across cohorts. To complement this evidence, it can, 23
The issue of the estimation of log-linearized Euler equations like equation (16) is discussed in Attanasio and Low (2004). The results we find are consistent with the findings of their simulations.
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TABLE III TOTAL CONSUMPTION EXPENDITURE: LOG SPECIFICATIONa
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
Income Equation Own shock ayy(t)
ayy(t − 2) Income/Consumption Shock ayc(t) ayc(t − 1) ayc(t − 2)
1 – – – – –
1 – 0.181 (4.936) −0.779 (3.957)
1 – – – – –
1 – 0.159 (1.982) −0.595 (1.426)
1 – – – – –
0.889 (0.200) −1.102 (0.249) 0.393 (0.229)
0.816 (0.155) −1.250 (0.206) 0.660 (0.199)
0.717 (0.162) −0.685 (0.216) 0.094 (0.191)
0.748 (0.146) −0.836 (0.177) 0.223 (0.139)
0.799 (0.191) −0.523 (0.253) −0.120 (0.214)
0.570 (0.187) −0.222 (0.211) 0.086 (0.152)
1 – −1.011 (0.008) – – 148.6 9.38 0.025
1 – −0.578 (0.324) −0.444 (0.330) 173.8
1 – −0.619 (0.125) – – 168.2 11.16 0.011
Consumption Equation Income/Consumption Shock acc(t) acc(t − 1) acc(t − 2) log L LR p-Value
1 – −0.601 (0.243) −0.411 (0.247) 100.9
1 – −1.011 (0.008) – – 97.2 7.4 0.060
1 – −0.647 (0.279) −0.368 (0.282) 153.3
(Continues)
O. P. ATTANASIO AND N. PAVONI
ayy(t − 1)
1 – 0.332 (14.452) −0.748 (10.305)
Excess Smoothnessb SE Excess Sensitivityc log L unrestricted model LR p-Value
Gross Earnings
Gross Earnings
Gross Earnings + Benefits
Gross Earnings + Benefits
Net Earnings + Benefits
Net Earnings + Benefits
−0.181 (0.100)
−0.226 (0.079)
−0.133 (0.050)
−0.141 (0.053)
−0.170 (0.066)
−0.048 (0.106)
101.9 2.0 0.368
97.9 1.4 0.497
153.5 0.4 0.819
149.9 2.58 0.275
174.4 1.24 0.538
168.6 0.8 0.670
a All data are in (first difference of) levels. SE given in parentheses. b Excess smoothness test computed as sum(axx(t − L)) − sum(ayx(t − L)) = 0, with L = 0 4; interest rate = 001. c Excess sensitivity test computed as LR test (restrictions: acy(t − 1) = acy(t − 2) = 0).
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TABLE III—Continued
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O. P. ATTANASIO AND N. PAVONI
therefore, be useful to consider the implications of the theory for the crosssectional variances of income and consumption. The test we develop here focuses on risk sharing within cohorts. For this purpose, the closed-form solution derived in Appendix B for the model with two independent shocks, one of which is assumed to be temporary and one permanent, is particularly useful. In this case, we have 1 (1 − λq) i ln xit + ln ξt−s p a aT s=0 t−1
(26)
ln cti =
γ +t 2
1 ap
2
1 − λq σv2p + aT
2
σv2T
q − t δ + μi + zt γ ln
The term μi allows for ex ante heterogeneity, which could capture distributional issues, the initial level of assets of individual i, or fixed effects that can be observable to the firm but may be unobservable to the econometrician. The term zt allows for aggregate shocks, which are assumed to be orthogonal to individual shocks and included in the information set of all agents in the economy. If we compute the cross-sectional variance at time t of both sides of equation (26), we have 2 1 i (27) Var(ln xit ) Var(ln ct ) = ap t−1 2 1 − λq i + Var ln ξt−s + Var(μi ) aT s=0 i i Cov(ln xt μ ) (1 − λq) i i + Cov(ln ξt−s μ ) +2 ap aT s where we have used the fact that Cov(ln xt ln ξt−s ) = 0 for all s.24 We start by i μi ) and Cov(ln xit μi ) are time invariant. The assuming that both Cov(ln ξt−s time invariance of these terms can be obtained by assuming constant μs across agents in the same group (as in our theoretical model). If we now take the first difference of equation (27), neglecting the individual indices, we have 2 2 1 1 − λq (28) Var(ln xt ) + Var(ξti ) Var(ln ct ) = ap aT 24 Here we are making use of the income process with two separate shocks typically assumed in the PIH literature, which we introduce in Appendix B. If we were to use the process with a single shock used in Section 3, equation (26) for log consumption would be much more complicated, as reported in Appendix B. It is easy to see, however, that the whole analysis goes through for Cov(ln xt ln ξt−s ) = 0, as long as it is time constant for all s ≥ 0.
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where Var(ln ct ) := Var(ln ct ) − Var(ln ct−1 ) and Var(ln xt ) is similarly defined. Under our assumptions, Var(ln yt ) = Var(ln xt + ln ξt ) = Var(ln xt ), which implies two things. First, we can replace the permanent component xt by total income yt in the first term of the right-hand side of equation (28). Second, without independent identification of the two components of income (xt and ξt ), only ap (= a) can be identified. Notice that equation (28) allows again the identification of the structural parameter a, which reflects the severity of the moral hazard problem. As noted by Deaton and Paxson (1994), under perfect risk sharing, the cross-sectional variance of consumption is constant over time.25 Under the PIH, as pointed out by Blundell and Preston (1998), the changes in the variance of consumption reflect changes in the variance of (permanent) income. Here, we consider a specific alternative to the perfect insurance hypothesis that implies that consumption variance grows, but less than the increase in the variance of permanent income. The empirical strategy we follow here is quite different from Blundell, Pistaferri, and Preston (2008). They studied the evolution of the cross-sectional variance of consumption growth,26 while we start from the specification for consumption levels in equation (26) to derive equation (28). We notice that, as we do not have longitudinal data, we cannot identify separately ap and aT , while Blundell and Preston (1998) did identify the proportion of permanent and transitory shocks that can be insured. The estimation of equation (28) also requires the identification of groups. Here the group implicitly defines the participants in a risk sharing arrangement and the test identifies the amount of risk sharing within that group. As with the estimation of the HRS system, the lack of truly longitudinal data and the use of time series of cross sections imply that the estimated variances (for income and consumption) have an error component induced by the variability of the sample. This is particularly important for the changes in the variance of income on the right-hand side: the problem induced is effectively a measurement error problem which induces a bias in the estimated coefficient. However, as explained in Appendix C, it is easy to obtain an expression for the bias implied by an ordinary least squares (OLS) estimator in finite samples and to correct it. To estimate the parameters in equation (28), we use the same sample we used for the HRS test, with the only difference being that we do not limit ourselves to the balanced pseudopanel, but use four cohorts, although the youngest and oldest are only used for part of the time period. Otherwise, the selection criteria used to form our sample are the same as above. 25 Attanasio and Szekely (2004) proposed a test of perfect risk sharing based on changes in the cross-sectional variances of marginal utilities. 26 This has important advantages, but it forces the use of the approximation Var(ct ) ≈ Var(ct ).
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O. P. ATTANASIO AND N. PAVONI TABLE IV VARIANCE BASED TEST
Nondurable Consumption
Nondurable Consumption per Adult Equiv.
Total Consumption
Total Consumption per Adult Equiv.
Gross earnings
0.0709 0.0133
0.0376 0.0154
0.0765 0.0177
0.0547 0.0196
Implied a
3.7556 0.0484
5.1571 0.0900
3.6144 0.0610
4.2746 0.0865
0.2357 0.0302
0.1476 0.0355
0.3019 0.0401
0.2495 0.0448
Implied a
2.0596 0.0447
2.6032 0.0747
1.8200 0.0492
2.0021 0.0634
Net earnings + benefits
0.2601 0.0351
0.1466 0.0413
0.3478 0.0463
0.2733 0.0519
Implied a
1.9608 0.0482
2.6121 0.0871
1.6957 0.0511
1.9129 0.0686
505
505
505
505
Individual Var.
Gross earnings + benefits
Number of observ.
The results are reported in Table IV. The four data columns in the table each report the slope coefficient of equation (28) and the implied a with the corresponding standard errors. The standard error of a is computed by the delta method. In the first two data columns, we use expenditure on nondurables and services as our definition of consumption, while in the last two we use total expenditure. In the third and last columns, the consumption figure is divided by the number of adult equivalents. The three panels correspond to the same three definitions of income we used for the HRS test. The first aspect to be noted is that all the slope coefficients are positive and statistically different from zero. Moreover, consistent with the theory, they all imply a value of a greater than unity. Finally, the results are affected only minimally by the consideration of adult equivalents. If we analyze the difference across income definitions, we find results that are consistent with the implications of the model and, by and large, with the evidence from the HRS approach. The coefficient on the changes in the variance of gross earnings is much smaller than the coefficient on the other income definitions. This is consistent with the evidence from Tables I and II, which showed more excess smoothness for this definition. Unlike in Tables I and II, however, the main difference in the size of the coefficient is between the first income definition on one side and the second and third on the other. With the HRS approach, the main difference was between the first and second on one side and the third on the other.
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Finally, if we look at the differences between the definitions of consumption that include durables and those that do not, we find that the coefficients are (except for the first income definition) larger for the former than the latter. Again, this is consistent with the evidence from the HRS approach, which finds less excess smoothness when durables are included in the definition of consumption, that is, some self-insurance mechanisms seem to be at work via durables. The consistency of the results obtained with the variance approach and those obtained with the HRS approach is remarkable because the two tests, as stressed above, focus on different aspects of risk sharing: the latter on insurance across groups and the former on insurance within groups. It is remarkable that both yield results that are in line with our model and indicate that the observed amount of risk sharing is between that predicted by a simple permanent income model and that predicted by perfect insurance markets. Comparing the magnitudes of the coefficients from the two approaches, we can obtain a measure of the different degrees of risk sharing possibilities that are available within cohorts as opposed to those available across cohorts. 5. CONCLUSIONS In this paper, we discuss the theoretical and empirical implications of a model where perfect risk sharing is not achieved because of information problems. A specific and important feature of our model is that in addition to the standard moral hazard problem, in the economy we study, agents have hidden access to the credit market. After characterizing the equilibrium of this model, we show how it can be useful to interpret individual data on consumption and income. Developing results in Abraham and Pavoni (2004), we show that in a competitive equilibrium of our model, agents typically obtain more insurance than in a Bewley setup. Moreover, we are able to construct examples in which we can get closed-form solutions for consumption. These results have more than an aesthetic value: in our empirical approach they allow us to give a structural interpretation to some of the empirical results in the literature and to those we obtain. In particular, in our model one should observe the so-called excess smoothness of consumption. Moreover, excess smoothness is distinct from the so-called excess sensitivity of income to consumption. Finally, we can map the excess-smoothness parameter into a structural parameter. The presence of excess smoothness follows from the fact that even in the presence of moral hazard and hidden assets, in general, a competitive equilibrium is able to provide some insurance over and above what individuals achieve on their own by self-insurance. This additional insurance is what generates excess smoothness in consumption, which can then be interpreted as a violation of the intertemporal budget constraint with a single asset. The equilibrium allocations generated by our model violate the IBC with a single asset because they neglect some state-contingent transfers the agents use to share risk.
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Tests of the risk-free intertemporal budget constraint become tests of market structure in our framework. For this reason, we start our empirical work using a test of excess smoothness that was first proposed as a test of the intertemporal budget constraint with a single asset by HRS. We extend their approach so that it can be applied to microdata. Because longitudinal data on consumption are rarely available, we work with time series of cross sections and synthetic cohort data. As we are forced to aggregate the consumption and income of individuals belonging to a given yearof-birth cohort, we necessarily lose some of the variability in idiosyncratic income and the possibility of studying the amount of risk sharing of these shocks. This is one of the reasons why, in addition to the extension of the HRS test, we propose an additional test, which uses the movements in the cross-sectional variances of consumption and income to identify the same structural parameters of our model. While related to the work of Deaton and Paxson (1994), Blundell and Preston (1998), and Blundell, Pistaferri, and Preston (2008), our approach is different in that it focuses on the variance in the level rather than changes of consumption. Moreover, as is the case for the version of the HRS test we present, we can give the coefficients we estimate a structural interpretation in terms of our moral hazard model. While many papers, starting with Campbell and Deaton (1989), have documented the ‘excess smoothness of consumption using aggregate time-series data, the evidence based on microdata is recent and very limited. Using our two different approaches and data from the United Kingdom, we forcefully reject both perfect risk sharing and the simple Bewley economy, while we do not reject the hypothesis of no excess sensitivity of consumption to income. More generally, our results are consistent with the model with moral hazard and hidden assets we considered. Particularly suggestive is the fact that when we consider income definitions that include smoothing mechanisms, such as social assistance and net taxes, we find less evidence of excess smoothness. Our results have obvious policy implications, as one could, in principle, be able to quantify in terms of welfare the insurance role played by the taxation system or unemployment insurance. Such computations would be immediately feasible from our analysis. We would also be able to perform accurate counterfactuals so as to evaluate the effects of a given policy. All normative issues, however, are left for future research. APPENDIX A A.1. Implementing the Efficient Allocation With Income Taxes In this section, we show that under some conditions the equilibrium allocation (e c y) can be described using a transfer scheme τ which is a function of income histories y t alone. This simplifies the analysis and allows us to describe the consumption allocation in terms of observables.
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Notice first that through yt = yt (θt ), the y component of the equilibrium allocation generates histories of income levels y t . Let us denote by y t (θt ) = (y1 (θ1 ) yt (θt )) this mapping. In general, yt (θt ) is not invertible, as it might be the case that for a positive measure of histories of shocks θt , we get the same history of incomes y t . A generalization of the argument used in Kocherlakota (2005), however, shows that it suffices to assume that the optimal plan of consumption c alone is y t -measurable. That is, that there exists a sequence of y t -measurable functions c ∗ such that for all t θt , we have c∗t (y t (θt )) = ct (θt ) We now show that under fairly general conditions, the implementation idea of Kocherlakota (2005) extends to the general case with hidden savings. Now notice that yt is y t -measurable by construction. As a consequence, from (9) is easy to see that the y t -measurability of c implies that τ is y t measurable as well. From the transfer scheme τ, we can hence obtain the new y t -measurable scheme τ∗ as follows: τ ∗t (y t (θt )) = τ t (θt ). Given τ∗ let T t−1 ∗ δ u(ct eˆ t )eˆ E t=1
:=
T t=1
δt−1 Θt
u c∗t (yˆ t (θt )) eˆ t (θt ) dμt (θt )
where c∗t (yˆ t (θt )) = τ ∗t (yˆ t (θt )) + y∗t (yˆ t (θt )) and the new mapping is induced by eˆ as follows: yˆt (θt ) = f (θt eˆ t (θt )) for all t θt . For any history of shocks θt a plan eˆ not only entails different effort costs, it also generates a different distribution over income histories y t , hence on transfers and consumption. This justifies our notation for the conditional expectation. We say that the equilibrium allocation (e c y) can be described with y t measurable transfers if the agent does not have an incentive to deviate from c ∗ e∗ given τ∗ . The incentive constraint in this case is T T t−1 ∗ ∗ ∗ t−1 (29) δ u(ct et )e ≥ E δ u(cˆt eˆ t )eˆ E t=1
t=1
where, as usual, the deviation path of consumption cˆ must be replicated by ˆ An important restriction in the deviations eˆ cona plan of risk-free bonds b. templated in constraint (29) is that they are required to generate attainable histories of ys, that is, histories of ys that can happen in an equilibrium allocation. The idea is that any off-the-equilibrium value for y t will detect a deviation with certainty. One can hence set the firm’s transfers to a very low value (perhaps minus infinity) in these cases, so that the agent never has an incentive to generate such off-the-equilibrium histories. Finally, suppose the agent chooses an effort plan eˆ so that the realized history yˆ t is attainable in equilibrium. This means both that there is a reporting
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strategy σˆ so that yˆ t = (y1 (σ) ˆ y2 (σ) ˆ yt (σ)) ˆ and that, given a consumption ˆ the utility the agent gets is E[ Tt=1 δt−1 u(cˆt et )|(e c y); σ] plan c, ˆ where the notation is that in the main text.27 This effectively completes the proof since the incentive constraint (4) guarantees that the agent chooses the truth-telling strategy which implies the equilibrium plans e and c as optimal for him. A.2. Proof of Proposition 1 We now show that, for the specification of preferences and production function we stated in Proposition 1, incentive compatibility fully characterizes the efficient allocation. In fact, we allow for a generic sequence of bond prices T −1 faced by both the agents and the firms in the economy. {qt }t=1 To avoid the use of the taxation principle, we consider transfer schemes that depend on the revelation plan σ. This notation also is more directly related to the Bewley model of Section 2.3. Recall that T < ∞ and consider the last period of the program. Using the budget constraint, assuming the agent declared history θT −1 so far, and has wealth bT after the realization of θT , his/her preferences over the report σT of θT can be represented as u(cT − eT ) = u(θT + τT (θT −1 σT ) + bT ) The key aspect to notice here is that since utility depends on the declaration σT only through the transfer, the agent will declare the productivity level that delivers the maximal transfer whenever possible. More precisely, for any value of eT (θT −1 θT ), the local incentive compatibility implies that for a sufficiently small ε > 0, we have u(θT + τ T (θT −1 θT ) + bT ) ≥ u(θT + τ T (θT −1 σT ) + bT )
for all σT ∈ [θ − ε θ + ε]
27 More in detail, notice that by definition, we have eˆ t (θt ) = g(θt yˆ t (θt )). Since yˆ t (θt ) is attainable, it can be induced from y by a “lie,” that is, there exists a σˆ such that yˆ t (θt ) = yt (σ(θ ˆ t )). But then eˆ t (θt ) = g(θt yt (σ(θ ˆ t ))) and from the definition of τ∗ , we have c∗t (ˆyt (θt )) = c∗t (yt (σ(θ ˆ t ))) = ˆ t )) which implies that ct (σ(θ T T t−1 ∗ t−1 E δ u(ct eˆ t )eˆ = E δ u(ct et )(e c y) σˆ t=1
t=1
for some σˆ ∈ Σ Finally, it is easy to see that c ∗ (and b∗ ≡ 0) is incentive compatible among the deviations that solve cˆ σt (σ(θt )) + qbˆ σt+1 (σ(θt )) − bˆ σt (σ(θt−1 )) = c∗t (σ(θt )) and limt→T qt−1 bˆ σt+1 (σ(θt )) ≥ 0 a.s. for all histories θt . A final remark: One can easily show that under the same conditions, e must also be xt -measurable. If e is not xt -measurable, it means that for at least two some θt and θ¯ t , we have et (θt ) = et (θ¯ t ) while f (θt et (θt )) = f (θt et (θ¯ t )) However, since u is decreasing in e effort incentive compatibility (at bt = 0) implies that τ s (θs ) = τ s (θ¯ s ) for some s ≥ t with τ not y s -measurable. This is contradiction to the fact that the equilibrium transfer scheme τ is y t -measurable.
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This condition applied to all θT ∈ Θ implies that a transfer payment τT which is invariant across θT s is the sole incentive compatible possibility. Now consider the problem in period T −1 We now show that, given θT −2 and θT −1 , an incentive compatible transfer scheme must induce a constant number for the present value of transfers τT −1 (θT −1 ) + qT −1 τT (θT −1 ) across θT −1 levels. Note that in our notation we used the fact that τ is constant across θT Consider a generic history θT −2 and suppose that τT −1 (θT −2 θT −1 )+qT −1 τT (θT −2 θT −1 ) < τT −1 (θT −2 θT −1 ) + qT −1 τT (θT −2 θT −1 ) for θ sufficiently close to θ . This cannot be incentive compatible. The agent who receives shock θT −1 would declare θT −1 (and change effort by the difference between the two productivity levels) and adjust his/her bond plan accordingly. We just have to show that there is a bond plan so that the agent is better off by declaring θT −1 . This is easy to see since τT does not depend on θT . More precisely, if cT −1 cT (θT ) is the consumption plan under θT −1 , then cT −1 + 1+qκ and cT (θT ) + 1+qκ comprise one of the conT −1 T −1 sumption paths attainable by deviating, where κ > 0 is the difference between the two net present values of transfers. A similar argument applies to the case where the inequality is reversed. In this case, the profitable deviation would be taken by the agent who receives shock θT −1 . The discounted value of transfer must hence be constant across θT −1 as well, that is, τT −1 (θT −1 ) + qT −1 τT (θT ) is θT −2 -measurable. Going backward, we get that for all t, the quantity NPVt :=
n T −t n=0
qt−1+s τ t+n (θt+n )
s=1
is at most θt−1 -measurable since it is a constant number for all continuation histories θt θt+1 θT following node θt−1 . If we consider the initial period (t = T n−1 1), (local) incentive compatibility implies that NPV1 (θT ) := n=1 ( s=1 qs )τn is one number a.s. for all histories θT . As we argued already in the main text while discussing equation (5), another necessary condition for incentive compatibility is the agent’s Euler equation. The problem of the firm facing a relaxed (local) incentive compatibility constraint hence reduces to the choice of the unique number NPV1 and the plan of bond holdings for the agent. Since the agent and the firm face the same sequence of bond prices, it is easy to see that the firm is indifferent across all bond plans. This implies that the only number NPV1 consistent with nonnegative profits and with maximizing the agent’s utility is zero. Moreover, this number can be obtained by setting τt ≡ 0 (equivalently, we could assume the transfer τ is set so that the agent chooses bt ≡ 0). It is now obvious that τt ≡ 0 is globally incentive compatible, implying that it must be optimal. We have hence shown that the optimal allocation corresponds to the bond economy allocation. Q.E.D.
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More extensively, in all cases, the optimal contract must solve the standard Euler equation at each node and the condition t−1 T NPV1 (θT ) := qs τ ∗t (θt ) t=1
s=1
t−1 = qs (c∗t (θt ) − y∗t (θt )) = 0 T
t=1
a.s. for all θT
s=1
But these are precisely the two defining properties for the self-insurance allocation: first, it must satisfy the Euler equation; second, it must satisfy the intertemporal budget constraint with the risk free bond, that is, the period zero net present value must be zero for all θT . Since the Euler equation (5) is always satisfied in our allocation, the only way to obtain a different allocation is that the transfer scheme τ be permitted to violate the agent’s period zero self-insurance intertemporal budget constraint for some history θT . The whole point of Proposition 1 is to demonstrate that this cannot be the case. A final remark: Although the proof uses finite time, by adapting the proof of Proposition 7 in Cole and Kocherlakota (2001), we can show the result for T = ∞ at least as long as u is a bounded function. REFERENCES ABOWD, J., AND D. CARD (1989): “On the Covariance Structure of Earnings and Hours Changes,” Econometrica, 57, 411–445. [1041] ABRAHAM, A., AND N. PAVONI (2004): “Efficient Allocations With Moral Hazard and Hidden Borrowing and Lending,” Working Paper 04-05, Duke University, Department of Economics. [1031,1037,1039,1061] (2005): “The Efficient Allocation of Consumption Under Moral Hazard and Hidden Access to the Credit Market,” Journal of the European Economic Association, 2005, 370–381. [1031] (2008): “Efficient Allocations With Moral Hazard and Hidden Borrowing and Lending: A Recursive Formulation,” Review of Economic Dynamics, 11, 781–803. [1031] ALESSIE, R., AND A. LUSARDI (1997): “Consumption, Savings, and Habit Formation,” Economic Letters, 55, 103–108. [1047] ALLEN, F. (1985): “Repeated Principal-Agent Relationship With Lending and Borrowing,” Economic Letters, 17, 27–31. [1036,1039] ATTANASIO, O. P. (2000): “Consumption,” in Handbook of Macroeconomics, ed. by J. B. Taylor and M. Woodford. Amsterdam: Elsevier. [1046,1047] ATTANASIO, O. P., AND M. BORELLA (2006): “Stochastic Components of Individual Consumption: A Time Series Analysis of Grouped Data,” Working Paper 12456, NBER. [1048,1049] ATTANASIO, O. P., AND H. W. LOW (2004): “Estimating Euler Equations,” Review of Economic Dynamics, 7, 405–435. [1055] ATTANASIO, O. P., AND N. PAVONI (2011): “Supplement to ‘Risk Sharing in Private Information Models With Asset Accumulation: Explaining the Excess Smoothness of Consumption’,” Econometrica Supplemental Material, 79, http://www.econometricsociety.org/ ecta/Supmat/7063_extensions.pdf; http://www.econometricsociety.org/ecta/Supmat/7063_data and programs.zip. [1030]
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ATTANASIO, O. P., AND E. M. SZEKELY (2004): “Wage Shocks and Consumption Variability in Mexico During the 1990s,” Journal of Development Economics, 73, 1–25. [1059] ATTANASIO, O. P., AND G. WEBER (1993): “Consumption Growth, the Interest Rate and Aggregation,” Review of Economic Studies, 60, 631–649. [1044,1046] (1995): “Is Consumption Growth Consistent With Intertemporal Optimization? Evidence From the Consumer Expenditure Survey,” Journal of Political Economy, 103, 1121–1157. [1046] BANKS, J., R. BLUNDELL, AND A. BRUGIAVINI (2001): “Risk Pooling, Precautionary Saving and Consumption Growth,” Review of Economic Studies, 68, 757–779. [1043] BEWLEY, T. (1977): “The Permanent Income Hypothesis: A Theoretical Formulation,” Journal of Economic Theory, 16, 252–292. [1035] BLUNDELL, R., AND I. PRESTON (1998): “Consumption Inequality and Income Uncertainty,” Quarterly Journal of Economics, 113, 603–640. [1059,1062] BLUNDELL, R., M. BROWING, AND C. MEGHIR (1994): “Consumer Demand and the Life-Cycle Allocation of Household Expenditures,” Review of Economic Studies, 61, 57–80. [1046] BLUNDELL, R., L. PISTAFERRI, AND I. PRESTON (2008): “Consumption Inequality and Partial Insurance,” Americal Economic Review, 98, 1887–1921. [1041,1043,1059,1062] BROWNING, M., AND T. F. CROSSLEY (2009): “Shocks, Stocks and Socks: Smoothing Consumption Over a Temporary Income Loss,” Journal of the European Economic Association, 7, 1169–1192. [1055] BROWNING, M., A. DEATON, AND M. IRISH (1985): “A Profitable Approach to Labor Supply and Commodity Demands over the Life-Cycle,” Econometrica, 53, 503–544. [1044] CAMPBELL, J. Y., AND A. DEATON (1989): “Why Is Consumption so Smooth?” Review of Economic Studies, 56, 357–374. [1027-1029,1046,1062] CAMPBELL, J. Y., AND N. G. MANKIW (1989): “Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence,” in NBER Macroeconomic Annual, ed. by O. Blanchard, and S. Fischer. Cambridge: MIT Press, 185–216. [1045] COLE, H. L., AND N. R. KOCHERLAKOTA (2001): “Efficient Allocations With Hidden Income and Hidden Storage,” Review of Economic Studies, 68, 523–542. [1036,1039,1066] CONSTANTINIDES, G. M., AND D. DUFFIE (1996): “Asset Pricing With Heterogeneous Consumers,” Journal of Political Economy, 104, 219–240. [1042] DEATON, A. (1985): “Panel Data From Time Series of Cross-Sections,” Journal of Econometrics, 30, 109–126. [1044,1048,1049] DEATON, A., AND C. PAXSON (1994): “Intertemporal Choice and Inequality,” Journal of Political Economy, 102, 437–467. [1059,1062] FUDENBERG, D., B. HOLMSTROM, AND P. MILGROM (1990): “Short-Term Contracts and LongTerm Relationships,” Journal of Economic Theory, 51, 1–31. [1036] GALI, J. (1991): “Budget Constraints and Time-Series Evidence on Consumption,” American Economic Review, 81, 1238–1253. [1028] GOLOSOV, M., AND A. TSYVINSKI (2007): “Optimal Taxation With Endogenous Insurance Markets,” Quarterly Journal of Economic, 122, 487–534. [1031,1032] HALL, R. E. (1978): “Stochastic Implications of the Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, 96, 971–987. [1048] HANSEN, L. P., W. ROBERDS, AND T. J. SARGENT (1991), “Time Series Implications of Present Value Budget Balance and of Martingale Models of Consumption and Taxes,” in Rational Expectations Econometrics, ed. by L. P. Hansen, and T. J. Sargent. Boulder: Westview, 121–161. [1027,1029] HECKMAN, J. J. (1974): “Life Cycle Consumption and Labor Supply: An Explanation of the Relationship Between Income and Consumption Over the Life Cycle,” American Economic Review, 64, 188–194. [1046] KOCHERLAKOTA, N. R. (2005): “Zero Expected Wealth Taxes: A Mirrlees Approach to Dynamic Optimal Taxation,” Econometrica, 74, 1587–1621. [1063]
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LIGON, E. (1998): “Risk Sharing and Information in Village Economies,” Review of Economic Studies, 65, 847–864. [1043] MEGHIR, C., AND L. PISTAFERRI (2004): “Income Variance Dynamics and Heterogeneity,” Econometrica, 72, 1–32. [1041] MIRRLEES, J. A. (1971): “An Exploration in the Theory of Optimum Income Taxation,” Review of Economic Studies, 38, 175–208. [1041] NALEWAIK, J. J. (2006): “Current Consumption and Future Income Growth: Synthetic Panel Evidence,” Journal of Monetary Economics, 53, 2239–2266. [1043] ROGERSON, W. (1985): “Repeated Moral Hazard,” Econometrica, 53, 69–76. [1043] WEST, K. D. (1988): “The Insensitivity of Consumption to News About Income,” Journal of Monetary Economics, 21, 17–34. [1028,1029,1046]
Dept. of Economics, University College London, Gower Street, London WC1E 6BT, United Kingdom and IFS and NBER;
[email protected] and Bocconi University and IGIER, via Roentgen 1, 20136 Milan, Italy, and University College London, and IFS, and CEPR;
[email protected]. Manuscript received March, 2007; final revision received March, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1069–1101
NONHOMOTHETICITY AND BILATERAL TRADE: EVIDENCE AND A QUANTITATIVE EXPLANATION BY ANA CECÍLIA FIELER1 The standard gravity model predicts that trade flows increase in proportion to importer and exporter total income, regardless of how income is divided into income per capita and population. Bilateral trade data, however, show that trade grows strongly with income per capita and is largely unresponsive to population. I develop a general equilibrium Ricardian model of trade that allows the elasticity of trade with respect to income per capita and with respect to population to diverge. Goods are of various types, which differ in their income elasticity of demand and in the extent to which there is heterogeneity in their production technologies. I estimate the model using bilateral trade data of 162 countries and compare it to a special case that delivers the gravity equation. The general model improves the restricted model’s predictions regarding variations in trade due to size and income. I experiment with counterfactuals. A positive technology shock in China makes poor and rich countries better off and middle-income countries worse off. KEYWORDS: International trade, income per capita, gravity equation, nonhomotheticity.
1. INTRODUCTION IT IS WELL KNOWN that poor countries trade much less than rich countries, both with each other and with the rest of the world. In 2000, for example, transactions to and from the twelve Western European countries alone accounted for 45% of international merchandise trade, and intra-Western European trade, for 16%. The fifty-seven African countries, in contrast, accounted for only 4.2% of world trade, and intra-African trade, a meager 0.2%. Doubling a country’s income per capita increases trade (average between imports and exports) as a share of income by 1.7% on average, while doubling a country’s population decreases its trade share by 2.0%. Despite these differences, standard gravity models of international trade predict that trade increases with importer and exporter total income, and ignore how total income is divided into income per capita and population. Protectionist policies and high transport costs are the usual explanations for the small volume of trade in poor countries. But even after controlling for tariffs and direct measures of trade costs, income per capita continues to have a significant, increasing effect on trade.2 Furthermore, this explanation does 1 This paper was the main chapter of my Ph.D. dissertation. I am grateful to my advisor, Jonathan Eaton, for his guidance and support. Two anonymous referees have suggested significant improvements to the paper. I also thank Jan De Loecker, Raquel Fernandez, Chris Flinn, Bo Honoré, Donghoon Lee, Debraj Ray, Esteban Rossi-Hansberg, Petra Todd, and Matt Wiswall, and several presentation attendees. 2 See Coe and Hoffmaister (1999), Limão and Venables (2001), and Rodrik (1998). Waugh (2010) estimated that to explain trade shares in the data, the cost of exporting from a low-income
© 2011 The Econometric Society
DOI: 10.3982/ECTA8346
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not take incentives into account: the low quality of infrastructure in poor countries may in part be due to these countries’ lack of incentives to trade. This paper takes an alternative view. I purposely abstract from differences in trade costs across countries and focus on two assumptions of gravity models that are inconsistent with microlevel evidence. The first is homothetic preferences. There is exhaustive evidence that the income elasticity of demand varies across goods and that this variation is economically significant.3 For example, spending on food, a low income-elastic good, ranged from 64% in Tanzania to less than 15% in Australia and North America in the early 1980s (Grigg (1994)). The second assumption is that production in poor and rich countries differs only in quantitative, not qualitative, aspects. This assumption is at odds with the theory of product cycle and with empirical evidence on technology diffusion.4 When a good is first invented, the argument goes, the technology to produce it differs greatly across countries, most of which do not know how to make it. At this stage, the good is generally produced in the typically highincome country where it was invented. As the product matures, methods to produce it become standardized and can be applied similarly to any country, including those where labor is cheap. So, in a cross section, poor countries should produce disproportionately more goods whose technologies have already diffused and are similar across countries. I propose an analytically tractable Ricardian model of trade that relaxes these two assumptions. Goods in the model are divided into types, which may differ in two respects: demand and technology. Poor households concentrate their spending on types with low income elasticity, and rich ones, on types with high income elasticity. The supply side setup is Ricardian: labor is the unique factor of production; markets are perfectly competitive, and comparative advantage arises from differences in technologies across goods and countries. The distribution of labor efficiency may be more variable for some types of goods than for others. Analogous to the product cycle theory, in general equilibrium, countries where overall productivity is low have low wages and produce less differentiated goods. Technologically advanced countries have high wages and produce goods whose technologies are more variable across countries. If there is only one type of good, the model reduces to Eaton and Kortum (2002; EK henceforth) and delivers the gravity equation. This special case thus
country would need to be approximately 4.5 times higher than the cost of exporting from the United States (Figure 2 in his paper). See also Markusen and Wigle (1990), who estimated the effect of trade barriers on trade across countries of different income levels, and Hunter and Markusen (1988), who estimated the effect of nonhomothetic preferences. 3 See Bils and Klenow (1998), Deaton (1975), Grigg (1994), and Hunter (1991). 4 See Comin and Hobijn (2006), Nabseth and Ray (1974), Romer (1990), and Vernon (1966).
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delivers the same patterns of trade as other gravity models, and it predicts the same elasticity of trade with respect to income per capita and to population.5 I estimate the model with one type (EK model) and with two types of goods, using data on bilateral merchandise trade flows in 2000. I use two data sets— one containing 162 countries and the other containing a subset of 19 Organization for Economic Cooperation Development (OECD) countries. The data report the total value of trade for each importer–exporter country pair. The EK model explains trade in the OECD sample very well, but not trade in the full sample with countries of different size and income levels. The new model, in turn, explains trade among OECD countries as well as EK and significantly improves upon EK in explaining the full sample. The estimated parameters of the model are such that production technologies are more variable in the type of good whose demand is more income-elastic. Hence, rich countries consume and produce these goods more intensively. And the variability in their production technologies generates large price dispersions, which give rich countries large incentives to trade. Poor countries, in contrast, produce and consume relatively more goods whose technologies are similar across countries. As a result, they trade little. So, the new model simultaneously explains the large volume of trade among rich countries and the small volume among poor countries, patterns that cannot be reconciled with the EK model. For example, trade among the 20 richest countries in the sample accounts on average for 27% of these countries’ incomes. Similarly, the new model predicts this trade to be 16% of income, while the EK model predicts that it is only 2%. Trade among the 20 poorest countries, in turn, is less than 2% of these countries’ incomes according to the data, and the new and the EK model. Trade deficits with rich countries are larger for middle- than for low-income countries in the data. To capture this moment, the estimated parameters imply that as a country’s income per capita grows, its demand for goods with high income elasticity increases before its supply. So, middle-income countries are net importers of these goods, rich countries are net exporters, and poor countries barely consume or produce them. Counterfactual simulations illustrate the welfare consequences of this pattern. A positive technology shock in China makes poor and rich countries better off and middle-income countries worse off. The shock shifts Chinese demand toward high income-elastic goods, while maintaining China’s specialization in low income-elastic goods. Two price changes ensue. First, the price of low elasticity goods (Chinese exports) decreases relative to wages in most countries, a change that benefits primarily poor countries, the largest consumers of low elasticity goods. Second, the price of high elasticity goods (Chinese imports) increases relative to low elasticity goods. This price change benefits net exporters of high elasticity goods—rich countries—and it hurts net 5 See Anderson and Van Wincoop (2003), Helpman and Krugman (1985), Redding and Venables (2004).
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importers—middle-income countries. A technology shock in the United States, a rich country, has the opposite effect of the shock in China. I extend the model to admit income inequalities within countries and intermediate inputs. With intermediate inputs, rich countries may absorb relatively more goods with heterogeneous technologies for two reasons: these goods have high income elasticity of demand (as before) and their production requires relatively more inputs themselves with heterogeneous technologies. Data on trade flows do not distinguish between these two possibilities. In an additional exercise, I estimate the model with importer fixed effects in lieu of structural assumptions on preferences. Supply side parameters do not change much with this specification, and again I find that rich countries consume relatively more goods with heterogeneous technologies.6 This paper relates to several other strands of literature. Linder (1961) also proposed a theory of trade where trade flows are determined by demand patterns, through nonhomothetic preferences. The proximity to a large consumer market for high-quality goods, Linder argued, gives firms in rich countries a comparative advantage in developing and producing these goods. Additionally, when exporting, these firms find more extensive markets for their highquality goods in other rich countries. Applying this same rationale to other income levels, Linder predicted that trade volumes are larger across countries with similar income levels. The new model also predicts that countries of similar income levels consume and produce the same types of goods: rich countries, high income-elastic goods; and poor countries, low income-elastic goods. In contrast to Linder, however, trade volumes depend on how differentiated products are. With the estimated parameters, rich countries trade a lot with each other because high income-elastic goods are more differentiated, while poor countries do not trade much because low income-elastic goods are less differentiated. At the microlevel, recent studies found that unit prices within commodity categories increase systematically with importer and exporter per capita income.7 If prices reflect quality differences, these findings suggest that highincome countries import and export higher quality goods—just as Linder predicted. But these findings are not easily mapped to the new model, where goods are not differentiated by quality levels.8 6
I thank one of the anonymous referees for suggesting the fixed-effects estimator. See Choi, Hummels, and Xiang (2009), Fieler (2011a), Hummels and Klenow (2005), and Schott (2004). Empirical support for the Linder hypothesis in aggregate trade flows is mixed. See, for example, Bergstrand (1990), Hallak (2010), and Leamer and Levinsohn (1995). 8 In principle, one could interpret the high income-elastic goods in the model as high-quality goods and the low income-elastic goods as low-quality. But here, as consumers get richer, the consumption of all goods increases in absolute terms, and only the relative consumption of low income-elastic goods decreases. With quality differentiation, it is more natural to assume that the absolute consumption of low-quality goods decreases with wealth, as consumers substitute these goods for higher quality ones. 7
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Previous models of trade with nonhomothetic preferences are highly stylized.9 In contrast to these models, the new model allows one to simultaneously analyze all directions of trade—North–North, North–South, and South– South—and to analyze data. I also introduce a new method for estimating the EK model.10 The usual regression approach is not applicable to the new model because of nonlinearities in the expression for trade flows. The new method explores the general equilibrium feature of the model and avoids some of the endogeneity problems of regression analysis (see Section 3). Its application extends EK to a larger data set—EK used only data on trade in manufactures among 19 OECD countries. The paper is organized as follows. I present the theory in Section 2 and the empirical analysis in Section 3. Section 4 exploits counterfactuals. Section 5 extends the model to include income inequalities within countries and intermediate inputs. Section 6 concludes. The Appendix in the Supplemental Material (Fieler (2011b)) generalizes the demand side setup of the model, and presents details of the data, robustness checks, and Monte Carlo simulations. 2. THEORY I present the model in Sections 2.1 and 2.2, solve it in Section 2.3, and explain its workings in Section 2.4. Section 2.5 shows that the EK model is a special case of the new model. 2.1. The Environment There are N countries, and goods are divided into S types, each with a continuum of goods. Goods of type τ ∈ {1 2 S} are denoted by jτ ∈ [0 1]. All consumers in the world choose the quantities of goods jτ , {q(jτ )}jτ ∈[01] , to maximize the same utility function, (1)
S
1/στ
(ατ )
τ=1
στ στ − 1
1
(στ −1)/στ
q(jτ )
djτ
0
S where ατ > 0 are weights and στ > 1 for all τ. I normalize τ=1 (ατ )1/στ = 1. Parameter στ is typically associated with its role as the elasticity of substitution, but here it also governs the income elasticity of demand. To see this, consider any two types of goods, τ = 1 2, and let xτ be the total spending on 9
See Flam and Helpman (1987), Markusen (1986), Matsuyama (2000), and Stokey (1991). For other methods, see Alvarez and Lucas (2007), Eaton and Kortum (2002), and Waugh (2010). 10
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goods of type τ. Then, from the first-order conditions, spending on type 1 relative to type 2 satisfies 1−σ1 x1 σ2 −σ1 α1 P1 (2) =λ 1−σ x2 α2 P2 2 where Pτ is the constant elasticity of substitution (CES) price index of goods of type τ = 1 2 and λ > 0 is the Lagrange multiplier associated with the consumer’s problem. It is strictly decreasing in the consumer’s total income. The term in parentheses governs the level of x1 /x2 . It increases with α1 and decreases with P1 . The term (λσ2 −σ1 ), in turn, governs how x1 /x2 changes with consumer income. If σ1 > σ2 , the ratio x1 /x2 is decreasing in λ and hence increasing in income. At all levels of income and prices, the income elasticity of demand for type 1 goods relative to the elasticity for type 2 equals σσ1 .11 So, util2 ity function (1) allows for consumers of different income levels to concentrate their spending on different types of goods. Parameter στ governs both the elasticity of substitution across goods of the same type and the income elasticity of demand for those goods. Nothing in the results, however, depends on the elasticity of substitution. Demand affects trade only through the allocation of spending across types because, within types, the share of each exporter in a country’s imports does not depend on the elasticity of substitution, only on technologies.12 Accordingly, throughout the paper, the only relevance of στ is as the income elasticity of demand. (See also Appendix A for a more general utility function.) 2.2. Technologies Labor, the unique factor of production, is perfectly mobile across types and goods, and it is immobile across countries.13 Technologies vary across goods and countries. Let zi (jτ ) be the efficiency of labor to produce good jτ in country i. Assuming constant returns to scale, the unit cost of producing good jτ in country i is ziw(jiτ ) , where wi is the wage in country i. Trade is subject to iceberg costs. Delivering one unit of good jτ from country i to country n requires the production of dni units. Hence, its total cost is pni (jτ ) =
dni wi zi (jτ )
dλ τ The income elasticity of demand of type τ is xwτ dx = (−στ w dw ), where w is the consumer dw income. 12 This result extends from Eaton and Kortum (2002), where there is only one type and σ does not enter the expression of trade flows. 13 Labor can be interpreted more generally in the theoretical model as an input bundle, including capital. I use the term labor throughout, however, because that is the interpretation used in the empirical analysis of Section 3 below. 11
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I normalize dii = 1 for all i and assume the triangle inequality, dni ≤ dnk dki for all i, k, and n. With perfect competition, the price of good jτ faced by consumers in country n is pn (jτ ) = min{pni (jτ ) : i = 1 N} Following EK, I employ a probabilistic representation of technologies to derive the distribution of prices. For any z ≥ 0, the measure of the set of goods jτ ∈ [0 1] such that zi (jτ ) ≤ z is equal to the cumulative distribution function of a Fréchet random variable, (3)
Fiτ (z) = exp(−Ti z −θτ )
where Ti > 0 for all countries i = 1 N, and θτ > 1 for all types τ = 1 S. These distributions are treated as independent across countries and types. Figure 1 shows four densities of Fréchet distributions. Given θτ , the countryspecific parameter Ti determines the level of the distribution: a larger Ti increases the measure of goods with large, efficient technologies zi (jτ ). Thus, the assumption that Ti does not depend on type τ, made just for parsimony, implies that a country that is generally efficient at making goods of one type is also efficient at making goods of other types. Parameters θτ are common to all countries but may differ across types. These parameters govern the spread of the distribution: the larger the θτ , the smaller the variability in labor efficiencies across goods and countries. In Figure 1, the
FIGURE 1.—Examples of Fréchet distributions.
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decrease in θ from 20 to 5 increases the dispersion of the distribution of technologies across goods for a fixed T . And, importantly, it increases the dispersion of technologies across countries: it shifts the density with T = 100 away from the density with T = 10. This property of the Fréchet distribution gives a dual role to parameters θτ in the model. First, the variability of technologies across goods governs comparative advantage within types. A greater dispersion in labor efficiencies (a smaller θτ ) generates a greater price dispersion and, consequently, a greater volume of trade. Trade is more intense in types where θτ is small. Second, the variability of labor efficiencies across countries governs comparative advantage across types. The mean of the Fréchet distribution helps illustrate this point. The cost of delivering one unit of good jτ from country i to ni (jτ ) country n relative to the cost of producing it domestically is ppnn = zzni (j(jττ)) dniwnwi . (jτ ) Taking the expectation over jτ , we get (4)
−1/θτ Ti dni wi E(pni (jτ )) = E(pnn (jτ )) Tn wn
Two factors control the cost of producing goods in country i relative to producing them in country n: the ratio of their effective wages ( dniwnwi ) and the ratio of technology parameters ( TTni ). Parameter θτ controls the relative importance of these two factors. As θτ increases, the term ( TTni )−1/θτ approaches 1 and effective wages swamp technology in determining costs. Poor countries tend to specialize in types where θτ is large because they have low wages. Rich countries, in turn, specialize in types where θτ is small because, in general equilibrium, these are the countries with large labor efficiencies, that is, large Ti ’s. Although the model is static, this production setup can be seen as arising from a product cycle if parameter θτ is interpreted as the age of goods of type τ. When θτ is small, goods of type τ have just been invented and methods to produce them vary greatly across countries. A good at this stage is produced in the typically high income country where it was invented. As θτ increases, methods to produce goods of type τ become standardized (less variable across countries) and production shifts to countries with low labor costs. 2.3. Equilibrium Following the literature, I do not distinguish between population and labor force. All countries have a continuum of individuals who supply inelastically one unit of labor. Let Li be the population of country i. Assume that (θτ + 1) > στ for all τ = 1 S, the well known necessary condition for a finite solution (see Eaton and Kortum (2002)). I denote the spending of an individual with small x and the spending of countries with capital X, and where needed, I use subscripts to specify type (τ), importer (n), and exporter (i).
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Given wages wi and trade costs dni , the distribution of technologies yields the distribution of prices, which, together with the utility function, yields the demand function. The spending of a typical consumer in country n on goods of type τ is (5)
1−στ τ xnτ = λ−σ n ατ Pnτ
where λn > 0 is the Lagrange multiplier associated with the consumer’s prob S lem. It is implicitly defined through the budget constraint, τ=1 xnτ = wn , as a continuous and strictly decreasing function of income wn . The CES price inτ )]1/(1−στ ) (Φnτ )−1/θτ , where is the dex is the same as in EK, Pnτ = [ ( θτ +1−σ θτ
N gamma function and Φnτ = i=1 Ti (dni wi )−θτ . The spending of a consumer in country n on goods of type τ from country i is (6)
xniτ =
Ti (dni wi )−θτ xnτ Φnτ
Country n’s imports from country i total (7)
Xni = Ln
S
xniτ
τ=1
By equating supply to demand, we get the labor market clearing conditions: (8)
Li wi =
N
Xni
for i = 1 N.
n=1
This completes the statement of the model. To summarize, an economy is defined by a set of N countries, each with its population Li , technology parameter Ti , and location implied by trade barriers {dni }ni≤N , and by a set of S types, each with its technology parameter θτ and preference parameters ατ and στ . Given wages w, the matrix of trade flows {Xni }ni≤N is given by equations (5)– (7). An equilibrium is a set of wages w ∈ Δ(N − 1) that satisfies the market clearing condition (8). 2.4. Income per capita and Trade Patterns This section explains how income per capita affects trade in the model. Consider the case with two types of goods, A and B, as in the empirical analysis of Section 3 below. If preferences were homothetic, the distribution of in-
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come across goods would be independent of income levels. But by equation (5), country n’s demand satisfies
(9)
1−σ XnA αA PnA A = (λn )σB −σA 1−σ XnB αB PnB B
which is the same as equation (2). Assuming σA > σB , the relative spending on , is decreasing in λ and increasing in wealth. goods of type A, XXnA nB Ultimately, however, we are interested in how this ratio affects trade. Let Xniτ be country n’s spending on goods of type τ from country i. From equation (9), country n’s imports from country i relative to its domestic consumption, Xni niA niB , is mostly determined by XXnnA if country n is rich and by XXnnB if it is poor. Xnn By equation (6), −θA XniA Ti dni wi = XnnA Tn wn
and
−θB XniB Ti dni wi = XnnB Tn wn
These are the expressions on the right-hand side of equation (4), raised to the power (−θτ ), so the conclusions drawn there follow: If θτ is large, the variability in production technologies across goods and countries is small, and consumers place more emphasis on the effective cost of labor ( dniwnwi ) than on technology parameters ( TTni ). To make this point clearer, suppose that θA < θB , as in the empirical results below, and that country n is poor. Then ( dniwnwi ) > 1 in general because wn is small and dni > 1. Hence, ( dniwnwi )−θB is close to zero because θB is large. ni niB Country n’s imports are then small, XXnn ≈ XXnnB = TTni ( dniwnwi )−θB ≈ 0. In words, poor countries have few incentives to trade because they consume goods whose technologies are similar across countries. Their lowest cost is typically attained domestically, with cheap labor and no trade costs. ni niA The scenario changes if country n is rich and XXnn ≈ XXnnA . Since θA is small, dni wi −θA the term ( wn ) is relatively close to 1 irrespective of whether ( dniwnwi ) is smaller or greater than 1. Then XXniA is largely determined by the ratio TTni . nnA Two effects follow. First, rich countries trade more than poor countries because their consumers put less emphasis on trade barriers and wages (dni wi ). Second, they trade more with other rich countries, whose technology parameters Ti are large. In accordance with the empirical evidence mentioned in the Introduction, the model predicts trade to be more intense among rich countries whenever σA > σB and θA < θB .
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2.5. A Special Case: The Eaton–Kortum Gravity Model The new model reduces to the EK model under two special cases. The easiest case arises when there is only one type of good: ατ = 1 for some τ. Production efficiencies are distributed as per EK (equation (3)) and the utility function becomes 1 στ q(jτ )(στ −1)/στ djτ στ − 1 0 which represents standard homothetic CES preferences. Country n’s imports from country i are then (10)
Xni = Xniτ =
Ti (dni wi )−θτ Xn Φnτ
where Xn = wn Ln is country n’s total income. This is the solution to the EK model.14 Trade flows depend on total income and not on its division into income per capita and population. The second case arises when θτ = θ for all τ = 1 S. Then the share of country i in country n’s consumption of type τ is Xniτ Ti (wi dni )−θ = N Xnτ Tk (wk dnk )−θ k=1
which does not depend on type τ. Hence, the share of country i in country n’s −θ i dni ) , as in equation (10). total consumption is the same, XXnin = N Ti (w T (w d )−θ k=1 k
k nk
This last case shows that nonhomothetic preferences alone do not modify trade flows. Consumers of different income levels may demand different types of goods, but if the distribution of technologies is equal across types, they source goods from the same countries. The converse is not true. If στ = σ for all τ, preferences are homothetic, but technologies may differ across types. Although I do not present the results, I estimate the model with this restriction and find the R2 to be about two-thirds closer to the unrestricted model than to the EK model. (Section 3.1.2 below discusses separately the effect of each parameter on trade.) 3. EMPIRICAL ANALYSIS I use data on bilateral merchandise trade flows in 2000 from the U.N. Comtrade data base (United Nations (2008)), and data on population and income 14 Eaton and Kortum (2002) considered only trade in manufactured products. So instead of country n’s total income, Xn , they have its manufacturing absorption.
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from the World Bank (2008). The data comprise 162 countries and account for 95% of world trade in 2000. There are 25,810 observations, each containing the total value of trade for an importer–exporter country pair. The list of countries and the details of the data and their compilation are supplied in Appendix B. Data specific to country pairs—distance between their most populated cities, common official language, and borders—are compiled by Mayer and Zignago (2006). The objective of this section is to evaluate qualitatively and quantitatively the ability of the model to explain bilateral trade flows. I consider only the special case with two types of goods, denoted by A and B, and I use two empirical specifications15 : the first is more efficient—it recovers simultaneously all parameters from the demand and the supply sides. The second uses importer fixed effects—it is more robust because it recovers the supply side parameters without putting structure into demand; demand parameters are estimated in a subsequent step. The first empirical specification is given in Section 3.1 and the second is given in Section 3.2. 3.1. Empirical Specification 1: Restricted Model I present the estimation procedure in Section 3.1.1, identification issues in Section 3.1.2, and results in Section 3.1.3. 3.1.1. Empirical Specification 1: Procedure Equations (5), (6), and (7) above imply that country n’s imports from country i equal (11)
Xni = Ln (xniA + xniB )
where, for τ = A B, xniτ =
Ti (dni wi )−θτ xnτ Φnτ
1−στ xnτ = (λn )−στ ατ Pnτ
Φnτ =
N
Ti (dni wi )−θτ
i=1
1/(1−στ )
θτ + 1 − στ Pnτ = (Φnτ )−1/θτ θτ 15
As a robustness check, I also estimate the model with three and with four types. In both cases, the trade flows predicted by the model do not change at all with respect to the two-type case, and the type-specific parameters ατ and θτ are not identified.
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(αA )1/σA + (αB )1/σB = 1, and the Lagrange multiplier λn is implicitly defined through the budget constraint of a typical consumer in country n, xnA + xnB = wn . Trade flows are a function of the set of N countries, each with its population Li , wage wi , technology parameter Ti and trade costs dni ; parameters θA and θB that control the spread of the distribution of technologies; utility parameters σA and σB that control the income elasticity of demand; and the weight of type A goods in preferences αA . I take the set of countries, their population Li , and wages wi from the data, and I estimate {dni }Nni=1 , {Ti }Ni=1 , θA , θB , αA , σA , and σB . Trade Barriers dni . Assume that trade costs take the form (12)
dni = 1 + {(γ1 + γ2 Dni + γ3 D2ni ) ∗ γborder ∗ γlanguage ∗ γtrade agreement }
for all n = i, and dnn = 1. The term in brackets is the proxy for geographic barriers, and the number 1 added to it is the production cost. Variable Dni is the distance (in thousands of kilometers) between countries n and i. So the term in parentheses is the effect of distance on trade costs. Parameter γborder equals 1 if countries n and i do not share a border, and it is a parameter to be estimated otherwise. If γborder is, say, 0.8, sharing a border reduces trade costs by 20%, but it does not affect production costs; if γborder > 1, sharing a border increases trade costs. Similarly, parameters γlanguage and γtrade agreement refer, respectively, to whether countries n and i share a language, and whether they have a trade agreement.16 Henceforth, Υ = {γ1 γ2 γ3 γborder γlanguage γtrade agreement } refers to the set of trade cost parameters and Υ˜ refers to the set of data on countries’ pairwise geopolitical characteristics—distance, common border, language, and trade agreement. Technology Parameters Ti . Given parameters {Υ θA θB αA σA σB }, data on population L, and geopolitical characteristics Υ˜ , equilibrium conditions (8) pin down a relation between technology parameters {Ti }Ni=1 and market clearing wages {wi }Ni=1 . One can either use technology parameters T to find market clearing wages w or use wages to find technology parameters. I use the latter 16 I use only the trade agreements that the World Trade Organization (WTO) lists as the best known: Association of Southeast Asian Nations (ASEAN), Common Market for Eastern and Southern Africa (COMESA), European Free Trade Association (EFTA), European Union, Mercosur, and North American Free Trade Agreement (NAFTA). Usually, an exponential functional form is assumed for trade costs, for example, dni = exp(γ1 + γ2 Dni + γ3 D2ni + γborder + γlanguage + γtrade agreement ), which facilitates log-linearizing regression models. In my estimation procedure, this convenience is useless and the choice between these two functional forms makes no difference in the empirical results. I chose equation (12) because, unlike the exponential function, its parameters are easily interpretable.
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approach. I take income per capita from the data as a proxy for wages.17 Then, for each guess of the parameters, I simulate the whole economy, generating trade flows Xni until I find technology parameters T that satisfy equilibrium conditions (8).18 This procedure reduces the number of parameters to be estimated by N = 162. Empirical Specification. Let zni = XXnniXi be the trade flow from country i to country n, normalized by the product of the two countries’ total income Xn Xi (= wn Ln wi Li ), and let z be the corresponding vector. After substituting trade costs of equation (12) and the implicit solutions for technology parameters Ti , the stochastic form of equation (11) becomes (13)
z = h(w L Υ˜ ; Υ θA θB αA σA σB ) + ε
The vector of trade flows z is a function h of the data {w L Υ˜ } and of the 11 parameters to be estimated {Υ θA θB αA σA σB }, plus the error term ε. I estimate equation (13) using non-linear least squares (NLLS).19 Anderson and Van Wincoop (2003) also estimated a trade model with NLLS by simulating the whole economy to get endogenous variables, and their discussion on the error terms holds here. The estimator is unbiased if ε is uncorrelated with the derivative of h with respect to w, L, and Υ˜ . This assumption holds if the error term is interpreted as measurement error. But error terms can enter the model in many other ways, about which the theory offers no guidance. In particular, one could add an error term to the specification of trade costs in equation (12) to account for the possibility that factors other than the geopolitical characteristics Υ˜ affect trade costs. Then standard econometric theory requires the error term to be uncorrelated with Υ˜ . But even if this condition is satisfied, a problem arises here because changes in trade costs dni affect equilibrium wages, which affect the estimates of technology parameters Ti , which in turn affect trade flows. Simulations suggest, however, that these general equilibrium effects of structural errors in trade costs are likely to be small—introducing large multiplicative shocks to trade costs yields very small changes to equilibrium relative wages. 17 From a theoretical viewpoint, it is easy to introduce the distinction between population and labor force by making the labor endowment of individuals in country i equal to a fraction βi < 1, corresponding to the labor force participation in country i. While this modification complicates the notation, its impact on the empirical results is nil. 18 Alvarez and Lucas (2007) proved the existence and uniqueness of equilibrium in the EK model. The new model satisfies standard conditions for existence (Mas-Colell, Whinston, and Green (1995, Chap. 17)), but I do not prove uniqueness. Still, I did not encounter any cases where the relation between w and T in the market clearing conditions was many-to-one or one-to-many. In Appendix D, I perform Monte Carlo simulations and find that the parameters are well identified, which suggests that the equilibrium is unique. The United States’ technology parameter Ti is normalized to 1. All Fortran programs are available in the Supplemental Material. 19 See Appendix C for the normalization of trade flows and scaling of observations.
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For the sake of comparison, EK estimated their model by introducing error terms to the trade cost specification and used instrumental variables or fixed effects to avoid endogeneity issues with wages. The estimation procedure above accounts for endogenous wages by simulating the whole economy. Its disadvantage is not correcting for potential endogeneity problems that arise from structural errors in trade costs.20 Identification of σ and θ. EK were not able to identify parameters θ and σ from trade data. In the EK model, parameter σ does not enter the expression for trade flows and, assuming exponential trade costs, parameter θ enters only as coefficients on trade costs. In the new model, parameters στ and θτ could in principle be identified, because they cannot be factored out of the expression for trade flows. But in practice, fixing different values for θA and for σA does not change the objective function. Parameters θA and θB are not separately identifiable from trade cost parameters Υ . A decrease in θA and θB increases the variance of the distribution of technologies, which in turn increases trade across all country pairs. This effect also arises if trade costs Υ decrease. Data on bilateral trade flows do not distinguish between these two changes—a decrease in θτ or in Υ . To recover the remaining parameters, however, I need to fix θA (or by symmetry, θB ). I set θA = 828, the median estimate in EK. Parameters σA and σB are not separately identifiable either. These parameters govern how spending across types changes with income per capita, but they play no individual role. As in the case of θτ , I need to fix σA to estimate the other parameters. I set σA = 5, an arbitrary value that satisfies the assumption σA < θA + 1. In Appendix C, I experiment with several values for θA and σA , and find that the objective function does not change, thus confirming that these parameters are not identified. The interpretation of the parameter estimates and of the results below is the same for all values of θA and σA in the Appendix. 3.1.2. Data and Parameter Identification This section explains intuitively the features of the data that allow for the identification of the parameters to be estimated: Υ , αA , σB , θB . Without loss of generality, let σA ≥ σB so that type A goods are more income-elastic and govern patterns of trade among rich countries. Given the heterogeneity of production technologies in type A, θA = 828, trade flows among rich countries provide information on trade costs Υ . Given Υ , the volume of trade among poor countries provides information on θB : the larger is θB , the smaller is the 20
The problem with using the EK estimation procedure here is not only that it is not applicable to the new model, but it is also not applicable to data sets with zero trade flows (such as the one I use) because it involves taking the logarithm of trade flows.
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heterogeneity in type B production technologies and the smaller is the volume of trade among poor countries. The volume of trade between rich and poor countries, in turn, provides information on preference parameter σB (σA is fixed to 5). If the income elasticity of demand is equal across types, σB = σA , the model predicts large volumes of trade between rich and poor countries because these countries specialize in different types of goods. If instead σB < σA , as the empirical results below indicate, demand patterns suppress trade among countries of different income levels. Rich countries demand relatively more type A goods, generally produced in rich countries, while poor countries demand more type B goods. Finally, parameter αA —the weight of type A goods in preferences—governs the size of sector A relative to sector B, thereby controlling the size of the “rich” and “poor” country groups above. Appendix D presents Monte Carlo experiments. I simulate data with random parameters, apply the optimization algorithm to simulated data, and find that the algorithm can recover the parameter values with precision. 3.1.3. Empirical Specification 1: Results I first present the results of the new model and then those of the EK model. The third column of Table I presents the parameter estimates.21 The model explains R2 = 42% of the data. Figure 2 plots the contribution of each importer to the objective function—observations in the graph sum to 58% (1–42%). Residuals tend to be larger for small countries, but excluding the 20 smallest countries in the sample does not change the results much.22 The new model introduces three parameters to the EK model: αA , σB , θB . Types A and B coexist in the economy, (αA )1/σA ∈ (0 1); type A has a higher income elasticity of demand, σA > σB (σA = 5 and σB = 129), and it presents more heterogeneous production technologies, θA < θB (θA = 828 and θB = 1434). Consumption of type A ranges from 88% of Japan’s gross domestic product (GDP) to 1 × 10−6 of the Democratic Republic of Congo’s, while supply ranges from 99.6% of Luxembourg’s GDP to 1 × 10−13 of the Democratic Republic of Congo’s. The hypothesis that σA > σB and θA < θB cannot be rejected at a 0.1% significance level. So rich countries produce and consume relatively more goods with heterogeneous technologies. As a result, they trade a lot, while poor countries trade little. For example, trade among the 20 richest countries in the sample accounts on average for 27% of these countries’ GDP, while trade among the remaining 142 countries accounts for only 16% of these countries’ GDP. The new model predicts that these numbers are 16% and 9%, respectively, while the EK model 21 Standard errors are clustered by importer and exporter. I use the formula in Cameron, Gelbach, and Miller (2011) for multi-way clustering. 22 Parameter estimates are within the 99% confidence interval of the original estimates, and the stylized facts of Figures 3, 4, and 5 continue to hold.
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NONHOMOTHETICITY AND BILATERAL TRADE TABLE I ESTIMATION RESULTSa EK Model
New Model
OECD Only
Full Sample
Specification 1
Specification 2
8.28
8.28
8.28 5.00
8.28
1.24 (0.10)
1.96 (0.08)
1.38 (0.04)
1.28 (0.07)
γ2
0.84 (0.23)
0.26 (0.09)
0.20 (0.03)
0.13 (0.03)
γ3
−0.21 (0.09)
0.00 (0.03)
−0.01 (0.00)
0.00 (0.00)
Border
0.98 (0.06)
0.85 (0.05)
0.99 (0.07)
0.94 (0.05)
Language
1.01 (0.10)
0.92 (0.06)
0.93 (0.05)
0.93 (0.05)
Trade agreement
0.91 (0.04)
1.27 (0.12)
1.22 (0.12)
1.24 (0.09)
θB
14.34 (0.95)
19.27 (3.89)
(αA )1/σA
0.82 (0.03)
σB
1.29 (0.10)
Normalized parameters θA σA Estimated parameters γ1
R2 Number of observations
74% 342
34% 25,810
42% 25,810
67% 25,810
a Standard errors in parentheses are clustered by importer and exporter.
(below) predicts 2% and 5%, respectively: it underestimates trade in general and reverses the order, predicting less trade among the rich than among the remaining countries. In Figures 3, 4, and 5, graph (a) refers to the data, graph (b) refers to the EK model, and graph (c) refers to the new model. Figure 3 plots countries’ ) against income per capita. Since the estimation trade share (i.e., imports+exports 2∗GDP procedure implies that observed and predicted incomes, on the x-axes, are the same, the graphs differ only because of differences between observed and estimated trade shares, on the y-axes. The data show an increasing, statistically and economically significant relation between trade share and income per capita. The model correctly predicts the increasing relation, but overestimates its magnitude. The slopes in Figure 3
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FIGURE 2.—The contribution of each importer in the objective function as a function of its total income.
imply that the richest country in the sample is expected to trade 16% more of its income than in the poorest country according to the data and 36% more according to the model.23 Figure 4 substitutes income per capita in the x-axis of Figure 3 with total income. A country’s total income contains little information on its trade share: the slope of the regression line in Figure 4(a) is small and statistically insignificant, and the R2 = 001. Similarly, the new model predicts a small slope and an R2 = 003. Size has two opposing forces in the model. First, large countries tend to trade less in general equilibrium. In a two-country world, for example, trade always represents a smaller fraction of the large country’s income than of the small country’s. Second, large countries tend to be richer and hence trade more. Figure 4(a) and (c) both display large variance in trade shares among countries of medium size. Countries with high income per capita and small population have large trade shares (e.g., Singapore and Hong Kong), while countries with low income per capita and large population have small trade shares (e.g., Pakistan and Bangladesh). Figure 5 illustrates countries’ choice of trading partners. For each country, I calculate the fraction of its trade that flows to or from one of the 20 richest 23 These countries are Luxembourg and the Democratic Republic of Congo, and the ratio of their income per capita is 538.
FIGURE 3.—Income per capita × trade share.
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FIGURE 4.—Total income × trade share.
NONHOMOTHETICITY AND BILATERAL TRADE
FIGURE 5.—Income per capita × trade with 20 richest countries.
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countries in the sample. Figure 5 plots this fraction against income per capita. Graph 5(a) refers to the data and graph 5(c) depicts both observations of the data (asterisks) and of the model (hollow diamonds). The observations of the model are well aligned with the data. In particular, both graphs show an increasing relation: rich countries trade relatively more with other rich countries. The EK Model. The EK model provides a good benchmark for the results above because the new model is built on EK and EK delivers the gravity equation. Using the procedure in Section 3.1.1 above, I estimate the EK model twice—once with the full sample and once with a subsample of the 19 OECD countries used by EK.24 Table I presents the results. EK used their model to study manufacturing trade among OECD countries, while I study trade across countries of different sizes and income levels. So, not surprisingly, the EK model predicts very well trade among OECD countries, but not trade in the full sample—the R2 decreases from 74% with the OECD sample to 34% with the full sample. Moreover, the EK and the new model yield similar predictions for the OECD sample, but not for the full sample.25 The EK model cannot reconcile the large volumes of trade among rich countries with the small volumes among poor countries. As (γˆ 1 γˆ 2 γˆ 3 ) changes from (124 084 −021) in the OECD sample to (196 026 −0001) in the full sample, trade costs increase for all OECD country pairs, and they increase by 80% on average. More generally, the model is not flexible enough to predict the increasing relationship between trade shares and income per capita (Figure 3). As a result, it systematically overestimates the share of poor countries in trade (Figure 5(b)). To correct for this shortcoming without overestimating trade among poor countries, the model underestimates trade shares for all countries.26 3.2. Empirical Specification 2: Importer Fixed Effects Trade flows in equation (11) can be rewritten as XnA Ti (dni wi )θB Xni XnA Ti (dni wi )θA zni = (14) + 1− = Xn Xi Xn Xi ΦnA Xn Xi ΦnB 24 With the EK model, after normalizing parameter θA and backing out the technology parameters Ti , trade flows are exclusively a function of trade cost parameters Υ . Parameters αA , σA , σB , and θB do not exist or do not affect trade flows in the EK model (see equation (10)). 25 The results of the new model with the OECD sample are not given in the table. Even though the new model adds three free parameters to EK, the R2 increases by only 0.5%, and the equiva1/σ lence of the two models cannot be ruled out since αA A is not statistically different from 1. 26 See Appendix C for a discussion on the weighting of observations. The finding that the EK model does not reconcile the large volumes of trade flows observed among large, rich countries with the small volumes of poor countries is robust to the choice of weights. But if more weight is placed on large relative to small countries, the model tends to overestimate trade among poor countries instead of underestimating trade among rich countries, as it does here.
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The share of country i in country n’s imports is a weighted average of its share is the share of type A in counin the two types of goods, where the weight XXnA n try n’s spending. The empirical specification below takes the weights XXnA as imn porter fixed effects, denoted by Fn = XXnA . The rest follows Section 3.1.1 above: n equation (12) specifies trade costs dni , and technologies Ti are implicitly defined through the market clearing conditions. Then (15)
z = g(w L Υ˜ ; F Υ θA θB ) +
Normalized trade flows z are a function g of the data on income per capita w, population L, and geopolitical characteristics Υ˜ , and of the 170 parameters to be estimated—the vector of N = 162 fixed effects F , six trade cost parameters Υ , and θA and θB —plus an error term . I estimate equation (15) using NLLS. If specification (13) is consistent, then specification (15) is also consistent. The first specification (13) is more efficient, and the second (15) is more robust because it makes fewer assumptions on demand: it does not specify how spending across types A and B varies with importer characteristics. As in specification (13), parameter θA is not identifiable in practice. I fix θA = 828 and obtain the same results with different values for θA in Appendix C. The last column of Table I presents the results. The estimates of Υ and θB do not change much from the previous estimates (third column).27 As before, rich countries produce and consume relatively more type A goods. These goods have more heterogeneous technologies (θA < θB ), and their share in spending (fixed effects) is increasing in income per capita, as shown in Figure 6. The latter pattern remains even after controlling for prices, which suggests that type A has a higher income elasticity of demand. I use the fixed effects to estimate demand parameters. From equation (5), Fn = [(λn )−σA αA PnA A ]/wn 1−σ
for all n = 1 N,
N τ )]1/(1−στ ) (Φnτ )−1/θτ , with Φnτ = i=1 Ti (dni wi )−θτ . The where Pnτ = [ ( θτ +1−σ θτ term (Φnτ )−1/θτ depends exclusively on parameters T , Υ , and θτ , estimated in equation (15). Then (16)
−1/θ −1/θ Fn = k wn ΦnA A ΦnB B ; αA σA σB
for all n = 1 N.
27 Testing for the equality of the supply side parameters {Υ θB } would be useful to check for the consistency of the model, but it is a complicated statistical problem. Standard tests do not apply because the covariance across the two estimators is unknown. Furthermore, the error terms in equations (13) and (15) are heteroskedastic and clustered by importer and exporter, making bootstrapping infeasible (without ad hoc structural assumptions on the distribution of error terms).
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(b)
ANA CECÍLIA FIELER FIGURE 6.—(a) The coefficient on log of per capita income is 0.12 (standard error 0.02) and it is 0.08 (0.02) with price controls. (b) Importer fixed effects and their estimates as a function of income per capita.
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Fixed effects Fn are a function k of wages wn , price terms (Φnτ )−1/θτ , and preference parameters αA , σA , and σB . I estimate {αA σA σB } nonlinearly from −1/θ −1/θ equation (16) using the first-stage estimates for Fn , ΦnA A and ΦnB B as data. As before, I fix σA = 5. Figure 6(b) compares the fixed effects of the first-stage estimation to their 1/σ predicted values. The parameter estimates are αA A = 068 and σB = 346. Hence, the fundamental results from specification (13) hold in the more robust specification (15): trade patterns in the data support the hypothesis that goods with more heterogeneous technologies have higher income elasticity of demand (θA < θB and σA > σB ). 4. COUNTERFACTUALS I now analyze counterfactuals. Since the model is highly stylized, the purpose of the exercise is not to pursue policy recommendations, but a better understanding of the model itself. Section 4.1 experiments with technology shocks and Section 4.2 experiments with changes in trade costs. The procedure is as follows. An economy is defined with the population of each country, taken from the data, and with the estimates from Section 3.1 of parameters T , αA , σA , σB , θA , θB , and of the matrix of trade costs dni through the estimate of Υ . Initially, wages in the data clear the market. For each counterfactual, I change the parameters that define the economy, solve the market clearing conditions (8) to get new wages, and recalculate utility in every country. 4.1. Technology Shocks A technology shock in country i is a unilateral increase in its technology parameter Ti . Its welfare impact depends on the net exports of the different types of goods because relative prices generally change. Figure 7 plots the production, demand, and net exports of type A goods as a fraction of GDP, against income per capita. Each observation corresponds to a country. The circles denote the share of type A in production, and the triangles denote its share in demand. Both curves are upward sloping because richer countries produce and consume relatively more type A goods. The crosses are the net exports of type A (production minus demand). They form a V-shaped curve: they are small for low- and high-income countries and large and negative for middle-income countries. Low-income countries produce and consume mostly type B; highincome countries produce and consume type A; middle-income countries consume more type A goods than they produce. Patterns for type B are implicit in Figure 7, since their demand and supply equal 1 minus the demand and supply of type A. Bilateral imbalances mostly explain the pattern in Figure 7. As the figure suggests, the trade deficit with rich countries is larger for middle- than for lowincome countries in the data and in the model. So even with symmetric trade
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FIGURE 7.—Production, demand, and net exports of type A goods.
costs, the model can generate bilateral imbalances to better fit the data. To check for this explanation, I eliminate bilateral imbalances from the data by imputing for each country pair the average between their imports and exports. When the model is estimated with these modified data, the lag between demand and supply of type A goods in Figure 7 largely (though not completely) disappears. This lag also implies, in general equilibrium, that rich countries are net exporters of high income-elastic goods, a prediction that is consistent with theoretical work on trade and nonhomothetic preferences.28 Between 1985 and 2000, China grew nearly four times relative to the rest of the world. To view the effects of continued growth in China, I increase China’s technology parameter TChina until its wages increase by 300% relative to the rest of the world. Chinese consumption of type A goods increases from 1% to 21% of GDP, and its production increases only from 1 × 10−7 to 3%. Two price changes ensue. First, the price of type B decreases relative to wages in most countries, because the productivity gains accrue mostly to goods of type B, in which China specializes. This price change benefits primarily poor countries, the largest consumers of type B goods. Second, the price of type A goods 28 See Fajgelbaum, Grossman, and Helpman (2009), Flam and Helpman (1987), Matsuyama (2000). Empirically, Choi, Hummels, and Xiang (2009), Fieler (2011a), and Schott (2004), among others, find that unit prices in trade increase with importer and exporter per capita income. If prices proxy quality levels, this suggests that high-income countries trade relatively more highquality goods, but it is not a finding about net exports.
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increases relative to type B, because their demand increases more than their supply. This price change benefits rich countries, net exporters of type A, and it hurts middle-income countries, the largest net importers of type A. In sum, the shock in China benefits poor and rich countries, and it hurts most middle-income countries. Wages in the 50 richest countries increase by 5.4% relative to the 50 poorest countries. The largest real wage increases occur in China’s rich and poor neighbors (e.g., Hong Kong (3.5%), Mongolia (2.1%), and Japan (1.0%)) and the largest real wage decreases occur in China’s middleincome neighbors (e.g., Malaysia (−0.6%) and Thailand (−0.3%)). This prediction is broadly consistent with the evolution of world income from 1980 through 2000, illustrated by Leamer (2007) and Leamer and Schott (2005): income per capita increased in rich and poor countries, and it largely stagnated in middle-income countries. Leamer and Schott conjectured that the growth of China and India hurt middle-income countries because these countries compete directly with goods produced in poor countries, while rich countries specialize in other goods. The model encompasses this mechanism and adds to it the possibility for poor countries to gain as consumers of Chinese and Indian goods. Next, I experiment with an increase in the United States’ technology parameter TUS that increases American wages by 25% relative to the rest of the world. Contrary to the shock in China, a shock in the United States decreases the price of type A goods relative to wages and to type B goods. It thus inverts the welfare effects of the Chinese shock: all middle-income countries benefit from the shock; most rich countries are made worse off and poor countries are left close to indifferent. The shock decreases nominal wages in the 30 richest countries in the sample by 1.1% relative to the rest of the world. The largest real wage increases occur in Mexico (0.3%) and in small, middle-income Central American countries (approximately 2.5%). Japan, Norway, and Switzerland experience small welfare losses. 4.2. Trade Barriers I consider two extreme changes in trade barriers: (i) eliminating trade barriers (dni = 1) and (ii) raising trade barriers to autarky levels (dni → ∞ for all n = i). As in standard models, small countries are the most affected by changes in trade costs, but different from standard models, after controlling for size, poor countries are less affected than rich countries. Eliminating trade barriers benefits all countries. Real wages increase by 16% in the United States and by more than 140% in some small countries. A move to autarky, in turn, makes all countries worse off. Real wages decrease by 0.1% in India and by 18% in Luxembourg. So, as others have found, changes in welfare are larger when the world moves to frictionless trade than to autarky (Waugh (2010)).
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5. EXTENSIONS I introduce into the model income inequality within countries in Section 5.1 and intermediate inputs in Section 5.2. 5.1. Income Inequality Income inequalities within countries affect demand patterns and, thereby, trade. For 121 countries in the sample, the World Bank (2008) provides data on the share of income held by each quintile of the population. I use these data to reestimate the model for this subset of countries. Instead of calculating demand for a single representative consumer in each country, I calculate it for five consumers, each representing one income quintile. A country’s total demand is the sum of these representative consumers weighted by 20% of the population. Introducing income inequality within countries does not change the results in Section 3. Parameter estimates practically do not change, and the R2 increases by 1% in the first specification (13) and it decreases by 0.2% in the second (15). Because of this small difference and because of the paucity of data on income distribution, Section 3 above presents only the results with no income inequality within countries.29 5.2. Intermediate Inputs The model of Section 2 can be extended to admit intermediate inputs, as in Eaton and Kortum (2002). Instead of only labor, assume that production requires a Cobb–Douglas bundle of inputs. The cost of each input bundle is ciτ = w
1− i
S
τ =1
βττ
S
(Piτ )βττ
for τ = 1 S,
τ =1
where βττ > 0 is the share of goods of type τ in the production of type τ, and
S (1 − τ =1 βττ ) > 0 is the labor share. Country n’s absorption of type τ is S βτ τ wn Lnτ + Ln xnτ (17) Xnτ = S τ =1 1− βτ τ
τ
=1
29
Data are available for only 121 countries, and even these countries only report the distribution of income sporadically. To collect data, I used reports from 1990 to 2007, giving priority to the information available in the year closest to 2000, the year of the trade data.
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where Lnτ is the labor of country n allocated to the production of type τ, and xnτ is the per capita final consumption of type τ in country n. Assum1−στ τ ing utility function (1), xnτ is given by equation (5): xnτ = λ−σ , where n ατ Pnτ λn is the Lagrange multiplier, implicitly defined through the budget constraint
S wn = τ=1 xnτ . Additionally, assuming the distribution of efficiencies in equation (3), country n’s imports of goods of type τ from country i are Ti (dni ciτ )−θτ Xnτ Φnτ
N −θτ where Φnτ = . The CES price indices are Pnτ = i=1 Ti (dni ciτ ) θτ +1−στ 1/(1−στ ) −1/θτ [ ( θτ )] (Φnτ ) . So the only change in trade shares XXniτ and in the nτ price indices is the substitution of wages wi for the cost of the input bundle ciτ . The market clearing conditions are N S wi Liτ = 1 − (19) βττ
Xniτ for i = 1 N and τ = 1 S, (18)
Xniτ =
τ =1
n=1
and (20)
Li =
S
Liτ
for i = 1 N.
τ=1
This completes the statement of the model with intermediate inputs. An equilibrium is a set of wages {wi }Ni=1 and a set of labor allocations {{Liτ }Ni=1 }Sτ=1 that satisfy the system of equations (19) and (20), where the terms Xniτ are given by equations (17) and (18). Empirically, there are a few differences between the model with and without intermediates. First, without intermediates, countries cannot trade more than their total income (which is clearly not a binding constraint in the empirical exercise of Section 3). Second, there are two potential sources of nonhomotheticity in demand in equation (17). To understand them, suppose again that there are only two types, A and B, and that type A presents more heterogeneous technologies (θA < θB ) so that, in equilibrium, countries with large technology parameters Ti have high wages and specialize in type A. Then rich countries consume relatively more type A goods either if type A is more income-elastic (σA > σB , as before) or if producing type A goods requires relatively more type > ββBA ), or both. Data on total trade flows do not distinA intermediates ( ββAA BB AB guish between these possibilities, so the parameters introduced by the model with intermediate goods are not identified in the data.30 30
See Yi (2003) for a study of intermediate goods in trade.
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6. CONCLUSION An integrated trade model, one that provides a single framework for trade among rich countries as well as trade among countries of different income levels, has long concerned economists. Generally speaking, North–North trade is explained through the differentiation of goods and services, while North– South trade is explained through differences in comparative advantage due to technologies or factor endowments. This paper proposes a model that delivers both North–North and North–South patterns. Trade among rich countries occurs primarily in differentiated goods, while trade of rich with poor countries occurs across sectors. A quantitative comparison of the predictions on trade flows of this integrated model to those of a gravity model shows the benefits of the integrated approach. Theoretical foundations of the gravity relationship are typically based on intraindustry trade of differentiated goods. So, not surprisingly, the EK gravity model explains trade among the rich OECD countries well but not trade among countries of different income levels. The new model, in turn, explains trade among OECD countries as well as EK and it explains trade in the sample with 162 countries much better than EK. For example, the model correctly predicts extensive trade among rich countries and scant trade among poor countries. I use the parameter estimates of the model to analyze counterfactuals. The model qualitatively predicts some effects from a technology shock in China that are often put forth by the popular press: the shift in Chinese demand and production toward luxury goods; the global decrease in the prices of low-end manufactures such as toys and textiles; the negative wage pressure experienced by textile industries in Malaysia, the Philippines, and Thailand; and the benefits accrued by Hong Kong and Singapore for selling high-end products, such as financial services, to China and other fast-growing East Asian economies. In the model, these changes benefit poor and rich countries, and they hurt middleincome countries—a prediction broadly consistent with the evolution of world income between 1980 and 2000, discussed in Leamer (2007) and Leamer and Schott (2005). Two features of the data, ignored throughout, are potentially useful in future research. First, data are available at the product level. Thus, the links between income elasticity of demand, and production and trade patterns in the model may be verifiable. A growing literature infers quality differences from unit prices and quantities within commodity categories, and finds systematic patterns of trade within categories.31 In contrast, I infer types of goods from aggregate data. Combining these micro and macro approaches may be fruitful. 31 See, for example, Hallak (2006), Hallak and Schott (2011), Khandelwal (2010), and Schott (2004).
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Second, the data are available for several years. I have emphasized throughout the analogy between product cycles and the variability in production technologies in the model. So a dynamic version of the model should be fit to study the effects of nonhomothetic preferences on technology diffusion and the evolution of trade and growth. REFERENCES ALVAREZ, F., AND R. E. LUCAS (2007): “General Equilibrium Analysis of the Eaton–Kortum Model of International Trade,” Journal of Monetary Economics, 54, 1726–1768. [1073,1082] ANDERSON, J., AND E. VAN WINCOOP (2003): “Gravity With Gravitas: A Solution to the Border Puzzle,” American Economic Review, 93, 170–192. [1071,1082] BERGSTRAND, J. H. (1990): “The Heckscher–Ohlin–Samuelson Model, the Linder Hypothesis and the Determinants of Bilateral Intra-Industry Trade,” The Economic Journal, 100, 1216–1229. [1072] BILS, M., AND P. KLENOW (1998): “Using Consumer Theory to Test Competing Business Cycle Models,” Journal of Political Economy, 106, 233–261. [1070] CAMERON, C. A., J. B. GELBACH, AND D. L. MILLER (2011): “Robust Inference With Multiway Clustering,” Journal of Business & Economic Statistics, 29, 238–246. [1084] CHOI, Y. C., D. HUMMELS, AND C. XIANG (2009): “Explaining Import Quality: The Role of the Income Distribution,” Journal of International Economics, 78, 293–303. [1072,1094] COE, D. T., AND A. W. HOFFMAISTER (1999): “North–South Trade: Is Africa Unusual?” Journal of African Economies, 8, 228–256. [1069] COMIN, D., AND B. HOBIJN (2006): “An Exploration of Technology Diffusion,” Working Paper 12314, NBER. [1070] DEATON, A. (1975): “The Measurement of Income and Price Elasticities,” European Economic Review, 6, 261–273. [1070] EATON, J., AND S. KORTUM (2002): “Technology, Geography, and Trade,” Econometrica, 70, 1741–1779. [1070,1073,1074,1076,1079,1096] FAJGELBAUM, P., G. M. GROSSMAN, AND E. HELPMAN (2009): “Income Distribution, Product Quality, and International Trade,” Working Paper 15329, NBER. [1094] FIELER, A. C. (2011a): “Quality Differentiation in International Trade: Theory and Evidence,” Mimeo, University of Pennsylvania. [1072,1094] (2011b): “Supplement to ‘Nonhomotheticity and Bilateral Trade: Evidence and a Quantitative Explanation’,” Econometrica Supplemental Material, 79, http://www.econometricsociety. org/ecta/Supmat/8346_data and programs.zip; http://www.econometricsociety.org/ecta/Supmat /8346_simulations.pdf. [1073] FLAM, H., AND E. HELPMAN (1987): “Vertical Product Differentiation and North–South Trade,” American Economic Review, 77, 810–822. [1073,1094] GRIGG, D. (1994): “Food Expenditure and Economic Development,” GeoJournal, 33, 377–382. [1070] HALLAK, J. C. (2006): “Product Quality and the Direction of Trade,” Journal of International Economics, 68, 238–265. [1098] (2010): “A Product-Quality View of the Linder Hypothesis,” Review of Economics and Statistics, 92, 453–466. [1072] HALLAK, J. C., AND P. SCHOTT (2011): “Estimating Cross-Country Differences in Product Quality,” Quarterly Journal of Economics, 126, 417–474. [1098] HELPMAN, E., AND P. KRUGMAN (1985): Market Structure and Foreign Trade. Cambridge: MIT Press. [1071] HUMMELS, D., AND P. KLENOW (2005): “The Variety and Quality of a Nation’s Exports,” American Economic Review, 95, 704–723. [1072]
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HUNTER, L. (1991): “The Contribution of Nonhomothetic Preferences to Trade,” Journal of International Economics, 30, 345–358. [1070] HUNTER, L., AND J. MARKUSEN (1988): “Per capita Income as a Determinant of Trade,” in Empirical Methods for International Economics, ed. by R. Feenstra. Cambridge: MIT Press. [1070] KHANDELWAL, A. (2010): “The Long and Short (of) Quality Ladders,” Review of Economic Studies, 77, 1450–1476. [1098] LEAMER, E. (2007): “A Flat World, a Level Playing Field, a Small World After All, or None of the Above? A Review of Thomas L. Friedman’s The World Is Flat,” Journal of Economic Literature, 45, 83–126. [1095,1098] LEAMER, E., AND J. LEVINSOHN (1995): “International Trade Theory: The Evidence,” in The Handbook of International Economics, Vol. III, ed. by G. Grossman and K. Rogoff. Amsterdam: Elsevier. [1072] LEAMER, E., AND P. SCHOTT (2005): “The Rich (and Poor) Keep Getting Richer,” Harvard Business Review, 83, 20. [1095,1098] LIMÃO, N., AND A. J. VENABLES (2001): “Infrastructure, Geographical Disadvantage, Transport Costs, and Trade,” The World Bank Economic Review, 15, 451–479. [1069] LINDER, S. (1961): An Essay on Trade and Transformation. New York: Wiley. [1072] MARKUSEN, J. R. (1986): “Explaining the Volume of Trade: An Eclectic Approach,” American Economic Review, 76, 1002–1011. [1073] MARKUSEN, J. R., AND R. M. WIGLE (1990): “Explaining the Volume of North–South Trade,” The Economic Journal, 100, 1206–1215. [1070] MAS-COLELL, A., M. D. WHINSTON, AND J. R. GREEN (1995): Microeconomic Theory. New York: Oxford University Press. [1082] MATSUYAMA, K. (2000): “A Ricardian Model With a Continuum of Goods Under Nonhomothetic Preferences: Demand Complementarities, Income Distribution, and North–South Trade,” Journal of Political Economy, 108, 1093–1120. [1073,1094] MAYER, T., AND S. ZIGNAGO (2006): “Notes on CEPII’s Distance Measures,” Mimeo, Centre d’Études Prospectives et d’Informations Internationales (CEPII). Available at http://www. cepii.fr/distance/noticedist_en.pdf. [1080] NABSETH, L., AND G. F. RAY (EDS.) (1974): The Diffusion of New Industrial Processes: An International Study. Cambridge: Cambridge University Press. [1070] REDDING, S. J., AND A. J. VENABLES (2004): “Economic Geography and International Inequality,” Journal of International Economics, 62, 53–82. [1071] RODRIK, D. (1998): “Trade Policy and Economic Performance in Sub-Saharan Africa,” Working Paper 6562, NBER. [1069] ROMER, P. (1990): “Endogenous Technological Change,” Journal of Political Economy, 98, S71–S102. [1070] SCHOTT, P. (2004): “Across-Product versus Within Product Specialization in International Trade,” Quarterly Journal of Economics, 119, 647–678. [1072,1094,1098] STOKEY, N. (1991): “The Volume and Composition of Trade Between Rich and Poor Countries,” Review of Economic Studies, 58, 63–80. [1073] UNITED NATIONS (2008): “UN Comtrade,” UN Commodity Trade Statistics Database, Statistics Division, United Nations. Available at http://comtrade.un.org/. [1079] VERNON, R. (1966): “International Investment and International Trade in the Product Cycle,” Quarterly Journal of Economics, 80, 190–207. [1070] WAUGH, M. (2010): “International Trade and Income Differences,” American Economic Review, 100, 2093–2124. [1069,1073,1095] WORLD BANK (2008): “World Development Indicators 2008,” Data Set, World Bank, Washington, DC. Available at http://web.worldbank.org/data. [1080,1096]
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YI, K. M. (2003): “Can Vertical Specialization Explain the Growth in the World Trade?” Journal of Political Economy, 111, 52–102. [1097]
Dept. of Economics, University of Pennsylvania, 3718 Locust Walk, 434 McNeil, Philadelphia, PA 19104-6297, U.S.A.;
[email protected]. Manuscript received January, 2009; final revision received June, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1103–1138
NATURE OR NURTURE? LEARNING AND THE GEOGRAPHY OF FEMALE LABOR FORCE PARTICIPATION BY ALESSANDRA FOGLI AND LAURA VELDKAMP1 One of the most dramatic economic transformations of the past century has been the entry of women into the labor force. While many theories explain why this change took place, we investigate the process of transition itself. We argue that local information transmission generates changes in participation that are geographically heterogeneous, locally correlated, and smooth in the aggregate, just like those observed in our data. In our model, women learn about the effects of maternal employment on children by observing nearby employed women. When few women participate in the labor force, data are scarce and participation rises slowly. As information accumulates in some regions, the effects of maternal employment become less uncertain and more women in that region participate. Learning accelerates, labor force participation rises faster, and regional participation rates diverge. Eventually, information diffuses throughout the economy, beliefs converge to the truth, participation flattens out, and regions become more similar again. To investigate the empirical relevance of our theory, we use a new county-level data set to compare our calibrated model to the time series and geographic patterns of participation. KEYWORDS: Female labor force participation, information diffusion, economic geography.
0. INTRODUCTION OVER THE TWENTIETH CENTURY, there was a dramatic rise in female labor force participation in the United States. Many theories of this phenomenon have been proposed. Some of them emphasize the role played by market prices and technological factors; others focus on the role played by policies and institutions, and a few recent ones investigate the role of cultural factors. All of them, however, focus on aggregate shocks that explain why the transition took place and abstract from the local interactions that could explain how the transition took place. 1
We thank seminar participants at Northwestern, the World Bank, Chicago GSB, University of Wisconsin–Madison, Minneapolis Federal Reserve, Princeton University, European University in Florence, University of Southern California, New York University, Boston University, Bocconi, Universitè Pompeu Fabra, Ente Einaudi, Boston Federal Reserve, and Harvard University and conference participants at the 2009 winter NBER EF&G Meetings, 2008 AEA, SITE, the 2007 NBER Summer Institute, the SED Conference, LAEF Households, Gender and Fertility Conference, the NBER Group on Macroeconomics across Time and Space, Midwest Macro Meetings, the NY/Philadelphia Workshop on Quantitative Macro, IZA/SOLE, and Ammersee. We especially thank Stefania Marcassa for excellent research assistance and Stefania Albanesi, Roland Benabou, Raquel Bernal, Jason Faberman, Jeremy Greenwood, Luigi Guiso, Larry Jones, Patrick Kehoe, Narayana Kocherlakota, Ellen McGrattan, Fabrizio Perri, Harald Uhlig, and the anonymous referees for comments and suggestions. Laura Veldkamp thanks Princeton University for their hospitality and financial support through the Kenen fellowship. © 2011 The Econometric Society
DOI: 10.3982/ECTA7767
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We use new data and theory to argue that women’s labor force participation decisions rely on information that is transmitted from one woman to another located nearby. The local nature of information transmission smooths the effects of changes in the environment and generates geographically heterogeneous, but locally correlated, reactions like those observed in our data. Our theory focuses on learning and participation of married women with young children, because this subgroup is responsible for most of the rise in participation. A crucial factor in mothers’ participation decisions is the effect of employment on their children. However, this effect is uncertain. The uncertainty makes risk-averse women less likely to participate. Learning resolves their uncertainty, causing participation to rise. In our overlapping generations model, women learn from their neighbors about the relative importance of nature (innate ability) and nurture (the role of maternal employment) in determining children’s outcomes (Section 1). Women inherit their parents’ beliefs and update them after observing the outcomes of neighboring women in the previous generation. Those outcomes reveal information about the effect of maternal employment only if the neighboring mothers were employed. Section 2 shows that higher local participation generates more information, which reduces uncertainty about the effect of maternal employment and makes participation of nearby women more likely. Thus, local participation snowballs and a gradual, but geographically concentrated rise in participation rates ensues. Using county-level U.S. data from 1940 through 2000, Section 3 documents how the growth rate of the women’s labor force varied over time and across counties.2 After the shift from an agricultural to an industrial economy separated the location of home and work, the female labor force grew slowly up through the post-war decades, accelerated during the 1970s and 1980s, and recently flattened out. Furthermore, this growth was uneven across geographic regions: High-participation rates emerged first in a few geographic centers and spread from there to nearby regions over the course of several decades. This process gave rise to significant spatial correlation across the participation rates of U.S. counties that is only marginally explained by common economic and demographic factors. This residual correlation slowly rose at the beginning of the period, peaked when aggregate labor force increased fastest, and finally declined as aggregate labor force stagnated. Sections 4 and 5 use moments of the labor force participation distribution across U.S. counties in 1940 to calibrate and simulate a dynamic learning model and explore its quantitative properties. The results are consistent with the S-shaped evolution of aggregate labor force, and the rise and fall in the spatial correlation of county-level participation rates. The model generates Sshaped dynamics because initially, when uncertainty is high, very few women 2 To our knowledge these county-level historical, demographic, economic and social data have not been explored before in economics research.
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participate in the labor market; information about the role of nurture diffuses slowly and beliefs are nearly constant. As information accumulates and the effects of labor force participation become less uncertain, more women participate, learning accelerates, and labor force participation rises more quickly. As uncertainty is resolved, beliefs converge to the truth and participation flattens out. The local nature of the learning process generates the rise and fall of spatial correlation in participation. Initially, female labor force participation is low everywhere and the minute differences are spatially uncorrelated. As women in some locations start working, their neighbors observe them and learn from them. Learning makes the neighbors more likely to work in the next generation, generating an increase in geographic heterogeneity and spatial correlation. Eventually as the truth about maternal employment is learned everywhere, heterogeneity and spatial correlation in local participation rates falls. Section 6 extends these results to a setting with multiple types of women. The model provides a simple framework for examining the transition dynamics and geography of a wide array of social and economic phenomena. Section 7 concludes by describing further extensions of the model that could capture other social and cultural transformations. Relationship to Other Theories Many recent papers have explored the rise in female labor force participation. Among these theories, some focus on changes that affect the costs or benefits of employment for all women: changes in wages, less discrimination, the introduction of household appliances, the less physical nature of jobs, or the ability to control fertility.3 In contrast, our theory focuses on why the participation of women with children rose so much faster than the aggregate participation rate. A complete understanding of the rise in participation requires both an explanation of what changed for all women and what made married mothers behave so differently. Another group of theories shares our focus on changes that affect mothers specifically, but unlike our theory, rely on aggregate shocks. For example, the decline in child care costs, medical innovation, or public news shocks are changes that spread quickly because there are no geographic barriers or distance-related frictions causing some regions to be unaffected.4 Obviously, 3 See Greenwood, Seshadri, and Yorukoglu (2005), Goldin and Katz (2002), Goldin (1990), and Jones, Manuelli, and McGrattan (2003) on the nature of jobs. 4 See Attanasio, Low, and Sanchez-Marcos (2008) and Del Boca and Vuri (2007) for child care costs, Albanesi and Olivetti (2007) for medical innovation, Fernández and Fogli (2009), Antecol (2000), and Fernández, Fogli, and Olivetti (2004) for the role of cultural change, and Fernández (2007) for an aggregate information-based learning theory. Note that this work was done independently and was published as Minneapolis Federal Reserve Staff Working Paper 386 prior to Fernández (2007).
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one can modify these theories to introduce geographic heterogeneity by adding income or preference heterogeneity.5 What is harder to explain is why the participation transition happened at different times in different places. The rates of change in participation were vastly different across counties, resulting in a rise and then a fall in the cross-county dispersion of participation rates. This is not a pattern that a typical aggregate shock generates. One would think that any local coordination motive (e.g., social pressure) or thick market externality (e.g., child care markets) could generate local differences in the speed of transition. But such a coordination model typically predicts a simultaneous switch from a low-participation to a high-participation outcome, unless there is some friction preventing perfect coordination. Our local information externality generates locally correlated behavior, while the imperfect nature of the information is the friction that prevents perfect economywide coordination. A third strand of related literature studies the geography of technology diffusion. While it does not discuss labor force participation, the learning process is similar to that in our model (see, e.g., Munshi (2004), Comin and Hobijn (2010), and Jovanovic (2009)). One way to interpret our message is that ideas about how technology diffuses should be applied to female labor force participation. In this case, the technology being learned about is outsourcing the care of one’s children. Of course, the spread of more traditional technologies such as washing machines and dishwashers could also explain the geographic diffusion of participation. But most technologies diffused throughout the country in the span of a decade or two.6 Part of the puzzle this paper wrestles with is isolating the information frictions that make learning about maternal employment so much slower than learning about consumer technologies. A key assumption that slows learning is that nurturing decisions in early childhood affect outcomes as an adult. Therefore, information about the value of nurturing is observed only with a generation-long delay. Facts about geographic heterogeneity do not prove that aggregate changes are irrelevant. Rather, they suggest such changes operate in conjunction with a mechanism that causes their effect to disseminate gradually across the country. We argue that this mechanism is the local transmission of information. Considering how beliefs react to changing circumstances and how these beliefs, in turn, affect participation decisions can help us understand and evaluate the ef5 See Fuchs-Schundeln and Izem (2007) for a static theory of geographic heterogeneity in labor productivity between East and West Germany. 6 For example, consider the refrigerator. In 1930, only 12% of households had one. By 1940, 63% did and by 1950, 86% did. Likewise, while only 3% of households had a microwave in 1975, 60% had one 10 years later. A few appliances, like the dishwasher and dryer, took longer to catch on. In these cases, early models were inefficient. Once a more efficient model came to market, adoption surged from below 10% to over 50% in one to two decades. See Greenwood, Seshadri, and Yorukoglu (2005).
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fects of many other important changes to the benefits and the costs of labor force participation. 1. THE MODEL In this section, we develop a theory in which the dramatic change in female labor force participation emerges solely as the result of local interactions. Because the bulk of the change came from married women with small children, we focus on their participation. We model local interactions that transmit information about the effect of maternal employment on children. The model makes two key assumptions. First, women were initially uncertain about the consequences of maternal employment on their children. The shift from agriculture to industry at the end of the 19th century changed the nature of work. In agriculture, women allocated time continuously between work and child-rearing. This was possible because home and work were in the same location. Industrialization required women who took jobs to outsource their child care. At that time, the effects of outsourcing were unknown. Women held beliefs about those effects which were very uncertain.7 The second key assumption is that learning happens only at the local level from a small number of observations, as in the Lucas (1972) island model. This allows learning to take place gradually, over the course of a century. In a richer model, this strong assumption could be relaxed. Section 6 sets up and simulates a model with multiple types where women need to observe others like themselves to learn their type-specific cost of maternal employment. For example, professionals do not learn from seeing hourly workers; urban mothers face different costs than rural ones. Instead of learning about what the cost of maternal employment is for the average woman, these women are learning about the difference between the average cost and the cost for their type. In this richer model, women can observe many more signals, as well as aggregate information like the true aggregate participation rate, and still learn slowly about the cost of maternal employment for their type. The results of the simple model below are nearly identical to this richer model. Preferences and Constraints Time is discrete and infinite (t = 1 2 ). We consider an overlapping generation economy made up of a large finite number of agents living for two periods. Each agent is nurtured in the first period, and consumes and has one child in the second period of her life. Preferences of an individual in family i born at 7 This is consistent with the decline in the labor market participation rate of married women observed during the turn of the century by Goldin (1995), and with the findings of Mammen and Paxson (2000), who documented a U-shaped relationship between women’s labor force rates and development in a cross section of countries.
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time t − 1 depend on their consumption cit , their labor supply nit ∈ {0 1}, and the potential wage of their child wit+1 : (1)
U=
1−γ wit+1 cit1−γ +β − Lnit 1−γ 1−γ
γ > 1
This utility function captures the idea that parents care about their child’s earning potential, but not about the choices they make.8 The last term, which captures the forgone leisure L if a woman chooses to work, is not essential for any theoretical results, but it is helpful later for calibration. The budget constraint of the individual from family i born at time t − 1 is (2)
cit = nit wit + ωit
where ωit ∼ N(μω σω2 ) is an endowment which could represent a spouse’s income. If the agent works in the labor force, nit = 1 and is zero otherwise. The key feature of the model is that an individual’s earning potential is determined by a combination of endowed ability and nurturing that cannot be perfectly disentangled. Endowed ability is an unobserved normal random variable ait ∼ N(μa σa2 ). If a mother stays home with her child, the child’s full natural ability is achieved. If the mother joins the labor force, some unknown amount θ of the child’s ability will be lost. Wages depend exponentially on ability: (3)
wit = exp(ait − nit−1 θ)
Of course, a child also benefits from higher household income when its mother joins the labor force. While this benefit is not explicitly modeled, θ represents the cost to the child of maternal employment, net of the gain from higher income. When we model beliefs, women will not rule out the possibility that employment has a net positive effect on their child’s development (θ < 0). Furthermore, Section 5.5 explores a model where all women initially believe that maternal employment is beneficial and shows that uncertainty alone can deter participation. Information Sets The constant θ determines the importance of nurture and is not known when making labor supply decisions. Women have two sources of information 8 Using utility over the future potential wage rather than recursive utility shuts down an experimentation motive where mothers participate so as to create information that their decedents can observe. Such a motive makes the problem both intractable and unrealistic. Most parents do not gamble with their children’s future just to observe what happens.
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about θ: beliefs passed down through their family, and the wage outcomes of themselves and their neighbors. Agents do not learn from aggregate outcomes. Young agents inherit their prior beliefs about θ from their parents’ beliefs. In the first generation, initial beliefs are identical for all families θi0 ∼ N(μ0 σ02 ) ∀i. Each subsequent generation updates these beliefs and passes down their updated beliefs to their children. To update beliefs at the beginning of time t, agents use both potential earnings and parental employment decisions for themselves and for J − 1 peers. We refer to w as the potential wage because it is observed, regardless of whether the agent chooses to work.9 Ability a is never observed, so θ can never be perfectly inferred from observed wages. But these potential wages are only informative about the effect of maternal employment on wages if a mother actually worked. Note from equation (3) that if nit−1 = 0, then wit only reflects innate ability and contains no information about θ. Since the content of the signals in the first period depends on the previous period’s participation rate, the model requires a set of initial participation decisions ni0 for each woman i. The set of family indices for the outcomes observed by agent i is Ji . Spatial location matters in the model because it determines the composition of the signals in this information set. Each agent i has a location on a two-dimensional map with indices (xi yi ). Signals are drawn uniformly from the set of agents within a distance d in each direction: Ji ∼ unif{[xi − d xi + d] × [yi − d yi + d]}J−1 . Agents use the information in observed potential wages to update their prior, according to Bayes’ law. Bayesian updating with J signals is equivalent to the following two-step procedure: First, run a regression of children’s potential wages on parents’ labor choices: W − μa = Nθ + εi where W and N are the J × 1 vectors {log wjt }j∈Ji and {nit−1 }j∈Ji , and εi is the regression residual. Let n¯ it be the sum of the labor decisions for the set of fam ilies that (i t) observes: n¯ it = j∈Ji nit . The resulting estimated coefficient θˆ is normally distributed with mean μˆ it = j∈Ji (log wjt − μa )njt /n¯ it and variance σˆ it2 = σa2 /n¯ it . Second, form the posterior mean as a linear combination of the estimated coefficient μˆ it and the prior beliefs μit−1 , where each component’s weight is its relative precision: (4)
μit =
σˆ it2 2 σit−1 + σˆ it2
μit−1 +
2 σit−1
σit2 + σˆ it2
μˆ it
9 This assumption could be relaxed. If wit were only observed once agent (i t) decided to work, then an informative signal about θ would only be observed if both nit = 1 and nit−1 = 1. Since this condition is satisfied less frequently, such a model would make fewer signals observed and make learning slower.
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Posterior beliefs about the value of nurturing are normally distributed θ ∼ N(μit σit2 ). The posterior precision (inverse of the variance) is the sum of the prior precision and the signal precision.10 Thus posterior variance is (5)
−2 σit2 = (σit−1 + σˆ it−2 )−1
The timing of information revelation and decision-making is
Equilibrium An equilibrium is a sequence of wages—distributions that characterize beliefs about θ, work, and consumption choices—for each individual i in each generation t such that the following four conditions are satisfied: First, taking beliefs and wages as given, consumption and labor decisions maximize expected utility (1) subject to the budget constraint (2). The expectation is conditioned on beliefs μit σit . Second, wages of agents born in period t − 1 are consistent with the labor choice of the parents, as in (3). Third, priors μit−1 and σit−1 are equal to the posterior beliefs of the parent, born at t − 1. Priors are updated using observed wage outcomes Jit , according to Bayes’ law (4). Fourth, distributions of elements Jit are consistent with the distribution of optimal labor choices ni(t−1) and each agent’s spatial location. 2. ANALYTICAL RESULTS In this section, we establish some cross-sectional and dynamic predictions of our theory that distinguish it from other theories. We begin by solving for the optimal participation decision. Substituting the budget constraint (2) and the 10 The fact that another woman’s mother chose to work is potentially an additional signal. But the information content of this signal is very low because the outside observer does not know whether this person worked because they were highly able, very poor, less uncertain, or had low expectations for the value of theta. Since these observations contain much more noise than wage signals and the binary nature of the working decision makes updating much more complicated, we approximate beliefs by ignoring this small effect. In Appendix D, we solve an extended model where women use this extra information. Over the 70-year simulation, the extra information increases participation by 2.4%.
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law of motion for wages (3) into expected utility (1) produces the following optimization problem for agent i born at date t − 1: exp((ait+1 − nit θ)(1 − γ)) (nit wit + ωit )1−γ (6) + βEait+1 θ max nit ∈{01} 1−γ 1−γ Taking the expectation over the unknown ability a and the importance of nurture θ delivers expected utilities from each choice. If a woman stays out of the labor force, her expected utility (EUO) is β (ωit )1−γ 1 2 2 + exp μa (1 − γ) + σa (1 − γ) EUOit = (7) 1−γ 1−γ 2 If she participates in the labor force, her expected utility is (8)
EUWit =
(wit + ωit )1−γ 1−γ β 1 + exp (μa − μit )(1 − γ) + (σa2 + σit2 )(1 − γ)2 1−γ 2
The optimal policy is to join the labor force when the expected utility from employment is greater than the expected utility from staying home (EUWit > EUOit ). Define Nit ≡ EUWit − EUOit to be the expected net benefit of labor force participation, conditional on information (μit σit ). 2.1. Comparative Statics: The Role of Beliefs, Wages, and Wealth Beliefs The key variable whose evolution drives the increase in labor force participation is beliefs, and particularly uncertainty. We begin by establishing two intuitive properties of labor force participation (both derived formally in Appendix A). First, a higher expected value of nurture reduces the probability that a woman will participate in the labor force, holding all else equal. The logic of this result appears in equation (8). Increasing the expected value of nurture decreases the net expected utility of labor force participation: ∂Nit /∂μit = −β times an exponential term, which is always nonnegative. Since −β < 0, a higher μit reduces the utility gain from labor force participation and, therefore, reduces the probability that a woman will participate. Second, greater uncertainty about the value of nurture reduces the probability that a woman will participate in the labor force, holding all else equal. More uncertainty about the cost of maternal employment on children makes labor force participation more risky. Participation falls because agents are riskaverse. Over time, as information accumulates and uncertainty falls, the net
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benefit of participating rises: ∂Nit /∂σit = (1 − γ)β times a nonnegative (exponential) term. Higher risk aversion makes (1 − γ) more negative and amplifies this effect. Thus, there are two ways our model could produce an increase in participation. First, women could have started with biased, pessimistic beliefs (low μ0 ) and participation rates would rise as women learned that participation is not as bad as they thought. This is the driving force in Fernández (2007). Instead, our calibration gives women unbiased beliefs about θ. Our women work more over time because they start out uncertain (high σ0 ) and learning reduces their uncertainty. It is possible that some force in the economy caused women around the world to be systematically deceived about the effect maternal employment has on their children, but the economic transition from agricultural work to the modern age and the new requirement that employed women outsource their children’s care undoubtedly created uncertainty. Wages Wages in our model have the standard role: Women work more if wages are higher. While other theories give wages and human capital a more central role (Olivetti (2006), Goldin and Katz (1999), Jones, Manuelli, and McGrattan (2003)), our baseline model holds the distribution of wages fixed. We explore the effects of a changing wage process in Section 5.5. Wealth Greater initial wealth ωit reduces the probability that a woman will participate in the labor force. Poorer women join the labor force before richer ones because poorer women have a higher marginal value of wage income. 2.2. Dynamic Properties One might think that the initial state after industrialization would be no women participating and no information being produced, and that this would be an absorbing state. The following result shows that zero participation is a state that can persist for many periods, but is exited each period with a small probability (proof given in Appendix A.2). RESULT 1: In any period where the labor force participation rate is zero ( j njt−1 = 0), there is a positive probability that at least one woman will work in the following period ( j njt ≥ 1). All it takes to escape a zero-participation state is for one extremely able woman to be born. She generates information that makes the women around her less uncertain about the effects of maternal employment. That information encourages these women to work. They, in turn, generate more information
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for women around them. Gradually, the information and participation disseminate. Condition (8) also suggests circumstances in which such a woman is likely to emerge. One example is a low endowment ωjt , which raises the marginal value of labor income. Depressions or wars, which reduce endowments by eliminating husbands’ incomes, can hasten the transition. Learning amplifies those kinds of shocks and causes them to persist long after their direct effects have disappeared. Shocks that cause more women to participate persist through their effects on the information that gets transmitted from generation to generation. S-Shaped Evolution of Participation Rates One of the hallmarks of information diffusion models is that learning is slow at first, speeds up, and then slows down again as beliefs converge to the truth. The concave portion of this S-shaped pattern can be explained by any theory. Because the participation rate is bounded above by 1, any shock to participation must eventually taper off. But many shocks to labor force participation would be strongest when they first hit. The interesting feature of this model is its prediction that participation first rises slowly and then speeds up. The information gleaned from observingothers’ labor market outcomes can be described as a signal with mean μˆ it = j∈Ji (log wjt − μa )njt /n¯ it and variance σˆ it2 = σa2 /n¯ it . Let ρ be the fraction of women who participate in the labor force. Then the expected precision of this signal is E[σˆ it−2 ] = ρNσa2 . A higher signal precision increases the expected magnitude of changes in beliefs. This conditional variance of t beliefs is the difference between prior variance and 2 − σit2 . Substituting in for posterior posterior variance: var(μit |μit−1 ) = σit−1 variance using equation (5) yields (9)
2 var(μit |μit−1 ) = σit−1 −
−2 it−1
σ
1 + σˆ it−2
Since ∂ var(μit |μit−1 )/∂σˆ it−2 > 0, the expected size of revisions is increasing in the precision of the observed signals and, therefore, in the fraction of women who work. This is the first force: As beliefs changed more rapidly, so did labor force participation early in the century. The concave part of the S-shaped increase in participation comes later, from convergence of beliefs to the truth. Over time, new information reduces pos2 (equation (5)). As posterior variance falls, beliefs terior variance: σit2 < σit−1 2 change less: ∂ var(μit |μit−1 )/∂σit−1 > 0. Endogenous Pessimism At the start of the transition, there is another force that suppresses participation: Women become more pessimistic about the benefits of maternal
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employment, on average ( i μit di rises). Women who have pessimistic beliefs (μit−1 > θ) do not participate and thus generate less information for their children than women with optimistic beliefs (μit−1 < θ). Since new information μˆ it is unbiased, on average, it moves beliefs toward the true θ (equation 4). Since the children of pessimistic women observe less new information, their posterior beliefs remain closer to their prior beliefs. The children of optimistic women revise their beliefs more, which brings them closer to the truth. Since pessimism is persistent and optimism is undone by learning, the average belief is pessimistic until information disseminates fully. 2.3. Geographic Properties The model produces two effects relating to geography: dispersion and spatial correlation in participation rates. Initial differences in participation rates come from random realizations of potential wages which create differences in beliefs across women. Our mechanism amplifies these small initial differences, because women who initially believe that maternal employment is not very costly join the labor force and generate more information for the women around them. Locations with high mean beliefs generate more information, which lowers the variance of their beliefs. Both high means and lower variance (less uncertainty) promote higher labor force participation rates. More participation feeds back by creating more information, which further reduces the uncertainty and risk associated with maternal employment. Local information diffusion creates a learning feedback mechanism that amplifies the effect of small differences in signal realizations. We formalize this local information effect in the following result (proven in Appendix A). Suppose that a woman has location (xi yi ). Define her neighborhood to be the set of agents whose outcomes are in her information set with positive probability: [xi − d xi + d] × [yi − d yi + d]. RESULT 2: A woman with an average prior belief who observes average signal draws in a neighborhood with a high participation rate at time t is more likely to participate at time t + 1, all else equal. Information diffusion makes cross-region dispersion in participation rates rise and then fall. All women have identical initial prior beliefs by assumption. Dispersion in beliefs is zero. In the limit as t → ∞, beliefs converge to the truth and their dispersion converges back to zero. In between, beliefs among women differ and, therefore, have positive dispersion. The rise and fall in belief dispersion is what creates a rise and fall in the dispersion of participation rates. 3. EMPIRICAL EVIDENCE: TIME SERIES AND GEOGRAPHIC To examine the transition in female labor force participation predicted by our model, we calibrate and simulate it. Before turning to those results, this
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section describes the data and the measures we use to compare the model to the data. It also presents direct evidence that changing beliefs plays a role in the transition. 3.1. Time Series Evidence We study the labor force participation behavior of white women over the period 1940–2005 using data from the U.S. Decennial Census and from the Census Bureau’s American Community Survey. Figure 1 reports the labor force participation rate in each decade for women between 25 and 34 years old.11 This implies that the data for each decade come from a distinct cohort of women. The increase is quite large: The fraction of women in the labor force rose from one-third in 1940 to nearly three-fourths in 2005. However, this increase in the aggregate rate hides large differences among subgroups of women. The increase comes mainly from the change in working behavior of married women with children. Women without children or unmarried women have always worked in large numbers: In 1940, their participation rate was already around 60%. On the other hand, the participation rate of married women with children at that time was only 10% and dramatically increased, reaching 62% in 2005. Therefore, to understand the large aggregate rise over the period, we need to understand what kept married women with
FIGURE 1.—Labor force participation among subgroups of women. Details of the data are in Appendix B.
11 We exclude women living in institutions. We also exclude individuals living on a farm or employed in agricultural occupations, since agricultural occupations may make working compatible with child-rearing. We also exclude residents of Alaska and Hawaii because they are not contiguous to the 48 states. Black women are excluded because racial as well as gender-related factors complicate their participation decisions. All observations are weighted using the relevant person weights.
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children out of the labor market at the beginning of the period and why their behavior changed so dramatically.12 Another interesting feature of the phenomenon that emerges from Figure 1 shows that the increase took place at different rates over the period: steady but slow in the first part of the sample, it significantly accelerated during the 1970s and 1980s, and recently flattened out, generating an S-shaped path. 3.2. Geographic Evidence The geographic predictions of our model are a distinctive feature: The rise of women’s labor force participation started in a few locations and gradually spread to nearby areas as information diffused. This section explores the geographic patterns of female labor force participation using county-level U.S. data. The data source is “Historical, Demographic, Economic, and Social Data: The United States, 1790–2000” produced by the Inter-University Consortium for Political and Social Research (2010). We start our analysis in 1940 because the wage data we need for our calibration begin only in 1940. There were 3107 U.S. counties in 1940. After eliminating counties with incomplete information over our entire sample period and excluding Hawaii and Alaska, 3074 counties remain. Our participation series is the number of working-age females in the civilian labor force divided by the total working-age female population. Appendix B and the Supplemental Material (Fogli and Veldkamp (2011)) contain details, sources, and summary statistics for all geographic data. Figure 2 maps the labor force participation rate for each U.S. county every 20 years. Darker colors indicate higher levels of female labor force participation (LFP). There are three salient features of the data. First, the levels of labor force participation are not uniform. While the average 1940 participation rate was 18.5%, there were counties with participation rates as low as 4.6% and as high as 50%. Second, the changes in participation rates are not uniform. While some areas increased their participation rate dramatically between 1940 and 1960 (for example, the Lake Tahoe region), others stayed stagnant until the 1980s and witnessed a surge in participation between 1980 and 2000 (for example, southern Minnesota). Third, there is spatial clustering: counties where the female participation rate is over 40% tend to be geographically close to other such counties. These counties are concentrated in the foothills of the southern Appalachians (Piedmont region), in the Northeast, Florida, Great Lakes, and West Coast. Central regions display much lower participation. To quantify the spatial features of the data and compare those features to the model, we use two statistics: cross-county dispersion and spatial correlation. 12 There were also changes in the composition of the population over the period: the fraction of married women with children (the group with the lowest participation rate) first increased and then decreased between 1940 and 2005. However, the reduction in the percentage of married women with children, from 53% in 1940 to 45% in 2005, was too small to account for the observed rise in the aggregate.
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FIGURE 2.—Female labor force participation rate by U.S. county.
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For each county i and time t, we first estimate LFPit = β1t + β2t controlsit + it . For a complete list of control variables and a discussion of sample selection, see Appendix B. For dispersion, we compute the standard deviation of the residuals across counties. This is a measure of geographic heterogeneity that is not attributable to observable economic features. For spatial correlation, we estimate correlation in the same residuals of all contiguous counties i and j: ιijd i j
N i j (10) I = ιijd 2j i
j
where N is the number of counties and ιijd = 1 if counties i and j are contiguous. This spatial correlation measure is also known as Moran’s I (Moran (1950)). It is a measure of local geographic similarity commonly used in fields such as geography, sociology, and epidemiology to measure spatial effects.13 We report both dispersion and correlation for each decade, and compare them to the model simulation results in Section 5. 3.3. Direct Evidence About Changes in Beliefs Survey Responses One empirical measure of beliefs is survey responses from 1930 through 2005. The precise wording of the survey question varies,14 but each one asks men and women whether they believe that a married woman—some are specific to a woman with children or preschool-aged children—should participate in the labor force. Support for participation with preschool aged children rose from 9% in 1936 to 58% in 2004. Of course, this does not prove that changes in beliefs caused participation to rise. It could be that people report more support for participation when they see participation rise. However, Farre and Vella (2007) showed that women who have more positive responses are more likely to work and more likely to have daughters who work. Causal or not, this is direct evidence that beliefs did change in the way the model predicts.15 13 While these other literatures frequently try to identify a causal relationship that drives spatial correlation, we make no such attempt here. In both the model and the data, issues like Manski (1993) reflection problems arise. We compare the contaminated moment in the model to the equivalent contaminated moments in the data. 14 Data are from the iPOLL data bank, maintained by the Roper Center for Public Opinion Research (http://www.ropercenter.uconn.edu/data_access/ipoll/ipoll.html). For wording of the survey questions and more data details, see the Supplemental Material. 15 In the model, we can ask agents whether they believe, based on their information set, that the average household’s utility would be higher if the mother worked. Consider an agent j with beliefs
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Ancestry Evidence At the center of the model is the idea that a key determinant of labor force participation is a belief inherited from one’s parents and influenced by one’s neighbors. An empirical literature identifies such an effect on female labor force participation. Fernández and Fogli (2009) studied second generation American women, and used differences in heritage to distinguish preferences and beliefs from the effect of markets and institutions. Also see Antecol (2000), Fortin (2005), and Alesina and Giuliano (2007), who showed that female labor force participation in the parents’ country of origin predicts participation of their American daughters. This effect intensifies for women who live in an ethnically dense neighborhood. While the effect could come from other cultural forces, it suggests that family and neighbors are central to participation decisions. 4. CALIBRATION To explore the quantitative predictions of our theory, we calibrate the economy to reproduce some key aggregate statistics in the 1940s and then compare its evolution over time and across regions with the data. Because we have census data every 10 years, we consider a period in the model to be 10 years. There are 3025 counties because this is the closest square number to the actual number of U.S. counties (3074). One hundred women live in each county. We focus on the dynamics generated by local interactions alone and abstract from changes due to wages wealth, and technology by holding the costs and benefits of maternal employment fixed over time. Table I summarizes our calibrated parameters. We construct initial 1930 participation to have a geographic pattern that resembles the U.S. data. This enables us to start with reasonable initial dispersion and spatial correlation. Initial participation rates affect subsequent local participation because they determine the probability of observing an informative signal. Appendices B and C offer additional detail about how national and local data sets were combined to infer the participation of married women with young children in each county, as well as the derivation of the calibration targets and initial conditions. μjt and σjt , who uses the mean of all the wage realizations he observes at time t (
wk /J) jt = to estimate average wage. He believes the average expected utility of working is EUW 2 )(1 − γ)2 ). ( k∈Jj wk /J + exp(μω ))1−γ /(1 − γ) + β/(1 − γ) exp((μa − μjt )(1 − γ) + 12 (σa2 + σjt jt = exp(μω (1 − γ))/(1 − γ) + β/(1 − He believes the expected utility of not working is EUO γ) exp(μa (1 − γ) + 12 σa2 (1 − γ)2 ). Therefore, he would answer yes to the survey equation if jt . The fraction of yes answers rises over time because uncertainty about nur jt ≥ EUO EUW k∈Jj
2 ture σit falls. In the calibrated model, the fraction of agents who respond yes rises from 3% in 1940 to 30% in 2000.
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A. FOGLI AND L. VELDKAMP TABLE I
PARAMETER VALUES FOR THE SIMULATED MODEL AND THE CALIBRATION TARGETS Mean log ability Std. log ability Mean log endowment Std. log endowment True value of nurture Outcomes observed Prior mean θ Prior std. θ Utility of leisure Risk aversion
μa σa μω σω θ J μ0 σ0 L γ
−0.90 0.57 −0.28 0.75 0.04 3–5 0.04 0.76 0.3 3
Women’s 1940 earnings distribution Women’s 1940 earnings distribution Average endowment = 1 Men’s 1940 earnings distribution Children’s test scores (Bernal and Keane (2011)) Growth of LFP in 1940s Unbiased beliefs Average 1940 LFP level 1940 LFP of women without kids Commonly used
Wages and Endowments The ability and endowment distributions in our model match the empirical distributions of annual labor income of full-time employed, married women with children under age 5 and their husbands. We match the moments for 1940, the earliest year for which we have wage data. Since we interpret women’s endowment ω as being husbands’ earnings and since earnings are usually described as log-normal, we assume ln(ω) ∼ N(μω σω2 ). We normalize the average endowment (not in logs) to 1 and use σω to match the dispersion of 1940 annual log earnings of husbands with children under 5. For the mean μa and standard deviation σa of women’s ability, we match the censored distribution of working women’s earnings in the first period of the model to the censored earnings distribution in the 1940 data. Our estimates imply that full-time employed women earn 81% of their husbands’ annual earnings, on average. True Value of Nurture Our theory is based on the premise that the effect of mothers’ employment on children is uncertain. This is realistic because only in the last 10 years have researchers begun to agree on the effects of maternal employment in early childhood. Harvey (1999) summarized studies on the effects of early maternal employment on children’s development that started in the early 1960s and flourished in the 1980s when the children of the women interviewed in the National Longitudinal Survey of Youth reached adulthood. She concluded that working more hours is associated with slightly less cognitive development and academic achievement before age 7. More recent work confirms this finding (Hill, Waldfogel, and Han (2005)). Combining Bernal and Keane’s (2011) estimates of the reduction in children’s test scores from full-time maternal employment of married women with estimates of the effect of these test scores on educational attainment and on expected wages (Goldin and Katz (1999)) delivers a loss of 4% of lifetime income from maternal employment (θ = 004).
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Information Parameters Without direct observable counterparts for our information variables, we need to infer them from participation data. Initial beliefs are assumed to be the same for all women and unbiased, implying μ0 = θ. Initial uncertainty σ0 is chosen to match women’s 1940 average labor force participation rate in the United States. Of course, 1940 participation decisions depend not only on initial uncertainty, but also on the number of signals that women use to update those beliefs J. Since J governs the speed of learning and, therefore, the speed of the transition, we choose J to match the aggregate growth in labor force participation between 1940 and 1950. In the model, the only difference between regions is their initial labor force participation rate. But in the data, some regions have higher population density than others. Since, logically, it should be easier to observe more people in a more densely populated area, we enrich the model by linking each region’s population density to the number of signals its inhabitants observe. In most regions, women observe four signals. But in the top 20% most densely populated regions, women observe five signals, and in the 20% least densely populated regions, they observe three signals. This is not an essential feature of the quantitative model. The results look similar when all women observe four signals. In the theory, the distance d governs the size of a woman’s neighborhood. Now that we have a county structure in our data and calibrated model, we interpret neighbors to be all residents of a woman’s own county and all neighboring counties (all counties that share a border or vertex). Preference Parameters Risk aversion γ is 3, a commonly used value. The value for leisure L is there to give women without children some reason not to participate. We calibrate L such that a woman who knows for sure that θ = 0 (because she has no child who could be harmed by her employment) participates with a 60% probability, just like women without children in 1940. The exogenous L parameter explains why some women without children do not work. Our theory explains the difference between women with and without children. 5. SIMULATION RESULTS This section compares the model’s predictions for labor force participation rates to the data—first the time series and then the geography. Finally, it examines wage and wealth predictions. 5.1. Time Series Results By itself, learning can generate a large increase in labor force participation (Figure 3). By 2010, our model predicts a 39% participation rate. While this
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FIGURE 3.—Aggregate level, cross-county heterogeneity, and spatial correlation of female labor force participation: data and calibrated model. See Section 3 for the construction of dispersion and spatial correlation measures.
falls short of the 62% rate observed in the 2005 data, the model is missing features like increasing wages, a decline in the social stigma associated with female employment, and changes in household durable technologies. One indicator of the size of these effects is the increase in the participation rate of women without children. While 60% of these women participated in 1940, 85% participated in 2005, a 25% increase. If the changes that affected all women were added to the learning effects specific to mothers of small children that this model captures, the results would more than account for the full increase, yet the results suggest that 50–66% of the increase in participation could be due to learning. Participation rises slowly at first, just as in the data, but the model does not match the sudden take off in the 1970s. Participation growth is governed by three key parameters. First, the number of signals observed J matters because more signals mean faster learning and faster participation growth. Second, the amount of noise in each signal σa matters because noisy signals slow down learning. Third, the initial degree of uncertainty matters because more uncertain agents weight new information more and thus their beliefs change quickly. This also speeds the transition. 5.2. Geographic Results The most novel results of our model are the geographic ones. These facts provide clues about how the female labor force transition took place. The right two panels of Figure 3 plot our two geographic measures, dispersion and spatial correlation, for the model and the data. The LFP dispersion measure captures the heterogeneity of participation rates across counties. In both the model and the data, the level of dispersion is similar and is humped-shaped; it rises then falls. This pattern is not unique
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to the United States. European participation also exhibits an S-shaped growth over time, and an increase and then decrease in dispersion. (See the Supplemental Material.) In the theory, dispersion rises because of the information externality: Regions that initially have high participation generate more informative signals that cause regional participation to rise more quickly. Regions with low participation have slower participation growth; with few women working, not enough information is being generated to cause other women to join the labor force. Learning does not create the heterogeneity in participation, rather it amplifies exogenous initial differences. Later in the century, dispersion falls. This happens because beliefs are converging to the truth. Since differences in beliefs generate dispersion, resolving those differences reduces dispersion. The second measure is spatial correlation, as defined in (10). Dispersion is necessary for there to be spatial correlation, because if all counties were identical, there could be no covariance in the participation rates of nearby counties. But dispersion and spatial correlation are not equivalent. It is possible to have very diverse counties, randomly placed on the map, that have high dispersion and no spatial correlation. Thus, spatial correlation measures how similar a location is to nearby locations and captures the strength of the information externalities. Spatial correlation rises, then falls. Initially, correlation is low because when few women work, information is scarce. When more women participate, they generate more information for those around them, encouraging other nearby women to participate and creating clusters of high participation. In the long run, this effect diminishes because once most information has diffused throughout the economy, the remaining cross-country differences are due to ability and endowments, which are spatially uncorrelated. The middle of the transition is when correlation is strongest. Nothing in the calibration procedure ensures that dispersion or spatial correlation looks like the data after 1930. Therefore, these patterns are supportive of the model’s mechanism. 5.3. Estimating a Geographic Panel Regression Another way to evaluate the model’s predictions is to estimate a regression where neighboring counties lagged labor force participation is an explanatory variable. To do this, we construct, for each county i, the average participation rate of its neighbors in the previous period (spatial lag) L¯ i(t−1) ≡ d(i j) LFP / j(t−1) j j d(i j), where d(i j) = 1 if counties i and j are contiguous and is 0 otherwise. Then we estimate the coefficients in an equation where county i’s time-t participation rate depends on their lagged participation and spatial lag, as well as time dummy variables γt , county demographic and economic characteristics xit , and county fixed effects αi : (11)
LFPit = ρ LFPi(t−1) +βL¯ i(t−1) + γt + φi xit + αi + it
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A. FOGLI AND L. VELDKAMP TABLE II DEPENDENT VARIABLEa : PARTICIPATION AT TIME t Data
Model
OLS
GMM
OLS
GMM
Participation at time t − 1
0.664* (0.010)
0.916* (0.062)
0.562* (0.007)
0.705* (0.072)
Neighbors’ participation at time t − 1
0.195* (0.011)
0.570* (0.103)
0.421* (0.011)
0.527* (0.115)
a The asterisk (*) denotes p-value < 0001. Robust standard errors are given in parentheses and are clustered at the county level. OLS reports estimates of ρ and β in (11). GMM reports estimates of ρ and β in the first difference of (11). Control variables xit include population density, urban and rural populations, education, and the average wage in each county and decade. Their coefficient estimates are reported in the Supplemental Material.
The second column of Table II reports the OLS estimates of ρ and β in (11). It uses our entire panel from 1940–2000. The key finding is that the coefficient on neighbors’ participation is economically and statistically significant. A 10% increase in the average participation rate of neighboring counties corresponds to a 1.9% increase in a county’s participation rate 10 years later. When we apply the same estimation procedure to the output of the model, the effect is stronger. Common problems with dynamic panel estimations are serial correlation in the residuals and a correlation between lagged dependent variables and the residuals, which contain unobserved fixed effects. Since both of these problems bias the coefficient estimates of lagged dependent variables, we use the Arellano and Bond (1991) generalized method of moments (GMM) approach to correct the bias. We use GMM to estimate a system of equations that includes the first difference of equation (11), with LFPi(t−2) LFPi(t−3) LFPi(t−4) , L¯ i(t−2) L¯ i(t−3) L¯ i(t−4) and the entire time series of controls xit as instruments. The third column of Table II reports the GMM estimates of ρ and β. The main result survives. In fact, the effect of neighboring counties more than doubles, suggesting a strong local externality. Performing the same estimation in the model produces strikingly similar coefficients. A 10% increase in neighboring counties’ participation increases participation an average of 5% in the following decade. This estimation appears to be unbiased. We cannot reject the null hypothesis of zero second-order serial correlation in residuals (p-value is 0.98). Furthermore, the Sargan test of overidentifying restrictions reveals that we cannot reject the null hypothesis that the instruments and residuals are uncorrelated (p-value is 0.35). See the Supplemental Material for full results, instrumental variable (IV) estimates, and further details. Of course, the Manski (1993) reflection problem prevents us from being able to interpret these findings as
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evidence of geographic causality. However, the fact that the model displays a geographic relationship similar to the data is supportive of our theory. 5.4. Selection Effects on Wealth and Wages The speed at which women switch from staying at home to joining the labor force depends not just on their location, but also on their socioeconomic status. Figure 4 shows the mean endowment and wage of a woman, relative to her husband, for employed women. In both the model and the data, wages are censored; they are only measured for the subset of women who participate. The model’s unconditional distribution of endowments and abilities is constant. What is changing is the selection of women who work. In other words, this is a selection effect. Early on, many women join the labor force because they are poor and desperate for income. Thus, employed women have small endowments. As women learn and employment poses less of a risk, richer women also join, raising the average endowment. This prediction distinguishes our theory from others. For example, since women with larger endowments can afford new appliances and child care first, technology-based explanations predict that richer women join first. Women’s relative wages declined early on. This finding is supported by O’Neill (1985), who documented a widening of the male–female wage gap in the mid-1950s to 1970s. She attributed it to the same selection effect that operates in our model: Early on, many women who work did so because they were highly skilled and could earn high wages. As learning made employment more attractive, less skilled women also worked, lowering the average wage. A curious result is that endowments fall and relative wages rise at the end of the sample. This small effect is not simulation error; extending the simulation a few more decades reveals that this is a persistent trend. It comes from belief dispersion, which acts like noise in an estimation and makes variables look
FIGURE 4.—Average endowment and relative wage for working women: belief dispersion for all women. Average relative wage is the woman’s wage divided by her husband’s wage (wit /ωit ), averaged over all employed women.
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less related. Starting in 1990, differences in women’s participation decisions are driven less by differences in beliefs, which are starting to converge, and more by differences in endowments and abilities. Thus, the endowment and wage selection effects become stronger again. This finding offers a warning about interpreting a wide range of statistics concerning female labor force participation. If there are significant changes in belief heterogeneity that affected participation over the 20th century, many estimated relationships between participation and other economic determinants of labor force participation will be biased. 5.5. Alternative Parameter Values and Model Timing Figure 5 shows that moderate differences in calibrated parameters do not overturn our results. Increasing the number of signals J speeds the transition but does not change the participation level that the model converges to (panel A). The exact value of the true θ, even a zero value, has only a modest effect on the participation rate that the model converges to (panel B). Replacing some of the initial uncertainty with pessimism (lowering σ, lowering μ0 ) slows learning initially (panel C). Even optimism can be offset with initial uncertainty. Furthermore, we can redefine a woman’s neighborhood d to be within 20–80 miles, with no perceptible differences in the results (panel D). Finally, feeding the time series of wages into the model has a negligible effect. Wage-based theories rely on mechanisms that raise labor supply elasticity to make wages matter. Our model has no such mechanism.
FIGURE 5.—Robustness exercises.
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We have also explored more significant changes to the model. One extension allows θ to fall over time. Another changes the model timing: Women spend 25 years growing up and 10 years having children under age 5. Both help to further slow learning.16 What the model cannot accommodate is a rapid flow of information. It would make the transition to high participation too fast. But that does not mean that women cannot observe a large number of signals. It does mean that as the number of signals rises, the noise in each of these signals must rise as well. Our calibration strategy pins down a rate of information flow in the first period of the model. Whether that amount of information is transmitted by observing 2 highly informative signals or 200 noisy ones does not matter. As long as we adjust the number of signals J to compensate for the expected amount of information each signal contains, the model delivers more or less the same prediction. The following extension of the model illustrates this point and also shows how the model can accommodate women who observe aggregate information, such as the average participation rate. 6. MODEL WITH MANY SIGNALS AND MULTIPLE TYPES OF WOMEN One concern with the model is that allowing women to observe many signals makes the transition too rapid (Figure 6, dotted line). To address this concern, we extend the model by introducing multiple types of women with different costs of maternal employment. This allows women to observe many signals— and even aggregate information—and still learn slowly. The idea is that each type of woman has a different cost θ and must observe other women of the same type to learn about their type’s cost. Professionals do not learn from seeing hourly workers. A female doctor who is on call all night
FIGURE 6.—Labor force participation with 20 signals per period. Allowing multiple types of women cancels out the effect of more signals. 16
Results from these two models are available upon request.
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does not learn about her θ from seeing the children of 9–5 workers, and urban mothers face different challenges and costs from rural ones. New research, magazine articles, or aggregate statistics contain no new information because the average cost of maternal employment is known. Each woman is now learning about how the cost of maternal employment for her type of woman differs from the average. The model setup is the same as the benchmark except that there are many types of women, indexed by ω. A woman of type ω has a cost of maternal ¯ σθ2 ), where the θ’s are independent and identically disemployment θω ∼ N(θ tributed (i.i.d.) across types. A woman’s type ω is publicly observable. The true cost of maternal employment for the average woman θ¯ is common knowledge. Simulation Results We use the same calibration as the benchmark model, except that there are now 5 types of women, with θω ’s equally spaced between 03 and 05. Each woman observes 20 signals and knows that the true mean of θ across all types was 04. The results in Figure 6 are similar to those of the benchmark model (Figure 3). The reason that the multitype version of the model looks so much like the original calibrated model is that the rate of information flow is the same. While women observe 20 signals, on average, only 4 are relevant for inferring their θω . The other 16 signals are not useful because they are about other types of women. Likewise, if there were 100 signals and 5 types of women, the resulting participation rates would look much like the benchmark model with 20 signals (Figure 6, dotted line). The more general point here is that the number of signals is not crucial for the model’s results. Instead, the rate of information flow is the key. Our calibration procedure pins down a rate of information flow by matching the speed of learning in the beginning of the sample. A model with many noisy signals or few informative ones will generate similar results as long as it implies a rate of information flow that is in line with the data. Of course, this leaves open the question of why the rate of information flow is so low. Many recent theories that use local learning (Amador and Weill (2006), Buera, Monge-Naranjo, and Primiceri (2011)), rational inattention (Sims (2003)), or information generated by economic activity (Veldkamp (2005), Kurlat (2009)) all require a meager rate of information flow to match the data. Future work could explore frictions that slow Bayesian learning. Perhaps social heterogeneity makes inference from others’ experiences difficult and slows down social learning. 7. CONCLUSION Many factors have contributed to the increase in female labor force participation over the last century. We do not argue that beliefs were the only relevant one. Rather, the model abstracts from other changes to focus on how the
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transition from low to high participation can be regulated by learning in a way that matches the time series and geographic data. Including local information transmission as part of the story of female labor force participation in the 20th century helps to explain its gradual dynamic and geographic evolution. This framework can be applied to study the geography of other types of economic and social change. One example is women’s career choices. Suppose that a high-intensity career for a mother brings with it a higher wage, but also more uncertain effects on children. Since the high-intensity career is more uncertain than a regular career, fewer women choose it early on. As more women choose high-intensity careers, others learn from them and the composition of careers changes. The growth in the fraction of employed women participating in high-intensity careers increases the average wage of working women. This could be one component of the explanation for a rise in female wages and its geographic patterns. Clearly, these social transformations involve a change in culture. Rather than seeing cultural change as a competing explanation, we see this framework as one that can potentially capture some of the flavor of cultural change. One way to view this paper is that a theory of information diffusion can change preferences. Another important feature of social behavior is the desire to fit in or coordinate with others. Using an objective function like that in beauty contest games Morris and Shin (2002), coupled with the geographic nature of information transmission, could provide a rich set of testable implications. Specifically, it could predict geographic patterns, such as the spread from urban to rural areas, in the types of cultural changes investigated by Greenwood and Guner (2010), Guiso, Sapienza, and Zingales (2006), and Bisin and Verdier (2001). Such work could help differentiate exogenous changes in preferences from information-driven changes in coordination outcomes. Another direction one could take this model is to interpret the concept of distance more broadly. Arguably, socioeconomic, ethnic, religious, or educational differences create stronger social barriers between people than physical distance does. If that is the case, the learning dynamics that arise within each social group may be quite distinct. If the initial conditions in these social groups differ, changes in labor force participation, career choice, or social norms may arise earlier in one group than in another. This model provides a vehicle for thinking about the diffusion of new behaviors, with uncertain consequences, among communities of people. APPENDIX A: PROOFS OF ANALYTICAL RESULTS A.1. Derivation of Comparative Statics Step 1. Define a cutoff wage w¯ such that all women who observe wit > w¯ choose to join the labor force. A woman joins the labor force when EUWit − EUOit > 0. Note that ∂Nit /∂wit = (nit wit + ωit )−γ > 0. Since Nit is monotoni-
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cally increasing in the wage w, there is a unique w¯ for each set of parameters, ¯ Nit = 0. such that at w = w, Step 2. Describe the probability of labor force participation. Let denote the cumulative density function (c.d.f.) for the unconditional distribution of wages in the population. This is a log-normal c.d.f. Since the log normal is unbounded and has positive probability on every outcome, its c.d.f. is therefore strictly increasing in its argument. Then the probability that a woman partici¯ which is then strictly decreasing in w. ¯ pates is 1 − (w), Step 3. Determine the effect of mean beliefs on labor force participation. Taking the partial derivative of the net utility gain from labor force participa¯ tion yields ∂Nit /∂μit = −β. By the implicit function theorem, ∂w/∂μ it > 0. ¯ ¯ ¯ ¯ Thus, ∂(1 − (w))/∂μit = (∂(1 − (w))/∂w)(∂w/∂μit ) < 0. Step 4. Calculate the effect of uncertainty on labor force participation. The benefit to participating is falling in uncertainty: ∂Nit /∂σit = (1 − γ)β × exp((μa − μit )(1 − γ) + 12 (σa2 + σit2 )(1 − γ)2 ). Since γ > 1, β > 0 by assumption, and the exponential term must be nonnegative, this means that ∂Nit /∂σit2 < 0. 2 ¯ As before, the implicit function theorem tells us that ∂w/∂σ it > 0. Thus, 2 2 ¯ ¯ ¯ ¯ ∂(1 − (w))/∂σ w)(∂ w/∂σ it = (∂(1 − (w))/∂ it ) < 0. A.2. Proof of Result 1: Zero Participation Is Not a Steady State For any arbitrary beliefs μjt and σjt , and endowment ωjt , there is some finite level of ability a∗ and an associated wage w∗ = exp(a∗ ), such that EUWit > EUOit > 0 ∀ajt ≥ a∗ . The fact that ajt is normally distributed means that Prob(ajt ≥ a∗ ) > 0 for all finite a∗ . Since woman j enters the labor force whenever EUWit > EUOit > 0 and this happens with positive probability, then njt = 1 with positive probability. Since this is true for all women j, it is also true that j njt ≥ 1 with positive probability. A.3. Proof of Result 2: Geographic Correlation Let α be the fraction of women who participate in family i’s region. The region a woman lives in does not affect her endowments or ability. Therefore, Nit+1 can be rewritten, using (7) and (8), as 1 2 Nit+1 = A − B exp (γ − 1)μit+1 + (γ − 1)2 σit+1 2 for positive constants A and B. Since a woman born at time t participates if Nit+1 > 0, it suffices to show that ∂Nit+1 /∂α > 0 for a woman with average prior beliefs and an average signal. The number of informative signals that a woman in family i, with an average signal draw, would see is n¯ it = αJ. Since beliefs and signals are unbiased by construction, then a woman with average prior beliefs has μit = θ and a
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woman with an average signal has μˆ it = θ. By equation (4), her posterior belief is μit+1 = θ, for any fraction α. Her posterior precision does depend on α: Ac2 , and the equation for n¯ it above, cording to equation (5), the definition of σˆ it+1 −2 −2 σit+1 = σit + αJ/σa . Since J and σa are both positive, posterior precision is 2 is decreasing in α, and Nit+1 is increasing in α. Thus, posterior variance σit+1 increasing in α. APPENDIX B: DATA—SOURCES AND DEFINITIONS County-Level Data Our county-level data set has information on a vast array of economic and socio-demographic variables for 3074 U.S. counties for each decade over the period 1940–2000. Most of the information comes from Census data and, in particular, from a data set called “Historical, Demographic, Economic, and Social Data: The United States, 1790–2000” (HDES) produced by the Interuniversity Consortium for Political and Social Research (series 2896). This data set is a consistency-checked and augmented version of the Integrated Public Use Microdata Series (IPUMS) produced by the Minnesota Population Center. We integrated this data set using several others, including the Census of Population and Housing, the County and City Data Book, the Census 2000 Summary Files, and IPUMS to obtain the most complete and homogeneous information at the county level for this span of time (United States Department of Commerce, Bureau of the Census, Census of Population and Housing (1980)). Sources and details about the construction of each single variable, together with summary statistics, are presented in the Supplemental Material. Benchmark Regression Our measures of dispersion and spatial correlation among county-level participation rates are calculated on the residuals from the year by year regression LFPit = β1t + β2t controlsit + it , where the list of controls includes population density, average education, a set of dummies for type of place (percentage of population that is urban, rural farm, and rural nonfarm), a set of dummies to capture demographic composition (percentage of population that is white, black, and from other races), and a set of dummies for industrial composition (percentage of establishments that are retail, services, manufacturing, and wholesale). One data issue we were concerned with was potential bias in our estimates from excluding counties with missing data. For this reason, we did not include manufacturing wages among the controls, since the information on wages is missing for a relevant number of counties in several years. Including wages, however, does not substantially alter our results. We also are missing the industrial composition for several counties in 1940 and, in particular, in the year
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2000. We therefore recalculate the residuals from the regression in the year 2000, excluding sectoral composition from the controls, and find that the spatial correlation measure rises from 0.38 to 0.45. It is possible that the spatial correlation increases because of spatial correlation in industrial sector composition that is now attributed to information. However, the correlation does not rise (in the first two significant digits) in the previous decades when industrial sectors are excluded. This suggests that the change in correlation is prevalently due to the sparser data available in 2000. Therefore, in Figure 3, we use the higher estimate, on the full sample of counties, as our measure of spatial correlation in the year 2000. Combining County and National Data We use two data sources to construct the labor force participation series: The IPUMS provides information about individuals’ participation decision, along with characteristics such as marital status and presence of children under 5 years of age, but it does not contain county-level geographic identifiers. The HDES allows us to compute participation rates by county, but only for all women over the age of 16. Since our theory focuses on the geographic patterns of participation of married mothers of young children, we need to use both sources to approximate the series we need. We first use the IPUMS data to calculate an adjustment factor for converting participation rates for all women over 16 years to participation rates for white, married women, aged 25–35, with a child under 5 years of age and not living on a farm or in an institution. We report the time series of the labor force participation of these two different samples of women in the last two columns of Table III. We calculate the adjustment factor for each decade as the ratio between these two columns. We then apply this adjustment factor to TABLE III FEMALE LABOR FORCE PARTICIPATION RATE, BY DECADE, FROM VARIOUS DATA SOURCES AND POPULATION SAMPLES IPUMS
1940 1950 1960 1970 1980 1990 2000
HDES All > 16
All > 16
White, 25–35, Nonfarm Married With Kid < 5
25.3 28.8 35.1 41.2 49.7 56.6 57.1
25.8 29.1 35.7 41.3 49.9 56.7 57.5
6.7 9.8 16.9 24.1 41.5 59.3 59.6
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convert, decade by decade, each county’s participation rate for all women in that county’s participation rate for mothers with children under 5. Finally, to control for economic and demographic differences across counties, we regress this county-level participation on our county-level control variables. The residuals from this regression form the data series that we use to compare with the geographic features of the model. To check for consistency between our two data sets, we compared the population-weighted average participation rate for all women (over 16 years) in all counties from the HDES data with the equivalent series from IPUMS. The second and third columns of Table III show that the two sources report almost identical aggregate participation rates. APPENDIX C: CALIBRATION Throughout, we look at women 25–54, with their own child younger than 5 living in the household. We use whites not living in an institution or on a farm, and not working in agriculture. ABILITIES: The distribution of women’s abilities is constructed so that their wages in the model match the distribution of women’s wages in the 1940 census data; σa = 057 is the standard deviation of log ability and μa = ln(earnings gap) − (σa2 )/2 is the mean of log ability. These parameters target the initial ratio between average earnings of working women and average earnings of all husbands (0.81 in the data), and target the standard deviation of log earnings of employed women in the data (0.53). A wage gap where women earn 81% of their husbands’ income is higher than most estimates. This is due to two factors. First, we do not require husbands to be full-time workers because we want to capture the reality that women’s endowments can be high or low. Second, poor women are more likely to be employed. By comparing only husbands of employed women to their wives, we are selecting poorer husbands. SELECTION EFFECTS IN THE MODEL: The distribution of observed wages in the data needs to be matched with the distribution of wages for employed women in the model. Employed women are not a representative sample. They are disproportionately high-skill women. The calibration deals with this issue by matching the truncated distribution of wages in the data to the same truncated sample in the model. In other words, we use the model to back out how much selection bias there is. ENDOWMENT DISTRIBUTION: The census provides husbands’ wages, starting in 1940. We select out only husbands with children under 5. In the model, the log endowment is normal. Therefore, we examine the log of 1940 wages. For husbands, mean(log_incwage_husb) = 704 and std(log_incwage_husb) = 073. Therefore, we set σω = 073. We normalize the mean husband’s endowment to 1 by choosing μω = −(σω2 )/2.
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TRUE VALUE OF NURTURE: To calibrate the θ parameter, we use microevidence on the effect of maternal employment on the future earnings of children. Our evidence on the effect of maternal employment comes from the National Longitudinal Survey of Youth (NLSY), in particular, the Peabody Picture Vocabulary Test (PPVT) at age 4 and the Peabody Individual Achievement Test (PIAT) for math and reading recognition scores measured at ages 5 and 6. One year of full-time maternal employment plus informal daycare reduces test scores by roughly 3.4% (Bernal and Keane (2011)). If a mother works from 1 year after birth until age 6, these 5 years of employment translate into a score reduction of 17%. The childhood test scores are significantly correlated with educational attainment at 18. A 1% increase in the math score at age 6 is associated with 0.019 years of additional schooling. A 1% increase in the reading test score at age 6 is associated with 0.025 additional school years. Therefore, 5 years of maternal employment translates into between 0.32 (17 × 0019) and 0.42 (17 × 0025) fewer years of school. The final step is to multiply the change in educational attainment by the returns to a college education. We use the returns to a year of college from 1940 to 1995 from Goldin and Katz (1999). Their estimates are the composition-adjusted log weekly wage for full-time/full-year, nonagricultural, white males. Those estimates are 0.1, 0.077, 0.091, 0.099, 0.089, 0.124, and 0.129 for the years 1939, 1949, 1959, 1969, 1979, 1989, and 1995. The average return to a year of college is 10%. Since maternal employment reduces education by 0.32–0.42 years, the expected loss in terms of foregone yearly log earnings is about 4%, or θ = 004. NUMBER OF SIGNALS: J is calibrated to get the aggregate labor force participation to rise from 6% in 1940 to 10% in 1950. Note that this is really pinning down a rate of information flow, rather than a number of signals. If signals had twice as much noise, but women observed twice as many of them, such a model could match this same calibration target. INITIAL PARTICIPATION IN 1930 (HETEROGENEOUS ACROSS REGIONS): We want to preserve some of the spatial information in our data. However, the model is on a square grid. Mapping irregular-sized U.S. counties onto this grid is a challenge. To do this, we used regions, which are larger than counties. Regions are constructed by taking the 48 contiguous states, computing the county centroid with the highest and the lowest longitude (the difference between the maximum and minimum lodist), and dividing the U.S. map into n vertical strips, each with width lodist/n. Then, for each strip, we compute the maximum and minimum latitude, and divide the strip into n boxes of equal height. We choose n = 10 because it is the largest possible number that does not result in there being boxes that contain no county centroids. In the model, we divide the evenly spaced agents into 100 regions of equal size and population. For each of these 100 regions, we assign the participation rate of the corresponding box on the U.S. map and assign agents randomly to participate or not. Each participates
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with a probability given by the regional participation rate. After calibrating initial participation, this regional aggregation structure is never used again and we compute statistics at the more local, county level. APPENDIX D: MODEL WITH LEARNING FROM OTHERS’ CHOICES To keep the model simple and tractable, we assumed that women do not draw any inference from the labor decisions of other women. They use the knowledge of whether J of their peers were nurtured to estimate the cost of maternal employment. But they do not take advantage of the fact that the mother’s employment decision reveals something about the mother’s beliefs, which is additional information about the true value of nurture θ. This section shows that our simplifying assumption is innocuous. Seeing other women’s labor force decisions does not significantly speed up learning for five reasons: (i) Participation is a binary choice. The binary nature of the signal eliminates much of its information. (ii) Early on, most women do not work and other women expect that the women they encounter will likely not work. Therefore, a woman who observes another woman not working early in the century gets very little new (unexpected) information. Observing working women is informative but it becomes commonplace only later in the century when most of the learning has been completed. (iii) Women observe the participation decisions of women from the previous cohort. Those women were less informed and less likely to work. (iv) The “noise” in women’s participation decisions is large. Women do not know others’ ability, do not know whether the mother was nurtured, and do not know how uncertain they were. Through all this noise, the belief about the mean is a weak signal. (v) The beliefs of others in your region are highly correlated with your own beliefs because people in the same region see common signals. A correlated signal contains less information than an independent one. To quantify these claims, we simulate an economy that is an approximation to the economy where women learn from the decisions of other women. To keep the linear Bayesian updating rules, we consider an economy where women observe additional normally distributed signals whose signal-to-noise ratio is the same as the information embedded in the participation decisions they observe. This is an upper bound on how much additional information comes from others’ decisions because normally distributed signals contain more information than any other kind of signal with the same signal-to-noise ratio (Cover and Thomas (1991)). To estimate the signal-to-noise ratio of women’s employment decisions, run a regression of participation on beliefs. Since the informativeness of women’s labor decisions changes over time, there should be a separate regression run for each decade. Compute the R2 . The signal-to-noise ratio, the ratio of the explained sum of squares (signal) to unexplained (noise), is R2 /(1 − R2 ). To construct a signal with the same amount of noise, first compute the cross-sectional
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FIGURE 7.—Labor force participation when women observe participation decisions of others.
variance of women’s beliefs. This is the total sum of squares. Multiply this variance by 1 − R2 to get the unexplained sum of squares. Create an m × 1 vector of i.i.d. normal random variables with mean zero and variance (1 − R2 ) var(μt ), where m is the number of women in the economy. Add this noise shock to the vector of women’s beliefs. Each woman in generation t + 1 sees a subset of the signals about generation t beliefs, where the subset is the signals with indices j ∈ Ji . The time series of labor force participation that results from simulating this model, with the same calibrated parameters as in Table I, appears in Figure 7. This approach generates a labor force participation rate that is only 5% higher at the end of the sample. Thus, learning from other women’s participation choices does speed up the increase in labor force participation by speeding up learning, but its effect is small. REFERENCES ALBANESI, S., AND C. OLIVETTI (2007): “Gender Roles and Medical Progress,” Working Paper 14873, NBER. [1105] ALESINA, A., AND P. GIULIANO (2007): “The Power of the Family,” Working Paper 13051, NBER. [1119] AMADOR, M., AND P.-O. WEILL (2006): “Learning From Private and Public Observations of Others’ Actions,” Mimeo, Stanford University, Stanford and UCLA. [1128] ANTECOL, H. (2000): “An Examination of Cross-Country Differences in the Gender Gap in Labor Force Participation Rates,” Labour Economics, 7, 409–426. [1105,1119] ARELLANO, M., AND S. BOND (1991): “Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations,” Review of Economic Studies, 58, 277–297. [1124] ATTANASIO, O., H. LOW, AND V. SANCHEZ-MARCOS (2008): “Explaining Changes in Female Labour Supply in a Life-Cycle Model,” American Economic Review, 98, 1517–1552. [1105] BERNAL, R., AND M. KEANE (2011): “Child Care Choices and Children’s Cognitive Achievement: The Case of Single Mothers,” Journal of Labor Economics (forthcoming). [1120,1134] BISIN, A., AND T. VERDIER (2001): “The Economics of Cultural Transmission and the Evolution of Preferences,” Journal of Economic Theory, 97, 298–319. [1129]
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BUERA, F. J., A. MONGE-NARANJO, AND G. E. PRIMICERI (2011): “Learning the Wealth of Nations,” Econometrica, 79, 1–45. [1128] COMIN, D., AND B. HOBIJN (2010): “An Exploration of Technology Diffusion,” American Economic Review, 100, 2031–2059. [1106] COVER, T., AND J. THOMAS (1991): Elements of Information Theory (First Ed.). New York: Wiley. [1135] DEL BOCA, D., AND D. VURI (2007): “The Mismatch Between Employment and Child Care in Italy: the Impact of Rationing,” Journal of Population Economics, 20, 805–832. [1105] FARRE, L., AND F. VELLA (2007): “The Intergenerational Transmission of Gender Role Attitudes and its Implications for Female Labor Force Participation,” Working Paper, Georgetown University. [1118] FERNÁNDEZ, R. (2007): “Culture as Learning: The Evolution of Female Labor Force Participation Over a Century,” Mimeo, New York University. [1105,1112] FERNÁNDEZ, R., AND A. FOGLI (2009): “Culture: An Empirical Investigation of Beliefs, Work, and Fertility,” American Economic Journal: Macroeconomics, 1, 146–177. [1105,1119] FERNÁNDEZ, R., A. FOGLI, AND C. OLIVETTI (2004): “Mothers and Sons: Preference Formation and Female Labor Force Dynamics,” Quarterly Journal of Economics, 119, 1249–1299. [1105] FOGLI, A., AND L. VELDKAMP (2011): “Supplement to ‘Nature or Nurture? Learning and Geography of Female Labor Force Participation’,” Econometrica Supplemental Material, 79, http://www.econometricsociety.org/ecta/Supmat/7767_data and programs.zip; http://www.econometricsociety.org/ecta/Supmat/7767_data description.pdf. [1116] FORTIN, N. (2005): “Gender Role Attitudes and the Labor Market Outcomes of Women Across OECD Countries,” Oxford Review of Economic Policy, 21, 416–438. [1119] FUCHS-SCHUNDELN, N., AND R. IZEM (2007): “Explaining the Low Labor Productivity in East Germany—A Spatial Analysis,” Working Paper, Harvard University. [1106] GOLDIN, C. (1990): Understanding the Gender Gap: An Economic History of American Women. Oxford, NY: Oxford University Press. [1105] (1995): “The U-Shaped Female Labor Force Function in Economic Development and Economic History,” in Investment in Human Capital, ed. by T. P. Schultz. Chicago: University of Chicago Press. [1107] GOLDIN, C., AND L. KATZ (1999): “The Returns to Skill in the United States Across the Twentieth Century,” Working Paper 7126, NBER. [1112,1120,1134] (2002): “The Power of the Pill: Oral Contraceptives and Women’s Career and Marriage Decisions,” Journal of Political Economy, 110, 730–770. [1105] GREENWOOD, J., AND N. GUNER (2010): “Social Change: The Sexual Revolution,” International Economic Review, 51, 893–923. [1129] GREENWOOD, J., A. SESHADRI, AND M. YORUKOGLU (2005): “Engines of Liberation,” Review of Economic Studies, 72, 109–133. [1105,1106] GUISO, L., P. SAPIENZA, AND L. ZINGALES (2006): “Does Culture Affect Economic Outcomes?” Journal of Economic Perspectives, 20, 23–48. [1129] HAINES, M. R., AND INTER-UNIVERSITY CONSORTIUM FOR POLITICAL AND SOCIAL RESEARCH (2010): “Historical, Demographic, Economic, and Social Data: The United States, 1790–2002,” Computer File, ICPSR02896-v3, Inter-University Consortium for Political and Social Research [distributor], Ann Arbor, MI (DOI:10.3886/ICPSR02896). [1116] HARVEY, E. (1999): “Short-Term and Long-Term Effects of Early Parental Employment on Children of the National Longitudinal Survey of Youth,” Developmental Psychology, 35, 445–459. [1120] HILL, J., J. WALDFOGEL, J. BROOKS-GUNN, AND W. HAN (2005): “Maternal Employment and Child Development: A Fresh Look Using Newer Methods,” Developmental Psychology, 41, 833–850. [1120] JONES, L., R. MANUELLI, AND E. MCGRATTAN (2003): “Why Are Married Women Working So Much?” Research Department Staff Report 317, Federal Reserve Bank of Minneapolis. [1105,1112]
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JOVANOVIC, B. (2009): “The Technology Cycle and Inequality,” Review of Economic Studies, 76, 707–729. [1106] KURLAT, P. (2009): “Lemons, Market Shutdowns and Learning,” Working Paper, MIT. [1128] LUCAS, R. (1972): “Expectations and the Neutrality of Money,” Journal of Economic Theory, 4, 103–124. [1107] MAMMEN, K., AND C. PAXSON (2000): “Women’s Work and Economic Development,” Journal of Economic Perspectives, 14, 141–164. [1107] MANSKI, C. (1993): “Identification of Endogenous Social Effects: The Reflection Problem,” Review of Economic Studies, 60, 531–542. [1118,1124] MORAN, P. (1950): “Notes on Continuous Stochastic Phenomena,” Biometrika, 37, 17–23. [1118] MORRIS, S., AND H. SHIN (2002): “The Social Value of Public Information,” American Economic Review, 92, 1521–1534. [1129] MUNSHI, K. (2004): “Social Learning in a Heterogeneous Population: Technology Diffusion in the Indian Green Revolution,” Journal of Development Economics, 73, 185–213. [1106] OLIVETTI, C. (2006): “Changes in Women’s Hours of Market Work: The Effect of Changing Returns to Experience,” Review of Economic Dynamics, 9, 557–587. [1112] O’NEILL, J. (1985): “The Trend in the Male–Female Wage Gap in the United States,” Journal of Labor Economics, 3, S91–S116. [1125] SIMS, C. (2003): “Implications of Rational Inattention,” Journal of Monetary Economics, 50, 665–690. [1128] UNITED STATES DEPARTMENT OF COMMERCE, BUREAU OF THE CENSUS, CENSUS OF POPULATION AND HOUSING (1980): “County Population by Age, Sex, Race, and Spanish Origin,” Computer File, ICPSR08108-v1, Inter-University Consortium for Political and Social Research [distributor], Ann Arbor, MI (DOI:10.3886/ICPSR08108). [1131] VELDKAMP, L. (2005): “Slow Boom, Sudden Crash,” Journal of Economic Theory, 124, 230–257. [1128]
Dept. of Economics, University of Minnesota, 1925 Fourth Street South, Minneapolis, MN 55455, U.S.A. and Federal Reserve Bank of Minneapolis; afogli@ umn.edu and Stern School of Business, New York University, 44 West Fourth Street, Suite 7180, New York, NY 10012, U.S.A. and NBER;
[email protected]. Manuscript received March, 2008; final revision received March, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1139–1180
MENU COSTS, MULTIPRODUCT FIRMS, AND AGGREGATE FLUCTUATIONS BY VIRGILIU MIDRIGAN1 Golosov and Lucas recently argued that a menu-cost model, when made consistent with salient features of the microdata, predicts approximate monetary neutrality. I argue here that their model misses, in fact, two important features of the data. First, the distribution of the size of price changes in the data is very dispersed. Second, in the data many price changes are temporary. I study an extension of the simple menu-cost model to a multiproduct setting in which firms face economies of scope in adjusting posted and regular prices. The model, because of its ability to replicate this additional set of microeconomic facts, predicts real effects of monetary policy shocks that are much greater than those in Golosov and Lucas and nearly as large as those in the Calvo model. Although episodes of sales account for roughly 40% of all goods sold in retail stores, the model predicts that these episodes do not contribute much to the flexibility of the aggregate price level. KEYWORDS: Menu costs, multiproduct firms, sales.
1. INTRODUCTION A WIDELY HELD VIEW in macroeconomics is that menu costs of price adjustment give rise to aggregate price inertia and thus provide a mechanism through which changes in monetary policy have real effects. This view lies at the heart of new-Keynesian analysis, which often cites menu costs as the microfoundation for the assumptions on price setting it makes. Golosov and Lucas (2007) recently challenged this view. They argued that when a standard menu-cost model is made consistent with the microprice data,2 money is nearly neutral. The key feature of the price data they emphasize is that the size of price changes is very large, nearly 10 percent on average. When they choose parameters governing an idiosyncratic productivity shock to produce this feature of the microprice data, the model predicts little stickiness in the aggregate price level. Their paper is viewed as a serious challenge to the key mechanism underlying new-Keynesian business cycle analysis. This paper revisits the Golosov and Lucas result. I argue that while their analysis captures some features of the microprice data, it does not produce two salient features of that data. The first is that in the data there is a large amount of heterogeneity in the size of price changes, so that relative to their 1 I am indebted to George Alessandria, Bill Dupor, Paul Evans, Patrick Kehoe, and Mario Miranda for valuable advice and support, as well as Ariel Burstein, Michael Dotsey, Mark Gertler, Narayana Kocherlakota, Ellen McGrattan, John Leahy, Martin Schneider, Takayuki Tsuruga, Ivan Werning, and Harald Uhlig for comments and suggestions. I acknowledge the James M. Kilts Center, GSB, University of Chicago, for making the data used in this study available. Any errors are my own. 2 Bils and Klenow (2004) and Klenow and Kryvtsov (2008).
© 2011 The Econometric Society
DOI: 10.3982/ECTA6735
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model’s predictions, the data have both many more small price changes and many more very large price changes. The second feature is that they abstract from sales, even though sales (or more generally, temporary price changes) account for a large majority of price changes. I add several ingredients to the Golosov and Lucas model to make it consistent with these two additional microfacts as well as the original facts that motivate their study. When I do so, I find that their near-neutrality result is overturned. In particular, my extended model has real effects of money that are of a similar magnitude to those in the popular Calvo model. To allow the model to produce very large price changes, I allow for a fattailed distribution of cost shocks and let the data pin down the shape of the distribution. To allow the model to produce small price changes, I assume economies of scope in price adjustment. In particular, I assume the retailer sells multiple goods and faces a single cost of changing the prices of these goods. Since the idiosyncratic shocks to the different goods are not perfectly correlated, at any point in time some goods’ prices may be far from their desired level while others may be close to their desired level. My model formalizes the ideas in Lach and Tsiddon (2007), who provided empirical evidence for this mechanism. The standard menu-cost model also does not produce the type of temporary price changes observed in the data. To produce such changes, the theory must, of course, provide a motive for retailers to change prices temporarily. A striking feature of temporary price changes is that almost 90% of the time, after a temporary price change, the nominal price returns exactly to the nominal price that prevailed before the change. This feature suggests the need for a nominal friction to account for such behavior. Existing studies have suggested a number of explanations for temporary price discounts.3 All of them, however, are about real prices and hence, they cannot explain the striking feature of the data: the nominal price, after a sale, typically returns exactly to the nominal pre-sale price. This latter feature is critical for understanding how nominal prices respond to nominal shocks. Indeed, as emphasized by Kehoe and Midrigan (2008), this feature of the data has an important effect on the model’s implications regarding monetary nonneutrality. Even though there is a large amount of high-frequency variation in prices associated with temporary price changes, there is much less low-frequency variation, which is ultimately what matters for how aggregate prices respond to low-frequency variation in monetary policy. Here instead of building a detailed model of why retailers offer temporary price discounts, I simply assume that these changes are a response to tem3 Search frictions (Butters (1977), Varian (1980), Burdett and Judd (1983)), demand uncertainty Lazear (1986), thick-market externalities (Warner and Barsky (1995)), loss-leader models of advertising (Chevalier, Kashyap, and Rossi (2003)), and intertemporal price discrimination (Sobel (1984)) to name a few.
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porary changes in wholesale costs. While this mechanism is simple, it is not without realism if one focuses on retail pricing: as documented by Eichenbaum, Jaimovich, and Rebelo (2011), most retail prices change in response to a change in costs. This also allows me to focus on the nominal rigidity used by Kehoe and Midrigan (2008) that allows the model to reproduce the striking feature of temporary price changes. To get the model to account for the striking reversion to the pre-sale price, I assume that retailers set two types of prices—posted prices and regular prices—both of which are costly to change. Although not specifically modeled, the idea behind these two costs is that posted prices are set by the retailer’s salespeople and regular prices are set by the headquarters. Recent studies find some evidence for such price-setting practices. For example, Zbaracki, Ritson, Levy, Dutta, and Bergen (2004), Zbaracki, Levy, and Bergen (2007) found that pricing is done at two levels: the upper management sets the overalpricing strategy of the store (the regular price), while the sales personnel has the discretion to deviate from the regular price by quoting the consumer a discounted price. My model has two main differences from the Golosov and Lucas model. First, my model produces more heterogeneity in the size of price changes. Second, I have many temporary price changes, while Golosov and Lucas do not. It turns out that in accounting for why my model produces much larger real effects of money than does the Golosov and Lucas model, the first difference is the most important. Consider first the role of heterogeneity in price changes. When heterogeneity is small, as it is in Golosov and Lucas, monetary policy has a strong selection effect that mitigates the real effects of money. When heterogeneity is large, as in my model, money policy has a very weak selection effect and money has much larger effects. Recall that the real effects of a given money shock are smaller the larger the response of the price level is to this shock. In the Golosov and Lucas model, prices respond strongly to a money shock; in my model, they do not. The reason that prices respond more strongly to a money shock in the Golosov and Lucas economy is the strong selection effect in which changes in monetary policy alter the mix of adjusting firms toward firms whose idiosyncratic shocks call for larger price increases.4 The strength of this selection effect depends on the measure of marginal firms whose desired price changes lie in the neighborhood of the adjustment thresholds. When the measure of marginal firms is relatively large, as in Golosov and Lucas, the implied distribution of price changes is bimodal with little size dispersion: most price changes are near 4 Also related is the Caplin and Spulber (1987) neutrality result. Other contributions in the menu-cost literature are Sheshinski and Weiss (1977), Caplin and Leahy (1991), Caballero and Engel (1993, 2007), Dotsey, King, and Wolman (1999), Danziger (1999), Burstein (2006), and Burstein and Hellwig (2008).
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the adjustment thresholds. In contrast, when the measure of marginal firms is small, as it is in my model, most firms are dispersed away from their adjustment thresholds, and hence the distribution of the size of price changes shows large dispersion. Accounting for the dispersion of the size of price changes is thus key to studying the aggregate implications of menu-costs economies. I show that in the data, the distribution of the size of price changes is, in contrast to the predictions of the Golosov–Lucas model, highly dispersed. Individual goods experience, over their lifetime, both very small and very large price changes. My modified menu-cost model produces these features of the data and because of this has a small selection effect. Indeed, my model’s predictions are fairly close to those of a constant-hazard Calvo model in which the selection effect is absent. Consider next the role of temporary price changes in accounting for my results. Briefly, in my model, retailers frequently adjust their prices for just a couple of periods when they experience idiosyncratic temporary cost shocks. During these periods, the retailers find it optimal to offset the effects of any monetary policy shock by adjusting its price to neutralize such shocks. Hence, all else equal, adding such temporary changes lowers the real effects of money shocks. It turns out that, as Kehoe and Midrigan emphasized, this effect is not large, even though the model accounts for the fact that about 40% of all goods are sold during such periods. Consider, finally, the robustness of my result. In my model I have focused on economies of scope in price setting as a mechanism to generate small price changes and fat-tailed distributions to generate large price changes. My result seems to be robust to alternative mechanisms to generating such heterogeneity in price changes. For example, a simple alternative way to generate small price changes is to make the menu-cost stochastic, as in Dotsey, King, and Wolman (1999) so that sometimes the menu-cost is so small that retailers change their price even if it is close to the desired price. For this alternative mechanism, I found the real effects of money to be similar to the model with economies of scope. I chose to focus on the model with economies of scope because there is some evidence that economies of scope are indeed a feature of the pricesetting technology in retail stores. In addition to the evidence of Lach and Tsiddon (2007), Levy, Bergen, Dutta, and Venable (1997) presented direct evidence of economies of scope by directly describing the technology of price adjustment in a large retail store. Moreover, I present additional evidence for economies of scope in this paper. I show that an individual good is more likely to experience a price change if other goods sold by the retailer show large desired prices. This latter feature of the data is inconsistent with the predictions of a menu-cost model without economies of scope. Although the main focus of the paper is, due to data availability, on a single grocery store, the results apply more generally to all sectors that comprise the U.S. consumer price index (CPI). To illustrate this point, this paper considers an extension of the analysis to recently available data from Nakamura
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and Steinsson (2008). I show that the strength of the selection effect in a model that accounts for the large mean and dispersion in the size of price changes in all sectors of the U.S. economy is small as well. Similarly, my results extend to an environment with capital accumulation and interest-elastic money demand. In addition to the work of Golosov and Lucas (2007), this paper is closely related to recent work by Klenow and Kryvtsov (2008) and Gertler and Leahy (2008). Both of these papers present parameterizations of economies in which menu costs generate aggregate price inertia almost as large as in Calvo-type models. In Klenow and Kryvtsov, the selection effect is weak because of the assumption of time-varying adjustment costs: most price changes occur in periods in which the realization of the adjustment cost is low. The economy studied by Gertler and Leahy is also characterized by a weak selection effect because of the assumption of a Poisson process for idiosyncratic shocks. The economy I study here shares elements of both of these approaches and, importantly, provides empirical evidence for the mechanism that underlies these results. 2. DATA Here I describe the data and the algorithm I use to distinguish between regular and temporary prices. I then report several salient features of the data that motivate the subsequent analysis. A. Description of the Data Set I use a data set of scanner price data, maintained by the Kilts Center for Marketing at the University of Chicago Graduate School of Business (GSB).5 The data set is a by-product of a randomized pricing experiment6 conducted by the Dominick’s Finer Foods7 retail chain in cooperation with the Chicago GSB. Nine years (1989–1997) of weekly store-level data on the prices of more than 9000 products for 86 stores in the Chicago area are available. The products available in this data base range from nonperishable food products to various household supplies, as well as pharmaceutical and hygienic products. Dominick’s sets prices on a weekly basis (it changes prices, if at all, only once a week),8 so I report all statistics at the weekly frequency. Data are available on the actual transaction prices faced by consumers, as well as quantity sold and 5
The data are available online in the Supplemental Material. Hoch, Dreze, and Purk (1994) discussed Dominick’s experiment in detail. 7 An earlier version of this paper also looked at data collected by AC Nielsen that also is available from the same source. The AC Nielsen data are constructed from an underlying panel of the purchasing history of households and subject to more measurement and time-aggregation problems. For this reason, I chose to eliminate this additional data set from the analysis in this paper. 8 Its pricing cycle is weekly, so price change, if at all, occurs only once during the week. See Dutta, Bergen, and Levy (2002, p. 1854). 6
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the wholesale cost of the good.9 Dominick’s sets prices on a chainwide basis, so prices across stores are highly correlated.10 I choose thus to work with the price of one single store, number 122, the store with the largest number of price observations that was part of the control zone and thus not subject to the pricing experiment. B. Algorithm to Identify Regular Prices To identify regular prices, I make use of an algorithm discussed in Kehoe and Midrigan (2008). The algorithm is based on the idea that a price is regular if the store charges it frequently in a window of time adjacent to that observation. The regular price is thus equal to the modal price in any given window surrounding a particular week provided the modal price is used sufficiently often. The algorithm is somewhat involved and I relegate a formal description to Appendix 1 in the Supplemental Material (Midrigan (2011)). Figure 1, reproduced from Kehoe and Midrigan (2008), presents several time series of the original price, pt , as well as the regular price, pRt , constructed using the algorithm. The algorithm does quite a good job of identifying the much more transitory price changes associated with temporary discounts from the more persistent price changes associated with a change in the regular price. Throughout this paper, I use the term “temporary prices” to denote prices that are not equal to the regular price in any given period. Although the algorithm I use is symmetric (it treats temporary price increases identically to temporary price decreases), most temporary price changes are associated with sales. To summarize, the algorithm I use is a filter that distinguishes high-frequency price variation from low-frequency price movements, much like the Hodrick– Prescott (HP) filter is used to identify trend from cycle. As with the Hodrick– Prescott filter, the algorithm is characterized by a parameter (the size of the window around the current observation used to compute the modal price: I use a 10-week window) that determines the relative frequency of the two types of price changes. Importantly, the results I report are not too sensitive to the exact length of the window (results for a 6- and 20-week window are virtually identical). Moreover, as is standard in business cycle research, I apply an identical filter to both my model and the data. Finally, I will report below a number of additional statistics that are independent of the filtering method I use so as to gauge the sensitivity of my results to alternative decompositions of high-/lowfrequency price changes. 9
The cost variable is an “average acquisition cost,” a weighted average of the cost of goods available in inventory. See Peltzman (2000) for a description of the cost data. 10 See Peltzman (2000) for a discussion of Dominick’s pricing practices. He noted that the retail price at Dominick’s stores is set by the chain’s head office and the correlation between price changes within a particular pricing zone are in the neighborhood of 0.8 to 0.9.
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FIGURE 1.—Example of the algorithm.
C. Motivating Facts Golosov and Lucas argued that the standard Calvo model vastly overstates the real effects of monetary shocks relative to a model that is consistent with the microdata on price changes. The Golosov and Lucas model, however, is inconsistent with three important features of the microdata. First, their model predicts little heterogeneity in the (absolute) size of price changes, while in the data there is a large amount of heterogeneity. Second, their model abstracts from temporary price changes, even though these changes account for the vast bulk of price changes in the data and periods with temporary discounts account for a sizeable fraction of goods sold. Third, I document a striking feature of temporary price changes: after a temporary price change, the price tends to revert to the exact nominal price (to the penny) in effect before the change. Here I briefly document these features of the data and use them to motivate my model.
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FIGURE 2.—Distribution of nonzero price changes: Dominick’s versus Golosov–Lucas model.
To document the heterogeneity of price changes, I present, in Figure 2, the distribution of (nonzero) price changes in the data and contrast it with the distribution implied by the Golosov and Lucas model (described in the next section). The two panels give the histogram for all price changes and regular price changes. Superimposed on these histograms is a solid line that gives the density implied by the Golosov and Lucas model. Clearly, the Golosov and Lucas model is unable to produce heterogeneity in the absolute size of price changes seen in the data. Specifically, it has both too few small price changes and too few large price changes relative to the data. For example, the standard deviation of the absolute size of price changes in the Golosov–Lucas model is 1.2%, while the corresponding number for regular prices in the data is 8.2%. To document the importance of temporary price changes, I note two facts. First, most price changes in the data are temporary price changes. Second, a sizeable fraction of goods are sold during periods in which a temporary price is chosen. The first fact is evident in Figure 1 for the selected series. For the data set as a whole, 96.5% of price changes are temporary. For the second fact, 40% of a store’s goods are sold during periods with a temporary price cut. Thus, in two senses, temporary price changes are an important feature of the data.
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To document the striking reversion of prices following temporary price changes, I note that 86% of the time, the nominal price following a temporary price change returns to the exact level it had before the change. Because of this feature of the data, although the frequency of all price changes is large (34% of prices change every week), the frequency of regular price changes is much smaller (2.9% per week). 3. A MENU-COST ECONOMY I extend the standard menu-cost model by adding several ingredients that allow the model to reproduce the features of data discussed above. I then calibrate the model and contrast its implications for the real effects of money with those of Golosov and Lucas. As I have documented, the standard menu-cost model produces too few small price changes as well as too few very large price changes. To allow the model to produce very large price changes, I follow Gertler and Leahy (2008) and assume a Poisson process for idiosyncratic productivity shocks. The frequency with which retailers receive these shocks is chosen so that the model matches a number of moments of the distribution of price changes in the data. To allow the model to produce small price changes, I assume economies of scope in price adjustment. In particular, I assume the retailer sells multiple goods and faces a single cost of changing the prices of these goods. I present evidence for both of these departures from the standard model in an empirical section (Section 3.C). To account for the pattern of temporary and regular price changes in the data, I make several additional assumptions. Retailers choose two prices for each good: a regular price, pRt , as well as a price to quote to the consumer (the posted price), pt . The latter is the price at which the consumer purchases the good and thus determines the store’s revenue. The regular price affects the retailer’s profits because every time the retailer posts a price that differs from the regular price, it must incur a fixed cost, κ. As a result, the posted price will deviate from the regular price infrequently, only when the benefit from doing so exceeds the fixed cost. I assume that changing prices is costly. Changing the regular price entails a fixed cost, φR . Similarly, changing the posted price requires a fixed cost, φ. These assumptions are motivated by evidence on the pricing practices of firms in the data. Zbaracki et al. (2004), Zbaracki, Levy, and Bergen (2007) provided evidence that prices are set at two levels: (a) the managerial level (e.g., by the headquarters) that sets list (regular) prices and (b) the sales force that is responsible for posted prices (including discounts, rebates, etc.). These authors found that managerial costs of changing list prices are substantially greater than the physical costs of changing posted prices (the costs of printing new price lists, etc.). Moreover, they also found that salespeople must coordinate departures of their prices from the regular (list) prices with the upper-
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level managers, activities that presumably involve a cost. See, for example, the following quote from a manager they interviewed11 : . . . I was a territory manager so I had no pricing authority. The only authority I had was to go to my boss and I would say, “OK, here is the problem I’ve got.” He would say “Fill out a request and we will lower the price for that account.” So this is how the pricing negotiations went. At that time I went up the chain to make any kind of adjustments I had to make. . . . My five guys have a certain level [of discount] they can go to without calling me. When they get to the certain point they have to get my approval. . . .”
A. Setup The economy is populated by a representative consumer, a unit measure of monopolistically competitive retailers, indexed by z, and a continuum of manufacturers. Each retailer sells N products, indexed by i = 1 N. Retailers purchase goods from manufacturers. I discuss the problem of the representative consumer, that of the retailer, and that of manufacturers, and then define an equilibrium for this economy. Consumers Consumers’ preferences are defined over leisure and a continuum of imperfectly substitutable goods purchased from retailers. Consumers sell part of their time endowment to the labor market, own shares in all firms, and trade state-contingent Arrow securities. Their problem is max
{ct (iz)}lt Bt+1
E0
∞
βt U(ct lt )
t=0
subject to 1 N 0
pt (i z)ct (i z) dz + Qt · Bt+1 ≤ Wt lt + Πt + Bt
i=1
where θ/(θ−1) N 1 (θ−1)/θ ct (i) ct = N i=1 1 γ/(γ−1) (γ−1)/γ ct (i) = [at (i z)ct (i z)] dz
0
Here z is an index over retailers, i is an index over goods, ct is an aggregator over the different goods, and ct (i) is an aggregator over the consumption 11
Zbaracki et al. (2004, p. 524).
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of good i purchased from the different retailers. I have in mind an economy where a good i sold by a retailer z is fairly substitutable with a good i sold by a retailer z (Saltines at Dominick’s versus Saltines at Cub Foods) so that γ is relatively high. In contrast, the different goods z are fairly imperfectly substitutable, so that θ is relatively low. I refer to an (i z) good–retailer combination as variety. I refer to at (i z) as the quality of the good. On one hand, a higher at increases the marginal utility from consuming that good. On the other hand, a higher at good is also more costly to sell, as I describe below. Changes in the quality of the good will thus make it optimal for the retailer to change its prices and will provide one source of price variation in the model, akin to the idiosyncratic productivity shocks assumed by Golosov and Lucas (2007). The assumption that idiosyncratic shocks affect both the cost at which a good is sold and the consumer’s preferences for the good is made for purely technical reasons. It allows me to reduce the dimensionality of the state space and thus the computational burden.12 The consumer’s budget constraint says that expenditure on goods and purchases of state-contingent securities must not exceed the consumer’s labor income, Wt lt , profits from ownership of firms, Πt , and the returns to last period’s purchases of state-contingent bonds, Bt . Here Bt+1 is a vector of statecontingent securities purchased at date t and Qt is a vector of prices of these securities. The formulation above implies that demand for any individual variety is ct (i z) = at (i z)
γ−1
pt (i z) Pt (i)
−γ
Pt (i) Pt
−θ ct
where Pt is the minumum expenditure necessary to deliver one unit of the final consumption good, ct , and Pt (i) is the price index for good i: Pt (i) = Pt =
12
1/(1−γ)
1
at (i z)
γ−1
0 N 1 Pt (i)1−θ N i=1
Pt (i z)
1−γ
dz
1/(1−θ)
A frequently employed alternative assumption that serves a similar purpose is that the menuγ−1 cost is proportional to at (e.g., Gertler and Leahy (2008)).
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Manufacturers Each variety (i z) is produced by a perfectly competitive sector. Firms in this sector hire labor lt (i z) and produce output according to yt (i z) =
lt (i z) et (i z)
where et (i z) is the inverse of that sector’s productivity. Manufacturers sell output to retailers at a price ωt (i z). Perfect competition implies that manufacturer’s profits are equal to 0: ωt (i z)yt (i z) − Wt lt (i z) = 0 so that ωt (i z) = et (i z)Wt ¯ 1} and is independent across retailers, but comI assume that et (i z) ∈ {e mon across varieties of goods sold by a given retailer, et (i z) = et (z). A retailer will thus synchronize its temporary price changes, as is the case in the data I describe below. This variable evolves over time according to the Markov transition probability ¯ t−1 (z) = 1) = α Pr(et (z) = e|e ¯ t−1 (z) = e) ¯ = ρ Pr(et (z) = e|e Retailers Retailers purchase goods from manufacturers at a unit price ωt (i z) and sell these goods to the consumers at a price pt (i z). The cost to the retailer of selling a good of quality at (i z) is at (i z)ωt (i z). I assume that shocks to at are correlated across the two varieties produced by a given retailer and that the log of at follows a random walk: log at (i z) = log at−1 (i z) + εta (i z) The retailer’s (nominal) profits from sales of good i are, therefore, Πt (i z) = [pt (i z) − at (i z)ωt (i z)]at (i z)γ−1 −θ −γ pt (i z) Pt (i) × ct Pt (i) Pt Notice that absent price adjustment frictions, the retailer’s optimal price is a constant markup over the unit cost: pt (i z) =
γ at (i z)ωt (i z) γ−1
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pt (iz) denote the actual markup the retailer charges, we Letting μt (i z) = at (iz)ω t (iz) can write the profits from selling good i as −θ 1−γ γ−1 γ Pt (i) Πt (i z) = ωt (i z) [μt (i z) − 1]μt (i z) Pt (i) ct Pt
This expression shows that the retailer’s profits from selling any individual good (conditional on its markup) are independent of the quality of the good, at (i z). The only effect of shocks to at (i z) is to alter the retailer’s markup in the presence of price adjustment costs and thus provide a source of lowfrequency price variation. It is this feature of the process for at , together with the assumption that it follows a random walk, that allows me to reduce the dimensionality of the state space. Price Adjustment Technology I assume that retailers set two prices: a regular price pRt and a posted (transactions) price pt . I assume that the adjustment costs apply to all goods the retailer sells. Whenever at least one of its posted prices pt (i) deviates from the regular price pRt (i), the retailer incurs a fixed cost κ. This cost is independent of the number of prices that deviate. Moreover, changing the regular price is also costly and entails a fixed cost φR . This cost is again incurred when at least one regular price is changed and is independent of the total number of regular prices that the retailer adjusts. Finally, changing posted prices is costly as well: the retailer pays a fixed cost φ every time at least one of its posted prices changes. Given these assumptions, the problem of any retailer z is ⎞ ⎛ N [pt (i z) − at (i z)ωt (i z)]ct (i z) − ⎟ ⎜ ⎟ ⎜ i=1 ⎟ ⎜ qt ⎜ φR × [any pRt (i z) = pRt−1 (i z)] − ⎟ max E0 ⎟ ⎜ pt (iz)pR (iz) t t ⎠ ⎝ φ × [any pt (i z) = pt−1 (i z)] − κ[any pRt (i z) = pt (i z)] − U
/P
where qt = U ct /Pt is the date 0 price of a date t Arrow security. The first term in c0 0 this expression denotes the profits the retailer makes and the last three terms reflect the cost associated with exercising the three different options. Retailer Decision Rules Kehoe and Midrigan (2008) discussed in detail the optimal decision rules in a (single-product) economy described by the assumptions I made above. Here I briefly summarize the workings of the model under the assumption that the manufacturer’s cost shocks, et (i z), are transitory: et (i z) = 1 most of the time, ¯ and leaves the low-value infrequently enters the low-value state et (i z) = e, state with high probability.
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Assuming that κ, the cost of deviating from the regular price, is relatively low, the retailer will find it optimal to use this option to respond to transitory changes in its wholesale cost, ωt (i z). Doing so entails paying a cost κ + φ in the period in which it initiates a price change, a cost κ in every subsequent period in which the posted price is below the regular price, and then a final cost φ to return its posted price to the old regular price. As long as φR is high relative to φ and κ, temporarily deviating from the regular price will be less expensive than undertaking two regular price changes (one down and another one up) to respond to this transitory cost shock. In contrast, a series of at shocks change the retailer’s desired price permanently: the retailer will choose to respond to such shocks by undertaking a regular price change at a cost φR . The alternative would be to deviate from the regular price, but such an option is too expensive given that the retailer would have to pay a cost κ in every period in which the posted price deviates from the regular price. Figure 3 illustrates this discussion by plotting a time series of simulated decision rules. The frequent transitory cost shocks (reflected in a lower desired price, the dotted line) lead the retailer to temporarily deviate from the regular
FIGURE 3.—Prices in an economy with temporary changes.
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price by changing the posted price only. In contrast, when the wholesale cost is in the high state and the retailer’s regular price is too far from the desired level (because of a change in at ), the retailer finds it optimal to undertake a regular price change. Because of discounting, it never pays off to change the regular price without using it. Hence in such periods, the posted price changes as well. Equilibrium I assume that nominal spending must be equal to the money supply: 1 N 0
pt (i z)ct (i z) dz = Pt ct = Mt
i=1
The money growth rate gt =
Mt Mt−1
evolves over time according to
log gt = ρm log gt−1 + εtm where εtm is an independent and identically distributed (i.i.d.) N(0 σm2 ) disturbance. An equilibrium is a collection of prices and allocations pt (i z) Pt (i) Pt Wt ct (i z) ct , and lt such that, taking prices as given, allocations and prices solve the consumer, retailer, and manufacturer’s problems and the goods, labor, and bond markets clear. B. Computing the Equilibrium I assume preferences of the form U(c l) = log(c) − l. This specification follows Hansen (1985) and Rogerson (1988) by assuming indivisible labor decisions implemented with lotteries. These assumptions ensure that the nominal wage is proportional to nominal spending, W = Pc, and thus proportional to the money supply, W = M. This closely follows Golosov and Lucas (2007) and ensures that shocks to the money supply translate one-for-one into changes in the nominal marginal cost and hence the retailer’s desired price. I consider a richer specification (in which I allow for interest-elastic money demand and capital accumulation) below. The quantitative analysis I report below assumes, for simplicity, that the cost of changing posted prices, φ, is equal to zero. This assumption considerably simplifies the notation below, but has little effect on my results. Appendix 3 in the Supplemental Material studies the economy with a nonzero cost of changing posted prices, φ > 0, and shows that calibration of the model to the data indeed requires a very small cost of changing posted prices.13 13 The only feature of the data that a model with φ = 0 misses is the fact that prices are sometimes sticky even during a sale. However, in the data, conditional on the retailer undertaking a
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The assumption that φ = 0 implies that the retailer’s state is fully characterized by the markup of its existing regular prices, pRt−1 (i), over the unit cost of selling the good, at (i)ωt (i), as well as the cost, to the manufacturer, of producing the good, et (i). Let μRt−1 (i) =
pRt−1 (i) at (i)ωt (i)
The aggregate state of this economy is characterized by the growth rate of money, g, and the distribution of retailers’ markups, μR−1 = {μRi−1 }, and efficiency levels, e. Let Λ denote this distribution. Let V R (μR−1 e; g Λ), V T (μR−1 e; g Λ), and V N (μR−1 e; g Λ) denote a firm’s value of (i) adjusting its regular and posted price (and selling at pt (i) = pRt (i) = pRt−1 (i) in that period), (ii) undertaking a temporary price change and leaving its regular price unchanged, pt (i) = pRt (i) = pRt−1 (i), and (iii) selling at the old regular price, pt (i) = pRt (i) = pRt−1 (i). Letting V = max(V R V T V N ) denote the envelope of these three options, the following system of functional equations characterizes the retailer’s problem (I impose here the log-utility specification above and substitute several equilibrium conditions to simplify the notation): N 1−γ R R ei (μR − 1)μR−γ Pˆ γ−1 − φR V (μ e; g Λ) = max i −1
i
μR i
i=1
+ βEV (μR −1 e ; g Λ )
V (μ e; g Λ) = max T
R −1
μi
N
ˆ γ−1 − κ e1−γ (μi − 1)μ−γ i i P
i=1
+ βEV (μ e ; g Λ ) R −1
V N (μR−1 e; g Λ) =
N
ˆ γ−1 − κ e1−γ (μRi−1 − 1)μR−γ i i−1 P
i=1
+ βEV (μR −1 e ; g Λ )
sale two periods in a row, the sale price changes from one period to another 67% of the time. Thus the model requires a very low φ to account for the fact that sales prices are, in fact, quite flexible. (This feature of the data has also been documented by Nakamura and Steinsson (2009).)
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where Pˆ = MP denotes the aggregate price level detrended by the money supply. The law of motion for the aggregate state is Λ = Γ (g Λ). As for ide R iosyncratic states, μR i−1 = μi g exp(ε )e if the retailer adjusts its price and = ia e μRi−1 g exp(ε )e otherwise. Here the retailer’s markup is eroded by the three ia
types of shocks: aggregate shocks to the money supply g, permanent shocks to the quality of the two goods, a, and transitory shocks to its wholesale cost, e. Solving for the equilibrium in this economy requires characterizing the objects V R (·), V T (·), and V N (·), as well as Γ (·), the law of motion for the distribution of markups. To solve this system of functional equations, I follow an approach developed by Krusell and Smith (1998)14 that restricts the aggregate state space to g, the growth rate of the money stock, and a single moment of the distribution Λ. In particular, I use Pˆ−1 , the last period’s aggregate price level (detrended by the money stock). The latter is a sufficient state variable in a (log-linearized) version of the Calvo pricing model. I show below that this state variable alone characterizes the evolution of aggregate variables in this menu-cost economy quite well. To implement the Krusell and Smith (1998) approach, I assume that the aggregate price level is a log-linear function of the two aggregate state variables log Pˆt = ς0 + ς1 log gt + ς2 log Pˆt−1 Given a guess for the coefficients in this expression, I solve the retailer’s problem using projection methods and a combination of cubic and linear spline interpolants.15 I then simulate retailer decision rules and use the simulated data to reestimate the coefficients in the postulated law of motion. These updated coefficients are used to recompute firm decision rules, simulate new data, and update the coefficients. Once these coefficients converge, the distance between actual (in simulations) and predicted (by the coefficients in the aggregate functions) aggregate time series is insignificant (the out-of-sample forecasts have an R2 in excess of 99.9%). Moreover, adding higher-order moments of Λ adds little precision.16 Notice that the accuracy of this first-moment approach is not inconsistent with my argument that higher-order moments of the distribution of desired price changes matter for the economy’s aggregate implications. The precision of the simple pricing rules reflects the fact that monetary disturbances induce little time series variation in these higherorder moments in the model.17 This is because the distribution of desired price 14 See also Klenow and Willis (2006) and Khan and Thomas (2007) for applications of this approach to models with nonconvexities. 15 See Miranda and Fackler (2002) for a detailed description of these methods as well as a toolkit of routines that greatly facilitate their implementation. The solution method is described in some more detail in the Supplemental Material. 16 These coefficients and the R2 are reported in Table III. 17 For example, the kurtosis of μR−1 in simulations ranges from a 10th percentile of 3.28 to a 90th percentile of 3.32.
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changes is mainly pinned down by the distribution of exogenous idiosyncratic shocks. The existence and continuity of the value functions can be established using standard theorems (Stokey and Lucas (1989)). Although these value functions are not concave, it can be shown that they satisfy K-concavity, a property introduced by Scarf (1959). In turn, K-concavity guarantees uniqueness of the optimal price functions. Aguirregabiria (1999) proved K-concavity in the context of a model similar to the one presented here in which the two control variables are each subject to a fixed cost of adjustment. Sheshinski and Weiss (1992) studied a special case of the economy presented here and also proved uniqueness of the optimal decision rules. C. Calibration and Parameterization I assume that the period is a week (the frequency at which Dominick’s sets prices). I set β = 0961/52 and γ = 3, a typical estimate of elasticity of substitution in grocery stores.18 As I show below, this value of γ allows the model to match well the response of quantities to temporary price cuts in the data. In particular, the model will be shown to account for the fraction of goods sold in periods with sales. I set θ = 1, consistent with the idea that different goods sold by a particular retailer are relatively less substitutable. However, the exact choice of θ plays little role since there are no aggregate good-specific shocks in the model. Finally, I assume that retailers sell N = 2 goods each. I allow for persistence in the growth rate of money given that most applied work assumes persistence in monetary policy.19 I calibrate the coefficients in the money growth rule by first projecting the growth rate of (monthly) M1 on current and 24 lagged measures of monetary policy shocks.20 I then fit an (autoregression) AR(1) process for the fitted values in this regression and obtain an autoregressive coefficient of 0.61 and standard deviation of residuals of σm = 00018. I then adjust these parameters to reflect the weekly frequency in my model economy. The rest of the parameters are calibrated to allow the model to match the microprice facts documented in the earlier section. I follow Gertler and Leahy 18 Nevo (1997), Barsky, Bergen, Dutta, and Levy (2000), and Chevalier, Kashyap, and Rossi (2003). 19 Christiano, Eichenbaum, and Evans (2005) reported that the growth rate of money increases persistently in response to an (identified) exogenous monetary policy shock and postulated a process for μt that is well approximated by an AR(1) with a (quarterly) persistence coefficient of 0.5. Alternatively, Smets and Wouters (2007) assumed an interest rate rule and also found evidence of inertia: their estimate of the coefficient on lagged interest rates in the interest rate rule is 0.8. 20 The results reported below use a new measure of shocks due to Romer and Romer (2004) that is available for 1969–1996. I also use the measure used by Christiano, Eichenbaum, and Evans (2005) and find very similar results. I thank Oleksiy Kryvtsov for sharing the Christiano, Eichenbaum, and Evans (2005) data with me.
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(2008) and assume that the permanent shocks to the good’s quality, at , arrive infrequently, according to a Poisson process21 : 0 with probability = 1 − pa ε˜ it = N(0 σa2 ) with probability = pa . Conditional on their arrival, the shocks are drawn from a Gaussian distribution with standard deviation σa . My specification of the process for shocks nests that of Golosov and Lucas (2007) (pα = 1), but is more flexible and allows me to account for the distribution of the size of price changes in the data. I assume that the permanent (quality) shocks are correlated across the goods sold by a given retailer. In particular, the actual innovations to a good’s quality, εit , depend on the underlying draws as εit = ε˜ it + χ mean(ε˜ it ), where χ is a parameter that governs the correlation of productivity shocks across the two goods. The eight parameters that I calibrate are σa —the standard deviation of permanent shocks, χ—the parameter governing the correlation of quality shocks across the two varieties sold by a retailer, α—the parameter governing the frequency of shocks to the manufacturer’s cost, ρ—the persistence of the low ¯ manufacturer cost state, e—the manufacturer’s cost in the low state (recall that the high state is normalized to 1), φR —the fixed cost of changing the regular prices, κ—the cost of deviating from the regular price, and pa —the frequency with which retailers experience a shock to the good’s quality. I choose these parameters so as to match the salient properties of the microprice data, specifically, moments of the distribution of the size of price changes, as well as statistics that capture the relative importance of high- versus lowfrequency price variation. I discuss these moments below. Specifically, in addition to the moment emphasized by Golosov and Lucas, the large mean size of price changes, I ask the model to capture (i) the large heterogeneity in the size of price changes, (ii) the frequency of posted and regular price changes, and (iii) the fact that after a sale, prices typically revert to the preexisting regular price. Data Moments The moments I use to calibrate the model are listed in the data column of Table I. Some of these moments were computed using the algorithm I discussed in the previous section. I emphasize that I apply an identical algorithm to the posted (transactions) price in the model and in the data so as to report statistics about regular prices. For example, when I report the frequency of “regular price changes,” this reflects the frequency of changes in the regular price 21 This assumption is a simpler alternative to the mixture of betas I have used in earlier work that produces very similar results.
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Moments
No Temp. Changes Golosov–Lucas
Data
Benchmark
058 034 0029 097
058 032 0025 096
0.58 0.053 0.045 0.29
0.58 0.046 0.044 0.06
047 086 025 022 037
045 089 022 022 040
0.41 0.00 0.02 0.01 0.01
0.30 0.00 0.00 0.00 0.00
020 011 003 005 009 013 021 089
020 010 003 005 009 014 020 091
0.11 0.11 0.03 0.05 0.09 0.15 0.22 0.94
0.11 0.11 0.10 0.10 0.11 0.11 0.12 –
061 070 032
073 073 029
0.89 0.86 0.47
0.89 0.87 0.47
21. Std. dev. size of price changes 22. Kurtosis price changes 23. Fraction changes < 1/2 mean 24. Fraction changes < 1/4 mean
018 315 036 019
015 165 036 034
0.08 2.97 0.28 0.07
0.01 1.06 0.00 0.00
25. Std. dev. size of regular price changes 26. Kurtosis regular price changes 27. Fraction regular price changes < 1/2 mean 28. Fraction regular changes < 1/4 mean
008 402 025 008
007 253 028 007
0.08 3.13 0.28 0.07
0.01 1.07 0.00 0.00
29. Fraction of price changes during a sale (within) 067 30. Fraction of changes in the sale price (between) 082
094 1
0.08 1
0.01 1
Used in calibration 1. Fraction of prices at annual mode 2. Frequency of price changes 3. Frequency regular price changes 4. Fraction of price changes that are temporary 5. Probability a temporary price spell ends 6. Probability temp. price returns to old regular 7. Fraction of periods with temporary prices 8. Fraction of periods with sales 9. Fraction of goods sold when sales 10. Mean size of price changes 11. Mean size of regular price changes 12. 10th percentile size regular price changes 13. 25th percentile size regular price changes 14. 50th percentile size regular price changes 15. 75th percentile size regular price changes 16. 90th percentile size regular price changes 17. Mean abs(fup − fdn) within store Additional moments 18. Frequency changes annual mode 19. Fraction prices at quarterly mode 20. Frequency changes quarterly mode
a Moments not included in the criterion function in the calibration are underlined.
identified by the algorithm, not the frequency with which retailers exercise the option of the regular price change in theory.22 Table I (rows 2–4) shows that 33% of goods experience a price change in any given week. Most of these price changes reflect temporary price discounts. 22 The two are very similar, since the algorithm does a very good job identifying changes in the theoretical regular price.
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The fraction of price changes that are temporary (in periods in which the actual price is not equal to the regular price) is 0.965. As a result, only 2.9% of goods experience a regular price change in any given week. Also notice (rows 5–9) that the fraction of periods in which the store charges a temporary price is equal to 0.25, and most (0.22 of the weeks) of these are periods with sales (the posted price is below the regular price). Sales are also periods in which the retailer sells a disproportionately larger amount of its goods: 37% of all goods are sold during sales. Temporary price changes are very transitory: conditional on the price being temporary at date t, it returns to a regular price the next week 47% of the time. When temporary changes do return, they do so to the old regular price 86% of the time. Rows 1 and 18–20 report additional statistics that capture the pattern of low-frequency price variation that are independent of the algorithm I use. The annual mode for any particular good accounts for 58% of all prices the retailer charges during the year.23 The annual mode also shows quite a bit of stickiness and changes from one year to another only 61% of the time. As Kehoe and Midrigan (2008) showed standard menu-cost models cannot account, simultaneously, for the high frequency of price changes in the data as well as these alternative measures of low-frequency nominal stickiness. For example, the standard model, when calibrated to match a frequency of price changes of 34% per week, predicts a fraction of prices at annual mode of only 23%, much less than the 58% in the data. The model I study here will be shown to fit all of these facts very well. I finally focus on the size of price changes (rows 10–16, 21–28), and again distinguish between changes in the posted price and changes in the regular price. A key feature of the data is that there is large heterogeneity in the size of price changes. One concern is that this heterogeneity reflects permanent good-level heterogeneity. To address these concerns, I report, following Klenow and Kryvtsov (2008), moments of the “standardized” distribution of price changes. In particular, I scale each price change by each good’s mean size of price changes, where a “good” represents a given manufacturer × product category. By construction, these data are free of good-specific heterogeneity. As in Klenow and Kryvtsov (2008), the mean size of price changes is very large: the retailer in question adjusts posted prices by 20% and regular prices by 11% on average.24 The standard deviation of price changes is, however, large as well: 18% for posted prices and 8% for regular prices. Given my focus on the distribution of the size of price changes, rows 12–16 report several percentiles of this distribution. I focus on regular prices because, 23 Hosken and Reiffen (2004), Kehoe and Midrigan (2008) and more recently Eichenbaum, Jaimovich, and Rebelo (2011) have also used modal statistics to characterize microprice data. 24 Because sales make up most of posted price changes, the statistics for sale prices are very similar to those for posted prices, and are not reported here.
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as shown by Kehoe and Midrigan (2008), it is the pattern of low-frequency price variation that mostly matters for the aggregate predictions of a menucost economy. Notice that 10% of all regular prices are less than 3% in absolute value, while 25% of all regular prices are less than 5% in absolute value. A lot of price changes are thus very small. Similarly, many price changes are also very large: the 75th percentile of this distribution is equal to 13%, while the 90th percentile is equal to 21%. Price changes within the store are strongly synchronized, especially for goods in the same product category, as I document in a subsequent section. Moreover, price changes also tend to be of the same sign. Table I (row 17) shows that the average (absolute value) of the difference between the fraction of regular price increases (fup) and decreases (fdn) in any given week is 0.89. This suggests that idiosyncratic shocks to the different goods are strongly correlated and I use this statistic to pin down the size of this correlation. Finally, rows 29 and 30 report that sales prices are highly flexible in the data. Conditional on the good being on sale and not returning to a regular price next period, the probability that the sale price changes is equal to 67%. Such price changes are large (18% on average) and dispersed (the standard deviation is 16%), just like all other price changes. Similarly, there is very little price rigidity in the sale price across different sales episodes: 82% of the time the store charges a price during a sale that is different than the price it has charged during the previous sale. The model is able to capture this flexibility, since I have assumed that the cost of changing posted prices, φ, is equal to 0. Criterion Function The criterion function I use to pin down the model’s parameters is the sum of the squared deviations of the moments in the model from those in the data, those listed in rows 1–17 in Table I. I target the frequency and average size of posted and regular price changes, the fraction of periods with temporary price changes, the fraction of goods sold during sales, and the fraction of prices at annual mode. In addition, I include the percentiles of the distribution of the size of regular price changes and require the model to account for the fact that price changes are typically of the same sign within a store. Benchmark Model Table II reports the parameter values used in the model (second column, benchmark). Quality (at ) shocks arrive infrequently, with pa = 0030. The volatility of these shocks, σa , is equal to 0.08 and thus fairly large. Shocks to the quality of a good sold by a given retailer are strongly correlated, χ = 084, implying a correlation of shocks equal to 0.53. Manufacturers’ costs change more frequently. The likelihood that the cost transits from a high to a low state is α = 0126. The low-cost state is much less persistent: the probability of transiting back to the high-cost state is ρ = 0524. Thus, retailers’ wholesale costs
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TABLE II PARAMETER VALUES
Benchmark
No Temp. Changes
Golosov–Lucas
BLS Calibration
Calibrated σa φR , relative to SS revenue χ pa α ρ e κ, relative to SS revenue
0080 0022 0845 0030 0126 0524 0741 0012
0100 0018 0999 0040 – – – –
0023 0038 – – – – – –
0112 0012 0466 0079
Assigned γ β (annual) ρm σm (%)
3 096 0884 0032
3 096 0884 0032
3 096 0884 0032
3 096 0610 0180
experience frequent, but temporary, price declines, leading retailers to offer temporary price discounts. The menu costs (expressed as a fraction of steady-state (SS) revenue) are φR = 0022 and κ = 0012. These are the costs the retailer pays every time it makes a regular price change or deviates from the regular price. Given that these events are relatively infrequent, adjustment costs are a much smaller share of the overall (across all periods) revenue of the retailer: the total resources used up by price changes are equal to 0.34% of revenue. For comparison, Levy et al. (1997) reported menu costs as large as 0.7% of revenue, while Zbaracki et al. (2004) reported price adjustment costs as large as 1.2% of revenue. The menu costs I estimate are thus by no means large relative to direct estimates in earlier work. The second column of Table I reports the moments in the model. The model does well at accounting for both high- and low-frequency price variation in the data. As in the data, posted prices change frequently (32% of the weeks), while regular prices much less so (2.5% of the weeks). Temporary price spells are transitory and last roughly 2 weeks. Most periods of temporary prices are periods with discounts (a good is on sale 22% of the time), and periods of sales account for a disproportionate amount of goods sold (40%). Finally, as in the data, temporary price changes revert to the old regular price most (86%) of the time. As a result, the frequent price changes in the model do not impart much low-frequency price variation: the retailer’s prices are equal to the annual mode 58% of the time, as in the data. The model also does a good job at reproducing the distribution of the size of regular price changes in the data. In particular, many price changes are small
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in absolute value: the 10th percentile is equal to 0.03 and the 25th percentile is equal to 0.05. Similarly, many price changes are large in absolute value: the 75th percentile is equal to 0.14, while the 90th percentile is equal to 0.20. All these numbers are very close to their counterparts in the data. Finally, the model accounts well for the mean size of regular and posted price changes, the key statistics emphasized by Golosov and Lucas. The fit is also good vis-à-vis the other moments that were not directly used in the criterion function: the fraction of prices at the quarterly mode and the frequency with which the quarterly mode changes, as well as the standard deviation and fraction of small posted and regular price changes. The model understates the kurtosis of price changes in the data (2.6 vs. 4 for regular prices), but since the kurtosis is very sensitive to a few outliers in the tails, I use the much more robust percentiles of the distribution to calibrate the model.25 To summarize, the model reproduces well salient features of the microprice data. This itself is not surprising, since I have explicitly targeted these facts when choosing parameters. Recall, however, that the question I ask in this paper is, “What are the aggregate implications of a model that accounts for the microlevel facts?” Since I have shown that my model is indeed consistent with the microdata, I can now study its aggregate implications. Before I do so, I discuss a number of additional experiments I conduct to isolate the role of the assumptions I have made above. Model Without Temporary Changes I report results from an economy in which the technology for changing prices is the same as in the standard menu-cost model. The retailer incurs a cost, φ, every time it changes its menu of posted prices. Permanent shocks to quality are the only source of idiosyncratic uncertainty. The difference between this economy and the benchmark economy is the absence of a motive for temporary price changes. I target the same set of moments as above, with the exception of those that characterize the low- versus high-frequency price variation (underlined in Table I). I choose the size of the menu-cost, φ, so that the model matches the fraction of prices at the annual mode. Targeting this statistic, Kehoe and Midrigan argued, is a more accurate method of comparing economies with and without temporary price changes. The alternative—targeting the frequency of posted price changes—would underpredict the real effects of money in the benchmark model, since most of these changes are temporary and do not allow retailers to respond to low-frequency changes in monetary policy. Similarly, targeting the frequency of regular price changes would overpredict the real effects of money, since temporary prices do respond to monetary policy shocks, albeit for only a short period. In contrast, the fraction of prices at the annual model is a robust 25 I thank an anonymous referee for the suggestion to no longer focus on this feature of the data.
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measure of low-frequency price variation in the model and more appropriate for measuring the response of prices to monetary shocks. Once again, the model does a good job capturing the heterogeneity in the size of price changes in the data. In contrast, it accounts less well for the pattern of temporal price variation. The frequency of posted price changes is only 0.053, much lower than in the data, while that of regular price changes (as identified by the algorithm) is equal to 0.045, higher than in the data. Golosov–Lucas Model I also report results from a version of the model without temporary price changes and in which the distribution of permanent shocks is Gaussian as opposed to Poisson, and there are no economies of scope. I choose the fixed cost to match the fraction of prices at the annual mode. The only additional parameter that remains to be chosen is the standard deviation of idiosyncratic shocks, which I choose, as did Golosov and Lucas, so as to match the average size of regular price changes. This economy fails to account for both the pattern of temporal price variation and the heterogeneity in the size of price changes. Price changes are now much more concentrated near the adjustment thresholds: the interquartile range is now only 2% (10% in the data). I show below that the model’s failure along this dimension explains the much lower real effects from monetary shocks that it predicts. Calvo Model I finally compare the predictions of the menu-cost economies to those of the Calvo model, as did Golosov and Lucas. I choose the adjustment hazard 1 − λ = 0055 so as to match the fraction of prices at the annual mode.26 4. AGGREGATE IMPLICATIONS I next turn to the models’ aggregate implications. My measures of the real effects of money are the volatility and persistence of HP-filtered consumption. I construct a measure of monthly consumption, as in the data, for comparability with other work. A. Business Cycle Statistics Table III shows that the standard deviation of consumption is 0.29% in the benchmark setup, 0.31% in the economy without temporary price changes, and as low as 0.07% in the Golosov and Lucas economy. In contrast, the Calvo 26 The alternative, of choosing λ so as to match the frequency of price changes in the menu-cost economy, produces similar results, since the frequency of price changes in the economy without temporary price changes is equal to 0.053. I adopt this alternative approach in a robustness section below.
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VIRGILIU MIDRIGAN TABLE III AGGREGATE IMPLICATIONS
Calvo
Business cycle statistics σ(C) (%) Serial correlation C
No Temp. Changes
Golosov–Lucas
a
Inflation dynamics var(π) intensive margin correlation (π, fractionally adjusted) Law of motion P b P(t − 1) g(t) R2
Benchmark
035 093
029 093
031 092
007 084
100 –
090 065
096 040
099 016
0946 −0668 1
0947 −0544 09997
0932 −0652 09998
0816 −0270 09808
a Business cycle statistics are computed for data sampled at monthly frequency. An HP (14400) filter is applied to the data. b Laws of motion are computed at the actual frequency (weekly) in the model.
model predicts a standard deviation of 0.35%. Thus, my benchmark economy that accounts for the salient features of the microprice data produces real effects of money that are 80% as large as in the Calvo model. In contrast, the Golosov and Lucas economy produces real effects of money that are 20% as large as in the Calvo model. Table III also reports the serial correlation of HP-filtered consumption, another measure of the real effects of money. Once again, the menu-cost models that do account for the heterogeneity in the size of price changes produce persistence that is as high as in the Calvo model (0.93). In contrast, the Golosov and Lucas model features somewhat less persistent business cycles (0.84). All of these economies produce little synchronization of price changes in response to monetary shocks: the fraction of inflation variance accounted for by the intensive margin (the mean size of price changes) is in excess of 90% (91% as reported by Klenow and Kryvtsov (2008) for U.S. data). This suggests that the extensive margin (the fraction of adjusting firms) accounts for little of the inflation variability, despite the fact that it is positively correlated with inflation (the correlation ranges from 0.16 in the Golosov–Lucas model to 0.65 in the benchmark model vs. 0.25 in the U.S. data). In both of these economies, idiosyncratic shocks are large and adjustment decisions are driven mostly by idiosyncratic rather than aggregate shocks. Notice also that these features of the data cannot be reproduced by a model in which (a) there are no price adjustment costs and (b) all shocks, including aggregate shocks, arrive infrequently. In such a model, prices would indeed change infrequently, only in periods with shocks. However, all prices would change in periods with money shocks and the model would predict, counterfactually, perfect synchronization of price changes at the aggregate level.
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I next explain the intuition behind the aggregate implications of the economies I study. Two separate forces account for the somewhat smaller variability of aggregate consumption in my benchmark economy vis-à-vis the Calvo setup. First, the menu-cost model features a selection effect due to the endogenous timing of price changes. Second, a large number of goods are sold during periods of sales in which prices are flexible and respond one-for-one to monetary shocks. To see the latter effect, notice that the price level for any individual good is Pt (i) =
1/(1−γ)
1
ωt (i z)1−γ μt (i z)1−γ dz
Mt
0
If, in response to a monetary shock, the retailer does not reset its price, its markup, μt , decreases, and hence the price index does not fully respond to the change in Mt . The weight, in the price index, on each individual price setter, depends on the amount sold by retailers, and hence is inversely proportional to its cost, as capturer by the ωt (i z)1−γ term. Because the low-cost retailers undertake temporary price discounts and thus react to monetary shocks, the aggregate price level becomes more flexible than in the absence of such temporary price changes. In the limit, if the only retailers that sell goods are those that sell on sale, the price level would respond one-for-one to monetary shocks. The experiments I report above allow me to disentangle the relative importance of these two effects. The difference between the benchmark economy and the economy without temporary price changes is the effect of sales. Clearly, this effect is fairly small (0.29% vs. 0.31%), as pointed out by Kehoe and Midrigan (2008), who addressed this issue in much detail. The reason for the small difference is that both economies are characterized by similar lowfrequency variability of prices. Even though prices change frequently in the benchmark economy, they usually return to the pre-sale price and respond to monetary policy shocks only temporarily. Hence, the stickiness of the regular price (at which almost 67% of all goods are sold) generates a nontrivial amount of stickiness in the aggregate price level. The difference between the real effects of money in the economies without temporary price changes and the Calvo model is solely accounted for by the endogenous timing of price changes in menu-cost economies. Clearly, accounting for the heterogeneity in the size of price changes has an important effect on the real effects of money the model predicts. I explain why this is the case below. B. Selection Effect I next focus on the key result of this paper: that the real effects of money are much greater in an economy with heterogeneity in the size of price changes. The key difference between my economy and that of Golosov and Lucas is what Golosov and Lucas (2007) refer to as the selection effect and Caballero and
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FIGURE 4.—Selection effect.
Engel (2007) refer to as the extensive margin effect. The endogenous timing of price changes implies that the mix of adjusters varies with the aggregate shock: in times of monetary expansion, adjusters are mostly firms whose idiosyncratic state is such that they need to raise prices. The strength of this effect critically depends on the shape of the distribution of desired price changes. I illustrate this using a heuristic example in Figure 4. The upper panels show the distribution of desired price changes in the absence of a monetary shock, in an economy with Gaussian (left panel) and Poisson shocks (right panel).27 In both cases I assume that the adjustment thresholds are equal to ±0.1: the shaded areas thus reflect the distribution of actual price changes. Notice how the distribution of the absolute size of actual price changes is much more dispersed in the Poisson economy and much more concentrated in the Gaussian economy. 27 This is an extreme example. In the benchmark economy, the distribution of desired price changes has no mass point at 0 because of the correlation in cost shocks across goods, as well as the fact that past shocks (aggregate and idiosyncratic to which the retailer has not yet responded) spread the measure of desired price changes away (but close) to 0.
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Absent a money shock, the distribution of desired price changes is symmetric: half of the retailers raise and lower their prices. After a monetary shock, the distribution of desired price changes shifts to the right: retailers would like, on average, to increase their prices. Notice in the lower panel of the figure that in the Gaussian economy, this shift in the distribution has a large impact on the distribution of retailers that change their prices: many more retailers are now increasing prices. This sizeable effect on the distribution arises from the fact with Gaussian shocks, the measure of retailers in the neighborhood of the adjustment thresholds is large. Hence changes in the money supply have a large effect on the identity of adjusting firms and thus on the aggregate price level. This selection effect is much weaker in the Poisson economy. Because distribution of price changes is much more dispersed, the measure of retailers in the neighborhood of adjustment thresholds is much smaller. Hence, the identity of retailers that adjust prices is virtually unchanged by the monetary shock: almost as many retailers find it optimal to lower their prices as before because the idiosyncratic shocks are large and the monetary shock does little to offset them. In the Dominick’s data, the distribution of price changes does not change much from periods of high to low inflation, as shown in Figure 5,28 thus providing additional support for the Poisson economy. This simple example abstracts from two features in the benchmark economy I study: (i) economies of scope generate many price changes that are very small and (ii) the adjustment thresholds are closer to 0 because a smaller menu cost is required to account for the frequency of price changes. Economies of scope flatten the adjustment hazard and thus weaken the strength of the selection effect even further. In the limit, if the number of goods sold by a retailer is sufficiently large, a good’s adjustment hazard is independent of its own desired price change; hence the economy resembles the constant-hazard Calvo setup much more. Similarly, smaller adjustment thresholds weaken the selection effect because most of the price increases caused by the monetary shock are small. All of these ideas can be summarized by recognizing that the response of the price level to a monetary shock in this economy can be approximated29 (in the limit, as m → 0) by p = f (x)h(x) dx + xf (x)h (x) dx m x x where f (x) is the distribution of desired price changes and h(x) is the adjustment hazard. The first term represents the fraction that change their prices— 28 I thank an anonymous referee for suggesting this test of the theory. Inflation is measured by the change in the CPI for food and beverages. High (low) corresponds to periods with inflation above (below) the mean. The figure reports the distribution of regular price changes; that for all price changes is very similar. 29 See Caballero and Engel (2007) for a derivation. Also see Burstein and Hellwig (2008) for a similar decomposition.
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FIGURE 5.—Distribution of (regular) price changes in high and low inflation periods.
present in Calvo and menu-cost economies alike. The second term captures the selection effect: variation in the adjustment hazard, h(x). This effect is larger when there is more mass, f (x), in the region of increasing hazard h (x) (more marginal firms) and when the marginal firms need larger price changes, x. C. Additional Experiments I next perform two experiments to gauge the sensitivity of my results. In particular, I study (i) a variation of my model without temporary price changes calibrated to the economy-wide data from the Bureau of Labor Statistics (BLS),30 and (ii) an extension of the model with capital, interest-elastic money demand, and a roundabout input–output structure, again calibrated to the aggregate data. BLS Data Table IV reports several moments of the distribution of the size of price changes in the BLS data. These statistics were computed by Nakamura and 30 See Kehoe and Midrigan (2008), who studied the BLS data through the lens of the economy with posted and regular prices.
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MENU COSTS, MULTIPRODUCT FIRMS, AND FLUCTUATIONS TABLE IV MOMENTS IN MODEL AND BLS DATA Model With Scope Economies
Model With Random Menu Cost
Moments
Data
Used in calibration 1. Fraction of prices at annual mode 2. Mean size of regular price changes 4. 25th percentile size regular price changes 5. 50th percentile size regular price changes 6. 75th percentile size regular price changes
0.75 0.11 0.03 0.07 0.13
0.75 0.09 0.03 0.07 0.13
0.75 0.09 0.03 0.09 0.16
Additional moments 1. Frequency of price changes 2. Frequency regular price changes 3. Frequency with which annual mode changes
0.22 0.07 0.64
0.10 0.09 0.70
0.10 0.07 0.70
Steinsson (2008) using the same algorithm I used above for the Dominick’s data.31 As in the Dominick’s data, price changes (I focus on regular prices) are large (11% on average) and dispersed: 25% of price changes are less than 3% in absolute value and another 25% are greater than 13% in absolute value. The frequency of posted price changes (22% per month) is greater than that of regular price changes (7% per month) and roughly 75% of prices are at the annual mode. I study two versions of a menu-cost economy using these data. The first has economies of scope and Poisson shocks, as above. In a second variation, I assume away the economies of scope, but introduce random menu costs, so that the menu-cost for any given good is equal to either 0, with probability pφ , or φ, with probability 1 − pφ . This is an alternative way to generate small price changes in the model. I choose parameters in each of these models so as to match the moments listed in rows 1–6 in Table IV and again target the fraction of prices at the annual mode. Table V reports the business cycle statistics in these economies. As earlier, the model with economies of scope predicts a volatility of consumption that is 85% as large as in the Calvo model with the same frequency of price changes. The economy with random costs also predicts real effects of money that are nearly as large. Thus, the conclusions I draw are not sensitive to the exact mechanism I use to generate small price changes in the model or to the fact that I have focused on the Dominick’s data. 31
The full set of statistics they computed is available in the online supplement titled “More Facts About Prices” (Nakamura and Steinsson (2008, Tables 24–28)). I only report aggregate statistics (computed as a weighted median of ELI-level statistics) here.
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VIRGILIU MIDRIGAN TABLE V AGGREGATE IMPLICATIONS, BLS CALIBRATION K and Inelastic M d
Benchmark
Businss cycle statistics σ(C) (%) Serial correlation C
Calvo
Economies of Scope
Random Menu Cost
Calvo
Economies of Scope
Calvo
Economies of Scope
045 093
039 092
042 093
046 085
033 080
058 089
055 088
075 085
055 080
075 089
070 088
0868 −0695 −0045
0811 −0482 −0056
0935 −0842 −0020
0924 −0792 −0026
σ(Y ) (%) Serial correlation Y Law of motion P P(t − 1) g(t) K(t − 1) R2
Add Intermediates
0897 −0777 – 1
0851 −0715 –
0859 −0775
0999
0999
1
0996
1
1000
Richer Business Cycle Dynamics I next extend the analysis to allow for capital accumulation, interest-elastic money demand, and use of intermediate inputs in production.32 I describe next the additional assumptions I make. Consumers: The consumer owns all the capital stock in this economy and rents it to manufacturers. The consumer chooses how much to invest in the capital stock and faces the budget constraint Pt [ct + it + ξit2 ] + Qt · Bt+1 ≤ Wt lt + Πt + Bt + Pt rtk kt it = kt+1 − (1 − δ)kt where it is investment, ξit2 is a quadratic capital adjustment cost, and rtk is the (real) rental rate of capital. The resource constraint for final goods now reads ct + it + ξit2 + nt = yt =
N 1 ct (i)(θ−1)/θ N i=1
θ/(θ−1)
where ct (i) is defined as earlier. Here the final good has four uses: consumption, investment, adjustment costs, and use as an intermediate input in production by manufacturing firms, nt .33 GDP in this economy is thus equal to yt − nt . 32
Basu (1995) and Nakamura and Steinsson (2009). The interpretation here, as is standard in the literature, is that the Dixit–Stiglitz aggregator represents the production function for a final good produced by firms in a perfectly competitive sector that buy intermediate goods i z from retailers. 33
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I follow Dotsey, King, and Wolman (1999) and assume a particular form for the money demand without explicitly modeling the source of the transactions demand for money: log
Mt = log ct − ηRt Pt
where Rt is the gross nominal risk-free rate. Manufacturers: I modify the production function to yt (i z) = cy [lt (i z)α kt (i z)1−α ]1−ν [nt (i z)]ν where cy is a constant of normalization, kt (i z) is the amount of capital hired by manufacturers in sector (i z), and nt (i z) is the amount of the intermediate good they purchase. Perfect competition implies that, since manufacturers buy the final good at price Pt , the price at which manufacturers sell the good to retailers is ωt (i z) = [Wt α (Pt rtk )1−α ]1−ν Ptν Retailers: The problem of the retailer is similar to that discussed earlier. I augment the aggregate state space to include the stock of capital, k. I approximate the law of motion for capital, real wages, and rental price of capital, since these affect the retailer’s profits. I solve the problem using a Krusell– Smith algorithm as above, now allowing aggregate variables to depend on the capital stock as well.34 Results: As in the exercise above, I assume a monthly frequency in the model, the frequency in the BLS data. I choose the size of the capital adjustment costs, ξ, to reproduce a volatility of investment that is three times greater than output, as in the data. The share of capital in value added, α, is equal to 0.33. I set the semielasticity of money demand equal to where η = 19, an estimate from Ireland (2009). I report results from two experiments: one without intermediate inputs and the other from an experiment in which ν is equal to 0.66, a value that implies a materials share of slightly below 50%, as in the data. Notice in Table V that my earlier results about the relative variability of output in the menu-cost models are robust to allowing for richer business cycle dynamics. The standard deviation of consumption (GDP) is 72% (73%) as large in the menu-cost model as in the Calvo model when I assume away intermediate inputs. When I do allow for intermediate inputs, the selection effect becomes slightly weaker, so the gap between menu costs and the Calvo model is bridged even further. The volatility of consumption is 0.55% in the menucost model (0.58% in the Calvo model), while that of GDP is equal to 0.70% (0.75%) in the Calvo model. 34
Once again, this approximate law of motion is very accurate. See Table V.
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The economy with capital and interest-elastic money demand alone suffers from the persistence problem discussed by Chari, Kehoe, and McGrattan (2002): the serial correlation of output is 0.80 versus 0.92 in the economy without capital and interest-elastic money. This problem is shared, however, by Calvo and menu-cost models alike. The key result of my paper—that the real effects of money in a menu-cost model with heterogeneity in the size of price changes are similar to those in the Calvo model—is invariant to the persistence problem. 5. EMPIRICS I have assumed economies of scope in price setting as well as infrequent cost shocks so as to reproduce the dispersion in the size of price changes. I next present empirical support for these assumptions using the Dominick’s data. A. Economies of Scope My results are robust to alternative mechanisms of generating the distribution of price changes in the data, as shown above. Nevertheless, I provide next some empirical support for economies of scope in price setting.35 My theory assumes economies of scope at the retail level (among different goods sold by the retailer). Hence in the data I attempt to establish evidence for economies of scope across different goods sold by Dominick’s. Synchronization One prediction of my model is that the retailer synchronizes its price changes, both regular and sale related, since the costs of such actions, φR and κ, are shared across goods. Table VI provides evidence that price changes within a store are synchronized. I report results from probit regressions that relate a good’s price adjustment decision to several measures of synchronization. These measures are the fraction of other prices that change in any given period in (a) the same manufacturer and product category, (b) the same manufacturer but a different product category, (c) the same product category but different manufacturers, and (d) all other goods within the store. I include these different partitions since different managers may be responsible for setting the prices of different goods and it is unclear ex ante at what level within the store the economies of scope are most important. These regressions also control for two good-specific variables: (i) the good’s p markup gap (the absolute log deviation of the good’s markup, μit = cit−1 , from it 35 Lach and Tsiddon (2007) and Levy et al. (1997) also provided empirical evidence of economies of scope in price adjustment.
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TABLE VI SYNCHRONIZATION IN PRICE CHANGESa All Price Changes I
Own markup gap Tempt−1
0.96 0.60
Regular Price Changes I
0.17 0.04
Initiate Sales
II
I
0.10 0.04
0.61 –
II
0.62 –
Fraction of price changes: Same manufacturer and category Same manufacturer, other category Same category, other manufacturers Storewide (all other goods)
Posted 0.83 0.05 0.01 0.27
Posted 0.03 −0.01 0.00 0.01
Regular 0.12 0.02 0.02 0.09
Posted 0.59 0.04 0.02 0.13
Sales 0.60 0.06 −0.01 0.18
Number of observations
450,182
450,182
450,182
364,541
364,541
a The marginal effect on the probability of price changes is reported. Observations are weighted (by each UPC’s revenue share). The fraction of price changes is computed for all UPCs other than the UPC in question, weighting incidences of price changes using each UPC’s revenue share. Observations are excluded if less than 5 products are available in a group in a particular period. All coefficients are statistically significantly different from 0 at the 1% level (unless underlined). Posted prices are the actual (transactions) prices set by the retailer.
its time series mean) and (ii) an indicator variable for whether the past price was a temporary price. I report in Table VI marginal effects of a change in the independent variables on the price adjustment decision.36 The markup gap is strongly correlated with a good’s likelihood of a price change: a 10% increase in the markup gap raises the probability of a price change by 9.6%. This suggests that pricing is indeed state-dependent, since in the Calvo model the adjustment hazard is constant.37 Price changes are synchronized, especially for goods in the same manufacturer and product category. When the fraction of all other changes increases from 0 to 1, the likelihood that the price in question also changes increases by 0.83. Synchronization is also present at the store level: an increase in the storewide fraction of price changes from 0 to 1 increases the likelihood of a price change for any individual good by 0.27. There is much less correlation between changes in the regular price of a good with all posted price changes for other goods: the marginal effects are close to 0.38 In contrast, there is much more synchronization of a given regular price change with other regular price changes. The strongest correlation is in a given manufacturer and product category (the marginal effect is 0.125), but there is also strong storewide synchronization (0.091). Hence there is little correlation between changes in regular prices and sales (which make up most price changes), but much stronger correlation between regular price changes. This 36 Standard errors, not reported here, are typically very small. I underline those coefficients that are not statistically significant from 0 at a 1% level. 37 Eichenbaum, Jaimovich, and Rebelo (2011) conducted a similar test of state dependence. 38 The markup is now computed using the old regular price.
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disconnect is evidence that temporary discounts and regular price changes are performed at different levels (consistent with assumption I make in the model) and there is little interaction among the two types of price changes. The last columns of Table VI focus on sales alone. Because these make up most of the price changes in the store, the evidence for synchronization is similar to that for all price changes. Direct Test of Economies of Scope Synchronization alone is necessary but not sufficient evidence of economies of scope, since it can arise for other reasons, including correlated shocks. A stronger test of my theory is one that recognizes that if economies of scope are indeed present, any individual good’s adjustment decision depends not only on that good’s desired price change, but also on the other goods’ desired price changes. This is the mechanism that generates small price changes in the model, since a good may experience a small price change as long as other goods need larger price changes. To test this mechanism, recall that in the presence of economies of scope, a good’s decision to adjust, for example, its regular price, depends on the vector of markups of all the goods the retailer sells: adjust good i
if V R (μR−1 ) > V TN (μR−1 )
A second-order approximation (around the optimal markup, so that first-order terms vanish by the envelope condition) to the above inequality yields adjust good i if α0 +α1 (ln μit −ln μ∗i )2 +α2 (1) (ln μjt −ln μ∗j )2 > φR j=i γ where μ∗i = γ−1 is the optimal markup. Because the menu-cost, φR , is shared by all goods, the adjustment decision is a function of the markup gap of all the goods, not of any individual good i. In contrast, absent economies of scope, the adjustment decision for any good i is a function of that good’s idiosyncratic state and not of the state of all other goods:
adjust good i
if α0 + α1 (ln μit − ln μ∗i ) > φR
In Table VII, I present results of regressions in which I test the null of no scope economies. Under this null, the price adjustment decision is independent of the markup gap of any good other than good i, that is, α2 = 0. I thus estimate probit regressions of equation (1). I once again combine goods into four nonoverlapping partitions, as above, to gauge the level of aggregation at which economies of scope are more important. These regressions, by using the markup gap directly, account for the possibility that unobserved correlated costs or preference shocks are responsible
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TABLE VII DIRECT TEST OF ECONOMIES OF SCOPEa All Price Changes
Regular Price Changes
Own markup gap Tempt−1
0.94 0.62
0.15 0.06
Average markup gap of other goods Same manufacturer and category Same manufacturer, other category Same category, other manufacturers Storewide (all other goods)
All 0.30 0.06 0.66 0.52
All 0.23 0.02 0.02 1.17
450,182
450,182
Number of observations
a The marginal effect on the probability of price changes is reported. Observations are weighted (by each UPC’s revenue share). The fraction of price changes is computed for all UPCs other than the UPC in question, weighting incidences of price changes using each UPC’s revenue share. Observations are excluded if less than 10 products are available in a given manufacturer/category group in a particular period. All coefficients are statistically significantly different from 0 at the 1% level (unless underlined).
for the synchronization in price changes. The markup gap for good i is indeed strongly correlated with the mean markup gap of all other goods within the store, but both are controlled for in these regressions. In other words, an economy with correlated cost shocks but no economies of scope would predict a coefficient α2 = 0, as evident in the decision rules above. Hence, this direct test allows me to determine whether synchronization in price changes is solely driven by correlated cost shocks or whether economies of scope play a role as well. As shown in Table VII, the average gap of all other goods is indeed an important determinant of any given good’s price adjustment decisions. This is especially true for regular price changes where the average markup gap of goods in the same manufacturer category has a marginal effect (0.23) on the price adjustment decision that is greater than that of that good’s own markup gap (0.15). Even stronger is the dependence on the average markup gap of all goods within the store (the marginal effect is 1.17). This evidence thus rejects the null hypothesis of no scope economies. It also suggests that, especially for regular prices, economies of scope are more important at the level of the store, rather than at the level of individual product categories. B. Evidence on Costs A second assumption I have made is that retailers infrequently receive relatively large cost shocks so that the distribution of their markup gap (absent a price change) has much mass near zero, while some markups are very far away from their optimal value. In other words, the distribution of markups in the model exhibits excess kurtosis.
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VIRGILIU MIDRIGAN TABLE VIII DISTRIBUTION OF THE MARKUP GAP AND CHANGES IN COSTSa Markup Gap
5% 10% 25% Median 75% 90% 95% Kurtosis Number of observations
Posted Prices
Regular Prices
Change in Costs
−031 −019 −007 000 006 015 021 766
−021 −013 −006 −001 005 012 018 930
−005 −002 −0003 0000 0004 002 005 2931
1,133,947
1,133,947
1,150,922
a Statistics weigh observations by revenue share of each product. I exclude observations with |gap| and |Delta cost| > 100%.
Such kurtosis is also present in the grocery store data I look at. I report in Table VIII a number of moments of the distribution of the markup gap for both posted and regular prices. I discuss the evidence for regular prices, although that for posted prices is very similar. Notice that the 25th percentile of the distribution of the markup gap is equal to −0.06, while the 75th percentile is equal to 0.05. Hence, a large number of desired price changes are fairly small. In contrast, the 5th percentile of this distribution is equal to −0.21, while the 95th percentile is equal to 0.18: hence a small number of goods experience very large markup gaps. As a result, the distribution of markup gaps shows large kurtosis: 7.66 for posted prices and 9.30 for regular prices. Table VIII also reports moments of the distribution of changes in the wholesale costs at which Dominick’s purchases its goods. The distribution of changes in costs has much mass near 0 (the 25th and 75th percentiles are −0.003 and 0.004, respectively) and since a small fraction of changes in costs are very large (the 1st and 99th percentiles are equal to ±0.12), the kurtosis of the distribution of changes in costs is equal to 29.39 C. Relationship to Other Work The key result in this paper is driven by the fact that the distribution of the size of price changes is highly dispersed. This is neither a new fact, nor specific to data from grocery stores. My contribution in this paper is to study the aggregate consequences of this dispersion, not document the facts per se. 39 See also the evidence in Eichenbaum, Jaimovich, and Rebelo (2011) using admittedly better cost data. They also found that costs (both high- and low-frequency measures) change infrequently.
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Klenow and Kryvtsov (2008) reported that 40% of price changes are less than 5% in absolute value in the BLS price data, a data set in which prices change by 9.5% on average. They also showed that heterogeneity in the size of price changes across sectors is, alone, insufficient to account for this large number of small price changes. Kashyap (1995) studied prices for products sold using retail catalogues and also documented that many price changes are small: 44% of price changes in his data set are less than 5% in absolute value. The kurtosis of price changes is 15.7 in his data. Kackmeister (2005) reported that 33% of price changes are less than 10% in absolute value in an environment where the average magnitude of price changes is 20% in a study of prices in retail stores. D. Limitations of My Analysis and Extensions My model misses several features of the data that are worth exploring in future work. First, I do not allow consumers to store the goods. To the extent to which they can do so, their ability to take advantage of sales would be greatly increased. Recall, however, that the model I study (through an appropriate choice of a demand elasticity) accounts for the fraction of goods that are sold during sales. Hence, although an extension of the model that allows for storability of goods would surely impart richer microlevel dynamics, my conjecture is that such an extension (appropriately parameterized to match the price and quantity facts) would not overturn my main results. Second, I assume that goods sold by different retailers are imperfectly substitutable, whereas in the data different retailers sell identical goods. An alternative would be a Hotelling-type setting in which consumers face fixed costs of reaching stores that sell identical goods. In such an environment, temporary shocks to demand would generate a motive for temporary price discounts.40 Combined with a rigidity in the regular price, as in Kehoe and Midrigan (2008), temporary changes would also revert to the pre-sale price, as in the data. An extension along these lines is also an exciting avenue for future research. 6. CONCLUSION Simple menu-cost models fail to account for two features of the microeconomic price data: the dispersion in the size of price changes and the fact that many price changes are temporary price discounts. I study an economy with economies of scope in price adjustment—a more flexible specification of the distribution of firm-level uncertainty—and a two-tier price setting technology (for regular and posted prices) that is capable of replicating these microlevel facts. 40
Warner and Barsky (1995).
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I find that in this economy the real effects of money are much greater than those in a simple menu-cost economy that fails to account for the microlevel facts. This result is primarily accounted for by the heterogeneity in the size of price changes in my model. When price changes are very dispersed in absolute value, the measure of marginal firms whose adjustment decisions are sensitive to aggregate shocks is small. Hence the selection effect is much weaker and the model produces real effects of a similar magnitude to those in Calvo-type models. REFERENCES AGUIRREGABIRIA, V. (1999): “The Dynamics of Markups and Inventories in Retailing Firms,” Review of Economic Studies, 66, 275–308. [1156] BARSKY, R., M. BERGEN, S. DUTTA, AND D. LEVY (2000): “What Can the Price Gap Between Branded and Private Label Products Tell Us About Markups?” Mimeo, University of Michigan. [1156] BASU, S. (1995): “Intermediate Goods and Business Cycles: Implications for Productivity and Welfare,” American Economic Review, 85, 512–531. [1170] BILS, M., AND P. KLENOW (2004): “Some Evidence on the Importance of Sticky Prices,” Journal of Political Economy, 112, 947–985. [1139] BURDETT, K., AND K. JUDD (1983): “Equilibrium Price Dispersion,” Econometrica, 51, 955–969. [1140] BURSTEIN, A. (2006): “Inflation and Output Dynamics With State-Dependent Pricing Decisions,” Journal of Monetary Economics, 53, 1235–1257. [1141] BURSTEIN, A., AND C. HELLWIG (2008): “Prices and Market Shares in a Menu Cost Model,” Mimeo, UCLA. [1141,1167] BUTTERS, G. (1977): “Equilibrium Distribution of Sales and Advertising Prices,” Review of Economic Studies, 44, 465–491. [1140] CABALLERO, R., AND E. ENGEL (1993): “Heterogeneity and Output Fluctuations in a Dynamic Menu-Cost Economy,” Review of Economic Studies, 60, 95–119. [1141] (2007): “Price Stickiness in Ss models: New Interpretations of Old Results,” Journal of Monetary Economics, 54, S100–S121. [1141,1165-1167] CAPLIN, A., AND J. LEAHY (1991): “State-Dependent Pricing and the Dynamics of Money and Output,” Quarterly Journal of Economics, 106, 683–708. [1141] CAPLIN, A., AND D. SPULBER (1987): “Menu Costs and the Neutrality of Money,” Quarterly Journal of Economics, 102, 703–725. [1141] CHARI, V. V., P. KEHOE, AND E. MCGRATTAN (2002): “Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?” Review of Economic Studies, 69, 533–563. [1172] CHEVALIER, J., A. KASHYAP, AND P. ROSSI (2003): “Why Don’t Prices Rise During Periods of Peak Demand? Evidence From Scanner Data,” American Economic Review, 93, 15–37. [1140, 1156] CHRISTIANO, L., M. EICHENBAUM, AND C. EVANS (2005): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113, 1–45. [1156] DANZIGER, L. (1999): “A Dynamic Economy With Costly Price Adjustments,” American Economic Review, 89, 878–901. [1141] DOTSEY, M., R. KING, AND A. WOLMAN (1999): “State-Dependent Pricing and the General Equilibrium Dynamics of Money and Output,” Quarterly Journal of Economics, 114, 655–690. [1141,1142,1171] DUTTA, S., M. BERGEN, AND D. LEVY (2002): “Price Flexibility in Channels of Distribution: Evidence From Scanner Data,” Journal of Economic Dynamics and Control, 26, 1845–1900. [1143]
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EICHENBAUM, M., N. JAIMOVICH, AND S. REBELO (2011): “Reference Prices and Nominal Rigidities,” American Economic Review (forthcoming). [1141,1159,1173,1176] GERTLER, M., AND J. LEAHY (2008): “A Phillips Curve With an Ss Foundation,” Journal of Political Economy, 116, 533–572. [1143,1147,1149,1156,1157] GOLOSOV, M., AND R. E. LUCAS, JR. (2007): “Menu Costs and Phillips Curves,” Journal of Political Economy, 115, 171–199. [1139,1143,1149,1153,1157,1165] HANSEN, G. (1985): “Indivisible Labor and the Business Cycle,” Journal of Monetary Economics, 16, 309–327. [1153] HOCH, S., X. DREZE, AND M. E. PURK (1994): “EDLP, Hi-Lo, and Margin Arithmetic,” Journal of Marketing, 58, 16–27. [1143] HOSKEN, D., AND D. REIFFEN (2004): “Patterns of Retail Price Variation,” RAND Journal of Economics, 35, 128–146. [1159] IRELAND, P. (2009): “On the Welfare Cost of Inflation and the Recent Behavior of Money Demand,” American Economic Review, 99, 1040–1052. [1171] KACKMEISTER, A. (2005): “Yesterday’s Bad Times Are Today’s Good Old Times: Retail Price Changes in the 1890s Were Smaller, Less Frequent, and More Permanent,” Finance and Economics Discussion Series, Federal Reserve Board. [1177] KASHYAP, A. (1995): “Sticky Prices: New Evidence From Retail Catalogues,” Quarterly Journal of Economics, 110, 245–274. [1177] KEHOE, P., AND V. MIDRIGAN (2008): “Temporary Price Changes and the Real Effects of Monetary Policy,” Working Paper 14392, NBER. [1140,1141,1144,1151,1159,1160,1165,1168,1177] KHAN, A., AND J. THOMAS (2007): “Inventories and the Business Cycle: An Equilibrium Analysis of (S, s) Policies,” American Economic Review, 97, 1165–1188. [1155] KLENOW, P., AND O. KRYVTSOV (2008): “State-Dependent or Time-Dependent Pricing: Does it Matter for Recent US Inflation?” Quarterly Journal of Economics, 123, 863–904. [1139,1143, 1159,1164,1176,1177] KLENOW, P., AND J. WILLIS (2006): “Real Rigidities and Nominal Price Changes,” Working Paper 06-03, FRB Kansas City. [1155] KRUSELL, P., AND A. SMITH (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106, 867–896. [1155] LACH, S., AND D. TSIDDON (2007): “Small Price Changes and Menu Costs,” Managerial and Decision Economics, 28, 649–656. [1140,1142,1172] LAZEAR, E. (1986): “Retail Pricing and Clearance Sales,” American Economic Review, 76, 14–32. [1140] LEVY, D., M. BERGEN, S. DUTTA, AND R. VENABLE (1997): “The Magnitude of Menu Costs: Direct Evidence From Large US Supermarket Chains,” Quarterly Journal of Economics, 112, 781–826. [1142,1161,1172] MIDRIGAN, V. (2011): “Supplement to ‘Menu Costs, Multiproduct Firms, and Aggregate Fluctuations’,” Econometrica Supplemental Material, 79, http://www.econometricsociety.org/ecta/ Supmat/6735_data and programs.zip; http://www.econometricsociety.org/ecta/Supmat/6735_ miscellaneous.pdf. [1144] MIRANDA, M., AND P. FACKLER (2002): Applied Computational Economics and Finance. Cambridge, MA: MIT Press. [1155] NAKAMURA, E., AND J. STEINSSON (2008): “Five Facts About Prices: A Reevaluation of Menu Cost Models,” Quarterly Journal of Economics, 123, 1415–1464. [1142,1143,1169] (2009): “Price Setting in Forward-Looking Customer Markets,” Mimeo, Columbia University. [1154,1170] NEVO, A. (1997): “Measuring Market Power in the Ready-to-Eat Cereal Industry,” Econometrica, 69, 265–306. [1156] PELTZMAN, S. (2000): “Prices Rise Faster Than They Fall,” Journal of Political Economy, 108, 466–502. [1144] ROGERSON, R. (1988): “Indivisible Labor, Lotteries and Equilibrium,” Journal of Monetary Economics, 21, 3–16. [1153]
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ROMER, C., AND D. ROMER (2004): “A New Measure of Monetary Shocks: Derivation and Implications,” American Economic Review, 94, 1055–1084. [1156] SCARF, H. (1959): “The Optimality of (S, s) Policities in the Dynamic Inventory Problem,” in Mathematical Methods for the Social Sciences, ed. by K. Arrow, S. Karlin, and P. Suppes. Cambridge, MA: MIT Press, 196–202. [1156] SHESHINSKI, E., AND Y. WEISS (1977): “Inflation and Costs of Price Adjustment,” Review of Economic Studies, 44, 287–303. [1141] (1992): “Staggered and Synchronized Price Policies Under Inflation,” Review of Economic Studies, 59, 331–359. [1156] SMETS, F., AND R. WOUTERS (2007): “Shocks and Frictions in US Business Cycles. A Bayesian DSGE Approach,” Working Paper 722, ECB. [1156] SOBEL, J. (1984): “The Timing of Sales,” Review of Economic Studies, 51, 353–368. [1140] STOKEY, N., AND R. E. LUCAS, JR. (1989): Recursive Methods in Economic Dynamics. Cambridge, MA: Harvard University Press. [1156] VARIAN, H. (1980): “A Model of Sales,” American Economic Review, 70, 651–659. [1140] WARNER, E., AND R. BARSKY (1995): “The Timing and Magnitude of Retail Store Markdowns: Evidence From Weekends and Holidays,” Quarterly Journal of Economics, 110, 321–352. [1140, 1177] ZBARACKI, M., M. RITSON, D. LEVY, S. DUTTA, AND M. BERGEN (2004): “Managerial and Customer Costs of Price Adjustment: Direct Evidence From Industrial Markets,” Review of Economics and Statistics, 86, 514–533. [1141,1147,1148,1161] ZBARACKI, M., D. LEVY, AND M. BERGEN (2007): “The Anatomy of a Price Cut: Discovering Organizational Sources of the Costs of Price Adjustment,” Mimeo, University of Western Ontario. [1141,1147]
Dept. of Economics, New York University, 19 West 4th Street, 6th Floor, New York, NY 10012, U.S.A.;
[email protected]. Manuscript received September, 2006; final revision received February, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1181–1232
EFFICIENT DERIVATIVE PRICING BY THE EXTENDED METHOD OF MOMENTS BY P. GAGLIARDINI, C. GOURIEROUX, AND E. RENAULT1 In this paper, we introduce the extended method of moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard generalized method of moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e., valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for a large time series dimension, but a fixed number of cross sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters. KEYWORDS: Derivative pricing, trading activity, GMM, information theoretic estimation, KLIC, identification, weak instrument, nonparametric efficiency, semiparametric efficiency.
1. INTRODUCTION THE GENERALIZED METHOD OF MOMENTS (GMM) was introduced by Hansen (1982) and Hansen and Singleton (1982) to estimate a structural parameter θ identified by Euler conditions, (1.1)
pit = Et [Mtt+1 (θ)pit+1 ]
i = 1 n ∀t
where pit , i = 1 n, are the observed prices of n financial assets, Et denotes the expectation conditional on the available information at date t, and Mtt+1 (θ) is the stochastic discount factor. Model (1.1) is semiparametric. The GMM estimates parameter θ regardless of the conditional distribution of the state variables. This conditional distribution, however, becomes relevant when the Euler conditions (1.1) are used for pricing derivative assets. Indeed, when the derivative payoff is written on pit+1 and a reliable current price is not available on the market, the derivative pricing requires the joint estimation of parameter θ and the conditional distribution of the state variables. 1 We thank D. Andrews, X. Chen, F. Corsi, R. Davidson, J. Fan, R. Garcia, J. Jackwerth, J. Jasiak, O. Linton, L. Mancini, A. Melino, C. Ortelli, O. Scaillet, M. Stutzer, F. Trojani, A. Vedolin, B. Werker, E. Zivot, a co-editor, and anonymous referees for helpful comments. The first author gratefully acknowledges financial support of the Swiss National Science Foundation through the NCCR FINRISK network. The second author gratefully acknowledges financial support of NSERC Canada and of the chair AXA/Risk Foundation: “Large Risks in Insurance.”
© 2011 The Econometric Society
DOI: 10.3982/ECTA7192
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The extended method of moments (XMM) estimator extends the standard GMM to accommodate a more general set of moment restrictions. The standard GMM is based on uniform conditional moment restrictions such as (1.1), that are valid for any value of the conditioning variables. The XMM can handle not only the uniform moment restrictions, but also local moment restrictions that are valid for a given value of the conditioning variables only. This leads to a new field of application of moment-based methods to derivative pricing, where the local moment restrictions correspond to Euler conditions for cross sectionally observed prices of actively traded derivatives. Such a framework accounts for the fact that the trading activity is much smaller on each derivative than on the underlying asset and that the characteristics of actively traded derivatives are not stable over time. More precisely, we explain how the XMM can be used to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed actively traded derivative prices and a time series of underlying asset returns. To illustrate the principle of XMM, let us consider the Standard and Poors (S&P) 500 index and its derivatives. Suppose an investor at date t0 is interested in estimating the price ct0 (h k) of a call option with time-to-maturity h and moneyness strike k that is currently not actively traded on the market. She has data on a time series of T daily returns of the S&P 500 index, as well as on a small cross section of current option prices ct0 (hj kj ), j = 1 n, of n highly traded derivatives. The XMM approach provides the estimated prices cˆt0 (h k) for different values of moneyness strike k and time-to-maturity h that interpolate the observed prices of highly traded derivatives and satisfy the hypothesis of absence of arbitrage opportunities. These estimated prices are consistent for a large number of dates T , but a fixed, even small, number of observed derivative prices n. To highlight the specificity of XMM with respect to GMM, we present in Section 2 an application to the S&P 500 index and its derivatives. First we show that the trading activity on the index option market is rather weak and the daily number of reliable derivative prices is small. Then we explain why the time series observations on the underlying index induce uniform moment restrictions, whereas the observed cross-sectional derivative prices correspond to local moment restrictions. The XMM estimator minimizes the discrepancy of the historical transition density from a kernel density estimator subject to both types of moment restrictions. The estimation criterion includes a Kullback– Leibler information criterion associated with the local moment restrictions to ensure the compatibility of the estimated derivative prices with the absence of arbitrage opportunities. The comparison of the XMM estimator and the standard calibration estimator for S&P 500 options data shows clearly that XMM outperforms the traditional approach. The theoretical properties of the XMM estimator are presented in Section 3 in a general semiparametric framework. We discuss the parameter identification under both uniform and local moment restrictions, and derive the efficiency bound. We prove that the XMM estimator is consistent for a fixed number n of cross-sectional observations associated with the local restrictions and a
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large number T (T → ∞) of time series observations associated with uniform restrictions. Moreover, the XMM estimator is asymptotically normal and semiparametrically efficient. Section 4 concludes. The set of regularity assumptions and the proofs of the theoretical results are gathered in Appendix A. Proofs of technical lemmas are given in Appendix B in the Supplemental Material (Gagliardini, Gouriéroux, and Renault (2011)). 2. THE XMM APPLIED TO DERIVATIVE PRICING 2.1. The Activity on the Index Option Market To maintain a minimum activity, the Chicago Board Options Exchange (CBOE) enhances the market of options on the S&P 500 index by periodically issuing new option contracts (Hull (2005, p. 187)). The admissible timesto-maturity at issuing are 1 month, 2 months, 3 months, 6 months, 9 months, 12 months, and so forth up to a maximal time-to-maturity. For instance, new 12-month options are issued every 3 months, when the options from the previous issuing attain the time-to-maturity of 9 months. This induces a cycle in the times-to-maturity of quoted options (see, e.g., Schwartz (1987, Figure 1), and Pan (2002, Figure 2)). For any admissible time-to-maturity, the options are issued for a limited number of strikes around the value of the underlying asset at the issuing date. This strategy restricts the number of options quoted on the market. Among these, only a few options are traded on a daily basis. This phenomenon is illustrated in Section 2.6, where we select the call and put options with a daily traded volume of more than 4000 contracts in June 2005. Since each contract corresponds to 100 options, 4000 contracts are worth between $5 million and $7 million U.S., on average. The daily number of the highly traded options varies between a minimum of 7 on June 10, 2005 and a maximum of 31 on June 16, 2005. The corresponding times-to-maturity and moneyness strikes also vary in time. For instance, on June 10, 2005 the actively traded times-to-maturity are 5, 70, and 135 days, while on June 16, 2005 the actively traded times-tomaturity are 1, 21, 46, 66, 131, 263, and 393 days. In brief, the number of highly traded derivatives on a given day is rather small. They correspond generally to puts for moneyness strikes below 1 and to calls otherwise. Moreover, the number of options, and the moneyness strikes and times-to-maturity vary from one day to another due to the issuing cycle and the trading activity. 2.2. Calibration Based on Current Derivative Prices The underlying asset features a regular trading activity, and the associated returns can be observed and are expected to be stationary. In contrast, the trading prices of a given option on two consecutive dates are not always observed and reliable. Therefore, the associated returns might not be computed. Even if these option returns were available, they would be nonstationary due to
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the issuing cycle discussed above. To circumvent the difficulty in modelling the trading activity and its potential nonstationary effect on prices, the standard methodology consists of calibrating the pricing operator daily.2 More precisely, let us assume that, at date t0 , the option prices ct0 (hj kj ), j = 1 n, are observed and that a parametric model for the risk-neutral dynamics of the relevant state variables implies a pricing formula ct0 (h k; θ) for the option prices at t0 . The number n of actively traded derivatives and their design hj kj depend on date t0 , but the time index is omitted for expository purposes. The unknown parameter θ is usually estimated daily by minimizing the least-squares criterion3 (e.g., Bakshi, Cao, and Chen (1997)): (2.1)
θˆ t0 = arg min θ
n [ct0 (hj kj ) − ct0 (hj kj ; θ)]2 j=1
This practice has drawbacks: First, the estimated parameters θˆ t0 are generally erratic over time. Second, the approximated prices cˆt0 (hj kj ) = ct0 (hj kj ; θˆ t0 ) are not compatible with the absence of arbitrage opportunities, since the approximated option prices cˆt0 (hj kj ) differ from the observed prices ct0 (hj kj ) for highly traded options.4 Third, even if the estimates can be shown to be consistent when n is large and the option characteristics (hj , kj ) are well distributed (see, e.g., Aït-Sahalia and Lo (1998)), in practice n is small and the estimates are not very accurate. 2.3. Semiparametric Pricing Model In general, the underlying asset is much more actively traded than each of its derivatives. For instance, the daily traded volume of a portfolio mimicking the S&P 500 index, such as the SPDR, is about $100 million U.S. It is possible to improve the above calibration approach by jointly considering the time series of observations on the underlying asset and the cross sectional data on its derivatives. For this purpose, the specification cannot be reduced to the risk-neutral dynamics only, but has to define in a coherent way the historical and risk-neutral dynamics of the variables of interest. Let us consider a discrete time model. We denote by rt the logarithmic return of the underlying asset between dates t − 1 and t. We assume that the information available to 2
Or calibrating the Black–Scholes implied volatility surface daily. Equivalently, a penalized least-squares (LS) criterion (Jackwerth (2000)), a local LS criterion under shape constraints (Aït-Sahalia and Duarte (2003)), a LS criterion under convolution constraints (Bondarenko (2003)), or an entropy criterion (Buchen and Kelly (1996), Stutzer (1996), Jackwerth and Rubinstein (1996)) in a nonparametric setting. 4 When the pricer provides a price different from the market price, the trader can have the misleading impression that she can profit from an arbitrage opportunity, which does not exist in reality. 3
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the investors at date t is generated by the random vector Xt of state variables, including the return rt as the first component. ASSUMPTION ON THE STATE VARIABLES: The process (Xt ) in X ⊂Rd is strictly stationary and time homogeneous Markov under the historical probability with transition density fXt |Xt−1 . We consider a semiparametric pricing model defined by the historical dynamics and a stochastic discount factor (s.d.f.) (Hansen and Richard (1987)). The historical transition density of process (Xt ) is left unconstrained. We adopt a parametric specification for the s.d.f. Mtt+1 (θ) = m(Xt+1 ; θ), where θ ∈ Rp is an unknown vector of risk premia parameters. This model implies a semiparametric specification for the risk-neutral distribution. For instance, the relative price at time t of a European call with moneyness strike k and time-to-maturity h written on the underlying asset is given by ct (h k) = E[Mtt+h (θ)(exp Rth − k)+ |Xt ] where Mtt+h (θ) = m(Xt+1 ; θ) · · · m(Xt+h ; θ) is the s.d.f. between dates t and t + h, and Rth = rt+1 + · · · + rt+h is the return of the underlying asset in this period. The option price depends on both the finite-dimensional s.d.f. parameter θ and the functional historical parameter fXt |Xt−1 . We are interested in estimating the pricing operator at a given date t0 , that is, the mapping that associates any European call option payoff ϕt0 (h k) = (exp Rt0 h − k)+ with its price ct0 (h k) at time t0 for any time-to-maturity h and any moneyness strike k. OBSERVABILITY ASSUMPTION: The data consist of a finite number n of derivative prices ct0 (hj kj ), j = 1 n, observed at date t0 , and T serial observations of the state variables Xt that correspond to the current and previous days t = t0 − T + 1 t0 . In particular, the state variables in the s.d.f. are assumed to be observable by the econometrician. The no-arbitrage assumption implies the moment restrictions for the observed asset prices. 2.4. The Moment Restrictions The moment restrictions are twofold. The constraints concerning the observed derivative prices at t0 are given by + ct0 (hj kj ) = E Mtt+hj (θ) exp Rthj − kj Xt = xt0 j = 1 n (2.2) The constraints concerning the risk-free asset and the underlying asset are (2.3)
E[Mtt+1 (θ)|Xt = x] = B(t t + 1) ∀x ∈ X E[Mtt+1 (θ) exp rt+1 |Xt = x] = 1 ∀x ∈ X
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respectively, where B(t t + 1) denotes the price at time t of the short-term riskfree bond. The conditional moment restrictions (2.2) are local, since they hold for a single value of the conditioning variable only, namely the value xt0 of the state variable at time t0 . This is because we consider only observations of the derivative prices ct0 (hj kj ) at date t0 . Conversely, the prices of the underlying asset and the risk-free bond are observed for all trading days. Therefore, the conditional moment restrictions (2.3) hold for all values of the state variables. They are called the uniform moment restrictions. The distinction between the uniform and local moment restrictions is a consequence of the differences between the trading activities of the underlying asset and its derivatives. Technically, it is the essential feature of the XMM that distinguishes this method from its predecessor GMM. There are n + 2 local moment restrictions at date t0 given by + E Mtt+hj (θ) exp Rthj − kj − ct0 (hj kj )Xt = xt0 = 0 (2.4) j = 1 n E Mtt+1 (θ) − B(t0 t0 + 1)|Xt = xt0 = 0 E Mtt+1 (θ) exp rt+1 − 1|Xt = xt0 = 0 Let us denote the local moment restrictions (2.2) as (2.5)
˜ ; θ)|X = x0 ] = 0 E[g(Y
˜ vector of relevant future valwhere Yt = (Xt+1 Xt+h¯ ) is the d-dimensional ¯ ues of the state variables, h is the largest time-to-maturity of interest, x0 ≡ xt0 , and the time index is suppressed. Similarly, the uniform moment restrictions (2.3) are written as (2.6)
E[g(Y ; θ)|X = x] = 0 ∀x ∈ X
Then the whole set of local moment restrictions (2.4) corresponds to (2.7)
E[g2 (Y ; θ)|X = x0 ] = 0
where g2 = (g g˜ ) . Since we are interested in estimating the pricing operator at a given date t0 , the value x0 is considered as a given constant. We assume that the s.d.f. parameter θ is identified from the two sets of conditional moment restrictions (2.5) and (2.6). We allow for the general case where some linear combinations η∗2 , say, of the components of θ are unidentifiable from the uniform moment restrictions (2.6) on the risk-free asset and the underlying asset, only. The identification of these linear combinations requires local moment restrictions (2.5) on the cross-sectional derivative prices at t0 . Intuitively, these linear combinations η∗2 correspond to risk premia parameters
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associated with risk factors that are not spanned by the returns of the underlying asset and the risk-free asset. This point is further discussed in Sections 2.7 and 3.2. 2.5. The XMM Estimator The XMM estimator presented in this section is related to the recent literature on the information-based GMM (e.g., Kitamura and Stutzer (1997), Imbens, Spady, and Johnson (1998)). It provides estimators of both the s.d.f. parameter θ and the historical transition density f (y|x) of Yt given Xt . By using the parameterized s.d.f., the information-based estimator of the historical transition density defines the estimated state price density for pricing derivatives. The XMM approach involves a consistent nonparametric estimator of the historical transition density f (y|x), such as the kernel density estimator T T xt − x xt − x 1 ˜ yt − y ˆ (2.8) K K K f (y|x) = ˜ hT hT hT hdT t=1 t=1 ˜ ˜ is the d-dimensional (resp., d-dimensional) where K (resp., K) kernel, hT is the bandwidth, and (xt yt ), t = 1 T , are the historical sample data.5 Next, this kernel density estimator is improved by selecting the conditional probability density function (p.d.f.) that is the closest to f (y|x), and satisfies the moment restrictions as defined below. θ) conDEFINITION 1: The XMM estimator (fˆ∗ (·|x0 ) fˆ∗ (·|x1 ) fˆ∗ (·|xT ) ˜ sists of the functions f0 f1 fT defined on Y ⊂ Rd and the parameter value θ that minimize the objective function
T f0 (y) 1 [f (y|xt ) − ft (y)]2 (2.9) dy + hdT log f0 (y) dy LT = T t=1 f (y|xt ) f (y|x0 ) subject to the constraints ft (y) dy = 1 t = 1 T (2.10) f0 (y) dy = 1 g(y; θ)ft (y) dy = 0 t = 1 T g2 (y; θ)f0 (y) dy = 0 The objective function LT has two components. The first component involves the chi-squared distance between the density ft and the kernel density 5 For expository purpose, the dates previous to t0 , at which data on (X Y ) are available, have been re-indexed as t = 1 T .
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estimator f (·|xt ) at any sample point xt for t = 1 T . The second component corresponds to the Kullback–Leibler information criterion (KLIC) between the density f0 and the kernel estimator f (·|x0 ) at the given value x0 . In addition to the unit mass restrictions for the density functions, the constraints include the uniform moment restrictions (2.6) written for all sample points and the whole set of local moment restrictions (2.7) at x0 . The combination of two types of discrepancy measures is motivated by computational and financial reasons. The chi-squared criterion evaluated at the sample points allows for closed form solutions f1 (θ) fT (θ) for a given θ (see Appendix A.2.1). Therefore, the objective function can be easily concentrated with respect to functions f1 fT , which reduces the dimension of the optimization problem. The KLIC criterion evaluated at x0 ensures that the minimizer f0 satisfies the positivity restriction (see, e.g., Stutzer (1996) and Kitamura and Stutzer (1997)). The positivity of the associated state price density at t0 guarantees the absence of arbitrage opportunities in the estimated derivative prices. The estimator of θˆ minimizes the concentrated objective function (2.11)
LcT (θ) =
T 1 E(g(θ)|xt ) V (g(θ)|xt )−1 E(g(θ)|x t) T t=1 exp(λ(θ) g2 (θ))|x0 − hdT log E
where the Lagrange multiplier λ(θ) ∈ Rn+2 is such that g2 (θ) exp(λ(θ) g2 (θ))|x0 = 0 (2.12) E for all θ, and E(g(θ)|x t ) and V (g(θ)|xt ) denote the expectation and variance of g(Y ; θ), respectively, with respect to (w.r.t.) the kernel estimator f (y|xt ). The first part of the concentrated objective function (2.11) is reminiscent of the conditional version of the continuously updated GMM (Ai and Chen (2003), ˆ are comAntoine, Bonnal, and Renault (2007)). The estimates θˆ and λ(θ) puted by writing the constrained optimization (2.11) and (2.12) as a saddlepoint problem (see, e.g., Kitamura and Stutzer (1997)) and applying a standard Newton–Raphson algorithm. Then the estimator of f (y|x0 ) is given by (2.13)
f ∗ (y|x0 ) =
ˆ ˆ g2 (y; θ)) exp(λ(θ) f (y|x0 ) ˆ g2 (θ))|x ˆ E[exp(λ( θ) ] 0
y ∈ Y
This conditional density is used to estimate the pricing operator at time t0 . DEFINITION 2: The XMM estimator of the derivative price ct0 (h k) is ˆ exp Rt h − k + fˆ∗ (y|x0 ) dy (2.14) cˆt0 (h k) = Mt0 t0 +h (θ) 0
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for any time-to-maturity h ≤ h¯ and any moneyness strike k. The estimator of the pricing operator density at time t0 up to time-to-maturity h¯ is ˆ fˆ∗ (y|x0 ). Mt0 t0 +h¯ (θ) The constraints (2.10) imply that the estimator cˆt0 (h k) is equal to the observed option price ct0 (hj kj ) when h = hj and k = kj , j = 1 n. Moreover, the no-arbitrage restrictions for the underlying asset and the risk-free asset are perfectly matched at all sample dates. The large sample properties of estimators θˆ and cˆt0 (h k) in Definitions 1 and 2 are examined in Section 3. The asymptotic analysis for T → ∞ corresponds to long histories of the state variables before t0 and a fixed number n of observed derivative prices at t0 , and is conditional on Xt0 = x0 and the option prices observed at t0 . The estimators θˆ and cˆt0 (h k) are consistent and asymptotically normal.6 The linear combinations of θ that are identifiable from the uniform moment restrictions (2.6) on the risk-free asset√and the underlying asset only are estimated at the standard parametric rate T . Any other direction η∗2 in the parameter space(see Section 2.4) and the derivative prices as well are estimated at the rate T hdT that corresponds to nonparametric estimation of conditional expectations given X = x0 . The estimators of derivative prices are asymptotically semiparametrically efficient for the informational content of the no-arbitrage restrictions.7 Finally, the XMM estimator in Definition 1 does not account for the restrictions induced by the time homogeneous first-order Markov assumption. While this could induce an efficiency loss, it has the advantage of providing computationally tractable estimators.8 The difficulty of accounting for the time-homogenous first-order Markov assumption is not specific to XMM. It also concerns the standard GMM when the researcher adopts the first-order Markov assumption for the state process. Such an assumption is used to ensure that the derivative prices are functions of a finite-dimensional state vector and to derive the optimal instruments. The sample end-point condition Xt0 = x0 is irrelevant for the asymptotic properties of the kernel density estimator (2.8) when the data are weakly serially dependent (see the mixing condition in Assumption A.5). 7 Asymptotically equivalent estimators are obtained by replacing the integral w.r.t. the kernel density estimator in (2.14) by the discrete sum involved in a kernel regression estimator. These latter estimators involve a smoothing w.r.t. X only, and are used in the application in Section 2.6. 8 The first-order Markov assumption implies the conditional independence of the future value Xt+1 and the past history Xt−1 given the current value Xt . This can be written as E[Ψ1 (Xt+1 )Ψ2 (Xt−1 )|Xt ] − E[Ψ1 (Xt+1 )|Xt ]E[Ψ2 (Xt−1 )|Xt ] = 0 for any integrable functions Ψ1 and Ψ2 . These restrictions are not conditional moment restrictions. The additional assumption of time homogeneity of the Markov process might be used to develop an adaptive estimation method. This extension is beyond the scope of our paper. 6
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2.6. Application to S&P 500 Options In this section we compare the XMM estimation and a traditional crosssectional calibration approach using the data on the S&P 500 options in June 2005 with daily trading volume larger than 4000 contracts (see Section 2.1). The OptionMetrics data base contains daily data, where the trades of an option with given time-to-maturity and strike are aggregated within the day. Even if the high frequency trading option prices are not available, this data base provides the closing bid/ask quotes. For a given day, these two quotes are rather close to each other, and to trading prices, for options with trading volume larger than 4000 contracts. The average of the closing bid/ask quotes is retained in the empirical analysis as a proxy for the trading price at the end of the day. 2.6.1. Cross-Sectional Calibration The cross-sectional calibration is based on a parametric stochastic volatility model for the risk-neutral distribution Q. We assume that under Q the S&P 500 return is such that (2.15)
1 rt = rft − σt2 + σt εt∗ 2
where rft is the risk-free rate between t − 1 and t, process (εt∗ ) is a standard Gaussian white noise, and σt2 denotes the volatility. The short-term risk-free rate rft is assumed to be deterministic. The volatility (σt2 ) is stochastic, is independent of shocks (εt∗ ) on returns, and follows an autoregressive gamma (ARG) process (Gouriéroux and Jasiak (2006), Darolles, Gouriéroux, and Jasiak (2006)), which is a discrete time Cox, Ingersoll, and Ross (CIR) (1985) process. Hence, the joint model for (rt σt2 ) is a discrete time analogue of the Heston (1993) model. The risk-neutral transition distribution of the stochastic volatility is noncentral gamma and is more conveniently defined through its conditional Laplace transform (2.16)
2 E Q [exp(−uσt+1 )|σt2 ] = exp[−a∗ (u)σt2 − b∗ (u)]
u ≥ 0
where E Q [·] denotes the expectation under Q, and functions a∗ and b∗ are defined by a∗ (u) = ρ∗ u/(1 + c ∗ u) and b∗ (u) = δ∗ log(1 + c ∗ u). Parameter ρ∗ , ρ∗ > 0, is the risk-neutral first-order autocorrelation of volatility process (σt2 ); parameter δ∗ , δ∗ ≥ 0, describes its (conditional) risk-neutral over-/ underdispersion; parameter c ∗ , c ∗ > 0, is a scale parameter. The conditional Laplace transform (2.16) is exponential affine in the lagged observation and the ARG process is discrete time affine. We refer to Gouriéroux and Jasiak (2006) for an in-depth discussion of the properties of the ARG process. The relative option prices are a function of the current value of volatility σt2 and of the three parameters θ = (c ∗ δ∗ ρ∗ ), that is, ct (h k) = c(h k; θ σt2 ),
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CALIBRATED PARAMETERS (CROSS-SECTIONAL APPROACH) FOR THE S&P 500 OPTIONS IN JUNE 2005a Day t0
1.6.05 2.6.05 3.6.05 6.6.05 7.6.05 8.6.05 9.6.05 10.6.05 13.6.05 14.6.05
Calibrated Parameters δˆ ∗t
ρˆ ∗t
cˆt∗ (×10−7 )
σˆ t0
Goodness of Fit RMSEt0
247303 005400 877199 168952 041250 108766 226166 315278 021085 002125
047588 099994 099999 099625 099998 081153 099106 096637 099909 099999
959115 942208 009906 657284 412201 125623 658317 679999 884000 101489
000347 000670 000480 000519 000563 000102 000501 000499 000555 000874
000161 000219 000114 0.00194 0.00071 0.00061 0.00191 0.00032 0.00035 0.00267
0
0
0
a Calibrated parameter θˆ , volatility σ ˆ t0 , and goodness of fit measure RMSEt0 for the first 10 trading days t0 of t0 June 2005. The calibration is performed using a Fourier transform approach to compute option prices. At each day t0 , the sample consists of the derivative prices at t0 of S&P 500 options with daily volume larger than 4000 contracts.
say. Function c is computed by Fourier transform methods as in Carr and Madan (1999) by exploiting the affine property of the joint process of excess returns and stochastic volatility.9 The volatility σt2 is calibrated daily jointly with θ by minimizing the mean squared errors as in (2.1).10 The calibrated parameter θˆ t0 and volatility σˆ t0 for the first 10 trading days of June 2005 are displayed in Table I. We also report the root mean squared errors nt0 ˆ ˆ t2 ) − ct (hj kj )]2 }1/2 as the goodness of fit RMSEt0 = { n1t j=1 [c(hj kj ; θt0 σ 0 0 0 measure. We observe that the calibrated parameters δˆ ∗t0 , ρˆ ∗t0 , and cˆt∗0 , and the goodness of fit measure vary in time and are quite erratic. The variation of the goodness of fit is due to a large extent to the small and time-varying number of derivative prices used in the calibration. 2.6.2. XMM Estimation In the XMM estimation, we consider the bivariate vector of state variables (2.17)
Xt = (˜rt σt2 )
9 Equations (5) and (6) in Carr and Madan (1999) are used to compute the option price as a function of time-to-maturity h and discounted moneyness B(t t + h)k (see Appendix B in the Supplemental Material). The term structure of risk-free bond prices is assumed to be exogeneous and is estimated at date t0 by cubic spline interpolation of market yields at available maturities. This motivates our preference for a specification with time-varying deterministic interest rate compared to one with constant interest rate. 10 It would also be possible to replace σt2 by the observed realized volatility and calibrate w.r.t. θ only. We have followed the standard approach, which provides smaller pricing errors.
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where r˜t := rt − rft is the daily logarithmic return from the S&P 500 index in excess of the risk-free rate and σt2 is an observable volatility factor. More specifically, σt2 is the one-scale realized volatility computed from 30-minute S&P 500 returns (e.g., Andersen, Bollerslev, Diebold, and Labys (2003)). The parameterized s.d.f. is exponential affine, (2.18)
2 − θ3 σt2 − θ4 r˜t+1 ) Mtt+1 (θ) = e−rft+1 exp(−θ1 − θ2 σt+1
where θ = (θ1 θ2 θ3 θ4 ) .11 This specification of the s.d.f. is justified by the fact that the corresponding semiparametric model for the risk-neutral distribution nests the parametric model used for cross-sectional calibration in Section 2.6.1 above. More precisely, the risk-neutral stochastic volatility model (2.15) and (2.16) can be derived from a historical stochastic volatility model of the same type and the exponential affine s.d.f. (2.18) with appropriate restrictions on parameter θ (see Section 3.2). For each trading day t0 of June 2005, we estimate by XMM the s.d.f. parameter θ = (θ1 θ2 θ3 θ4 ) and five option prices ct0 (h k) at a constant time-tomaturity h = 20 days and moneyness strikes k = 096 098 1 102 104. The options are puts for k < 1 and calls for k ≥ 1. The estimator is defined as in Section 2.5, using the current and previous T = 1000 daily historical observations on the state variables, and the derivative prices of the actively traded S&P 500 options at t0 . We use a product Gaussian kernel and select two bandwidths for the state variables according to the standard rule of thumb (Silverman (1986)). The estimation results for the first 10 trading days of June 2005 are displayed in Table II. A direct comparison of the estimated structural parameters given in Tables I and II is difficult, since the parameters in Table I concern the risk-neutral dynamics whereas those in Table II concern the s.d.f. For this reason, we focus in Sections 2.6.3 and 2.6.4 below on a direct comparison of estimated option prices. Nevertheless, the following two remarks are interesting. First, the XMM estimated parameters are much more stable over time than the calibrated parameters, since they use the same historical information on the underlying asset. Second, from the computational point of view, the calibration method is rather time-consuming since it requires the inversion of the Fourier transform for all option prices at each evaluation of the criterion function and its partial derivatives w.r.t. the parameters, and at each step of the optimization algorithm. On the contrary, the full XMM estimation takes about 1 minute on a standard computer. 11
The option prices at time-to-maturity h depend on the pattern of the deterministic short rate between t and t + h by means of the bond price B(t t + h) only. The term structure of bond prices is estimated from market yields as in Section 2.6.1.
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ESTIMATED S.D.F. PARAMETERS AND OPTION PRICES (XMM APPROACH) FOR THE S&P 500 OPTIONS IN JUNE 2005a Day t0
1.6.05 2.6.05 3.6.05 6.6.05 7.6.05 8.6.05 9.6.05 10.6.05 13.6.05 14.6.05
Option Prices (×10−2 )
s.d.f. Parameters θˆ 1 (×10−6 )
θˆ 2
θˆ 3
θˆ 4
k = 096
k = 098
k=1
k = 102
k = 104
0910 0897 0900 0892 0888 0891 0890 0878 0864 0851
−0059 −0057 −0054 −0056 −0056 −0063 −0063 −0063 −0063 −0063
0121 0118 0115 0118 0117 0124 0124 0125 0124 0125
0860 0860 0860 0860 0860 0860 0860 0860 0860 0860
0269 0215 0229 0155 0218 0164 0223 0110 0206 0250
0631 0584 0539 0496 0550 0531 0574 0495 0553 0588
1356 1342 1282 1270 1227 1288 1280 1326 1253 1295
0651 0686 0637 0622 0514 0628 0607 0701 0593 0580
0162 0204 0169 0210 0147 0186 0220 0233 0193 0133
a Estimated s.d.f. parameter θˆ and relative option prices cˆ (h k) at time-to-maturity h = 20 for the first 10 trading t0 days t0 of June 2005. The option prices correspond to puts for k < 1 and to calls for k ≥ 1. The estimation is performed using XMM. At each day t0 , the sample consists of the current and previous T = 1000 observations on the state variables, and the derivative prices at t0 of S&P 500 options with daily volume larger than 4000 contracts.
2.6.3. Static Comparison of Estimated Option Prices In Figure 1, we display the estimated relative prices of S&P 500 options on June 1 and June 2, 2005 as a function of the discounted moneyness strike k˜ = B(t t + h)k and time-to-maturity h. The results for June 1 (resp., June 2) are displayed in the upper (resp., lower) panels. In the right panels, the solid lines correspond to XMM estimates and in the left panels correspond to calibrated estimates. Circles correspond to the observed prices of highly traded options. On June 1, there are four highly traded times-to-maturity, which are 12, 57, 77, and 209 days. For the longest time-to-maturity h = 209, there is only one actively traded call option (resp., one put for h = 57 and two calls for h = 77). All remaining highly traded options correspond to time-to-maturity 12. When both the put and call options with identical moneyness strike and time-to-maturity are actively traded, we select the put if k˜ < 1 and select the call otherwise. This is compatible with the procedure suggested by Aït-Sahalia and Lo (1998). After applying this procedure, we end up with 11 highly traded options on June 1 and 8 highly traded options for the 3 times-to-maturity on June 2, 2005. As discussed in Section 2.2, the calibration approach may produce different values for observed and estimated option prices, while, by definition, these prices coincide within the XMM approach. Both calibration and XMM can also be used to price puts and calls that do not correspond to highly traded times-to-maturity and highly traded moneyness strikes. To show this, Figure 1 includes (dashed lines) the times-to-maturity 120 for June 1, 2005 and times-to-maturity 119 for June 2, 2005, which are not traded. The calibration method assumes a parametric risk-neutral model, whereas the XMM
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FIGURE 1.—Estimated call and put prices for S&P 500 options on June 1 and June 2, 2005. In the upper right panel, the solid lines correspond to estimated relative option prices as a function of discounted moneyness strike B(t t + h)k for the highly traded times-to-maturity h = 12 57 77 209 on June 1, 2005, obtained by XMM. The dashed line corresponds to XMM estimated prices for the nontraded time-to-maturity h = 120. The price curves correspond to puts if B(t t + h)k < 1 and to calls otherwise. In the upper left panel, the solid and dashed lines are the price curves obtained by the parametric pricing model (2.14) and (2.15) with the calibrated parameters in Table I for times-to-maturity h = 12 57 77 209 and h = 120, respectively. In both panels, circles correspond to observed S&P 500 option prices with daily trading volume larger than 4000 contracts. The two lower panels correspond to June 2, 2005 with highly traded times-to-maturity h = 11 31 208 and nontraded time-to-maturity h = 119.
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risk-neutral model is semiparametric. Any specification error in the fully parametric pricing model is detrimental for the calibration approach, while the XMM approach is more flexible. An advantage of XMM is that, by construction, the estimated derivative prices coincide with the market prices for highly traded options, while the calibrated prices differ from the observed prices, and these discrepancies can be large. This is not due to the specific parametric risk-neutral model that was used. It would also occur if a more complicated parametric model, or the nonparametric approach proposed in Aït-Sahalia and Lo (1998), were used. By construction, the specific risk-neutral stochastic volatility model that underlies the calibration method produces smooth symmetric option pricing functions w.r.t. the log moneyness (see Gouriéroux and Jasiak (2001, Chap. 13.1.5)), with similar types of curvature when the time-tomaturity increases. In contrast, the XMM produces option pricing functions with different skew according to the time-to-maturity, which means that the method captures the leverage effect and its term structure. 2.6.4. Dynamic Comparison of Estimated Option Prices Let us now consider the dynamics of the option pricing function. In Figure 2, we display the time series of Black–Scholes implied volatilities at a fixed timeto-maturity h = 20 and moneyness strikes k = 096 1 104 106 for all trading days in June 2005. The XMM implied volatilities are indicated by circles and the calibrated implied volatilities are marked by squares. The XMM implied volatilities are more stable over time because of the use of the historical information on the underlying asset. Since most of the highly traded options have moneyness strikes close to at-the-money, the calibration approach is rather sensitive to infrequently observed option prices with extreme strikes. The sample means of the two time series of implied volatilities in each panel of Figure 2 are different. In particular, for k = 096 and k = 1, the XMM implied volatilities are, on average, larger than the calibrated ones and smaller for k = 104 and k = 106. The reason is that the XMM approach captures the smirk, that is, the skewness, in the implied volatility curve, while the calibrated model can reproduce either a smile or a flat pattern only. 2.7. Discussion and Possible Extensions We have illustrated above how to implement the XMM estimator for derivative pricing in a two-factor model with exponential affine s.d.f. In practice, the econometrician has to select the set of observable underlying factors and the set of asset prices, including reliable derivative prices, which are used for defining the uniform and local moment restrictions. There exist arguments for introducing more factors. (i) For instance, if the risk-free interest rate is assumed to be stochastic, additional factors can be the risk-free rate, its realized volatility, and the realized covolatility between the index return and the risk-free rate. This would lead to a five-factor model.
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FIGURE 2.—Time series of implied volatilities of S&P 500 options in June 2005. In all panels, annualized implied volatilities at time-to-maturity h = 20 and moneyness strike k = 096 1 104 106, respectively, are displayed for each trading day in June 2005. Circles are implied volatilities computed from option prices estimated by the XMM approach; squares are implied volatilities from the cross-sectional calibration approach. The ticks on the horizontal axis correspond to Mondays.
(ii) If the stochastic volatility follows a Markov process of order q larger than 1, the first-order Markov assumption is recovered by introducing both the current 2 2 and the lagged volatilities σt2 and σt−1 σt−q+1 in the vector Xt . Pragmatic arguments such as tractability and robustness suggest limiting the number of factors, for example, to a number smaller than or equal to 3. The XMM approach can be easily extended to account for a finite number of cross sections of reliable observed derivative prices, for example, the last 10 trading days.12 This is achieved by including in criterion (2.9) a local KLIC component for each cross section. However, there is no a priori reason to overlook any reliable price data. Thus, we could propose to consider the prices of actively traded derivatives for all previous dates and not only for a finite number of dates. By using this additional information, one may expect to improve the convergence rate of the s.d.f. parameters that are not identifiable from the uniform restrictions (2.3) and possibly achieve the parametric convergence rate for more θ parameters, although not necessarily for all of them. For instance, when the s.d.f. involves risk premium parameters for extreme risk, the associated estimators could admit a smaller rate, since the derivatives with large 12
Or the 10 past trading days with the values of the state variable closest to x0 .
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strike are very infrequently traded. In any case, the estimated option prices themselves will still have a nonparametric rate of convergence since the pricing model is semiparametric. When an infinite number of local conditional moment restrictions corresponding to the sparse characteristics of traded derivatives are considered, we might the local moment restrictions to aggregate get the uniform restriction h∈H k∈K It (h k)E[Mtt+h (θ)(exp Rth − k)+ − ct (h k)|Xt ] = 0, where It (h k) is the activity indicator that is equal to 1 if option (h k) is actively traded at date t and is equal to 0 otherwise, and the sums are over some sets H of times-to-maturity and K of moneyness strikes. The analysis of the statistical properties of the associated estimators will heavily depend on the choice of sets H and K, and on the assumptions concerning the trading activity. A model for trading activity would have to account for the effect of periodic issuing of options by the CBOE, the observed activity clustering, the activity due to dynamic arbitrage strategies applied by some investors, and the activity when some investors use the options as standard insurance products. The introduction of a realistic trading activity model would lead to a joint estimation of pricing and activity parameters, provide estimated option prices, and allow for the prediction of future activity. Accounting for the option data at all dates is an important extension of our work which is left for future research. Let us finally discuss the central question of identification. We pointed out in Section 2.4 that some s.d.f. parameters may be unidentifiable when the uniform moment restrictions (2.3) for the underlying asset and the risk-free asset only are considered. Intuitively, a lack of identification has to be expected whenever the returns of the underlying asset and the risk-free asset do not span the set of state variables, and there are at least as many unknown s.d.f. parameters as state variables. In such a framework, there may exist some directions of change in parameter θ that produce a change in the s.d.f. Mtt+1 (θ) orthogonal to the underlying asset and risk-free asset gross returns. In Section 3.2, we formalize this intuitive argument for the specific stochastic volatility data generating process (DGP) used in the application and prove lack of identification from uniform moment conditions. It could be proposed to introduce parameter restrictions to solve the identification problem. For instance, the restriction θ3 = 0 in the s.d.f. (2.18) yields identification of the full s.d.f. parameter vector from the uniform moment conditions (2.3). From the analysis in Section 3.2, such a restriction is equivalent to imposing an a priori level of the risk premium parameter θ2 for stochastic volatility. In this case, the local moment restrictions from the derivative prices are noninformative for the estimation of θ, but still contribute to efficient estimation of the derivative prices. While imposing identifying restrictions simplifies the estimation problem, such an approach has some limitations. First, the degree and the directions of underidentification are not known a priori since they depend on the unknown DGP. Second, an ad hoc parametric restriction is likely to be misspecified and can lead to mispricing. Under the maintained
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hypothesis that the researcher knows the correct degree of underidentification, the validity of the selected identifying restrictions can be tested by a standard specification test for conditional moment restrictions. The difficulties in selecting the correct identifying restrictions are overcome by the XMM estimator, which is robust to the lack of identification from the uniform moment restrictions. 3. THEORETICAL PROPERTIES OF XMM Let us now examine the theoretical properties of the XMM estimator introduced in Section 2 for derivative pricing. First we discuss the parameter of interest and the moment restrictions. Next we derive the identification conditions and explain how they differ from the standard GMM conditions. Then we introduce instruments, define a convenient notion of efficiency called kernel nonparametric efficiency, and derive the optimal instruments. Finally, we prove the kernel nonparametric efficiency of the XMM estimator. 3.1. The Parameter of Interest Let us consider a semiparametric estimation of the conditional moments (3.1)
E0 [a(Y ; θ0 )|X = x0 ]
subject to the uniform and local conditional moment restrictions (3.2)
E0 [g(Y ; θ0 )|X = x] = 0 ∀x ∈ X P0 -a.s.
(3.3)
˜ ; θ0 )|X = x0 ] = 0 E0 [g(Y
where θ0 is the true parameter value and E0 [·] denotes the expectation under the true DGP P0 . The unknown parameter value θ0 is in set Θ ⊂ Rp , variables ˜ (X Y ) are in X × Y ⊂ Rd × Rd , and x0 is a given value in X . The moment functions g and g˜ are given vector-valued functions on Y × Θ. Function a on Y × Θ has dimension L. In the derivative pricing application in Section 2, vector a(Y ; θ) is the product of s.d.f. Mtt+h (θ) and payoff (exp Rth − k)+ on the L derivatives of interest. Vector functions g and g˜ define the uniform moment restrictions on the risk-free asset and underlying asset, and the local moment restrictions on observed derivative prices (see (2.5) and (2.6)). The conditional moment of interest (3.1) can be viewed as an additional parameter β0 in B ⊂ RL identified by the local conditional moment restrictions (3.4)
E0 [a(Y ; θ0 ) − β0 |X = x0 ] = 0
Thus, all parameters to be estimated are in the extended vector θ0∗ = (θ0 β0 ) that satisfies the extended set of uniform and local moment restrictions (3.2),
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(3.3), and (3.4) (see Back and Brown (1992) for a similar extension of the parameter vector). In this interpretation, β0 is the parameter of interest, while θ0 is a nuisance parameter. 3.2. Identification Let us now consider the semiparametric identification of extended parameter θ0∗ . From moment restriction (3.4), it follows that β0 is identified if θ0 is identified. Thus, we can restrict the analysis to the identification of θ0 . 3.2.1. The Identification Assumption ASSUMPTION 1: The true value of parameter θ0 is globally semiparametrically identified: E0 [g(Y ; θ)|X = x] = 0
∀x ∈ X P0 -a.s.
˜ ; θ)|X = x0 ] = 0 E0 [g(Y
θ∈Θ
⇒
θ = θ0
ASSUMPTION 2: The true value of parameter θ0 is locally semiparametrically identified:
∂g E0 (Y ; θ0 ) X = x α = 0 ∂θ α ∈ Rp ∀x ∈ X P0 -a.s.
∂g˜ (Y ; θ0 ) X = x0 α = 0 E0 ∂θ
⇒
α = 0
We need to distinguish the linear transformations of θ0 (α θ0 , say, where α ∈ Rp ) that are identifiable from the uniform moment restrictions (3.2) alone from the linear transformations of θ0 that are identifiable only from both uniform and local moment restrictions (3.2) and (3.3). The former transformations are called full-information identifiable, while the latter are called fullinformation unidentifiable. Let us consider the linear space
∂g ∗ p J = α ∈ R : E0 (Y ; θ0 ) X = x α = 0 ∀x ∈ X P0 -a.s. ∂θ of dimension s∗ , say, 0 ≤ s∗ ≤ p. The full-information identified transformations are α θ0 with α ∈ (J ∗ )⊥ , while the full-information unidentifiable transformations are α θ0 with α ∈ Rp \ (J ∗ )⊥ . There exist parameterizations of the moment functions such that p − s∗ components of the parameter vector are
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full-information identifiable and s∗ components are full-information unidentifiable. Indeed, let us consider a linear change of parameter ∗ R1 θ η1 (3.5) = η∗ = η∗2 R2 θ where R∗ = [R1 R2 ] is an orthogonal (p p) matrix, R1 is a (p p − s∗ ) matrix whose columns span (J ∗ )⊥ , and R2 is a (p s∗ ) matrix whose columns span J ∗ . ∗ The matrices R1 and R2 depend on the DGP P0 . The subvectors η∗1 ∈ Rp−s and ∗ η∗2 ∈ Rs are full-information identifiable and full-information underidentified, ∂g˜ respectively. Assumption 2 is equivalent to matrix E0 [ ∂θ (Y ; θ0 )|X = x0 ]R2 having full column rank. Thus, a necessary order condition for local identification is that the number of local moment restrictions is larger than or equal to s∗ , that is, the dimension of the linear space J ∗ characterizing the full-information unidentifiable parameters. The standard GMM considers only uniform moment restrictions and assumes full-information identification for the full vector θ0 under the DGP P0 . To illustrate the difference with our setting, let us assume that the DGP P0 is in the parametric stochastic volatility model which is compatible with the parametric risk-neutral specification of Section 2.6.1 and the semiparametric model of Section 2.6.2. Specifically, the true historical distribution of Xt = (˜rt σt2 ) is such that (3.6)
r˜t = γ0 σt2 + σt εt
where r˜t = rt − rft is the underlying asset return in excess of the deterministic risk-free rate, the shocks (εt ) are IIN(0 1), and the volatility (σt2 ) follows an ARG process independent of (εt ) with parameters ρ0 ∈ [0 1), δ0 > 0, and c0 > 0. The transition of (σt2 ) is characterized by the conditional Laplace 2 transform E0 [exp(−uσt+1 )|σt2 ] = exp[−a0 (u)σt2 − b0 (u)], u ≥ 0, where a0 (u) = ρ0 u/(1 + c0 u) and b0 (u) = δ0 log(1 + c0 u). The s.d.f. is given by (3.7)
2 Mtt+1 (θ) = e−rft+1 exp(−θ1 − θ2 σt+1 − θ3 σt2 − θ4 r˜t+1 )
with true parameter value θ0 = (θ10 θ20 θ30 θ40 ) . Then the true risk-neutral distribution Q is: (3.8)
1 r˜t = − σt2 + σt εt∗ 2
where εt∗ ∼ IIN(0 1) and (σt2 ) follows an ARG process independent of (εt∗ ) with parameters ρ∗0 , δ∗0 , and c0∗ , that are functions of ρ0 , δ0 , c0 , and θ0 given in (A.24). The no-arbitrage restrictions (2.3) written under the DGP P0 are satisfied for the true value of the s.d.f. parameter θ0 if and only if (see Appendix A.3.1) (3.9)
θ10 = −b0 (λ02 )
θ30 = −a0 (λ02 )
θ40 = γ0 + 1/2
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where λ02 = θ20 + γ02 /2 − 1/8. The restrictions on θ10 and θ30 fix the predetermined component of the s.d.f., while the last equality in (3.9) relates θ40 with the volatility-in-mean parameter γ0 . Parameter θ20 is unrestricted. The econometrician adopts a semiparametric framework with parametric s.d.f. (3.7), and uses the uniform and local moment restrictions (2.2) and (2.3) to identify θ0 . From equations (3.9), parameter θ40 is full-information identified, while parameters θ10 , θ20 , and θ30 are full-information unidentifiable. Thus, the uniform moment restrictions (2.3) on the risk-free asset and the underlying asset are insufficient to identify the full parameter vector θ0 . The linear space defining the full-information identifiable transformations is characterized next. PROPOSITION 1: When the DGP P0 is compatible with (3.6)–(3.8), the following conditions hold: (i) The full-information identifiable transformations are θ0 α with α ∈ (J ∗ )⊥ , where
exp rft+1 Q ∗ 4 ξt+1 αXt = x = 0 ∀x ∈ X J = α ∈ R : E0 exp rt+1 2 σt2 r˜t+1 ) . The linear space J ∗ has dimension s∗ = 1 and is and ξt+1 = (1 σt+1 δ0 c0 0 0 (λ02 ) 1 − da (λ02 ) 0) = (− 1+c 1 − (1+cρ0λ0 )2 0) . spanned by vector r2 = (− db du du λ0 0 2
0 2
˜ ; θ0 )/∂θ |X = x0 ]r2 are given by (ii) The elements of the (n 1) vector E0 [∂g(Y 2 2 |Xt = x0 (1 − ρ∗0 ) CovQ0 σtt+h BS kj σtt+h j j 2 2 |Xt = x0 + ρ∗0 CovQ0 σt+h BS kj σtt+h j j
2 2 2 = σt+1 + · · · + σt+h is the integrated volatility between for j = 1 n, where σtt+h 2 t and t + h, and BS(k σ ) is the Black–Scholes price of a call option with time-tomaturity 1, moneyness strike k, and volatility σ 2 . Hence, the true value of the s.d.f. parameter θ0 is locally semiparametrically identified by the conditional moment restrictions (2.2) and (2.3).
The characterization of linear space J ∗ in Proposition 1(i) formalizes the intuitive spanning argument given in Section 2.7 to explain the lack of iden 2 2 r2 = σt+1 − E0Q [σt+1 |σt2 ] (see (A.27)), the linear combinatification. Since ξt+1 tion of state variables that is unspanned by the returns of the underlying asset and the risk-free asset is the risk-neutral unexpected volatility. The dimension of the subspace of vectors α ∈ R4 associated with the full-information identifiable transformations α θ0 is 3, while a full-information unidentifiable transformation is η∗2 = R2 θ0 , where R2 = r2 /r2 . Since the Black–Scholes price is an increasing function of volatility and the integrated volatility is stochastically increasing w.r.t. the spot volatility under Q conditionally on
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Xt = x0 (see Lemma A.4 in Appendix A.3), from Proposition 1(ii), the vec∂g˜ tor E0 [ ∂θ (Y ; θ0 )|X = x0 ]R2 has strictly positive elements. Thus, the full vector of parameters θ becomes identifiable when the local moment restrictions (2.2) from the observed derivative prices are taken into account. 3.2.2. Admissible Instrumental Variables It is also useful to discuss a weaker notion of identification, based on a given matrix of instruments Z = H(X). The uniform conditional moment restrictions (3.2) imply the unconditional moment restrictions E0 [Z · g(Y ; θ0 )] =: E0 [g1 (X Y ; θ0 )] = 0. Let us denote the whole set of local conditional moment restrictions at x0 as E0 [g2 (Y ; θ0 )|X = x0 ] = 0, where g2 = (g g˜ ) . Thus, parameter θ0∗ = (θ0 β0 ) satisfies the moment restrictions (3.10)
E0 [g1 (X Y ; θ0 )] = 0
(3.11)
E0 [g2 (Y ; θ0 )|X = x0 ] = 0
(3.12)
E0 [a(Y ; θ0 ) − β0 |X = x0 ] = 0
DEFINITION 3: The instrument Z is admissible if the true value of the parameter θ0 is globally semiparametrically identified by the moment restrictions (3.10) and (3.11), and is locally semiparametrically identified by their differential counterparts. Let us introduce the linear change of parameter η1 R1Z θ (3.13) η = = η2 R2Z θ where R = [R1Z R2Z ] is an orthogonal (p p) matrix and R2Z is a (p sZ ) matrix 1 whose columns span the null space JZ = Ker E0 [ ∂g (X Y ; θ0 )]. The subvector ∂θ of parameters η1 is locally identified by the unconditional moment restrictions (3.10), whereas the subvector of parameters η2 is identifiable only from both sets of restrictions (3.10) and (3.11). 3.3. Kernel Moment Estimators Let Z be a given admissible instrument. Let us introduce a GMM-type estimator of θ0∗ that is obtained by minimizing a quadratic form of sample counterparts of moments (3.10)–(3.12). The kernel density estimator fˆ(y|x) in (2.8) is used to estimate the conditional moments in (3.11) and (3.12), 2 (Y ; θ)|x0 ] := g2 (y; θ)fˆ(y|x0 ) dy E[g
T t=1
xt − x0 g2 (yt ; θ)K hT
T xt − x0 K hT t=1
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and similarly for E0 [a(Y ; θ) − β|X = x0 ]. T ) of parameter θ0∗ = θT∗ = ( θT β DEFINITION 4: A kernel moment estimator (θ0 β0 ) based on instrument Z is defined by θT∗ =
arg min
θ∗ =(θ β ) ∈Θ×B
where gT (θ∗ ) =
√
QT (θ∗ )
QT (θ∗ ) = gT (θ∗ ) Ω gT (θ∗ )
2 (Y ; θ)|x0 ] 1 (X Y ; θ)] T hdT E[g T E[g
; θ) − β|x0 ] T hdT E[a(Y
and E[·|x 0 ] denote a historical sample average and a kernel estimator of E[·] the conditional moment, respectively, and Ω is a positive definite weighting matrix. The empirical moments in gT (θ∗ ) have different rates of convergence that are parametric and nonparametric. This explains why the asymptotic analysis is different from the standard GMM. To derive the large sample properties of the kernel moment estimator θT∗ , we prove the weak convergence of a suitable empirical process derived from sample moments gT (θ∗ ) (see Lemma A.1 in Appendix A.1.2). The definition of the empirical process is in the spirit of Stock and Wright (2000), but the choice of weak instruments is different.13 The various rates of convergence in gT (θ∗ ) can be disentangled by using the trans formed parameters (η1 η2 ) defined in (3.13). PROPOSITION 2: Under Assumptions A.1–A.24 in Appendix A.1, the kernel moment estimator θˆ T∗ based on instrument Z is consistent and asymptotically normal, ⎛ √T ( η1T − η10 ) ⎞ d ⎝ T hdT ( η2T − η20 ) ⎠ −→ N(B∞ (J0 ΩJ0 )−1 J0 ΩV0 ΩJ0 (J0 ΩJ0 )−1 ) T − β0 ) T hdT (β as T → ∞, where (η10 η20 ) is the true value of the transformed structural pa√ ¯ 0 ΩJ0 )−1 J0 Ωb0 , with c¯ := lim T hd+2m rameter, the bias is B∞ = − c(J ∈ [0 ∞) T 13 The local moment restrictions (3.11) can be approximately written as E0 [g2 (Y ; θ0 )|X = x0 ] 0 )/[hdT fX (x0 )] and fX denotes the unconditional p.d.f. E0 [ZT g2 (Y ; θ0 )] = 0, where ZT = K( X−x hT of X. The “instrument” ZT is weak in the sense of Stock and Wright (2000). However, it depends on T , and the rates of convergence of the estimators differ from the rates of convergence in Stock and Wright (2000).
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and m ≥ 2 the order of differentiability of the p.d.f. fX of X, and matrices J0 and V0 , and vector b0 are given in (A.2) and (A.4) in Appendix A.1, and depend on kernel K and instrument Z. 3.4. Kernel Nonparametric Efficiency Bound The class of kernel moment estimators helps us define a convenient notion of efficiency for estimating β0 = E0 (a|x0 ) := E0 [a(Y ; θ0 )|X = x0 ]. Let us consider a scalar parameter β0 and derive the optimal weighting matrix, admissible instruments and bandwidth. We assume that kernel K is a product kernel d K(u) = i=1 κ(ui ) of order m. 3.4.1. Efficiency Bound With Optimal Rate of Convergence Let us consider a bandwidth sequence hT = cT −1/(2m+d) , where c > 0 is a constant. From Proposition 2, it follows that estimator βˆ T achieves the optimal d-dimensional nonparametric rate of convergence T −m/(2m+d) , and its asymptotic mean squared error (MSE) constant M(Ω Z c a) > 0 depends on the weighting matrix Ω, instrument Z, bandwidth constant c, and moment function a. DEFINITION 5: The kernel nonparametric efficiency bound M(x0 a) for estimating β0 = E0 (a|x0 ) is the smallest possible value of M(Ω Z c a) that corresponds to the optimal choice of weighting matrix Ω, admissible instrument Z, and bandwidth constant c. PROPOSITION 3: Let Assumptions 1, 2, and A.1–A.25 in Appendix A.1 hold. (i) There exist an optimal weighting matrix Ω∗ (a) and an optimal bandwidth constant c ∗ (a). They are given in (A.9) and (A.10) in Appendix A.1.6. (ii) Any instrument ∂g ∗ (Y ; θ0 ) X W (X) (3.14) Z = E0 ∂θ where W (X) is a positive definite matrix, P0 almost surely (a.s.), is optimal, independent of a. It is easily verified that JZ ∗ = J ∗ for any instrument Z ∗ in (3.14). COROLLARY 4: An admissible instrument Z is optimal if and only if the corresponding unconditional moment restrictions (3.10) identify all full-information identifiable parameters. The set of optimal instruments is independent of the selected kernel.
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Since we focus on the estimation of local conditional moment β0 , the set of optimal instruments is larger than the standard set of instruments for efficient estimation of a structural parameter θ0 identified by (3.2). While in the standard framework, W (X) = V0 [g(Y ; θ0 )|X]−1 is the efficient weighting matrix for conditionally heteroskedastic moment restrictions (e.g., Chamberlain (1987)), any choice of a positive definite matrix W (X) is asymptotically equivalent for estimating β0 .14 The expression of the kernel nonparametric efficiency bound is easily written in terms of the transformed parameters (η∗1 η∗2 ) defined in (3.5). PROPOSITION 5: Under Assumptions 1, 2 and A.1–A.25 in Appendix A.1, the kernel non-parametric efficiency bound M(a x0 ) is given by the lower right element of the matrix −1 −1 1 w2 ∗ ∗2m 2 J0 Σ0 + c wm b(x0 )b(x0 ) (3.15) J0∗ c ∗d fX (x0 ) where w2 = Rd K(u)2 du, wm = R vm κ(v) dv, c ∗ = c ∗ (a) is the optimal bandwidth constant given in (A.9) in Appendix A.1.6, ⎞ ⎛ ∂g2 E0 0 x 0 ⎟ ⎜ ∂η∗2 ⎟ J0∗ = ⎜ ⎠ ⎝ ∂a −1 E0 x0 ∗ ∂η2 V0 (g2 |x0 ) Cov0 (g2 a|x0 ) Σ0 = V0 (a|x0 ) Cov0 (a g2 |x0 ) and
m 1 1 Δ (E0 (g2 |x)fX (x)) − E0 (g2 |x)Δm fX (x) b(x) = m! fX (x) Δm (E0 (a|x)fX (x)) − E0 (a|x)Δm fX (x) d with Δm := i=1 ∂m /∂xm i and all functions evaluated at θ0 .
The matrix in (3.15) resembles the GMM efficiency bound for estimating parameters (η∗2 β) from orthogonality conditions based on function (g2 a − β) with η∗1 known. Since moment restrictions (3.11) and (3.12) are conditional on X = x0 , the variance of the orthogonality conditions is replaced by the asymp1 w2 2 totic MSE matrix c∗d Σ + c ∗2m wm b(x0 )b(x0 ) of the kernel regression estifX (x0 ) 0 mator at x0 (up to a scale factor), and the expectations in the Jacobian matrix 14 In the application to derivative pricing, the state variables process is Markov. The instrument in (3.14) is optimal not only in the class of instruments written on Xt , but also in the class of instruments Zt = H(Xt Xt−1 ) written on the current and lagged values of the state variables, since the past is not informative once the present is taken into account.
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J0∗ are conditional on X = x0 . In particular, the efficiency bound depends on the likelihood of observing the conditioning variable close to x0 by means of fX (x0 ). The estimation of the full-information identified parameter η∗1 is irrelevant for the efficiency bound of β since the estimation of η∗1 achieves a faster parametric rate of convergence. 3.4.2. Bias-Free Kernel Nonparametric Efficiency Bound If we restrict ourselves to asymptotically unbiased estimators for practical purposes, the bandwidth sequence has to be such that c¯ = lim T hT2m+d = 0. T − β0 ) can When c¯ = 0 in Proposition 2, the asymptotic variance of T hdT (β 2 be written as w V (Z Ω a), where V (Z Ω a) is independent of the selected kernel K and bandwidth hT . DEFINITION 6: The bias-free kernel nonparametric efficiency bound B (a x0 ) is the smallest asymptotic variance V (Z Ω a) corresponding to the optimal choice of the weighting matrix Ω and of the instrument Z. COROLLARY 6: (i) There exists an optimal choice of the weighting matrix Ω and of the instruments Z. The optimal instruments are given in (3.14) and the optimal weighting matrix is given in Appendix A.1.7. (ii) The bias-free kernel nonparametric efficiency bound B (x0 a) is 1 V0 (a) − Cov0 (a g2 )V0 (g2 )−1 Cov0 (g2 a) B (x0 a) = fX (x0 )
∂a ∂g2 −1 + E0 R2 − Cov0 (a g2 )V0 (g2 ) E0 R2 ∂θ ∂θ
−1 ∂g2 ∂g2 × R2 E0 R2 V0 (g2 )−1 E0 ∂θ ∂θ
∂a ∂g2 − R2 E0 V0 (g2 )−1 Cov0 (g2 a) × R2 E0 ∂θ ∂θ where all moments are conditional on X = x0 and evaluated at θ0 . Since the expression of B (x0 a) is a quadratic function of a, this formula holds for vector moment functions a as well. When the parameter θ0 itself is full-information identifiable, the bias-free kernel nonparametric efficiency bound becomes (3.16)
B (x0 a) =
1 V0 (a|x0 ) fX (x0 )
− Cov0 (a g2 |x0 )V0 (g2 |x0 )−1 Cov0 (g2 a|x0 )
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Since the conditional moment of interest is also equal to E0 (a|x0 ) = E0 [a(Y ; θ0 ) − Cov0 (a g2 |x0 )V0 (g2 |x0 )−1 g2 (Y ; θ0 )|x0 ], the bound (3.16) is simply the variance–covariance matrix of the residual term in the affine regression of a on g2 performed conditional on x0 . A similar interpretation was already given by Back and Brown (1993) in an unconditional setting (see also Brown and Newey (1998)) and extended to a conditional framework by Antoine, Bonnal, and Renault (2007). In the general case, the efficiency bound B (x0 a) balances the gain in information from the local conditional moment restrictions and the efficiency cost due to full-information underidentification of θ0 . 3.4.3. Illustration With S&P 500 Options Let us derive the bias-free kernel nonparametric efficiency bounds for derivative price estimation when the DGP P0 is the stochastic volatility model (3.6) and (3.7) with parameters given by γ0 = 0360, ρ0 = 0960, δ0 = 1047, c0 = 365 × 10−6 , θ10 = 0460 × 10−6 , θ20 = −0060, θ30 = 0115, and θ40 = 0860. The risk-free term structure is as in Sections 2.6.1 and 2.6.2. The ARG parameters ρ0 , δ0 , and c0 are set to match the stationary mean, variance, and first-order autocorrelation of the realized volatility σt2 of the S&P 500 index in the period from June 1, 2001 to May 31, 2005. The risk premia parameters θ20 and θ40 for stochastic volatility and underlying asset return, respectively, correspond to the XMM estimates obtained in Section 2.6.2 for the S&P 500 options on June 1, 2005 (see Table II). Parameters γ0 , θ10 , and θ30 are then fixed by the no-arbitrage restrictions (3.9). At a current date t0 , the prices of n = 11 actively traded call options are observed with the same times-to-maturity and moneyness strikes as the S&P 500 put and call options with daily traded volume larger than 4000 contracts on June 1, 2005 (see Section 2.6.3). The current values of the state variables are the return and the realized volatility of the S&P 500 index on June 1, 2005. Let us focus on the time-to-maturity h = 77 days. The bias-free kernel nonparametric efficiency bound on the call option prices is displayed in Figure 3. The dashed line represents the theoretical call prices E(a(k)|xt0 ) computed under the DGP P0 and the dashed lines represents 95% asymptotic confidence intervals E(a(k)|xt0 ) ± 196 √w 2 B (xt0 k)1/2 as a function of moneyness k. The T hT
circles indicate the theoretical prices of the observed derivatives at time-tomaturity 77 days. To better visualize the pattern of the kernel nonparamet ric efficiency bound as a function of the moneyness strike, w2 /T h2T is set 10 times larger than the value implied by the sample size and the bandwidths used in the empirical application.15 The width of the confidence interval for derivative price E(a(k)|xt0 ) depends on moneyness strike k. This width is zero when 15 For expository purposes we consider symmetric confidence bands. These bands have to be truncated at zero to account for the positivity of derivative prices and get asymmetric bands. However, with the correct standardization, the truncation effect is negligible and arises only for large strikes. Moreover, the bands in Figure 3 are pointwise bands. Corollary 6 can be used to get
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FIGURE 3.—Bias-free kernel nonparametric efficiency bound on June 1, 2005 time-to-maturity 77. The dashed line corresponds to the relative call price E(a(k)|xt0 ) at time-to-maturity h = 77 and the solid lines correspond to pointwise 95% symmetric confidence intervals
E(a(k)|xt0 ) ± 196 √w 2 B(xt0 k)1/2 . The value of T hT
w2 /T h2T is set 10 times larger than in the
empirical application.
k corresponds to the moneyness strikes of the two observed calls at time-tomaturity h = 77. The width of the confidence interval is close to zero when the derivative is deep in-the-money or deep out-of-the-money. Indeed, for moneyness strikes approaching zero or infinity, the kernel nonparametric efficiency bound goes to zero, since the option price has to be equal to the underlying asset price or equal to zero, respectively, by the no-arbitrage condition.16 The confidence intervals in Figure 3 are generally larger for moneyness values k < 1 compared to k > 1. This is due to the different informational content of the set of observed option prices for the different moneyness strikes at the timeto-maturity of interest. Finally note that when the confidence intervals and the bid–ask intervals do not intersect, some mispricing by the intermediaries or the model exists. 3.5. Asymptotic Normality and Efficiency of the XMM Estimator When the optimal instruments and optimal weighting matrix are used, the kernel moment estimator of β0 in Definition 4 is kernel nonparametrically efellipsoidal confidence sets for joint estimation of several derivative prices. It could also be possible to derive functional confidence sets for the entire pricing schedule as a function of moneyness strike k and for given time-to-maturity h, by considering cˆt0 (h k) as a stochastic process indexed by k. These developments are beyond the scope of this paper. 16 However, the relative accuracy can be poor in these moneyness regions.
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ficient. However, in application to derivative pricing, this estimator does not ensure a positive estimated state price density. This is a consequence of the quadratic GMM-type nature of the kernel moment estimators. The positivity of the estimated state price density is achieved by considering the informationbased XMM estimator defined in Section 2.5. In the general setting, the XMM estimator of β0 is ˆ fˆ∗ (y|x0 ) dy Eˆ ∗ (a|x0 ) = a(y; θ) where θˆ is defined in (2.11) and (2.12), and fˆ∗ (y|x0 ) is defined in (2.13). The large sample properties of the XMM estimator of β0 are given in Proposition 7 for a bandwidth sequence that eliminates the asymptotic bias. d+2m PROPOSITION 7: Suppose the bandwidth is such that c¯ = lim = 0. Then T hT d ∗ the XMM estimator Eˆ (a|x0 ) is consistent, converges at rate T hT , is asymptotically normal, and is bias-free kernel nonparametrically efficient: T hdT ˆ ∗ d (E (a|x0 ) − E0 (a|x0 )) → N(0 B (x0 a)) w
The XMM estimator of β0 is asymptotically equivalent to the best kernel moment estimator of β0 with optimal instruments and weighting matrix. By considering the constrained optimization of an information criterion (see Definition 1), the XMM estimator can be computed without making explicit the optimal instruments and weighting matrix. The asymptotic distribution of the XMM estimator of θ0 is given for the transformed parameter (η∗1 η∗2 ) in (3.5). COROLLARY 8: Suppose the bandwidth is such that c¯ = lim T hd+2m = 0. The T XMM estimators ηˆ ∗1 and ηˆ ∗2 are asymptotically equivalent to the kernel moment estimators with optimal weighting matrix and instrument Z as in (3.14) with W (X) = V0 [g(Y ; θ0 )|X]−1 . In particular, the rates of convergence for the full-information identifiable √ parameter η∗1 and the full-information unidentifiable parameter η∗2 are T and T hdT , respectively. The different rates of convergence of full-information identifiable and full-information unidentifiable parameters are reflected in the empirical results on the S&P 500 option data displayed in Table II in Section 2.6. If the DGP P0 is the stochastic volatility model (3.6) and (3.7), then parameter θ40 is full-information identifiable (see Section 3.2) and its estimates are very stable over time, whereas the estimates of the full-information unidentifiable parameters θ10 , θ20 , and θ30 feature a larger time variability. Proposition 7 and Corollary 8 extend the first-order asymptotic equivalence between
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information-based and quadratic GMM estimators (see Kitamura, Tripathi, and Ahn (2004), Kitamura (2007)) to a setting that includes local conditional moment restrictions and allows for full-information unidentifiable parameters. 4. CONCLUDING REMARKS The literature on joint estimation of historical and risk-neutral parameters is generally based on either maximum likelihood- (ML) or GMM-type methods. A part of this literature relies on uniform moment restrictions from a time series of spot prices only and implicitly assumes that the risk premia parameters are full-information identified (e.g., Bansal and Viswanathan (1993), Hansen and Jagannathan (1997), Stock and Wright (2000)). This assumption is not valid when some risk premia parameters can be identified only from option data. Another part of the literature exploits time series of both spot and option prices (e.g., Duan (1994), Chernov and Ghysels (2000), Pan (2002), Eraker (2004)). However, the activity on derivative markets is rather weak, and these approaches typically rely either on artificial option series, which are approximately near-the-money and at short time-to-maturity, or on option quotes of both actively and less actively traded options, or on ad hoc assumptions on characteristics of time-varying options. In this paper, we introduce a new XMM estimator of derivative prices using jointly a time series of spot returns and a few cross sections of derivative prices. We argue that these two types of data imply different types of conditional moment restrictions that are uniform and local, respectively. First, the XMM approach allows for consistent estimation of the s.d.f. parameters θ even if they are full-information unidentifiable. Second, the XMM estimator of the pricing operator at a given date is consistent for a fixed number of cross sectionally observed derivative prices. Third, the XMM estimator is asymptotically efficient. These results are due to both the parametric s.d.f. and the deterministic relationships between derivative prices that hold in a no-arbitrage pricing model with a finite number of state variables. The application to the S&P 500 options shows that the new XMM-based calibration approach outperforms the traditional cross-sectional calibration approach while being easy to implement. In particular, the XMM estimated option prices are compatible with the observed option prices for highly traded derivatives and are more stable over time. Finally, the asymptotic results that have been derived for the XMM estimator can be used to develop test procedures. Tests of correct specification of the parametric s.d.f. can be based on the minimized XMM objective function. Overidentification tests could also be defined by increasing the number of local restrictions, that is, the number of observed option prices in the application to derivative pricing, and comparing the XMM estimators with and without additional local restrictions. This would extend the standard overidentification tests introduced by Hansen (1982) and Szroeter (1983) for full-information identifiable parameters.
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The XMM approach can be applied to other financial markets with heterogenous trading activity. The T-bond market, for instance, has features similar to the index derivatives market: regular issuing of standardized products and small number of times-to-maturity which are highly traded daily. The XMM approach allows for a first correction of alternative estimation and pricing methods that assume a high activity for all assets proposed on the markets. It will have to be completed by a model that describes the activity on financial markets, which is a challenging topic for future research. APPENDIX A: PROOFS In this appendix, we give the proofs of Propositions 2, 3, and 5 (Section A.1), Proposition 7 (Section A.2), and Proposition 1, as well as of other results of Section 3.2 concerning the parametric stochastic volatility model (Section A.3). A.1. Asymptotic Properties of Kernel Moment Estimators We use the following notation. The symbol ⇒ denotes weak convergence in the space of bounded real functions on set Θ ⊂ Rp , equipped with the uniform metric (see, e.g., Andrews (1994)). The Frobenius norm of matrix A is 1/2 α = (α1 αd ) ∈ Nd and vector x ∈ Rd , A = [Tr(AA )]d Forα a multiindex α α α1 α we set |α| := i=1 αi , x := x1 · · · · · xd d , and ∂|α| f/∂xα := ∂|α| f/∂x1 1 · · · ∂xd d Symbol f ∞ denotes the sup-norm f ∞ = supx∈X f (x) of a continuous function f defined on set X We denote by C m (X ) the space of functions f on X that are continuously differentiable up to order m ∈ N, Dm f ∞ := d m |α| α m m 2 |α|=m ∂ f/∂x ∞ , and Δ f := i=1 ∂ f/∂xi . Furthermore, L (FY ) denotes the Hilbert space of real-valued functions, which are square integrable w.r.t. the distribution FY of random variable (r.v.) Y , and · L2 (FY ) is the corresponding L2 -norm. Linear space Lp (X ), p > 0, of p-integrable functions w.r.t. Lebesgue measure on set X is defined similarly. We denote by g2∗ the function defined by g2∗ (y; θ) = (g2 (y; θ) a(y; θ) ) . Finally, all functions of θ are evaluated at θ0 , when the argument is not explicit, and the expectation E[·] is w.r.t. the DGP P0 . A.1.1. Regularity Assumptions Let us introduce the following set of regularity conditions: ASSUMPTION A.1: The instrument Z is given by Z = H(X), where H is a matrix function defined on X and is continuous at x = x0 ASSUMPTION A.2: The true value of the parameter θ0 ∈ Rp is globally identified with instrument Z, that is, E[g1 (X Y ; θ)] E[g2 (Y ; θ)|X = x0 ] = 0 θ ∈ Θ ⇒ θ = θ0
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where g1 (X Y ; θ) = Z · g(Y ; θ). ASSUMPTION A.3: The true value of the parameter θ0 is locally identified with instrument Z, that is, the matrix ⎞
⎛ ∂g1 ⎜ E ∂θ (X Y ; θ0 ) ⎟ ⎜ ⎟ ⎝ ⎠ ∂g2 E (Y ; θ0 ) X = x0 ∂θ has full column rank. ASSUMPTION A.4: The parameter sets Θ ⊂ Rp and B ⊂ RL are compact and the true parameter θ0∗ = (θ0 β0 ) is in the interior of Θ × B, where β0 = E[a(Y ; θ0 )|X = x0 ]. ˜
ASSUMPTION A.5: The process {(Xt Yt ) : t ∈ N} on X × Y ⊂ Rd ×Rd is strictly stationary and geometrically strongly mixing. ASSUMPTION A.6: The stationary density fX of Xt is in class C m (X ) for some m ∈ N, m ≥ 2, and is such that fX ∞ < ∞ and Dm fX ∞ < ∞. ASSUMPTION A.7: For t1 < t2 , the stationary density ft1 t2 of (Xt1 Xt2 )is such that supt1
ft1 t2 t3 t4 ∞ < ∞
d ASSUMPTION A.8: The product kernel K(u) = i=1 κ(ui ), u ∈Rd , is such that d (i) Rd K(u) du= 1, (ii) K is bounded, limu→∞ lu K(u) = 0, Rd |K(u)| du < 2 2 ∞, and w := Rd K(u) du < ∞, and (iii) R v κ(v) dv = 0 for any l ∈ N such that l < m and R κ(v)|v|m dv < ∞. → ASSUMPTION A.9: The bandwidth hT is such that T 1/2 hdT → ∞ and T hd+2m T c¯ ∈ [0 ∞), as T → ∞. ASSUMPTION A.10: The matrices S0 = V [g1 (Xt Yt ; θ0 )]
Σ0 = V [g2∗ (Yt ; θ0 )|Xt = x0 ]
exist and are positive definite. ASSUMPTION A.11: For any θ ∈ Θ: E[g1 (Xt Yt ; θ)4 ] < ∞, E[g2∗ (Yt ; θ)4 ] < ∞.
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ASSUMPTION A.12: For any θ ∈ Θ, the function x −→ ϕ(x; θ) = E[g2∗ (Yt ; θ)|Xt = x]fX (x) is in class C m (X ) such that supθ∈Θ Dm ϕ(·; θ)∞ < ∞ and ∂|α| ϕ/∂xα is uniformly continuous on X × Θ for any α ∈ Nd with |α| = m ASSUMPTION A.13: For any θ τ ∈ Θ, the function E[g2∗ (Yt ; θ)g2∗ (Yt ; τ) | Xt = ·]fX (·) is continuous at x = x0 ASSUMPTION A.14: For any θ ∈ Θ, supt1
Xt1 = · Xt2 = · Xt3 = · Xt4 = · ft1 t2 t3 t4 (· · · ·) < ∞ ∞
ASSUMPTION A.16: There exists a basis of functions {ψj : j ∈ N} in L2 (FY ), where FY is the stationary cumulative distribution function (c.d.f.) of Yt such that ψj L2 (FY ) = 1, j ∈ N, and ∗ 2
g (y; θ) =
∞
cj (θ)ψj (y)
y ∈ Y
j=1
for any θ ∈ Θ, where {cj (θ) : j ∈ N} is a sequence of coefficient ∞ vectors. Moreover, there exist r > 2 and a sequence {λj > 0 : j ∈ N} such that j=1 λj < ∞, and ∞
2/r λj E Zt ψj (Yt )r + E[ψj (Yt )2 |Xt = x0 ] < ∞
j=1 ∞ 1 cj (θ)2 = 0 J→∞ θ∈Θ λ j j=J
lim sup
ASSUMPTION A.17: The function x −→ ϕj (x) = E[ψj (Yt )2 |Xt = x]fX (x) is in class C 2 (X ) for any j ∈ N such that supj∈N ϕj ∞ < ∞ and supj∈N D2 ϕj ∞ < ∞. ASSUMPTION A.18: The functions ψj are such that supj∈N E[|ψj (Yt )|r¯ ] < ∞ for r¯ > 2 and sup sup E ψj Yt1 ψj Yt2 Xt1 = · Xt2 = · ft1 t2 (· ·) ∞ < ∞ j∈N t1
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ASSUMPTION A.19: The moment function θ −→ (E[g1 (Xt Yt ; θ)] E[g2∗ (Yt ; θ)|Xt = x0 ] ) is continuous on Θ. ASSUMPTION A.20: The weighting matrix Ω is positive definite. ASSUMPTION A.21: Function g2∗ (y; θ) is twice continuously differentiable w.r.t. (y θ) ∈ Y × Θ. ASSUMPTION A.22: There exist γ1 γ2 > 1 and τ > 2, such that τ γ1
∂g1 ∂g1 E ∂θ (Xt Yt ; θ0 ) < ∞ E sup ∂θ (Xt Yt ; θ) < ∞ θ∈Θ γ2
2 ∂ g1 E sup (Xt Yt ; θ) < ∞ i j = 1 p θ∈Θ ∂θi ∂θj ∂g∗
ASSUMPTION A.23: The function E[ ∂θ2 (Yt ; θ0 )|Xt = ·]fX (·) is continuous at x0 and δ
∗ ∂g2 (Y ; θ ) E ∂θ t 0 < ∞ for δ > 2. Moreover, the functions
2 ∗ ∂ g2 E sup (Yt ; θ) Xt = · fX (·) θ∈Θ ∂θi ∂θj
i j = 1 p
are bounded. ASSUMPTION A.24: The functions ∗ ∂g2 ∂g1 (Xt Yt ; θ) E (Yt ; θ)|Xt = x0 θ −→ E ∂θ ∂θ
2 ∂ g1 (Xt Yt ; θ) i j = 1 p θ −→ E ∂θi ∂θj are continuous on Θ.
(Y ; θ0 )|X)W (X), where W (X) is ASSUMPTION A.25: The matrix Z ∗ = E( ∂g ∂θ a positive definite matrix P0 -a.s., is an instrument that satisfies Assumptions A.1, A.10, A.11, A.16, A.19, A.22, and A.24. Due to the different rates of convergence of the empirical moments in gT (θ∗ ) in Definition 4, it is not possible to follow the standard approach for the GMM framework to derive the asymptotic properties of θ∗T . In particular, to prove consistency, we cannot rely on the uniform a.s. convergence of the criterion
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QT to a limit deterministic function. Indeed, after dividing QT by T , the part of the criterion that involves the local conditional moment restrictions is asymptotically negligible. The corresponding limiting criterion is not uniquely minimized at θ0 in the full-information underidentified case. To prove consistency and asymptotic normality of θ∗T , we follow an alternative approach that relies on empirical process methods (see Stock and Wright (2000) for a similar approach). The empirical process of the sample moment restrictions is (A.1)
gT (θ∗ ) − mT (θ∗ ) =: T −1/2 ΨT (θ) =
T
gtT (θ)
θ ∈ Θ
t=1
where
√ mT (θ ) = T E[g1 (X Y ; θ)] T hdT E[g2 (Y ; θ)|x0 ] T hdT E[a(Y ; θ) − β|x0 ] ∗
Due to the linearity of gT and mT w.r.t. β, the empirical process ΨT is function of parameter θ, but not of parameter β. Moreover, the triangular array gtT (θ) is not zero mean, because of the bias term in the nonparametric component. Assumptions A.1 and A.4–A.19 are used to prove the weak convergence of the empirical process ΨT to a Gaussian stochastic process on Θ (see Lemma A.1 in Section A.1.2). In particular, Assumption A.5 on weak serial dependence, Assumptions A.6 and A.7 on the smoothness of the stationary density of (Xt ), Assumptions A.8 and A.9 on the kernel and the bandwidth, and Assumptions A.10–A.15 on the moment functions yield the asymptotic normality of the finite-dimensional distributions of process ΨT . Assumption A.16 disentangles the dependence of g2∗ (y; θ) on y and θ. It is used, together with Assumptions A.17 and A.18, to prove the stochastic equicontinuity of process ΨT along the lines of Andrews (1991). The weak convergence of empirical process ΨT is combined with Assumption A.2 on global identification, Assumption A.19 on the continuity of the moment functions, and Assumption A.20 on the weighting matrix to prove the consistency of kernel moment estimator θ∗T (see Section A.1.3). Assumptions A.21–A.24 concern the first- and second-order derivatives of the moment functions w.r.t. the parameter θ. These assumptions are used, together with the local identification Assumption A.3 and Assumption A.4 of interior true parameter value, to derive an asymptotic expansion for θ∗T in terms of ΨT (θ0 ). From the asymptotic normality of ΨT (θ0 ) (Lemma A.1), we deduce the asymptotic normality of kernel moment estimator θ∗T (see Section A.1.4). Finally, Assumption A.25 is used to establish the asymptotic results for the kernel moment estimators with optimal instruments (see Section A.1.6).
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Let us now discuss the bandwidth conditions in Assumption A.9. The con→ c¯ ∈ [0 ∞) is standard in nonparametric regression analysis. dition T hd+2m T When c¯ > 0, the bandwidth features the optimal d-dimensional rate of convergence, whereas when c¯ = 0, the asymptotic bias becomes negligible. Condition T 1/2 hdT → ∞ is stronger than the standard condition T hdT → ∞. Such a stronger bandwidth condition is necessary to ensure negligible second-order terms in the asymptotic expansion of the kernel moment estimator. Indeed, in the full-information underidentified case, somelinear combinations of parameter θ0 are estimated at a nonparametric rate√ T hdT , whereas other linear combinations are estimated at a parametric rate T . Thus, we need to ensure that the second-order term with the smallest rate of convergence is negligible w.r.t. the first-order term with the largest rate of convergence: √ (1/ T hdT )2 = o(1/ T ) ⇐⇒ T 1/2 hdT → ∞ The bandwidth conditions in Assumption A.9 can be satisfied when d < 2m. In particular, a kernel of order m = 2 is sufficient when the dimension d < 4. A.1.2. Asymptotic Distribution of the Empirical Process The asymptotic distribution of the empirical process ΨT is given in Lemma A.1 (below), which is proved in Appendix B in the Supplemental Material. The proof uses consistency and asymptotic normality of kernel estimators (e.g., Bosq (1998)), the Liapunov central limit theorem CLT (Billingsley (1965)), results on kernel M estimators (Tenreiro (1995)), weak convergence of empirical processes (Pollard (1990)), and a proof of stochastic equicontinuity similar to Andrews (1991). LEMMA A.1: Under Assumptions A.1 and A.4–A.19, ΨT ⇒ b + Ψ , where Ψ (θ), θ ∈ Θ, denotes the zero-mean Gaussian stochastic process on Θ with covariance function given by V0 (θ τ) = E[Ψ (θ)Ψ (τ) ] 0 S0 (θ τ) = 0 w2 Σ0 (θ τ)/fX (x0 )
for θ τ ∈ Θ
with S0 (θ τ) =
∞
Cov[g1 (Xt Yt ; θ) g1 (Xt−k Yt−k ; τ)]
k=−∞
Σ0 (θ τ) = Cov[g2∗ (Yt ; θ) g2∗ (Yt ; τ)|Xt = x0 ]
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and the continuous function b is given by lim T hd+2m 1 T wm b(θ) = m! fX (x0 ) " ! 0 ϕ(x0 ; θ) m θ ∈ Θ × Δ fX (x0 ) Δm ϕ(x0 ; θ) − fX (x0 ) with ϕ(x; θ) := E[g2∗ (Yt ; θ)|Xt = x]fX (x) and wm := R vm κ(v) dv. In particu√ d ¯ 0 V0 ), where lar, ΨT (θ0 ) −→ N( cb " ! 0 wm 1 ϕ(x0 ; θ0 ) m (A.2) b0 = Δ fX (x0 ) m! fX (x0 ) Δm ϕ(x0 ; θ0 ) − fX (x0 ) S0 0 V0 = 0 w2 Σ0 /fX (x0 ) matrices S0 and Σ0 are defined in Assumption A.10, and c¯ := lim T hd+2m . T The block diagonal elements of matrix V0 are the standard asymptotic variance–covariance matrices of sample average and kernel regression estimators, respectively. The bias function b(θ) is zero for the unconditional moments and is equal to the kernel regression bias for the conditional moments. Lemma A.1 implies that unconditional and conditional empirical moment restrictions are asymptotically independent and that the convergence is uniform w.r.t. θ ∈ Θ. A.1.3. Consistency of Kernel Moment Estimators By using the weak convergence of process ΨT (θ) from Lemma A.1, the continuous mapping theorem, and Assumptions A.4 and A.19, we get QT (θ∗ ) = √ mT (θ∗ ) ΩmT (θ∗ ) + Op ( T ) uniformly in θ∗ ∈ Θ × B. From global identification Assumption A.2 and from Assumption A.20 on the weighting matrix, we have inf
θ∗ ∈Θ×B:θ∗ −θ0∗ ≥ε
mT (θ∗ ) ΩmT (θ∗ ) ≥ CT hdT
√ for any ε > 0 and a constant C = Cε > 0. Then, by using T = o(T hdT ) from Assumption A.9, we get infθ∗ ∈Θ×B:θ∗ −θ0∗ ≥ε QT (θ∗ ) ≥ 12 CT hdT with probability approaching 1. Since T hdT → ∞ from Assumption A.9, we conclude that the minimizer θ∗T of QT is such that P[ θ∗T − θ0∗ ≥ ε] → 0 as T → ∞ for any ε > 0 (see Appendix B in the Supplemental Material for a detailed derivation).
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A.1.4. Asymptotic Distribution of Kernel Moment Estimators θ∗T )/∂θ∗ = 0 and a mean-value expanFrom the first-order condition ∂QT ( sion, we have (A.3)
∂ gT ∗ ∂ gT ∗ ∗ ∂ gT ∗ ∗ ( θ )Ω g (θ ) + (θT )Ω ∗ (# θT )(θT − θ0∗ ) = 0 T T 0 ∗ ∗ ∂θ ∂θ ∂θ
θ∗T and θ0∗ componentwise. Compared to the standard where # θ∗T is between GMM framework, we have to disentangle the linear transformations of θ0 with parametric and nonparametric convergence rates. Let us introduce the invertible (p + L p + L) matrix RT =
T −1/2 R1Z 0
(T hdT )−1/2 R2Z 0
0 (T hdT )−1/2 IdL
where the matrices R1Z and R2Z are defined by the linear change of parameter from θ to η = (η1 η2 ) in (3.13). By premultiplying equation (A.3) by matrix RT , we get RT
∂ gT ∗ (θ )Ω gT (θ0∗ ) ∂θ∗ T
⎛ √T ( η1T − η10 ) ⎞ ∂ g ∂ g T ( θ∗T )Ω ∗ (# θ∗T )RT ⎝ T hdT ( η2T − η20 ) ⎠ = 0 +RT ∂θ ∂θ d T hT (βT − β0 ) T ∗
We have the following Lemma A.2 which is proved in Appendix B in the Supplemental Material using the uniform law of large numbers (ULLN) and the CLT for mixing processes in Potscher and Prucha (1989) and Herrndorf (1984), respectively. LEMMA A.2: Under Assumptions A.1–A.24, we have plimT →∞ ∂ gT #∗ plimT →∞ ∂θ ∗ (θT )RT = J0 , where matrix J0 is given by
(A.4)
⎛ ∂g 1 E R1Z ⎜ ∂θ ⎜ ⎜ 0 J0 = ⎜ ⎜ ⎜ ⎝ 0
⎞ 0 ∂g2 E x 0 R2Z ∂θ ∂a E x0 R2Z ∂θ
0
⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎠ −IdL
∂ gT ∂θ∗
( θ∗T )RT =
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⎛ ∂g 1 E ⎜ ∂η1 ⎜ ⎜ 0 =⎜ ⎜ ⎜ ⎝ 0
⎞ 0
0
∂g2 E x0 ∂η 2 ∂a E x0 ∂η2
⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎠ −IdL
From the local identification Assumption A.3, matrix J0 is full rank. Thus √ d T − β0 ) (A.5) T ( η1T − η10 ) T hT ( η2T − η20 ) T hdT (β = −(J0 ΩJ0 )−1 J0 Ω gT (θ0∗ ) + op (1) √ d Since gT (θ0∗ ) = ΨT (θ0 ) −→ N( cb0 V0 ) from Lemma A.1, Proposition 2 follows. A.1.5. Optimal Weighting Matrix for Given Instrument and Bandwidth for some constant c > 0, When the bandwidth is such that hT = cT −1/(2m+d) √ d T ( η − η ) T h ( η from Proposition 2 the asymptotic MSE of ( 1T 10 2T − T d −1 −1 η20 ) T hT (βT − β0 ) ) is (J0 ΩJ0 ) J0 ΩM0 ΩJ0 (J0 ΩJ0 ) , where M0 := V0 + c 2m+d b0 b0 . The optimal weighting matrix for given instrument Z and bandwidth constant c is (A.6)
Ω = M0−1 = (V0 + c 2m+d b0 b0 )−1
The corresponding minimal MSE is (J0 M0−1 J0 )−1 Since M0 and J0 are block diagonal w.r.t. η1 and (η2 β ) , the associated asymptotic MSE of the estimator of β is (A.7)
M(Z c a) = e J0Z
w2 2 Σ0 + c 2m wm b(x0 )b(x0 ) d c fX (x0 )
−1
−1 J0Z
e
m
1 Δ ϕ(x;θ0 ) 0) where e = (0L×sZ IdL ) , b(x) = m! ( fX (x) − ϕ(x;θ Δm fX (x)) and fX (x)2 ⎞ ⎛ ⎞ ⎛ ∂g2 ∂g2 E 0 x E R x 0 0 0 2Z ⎟ ⎜ ⎟ ⎜ ∂θ ⎟ = ⎜ ∂η2 ⎟ J0Z =: ⎜ ⎠ ⎝ ⎠ ⎝ ∂a ∂a R −Id E x x E −Id 0 2Z L 0 L ∂θ ∂η2
A.1.6. Proof of Propositions 3 and 5 The optimal instrument Z ∗ and bandwidth constant c ∗ are derived by minimizing function M(Z c a) in (A.7) w.r.t. Z and c. In the standard GMM
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framework with full-information identified parameter θ0 , the matrix R2Z is empty and M(Z c a) is independent of Z. Any admissible instrument is optimal for estimating the conditional moment β, and c ∗ corresponds to the usual optimal bandwidth constant for kernel regression estimation (e.g., Silverman (1986)). In the full-information underidentified case, the optimal instrument and bandwidth selection problems are nonstandard, as seen below. A.1.6.1. Optimal Instruments. Let us first prove that instrument Z ∗ in (3.14) is admissible, that is, satisfies Assumptions A.2 and A.3. We have, for any vector α ∈ Rp ,
(A.8)
∂g1 E (X Y ; θ0 ) α = 0 ∂θ
∂g ∂g (Y ; θ0 )X W (X)E ⇐⇒ E E (Y ; θ ) X α =0 0 ∂θ ∂θ ∂g (Y ; θ0 ) X W (X) ⇐⇒ α E ∂θ ∂g ×E (Y ; θ ) X α=0 (P0 -a.s.) 0 ∂θ ∂g ⇐⇒ E (Y ; θ0 ) X α = 0 (P0 -a.s.). ∂θ
Thus, Assumption A.3 follows from Assumption 2 in the text. Then Assumption A.2 is also satisfied if Θ is taken small enough, which is sufficient for the validity of the asymptotic results. From Assumption A.25, the asymptotic properties in Proposition 2 apply for the kernel moment estimators with instrument Z∗ Let us now prove that instrument Z ∗ is optimal. In the expression of M(Z c a) in (A.7), instrument Z affects matrix J0Z only, and the matrix J0Z # are depends on Z through JZ = KerE[∂g1 (X Y ; θ0 )/∂θ ] only. If Z and Z # two admissible instruments such that JZ ⊂ JZ# , then M(Z c a) ≤ M(Z c a). Since J ∗ ⊂ JZ for any admissible instrument Z, Z ∗ is an optimal instrument if J ∗ = JZ ∗ . The latter equality follows from (A.8). Since J ∗ = JZ ∗ , the ranges of matrix R2 defined in (3.5) and matrix R2Z ∗ coincide. From (A.7), the asymptotic MSE for (any) optimal instrument Z ∗ becomes
∗ 0
M(c a) = e J
1 w2 2 Σ0 + c 2m wm b(x0 )b(x0 ) c d fX (x0 )
−1 −1 J0∗ e
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where
⎞ ⎛ ∂g2 E 0 x 0 ⎟ ⎜ ∂η∗2 ⎟ J0∗ = ⎜ ⎝ ⎠ ∂a −IdL E x0 ∗ ∂η2
A.1.6.2. Optimal Bandwidth. In the rest of this proof, we assume L = dim(a) = 1. Then function M(c a) is scalar and the first-order condition for minimizing M(c a) w.r.t. c is given by
∂M(c a) = e (J0∗ Σ˜ −1 J0∗ )−1 J0∗ Σ˜ −1 ∂c w2 d 2m−1 2 Σ0 + 2mc × − d+1 wm b(x0 )b(x0 ) c fX (x0 ) −1 ∗ ∗ ˜ −1 ∗ −1 ˜ × Σ J0 (J0 Σ J0 ) e = 0 2 b(x0 ) b(x0 ) and e = (0 1) ∈ Rs × R. The solution with Σ˜ := c1d fXw(x ) Σ0 + c 2m wm 0 c ∗ (a) is c ∗ (a) = ξ1/(2m+d) , where ξ = ξ(a) satisfies the equation ∗
2
(A.9)
ξ=
w2 d 2 f (x ) 2mwm X 0 ∗ × e (J0 A(ξ)−1 J0∗ )−1 J0∗ A(ξ)−1 Σ0 A(ξ)−1 J0∗ (J0∗ A(ξ)−1 J0∗ )−1 e /e (J0∗ A(ξ)−1 J0∗ )−1 J0∗ × A(ξ)−1 b(x0 )b(x0 ) A(ξ)−1 J0∗ (J0∗ A(ξ)−1 J0∗ )−1 e 2
where A(ξ) := fXw(x ) Σ0 + ξb(x0 )b(x0 ) . The optimal bandwidth sequence is 0 hT = c ∗ (a)T −1/(2m+d) . Then, from (A.6), the optimal weighting matrix becomes (A.10)
Ω∗ (a) = (V0 + c ∗2m+d b0 b0 )−1
where V0 is defined in (A.2), with optimal instrument Z ∗ . This concludes the proof of Proposition 3. A.1.6.3. Kernel Nonparametric Efficiency Bound. The kernel nonparametric efficiency bound is M(c ∗ (a) a) and is equal to M(x0 a) in Proposition 5. A.1.7. Proof of Corollary 6 The proof of Corollary 6 is similar to the proof of Propositions 3 and 5, and is given in Appendix B in the Supplemental Material. In particular, the optimal weighting matrix is Ω = V0−1
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A.2. Asymptotic Properties of the XMM Estimator A.2.1. Concentration With Respect to the Functional Parameters We first concentrate the estimation criterion in Definition 1 w.r.t. the functional parameters. Let us introduce the Lagrange multipliers λ, μ, λt , and μt , t = 1 T . The Lagrangian function is given by (A.11)
LT =
T 1 [f (y|xt ) − ft (y)]2 dy T t=1 f (y|xt ) + hdT log[f0 (y)/f (y|x0 )]f0 (y) dy
T 1 d −2 ft (y) dy − 1 − hT μ μt f0 (y) dy − 1 T t=1 T 1 d g(y; θ)ft (y) dy − hT λ g2 (y; θ)f0 (y) dy −2 λ T t=1 t The first-order conditions w.r.t. the functional parameters ft , t = 1 T , and f0 yield (A.12) (A.13)
ft (y) = f (y|xt ) + μt f (y|xt ) + λt g(y; θ)f (y|xt ) f0 (y) = f (y|x0 ) exp(λ g2 (y; θ) + μ − 1)
t = 1 T
The Lagrange multipliers λt μt , t = 1 T , and μ can be deduced from the constraints. We get (A.14)
λt = −V (g(θ)|xt )−1 E(g(θ)|x μt = −λt E(g(θ)|x t ) t ) exp(1 − μ) = E exp(λ g2 (θ))|x0
where E(·|x) and V (·|x) denote the conditional expectation and the conditional variance w.r.t. the kernel density estimator f (·|xt ), respectively. By replacing (A.14) into (A.12) and (A.13), we get the concentrated functional parameters (A.15)
−1 ft (y; θ) = f (y|xt ) − E(g(θ)|x t ) V (g(θ)|xt ) × g(y; θ) − E(g(θ)|x t = 1 T t ) f (y|xt )
f0 (y; θ λ) =
exp(λ g2 (y; θ)) f (y|x0 ) g (θ))|x ] E[exp(λ 2 0
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The concentrated Lagrangian becomes (A.16)
LcT (θ λ) =
T 1 E(g(θ)|xt ) V (g(θ)|xt )−1 E(g(θ)|x t) T t=1 exp(λ g2 (θ))|x0 − hdT log E
The XMM estimator θT is the minimizer of LcT (θ) = LcT (θ λ(θ)), where c 2 (θ) exp(λ(θ) g2 (θ))|x0 ] = 0 for any λ(θ) = arg maxλ LT (θ λ) is such that E[g θ [see (2.11) and (2.12)]. The estimator f ∗ (·|x0 ) of f (·|x0 ) is obtained from λT = λ( θT ) (see (2.13)). (A.15) by replacing θ by θT and λ by A.2.2. Asymptotic Expansions λT . A.2.2.1. Asymptotic Expansion of Estimator θT and Lagrange Multiplier λT is deThe asymptotic expansion of estimator θT and Lagrange multiplier rived by expanding the criterion (A.16) around θ = θ0 and λ = 0. We have to distinguish between the linear combinations of θT converging at a parametric rate and those converging at a nonparametric rate. For this purpose, we use the change of parameterization in (3.5) from θ to η∗ = (η∗1 η∗2 ) The asymptotic ∗2T , and expansion for the estimators of the transformed parameters η ∗1T and η the Lagrange multiplier λT are given in Lemma A.3. LEMMA A.3: (i) The asymptotic expansions of η ∗1T and η ∗2T are given by −1
√ ∂g ∂g ∗ ∗ −1 X V (g|X) E X R1 T ( η1T − η10 ) − R1 E E ∂θ ∂θ T 1 ∂g xt V (g|xt )−1 g(yt ; θ0 ) ×√ R1 E ∂θ T t=1 and (A.17)
T hdT ( η∗2T − η∗20 )
−1 ∂g2 ∂g2 x0 V (g2 |x0 )−1 E − R2 E x 0 R2 ∂θ ∂θ ∂g2 −1 d x0 V (g2 |x0 ) × R2 E T hT g2 (y; θ0 )f (y|x0 ) dy ∂θ
respectively. (ii) The asymptotic expansion of λT is λT −V (g2 |x0 )−1 (Id − M) g2 (y; θ0 )f (y|x0 ) dy (A.18)
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2 where M is the orthogonal projection matrix on the column space of E( ∂g |x0 )R2 ∂θ −1 for the inner product defined by V (g2 |x0 ) : −1 ∂g2 ∂g2 ∂g2 −1 x0 V (g2 |x0 ) E x0 R2 R2 E x0 R2 (A.19) M = E ∂θ ∂θ ∂θ ∂g2 x0 V (g2 |x0 )−1 × R2 E ∂θ
The asymptotic expansion of η ∗1T comes from the part of the criterion (A.16) that involves the uniform moment restrictions, since the part that involves the local moment restrictions is asymptotically negligible for η ∗1T . The asymptotic ∗ expansion of η 2T involves the local moment restrictions only, since the uniform moment restrictions are not informative for parameter η2 . The term log E(exp(λ g2 (θ))|x0 ) in (A.16) is induced by the KLIC component and has a weighting factor hdT (see Definition 1). This weighting factor ensures that the contribution of the discrepancy measure associated with the local restrictions at x0 is asymptotically the same as for a kernel moment estimator with optimal weighting matrix and instruments. A.2.2.2. Asymptotic Expansion of f ∗ (·|x0 ). Using f ∗ (y|x0 ) = f0 (y; θT λT ), from (A.15) we have θT ) 1 + λT g2 (y; f (y|x0 ) 1 + λT E(g2 (θT )|x0 ) 2 ( θT ) − E(g θT )|x0 ) f (y|x0 ) 1 + λT g2 (y;
f ∗ (y|x0 )
λT g2 (y; θ0 )f (y|x0 ) f (y|x0 ) + Then, from (A.18) we get (A.20)
f ∗ (y|x0 ) f (y|x0 ) − f (y|x0 )g2 (y; θ0 ) V (g2 |x0 )−1 × (Id − M) g2 (y; θ0 )f (y|x0 ) dy
∗ (a|x0 ). We have A.2.2.3. Asymptotic Expansion of E ∗ (a|x0 ) = a(y; E θT )f ∗ (y|x0 ) dy
a(y; θ0 )f (y|x0 ) dy +
+
∂a (y; θ0 )f (y|x0 ) dy ( θT − θ0 ) ∂θ
a(y; θ0 )[f ∗ (y|x0 ) − f (y|x0 )] dy
EFFICIENT DERIVATIVE PRICING
1225
∂a x0 R2 ( η∗2T − η∗20 ) ∂θ + a(y; θ0 ) f (y|x0 ) − f (y|x0 )
E(a|x0 ) + E
− f (y|x0 )g2 (y; θ0 ) V (g2 |x0 )−1 × (Id − M) g2 (y; θ0 )f (y|x0 ) dy dy where the last asymptotic equivalence comes from (A.20) and the fact that the contribution of η ∗1T − η∗10 is asymptotically negligible. Then, from (A.17) we get ∂a ∗ x0 E (a|x0 ) E(a|x0 ) − E ∂θ
−1 ∂g2 ∂g2 −1 × R2 R2 E x0 V (g2 |x0 ) E x0 R2 ∂θ ∂θ ∂g2 −1 g2 (y; θ0 )f (y|x0 ) dy x0 V (g2 |x0 ) × R2 E ∂θ + a(y; θ0 )[f (y|x0 ) − f (y|x0 )] dy − Cov(a g2 |x0 )V (g2 |x0 )−1 (Id − M) × g2 (y; θ0 )f (y|x0 ) dy We deduce the asymptotic expansion (A.21)
∗ (a|x0 ) − E(a|x0 ) E a(y; θ0 )δf (y|x0 ) dy −1
− Cov(a g2 |x0 )V (g2 |x0 )
g2 (y; θ0 )δf (y|x0 ) dy
∂g2 ∂a −1 R x x − Cov(a g |x )V (g |x ) E − E 0 2 2 0 2 0 0 R2 ∂θ ∂θ
−1 ∂g2 ∂g2 −1 × R2 E x0 V (g2 |x0 ) E x0 R2 ∂θ ∂θ ∂g2 −1 x0 V (g2 |x0 ) g2 (y; θ0 )δf (y|x0 ) dy × R2 E ∂θ
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P. GAGLIARDINI, C. GOURIEROUX, AND E. RENAULT
where δf (y|x0 ) := f (y|x0 ) − f (y|x0 ). A.2.3. Asymptotic Distribution of the XMM Estimator (Proofs of Proposition 7 and Corollary 8) ∗ (a|x0 ). In Let us derive the asymptotic distribution of the estimator E the asymptotic expansion (A.21), the first two terms in the right-hand side (RHS) correspond to the residual of the regression of a(y; θ0 )δf (y|x0 ) dy on g2 (y; θ0 )δf (y|x0 ) dy. This residual is asymptotically independent of the third term in the RHS. Thus,√ from the asymptotic normality of integrals d T hd ∗ (a|x0 ) − E(a|x0 )] −→ N(0 B (x0 a)), of kernel estimators, we get w T [E where B (x0 a) is given in Corollary 6. This proves Proposition 7. Corollary 8 is proved by checking that the asymptotic expansion for the XMM estimator θˆ T in Lemma A.3(i) corresponds to the asymptotic expansion for the = 0, Ω = V0−1 , and kernel moment estimator in (A.5) with c¯ = lim T hd+2m T (Y ; θ0 )|X]V [g(Y ; θ0 )|X]−1 . The details of the derivation are given Z = E[ ∂g ∂θ in Appendix B in the Supplemental Material. A.3. A Parametric Stochastic Volatility Model In this appendix, we consider the parametric stochastic volatility model introduced in Section 3.2. We first derive equations (3.9) from the no-arbitrage conditions, then we give the dynamics under the risk-neutral distribution, and finally we prove Proposition 1. A.3.1. Proof of Equations (3.9) Let us assume that the DGP P0 is compatible with the historical dynamic of the state variables Xt = (˜rt σt2 ) given in (3.6) and the s.d.f. (3.7) with true parameter value θ0 . The restrictions implied by the no-arbitrage assumption for the risk-free asset and the underlying asset are given by (A.22)
E0 [Mtt+1 (θ0 ) exp rft+1 |xt ] = 1 E0 [Mtt+1 (θ0 ) exp rt+1 |xt ] = 1
∀xt = (˜rt σt2 )
respectively. Let us first consider the no-arbitrage restriction for the risk-free asset. We have E0 [Mtt+1 (θ0 )erft+1 |xt ] 2 = E0 [exp(−θ10 − θ20 σt+1 − θ30 σt2 − θ40 r˜t+1 )|xt ]
(θ40 )2 2 0 0 2 0 0 = exp(−θ1 − θ3 σt )E0 exp − θ2 + θ4 γ0 − σt+1 xt 2
1227
EFFICIENT DERIVATIVE PRICING
(θ0 )2 = exp − θ10 + b0 θ20 + θ40 γ0 − 4 2
(θ0 )2 − θ30 + a0 θ20 + θ40 γ0 − 4 σt2 2 2 where we have integrated r˜t+1 conditional on σt+1 and used the definition of the ARG process. Since the RHS has to be equal to 1 for any admissible value of σt2 , we get (θ40 )2 (θ40 )2 0 0 0 0 0 0 (A.23) θ1 = −b0 θ2 + θ4 γ0 − θ3 = −a0 θ2 + θ4 γ0 − 2 2
Let us now consider the no-arbitrage restriction for the underlying asset. By 2 −θ30 σt2 −(θ40 −1)˜rt+1 )|xt ], using E0 [Mtt+1 (θ0 ) exp rt+1 |xt ] = E0 [exp(−θ10 −θ20 σt+1 we get the same conditions as in (A.23) by replacing θ40 with θ40 − 1: (θ40 − 1)2 0 0 0 θ1 = −b0 θ2 + (θ4 − 1)γ0 − 2 (θ0 − 1)2 θ30 = −a0 θ20 + (θ40 − 1)γ0 − 4 2 Since functions a0 and b0 are one-to-one, we get θ20 + (θ40 − 1)γ0 − θ20 + θ40 γ0 −
(θ40 )2 2
(θ40 −1)2 2
=
, that is, θ40 = γ0 + 12 Equations (3.9) follow.
A.3.2. Risk-Neutral Distribution The risk-neutral distribution Q is characterized by the conditional Laplace 2 2 ) , which is given by E0Q [exp(−u˜rt+1 − vσt+1 )|xt ] = transform of (˜rt+1 σt+1 −rft+1 2 E0 [Mtt+1 (θ0 ) exp(−u˜rt+1 − vσt+1 )|xt ]. In Appendix B in the Supplemene tal Material, we show that, under the risk-neutral distribution Q, the underlying asset return follows an ARG stochastic volatility model with adjusted risk premium parameter γ0∗ = −1/2 and volatility parameters (A.24)
ρ0 [1 + c0 (θ + γ02 /2 − 1/8)]2 c0 c0∗ = 0 1 + c0 (θ2 + γ02 /2 − 1/8)
ρ∗0 =
0 2
δ∗0 = δ0
A.3.3. Proof of Proposition 1 Let us consider XMM estimation with parametric s.d.f. Mtt+1 (θ) = e−rft+1 × e . This is a well specified s.d.f., with true parameter value θ0 satisfying the restrictions in equations (3.9). The econometrician uses the 2 −θ σ 2 −θ r˜ −θ1 −θ2 σt+1 3 t 4 t+1
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P. GAGLIARDINI, C. GOURIEROUX, AND E. RENAULT
uniform and local conditional moment restrictions (2.2) and (2.3) to identify θ0 . We first derive subspace J ∗ ⊂ R4 and matrix R2 . A.3.3.1. Derivation of Subspace J ∗ and Matrix R2 . The null space J ∗ associated with the uniform conditional moment restrictions is the linear space of vectors α ∈ R4 such that
exp rft+1 ∂Mtt+1 (A.25) E0 (θ0 ) xt α = 0 ∀xt exp rt+1 ∂θ By using ∂Mtt+1 (θ0 )/∂θ = Mtt+1 (θ0 )ξt+1 , we get the characterization of J ∗ given in Proposition 1(i). Moreover, since θ0 satisfies the no-arbitrage restrictions (A.22), we deduce that any vector θ = θ0 + αε, where ε is small and α satisfies (A.25), is such that E0 [Mtt+1 (θ) exp rft+1 |xt ] = 1 and E0 [Mtt+1 (θ) exp rt+1 |xt ] = 1 ∀xt at first order in ε. Therefore, the vectors in J ∗ are the directions dθ = θ − θ0 of infinitesimal parameter changes that are compatible with no arbitrage. From Section A.3.1, the parameter vectors θ compatible with no arbitrage are characterized by the nonlinear restrictions θ1 = −b0 (λ2 ), θ3 = −a0 (λ2 ), and θ4 = γ0 + 1/2, where λ2 := θ2 + γ02 /2 − 1/8. Thus, the tangent set at θ0 is spanned by the vector dθ1 dθ2 dθ3 dθ4 (A.26) α = dθ2 dθ2 dθ2 dθ2 θ=θ0 da0 0 db0 0 (λ ) 1 − (λ ) 0 = r2 = − du 2 du 2
where λ02 := θ20 + γ02 /2 − 1/8. We deduce that the linear space J ∗ has dimension dim(J ∗ ) = 1 and is spanned by the vector r2 . The orthogonal matrix R2 is given by R2 = r2 /r2 . ˜ ; θ0 )/∂θ |X = x0 ]r2 and Proof of LoA.3.3.2. Derivation of Vector E0 [∂g(Y ˜ ; θ0 )/∂θ |X = cal Identification. The jth element of the (n 1) vector E0 [∂g(Y x0 ]r2 is given by
∂g˜ j (Y ; θ0 ) E0 X = x0 r2 ∂θ
∂Mtt+hj (θ0 ) Rth + j = E0 (e − kj ) Xt = x0 r2 j = 1 n ∂θ h By using ∂Mtt+h (θ0 )/∂θ = Mtt+h (θ0 ) l=1 ξt+l , r2 = (−δ∗0 c0∗ 1 −ρ∗0 0) from (A.24), and (A.27)
2 2 2 r2 = σt+1 − (ρ∗0 σt2 + δ∗0 c0∗ ) = σt+1 − E0Q [σt+1 |σt2 ] ξt+1
EFFICIENT DERIVATIVE PRICING
we get
E0
1229
∂g˜ j (Y ; θ0 ) X = x 0 r2 ∂θ
=
hj
B(t0 t0 + hj )
l=1
2 R 2 2 − E0Q [σt+l |σt+l−1 ])(e thj − kj )+ |Xt = x0 × E0Q (σt+l By conditioning on the volatility path and using the Hull–White (1987) formula, 2 2 2 − E0Q [σt+l |σt+l−1 ])(eRth − k)+ |Xt = x0 B(t0 t0 + h)E0Q (σt+l 2 2 2 2 − E0Q [σt+l |σt+l−1 ]) BS(k σtt+h )|Xt = x0 = E0Q (σt+l 2 where BS(k σ 2 ) is the Black–Scholes price for time-to-maturity 1, and σtt+h 2 is the integrated volatility between t and t + h. Moreover, since E0Q [σt+l − Q Q Q 2 2 2 2 2 E0 [σt+l |σt+l−1 ]|Xt ] = E0 [σt+l − E0 [σt+l |σt+l−1 ]|Xt+l−1 ] = 0 by iterated expectation and the Markov property under Q, we have 2 2 2 2 E0Q (σt+l − E0Q [σt+l |σt+l−1 ]) BS(k σtt+h )|Xt = x0 2 2 2 2 = CovQ0 σt+l − E0Q [σt+l |σt+l−1 ] BS(k σtt+h )|Xt = x0 2 2 2 = CovQ0 [σt+l − ρ∗0 σt+l−1 BS(k σtt+h )|Xt = x0 ]
Thus, by using
hj l=1
2 2 2 2 (σt+l − ρ∗0 σt+l−1 ) = (1 − ρ∗0 )σtt+h − ρ∗0 σt2 + ρ∗0 σt+h , we get j j
E0 [∂g˜ j (Y ; θ0 )/∂θ |X = x0 ]r2 2 2 = (1 − ρ∗0 ) CovQ0 σtt+h BS(kj σtt+h )|Xt = x0 j j 2 2 BS(kj σtt+h )|Xt = x0 + ρ∗0 CovQ0 σt+h j j Finally, let us prove that E0 [∂g˜ j (Y ; θ0 )/∂θ |X = x0 ]R2 > 0 for j = 1 n, which implies the local identification of θ0 (Assumption 2). We have 2 2 CovQ0 (σtt+h BS(k σtt+h )|Xt = x0 ) > 0 for any h k > 0, since the Black– Scholes price is strictly increasing w.r.t. the volatility and the risk-neutral dis2 tribution of σtt+h given Xt = x0 is nondegenerate. Moreover, we have the following lemma. 2 LEMMA A.4: The integrated volatility σtt+h is stochastically increasing in the 2 spot volatility σt+h under the conditional risk-neutral distribution Q given Xt = x0 , 2 2 that is, P0Q [σtt+h ≥ z|σt+h = s Xt = x0 ] is increasing w.r.t. s, for any z.
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Faculty of Economics, Università della Svizzera Italiana (USI), Via Buffi 13, CH-6900 Lugano, Switzerland and Swiss Finance Institute; patrick.gagliardini@ usi.ch, Laboratoire de Finance Assurance, 15, Boulevard Gabriel Péri, 92245 Malakoff Cedex, France, Dept. of Economics, University of Toronto, 150 St. George St., Toronto, Ontario M5S 3G7, Canada, and CEPREMAP; Christian.Gourieroux@ ensae.fr, and Dept. of Economics, University of North Carolina at Chapel Hill, 107 Gardner Hall, CB 3305, NC 27599-3305, U.S.A. and CIRANO-CIREQ (Montreal);
[email protected]. Manuscript received May, 2007; final revision received November, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1233–1275
PERSISTENT PRIVATE INFORMATION BY NOAH WILLIAMS1 This paper studies the design of optimal contracts in dynamic environments where agents have private information that is persistent. In particular, I focus on a continuoustime version of a benchmark insurance problem where a risk-averse agent would like to borrow from a risk-neutral lender to stabilize his utility. The agent privately observes a persistent state variable, typically either income or a taste shock, and he makes reports to the principal. I give verifiable sufficient conditions showing that incentive-compatible contracts can be written recursively, conditioning on the report and two additional state variables: the agent’s promised utility and promised marginal utility of the private state. I then study two examples where the optimal contracts can be solved in closed form, showing how persistence alters the nature of the contract. Unlike the previous discretetime models with independent and identically distributed (i.i.d.) private information, the agent’s consumption under the contract may grow over time. Furthermore, in my setting the efficiency losses due to private information increase with the persistence of the private information, and the distortions vanish as I approximate an i.i.d. environment. KEYWORDS: Dynamic contracting, continuous time, stochastic control, principalagent model.
1. INTRODUCTION PRIVATE INFORMATION is an important component of many economic environments and in many cases that information is persistent. Incomes of individuals from one period to the next are highly correlated and can be difficult for an outsider to verify. Similarly, a worker’s skill at performing a task, a manager’s ability in leading a firm, and a firm’s efficiency in producing goods are all likely to be highly persistent and contain at least some element of private information. While models of private information have become widespread in many areas of economics, with very few exceptions, previous research has focused on situations where the private information has no persistence. But estimates suggest that idiosyncratic income streams and individual skills, which likely have large private information components, are highly persistent.2 In this paper I analyze a benchmark dynamic contracting model with persistent private information. Working in continuous-time, I develop general sufficient conditions which allow for a recursive representation of incentive-compatible contracts. This allows me to develop a characterization of such contracts, which has been lacking, and makes a range of new and empirically relevant applications amenable for analysis. I use my characterization to explicitly solve for optimal contracts 1 I thank Dilip Abreu, Katsuhiko Aiba, Narayana Kocherlakota, Rui Li, Chris Sleet, and Ted Temzelides for helpful comments. I also especially thank the co-editor and two anonymous referees for comments that greatly improved the paper. Financial support from the National Science Foundation is gratefully acknowledged. 2 See, for example, Storesletten, Telmer, and Yaron (2004) and Meghir and Pistaferri (2004).
© 2011 The Econometric Society
DOI: 10.3982/ECTA7706
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in some key examples, showing how increasing the persistence of the private information adversely affects the degree of risk sharing. In particular, I focus on a continuous-time dynamic contracting relationship between a risk-neutral principal and a risk-averse agent, where the agent privately observes a payoff-relevant persistent state variable. I start with a relatively general setting, which maps into applications similar to classic papers in the literature. One version of the model is a classic insurance problem along the lines of the Green (1987) and Thomas and Worrall (1990), where a riskaverse agent would like to borrow from a risk-neutral lender to stabilize his income stream. The income stream is private information to the borrower and is persistent. Another version of the general model is an insurance problem similar to Atkeson and Lucas (1992), where the agent is hit with privately observed preference shocks which alter his marginal utility of consumption. My general characterization, which applies to both applications, adapts the continuoustime methods developed in Williams (2009) for hidden action problems to handle models with private information. One of the main contributions of this paper is to provide a set of necessary and sufficient conditions which ensure that a contract is incentive-compatible. To do so, I apply a stochastic maximum principle due to Bismut (1973, 1978) to derive the agent’s optimality conditions facing a given contract, which in turn provide conditions that an incentive-compatible contract must satisfy. I show that such contracts must be history dependent, but this dependence is encapsulated in two endogenous state variables: the agent’s promised utility and promised marginal utility under the contract. The use of promised utility as a state variable in contracting is now well known, but when there is persistent private information, incentive compatibility requires more conditioning information.3 The difficulty with persistent information is that deviations from truth-telling alter agents’ private evaluations of future outcomes; put differently, agents’ continuation types are no longer public information. I show that in my setting, incentive compatibility can be ensured by conditioning on a discounted expected marginal utility state variable, which captures the expected future shadow value of the private information. The discount rate in this state variable includes a measure of the persistence. More persistent information is longer lived and so has a greater impact on future evaluations, while as the persistence vanishes, this additional state variable becomes unnecessary. The other main contribution of the paper is to explicitly solve for optimal contracts in some special cases of interest. In particular, I study a hidden en3 The use of promised utility as a state variable follows the work of Abreu, Pearce, and Stacchetti (1986, 1990) and Spear and Srivastrava (1987). It was used by Green (1987) and Thomas and Worrall (1990) in settings like mine, and by Sannikov (2008) in a continuous-time model. Similar marginal utility states have been used by Kydland and Prescott (1980) and in related contexts by Werning (2001), Abraham and Pavoni (2008), and Kapicka (2006), as well as my earlier paper (Williams (2009)).
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dowment model where the agent has exponential utility and a model with permanent private taste shocks where the agent has power utility over consumption. The hidden endowment model highlights the relationship between the persistence of the private information and the magnitude of the distortions this information causes. The largest distortions occur when the private information is permanent. In this case, the optimal contracts entail no risk sharing, consisting only of a deterministic transfer which changes the time profile of the agent’s consumption. However, as information becomes less persistent, risk sharing increases and the agent’s consumption becomes more stable. In particular, in the i.i.d. limit, I obtain efficiency and complete stabilization of consumption. This result differs from the discrete-time i.i.d. models in the literature, which clearly have nonzero distortions. But I show that as the period length shrinks in a discrete-time model, the distortions shrink as well. Thus the existing results are consistent with my findings. The private taste shock model has much of the same structure, but it becomes intractable analytically outside the case of permanent shocks. My results differ in some fundamental ways from those in the literature. In particular, Rogerson (1985) and Golosov, Kocherlakota, and Tsyvinski (2003) showed in discrete time that an “inverse Euler equation” governs consumption dynamics in private information models. Technically, this implies that the inverse of the agent’s marginal utility of consumption is a martingale. Closely related are the “immiseration” results of Thomas and Worrall (1990) and Atkeson and Lucas (1992) which imply that the agent’s promised utility tends to minus infinity under the optimal contract. In my examples, these results fail—the agent’s promised utility follows a martingale and consumption has a positive drift under the optimal contract. As I discuss in more detail below, these differences rely at least partly on differences in the environments. In the discrete analogue of my model, when deciding what to report in the current period, the agent trades off current consumption and future promised utility. In my continuous-time formulation, the agent’s private state follows a process with continuous paths and the principal knows this. Thus in the current period the agent only influences the future increments of the reported state.4 Thus current consumption is independent of the current report and all that matters for the reporting choice is how future transfers are affected. As the reporting problem and, hence, the incentive constraints become fully forward-looking, optimal allocations no longer involve the balance of current and future distortions that the inverse Euler equation embodies. By contrast, in a moral hazard setting with hidden effort such as Williams (2009), an inverse Euler equation does hold. When effort is costly, 4
As discussed below, this is a requirement of absolute continuity. If the agent were to report a process which jumped discretely at any instant, the principal would be able to detect that his report was a lie.
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deviations from the behavior specified by the contract have instantaneous utility effects which are absent in the hidden information model. Thus the inverse Euler equation is sensitive to the source of the information frictions. Besides my own previous work, the closest paper in terms of technique is DeMarzo and Sannikov (2006), who also studied a continuous-time hidden information problem.5 My conversion of the reporting problem to a hidden action problem follows their approach. However, they focused on the case where agents are risk-neutral and there is no persistence in information. My paper is more general along these dimensions. One particular difference is that given the risk neutrality, they can define cash flows—their source of private information—as increments of a Brownian motion with constant drift. Thus private information in their case is i.i.d. However, in my environment it is more natural, and more consistent with the literature, to define the private state as the level of a diffusion process. As all diffusions are persistent, private information is always persistent in my environment.6 In addition, there are a few recent papers which studied models with persistent private information. In a discrete-time setting, Battaglini (2005) and Tchistyi (2006) characterized contracts between risk-neutral agents when the unobserved types switch according to a Markov chain. Their results rely heavily on risk neutrality and thus do not extend to the classic insurance issues which I address here. In their analysis of disability insurance, Golosov and Tsyvinski (2006) studied a particularly simple form of persistent information, where the type enters an absorbing state. Again, their results heavily exploit this particular structure. More closely related to the present paper, Zhang (2009) studied a continuous-time hidden information model with two types that switch according to a continuous-time Markov chain. His model is a continuous-time version of Fernandes and Phelan (2000), who considered a relatively general approach for dealing with history dependence in dynamic contracting models. Both Zhang (2009) and Fernandes and Phelan (2000) allow for persistence in the hidden information, at the cost of a state space which grows with the number of types. In contrast, my approach deals with a continuum of types and requires two endogenous state variables. This is achieved by exploiting the continuity of the problem to introduce a state variable which captures the shadow value (in marginal utility terms) of the hidden information. Thus my approach is analogous to a first-order approach to contracting.7 Kapicka (2006) considered a related first-order approach in a discrete-time model. As I discuss in more detail in Section 8, we both use similar ideas to 5 Biais, Mariotti, Plantin, and Rochet (2007) studied a closely related discrete-time model and its continuous-time limit. 6 As I discuss in Section 2.2 below, the i.i.d. case can be approximated by considering a limit of a mean reverting process as the speed of mean reversion goes to infinity. 7 See Williams (2009) for more discussion of the first-order approach to contracting in continuous time.
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arrive at recursive representations of contracts, but our state variables differ. Kapicka (2006) took the agent’s promised marginal utility of consumption as his additional state variable, while in my approach, the marginal utility of the hidden state (discounted by the persistence of the private information) is the relevant state variable. Although these state variables agree in some cases, I present an example below where they differ. In addition, my proof of the applicability of the first-order approach is more complete and direct, and I show how to verify my sufficient conditions in applied settings. Our implications for contracts differ as well. Kapicka (2006) showed how optimal contracts may significantly distort the agent’s consumption–savings decision. However, I show that when shocks are permanent, this channel need not be distorted. In my example, the only way the principal can ensure truthful revelation is to make the transfer independent of the report. Such a contract does not distort the agent’s intertemporal decisions, but it clearly does not insure the agent either. The persistence of private information gives the contracting problem an adverse selection component as well. In a continuous-time setting, Sannikov (2007) and Cvitani´c and Zhang (2007) studied contracting problems with adverse selection. Both focused on cases where there are two possible fixed types of agents and, apart from the agent’s type, there is no persistence in private information. While I do not emphasize the screening component of contracts, my methods could be applied to screening models where agents’ types vary over time. The rest of the paper is organized as follows. The next section describes the model and discusses a key change of variables which is crucial for analysis. Section 3 analyzes the agent’s reporting problem when facing a given contract, deriving necessary optimality conditions. The two endogenous state variables arise naturally here. Section 4 characterizes truthful reporting contracts, presenting sufficient conditions which ensure incentive compatibility. Section 5 presents a general approach to solve for optimal contracts. Then in Section 6, I consider a hidden endowment example where the optimal contract can be determined in closed form. I show how persistence matters for the contract and contrast our results with some of the others in the literature. Section 8 studies a related model with private taste shocks, which can also be solved explicitly with permanent shocks. Finally, Section 9 offers some brief concluding remarks. 2. THE MODEL 2.1. Overview The model consists of a risk-averse agent who privately observes a state variable which affects his utility and who receives transfers from a risk-neutral principal to smooth his consumption. The privately observed state variable can be an endowment process as in Thomas and Worrall (1990) or a taste shock as in Atkeson and Lucas (1992). If the principal could observe the private state, then he could absorb the risk and fully stabilize the agent’s utility. However, with the
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state being the private information of the agent, the principal must rely on the agent’s reports. Under the full-information contract, the agent would have an incentive to lie, for example, reporting that his income is lower than it really is. Thus the key problem is to design a contract which provides the agent the incentive to truthfully report his information. Relative to the literature, my key innovation is to allow for persistence in the agent’s privately observed state. 2.2. Basic Layout I start by considering a finite horizon [0 T ] and later let T → ∞. I use a plain letter to denote a whole path of a variable; thus, for instance, b = (bt )Tt=0 . I suppose that the privately observed variable b is given by a Markov diffusion process defined on a probability space with a Brownian motion W , which evolves as dbt = μ(bt ) dt + σ dWt I assume that σ > 0 is constant and that the drift is affine, μ(b) = μ0 − λb with λ ≥ 0. All the results in the paper could be easily extended to the more general case where the drift μ : R → R is twice continuously differentiable, (weakly) decreasing, and (weakly) concave. While this extension may be useful in applications, it adds little conceptually to the issues at hand and only leads to more cumbersome notation.8 With λ = 0, b is a Brownian motion with drift, so its increments are i.i.d. and shocks have permanent effects.9 On the other hand, with λ > 0, b is an Ornstein–Uhlenbeck process (see Karatzas and Shreve (1991)). This is a continuous-time version of a stationary Gausssian autoregressive process with the properties μ0 μ0 −λt + b0 − e E(bt |b0 ) = λ λ Cov(bt bs |b0 ) =
σ 2 −λ|s−t| − e−λ(s+t) e 2λ
Thus μ0 /λ gives the mean of the stationary distribution, while λ governs the rate of mean reversion and hence the persistence of the process. As mentioned 8 Note that I allow b to affect utility in a general way, so b may be a transformation of an underlying state variable. 9 As noted above, if we were to follow DeMarzo and Sannikov (2006), then bt would be the cumulative state process and, hence, i.i.d. increments would correspond to an i.i.d. state variable. However, apart from the risk-neutral case they consider, defining utility over the increments of a diffusion process is problematic.
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above, the private√state cannot be i.i.d., but we can approximate an i.i.d. process by setting σ = σ¯ λ and μ0 = μλ ¯ for some σ ¯ μ¯ > 0, and letting λ → ∞. The limit is an i.i.d. normal process with mean μ¯ and variance σ¯ 2 /2. The key issue in this model, of course, is that the agent alone observes bt , while the principal only observes the agent’s report of it, which I denote yt . To simplify matters, I assume that the agent cannot overreport the true state, so yt ≤ bt . In most settings, the relevant incentive constraint guards against underreporting, so such a constraint is not overly restrictive.10 When the private state variable is the agent’s endowment, this assumption can be motivated by requiring the agent to deposit a portion of his endowment in an account which the principal observes. The good is nonstorable and the agent cannot manufacture additional endowments, so the deposit (or report) is a verifiable statement of at most the entire endowment. The fact that the agent’s private state has continuous sample paths is key to our analysis. This continuity allows us to use local information to summarize histories, it gives structure to the types of reports the agent can make, and together with the concavity of utility, it allows us to map local optimality conditions into global ones. Other types of processes which allow for jumps may be useful in some applications (such as Zhang (2009)). But allowing for jumps is more complex, as the contract would require more global information. Models with continuous information flows provide a useful benchmark and are natural in many applications. When the private state is the hidden endowment, similar continuous specifications have been widely used in asset pricing following Breeden (1979). Although lumpiness may be important for some types of income, business or financial income may involve frequent flows which may be difficult for an intermediary to observe. Similarly, when the private state is a preference shock, we focus not on lumpiness due to, say, discrete changes in household composition, but rather on more frequent, smaller fluctuations in tastes or skills. The agent’s information can be summarized by the paths of W , which induce Wiener measure on the space C[0 T ] of continuous functions of time. The agent’s reporting strategy is thus a predictable mapping y : C[0 T ] → C[0 T ]. Denote this mapping y(ω) and its time t component yt (ω) which is measurable with respect to Bt , the Borel σ-algebra of C[0 T ] generated by {Ws : s ≤ t}. The principal observes y only and thus his information at date t can be represented via Yt , the Borel σ-algebra of C[0 T ] generated by {ys : s ≤ t}. I assume that the private state is initialized at a publicly known value b0 and that the principal knows the process which the state follows (i.e., he knows μ and σ), but he does not observe the realizations of it. 10 In our setting, this restriction facilitates analyzing the incentive constraints. Removing the restriction at the outset is difficult due to the linearity of agent’s reporting problem.
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As the principal obtains continuous reports from the agent, the agent is not free to choose an arbitrary reporting strategy. In particular, based on the rey ports, the principal can construct a process Wt which evolves as (1)
y
dWt =
dyt − μ(yt ) dt σ y
Under a truthful reporting strategy yt = bt , clearly we have Wt = Wt . Thus the y agent is restricted to reporting strategies which ensure that Wt is a Brownian motion with respect to the principal’s information set. If he were to choose y a reporting strategy which did not make Wt a Brownian motion, say, for instance, he reported a constant endowment for a strictly positive length of time (which has probability zero), then the principal would detect this lie and would be able to punish him. Formally, the agent’s report y must be absolutely continuous with respect to his true state b. Hence via the Girsanov theorem and related results (see Chapters 6 and 7 of Liptser and Shiryaev (2000), for example), this means that the agent’s reporting process is equal to the true state process plus a drift, dyt = dbt + Δt dt where Δt is a process adapted to the agent’s information set. Since the agent can report (or deposit) at most his entire state, we must have Δt ≤ 0. Integrating this evolution and using y0 = b0 , we see that the report of the private state is equal to the truth plus the cumulative lies, t yt = bt + Δs ds ≡ bt + mt 0
where we define mt ≤ 0 as the “stock of lies.” With this notation, we can then write the evolution of the agent’s reporting and lying processes as (2)
dyt = [μ(yt − mt ) + Δt ] dt + σ dWt
(3)
dmt = Δt dt
with y0 = b0 and m0 = 0. The principal observes yt , but cannot separate Δt from Wt and thus cannot tell whether a low report was due to a lie or a poor shock realization. Moreover, the stock of lies mt is a hidden state which is unobservable to the principal, but influences the evolution of the observable report state. In our environment, a contract is a specification of payments from the principal to the agent conditional on the agent’s reports. The principal makes payments to the agent throughout the period s : C[0 T ] → C[0 T ] which are adapted to his information {Yt }, as well as a terminal payment ST : C[0 T ] → R which is YT -measurable. This is a very general representation, allowing almost
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arbitrary history dependence in the contract. We will study the choice of the optimal contract s by the principal, finding a convenient representation of the report history. 2.3. A Change of Variables As the contract is history dependent, the agent’s utility at any given time will, in general, depend on the whole past history of reports. This makes direct analysis difficult, as standard dynamic programming methods are not applicable. This lack of a recursive structure is well known in contracting models, and the seminal works of Spear and Srivastrava (1987), Green (1987), and Abreu, Pearce, and Stacchetti (1990) show how to make the problem recursive by enlarging the state space. We use the agent’s optimality conditions to derive such additional necessary state variables. To do so, as in Williams (2009), I follow Bismut (1978) and change state variables to focus on the distribution of observed reports. Rather than directly analyzing the agent’s choice of what to report at each date, it is easier to view the agent as choosing a probability distribution over the entire reporting process that the principal observes. Alternative reporting strategies thus change the distribution of observed outcomes. A key simplification of our continuous-time setting is the natural mapping between probability measures and Brownian motions. The following calculations rely on standard stochastic process results, as in Liptser and Shiryaev (2000). To begin, fix a probability measure P0 on C[0 T ] and let {Wt 0 } be a Wiener process under this measure, with {Ft } the completion of the filtration generated by it. Under this measure, the agent reports that his private state follows a martingale, (4)
dyt = σ dWt 0
with y0 given. Different reporting choices alter the distribution over observed outcomes in C[0 T ]. In particular, for any feasible lying choice Δ = {Δt }, define the family of Ft -predictable processes t μ(ys − ms ) + Δs dWs0 Γt (Δ) = exp (5) σ 0 2 t μ(ys − ms ) + Δs 1 − ds 2 0 σ Γt is an Ft martingale with E0 [ΓT (Δ)] = 1, where E0 represents expectation with respect to P0 . Thus, by the Girsanov theorem, we define a new measure PΔ via dPΔ = ΓT (Δ) dP0
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and the process Wt Δ defined by t μ(ys − ms ) + Δs Δ 0 (6) ds Wt = Wt − σ 0 is a Brownian motion under PΔ . For the truthful reporting strategy Δ∗t ≡ 0, we denote PΔ∗ = P ∗ , with corresponding Brownian motion Wt ∗ Hence each effort choice Δ results in a different Brownian motion, similar to what we constructed y above in (1). However, the Brownian motion Wt is based on the principal’s observations of y, assuming truthful reporting. Here the Brownian motion Wt Δ is based on the agent’s (full) information, as a means to depict the distribution PΔ over observed reports. The density process Γt captures the effect of the agent’s lies on observed outcomes, and we take it rather than yt as the key state variable. Although the contract depends on the entire history of reports, this density allows us to average over that history in a simple manner. Using (5), the density evolves as (7)
dΓt =
Γt [μ(yt − mt ) + Δt ] dWt 0 σ
with Γ0 = 1. The stock of lies mt is the other relevant state variable for the agent, and it is convenient to analyze its evolution under the transformed probability measure. Thus we take zt = Γt mt , which captures covariation between reports and the stock of lies, as the relevant hidden state variable. Simple calculations show that it follows zt dzt = Γt Δt dt + [μ(yt − mt ) + Δt ] dWt 0 (8) σ with z0 = 0. By changing variables from (y m) to (Γ z), the agent’s reporting problem is greatly simplified. Instead of depending on the entire past history of the state yt , the problem only depends on contemporaneous values of the states Γt and zt . The history y is treated as an element of the probability space. This leads to substantial simplifications, as I show below. 3. THE AGENT’S REPORTING PROBLEM In this section, I derive optimality conditions for an agent facing a given contract. The agent’s preferences take a standard time additive form, with discount rate ρ, smooth concave flow utility u(s b), and terminal utility U(s b) defined over the payment and the private state. Below I focus on the special cases where b is the agent’s endowment, so u(s b) = v(s + b), and where b is the log of a taste shock, so u(s b) = v(s) exp(−b) where v(s) ≤ 0. I suppose that the agent has the option at date zero to reject a contract and remain in autarky. This gives a participation constraint that the utility the agent achieves under the contract must be greater than his utility under autarky, denoted V a (b0 ).
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However, after date zero, both the parties are committed to the contract and cannot leave it. The agent’s preferences for an arbitrary reporting strategy {Δt } can be written T −ρt −ρT V (y; s) = EΔ e u(st (y) yt − mt ) dt + e U(ST (y) yT − mT )
0 T
= E0
e−ρt Γt u(st (y) yt − mt ) dt
0 −ρT
+e
ΓT U(ST (y) yT − mT )
Here the first line uses the contract, substitutes y − m for b, and takes the expectation with respect to the measure PΔ over reporting outcomes. The second line uses the density process defined above. The agent takes the contract s and the evolution (7)–(8) as given, and solves sup V (y; s) {Δt ≤0}
Under the change of variables, the agent’s reporting problem is a control problem with random coefficients. As in Williams (2009), I apply a stochastic maximum principle from Bismut (1973, 1978) to derive the agent’s necessary optimality conditions. I first derive all conditions using the new variables (Γ z) and then express everything in terms of the original states (y m). Analogous to the deterministic Pontryagin maximum principle, I define a (current-value) Hamiltonian function which the optimal control will maximize. As in the deterministic theory, associated with the state variables are co-state variables which have specified terminal conditions. However, to respect the stochastic information flow and satisfy the terminal conditions, the co-states are now pairs of processes which satisfy backward stochastic differential equations. Thus I introduce (q γ) and (p Q) as the co-states associated with the state Γ and with the state z, respectively. In each pair, the first co-state multiplies the drift of the state, while the second multiplies the diffusion term. Thus the Hamiltonian is given by
H(Γ z) = Γ u(s(y) y − z/Γ ) + (Γ γ + Qz)(μ(y − z/Γ ) + Δ) + pΓ Δ The Hamiltonian captures the instantaneous utility flows to the agent. Here we see that a lie Δ affects the future increments of the likelihood of outcomes Γ , as well as the hidden state variable z. Moreover, these state variables matter for the agent’s future utility evaluations, both through the direct dependence of
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the utility function on the hidden state and through the change in the likelihood of alternative reporting paths. The agent’s optimal choice of his report perturbation Δ ≤ 0 is given by maximizing the Hamiltonian, and the evolution of the co-state variables is given by differentiating the Hamiltonian. To simplify matters, I invoke the revelation principle and focus on contracts which induce truthful revelation. Thus we have Δt ≡ 0, mt ≡ 0, and yt = bt . As the Hamiltonian H is linear Δ and Γ ≥ 0, so as to have a truthful current report Δ = 0 be optimal, we thus require (9)
γ + Qm + p ≥ 0
Moreover, given truthful reporting in the past (so m = 0), it must be optimal to report truthfully in the present, and thus we can strengthen this to (10)
γ + p ≥ 0
Under truthful revelation, the co-state variables evolve as (11)
∂H(Γ z) dqt = ρqt − dt + γt σ dWt 0 ∂Γ = [ρqt − u(st yt )] dt + γt σ dWt ∗
(12)
qT = U(sT yT ) ∂H(Γ z) dt + Qt σ dWt 0 dpt = ρpt − ∂z = [ρpt − λγt + ub (st yt )] dt + Qt σ dWt ∗ pT = −Ub (ST yT )
Here (11) and (12) carry out the differentiation, evaluate the result under truthful revelation, and change the Brownian motion as in (6). Details of the derivations are provided in Appendix A.1. Notice that pt and qt solve backward stochastic differential equations, as they have specified terminal conditions but unknown initial conditions. Below I show that these co-state processes encode the necessary history dependence that truthful revelation contracts require. In effect, the principal is able to tune the coefficients γt and Qt in these state variables. That is, incentive-compatible contracts can be represented via specifications of st (y) = s(t yt qt pt ), γt = γ(t yt qt pt ), and Qt = Q(t yt qt pt ) for some functions s, γ, and Q. Notice as well that the co-state evolution equations do not depend on Γt or zt and thus the entire system consists of (2) with Δt = mt = 0, (11), and (12).
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To help interpret the co-state equations, consider first the co-state (11). As can be easily verified, we can write its solution as T −ρ(τ−t) −ρ(T −t) (13) e u(sτ yτ ) dτ + e U(sT yT ) Ft qt = E∗ t
where E∗ is the expectation with respect to P ∗ . Thus q0 = V (y) and the agent’s optimal utility process becomes a state variable for the contracting problem. Using utility as a state variable is a well known idea in the literature following Abreu, Pearce, and Stacchetti (1986) and Spear and Srivastrava (1987), and has been widely used in contexts like ours following Green (1987) and Thomas and Worrall (1990). In environments without persistent private information, the promised utility encapsulates the necessary history dependence. However, with persistent private information, additional information is necessary. The agent’s decisions at any date now may depend on both his previous history of reports as well as the true history of his private state. Put differently, with persistent information, a lie in the current period affects both the principal’s expectations of future realizations of the hidden state and the agent’s expectations of his own future payoffs. However, as I show below, when the agent’s utility function is concave in the hidden state, local information suffices to capture the additional dependence. The variable pt , which we call the promised marginal utility state, gives the marginal value in utility terms of the hidden state variable mt evaluated at mt = 0. Thus it captures the marginal cost of not lying. In particular, suppose that (10) binds almost everywhere.11 Then (12) becomes y
dpt = [(ρ + λ)pt + ub (st yt )] dt + Qt σ dWt It is easy to verify that the solution of this equation is T pt = −E∗ (14) e−(ρ+λ)(τ−t) ub (sτ yτ ) dτ + e−(ρ+λ)(T −t) Ub (sT yT ) Ft t
Thus pt is the negative of the agent’s optimal marginal utility (of the private state bt ) under the contract. As noted in the Introduction, similar state variables have been used in related contexts by Werning (2001), Abraham and Pavoni (2008), Kapicka (2006), and Williams (2009).12 Here the persistence of the endowment effectively acts as an additional discount, since larger λ means 11 Here we make this assumption only to help interpret the variable pt , but it is not necessary for our results. We discuss this condition in more detail in Section 5.3 below. 12 Most of these references use either current or discounted future marginal utility of consumption as the additional state variable. The marginal utility of the private state is not necessarily the same as the marginal utility of consumption, as we show below.
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faster mean reversion which shortens the effective life of the private information. In the limit as λ → ∞, we approximate an i.i.d. process and pt converges to an i.i.d. random variable. Thus in the i.i.d. case, a contract only needs to condition on the utility process, while persistent private information requires the marginal utility process as well. Notice that γt , the loading of the utility process on the Brownian motion, is a key means to induce truthful revelation. When bt is the agent’s endowment, a risk-neutral principal who directly observes bt will fully stabilize the agent’s consumption, and thus set γt = 0. (This should be clear intuitively, but will be shown explicitly below.) But when the agent has private information, the principal induces truthful revelation by making the agent’s utility vary with his report. In particular, (10) implies γt ≥ −pt ≥ 0, so that promised utility qt increases with larger realizations of the private state. With persistent private information, the principal also chooses Qt , the loading of the promised marginal utility state on the endowment shocks. Thus when considering lying, the agent must balance the higher utility promise with the effects on his marginal utility promise. 4. TRUTHFUL REVELATION CONTRACTS I now characterize the class of contracts which are individually rational and can induce the agent to truthfully report his private information. Instead of considering a contract as a general history-dependent specification of s(y) as above, I now characterize a contract as processes for {st γt Qt } conditional on the states {yt qt pt }. In this section I provide conditions on these processes that a truthful revelation contract must satisfy. In standard dynamic contracting problems, like Thomas and Worrall (1990), contracts must satisfy participation, promise-keeping, and incentive constraints. Similar conditions must hold in our setting. I have already discussed the participation constraint, which must ensure that the agent’s utility under the contract is greater than under autarky. This simply puts a lower bound on the initial utility promise (15)
q0 ≥ V a (y0 )
where we use that y0 = b0 . The promise-keeping constraints in our environment simply state that the contract is consistent with the evolution of the utility and marginal utility state variables (11)–(12). The incentive constraint is the instantaneous truth-telling condition (10) which ensures that if the agent has not lied in the past, he will not do so in the current instant. However, to be sure that a contract is fully incentive-compatible, we must consider whether the agent may gain by lying now and in the future. Such “double deviations” can be difficult to deal with in general (see Kocherlakota (2004) and Williams (2009), for example), but I show that in our setting, they can be ruled out by some further restrictions on Qt under the contract.
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I now provide the key theoretical results that characterize the class of truthful revelation contracts. The theorem adapts my previous results from Williams (2009), which in turn build on Schattler and Sung (1993) and Zhou (1996). However, the setting here leads to a more direct proof, which is contained in Appendix A.2. The idea of the proof is to use the representations of the agent’s utility and marginal utility processes under the contract, along with the concavity assumptions on the primitives, to bound the gain from deviating from truth-telling. THEOREM 4.1: Assume that the agent’s utility functions u and U are twice differentiable, increasing, and concave in b, and that λ ≥ 0. Then we have the following results: (a) Any truthful revelation contract {st γt Qt } satisfies (i) the participation constraint (15), (ii) the promise-keeping constraints (11)–(12), and (iii) the incentive constraint (10). (b) Suppose λ = 0 and u is three times differentiable in b with ubbb ≥ 0. Then any contract satisfying conditions (i)–(iii) and insuring that the conditions (16)
Qt ≤ −
ubb (st yt ) 2λ
and (17)
T
Qt ≤ −E∗
−ρ(τ−t)
e
[ubb (sτ yτ ) + 2λQτ ] dτ Ft
t
hold for all t is a truthful revelation contract. (c) Suppose λ = 0 and u is three times differentiable in b with ubbb ≥ 0. Then any contract satisfying conditions (i)–(iii) and insuring that the condition T (18) e−ρ(τ−t) ubb (sτ yτ ) dτ Ft Qt ≤ −E∗ t
holds for all t is a truthful revelation contract. Since u is concave in b, the right sides of the inequalities (16) and (18) are positive, and (16) implies that the right side of (17) is positive as well. Thus Qt ≤ 0 for all t is a stronger, but simpler, sufficient condition. In particular, the incentive constraint (10) and Qt ≤ 0 together are sufficient to ensure that the agent’s optimality condition (9) holds, since mt ≤ 0. However the sufficient conditions (16)–(17) or (18) in the theorem are weaker in that they allow some positive Qt settings. We show below that these conditions, but not the stronger Qt ≤ 0 condition, are satisfied in our examples. Intuitively, the sufficient conditions trade off level and timing changes in the agent’s consumption profile under the contract. As discussed above, the
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contract links the volatility of the agent’s promised utility to his report, spreading out continuation utility so as to insure truthful revelation. However, the promised marginal utility state allows the contract to condition the spread of continuation utilities, making the volatility of future continuation utility depend on the current report. In particular, if an agent were to lie and set Δt < 0 for some increment δ > 0 of time, his promised future utility qt+δ would be lowered as δγt Δt < 0. If the stronger sufficient condition Qt < 0 holds, then the lie would also increase (i.e., make less negative) the promised marginal utility state pt+δ as δQt Δt > 0. Since this state variable gives the negative of marginal utility, this is associated with a reduction in expected marginal utility. This roughly means that after a lie, an agent could expect lower lifetime consumption overall (from the fall in q), but consumption would grow in the future (from the reduction in future marginal utility). Moreover, from the incentive constraint (10), we see that the current lie would lead to smaller utility dispersion in the future as γt+δ is now bounded by the smaller quantity −pt+δ . By using the concavity of utility and the continuity of sample paths, we show that contracts which appropriately balance these effects are incentive-compatible. In some cases the sufficient conditions (16)–(18) may be overly stringent or difficult to verify. To be sure that the contract does indeed ensure truthful revelation, we then must re-solve the agent’s problem facing the given contract. This is similar to the ex post incentive compatibility checks that Werning (2001) and Abraham and Pavoni (2008) conducted. Re-solving the agent’s problem typically would require numerical methods, as indeed would finding the contract itself. In the examples below, we verify that the sufficient conditions do hold, but we also illustrate how to analytically re-solve the agent’s problem. 5. OPTIMAL CONTRACTS ON AN INFINITE HORIZON I now turn to the principal’s problem of optimal contract design over an infinite horizon. Formally, I take limits as T → ∞ in the analysis above. Thus we no longer have the terminal conditions for the co-states in (11) and (12); instead we have the transversality conditions limT →∞ e−ρT qT = 0 and limT →∞ e−ρT pT = 0. For the purposes of optimal contract design, we can effectively treat the backward equations that govern the dynamics of promised utility and promised marginal utility as forward equations under the control of the principal. Thus the principal’s contract choice is a standard stochastic control problem. 5.1. Basic Layout As is standard, I suppose that the principal’s objective function is the expected discounted value of the transfers ∞ −ρt e st dt J = Ey 0
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where he discounts at the same rate as the agent. The principal’s problem is to choose a contract to minimize J subject to satisfying (i) the participation constraint (15), (ii) the promise-keeping constraints (11)–(12), and (iii) the incentive constraint (10). We focus on the relaxed problem and do not impose the sufficient conditions from Theorem 4.1, but instead check ex post whether they are satisfied (and if not, we directly check whether the contract is incentive compatible). While the participation constraint bounds q0 , p0 is effectively free and so we treat it as a choice variable of the principal. Thus a contract consists of {st γt Qt } and values for q0 and p0 . I focus on the dynamic programming approach to the principal’s problem.13 Abusing notation slightly, denote the principal’s value function J(y q p), and let Jy (y q p) be its partial derivatives. Via standard arguments (see, e.g., Yong and Zhou (1999)), the value function satisfies the Hamilton–Jacobi– Bellman (HJB) equation
(19) ρJ = min s + Jy μ(y) + Jq [ρq − u(s y)] {sγ≥−pQ}
+ Jp [ρp + γμ (y) + ub (s y)] +
σ2 [Jyy + Jqq γ 2 + Jpp Q2 + 2(Jyq γ + Jyp Q + Jpq γQ)] 2
where we suppress the arguments of J and its derivatives. Given the solution to (19), the initial value p0 is chosen to maximize J(y0 q0 p0 ). 5.2. Full Information As a benchmark, we consider first the full-information case where the principal observes all state variables. His problem in this case is to maximize J subject only to the participation constraint. It is possible to include this as a constraint at date zero, but to make the analysis more comparable with the previous discussion, we include qt as a state variable. The principal’s costs depend on yt and qt , which govern the amount of utility he must deliver to satisfy the participation constraint, but there are no direct costs associated with the marginal utility state pt . Denoting the full-information value function J ∗ (y q), we see that it solves an HJB equation similar to (19):
σ2 ∗ ∗ ∗ ∗ ∗ 2 ∗ ρJ = min s + Jy μ(y) + Jq [ρq − u(s y)] + [Jyy + Jqq γ + 2Jyq γ] {sγ} 2 13 Cvitani´c, Wan, and Zhang (2009) used a stochastic maximum principle approach to characterize contracts in a moral hazard setting. In an earlier draft, I used the maximum principle to provide some partial characterizations of the dynamics of optimal contracts.
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The optimality conditions are (20)
Jq∗ us (s y) = 1
∗ ∗ γ = −Jyq /Jqq
The first condition balances the instantaneous costs of payments with their impact on lessening future transfers. Since γ governs both the volatility of the utility promise q and its covariation with the report y, the second condition balances these effects. Letting s(y q) be the solution of the first optimality condition, the HJB becomes ∗ 2 (Jyq )
σ2 ∗ (21) Jyy − ∗ ρJ ∗ = s(y q) + Jy∗ μ(y) + Jq∗ ρq − u(s(y q) y) + 2 Jqq In general, the agent’s consumption varies with the state y. However, in the hidden endowment case, we obtain the standard result that a risk-neutral principal absorbs all the risk, completely stabilizing the agent’s consumption. In ¯ particular, when u(s b) = v(s + b), let c(q) = v−1 (ρq) be the constant consumption consistent with promised utility q. Then it is straightforward to verify that the solution to (21) is given by (22)
J ∗ (y q) =
¯ c(q) + j(y) ρ
where j(y) solves the second-order ordinary differential equation (ODE) (23)
1 ρj(y) = −y + j (y)μ(y) + j
(y)σ 2 2
Below we provide an explicit solution of this ODE for a particular parameterization. Thus the optimal full-information contract indeed calls for complete ¯ t ) − yt and γt = 0. Together these imply that constabilization, setting st = c(q ¯ 0 ) and sumption and promised utility are constant under the contract ct = c(q qt = q0 for all t. 5.3. Hidden Information We now return to the hidden information problem, where it is difficult to characterize the general case in much detail. We now assume that the incentive constraint (10) binds, so that γ = −p. We relax this condition in an example below and verify that it holds. However, we also conjecture that it holds more generally. Under full information, the principal need not worry about incentives and can provide constant promised utility. With hidden information, utility must vary to provide incentives. But because the agent is risk averse, the principal will typically find it optimal to induce as little volatility in the agent’s utility as possible so as to induce truthful revelation, and thus he will set γ at the lowest feasible level.
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The γ choice is constrained, while the other optimality conditions from (19) are (24)
Jq us (s y) − Jp ubs (s y) = 1
(25)
Jpp Q + Jyp − pJqp = 0
Relative to the full-information case, the first-order condition for the payment (24) has an additional term coming from the effect of consumption on the marginal utility state pt . That is, payments affect both the promised utility and promised marginal utility, and hence impact the agent’s future valuations. The condition (25) for Qt balances the effects of the variability of pt with the covariations of pt with yt and qt . We can use these optimality conditions and the HJB equation to solve for a candidate optimal contract. Then we can check whether the sufficient conditions from Theorem 4.1 hold. If not, we would directly solve the agent’s reporting problem given the contract to check incentive compatibility. We follow this procedure below. 6. A CLASS OF HIDDEN ENDOWMENT EXAMPLES In this section, we study a class of examples which allows for explicit solutions. We suppose b is the agent’s endowment and that the agent has exponential utility u(s b) = − exp(−θ(s + b)) As is well known, exponential utility with linear evolution often leads to explicit solutions, and this is once again the case here. We fully solve for the optimal contract and verify that it is indeed incentive compatible. Then we discuss how persistence affects the contract and compare our results to those in the literature. Some of the intermediate calculations for this section are provided in Appendix A.3. 6.1. Full Information As a benchmark, we first consider the full-information case from Section 5.2. Inverting the agent’s utility function, consumption under the contract is ¯ c(q) =−
log(−ρq) θ
It is easy to verify that the solution of the ODE (23) for the principal’s cost is j(y) = −
μ0 y − ρ(λ + ρ) λ + ρ
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We then have that the full-information cost function is J ∗ (y q) = −
μ0 y log(−ρq) − − ρθ ρ(λ + ρ) λ + ρ
Thus the principal’s full-information cost is linear and decreasing in the endowment y as well as being linear in the consumption equivalent of the promised utility q, which here implies that the cost is logarithmic in q. 6.2. Persistent Endowment We now suppose that the agent’s endowment is private information and is persistent. We show that the contract is similar to the full-information case, but now it may depend on the promised marginal utility pt . However, when the endowment is permanent, the ratio pt /qt is necessarily constant. This follows from the proportionality of utility and marginal utility with exponential preferences, since (13) and (14) with λ = 0 imply pt = θqt . Thus we can dispense with pt as a separate state variable. When λ > 0, the pt /qt ratio may vary over time. However, we show that under the optimal contract, it is indeed constant at a level which depends on λ. This is clearly a very special case of our general results, as typically both pt and qt would be necessary state variables. However, this special case facilitates explicit calculations. We also show that the degree of risk sharing increases as the endowment becomes less persistent. There is no risk sharing at all in the permanent case, while in the i.i.d. limit, the contract allocation converges to full information with complete consumption stabilization. Thus the distortions inherent in the discrete-time i.i.d. environments of Thomas and Worrall (1990) and others vanish in continuous time, as we discuss below. 6.2.1. The Optimal Contract With an Arbitrary Initial Condition We first consider contracts with an arbitrary initial condition for the marginal utility state p0 . Later, we consider the choice of the initial value. Thus first consider the principal’s problem given the states (y q p). We show in Appendix A.3.1 that the principal’s cost function can be written (26)
J(y q p) = j0 + j1 y − j2 log(−q) + h(p/q)
for some constants (j0 j1 j2 ) and some function h which solves a second-order ODE given in the Appendix. Letting k = p/q, in the permanent endowment case, kt = θ for all t, but when λ > 0, then kt becomes a state variable. We also show in Appendix A.3.1 that the agent’s consumption under the contract c and the diffusion coefficient Q on marginal utility can be written c(q k) = −
ˆ log(−qc(k)) θ
ˆ Q(q k) = −qQ(k)
PERSISTENT PRIVATE INFORMATION
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ˆ ˆ for functions c(k) and Q(k) which we provide there. The consumption and cost functions are similar to the full-information case, but they depend on the ratio of the utility and marginal utility promises k, which in general will vary over time. Using the previous results, the dynamics of the co-states can be written (27)
ˆ t )]qt dt − σpt dWt dqt = [ρ − c(k ˆ t ) dWt ˆ t )qt ] dt − σqt Q(k dpt = [(ρ + λ)pt − θc(k
Here we presume truthful revelation, so the principal effectively observes the y true shocks Wt = Wt . Applying Ito’s lemma gives the dynamics of the ratio: (28)
ˆ t )) dt ˆ t )(kt − θ) + λkt + σ 2 kt (k2t − Q(k dkt = c(k ˆ t )) dWt + σ(k2t − Q(k
When λ = 0, this ratio is necessarily constant, but now it may evolve stochastically. But we show next that if the ratio is initialized optimally, it remains constant. 6.2.2. The Optimal Initial Condition Recall that the marginal utility state pt is an endogenous, backward stochastic differential equation. Thus its initial condition is not specified, but is instead free to be chosen by the principal. While the optimal choice of k0 for λ > 0 is difficult to establish analytically, in Appendix A.3.1 we verify numerically that ρθ . This result is intuitive, as it is proporthe optimal initial condition is k∗0 = ρ+λ tional to the ratio of the discount rates in the utility and marginal utility state variables. Note that although the choice of initial condition required numerical methods, conditional on it, the rest of our results are analytic. Given k∗0 , the ˆ ∗ ) = (k∗ )2 , which in ˆ ∗0 ) = ρ and Q(k expressions in the Appendix then imply c(k 0 0 turn imply c(q k∗0 ) = −
log(−ρq) ¯ = c(q) θ
Q(q k∗0 ) = −q(k∗0 )2
Moreover, from (28) we see that the promised utility and marginal utility states remain proportional throughout the contract: kt = k∗0 for all t. The principal’s cost function is nearly the same as the full-information case, with the only change being an additional additive constant term. In particular, using the expressions in Appendix A.3.1 and letting p∗0 = k∗0 q0 , we have J(y0 q0 p∗0 ) = J ∗ (y0 q0 ) +
σ 2θ 2(ρ + λ)2
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The additional cost of not observing the endowment is increasing in the local variance of the endowment, which is a measure of the “size” of the private information. The cost is also increasing in the agent’s risk aversion parameter θ, reflecting the additional expected payments which must be made to compensate for risk, and decreasing in rate of time preference, reflecting the effective horizon of the payments. Finally, the additional cost declines as λ increases and the endowment becomes less persistent. The optimal contract has striking implications for the dynamics of consumption and utility. In particular, consumption is the same function of promised utility whether the agent’s endowment is private information or not. However, in the full-information case, promised utility is constant, while with private information it varies. In fact, using cˆ = ρ and pt = k∗0 qt in (27), we have (29)
dqt = −
σρθ qt dWt ρ+λ
and thus promised utility follows a martingale. This equation can be solved explicitly, leading to an explicit solution for consumption as well: σ 2 ρ2 θ 2 σρθ Wt t− qt = q0 exp − 2(ρ + λ)2 ρ+λ
¯ 0) + ct = c(q
σ 2 θρ2 σρ Wt t+ 2 2(ρ + λ) ρ+λ
Therefore, consumption follows a Brownian motion with drift, growing over time so as to provide risk compensation for the increasing variability. As consumption tends to grow over time, promised utility tends toward its upper bound of zero. The volatility of consumption provides one measure of the degree of risk sharing that the contract provides. When λ = 0, the endowment and consumption are both Brownian motions with drift, having the same local variance σ 2 . Thus when the endowment is permanent, the contract provides no risk sharing. It only alters the time path of consumption, providing more consumption up front in exchange for a lower average growth rate. However, as information becomes less persistent, the local variance of consumption falls and the amount of risk sharing increases. √ In particular, in the limit approaching an i.i.d. endowment (with σ = σ¯ λ and λ → ∞), the optimal contract with private information converges to the efficient, full-information allocation. The results are illustrated in Figure 1, which plots the distributions of consumption under autarky and under the contract with two different values of λ. In each case we show the mean of the distribution along with 1 and 2 standard deviation bands. We scale the drift and diffusion parameters with λ to main-
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FIGURE 1.—Distributions of consumption under the optimal contract and under autarky for two different values of λ.
tain a constant unconditional mean and variance.14 The top two panels plot the distributions of the endowment, and hence consumption under autarky. The endowment is stationary, so after being initialized at a common point (the unconditional mean), the variance of the distribution grows until it reaches the stationary distribution. Clearly, with less persistence (larger λ), this convergence is much faster. The bottom two panels plot the consumption distribution under the optimal contract. Here we see that early on the distribution of consumption under the contract is tighter than under autarky, reflecting the risk sharing properties of the contract. Clearly, with less persistence, there is more risk sharing, so the consumption distribution is even more compressed. However, unlike the autarky case, consumption is nonstationary under the contract. Consumption has an upward trend (which is difficult to see in the figure) and its distribution fans out over time. 14
In particular, we choose θ = 1, ρ = 01, σ¯ = 025, and μ¯ = 1.
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6.3. Verifying Incentive Compatibility In Appendix A.3.2, we verify that the sufficient conditions (16) and (17) from Theorem 4.1 hold. Thus we know the contract derived above is indeed the optimal incentive-compatible contract. However, directly verifying that the contract is incentive compatible helps to understand how the contract provides incentives for revelation. When the λ = 0, this is straightforward. If the agent were to deviate from truth-telling and report a lower endowment, then he would receive a permanently lower promise of future utility and hence future consumption. This would exactly balance the increased consumption he would gain due to his lower transfer payment to the principal. Since we allow the possibility of lying, we distinguish between the true eny dowment shock Wt and the principal’s prediction of it Wt . Under the optimal contract and an arbitrary reporting strategy, we have ct = s(qt yt ) + yt − mt log(−ρqt ) − yt + yt − mt θ σ 2θ y ¯ 0) + t + σWt − mt = c(q 2 t σ 2θ ¯ 0) + t + [dyτ − μ0 dτ] − mt = c(q 2 0 t 2 σ θ ¯ 0) + t + [dbτ + (Δτ − μ0 ) dτ] − mt = c(q 2 0 =−
¯ 0) + = c(q
σ 2θ t + σWt 2
where we have used bt = b0 + μ0 t + σWt and the definition of mt . Thus the agent’s consumption is independent of his reporting process yt , depending only on his initial utility promise, a deterministic transfer, and the underlying shocks to his endowment. Put differently, the contract ensures that the agent receives the same consumption whatever he reports. Thus the contract is consistent with truthful revelation, although reporting truthfully is by no means the unique best response of the agent. When λ > 0, directly verifying incentive compatibility requires fully solving the agent’s problem facing the contract. These more complicated calculations are carried out in Appendix A.3.2. 7. RELATIONSHIP WITH PREVIOUS LITERATURE 7.1. General Discussion My results in the previous section are in sharp contrast to the discrete-time models with i.i.d. private information. As discussed above, Thomas and Wor-
PERSISTENT PRIVATE INFORMATION
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rall (1990) showed that the optimal contract leads to immiseration, with the agent’s promised utility tending toward minus infinity. But under the optimal contract here, consumption grows on average over time. In both our setting and theirs, the distribution of promised utility fans out over time so as to provide incentives. However, in our setting, the Brownian motion shocks driving the endowment process have a distribution which naturally fans out over time. Therefore, by linking the agent’s promised utility (and hence consumption) to the reported endowment shocks, the principal is able to achieve the fanning out of the utility distribution which is required to provide incentives, yet still have consumption increase over time. The speed at which the utility distribution fans out depends on the persistence of the information. As the endowment becomes less persistent, the effective life of the private information is shorter, and smaller increases in the utility dispersion are required to ensure truthful reporting. To provide t a bit more detail, note that we can define tthe cumulative endowment Yt = 0 yu du and the cumulative transfers St = 0 su du as ordinary (not stochastic) integrals. By requiring Yt and St to thus be absolutely continuous, we are able to define utility flows. If Yt were of unbounded variation, t over their t Y say by specifying Yt = 0 yu du + 0 σu dWu , then it would be unclear how to define utility over the flow, except in the risk-neutral case studied by DeMarzo and Sannikov (2006). While this technical difference may seem relatively minor, it implies that as we approach the i.i.d. limit, the information frictions vanish. In particular, our limit result is driven by the fact√that the diffusion term in (29) vanishes as λ → ∞, since σ increases with λ. If σ increased with λ, then there would be nonzero frictions in the limit, but the variance of the endowment process would explode. The efficient limit result is also implicit in Thomas and Worrall (1990), a point we make explicit in the next section. In the course of establishing that as the discount factor tends to unity, the contract converges to the first best (their Proposition 4), they showed that the deviations from efficiency are tied to the cost of inducing the efficient actions for one period. Rather than letting discount rates decline as in their limit, our results effectively let the period length shrink to approach continuous-time. But the consequence is the same: the inefficiency vanishes. Thus as periods become shorter, deviations from efficiency are sustainable if either the cumulative endowment has unbounded variation, which causes conceptual problems, or the endowment is persistent, which is quite natural. We illustrate the efficiency of the continuous-time limit in Thomas and Worrall (1990) in the next section. As a final comparison, note that although the inverse Euler equation does not hold in our environment, it does hold in the continuous-time moral hazard model in Williams (2009). In the moral hazard (hidden effort) setting, the contract specifies a consumption payment to the agent conditional on realized output. However, effort is costly to the agent and thus the incentive constraints reflect the instantaneous effect of the information friction. By contrast, in the
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hidden information model, lying has no instantaneous effect on the agent: it only affects his future expected transfers. Thus the incentive constraints are fully forward-looking, implying constraints on the evolution of promised utility. Thus whether the inverse Euler equation holds does not depend on whether time is continuous or discrete, but it is sensitive to the source of the information friction. 7.2. Efficiency in a Limit of Thomas and Worall In this section, we demonstrate that as the period length shrinks to zero in the discrete-time model of Thomas and Worrall (1990), the contract converges to the efficient allocation. As discussed previously, such a limit is implicit in their paper; here we make the argument explicit for the case of exponential preferences and binary endowment realizations. In particular, suppose now that time is discrete with interval ε between periods. The agent’s endowment realization bt each period is i.i.d. and can take on values b1 with probability 05 and b2 > b1 with probability 05. The restriction to equal probabilities of high and low realizations helps preserve the link with our continuous-time model, where, over short intervals, upward and downward movements are equally likely. The agent has exponential preferences with parameter θ as above and discount factor α = exp(−ρε). Each period, upon receiving a report of bi , the contract specifies a transfer si and a utility promise (now relative to autarky) of qi for the next period. To simplify some notation, ¯ let u¯ = 05[u(b1 ) + u(b2 )], and define the value of autarky as A = uε/(1 − α). In their Proposition 6, Thomas and Worrall (1990) showed that in this setting, the optimal contract can be represented as a set of numbers 0 ≤ a2 ≤ a1 and d1 ≥ d2 ≥ 0 which satisfy exp(−θsi ) = −ai (q + A) qi = di q + (di − 1)A 05 05 + = −A a1 a2 05 05 + = 1 d1 d2 along with the (binding) downward incentive constraint u(s2 + b2 )ε + αq2 = u(s1 + b2 )ε + αq1 and the promise-keeping constraint 05[u(s1 + b1 )ε − u(b1 )ε + αq1 ] + 05[u(s2 + b2 )ε − u(b2 )ε + αq2 ] = q
PERSISTENT PRIVATE INFORMATION
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To solve for the optimal contract, we first substitute for si and qi in terms of ai and di , then use the normalizations of ai and di to eliminate a2 and d2 . Finally, we combine the incentive and promise-keeping constraints to obtain a single equation in a1 : 05 05α ¯ 1 ε) = 0 − exp(−θb2 ) (30) + a1 ε + − (1 + ua 05 u¯ 05α + 1− ¯ 1ε 1 − α a1 ε 1 + ua The solution of equation (30), which is a cubic equation in a1 , determines the optimal contract. After some simplification, it can be written 05
(1 − α) (1 − α)2 a1 + exp(−θb2 )(1 − 05α) + (1 + 05(2 − α))u¯ 2 ε ε
¯ ¯ + u[exp(−θb − 15α)]a21 2 )(2 − 15α) + u(25 3 ¯ + u¯ 2 [exp(−θb2 ) + u]εa 1 = 0
We now find the limiting solution as the period length shrinks. We use the facts that limε→0 (1−α) = ρ and limε→0 α = 1, and note that the cubic term is ε of order ε, so it vanishes in the limit. Therefore, the limit contract solves the quadratic equation ¯ ¯ 21 = 0 05ρ2 + [05 exp(−θb2 ) + 15u]ρa 1 + [15 exp(−θb2 ) + u]a The cost-minimizing root of this equation is then a1 = ρ exp(θb1 ) Then note that utility in state 1 under the contract is u(s1 + b1 ) = − exp(−θb1 ) exp(θs1 ) = exp(−θb1 )a1 (q + A) = ρ(q + A) with the same result for state 2. So consumption and utility are completely smoothed under the contract, and promised utility remains constant (d1 = d2 = 1). Thus the continuous-time limit of the discrete-time i.i.d. model of Thomas and Worrall (1990) agrees with the i.i.d. limit of our continuoustime persistent information model above, with both implying efficiency in the limit. 8. A PRIVATE TASTE SHOCK EXAMPLE We now turn to an example where the agent has private information about a preference shock which affects his marginal utility of consumption. Similar
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discrete-time models have been studied by Atkeson and Lucas (1992) with i.i.d. shocks, and Kapicka (2006) with persistent private information. We now suppose that b is the logarithm of a private taste shock and that the agent has power utility over consumption, u(s b) = exp(−b)
s1−θ 1−θ
where the coefficient of relative risk aversion is θ > 1.15 Clearly this specification satisfies all of our assumptions, as ub > 0, ubb < 0, and ubbb > 0. This example is similar to the previous one, with some differences that we discuss below. For simplicity, we focus here on the permanent shock case, as this allows for explicit solutions. Since ub = −u in this model, (11)–(12) imply that p = q; thus we can dispense with p as a separate state variable. We also assume that there is no trend in the taste shocks and E(exp(−b)) = 1, which 2 implies that μ0 = σ2 . 8.1. Full Information As before, we begin with the full-information benchmark. Using the parameterization above, we show in Appendix A.3.3 that the optimality conditions (20) and the HJB equation (21) can be solved explicitly: 1 ∗ y ((1 − θ)q)1/(1−θ) J (y q) = A exp 1−θ 1 1/θ y ((1 − θ)q)1/(1−θ) s(y q) = A exp 1−θ where
σ 2 (θ − 1) A≡ ρ+ 2θ2
θ/(1−θ)
The flow utility is then proportional to promised utility q and independent of y: 1−θ 1 exp(−y) 1/θ 1/(1−θ) A exp y ((1 − θ)q) u(s(y q) y) = 1−θ 1−θ σ 2 (θ − 1) q = ρ+ 2θ2 15
We restrict attention to θ > 1 so that u < 0. Our results hold more generally.
PERSISTENT PRIVATE INFORMATION
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Instead of fully stabilizing consumption and utility, as the full-information contract did in the hidden endowment example, here the optimal contract offsets the impact of the taste shocks on the current utility flow. Since the taste shocks are stochastic, promised utility is then no longer constant. In particular, from (20), we have (31)
γ=−
∗ Jyq ∗ qq
J
1 = − q θ
Therefore, the evolution of promised utility (11) is dqt = −
σ 2 (θ − 1) σ qt dt − qt dWt 2θ2 θ
Thus since qt < 0 and θ > 1, promised utility has a positive drift and thus tends to increase over time. The taste shocks get translated into fluctuations in promised utility, and the larger is the degree of risk aversion θ, the smaller is the volatility of promised utility. 8.2. Private Information When the agent’s taste shocks are private information, the principal must provide incentives for truthful revelation. He does this by making promised utility respond more to new observations than it does under full information. In particular, the incentive constraint (10) here becomes γ = −p = −q, again assuming the constraint binds. Since θ > 1, comparison with (31) shows that an incentive-compatible contract clearly provides less insurance against taste shocks than the first best contract. As we show in Appendix A.3.3, the solution of the HJB equation (19) is very similar to the full-information case, with a different leading constant: 1 J(y q) = B exp y ((1 − θ)q)1/(1−θ) (32) 1−θ 1 1/θ s(y q) = B exp y ((1 − θ)q)1/(1−θ) 1−θ where B ≡ (ρ)θ/(1−θ) θ Moreover, since θ > 1 and 1−θ < 0, we see that B > A. Thus, as one would expect, the principal’s costs J(y q) of delivering a given level of promised utility q given a truthful report of y are higher under private information than with full information. In addition, the agent obtains a larger transfer s under private information, as he must be compensated for bearing additional risk. Once again,
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the flow utility is proportional to promised utility q, but now with proportionality factor ρ. This implies that under the optimal contract, promised utility is a martingale, (33)
dqt = −σqt dWt
This is just like the hidden endowment case (29) with λ = 0. Once again, with permanent shocks there is a complete lack of risk sharing, as the full magnitude of the shocks to the private state get translated into shocks to promised utility. In fact, the contract can be implemented via a constant payment st = s¯ , with s¯ chosen to deliver the initial promised utility q0 . In turn, this insures that qt =
s¯1−θ exp(−yt ) (1 − θ)ρ
which satisfies (33) and when substituted into (32), gives indeed that s(y q) = s¯. Thus the agent’s consumption is independent of his reports yt and there is no risk sharing. As above, we could verify the sufficient condition (18) that ensures implementability, but the constancy of the payment st yields it directly. As the payment is independent of the report, it is clearly (weakly) consistent with truthful revelation. Although direct comparisons are a little difficult given the differences in the setups, my results differ substantially from Kapicka (2006). We both use similar ideas, using a first-order approach to contracting to justify a recursive representation of a contract with additional endogenous state variables. However, our state variables differ. Kapicka (2006) took the agent’s promised marginal utility of consumption as his additional state variable, while in my approach, the marginal utility of the hidden state is the relevant state variable. With the multiplicative permanent taste shocks in this example, the promised marginal utility of the hidden state is equal to the level of promised utility, and hence is superfluous. More generally, my state variable captures the lifespan of the private information, as the promised marginal utility is discounted by the persistence of the hidden state. Our implications for contracts differ as well. In a numerical example similar to the one in this section, Kapicka (2006) showed how the contract may significantly distort the evolution of the agent’s marginal utility of consumption. This leads to a sizeable “intertemporal wedge” that distorts the savings margin of the type discussed by Golosov, Kocherlakota, and Tsyvinski (2003) among others. But in our case, where we obtain explicit analytic solutions for the optimal contract, consumption is deterministic. This implies that the marginal utility of consumption is a constant multiple of promised utility and so is a martingale. Thus we obtain a standard Euler equation with no intertemporal wedge. The reasons for these differences are largely the same as in the hidden endowment example above. Kapicka’s results are driven by the interaction between current transfers and future promises, with the response of current consumption to a
PERSISTENT PRIVATE INFORMATION
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report playing a key role. But as we have seen, a lie cannot instantaneously affect the agent’s consumption in continuous-time, so all incentive provision is loaded on promised utility. Since the shocks are permanent, a current lie does not affect the evolution of future reports. In this example, this means that a lie effectively has a multiplicative, scale effect on the reported taste shock process, and hence on utility. With the preferences in this example, future behavior is unaffected by a permanent multiplicative scale effect on utility. Thus the only way the principal can ensure truthful revelation is to make the transfer independent of the report. Such a contract does not lead to an intertemporal wedge, but it clearly does not insure the agent against the taste shocks either. Clearly some of my results are special to the permanent shock case, as with persistent but not permanent shocks a current lie would influence future reports. Kapicka (2006) considered intermediate cases of persistent but not permanent shocks, and showed numerically how persistence affects the results. While we were able to do so analytically in the hidden endowment example above, such results in this setting would require numerical methods. 9. CONCLUSION In this paper I have developed some relatively general methods to study contracting problems with persistent private information. I have shown how the methods can lead to explicit solutions in examples. By casting the model in continuous-time, I was able to use powerful tools from stochastic control. These allowed me to deduce that an optimal contract must condition on two additional endogenous state variables, the agent’s promised utility and promised marginal utility under the contract. While the use of promised utility as a state variable is now widely applied, the marginal utility state variable is more novel, although it does have some precedents in the literature. My main results use the representation of the contract derived from analyzing the agent’s necessary conditions for optimality. The key theoretical results in the paper provide some sufficient conditions which guarantee that the contract does indeed provide incentives for truthful revelation. In my examples, I show that these conditions are verified, but in other settings, they may be more difficult to establish. In such cases, my main results could still be used, although one may then have to verify directly incentive compatibility. Apart from special cases like my examples, such solutions would require numerical methods. This is not surprising, as typical dynamic decision problems, not to mention the more complex contracting problems we consider here, require numerical solution methods. In any case, there is an array of well developed numerical methods for solving partial differential equations like those which result from our analysis. In addition to laying out a framework, I have established some substantive results on the nature of continuous-time contracts with persistent private information. The examples highlight the close link between the persistence of the
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private information and the size of efficiency losses this information causes. In particular, when shocks are permanent, distortions are the largest and risk sharing may break down, while in the i.i.d. limit, we obtain efficiency. Although this efficient limit differs from the discrete-time i.i.d. models in the literature, it is inherent in those models as period length shrinks. Correspondingly, I have shown that the “inverse Euler equation” of Rogerson (1985) and Golosov, Kocherlakota, and Tsyvinski (2003) need not hold, and the related immiseration results of Thomas and Worrall (1990) may fail. Both of these are sensitive to the details of the contracting problem, depending on how deviations from the contracted behavior affect utility and the state evolution. APPENDIX A.1. Derivation of the Co-State Evolution Given the change of variables, the evolution of the co-states follows from the maximum principle in Bismut (1973, 1978). The maximum principle also requires some smoothness and regularity conditions, and a linear growth condition on μ: |μ(y)| ≤ K(1 + |y|)
for some K
All of these conditions hold under the affine specification that we focus on. The text spells out the general co-state evolution in the lines preceding (11)– (12). Carrying out the differentiation, we have (suppressing arguments of functions) ∂H(Γ z) z z = u + γ(μ + Δ) + pΔ + ub + (Γ γ + Qz)μ 2 ∂Γ Γ Γ ∂H(Γ z) z = −ub + Q(μ + Δ) − γ + Q μ ∂z Γ Thus under truthful revelation Γ = 1 and z = Δ = 0, we have ∂H(Γ z) = u + γμ ∂Γ ∂H(Γ z) = −ub + Qμ − γμ ∂z Substituting the above evolution and using the change of measure (6) gives dqt = [ρqt − u(st yt ) − γt μ(yt )] dt + γt σ dWt 0 = [ρqt − u(st yt )] dt + γt σ dWt ∗ dpt = [ρpt + ub (st yt ) − Qμ(yt ) + γt μ (yt )] dt + Qt σ dWt 0 = [ρpt + ub (st yt ) + γt μ (yt )] dt + Qt σ dWt ∗
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Then using μ = −λ, we have (11)–(12) in the text. A.2. Proof of Theorem 4.1 The necessity of the conditions is obvious, as they were derived from the agent’s necessary optimality conditions. For the converse, we first use the representations of the agent’s utility and marginal utility under the contract. Then we evaluate the potential gain from deviating from truth-telling, which involves calculating expected utility under an alternative reporting strategy. Using the concavity of u and U, we bound the utility difference by a linear approximation. We then use the incentive constraint to further bound the utility gain. The last steps of the proof hold fixed the alternative report and show that even if the agent had lied in the past, under the stated sufficient conditions, his utility gain from lying in the future is negative. Although it is natural for discussion and computation to work with the discounted utility and marginal utility processes {qt pt } defined in (11)–(12), it is easier here to work with the undiscounted processes q˜ t = e−ρt qt and p˜ t = e−ρt pt . Then making this substitution into (11) and integrating gives (A.1)
−ρT
e
T
U(ST yT ) = q˜ T = q0 −
−ρt
e
T
u(st yt ) dt +
0
e−ρt γt σ dWt ∗
0
Further, using (12) and (3) along with the substitution for p˜ t gives (A.2)
T
p˜ T mT = 0
e−ρt [pt Δt − λmt γt + mt ub (st yt )] dt
T
+
e−ρt Qt mt σ dWt ∗
0
For an arbitrary reporting policy Δ that results in yˆ = y − m, from (6) we have (A.3)
σ dWt ∗ = σ dWt Δ + [μ(yt − mt ) + Δt − μ(yt )] dt
Now for this arbitrary reporting policy, we wish to compute the gain of deviating from the truthful reporting strategy y (which gives promised utility q0 ), (A.4)
T
e−ρt [u(st yt − mt ) − u(st yt )] dt
ˆ − q0 = EΔ V (y) 0
T
+
−ρt
e 0
γt σ dWt
∗
+ EΔ e−ρT [U(ST yT − mT ) − U(ST yT )]
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NOAH WILLIAMS
where we have used (A.1). Now by the concavity of U, we have
(A.5) EΔ e−ρT [U(ST yT − mT ) − U(ST yT )] ≤ −EΔ [e−ρT Ub (ST yT )mT ] = EΔ [p˜ T mT ] Hence, combining (A.5) and (A.2) with (A.4), we get (A.6)
ˆ − q0 V (y) ≤ EΔ
T
−ρt
e
[u(st yt − mt ) − u(st yt ) + mt ub (st yt )] dt
0
T
+ EΔ
e−ρt [pt Δt − λmt γt ] dt
0
T
+
−ρt
e
σ[γt + Qt mt ] dWt
∗
0
T
= EΔ
e−ρt [u(st yt − mt ) − u(st yt ) + mt ub (st yt )] dt
0
T
+ EΔ
e−ρt pt Δt − λmt γt
0
+ (γt + Qt mt )(μ(yt − mt ) + Δt − μ(yt )) dt
Here the equality uses the change of variables from (A.3) and the fact that the stochastic integral with respect to W Δ has expectation zero with respect to PΔ . Next we use the incentive constraint (10) to eliminate the pt Δt and γt Δt terms from (A.6), as their sum is bounded by zero. In addition, the functional form for μ gives γt (μ(yt − mt ) − μ(yt ) − λmt ) = 0
and
Qt mt (μ(yt − mt ) − μ(yt )) = λQt m 2 t
Using these results and regrouping terms in (A.6), we have (A.7)
T
ˆ − q0 ≤ EΔ V (y)
e−ρt [u(st yt − mt ) − u(st yt )
0
+ mt ub (st yt ) + λQt m + Qt mt Δt ] dt 2 t
PERSISTENT PRIVATE INFORMATION
T
= EΔ
1267
e−ρt [u(st yˆt ) − u(st yˆt + mt )
0
+ mt ub (st yˆt + mt )] dt
T
+ EΔ
−ρt
e
(λQt m + Qt mt Δt ) dt 2 t
0
ˆ By the concavity of u, we where the equality follows from the definition of y. know that the sum of the first three terms in (A.7) is negative, but to bound the entire expression, the terms in mt on the last line cause some additional complications. However, when Qt ≤ 0, then we have both Qt m2t ≤ 0 and Qt mt Δt ≤ 0. Thus the utility gain from lying is bounded by zero and the contract ensures truthful revelation. We now develop the weaker sufficient conditions stated in the theorem which allow Qt > 0. Since the expectation in (A.7) is calculated with respect to PΔ , we can treat yˆ as fixed and vary m. To make clear that we are fixing the measure over yˆ (but now varying Δ), we denote Pyˆ = PΔ for ˆ In other words, we have changed the meathe particular Δ resulting in y. ˆ and we now fix sure in calculating utility under the reporting strategy y, that measure but vary m. Thus we are implicitly varying the truth y, but we have eliminated the dependence of the problem on y. That is, we now solve T sup Eyˆ e−ρt [u(st yˆt ) − u(st yˆt + mt ) {Δt }
0
+ mt ub (st yˆt + mt ) + λQt m + Qt mt Δt ] dt 2 t
subject to (3) and m0 = 0. By (A.7), we know that this maximized value gives ˆ − q0 . an upper bound on V (y) The Hamiltonian for this relatively standard stochastic control problem, with co-state ξ associated with m, is (A.8)
ˆ − u(s yˆ + m) + mub (s yˆ + m) H ∗ (m Δ) = u(s y) + λQm2 + QmΔ + ξΔ
Thus the optimality condition for truth-telling (Δt = 0) is (A.9)
Qt mt + ξt ≥ 0
Unlike the incentive constraint (10), we want to be sure that (A.9) holds for all mt . That is, we provide conditions to ensure that even if the agent had lied in the past, he will not want to do so in the future.
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NOAH WILLIAMS
From Haussmann (1986), we know that (A.9) is sufficient for optimality when the maximized Hamiltonian H ∗ (m 0) is concave in m. Note that we have ∗ (m 0) = ubb (s yˆ + m) + mubbb (s yˆ + m) + 2λQ Hmm
= ubb (s y) + mubbb (s y) + 2λQ Now using the assumption ubbb ≥ 0, we know mubbb ≤ 0. Thus we have ∗ (m 0) ≤ ubb (s y) + 2λQ Hmm
Then the required condition (16) in part (b) of the theorem guarantees that ∗ H ∗ (m 0) is concave. When λ = 0, Hmm (m 0) is bounded above by ubb (s y). So for part (c), the concavity of u assures that H ∗ (m 0) is concave without any further restrictions. We find the evolution of the co-state ξ by differentiating the Hamiltonian: dξt = [ρξt − Hm∗ (m 0)] dt + ηt dWt
yˆ yˆ
= [ρξt − mt (ubb (st yˆt + mt ) + 2λQt )] dt + ηt dWt ξT = 0 As in (13) and (14), the solution of this can be written
T
−ρ(τ−t)
ξt = Eyˆ
e
ms (ubb (sτ yˆτ + mτ ) + 2λQτ ) dτ Ft
t
But the optimality condition (A.9) ensures that Δτ = 0 and thus mτ = mt for τ ≥ t. So then we have
T
ξt = mt Eyˆ
−ρ(τ−t)
e
(ubb (sτ yˆτ + mt ) + 2λQτ ) dτ Ft
t
Substituting this into (A.9) gives
T
Qt mt + mt Eyˆ
e−ρ(τ−t) (ubb (sτ yˆτ + mt ) + 2λQτ ) dτ Ft ≥ 0
t
When mt = 0, this condition clearly holds, which is to say that if the agent never lied, he will not find it optimal to do so. But when mt < 0, this implies (A.10)
Qt ≤ −Eyˆ
T
−ρ(τ−t)
e t
(ubb (sτ yˆτ + mt ) + 2λQτ ) dτ Ft
PERSISTENT PRIVATE INFORMATION
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But since ubbb ≥ 0 and mt < 0, we know −ubb (sτ yˆτ + mt ) ≥ −ubb (sτ yˆτ ). Therefore, T Qt ≤ −Eyˆ e−ρ(τ−t) (ubb (sτ yˆτ ) + 2λQτ ) dτ Ft t
is sufficient to guarantee that (A.10) holds. Since yˆ was arbitrary, this is the same as (17) in part (b) of the theorem. Clearly, setting λ = 0 gives (18) in part (c). Thus we have shown that Δt = mt = 0 for all t is the solution of the control problem above, under the conditions (16) and (17), or (18) when λ = 0. Clearly ˆ − q0 ≤ 0. with Δt = mt = 0, the maximized objective in (A.7) is zero, so V (y) Thus the utility gain from lying is bounded by zero, and the contract ensures truthful revelation. A.3. Calculations for the Examples A.3.1. Calculations for the Hidden Endowment Examples We first verify the guess (26) of the form of the value function. From the optimality conditions (24)–(25) and the form of the guess, we get log θ log(Jq + θJp ) + −y θ θ log θ log(j2 + h (k)(k − θ)) − log(−q) + − y = θ θ
pJqp h (k) ˆ Q= + k ≡ −qQ(k) = −qk
Jpp h (k) s=
Then substituting these into the HJB equation (19), we get j0 =
μ0 log θ − ρθ ρ(ρ + λ)
j1 = − j2 =
1 ρ+λ
1 ρθ
while h(k) satisfies the second-order ODE, 1 1
+ h (k)(k − θ) + λh (k)k (A.11) ρh(k) = log θ ρθ h (k)2 σ 2 k2 1 − + 2 ρθ h
(k)
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FIGURE A.1.—The function h(k) from the principal’s cost function with λ = 005. The dotted lines show the optimal choice k∗0 = ρθ/(ρ + λ) and its corresponding value h(k∗0 ).
Thus we have verified the guess. The agent’s consumption under the contract can then be written: c(q k) =
ˆ log(−qc(k)) log(1/ρ + θh (k)(k − θ)) log(−q) − ≡− θ θ θ
While it does not seem feasible to solve for h(k) analytically, it is relatively simple to solve the ODE (A.11) numerically.16 To find the optimal initial condition, we numerically solve for h and then pick the minimum. We have done this ρθ for a range of different parameterizations, all yielding the same result k∗0 = ρ+λ . One numerical example is illustrated in Figure A.1, which plots h(k) when λ = 005.17 The minimum clearly occurs at k∗0 , with the cost function increasing 16 It is relatively simple, since the equation is a second-order ODE. However, the equation has singularities which complicate matters somewhat. In practice, solving (A.11) as an implicit ODE has worked best. 17 The preference parameters are θ = 1 and ρ = 01. For the endowment, we scale the drift and diffusion so that we approximate an i.i.d. N(1 0152 /2) process as λ → ∞. Thus we set √ σ = 015 λ and μ0 = λ.
PERSISTENT PRIVATE INFORMATION
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rapidly for larger k and quite slowly for smaller k. Similar results were found for a range of different parameterizations. Note as well that since h (k∗0 ) = 0, ˆ ∗ ) = (k∗ )2 . For the principal’s cost, note that at k∗ ˆ ∗0 ) = ρ and Q(k we have c(k 0 0 0 we have 1 σ 2θ 1 ∗ h(k0 ) = log + ρθ ρθ 2(ρ + λ)2 A.3.2. Calculations That Verify Incentive Compatibility We now verify that the sufficient conditions (16) and (17) from Theorem 4.1 hold. Under the optimal contract, we have ¯ t )) = θ2 ρqt ubb (st yt ) = v
(yt + st ) = −θ2 exp(−θc(q In addition, we have ∗ 2 t 0
Qt = −(k ) q = −θ
2
ρ ρ+λ
2 qt
Thus since for all λ > 0, 2 ρ ρ < ρ+λ 2λ we have that (16) holds with a strict inequality when λ > 0. (The condition is irrelevant when λ = 0.) Moreover, ∞ −ρ(τ−t)
−Ey e [u (yτ + sτ ) + 2λQτ ] dτ Ft t
= −θ
2
2λρ2 ρ− (ρ + λ)2
= −θ2
ρ2 + λ2 qt (ρ + λ)2
≥ −θ2
ρ2 qt = Qt (ρ + λ)2
∞
e−ρ(τ−t) Ey [qτ |Ft ] dτ
t
where we use the fact that qt is a martingale and the expression for Qt above. Thus (17) holds for all λ ≥ 0 (with strict inequality when λ > 0) under the optimal contract. To directly verify incentive compatibility when λ > 0, we now re-solve the agent’s problem given a contract. In particular, the contract specifies a payment s(yt qt ) = −log(−ρqt )/θ − yt , so that under an arbitrary reporting strategy, the
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agent’s consumption is ct = −log(−ρqt )/θ − mt . Clearly, the agent’s problem depends on the endogenous state variable qt , which under his information set evolves as σρθ y qt dWt ρ+λ ρθ qt σ dWt + [μ(yt − mt ) + Δt − μ(yt )] dt =− ρ+λ
dqt = −
=−
σρθ ρθ qt (λmt + Δt ) dt − qt dWt ρ+λ ρ+λ
where we have changed measure from the principal’s to the agent’s information set. The agent’s problem is then to maximize his utility over reporting strategies subject to this law of motion for qt and the evolution (3) for mt . The problem no longer depends directly on the level of the endowment yt . Letting V (q m) denote his value function, we see that it satisfies the HJB equation
ρθ ¯ − m] − Vq q(λm + Δ) ρV = max − exp −θ[c(q) Δ≤0 ρ+λ 2 σρθ 1 + Vm Δ + Vqq q 2 ρ+λ Truth-telling is optimal if the following analogue of the incentive constraint (9) holds: −Vq
ρθ q + Vm ≥ 0 ρ+λ
It is easy to verify that the following function satisfies the HJB equation with Δ = 0: V (q m) =
q exp(θm)(ρ + λ) ρ + λ + θλm
Moreover, with this function, we have −Vq
ρθ q exp(θm)θ2 λ2 m q + Vm = ≥ 0 ρ+λ (ρ + λ + θλm)2
so that truth-telling is indeed optimal no matter what m is. In particular, we assumed that m0 = 0 and since the agent never lies, then mt = 0 and V (qt 0) = qt for all t. Thus the contract does indeed provide incentives for truthful revelation.
PERSISTENT PRIVATE INFORMATION
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A.3.3. Calculations for the Taste Shock Example First we solve the HJB equation (21) in the full-information case. Under our parameterization, this can be written ρJ ∗ = e−(1/θ)y (Jq∗ )1/θ + Jy∗ μ0 + Jq∗ ρq ∗ 2 (Jyq ) 1 −(1/θ)y ∗ 1/θ σ 2 ∗ e Jyy − ∗ (Jq ) + − 1−θ 2 Jqq Then we guess that the solution is of the form 1 y ((1 − θ)q)1/(1−θ) J ∗ = A exp 1−θ Carrying out the appropriate differentiation and substituting the expres1 y)((1 − sion into the HJB equation, we see that there is a common exp( 1−θ 1/(1−θ) factor on both sides of the equation. Collecting constants, we then θ)q) have that A solves θ ρ σ2 1 1 μ0 = + + A1/θ − A −ρ + 2 2 1−θ 1−θ 2 (1 − θ) θ(1 − θ) 1−θ Solving for A, simplifying, and using the value of μ0 yields the expression in the text. The solution of the hidden information case proceeds in much the same way, only now with a different γ. In particular, the HJB equation (19) can now be written ρJ = e−(1/θ)y Jq1/θ + Jy μ0 + Jq ρq −
1 −(1/θ)y 1/θ σ 2 e Jq + [Jqq q2 + Jyy − 2Jyq q] 1−θ 2
Then we guess that the solution is again of the form 1 y ((1 − θ)q)1/(1−θ) J = B exp 1−θ 1 Once again, there will be a common exp( 1−θ y)((1 − θ)q)1/(1−θ) factor on both sides of the equation. Collecting constants, we then have that B solves ρ σ2 θ 1 2 μ0 + + + − B −ρ + 1−θ 1−θ 2 (1 − θ)2 (1 − θ)2 (1 − θ)2
=
θ B1/θ 1−θ
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NOAH WILLIAMS
Solving for B, simplifying, and using the value of μ0 yields the expression in the text. REFERENCES ABRAHAM, A., AND N. PAVONI (2008): “Efficient Allocations With Moral Hazard and Hidden Borrowing and Lending,” Review of Economic Dynamics, 11, 781–803. [1234,1245,1248] ABREU, D., D. PEARCE, AND E. STACCHETTI (1986): “Optimal Cartel Equilibria With Imperfect Monitoring,” Journal of Economic Theory, 39, 251–269. [1234,1245] (1990): “Toward a Theory of Discounted Repeated Games With Imperfect Monitoring,” Econometrica, 58, 1041–1063. [1234,1241] ATKESON, A., AND R. E. LUCAS (1992): “On Efficient Distribution With Private Information,” Review of Economic Studies, 59, 427–453. [1234,1235,1237,1260] BATTAGLINI, M. (2005): “Long-Term Contracting With Markovian Consumers,” American Economic Review, 95, 637–658. [1236] BIAIS, B., T. MARIOTTI, G. PLANTIN, AND J.-C. ROCHET (2007): “Dynamic Security Design: Convergence to Continuous Time and Asset Pricing Implications,” Review of Economic Studies, 74, 345–390. [1236] BISMUT, J. M. (1973): “Conjugate Convex Functions in Optimal Stochastic Control,” Journal of Mathematical Analysis and Applications, 44, 384–404. [1234,1243,1264] (1978): “Duality Methods in the Control of Densities,” SIAM Journal on Control and Optimization, 16, 771–777. [1234,1241,1243,1264] BREEDEN, D. (1979): “An Intertemporal Asset Pricing Model With Stochastic Consumption and Investment Opportunities,” Journal of Financial Economics, 7, 265–296. [1239] CVITANIC´ , J., AND J. ZHANG (2007): “Optimal Compensation With Adverse Selection and Dynamic Actions,” Mathematics and Financial Economics, 1, 21–55. [1237] CVITANIC´ , J., X. WAN, AND J. ZHANG (2009): “Optimal Compensation With Hidden Action and Lump-Sum Payment in a Continuous-Time Model,” Applied Mathematics and Optimization, 59, 99–146. [1249] DEMARZO, P. M., AND Y. SANNIKOV (2006): “Optimal Security Design and Dynamic Capital Structure in a Continuous-Time Agency Model,” Journal of Finance, 61, 2681–2724. [1236,1238, 1257] FERNANDES, A., AND C. PHELAN (2000): “A Recursive Formulation for Repeated Agency With History Dependence,” Journal of Economic Theory, 91, 223–247. [1236] GOLOSOV, M., AND A. TSYVINSKI (2006): “Designing Optimal Disability Insurance: A Case for Asset Testing,” Journal of Political Economy, 114, 257–279. [1236] GOLOSOV, M., N. KOCHERLAKOTA, AND A. TSYVINSKI (2003): “Optimal Indirect and Capital Taxation,” Review of Economic Studies, 70, 569–587. [1235,1262,1264] GREEN, E. J. (1987): “Lending and the Smoothing of Uninsurable Income,” in Contractual Arrangements for Intertemporal Trade, ed. by E. C. Prescott and N. Wallace. Minneapolis: University of Minnesota Press, 3–25. [1234,1241,1245] HAUSSMANN, U. G. (1986): A Stochastic Maximum Principle for Optimal Control of Diffusions. Essex, U.K.: Longman Scientific & Technical. [1268] KAPICKA, M. (2006): “Efficient Allocations in Dynamic Private Information Economies With Persistent Shocks: A First Order Approach,” Working Paper, University of California, Santa Barbara. [1234,1236,1237,1245,1260,1262,1263] KARATZAS, I., AND S. E. SHREVE (1991): Brownian Motion and Stochastic Calculus (Second Ed.). New York: Springer-Verlag. [1238] KOCHERLAKOTA, N. (2004): “Figuring Out The Impact of Hidden Savings on Optimal Unemployment Insurance,” Review of Economic Dynamics, 7, 541–554. [1246] KYDLAND, F. E., AND E. C. PRESCOTT (1980): “Dynamic Optimal Taxation, Rational Expectations and Optimal Control,” Journal of Economic Dynamics and Control, 2, 79–91. [1234]
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LIPTSER, R. S., AND A. N. SHIRYAEV (2000): Statistics of Random Processes (Second Ed.), Vol. I. Berlin: Springer. [1240,1241] MEGHIR, C., AND L. PISTAFERRI (2004): “Income Variance Dynamics and Heterogeneity,” Econometrica, 72, 1–32. [1233] ROGERSON, W. P. (1985): “Repeated Moral Hazard,” Econometrica, 53, 69–76. [1235,1264] SANNIKOV, Y. (2007): “Agency Problems, Screening and Increasing Credit Lines,” Working Paper, UC Berkeley. [1237] (2008): “A Continuous-Time Version of the Principal-Agent Problem,” Review of Economic Studies, 75, 957–984. [1234] SCHATTLER, H., AND J. SUNG (1993): “The First-Order Approach to the Continuous-Time Principal-Agent Problem With Exponential Utility,” Journal of Economic Theory, 61, 331–371. [1247] SPEAR, S., AND S. SRIVASTRAVA (1987): “On Repeated Moral Hazard With Discounting,” Review of Economic Studies, 54, 599–617. [1234,1241,1245] STORESLETTEN, K., C. I. TELMER, AND A. YARON (2004): “Cyclical Dynamics in Idiosyncratic Labor Market Risk,” Journal of Political Economy, 112, 695–717. [1233] TCHISTYI, A. (2006): “Security Design With Correlated Hidden Cash Flows: The Optimality of Performance Pricing,” Working Paper, NYU Stern School of Business. [1236] THOMAS, J., AND T. WORRALL (1990): “Income Fluctuation and Asymmetric Information: An Example of a Repeated Principal-Agent Problem,” Journal of Economic Theory, 51, 367–390. [1234,1235,1237,1245,1246,1252,1256-1259,1264] WERNING, I. (2001): “Optimal Unemployment Insurance With Unobservable Saving,” Working Paper, MIT. [1234,1245,1248] WILLIAMS, N. (2009): “On Dynamic Principal-Agent Models in Continuous Time,” Working Paper, University of Wisconsin–Madison. [1234-1236,1241,1243,1245-1247,1257] YONG, J., AND X. Y. ZHOU (1999): Stochastic Controls. New York: Springer-Verlag. [1249] ZHANG, Y. (2009): “Dynamic Contracting With Persistent Shocks,” Journal of Economic Theory, 144, 635–675. [1236,1239] ZHOU, X. Y. (1996): “Sufficient Conditions of Optimality for Stochastic Systems With Controllable Diffusions,” IEEE Transactions on Automatic Control, AC-41, 1176–1179. [1247]
Dept. of Economics, University of Wisconsin–Madison, William H. Sewell Social Science Building, 1180 Observatory Drive, Madison, WI 53706-1393, U.S.A.;
[email protected]. Manuscript received January, 2008; final revision received December, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1277–1318
RECURSIVE METHODS IN DISCOUNTED STOCHASTIC GAMES: AN ALGORITHM FOR δ → 1 AND A FOLK THEOREM BY JOHANNES HÖRNER, TAKUO SUGAYA, SATORU TAKAHASHI, AND NICOLAS VIEILLE1 We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players’) equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period’s state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting. KEYWORDS: Stochastic games.
1. INTRODUCTION DYNAMIC GAMES ARE DIFFICULT TO SOLVE. In repeated games, finding some equilibrium is easy, as any repetition of a stage-game Nash equilibrium will do. This is not the case in stochastic games. The characterization of even the most “elementary” equilibria for such games, namely (stationary) Markov equilibria, in which continuation strategies depend on the current state only, turns out to be often challenging. Further complications arise once the assumption of perfect monitoring is abandoned. In that case, our understanding of equilibria in repeated games owes an invaluable debt to Abreu, Pearce, and Stacchetti (1990), whose recursive techniques have found numerous applications. Similar ideas have been developed in stochastic games by Mertens and Parthasarathy (1987, 1991) to establish the most general equilibrium existence results to date (see also Solan (1998)). Those ideas have triggered a development of numerical methods (Judd, Yeltekin, and Conklin (2003)), whose use is critical for applications in which postulating a high discount factor appears too restrictive. In repeated games, more tractable characterizations have been achieved in the case of low discounting. Fudenberg and Levine (1994, hereafter FL) and Fudenberg, Levine, and Maskin (1994, hereafter FLM) showed that the set of (perfect public) equilibrium payoffs can be characterized by a family of static constrained optimization programs. Based on this and other insights, they de1 This paper is the result of a merger between two independent projects: Hörner and Vieille developed the algorithm, and Sugaya and Takahashi established the folk theorem by a direct argument. We thank Katsuhiko Aiba, Eduardo Faingold, Drew Fudenberg, Yingni Guo, Tristan Tomala, and Yuichi Yamamoto for helpful discussions. J. Hörner gratefully acknowledges financial support from NSF Grant SES 092098. N. Vieille gratefully acknowledges financial support from Fondation HEC.
© 2011 The Econometric Society
DOI: 10.3982/ECTA9004
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rived sufficient conditions for a folk theorem for repeated games with imperfect public monitoring. The algorithm developed by FL has proved to be useful, as it allows identification of the equilibrium payoff set in interesting cases in which the sufficient conditions for the folk theorem fail. This is the case, for instance, in the partnership game of Radner, Myerson, and Maskin (1986). More importantly, FL’s algorithm can accommodate both long-run and shortrun players, which is essential for many applications, especially in macroeconomics and political economy, in which consumers or voters are often modeled as nonstrategic (see Mailath and Samuelson (2006, Chaps. 5 and 6)). This paper extends these results to stochastic games. More precisely, it provides an algorithm that characterizes the set of perfect public equilibrium payoffs in stochastic games with finitely many states, signals, and actions, in which states are observed and monitoring is imperfect but public, under the assumption that the limiting set of equilibrium payoffs (of the long-run players) is independent of the initial state. This assumption is satisfied, in particular, if the Markov chain over states defined by any Markov strategy profile is irreducible. This algorithm is a natural extension of FL and indeed reduces to it if there is only one state. The key to this characterization lies in the linear constraints to impose on continuation payoffs. In FL, each optimization program is indexed by a direction λ ∈ RI that specifies the weights on the payoffs in the objective of the I players. Trivially, the boundary point v of the equilibrium payoff set that maximizes this weighted average in the direction λ is such that, for any realized signal y, the continuation payoff w(y) must satisfy the property that λ · (w(y) − v) ≤ 0. Indeed, one of the main insights of FL is that once discounting is sufficiently low, attention can be restricted to this linear constraint for each λ so that the program itself becomes linear in w and, hence, considerably more tractable. In a stochastic game, it is clear that the continuation payoff must be indexed not only by the public signal, but also by the realized state, and since the stage games that correspond to the current and to the next state need not be the same, there is little reason to expect this property to be preserved for each pair of states. Indeed, it is not. One might then wonder whether the resulting problem admits a linear characterization at all. The answer is affirmative. When all players are long-run, this characterization can be used to establish a folk theorem under assumptions that parallel those invoked by FLM. In stochastic games, note that state transitions might be affected by actions taken, so that, because states are observed, they already provide information about players’ past actions. Therefore, it is natural to impose rank conditions at each state on how actions affect the joint distribution over the future state and signal. This is weaker than requiring such conditions on signals alone and, indeed, it is easy to construct examples where the folk theorem holds without any public signals. This folk theorem also generalizes Dutta’s (1995) folk theorem for stochastic games with perfect monitoring. Unlike ours, Dutta’s ingenious proof is constructive, extending ideas developed by Fudenberg and Maskin (1986) for the
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case of repeated games with perfect monitoring. However, his assumptions (except for the monitoring structure) are the same as ours. In independent and simultaneous work, Fudenberg and Yamamoto (2010) provided a different, direct proof of the folk theorem for stochastic games with imperfect public monitoring under irreducibility, without a general characterization of the equilibrium payoff set. Their rank assumptions are stronger, as discussed in Section 5. Finally, our results also imply the average cost optimality equation from dynamic programming, which obtains here as a special case of a single player. The average cost optimality equation is widely used in operations research, for instance, in routing, inventory, scheduling, and queueing problems, and our results might thus prove useful for game-theoretic extensions of such problems, as in inventory or queueing games. Of course, stochastic games are also widely used in economics. They play an important role in industrial organization (among many others, see Ericson and Pakes (1995)). It is hoped that methods such as ours might help provide integrated analyses of questions whose treatment had to be confined to simple environments so far, such as the role of imperfect monitoring (Green and Porter (1984)) and of business cycles (Rotemberg and Saloner (1986)) in collusion. Rigidities and persistence play an important role in macroeconomics as well, giving rise to stochastic games. See Phelan and Stacchetti (2001), for example. In Section 6, we apply our results to a simple political economy game and establish a version of the folk theorem when some players are short-run. 2. NOTATION AND ASSUMPTIONS We introduce stochastic games with public signals. At each stage, the game is in one state and players simultaneously choose actions. Nature then determines the current reward (or flow payoff) profile, the next state, and a public signal as a function of the current state and the action profile. The sets S of possible states, I of players, Ai of actions available to player i, and Y of public signals are assumed finite.2 Given an action profile a ∈ A := i Ai and a state s ∈ S, we denote by r(s a) ∈ RI the reward profile in state s given a, and denote by p(t y|s a) the joint probability of moving to state t ∈ S and of receiving the public signal y ∈ Y . (As usual, we can think of r i (s a) as the expectation given a of some realized reward that is a function of a private outcome of player i and the public signal only.) We assume that at the end of each period, the only information publicly available to all players consists of nature’s choices: the next state together with
×
2 For notational convenience, the set of available actions is independent of the state. See, however, footnote 12. All results extend beyond that case.
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the public signal. When properly interpreting Y , this includes the case of perfect monitoring and the case of publicly observed rewards. Note, however, that this fails to include the case where actions are perfectly monitored, yet states are not disclosed. In such a case, the natural “state” variable is the (common) posterior belief of the players on the underlying state. Thus, in the stochastic game, in each period n = 1 2 , the state is observed, the stage game is played, and the corresponding public signal is then revealed. The stochastic game is parameterized by the initial state s1 , and it will be useful to consider all potential initial states simultaneously. The public history at the beginning of period n is then hn = (s1 y1 sn−1 yn−1 sn ). We set H1 := S, the set of initial states. The set of public histories at the beginning of period n is therefore Hn := (S × Y )n−1 × S. We let H := n≥1 Hn denote the set of all public histories. The private history for player i at the beginning of period n is a sequence hin = (s1 ai1 y1 sn−1 ain−1 yn−1 sn ), and we similarly define H1i := S, Hni := (S × Ai × Y )n−1 × S and H i := n≥1 Hni . Given a stage n ≥ 1, we denote by sn the state, by an the realized action profile, and by yn the public signal in period n. We often use the same notation to denote both these realizations and the corresponding random variables. A (behavior) strategy for player i ∈ I is a map σ i : H i → Δ(Ai ). Every pair of initial state s1 and strategy profile σ generates a probability distribution over histories in the obvious way and thus also generates a distribution over sequences of the players’ rewards. Players seek to maximize their payoffs, that is, average discounted sums of their rewards, using a common discount factor δ < 1. Thus, the payoff of player i ∈ I if the initial state is s1 and the players follow the strategy profile σ is defined as (1 − δ)
+∞
δn−1 Es1 σ [r i (sn an )]
n=1
We consider a special class of equilibria. A strategy σ i is public if it depends on the public history only and not on the private information. That is, a public strategy is a mapping σ i : H → Δ(Ai ). A perfect public equilibrium (hereafter, PPE) is a profile of public strategies such that, given any period n and public history hn , the strategy profile is a Nash equilibrium from that period on. Note that this class of equilibria includes Markov equilibria, in which strategies only depend on the current state and period. In what follows though, a Markov strategy for player i will be a public strategy that is a function of states only, that is, a function S → Δ(Ai ).3 For each Markov strategy profile α = (αs )s∈S ∈ ( i∈I Δ(Ai ))S , we denote by qα (t|s) := p(t × Y |s αs ) the transition probabilities of the Markov chain over S induced by α.
×
3 In the literature on stochastic games, such strategies are often referred to as stationary strategies.
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Note that the set of PPE payoffs is a function of the current state only, and does not otherwise depend on the public history or on the period. Perfect public equilibria are sequential equilibria, but it is easy to construct examples that show that the converse is not generally true. What we characterize, therefore, is a subset of the sequential equilibrium payoffs. We denote by Eδ (s) ⊂ RI the (compact) set of PPE payoffs of the game with initial state s ∈ S and discount factor δ < 1. All statements about convergence of or equality between sets are understood in the sense of the Hausdorff distance d(A B) between sets A and B. Because both state and action sets are finite, it follows from Fink (1964) and Takahashi (1962) that a (perfect public) equilibrium always exists in this setup. Our main result does not apply to all finite stochastic games. Some examples of stochastic games, in particular those involving absorbing states, exhibit remarkably complex asymptotic properties. See, for instance, Bewley and Kohlberg (1976) or Sorin (1986). We shall come back to the implications of our results for such games. Our main theorem makes use of the following assumption. ASSUMPTION A: The limit set of PPE payoffs is independent of the initial state: for all s t ∈ S, lim d(Eδ (s) Eδ (t)) = 0 δ→1
This is an assumption on endogenous variables. A stronger assumption on exogenous variables that implies Assumption A is irreducibility: For any (pure) Markov strategy profile a = (as ) ∈ AS , the induced Markov chain over S with transition function qa is irreducible. Actually, it is not necessary that every Markov strategy gives rise to an irreducible Markov chain. It is clearly sufficient if there is some state that is accessible from every other state regardless of the Markov strategy. Another class of games that satisfy Assumption A, although they do not satisfy irreducibility, is the class of alternating-move games (see Lagunoff and Matsui (1997) and Yoon (2001)). With two players, for instance, such a game can be modeled as a stochastic game in which the state space is A1 ∪ A2 , where ai ∈ Ai is the state that corresponds to the last action played by player i, when it is player −i’s turn to move. (Note that this implies perfect monitoring by definition, as states are observed.) Note also that by redefining the state space to be S × Y , one may further assume that only states are disclosed. That is, the class of stochastic games with public signals is no more general than the class of stochastic games in which only the current state is publicly observed. However, the Markov chain over S × Y with transition function q˜ a (t z|s y) := p(t z|s as ) need not be irreducible even if qa (t|s) is.
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3. AN ALGORITHM TO COMPUTE EQUILIBRIUM PAYOFFS 3.1. Preliminaries: Repeated Games As our results generalize the algorithm of FL, it is useful to start with a reminder of their results and to examine, within a specific example, what difficulties a generalization to stochastic games raises. Recall that the set of (perfect public) equilibrium payoffs must be a fixed point of the Bellman–Shapley operator (see Abreu, Pearce, and Stacchetti I (1990)). Define the one-shot game Γδ (w), where w : Y → R , with action sets i A and payoff function (1 − δ)r(a) + δ y∈Y p(y|a)w(y). Note that if v is an equilibrium payoff vector of the repeated game, associated with action profile α ∈ i∈I Δ(Ai ) and continuation payoff vectors w(y), as a function of the initial signal, then α must be a Nash equilibrium of Γδ (w) with payoff v: v = (1 − δ)r(α) + δ (1) p(y|α)w(y)
×
y∈Y
Conversely, if we are given a function w such that w(y) ∈ Eδ for all y and a Nash equilibrium α of Γδ (w) with payoff v, then we can construct an equilibrium of the repeated game with payoff v in which the action profile α is played in the initial period. Therefore, the analysis of the repeated game can be reduced to that of the one-shot game. The constraint that the continuation payoff lies in the (unknown) set Eδ complicates the analysis significantly. FL’s key observation is that this constraint can be replaced by linear constraints for the sake of asymptotic analysis (as δ → 1). If α is a Nash equilibrium of the one-shot game Γδ (w) with payoff v ∈ Eδ , then by subtracting δv on both sides of (1) and dividing through by 1 − δ, we obtain p(y|α)x(y) v = r(α) + y∈Y
where, for all y, x(y) :=
δ (w(y) − v) 1−δ
or w(y) = v +
1−δ x(y) δ
Thus, provided that the equilibrium payoff set is convex, the payoff v is also ˜ in Eδ˜ for all δ˜ > δ, because we can use as continuation payoff vectors w(y) the rescaled vectors w(y) (see Figure 1). Conversely, provided that the normal vector to the boundary of Eδ varies continuously with the boundary point, then any set of payoff vectors w(y) that lie in one of the half-spaces defined by this normal vector (i.e., such that λ · (w(y) − v) ≤ 0 or, equivalently, λ · x(y) ≤ 0) must also lie in Eδ for discount factors close enough to 1. In particular, if we
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FIGURE 1.—Continuation payoffs as a function of the discount factor.
seek to identify the payoff v that maximizes λ · v on Eδ for δ close enough to 1, given λ ∈ RI , it suffices to compute the score k(λ) := sup λ · v xvα
such that α is a Nash equilibrium with payoff v of the game Γ (x) whose payoff function is given by r(a) + y∈Y p(y|a)x(y), and subject to the linear constraints λ · x(y) ≤ 0 for all y. Note that the discount factor no longer appears in this program. We thus obtain a half-space H(λ) := {v ∈ RI : λ · v ≤ k(λ)} that contains Eδ for every δ. This must be true for all vectors λ ∈ RI . Let H := λ∈RI H(λ). It follows, under an appropriate dimensionality assumption, that the equilibrium payoff set can be obtained as the intersection of these half-spaces (see FL, Theorem 3.1):
H = lim Eδ δ→1
3.2. A Stochastic Game Our purpose is to come up with an algorithm for stochastic games that would generalize FL’s algorithm. To do so, we must determine the appropriate constraints on continuation payoffs. Let us attempt to adapt FL’s arguments to a specific example. EXAMPLE 1: There are two states, i = 1 2, and two players. Each player only takes an action in his own state: player i chooses L or R in state i. Actions
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FIGURE 2.—Rewards and transitions in Example 1.
are not observable, but do affect transitions, so that players learn about their opponents’ actions via the evolution of the state. If action L (R) is taken in state i, then the next state is again i with probability pL (pR ). Let us pick here pL = 1 − pR = 1/10. Rewards are given in Figure 2 (transition probabilities to states 1 and 2, respectively, are given in parentheses). Player i has a higher reward in state j = i, independently of the action. Moreover, by playing L, which yields him the higher reward in his own state, he maximizes the probability to switch states. Thus, playing the efficient action R requires intertemporal incentives, which are hard to provide absent public signals. Constructing an equilibrium in which L is not always played appears challenging, but not impossible: playing R in state i if and only if the state was j = i in the previous two periods (or since the beginning of the game if fewer periods have elapsed) is an equilibrium for some high discount factor (δ ≈ 0823). So there exist equilibrium payoffs above 1. In analogy with (1), we may now decompose the payoff vector in state s = 1 2 as vs = (1 − δ)r(s αs ) + δ (2) p(t|s αs )wt (s) t
where t is the next state, wt (s) is the continuation payoff then, and p(t|s αs ) is the probability of transiting from s to t given action αs at state s. Fix λ ∈ RI . If vs maximizes the score λ · vs in all states s = 1 2, then the continuation payoff in state t gives a lower score than vt , independently of the initial state: for all s t, (3)
λ · (wt (s) − vt ) ≤ 0
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Our goal is to eliminate the discount factor. Note, however, that if we subtract δvs on both sides of (2) and divide by 1 − δ, we obtain (4)
vs = r(s αs ) +
p(t|s αs )
t
δ (wt (s) − vs ) 1−δ
and there is no reason to expect λ · (wt (s) − vs ) to be negative unless s = t (compare with (3)). Unlike the limiting set of feasible payoffs as δ → 1, the set of feasible rewards does depend on the state (in state 1, it is the segment [(1 1) (0 3)]; in state 2, the segment [(1 1) (3 0)]; see the right panel of Figure 2), and so the score λ · wt (s) in state t might exceed the maximum score achieved by vs in state s. Thus, defining x by xt (s) :=
δ (wt (s) − vs ) 1−δ
we know that λ · xs (s) ≤ 0 for all s, but not the sign of λ · xt (s), t = s. On the one hand, we cannot restrict it to be negative: if λ · x2 (1) ≤ 0, then because also λ · x1 (1) ≤ 0, by considering λ = (1 0), player 1’s payoff starting from state 1 cannot exceed his highest reward in that state (i.e., 1). Yet we know that some equilibria yield strictly higher payoffs. On the other hand, if we impose no restrictions on xt (s), s = t, then we can set vs as high as we wish in (4) by picking xt (s) large enough. The value of the program to be defined would be unbounded. What is the missing constraint? We do know that (3) holds for all pairs (s t). By adding up these inequalities for (s t) = (1 2) and (2 1), we obtain (5)
λ · (w1 (2) + w2 (1) − v1 − v2 ) ≤ 0
or, rearranging,
λ · (x1 (2) + x2 (1)) ≤ 0 Equation (5) has a natural interpretation in terms of s-blocks, as defined in the literature on Markov chains (see, for instance, Nummelin (1984)). When the Markov chain (induced by the players’ strategies) is communicating, as it is in our example, we might divide the game into the subpaths of the chain between consecutive visits to a given state s. The score achieved by the continuation payoff once state s is revisited on the subpath (s1 sk ) (where s1 = sk = s) cannot exceed the score achieved by vs , and so the difference in these scores, k−1 as measured by the sum λ · j=1 xsj+1 (sj ), must be negative. Note that the irreducibility assumption also guarantees that the limit set of feasible payoffs F (as δ → 1) is independent of the initial state, as shown in the right panel of Figure 2.4 4 This set can be computed by considering randomizations over pure Markov strategies (see Dutta (1995, Lemma 1)).
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To conclude, we obtain the program sup λ · v vxα
over v ∈ R2 , {xt (s) ∈ R2 : s t = 1 2}, and α = (αs )s=12 such that, in each state s, αs is a Nash equilibrium with payoff v of the game whose payoff function is given by r(s as ) + t p(t|s as )xt (s), and such that λ · x1 (1) ≤ 0, λ · x2 (2) ≤ 0, and λ · (x1 (2) + x2 (1)) ≤ 0. Note that this program already factors in our assumption that equilibrium payoffs can be taken to be independent of the state. It follows from Theorem 1 in the next section that this is the right program. Perhaps it is a little surprising that the constraints involve unweighted sums of vectors xt (s), rather than, say, sums that are weighted by the invariant measure under the equilibrium strategy. Lemma 1 below provides a link between the constraints and such sums. What should come as no surprise, though, is that generalizing these constraints to more than two states involves considering all cycles, or permutations, over states (see Constraint 2(ii) below). 3.3. The Characterization Given a state s ∈ S and a map x : S × Y → RS×I , we denote by Γ (s x) the one-shot game with action sets Ai and payoff function p(t y|s as )xt (s y) r(s as ) + t∈S y∈Y
where xt (s y) ∈ RI is the tth component of x(s y). Given λ ∈ RI , we denote by P (λ) the maximization program sup λ · v vxα
where the supremum is taken over all v ∈ RI , x : S × Y → RS×I , and α = (αs ) ∈ ( i∈I Δ(Ai ))S such that the following constraints hold:
×
CONSTRAINT 1: For each s, αs is a Nash equilibrium with payoff v of the game Γ (s x). T ⊆ S, for each permutation φ : T → T and each CONSTRAINT 2: For each map ψ : T → Y , we have λ · s∈T xφ(s) (s ψ(s)) ≤ 05 5 When Y = ∅, the domain of x is S, and the definition should be understood without reference to ψ.
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Denote by k(λ) ∈ [−∞ +∞] the value of P (λ). We will prove that the feasible set of P (λ) is non-empty, so that k(λ) > −∞ (Proposition 1), and that the value of P (λ) is finite, so that k(λ) < +∞ (Section 3.4). It is also useful to introduce the program P (λ α) parameterized by α, where the supremum is taken over all v and x, with α fixed. Let k(λ α) denote the value of P (λ α). Observe that P (λ α) is a linear program and that k(λ) = supα k(λ α). We define H(λ) := {v ∈ RI : λ · v ≤ k(λ)} and set H := λ∈RI H(λ). Note that H is convex. Let S 1 denote the set of λ ∈ RI of norm 1.6 Observe that H(0) = RI , that H(λ) = H(cλ) for every λ ∈ RI , and c > 0. Hence H is also equal to λ∈S1 H(λ). Our main result is a generalization of FL’s algorithm to compute the limit set of payoffs as δ → 1. THEOREM 1—Main Theorem: Assume that H has nonempty interior. Under Assumption A, Eδ (s) converges to H as δ → 1 for any s ∈ S. Note that with one state only, our optimization program reduces to the algorithm of FL. The proof of Theorem 1 is organized into two propositions, stated below and proved in the Appendix. Note that these propositions do not rely on Assumption A. PROPOSITION 1: For every δ < 1, we have the following conditions: (i) k(λ) ≥ mins∈S maxw∈Eδ (s) λ · w for every λ ∈ S 1 . (ii) H ⊇ s∈S Eδ (s). We note that it need not be the case that H ⊇ Eδ (s) for each s ∈ S. PROPOSITION 2: Assume that H has nonempty interior and let Z be any compact set contained in the interior of H. Then Z ⊆ Eδ (s) for every s ∈ S and δ large enough. The logic of the proof of Proposition 2 is inspired by FL and FLM, but differs in important respects. We here give a short and overly simplified account of the proof, which nevertheless contains some basic insights. Let a payoff vector v ∈ Z and a direction λ ∈ S 1 be given. Since v is interior to H, we have λ · v < k(λ), and there thus exists x = (xt (s y)) such that v is a Nash equilibrium payoff of the one-shot game Γ (s x) and all inequality constraints on x hold with strict inequalities. For high δ, we use x to construct equilibrium continuation payoffs w adapted to v in the discounted game, with the interpretation that xt (s y) is the normalized (continuation) payoff increment, should (t y) occur. Since we have no control over the sign of λ · xt (s y), the one-period argument that is familiar from 6
Throughout, we use the Euclidean norm.
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repeated games does not extend to stochastic games. To overcome this issue, we instead rely on large blocks of stages of fixed size. Over such a block, and thanks to the inequalities Constraint 2 satisfied by x, we prove that the sum of payoff increments is negative. This in turn ensures that the continuation payoff at the end of the block is below v in the direction λ. Since H is convex, it follows from these two propositions that
H = lim δ→1
Eδ (s)
s∈S
This statement applies to all finite stochastic games with observable states and full-dimensional H, whether they satisfy Assumption A or not. Theorem 1 then follows, given Assumption A. 3.4. Finiteness of P (λ) It is instructive, and useful for the sequel, to understand why the value of P (λ) is finite for each possible choice of λ. To see this, we rely on the next lemma, which we also use in other places. LEMMA 1: Let q be an irreducible transition function over a finite set R, with invariant measure μ. Then, for each T ⊆ R and every one-to-one map φ : T → T , there is πTφ ≥ 0 such that the following statement holds. For every (xt (s)) ∈ RR×R , we have μ(s) q(t|s)xt (s) = πTφ xφ(s) (s) s∈R
t∈R
T ⊆R φ:T →T
s∈T
where the sum ranges over all one-to-one maps φ : T → T . In words, for any payoff increments (xt (s)), the expectation of these increments (with respect to the invariant measure) is equal to some fixed conical combination of their sums over cycles. This provides a link between Constraints 1 and 2. PROOF OF LEMMA 1: We exploit an explicit formula for μ. Given a state s ∈ R, an s-graph is a directed graph g with vertex set R and the following two properties: • Any state t = s has outdegree 1, while s has outdegree 0. • The graph g has no cycle. Equivalently, for any t = s, there is a unique longest path starting from t, and this path ends in s. We identify such a graph with its set of edges. The weight of
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any such graph g is defined to be q(g) := (tu)∈g q(u|t). Let G(s) denote the set of s-graphs. Then the invariant measure is given by q(g) g∈G(s)
μ(s) =
q(g)
t∈R g∈G(t)
(see, e.g., Freidlin and Wentzell (1991)). Thus, we have
μ(s)
s∈R
q(t|s)xt (s) =
q(g)q(t|s)xt (s)
s∈R g∈G(s) t∈R
t∈R
q(g)
t∈R g∈G(t)
Let Ω be the set of triples (s g t), such that g ∈ G(s). Define an equivalence relation ∼ over Ω by (s g t) ∼ (s g t ) if g ∪ {(s → t)} = g ∪ {(s → t )}. Observe that for each (s g t) ∈ Ω, the graph g ∪ {(s → t)} has exactly one cycle (which contains s → t) and all vertices of R have outdegree 1. Let C ⊆ Ω be any equivalence class for ∼. Let s1 → s2 → · · · → sk = s1 denote the unique, common cycle of all (s g t) ∈ C. Define T := {s1 sk−1 } and denote by φ : T → T the map that associates to any state u ∈ T its successor in the cycle. Observe that the product q(g)q(t|s) is independent of (s g t) ∈ C; we denote it ρCTφ . It is then readily checked that
q(g)q(t|s)xt (s) = ρCTφ
xφ(u) (u)
u∈T
(sgt)∈C
The result follows by summation over equivalence classes.
Q.E.D.
This lemma readily implies that the value of P (λ) is finite for each direction λ. Indeed, let a direction λ ∈ RI and a feasible point (v x α) in P (λ) be given, and recall that qα is the transition function of the Markov chain induced by α. Let R ⊆ S be an arbitrary recurrent set for this Markov chain and denote by μ ∈ Δ(R) the invariant measure over R. For each s t ∈ R, let f (s t) be a signal y that maximizes λ · xt (s y). Then Lemma 1 implies that μ(s)r(s αs ) + λ · μ(s) qα (t|s)xt (s f (s t)) λ·v ≤λ· s∈R
=λ·
s∈r
s∈R
μ(s)r(s αs )
t∈S
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+
πTφ λ ·
T ⊆R φ:T →T
≤λ·
xφ(s) s f (s φ(s))
s∈T
μ(s)r(s αs )
s∈R
It thus follows that μ(s)r(s αs ) λ·v≤λ· s∈S
for every invariant (probability) measure of the Markov chain induced by α. 4. CONNECTION TO KNOWN RESULTS Our characterization includes as special cases the characterization obtained by FL for repeated games, as well as the average cost optimality equation from dynamic programming in the case of one player. In this section, we explain these connections in more detail. 4.1. Another Generalization of FL’s Algorithm The specific form of the optimization program P (λ) is intriguing and calls for some discussion. We here elaborate upon Section 3.2. Perhaps a natural generalization of FL’s algorithm to the setup of stochastic games would have been the following. Given a direction λ ∈ S 1 , for each initial state s ∈ S, consider the highest PPE payoff vs ∈ RI in the direction λ, when starting from s. Again, for each initial state s, there are a mixed action profile αs and continuation PPE payoffs wt (s y) to be interpreted as continuation payoffs in the event that the signal is y and the next state is t, and such that the following statements hold: with payoff vs of the game with (a) For each s, αs is a Nash equilibrium payoff function (1 − δ)r(s as ) + δ ty p(t y|s as )wt (s y). (b) For any two states s t ∈ S and any signal y ∈ Y , λ · wt (s y) ≤ λ · vt . Mimicking FL’s approach, we introduce the program, denoted P˜ (λ δ), sup(vs )wα mins λ · vs , where the supremum is taken over all ((vs ) w α) such that (a) and (b) hold.7 As we discussed in Section 3.2 and unlike in the repeated game framework, the value of P˜ (λ δ) does depend on δ. The reason δ (wt (s y) − vs ), the inequality λ · xt (s y) ≤ 0 is that when setting xt (s y) = 1−δ need not hold (note that xt (s y) involves wt (s y) and vs , while (b) involves wt (s y) and vt ). 7 Taking the minimum (or maximum) over s in the objective function has no effect asymptotically under Assumption A. We choose the minimum for the convenience of the discussion.
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To obtain a program that does not depend on δ, we relax the program P˜ (λ δ) as follows. Observe that for any sequence s1 sk of states such that sk = s1 and for each yj ∈ Y (1 ≤ j < k), we have λ·
k−1 j=1
δ wsj+1 (sj yj ) − vsj λ· xsj+1 (sj yj ) = 1−δ j=1 k−1
δ = λ· wsj+1 (sj yj ) − vsj+1 ≤ 0 1−δ j=1 k−1
where the second equality holds since s1 = sk and the final inequality holds by (b). These are precisely the averaging constraints which appear in our linear program P (λ). That is, the program P (λ) appears as a discount-independent relaxation of the program P˜ (λ δ).8 This immediately raises the issue of whether this is a “meaningful” relaxation of the program. We here wish to suggest that this is the case, by arguing that the two programs P (λ) and P˜ (λ δ) have asymptotically the same value, as δ → 1. To show this claim, we start with a feasible point (v x α) in P (λ) and construct a feasible point ((vs ) w α) in P˜ (λ δ) such that the norm of vs − v is on the order of 1 − δ for each s ∈ S. To keep the discussion straightforward, we assume that state transitions are independent of actions and signals: p(t y|s a) = q(t|s) × π(y|s a).9 Set xt (s y). Note that for each state s ∈ S, αs is a Nash first w˜ t (s y) := v + 1−δ δ equilibrium with payoff v of the one-shot game with payoff function (1 − δ)r(s as ) + δ ty p(t y|s as )w˜ t (s y). The desired continuation payoff vec˜ That is, we tor w is obtained by adding an action-independent vector to w. set wt (s y) := w˜ t (s y) + Ct (s) for some vectors Ct (s) ∈ RI . For each choice of (Ct (s)), the profile αs is still a Nash equilibrium of the one-shot game when ˜ and it yields a payoff of continuation payoffs are given by w instead of w, p(t y|s αs )wt (s y) vs := (1 − δ)r(s αs ) + δ = v+δ
t∈S y∈Y
q(t|s)Ct (s)
t∈S
We thus need to check that there exist vectors Ct (s) with norms on the order of 1 − δ such that the inequalities λ · wt (s y) ≤ λ · vt hold for all (s t y). Setting 8 Note that ((vs ) x α) does not define a feasible point in P (λ), since feasibility in P (λ) requires that payoffs from different initial states coincide. This is straightforward to fix. Let s¯ be a state that minimizes λ · vs . Set v := vs¯ , and x¯ t (s y) := xt (s y) + vs¯ − vs for each s t y. Then ¯ α) is a feasible point in P (λ). (v x 9 The proof in the general case is available upon request.
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ct (s) := λ · Ct (s), basic algebra shows that the latter inequalities are satisfied as soon as the inequalities ct (s) − δ (S ) q(u|t)cu (t) ≤ lt (s) u∈S
hold for all pairs (s t) of states, where lt (s) := miny∈Y λ · (v − w˜ t (s y)). We are left to show that such values of ct (s) can be found. Feasibility of (v x α) in P (λ) implies that s∈T lφ(s) (s) ≥ 0 for every T ⊆ S and every permutation φ over T (this is a simple rewriting of Constraint 2). As shown in Appendix A.5 (see Claim 3), this implies that there a vector exists ∗ l∗ = (lt∗ (s)) ∈ RS×S such that lt∗ (s) ≤ lt (s) for every s t ∈ S, and s∈T lφ(s) (s) = 0 for every T ⊆ S and every permutation φ over T . Given δ < 1 and t ∈ S, we denote by μδt ∈ Δ(S) the δ-discounted occupation measure of the Markov chain with initial state t. That is, μδt (s) is the expected discounted frequency of visits to state s when starting from state t. Formally, μδt (s) := (1 − δ)
+∞
δn−1 Pt (sn = s)
n=1
where Pt (sn = s) is, the probability, that the Markov chain visits state s in stage n. Elementary algebra then shows that the vector ct (s) := Eμδt [ls∗ (s)] solves ct (s) − δ
q(u|t)cu (t) = lt∗ (s)
u∈S
for each s t, and is, therefore, a solution to (S ).10 We conclude by arguing briefly that the norm of Ct (s) is on the order of 1−δ, as desired. Because lt (s) = miny∈Y λ · (v − w˜ t (s y)) = − 1−δ maxy∈Y λ · xt (s y), δ lt (s) is on the order of 1 − δ. It follows from the proof of Claim 3 that l∗ is a solution for a linear program whose constraints are linear in l = (lt (s)). Thus l∗ and ct (s) = Eμδt [ls∗ (s)] are also on the order of 1 − δ. It then suffices to choose Ct (s) on the order of 1 − δ such that λ · Ct (s) = ct (s). 4.2. Dynamic Programming It is sometimes argued that the results of Abreu, Pearce, and Stacchetti (1990) can be viewed as generalizations of dynamic programming to the case of multiple players. In the absence of any payoff-relevant state variable, this claim is difficult to appreciate within the context of repeated games. By focusing on The computation uses the identity Eμδt [f (s)] = (1−δ)f (t)+δ holds for any map f : S → R. 10
u∈S
q(u|t)Eμδu [f (s)] which
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one-player games, we show that, indeed, our characterization reduces to the optimality equation of dynamic programming. We here assume irreducibility. Irreducibility implies Assumption A, which in turn implies that the set of limit points of the family {vδ (s)}δ<1 as δ → 1 is independent of s, where vδ (s) is the value of the Markov decision process with initial state s. COROLLARY 1: In the one-player case with irreducible transition probabilities, the set H is a singleton {v∗ }, with v∗ = limδ→1 vδ (s) for each s ∈ S. Moreover, there is a vector x∗ ∈ RS such that ∗ ∗ ∗ (6) p(t × Y |s as )xt v + xs = max r(s as ) + as ∈A
t∈S
holds for each s ∈ S, and v = v∗ is the unique value solving (6) for some x ∈ RS . This statement is the so-called average cost optimality equation. See Hoffman and Karp (1966), Sennott (1999), or Kallenberg (2002). To get some intuition for this corollary, note first that with one player, signals become irrelevant and we might ignore them. Consider then the direction λ = 1. To maximize the player’s payoff, we should increase the values of xt (s) as much as possible. So conjecture for a moment that all the Constraints 2 bind: for all T ⊆ S and permutations φ : T → T , s∈T xφ(s) (s) = 0. Let us then set x∗t := xt (¯s), for some fixed state s¯ ∈ S. Note that for all s t ∈ S, xt (s) = −xs¯ (t) − xs (¯s) = xt (¯s) − xs (¯s) = x∗t − x∗s where the first two equalities use the binding constraints. Because as is a Nash equilibrium of the game Γ (s x) with payoff v∗ , we have ∗ p(t × Y |s as )xt (s) v = max r(s as ) + as ∈A
t∈S
Using xt (s) = x∗t − x∗s gives the desired result. The proof below verifies the conjecture and supplies missing steps. PROOF OF COROLLARY 1: By Proposition 1(i), the common set of limit points of {vδ (s)}δ<1 is a subset of H, so that H = ∅. We first argue that H is a singleton. The set H is uniquely characterized by the two values k(λ), λ ∈ {−1 +1}. Since H = ∅, we have k(+1) ≥ −k(−1). Note now that any pair (v x) that is feasible in P (λ α) is also feasible in P (λ a), for any a ∈ AS such that αs (as ) > 0 for each s. Consequently, we need only look at pure Markov strategies so as to compute k(+1) and k(−1). Let a = (as ) ∈ AS be any pure Markov strategy which induces an irreducible Markov chain over S with invariant measure μ. Let (v+ x+ ) be a feasible pair
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in P (+1 a) and let (v− x− ) be a feasible pair in P (−1 a). By Lemma 1, we have
μ(s)r(s as ) + πTφ x+φ(s) s f + (s φ(s)) v+ ≤ T ⊆S φ:T →T
s∈S
≤
μ(s)r(s as ) +
s∈T
πTφ
T ⊆S φ:T →T
s∈S
x−φ(s) s f − (s φ(s)) ≤ v−
s∈T
+ − + − since s∈T xφ(s) (s f (s φ(s))) ≤ 0 ≤ s∈T xφ(s) (s f (s φ(s))), where y = + + − f (s t) maximizes xt (s y) and y = f (s t) minimizes x−t (s y). Therefore, k(+1) ≤ −k(−1) and H is a singleton. This implies, in particular, that v∗ := limδ→1 vδ (s) exists for each s ∈ S. We use the following claim. CLAIM 1: If a vector (xt (s)) ∈ RS×S satisfies and all permutations φ : T → T , and (7)
v∗ ≤ r(s as ) +
s∈T
xφ(s) (s) = 0 for all T ⊆ S
p(t × Y |s as )xt (s)
t∈S
for some a = (as ) ∈ AS and all s ∈ S, then the inequality (7) holds with equality for each s. PROOF: Assume to the contrary that the inequality (7) is strict for some s¯ ∈ S. Let μ ∈ Δ(S) be the invariant measure of the Markov chain induced by a over S. By Lemma 1, v∗ <
μ(s)r(s as )
s∈S
But this implies that, for all δ high enough, the δ-discounted payoff induced by a exceeds vδ (s), which is a contradiction. Q.E.D. Pick a∗ ∈ AS and (xt (s y)) such that (v∗ x) is feasible in P (+1 a∗ ). Pick a ∗ ∗ S×S vector such that x∗t (s) ≥ xt (s y) for all s t ∈ S and y ∈ Y , x =∗ (xt (s)) ∈ R and s∈T xφ(s) (s) = 0 for all (T φ).11 We claim that, for each s ∈ S, we have (8)
11
∗ v = max r(s as ) + p(t × Y |s as )xt (s) ∗
as ∈A
t∈S
See Claim 3 in Appendix A.5 for the existence of x∗ .
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Observe first that, for each given s, v∗ = r(s a∗s ) + p(t y|s a∗s )xt (s y) t∈S y∈Y
≤ r(s a∗s ) +
p(t × Y |s a∗s )x∗t (s)
t∈S
Thus, the left-hand side in (8) does not exceed the right-hand side. Let now a state s ∈ S and an action profile as ∈ A be given that satisfy the inequality (9) p(t × Y |s as )x∗t (s) v∗ ≤ r(s as ) + t∈S
Then, by Claim 1 applied to a := (a∗−s as ), (9) holds with equality. This proves that (8) holds for each s ∈ S. Since s∈T x∗φ(s) (s) = 0 for all (T φ), there is a vector x∗ ∈ RS such that x∗t (s) = x∗t − x∗s for each s t ∈ S. Uniqueness of v∗ follows at once. Assume indeed that (6) holds for some (v x). For each s ∈ S, let as be an action that maximizes the right-hand side of (6) and let a := (as ). Set x˜ t (s y) := xt − xs for each s t ∈ S and y ∈ Y . ˜ a) is feasible in P (+1) and in P (−1) as well. Hence, −k(−1) ≤ Then (v x v ≤ k(+1), so that v = v∗ . Q.E.D. 4.3. Interpretation of the Variables xt (s y) The variables xt (s y) are not continuation payoffs per se. Rather, they are payoff differences that account for both the signal and the possible change of states. In the case of a repeated game, they reduce to a variable of the sigδ nal alone (in the notation of FL, they are then equal to 1−δ (w(y) − v)). This variable reflects how the continuation payoff adjusts from the current to the following period to provide the appropriate incentives as a function of the realized signal. In the case of dynamic programming, these variables collapse to a function xt (s). This is the relative value function, as it is known in stochastic dynamic programming, and it captures the value of the Markov decision process in state t relative to state s. It can be further decomposed into a difference x∗t − x∗s for some function x∗ that only depends on the current state. While there is no reason to expect the system of inequalities Constraint 2 to simplify in general, there are some special cases in which it does. For instance, given our discussion above, we might suspect that the payoff adjustments required by the provision of incentives, on one hand, and by the state transitions, on the other, can be disentangled whenever transitions are uninformative about actions, conditional on the signals. Indeed, both in the case of action-independent transitions (studied in Section 7) and in the case of perfect
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˜ y) monitoring, we can show that these variables can be separated as xˆ t (s) + x(s for some function xˆ that only depends on the current and the next states, and some function x˜ that only depends on the current state and the realized signal. 5. THE FOLK THEOREM FLM established a folk theorem for repeated games with imperfect public monitoring when the signal distribution satisfies some rank condition. In this section, we extend their folk theorem to stochastic games. We derive our folk theorem by investigating the programs P (λ) under a similar rank condition and relating scores k(λ) to feasible sets and to minmax payoffs. We do not rely on Assumption A in this section. Rather, it is more convenient to impose state independence on limiting feasible sets and minmax values when necessary. Let Fδ (s) be the convex hull of the set of feasible payoffs of the game with initial state s ∈ S and discount factor δ < 1. The set Fδ (s) is compact and converges to F(s) as δ → 1, where F(s) is the set of limit-average feasible payoffs with initial state s (see, for instance, Dutta (1995, Lemma 2)). Let also miδ (s) be player i’s minmax payoff in the game with initial state s and discount factor δ, defined as m (s) := min max(1 − δ) i δ
σ −i
σi
∞
δn−1 Esσ [r i (sn an )]
n=1
where the minimum and the maximum are taken over public strategies σ −i and σ i , respectively. As δ → 1, miδ (s) converges to mi (s), where mi (s) is player i’s limit-average minmax payoff with initial state s (see Mertens and Neyman (1981) and Neyman (2003)). Define the intersection of the sets of feasible and individually rational payoffs: F ∗ := {v ∈ F(s) : vi ≥ mi (s) ∀i ∈ I} s∈S
For a given state s ∈ S and an action profile α−i = (αj )j =i , let Π i (s α−i ) be the |Ai | × |S × Y | matrix whose (ai (t y))th component is given by p(t y|s ai α−i ). For s ∈ S and α = (αi ), we stack two matrices vertically: i Π (s α−i ) ij Π (s α) := Π j (s α−j )
×
An action profile α ∈ i Δ(Ai ) has individual full rank for player i in state s if Π i (s α−i ) has rank |Ai |; the profile α has pairwise full rank for players i and j in state s if Π ij (s α) has rank |Ai | + |Aj | − 1. Note that |Ai | + |Aj | − 1 is the highest possible rank since Π ij (s α) always has at least one nontrivial linear relation among its row vectors.
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ASSUMPTION F1: Every pure action profile has individual full rank for every player in every state. ASSUMPTION F2: For each state s and each pair (i j) of distinct players, there exists a mixed action profile that has pairwise full rank for players i and j in state s. The assumptions are the obvious generalizations of the assumptions of individual and pairwise full rank made by FLM. Note that Assumptions F1 and F2 are weaker than the rank assumptions of Fudenberg and Yamamoto (2010). Fudenberg and Yamamoto required that players can statistically identify each others’ deviations via actual signals y, whereas we allow players to make inferences from observed states as well. Example 1 in Section 3.2 illustrates the difference. In this example, there are no public signals, so Fudenberg and Yamamoto’s rank assumptions are not satisfied. On the other hand, Assumptions F1 and F2 are satisfied if pL = pR .12 If pL = pR , then incentives cannot be provided for players to play R, so that the unique PPE payoff is (1 1). With the above assumptions, we characterize k(λ) in terms of feasible sets and minmax payoffs. Let ei denote the ith coordinate basis vector in RI . LEMMA 2: Under Assumptions F1 and F2, we have the following situations: (i) If λ ∈ S 1 and λ = −ei for any i, then k(λ) = mins∈S maxw∈F(s) λ · w. (ii) k(−ei ) = − maxs∈S mi (s) for any i. While this lemma characterizes the value of the optimization program for each direction λ under Assumptions F1 and F2, the algorithm can be adapted to the purpose of computing feasible sets and minmax payoffs (in public strategies) without these assumptions. In the first case, it suffices to ignore the incentive Constraint 1 in the program P (λ) and to take the intersection of the resulting half-spaces. In the second case, it suffices to focus on the incentives of the minmaxed player i and to take as a direction λ the coordinate vector −ei . The proofs follow similar lines; details are available from the authors. Combined with Proposition 2, Lemma 2 implies the following folk theorem, which extends both the folk theorem for repeated games with imperfect public monitoring by FLM and the folk theorem for stochastic games with observable actions by Dutta (1995).13 12
When the set of available actions depends on the state, as in Example 1, the definitions of full rank must be adjusted in the obvious way. Namely, we say that α has individual full rank for player i at state s if the rank of Π i (s α−i ) is no less than the number of actions available to player i at state s. A similar modification applies to pairwise full rank. 13 Note that Dutta (1995, Theorem 9.3) showed that full dimensionality can be weakened to payoff asymmetry if mixed strategies are observable, an assumption that makes little sense under imperfect monitoring.
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FIGURE 3.—The limit set of PPE payoffs in Example 1.
THEOREM 2—Folk Theorem: Under Assumptions F1 and F2, it holds that H = F ∗ . In particular, if F(s) = F and mi (s) = mi for all s ∈ S, and F ∗ = {v ∈ F : vi ≥ mi ∀i ∈ I} has nonempty interior, then Eδ (s) converges to F ∗ as δ → 1 for any s ∈ S. It is known that under irreducibility, F(s) and mi (s) are independent of state s. In Example 1 in Section 3.2, irreducibility is satisfied if pL pR = 1. Dutta (1995) provided somewhat weaker assumptions on the transition function that guarantee the state-independence property. But these assumptions are not necessary: the limit set of feasible payoffs is independent of the initial state as long as at least 1 of these probabilities is strictly less than 1, and the minmax payoff is always independent of the initial state. To illustrate this folk theorem, let, for instance, pL = 1 − pR = 1/10 in Example 1, so that playing action L gives player i both his highest reward in that state and the highest probability of transiting to the other state in which his reward is for sure higher than in his own state. The folk theorem holds here. The set of PPE payoffs is then the convex hull of (1 1), (3/2 3/2), (55/28 1), and (1 55/28). See Figure 3.
6. SHORT-RUN PLAYERS It is trivial to extend the algorithm to the case in which some players are short-run. Following FL, suppose that players i = 1 L, L ≤ I, are long-run players whose objective is to maximize the average discounted sum of rewards,
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with discount factor δ < 1. Players j ∈ SR := {L + 1 I} are short-run players, each representative of which plays only once. For each state s ∈ S, let
× L
B(s) :
i=1
× I
Δ(Ai ) →
Δ(Aj )
j=L+1
be the correspondence that maps any mixed action profile (α1 αL ) for the long-run players to the corresponding static equilibria for the short-run players in state s. That is, for each α ∈ graph B(s), and each j > L, αj maximizes r j (s · α−j ). The characterization goes through if we “ignore” the short-run players and simply modify Constrain 1 by requiring that v be a Nash equilibrium payoff of the game Γ (s x) for the long-run players, achieved by some αs ∈ graph B(s) for each s. 6.1. An Example We now provide an illustration of the algorithm that attempts to tread the thin line between accessibility and triviality. Consider the following game, loosely inspired by Dixit, Grossman, and Gul (2000) and Phelan (2006). EXAMPLE 2: There are two parties. In any given period, one of the two parties is in power. Party i = 1 2 is in power in state i. Only the party in power and the households take actions. Households can produce (P) at cost c with value 1, or not produce (NP). There is a continuum of households and, therefore, we treat them as one short-run player. The government in power can either honor (H) or confiscate (C). Honoring means choosing a fixed tax rate τ and, therefore, getting revenues τμi , where μi is the fraction of households who produce in state i. By confiscating, the government appropriates all output. This gives rise to the payoff matrix given by Figure 4. It is assumed that 1 − τ > c > 0. Actions are not observed, but parties that honor are more likely to remain in power. More precisely, if the state is i and the realized action is H, the state remains i with probability pH ; if the action is C, it remains the same with probability pL , with 0 < pL < pH < 1. Note that, given the households’ preferences, the best-reply correspondence in state i is ⎧ i ⎨ [0 1] if α = c/(1 − τ), B(i)(αi ) = {0} if αi < c/(1 − τ), ⎩ {1} if αi > c/(1 − τ),
FIGURE 4.—A political game.
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where αi is the probability that party i plays H. The feasible set F is independent of the initial state and is equal to the convex hull of the payoff vectors ¯ 0), and (0 v), ¯ where (0 0), ((1 − c)/2 (1 − c)/2), (v c c 1 − pL H p + 1− pL (1 − c) p¯ := v¯ := 2 − p¯ − pL 1−τ 1−τ These expressions are intuitive. Note that the highest symmetric payoff involves both parties confiscating as much as possible, subject to the constraint that the households are still willing to produce. The payoff from confiscating at this rate is 1 − c, and since time in power is equally spent between both parties, the resulting payoff is ((1 − c)/2 (1 − c)/2). Consider next the case in which one party always confiscates, so that households never produce in that state, while the other party confiscates at the highest rate consistent with all the households producing. The invariant distribution assigns probability (1 − pL )/(2 − p¯ − pL ) to that state and the asymmetric payoff follows. Finally, the minmax payoff of each party is zero, which is an equilibrium payoff. Theorem 1 applies to this game. Let us compute the equilibrium payoff set as δ → 1. The optimization program with weights λ = (λ1 λ2 ) involves eight variables xit (s), i = 1 2, s t = 1 2, satisfying the constraints (10)
λ · x1 (1) ≤ 0
λ · x2 (2) ≤ 0
λ · (x2 (1) + x1 (2)) ≤ 0
in addition to the requirement that αi be a Nash equilibrium of the game Γ (i x), i = 1 2. Consider a vector λ > 0. Note that the constraints (10) must bind: Indeed, because player i does not make a choice in state −i, the Nash equilibrium requirements (i.e., Constraint 1 in program P (λ)) give us at most three constraints per player (his preference ordering in the state in which he takes an action and the fact that he gets the same payoff in the other state). In addition to the three constraints (10), this gives us nine constraints, and there are eight variables xit (s). We can check that if one of the three constraints is slack, we can increase the payoff of a player by changing the values of xit (s), while satisfying all binding constraints. Observe now that we must have μi ∈ {0 1}, i = 1 2. Indeed, suppose that μi ∈ (0 1) for some i, so that αi ≥ c/(1 − τ). Then we can increase μi and decrease xii (i) xii (−i) so as to keep player i’s payoff constant, while keeping him indifferent between both actions (which he must be since μi ∈ (0 1)). Given these observations, this now becomes a standard problem that can be solved by enumeration. Note, however, that we do not need to consider the case α1 > c/(1 − τ) α2 > c/(1 − τ): if this were the case, we could decrease both of these probabilities in such a way as to maintain the relative time spent in each state the same, while increasing both players’ payoffs in their own state (because μi = 1 is feasible as long as αi ≥ c/(1 − τ)).
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FIGURE 5.—The limit set of equilibrium payoffs in Example 2.
It is not hard to guess that the optimal action profile for λ1 = λ2 > 0 is to set αi = c/(1 − τ), i = 1 2 (and, as mentioned, μi = 1), and we obtain the highest feasible symmetric payoff. If we consider a coordinate direction, say λ1 > 0 λ2 = 0, it is intuitive that households will not produce in state 2, and party 2 will confiscate with sufficient probability for such behavior to be optimal, and party 1 will confiscate at the highest rate consistent with households producing. Party 1 must not be willing either to confiscate for sure (which increases his reward when he does) or to honor for sure (which increases the fraction of time spent in his state, when his reward is positive), and this means that his payoff cannot exceed H p − τpL − (1 − τ) (1 − pL )(pH − τpL ) v˜ := min pH − pL (2 − pL )(pH − pL ) assuming this is positive. It follows that the limit equilibrium payoff set is given by H = {v ∈ F : 0 ≤ vi ≤ v˜ ∀i = 1 2} if v˜ > 0, and the singleton payoff is (0 0) ˜ the payoff set shrinks otherwise. Note that from the first argument defining v, to (0 0) as pH → pL , as is to be expected: if confiscation cannot be statistically detected, households will prefer not to produce. See Figure 5 for an illustration in the case in which v˜ > (1 − c)/2. 6.2. A Characterization With Full Rank The equilibrium payoff set obtained in Example 2 is reminiscent of the results of FL for repeated games with long-run and short-run players. We provide here an extension of their results to the case of stochastic games, but maintain full rank assumptions for long-run players. One of the main insights of FL, which is an extension of Fudenberg, Kreps, and Maskin (1990), is that achieving some payoffs may require the long-run
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players to randomize on the equilibrium path, so as to induce short-run players to take particular actions. This means that long-run player i must be indifferent between all actions in the support of his mixture, and so continuation payoffs must adjust so that his payoff cannot exceed the one he would achieve from the action in this support that yields him the lowest payoff. For a direction λ = ±ei , the pairwise full rank condition ensures that the above requirement is not restrictive, that is, we only need to care about the feasibility given that short-run players take static best responses. However, for the coordinate directions λ = ±ei , since we cannot adjust long-run player i’s continuation payoffs without affecting the score, the optimal mixed action of player i is determined by taking into account both the effect on the short-run players’ actions and the cost of letting player i take that action. The same applies here. We first deal with non-coordinate directions. To incorporate short-run players’ incentives, we define Fδ (s) as the convex hull of the set of long-run players’ feasible payoffs of the game with initial state s and discount factor δ, where we require that the players play actions from graph B(t) whenever the current state is t. The set F(s) of long-run players’ limit-average feasible payoffs is defined similarly. The set Fδ (s) converges to F(s) as δ → 1. ASSUMPTION F3: For each state s and pair (i j) of long-run players, every mixed action profile in graph B(s) has pairwise full rank for players i and j in state s. Similarly to Lemma 2, we can make the following statements: LEMMA 3: Under Assumption F3, we have the following conditions. i k(λ) = mins∈S maxw∈F(s) λ · w. (i) If λ ∈ S 1 and λ = ±e for anyi i, then (ii) H = {v ∈ s∈S F(s) : −k(−e ) ≤ vi ≤ k(ei ) ∀i = 1 L}. Theorem 1 applies here. Namely, if Assumption A is satisfied and H has nonempty interior, then Eδ (s) converges to H as δ → 1 for every s ∈ S. Lemma 3 leaves open how to determine k(±ei ). Under Assumption F3, since every mixed action profile has individual full rank, we can control each long-run player’s continuation payoffs and induce him to play any action. This implies that, without loss of generality, in program P (±ei ), we can ignore the incentives of long-run players other than player i without affecting the score. We can obtain a further simplification if Assumption F3 is strengthened to full monitoring. In this case, k(ei ) is equal to the value of the program Qi+ , sup vi vi x¯ i α
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where the supremum is taken over vi ∈ R, x¯ i : S → RS , and α = (αs ) ∈ s graph B(s) such that (i) for each s, i i i −i i −i i r (s as αs ) + p(t × A|s as αs )x¯ t (s) v = min
×
ais ∈support(αis )
t∈S
and for each T ⊆ S and for each permutation φ : T → T , we have (ii) i i ¯ x s∈T φ(s) (s) ≤ 0. See Section A.4 for a proof. Here v is equal to the minimum of player i’s payoffs, where ais is taken over the support of player i’s mixed action αis . If vi were larger than the minimum, then we would not be able to provide incentives for player i to be indifferent between all actions in the support of αis . Note that this program is simpler than P (ei ) in that player i’s choice is restricted to the support of αis and that x¯ i depends on the current and next states, but is independent of realized actions. Similarly, −k(−ei ) is equal to the solution for the program Qi− , inf vi
vi x¯ i α
where the infimum is taken over vi ∈ R, x¯ i : S → RS , and α = (αs ) ∈ s graph B(s) such that (i) for each s, i i i −i i −i i p(t × A|s as αs )x¯ t (s) v = max r (s as αs ) +
×
ais ∈Ai
t∈S
and (ii) for each T ⊆ S and for each permutation φ : T → T , we have ¯ iφ(s) (s) ≥ 0. These programs reduce to FL, Theorem 6.1(ii) if there is s∈T x only one state. See also Section 4.2 for a discussion of the related one-player case. 7. ACTION-INDEPENDENT TRANSITIONS Our characterization can be used to obtain qualitative insights. We here discuss the case in which the evolution of the state is independent of players’ actions. To let the transition probabilities of the states vary, while keeping the distributions of signals fixed, we assume that the transition probabilities can be written as a product q(t|s) × π(y|s a). In other words, and in any period, the next state t and the public signal y are drawn independently, and the action profile only affects the distribution of y. In this setup, it is intuitively clear that the (limit) set of PPE payoffs should only depend on q through its invariant measure, μ. If the signal structure is rich enough and the folk theorem holds, this is straightforward. Indeed, the (limit) set of feasible payoffs is equal to the convex hull of { s∈S μ(s)r(s a) : a ∈ A} and a similar formula holds for the (limit) minmax.
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We prove below that this observation remains valid even when the folk theorem fails to hold. The proof is based on our characterization, and providing a direct proof of this fact does not seem to be an easy task. In this subsection, we fix a signal structure π : S × A → Δ(Y ) and consider two transition functions, denoted q and (with an abuse of notation) p. To stress the dependence of the analysis on the transition function and given a (unit) direction λ ∈ S 1 , we denote by Pq (λ) the optimization program P (λ) when the transition probability over states is set to q : S → Δ(S). We also denote by kq (λ) the value of Pq (λ) and denote by H(q) the intersection of all half-spaces {v ∈ RI : λ · v ≤ kq (λ)} obtained by letting λ vary. PROPOSITION 3: Let p and q be two irreducible transition functions over S, with the same invariant measure μ. Then H(p) = H(q). This implies that for the purpose of studying the limit equilibrium payoff set in the case of transitions that are action-independent, we might as well assume that the state is drawn (i.i.d.) across periods.14 In that case, the stochastic game can be viewed as a repeated game in which player i’s actions in each period are maps from S to Ai . The program P (λ α) can then be shown to be equivalent to the corresponding program of FL. One direction is a simple change of variable; the other direction relies on Lemma 1. 8. CONCLUDING COMMENTS This paper shows that recursive methods developed for the study of repeated games can be generalized to stochastic games and can be applied to obtain qualitative insights as δ → 1. This, of course, leaves many questions unanswered. First, as mentioned, the results derived here rely on δ → 1. While not much is known in the general case for repeated games either, there still are a few results available in the literature (on the impact of the quality of information, for instance). It is then natural to ask whether those results can be extended to stochastic games. Within the realm of asymptotic analysis, it also appears important to generalize our results to broader settings. The characterizations of the equilibrium payoff set and the folk theorem established in this paper rely on some strong assumptions. The set of actions, signals, and states are all assumed to be finite. We suspect that the characterization can be generalized to the case of 14 One direction is intuitive: when transitions are action-independent, we can construct equilibria in which play in periods 1 k + 1 2k + 1 proceeds independently from play in periods 2 k + 2 2k + 2 and so on. Hence, the equilibrium payoff set with transition q and discount δ “contains” the equilibrium payoff set with transition qk and discount δk (note, however, that we are not comparing sets for a common initial state). Hence, taking δ to 1 faster than we take k to infinity makes it plausible that the limit payoff set in the i.i.d. case is included in the limit set given q.
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richer action and signal sets, but extending the result to richer state spaces raises significant challenges. Yet this is perhaps the most important extension: even with finitely many states, beliefs about those states affect the players’ incentives when states are no longer common knowledge, as is often the case in applications in which players have private information, and these beliefs must therefore be treated as state variables themselves. Finally, there is an alternative possible asymptotic analysis that is of significant interest. As is the case with public signals (see the concluding remarks of FLM), an important caveat is in order regarding the interpretation of our limit results. It is misleading to think of δ → 1 as periods growing shorter, as transitions are kept fixed throughout the analysis. If transitions are determined by physical constraints, then these probabilities should be adjusted as periods grow shorter. As a result, feasible payoffs will not be independent of the initial state. It remains to be seen whether such an analysis is equally tractable. APPENDIX: PROOFS A.1. Proof of Proposition 1 Let δ < 1 be given. To show the first statement, for each state s, choose an equilibrium payoff vs that maximizes the inner product λ · w for w ∈ Eδ (s). Consider a PPE σ with payoffs (vs ). For each initial state s, let αs = σ(s) be the mixed moves played in stage 1 and let wt (s y) be the continuation payoffs following y, if in state t at stage 2. Note that λ · wt (s y) ≤ λ · vt for y ∈ Y and s t ∈ S. By construction, for each s ∈ S, αs is a Nash equilibrium, with payoff vs , of the game with payoff function p(t y|s as )wt (s y) (1 − δ)r(s as ) + δ t∈S y∈Y δ (wt (s y) − vs ) Observe that αs is a Nash equilibrium of Set xt (s y) := 1−δ the game Γ (s x), with payoff vs . Next, let s¯ be a state that minimizes λ · vs and let x˜ t (s y) := xt (s y) + vs¯ − vs for each s t ∈ S, y ∈ Y . Then αs is a Nash ˜ with payoff vs¯ . On the other hand, for each T φ ψ, we equilibrium of Γ (s x), have
δ λ· λ· xφ(s) (s ψ(s)) = wφ(s) (s ψ(s)) − vs 1−δ s∈T s∈T
= and hence λ·
s∈T
δ λ· wφ(s) (s ψ(s)) − vφ(s) ≤ 0 1−δ s∈T
x˜ φ(s) (s ψ(s)) = λ ·
s∈T
xφ(s) (s ψ(s)) + λ ·
s∈T
(vs¯ − vs ) ≤ 0
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˜ α) is feasible in P (λ). Thus Therefore, (vs¯ x k(λ) ≥ λ · vs¯ = min max λ · w s∈S w∈Eδ (s)
The second statement follows immediately from the first statement. A.2. Proof of Proposition 2 Since Z is a compact set contained in the interior of H, there exists η > 0 such that Zη := {z ∈ RI : d(z Z) ≤ η} is also a compact set contained in the interior of H. Zη is a “smooth approximation” of Z such that the curvature of the boundary of Zη is at most 1/η. We start with a technical statement. We say that (v x) is feasible in P (λ) if (v x α) is feasible in P (λ) for some α. LEMMA 4: There are ε0 > 0 and a bounded set K ⊂ RI × RS×Y ×S×I of (v x) such that the following statement holds. For every z ∈ Zη and λ ∈ S 1 , there exists (v x) ∈ K such that (v x) is feasible in P (λ) and λ · z + ε0 < λ · v. PROOF: Given z ∈ Zη and since Zη is contained in the interior of H, we have λ · z < k(λ) for every λ ∈ S 1 . Therefore, there exists a feasible pair (v x) in P (λ) such that λ · z < λ · v. The conclusion of the lemma states that (v x) can be chosen within a bounded set, independently of λ and of z.15 Choose ε˜ > 0 such that λ · z + ε˜ < λ · v and define x˜ t (s y) = xt (s y) − ελ ˜
v˜ = v − ελ ˜
for each s t ∈ S and y ∈ Y . Observe that for each s ∈ S, v is an equilibrium pay˜ Note in addition that s∈T λ · x˜ φ(s) (s ψ(s)) ≤ −ε|T off of the game Γ (s x). ˜ |< 0 for each T φ, and ψ. Therefore, for every z˜ close enough to z, for every λ˜ ˜ and λ˜ · z˜ < λ˜ · v. ˜ x) ˜ is feasible in P (λ) ˜ The result close enough to λ, the pair (v Q.E.D. then follows, by compactness of Zη and of S 1 . In the sequel, we let κ0 > 0 be such that x ≤ κ0 and z − v ≤ κ0 for every (v x) ∈ K and z ∈ Zη . Choose n ∈ N such that ε0 (n − 1)/2 > 2κ0 |S|. Set ε := ε0
n−1 − 2κ0 |S| > 0 2
15 Note, however, that for given z and λ, the set of feasible pairs (v x) in P (λ) such that λ · z < λ · v is typically unbounded.
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Next, choose δ¯ < 1 to be large enough so that (n/2)2 (1−δ) ≤ |S| and 1−δn−1 ≥ ¯ Finally, set (n − 1)(1 − δ)/2 for every δ ≥ δ. κ :=
2(n − 1) κ0 δ¯ n−1
Given a map w : Hn → RI which associates a payoff vector, to any history of length n, we denote by Γ n (s w; δ) the δ-discounted, (n − 1)-stage game with final payoffs w and initial state s. The following proposition is essential for the proof of Proposition 2. PROPOSITION 4: For every direction λ ∈ S 1 , every z ∈ Zη , and every discount ¯ there exist continuation payoffs w : Hn → RI such that the following factor δ ≥ δ, conditions hold: C1. For each s, z is a PPE payoff of the game Γ n (s w; δ). C2. We have w(h) − z ≤ (1 − δ)κ for every h ∈ Hn . C3. We have λ · w(h) ≤ λ · z − (1 − δ)ε for every h ∈ Hn . PROOF: Let λ, z, and δ be given as stated. Pick a feasible point (v x α) in P (λ) such that (v x) ∈ K and λ · z + ε0 < λ · v. Given s t ∈ S and y ∈ Y , set φt (s y) = v +
1 − δ¯ xt (s y) δ¯
For each history h of length not exceeding n, we define w(h) by induction on the length of h. The definition of w follows FL. If h = (s1 ), we set w(h) = z. For k ≥ 1 and hk+1 ∈ Hk+1 , we set (A.11)
¯ − δ) δ − δ¯ δ(1 w(hk ) − v w(hk ) + φsk+1 (sk yk ) + w(hk+1 ) = ¯ ¯ δ(1 − δ) δ(1 − δ) δ¯
where hk is the restriction of hk+1 to the first k stages. Let k ≥ 1 and hk ∈ Hk . As in FL, using (A.11), αsk is a Nash equilibrium of the one-shot game with payoff function (1 − δ)r(sk a) + δ
p(t y|sk a)w(hk y t)
t∈S y∈Y
with payoff w(hk ). By the one-shot deviation principle, it follows that the Markov strategy profile α is a PPE of the game Γ n (s1 w; δ), with payoff z (for each initial state s1 ). This proves C1.
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We now turn to C2. Let hn ∈ Hn be given and let hk (k ≤ n) denote the restriction of hn to the first k stages. Observe that δ − δ¯ (w(hk ) − v) ¯ δ(1 − δ) ¯ − δ) δ(1 + φsk+1 (sk yk ) − v + ¯ δ(1 − δ)
w(hk+1 ) − v =
1 (w(hk ) − v) δ¯
1−δ 1 (w(hk ) − v) + xsk+1 (sk yk ) δ δ
=
for any k ≤ n − 1. Therefore, w(hn ) − v =
1
δ
(z − v) + n−1
1−δ 1 xs (sk yk ) δ k=1 δn−1−k k+1 n−1
and we get 1 − δn−1 1 − δ k−1 (z − v) + δ xsk+1 (sk yk ) δn−1 δn−1 k=1 n−1
(A.12)
w(hn ) − z =
Hence w(hn ) − z ≤
n−1 1 − δn−1 1 − δ k−1 z − v + δ xsk+1 (sk yk ) n−1 n−1 δ δ k=1
2(1 − δn−1 ) κ0 δn−1 2(n − 1)(1 − δ) ≤ κ0 = (1 − δ)κ δ¯ n−1
≤
This proves C2. We finally prove that C3 holds as well. The proof makes use of the following lemma. LEMMA 5: Let x1 xm ∈ [−1 1] be such that m k−1 m/2 2 xk ≤ (1−δ1−δ ) . k=1 δ
m k=1
xk ≤ 0. Then
m PROOF: If m is even, the sum k=1 δk−1 xk is highest if xk = 1 for k ≤ m2 , and xk = −1 for k > m2 . If m is odd, the sum is maximized by setting xk = 1 for k < m+1 , xk = 0 for k = m+1 , and xk = −1 for k > m+1 . In both cases, the sum 2 2 2 1−2δm/2 +δm . Q.E.D. is at most 1−δ
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Set m0 = 0, l1 + 1 := min{k ≥ 1 : sm = sk for some m > k} (min ∅ = +∞), and m1 + 1 := max{k ≤ n : sk = sl1 +1 }. Next, as long as lj < +∞, define lj+1 + 1 := min{k ≥ mj + 1 : sm = sk for some m > k} and mj+1 + 1 := max{k ≤ n : sk = slj+1 +1 }. Let J be the largest integer j with lj < +∞. mj Since slj +1 = smj +1 , we have λ · k=l x (sk yk ) ≤ 0 for each j ≤ J. By j +1 sk+1 Lemma 5, this implies mj J
2 κ 0 lj δ 1 − δ(mj −lj )/2 1 − δ j=1 J
δk−1 λ · xsk+1 (sk yk ) ≤
j=1 k=lj +1
2 κ0 lj /2 δ − δmj /2 = 1 − δ j=1 J
The sum which appears in the last line is of the form J (uj − vj )2 j=1
with 1 ≥ u1 ≥ v1 ≥ · · · ≥ uJ ≥ vJ ≥ δn/2 . Such a sum is maximized when J = 1, u1 = 1, and v1 = δn/2 . It is then equal to (1 − δn/2 )2 . On the other hand, there are at most |S| stages k with mj < k ≤ lj+1 for some j. Therefore,
2 1 − δ k−1 1−δ κ0 δ λ · xsk+1 (sk yk ) ≤ |S|κ0 n−1 + n−1 1 − δn/2 n−1 δ δ δ k=1 n−1
2 1−δ κ0 n ≤ |S|κ0 n−1 + n−1 (1 − δ)2 δ δ 2 ≤ 2|S|κ0
1−δ δn−1
Substituting this inequality into (A.12), we obtain 1 − δn−1 1 − δ k−1 λ · w(hn ) ≤ λ · z + λ · (z − v) + δ λ · xsk+1 (sk yk ) δn−1 δn−1 k=1 n−1
n−11−δ 1−δ + 2|S|κ0 n−1 2 δn−1 δ 1−δ = λ · z − ε n−1 δ ≤ λ · z − ε(1 − δ) ≤ λ · z − ε0
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Q.E.D.
as claimed.
¯¯ ≤ η and (1− δ)κ ¯¯ 2 ≤ 2εη. The next Let δ¯¯ < 1 be large enough so that (1− δ)κ lemma follows since the curvature of Zη is uniformly bounded from above. ¯¯ there exists a direction λ ∈ S 1 such LEMMA 6: For every z ∈ Zη and every δ ≥ δ, I that if w ∈ R satisfies w − z ≤ (1 − δ)κ and λ · w ≤ λ · z − (1 − δ)ε, then we have w ∈ Zη . PROOF: By the definition of Zη , for each z ∈ Zη , there exists z0 ∈ Z such that z − z0 ≤ η. Let λ := (z − z0 )/z − z0 . (If z0 = z, then take any unit vector.) Then, for any w, we have w − z0 2 = z − z0 2 + 2(z − z0 ) · (w − z) + w − z2 ≤ z − z0 2 − 2(1 − δ)εz − z0 + (1 − δ)2 κ2 The last expression is a quadratic form, which is maximized when z − z0 = 0 or z − z0 = η. Therefore, w − z0 2 ≤ max{(1 − δ)2 κ2 η2 − 2(1 − δ)εη + (1 − δ)2 κ2 } ≤ η2 ¯¯ Thus w − z ≤ η, hence w ∈ Z . because δ ≥ δ. 0 η
Q.E.D.
¯¯ For any z ∈ Z , ¯ δ}. We here prove Proposition 2. Fix any δ ≥ max{δ η i we construct a public strategy σ : H → i Δ(A ) and continuation payoffs w : H → Zη inductively as follows. Set w(h) = z ∈ Zη for any h = (s1 ) ∈ H1 . For k ≥ 1 and h ∈ H(n−1)(k−1)+1 , given that w(h) ∈ Zη , by Proposition 4, there exist continuation payoffs wh : Hn → RI that satisfy C1 (that is, there exists a PPE of Γ n (s(n−1)(k−1)+1 wh ; δ), with payoff w(h)), C2, and C3. Let σ prescribe the PPE of Γ n (s(n−1)(k−1)+1 wh ; δ) for the block of periods between (n − 1)(k − 1) + 1 and (n − 1)k. For any h˜ ∈ H(n−1)k+1 whose restriction to the ˜ where n h˜ is the ˜ = wh (n h), first (n − 1)(k − 1) + 1 periods is equal to h, let w(h) restriction of h˜ to the last n periods. It follows from C2, C3, and Lemma 6 that ˜ ∈ Zη . By the one-shot deviation principle and C1, for any initial state s, w(h) the constructed strategy σ is a PPE of the whole (infinite-horizon) game, with payoff z. Thus Z ⊂ Zη ⊆ Eδ (s) for any s ∈ S.
×
A.3. Proof of Lemma 2 We prove the two statements in turn. We start with the first one and consider the two cases λ = ±ei for any i and λ = ei . Suppose that λ = ±ei for any i. Let δ < 1 be given. Then there exists a pure Markov strategy profile (as ) ∈ AS with payoff (vs ) such that for each state s,
RECURSIVE METHODS IN DISCOUNTED STOCHASTIC GAMES
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vs maximizes the inner product λ · w for w ∈ Fδ (s). By construction, for each s ∈ S, vs = (1 − δ)r(s as ) + δ t p(t × Y |s as )vt . Fix ε > 0 arbitrarily. By Assumption F2 and Lemma 6.2 of FLM, there exists an open and dense set of action profiles, each of which has pairwise full rank for all pairs of players. Therefore, for each s ∈ S, there exist vˆ s , αˆ s , and wˆ t (s) such that λ · wˆ t (s) ≤ λ · vˆ t for t ∈ S, vˆ s = (1 − δ)r(s αˆ s ) + δ t p(t × Y |s αˆ s )wˆ t (s), λ · vˆ s ≥ λ · vs − ε, and αˆ s has pairwise full rank for all pairs of players in state s. Similarly to the proof of Proposition 1, let s¯ be a state that minimizes λ · vˆ s . δ Set x˜ t (s) := 1−δ (wˆ t (s) − vˆ s ) + vˆ s¯ − vˆ s for s t ∈ S. Then we have vˆ s¯ = r(s αˆ s ) +
p(t × Y |s αˆ s )x˜ t (s)
t∈S
for each s ∈ S and λ· x˜ φ(s) (s) ≤ 0 s∈T
for each T ⊆ S, for each permutation φ : T → T . For each s ∈ S, although αˆ s ˜ since αˆ s satisfies pairwise full rank, may not be a Nash equilibrium of Γ (s x), there exists xˆ t (s y) for y ∈ Y and t ∈ S such that αˆ s is a Nash equilibrium of ˆ with payoff vˆ s¯ , and such that λ · xˆ t (s y) = 0 for each y ∈ Y and Γ (s x˜ + x) t ∈ S. With xt (s y) := x˜ t (s) + xˆ t (s y), we have λ· xφ(s) (s ψ(s)) ≤ 0 s∈T
for each T ⊆ S, each permutation φ : T → T , and each map ψ : T → Y . It follows that k(λ) ≥ λ · vˆ s¯ ≥ min max λ · w − ε s∈S w∈Fδ (s)
Since ε > 0 and δ < 1 are arbitrary, k(λ) ≥ sup min max λ · w ≥ min max λ · w δ<1
s∈S w∈Fδ (s)
s∈S w∈F(s)
Conversely, we have max λ · w = sup λ ·
w∈F(s)
α
μαs (t)r(t αt )
t∈S
where μαs (t) is the (expected) long-run frequency of state t under the Markov chain induced by α with initial state s and where the supremum is taken over
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all Markov strategy profiles α = (αt )t∈S . Thus μα (t)r(t αt ) min max λ · w ≥ sup min λ · s∈S w∈F(s)
α
μα
t∈S
where the minimum is taken over all invariant measures μα of the Markov chain induced by α. By Lemma 1, for each α, we have λ· μα (t)r(t αt ) ≥ k(λ α) t∈S
for any invariant measure μα . Hence, min max λ · w ≥ sup k(λ α) = k(λ) s∈S w∈F(s)
α
Suppose next that λ = ei for some i. Let δ < 1 be given. Then there exists a pure Markov strategy profile (as ) ∈ AS with payoff (vs ) such that for each i state s, vsi maximizes w for w ∈ Fδ (s). By construction, for each s ∈ S, vs = (1 − δ)r(s as ) + δ t p(t × Y |s as )vt and player i maximizes his own payoff. δ Let s¯ be a state that minimizes vsi . Set x˜ t (s) := 1−δ (vt −vs )+vs¯ −vs for s t ∈ S. Then we have vs¯ = r(s as ) + t p(t × Y |s as )x˜ t (s) for each s ∈ S without affecting player i’s incentives. For each s ∈ S, although players other than i may ˜ by Assumption F1, there exists xˆ t (s y) not maximize their payoffs in Γ (s x), ˆ with for y ∈ Y and t ∈ S such that as is a Nash equilibrium of Γ (s x˜ + x) payoff vs¯ , and such that xˆ it (s y) = 0 for each y ∈ Y and t ∈ S. Thus we have k(ei ) ≥ supδ<1 mins∈S maxw∈Fδ (s) wi ≥ mins∈S maxw∈F(s) wi . The other direction of inequality is similar to that in the previous case. This concludes the proof of the first statement. Last, consider λ = −ei for some i. Let δ < 1 be given. Then there exists a for minmaxing Markov strategy profile (ais α−i s ) for player i. By construction, i −i i each s ∈ S, miδ (s) = (1 − δ)r i (s ais α−i ) + δ p(t × Y |s a α )m (t) and s s s δ t player i maximizes his own payoff. δ Let s¯ be a state that maximizes miδ (s). Set x˜ it (s) := 1−δ (miδ (t) − miδ (s)) + i i i i mδ (¯s) − mδ (s) for s t ∈ S. Then we have mδ (¯s) = r (s ais α−i s ) + t p(t × i ˜ Y |s ais α−i ) x (s) for each s ∈ S without affecting player i’s incentives. For s t each s ∈ S, although players other than i may not maximize their payoffs ˜ by Assumption F1, there exists xˆ t (s y) for y ∈ Y and t ∈ S such in Γ (s x), ˆ with player i’s payoff miδ (¯s), that as is a Nash equilibrium of Γ (s x˜ + x) i and such that xˆ t (s y) = 0 for each y ∈ Y and t ∈ S. (If (ais α−i s ) does not have individual full rank, apply Lemma 6.3 of FLM to approximate (ais α−i s ) by a sequence of action profiles with individual full rank and the best response property for player i, as we did in the case of λ = ±ei .) Thus we have k(−ei ) ≥ supδ<1 (− maxs miδ (s)) ≥ − maxs mi (s).
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Conversely, we have mi (s) ≤ inf sup α−i
αi
μαs (t)r i (t αt )
t∈S
where αi and α−i are Markov strategies, and μαs (t) is the long-run frequency of state t under the Markov chain induced by α with initial state s. Thus max mi (s) ≤ inf sup max μα (t)r i (t αt ) s∈S
α−i
αi
μα
t∈S
where the maximum is taken over all invariant measures μα of the Markov chain induced by α. Let (α˜ i α−i ) be an arbitrary Markov strategy and let ε > 0. Then there exists (v x) that is feasible in P (−ei (α˜ i α−i )) and −vi > k(−ei α˜ i α−i ) − ε. It follows from the incentive constraints in P (−ei (α˜ i α−i )) that for each s ∈ S, we have i vi ≥ r i (s αis α−i p(t y|s αis α−i s )+ s )xt (s y) t∈Sy∈Y
By Lemma 1, it thus follows that for each (α˜ i α−i ), i μαi α−i (s)r i (s αis α−i ˜ i α−i ) + ε s ) ≤ vi < −k(−e α s∈S
for any αi and any invariant measure μαi α−i . Since (α˜ i α−i ) and ε > 0 are arbitrary, max mi (s) ≤ inf inf(−k(−ei α˜ i α−i )) = −k(−ei ) s∈S
α−i α˜ i
This concludes the proof of the second statement. A.4. Proof for the Characterization With Full Monitoring We first show that P (ei ) and Qi+ are equivalent. Take any (v x α) that is feasible in P (ei ). We have −i i −i i i −i ) + α−i vi = r i (s ais α−i s s (as )p(t as |s as as )xt (s as as ) t∈S a−i ∈A−i s
for every s ∈ S and ais ∈ support(αis ). For each s, set x˜ it (s) :=
max
max xit (s ais a−i s )
−i ais ∈support(αis ) a−i s ∈A
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HÖRNER, SUGAYA, TAKAHASHI, AND VIEILLE
and
v˜ := i s
min
ais ∈support(αis )
−i s
r (s a α ) + i
i s
−i s
p(t × A|s a α )x˜ (s) i s
i t
t∈S
Next, set x¯ it (s) := x˜ it (s) + vi − v˜ si . Since v˜ si ≥ vi for every s ∈ S, we have that (vi x¯ i α) is feasible in Qi+ . Conversely, take any (vi x¯ i α) that is feasible in Qi+ . For each s ∈ S and −i as ∈ A−i , we set ¯ it (s) + vi xit (s ais a−i s ) := x i i −i i −i i − r (s as αs ) + p(u × A|s as αs )x¯ u (s) u∈S
if ais ∈ support(αis ) (note that the right-hand side is independent of a−i s ) i −i i i i and we let xit (s a a ) be sufficiently small if a ∈ / support(α ). Since v ≤ s s s s i −i i i i ¯ ) + p(u × A|s a α ) x (s) for every a ∈ support(α ), we r i (s ais α−i s s s u s s u∈S have that (vi v−i xi x−i α) is feasible in P (ei ). (v−i and x−i are chosen to give incentives for players −i to play α−i .) The equivalence between P (−ei ) and Qi− follows similarly. A.5. Proof of Proposition 3 We let a unit direction λ ∈ S 1 and a Markov strategy profile α = (αs ) be given. Consider programs Pp (λ α) and Pq (λ α) with fixed α. We prove that for each feasible pair (v x) in Pp (λ α), there is z : S × Y → RS×I such that (v z) is a feasible pair in Pq (λ α). This implies the result. The proof uses the following two observations. Denote by P (resp. Q) the S × S matrix whose (s t)th entry is p(t|s) (resp. q(t|s)). CLAIM 2: The ranges of the two linear maps (with matrices) Id − P and Id − Q are equal. PROOF: Assume that q is aperiodic. Let there be given a vector ξ ∈ RS in the range of Id − P. Hence, there is ψ ∈ RS such that ξ = ψ − Pψ. Hence, μ ξ = 0. Since q is an irreducible transition function with invariant measure μ, the vector Qn ξ converges as n → ∞ to the constant vector, whose components are all equal to μ ξ = 0. Moreover, the convergence takes place at an +∞ exponential speed. Therefore, the vector ψ˜ := n=0 Qn ξ is well defined and ˜ This proves the result.16 solves ψ˜ = ξ + Qψ. Q.E.D. 16 The proof in the case where q is periodic is similar. As in the aperiodic case, the infinite sum in ψ˜ is well defined.
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S×S be given such that the inequality CLAIM 3: Let a vector c = (ct (s)) ∈ R s∈T cφ(s) (s) ≤ 0 holds for every T ⊆ S and every permutation φ over T . Then thereexists a vector c ∗ = (ct∗ (s)) ∈ RS×S such that (i) c ∗ ≥ c (componentwise) and ∗ (s) = 0 for every T ⊆ S and every permutation φ over T . (ii) s∈T cφ(s)
PROOF: Consider the linear program supg Tφ s∈T gφ(s) (s), where the first sum ranges over all (T φ) such that T ⊆ S and φ is a permutation over T , and the supremum is taken over all g such that (i) g ≥ c and (ii) s∈T gφ(s) (s) ≤ 0 for every T ⊆ S and every permutation φ over T . This program is bounded and c is a feasible point, hence there is an optimal solution c ∗ . We claim that the value of the program is 0. Assume to the contrary that ∗ (A.13) cφ(s) (s) < 0 s∈T
for some T and some φ. It must then be the case that for each s ∈ T there some permutation φs over Ts with exists some set Ts ⊆ S containing s and φs (s) = φ(s) and such that the constraint t∈Ts cφ∗ s (t) (t) ≤ 0 is binding. (Other∗ wise indeed, a small increase in the value of cφ(s) (s) would preserve feasibility ∗ ∗ of c and improve upon c .) In particular, ∗ cφ∗ s (t) (t) = −cφ(s) (s) t∈Ts t =s
When summing over s ∈ T , and by (A.13), we obtain (A.14) cφ∗ s (t) (t) > 0 s∈T t∈Ts t =s
We claim that the directed graph over S with edge set s∈T t∈Ts t =s (t φs (t)) (where any edge that appears more than once is repeated) is a collection of disjoint cycles. Hence (A.14) contradicts the fact that c ∗ is feasible. To see this last claim, observe that for each state s ∈ T , the edges (t φs (t)), where t ∈ Ts , t = s, form a directed path from φ(s) to s. Hence, the union over s of these paths is a union of disjoint cycles. Q.E.D. We turn to the proof of Proposition 3. Let (v x) be a feasible pair in Pp (λ α). For (s t) ∈ S, we set ct (s) := maxy∈Y λ · xt (s y). Apply Claim 3 to ∗ (s) = 0 for every T ⊆ S and every permutation obtain (ct∗ (s)). Since s∈T cφ(s) φ over T , there is a vector c¯∗ ∈ RS such that ct∗ (s) = c¯t∗ − c¯s∗ for every s t ∈ S. ¯ For s t ∈ Next, by Claim 2, there is d¯ ∈ RS such that (Id − P)c¯∗ = (Id − Q)d. S, set dt (s) := d¯t − d¯s . Observe that, by construction, one has t∈S p(t|s)ct∗ (s) = t∈S q(t|s)dt (s) for each s ∈ S.
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HÖRNER, SUGAYA, TAKAHASHI, AND VIEILLE
Finally, we set zti (s y) := λi dt (s) +
p(u|s)(xiu (s y) − λi cu∗ (s))
u∈S
for any i s t y. We claim that (v z) is feasible in Pq (λ α), as desired. By construction, we have λ · zt (s y) = dt (s) + p(u|s)(λ · xu (s y) − cu∗ (s)) ≤ dt (s) u∈S
hence the vector z satisfies all linear constraints in Pq (λ α). On the other hand, and for each y, we have q(t|s)zti (s y) = p(u|s)xiu (s y) t∈S
u∈S
+λ
i
=
q(t|s)dt (s) −
t∈S
∗ u
p(u|s)c (s)
u∈S
p(t|s)xit (s y)
t∈S
Hence the equality q(t|s)π(y|s αs )zti (s y) = p(t|s)π(y|s αs )xit (s y) t∈Sy∈Y
t∈Sy∈Y
holds for any s ∈ S. This concludes the proof of Proposition 3. REFERENCES ABREU, D., D. PEARCE, AND E. STACCHETTI (1990): “Toward a Theory of Discounted Repeated Games With Imperfect Monitoring,” Econometrica, 58, 1041–1063. [1277,1282,1292] BEWLEY, T., AND E. KOHLBERG (1976): “The Asymptotic Theory of Stochastic Games,” Mathematics of Operations Research, 1, 197–208. [1281] DIXIT, A., G. M. GROSSMAN, AND F. GUL (2000): “The Dynamics of Political Compromise,” Journal of Political Economy, 108, 531–568. [1299] DUTTA, P. K. (1995): “A Folk Theorem for Stochastic Games,” Journal of Economic Theory, 66, 1–32. [1278,1285,1296-1298] ERICSON, R., AND A. PAKES (1995): “Markov Perfect Industry Dynamics: A Framework for Empirical Work,” Review of Economic Studies, 62, 53–82. [1279] FINK, A. M. (1964): “Equilibrium in a Stochastic n-Person Game,” Journal of Science of the Hiroshima University, Ser. A-I, 28, 89–93. [1281] FREIDLIN, M. I., AND A. D. WENTZELL (1991): Random Perturbations of Dynamical Systems. New York: Springer-Verlag. [1289] FUDENBERG, D., AND D. LEVINE (1994): “Efficiency and Observability With Long-Run and Short-Run Players,” Journal of Economic Theory, 62, 103–135. [1277]
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FUDENBERG, D., AND E. MASKIN (1986): “The Folk Theorem for Repeated Games With Discounting or With Incomplete Information,” Econometrica, 54, 533–554. [1278] FUDENBERG, D., AND Y. YAMAMOTO (2010): “The Folk Theorem for Irreducible Stochastic Games With Imperfect Public Monitoring,” Mimeo, Harvard University. [1279,1297] FUDENBERG, D., D. KREPS, AND E. MASKIN (1990): “Repeated Games With Long-Run and Short-Run Players,” Review of Economic Studies, 57, 555–573. [1301] FUDENBERG, D., D. LEVINE, AND E. MASKIN (1994): “The Folk Theorem With Imperfect Public Information,” Econometrica, 62, 997–1040. [1277] GREEN, E. J., AND R. H. PORTER (1984): “Noncooperative Collusion Under Imperfect Price Information,” Econometrica, 52, 87–100. [1279] HOFFMAN, A. J., AND R. M. KARP (1966): “On Nonterminating Stochastic Games,” Management Science, 12, 359–370. [1293] JUDD, K. L., S. YELTEKIN, AND J. CONKLIN (2003): “Computing Supergames Equilibria,” Econometrica, 71, 1239–1254. [1277] KALLENBERG, L. (2002): “Finite State and Action MDPs,” in Handbook of Markov Decision Processes Methods and Applications, ed. by E. A. Feinberg and A. Shwartz. Norwell, MA: Kluwer Academic, 21–87. [1293] LAGUNOFF, R., AND A. MATSUI (1997): “Asynchronous Choice in Repeated Coordination Games,” Econometrica, 65, 1467–1477. [1281] MAILATH, G. J., AND L. SAMUELSON (2006): Repeated Games and Reputations: Long-Run Relationships. New York: Oxford University Press. [1278] MERTENS, J.-F., AND A. NEYMAN (1981): “Stochastic Games,” International Journal of Game Theory, 10, 53–66. [1296] MERTENS, J.-F., AND T. PARTHASARATHY (1987): “Equilibria for Discounted Stochastic Games,” Discussion Paper 8750, C.O.R.E. [1277] (1991): “Nonzero-Sum Stochastic Games,” in Stochastic Games and Related Topics, ed. by T. E. S. Raghavan, T. S. Ferguson, T. Parthasarathy, and O. J. Vrieze. Boston: Kluwer Academic, 145–148. [1277] NEYMAN, A. (2003): “Stochastic Games: Existence of the Minmax,” in Stochastic Games and Applications, ed. by A. Neyman and S. Sorin. NATO ASI Series. Dordrecht, The Netherlands: Kluwer Academic, 173–193. [1296] NUMMELIN, E. (1984): General Irreducible Markov Chains and Non-Negative Operators. Cambridge: Cambridge University Press. [1285] PHELAN, C. (2006): “Public Trust and Government Betrayal,” Journal of Economic Theory, 130, 27–43. [1299] PHELAN, C., AND E. STACCHETTI (2001): “Sequential Equilibria in a Ramsey Tax Model,” Econometrica, 69, 1491–1518. [1279] RADNER, R., R. MYERSON, AND E. MASKIN (1986): “An Example of a Repeated Partnership Game With Discounting and With Uniformly Inefficient Equilibria,” Review of Economic Studies, 53, 59–69. [1278] ROTEMBERG, J. J., AND G. SALONER (1986): “A Supergame-Theoretic Model of Price Wars During Booms,” American Economic Review, 76, 390–407. [1279] SENNOTT, L. I. (1999): Stochastic Dynamic Programming and the Control of Queuing Systems. New York: Wiley Interscience. [1293] SOLAN, E. (1998): “Discounted Stochastic Games,” Mathematics of Operations Research, 23, 1010–1021. [1277] SORIN, S. (1986): “Asymptotic Properties of a Non-Zero Sum Stochastic Game,” International Journal of Game Theory, 15, 101–107. [1281] TAKAHASHI, M. (1962): “Stochastic Games With Infinitely Many Strategies,” Journal of Science of the Hiroshima University, Ser. A-I, 26, 123–134. [1281] YOON, K. (2001): “A Folk Theorem for Asynchronously Repeated Games,” Econometrica, 69, 191–200. [1281]
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Dept. of Economics, Yale University, 30 Hillhouse Avenue, New Haven, CT 06520, USA;
[email protected], Dept. of Economics, Princeton University, Fisher Hall, Princeton, NJ 08544, U.S.A.; tsugaya@ princeton.edu, Dept. of Economics, Princeton University, Fisher Hall, Princeton, NJ 08544, U.S.A.; satorut@ princeton.edu, and HEC Paris, 78351 Jouy-en-Josas, France;
[email protected]. Manuscript received December, 2004; final revision received August, 2010.
Econometrica, Vol. 79, No. 4 (July, 2011), 1319–1321
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Prospective contributors are invited to submit titles and abstracts of their papers by May 4, 2011 at the conference website: https://editorialexpress.com/conference/NAWM2012 Authors that submit complete papers are treated favorably. JONATHAN LEVIN Program Committee Chair
Econometrica, Vol. 79, No. 4 (July, 2011), 1323
FORTHCOMING PAPERS THE FOLLOWING MANUSCRIPTS, in addition to those listed in previous issues, have been accepted for publication in forthcoming issues of Econometrica. EATON, JONATHAN, SAMUEL KORTUM, AND FRANCIS KRAMARZ: “An Anatomy of International Trade: Evidence From French Firms.” ECKSTEIN, ZVI, AND OSNAT LIFSHITZ: “Dynamic Female Labor Supply.” GOSSNER, OLIVIER: “Simple Bounds on the Value of a Reputation.” LEE, JIHONG, AND HAMID SABOURIAN: “Efficient Repeated Implementation.” MCLENNAN, ANDREW, PAULO K. MONTEIRO, AND RABEE TOURKY: “Games With Discontinuous Payoffs: A Strengthening of Reny’s Existence Theorem.” PERRIGNE, ISABELLE, AND QUANG VUONG: “Nonparametric Identification of a Contract Model With Adverse Selection and Moral Hazard.” ROBIN, JEAN-MARC: “On the Dynamics of Unemployment and Wage Distributions.”
© 2011 The Econometric Society
DOI: 10.3982/ECTA794FORTH
Econometrica, Vol. 79, No. 4 (July, 2011), 1325
SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES FOR MONOGRAPHS IN ECONOMIC THEORY, a PDF of the manuscript should be emailed to Professor George J. Mailath at
[email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor George J. Mailath Department of Economics 3718 Locust Walk University of Pennsylvania Philadelphia, PA 19104-6297, U.S.A. For monographs in theoretical and applied econometrics, a PDF of the manuscript should be emailed to Professor Rosa Matzkin at
[email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor Rosa Matzkin Department of Economics University of California, Los Angeles Bunche Hall 8349 Los Angeles, CA 90095 They must be accompanied by a letter of submission and be written in English. Authors submitting a manuscript are expected not to have submitted it elsewhere. It is the authors’ responsibility to inform the Editors about these matters. There is no submission charge. The Editors will also consider proposals consisting of a detailed table of contents and one or more sample chapters, and can offer a preliminary contingent decision subject to the receipt of a satisfactory complete manuscript. All submitted manuscripts should be double spaced on paper of standard size, 8.5 by 11 inches or European A, and should have margins of at least one inch on all sides. The figures should be publication quality. The manuscript should be prepared in the same manner as papers submitted to Econometrica. Manuscripts may be rejected, returned for specified revisions, or accepted. Once a monograph has been accepted, the author will sign a contract with the Econometric Society on the one hand, and with the Publisher, Cambridge University Press, on the other. Currently, monographs usually appear no more than twelve months from the date of final acceptance.
© 2011 The Econometric Society
DOI: 10.3982/ECTA794SUM
