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CONTENTS An Economic Model of Friendship: Homophily, Minorities, and Segregation . . . . . . . . . . . . . . . . . . . ...

Author: Stephen Morris

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CONTENTS An Economic Model of Friendship: Homophily, Minorities, and Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . ORIANA BANDIERA, IWAN BARANKAY, AND IMRAN RASUL: Social Connections and Incentives in the Workplace: Evidence From Personnel Data . . . . . . . . . . . . . . . . . . . . . . . . . . RODNEY J. GARRATT, THOMAS TRÖGER, AND CHARLES Z. ZHENG: Collusion via Resale . . . CHRISTIAN HELLWIG AND GUIDO LORENZONI: Bubbles and Self-Enforcing Debt . . . . . . . . . ARNAUD COSTINOT: An Elementary Theory of Comparative Advantage . . . . . . . . . . . . . . . . EMMANUEL GUERRE, ISABELLE PERRIGNE, AND QUANG VUONG: Nonparametric Identification of Risk Aversion in First-Price Auctions Under Exclusion Restrictions . . . . . . . JUSHAN BAI: Panel Data Models With Interactive Fixed Effects . . . . . . . . . . . . . . . . . . . . . . . SERGIO CURRARINI, MATTHEW O. JACKSON, AND PAOLO PIN:

1003 1047 1095 1137 1165 1193 1229

NOTES AND COMMENTS: FEDERICO ECHENIQUE AND IVANA KOMUNJER: Testing Models With Multiple Equi-

libria by Quantile Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 JÖRG STOYE: More on Confidence Intervals for Partially Identified Parameters . . . 1299 OLIVIER GOSSNER, EHUD KALAI, AND ROBERT WEBER: Information Independence and Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317 ANNOUNCEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 FORTHCOMING PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES . . . . . . . . . . 1337

VOL. 77, NO. 4 — July, 2009

An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics Founded December 29, 1930 Website: www.econometricsociety.org EDITOR STEPHEN MORRIS, Dept. of Economics, Princeton University, Fisher Hall, Prospect Avenue, Princeton, NJ 08544-1021, U.S.A.; [email protected] MANAGING EDITOR GERI MATTSON, 2002 Holly Neck Road, Baltimore, MD 21221, U.S.A.; mattsonpublishingservices@ comcast.net CO-EDITORS DARON ACEMOGLU, Dept. of Economics, MIT, E52-380B, 50 Memorial Drive, Cambridge, MA 021421347, U.S.A.; [email protected] WOLFGANG PESENDORFER, Dept. of Economics, Princeton University, Fisher Hall, Prospect Avenue, Princeton, NJ 08544-1021, U.S.A.; [email protected] JEAN-MARC ROBIN, Maison des Sciences Economiques, Université Paris 1 Panthéon–Sorbonne, 106/112 bd de l’Hôpital, 75647 Paris Cedex 13, France and University College London, U.K.; [email protected] LARRY SAMUELSON, Dept. of Economics, Yale University, 20 Hillhouse Avenue, New Haven, CT 065208281, U.S.A.; [email protected] JAMES H. STOCK, Dept. of Economics, Harvard University, Littauer M-24, 1830 Cambridge Street, Cambridge, MA 02138, U.S.A.; [email protected] HARALD UHLIG, Dept. of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, U.S.A.; [email protected] ASSOCIATE EDITORS YACINE AÏT-SAHALIA, Princeton University JOSEPH G. ALTONJI, Yale University JAMES ANDREONI, University of California, San Diego JUSHAN BAI, Columbia University MARCO BATTAGLINI, Princeton University PIERPAOLO BATTIGALLI, Università Bocconi DIRK BERGEMANN, Yale University XIAOHONG CHEN, Yale University VICTOR CHERNOZHUKOV, Massachusetts Institute of Technology J. DARRELL DUFFIE, Stanford University JEFFREY ELY, Northwestern University HALUK ERGIN, Washington University in St. Louis MIKHAIL GOLOSOV, Yale University FARUK GUL, Princeton University JINYONG HAHN, University of California, Los Angeles PHILIP A. HAILE, Yale University MICHAEL JANSSON, University of California, Berkeley PHILIPPE JEHIEL, Paris School of Economics and University College London PER KRUSELL, Princeton University and Stockholm University FELIX KUBLER, University of Zurich

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Econometrica, Vol. 77, No. 4 (July, 2009), 1003–1045

AN ECONOMIC MODEL OF FRIENDSHIP: HOMOPHILY, MINORITIES, AND SEGREGATION BY SERGIO CURRARINI, MATTHEW O. JACKSON, AND PAOLO PIN1 We develop a model of friendship formation that sheds light on segregation patterns observed in social and economic networks. Individuals have types and see typedependent benefits from friendships. We examine the properties of a steady-state equilibrium of a matching process of friendship formation. We use the model to understand three empirical patterns of friendship formation: (i) larger groups tend to form more same-type ties and fewer other-type ties than small groups, (ii) larger groups form more ties per capita, and (iii) all groups are biased towards same-type relative to demographics, with the most extreme bias coming from middle-sized groups. We show how these empirical observations can be generated by biases in preferences and biases in meetings. We also illustrate some welfare implications of the model. KEYWORDS: Networks, homophily, segregation, friendships, social networks, integration, diversity, minorities.

1. INTRODUCTION THE NETWORK STRUCTURE of social interactions influences a variety of behaviors and economic outcomes, including the formation of opinions, decisions on which products to buy, investment in education, access to jobs, and social mobility, just to name a few. The extent to which a society is segregated across different groups can be critical in determining things like how quickly information diffuses and the extent to which there is underinvestment in human capital, among other things. In this paper we examine a fundamental and pervasive phenomenon of social networks which is known as “homophily.”2 This term refers to a tendency of various types of individuals to associate with others who are similar to themselves. Homophily has been documented quite broadly, across characteristics such as age, race, gender, religion, and profession, and is generally a quite strong and robust observation (see McPherson, Smith-Lovin, and Cook (2001) for an overview of research on homophily). Given the importance of social networks, developing models that help us to understand homophily is essential. 1 This paper was presented as the Fisher–Schultz Lecture at the 2007 European Meetings of the Econometric Society in Budapest. We gratefully acknowledge financial support from the Center for Advanced Studies in the Behavioral Sciences, the Guggenheim Foundation, the Abdus Salam International Centre for Theoretical Physics, the NSF under Grant SES–0647867, and the EUSTREP project 516446. We thank Daron Acemoglu, Bernd Beber, Marika Cabral, Xiaochen Fan, Ben Golub, Steven Goodreau, and Carlos Lever, as well as a co-editor and three anonymous referees for helpful comments and suggestions. We thank James Moody for making available the network data from the Add Health data set. 2 The etymology of the term is simple: homo = self and philìa = love. Homophily is a term coined by Lazarsfeld and Merton (1954).

© 2009 The Econometric Society

DOI: 10.3982/ECTA7528

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We begin our analysis by providing some detailed observations about patterns of homophily. Specifically, we identify three patterns of homophily in the Adolescent Health data set (described in detail below), which examines friendship patterns in a representative sample of U.S. high schools.3 We then build a model of friendship formation and show that the model can generate the observed patterns of homophily, and we trace the different patterns of observed homophily to different aspects of the model. The three main observations that we point out from the data are summarized as follows. Consider a high school and the patterns of friendship within it. In particular, let the type of an individual be his/her race. The first empirical observation is a basic one, which we refer to as relative homophily: larger groups (measured as a fraction of the population of their respective schools) form a greater fraction of their friendships with people of their same type. This observation echoes previous studies. The second main empirical observation that we document is that larger groups form significantly more friendships per capita. More specifically, members of a group that comprise a small minority in a school form roughly six friendships per capita, while members of groups that comprise large majorities (close to 100 percent of a school) form on average more than eight friendships. The third main empirical observation that we document concerns a tendency for groups to form same-type friendships at rates that exceed the relative fractions in the population. That is, groups tend to “inbreed” (a term from the literature on homophily). More specifically, the new observation is that this inbreeding is essentially absent for groups that comprise very small or very large fractions of their school, while the inbreeding is very significant for groups that comprise a middle-ranged fraction of their school. With these three observations in mind, we develop a simple model of friendship formation. In the model, individuals come into a society and form friendships when they enter. Friends are met through a process of random matching, and each matching entails a fixed cost. There are diminishing returns to forming friendships, so eventually an individual exits the process. The critical determinants of an individual’s strategy are his or her preferences over the mix of types of friends that he or she forms as well as the mix that he or she faces in the matching process. The implications of the model are summarized as follows. We begin by noting that if agents’ preferences over friendships are insensitive to type, so that agents only care about total number of friends and not the composition of types, then all agents form the same number of friendships under any matching process such that all agents meet the same expected number of friends per 3 The National Longitudinal Survey of Adolescent Health (commonly referred to as Add Health) is a program project designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris, and funded by Grant P01-HD31921 from the National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Persons interested in obtaining data files from Add Health should contact Add Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC 27516-2524 ([email protected]).

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unit of search. Next, we note that any matching process such that types are matched in frequencies in proportion to their relative stocks in the matching process cannot generate the observed evidence of generalized inbreeding, regardless of type sensitivity of preferences. Given these two observations, we explore the implications of type sensitivity of preferences and bias in matching. We first examine type sensitivity of preferences and show that if agents see higher marginal returns when forming a mix of friendships that is biased toward same type, then larger groups form more friendships per capita in equilibrium. Second, we show that a bias in the matching process, so that all groups can simultaneously be meeting their own type at rates faster than their fraction in the matching pool, generates inbreeding homophily patterns that match the observed patterns (where middle-sized groups are the most inbred). Such a bias in matching could be attributed to a variety of factors such as tracking and membership in various clubs and activities, as well as meeting friends through friends.4 We then fit a parametric version of the model to the data, and show that a model with both type-sensitive preferences and a matching bias fits all of the observations, while just type-sensitive preferences generate the observation of larger groups forming more per capita friendships but having no inbreeding homophily and just a matching bias does the reverse. We conclude the analysis with a brief consideration of welfare issues. While the model is too stark to take seriously for policy implications, the welfare analysis points out that average welfare depends in sensitive ways on the structure of preferences and matching. This suggests that it is vital to build richer models of friendship formation so as to better understand homophily and self-segregation within schools (and other applications), and to develop well founded policies. Regarding contributions to the literature, homophily has been noted throughout history and is seen in such adages as “birds of a feather flock together,” which dates to at least the seventeenth century.5 In addition, as mentioned above, there is a large literature examining homophily along a wide variety of dimensions including ethnicity (e.g., see Fong and Isajiw (2000) and Baerveldt, Van Duijn, Vermeij, and Van Hemert (2004) for examples of studies focusing on ethnicity, and McPherson, Smith-Lovin, and Cook (2001) for a wider survey). Researchers are well aware that homophily patterns could be influenced by both the meeting process and preferences. For example, Moody (2001, p. 680) noted that “friendship segregation results from the multilevel influences of mixing opportunity and individual preference.” However, our model seems to be the first systematic investigation of the roles of preferences and matchings in determining the emerging patterns of homophily in social 4

See Feld (1981) for some discussion of such factors. Lazarsfeld and Merton (1954) attributed this quote to Burton (1621), but a version of it appears in Holland’s (1601) translation of Livy’s Roman History and seems to come from folklore before that. 5

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ties.6 In terms of the specific observations that we examine, the contributions are as follows. The observation regarding relative homophily, and pointing out that this essentially follows from having friendships balance across types, is something that has roots in what is known as contact theory. For example, Blau (1977) (see also Allport (1954)) pointed out that since each cross-group friendship must involve a member from each group, then smaller groups must have more cross-group friendships on a per capita basis. In our model, however, overall relative homophily is not implied by this alone; it also requires that preferences have diminishing returns. The second observation that larger groups form more relationships is an empirical finding that has been noted before, for instance, by Marsden (1987) in a study of advice networks. Showing how this, at least in the context of our model, is generated by type-sensitive preferences is new. As we show, absent sensitivity to types in preferences, to explain the larger number of friendships formed by larger groups, the matching process would need to be one of increasing returns so that types that make up a larger fraction of a school or society systematically meet more people per unit of time. We discuss such an alternative mechanism in Section 7.3. The third observation that inbreeding has a specific pattern, and is maximal for middle-sized groups also appears to be new, as does the explanation that this is tied to biases in matching processes.7 The welfare analysis that accompanies the model is new as well. Finally, this paper is related to the search literature (e.g., see Diamond (1982), Mortensen and Pissarides (1994), and Acemoglu and Shimer (1999), among others). Our analysis differs from most of that literature in allowing for each agent to form many relationships, and so results in a many-to-many matching. We also examine biased meeting processes, so that different types meet at different rates. Although this leads to some increased generality, it also pushes us to simplify on other fronts. In particular, we focus on settings where agents accept all matches so that their decision variable are stopping times.8 We also provide (in Appendix A) some foundations for our continuum model in terms of viewing it as the limit of a large finite matching model. 6

In fact, even more generally, most models of network formation fall into either a class of random graph and/or statistical models governed by random attachment or game theoretic models governed by preferences. See Jackson (2006) for some discussion and background. 7 There is some prior evidence that the percentage of cross-group links can vary nonlinearly and nonmonotonically with measures of population heterogeneity (e.g., see Blalock (1982) and Moody (2001)), although not in the explicit form found here. There have been explanations proposed for why inbreeding might vary with population mixes, for instance, based on power differences and competition (e.g., Giles and Evans (1986)). Moody (2001) suggested that when two groups are close to a majority, their tendency to inbreed increases because of competing groups concerned about sociological factors, which are absent from our preference-based model. 8 In the Supplemental Material (Currarini, Jackson, and Pin (2009)), we present some results on an extension of the model where agents can experience satiation with respect to some types and not others, which leads to substantial complications in the model.


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We begin the paper by defining homophily indices; then we present observations on the Add Health data, the model, and the analysis. 2. MEASURING HOMOPHILY Before taking a closer look at homophily patterns, we define some indexes of homophily. There are K different types of agents. Let Ni denote the number of type i Ni individuals in the population and let wi = N be the relative fraction of type i in the population, where N = k Nk . Let si denote the average number of friendships that agents of type i have with agents who are of the same type and let di be the average number of friendships that type i agents form with agents of types different than i. The obvious basic homophily index, Hi , measures the fraction of the ties of individuals of type i that are with that same type. DEFINITION 1: The homophily index is defined by Hi =

si si + di

When the index Hi is increasing with relative group size, then we say that the pattern of friendships satisfies relative homophily. DEFINITION 2: A profile (s d) = (s1 d1 s2 d2 sK dK ) satisfies relative homophily if wi > wj implies Hi > Hj . As mentioned above, the index Hi is partly determined by relative populations, and so a benchmark is the relative size of a group, which corresponds to the expectation of the fraction of same-type friendships in a world where friendships are distributed uniformly at random among pairs of agents in the population. DEFINITION 3: The profile (s d) = (s1 d1 s2 d2 sK dK ) satisfies baseline homophily if for all i, Hi = wi Note that baseline homophily immediately implies relative homophily, while the reverse is not true. The observed tendency of friendships to be biased towards own types beyond the effect of relative population sizes has been referred to in the sociological literature as “inbreeding homophily” (see, e.g., Coleman (1958), Marsden (1987), and McPherson, Smith-Lovin, and Cook (2001)).

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DEFINITION 4: The profile (s d) satisfies inbreeding homophily for type i if Hi > w i One can also have the reverse condition where friendships are biased toward having different-type friendships. DEFINITION 5: The profile (s d) satisfies heterophily for type i if Hi < w i Generally, there is a difficulty in simply measuring homophily according to Hi . For example, consider a group that comprises 95 percent of a population. Suppose that its same-type friendships are 96 percent of its friendships. Compare this to a group that comprises 5 percent of a population and has 96 percent of its friendships being same type. Although both have the same homophily index, they are very different in terms of how homophilous they are relative to how homophilous they could be. Comparing the homophily index, Hi , to the baseline, wi , provides some information, but even that does not fully capture the idea of how biased a group is compared to how biased it could potentially be. To take care of this we use the measure developed by Coleman (1958) that normalizes the homophily index by the potential extent to which a group could be biased.9 DEFINITION 6: The inbreeding homophily of type i is IH i =

Hi − w i 1 − wi

This index measure the amount of bias with respect to baseline homophily as it relates to the maximum possible bias (the term 1 − wi ). It can be easily checked that we have inbreeding homophily for type i if and only if IH i > 0, and have inbreeding heterophily for type i if and only if IH i < 0. The index of inbreeding homophily is 0 if there is pure baseline homophily and is 1 if a group completely inbreeds.1011 9 As will be clear from the data, this also corrects an issue that the raw homophily index is heteroskedastic when viewed relative to group fraction wi , while this normalized index has a more constant variance as a function of wi . 10 One could also define a heterophily index, which would be (si /(si + di ) − wi )/(−wi ), reflecting the extent to which a group is outgoing. It would be 0 at baseline homophily and 1 if a group only formed different-type friendships. 11 The measures Hi and IH i have slight biases in small samples. For example, suppose that there was no bias in the friendship formation process so that we are in a “baseline” society. Then the expected fraction of same type meetings of type i is (Ni − 1)/(N − 1). Thus, the expected

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3. PATTERNS OF FRIENDSHIPS AND HOMOPHILY Table I presents friendships in an American high school taken from the Add Health data set.12 Table I records the percentage of friendships by race. Cell ij indicates the percentage of all agents of type j’s friendships that are with members of race i. Columns sum to 100 percent. If friendships were formed purely at random, we would observe a first row centered around 51 percent, a second centered around 38 percent, and so forth. The numbers on the main diagonal record the percentage of total friendships that are of same type.13 These values are substantially higher than the relative group’s size for the two larger groups (whites and blacks), and lower for the smallest group (Hispanics), reflecting inbreedTABLE I PERCENTAGE OF LINKS ACROSS ETHNICITIES IN AN AMERICAN SCHOOLa Ethnicity of Students Percent of Friends by Ethnicity

White Black Hispanic Others

White n = 131 % = 51

Black n = 96 % = 38

Hispanic n = 13 %=5

Others n = 15 %=6

85 4 4 7

7 85 6 2

47 46 2 5

74 11 4 11

a From Add Health 1994 Data.

value of Hi − wi in a baseline society is −(N − Ni )/(N(N − 1)), which vanishes as N becomes large. The expected value of IH i is then −1/(N − 1), which is independent of i, vanishing in N, and slightly negative. Given that we are finding significantly positive values of IH i and that the N’s vary from 30 to 5000, this is not a problem in our analysis. 12 The data from Add Health were collected over several years starting in 1994 from a carefully stratified sample of high schools and middle schools (to vary by size, location, include public and private, vary racial composition, and socio-economic backgrounds). There are behavioral and demographic data in the data set from 112 schools; here we use the data from 84 schools for which extensive network information was obtained. The data are based on student interviews. The friendship data were based on reports of friendships by each student. Students were shown a list of all the other students in the school and permitted to name up to five friends of each sex. Only 3 percent nominated 10 friends, and only 24 percent hit the constraint on one of the sexes, and so the constraints do not seem to be a substantial measurement issue (see Moody (2001) for more discussion). The data include information about how much interaction there is between individuals, which we do not make use of as it does not add much to our analysis. Here a tie is present if either student mentioned the other as a friend. Students could also identify other students with whom they had sentimental relations, which are not reported among friendships. The attribution of race is based on the self-reported classification. 13 Table I provides a view similar to Table 3 in Baerveldt, Van Duijn, Vermeij, and Van Hemert (2004), who analyzed interracial friendships in a Dutch high school, obtaining a similar structure.

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FIGURE 1.—Friendship network in a U.S. school. Colors identify races: white = whites; grey = blacks; light grey = Hispanics; black = others.

ing homophily for both whites and blacks, and some heterophily for Hispanics. The IH index of inbreeding homophily is 069 for whites (whose relative population is 51 percent), 076 for blacks (relative population 38 percent), and −011 for Hispanics (2 percent of population). This nonmonotonic trend of the IH index is discussed in Sections 7 and 8. The network of friendships by race in this high school is pictured in Figure 1. Figure 1 illustrates the tendency of white and black students to form distinct communities (high levels of inbreeding homophily), and the different behavior of Hispanic students (less numerous), who integrate more with other races and fail to form an independent community.14 We now turn to examining the three observations mentioned in the Introduction.15 3.1. Relative Homophily In each of the 84 high schools there are four different racial categories, leading to 336 potential observations. There are 31 cases where there are no students of a given racial category, and so we have 305 total observations. Each observation is then a race within a particular school. 14

The figure was drawn with the Pajek program, which uses an algorithm to cluster nodes together that are more densely connected. 15 See Currarini, Jackson, and Pin (2008) for more detailed analysis of these data.


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FIGURE 2.—The homophily index Hi plotted as a function of the fraction of the population wi .

Figure 2 shows that relative homophily is pervasive in the Add Health schools. The horizontal axis lists the fraction wi of a group i in its school and the vertical axis lists the (nonnormalized) homophily index Hi . The 45 degree line provides the baseline homophily benchmark. Beyond the clear positive tendency of relative homophily,16 we also note that the baseline homophily line acts as a lower envelope of the data, which is evidence of pervasive inbreeding homophily, as analyzed below.17 3.2. Group Size and the Total Number of Friends Next, we examine how relative group size relates to the average number of friendships formed by group members. In Figure 3, there is a clear positive relationship between the size of racial groups and the total number of per capita friends that they form.18 This aspect of friendship formation does not appear in Blau’s (1977) analysis of opportunity driven homophily or in much of the literature that follows. Marsden (1987) does remark on such a pattern in his empirical study of the “discussion 16 The slope of a regression line through the data is 0.99 with a t-statistic of 31. We fit a nonlinear curve below. 17 Figure 2 is very similar to Figure 6 in Echenique and Fryer (2007), who focused on defining segregation measures, and who developed a measure based on the spectral decomposition of the friendship matrix and tested it on the same data. Thus, the same pattern exists for other measures as well. 18 The coefficient of a regressed line is 1.9 with a t-statistic of 5.0 and the intercept is 5.9 with a t-statistic of 40.

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FIGURE 3.—Larger groups make more friends: per capita friendships si + di plotted as a function of the fraction of the population wi .

network” based on a national U.S. survey, but did not discuss a connection between such a pattern and other aspects of friendship formation. 3.3. Inbreeding Homophily As mentioned above, another feature of the data is that there is inbreeding homophily for most groups. Moreover, there is a significant and distinctive pat-

FIGURE 4.—Patterns of inbreeding homophily IH i by relative group size and race.


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tern to IH i as a function of relative group size. Figure 4 illustrates inbreeding homophily as a function of group size (and also as a function of race).19 Figure 4 indicates two clear patterns. First, there is inbreeding homophily for most groups, with some exceptions for the smallest groups. Second, there is a nonlinear and nonmonotone relationship, as inbreeding homophily is near 0 at the extremes and significantly positive (reaching a maximum of almost 0.8) near the middle. 4. A NESTED SET OF SEARCH-BASED MODELS OF FRIENDSHIP FORMATION We now examine a search-/matching-based model of friendship formation.20 4.1. Agents, Types, and Payoffs Agents come in a finite variety of types T = {1 K}. These might correspond, for example, to ethnicities, religious affiliations, professions, age, or some combinations of traits. The utility of an agent depends on the numbers of his or her friends21 who are of his or her same type and who are of different types. For the purposes of this model, the agent only distinguishes between same and different, and does not distinguish in any finer way among differences.22 The total utility to an agent of type i who has si same-type friends and di different-type friends is U(si di ). The function U is not indexed by the type of the agent. Thus agents’ base preferences are the same across types, but their resulting outcomes may differ depending on the composition of the society that they face. For convenience, we allow agents to form fractional friendships. This allows us to treat U as strictly increasing and continuous, with continuous first- and second-order partial derivatives.23 Let Us (si di ) denote the partial derivative 19 In a regression of IH i versus wi and wi2 , we find a coefficient of 2.1 on wi and a coefficient of −22 on wi2 , with t-statistics of 16 and −15, respectively. The intercept term is 0.04 and is insignificant. Running a regression of the inbreeding homophily index on the variable wi (1 − wi ) and forcing a zero intercept, we obtain a coefficient of 2.1 (standard error 0.07), with a t-value of 29. 20 Such search/matching models have been used in a variety of contexts (e.g., labor markets as in Mortensen and Pissarides (1994)). The model introduced here is distinct in that agents form many friendships. 21 The word “friend” is used throughout, but this might also correspond to some other sort of relationship, depending on the application. 22 This is consistent with empirical evidence. For example, Marsden (1988) did not find any significant distinction between friendships and races after accounting for homophily. As McPherson, Smith-Lovin, and Cook (2001, footnote 7) pointed out, “The key distinction appears to be same–different, not any more elaborated form of stratification.” 23 The case in which agents have satiated preferences is also of interest in various applications and is treated in the Supplemental Material (Currarini, Jackson, and Pin (2009)).

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of U with respect to s evaluated at (si di ) and use similar standard notation for other partial derivatives. U is strictly concave in si and is strictly concave in di . We maintain an assumption that the utility function U exhibits overall diminishing returns to friendships; that is, U(as ad) < aU(s d) for all (s d) and a > 1. Diminishing returns is a standard decreasing returns to scale condition, here applied to preferences for friendship. Geometrically, it requires that the marginal utility of friendships decreases along any ray going out of the origin in the space (s d).24 This condition is equivalent to the condition on the total derivative, (1)

sUs (as ad) + dUd (as ad) < sUs (s d) + dUd (s d)

for all (s d) and a > 1. Diminishing returns to friendships is satisfied if s and d are substitutes in agents’ preferences, given the strict concavity of U in each of s and d. It is also satisfied when s and d are complements, as, for instance, in Cobb–Douglas utility functions in which the sum of powers is less than 1. Finally, to ensure that agents will not form an infinite number of friendships, we assume that marginal utilities vanish as s and d go to infinity. More precisely, for all ε > 0 and s , there exists d such that Us (s d) < ε and Ud (s d) < ε for all s > s and d > d (and similarly for all ε > 0 and d ). Before moving on, let us mention a few illustrative examples that could provide a basis for various formulations of the utility function U(· ·). • Information Networks: Individuals receive information from their friends. For instance, the information they learn from a same-type individual could be more correlated with their own information than that of different types. However, a same-type individual could be easier to communicate with.25 • Professional Networks: Here, same-type individuals are easier to communicate with, but offer less creative synergy than different-type individuals.26 • Purely Social Networks: One shares more interests with same-type individuals than with different-type individuals.27 • Risk Sharing Networks: A same-type individual’s income is more highly correlated with own income, which offers less potential for risk-sharing than different-type individuals. However, same types are closer geographically or 24 This property is weaker than overall concavity of U. It implies concavity of U if U is quasiconcave and homothetic (see Friedman (1973)). 25 For instance, this could concern information such as job market information, as in the literature surveyed by Ioannides and Datcher Loury (2004). 26 For a discussion, see Page (2007). 27 For an example of social- or status-based preferences, see Watts (2007).


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socially than different types, and thus are easier to arrange a viable (and selfenforcing) risk-sharing agreement with.28 4.2. A Matching Process We examine a continuous-time matching process with a new inflow of agents of mass Ni of type i per unit of time. For each unit of time in the matching process, regardless of the outcome, an agent incurs an opportunity cost c > 0.29 Thus the cost of an agent of type i who stays in the matching process for ti units of time is ti c. If agents of type i stay for ti units of time in the matching process, then the stock of type i’s involved in the matching process in a steady state will be Mi = ti Ni . Thus the relative stock of a given type of agent increases by either increasing their inflow or by increasing how long they stay in the matching process. Agents accumulate friendships over time in the matching process. Given that their utility is increasing in friends of each type, it is a dominant strategy for them to accept friendships with any agents whom they meet. Thus, their main decision is simply when to exit the matching process. The mix of friendships that agents form is determined by the stocks of various types in the matching process. For any given agent of type i who searches for a new friend, the agent meets another agent of type i with probability qi and an agent of different type with a probability 1 − qi . These probabilities are endogenous to equilibrium, determined by the steady-state stocks of agents in the matching process. Generally, the matching process can be described by an n × n matrix q ∈ [0 1]n×n , where qij is the fraction of i’s meetings per unit of time that are with type j and the matrix is row stochastic so that j qij = 1. These relative meeting probabilities depend on the stocks of agents in the society, so a matching process is described by a function F : Rn+ → [0 1]n×n , where q = F(M1 Mn ) is the matching that occurs as a function of the relative stocks of agents in the society. To be well defined, the matching process needs to balance, so that the number of meetings where an i meets a j is the same as those where a j meets an i. A matching process F is balanced at a given M if q = F(M1 Mn ) is such that Mi qij = qji Mj for all i and j. The canonical matching process is one where agents meet each other in proportion to their relative stocks. We call that the unbiased matching process, and it is such that qij = Mj /M, where M = k Mk . 28

See Bramoullé and Kranton (2007) for more discussion of such trade-offs in a risk-sharing environment. 29 Adding discounting complicates the expressions without adding any insight.

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Given that agent’s preferences only depend on own and other types, we let qi = qii and then 1 − qi = j=i qij . Thus, if an agent of type i stays in the matching for a time ti , then he or she forms a total of ti friendships, with proportions qi ti of them being of same type and (1 − qi )ti being of different type. We remark that in this model the homophily and segregation of types that results never involve an individual being rejected as a friend by another individual.30 Instead, it is governed by the relative sizes of populations, agents’ preferences and the implications for their matching decisions, and feedback in the matching process which governs the mix of friendships that individuals are faced with forming. Two simplifications that we have made in the model are the continuous time matching and the fact that the mix of types that agents meet is deterministic. These are both made for convenience. In Appendix A we provide a derivation of a random-matching process with a continuum of agents and show how taking limits as time between matchings goes to 0 can justify our formulation, at least for the case of an unbiased matching process. 4.3. An Agent’s Decision Problem Given the matching probabilities qi and 1 − qi , an agent of type i solves the problem (2)

max U(qi ti (1 − qi )ti ) − cti ti

The following lemma states a straightforward but useful necessary condition for an optimum. LEMMA 1: An optimal choice of the time spent in the matching ti for an agent of type i given matching probabilities qi and 1 − qi satisfies the condition (3)

qi Us (qi ti (1 − qi )ti ) + (1 − qi )Ud (qi ti (1 − qi )ti ) = c

In what follows we maintain the assumption that Us (0 0) > c and Ud (0 0) > c, so that agents will at least form some minimal number of friendships regardless of the matching process that they face. Note that even though agents’ strategies boil down to choosing a length of time to spend in the matching process, their choices will depend both on their preferences and on the mix of stocks of agents in the matching process. Most importantly, the stocks of agents in the matching process will depend on the preferences, and so there can be a feedback effect that determines the equilibrium. This is critical to understanding the model, since, for example, if there is a bias 30 Such rejection can result with satiated preferences, as detailed in the Supplemental Material (Currarini, Jackson, and Pin (2009)).


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in agents’ preferences toward same-type friendships, then a larger group will face a more attractive matching, leading them to stay in the matching process longer, leading to an even more biased matching, and so forth. This endogeneity is captured in the definition of steady-state equilibrium.

4.4. Steady-State Equilibrium A steady-state equilibrium accounts for the relation between players’ strategies and the resulting matching process. A steady-state equilibrium of the system for a given set of inflows Ni and a utility function U is a specification of strategies for each type such that the resulting stocks of agents at each point in their search process lead to the relative meeting probabilities that justify the strategies and such that the outflow of agents is the same as the inflow. More specifically, a steady-state equilibrium is a specification of a triple (ti Mi qi ) for each i, representing, respectively, the time spent matching for the type, the stock of the type in the matching, and the relative probability of meeting the type in the matching that satisfy the following conditions, where there is at least one i such that ti > 0: (i) Agents optimize given meeting probabilities: ti solves (2) given qi for each i. (ii) Strategies determine stocks: Mi = Ni ti for each i. (iii) Stocks determine meeting probabilities: q = F(M1 Mn ). (iv) The matching process is balanced so that qij Mi = qji Mj for each i and j. The restriction that ti > 0 for at least one i means that a steady-state equilibrium is a nontrivial equilibrium. The trivial situation where no agents enter the matching process expecting that no other agents will enter could potentially be thought of as an equilibrium if one allows for an empty matching process, but it would be unstable since all agents strictly prefer to enter the matching if there are any other agents entering the matching. Steady-state equilibrium under the above definition precludes such degenerate situations directly. In the case of an unbiased matching process, where qij = Mj /M, it is clear that (iv) is satisfied, and so for such processes, one then needs only to verify that (i) and (ii) are satisfied. We do not include an explicit requirement that the inflows match the outflows, as this is implied by (ii). Since the flow of agents of type i who are exiting at any given time is Mi /ti , the exiting amount will be Ni under (ii). Note also that in equilibrium qi coincides with the homophily index Hi defined in Section 2, since si = ti qi and si + di = ti qi + ti (1 − qi ) = ti , and so Hi = si /(si + di ) = ti qi /ti = qi . The following example illustrates the definition and shows the feedback effects of strategies affecting stocks and stocks affecting strategies.

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4.5. An Example of a Steady-State Equilibrium Consider preferences represented by (4)

U(s d) = (s + γd)α

where 0 < α < 1 captures diminishing returns, and the parameter γ ≥ 0 determines the relative bias in preferences between same and different types. If γ < 1, then an agent sees higher marginal benefits from same-type friendships than different-type friendships, and if γ > 1, then the reverse is true. It is then easy to verify that an agent’s optimal time in the matching is (5)

1/(1−α) α (γ + (1 − γ)qi )α/(1−α) ti = c

For the case of α = 12 , two types, and an unbiased matching process, we compute the steady-state equilibrium. Condition (iii) of steady-state equilibrium implies that w i ti qi = qj w j tj Substituting yields wi qi ((1 − γ)(1 − qi ) + γ) = (1 − qi )((1 − γ)qi + γ) 1 − wi Given that wi ∈ [0 1], there is a unique solution for qi ∈ [0 1], which is31

(6)

2 wi wi 2 −1+ − (1 − 2γ ) + 4γ 2 (1 − γ 2 ) (1 − 2γ) 1 − wi 1 − wi qi = wi 2 − 1 (1 − γ) 1 − wi

Figure 5 shows how qi depends on wi , when γ < 1 and γ > 1. To see a full solution, we can specialize to let γ = 1/2, w1 = 2/3 and w2 = 1/3. Then ti =

1 (1 + qi ) 8c 2

31 Equation (6) is defined, at the limit, for wi = 1/2 and wi = 1, and the result are, respectively, qi = 1/2 and qi = 1, as expected.


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FIGURE 5.—The behavior of qi as a function of wi when U(si di ) = si + γdi . On the left-hand side, γ = 15 ; on the right-hand side, γ = 2. The straight line is the identity.

It follows that q1 = 2t1 /(2t1 + t2 ) and q2 = t2 /(2t1 + t2 ). Solving simultaneously for t1 and t2 leads to the unique solution √ √ 3− 3 3 t1 = 2 and t2 = (7) 8c 8c 2 and (8)

q1 =

√

3 − 1 and q2 = 2 −

√ 3

Here we see that q1 > w1 and q2 < w2 , and so the relative stocks are biased toward the larger group and the stocks differ from the flows. This reflects the feedback effect—the larger group finds the matching more attractive and spends more time matching—and illustrates the feedback between strategies and the determination of the stocks in the matching. 5. SOME BENCHMARK RESULTS We begin with some benchmark results. First, we show that if preferences are type-neutral, then all equilibria involve all types forming the same total number of friendships. That is, if preferences are not sensitive to the ratio of same-type to different-types friendships, then all types form the same number of friendships. We say that U is type-neutral if U(s d) = U(s d ) whenever s + d = s + d . When preferences are not type-neutral, we say that they are type-sensitive. PROPOSITION 1: Let U be type-neutral. Then for any matching process and any steady-state equilibrium, all types of agents form the same total number of friendships.

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Proofs of this and other propositions are given in the Appendix. Note that an implication of Proposition 1 is that if the matching process is unbiased, then there will be baseline homophily, since all types spend the same time in the matching process and then form friendships in proportion to their relative inflows. Next we show that without some bias in the matching process, it is impossible to have all groups be homophilous at once. Simply by balance, if one group is homophilous, then another group must be heterophilous. PROPOSITION 2: If the matching process is unbiased, then for any preferences and any steady-state equilibrium, if some type is homophilous so that IH i > 0 for some i, then there exists at least one other type j that is heterophilous so that IH j < 0. These two results show that in the context of our model, it is impossible to generate differences in the number of friendships formed without some type sensitivity to preferences, and it is impossible to generate homophily for all groups without some bias in the matching. Thus, matching the observed patterns in the data requires type sensitivity in preferences and a bias in the matching process. In Section 7.3 we discuss how these results depend on the constraints imposed by our model and how alternative models are also consistent with the same data. 6. UNBIASED MATCHING WITH TWO TYPES Given Propositions 1 and 2, it follows that type-sensitive preferences and biased matching are needed in our model to generate equilibria consistent with the data. It is informative to introduce these one at a time, since each one can explain specific patterns of the data independently of the other. We start by examining type sensitivity of preferences in the context of an unbiased matching. In this section we focus on the case in which there are only two types. The case of two types provides much of the basic intuitions and insights, making the analysis more transparent. There are some complications when we move to more types that we explore in Section 8. 6.1. Relative Homophily and Equilibrium Conditions We begin by showing that a steady-state equilibrium exists and that equilibrium conditions alone imply relative homophily. PROPOSITION 3—Relative Homophily: There exists a steady-state equilibrium and in all steady-state equilibria, both types are active. Moreover, if Ni > Nj , then there exists an equilibrium where qi > 1/2 and relative homophily is satisfied (regardless of the type sensitivity of preferences).


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Thus, relative homophily is implied directly by equilibrium conditions, independently of how agents’ preferences depend on the mix of friends, at least in equilibria where the relative stocks are ordered as the relative inflows (which always exist and in a general class of settings are unique, as discussed below). Simple accounting for friendships in equilibrium implies that cross-group friendships must add up, so Ni di = Nj dj and if Ni > Nj , then di < dj . This does not directly imply that relative homophily is satisfied, as it is conceivable that si is proportionally even smaller than sj , but this is ruled out by diminishing returns to scale, as then the group j would be facing lower overall returns to forming friendships (as they would be forming more of each type), but are facing the same costs. This idea that relative group sizes are really at the heart of relative homophily, without any biases in meetings or preferences (beyond relative population proportions), is relatively straightforward and has been noted before. For example, Blau (1977) pointed out how relative group sizes affect the opportunities of members of various groups to form ties. In a case of two groups, Blau observed that, due to the reciprocal nature of ties, there will be more cross-type friendships per capita for the smaller group than for the larger group. One does need some added structure (here diminishing returns to scale) to tie down same-type friendships and show that this implies relative homophily, but the basic accounting of cross-type friendships is a main driver. While Proposition 3 states the existence of equilibrium and that relative homophily is exhibited by all equilibria, it leaves open the possibility that there can be multiple equilibria.32 While there are examples of multiple equilibria for some special preference specifications, for many reasonable classes of preferences, equilibrium is unique. In particular, if preferences are such that t(q) t (q) < 1−q when q < 1, then there is a unique steady-state equilibrium.33 CLAIM 1: If preferences are such that t(q)(1 − q) is decreasing in q, then there is a unique steady-state equilibrium. Claim 1 is shown in the Appendix. Note that t(q)(1 − q) is decreasing in q for all preferences in the class in Section 4.5, provided that α < γ. The existence of multiple equilibria can occur when preferences are biased enough toward same type so that one can start at one equilibrium, increase the stock of a given type, and have that type respond by increasing its friendship formation enough (and the other type decreasing 32 Multiple equilibria occur in many search and matching models, given the externalities present. For example, see Diamond (1982) and the literature that followed for more discussion of such issues. 33 This presumes that t has a derivative, which is true for many preference specifications. More generally, all that is needed is that t(q)(1 − q) is a decreasing function of q, which is equivalent to the stated condition when t is differentiable.

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enough) so that we generate the higher stock of that type in the matching. The above condition rules out this possibility. 6.2. Same-Type Bias, Numbers of Friends, and Homophilous Behavior Propositions 1 and 2 imply that although equilibrium conditions imply relative homophily, the empirical observations of inbreeding homophily and increasing numbers of friends with group size require additional structure. Moreover, the relative homophily observed in Section 3.1 was such that larger groups formed fewer different-type and more same-type friendships than smaller groups. To derive the full empirical evidence of Sections 3.1–3.3, the model must include some bias in preferences, as we see from Proposition 1. We now turn to the implications of preferences that are biased to have higher returns from mixes that are heavier in same type. DEFINITION 7: Preferences exhibit a same-type bias if (9)

∂ ∂ U(qt (1 − q)t) > U((1 − q)t qt) ∂t ∂t

for q > 1/2. This states that the return to additional friendships (in proportion to the current mix) is larger when the mixture is biased toward same type, which is reminiscent of conditions on technological change in production functions. Note that this condition does not require that agents always prefer to substitute away from different types and toward same types. It simply requires that faced with a given mix or its reciprocal, agents get a higher marginal return from the one with more same types. With such a same-type bias in preferences, we can then derive a result that links larger groups (as a percentage of population) to greater numbers of friendships. PROPOSITION 4—Increasing Numbers of Friends and Friendship Mixes With Biased Preferences: Let preferences exhibit same-type bias, and let there be two groups i and j such that Ni > Nj . Then there exists an equilibrium such that the larger group forms more total friendships per capita than the smaller group.34 Moreover, in equilibrium the larger group forms more same-type friendships and fewer different-type friendships per capita than the smaller group, and the larger group exhibits inbreeding homophily while the smaller group exhibits inbreeding heterophily. 34

Again, when preferences are such that t(q)(1 − q) is decreasing, then this is unique.

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The proof is based on the fact that under same-type bias, facing more same types in the matching (a higher qi ) implies higher marginal returns from friendships. This implies that the larger group will spend more time in the matching process and form more total friendships per capita. This biases the stock of the larger group in the matching process to be larger than their relative flow, so that qi > wi for the larger group. This leads to inbreeding homophily for the larger group and heterophily for the smaller group. Note that while Proposition 4 shows that the larger group in an equilibrium will form a larger number of friendships, it does not state that if we look across equilibria, then the friendships will be ordered by group size. For example, the proposition implies that in a population that is 70 percent type 1’s and 30 percent type 2’s, the type 1’s will form more friendships in equilibrium than the type 2’s. It does not state how those numbers of friendships will compare with another society that is 80 percent types 1’s and 20 percent type 2’s. The data have those extra comparisons, where we see not only that a 70 percent group forms more friendships on average than a 30 percent group, but also that this trend is generally increasing by group size, even when comparing across different populations (that is, different schools). To draw this extra conclusion, we need additional requirements, which amount to the condition that ensured uniqueness of equilibrium plus a strengthening of the same-type bias condition. These conditions ensure a monotone relationship between the equilibrium qi and wi , and a monotone relationship between ti and qi , which coupled together ensure comparability of the total number of friendships across equilibria. DEFINITION 8: Preferences exhibit a strong same-type bias if q)t) is increasing in q.

∂ ∂t

U(qt (1 −

This strong same-type bias condition states that the marginal returns from scaling up the number of total friends that an individual has is increasing as the mix becomes more biased toward own type. This condition ensures that t(q) is an increasing function of q. PROPOSITION 5: If preferences exhibit a strong same-type bias and are such that t(q)(1 − q) is decreasing in q, then the unique steady-state equilibrium number of friendships formed per capita by a group i, ti , is increasing in the group’s relative size, wi . Both of these conditions are satisfied by the preferences that appeared in Section 4.5, provided that γ < 1 to ensure strong same-type bias and α < γ to ensure that t(q)(1 − q) is decreasing in q. 7. INBREEDING HOMOPHILY AND THE MATCHING PROCESS Proposition 4 shows that the model with preference bias leads to the observed relation between group size and total number of friends, and exhibits

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inbreeding homophily for the larger group. However, it also predicts that the smaller group necessarily should be heterophilous, which is not consistent with the empirical evidence discussed in Section 3, where we saw inbreeding homophily for most groups and, moreover, the largest inbreeding homophily for middle-sized groups.35 This was shown more generally in Proposition 2, but to get a fuller understanding of this, consider the inbreeding homophily index si − wi qi − wi t = i IH i = 1 − wi 1 − wi Since the relative stocks must add up in equilibrium with an unbiased matching process (from (iii) and (iv) in the definition of equilibrium), so that q1 = 1 − q2 (when there are two groups), it follows that (10)

IH 2 =

1 − q1 − (1 − w1 ) w1 − q1 = 1 − (1 − w1 ) w1

Thus, IH 2 < 0 if and only if IH 1 > 0. For one group to have inbreeding homophily, the other group must be heterophilous. Proposition 2 and the above discussion clarify that some form of bias in the matching process is needed to generate the full spectrum of observables. In the next section, we consider a specific form of such a matching bias (allowing all types to meet their own types at a frequency above their proportion in the matching process) that can generate the observed inbreeding homophily patterns. 7.1. A Biased Matching Process We examine a bias in the matching process which works in favor of sametype matches. Individuals who have preferences with a same-type bias would tend to gain by biasing their search so that it yields a higher ratio of same type compared to their stock in the matching process.36 This could be done in various ways, including meeting friends through friends, as well as joining clubs or taking part in activities that are biased toward own type. Let us emphasize that the unbiased matching process considered in the previous sections could still exhibit a bias in steady-state equilibrium away from relative population frequencies in that the qi ’s could differ significantly from 35 When there are only two types and they are such that Ni = Nj , then our model exhibits, by symmetry, qi = qj = 1/2 = wi = wj and so baseline homophily for both groups. 36 Such biases in matching processes have appeared in the search literature under the title “directed search,” as in Acemoglu and Shimer (1999).


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the wi ’s. What is different about a bias in the matching process is that the interplay between the matchings of different types is no longer a simple linear relationship. Now one can have situations where all types are meeting same types at a rate faster than their relative stocks in the matching process. So, there can be an inherent same-type bias in the matching process beyond the feedback and equilibrium effects. To get an impression of how this works, consider a tractable but very flexible example of a matching process F which is specified for the case of two types and implicitly defined so that37 (11)

q1β + q2β = 1

When β = 1, this is the unbiased (uniformly random) meeting technology that we examined above.38 However, when β > 1, this leads to higher qi than under uniform meeting, and actually leads to the highest impact for the smallest groups. Note that these groups have the most incentive to meet their own types, as they are not meeting them naturally. Note that even though Figure 4 shows the greatest homophily for middlesized groups, that does not mean that the bias in meeting has to be greatest for them. In fact, what we saw from the model without any meeting bias was that small groups would end up with negative homophily measures. Thus, to match the observed data, small groups need to have the largest bias in meetings, as the equilibrium balance conditions without any bias would force them to have quite negative homophily measures. This is in line with the β exponent in (11). With such a matching process, existence can be established via an argument similar to that of the proof of Proposition 3, as can a version of Proposition 4, as we show in the Appendix (see Proposition 9). In terms of Proposition 2, if we examine preferences that are type independent, then it is true that each type will form the same number of friendships. However, baseline homophily is no longer implied. For example, even if w1 = w2 = 1/2, it is possible that qi > wi for each i, as the meeting process can now be biased to same type and inbreeding homophily can be satisfied for all types. To get a feel for how the extended model works, we fit a special case of it to the Add Health data. 37 Note that in the case of two types, qij = 1 − qii and so (iv) becomes Mi (1 − qi ) = Mj (1 − qj ). F is then defined so q1 and q2 simultaneously satisfy the two equations (11) and Mi (1 − qi ) = Mj (1 − qj ). 38 The formal derivation appearing in Appendix A, which justifies our model as a limit of well defined meeting processes, is based on a completely uniform random-mixing process as in AlósFerrer (1999). Extending that justification for the processes here requires extending the AlósFerrer (1999) results to a more general set of processes where meeting probabilities are biased. This appears to work, but working that out is a project in itself that would take us well beyond the scope of this paper. For the purposes of this section, it is enough to verify that it works for the examples in question, which one can do directly.

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FIGURE 6.—Fitted model and data: left, inbreeding homophily by relative group size; right, number of friends by relative group size. Parameter values are α = 02, β = 2, γ = 08, and c = 004.

7.2. Fitting the Model to the Data Reconsider the utility function from the example in Section 4.5: U(s d) = (s + γd)α = (tq + γt(1 − q))α Let the matching process be governed by (11). We fit the data by numerically estimating the equilibrium for each case on a grid of values for α, β, and γ. The grid was such that α, γ, and 1/β were on a grid between 0 and 1 with increments of 0.1, and c was on a grid between 0 and 0.1 with increments of 0.01. We then examine the generated curves from each profile of parameters in the grid and compare them to the best-fit curves in the data. We choose the parameter values that lead to the minimum distance between the equilibrium curves and best-fit curves.39 We find best-fit parameters of α = 02 β = 2 γ = 08, and c = 004. The fitted curves are pictured in Figure 6. The picture shows the behavior of the inbreeding homophily index IH i = (q(wi ) − wi )/(1 − wi ), whose shape is not far from the scatter. 7.3. On the Role of Preferences and Opportunities Let us discuss the implications of our findings regarding the roles of preferences and opportunities in generating homophily in social networks.40 As we pointed out in Propositions 1 and 2, a bias in the matching process (in favor 39 Distance was computed numerically in Euclidean distance (square root of squared distances), using the best-fit curves from footnotes 18 and 19. The distance between curves was measured as the average distance on a grid of 200 values of w and was summed across the two curves. 40 For more background discussion on sources of homophily, see the survey by McPherson, Smith-Lovin, and Cook (2001).


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FIGURE 7.—Top, Inbreeding homophily; bottom, numbers of friendships. For all cases, α = 02 and c = 004; for biased preferences only (grey solid), β = 1 and γ = 08; for biased matchings only (grey dashed), β = 2 and γ = 1; for full bias (black), β = 2 and γ = 08.

of one’s own type) is needed to generate inbreeding homophily for all groups, and having type-sensitive preferences (such that agents care about more than just total number of friends) is necessary, in the context of our model, to derive a relationship between group size and numbers of friends. Figure 7 illustrates these properties relative to the fitting exercise from Section 7.2, where γ = 1 means that preferences are type-neutral, while β = 1 is the case of unbiased matching. We see that a bias in meetings generates the observed inbreeding homophily patterns regardless of the type sensitivity of preferences, but does not generate the observed friendship relationship without type sensitivity in preferences; at the same time, type sensitivity of preferences generates the observed friendship relationship regardless of the bias in matching and does not generate the observed inbreeding homophily without a bias in matching. While our model results point to a role of both preferences and the matching process in generating the observed friendship patterns in American high schools, there are alternative ways to explain the same evidence that hinge only on matching opportunities. “Increasing returns” in the matching process, so that larger groups face lower costs/higher returns from search (e.g., similar to search models initiated by Diamond (1982) and used extensively in the literature on job market search) could lead to predictions similar to the type sensitivity of preferences. The concept of increasing returns to matching is different from the bias in matching that we have examined, which lead to a change in

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the mix of types met: increasing returns lead to faster overall rates of meeting for larger groups, while our bias affects only the relative mix of types that an agent meets, not the rate at which an agent is matched. To see how increasing returns to matching could generate differences in relative number of friendships formed even with type-neutral preferences, let τi denote the expected number of friends met by type i per unit of search. The necessary condition for optimality, (i), in a case of type-neutral preferences becomes τi U (ti ) = c, from which it is clear that different types could then have different numbers of friendships formed per capita (and note also that the same could be obtained by assuming heterogeneous costs).41 There are instances where such differences in rates of encounters are natural; for example, in a case with cultural or linguistic barriers in communication. However, many factors which lead to differences in encounter rates (e.g., linguistic or cultural barriers) also tend to enter into the benefits of a friendship, and an explanation for the data built entirely on increasing returns would require the meeting process to be sensitive to type while the agents in the society are type blind. Nevertheless, understanding the relative roles for increasing returns to matching and type sensitivity in preferences is an important question for future research. To more fully identify the potential role of increasing returns to matching requires having information about the meeting process, which is not in the data considered here. 8. MANY TYPES In this section we discuss the extension of the results from two types to many types. For these points, we return to the model with meetings in proportion to stocks. The same-type bias condition that was used to establish the positive relation between total number of friends and group size and the positive relation between inbreeding homophily and group size is strengthened to the strong same-type bias condition. The same-type bias condition can only compare the preferences for two groups that face matchings where their relative stocks sum to 1. With more groups, the same-type bias condition does not make comparisons across groups, while the stronger version does. This provides an analog to Proposition 4 for the case of more than two types. PROPOSITION 6: Let there be more than two types and let U satisfy strong sametype bias. Then there exists a steady-state equilibrium such that Ni > Nj implies ti > tj and qi > qj (larger groups make more friends). The largest group displays inbreeding homophily, while the smallest group displays inbreeding heterophily. Moreover, if Ni > Nj , then IH i > IH j (larger groups display larger inbreeding homophily). 41 As suggested by an anonymous referee, a positive relation between group size and rate of encounters can be generated by assuming that friends of the same type are met at higher overall (rather than relative) rates than friends of different types.


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Proposition 6 can be explained as follows. Diminishing overall returns to friendship and strong same-type bias imply that larger groups form more friends in equilibrium. By staying longer in the search process, members of larger groups are found more often than members of smaller groups, which implies a larger index of inbreeding homophily. To fully understand this, it is useful to outline the proof. The fact that in equilibrium a higher q implies a higher total number of friends, t(q), follows the same steps as in the proof of Proposition 4, using strong same-type bias. The steady-state condition (iii) then implies that the term tw is constant across types. Therefore, the terms q t(q) and wq move together across types. Interestingly, in equilibrium the effect of group size on total number of friends and on inbreeding homophily balance across groups. The fixed point argument used in the proof shows that there exist a steady-state equilibrium in which w and q are positively related, from which the conclusion that larger groups display larger inbreeding homophily follows. For the many type case, we do not have a simple sufficient condition that guarantees uniqueness. We do, however, provide sufficient conditions to ensure that all equilibria will exhibit the properties of Proposition 6 (see Appendix C). The issue of whether or not we can order the equilibrium ti ’s across different equilibria cannot be established in the same manner as it was in the case of two types (e.g., Proposition 5). To understand this, consider a group that is 60 percent of the population. It can be very different in a society with two groups that are each 20 percent of the population than in a population with one other group that comprises 40 percent of the population. In particular, it might have a larger relative stock in the population with two smaller groups (who find their matchings less attractive and exit relatively earlier under a strong sametype bias) than with one larger group. In a society of three groups with relative inflowing weights of w = (06 02 02) this could lead to a higher stock and more friends for the largest group than in a society that has only two groups with relative inflows of w = (061 039) (which is then similar to a three group society with a very tiny third group). As for the case of two types, the result of Proposition 6 does not fit the observed patterns of inbreeding homophily: the smallest group still needs to have inbreeding heterophily and inbreeding homophily is monotone in relative group size. Again, an additional bias in the matching can reverse this. 9. WELFARE The model we have developed can be used to explore welfare issues, since preferences are explicit in the model. This contrasts with much of the literature on this topic, which has implicitly or explicitly adopted the view that an even mix or maximal diversity is desirable per se (see, for instance, the influential paper by Moody (2001)). This is not to say that such a view cannot be justified,

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but more simply that such a justification should be based on some foundation and that a model such as ours can begin to provide the foundation for a welfare analysis. At the same time that we emphasize a need for a foundational model for a welfare analysis, we also emphasize that our model is too simplistic to provide a welfare analysis that we can rely upon. In particular, our model misses many important aspects of the benefits of diversity, especially in a learning environment, as mixtures only enter preferences in direct friendships. We also emphasize that it does not directly account for the behavioral influences, diffusion, and other implications of social structure. The reason for examining a welfare analysis in the context of this model is rather to derive some simple comparative statics across groups, and then also to point out that the equilibrium effects are such that small changes in preference specifications can lead to large differences in welfare conclusions. This suggests that more exacting analyses are needed to fully understand optimal policies in programs that might affect, for instance, racial mixes in schools or the extent to which there is tracking and other aspects within school that constrain the meeting process. For the purposes of the welfare analysis, we examine a class of preferences that satisfy the following condition (and, again, this is satisfied by the preferences in Section 4.5): We say that U is homogeneous of degree α ∈ (0 1) if U(ks kd) = kα U(s d) for all (s d). Note that homogeneity of degree α does not preclude having a same-type bias in preferences. It is simply an assumption about how utility varies as the scale of friends is varied: the term “homogeneity” does not refer to comparisons of types, but to a standard mathematical property. PROPOSITION 7: If U is homogeneous of degree α ∈ (0 1), then the steadystate equilibrium average utility of group i, U(si di ) = U(qi ti (1 − qi )ti ), is proportional to ti (qi ), the optimal total number of friends of type i. Thus, arg maxq t(q) = arg maxq U(qt (1 − q)t). Proposition 7 shows that in the case of homogeneous preferences, a given type’s welfare is directly proportional to the number of friends it forms per capita. This has an immediate implication that larger groups will have higher equilibrium welfare (under the conditions of our early propositions having strong same-type bias) than smaller groups. To get some feeling for what this implies for total welfare, we explore the case where β = 1 and then return to discuss changes for β > 1. PROPOSITION 8: IfU is homogeneous of degree α ∈ (0 1) and β = 1, then average total welfare i wi Ui equates (wi /qi )Ui for all i = 1 K. Therefore, maximizing average welfare corresponds to maximizing (w1 /q1 )U1 , which is equal to (c/α)t1 · w1 /q1 .


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Propositions 7 and 8 taken together then show that aggregate welfare is proportional to the number of friends ti formed in equilibrium divided by the term qi /wi . Both the optimal total number of friends ti and qi /wi increase with group size wi (Proposition 6).42 The change in aggregate welfare triggered by a change in relative populations is the result of these two effects. If the population of type i has increased its weight, the increase in total friends formed by i-type agents has a positive effect, while the increase of homophily of itype agents has the effect of inhibiting the total friendships made by the other types, thus having a negative effect on welfare. The term ti · wi /qi highlights the trade-off between these effects. In a society with two groups, i and j, the effect of increasing the size of group i (locally) depends on the relative magnitudes of the derivatives of ti and of qi /wi with respect to wi . If the increase in the number of friends outweighs the increase in homophilous behavior, the total welfare increases, and not otherwise. Another way to look at the welfare effect of changing relative populations is to study the change in w1 U1 + w2 U2 . This change is proportional to w1 t(q1 (w1 )) + (1 − w1 )t(1 − q1 (w1 )), where q(wi ) is an equilibrium function of wi . Taking the derivative with respect to w1 yields t(q1 (w1 )) − t(1 − q1 (w1 )) + w1 t (q1 (w1 ))q − (1 − w1 )t (1 − q1 (w1 ))q So lowering the number of agents at a lower utility has a direct effect for those at the higher utility, but then also has indirect effects on both of the utility levels. When we change to β > 1, we will still have a similar trade-off, but the specifics of the trade-off can be affected. The utility function studied in Section 4.5 can be used to illustrate how simple variations in preferences can lead to large changes in welfare conclusions. For simplicity, we consider the case where β = 1, but this also holds for β > 1. In particular, when preferences are homophilous (γ < 1), we obtain complete segregation, while heterophilous preferences (γ > 1) yield an optimal solution with two equally numerous groups, and when preferences are type-neutral (γ = 1), the mix is irrelevant. What drives these extreme conclusions is the fact that same-type and different-type friends are substitutes, just perhaps not in a one-to-one ratio. Thus, these preferences lack a specific taste for diversity, and a constant marginal rate of substitution drives us to corner solutions. If instead, there is some preference for diversity, √ the welfare conclusions √ change. Consider, for instance, U(s d) = s + γ d (we examine this form 42

Proposition 6 is stated in terms of IH i rather than qi /wi , but the results are proven for both.

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since it allows an explicit calculation of equilibrium). Now an individual has an explicit taste to have friendships of both types. Here the optimal t is (1 − γ 2 )q + γ 2 + 2γ q(1 − q) t(q) = 4c 2 It can be checked that t(q) has a maximum at q = 1/(1 + γ 2 ). If 0 < γ < 1, then 12 < 1/(1 + γ 2 ) < 1. Figure 8 shows (numerical approximation) results for total welfare for the case of two types. Even when same-type bias is present, but not too strong (for values of γ not below 12 ), an equal split of the population gives the optimal welfare. For higher homophily (lower γ), the optimal welfare is reached with an unequal distribution, which, however, never reaches full segregation.43 In Appendix D we illustrate the problem of organizing more than two groups for the simple case of log utilities. Although those preferences are quite special, they allow for explicit analytic solutions in the more complex case of many types and provide some insight on the trade-offs in that case.

FIGURE 8.—The two plots show the welfare of the two types i (higher curve) and j (lower curve), and the aggregate welfare (middle curve) versus wi (plotted only from 05 to 1, by symme√ √ try) when U(s d) = s + γ d. The left-hand frame is for γ 2 = 03, where the optimal aggregate welfare is for wi = 05; the right-hand frame is for γ 2 = 02, where the optimal aggregate welfare is for wi 088.

43 We checked numerically that the same qualitative behavior happens for a biased matching like the one discussed in Section 7. What happens is that as β increases above 1, the γ that fixes a threshold between an optimal split, which is equal or unequal, also decreases below 1/2 (e.g., for β = 2, this threshold is at γ 0379).


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10. CONCLUDING REMARKS In this paper, we started from some basic observation regarding friendship patterns in a selected sample of American high schools: larger racial groups form more friends per capita, and all groups display inbreeding homophily, where the highest levels are for middle-sized groups. We then studied a searchbased model of friendship formation, trying to identify the role of opportunity and preferences in generating the observed data. We have shown that if all types meet the same number of other agents per unit of time, then generating differences in per capita friendships in our model requires having preferences depend on more than just total number of friends. So, without differences in meeting rates across type, to generate the observed data, preferences need to be sensitive to types. On the other hand, we find that the observed inbreeding homophily patterns can only be generated (in the context of our model) with some bias in the meeting process in favor of own type. Thus, according to our model’s results, both type-sensitive preferences and biased opportunities play a role in friendship formation. As we discussed in Section 7.3, these conclusions, and in particular the role played by preferences, depend on the assumption that all types meet the same number of friends per unit of time. If increasing returns to matching are introduced, so that members of larger groups meet more people in total per unit of time (rather than just meeting more of their own type per unit of time) than members of smaller groups, then the structure of the matching process alone can lead to patterns of friendship formation similar to those observed here. Even though there may not be obvious explanations for why the matching process should be based on type when agents ignore type, such potential alternative explanations point to the importance of obtaining more detailed data on the matching process that leads to friendship formation, and of developing richer models of friendship formation to help disentangle the relative roles of preferences and matching, which are important subjects for further research. Finally, our analysis has also shown the sensitivity of overall welfare to details of the setting. Generally, this suggests that more attention should be paid to modeling the homophily and the patterns of social ties that emerge from variations on matching processes, preferences, and equilibrium conditions, especially given the importance of social structure in many applications. Finally, our analysis suggests that the presence of more than two groups in a society or community opens the way to equilibrium phenomena that do not arise when there are only two groups. APPENDIX A: THE MATCHING PROCESS In this appendix, we provide a foundation for the matching process that justifies our definition of steady-state equilibrium with a continuum of agents. We

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proceed as follows. First, we show that if agents were to stay an integer number of periods, then there exists a discrete-time random-matching process over the continuum so that the process would be in steady state, and the outgoing agents would have a distribution of realized meetings so that the number of same types (and hence different types) that they meet would be governed by a binomial distribution. Next, by taking a limit as the number of periods becomes large, we have a process where the outgoing distribution will converge (in the sense of weak convergence of measures or convergence in distribution) to a Dirac measure with weight 1 on the expected proportion of meetings with own type, pi . Finally, we take this limit of large periods by subdividing the matches so that each period an agent makes ε new friends, where ε → 0, and so many matches are needed to reach ti total friends. Consider a discrete-time matching process with a measure of new types i entering in each period Ni and staying an integer Ti number of periods. This leads to masses Mi (Ti ) = Ni Ti of agents of type i in the matching at any given period and a total mass of M(T ) = i Mi (Ti ) of agents in the matching, where T = (T1 Tn ). Let pi (T ) = Mi (Ti )/M(T ) be the proportion of type i’s in the matching. Agents are labeled from [0 M(T )] with a label a. The matching type of an agent a currently in the matching is a vector θa = (ia ta pa ), which indicates, the type of the agent ia , how many periods the agent has been in the matching ta , and the proportion of same types that the agent has been matched with to date pa . Given T , there is a finite set of possible θ’s, denoted by Θ(T ).44 A matching is a Borel measurable bijection σ : M(T ) → M(T ) with the property that σ 2 (a) = a = σ(a). A random-matching scheme is a probability space with a countably additive probability measure on the space of matchings. Let μ be a Lebesgue measure and let Iθ (a) be an indicator function which states whether agent a is of type θ. Let us begin a process with a fraction of ν(θ) of type θ’s. LEMMA 2 —Alós-Ferrer (1999, Proposition 4.4)45 : There exists a matching scheme with probability measure P such that the following properties hold: P1. μ(σ(E)) = μ(E) for all Borel E, P-almost surely. P2. For all a, P(σ(a) is of type θ) = ν(θ). P3. For all θ and θ , Iθ (a)Iθ (σ(a)) dμ = ν(θ)ν(θ ) P-almost surely. a

44

Given that we start the process with a measurable mapping from agents to matching types, Lemma 4.1 in Alós-Ferrer (1999) allows us to relabel agents in a measurable way, so that we can effectively simply partition the interval into subintervals by collecting agents of given matching types together. 45 See also Duffie and Sun (2007).


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P4. The matching is atomless, so that P(σ(a) = b) = 0 for all a and b. Lemma 2 provides a random matching that is measure-preserving (P1), is atomless (which can be seen as a minimal form of anonymity) (P4), has a distribution over matches for each agent that is proportional to the relative fraction of matching types in the population (P2), and is such that a conclusion equivalent to that of the law of large numbers holds, so that the measure of type i’s who mix with type j’s is proportional to the product of the proportions in the population (P3). Thus, Lemma 2 allows for a matching that operates as if the law of large numbers holds for the continuum. Now, given T , set ν((i t p)) = ((Mi (T ))/Ti )Bpi (T )t (p), where Bpi (T )t (p) is the probability of having a fraction of p same types out of t draws with a probability of pi (T ) on each draw when the matches follow a binomial distribution. This gives the binomial distribution over possible sequences of realized matches for those of type i who are in their tth period of matching, who make up a fraction (Mi (T ))/Ti of the overall set of type i’s. Given that we start the types in proportion to what their realized frequency should be under the binomial distribution, it then follows from Lemma 2 (especially (P3)) that we can find a random matching that gives back the same proportions over types as the new matches and the resulting new matching types will have in a realization that is governed by the binomial distribution. LEMMA 3: There exists a random-matching process that satisfies (P1)–(P4), such that if ν is as described above, then we are in steady state so that ν is the outflowing distribution of matching types. What we have shown is that we can find a discrete-time random-matching process, so that if each type stayed for an integer number of periods, then we could find a steady-state equilibrium where the outflow of agents of a given type would have a distribution of realized fractions of matchings with same types that matches a binomial distribution. Next, we take a limit, letting the number of periods become large for each i. We do this as follows. Instead of making a unit of friendship in each period, consider a setting where agents get an ε of friendship from each match, where ε > 0. Then let Ti (ε) ∈ arg min |T ε − ti | T

be the number of periods that an agent of type i would have to stay so as to accumulate a total amount ti of friendships, where ti is the desired number from the text. For each ε, we have a well defined T (ε) and a resulting steady-state matching process where, in each period, there is an inflow and an outflow of a mass (N1 Nn ) of the various types of agents, and where the outgoing agents of type i have a distribution over the fraction of same types that they

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met which is governed by a binomial distribution on draws with Ti (ε) draws with probability pi (T (ε)) of meeting own type on each draw. Note that as ε → 0, it follows that each Ti (ε)ε → ti and that the above described distribution over realized meetings converges (in the sense of weak convergence of measures) to the Dirac measure with mass 1 on pi . This follows from standard results concerning the limit of a sequence of binomial distributions (e.g., see Billingsley (1968)). APPENDIX B: PROOFS PROOF OF PROPOSITION 1: If preferences are type-neutral we can write any type i’s utility as strictly concave function U(ti ), and then (i) in the definition of a steady-state equilibrium implies that for each group i, U (ti ) = c Then strict concavity of U implies that ti = tj for all groups i and j, for any matching process in our model. Q.E.D. PROOF OF PROPOSITION 2: If the matching process is unbiased, then for all types, the probability of meeting type i is equal to qi = Mi /M. This implies that q = i i i wi = 1, so that if qi > wi for some type i, then qj < wj for at least one type j. Q.E.D. PROOF OF PROPOSITION 3: Without loss of generality, let Ni ≥ Nj . An equilibrium must specify qi , Mi , and ti for each type such that (i)–(iii) are satisfied (noting that (iv) is satisfied automatically with an unbiased meeting process). Let t(q) be the function that assigns the unique optimal t to any q ∈ (0 1), as in Lemma 1.46 As argued following the proposition, it is easy to see that steady-state conditions (ii) and (iii) require that Nj dj = Ni di or Nj tj (qj )(1 − qj ) = Ni ti (qi )(1 − qi ). Given that qj = 1 − qi , it follows that a necessary condition for an equilibrium is (12)

t(qi )(1 − qi ) Nj = Ni t(1 − qi )qi

We claim that if we find a qi that satisfies (12), then the specification of t(qi ) Mi = Ni t(qi ) qi together with t(1 − qi ) Mj = Nj t(1 − qi ) qj = 1 − qi forms an equilibrium. The fact that (i) and (ii) are satisfied follows directly from the definition of t and the way in which the M’s are defined. So let us 46 The uniqueness follows from the conditions on preferences that imply that qi Us (qi ti (1 − qi )ti ) + (1 − qi )Ud (qi ti (1 − qi )ti ) is decreasing in ti .


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check that (iii) is also satisfied. Let us verify that qi = Mi /(Mi + Mj ) for each i so that (iii) holds. It is enough to check this for i, given that qj = 1 − qi . By (12), it follows that Nj t(1 − qi )qi = Ni t(qi )(1 − qi ) or (Nj t(1 − qi ) + Ni t(qi ))qi = Ni t(qi ) Then (Mj + Mi )qi = Mi which implies the desired conclusion. So, to establish existence of an equilibrium, we show that there exists a qi that satisfies (12). When qi = 12 , it follows that (t(qi ))/(t(1 − qi )) = 1. Thus, if Ni = Nj , then qi = 12 is an equilibrium. Next, note that as qi increases, the continuous function (t(qi )(1 − qi ))/(t(1 − qi )qi ) converges to 0 as qi → 1.47 Thus, when Ni > Nj , there always exists a solution to (12) such that 1 > qi > 1/2. Q.E.D. PROOF OF CLAIM 1: If t(qi )(1 − qi ) is a decreasing function, then t(1 − qi )qi is an increasing function. Thus, the expression on the right-hand side of (12) is a decreasing function and the equality can hold at most at one value of q. Q.E.D. PROOF OF PROPOSITION 4: Consider the equilibrium whose existence is proven in Proposition 3. This equilibrium satisfies relative homophily and is such that qi > 1/2. The same-type biased preference condition (9) implies that (13)

∂ ∂ U(qi tj (1 − qi )tj ) > U((1 − qi )tj qi tj ) ∂t ∂t

Then optimization by j implies that (14)

∂ ∂ U((1 − qi )tj qi tj ) = U(qj tj (1 − qj )tj ) = c ∂t ∂t

Equations (13) and (14) then imply that (15) 47

∂ U(qi tj (1 − qi )tj ) > c ∂t

Note that t is bounded above and is bounded away from 0 under our preference conditions.

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The condition of overall diminishing returns to friendship then implies that si + di = ti > sj + dj = tj (given that the left-hand side of (15) is equal to c at the optimal choice ti for type i). Therefore, (16)

ti =

dj di > tj = 1 − qi qi

which implies that (17)

dj Ni wi qi > = = 1 − qi di Nj wj

Since qi + qj = 1 and wi + wj = 1, (17) implies that qi /qj > wi /wj , qi > wi (inbreeding homophily) and qj < wj (inbreeding heterophily). The fact that si > sj is implied by the equilibrium condition Ni di = Nj dj (which implies di < dj ) and by the fact that ti > tj . Q.E.D. PROOF OF PROPOSITION 5: We first show that if U exhibits strong same-type bias, then t(q) is increasing in q. By strong same-type bias, it follows that (18)

∂ ∂ U(

qt (1 −

q)t) > U(qt (1 − q)t) ∂t ∂t

when

q > q. The assumption of overall diminishing returns to friendship of U implies that t(

q) > t(q). Next, the fact that (1 − q)t(q) is decreasing implies that the expression on the right-hand side of (12) is decreasing in q. Thus, since as wi increases, the unique solution of qi to (12) increases, the left-hand side of (12) is simply (1 − wi )/wi . This in turn implies that the steady-state equilibrium value of ti increases, as claimed. Q.E.D. PROOF OF PROPOSITION 6: Note that by the proof of Proposition 5, we know that t(q) is increasing in q. Let (N1 N2 NK ) be ordered in nondecreasing order. We show the existence of an equilibrium that satisfies the statement of the proposition by showing the existence of an interior fixed point of an appropriately constructed function F that maps the (K − 1)-dimensional simplex into itself. F is constructed as follows. Let F1 denote the function that maps a vector q = (q1 q2 qK ) of the simplex to the vector F 1 (q) = (t(q1 ) t(q2 ) t(qK )). Let F 2 be the continuous function that orders any K-vector in nondecreasing order. Let F 3 be the continuous function defined by N1 x1 N2 x2 NK xK 3 (19) F (x1 x2 xK ) = Ni xi Ni xi Ni xi


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Note that for any q = (q1 q2 qK ), F(q) ≡ F 3 ◦ F 2 ◦ F 1 (q) is ordered in nondecreasing order and the sum of its elements is 1. F is a continuous function from the simplex to itself and therefore possesses a fixed point by Brower’s theorem. Note that at the fixed point, F 2 preserves the ordering of the vector q because t(q) is increasing, so that F 2 applied to the fixed point is just the identity function. We conclude the proof by showing that such a fixed point is interior. Suppose not, so that there is at least one qi such that qi = 0. By the assumption that Us (0 0) > c and Ud (0 0) > c, F 3 ◦ F 2 ◦ F 1 has all positive values (because F 1 has all positive values), which implies that qi = 0 cannot be an element of the fixed point. Finally, note that given the steady-state condition ti wi /qi = tj wj /qj , the vector (q1 /w1 q2 /w2 qK /wK ) must be ordered in nondecreasing order (given that the vector (t1 t2 tK ) is nondecreasing, as shown above). Given that qi = 1 and wi = 1, we conclude that qK /wK > 1 while q1 /w1 < 1. Also, since wi > wj implies that qi /wi > qj /wj , it also follows easily that (qi − wi )/(1 − wi ) > (qj − wj )/(1 − wj ) (simply noting that this can be rewritten as (qi /wi − 1)/(1/wi − 1) > (qj /wj − 1)/(1/wj − 1)). Q.E.D. PROOF OF PROPOSITION 7: The condition for the optimal stopping implies that c=

∂ U(qt (1 − q)t) ∂t

This, together with the homogeneity of degree α, implies that t(q) = α

U(qt (1 − q)t) c

which in turn implies for each type i, (20)

Ui (qi t(qi ) (1 − qi )t(qi )) =

ct(qi ) α

so that the optimal number of friends given qi is proportional to the maximized level of utility up to the constant αc . Q.E.D. PROOF OF PROPOSITION 8: The equations in conditions (ii) and (iii) for steady-state equilibrium imply that there exists an M such that for all i M=

Ni ti qi

for each i. This implies that Ni ti Nj tj = qi qj

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for every i and j. Therefore, in equilibrium, w i Ui w j Uj = qi qj Maximizing the aggregate (or average) utility in society then amounts to maximizing qj w1 cw1 = w i Ui = w 1 U1 1 + U1 = t1 q q αq1 1 1 i j≥2 which establishes the claim.

Q.E.D.

PROPOSITION 9: Consider a biased matching process in a society with two types described by q2 = f (q1 ), where f is continuous and decreasing. There exists an equilibrium and one such that q1 > q2 . Moreover, if preferences satisfy strong same-type bias, then there exists an equilibrium where larger groups make more total friends. PROOF: In equilibrium it follows that (21)

N2 t(q2 )(1 − q2 ) = N1 (t(q1 )(1 − q1 ))

or (22)

N2 t(q1 )(1 − q1 ) = N1 t(q2 )(1 − q2 )

Let the bias be q2 = f (q1 ), with f (1) → 0 and f (0) → 1. Then we rewrite (22) as (23)

N2 t(q1 )(1 − q1 ) = N1 t(f (q1 ))(1 − f (q1 ))

Let qˆ be a fixed point of f . Then q1 = q2 = qˆ is an equilibrium for N1 = ˆ the term on the rightN2 . If we let qi increase and tend to 1 starting from q, hand side of (23) tends to zero, since t(q1 ) is bounded and t(0) > 0 under our assumptions. So there exists an equilibrium with q1 > qˆ when N1 > N2 . ˆ and q1 > qˆ implies that Since f is decreasing, it has a unique fixed point q, q2 = f (q1 ) < q1 To conclude the theorem, apply the first part of the proof of Proposition 5, showing that strong same-type bias implies that t(q) is increasing, to the equiQ.E.D. librium where q1 > q2 .


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APPENDIX C: RULING OUT NON-WELL -ORDERED EQUILIBRIA In this section, we establish conditions under which the results of Proposition 6 apply to all steady-state equilibria. Recall that U is homogeneous of degree α ∈ (0 1) if U(ks kd) = kα U(s d) for all (s d). PROPOSITION 10: Let U(s d) be homogeneous of degree α ∈ (0 12 ]. In every steady-state equilibrium, Ni > Nj implies qi > qj . If preferences also exhibit sametype bias, then ti = si + di > tj = sj + dj whenever Ni > Nj , and i exhibits more inbreeding homophily than j (IH i > IH j ).48 PROOF: From condition (iii) of equilibrium, it follows that if Ni > Nj , then ti /qi < tj /qj . If ∂ qt /∂q < 0, it follows that qi > qj . Additionally, ∂ qt /∂q = ∂t ∂t ( ∂q q − t)/q2 < 0 if and only if ∂q · qt < 1. Equation (3) and the fact that U is homogeneous of degree α imply that (24)

t(q) = α

U(qt (1 − q)t) c

We take the derivative of (24) with respect to q, obtaining

∂t = αt Us (qt (1 − q)t) − Ud (qt (1 − q)t) ∂q (25) ∂t ∂ U(qt(1 − q)t) c + ∂q ∂t from which, since (26)

∂ ∂t

U(qt (1 − q)t) = c, it follows that

∂t α t(Us (qt (1 − q)t) − Ud (qt (1 − q)t)) = ∂q 1 − α c

From (26) it follows that ∂t q α q(Us (qt (1 − q)t) − Ud (qt (1 − q)t)) · = ∂q t 1−α c 48 The next example shows that even with two types, if U is homogeneous of degree α > 1/2, we can have equilibria with Ni > Nj , but qi < qj and ti < tj . Consider U(s d) = s09 + d 09 /2, 2N1 = N2 and c = 09. Given homogeneity, it is possible to express explicitly

t(q) = (q09 + (1 − q)09 )10 We find numerically two steady-state equilibria: (i) a non-well-ordered equilibrium in which q1 062, q2 038, t1 023, and t2 007; (ii) a well ordered equilibrium in which q1 0001, q2 0999, t1 00001, and t2 100.

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α qUs (qt (1 − q)t) + (1 − q)Ud (qt (1 − q)t) 1−α c α ≤ 1 = 1−α
0, while the second inequality comes from the fact that α ≤ 12 . The last part of the proof follows from Proposition 6. Q.E.D. APPENDIX D: WELFARE ANALYSIS WITH MANY TYPES AND LOGARITHMIC PREFERENCES We use a logarithmic form of preferences to investigate alternative policies when there are more than two groups. Let U(s d) = a log s + b log(s + d) + g log d, where a, b, and g are nonnegative. Note that in this case U(s d) is not homogeneous.49 Consider the case in which a = 1, b > 0, and g = 0. The utility of type i agents as a function of wi is then written as 1+b 1+b + b log − 1 − b Ui (wi ) = log wi c c The utility of type i agents is indifferent to a split of type j into more types, as long as wi is kept constant. There is no externality on other types if a type splits into more subgroups. The type i utility is always increasing in wi . If the number of types is K = 2, then the expected average welfare wi U(wi ) + (1 − wi )U(1 − wi ) is 1+b 1+b + b log wi log wi c c 1+b + (1 − wi ) log (1 − wi ) − 1 − b c which is convex, always achieves a minimum at wi = 12 , and achieves a maximum in the completely segregated case where wi = 0 or wi = 1. 49 The log function can be considered a case of homogeneity α at the limit where α → 0. It is, however, easy to check that not all our results concerning homogeneous functions hold in the logarithmic case. Equation (3) implies that for any type i, the optimal number of friends is ti = (a + b + g)/c, independently of qi . Condition (iii) of steady-state equilibrium implies that for any type i,

Ni ti Ni qi = K = K = wi N t j j j j Nj The steady-state equilibrium is then easy to compute. For any type i, si = qi ti = wi (a + b + g)/c and di = (1 − qi )ti = (1 − wi )(a + b + g)/c. U(s d) trivially exhibits baseline homophily (qi = wi for any type i).


1043

Consider a case in which a = 1, b = 0, and g > 0. The utility function of type i agents depends on wi and is 1+g 1+g + g log (1 − wi ) − 1 − g Ui (wi ) = log wi c c There is still no externality if another type is split into more types. It is easy to 1 check that Ui (wi ) is increasing in wi up to the point where wi = 1+g and then is decreasing. If g is a positive integer, then the social optimum would be to split g + 1 types (if available) equally in the matching process. If we consider only K = 2 types, then the expected average welfare is 1+g (wi + g(1 − wi )) log wi c 1+g + (wi g + (1 − wi )) log (1 − wi ) − 1 − g c The previous function always has a critical point in wi = 12 . Its second derivative with respect to wi is −g + (1 + g)wi (1 − wi ) wi2 (1 − wi )2 which is negative for any wi ∈ (0 1) only if g > 13 . If g ≥ 13 , then the average welfare obtains its maximum for wi = 12 . If instead g < 13 , then wi = 12 is a local minimum for the aggregate welfare, whose functional form is symmetric and bimodal, as in Figure 8. Figure 9 shows the two cases.

FIGURE 9.—Average aggregate welfare when U(s d) = log s + g log d. The left-hand side is for g = 12 ; the right-hand side is for g = 15 .

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S. CURRARINI, M. O. JACKSON, AND P. PIN REFERENCES

ACEMOGLU, D., AND R. SHIMER (1999): “Holdups and Efficiency With Search Frictions,” International Economic Review, 40, 827–849. [1006,1024] ALLPORT, W. G. (1954): The Nature of Prejudice. Cambridge, MA: Addison-Wesley. [1006] ALÓS-FERRER, C. (1999): “Dynamical Systems With a Continuum of Randomly Matched Agents,” Journal of Economic Theory, 86, 245–267. [1025,1034] BAERVELDT, C., M. A. J. DUIJN, L. VERMEIJ, AND D. A. HEMERT (2004): “Ethnic Boundaries and Personal Choice: Assessing the Influence of Individual Inclinations to Choose Intra-Ethnic Relationships on Pupils’ Networks,” Social Networks, 26, 55–74. [1005,1009] BILLINGSLEY, P. (1968): Convergence of Probability Measures. New York: Wiley. [1036] BLALOCK, JR., H. M. (1982): Race and Ethnic Relations. Englewood Cliffs, NJ: Prentice-Hall. [1006] BLAU, P. M. (1977): Inequality and Heterogeneity: A Primitive Theory of Social Structure. New York: Free Press. [1006,1011,1021] BRAMOULLÉ, Y., AND R. KRANTON (2007): “Risk Sharing Across Communities,” The American Economic Review, 97, 70–74. [1015] BURTON, R. (1621): The Anatomy of Melancholy (reprinted in 1927). New York: Farrar and Rinehart. [1005] COLEMAN, J. (1958): “Relational Analysis: The Study of Social Organizations With Survey Methods,” Human Organization, 17, 28–36. [1007,1008] CURRARINI, S., M. O. JACKSON, AND P. PIN (2008): “Homophilous Behaviour in High School Frienships,” Mimeo, Stanford University. [1010] (2009): “Supplement to ‘An Economic Model of Friendship: Homophily, Minorities, and Segregation’,” Econometrica Supplemental Material, 77, http://econometricsociety.org/ecta/ Supmat/7528_data.pdf. [1006,1013,1016] DIAMOND, P. A. (1982): “Aggregate Demand Management in Search Equilibrium,” The Journal of Political Economy, 90, 881–894. [1006,1021,1027] DUFFIE, D., AND Y. SUN (2007): “Existence of Independent Random Matching,” The Annals of Applied Probability, 17, 386–419. [1034] ECHENIQUE, F., AND R. G. FRYER, JR. (2007): “A Measure of Segregation Based on Social Interactions,” Quarterly Journal of Economics, 122, 441–485. [1011] FELD, S. L. (1981): “The Focused Organization of Social Ties,” The American Journal of Sociology, 86, 1015–1035. [1005] FONG, E., AND W. W. ISAJIW (2000): “Determinants of Friendship Choices in Multiethnic Society,” Sociological Forum, 15, 249–271. [1005] FRIEDMAN, J. W. (1973): “Concavity of Production Functions and Non-Increasing Returns to Scale,” Econometrica, 41, 981–984. [1014] GILES, M. W., AND A. EVANS (1986): “The Power Approach to Intergroup Hostility,” Journal of Conflict Resolution, 30, 469–486. [1006] HOLLAND, P. (1601): The Historie of the World. Commonly Called, The Naturall Historie of C. Plinius Secundus. London: Adam Islip. [1005] IOANNIDES, Y. M., AND L. DATCHER LOURY (2004): “Job Information Networks, Neighborhood Effects, and Inequality,” Journal of Economic Literature, 42, 1056–1093. [1014] JACKSON, M. O. (2006): “The Economics of Social Networks,” in Advances in Economics and Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society, Vol. 1, ed. by R. Blundell, W. Newey, and T. Persson. Cambridge, U.K.: Cambridge University Press, Chapter 1. [1006] LAZARSFELD, P. F., AND R. K. MERTON (1954): “Friendship as a Social Process: A Substantive and Methodological Analysis,” in Freedom and Control in Modern Society, ed. by M. Berger. New York: Van Nostrand, 18–66. [1003,1005] MARSDEN, P. V. (1987): “Core Discussion Networks of Americans,” American Sociological Review, 52, 122–131. [1006,1007,1011]


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(1988): “Homogeneity in Confiding Relations,” Social Networks, 10, 57–76. [1013] MCPHERSON, M., L. SMITH-LOVIN, AND J. M. COOK (2001): “Birds of a Feather: Homophily in Social Networks,” Annual Review Sociology, 27, 415–44. [1003,1005,1007,1013,1026] MOODY, J. (2001): “Race, School Integration, and Friendship Segregation in America,” American Journal of Sociology, 107, 679–716. [1005,1006,1009,1029] MORTENSEN, D. T., AND C. A. PISSARIDES (1994): “Job Creation and Job Destruction in the Theory of Unemployment,” Review of Economic Studies, 61, 397–415. [1006,1013] PAGE, S. E. (2007): The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies. Princeton, NJ: Princeton University Press. [1014] WATTS, A. (2007): “Formation of Segregated and Integrated Groups,” International Journal of Game Theory, 35, 505–519. [1014]

Dipartimento di Scienze Economiche, Università di Venezia, San Giobbe 873, 30121 Venice, Italy and School for Advanced Studies in Venice (SSAV), Venice, Italy; [email protected], Dept. of Economics, Stanford University, 579 Serra Mall, Stanford, CA 943056072, U.S.A. and Santa Fe Institute, Santa Fe, NM 87501, U.S.A.; jackson@ stanford.edu, http://www.stanford.edu/~jacksonm/, and Dipartimento di Economia Politica, Università degli Studi di Siena, Piazza San Francesco 7, 53100 Siena, Italy and Max Weber Programme, European University Institute, 50014 San Domenico di Fiesole, Italy; [email protected]. Manuscript received October, 2007; final revision received August, 2008.


SOCIAL CONNECTIONS AND INCENTIVES IN THE WORKPLACE: EVIDENCE FROM PERSONNEL DATA BY ORIANA BANDIERA, IWAN BARANKAY, AND IMRAN RASUL1 We present evidence on the effect of social connections between workers and managers on productivity in the workplace. To evaluate whether the existence of social connections is beneficial to the firm’s overall performance, we explore how the effects of social connections vary with the strength of managerial incentives and worker’s ability. To do so, we combine panel data on individual worker’s productivity from personnel records with a natural field experiment in which we engineered an exogenous change in managerial incentives, from fixed wages to bonuses based on the average productivity of the workers managed. We find that when managers are paid fixed wages, they favor workers to whom they are socially connected irrespective of the worker’s ability, but when they are paid performance bonuses, they target their effort toward high ability workers irrespective of whether they are socially connected to them or not. Although social connections increase the performance of connected workers, we find that favoring connected workers is detrimental for the firm’s overall performance. KEYWORDS: Favoritism, managerial incentives, natural field experiment.

1. INTRODUCTION THIS PAPER EXPLORES the effects of social relationships between individuals within a firm on the productivity of individuals and on the firm’s overall performance. The idea that human relations affect behavior in the workplace has been long discussed in the sociology literature (Mayo (1933), Barnard (1938), Roethlisberger and Dickson (1939), and Roy (1952)). Economists have joined this debate relatively recently, due both to the burgeoning theoretical literature on how social relations and social preferences matter for economic behavior, in general, and to the increasing availability of personnel data, in particular. In the context of firms, much of the literature—theoretical and empirical— has studied the effect of social relations within a single tier of the firm hierarchy, such as among managers or among workers.2 However, it is reasonable to 1 Financial support from the ESRC is gratefully acknowledged. We thank a co-editor, three anonymous referees, Simon Burgess, Arnaud Chevalier, Paul Grout, Costas Meghir, Antonio Merlo, Paul Oyer, Torsten Persson, Canice Prendergast, Kathryn Shaw, David Stromberg, John Van Reenen, Fabrizio Zilibotti, and numerous seminar and conference participants for comments and suggestions that have helped improve the paper. Brandon R. Halcott provided excellent research assistance. We thank all those involved in providing the data. This paper has been screened to ensure no confidential information is revealed. All errors remain our own. 2 Lazear (1989), Kandel and Lazear (1992), and Rotemberg (1994) developed models incorporating social concerns into the analysis of behavior within firms. While they emphasized that individuals have social concerns for others at the same tier of the firm hierarchy, their analysis is equally applicable across tiers of the hierarchy. Bewley (1999) offered extensive evidence from interviews with managers arguing that concerns over fair outcomes for workers and the morale of employees are important determinants of their behavior.


DOI: 10.3982/ECTA6496

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O. BANDIERA, I. BARANKAY, AND I. RASUL

expect that such social connections might also span across layers of the hierarchy, in particular between managers and their subordinates, and this is likely to have important consequences for individual and firm performance, the optimal design of compensation schemes, and the structure of organizations (Prendergast and Topel (1996)).3 In general, social connections between managers and workers can help or harm firm performance. On the one hand, social connections may be beneficial to firm performance if they allow managers to provide nonmonetary incentives to workers or help reduce informational asymmetries within the firm. On the other hand, managers may display favoritism toward workers they are socially connected with, and this might be detrimental to the firm’s overall performance.4 In this paper, we present evidence on whether the existence of social connections between workers and managers affects the performance of connected workers and that of the firm as a whole. The firm we study is a leading producer of soft fruit in the United Kingdom. We focus on the behavior of individuals at two tiers of the firm hierarchy—workers and managers. The main task of the workers is to pick fruit, whereas managers are responsible for logistics. Two key features of this setting are that workers are paid piece rates and that managerial effort is complementary to worker effort and can be targeted to individual workers. Taken together, these features imply managers can significantly influence a worker’s productivity and hence his earnings. Managers and workers are all hired for one fruit picking season. They are university students from eight Eastern European countries and are thus of similar ages and backgrounds. In addition, they live on the farm site for the entire duration of their stay. Both features increase the likelihood of managers and workers forming strong social connections with each other. To measure social connections we exploit three sources of similarity between managers and workers—whether they are of the same nationality, whether they live in close proximity to each other on the farm, and whether they arrived at a similar time on the farm. Our underlying assumption is that individuals are more likely to befriend others if they are of the same nationality, if they are neighbors, or if they share early experiences in a new workplace. To identify the effect of social connections we exploit two sources of variation. First, the organization of the workplace is such that the allocation of 3 A related theoretical literature emphasizes the inefficiencies that arise from collusion between managers and workers (Tirole (1986), Kofman and Lawarrée (1993)) influence activities and other forms of rent seeking behavior by workers (Milgrom (1988), Milgrom and Roberts (1990)). 4 Both the positive and negative effects of social connections have been stressed in the organizational behavior and sociology literatures. Examples of such work includes that on the effect of manager–subordinate similarity on subjective outcomes such as performance evaluations and job satisfaction (Tsui and O’Reilly (1989), Thomas (1990), Wesolowski and Mossholder (1997)), and on how social networks within-firm influence within-firm promotions (Podolny and Baron (1997)).

SOCIAL CONNECTIONS AND INCENTIVES

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workers to managers changes on a daily basis. We exploit this variation to identify the effect of social connections from the comparison of the performance of a given worker on days when he is socially connected to his manager and days when he is not. Exploiting the within worker variation allows us to separate the effect of social connections from the effect of unobservable individual traits, such as ability, that make workers more likely to befriend managers and to have higher performance regardless of their connections. Similarly, as we observe the same manager managing both workers she is socially connected to and workers she is not connected to, we are also able to control for time invariant sources of unobserved manager heterogeneity that affect the productivity of connected and unconnected workers alike, such as their management style or motivational skills.5 Second, we designed and implemented a field experiment to exogenously vary the strength of managerial incentives. In the experiment we changed the managerial compensation scheme from fixed wages to the same level of fixed wages plus a performance bonus that is increasing in the average productivity of the workers on the field that day. Workers were paid according to the same compensation scheme—piece rates—throughout. The experiment allows us to identify whether and how the effect of social connections between the same managers and workers changes once managers are given performance pay, and thus provides an ideal counterfactual to assess the effect of social connections on the overall firm’s performance. To be precise, if the managers’ behavior toward connected workers changes once their interests are more closely aligned with the firm’s, their previous behavior under fixed wages could have not been maximizing the firm’s average productivity. We provide further evidence on this issue by quantifying the net effect of social connections on the firm’s overall performance. Our main findings are as follows. First, when managers are paid fixed wages, the productivity of a given worker is 9% higher when he is socially connected to his manager, relative to when he is not, all else equal. As workers are paid piece rates, this translates into the same proportionate change in earnings. Second, when managers are paid performance bonuses that tie their pay to the average productivity of workers they manage, being socially connected to the manager has no effect on workers’ productivity. Third, the introduction of managerial performance pay significantly decreases the productivity of low ability workers when they are connected to their manager relative to when they were connected to their manager and she was paid a fixed wage. The introduction of managerial performance pay increases the productivity of high ability workers, especially when they are not connected 5

Our empirical strategy is informed by the evidence that individual “styles” of managers affect firm performance over and above firm level characteristics themselves (Bertrand and Schoar (2003), Malmendier and Tate (2005)).

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to their managers. These findings indicate that when managers face low powered incentives, they favor the workers they are socially connected to, regardless of the workers’ ability. In contrast, when they face high powered incentives, managers favor high ability workers regardless of the workers’ connection status. Fourth, an increase in the level of social connections between managers and workers has a detrimental effect on the firms’ average productivity when managers are paid fixed wages and has no effect when managers are paid performance bonuses. In this setting, social connections are therefore detrimental for the firm because their existence distorts the allocation of managerial effort in favor of lower ability workers. To the best of our knowledge, this paper is the first to provide quantitative evidence on the productivity effect of social connections across tiers of the firm hierarchy. The paper builds on our earlier findings on the effects of the introduction of high powered managerial incentives in the same setting (Bandiera, Barankay, and Rasul (2007)). The earlier paper focuses on the allocation of managerial effort under performance pay, irrespective of the social connections within the firm. The current paper provides novel evidence on the importance of social connections in a firm setting, a previously unexplored determinant of managers’ behavior. The paper contributes to the growing empirical evidence on the interplay between social networks and individual and firm performance. This literature has explored how the response of workers to incentives depends on their social connections with their co-workers at the same tier of the firm hierarchy (Bandiera, Barankay, and Rasul (2005)), and how the demographic differences between managers and their subordinates affect the subordinates’ rate of quits, dismissals and promotions (Giuliano, Levine, and Leonard (2005)).67 Our paper also relates to the literature on employee and employer discrimination (Becker (1957)), and in particular to the findings of Black and Strahan (2001), who exploited a deregulation in product markets to show that when competition is low, firms favor male over female employees, both in terms of wages and promotion prospects. 6 Another branch of this literature has explored the effects of the CEO or managerial board of firms being socially connected to those outside of the firm, such as local politicians and bureaucrats (Bertrand, Kramarz, Schoar, and Thesmar (2005), Kramarz and Thesmar (2005), Mian and Khwaja (2005)). In nonfirm settings, Garicano, Palacios, and Prendergast (2005) presented evidence from soccer matches on how referees favor home teams in order to satisfy the crowds in the stadium. Laband and Piette (1994) showed that journal editors use professional contacts to identify high impact papers. In that context, favoritism thus reduces informational asymmetries and is efficiency enhancing in the market for scientific knowledge. 7 Fehr and Fischbacher (2002) provided an overview of the laboratory evidence on social preferences in workplace environments. One branch of this stems from Akerlof (1982) and Akerlof and Yellen (1988), who viewed the labor relation as a partial gift exchange. A separate branch of this experimental literature presents evidence that workers care about their pay relative to other workers (Charness and Kuhn (2005)).


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Finally, it is important to stress from the outset that as in all the studies using detailed data from one particular firm, precision comes at the cost of generality. In the last section we highlight the key characteristics of this workplace and discuss the external validity of our findings. The paper is organized as follows. Section 2 describes our context and experimental research design. Section 3 develops a theoretical framework to highlight the central forces at play when social connections can have potentially positive and negative effects on worker productivity. Section 4 describes the data and the identification strategy. Section 5 presents the main results on the effect of social connections on worker productivity under each managerial incentive scheme. Section 6 then explores whether there are heterogeneous effects of social connections across workers of different ability to derive implications for the firm’s overall productivity. Section 7 concludes. Proofs and further robustness checks are given in the Appendix. 2. THE CONTEXT AND EXPERIMENTAL DESIGN 2.1. Context We analyze the behavior of managers and workers in the fruit picking division of a leading U.K. producer of soft fruit during the 2003 season. Workers and managers are hired from eight countries in Eastern Europe on seasonal contracts that last between three and six months.8 To be recruited, individuals must be full-time university students and have at least one year remaining before graduation. Two features of the work environment increase the likelihood of individuals forming strong social connections to each other: (i) workers and managers are of similar ages and have similar socioeconomic backgrounds; (ii) they live and work on the farm site for the entire duration of their stay, which on average is 100 days. The workers’ primary task is to pick fruit. They typically pick on two or three different fields each day. At the start of a field-day the manager allocates each worker to a row of fruit to be picked. Once a worker clears this row, the manager is responsible for reallocating the worker to another row within the field. This process continues until all fruit within the field is picked. As each worker picks on his own row, his productivity is independent of the efforts of other workers on the same field-day, so that there are no complementarities between workers arising from the production technology. Workers do not choose how many hours to work—all workers are present on the field-day for the number of hours it takes to pick all the available fruit. Once a field is picked, workers and managers move on to other fields. As explained below, the match of 8 Their work permit allows them to work on other U.K. farms subject to the approval of the permit agency. Their outside option to employment at the farm is therefore to return home or to move to another farm during the season. Few workers are hired for consecutive seasons and workers are not typically hired from the local labor market.

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workers to managers can change across fields on the same day. The only choice variable of workers is how much effort to exert into picking.9 Workers are paid a piece rate per kilogram of fruit picked. Each worker’s pay is thus related to his productivity, which is an increasing function of his effort, the quantity of fruit available on the rows he is assigned, and the managerial effort targeted toward him. Managers are each assigned a group of around 20 workers, and their task is to monitor the quality of fruit picking and to organize the field logistics for this group. Managers on the same field focus on their assigned group of workers and work independently of each other. Managers control quality on three dimensions—that all ripe fruit is picked, that fruit is not damaged, and that fruit is correctly classified by size. Field logistics include the allocation of workers to rows and organizing the movement of fruit from the field to the packaging plant.10 The key choice variables of each manager are the allocation of workers to rows and the allocation of effort among her workers. Managers are responsible for allocating workers to rows at the start of the field-day and for reallocating workers to new rows once they have finished picking the row they were originally assigned. How the manager matches workers to rows is important because there is considerable variation in the quantity of fruit across rows within a field. Some of this is due to the natural variation in fruit quantity on different plants. This variation also stems from some rows being closer to pillars that support the plastic covering over the field. Rows close to pillars are harder to pick, air circulation is worse, and hence heat tends to accumulate. These factors reduce the marginal productivity of worker’s effort in these rows, other things equal. The manager chooses how to allocate her effort across workers along two dimensions. First, if several workers finish picking their rows at the same time, the manager has to decide who to reallocate to a new row first. Second, workers place the fruit they have picked into crates. Once these are full, managers have to ensure that new empty crates are provided to workers and that full crates are removed from the field and shipped to the packaging plant. If several workers simultaneously fill their crates, the manager chooses who to help first. In this environment, managerial effort is therefore complementary to worker’s effort. The effort costs to the manager are considerable because the workers she is responsible for are dispersed over a large area. The median field size is such 9 Work is offered on a casual basis with no daily guarantee of employment. In practice, managers manage each day, and workers are engaged in picking tasks every other day. On other days workers are asked to perform nonpicking tasks such as planting or weeding, or may be left unemployed for the day. Over the season, individuals are not observed moving across tiers of the hierarchy from picking tasks to managerial tasks or vice versa. 10 A separate group of individuals, called field runners, are responsible for physically moving fruit from the field to the packaging plant. They do not themselves pick fruit nor do they manage workers.


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that each manager has to cover an area of around one hectare. To ensure she is aware of which workers need to be reallocated to new rows and which need their crates replaced, managers need to continuously walk around the field.11 Social connections between managers and workers can have two effects. First, if a manager is concerned about the pay of the workers she is socially connected to, she can allocate more of her effort toward them, thus increasing their productivity and their earnings. The effect of managerial effort on worker productivity can be substantial. Assuming that workers pick at a constant speed, if the manager slacks for 5 minutes every hour and a worker is left to wait for a new crate for the same time, his productivity would be 5/60 = 8% lower. Second, a manager might be better informed about the ability or skills of workers she is socially connected to or be able to exert stronger social pressure on them to work hard, both of which generate a difference in the allocation of managerial effort between connected and unconnected workers.12 We now discuss the two features of this work environment that allow us to assess whether social connections shape the managers’ effort allocation choice and, as a consequence, workers’ earnings, and how this depends on the compensation scheme in place for managers. 2.2. Key Feature 1: Natural Variation The production technology is such that the demand and supply of picking labor varies across field-days. On any given field-day the demand for labor depends on the size of the field, on the orders received from supermarkets for the specific variety of fruit grown on that field, and on the number of plants that have reached maturity. This varies over time and declines during the life cycle of the field. The supply of labor depends on the demand for picking labor on other fields, which varies for the same reasons, and the demand for nonpicking tasks such as planting and weeding.13 Due to this natural variation, the number of workers and managers varies within the same field across different days and across different fields within the same day. Importantly for our study, this also implies that the same worker can be supervised by different managers on different field-days. In particular, a worker can be supervised by a manager he is socially connected to on some 11 The disposition of plants in the field is such that it is not practical for workers to retain a stock of empty crates. 12 In principle, a manager could boost connected workers’ productivity by letting them slack on quality, namely by allowing connected workers to leave hard to reach fruits on the plants. In practice, however, this is unlikely to be the case for two reasons. First, damaged or misclassified fruit is identified at the packing stage of the production process. The monitoring system in place then allows senior management to attribute fruit quality to individual managers. Second, a permanent employee of the farm checks that no ripe fruit is left on the plants at the end of each day. 13 The fruit is planted some years in advance, so the quantity of fruit to be picked is given. The order in which fields are picked is decided at the start of the season.

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field-days and by another manager that he is not socially connected to on others. Managers and workers are allocated to fields by a higher tier permanent employee of the farm, who we refer to as the chief operating officer (COO).14 2.3. Key Feature 2: The Experimental Research Design We designed and implemented a field experiment in which we exogenously changed the compensation scheme of the managers and the COO. At the start of the 2003 season, the managers and the COO were paid a fixed wage. Midway through the 2003 season, we added a performance bonus to the same level of fixed wages. The experiment left the compensation scheme of the workers unchanged—workers were paid piece rates throughout the 2003 season.15 The bonus payment was awarded on field f and day t if the workers’ average productivity on the field-day, Y f t , exceeded an exogenously fixed threshold Y ∗ . Conditional on reaching the threshold, the total monetary value of the bonus payment available to the managers, B(Y f t ), increases at an increasing rate in the average field-day productivity. The personnel software does not allow the exact match between workers and managers within the field-day to be recorded, but it does record the identity of all the managers and all the workers on the field-day. Each manager then obtains an equal share of the bonus payment generated on the field-day. If there are Mf t managers present, each obtains a payment of (1/Mf t )B(Y f t ).16 14 Section 4 makes precise the underlying assumptions that allow us to exploit this source of variation to identify the effects of social connections in the workplace. In the Appendix we present evidence in support of these identifying assumptions. 15 The change was announced to the COO and managers a week in advance of the actual change. During this week, we spent time going through numerical examples with management to make sure they understood how the performance bonus would be calculated. Workers were not informed of the change in managerial compensation, but given that managers and workers live on the farm, they are likely to have understood the change over time. 16 The bonus payment schedule is piecewise linear: ⎧ 0 if Y ∗ > Y f t , ⎪ ⎪ ⎨ a1 + b1 Y f t if Y ∗ + c1 > Y f t ≥ Y ∗ , B(Y f t ) = ⎪ a + b2 Y f t if Y ∗ + c2 > Y f t ≥ Y ∗ + c1 , ⎪ ⎩ 2 a3 + b3 Y f t if Y f t > Y ∗ + c2 ,

where ai , bi , and ci are constants such that a3 < a2 < a1 , b3 > b2 > b1 , c2 > c1 . This reflects the fact that the marginal cost of supplying managerial effort is increasing. The parameters ai , bi , and ci are set such that B(Y f t ) is a continuous and convex function. The values of ai , bi , ci , and Y ∗ cannot be provided due to confidentiality. Finally, we note that since the bonus payment is shared by all managers on a field, free-riding might reduce the strength of incentives. Three features of this context make free-riding unlikely: (i) there are between two and four managers on most fielddays; (ii) managers can monitor each other on the field; (iii) managers interact repeatedly both on and off the field.


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The daily bonus payment that accrues to the COO for any given field is 1.5 times that which accrues to a manager on the field. Moreover, since the COO is responsible for every field operated on a given day, his bonus equals the sum across all fields operated that day and is therefore equal to 15 f (1/Mf t )B(Y f t ). The introduction of the bonus might affect managers’ behavior through three channels: (i) because they now have a stake in the firm’s productivity, (ii) because the COO might exert more pressure to maximize his bonus payments, and (iii) because of increased competition for managerial jobs. Indeed, given that the quantity of fruit to be picked is constant, if the introduction of the bonus increases productivity, the demand for picking and managerial labor might fall as fewer workers are needed to pick the same amount. All three channels lead managers to take actions that increase the firm’s productivity. Finally, to avoid multitasking concerns (Holmstrom and Milgrom (1991)), the performance bonus was not awarded if the quality of fruit picking declined.17 The fraction of field-days on which the bonus was earned varies from 20 to 50% across managers. The ex post monetary value of the performance bonus to managers is substantial. Averaged across all field-days worked under the bonus, managerial hourly earnings increased by 7%. Conditional on obtaining the bonus, managerial hourly earnings increased by 25%. The true expected hourly earnings increase to managers of the performance bonus lies between these bounds.1819 To identify whether managers allocate more effort to workers they are socially connected to, we compare the productivity of the same worker on fielddays in which he is socially connected to his manager and his productivity on field-days in which he is not socially connected to his manager. We exploit the exogenous variation in managerial incentives our research design provides to identify whether the effects of social connections depend on the managerial in17

Quality is defined along two dimensions: (i) the quantity of damaged fruit; (ii) classification as either suitable for market or supermarket, largely based on the size of each fruit. If the percentage of damaged or misclassified fruit rose by more than 2% from a preestablished norm, then the bonus was not awarded. 18 Given that (i) managers are from Eastern Europe, (ii) their base pay is 20% higher than the U.K. minimum wage, and (iii) most individuals save earnings to spend later in their home country, these increases in hourly earnings translate into large increases in real income. As of January 2003, gross monthly earnings at the U.K. minimum wage (€1105) are 5 times as high as at the minimum wage in Poland (€201), where 40% of managers come from, and almost 20 times higher than in Bulgaria (€56), where 30% of managers come from. 19 The managers were unaware they were taking part in an experiment and that the data would be used for scientific research. As such, our experiment is a natural field experiment according to the taxonomy of Harrison and List (2004). The managers were, however, aware that productivity data were recorded and kept by the farm owner, and that the data would be analyzed to improve the firms’ overall efficiency.

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centive scheme in place. The comparison allows us to establish whether social connections are beneficial or detrimental to the firm’s overall performance as explained in the next section. 3. THEORETICAL FRAMEWORK We present a simple theoretical framework to illustrate the effect of social connections on the level and allocation of managerial effort across workers, and on firm productivity. The framework illustrates that the existence of social connections weakly increases managerial effort, but also changes the allocation of effort in favor of workers the manager is socially connected to. The net effect on the firm’s aggregate productivity is ambiguous. The framework makes precise how we can sign the allocation effect by exploiting the exogenous change in the strength of managerial incentives. 3.1. Technology and Incentives We assume production requires one manager and two workers in any given field. Workers pick fruit and the manager organizes logistics for each worker. The manager chooses her level of effort and how to allocate it between the workers. To make matters concrete, the managerial effort directed toward a worker can be thought of as the effort devoted to ensure he is allocated a new row of fruit as soon as he is done picking the current one or the effort devoted to ensure he does not have to wait for his crates to be replaced. For simplicity and without loss of generality, we do not model workers’ effort choices. Also, for simplicity we assume the manager’s effort targeted toward worker i affects worker i alone. The output of worker i is then given by yi = θi ki mi , where θi measures his innate ability, mi is the managerial effort targeted toward him, and ki > 0 is a measure of the strength of the complementarity between the manager’s and worker’s efforts. We assume the two workers have ability levels θ and 1, with θ > 1, and index them with subscripts h and l for high and low ability, respectively.20 ¯ > 1) and low (m = 1). The Managerial effort takes two values, high (m = m disutility of effort to the manager, C(m), equals 0 if effort is low and is greater than 0 if effort is high.21 20 The qualitative results are unchanged if we allow workers to also choose effort. The qualitative results are also unchanged if we allow mi to have a positive spillover effect on the output of the nontargeted worker j, as long as the direct effect of mj on the productivity of worker j is sufficiently stronger than the effect of mi on j. 21 The assumption reflects the fact that in our setting the manager’s cost of effort depends on total effort rather than on the identity of the workers targeted. Namely the cost of moving around the field to identify which crates to replace and which workers to reallocate does not depend on the ability of the worker who gets reallocated.


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The productivity of worker i is measured as the kilograms of fruit picked per hour. As all workers in the field work for the same time, we normalize hours to 1 so output and productivity coincide. Total output is i yi as there are no spillovers across workers or complementarities in production. As in our empirical setting, we assume worker i’s pay, pWi , equals his productivity, yi , to reflect the fact that workers are paid piece rates, and hence their earnings are a linear function of their productivity. The manager’s com pensation schedule is pM = f + bY , where f is a fixed wage and Y = i yi is the aggregate output of her subordinates. The parameter b ≥ 0 captures the strength of managerial incentives, namely the variable component of managerial pay which is linearly related to aggregate worker productivity.

3.2. Social Connections Social connections can affect in reduced form both agents’ preferences and the production technology. To capture the first channel, we follow Prendergast and Topel (1996) in assuming the manager’s utility depends on her pay and the pay of her subordinates, that is, (1)

uM = pM +

σi pWi

i∈(hl)

where σi measures the social connection between the manager and worker i. We assume that σi = σ > 0 if worker i is connected to the manager while σi = 0 if he is not. These preferences can be seen to represent managers’ altruism toward their subordinates, but also as the reduced form of a model in which the manager cares about the connected workers’ earnings because she receives kickbacks from them.22 If social connections ameliorate the moral hazard problem between the manager and workers, or if they help foster cooperation or improve communication between managers and workers, they affect workers’ productivity directly. To capture this second channel, we assume the strength of the complementarity between managerial and worker effort depends on their social connections. That is, given worker i’s productivity, yi = θi ki mi , we assume ki = k > 1 if worker i is connected to the manager (σi = σ), while ki = 1 if he is not (σi = 0). 22 We focus on whether managers and workers are socially connected or not, rather than on the strength of the social connection. What matters for the analysis is that managers may be connected to a greater extent to some workers than to others. We also focus on the case in which σ ≥ 0. A negative weight could be interpreted as the manager being spiteful toward the worker.

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3.3. The Manager’s Effort and Allocation Choices The manager chooses (mh ml ), namely how much effort to allocate to the high and low ability worker, to maximize her utility, as given in (1). Substituting for the manager’s and workers’ pay, the manager’s problem is (2)

max (b + σh )θk(σh )mh + (b + σl )k(σl )ml − C(mh + ml )

mh ml

The two propositions below describe the effect of social connections on managerial effort and, hence, on the firm’s productivity, respectively. All proofs are given in the Appendix. PROPOSITION 1: Social connections weakly increase the level of managerial effort and might alter the allocation of effort in favor of the worker the manager is socially connected to. The existence of social connections implies σi > 0 and k(σi ) > 1. Thus social connections raise the marginal benefit of effort both because the manager internalizes the effect of her effort on the connected worker earnings and because the marginal effect of managerial effort on worker’s productivity is higher when the manager and the worker are socially connected. Other things equal, social connections therefore have an unambiguous effect on the level ¯ when she is soof effort, namely a manager is more likely to choose m = m, cially connected to one or both workers. This follows immediately from the fact that since both the production and the cost of effort functions are linear in (mh ml ), and there are no spillovers across workers, the manager’s utility function (2) is linear in (mh ml ). This implies that, regardless of the level of effort chosen, the manager will target only the worker who yields the highest marginal benefit.23 Social connections, however, affect the allocation of managerial effort. Indeed, the marginal benefit of targeting worker i is equal to (b + σi )k(σi ); thus, other things being equal, the manager is more likely to target worker i if σi > 0 and k(σi ) > 0. Therefore, social connections also have an allocation effect on managerial effort. In the Appendix we show that (i) if the manager is connected to the high ability worker, she always targets him; (ii) if the manager is connected to the low ability worker, there exists a set of parameters for which she targets her effort toward him. In this second case, social connections distort the allocation of managerial effort toward low ability workers and might therefore be detrimental to the firm overall. Whether social connections are beneficial or detrimental for the ¯ if and only if {max[θkh (b + σh ) kl (b + σl )]}(m ¯ − 1) > c. The rightThe manager chooses m hand side is increasing in (σh σl ). In a model with continuous managerial effort and convex costs, the effect of social connections on managerial effort would be strictly positive. 23


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firm in this case depends on the sign and relative magnitude of the level and allocation effects as described in the following result. PROPOSITION 2: If the manager is connected to the high ability worker, social connections have an unambiguously positive effect on the firm’s productivity. If the manager is connected only to the low ability worker, the effect of social connections on the firm’s productivity is ambiguous. It is more likely to be negative if the complementarity with the connected worker is low or if the difference in workers’ ability is large. To summarize, the existence of social connections has both a level effect and an allocation effect on managerial effort. As the firm’s productivity is increasing in managerial effort and social connections weakly increase effort, the level effect of social connections is always weakly positive. The sign of the allocation effect is, however, ambiguous. If the manager is connected only to the high ability worker, she targets him and the allocation effect is positive. In the Appendix we provide the precise conditions under which if the manager is connected only to the low ability worker, she targets him and the allocation effect is negative.24 3.4. Testing for a Negative Allocation Effect The framework above makes precise that social connections can be detrimental for the firm only if their existence distorts the allocation of managerial effort in favor of low ability workers who are connected to managers. We now show that an exogenous change in the strength of managerial incentives b can be used to test whether the allocation effect is negative. The test relies on two sources of variation: the variation in the strength of managerial incentives and the variation in social connections. The first stems from the fact that the productivity of the same workers is measured both when managerial incentives are low powered (fixed wage) and when they are high powered (bonus scheme). The second stems from the fact that the productivity of the same workers is observed both on field-days when they are socially connected to managers and on field-days in which they are not while their co-workers are. An increase in b increases both the marginal benefit of effort and the relative weight the manager places on productivity vis-à-vis the utility of the connected workers. Increasing b can thus affect both the level and the allocation of managerial effort. Our empirical test is then based on a revealed preference argument: if the manager changes her effort allocation from the low to the high ability worker when she has a larger stake in the firm’s productivity, namely 24 Note that since manager’s pay is increasing in productivity, social connections affect the wage bill. Thus even if social connections increase the firm’s productivity, they might reduce the firm’s profits.

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when the performance bonus is in place, the allocation of managerial effort across workers under the wage regime could have not been maximizing productivity. The test then consists of measuring the effect of social connections on connected workers of different ability by managerial incentive scheme and testing whether the manager reallocates effort from low to high ability workers as a result of the increase in the strength of her incentives, b. If the allocation effect is indeed detrimental to productivity, the allocation of effort should depend on the managerial incentive scheme as follows. First, when the manager is paid a fixed wage, she should target connected workers regardless of their ability. Hence, the productivity of both low and high ability workers should be higher when they are connected to their manager compared to when they are not. Second, when the manager is paid performance bonuses, she should target high ability workers regardless of their connections. Hence the introduction of the bonus should (i) strictly increase the productivity of high ability workers on field-days in which they are not connected and (ii) strictly decrease the productivity of the low ability workers on field-days in which they are connected. Finding evidence consistent with these predictions would provide support for the hypothesis that social connections have a negative allocation effect on productivity against the joint alternative hypotheses that the allocation effect is nonnegative or that the increase in b is not sufficiently large to change managerial behavior. Finally, we present evidence on the net effect of social connections on productivity by incentive scheme. This allows us to gauge whether any beneficial levels effect of social connections on managerial effort more than offset the distortionary effects they have on the allocation of managerial effort. 4. DATA AND DESCRIPTIVES 4.1. Data Sources Our primary data source is the firm’s personnel records. These contain three types of information. First, they list each worker’s productivity on every fieldday they pick fruit. Productivity is defined as the kilograms of fruit picked per hour and is electronically recorded with little measurement error. Second, while they do not contain information on the exact worker–manager match, the data identify all the workers and managers present on each field-day. On most field-days there are between 40 and 80 workers, and between 2 and 4 managers, so we are able to build a measure of the probability that a given worker–manager pair is matched. Finally, the personnel records contain information on each individual’s nationality, date of arrival, and accommodation location on the farm, which we use to measure social connections as described below. Throughout, we analyze data on the main fruit type and focus on the main farm site during the peak picking season from May 1st until August 31st 2003.


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As part of our experimental design, the change in managerial incentives occurred midway through the peak season—June 27th—so there are 43 days in the prebonus period and 51 days postbonus. To ensure that changes in field composition do not drive the results, we focus on fields that were picked at least one week either side of the change in managerial incentives. Note that a given field is not picked on every day and more than one field is picked on any given day. To ensure our estimates are not contaminated by changes in the composition of the workforce over the season, we restrict the sample to individuals who work at least one week either side of the change in managerial incentives. The final sample then contains 10,148 worker-field-day productivity observations from 241 field-days. This covers 144 workers, 10 managers, 13 fields, and 94 days.25 4.2. Measuring Social Connections We measure social connections between managers and workers along three dimensions: nationality, time of arrival on the farm, and the location on the farm where individuals reside during the season. The first measure defines a worker and manager to be connected if they are of the same nationality, based on the assumption that people are more likely to befriend others who come from the same country and share the same mother tongue. As individuals are hired from eight Eastern European countries, we observe considerable variation along this dimension.26 The second measure of social connections is based on the time that individuals arrive on the farm. This varies across individuals for reasons that are exogenous to the worker’s performance, such as their university term dates in their home countries and the date on which their work permit is issued. On arrival, individuals are consecutively assigned a worker number and then attend an induction program with others who have arrived at a similar time. Hence the first group of people that each individual is exposed to, and may form social ties with, are those who arrive on a similar date. If two individuals have a worker number within the same 10 digit window, we define the two to be socially connected through their arrival cohort. The third measure of social connections is based on the geographic location where individuals live during their stay on the farm. Each worker lives in a caravan with up to five others, and each caravan is assigned a unique number. On the main farm site, caravans are arranged around a communal space and 25 Fields are located on two sites on the farm, of which we only use the largest for the analysis as fruit in the smaller site began to ripen only after the introduction of the managerial performance bonus scheme. 26 Among workers, the most common nationalities are Polish (35%), followed by Ukrainian (29%) and Bulgarian (10%). Among managers, 40% are Polish, 30% are Bulgarian, and the others are Lithuanian.

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numbered consecutively from 1 to 46. We define two individuals to be socially connected through their living site if they live within five caravan numbers of each other. The underlying assumption is that individuals are more likely to form social ties with their neighbors.27 While we do not have direct information on the social relations between managers and workers, we can provide evidence that the three measures of similarity—nationality, arrival cohort, and neighborhood—are predictors of friendship in this setting. In 2004, that is one year after the season we analyze here, we administered a worker survey to workers in the same farm to collect information about friendship links. Using those data, in Bandiera, Barankay, and Rasul (2009) we found that the odds of a worker j to be named as a friend by another worker i if they are of the same nationality is 14.7 times larger than the odds of worker j being named by i if they are of different nationalities. The corresponding figures for arrival cohort and geographical neighborhood are 14.3 and 9.7. These odds are all significantly different from 1, and the results are robust to conditioning on a host of other controls for the similarity in observables between workers. Based on these three criteria of similarity, most workers are connected to at least one manager along at least one dimension. Of the 10,148 worker-fieldday observations in our sample, 8884 correspond to workers who are socially connected to managers. We therefore identify the causal effect of social connections on worker performance from the observed within worker variation in productivity. In other words, instead of comparing the productivity of workers who are connected to the productivity of workers who are not, we identify the effect of social connections by comparing the productivity of the same worker on field-days in which he is connected to his manager, to his productivity on field-days in which he is not connected. Since workers who are never connected to any manager do not contribute to these estimates, we restrict the sample to the 8884 worker-field-day observations of workers who are socially connected to at least one manager on the farm.28 To measure whether a worker is connected to his manager on any given fieldday we first define cij = 1 if worker i and manager j are connected along any dimension and = 0 otherwise. Second, we note that while each worker is assigned to only one manager, we do not know the exact match of workers to managers within the field. On most field-days there are between 2 and 4 managers and between 40 and 80 workers present. Given Mf t managers present on the field-day, we can compute the probability that worker i is connected to his manager as the share of managers worker i is connected to on the field-day, 27 There are no opportunities for workers themselves to choose their caravan or worker numbers. 28 Unconnected workers are, however, not significantly different from connected workers on observables such as age, gender, and previous work experience.

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Cif t = ( j cij )/Mf t , where the summation in the numerator is over all managers j on field-day f t.29 4.3. Descriptives Table I reports descriptive statistics for our variable of interest Cif t , the share of managers on field-day f t that are socially connected to worker i. The first row shows that, on average, the share is .425 when managers are paid fixed wages and .412 when managers are paid performance bonuses. The fact that the shares are almost identical under the two compensation schemes suggests that the process by which managers and workers are allocated to fields is orthogonal to the compensation scheme in place. The empirical analysis exploits the variation in social connections within a worker over time. Table I shows that, reassuringly, at least one-third of the overall variation in social connections arises from variation within a worker TABLE I DESCRIPTIVES ON THE SOCIAL CONNECTIVITY BETWEEN WORKERS AND MANAGERS BY MANAGERIAL INCENTIVE SCHEMEa Managerial Incentive Scheme Share of Managers

Fixed Wages

Performance Bonus

Connected to i (Cif t )

.425 (.297) [.196]

.412 (.300) [.154]

Who are the same nationality as i

.304 (.344) [.145]

.285 (.319) [.111]

Who are in the same living area as i

.139 (.116) [.167]

.138 (.172) [.138]

Who are from the same arrival cohort as i

.038 (.076) [.079]

.056 (.101) [.074]

a All variables are defined at the worker-field-day level. Means, standard deviation between workers is in parentheses, and standard deviation within worker is in brackets. A manager and worker are defined to be resident in the same living area if they live within five caravans from each other on the farm. A manager and worker are defined to be in the same arrival cohort if they have identification numbers within the same 10 digit window. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. The sample is restricted to the 129 workers who work for at least one week under both incentive schemes and are connected to at least one manager on at least one dimension. On average, each worker is observed picking on 41 field-days when managers are paid fixed wages and 30 field-days when managers are paid a performance bonus. Overall there are 5137 worker-field-day observations when managers are paid fixed wages, and 3747 worker-field-day observations when managers are paid a performance bonus.

29 The median number of managers and workers is 3 and 59, respectively. Field-days with less than 4 managers account for 83% of the sample.

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O. BANDIERA, I. BARANKAY, AND I. RASUL TABLE II WORKER PRODUCTIVITY (KG/HR), BY SOCIAL CONNECTIVITY TO MANAGERS AND MANAGERIAL INCENTIVE SCHEMEa Managerial Incentive Scheme:

Panel A: All Workers Unconnected on field-day (DCif t = 0) Connected on field-day (DCif t = 1) Difference Panel B: Low Ability Workers Unconnected on field-day (DCif t = 0) Connected on field-day (DCif t = 1) Difference Panel C: High Ability Workers Unconnected on field-day (DCif t = 0) Connected on field-day (DCif t = 1) Difference

Fixed Wages

Performance Bonus

Difference

721 (211) 898 (345)

952 (600) 970 (527)

230*** (535) 712 (272)

177*** (352)

179 (750)

1596*** (609)

610 (248) 737 (173)

660 (342) 677 (212)

506 (423) −603** (211)

127*** (287)

161 (368)

111** (435)

776 (259) 1032 (519) 256*** (518)

1079 (672) 1161 (668) 815 (899)

303*** (609) 128*** (338) 175** (719)

a Means, standard errors are given in parentheses. *** denotes significance at 1%, ** at 5%, and * at 10%. All variables are defined at the worker-field-day level. The standard errors on the differences and difference-in-difference are estimated from running the corresponding least squares regression, allowing the standard errors to be clustered by worker. Productivity is measured as the number of kilograms of fruit picked per hour by the worker on the field-day. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. A worker is defined to be unconnected on the field-day if she is not socially connected to any of her managers that field-day. A worker is defined to be connected on the field-day if she is socially connected to at least one of her managers. Low (high) ability workers are those whose average productivity under the bonus is below (above) the median average productivity.

over field-days. This is true under both managerial incentive schemes, and along each dimension that defines social connections.30 Throughout we analyze the effect of social connections on worker productivity because, in our setting, productivity can be directly affected by managers’ behavior and it determines workers’ earnings given that they are paid piece rates. Table II presents descriptive evidence on productivity by connection status, managerial incentive scheme, and workers’ ability. For ease of exposition, 30 For the variance decomposition to sum to the total variance in an unbalanced panel, it is necessary to weight the between component by the number of workers on the field-day.


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we employ a discrete measure of social connections, DCif t , which is equal to 1 if worker i is connected to at least one manager on field-day f t and equal to 0 otherwise. To analyze whether the effect of connections differs by workers’ ability, we rank workers according their average productivity when managers are paid bonuses and use the median to split them into two ability groups.31 Panel A of Table II pools all workers and illustrates that when managers are paid fixed wages, worker productivity is 7.21 kg/hr when workers are unconnected and rises significantly to 8.98 kg/hr when workers are managed by individuals they are socially connected to. In contrast, when managers are paid bonuses, the average worker’s productivity is no different on field-days when he is socially connected to field-days when he is socially unconnected. The unconditional difference-in-difference in workers’ productivity by their social connections to managers and across managerial incentive scheme is 1.59 kg/hr, and is significantly different from zero. As workers are paid piece rates, differences in worker productivity by social connectivity to managers and managerial incentive scheme translate into similar differences in worker earnings. This is quantitatively important both in percentage terms and in absolute terms when aggregated over the season.32 As highlighted by the theoretical framework in Section 3, the evidence in panel A is consistent with two interpretations of managers’ behavior following the introduction of the bonus: either managers exert more effort and target all workers regardless of their connection status, or managers exert more effort and reallocate it from connected workers toward high ability workers to maximize average productivity. To distinguish between these interpretations, panels B and C provide evidence on the effect of social connections on workers who are below and above the median level of ability, respectively. Three points are of note. First, when managers are paid fixed wages, the first data column in panels B and C shows that social connections increase worker productivity for both groups, suggesting that managers target connected workers regardless of their ability level. Second, panel B shows that the introduction of the bonus does not affect the productivity of low ability workers in days in which they are not connected, 31 The theoretical framework makes clear that the probability of social connections affecting the managers’ allocation decisions is decreasing in the strength of managerial incentives. Thus if social connections affect productivity only when managers are paid fixed wages, the productivity under the bonus better reflects a worker’s true ability. It is important to note that there is little churning of workers in this ranking—the rank correlation between workers’ average productivity when managers are paid wages and when managers are paid bonuses is .69. 32 The average worker picks on two to three fields per day and stays on the farm for 100 days. A back of the envelope calculation suggests that over the course of a season, a worker would earn £500 more if managers were always paid a fixed wage and workers were always managed by individuals they are socially connected to. Given that workers in our sample live in Eastern Europe and much of their earnings are saved to spend in their home country, the real value of these differences is substantial.

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while it reduces their productivity by around 8% on connected field-days. This is consistent with the view that managers target low ability connected workers when paid fixed wages, but stop engaging in such behavior when paid bonuses. Consequently, the productivity of low ability workers when managed by those they are connected to significantly falls as managers interests become more aligned with those of the firm. Third, panel C shows the introduction of managerial performance bonuses increases the productivity of high ability workers both on field-days in which they are unconnected and on field-days in which they are connected. The effect on unconnected field-days is more than twice as large as the effect on connected field-days, and the difference-in-difference is positive and precisely estimated. This indicates that the introduction of the bonus increases the productivity of high ability workers because managers’ efforts are both higher and more likely to be targeted toward them. Overall, the evidence in Table II indicates that managers target connected workers irrespective of their ability when paid fixed wages, whereas they reallocate their efforts in favor of high ability workers when paid performance bonuses irrespective of whether they are socially connected to them or not. This suggests that social connections distort the allocation of managerial effort across workers and that this effect is detrimental to the firm’s productivity as managers stop targeting connected workers when their interests become more closely aligned with the firms’. In the remainder of the paper, we present formal evidence to shed light on whether these descriptive results are robust to controlling for other determinants of productivity. In doing so, we make precise the underlying identifying assumptions required to interpret this evidence as causal and present evidence in support of these identifying assumptions. 5. SOCIAL CONNECTIONS AND WORKER PRODUCTIVITY 5.1. Methodology The empirical analysis proceeds in two stages. First, we estimate the effect of social connections on the average worker by managerial incentive scheme. Next, we allow the effect of social connections and managerial incentives to differ across workers. To identify whether social connections affect worker’s productivity and how this depends on the managerial compensation scheme in place, we estimate the panel data regression (3)

yif t = αi + λf + γ0 (1 − Bt ) × Cif t + γ1 (Bt × Cif t ) + ρBt + τdk (Bt × Dkid ) + μs Ssf t k

d∈Nk

s∈Mf t

+ δXif t + ηZf t + ϕt + uif t


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where yif t is worker i’s log productivity on field f and day t. The worker fixed effects αi account for permanent productivity differences across workers, such as those arising from innate ability or motivation, and the field fixed effects λf capture permanent productivity differences across fields, such as those arising from soil quality.33 Cif t is the log of the share of managers worker i is socially connected to on the field-day. Bt is a dummy variable equal to 1 after the performance bonus is introduced (June 27th) and equal to 0 otherwise. The parameters of interest throughout are γ0 , which measures the effect of social connections when managers are paid a fixed wage, and γ1 , which measures the effect of social connections when managers are paid performance bonuses. The null hypothesis is that social connections do not affect productivity, so γ0 = γ1 = 0. Since connectivity is defined along the lines of nationality, living site, and arrival cohort, γ0 and γ1 might be biased if, for example, the introduction of the bonus has different effects on workers of different nationalities. This is because the connection measure Cif t would then also be picking up any differential effect of the performance bonus by worker nationality. Obviously, similar concerns arise if workers are differentially affected on the basis of their living site or time of arrival on the farm once managerial performance bonuses are introduced. To address these concerns, we control for a set of interactions between the performance bonus dummy Bt and the complete set of nationality, arrival cohort, and living site dummies. To do so we define a dummy variable Dkid = 1 if worker i is of type d along dimension k and = 0 otherwise, and Nk denotes the total number of types along dimension k. For example, when k is nationality, Dkid = 1 when the worker is of nationality d, and Nk = 8 as this is the number of different nationalities in our data. These interactions flexibly control for any heterogenous effect on workers of the change in managerial incentives along these dimensions. Hence we estimate the effect of the within worker variation in social connectivity conditional on any heterogeneous effects between workers that may arise as managers respond to the introduction of performance bonuses along other margins apart from those arising from social connections with their subordinates. Ssf t is a dummy equal to 1 if manager s works on field f on day t and equal to 0otherwise, and Mf t is the set of managers who work on the field-day. Hence s∈Mf t μs Ssf t in (3) corresponds to a full set of manager dummies. These control for time invariant traits of each manager, such as their ability to motivate workers and their management style, that affect the performance of managed workers. These allow us to address the concern that there are unobservable 33

If this specification is estimated only with worker fixed effects, they explain 25% of the variation in worker productivity, suggesting there is considerable heterogeneity across workers. Estimating the specification conditional on only field fixed effects explains 11% of the overall variation. Estimating the specification conditional only on manager fixed effects explains 3.5% of the overall variation.

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managers’ characteristics that drive both their social connections and the performance of their subordinates. Xif t is the worker’s picking experience, defined as the cumulative number of field-days they have picked fruit on the farm. Zf t captures time-varying field characteristics. This includes the field’s life cycle, defined as the nth day the field is picked divided by the total number of days the field is picked over the season. This captures the natural within field trend in productivity as fields deplete over time. We also include a time trend t to capture learning by farm management and any aggregate trends in productivity.34 We also note that the social connections between a worker and his managers are unlikely to be identically and independently distributed within a worker over field-days. We therefore adopt a conservative strategy in estimating standard errors and allow the disturbance terms uif t to be clustered by worker throughout.35 The parameters of interest (γ0 γ1 ) identify the causal effect of social connections on worker productivity under each managerial incentive scheme by comparing the productivity of a given worker on field-days when he is socially connected to his manager to his productivity on field-days when he is unconnected. The validity of the identification strategy and the causal interpretation given to the results relies on two key assumptions. The first is that unobserved determinants of workers’ allocation to managers are orthogonal to the managerial incentive scheme in place. The second is that any effect of social connections on individual productivity that is unrelated to the managerial incentive scheme in place remains unchanged over time. We provide detailed evidence in support of both of these identifying assumptions in Section 5.3. 5.2. Baseline Results Table III presents estimates of our baseline specification (3). In column 1 we measure social connections with the dummy variable DCif t that equals 1 if worker i is connected to any of the managers in field-day f t and equals to 0 otherwise. This is the variable used for the previous descriptive evidence in Table II. The results show that the pattern of unconditional differences in worker productivity by social connections and managerial incentive scheme is robust to conditioning on a rich set of determinants of worker productivity. In particular, column 1 shows that when managers are paid a fixed wage, the average worker has significantly higher productivity on field-days on which he is 34 As fields are operated on at different parts of the season and not all workers pick each day, the effects of the field life cycle and workers’ picking experience can be separately identified from that of the time trend. The average field life cycle is not significantly different under the two managerial compensation schemes. 35 Clustering the disturbance terms uif t by field-day—say because workers on the same fieldday face common productivity shocks—leads to the standard errors on the parameters of interest, γ0 and γ1 , being considerably smaller than those we report.

TABLE III SOCIAL CONNECTIONS AND MANAGERIAL INCENTIVESa

1. Any Managers Connected To

049** (019) 016 (032)

Difference-in-difference estimate

033 (034)

Interactions of nationality × performance bonus dummy Interactions of living site × performance bonus dummy Interactions of arrival cohort × performance bonus dummy Interactions of worker fixed effect × performance bonus dummy Field-date fixed effects Adjusted R-squared Number of observations (worker-field-day)

3. Heterogeneous Effects of the Bonus on Workers

4. Field-Date Fixed Effects

158*** (040) −083 (088)

143*** (040) −079 (088)

106** (045) −060 (061)

241*** (095)

222** (096)

167** (075)

Yes [.147] Yes [.000] Yes [.000] No No

Yes [.042] Yes [.000] Yes [.000] No No

No No No Yes [.000] No

4355 8884

4361 8884

4479 8884

No No No Yes [.000] Yes 5817 8884

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a Dependent variable = log of worker’s productivity (kilograms picked per hour on the field-day). *** denotes significance at 1%, ** at 5%, and * at 10%. In columns 1 to 3 the standard errors allow for clustering at the worker level. In column 4 standard errors are clustered at the field-date level. All specifications control for worker, field, and manager fixed effects. The other controls included in specifications 1 to 3 include the managerial performance bonus dummy, the worker’s picking experience, the field life cycle, a time trend, and interactions between the performance bonus dummy and the worker’s nationality, arrival cohort, and living site. The field life cycle is defined as the nth day the field is picked divided by the total number of days the field is picked over the season. In column 4 these interactions are replaced by interactions of the worker fixed effect and the performance bonus dummy, and a series of field-date fixed effects and hence the field life cycle and time trend are dropped from this specification. All continuous variables are in logarithms. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. All sample workers are connected to at least one manager on at least one field-day and work at least one week under each incentive scheme. In column 1 a worker is defined to be unconnected on the field-day if she is not socially connected to any of her managers that field-day, and the worker is defined to be connected on the field-day if she is socially connected to at least one of her managers. The difference-in-difference estimate is the difference in the effect of social connections on worker productivity by managerial incentive scheme. At the foot of each column we report the p-value on the F -test on the joint significance of the interaction terms with the performance bonus dummy.


Any managers connected to i, fixed wages for managers (DCif t ) Any managers connected to i, performance bonus for managers (DCif t ) Share of managers connected to i, fixed wages for managers (Cif t ) Share of managers connected to i, performance bonus for managers (Cif t )

2. Share of Managers Connected To

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socially connected to his managers ( γ0 > 0). When managers are paid performance bonuses, there is no effect on the average worker’s productivity of being socially connected to her managers on the field-day ( γ1 = 0). The magnitude of γ0 implies that when managers are paid a fixed wage, being connected to at least one manager on the field, increases productivity by 5% for the average worker, whereas there is no such effect when managers are paid performance bonuses, although the difference in the effects is not statistically significant. In column 2 we measure social connections by the share of managers on field-day f t that are connected to worker i by either nationality, living site, or arrival cohort. Compared to the dummy variable DCif t , this is a more precise measure as it distinguishes between field-days in which a worker is more likely to be connected to his manager. The pattern of coefficients is the same as in column 1 but the implied magnitude of the effect is larger. Evaluating at the mean, the magnitude of γ0 implies that when managers are paid a fixed wage, the productivity of a worker on field-days when he is socially connected to all the managers on the field relative to his productivity on field-days when he is socially unconnected to managers will be .642 kg/hr higher, other things being equal. Relative to a baseline average worker productivity of 7.21 kg/hr when managers are paid fixed wages and workers are not connected, this represents a 9% increase of productivity on connected days. Since workers are paid piece rates based on productivity, earnings increase by the same percentage.3637 Taken together, this pattern of results suggests the effect of social connections in the workplace is for managers to favor workers they are connected to when their incentives are low powered. At the foot of columns 1 and 2 we report the implied difference-in-difference estimate, ( γ0 − γ1 ). In line with the descriptive evidence, this is positive in both cases and significantly different 36 The difference between the estimated γ0 parameters in columns 1 and 2 lends support to the idea that managers and workers do not choose who they work with, even within a field-day. Namely, if managers favor socially connected workers and workers could sort across managers within the field, workers should assign themselves to a manager they are socially connected to if such a manager were present. In that case, however, the effect of being connected to one manager should be no different than being connected to two or more. The fact that the implied effect of being connected to all managers (from column 2) is almost double the effect of being connected to at least one (from column 1) indicates that workers cannot assign themselves to a manager who they are connected to. 37 While these baseline results focus on the effects of social connections on worker productivity, we also explored whether the strength of social ties between a worker and his managers affect worker productivity. We can define the strength of the social tie as the number of dimensions along which the two are connected, k DCifk t . We find that a worker’s productivity is monotonically increasing in the number of dimensions along which he is connected to his managers when his managers are paid a fixed wage and that there is no such effect under performance bonuses. However, these results should be interpreted with caution because, given that each dimension of connectivity is orthogonal to the others, there are only 5% of observations from which the effects of being connected along strictly more than one dimension can be identified.


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from zero at the 1% significance level when using the continuous measure of social connections. The pattern of coefficients helps rule out three alternative hypotheses of why social connections may matter in this workplace. First, if workers were always assigned to socially connected managers when productivity on the field is exogenously higher, connections should have the same positive effect unγ1 > 0. Second, when they are on the field-day der both schemes, that is, γ0 = with managers they are socially connected to, if workers prefer to socialize with their managers, connections should have the same negative effect under both γ1 < 0. Third, the pattern of coefficients allows us to schemes, that is, γ0 = rule out the hypothesis that the effect of social connections is driven by workers’ rather than managers’ behavior. Indeed, if workers were to internalize the effect of their effort on their manager’s pay when socially connected to her, we would observe workers exerting more effort when this actually affects the manager’s pay, namely when the manager is paid the performance bonus, that γ1 . is γ0 = 0 < A concern with these results is that the difference-in-difference estimate of social connections might be picking up heterogeneous effects of the managerial bonus scheme across workers that are unrelated to workers’ social connections. To account for this, we introduce a complete set of interactions between each worker’s fixed effect and the performance bonus dummy. This flexibly captures any differential effects across workers of the change in managerial incentives. The result, reported in column 3, shows that the magnitude and significance of the parameters of interest are similar to those in the baseline estimates.38 Finally, we address the concern that there may exist field-day factors that create a spurious correlation between social connections and productivity. For example, managers might lobby the COO to be allocated workers they are connected to on field-days when productivity is exogenously higher. Also, the effect of social connections might depend on the field-day piece rate, which is on average lower when managers are paid performance bonuses as shown in Bandiera, Barankay, and Rasul (2007). To address this concern, the final specification includes field-day fixed effects. The effects of social connections Cif t under each managerial incentive scheme are then identified off the variation across workers in the same field-day in the level of their social connections in deviation from the workers’ average level of social connections under each managerial compensation scheme. The result in column 4 shows the previous results to be robust to conditioning on factors that vary across field-days, such as managers lobbying for workers, field conditions, the hours worked on the field-day, or the level of the piece rate for workers.39 38 The results are robust to controlling for a complete set of worker-field dummies that allow the productivity of each worker to differ across fields. 39 To ensure the estimates do not capture the effect of the composition of the workforce changing over the season, throughout we restrict the sample to workers who work at least one week

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5.3. Evidence in Support of the Identifying Assumptions We have identified the causal effect of social connections on worker productivity under each managerial incentive scheme by comparing the productivity of a given worker on field-days when he is socially connected to his manager to his productivity on field-days when he is unconnected. The validity of the identification strategy and the causal interpretation given to the results relies on two assumptions. The first is that unobserved determinants of workers’ allocation to managers are orthogonal to the managerial incentive scheme in place. As discussed in Section 2.2, the within worker variation in social connections is exogenous to the behavior of workers and managers because the allocation of individuals to fields is determined by the COO, based on the demand for labor for picking and nonpicking tasks across fields. Nevertheless, workers’ allocation to managers might still depend on factors that affect performance and are observable to the COO. Alternatively, workers and managers might engage in behaviors to influence their assignment to each other. To provide support for this identifying assumption, the Appendix presents evidence that the allocation rules do not change with the change in managerial incentives. We show that (i) compared to workers who are not connected to any manager, connected workers are equally likely to be selected to pick or to be selected to work on any task as opposed to stay unemployed on a given day, regardless of the incentive scheme in place; (ii) field-day and worker-field-day specific determinants of productivity do not predict the level of social connections Cif t differentially under the two managerial incentive schemes; (iii) all managers are equally likely to be assigned connected workers under both incentive schemes; (iv) we exploit the fact that some dimensions of connectivity, such as nationality, are more easily observable to the COO than others, such as time of arrival. If the COO systematically assigns workers to managers on the basis of their connections, we should find the effect of social connections to be mostly driven by dimensions that are easier to observe. We find no evidence to support this assertion. The second underlying identifying assumption is that the effect of social connections on individual productivity does not change over time for reasons other than the change in managerial incentives. This ensures that there are no timevarying unobservables that (i) are correlated with the introduction of the bonus and (ii) determine the effect of social connections on productivity. The Appendix shows two pieces of evidence in support of this assumption. First we show that the effect of social connections does not depend on the time of the season, the field life cycle or the workers’ tenure. Rather, the effect either side of the change in managerial incentives. The results are robust to less conservative sample definitions, namely to including all workers on payroll or only workers who have worked at least one day either side of the change in managerial incentives.


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of social connections changes discontinuously when the bonus is introduced.40 Second, we analyze the effect of a placebo bonus on productivity in a different season (2004) and in different fields within the 2003 season, neither of which were subject to the introduction of managerial performance pay. Reassuringly, the effect of connections on productivity does not change with the introduction of the placebo bonus, suggesting the effect of social connections does not change spuriously at a specific point in time or in the field life cycle that happens to coincide with the introduction of the performance bonus in 2003. 6. THE EFFECT OF SOCIAL CONNECTIONS ON THE FIRM’S PERFORMANCE We have presented evidence that the average worker benefits from being connected to his manager—in terms of his productivity and hence earnings— only when managers are subject to low powered incentives. In contrast, there are no such benefits to being socially connected to managers when they are paid performance bonuses. As highlighted by the theoretical framework in Section 3, two interpretations on managerial behavior are consistent with the findings. First it might be that when managers are paid wages, they only devote effort to connected workers, whereas when they are paid bonuses they increase the overall level of effort and devote some to every worker. Second, it might be that when managers are paid wages, they only devote effort to connected workers, whereas when they are paid bonuses they reallocate their, possibly higher, effort toward high ability workers irrespective of whether they are socially connected to them or not. Distinguishing between these interpretations is key to assess the effect of social connections on the firm’s overall productivity. If the data support the first interpretation, the implication is that social connections do not distort the allocation of managerial effort and have a positive effect on firm’s productivity because they increase managerial effort. To the contrary, if the data support the second interpretation, the interpretation is that social connections lead to a misallocation of managerial effort that decreases the firm’s productivity. The theoretical framework makes clear that such an allocation effect might reduce the firm’s productivity when the manager is connected to low ability workers and targets them instead of the high ability workers. To assess whether this is the case, we proceed in two stages. In Section 6.1 we assess whether the allocation effect is negative, namely we estimate the effect of social connections on connected workers of different ability by managerial incentive scheme and we test whether the manager reallocates effort from low to high ability workers 40 A related concern is that the overall effect of social connections through time might mask heterogeneous effects for the three different components of the connection measure. Further specifications, not reported for reasons of space, reveal that the effect of each component does not vary with time and changes discontinuously when bonuses are introduced.

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as a result of the increase in the strength of her incentives. In Section 6.2 we present evidence on the overall effect of social connections on the firm’s performance, so as to assess whether the positive incentive effect prevails over the (possibly) negative allocation effect. 6.1. The Allocation Effect 6.1.1. Quantile Regression Estimates To explore whether the effects of social connections are heterogeneous across workers, we use quantile regression methods to estimate the conditional distribution of the log of productivity of worker i on field f on day t, yif t , at different quantiles, θ. We therefore estimate the specification (4)

Quantθ (yif t |·) = αθ Bt + βθ Cif t + γθ (Bt × Cif t ) + λθf + μθs Ssf t + δθ Xif t + ηθ Zf t + ϕθ t s∈Mf t

where all variables are as previously defined and bootstrapped standard errors based on 1000 replications are calculated throughout. The effect of the managerial performance bonus on unconnected field-days at the θth conditional quantile of log worker productivity is measured by αθ . The corresponding effect on connected field-days is given by αθ + γθ Cif t . Since the connection variable Cif t is continuous, we define worker i to be connected on field-day f t when the share of managers he is connected to is higher than a given threshold, C¯ if t , and we experiment with alternative values of the threshold. The effect of social connections when managers are paid fixed wages and bonuses is captured by βθ and βθ + γθ , respectively. The estimates of αθ and (αθ + γθ Cif t ) at different conditional quantiles of worker productivity allow us to distinguish between the two interpretations given above. Indeed, if social connections do not distort the allocation of effort but rather the manager targets connected workers before the bonus and exerts additional effort to target all workers after the bonus, we should observe the effect of the bonus to be nonnegative and stronger on field-days when the worker is not connected, that is, αθ > αθ + γθ C¯ if t ≥ 0 for all θ. In contrast, if social connections distort the allocation of effort and this is detrimental to the firm’s productivity, we should observe that managers reallocate effort from low ability workers when connected to high ability workers regardless of their connection status. This implies (i) the introduction of the bonus strictly decreases the productivity of workers in the left tail of the productivity distribution on field-days in which they are connected and has no effect when they are not connected, namely αθ + γθ C¯ if t < 0 and αθ = 0 for low θ; (ii) the introduction of the bonus strictly increases the productivity of workers in the right tail of the productivity distribution on field-days in which they

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SOCIAL CONNECTIONS AND INCENTIVES TABLE IV QUANTILE REGRESSION ESTIMATESa Quantile of Worker Productivity 10th

25th

50th

75th

90th

Performance bonus for managers (αθ ) Share of managers connected to i (βθ ) Performance bonus for managers × share of managers connected to i (γθ )

−093 (078) 013 (067) −228** (104)

001 (037) 074** (032) −175*** (050)

100** (034) 144*** (030) −179*** (047)

214*** (040) 272*** (036) −168*** (057)

.312*** (.041) .392*** (.037) −.233*** (.060)

Implied effect of performance bonus for managers on connected field-days (αθ + γθ Cif t ) (evaluated at mean of Cif t )

−166** (069)

−055* (033)

042 (031)

160*** (035)

.237*** (.035)

8884

8884

Number of observations (worker-field-day)

8884

8884

8884

a Dependent variable = log of worker’s productivity (kilograms picked per hour on the field-day). *** denotes significance at 1%, ** at 5%, and * at 10%. Standard errors are reported in parentheses. All specifications control for field and manager fixed effects. The other controls included in each specification are the worker’s picking experience, the field life cycle, and a time trend. The field life cycle is defined as the nth day the field is picked divided by the total number of days the field is picked over the season. All continuous variables are in logarithms. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. The implied effect of the bonus on connected days is computed assuming that the share of managers the worker is connected to is at the sample mean.

are not connected and has a weakly positive effect when they are connected, namely αθ > αθ + γθ C¯ if t ≥ 0 for high θ. Table IV reports the estimates of αθ , βθ , γθ , and αθ + γθ C¯ if t from specification (4) at various quantiles, and is where C¯ if t is set at the sample mean. Two points are of note. First, the effect of the managerial bonus on unconnected field-days, αθ , is zero at the bottom two quantiles, and is positive and increasing in θ for the top quantiles. Second, the effect of the managerial bonus on connected field-days, αθ + γθ C¯ if t , is negative and significant in the first two quantiles, and is positive and significant in the last two quantiles.41 Taken together, these results provide evidence against the interpretation that after the introduction of the bonus, managers exert extra effort and target all 41

As the effect of the bonus on both connected and unconnected field-days is increasing in θ, the evidence is not consistent with the hypothesis that workers work harder to increase the probability of being selected to pick once the bonus is introduced. If such “rat race” effects were responsible for the productivity increase, we should observe workers at the margin of being selected to be most affected.

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workers regardless of their connection status. Rather the data suggest the introduction of the bonus strictly decreases the productivity of workers in the left tail of the productivity distribution on field-days in which they are connected and has no effect when they are not connected. The findings unambiguously provide support to the interpretation that social connections distort the allocation of effort when managers are paid fixed wages and that this allocation effect is detrimental for the firm’s productivity.42 6.1.2. Fixed Effects Estimates To complement the quantile regression evidence, we estimate the productivity effect of social connections, of the managerial bonus, and of their interaction, individually for each worker. To do so, we estimate the panel data specification (5)

yif t =

129

Λif t Di + λf + ρXif t + ηZf t + ϕt +

i=1

μs Ssf t + uif t

s∈Mf t

where Di equals 1 for worker i, and is 0 otherwise, and all other variables are as previously defined. To explore heterogeneous effects across workers, we define (6)

Λif t = αi [DCif t × (1 − Bt )] + βi [DCif t × Bt ] + γi [(1 − Bt ) × (1 − DCif t )] + δi [Bt × (1 − DCif t )]

For each worker we therefore estimate four parameters that capture his residual productivity on field-days when he is (i) connected and managers are paid wages (αi ); (ii) connected and managers are paid bonuses (βi ); (iii) unconnected and managers are paid fixed wages (γi ); (iv) unconnected and managers are paid bonuses (δi ). Figure 1 shows the kernel density estimates of the four estimates of residual productivity. Panels (a) and (b) show the effect of being socially connected to managers for a given managerial compensation scheme. Panel (a) shows that when managers are paid fixed wages, the entire distribution of conditional productivity shifts to the right on field-days in which workers are connected compared to when they are not connected. The p-value of the Kolmogorov–Smirnov test for the null of equality of distributions is .01. Panel (b) shows that when managers are paid performance bonuses, the distributions of conditional productivity on connected and unconnected field-days overlap. In this case, the Kolmogorov–Smirnov test fails to reject the null at 42 The results are qualitatively unchanged if we set the threshold C¯ if t to (i) the sample minimum of Cif t , so that a worker is defined to be connected as long as he is connected to one of the managers; (ii) the sample maximum of Cif t , so that a worker is defined to be connected only if he is connected to all managers on the field-day.


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FIGURE 1.—Worker’s fixed effects, by connection status and managerial incentive scheme. Residual productivity is the worker fixed effect in the regression of log productivity on worker experience, field life cycle, trend, field fixed effects, and manager fixed effects. For each worker we estimate a fixed effect in each of the four possible states: when managers are paid wages and the worker is connected, when managers are paid wages and the worker is not connected, when managers are paid bonuses and the worker is connected, and when managers are paid bonuses and the worker is not connected.

conventional levels of significance. Panels (a) and (b) together confirm the previous findings that being connected increases workers’ conditional productivity when managers are paid a fixed wage, while it has no discernible effect when managers are paid performance bonuses. To assess whether social connections distort the managers’ allocation of effort and whether this is detrimental for productivity, we analyze whether managers reallocate effort from low ability workers on connected field-days to high ability workers on unconnected field-days after the introduction of the bonus. In line with the quantile regression estimates, panel (c) shows that on fielddays when the worker is connected, the distribution of conditional productivity has a thicker left tail and the p-value of the Kolmogorov–Smirnov test is .04. Panel (c) thus indicates that the introduction of the managerial performance bonus reduces the productivity of low ability workers who were previously tar-

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geted when managers were paid wages. Note also that on field-days when the worker is connected, the distribution of conditional productivity has a higher variance when managers are paid bonuses compared to when they are paid fixed wages. This illustrates that social connections reduce the variation in productivity naturally arising from differences in worker ability when managers are paid fixed wages but not when they are paid bonuses. Finally, panel (d) illustrates the effect of the introduction of the bonus on unconnected field-days. In line with the previous estimates, on unconnected fielddays the distribution of conditional productivity has a thicker right tail when managers are paid bonuses compared to when they are paid fixed wages— the p-value of the Kolmogorov–Smirnov test is .06. This is consistent with the introduction of the bonus increasing the productivity of high ability workers who were previously untargeted on field-days when they were unconnected and managers were paid wages. Overall, panels (c) and (d) are in line with the quantile regressions results and provide support to the interpretation that when managers are paid fixed wages, they favor connected workers and that this allocation of managerial effort is detrimental for the firm’s overall productivity. 6.1.3. Further Evidence The balance of evidence indicates that following the introduction of the bonus, managers reallocate effort from low ability connected workers to high ability unconnected workers, rather than devoting more effort toward all workers. This is consistent with the characteristics of the production technology in our setting. Indeed, as each manager is responsible for 20 workers distributed over one hectare, it is often impossible for her to target her effort toward all workers simultaneously. The technology is such that managerial effort is a rival good in that if the manager decides to target her effort toward one worker, she necessarily does so at the expense of another worker. A testable implication of this property is that if favors are rival, the effect of social connections on the productivity of worker i should be smaller when the share of his co-workers who are also connected to managers increases. In short, if few workers are connected, the manager can devote all of her time to favor them. If more workers are connected, the manager needs to spread her favors more thinly. To check for this, we reestimated our baseline specification (3), allowing the effect of social connections on the productivity of worker i on field-day f t to vary with the share of workers who are also connected to a manager on field-day f t. We find that when managers are paid fixed wages, social connections increase the productivity of a connected worker by 15% if one-quarter of the workers on the field are also connected, by 7% if half of the workers on the field are also connected, and have no effect if more than two-thirds of the workers on the field are also connected. In line with previous evidence, neither the connection status of worker i nor the share of connected


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workers on the same field-day affect productivity after the introduction of managerial performance pay. 6.2. The Overall Effect on Average Field-Day Productivity The evidence above indicates that when managers are paid fixed wages, social connections distort the allocation of managerial effort in favor of connected workers at the expense of high ability workers. This, together with the fact that when they are paid performance bonuses they allocate effort to high ability workers only, indicates that the allocation effect is detrimental for the firm’s performance. We now present evidence on the overall effect of social connections, namely we investigate whether the extent to which social connection increase managerial effort is sufficient to overcome the negative allocation effect. To do so, we aggregate the data at the field-day level and estimate a specification analogous to (3): (7)

y¯f t = κ0 (1 − Bt ) × C¯ f t + κ1 (Bt × C¯ f t ) + ρBt + μs Ssf t + ηZf t + ϕt + λf + uif t s∈Mf t

where y¯f t is the log of the average productivity on field-day f t, C¯ f t is the share of managers worker i is socially connected to, averaged across all workers on field-day f t, Bt is a dummy variable equal to 1 afterthe performance bonus is introduced (June 27th) and equal to 0 otherwise, s∈Mf t μs Ssf t is a full set of managers dummies, Zf t is the field-life cycle, and t is a time trend, as in the baseline specification (3). The parameters of interest are κ0 , which measures the effect of social connections on average productivity when managers are paid a fixed wage, and κ1 , which measures the effect of social connections when managers are paid performance bonuses. The findings, reported in Table V, indicate that the overall effect of social connections is negative when managers are paid fixed wages and is zero when they are paid performance bonuses. An increase in average social connections by 1 standard deviation reduces average productivity by 8% unconditionally and by 5% in the full conditional specification (7) when mangers are paid fixed wages. The effect of social connections under the bonus is small and not significantly different than zero. Given our previous finding that the allocation effect is negative when managers are paid wages and is zero when they are paid bonuses, the results in Table V imply that the effect of social connections on managerial effort might be positive, albeit not very large, in the fixed wage regime only. The next section discusses how the finding that social connections are detrimental to the overall firm productivity might depend on specific features of the context we study.

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O. BANDIERA, I. BARANKAY, AND I. RASUL TABLE V THE EFFECT OF SOCIAL CONNECTIONS ON AVERAGE PRODUCTIVITY

Average connections (C¯ if t ), fixed wages for managers Average connections (C¯ if t ), performance bonus for managers Manager fixed effects Time controls Field fixed effects Adjusted R-squared Number of observations (field-day)

1. Unconditional

2. Managers FE

3. Time Controls

4. Field FE

−779*** (201) 206 (509)

−544** (255) 149 (529)

−675*** (238) 122 (484)

−535** (250) 031 (480)

No No No 944 241

Yes No No 943 241

Yes Yes No 948 241

Yes Yes Yes 909 241

a Dependent variable = log of average field-day productivity (kilograms picked per hour on the field-day). *** denotes significance at 1%, ** at 5%, and * at 10%. AR(1) regression estimates are reported. Panel corrected standard errors are calculated using a Prais–Winsten (AR(1)) regression, allowing the error terms to be field specific heteroskedastic, and contemporaneously correlated across fields. The autocorrelation process is assumed to be specific to each field. Each field-day observation is weighted by the log of the number of workers present. All specifications control for the bonus dummy that is equal to 1 when the managerial performance bonus scheme is in place and equal to 0 otherwise. The dependent variable is the log of the average productivity at the field-day level. The average connection variable is computed as the average of the social connection measure for all workers on the field-day. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. Time controls included in columns 3 and 4 are a linear time trend and the field life cycle, defined as the nth day the field is picked divided by the total number of days the field is picked over the season.

7. DISCUSSION We have provided evidence on the interplay between social connections, incentives, and productivity within a firm. We show that in a setting where managerial effort can be targeted to affect the productivity and earnings of individual workers, the existence of social connections between individuals at different tiers of the firm hierarchy affects individual and firm performance. We find that managers target connected workers, but only when their monetary incentives are low powered. After the introduction of managerial performance pay, managers stop favoring low ability workers they are socially connected to and target their effort toward high ability workers instead. The results indicate that while social connections can increase the productivity of connected workers, their effect on the allocation of managerial effort is detrimental for the firm’s productivity overall under the fixed wage regime. Our results bring new evidence to the small but growing literature that highlights the importance of social relationships in the workplace. Our findings indicate that managerial behavior is shaped by both social connections with subordinates and monetary incentives. Both factors are key to explaining the success of existing incentive structures and to guide the design of optimal compensation schemes for both workers and managers.


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The use of detailed personnel data combined with the purely exogenous variation created by our natural field experiment allows us to precisely identify the causal effect of social connections between workers and managers on the performance of individual workers and on the firm’s performance overall. Precision, however, inevitably comes at the cost of a loss of generality, because the firm we study, as any other, has unique features that shape social connections between workers and managers, and their effect on productivity. The following features of this work environment are particularly noteworthy for the external validity of this study. First, there are two characteristics of this firm that have opposite effects on the probability that social connections form and are strong enough to affect behavior. On the one hand, the fact that managers and workers are of similar ages and backgrounds, and live on the farm site for the entire duration of their stay increase the likelihood that they form strong social connections with one another. On the other hand, as managers and workers are employed on short term seasonal contracts, this might prevent the formation of meaningful longrun social ties relative to other settings. Second, in our setting all the actions managers can take to help connected workers—allocating them to better rows, reallocating them quickly to new rows, providing them with new crates as soon as needed—are costlessly observed by others on the field. To the extent that favoritism is disapproved of by unconnected workers, the fact that these actions are observable by all workers reduce managers’ ability to favor their friends. We thus expect the effect of social connections and favoritism to be stronger in settings where favoritism can be more easily disguised. Third, the specific form that the effects of social connections take depends on the technology and incentive schemes in the workplace. In our context workers are paid piece rates and managers can undertake actions that improve the productivity and hence earnings of connected workers. In other contexts in which workers are paid fixed wages, social connections might be exploited to allow subordinates to slack, allocating subordinates to more desirable positions, or helping subordinates be promoted. Moreover, in our context workers’ productivity is precisely measured, so there is also no scope for managers to show favoritism through subjective evaluations of workers. In general, managers will have more margins along which to favor workers and all such activities will affect the firm’s overall performance. Perhaps the most important consideration is that in other settings, the allocative effect of social connections on managerial effort might be beneficial. As emphasized throughout, social connections can reduce informational asymmetries, facilitate joint problem solving, and provide managers the ability to motivate workers through social rewards and punishments. In our context, the tasks workers undertake are relatively simple and so any potential benefits that social connections have for problem solving or improved communication more generally, are likely to be small. In other settings, the productivity enhancing

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effects of social connections might dominate the inefficiency due to favoritism. For example Ichniowski and Shaw (2005) presented evidence from steel finishing lines—a relatively complex task that involves problem solving—of such positive effects of improved communication within and between tiers of the firm hierarchy.43 The fact that managers devote effort to increase the productivity of connected workers, even when they are paid fixed wages, suggests that social connections between managers and workers can provide an alternative, and possibly cheaper, mechanism to the provision of monetary incentives. It may thus be in a firm’s best interests to foster social ties between management and workers. Indeed many firms are observed devoting resources toward such bonding exercises. On a related note, the fact that managers behave as if they derive utility from helping connected workers implies that being socially connected to their subordinates lowers the managers’ participation constraint and thus the firm’s wage bill may be reduced. However, this strategy may be suboptimal if it leads to the self-selection of lower quality managers to the firm over time.44 More generally, our findings provide support for the idea that interplays between social relationships and incentives within firms need to be taken into account in order to understand how individuals respond to a given set of incentives and to understand the optimal set of incentives within an organization. Differences in the social organization of the workplace might therefore explain part of the productivity differences among otherwise observationally similar firms. APPENDIX A.1. Proofs PROOF OF PROPOSITION 1: As the manager’s payoff function (b + σh ) × θk(σh )mh + (b + σl )k(σl )ml − C(mh + ml ) is linear in (mh ml ), the manager will target only the worker that yields the highest marginal benefit.45 The 43 Related work, Nagin, Rebitzer, Sanders, and Taylor (2002) presented evidence from a field experiment in a call center that exogenously varied the probability that employees would be monitored by managers. Their results suggest that management’s “perceived empathy and fairness” in dealing with employees may play an important role in reducing workplace opportunism. Other beneficial effects of social capital within firms have also been discussed in the sociology literature. These include potentially better hiring outcomes through the use of referrals by current employees (Fernandez, Castillo, and Moore (2000)). 44 Social connections within firms are just one alternative to using monetary incentives to solve agency problems. There is a growing theoretical and empirical literature on the relationship between intrinsic and extrinsic motivation (Frey and Oberholzer-Gee (1997), Kreps (1997), Bénabou and Tirole (2003)). 45 This property would of course be retained if workers also chose their effort level. The analysis would however be more cumbersome as the worker’s effort level would depend on the manager’s and vice versa.


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¯ − 1) > c. ¯ if and only if {max[θkh (b + σh ) kl (b + σl )]}(m manager chooses m The right-hand side is increasing in (σh σl ), which proves the first part of the proposition, that the existence of social connections weakly increases the level of managerial effort. The manager allocates her effort to worker i if and only if (b + σi )θk(σi ) ≥ (b + σj )k(σj ). There are two cases to consider. If σh = σ and σl ∈ {σ 0}, that is, if the manager is socially connected to both workers or only the high ability worker, she targets the high ability worker for all other parameter values. If σh = 0 < σl = σ, that is, if the manager is connected only to the low ability bθ . This proves worker, she targets the high ability worker if and only if k < b+σ the second part of the proposition, that there exists a part of the parameter space in which the existence of social connections alters the allocation of managerial effort in favor of the worker the manager is connected to. Q.E.D. PROOF OF PROPOSITION 2: As m is weakly increasing in (σh σl ) and the firm’s average productivity, 12 (θk(σh )mh + k(σl )ml ), is increasing in m, the levels effect of social connections is weakly positive. For any given level of effort, if σh = σ and σl ∈ {σ 0}, then the manager targets the high ability worker only and the allocation effect is positive as productivity is equal to θkmh > θmh . This proves the first part of the proposition that if the manager is connected to the high ability worker, the effect of social connections on the firm’s productivity is unambiguously positive. bθ < k < θ, the allocation effect is negative, namely If σh = 0 < σl = σ and b+σ the manager targets the low ability worker although targeting the high ability worker would yield higher output.46 The net effect of the existence of social connections on the firm’s productivity then depends on the balance between the levels effect and the allocation effect. If m(σh = 0 σl = 0) = m(σh = 0 σl = σ), namely if the manager chooses the same level of effort regardless of social connections, the allocation effect dominates and the effect of social connections is unambiguously negative. If m(σh = 0 σl = 0) < m(σh = 0 σl = σ), ¯ when the firm’s productivity is equal to θ when σh = 0 σl = 0 and equal to km σh = 0 σl = σ. This proves the second part of the proposition that if the manager is connected only to the low ability worker and targets him, social connecQ.E.D. tions reduce firm’s productivity if k < mθ¯ . A.2. Identification: The COO’s Allocation Algorithm We present evidence in support of the first identifying assumption that the allocation algorithm used by the COO to assign workers to tasks and workers to managers does not change with the introduction of managerial performance bonuses. We proceed in four steps. First, we compare the allocation of 46 Total output is equal to θm when the manager is not connected to either worker and is equal to km when she is connected to the low ability worker. Thus, for any level of m, output is lower when the manager is connected.

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the connected workers we focus on for our main analysis to the allocation of unconnected workers. Second, we analyze the determinants of the level of social connections Cif t and test whether their effect changes after the change in incentive scheme. Third, we test whether the COO is more likely to assign connected workers to some managers rather than others and whether this changes after the change in incentive scheme. Finally, we exploit the fact that some dimensions of connectivity, such as nationality, are more easily observable to the COO than others, such as time of arrival. If such sorting biases the estimates, we should find the effect of social connections to be mostly driven by dimensions that are easier to observe. A.2.1. Connected versus Unconnected Workers We estimate the probability of a given worker being selected into employment by the COO while controlling for farm level variables that affect the probability of being hired independently of the incentive scheme in place. These farm level variables measure the supply and demand of labor. We measure labor supply using personnel records on the number of workers available for hire on the farm on any given day. We measure the demand for labor using the total daily fruit yield on each site on the farm. The total yield is orthogonal to the incentive scheme as it is determined by planting decisions taken one or two years earlier. Fields are located on two sites, of which we use the largest for the analysis as fruit in the smaller site begins to ripen only after the introduction of the performance bonus scheme. However, as both sites hire workers from the same pool, we control for yields in each site separately. We then estimate the following conditional logit model, where observations are grouped by worker: (8)

Pr(pit = 1) = Λ(Bt Bt × Ci XtD XtS Xit )

pit = 1 if worker i is selected by the COO to pick on day t on the main site and = 0 if they are assigned to nonpicking tasks. Bt is the performance bonus dummy, Ci is a dummy variable equal to 1 if worker i is socially connected to any of the managers along any dimension of nationality, arrival cohort, and living site, and equal to 0 otherwise. XtD and XtS proxy the demand and supply of labor on day t. To allow for workers’ previous performance to affect their probability of being selected, Xit measures worker i’s productivity on the last day she picked, in percentage deviation from the mean productivity on that day, to remove the effects of factors that determine the productivity of all workers and are beyond the worker’s control.47 47 We first take the deviation of the worker’s productivity from the field average productivity on each field he picked on the day he was last selected to pick, and then calculate a weighted average of this across all fields he worked on where the weights are based on the number of pickers on the field.

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SOCIAL CONNECTIONS AND INCENTIVES TABLE A.I SOCIAL CONNECTIONS, MANAGERIAL INCENTIVES, AND THE COO’S ALLOCATION ALGORITHMa Probability of Being Selected to Pickb

Performance bonus for managers Performance bonus for managers × worker i is socially connected Total yield in site 1 Total yield in site 2 Number of workers available to pick fruit Worker i’s previous deviation from mean productivity Log-likelihood Number of observations (worker-day)

134 (495) 524 (214) 224*** (153) 883*** (036) 380*** (037) 116* (091) −5186.8 15,551

Probability of Being Unemployedc

204* (764) 605 (253) 802*** (057) 800*** (032) 183*** (178) 107 (107) −3208.5 9808

a Standard errors are given in parentheses, clustered by worker. *** denotes that the log odds ratio is significantly different from 1 at 1%, ** at 5%, and * at 10% levels. Conditional logit estimates are reported where observations are grouped by worker. All continuous variables are divided by their standard deviations so that one unit increase can be interpreted as increase by one standard deviation. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. “Total yield” on the site is the total kilograms of the fruit picked on the site-day. The “number of workers available to pick fruit” is the total number of individuals that are on the farm that day and are available for fruit picking. “Worker i’s previous deviation from mean productivity” is defined on the last day the worker was selected to pick. We first take the deviation of the worker’s productivity from the field average productivity on each field he picked on the day he was last selected to pick, and then calculate a weighted average of this across all fields he worked on where the weights are based on the number of pickers on the field. Worker i is defined to be unemployed on day t if she is present on the farm but is not assigned to any paid tasks. b Dependent variable = 1 if worker i is chosen to pick on day t in main site, =0 if worker is assigned to nonpicking tasks. c Dependent variable = 1 if worker i is unemployed on day t , =0 if assigned to nonpicking tasks.

All continuous variables are divided by their standard deviations so that one unit increase can be interpreted as an increase of 1 standard deviation. We report odds ratios throughout; standard errors are calculated using the delta method. Data column 1 of Table A.I shows that, other things being equal, there is no differential effect on socially connected or unconnected workers of being selected to pick fruit after the introduction of the managerial performance bonus. Namely, the coefficient is not significantly different from 1. The other coefficients show that, as expected, workers are more likely to be assigned to fruit picking tasks on days in which the fields on the main site bear more fruit (namely, the coefficient of XtD is significantly larger than 1) and on days in which they face less competition from other workers (namely, the coefficient of XtS is significantly smaller than 1).

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Conditional on not being selected to pick on the main site on a given day, a worker can either be assigned to other tasks on the main site, to work on the other site, or be left unemployed for the day. The next specification checks whether the assignment of workers to nonpicking tasks varies differentially by socially connected and unconnected workers when the performance bonus is introduced. The result in data column 2 again shows there to be no such differential effect of the COO’s decision across workers based on their social connection to managers. The pattern of other coefficients confirms that the introduction of the bonus scheme significantly raises the probability of being unemployed. As expected, the probability of being unemployed for the day is lower when yields are higher and when the stock of available workers is lower. A.2.2. Determinants of Social Connections To provide evidence that field-day and worker-field-day specific determinants of productivity do not predict the level of social connections Cif t differently under the two managerial incentive schemes, we estimate regressions of the form (9)

Cif t = αi + λf + υBt + [(φ0 + φ1 Bt ) × Xif t ] + [(ϕ0 + ϕ1 Bt ) × Zf t ] + μs Ssf t + uif t s∈Mf t

where Bt is the bonus dummy, Xif t captures worker i’s time-varying characteristics, and Zf t captures several time-varying field characteristics, such as the field life cycle, a time trend, worker’s tenure on the farm, the number of workers on the field, the number of managers on the field, the total man hours worked on the field, and the total kilos of fruit picked. Our identifying assumption requires φ1 = ϕ1 = 0. It is reassuring that we fail to reject the null of zero coefficients in all cases.48 A.2.3. The Allocation of Workers to Managers Finally, we present evidence on whether managers can influence the composition of the group of workers they are allocated to and, in particular, whether 48 Three other pieces of evidence also suggest that farm operations do not change over the two halves of the season. First, the ratio of workers to managers does not change significantly, remaining at 20 throughout. Second, at the field-day level, the average share of workers who are socially connected to managers does not change significantly over the two halves of the season, neither does the variation in this share between fields on the same day. This suggests workers do not become sorted into fields by social connections over time. Third, using the estimated worker fixed effect from (3), αi , as a measure of a worker’s ability, we find that groups of workers on the field-day are equally heterogeneous before and after the change in managerial incentives. Hence there is no evidence the COO sorts workers differently by ability into fields postbonus.


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the composition of workers they are assigned differs after the change in managerial incentives. To begin with we note that Table I indicates that workers are equally likely to be connected to managers under both schemes. This is inconsistent with the hypothesis that managers can affect the share of connected workers in their group and choose a different share after the introduction of the bonus. To provide further evidence on this point we test whether some managers are significantly more likely to be assigned connected workers and whether this changes after the introduction of the bonus. Note that since some nationalities are more numerous than others, some managers are mechanically connected to more workers. However, we are interested in establishing whether different managers are more or less likely to be assigned to workers they are connected to, regardless of the total number of workers they are connected to. To do so, we construct a data set at the manager-field-day level and estimate the following, (10) μs Ssf t + vs (Ssf t × Bt ) + ζsf t Zsf t = s∈Mf t

s∈Mf t

where Zmf t is the log of the ratio of the number of workers connected to manager m present on field-day f t over the total number of workers connected to manager m who are working on day t. The numerator thus represents the number of connected workers the manager is assigned to, whereas the denominator is the number of workers the managers could have potentially been assigned to on day t. All other controls are as previously defined. We test two hypotheses: (i) H0 : μs = 0 for all s, namely all managers are equally likely to be assigned connected workers when they are paid fixed wages, and (ii) H0 : vs = 0 for all s, namely the allocation does not change after the introduction of the bonus. The p-values for these hypotheses are .64 and .94, respectively, so we cannot reject either hypothesis. Hence there is no evidence the COO treats managers differently before and after the bonus or that some managers are more able to be allocated to connected workers while others are not. A.2.4. Observability of Social Connections A final check on whether the COO intentionally sorts managers and workers into fields on the basis of their social connections is based on the intuition that some dimensions of connectivity, such as nationality, are more easily observable to the COO than others, such as time of arrival. If such sorting biases the estimates, we should find the effect of social connections to be mostly driven by dimensions that are easier to observe. In Table A.II we estimate a specification analogous to (3) that separately controls for each dimension of social connectivity. To compare the magnitudes of the coefficients, we consider the implied effect on worker productivity of a 1 standard deviation increase in each of the connectivity measures from its

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O. BANDIERA, I. BARANKAY, AND I. RASUL TABLE A.II SOCIAL CONNECTIONS AND MANAGERIAL INCENTIVES Type of Social Connection

Share of managers of same nationality as i, fixed wages for managers Share of managers of same nationality as i, performance bonus for managers Share of managers living in same area as i, fixed wages for managers Share of managers living in same area as i, performance bonus for managers Share of managers of same arrival cohort as i, fixed wages for managers Share of managers of same arrival cohort as i, performance bonus for managers Interactions of nationality × performance bonus dummy Interactions of living site × performance bonus dummy Interactions of arrival cohort × performance bonus dummy Interactions of worker fixed effect × performance bonus dummy Field-date fixed effects Adjusted R-squared Number of observations (worker-field-day)

162*** (045) −075 (134) 087* (049) −070 (071) 309*** (088) −079 (142) Yes [.068] Yes [.000] Yes [.000] No No .4366 8884

a Dependent variable = log of worker’s productivity (kilograms picked per hour on the field-day). *** denotes significance at 1%, ** at 5%, and * at 10%. Standard errors are reported in parentheses and allow for clustering at the worker level. All specifications control for worker, field, and manager fixed effects. The other controls include the managerial performance bonus dummy, the worker’s picking experience, the field life cycle, a time trend, and interactions between the performance bonus dummy and the worker’s nationality, arrival cohort, and living site. The field life cycle is defined as the nth day the field is picked divided by the total number of days the field is picked over the season. All continuous variables are in logarithms. At the foot of the column we report the p-value on the F -test on the joint significance on the interaction terms with the performance bonus dummy.

mean. We find that when managers are paid a fixed wage, the productivity of a given worker is 4.6%, 1.5%, and 3.3% higher when the share of managers he is connected to by nationality, living site, and arrival cohort, respectively, is 1 standard deviation higher. The magnitude of the effect is thus similar for dimensions that can be observed—nationality—and for dimensions that cannot be easily observed—arrival cohort. If we compare the effect of the same change across different connection measures, the magnitude of the effect is actually the largest for the dimension that is least observable—arrival cohort. This provides further evidence against the hypothesis that the COO sorts managers and workers into fields on the basis of their social connections. Finally, social connections along any dimension do not affect worker productivity when managers are paid a performance bonus.49 49 We chose to measure social connections along the dimensions of nationality, living site, and time of arrival in order to capture social links that form for different reasons and, indeed, the


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A.3. Identification: Time Effects We now present evidence in support of the second identifying assumption that any effect of social connections on individual productivity unrelated to the managerial incentive scheme in place remains unchanged over time. If not, γ1 may simply pick up that the then in the baseline specification (3), γ0 and effect of social connections naturally dies out over time, rather than because managers change their behavior when they are paid performance bonuses. For example, managers may initially favor some workers in order to befriend them. Similarly workers may initially work hard under some managers in order to befriend them. This would explain the pattern of coefficients we find in the data and then suggest social connections do not distort managerial effort in the long run. In Table A.III we analyze whether the effects of social connections on worker productivity naturally disappear over time. In column 1 we split both the preand postperformance bonus periods into halves and allow the effect of connections to change within the pre- and postbonus periods. Intuitively, if the effect of social connections were naturally declining over time, we would expect it to be higher in the first half of the prebonus period than in the second half, and again higher in the first half of the postbonus period than in the second half. Column 1 shows that, in contrast, there is no change in the effect of social connections within each period. Rather, the effect of social connections on worker productivity disappears discontinuously with the introduction of the performance bonus for managers. γ1 might pick up that later in the field life A second concern is that γ0 and cycle there is less variation in the fruit available across different rows and so managers have no means by which to favor connected workers, even though they prefer to do so. To check for this, in column 2 we allow the effect of social connections to vary with a field specific time trend—the field life cycle. We do not find statistically significant evidence that the effect of social connections diminishes within a field over time. A third time related concern is that the true social ties between a worker and his managers are measured with error using Cif t which is based on three particular dimensions. This measurement error is nonclassical because it increases over time if workers learn they are better off being socially connected to managers and so invest more into forming social ties with managers over time, irrespective of whether they are of the same nationality, living site, and arrival cohort. If so, we should find the effect of Cif t to diminish with the time the worker has spent on the farm. In column 3 we allow the effect of social connections to vary with a worker specific time trend—the number of days the worker has been present on the farm. There is no significant evidence of such effects, although the interaction terms are not precisely estimated. correlation among the three measures is very low. In line with this we find their estimated effect on productivity to be the same, regardless of whether they are included together or one at a time.

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TABLE A.III ROBUSTNESS OF RESULTS TO TIME EFFECTSa 2. Field Specific Time Trend

3. Worker Specific Time Trend

165*** (043) −037 (092) −018 (076)

188*** (067) −003 (116)

269*** (094) 345 (511)

4. Placebo Bonus Based on Field Life Cycle

5. Placebo Bonus Based on 2004 Season


Share of managers connected to i, fixed wages for managers Share of managers connected to i, performance bonus for managers Share of managers connected to i, fixed wages for managers × 2nd quarter dummy (31st May) Share of managers connected to i, performance bonus for managers × 4th quarter dummy (29th July) Share of managers connected to i, fixed wages for managers × field life cycle Share of managers connected to i, performance bonus for managers × field life cycle Share of managers connected to i, fixed wages for managers × days on farm for worker i Share of managers connected to i, performance bonus for managers × days on farm for worker i Share of managers connected to i, placebo bonus based on field life cycle = 0 Share of managers connected to i, placebo bonus based on field life cycle = 1

1. Farm Specific Time Trend

−133 (099) −089 (141) −249 (200) −047 (035) −109 (132) −087 (081) −033 (138) (Continues)

TABLE A.III—Continued 1. Farm Specific Time Trend

2. Field Specific Time Trend

3. Worker Specific Time Trend

4. Placebo Bonus Based on Field Life Cycle

Interactions of nationality × performance bonus dummy Interactions of living site × performance bonus dummy Interactions of arrival cohort × performance bonus dummy Adjusted R-squared Number of observations (worker-field-day)

201* (109) 215*** (033) Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

.4374 8884

.4524 8884

.4368 8884

.6260 1584

.4532 2692

a Dependent variable = log of worker’s productivity (kilograms picked per hour on the field-day). *** denotes significance at 1%, ** at 5%, and * at 10%. Standard errors are reported in parentheses and allow for clustering at the worker level. All specifications control for worker, field, and manager fixed effects. The other controls included in each specification are the managerial performance bonus dummy, the worker’s picking experience, the field life cycle, and a time trend. The field life cycle is defined as the nth day the field is picked divided by the total number of days the field is picked over the season. All continuous variables are in logarithms. A manager and given worker i are defined to be connected if they are either of the same nationality, live in the same area, or are in the same arrival cohort. All sample workers are connected to at least one manager on at least one field-day and work at least one week under each incentive scheme. In column 1 the 2nd quarter dummy is defined to be equal to 0 before May 31st and equal to 1 thereafter. The 4th quarter dummy is defined to be equal to 0 before July 29th and equal to 1 thereafter. These dummy variables split the pre- and postbonus periods equally into two halves. In column 3 the days on the farm for a worker are defined as the number of days elapsed since the worker first arrived on the farm. In column 4 the placebo bonus dummy based on the field life cycle is defined to be 0 if the field is less than .53 of the way though its life cycle, and 1 otherwise. In this column the sample is restricted to fields that are only operated in the period when managers are paid a performance bonus (after June 27th). In column 5, the sample covers the same period of time (May 1st to Aug 31st) in the following year—2004—when managers were paid wages throughout. The placebo bonus is equal to 1 after June 27, 2004. The interaction terms at the foot of the table are defined with respect to the placebo bonus dummy variable in columns 4 and 5.


Share of managers connected to i, placebo bonus 2004 = 0 Share of managers connected to i, placebo bonus 2004 = 1

5. Placebo Bonus Based on 2004 Season

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Overall, the evidence in columns 1–3 indicates that the effect of social connections does not decline smoothly with time, field specific trends, or worker specific trends. Rather there is a discontinuous effect of social connections on worker performance at the time when managerial performance bonuses were introduced. Given that we had full control over the timing of this change, our experimental research design ensures that the exact date on which the managerial incentive schemes changed is uncorrelated with any determinants of individual productivity. To provide further support, columns 4 and 5 report the results of two placebo tests. Column 4 uses fields that were picked only after the introduction of the bonus and are therefore excluded from our main sample. Given that in our sample the bonus is introduced when the average (and median) field is half the way through its life cycle, we define a placebo bonus dummy to be equal to 0 if the field is in the first half of its life cycle and equal to 1 if it is in the second half. The results in column 4 indicate that social connections have no effect on worker productivity either side of the placebo dummy, thus ruling out that our previous results were due to the effect of social connections naturally disappearing once fields have reached half of their life cycle. Column 5 uses data from the same tasks in the same farm one year later, namely in 2004, when the managers were paid fixed wages throughout the season. We define the placebo bonus dummy to be equal to 0 before the date bonuses were introduced in 2003 (June 27th) and equal to 1 thereafter. All variables are defined as in (3) and the sample is selected according to the same criteria.50 It is reassuring that column 5 shows that the effect of social connections during the entire 2004 season is of similar magnitude to the effect before the introduction of the bonus in 2003. In other words, in 2004 when managers are paid fixed wages throughout the season, they appear to allocate more effort toward connected workers throughout the season. REFERENCES AKERLOF, G. A. (1982): “Labor Contracts as a Partial Gift Exchange,” Quarterly Journal of Economics, 97, 543–569. [1050] AKERLOF, G. A., AND J. L. YELLEN (1988): “Fairness and Unemployment,” American Economic Review Papers and Proceedings, 78, 44–49. [1050] BANDIERA, O., I. BARANKAY, AND I. RASUL (2005): “Social Preferences and the Response to Incentives: Evidence From Personnel Data,” Quarterly Journal of Economics, 120, 917–962. [1050] (2007): “Incentives for Managers and Inequality Among Workers: Evidence From a Firm Level Experiment,” Quarterly Journal of Economics, 122, 729–774. [1050,1071] (2008): “Social Capital in the Workplace: Evidence on Its Formation and Consequences,” Labor Economics, 15, 724–748. [1062] 50 Namely, we select workers that work at least one week on either side of the placebo bonus and fields that are operated for at least one week either side. We restrict the sample to the peak picking season (May 1 to August 31) and to the main site.


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BARNARD, C. (1938): The Functions of the Executive. Cambridge, MA: Harvard University Press. [1047] BECKER, G. S. (1957): The Economics of Discrimination. Chicago: University of Chicago Press. [1050] BÉNABOU, R., AND J. TIROLE (2003): “Intrinsic and Extrinsic Motivation,” Review of Economic Studies, 70, 489–520. [1082] BERTRAND, M., AND A. SCHOAR (2003): “Managing With Style: The Effect of Managers on Firm Policies,” Quarterly Journal of Economics, 118, 1169–1208. [1049] BERTRAND, M., F. KRAMARZ, A. SCHOAR, AND D. THESMAR (2005): “Politically Connected CEOs and Corporate Outcomes: Evidence From France,” Mimeo, University of Chicago. [1050] BEWLEY, T. F. (1999): Why Wages Don’t Fall During a Recession. Cambridge, MA: Harvard University Press. [1047] BLACK, S. E., AND P. STRAHAN (2001): “The Division of Spoils: Rent Sharing and Discrimination in a Regulated Industry,” American Economic Review, 91, 814–831. [1050] CHARNESS, G., AND P. KUHN (2005): “Pay Inequality, Pay Secrecy, and Effort: Theory and Evidence,” Mimeo, University of California, Santa Barbara. [1050] FEHR, E., AND U. FISCHBACHER (2002): “Why Social Preferences Matter—The Impact of NonSelfish Motives on Competition, Cooperation and Incentives,” Economic Journal, 112, C1–C33. [1050] FERNANDEZ, R. M., E. J. CASTILLO, AND P. MOORE (2000): “Social Capital at Work: Networks and Employment at a Phone Center,” American Journal of Sociology, 105, 1288–1356. [1082] FREY, B. S., AND F. OBERHOLZER-GEE (1997): “The Cost of Price Incentives: An Empirical Analysis of Motivation Crowding-Out,” American Economic Review, 87, 746–755. [1082] GARICANO, L., I. PALACIOS, AND C. PRENDERGAST (2005): “Favoritism Under Social Pressure,” Review of Economics and Statistics, 87, 208–216. [1050] GIULIANO, L., D. I. LEVINE, AND J. LEONARD (2005): “Do Race, Gender, and Age Differences Affect Manager-Employee Relations? An Analysis of Quits, Dismissals, and Promotions at a Large Retail Firm,” Mimeo, University of California, Berkeley. [1050] HARRISON, G., AND J. LIST (2004): “Field Experiments,” Journal of Economic Literature, 152, 1009–1055. [1055] HOLMSTROM, B., AND P. MILGROM (1991): “Multitask Principal–Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design,” Journal of Law, Economics and Organization, 7, 24–52. [1055] ICHNIOWSKI, C., AND K. L. SHAW (2005): “Connective Capital: Building Problem-Solving Networks Within Firms,” Mimeo, Stanford University. [1082] KANDEL, E., AND E. LAZEAR (1992): “Peer Pressure and Partnerships,” Journal of Political Economy, 100, 801–813. [1047] KOFMAN, F., AND J. LAWARREE (1993): “Collusion in Hierarchical Agency,” Econometrica, 61, 629–656. [1048] KRAMARZ, F., AND D. THESMAR (2005): “Social Networks in the Boardroom,” Discussion Paper 5496, CEPR. [1050] KREPS, D. M. (1997): “The Interaction Between Norms and Economic Incentives: Intrinsic Motivation and Extrinsic Incentives,” American Economic Review Papers and Proceedings, 87, 359–364. [1082] LABAND, D. N., AND M. J. PIETTE (1994): “Favoritism versus Search for Good Papers: Empirical Evidence Regarding the Behavior of Journal Editors,” Journal of Political Economy, 102, 194–203. [1050] LAZEAR, E. P. (1989): “Pay Equality and Industrial Politics,” Journal of Political Economy, 97, 561–580. [1047] MALMENDIER, U., AND G. TATE (2005): “CEO Overconfidence and Corporate Investment,” Journal of Finance, 60, 2661–2700. [1049] MAYO, E. (1933): The Human Problems of an Industrial Civilization. New York: Macmillan. [1047]

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MIAN, A. R., AND A. I. KHWAJA (2005): “Do Lenders Favor Politically Connected Firms? Rent Provision in an Emerging Financial Market,” Quarterly Journal of Economics, 120, 1371–1411. [1050] MILGROM, P. R. (1988): “Employment Contracts, Influence Activities, and Efficient Organization Design,” Journal of Political Economy, 96, 42–60. [1048] MILGROM, P. R., AND J. ROBERTS (1990): “The Efficiency of Equity in Organizational Decision Processes,” American Economic Review Papers and Proceedings, 80, 154–159. [1048] NAGIN, D., J. B. REBITZER, S. SANDERS, AND L. J. TAYLOR (2002): “Monitoring, Motivation, and Management: The Determinants of Opportunistic Behavior in a Field Experiment,” American Economic Review, 92, 850–873. [1082] PODOLNY, J. M., AND J. N. BARON (1997): “Resources and Relationships: Social Networks and Mobility in the Workplace,” American Sociological Review, 62, 673–693. [1048] PRENDERGAST, C., AND R. H. TOPEL (1996): “Favoritism in Organizations,” Journal of Political Economy, 104, 958–978. [1048,1057] ROETHLISBERGER, F. J., AND W. DICKSON (1939): Management and the Worker. Cambridge: Harvard University Press. [1047] ROTEMBERG, J. J. (1994): “Human Relations in the Workplace,” Journal of Political Economy, 102, 684–717. [1047] ROY, D. (1952): “Quota Restriction and Goldbricking in a Machine Shop,” American Journal of Sociology, 57, 427–442. [1047] TIROLE, J. (1986): “Hierarchies and Bureaucracies: On the Role of Collusion in Organizations,” Journal of Law, Economics and Organizations, 2, 181–214. [1048] THOMAS, D. A. (1990): “The Impact of Race on Managers’ Experiences of Developmental Relationships,” Journal of Organizational Behavior, 11, 479–492. [1048] TSUI, A. S., AND C. A. OREILLY (1989): “Beyond Simple Demographic Effects: The Importance of Relational Demography in Superior-Subordinate Dyads,” Academy of Management Journal, 32, 402–423. [1048] WESOLOWSKI, M. A., AND K. W. MOSSHOLDER (1997): “Relational Demography in SupervisorSubordinate Dyads: Impact on Subordinate Job Satisfaction, Burnout, and Perceived Procedural Justice,” Journal of Organizational Behavior, 18, 351–362. [1048]

Dept. of Economics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, U.K.; [email protected], The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104, U.S.A.; [email protected], and Dept. of Economics, University College London, Drayton House, 30 Gordon Street, London WC1E 6BT, U.K.; [email protected]. Manuscript received May, 2006; final revision received December, 2008.


COLLUSION VIA RESALE BY RODNEY J. GARRATT, THOMAS TRÖGER, AND CHARLES Z. ZHENG1 The English auction is susceptible to tacit collusion when post-auction interbidder resale is allowed. We show this by constructing equilibria where, with positive probability, one bidder wins the auction without any competition and divides the spoils by optimally reselling the good to the other bidders. These equilibria interim Pareto-dominate (among bidders) the standard value-bidding equilibrium without requiring the bidders to make any commitment on bidding behavior or postbidding spoil division. KEYWORDS: English auction, second-price auction, collusion, sunspots, resale.

1. INTRODUCTION IN PRIVATE-VALUE ENGLISH AUCTIONS that ban resale, it is a dominant strategy for each participant to bid up to her use value. With resale allowed, value bidding remains an equilibrium outcome, but there is no dominant strategy. Resale opens the possibility that some bidders will optimally drop out at a price below their use values. They prefer to let a competitor win and buy from her in the resale market. The existence of non-value-bidding equilibria is important because the celebrated advantages of the English auction, in particular efficiency, are based on value bidding and because resale is possible in most applications. In this paper we construct a family of nonvalue-bidding equilibria for an English auction that allows interbidder resale. Such equilibria exist in any independent private value environment (symmetric or asymmetric) for any number of bidders (Proposition 1). Each equilibrium in this family is identified by the choice of a designated bidder and a threshold type, below which all bidders, except the designated bidder, bid zero. All bidders with types above the threshold bid up to their values. In cases where the designated bidder wins the initial auction and has a sufficiently low type, she will offer the item for resale instead of consuming it. Because the determination of the designated bidder does not depend upon her type and the resale market retains information asymmetry, the final outcome may be inefficient. Since a designated bidder may win the initial auction at a low price, such equilibria provide an opportunity for a form of tacit collusion among the bidders. By using a publicly observed randomizing device (or sunspot) to choose the designated bidder, the surplus can be distributed in a way that makes every bidder of every type better off than under the value-bidding equilibrium; that 1 We thank Subir Bose, Paul Heidhues, George Mailath, Leslie Marx, Tymofiy Mylovanov, Greg Pavlov, Larry Samuelson, and three anonymous referees for helpful comments. We are particularly grateful to Dan Levin for suggesting that we investigate the collusive properties of our equilibrium construction.


DOI: 10.3982/ECTA7504

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R. J. GARRATT, T. TRÖGER, AND C. Z. ZHENG

is, the value-bidding equilibrium is interim (bidder) Pareto dominated (Proposition 2).2 The recommendation made by the sunspot device is not binding. Once the sunspot picks a designated bidder, it is in the interest of each bidder to bid accordingly in the initial auction based on the expectation that others will follow their assigned roles. Previous models of collusion in second-price and English auctions (e.g., Graham and Marshall (1987), Mailath and Zemsky (1991), Marshall and Marx (2007)) rely on pre-auction communication, in which every colluding bidder reports her type to the bidding ring. By communicating, the colluding bidders determine side payments and designate a single bidder to participate in the auction and win at a low price. These papers specify mechanisms that achieve efficient collusion as an equilibrium. However, the proposed use of pre-auction communication is problematic because it is usually illegal and participating bidders risk being detected. Moreover, the proposed collusive schemes require the nondesignated bidders in the bidding ring to bid below their values in the actual auction. Without resale, such bidding strategies are weakly dominated, and bidders may not be willing to play them. To make the expectation of a dominated strategy credible, the colluding bidders might require a commitment device.3 By introducing the possibility of resale after an English or second-price auction, our paper rationalizes collusion without pre-auction communication or dominated strategies. Instead of pre-auction communication of private information, a publicly observable sunspot selects a designated bidder in a manner commonly known to the colluding bidders, and the final owner of the good is decided through a resale mechanism. Before the auction, no one commits to what she will do in the auction or at resale. During the auction, colluding bidders do not bid up to their values; however, we prove that such strategies are not weakly dominated given the option for resale (Appendix B). The winner of the auction chooses a resale mechanism that is optimal for her given the posterior beliefs after the auction, and only after the initial auction has ended can she commit to the rules of her resale mechanism. McAfee and McMillan (1992, p. 587) noted that in practice a bidding ring’s own “knockout auction” often happens after rather than before the legitimate auction. This practice is well represented in our equilibria. 2

Readers who are familiar with U.S. litigation history might draw some parallels between our proposed use of a sunspots variable and the famous phases-of-the-moon bidding ring that was operated by electrical equipment suppliers in the 1950s. The phases-of-the-moon scheme earned its designation because it involved an explicit two-week rotation to determine the low bidder. See Smith (1961). 3 Sustaining collusion in first-price auctions is more difficult than in second-price auctions because nondesignated bidders have a strict incentive to overbid the designated bidder whose bid in the main auction is below their value; see McAfee and McMillan (1992) and Marshall and Marx (2007).

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Our Pareto-improving equilibria are, in contrast to the previously proposed collusive schemes, not ex post efficient.4 This is consistent with an impossibility result in Lopomo, Marshall, and Marx (2005), which shows that inefficiency is a quite general feature of Pareto-improving equilibria in English auctions without pre-auction communication. From the viewpoint of antitrust authorities, inefficient collusive equilibria are important precisely because of the distortion; efficient collusion involves merely a pure transfer from the seller to the bidders. Blume and Heidhues (2004) completely characterized the Bayesian Nash equilibria for the second-price auction with three or more bidders with a common type space. These equilibria are valid for English auctions and have the same bidding structure as our equilibria. However, because there is no resale market, the Blume–Heidhues equilibria are in dominated strategies. Moreover, in some environments none of these equilibria Pareto-dominates the value-bidding equilibrium, even if we give everyone a chance to be the designated bidder through sunspot coordination. This is because high-value bidders strictly prefer the value-bidding equilibrium; we show this for all symmetric environments with strictly concave value distributions (Proposition 3). But with resale, there always exists an equilibrium that makes every type of every bidder strictly better off than the value-bidding equilibrium. In contrast to much of the earlier literature on auctions with resale, our model allows for any number of asymmetric bidders. With multiple bidders at the resale stage, the optimal resale mechanism typically is no longer a takeit-or-leave offer, as often assumed, but is an optimal auction as derived by Myerson (1981). A technical novelty of our paper is that we avoid complicated explicit computations of bidders’ resale payoffs. We identify several structural properties of the resale continuation game that facilitate our perfect Bayesian equilibria for the entire auction-with-resale game. These structural properties do not appear to be specific to the Myerson optimal auction, suggesting that our qualitative results extend to other forms of the resale market.5 Our result that the value bidding equilibrium is interim (bidder) Pareto dominated is based on two regularity properties of resale that are satisfied in our equilibria. The properties ensure that even when the threshold is arbitrarily close to zero, a nonvanishing fraction of the bidder types below the threshold still engage in actual resale trade. The proof begins with the observation that our equilibria interim Pareto-dominate the value-bidding equilibrium if the prior type distributions are uniform. Then we extend this dominance rela4 The payoff gains in our equilibria relative to value bidding can still be substantial (see Table I, Section 5). 5 For instance, the period-2 seller may be restricted to use a second-price or English auction with an optimal reserve price, or the period-2 seller may use an English auction with the right to reject all bids (Haile (2003)), or, in two-bidder environments, a random draw may specify which bidder has the right to propose a resale price (Calzolari and Pavan (2006)).

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tion to arbitrary type distributions provided that the thresholds in our equilibria are sufficiently small. Sufficiently small thresholds allow us to approximate the expected payoffs by the ones in the uniform-distribution case. The aforementioned regularity properties imply that a bidder’s gain from trade at resale outweighs the error of the approximation. The threshold-bidding strategies in our equilibria are built upon Garratt and Tröger (2006) for English and second-price auctions. However, there are nontrivial differences. In Garratt and Tröger, the bidder who will become the reseller has no private information and the other bidders have identical prior value distributions. In contrast, in this paper every bidder has private information and bidders’ value distributions can differ. Our extension of the equilibrium construction to the case of asymmetric bidders is made possible by conditioning the designated bidder’s bid on the identities of the bidders who stay in the auction. This information is not available in a sealed-bid format. Hence, in environments with three or more bidders, our equilibrium construction applies to second-price auctions only under an additional symmetry assumption (Remark 3). Recent works by Lebrun (2007) and Hafalir and Krishna (2008) compared revenue in first- and second-price auctions with resale in two-bidder models. Hafalir and Krishna showed that in a two-bidder asymmetric model there exists a “general revenue ranking” in favor of first-price auctions, provided bidders play the value-bidding equilibrium in the second-price auction. Lebrun showed that, depending on the selected equilibria, this ranking does not necessarily hold when behavior (mixed) strategies are allowed. Our construction of equilibria for second-price auctions that interim Pareto-dominate value bidding does not challenge the revenue ranking established by Hafalir and Krishna. Because the value-bidding equilibrium is efficient, it yields a seller revenue that is necessarily greater than in any interim Pareto-dominating equilibrium. Collusive equilibria have been constructed for multiunit auctions by Milgrom (2000), Brusco and Lopomo (2002), and Engelbrecht-Wiggans and Kahn (2005); however, resale does not play a role.6 In these multiunit environments bidders signal their preferences in early rounds and then optimally abstain from bidding on other bidders’ preferred items. Interestingly, the open aspect of the ascending English auction is essential in their construction, as it is here in the case of ex ante asymmetric bidders. 2. MODEL We consider environments with n ≥ 2 risk-neutral bidders pursuing a single indivisible private good. Bidder i ∈ N := {1 n} has a privately known use value, or type, ti ∈ Ti = [0 t i ] (t i > 0) for the good. The type space is denoted by 6

Pagnozzi (2007) analyzed multiunit auctions with resale in a complete-information model.


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T := T1 × · · · × Tn .7 From the viewpoint of the other bidders, ti is independently distributed according to a probability distribution with cumulative distribution function Fi , called prior belief, with support Ti and Lipschitz continuous positive density fi . We consider a two-period interaction, which begins after each bidder i has privately observed her use value ti . In period 1, the good is offered via an English auction that is modeled as in Milgrom and Weber (1982). The auctioneer continuously raises the current price beginning at 0; Remark 1 extends the equilibrium construction to an English auction that starts at a positive price. Initially, all bidders are “in” or “active.” As the current price rises, each bidder can irreversibly “drop out” at any point. When somebody drops out, other bidders may react by dropping out at the same price. Dropout decisions are publicly observed. The auction ends at the first current price where at most one bidder is active. The ending current price is called the auction price. If one bidder is active at the end, this is the auction winner; if all active bidders drop out at the auction price, there is a tie among the previously active bidders and any tieing bidder becomes the winner with equal probability. The auction winner either consumes the good in period 1, thereby ending the game, or becomes the period-2 seller. The period-2 seller proposes a sales mechanism to the losing bidders, called period-2 buyers. A sales mechanism is any game form to be played by the period-2 buyers. The mechanism is played if it is accepted by all period-2 buyers; otherwise, the period-2 seller consumes the good in period 2.8 Observe that the period-2 seller faces no restrictions: she can choose an arbitrary sales mechanism. No bidder can pre-commit in period 1 to use a particular mechanism in period 2. Every bidder’s discount factor is δ ∈ (0 1]. From the viewpoint of period 1, the payoff of type ti of bidder i is ti (q1 + δq2 ) − p1 − δp2 , where qk (k = 1 2) denotes the probability of her consuming the good in period k and pk denotes her net expected monetary payment in period k. From the viewpoint of period 2, the bidders only care about their period-2 payments and period-2 allocation probabilities. 2.1. Histories, Strategies, Beliefs, and Period-2 Outcome A nonterminal history lists the observed dropout decisions and the current price at any point before the period-1 auction ends. The set of nonterminal histories where bidder i ∈ N is active is denoted Hi . A terminal history lists the observed dropout decisions up to the end of the auction, the auction price, and the winner. The set of terminal histories is denoted Hterm . The set of all histories is denoted H = H1 ∪ · · · ∪ Hn ∪ Hterm . 7

We shall use boldface letters to denote multidimensional quantities. We assume that there is no further resale after the reseller’s resale mechanism. However, for certain prior beliefs, this assumption can be weakened to allow the winner of the resale mechanism to offer further resale à la Zheng (2002). 8

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A bidding strategy profile (βi (· | h))i∈Nh∈Hi determines, for any bidder i ∈ N, any history h ∈ Hi , and any type ti ∈ Ti , the current price b = βi (ti | h) at which bidder i plans to drop out. We call b the bid of type ti of bidder i at h. If b is smaller than or equal to the current price at h, then b is interpreted as dropping out immediately at h. Observe that if at h a bidder other than bidder i bids b < b, then the plan to drop out at b becomes irrelevant once the current price b is reached. A resale decision profile is a vector (γh )h∈Hterm , where γh (t) equals resale if type t of the winner offers the good for resale at the terminal history h and where γh (t) equals consume if the winner consumes the good in period 1. A period-1 belief profile (Ah )h∈H assigns to each history h a probability distribution Ah on T that represents the (common) belief at h about the bidders’ types. A period-2 belief profile (Gh )h∈Hterm assigns to each terminal history h a probability distribution Gh on T that represents the (common) belief about the bidders’ types if the winner offers the good for resale. Using the revelation principle, we can describe a period-2 outcome as a vector (Pi (t) Qi (t))i∈Nt∈T such that, for any bidder i and type profile t ∈ T, the number Pi (t) denotes the expected net period-2 monetary transfer of bidder i (t) denotes the probability that bidder i consumes the good in period 2, and Qi where k∈N Qk (t) = 1 and k∈N Pk (t) = 0. 2.2. The Equilibrium Concept An equilibrium consists of a bidding profile (βi (· | h))i∈Nh∈Hi , a resaledecision profile (γh )h∈Hterm , a period-1 belief profile (Ah )h∈H , a period-2 belief profile (Gh )h∈Hterm , and a family of period-2 outcomes (Pih (t) Qih (t))i∈Nt∈Th∈Hterm with the following properties: (a) For all h ∈ Hterm , the period-2 outcome (Pih (t) Qih (t))i∈Nt∈T is induced by a perfect Bayesian equilibrium of the period-2 continuation game, given the period-2 seller from terminal history h and the (commonly known) period-2 belief Gh . (b) The period-1 belief profile (Ah )h∈H obeys Bayes’ rule with respect to the bidding profile (βi (· | h))i∈Nh∈Hi . (c) For all h ∈ Hterm , j ∈ N, and tj ∈ Tj , if bidder j with type tj is the period-1 winner at history h, then γh (tj ) equals resale (resp., consume) if tj is strictly smaller (resp., greater) than j’s discounted period-2 payoff, given the period-2 belief Gh and the expected period-2 outcome (Pih (t) Qih (t))i∈Nt∈T . (d) The period-2 belief profile (Gh )h∈H obeys Bayes’ rule with respect to the period-1 belief profile (Ah )h∈H and the resale-decision profile (γh )h∈Hterm . (e) For all i ∈ N, ti ∈ Ti , and h ∈ Hi , the bid βi (ti | h) maximizes the expected payoff of type ti of bidder i at the history h, given the belief Ah , provided that everyone else abides by the bidding profile, that bidder i abides by her bidding strategy after additional bidders drop out, and that the resale-decision profile and family of period-2 outcomes are implemented.


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We construct equilibria where period-1 beliefs and period-2 beliefs are stochastically independent across bidders, and where any belief about bidder i’s (i ∈ N) type is derived from the prior belief and the information that her type lies in a (possibly degenerate) interval Ji ⊆ Ti . Let J denote the set of interval products J = J1 × · · · × Jn ⊆ T. We identify any belief with an element of J . That is, we will treat J variably as a product of intervals or as a cumulative distribution function on T and will treat Ji variably as an interval or a cumulative distribution function on Ti . 3. THE PERIOD-2 CONTINUATION GAME In view of equilibrium condition (a), we select a perfect Bayesian equilibrium of the period-2 continuation game for any period-2 seller j ∈ N and any belief J = J1 × · · · × Jn ∈ J . We use a well known selection: the mechanism proposed by the period-2 seller is Myerson’s (1981) auction that is optimal for period-2 seller j based on the period-2 belief profile J, given the assumption that resale after period 2 is impossible.9 Denote the period-2 outcome implemented by jJ jJ this auction by (Pi (t) Qi (t))i∈Nt∈T for any (j J) specified above.10 Notation is needed to describe the properties of the Myerson optimal auction N, let J−i = outcome that are used in the following analysis. For all i ∈ J denote the marginal distribution induced by J on T := −i k=i k k=i Tk . The probability that type ti ∈ Ti of bidder i consumes the good in period 2 (by keeping it if i = j and obtaining it if i = j) is denoted jJ Qi (t) dJ−i (t−i ) qij (ti J) = T−i

and her period-2 expected payoff is denoted jJ lij (ti J) = ti qij (ti J) − Pi (t) dJ−i (t−i ) T−i

For the period-2 seller, we will use the shortcuts qj (tj J) = qjj (tj J) and wj (tj J) = ljj (tj J). 9 Our period-2 environment differs from Myerson’s environment insofar as the period-2 seller may be privately informed about her type. This plays no role because, by assumption, the period-2 seller is not a player in her sales mechanism, so that the period-2 buyers’ beliefs about the seller’s type have no impact on their behavior. If we did allow mechanisms where the period-2 seller is a player, the Myerson optimal auction outcome would still be a “strong solution” as defined by Myerson (1983); see Mylovanov and Tröger (2008). 10 Observe that the period-2 outcome includes payments and allocation probabilities for types / Ji occur with probability 0. We assume that any not in J. The period-2 seller believes that types ti ∈ type ti > sup Ji obtains the good with the same probability as type sup Ji and any type ti < inf Ji obtains the good with probability 0.

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A well known implication of the incentive compatibility of the period-2 outcome, proved via the envelope theorem in integral form (Milgrom and Segal (2002)), is the following envelope formulas: for all ti ti ∈ Ti (ti < ti ) and tj tj ∈ Tj (tj < tj ), ti lij (ti J) − lij (ti J) = (1) qij (x J) dx ti

(2)

j

wj (tj J) − wj (t J) =

tj tj

qj (x J) dx

Period-2 payments can be defined such that the ex post participation conditions are satisfied: for all t ∈ T, j ∈ N, and i = j, jJ

jJ

jJ

jJ

(3)

tj ≤ tj Qj (t) − Pj (t)

(4)

0 ≤ ti Qi (t) − Pi (t)

If there are expected gains from trade between the period-2 seller and all buyers, then by using an optimal auction, the period-2 seller j captures a nonzero share of the gains: for all tj ∈ Tj , (5)

[∀i ∈ N \ {j} max Ji > tj ] ⇒ wj (tj J) > tj

A further straightforward property of the Myerson optimal auction outcome is that the period-2 seller’s period-2 payoff is continuous in the period-2 belief about her type (in fact, the payoff is independent of the belief). To state this property, let [0 x] × J−j denote the belief that bidder j’s type is at most x and other bidders’ types are in J−j . For all j ∈ N, tj ∈ Tj , and J ∈ J , the map (6)

x → wj (tj [0 x] × J−j ) is continuous on Tj

The next property—that a bidder’s period-2 payoff as a buyer is never larger than as a seller—follows from the fact that the bidder’s type is a lower bound for her period-2 payoff as a seller and an upper bound for her period-2 payoff as a buyer. For all i ∈ N, ti ∈ Ti , j ∈ N \ {i}, and J ∈ J , (7)

wi (ti J) ≥ lij (ti J)

Another property of the Myerson optimal auction outcome is proved in Appendix A: each bidder as a period-2 seller consumes the good with a weakly higher probability than obtaining it as a period-2 buyer, given the same period-2 beliefs. LEMMA 1: Let i ∈ N, ti ∈ Ti , j ∈ N \ {i}, and J ∈ J . Then (8)

qi (ti J) ≥ qij (ti J)


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Our equilibrium construction in the next section applies to any resale market that satisfies properties (1)–(8). None of these properties appears specific to the Myerson optimal auction outcome. In particular, (5) simply reflects the facts that the period-2 seller has some bargaining power and there is some resale trade. Property (7) is implied by (3) if n = 2; otherwise, (7) essentially provides an upper bound on an auction loser’s ability to extract rents if two other bidders trade the good in period 2. Property (8) reflects the basic intuition that private information leads to less trade than efficiency requires. 4. EQUILIBRIA FOR ENGLISH AUCTIONS WITH RESALE In this section, we construct a family of equilibria for the English auction with resale. In each equilibrium, one of the bidders, say bidder 1, is commonly known to be the designated bidder of the period-1 auction. Bidding strategies depend upon a threshold t ∗ , which can take on any value in the interval (0 mini∈N t i ]. Every bidder with type above t ∗ bids her own type. All designated-bidder types below t ∗ bid more than 0 and not more than t ∗ . Any other bidder with a type below t ∗ drops out at the beginning of the auction. If someone with type above t ∗ wins, resale does not occur. Otherwise, the designated bidder wins at price 0; if her type is sufficiently low, she offers the good for resale in period 2 according to a continuation equilibrium described in Section 3. Since the selection of the designated bidder does not depend upon her type and informational asymmetries remain at resale, these equilibria are inefficient, contrary to the value-bidding equilibrium of English auctions.11 Before we present the equilibria (Proposition 1), we state some results that are used to specify the period-1 strategy for bidder 1. First, we establish that bidder 1’s resale decision is defined by a nonzero cutoff (Lemma 2) such that lower types of bidder 1 prefer offering resale to immediate consumption and higher types have the reverse preference. Second, for each bidder i = 1, we define a price at which type t ∗ is indifferent between winning the period-1 auction and waiting for resale, and we show that these prices are strictly positive (Lemma 3). These prices become the bids for all types below t ∗ of bidder 1 who offer resale. Fix a threshold t ∗ > 0. For any x ∈ T1 , let J∗x = [0 x] × [0 t ∗ ]n−1 denote the belief resulting from the prior belief and the information that bidder 1’s type is below x and the other bidders’ types are below t ∗ . To find a cutoff type between consumption and offering resale, let (9)

τ∗ := sup{x ∈ T1 |δw1 (x J∗x ) > x}

if

δ < 1

We define τ∗ = t 1 if δ = 1. 11 Haile (1999) proved that when resale after an English or second-price auction is allowed, the efficient value-bidding equilibrium remains valid.

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LEMMA 2: Let t ∗ be a threshold. Then 0 < τ∗ < t ∗ if δ < 1. For all t1 ∈ T1 , (10)

t1 ≥ δw1 (t1 J∗τ∗ ) if t1 > τ∗

(11)

t1 ≤ δw1 (t1 J∗τ∗ ) if t1 < τ∗

(12)

τ∗ → t ∗

as δ → 1

δ = 1

For all bidders i = 1, let b∗i denote the price that makes type t ∗ of bidder i indifferent between (i) winning the auction at price b∗i and consuming the good, and (ii) participating in a resale market where bidder 1 is the period-2 seller and the period-2 belief is J∗τ∗ : (13)

b∗i := t ∗ − δli1 (t ∗ J∗τ∗ )

The following lemma provides bounds for b∗i . LEMMA 3: Let t ∗ be a threshold. Then 0 < b∗i < t ∗ for all bidders i = 1. The bidding strategy for designated-bidder types below t ∗ is as follows. If her type is below τ∗ , then at any history h ∈ H1 she bids maxi∈S1 (h) b∗i , where S1 (h) denotes the set of bidders other than 1 who are active at the history h, and she will offer resale if she wins. If her type is between τ∗ and t ∗ , then she bids t ∗ and she does not offer resale if she wins. PROPOSITION 1: For any threshold t ∗ , there exists an equilibrium with properties (i)–(iv). (i) For any history h ∈ H1 , type t1 ∈ T1 of bidder 1 bids ⎧ max b∗i if t1 ≤ τ∗ , ⎪ ⎨ i∈S 1 (h) β1 (t1 | h) = t ∗ (14) if τ∗ < t1 ≤ t ∗ , ⎪ ⎩ if t1 > t ∗ . t1 (ii) For all bidders i = 1 and any history h ∈ Hi , type ti ∈ Ti of bidder i bids 0 if ti ≤ t ∗ , βi (ti | h) = (15) ti if ti > t ∗ . (iii) Let hˆ ∈ Hterm denote the history where all bidders other than bidder 1 have dropped out at the beginning of the auction. Bidder 1’s resale decision at history hˆ is Resale if t1 ≤ τ∗ , (16) γhˆ (t1 ) := Consume if t1 > τ∗ .


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ˆ the period-2 outcome is the Myerson optimal auction outcome (iv) At history h, given the period-2 seller (bidder 1) and the period-2 belief [0 τ∗ ] × [0 t ∗ ]n−1 . Any equilibrium that satisfies properties (i)–(iv) is called a t ∗ -equilibrium. Proposition 1 is proved in Appendix A. Here we explain heuristically, for the case of δ < 1, why a bidder i = 1, who is not the designated bidder, would abide by the equilibrium strategy of dropping out of the auction if her type is below the threshold t ∗ . Bidder i can deviate in at least two ways. Either she can try to outbid the designated bidder (bidder 1) and consume the good upon winning it or she can try to outbid bidder 1 and offer resale upon winning. To explain why both kinds of deviation are unprofitable, let us suppose that all bidders other than bidder i have types below t ∗ and that bidder 1’s type is below τ∗ . Otherwise, bidder i must bid at least t ∗ to win the auction and cannot resell at a profit. If bidder i manages to outbid bidder 1 and consumes the good upon winning, then her payoff is equal to ti − b∗i , because bidder 1 bids up to b∗i against bidder i and everyone else quits at zero price. This payoff is represented by the slope-1 straight line, labeled X, in Figure 1. If bidder i plays the equilibrium strategy of dropping out of the auction and trying to buy the good at resale, then her expected payoff is equal to δli1 (ti J∗τ∗ ); that is because bidder 1 wins the good at zero price and hence the post-auction belief is J∗τ∗ . This expected payoff is represented by the curve labeled Y in Figure 1. Note that curve Y and line X intersect at the threshold t ∗ . This is because bidder 1’s highest bid b∗i against a deviant bidder i, defined by (13), makes bidder i of type t ∗ indifferent between consuming now and buying at resale. An important point is that curve Y is everywhere less steep than line X, so that bidder i with types below t ∗ prefers abiding by the equilibrium and waiting for resale to outbidding bidder 1 and consuming the good. This follows from the envelope formula (1): When ti changes from t ∗ , the payoff from consuming now, ti − bi (t ∗ ), changes at the rate 1, while the payoff from buying at resale, δli1 (ti J∗τ∗ ), changes at the rate δqi1 (ti J∗τ∗ ) < 1.

FIGURE 1.—X, consume now; Y , buy at resale; Z, win now and offer resale.

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Now consider the deviation where bidder i outbids the designated bidder and offers the good for resale upon winning. If bidder i manages to do that, then her role in period 2 is switched from a bidder to a seller and the postauction beliefs J∗τ∗ are unchanged. Her expected payoff at the start of period 2 will be equal to wi (ti J∗τ∗ ), so her present expected payoff from such a deviation is equal to δwi (ti J∗τ∗ ) − b∗i . This payoff is represented by the curve labeled Z in Figure 1. Note that curve Z is below curve X for all ti ≥ t ∗ . This is because a reseller whose type is above t ∗ cannot profit from resale, since everyone else’s type is below t ∗ . A crucial observation is that, for ti < t ∗ , the curve Z is everywhere steeper than the curve Y . As Z is below Y at t ∗ , that means Z is always below Y for types below t ∗ , that is, bidder i with types below t ∗ would rather be a period-2 buyer than a period-2 seller. To verify these relationships, recall the definition of b∗i given in (13). We have Z ∗ := δwi (t ∗ J∗τ∗ ) − b∗i ≤ wi (t ∗ J∗τ∗ ) − b∗i = t ∗ − b∗i = δli1 (t ∗ J∗τ∗ ) =: Y ∗ i By the envelope formulae (1) and (2), ∂w (ti J∗τ∗ ) = qi (ti J∗τ∗ ) and ∂l∂ti1i (ti J∗τ∗ ) = ∂ti qi1 (ti J∗τ∗ ). By the inequality (8), qi (ti J∗τ∗ ) ≥ qi1 (ti J∗τ∗ ). Thus, the expected payoff from winning now and offering resale, δwi (ti J∗τ∗ ) − b∗i , decreases from level Z ∗ faster than the expected payoff from buying at resale, δli1 (ti J∗τ∗ ), decreases from level Y ∗ . As Z ∗ ≤ Y ∗ , the claim is established. We provide some remarks on Proposition 1.

REMARK 1: Proposition 1 can be extended to the case where the English auction in period 1 has a reserve price r > 0. Amend the English auction as follows. The auction starts with a current price lower than r (say zero price) that corresponds to “no sale.” If someone drops out at no sale, then the price clock pauses to give others a chance to drop out. Once no more bidders drop out at no sale, the price clock jumps to the reserve price r. An equilibrium can be constructed for any threshold t ∗ > r, provided the discount factor is sufficiently close to 1. Let tˆ ∈ T1 be the type of bidder 1 such that her expected payoff (for the entire auction-resale game) is zero if she wins the good at price r and offers the good for resale, given the belief that the types in [0 t ∗ ] of other bidders participate in the resale market. Suppose δ is sufficiently close to 1, so that tˆ < r. According to the equilibrium, bidder 1 drops out at no sale if and only if her type is below tˆ. Once bidder 1 has dropped out at no sale, other bidders play the value-bidding equilibrium; if bidder 1 does not drop out at no sale, then the bidders’ subsequent actions are analogous to the equilibria described in Proposition 1, where “dropping out at zero price” is replaced by “dropping out at no sale.” Resale occurs given the belief that


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(i) bidder 1’s type is distributed on [tˆ τ] for some τ ∈ (r t ∗ ) and (ii) the other bidders’ types are distributed on [0 t ∗ ]. REMARK 2: The t ∗ -equilibrium construction makes essential use of the transparent dynamic nature of an English auction, because the designated bidder’s dropout price depends on the set of the other bidders who have not dropped out (the upper branch of (14)). This dependence is important in our construction because, by (13), bidders drawn from different distributions need different prices b∗i to be kept obedient to the threshold t ∗ . For exactly this reason, the t ∗ -equilibrium construction does not generally extend to second-price auctions. The construction does extend if b∗2 = · · · = b∗n , which holds if bidders 2 to n are ex ante symmetric. REMARK 3: The t ∗ -equilibria are not the only equilibria that differ from the value-bidding equilibrium. There also exist “extreme equilibria,” where bidder 1’s bid is so high that all types of all other bidders find it optimal to drop out at the beginning of the auction.12 Extreme equilibria are conceptually simpler than t ∗ -equilibria; however, there are practical reasons why extreme equilibria might not be played. First, in an extreme equilibrium, the good is always sold at zero price at the initial auction. This would make a regulator suspicious of collusion, which the bidders may want to avoid. Second, if low-type bidders have a budget constraint that prevents them from staying active up to very high prices, then a designated bidder’s bidding strategy in an extreme equilibrium is not credible.13 Third, extreme equilibria cannot, generally, be used to obtain the interim Pareto-dominance property described below (see Section 5.1 for an example). REMARK 4: One may drop the assumption that the period-2 seller can prevent further resale transactions. Suppose that, beginning with the period-2 seller, each current owner of the good designs a sales mechanism, knowing that the next owner will design her own sales mechanism, and so on. This amounts to using Zheng’s (2002) repeated-resale game to describe period 2 of our model. For a certain class of period-2 beliefs,14 Zheng’s result shows that there exists a period-2 perfect Bayesian continuation equilibrium such that the final outcome is still the Myerson optimal auction outcome intended by the period-2 seller (though this outcome is achieved via intermediate sales mechanisms different from Myerson’s). Accordingly, for certain prior beliefs, our t ∗ -equilibrium construction extends to the repeated-resale market without any 12 Zheng (2000, Section 5.2) constructed an extreme equilibrium in a second-price-auction-type mechanism with reserve prices. See also Garratt and Tröger (2006, Section 4). 13 Brusco and Lopomo (2008) made this point previously in a no-resale model. 14 Mylovanov and Tröger (2009) characterized the class of beliefs such that Zheng’s construction applies.

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change; this is true, in particular, for symmetric environments (i.e., where prior beliefs are identical across bidders). One may conjecture that the properties (1)–(8) hold for the repeated-resale market for a larger class of prior beliefs, but proving this hinges on first solving Zheng’s repeated-resale game for the corresponding beliefs. 5. THE INTERIM PARETO DOMINANCE OF COLLUSION We assume that a sunspot (à la Shell (1977) and Cass and Shell (1983)) with n equally likely states is commonly observed before the period-1 auction starts (and after the bidders have been privately informed). This extends the game so that actions can depend on the realization of the sunspot state. Given any strategy profile, the payoff of any type of a given bidder is defined via the expectation over the n sunspot states. We call an equilibrium in the extended game Pareto improving if it interim Pareto-dominates the value-bidding equilibrium, that is, if every type of every bidder is strictly better off than in the value-bidding equilibrium. We show that Pareto-improving equilibria exist in any environment. For any threshold t ∗ , we define a t ∗ -collusive equilibrium: if the realized sunspot state is j = 1 n, then a t ∗ -equilibrium is played, with bidder j taking the role of the designated bidder. Clearly this constitutes an equilibrium. PROPOSITION 2: The t ∗ -collusive equilibria are Pareto improving for all t ∗ sufficiently close to 0. Note the generality of this result. A Pareto-improving equilibrium exists in any symmetric or asymmetric environment; in particular, strong bidders can gain from colluding with weak bidders. Moreover, the discount factor does not have to be close to 1; the result applies to any nonzero discount factor. The proof of Proposition 2 utilizes two regularity properties of the period-2 outcome when the period-2 belief is concentrated on types close to 0. These properties, which are stated in Lemma 5 and Lemma 6, do not appear to be specific to the Myerson optimal auction outcome; the conclusion of Proposition 2 holds for any resale market outcome that satisfies these properties. The first step toward the proof of Proposition 2 is to observe that it is sufficient to focus the payoff comparison on the types in the interval [0 t ∗ ]. LEMMA 4: In a t ∗ -collusive equilibrium, if type t ∗ of a given bidder is strictly better off than in the value-bidding equilibrium, then all types above t ∗ of this bidder are strictly better off. PROOF: Consider any type ti ≥ t ∗ of a bidder i ∈ N. Her payoff in the t ∗ collusive equilibrium can be different from her value-bidding equilibrium payoff only in the event that the highest type among the other bidders is less than


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(1) ≤ t ∗ ). Then bidder i’s payoff in the value-bidding equilibor equal to t ∗ (t−i (1) b∗i rium is ti − t−i and her payoff in the t ∗ -collusive equilibrium is either ti − n−1 n or ti − n−1 t ∗ , depending on whether the sunspot variable selects a designated n bidder j = i with type tj < τ∗ or not. Either way, the payoff difference is indeQ.E.D. pendent of ti .

The idea behind using t ∗ close to 0 in Proposition 2 is to make the interval of relevant types [0 t ∗ ] small, so that approximations of payoff comparisons can be obtained using first-order Taylor expansions of the prior belief distributions, (17)

Fi (x) = fi (0)x + hi (x)

(x ≥ 0 i = 1 n)

where hi (x)/x → 0 as x → 0. The Taylor expansions (17) are exact (hi (x) = 0) if the priors are (possibly asymmetric) uniform distributions. The uniform-priors example captures the rough idea of why Proposition 2 is correct. In the value-bidding equilibrium with uniform priors, it is well known that the payoff of type ti ≤ t ∗ of bidder i is ti

tn (18) Fk (x) dx = fk (0) i Uivaluniform (ti ) = n 0 k=i k=i For the t ∗ -collusive equilibrium payoff with uniform priors, denoted Ui∗uniform (ti ), we obtain a lower bound by not counting the gains from resale trade: 1

(19) fk (0)(t ∗ )n−1 ti Ui∗uniform (ti ) ≥ n k=i because with probability 1/n, bidder i is the designated bidder, in which case she gets the good for free if all others have valuations below t ∗ . Clearly, (18) and (19) imply (20)

Ui∗uniform (ti ) > Uivaluniform (ti ) for all ti ∈ (0 t ∗ )

The strict inequality in (20) also holds at ti = 0 and ti = t ∗ , because in a t ∗ collusive equilibrium, type 0 makes a profit as a period-2 seller with positive probability and type t ∗ gets an information rent as a period-2 buyer with positive probability. Thus, in the uniform-priors example, t ∗ -collusive equilibria are Pareto improving for all t ∗ . Using the Taylor expansion (17), we generalize this result to arbitrary prior distributions and small t ∗ via several lemmas that are proved in Appendix A. For all i ∈ N, let τ∗i be defined analogously to τ∗ , with bidder i instead of bidder 1 taking the designated-bidder role. Lemma 5 states that, for small t ∗ , a nonvanishing fraction of the designated-bidder types below t ∗ offer the good for resale on the equilibrium path of a t ∗ -equilibrium.

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LEMMA 5: There exists 0 < θ < 1 such that, for all t ∗ sufficiently close to 0, (21)

∀i ∈ N: τ∗i > θt ∗

The proof uses the fact that the period-2 seller, if she believes that each buyer’s type belongs to [0 t ∗ ], is free to make a take-it-or-leave-it fixed-price offer at t ∗ /2 to any buyer. This lower bound on what the period-2 seller can achieve ensures that all types that are sufficiently small relative to t ∗ offer resale, thus bounding τ∗i from below. For all i ∈ N, let J∗i denote the belief resulting from the prior belief and the information that bidder i’s type is below τ∗i and the other bidders’ types are below t ∗ . Lemma 6 states that, given the period-2 belief J∗i for any small t ∗ , a nonvanishing fraction of the period-2 buyer types below t ∗ buy the good with a nonvanishing probability; if δ = 1, the probability is conditional on the seller’s type being below t ∗ . LEMMA 6: There exist 0 < ξ < 1 and ε > 0 such that for all t ∗ sufficiently close to 0, ⎧ ∗j ⎨ qij (ti J ) if δ < 1, ∀i ∈ N j ∈ N \ {i} ti ∈ [ξt ∗ t ∗ ]: ε < qij (ti J∗j ) (22) ⎩ if δ = 1. Fj (t ∗ ) The intuition behind this result is that because the conclusion (22) holds for any t ∗ in the uniform-priors example, one can use the Taylor expansion (17) to show the conclusion for arbitrary priors and small t ∗ . The proof uses the assumption that the prior densities are Lipschitz continuous. This implies that the virtual valuation functions (Myerson (1981)) for the period-2 beliefs about the period-2 buyers’ types are strictly increasing if t ∗ is small. According to the Myerson optimal auction outcome, the good is then resold to the period-2 buyer with the highest virtual valuation, unless the period-2 seller’s type is higher. This allocation rule yields explicit formulas for the buyer-allocation probabilities qij (ti J∗j ). The lower bound (22) is obtained via approximations for the virtual valuation functions that are obtained using (17). The next lemma provides an approximation result for payoffs in the valuebidding equilibrium.15 LEMMA 7: The payoff of type ti ≤ t ∗ of bidder i ∈ N in the value-bidding equilibrium is

1 fk (0) ti n + O((t ∗ )n ) Uival (ti ) = n k=i 15 For any k ≥ 0, we will use O ((t ∗ )k ) to denote any function h(x t ∗ ) (or h(t t ∗ ) or h(ti t ∗ )) such that supx∈[0t ∗ ] |h(x t ∗ )|/(t ∗ )k → 0 as t ∗ → 0.


1111

The next lemma states that, given the period-2 belief J∗i for any small t ∗ , a nonvanishing fraction of the period-2 seller types below t ∗ sell the good with a nonvanishing probability. LEMMA 8: Let θ be as in Lemma 5. There exist 0 < ξ < θ and ε > 0 such that, for all t ∗ sufficiently close to 0, (23)

∀j ∈ N tj ∈ [0 ξt ∗ ]: qj (tj J∗j ) < 1 − ε

To prove this, we observe that the upper bound (23) follows from the lower bound (22) because buyer- and seller-allocation probabilities add up to 1 in expectation over all types. Let Uijb (ti ) denote the payoff of type ti ≤ t ∗ of bidder i ∈ N in a t ∗ -equilibrium where j = i is the designated bidder. Using the lower bound on the trading probability (22) and the envelope formula (1), we obtain a lower bound for her payoff. LEMMA 9: Let ξ and ε be as in Lemma 6. For all sufficiently small t ∗ and ti ∈ [0 t ∗ ],

Fk (t ∗ ) Uijb (ti ) ≥ δε max{0 ti − ξt ∗ }Fj (θt ∗ ) k∈{ij} /

Let Uis (ti ) denote the payoff of type ti ≤ t ∗ of bidder i ∈ N in a t ∗ -equilibrium where i is the designated bidder. Using the upper bound on the no-trade probability (23) and the envelope formula (2), we obtain a lower bound for her payoff. LEMMA 10: Let ξ and ε be as in Lemma 8. For all sufficiently small t ∗ and ti ∈ [0 t ∗ ],

Fk (t ∗ ) Uis (ti ) ≥ (ti + δε max{0 ξt ∗ − ti }) k=i

Let Ui∗ (ti ) denote the payoff of type ti ≤ t ∗ of bidder i ∈ N in a t ∗ -collusive equilibrium. Combining Lemma 9 and Lemma 10, we get, for all sufficiently small t ∗ and ti ∈ [0 t ∗ ], (24)

Ui∗ (ti ) ≥

1 (ti + δε max{0 ξt ∗ − ti }) Fk (t ∗ ) n k=i +

n−1 δε max{0 ti − ξt ∗ }Fj (θt ∗ ) Fk (t ∗ ) n k∈{ij} /

1112


1 1 n−1 ti + δε max{0 ξt ∗ − ti } + δε max{0 ti − ξt ∗ }θ = n n n

∗ n−1 ∗ n−1 · fk (0)(t ) + O((t ) )

(17)

(25)

k=i

=

(26)

k=i

fk (0)g

ti (t ∗ )n + O((t ∗ )n ) t∗

where 1 1 n−1 δε max{0 x − ξ}θ g(x) = x + δε max{0 ξ − x} + n n n (x ∈ [0 1]) Because g(0) > 0, g(1) > 1/n, and x > xn if 0 < x < 1, ∀x ∈ [0 1]: g(x) >

xn n

Hence, combining (26) with Lemma 7 gives Ui∗ (ti ) − U val (ti )

xn g(x) − ≥ f (0) min +O(1) k x∈[01] (t ∗ )n n k=i

>0

Hence, for sufficiently small t ∗ , min (Ui∗ (ti ) − U val (ti )) > 0

ti ∈[0t ∗ ]

This completes the proof of Proposition 2. We provide some remarks on Proposition 2. REMARK 5: The uniform-priors example shows that the gains from playing a Pareto-improving equilibrium can be quite large. Table I shows the gains to a bidder with type t ∗ = 09 in an environment with Fi (i ∈ N) uniform on [0 1], for various numbers of bidders n. The gains to type t ∗ are the minimum gains over all types in this example. REMARK 6: Proposition 2 extends to an English auction with a small reserve price. This follows from Remark 1, by continuity. For larger reserve prices r, the question is whether bidders can collude so that the payoff of any bidder type above r is larger than in the value-bidding equilibrium with reserve price r (where bidders with types below r abstain). We have three results for environments where the prior beliefs Fi (i ∈ N) are uniform on [0 1]. First, if n = 2,


1113

TABLE I Fi (t) = t, t ∗ = 09 n

Uival (09)

Ui∗ (09)

% Increase

2 5 10

0.405 0.1181 0.03487

0.50625 0.19043 0.06277

25 61.24 80.01

then a Pareto improving equilibrium exists for any reserve price below 1. Second, if n ≥ 4, then a Pareto-improving equilibrium exists if the optimal reserve price under value bidding, 1/2, is used. Third, a Pareto-improving equilibrium exists for any reserve price arbitrarily close to 1 if n is sufficiently large. Proposition 2 establishes that the bidders can always achieve some, possibly small, Pareto improvement over value bidding. This raises two questions. First, is the restriction to small t ∗ needed for this result? Second, is resale trade needed? 5.1. Not All t ∗ -Collusive Equilibria Are Pareto Improving In this section we provide an example showing that the Pareto improvement can break down if t ∗ is not sufficiently small. We consider a symmetric twobidder environment without discounting (δ = 1). Let t ∗ = 1.16 We will construct a prior belief F := F1 = F2 with support [0 1] such that the payoff of type t ∗ = 1 of any bidder in a 1-collusive equilibrium is smaller than in the value-bidding equilibrium.17 Hence, by continuity, a positive mass of types prefers value bidding. Let t˜ denote a random variable with cumulative distribution function F . In the value-bidding equilibrium, type 1 of any bidder obtains the payoff 1 − E[t˜], where E[·] denotes the expected-value operator. In a 1-collusive equilibrium, her payoff is 1 1 + 1 − E[p∗ (t˜)] 2 2 16 The example can be easily generalized to show that for any t ∗ > 0 there exists an F such that a t ∗ -collusive equilibrium is not Pareto improving. 17 Obtaining a Pareto improvement remains impossible if arbitrary probabilities are allowed for the sunspot states. Let u1 and u2 denote the two bidders’ type-t ∗ payoffs in a t ∗ -equilibrium. Let uval denote the type-t ∗ payoff in the value-bidding equilibrium. Let σ denote the sunspot probability that bidder 1 is the designated bidder. If (u1 + u2 )/2 < uval , then we cannot have that both bidder 1 is better off (σu1 + (1 − σ)u2 ≥ uval ) and bidder 2 is better off ((1 − σ)u1 + σu2 ≥ uval ). To see this, add the inequalities.

1114


where p∗ (t) (t ∈ [0 1]) denotes an optimal resale price of type t of any bidder given the period-2 belief [0 1] about the other bidder. Thus, type 1 strictly prefers the value-bidding equilibrium if (27)

E[p∗ (t˜)] > 2E[t˜]

We construct a distribution F such that (27) holds. F is piecewise linear with a single kink at some point α ∈ (0 1/3). We compute an explicit solution for the resale price function p∗ . This allows us to verify (27). For all t ∈ [0 1], let18 ⎧ 1−α ⎪ ⎨ t if 0 ≤ t ≤ α, α F(t) := α ⎪ ⎩1 − α + (t − α) if α ≤ t ≤ 1. 1−α Straightforward calculations show that E[t˜] = α. Let t ∈ (0 1). To find the optimal resale price p∗ (t) for a type-t seller, observe that p = p∗ (t) maximizes the period-2 payoff π(p t) := (p − t)(1 − F(p)) (written net of own type t) among all p ∈ [0 1]. Suppose that p∗ (t) < α. Because p∗ (t) > t > 0, the first-order condition ∂ 1−α π(p t) (2p∗ (t) − t) 0= =1− ∂p α ∗ p=p (t) holds. This implies p∗ (t) = (28)

t

Because the designated bidder gets the good for free if all others have types below t ∗ , 1 1 it ∗ u = F(t ∗ )n−1 t ∗ + n k=1 k n n

t

t∗

F(t)n−1 dt

It is well known that t F(t)n−1 dt uval = 0

Strict concavity of F implies ∀0 < t < t ∗ :

F(t) F(t ∗ ) > ∗ t t

Therefore, uval −

t t∗

F(t)n−1 dt > =

F(t ∗ )n−1 (t ∗ )n−1 n 1

n

k=1

it ∗ k

u

t∗

t n−1 dt = 0

−

F(t ∗ )n−1 1 ∗ n (t ) (t ∗ )n−1 n

t t∗

F(t)n−1 dt

implying (29). In a no-resale collusive equilibrium based on a probability distribution D, the payoff of type t of bidder k ∈ N is ∗ D ∗ uk := uit k dD(i t ) Suppose that the highest possible type of each bidder is at least as well off as val for all k ∈ N. Then in the value-bidding equilibrium: uD k ≥u 1 D ≤ u = n k=1 k n

u

val

a contradiction.

1 it ∗ (29) uk dD(i t ∗ ) < uval n k=1 n

Q.E.D.


1117

The proof works by showing that, in any no-resale collusive equilibrium, the payoff of the highest type averaged over all bidders is smaller than her valuebidding payoff. Hence, somebody must be worse off in any no-resale collusive equilibrium. Of course, by continuity, the result can be extended to types close to the highest possible type. Blume and Heidhues (2004) showed that in environments where the priors (Fi )i∈N have a common support and there are at least three bidders, the noresale t ∗ -equilibria with t ∗ ranging in [0 maxi∈N t¯i ] are the only equilibria of the second-price auction without resale.19 Thus, the set of no-resale collusive equilibria spans all of the possible equilibrium utility profiles in this environment. Hence we have the following additional result. COROLLARY 1: Suppose that the prior belief is strictly concave and identical for all of the n ≥ 3 bidders. Then, assuming no resale, the value-bidding equilibrium of the second-price auction is not Pareto dominated by any equilibrium of the second-price auction with a public randomization device.

APPENDIX A: PROOFS OF PROPOSITION 1 AND THE LEMMAS USED TO ESTABLISH PROPOSITION 2 PROOF OF LEMMA 1: Following Myerson (1981, pp. 68–69), we define weakly increasing functions c i : Ti → R (i ∈ N) such that, given any type profile t ∈ T, the period-2 seller j ∈ N optimally assigns the good with equal probability to any one of the buyers in the set20,21 i ∈ N \ {j}|tj < c i (ti ) = max c k (tk ) k∈N\{j}

and consumes the good if the set is empty. Hence, for all i ∈ N, ti ∈ Ti , and j ∈ N \ {i}, qij (ti J) ≤ Pr[c i (ti ) > t˜j ] · Pr c i (ti ) ≥ max c k (t˜k ) (30) k∈N\{ij}

19

They also showed that with any positive reserve price only the value-bidding equilibrium (t = 0) remains. This is in contrast to the “with resale” case where t ∗ -equilibria with t ∗ > 0 are robust to reserve prices (Remark 1). 20 Writing “tj < · · ·” instead of “tj ≤ · · ·,” we deviate from Myerson’s original definition while retaining optimality for the seller. 21 The domain of c i in Myerson’s original definition is Ji . For all i ∈ N, we extend c i to Ti via c i (ti ) = −∞ if ti < min Ji and c i (ti ) = c i (max Ji ) if ti > max Ji ; compare footnote 10. ∗

1118


where (t˜1 tñ ) denotes a random vector with distribution J. Similarly, qi (ti J) = Pr ti ≥ max c k (t˜k ) (31) k∈N\{i}

= Pr[ti ≥ c j (t˜j )] · Pr ti ≥ max c k (t˜k ) k∈N\{ij}

From Myerson’ construction, c i (ti ) ≤ ti and c i (t˜j ) ≤ t˜j . Hence, (30) and (31) yield (8). Q.E.D. PROOF OF LEMMA 2: For any x, the function φ(x t) := δw1 (t J∗x ) − t is Lipschitz continuous in t, where the Lipschitz constant can be chosen independently of x by (2). By (6), for any t the function φ(x t) is continuous in x. Hence, φ is continuous. By (5), φ(0 0) > 0. Moreover, for all t ≥ t ∗ , w1 (t J∗t ) = t by (3) and (4); hence, φ(t t) ≤ 0, with the strict inequality holding for δ < 1. Thus, φ(τ∗ τ∗ ) = 0, τ∗ > 0, and, if δ < 1, then τ∗ < t ∗ . To prove (10) and (11), observe that (2) together with q1 (s J∗τ∗ ) ≤ 1 (s ∈ T1 ) implies that φ(τ∗ t) is weakly decreasing in t. To prove the limit result (12), consider the correspondence ψ : δ → {t ∈ [0 t ∗ ]|φ(t t) ≤ 0} Because φ is continuous, ψ is upper hemicontinuous. From (5), ψ(1) = {t ∗ }. Q.E.D. Hence, τ∗ = min ψ(δ) → min ψ(1) = t ∗ as δ → 1, δ = 1. PROOF OF LEMMA 3: By Lemma 2, τ∗ > 0. Hence, (3) and (4) imply li1 (t ∗ J∗τ∗ ) < t ∗ , so (13) implies 0 < b∗i . By (13), the remaining claim b∗i < t ∗ is implied by the claim li1 (t ∗ J∗τ∗ ) > 0, which we establish now. Suppose that li1 (t ∗ J∗τ∗ ) ≤ 0. Then li1 (ti J∗τ∗ ) ≤ 0 for all ti < t ∗ by (1). However, li1 (ti J∗τ∗ ) ≥ 0 1J∗∗ by (4). Hence, li1 (ti J∗τ∗ ) = 0. Thus, qi1 (ti J∗τ∗ ) = 0 by (1). Hence, Qi τ (t) = 0 1J∗∗ for almost all t ∈ J∗τ∗ . Thus, Pi τ (t) = 0 by (4). Because probabilities sum up 1J∗∗ 1J∗∗ to 1 and payments sum up to 0, Q1 τ (t) = 1 and P1 τ (t) = 0 for almost all t ∈ J∗τ∗ . Hence, w1 (t1 J∗τ∗ ) = t1 for almost all t1 ≤ τ∗ , contradicting (5). Q.E.D. PROOF OF PROPOSITION 1: We begin with a complete description of a deperiod-1 beliefs and period-2 beliefs. For any i ∈ N and p ≥ 0, let Hip note the set of histories with current price p (equal to auction price if the d denote the set of hishistory is terminal) where bidder i is active; let Hip tories where bidder i has dropped out at price p (while the current price is greater than or equal top). Observe that, for any i ∈ N, these sets cover the a d ∪ Hip ). Hence, to describe the equilibrium set of all histories: H = p≥0 (Hip period-1 belief profile (Ah )h∈H , it is sufficient to specify, for all i ∈ N, p ≥ 0, a d and h ∈ Hip ∪ Hip , the period-1 belief Ahi about bidder i at history h. In the

1119

COLLUSION VIA RESALE TABLE II PERIOD -1 BELIEFS ABOUT BIDDER i AT ANY HISTORY WHERE BIDDER i HAS DROPPED OUT

i

≥2 ≥1 ≥1 =1 =1

i’s Dropout Price p

Ahi at h ∈ Hdip

≤ t∗ ∈ (t ∗ t i ] > ti < t∗ = t∗

[0 t ∗ ] {p} {t i } [0 τ∗ ] [τ∗ t ∗ ]

following Tables II–IV, we shall identify the posterior distributions Ahi and Ghi with their supports. Let i ∈ N. At the initial history h0 , all bidders are active and Ah0 i = Fi . Table II provides information on period-1 beliefs about bidder i at any history where bidder i has dropped out at price p. Table III provides information on period-1 beliefs about bidder i at any noninitial history where bidder i is active. Let Hˆ denote the set of histories such that bidder 1 is active while, according to β1 , she would not be active if her type were less than or equal to τ∗ . Equilibrium condition (b) can be verified in a straightforward manner using (14), (15), Table II, and Table III. For example, if bidder 2 drops out at the initial history h0 , then (15) implies the period-1 belief [0 t ∗ ]. The event that she drops out at a price in (0 t ∗ ] has probability 0; hence, Bayes’ rule allows an arbitrary period-1 belief such as [0 t ∗ ]. Next we specify period-2 beliefs for any terminal history h ∈ Hterm . Let ω(h) ∈ N denote the winner at the terminal history h. Period-2 beliefs about losing bidders are identical to the beliefs at the end of the auction, because the TABLE III PERIOD -1 BELIEFS ABOUT BIDDER i AT ANY NONINITIAL HISTORY WHERE BIDDER i IS ACTIVE Current Price p

Ahi at h ∈ Haip

≥2 ≥1 ≥1

≤ t∗ ∈ [t ∗ t i ] > ti

=1

< t∗

[t ∗ t i ] if h = h0 [p t i ] {t i } [τ∗ t 1 ] if h ∈ Hˆ T1 if h ∈ / Hˆ

i

1120

R. J. GARRATT, T. TRÖGER, AND C. Z. ZHENG TABLE IV

PERIOD -2 BELIEFS ABOUT THE PERIOD -1 WINNER AND THE RESALE-DECISION PROFILE

i

Auction

Ghi at h ∈ (Haip ∪ Hdip )∩ Hterm ,

Price p

i = ω(h)

≥2

≤ t∗

≥1 ≥1 =1

∈ (t ∗ t i ] > ti ≤ t∗

a if h ∈ Hip d if h ∈ Hip

γh (ti ) =

Resale if ti ≤ x Consume if ti > x

[0 τh ] [0 τh ]

x = τh x = τh

Any belief {t i } [0 τ∗ ]

Any optimal x < p Any optimal x x = τ∗

decision whether to offer the good for resale is not made by the losing bidders: Ghi = Ahi

for all h ∈ Hterm and i = ω(h)

In the no-discounting case δ = 1, the same equation Ghi = Ahi is assumed for the winner i = ω(h) and γh (ti ) = Resale, so that equilibrium conditions (c) and (d) are clearly satisfied. Suppose that δ < 1. Consider a terminal history h where any bidder i ≥ 2 a d ∪ Hip ) ∩ Hterm and i = ω(h). As in the wins at a price p ≤ t ∗ , that is, h ∈ (Hip ∗ proof of Lemma 2, there exists τh ∈ (0 t ) such that δwi (τh Gh ) = τh

if

Ghi = [0 τh ]

Table IV specifies, for any δ < 1, information on the period-2 beliefs about the period-1 winner and the resale-decision profile, where h can be any history in a d ∪ Hip ) ∩ Hterm . (Hip Equilibrium condition (d) can be verified using Table IV. To understand the a , the belief about bidder i is [t ∗ t i ] row with h , observe that at history h ∈ Hip by Table III. According to γh , no type in [t ∗ t i ] chooses Resale at h. Hence, Bayes’ rule allows an arbitrary period-2 belief such as [0 τh ]. Verifying equilibrium condition (d) for the next row is analogous to Lemma 2. To verify equilibrium condition (d) for the row with Any belief, observe that by Table II losing bidders are believed to have types less than or equal to p, and by the second row of Table III the winner is believed to have a type greater than or equal to p. Because none of the winning types chooses Resale, Bayes’ rule allows for any belief. To verify equilibrium condition (d) for the next row, observe that either γh (t i ) = Resale or γh (t i ) = Consume. In the Resale case, Ghi = {t i } follows from Ahi = {t i } (the third row of Table III) by Bayes’ rule. In the Consume case, the event that bidder i offers resale has probability 0, so Bayes’ rule allows for any period-2 belief. To verify equilibrium condition (d) for the last row of Table IV, we distinguish two cases. If Ah1 ∈ {T1 [0 τ∗ ]}, then, according to γh , Bayes’ rule im-


1121

plies Gh1 = Ah1 ∩ [0 τ∗ ] = [0 τ∗ ]; if Ah1 ∈ {[τ∗ t 1 ] [τ∗ t ∗ ]}, then the event that bidder 1 chooses Resale has probability 0, so Bayes’ rule allows for any belief, such as Gh1 = [0 τ∗ ]. The proof of equilibrium condition (c) for the first two rows of Table IV is analogous to the corresponding argument in the proof of Lemma 2. To verify equilibrium condition (c) for the row with Any belief, observe that type p of the winner strictly prefers Consume because δ < 1 and the highest type among the losing bidders is less than or equal to p by Table II. The exact value of the optimal cutoff type x in this row and the next one plays no role. Equilibrium condition (c) for the last row of Table IV follows from Lemma 2. Equilibrium condition (a) follows by construction of the Myerson optimal auction outcome because all beliefs in Table IV belong to J . To verify equilibrium condition (e), write b hiti b if, at history h, the expected payoff of type ti of bidder i when she bids b is not smaller than when she bids b, given that other bidders stick to their candidate equilibrium strategies from h onward and bidder i sticks to her candidate equilibrium strategy after additional bidders drop out. We write b hiti b if b hiti b and b hiti b . We will sometimes omit the lower indices. First we consider bidder 1 types above the threshold t ∗ : (32)

∀h ∈ H1 t1 ≥ t ∗ b ≥ 0: t1 h1t1 b

We prove (32) for the initial history h = h0 ; other histories are treated similarly. Fix b ≥ 0. One of the Events E1 –E4 occurs; in each event, the payoff from bid t1 is not lower than from bid b. EVENT E1—Some Bidder Not Equal to 1 Bids Less Than min{b t1 }: Then the bid b leads to the same ending history (thus, same payoff) as the bid t1 , because bidder 1’s bid at the initial history h0 becomes irrelevant. EVENT E2—All Bidders Not Equal to 1 Bid Greater Than b: Then the highest type among the bidders not equal to 1 is some t > max{b t ∗ }. With discounting (δ < 1), the bid b yields the payoff 0, because by Table IV the winner consumes the good. Without discounting (δ = 1), the good is offered for resale, but the bidders’ period-2 payoffs add up to at most max{t1 t } (equal to the highest type among all bidders) and the winner obtains a period-2 payoff of greater than or equal to t by (3), implying that bidder 1 obtains from bid b a payoff of less than or equal to max{t1 − t 0} by (4). But the bid t1 yields a payoff equal to max{t1 − t 0}, because bidder 1 pays t if she wins. Hence, t1 is weakly better than b. EVENT E3—All Bidders Not Equal to 1 Bid Greater Than min{b t1 } and the Minimum Bid Among These Bidders Is t ≤ b: Then t1 < b. Hence, t1 < t ≤ b and the highest type among the bidders not equal to 1 is t > t ∗ . The bid b

1122


of bidder 1 becomes irrelevant once the current price t is reached, because a bidder has dropped out. Either bidder 1 wins at price t or, given her candidate equilibrium strategy (14), she drops out at price t . In both cases, payoffs are bounded due to (3) and (4). If she wins, then the bidders’ period-2 payoffs add up to at most t ; hence bidder 1’s payoff is less than or equal to 0 due to the auction price t . If she loses at price t , then her payoff is 0. In any case, the bid b yields a payoff less than or equal to 0, so that the bid t1 is weakly better. EVENT E4 —At Least One Bidder Not Equal to 1 Bids min{b t1 } and All Others Bid More: The probability of E4 is positive (and hence E4 is payoffrelevant) only if b = 0. If all bidders not equal to 1 bid 0, then bidder 1’s type t1 ≥ t ∗ is the highest among all bidders, implying that bidder 1’s payoff from the bid b = 0 is less than or equal to t1 (whether or not she wins the tie at 0), so that the bid t1 , which yields payoff t1 , is weakly better. If some bidders not equal to 1 bid more than 0, then the highest type among bidders not equal to 1 is some t > t ∗ , implying that bidder 1’s payoff from bid 0 is less than or equal to max{t1 − t 0} by (3) and (4), while her payoff from bid t1 equals max{t1 − t 0}. This completes the proof of (32). For bidder 1 types below the threshold, any bid between 0 and the threshold is optimal, (33)

∀h ∈ H1 t1 ≤ t ∗ b ∈ (0 t ∗ ] b ≥ 0: b h1t1 b

The proof of (33) uses similar arguments as the proof of (32). The only essentially new aspects are that the proof of b 0 uses the property (7), and that bidder 1 is indifferent in the range (0 t ∗ ] where no other bidder is expected to drop out. We omit the details. Turning to the nondesignated bidder types above the threshold, observe that value bidding is at least as good as any bid above the designated bidder’s competing bid: (34)

∀i = 1 h ∈ Hi ti ≥ t ∗ b > b∗i : ti hiti b

The proof of (34) uses similar arguments as the proof of (32). One defines Events E1 –E4 analogous to E1 –E4 , with bidder 1 replaced by bidder i and t1 replaced by ti . One of the Events E1 –E4 occurs; in each event, the payoff from bid ti is not lower than from bid b. The only essentially new aspect is that Event E4 has positive probability (and, hence, is payoff-relevant) only if b = t ∗ and δ < 1, in which case bidder 1 bids t ∗ . Consider Event E4 . Suppose that n = 2. Then the bid b = t ∗ ends the auction with a tie between bidder 1 and bidder i = 2. If bidder 1 wins the tie, then, by Table II and Table IV, bidder 1 consumes the good so that bidder i obtains the payoff 0; if bidder i wins the tie, then she obtains ti − t ∗ . In any case, the bid ti is weakly better than b. Suppose


1123

that n > 2. Then bidder i loses with bid b and obtains the payoff 0, so that the bid ti is weakly better. This completes the proof of (34). For nondesignated-bidder types below the threshold, any bid between the designated bidder’s competing bid and t ∗ is at least as good as any bid above t∗: (35)

∀i = 1 h ∈ Hi ti ≤ t ∗ b ∈ (b∗i t ∗ ] b > t ∗ : b hiti b

To prove (35), first consider the initial history h = h0 . If at least one bidder not equal to i has a type above t ∗ , bidder i obtains payoff 0 with any b; otherwise all bidders not equal to i have types below t ∗ . If n > 2, then some bidder drops out at price 0, so that bidder i’s bid at the initial history becomes irrelevant and any b > 0 leads to the same ending history. Suppose that n = 2. If t1 ≤ τ∗ , then bidder 1 drops out at price b∗i , which ends the auction, so that all b lead to the same ending history. If t1 > τ∗ , then bidder 1 drops out at price t ∗ and bidder i obtains payoff 0 with any b. Arguments are similar for h = h0 . This completes the proof of (35). For nondesignated-bidder types above the threshold, value bidding is at least as good as any bid below the designated bidder’s competing bid: (36)

∀i = 1 h ∈ Hi ti ≥ t ∗ b ≤ b∗i :

ti hiti b

We prove (36) at h = h0 (other histories are treated similarly). Also, we do not consider the bid b = b∗i ; its treatment combines the arguments used to prove (34) with the arguments below, depending on whether or not bidder i wins the tie against bidder 1. Suppose first that n = 2. One of the Events E5 –E7 occurs; in each event, the payoff from bid ti is not lower than from bid b. EVENT E5—Bidder 1 Bids b∗2 : Then with bid ti , bidder i wins at price b∗2 , the period-2 belief is J∗τ∗ , and bidder i’s payoff is max{δwi (ti J∗τ∗ ) ti } − b∗i = ti − b∗i With any bid b < b∗i , bidder i loses, the period-2 belief is J∗τ∗ , and her payoff is (1)

δli1 (ti J∗τ∗ ) ≤ δ(ti − t ∗ ) + δli1 (t ∗ J∗τ∗ ) (13)

= δ(ti − t ∗ ) + δ(t ∗ − b∗i ) = δ(ti − b∗i ) ≤ ti − b∗i

Hence, the bid ti is weakly better. EVENT E6 —Bidder 1 Bids t ∗ : This event has positive probability only if δ < 1, in which case bidder 1 consumes the good if she wins, so that any bid b < b∗i yields the bidder-i payoff 0. But the bid ti yields the payoff ti − t ∗ ≥ 0.

1124


EVENT E7 —Bidder 1 Bids Greater Than t ∗ : Then the proof that ti is weakly better than any bid b < b∗i uses similar arguments as the proof of (32). Now suppose that n ≥ 3. Consider first the bid b = 0. If all bidders other than 1 and i bid 0, then the proof that bid ti is weakly better is analogous to the treatment of Event E5 . If some bidder other than 1 or i does not bid 0, then this bidder bids more than t ∗ and this case is analogous to the treatment of Event E7 . Now consider bids b > 0. If some bidder other than 1 or i bids 0, then all bids b > 0 (in particular, the bid ti ) yield the same ending history and hence same payoff (because the initial-history bid becomes irrelevant). Otherwise, all bidders other than 1 and i bid strictly more than t ∗ , so that any bid b < b∗i yields the payoff 0 and the bid ti is weakly better (similar arguments as in the proof of (32)). This completes the proof of (36). For nondesignated-bidder types below the threshold, the bid 0 is at least as good as any bid below the designated bidder’s competing bid: (37)

∀i = 1 h ∈ Hi ti ≤ t ∗ b < b∗i :

0 hiti b

To prove (37), observe that the bid 0 yields the same period-2 belief as any bid b < b∗i in any event, and bidder i always loses the auction. Hence, payoffs are the same. Finally, for nondesignated-bidder types below the threshold, the bid 0 is at least as good as any bid between the designated bidder’s competing bid and t ∗ : (38)

∀i = 1 h ∈ Hi ti ≤ t ∗ b ∈ [b∗i t ∗ ]:

0 hiti b

We prove (38) at h = h0 (other histories are treated similarly). Also, we do not consider the bid b = b∗i ; its treatment combines the arguments used to prove (37) with the arguments below, depending on whether or not bidder i wins the tie against bidder 1. Suppose first that n = 2. One of the Events E5 –E7 defined above occurs; in each event, the payoff from bid ti is not lower than from bid b. Suppose Event E5 occurs. Then with bid b, bidder i wins at price b∗2 , the period-2 belief is J∗τ∗ , and bidder i’s payoff is Uˆ i (ti ) = max{δwi (ti J∗τ∗ ) ti } − b∗i With bid 0, bidder i loses, the period-2 belief is J∗τ∗ , and her payoff is Ui∗ (ti ) = δli1 (ti J∗τ∗ ) Observe that (13) Uˆ i (t ∗ ) = t ∗ − b∗i = Ui∗ (t ∗ )


1125

From (1), Ui∗ is Lipschitz continuous and hence is differentiable almost everywhere, and the derivative is Ui∗ (ti ) = δqi1 (ti J∗τ∗ ) Similarly, the derivative of Uˆ i is Uˆ i (ti ) ≥ δqi (ti J∗τ∗ )

Hence,

t∗

Ui∗ (ti ) = Ui∗ (t ∗ ) −

Ui∗ (s) ds

ti

= Uˆ i (t ∗ ) − δ

t∗

qi1 (s J∗τ∗ ) ds

ti

(8)

≥ Uˆ i (t ∗ ) − δ

≥ Uˆ i (t ∗ ) −

t∗

qi (s J∗τ∗ ) ds

ti

t∗

Uˆ i (s) ds

ti

= Uˆ i (ti ) Hence, the bid 0 is weakly better. If Event E6 or Event E7 occurs, bidder i’s payoff is 0 with either bid 0 or any b ∈ (b∗i t ∗ ]. Now suppose that n ≥ 3. If all bidders other than 1 and i bid 0, then the proof that bid 0 is weakly better than b ∈ (b∗i t ∗ ] is analogous to the treatment of Event E5 . If some bidder other than 1 or i does not bid 0, then bidder i’s payoff is 0 anyway. Q.E.D. PROOF OF LEMMA 5: If δ = 1, the result is obvious because τ∗i = t i . Let δ < 1 and let δ 1 1 (39) θ < min 4 1 − δ 12 Let i = 1 (the proof is analogous for other bidders). From (17), F2 (t ∗ /2)/ F2 (t ∗ ) → 1/2 as t ∗ → 0. Hence, for all t ∗ sufficiently close to 0, (40)

1−

F2 (t ∗ /2) 1 > F2 (t ∗ ) 3

1126


A lower bound for the period-2 payoff of any type x ≤ θt ∗ of bidder 1 is the payoff from a take-it-or-leave-it fixed-price offer at t ∗ /2 to bidder 2: ∗ t F2 (t ∗ /2) 1 t∗ (39)(40) ∗ − x > x + w1 (x Jx ) ≥ x + 1 − F2 (t ∗ ) 2 34 Therefore, for all x ≤ θt ∗ , δw1 (x J∗x ) − x >

(39) δ ∗ t − (1 − δ)x ≥ 0 12

implying τ∗ ≥ θt ∗ by (9).

Q.E.D.

PROOF OF LEMMA 6: For any threshold t ∗ , i ∈ N, and ti ∈ [0 t ∗ ], let22 (41)

Vi (ti ) := ti −

Fi (t ∗ ) − Fi (ti ) fi (ti )

Step 1. If t ∗ is sufficiently close to 0, then Vi is strictly increasing. To show this, consider any t t such that 0 ≤ t < t ≤ t ∗ . Then (42)

Vi (t ) − Vi (t) = t − t −

Fi (t ∗ ) − Fi (t ) Fi (t ∗ ) − Fi (t) + fi (t ) fi (t)

= t − t −

Fi (t ∗ ) − Fi (t ) Fi (t ∗ ) − Fi (t) + fi (t ) fi (t )

−

Fi (t ∗ ) − Fi (t) Fi (t ∗ ) − Fi (t) + fi (t ) fi (t)

= t − t +

Fi (t ) − Fi (t) fi (t ) − fi (t) ∗ + (F (t ) − F (t)) i i fi (t ) fi (t)fi (t )

fi (t ) − fi (t) fi (t)fi (t ) L ∗ ≥ (t − t) 1 − Fi (t ) fi (t)fi (t )

≥ t − t + (Fi (t ∗ ) − Fi (t))

where L is a Lipschitz constant for fi . Let t ∗ be so close to 0 that fi (0)/2 < fi (s) < 2fi (0) for all s ∈ [0 t ∗ ]. Then t∗ Fi (t ∗ ) = 0 fi (s) ds < 2fi (0)t ∗ and (42) implies ∗ 4L Vi (t ) − Vi (t) ≥ (t − t) 1 − 2fi (0)t fi (0)2 22

This is the virtual valuation function (cf. Myerson (1981)) for the belief [0 t ∗ ] about bidder i.


1127

The right-hand side is strictly positive if t ∗ < fi (0)/8L, showing that Vi is strictly increasing. The next step is to provide lower bounds for the virtual valuation functions Vi (i ∈ N) if t ∗ is small. To this end, define (43)

κi (t ∗ ) :=

fi (0) min∗ fi (x)

x∈[0t ]

Observe that κi (t ∗ ) → 1 as t ∗ → 0. Hence, if t ∗ is small, the lower bound established in Step 2 approximates the virtual valuation function for a uniform distribution on [0 t ∗ ]. Step 2. For all i ∈ N, t ∈ [0 t ∗ ], and any threshold t ∗ , Vi (t) ≥ t − κi (t ∗ )(t ∗ − t) + O(t ∗ ) (cf. footnote 15). Using (17) and (41), (44)

Vi (t) = t − =t−

fi (0)t ∗ + hi (t ∗ ) − fi (0)t − hi (t) fi (t) fi (0) ∗ (t − t) + h1i (t t ∗ ) fi (t)

≥ t − κi (t ∗ )(t ∗ − t) + h1i (t t ∗ ) where (45)

h1i (t t ∗ ) :=

hi (t ∗ ) − hi (t) fi (t)

Observe that (17) implies sup |hi (x)|

x∈[0t ∗ ]

t

∗

|hi (x)| → 0 as t ∗ → 0 x x∈[0t ∗ ]

≤ sup

Hence, defining f i (t ∗ ) := minx∈[0t ∗ ] fi (x), sup |h1i (x t ∗ )|

x∈[0t ∗ ]

t∗

sup |hi (x)| |hi (t ∗ )| x∈[0t ∗ ] 1 ≤ + f i (t ∗ ) t∗ t∗ →0

as

t ∗ → 0

The next step is to provide formulas for period-2-buyer allocation probabilities. From now on, suppose that t ∗ is sufficiently close to 0 so that the conclusion of Step 1 holds.

1128


Step 3. For any j ∈ N, i ∈ N \ {j}, and ti ∈ [0 t ∗ ], (46)

Fj (Vi (ti )) Fk Vk−1 (Vi (ti )) qij (ti J ) = Fj (τj∗ ) k∈N\{ij} Fk (t ∗ ) ∗j

where Vk−1 (x) := 0 for all x < Vk (0). According to the Myerson optimal auction outcome given the period-2 seller j ∈ N and the belief J∗j , the good is assigned to the buyer with the highest virtual valuation, unless j’s use value is higher (cf. Myerson (1981)). From this the allocation probabilities (46) are straightforward. Step 4.—Proof of (22). Define ξ = 4/5. Using Step 2, for any i ∈ N, 1 3 Vi (ξt ∗ ) 4 ≥ − κi (t ∗ ) + O(1) → t∗ 5 5 5

as

t ∗ → 0

Thus, using Step 1, if t ∗ is sufficiently close to 0, then (47)

∀ti ∈ [ξt ∗ t ∗ ]: Vi (ti ) ≥

t∗ 2

Using (46), for any j = i, if δ < 1, then ∗j

qij (ti J )

= (47)τ∗j 0 as t ∗ → 0 fk (0) + O(1) 2 k=i

1129


If δ = 1, similar arguments show that qij (ti J∗j )/Fj (t ∗ ) → 1/2n−1 because Q.E.D. τ∗j = t j . Hence, we can choose any ε < 1/2n−1 . PROOF OF LEMMA 7: Using the envelope theorem,

ti

U (ti ) = val i

0 (17)

0

=

Fk (x) dx

k=i ti

=

(fk (0)x + hk (x)) dx k=i

ti

0

n−1

fk (0)x

+ h (x) dx 1

k=i

1 fk (0) ti n + = n k=i

h1 (x) where n−1 → 0 as x → 0 x

ti

h1 (x) dx

0

Let ε > 0. If t ∗ is sufficiently small, then |h1 (x)| ≤ εxn−1 for all x ≤ t ∗ . Therefore, ti ti ti 1 1 h (x) dx ≤ |h (x)| dx ≤ ε xn−1 dx ≤ ε(t ∗ )n 0

0

0

Q.E.D.

which completes the proof. PROOF that (48)

OF

LEMMA 8: First let δ < 1. Let ξ < θ and ε > 0 be so close to 0

1 − (n − 1)ε(1 − ξ) < (1 − ξ/θ)(1 − ε)

Because probabilities add up to 1, (49)

τ∗j

qj (tj J∗j )

0

dFj (tj ) + Fj (t ∗ ) i=j

t∗

qij (ti J∗j )

0

dFi (ti ) = 1 Fi (t ∗ )

Using Lemma 5, (50)

lim inf ∗

t →0

Fj (τ∗j ) − Fj (ξt ∗ ) ∗j

Fj (τ )

≥ lim inf ∗

t →0

Fj (θt ∗ ) − Fj (ξt ∗ ) ∗

Fj (θt )

ξ = 1− θ

(17)

1130


It is well known that the incentive compatibility constraints for period 2 imply (∗) the functions qj (· J∗j ) are weakly increasing. We find 1−

ξ θ

lim sup qj (ξt ∗ J∗j ) t ∗ →0

Fj (τ∗j ) − Fj (ξt ∗ )

(50)

≤ lim inf ∗

Fj (τ∗j )

t →0

≤ lim sup

t ∗ →0

(∗)

τ∗j

qj (tj J∗j )

≤ lim sup

t ∗ →0

t ∗ →0

Fj (τ∗j ) − Fj (ξt ∗ ) Fj (τ∗j )

0

(49)

= 1 − lim inf ∗

t →0

≤ 1 − lim inf ∗

t →0

≤ 1 − ε lim inf ∗

t →0

= 1 − ε lim inf ∗

t →0

t∗ ξt ∗

i=j

qj (ξt ∗ J∗j )

dFj (tj ) Fj (τ∗j )

qij (ti J∗j )

0

i=j

(22)

t∗

lim sup qj (ξt ∗ J∗j )

i=j

qij (ti J∗j ) t∗ ξt ∗

dFi (ti ) Fi (t ∗ )



Fi (t ∗ ) − Fi (ξt ∗ ) i=j

Fi (t ∗ )

(17)

= 1 − (n − 1)ε(1 − ξ) ξ (48) (1 − ε) < 1− θ Dividing both sides by 1 − ξ/θ yields (23). Now let δ = 1. Define ξ < 1 and ε > 0 as in (48) with θ = 1. Using (3) and (4), qj (tj J∗j ) = 1 for all tj > t ∗ . Hence, because τ∗j = t j and probabilities sum up to 1,

t∗

∗j

∗

qj (tj J ) dFj (tj ) + (1 − Fj (t )) + 0

i=j

0

t∗

qij (ti J∗j )

dFi (ti ) = 1 Fi (t ∗ )


1131

Rearranging yields

t∗

(51) 0

dFj (tj ) + qj (tj J ) Fj (t ∗ ) i=j

t∗

∗j

0

qij (ti J∗j ) dFi (ti ) = 1 Fj (t ∗ ) Fi (t ∗ )

Similar to the cases δ < 1, we obtain (23) because (1 − ξ) lim sup qj (ξt ∗ J∗j ) t ∗ →0

(17)

= lim sup

Fj (t ∗ ) − Fj (ξt ∗ ) Fj (t ∗ )

t ∗ →0

(∗)

t∗

≤ lim sup

t ∗ →0

qj (tj J∗j )

0

(51)

= 1 − lim inf ∗

t →0

i=j

(22)

≤ 1 − ε lim inf ∗

t →0

t∗

0

i=j

qj (ξt ∗ J∗j )

dFj (tj ) Fj (t ∗ )

qij (ti J∗j ) dFi (ti ) Fj (t ∗ ) Fi (t ∗ ) t∗

ξt ∗


Q.E.D.

PROOF OF LEMMA 9: If ti < ξt ∗ , the claim Uijb (ti ) ≥ 0 is immediate from (4). Let ti > ξt ∗ . Using (1) and lij (0 J∗j ) ≥ 0 (from (4)), if δ < 1, then 1 1 U b (ti ) ≥ δ Fj (τ∗j ) Fk (t ∗ ) ij k∈{ij} /

(22)

≥ δ

ti

qij (x J∗j ) dx

0

ti ξt ∗

ε dx = δε(ti − ξt ∗ )

which together with Lemma 5 implies the claim. If δ = 1, similar arguments imply k∈{ij} /

1 U b (ti ) ≥ εFj (t ∗ )(ti − ξt ∗ ) Fk (t ∗ ) ij

Q.E.D.

PROOF OF LEMMA 10: If ti > ξt ∗ , the claim Uis (ti ) ≥ ti is immediate from equilibrium condition (c). Let ti ≤ ξt ∗ . First let δ < 1. Then ti < θt ∗ ≤ τ∗i by Lemma 5. Using (2) and the fact that δwi (τ∗i J∗i ) = τ∗i (from the definition

1132


of τ∗i ), 1 U s (ti ) = τ∗i − δ Fk (t ∗ ) i

k=i

τ∗i

qi (x J∗i ) dx

ti

∗i

∗i

= τ − δ(τ − ti ) + δ

τ∗i

(1 − qi (x J∗i )) dx

ti (23)

≥ ti + δε(ξt ∗ − ti )

as was to be shown. If δ = 1, then the claim follows from analogous arguments with τ∗i replaced by t ∗ . Q.E.D. APPENDIX B: t ∗ -EQUILIBRIUM STRATEGIES ARE UNDOMINATED We shall show that the strategies used in t ∗ -equilibria are undominated. To do so we utilize the “type-player interpretation” of the auction-with-resale game, where different types of a bidder represent different players (see, e.g., Osborne and Rubinstein (1994, p. 26)).23 A pure strategy s for type ti ∈ Ti of bidder i is undominated if, for any strategy s = s, either s yields the same (expected) payoff as s no matter what other bidders do, or (∗) there exists a profile of strategies s−i for the bidders other than i such that s yields a strictly higher payoff than s against s−i . Assume that there are n = 2 bidders, so that the period-2 seller makes a fixedprice take-it-or-leave-it offer according to the Myerson optimal auction outcome; at the end of this section we comment on the extension of the arguments to environments with n ≥ 3 bidders. A pure strategy s for type ti of bidder i is described by a bid βi (ti ) and a resale price function ri (ti ·), where ri (ti p) ∈ [0 ∞) ∪ {Consume} denotes her resale price or consumption decision when she wins the auction at price p. The domain of the function ri can be restricted to the set {(ti p)|ti ∈ [0 t i ] p ∈ [0 βi (ti )]}.24 Any resale price is accepted by all losing-bidder types greater than or equal to this price.

23 This corresponds to the natural viewpoint that the bidders select their strategies after they have learned their private information. 24 We do not have to specify a bidder’s resale price if she wins at a price higher than her bid, because under this circumstance different resale prices yield equivalent strategies.


1133

The t ∗ -equilibrium strategies can be defined such that, for all p,25 (52)

ri (ti p) ∈

(ti /δ t −i ) ∪ {Consume} (ti t −i ) ti

if δ < 1, if δ = 1 and ti < t −i if δ = 1 and ti ≥ t −i

Proposition 4 shows that pure strategies satisfying (52) are undominated, except possibly for type-0 bidders. The proof proceeds in two steps. First, we show that a strategy of any type of bidder i that involves a (nonrandomized) bid b cannot be weakly dominated by an alternative strategy where the bidder submits a bid other than b. This is true because assuming the other bidder bids so aggressively that bidder i must wait for a resale offer, it may be the case that a favorable resale offer is made only if bidder −i wins at price b, making the bid b uniquely optimal. In the second step we show that a strategy of any type of bidder i that involves a bid b and a certain resale price x = Consume if the bidder wins at a given price p ≤ b cannot be weakly dominated by a strategy where bidder i sticks to the bid b, but changes her resale behavior upon winning at p. Here we use the assumption that bidder −i’s valuation density is positive at x. Indeed, if, for some small ε > 0, types [x x + ε] of bidder −i bid p and all other types bid more than b, then it is uniquely optimal to offer resale at price x upon winning at p (and the resale decisions when winning at a price that is unequal to p are irrelevant). The argument is slightly different if x = Consume. PROPOSITION 4: Let n = 2. For all types ti = 0 of all bidders i, any pure strategy satisfying (52) is undominated. PROOF: Consider any pure strategy s for type ti of bidder i consisting of a bid b and a resale price function ri (ti ·). Let V (b) denote the set of all (pure or mixed) strategies of type ti of bidder i where she bids b with certainty. / V (b). Define strategies s−i = (β−i (t−i ) r−i (t−i ·)) Step 1.—(∗) holds if s ∈ for the various types t−i ∈ T−i of bidder −i as follows. Bidder −i of type t−i bids β−i (t−i ) > max{b t−i }. If she wins at any price p = b, then she chooses a resale price r−i (t−i p) optimally given the belief that bidder i’s type is in [ti − ε t i ], where ε > 0 is defined below.26 She chooses r−i (t−i b) optimally given the belief that bidder i’s type is in [0 ti ]. 25 If δ = 1 and ti ≥ t −i , all resale prices greater than or equal to ti yield the same payoff as Consume, so we can fix the price at ti . Otherwise, any strategy not satisfying (52) is dominated. If δ < 1, any price less than or equal to ti /δ is no better, and sometimes worse, than consuming the good. If δ = 1, any price less than or equal to ti is no better, and sometimes worse, than any price in (ti t −i ). Any price greater than or equal to t −i will be accepted with probability 0 from an ex ante viewpoint. 26 The argument works with ε = 0 if ti < t i . But the belief {t i } implies r−i (t−i p) = t i if t−i is small, which violates (52).

1134


Now consider bidder i of type ti who is bidding against s−i . Any bid that is unequal to b yields a payoff that is less than or equal to ε, while the bid b yields a positive expected payoff uˆ > 0 because with positive probability bidder i gets ˆ the strategy s is strictly better a resale offer priced below ti . Choosing ε < u, than any strategy not in V (b). This completes Step 1. If δ = 1 and ti ≥ t −i , we are done because V (b) = {s}. From now on, assume that δ < 1 or ti < t −i . Given any p ≤ b, let W (p b) denote the set of all (pure or mixed) strategies where type ti of bidder i chooses the resale price ri (ti p) with certainty when she wins at price p. Step 2.—(∗) holds if s ∈ V (b) \ W (p b). First assume that ri (ti p) =: x = Consume. Observe that ti < t −i by (52). Define strategies s−i = (β−i (t−i ) r−i (t−i ·)) such that β−i (t−i ) = p if t−i ∈ [x x + ε] and β−i (t−i ) > b otherwise, where ε > 0 is chosen so small that (53)

y−

F−i (x + ε) − F−i (y) > ti f−i (y)

∀y ∈ [x x + ε]

Resale prices r−i (t−i ·) are arbitrary. Against s−i , if type ti of bidder i wins at price p, then the resale price x is uniquely optimal for her (her period-2 payoff from resale price y is strictly decreasing for y ∈ [x x + ε], which can be seen by computing the derivative of the payoff and using (53)). Because the price x > ti /δ is accepted with certainty, it is also better than Consume. Hence, s is strictly better than any s . Now assume that ri (ti p) = Consume. Define strategies s−i = (β−i (t−i ) r−i (t−i ·)) such that β−i (t−i ) = p if t−i ∈ [0 ε] and β−i (t−i ) > b otherwise, where ε < ti . Resale prices r−i (t−i ·) are arbitrary. If type ti of bidder i wins at price p against s−i , then Consume is better than offering resale. Hence, s is strictly better than any s . This completes Step 2. Steps 1 and 2 show that s is undominated because27 {s} = W (p b) ∩ V (b) Q.E.D. p≤b

Now consider environments with n ≥ 3 bidders. The crucial complication in extending Proposition 4 occurs in Step 1 of the proof. Below we outline a proof of the fact that strategies involving arbitrary bid functions are undominated, as long as the resale behavior is appropriately restricted in a manner similar to (52). Any type of a given bidder must choose a planned dropout price (bid) at each history during the English auction. First, one shows that a strategy that involves 27 Observe that, for the verification of (∗) in Steps 1 and 2, the strategy profile s−i can itself be taken to be a profile of strategies satisfying (52). In this sense, any pure strategy satisfying (52) survives any iterated elimination of dominated strategies.


1135

some bid b at the initial history cannot be weakly dominated by a strategy that involves a bid different from b at the initial history. The resale mechanisms involved in this argument are second-price auctions with reserve prices similar to the resale prices used in the two-bidder proof. Next, a strategy that involves a bid b at the initial history and a bid b at the history h , which is reached when a certain bidder drops out at a certain price, cannot be dominated by a strategy where she submits the bid b at the initial history and a bid that is unequal to b at h ; to show this, one constructs strategies of the other bidders such that history h is reached and then, as before, the remaining active rivals bid so aggressively that the bidder must wait for a resale offer. Continuing inductively to the end of the auction, one sees that a strategy cannot be dominated by any strategy that involves a bidding structure different from the one used in the original strategy. REFERENCES BLUME, A., AND P. HEIDHUES (2004): “All Equilibria of the Vickrey Auction,” Journal of Economic Theory, 114, 170–177. [1097,1115,1117] BRUSCO, S., AND G. LOPOMO (2002): “Collusion via Signalling in Simultaneous Ascending Bid Auctions With Heterogeneous Objects, With and Without Complementarities,” Review of Economic Studies, 69, 407–436. [1098] (2008): “Budget Constraints and Demand Reduction in Simultaneous Ascending-Bid Auctions,” The Journal of Industrial Economics, 56, 113–142. [1107] CALZOLARI, G., AND A. PAVAN (2006): “Monopoly With Resale,” RAND Journal of Economics, 37, 362–375. [1097] CASS, D., AND K. SHELL (1983): “Do Sunspots Matter?” Journal of Political Economy, 91, 193–227. [1108] ENGELBRECHT-WIGGANS, R., AND C. M. KAHN (2005): “Low-Revenue Equilibria in Simultaneous Ascending-Bid Auctions,” Management Science, 51, 508–518. [1098] GARRATT, R., AND T. TRÖGER (2006): “Speculation in Standard Auctions With Resale,” Econometrica, 74, 753–769. [1098,1107] GRAHAM, D. A., AND R. C. MARSHALL (1987): “Collusive Bidder Behavior at Single-Object Second-Price and English Auctions,” Journal of Political Economy, 95, 1217–1239. [1096] HAFALIR, I., AND V. KRISHNA (2008): “Asymmetric Auctions With Resale,” American Economic Review, 98, 87–112. [1098] HAILE, P. (1999): “Auctions With Resale,” Mimeo, University of Wisconsin, Madison. [1103] (2003): “Auctions With Private Uncertainty and Resale Opportunities,” Journal of Economic Theory, 108, 72–100. [1097] LEBRUN, B. (2007): “First-Price and Second-Price Auctions With Resale,” Mimeo. [1098] LOPOMO, G., R. C. MARSHALL, AND L. M. MARX (2005): “Inefficiency of Collusion at English Auctions,” Contributions to Theoretical Economics, 5, Article 1. [1097] MAILATH, G. J., AND P. ZEMSKY (1991): “Collusion in Second Price Auctions With Heterogeneous Bidders,” Games and Economic Behavior, 3, 467–486. [1096] MARSHALL, R. C., AND L. M. MARX (2007): “Bidder Collusion,” Journal of Economic Theory, 133, 374–402. [1096] MCAFEE, R. P., AND J. MCMILLAN (1992): “Bidding Rings,” American Economic Review, 82, 579–599. [1096] MILGROM, P. R. (2000): “Putting Auction Theory to Work: The Simultaneous Ascending Auction,” Journal of Political Economy, 108, 245–272. [1098]

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MILGROM, P. R., AND I. SEGAL (2002): “Envelope Theorems for Arbitrary Choice Sets,” Econometrica, 70, 583–601. [1102] MILGROM, P. R., AND R. J. WEBER (1982): “A Theory of Auctions and Competitive Bidding,” Econometrica, 50, 1089–1122. [1099] MYERSON, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6, 58–73. [1097,1101,1110,1117,1126,1128] (1983): “Mechanism Design by an Informed Principal,” Econometrica, 51, 1767–1797. [1101] MYLOVANOV, T., AND T. TRÖGER (2008): “Optimal Auction Design and Irrelevance of Privacy of Information,” Mimeo, University of Bonn. [1101] (2009): “A Characterization of the Conditions for Optimal Auction With Resale,” Economic Theory (forthcoming). [1107] OSBORNE, M., AND A. RUBINSTEIN (1994): A Course in Game Theory. Cambridge, MA: MIT Press. [1132] PAGNOZZI, M. (2007): “Should Speculators Be Welcomed in Auctions?” Working Paper 176, Centre for Studies in Economics and Finance, University of Salerno. [1098] SHELL, K. (1977): “Monnaie et Allocation Intertemporelle,” Mimeo, Seminaire Roy-Malinvaud, Centre National de la Recherche Scientifique, Paris. [1108] SMITH, R. A. (1961): “The Incredible Electrical Conspiracy (Part II),” Fortune, 63, 161–164, 210–224. [1096] ZHENG, C. (2000): “An Optimal Auction When Resale Cannot Be Prohibited,” Discussion Paper 1303, CMS-EMS, Northwestern University. [1107] (2002): “Optimal Auction With Resale,” Econometrica, 70, 2197–2224. [1099,1107]

Dept. of Economics, University of California at Santa Barbara, Santa Barbara, CA 93106-9210, U.S.A.; [email protected], Dept. of Economics, University of Berne, Schanzeneckstr. 1, 3001 Berne, Switzerland; [email protected], and Dept. of Economics, Iowa State University, Ames, IA 50011, U.S.A.; czheng@ iastate.edu. Manuscript received October, 2007; final revision received September, 2008.


BUBBLES AND SELF-ENFORCING DEBT BY CHRISTIAN HELLWIG AND GUIDO LORENZONI1 We characterize equilibria with endogenous debt constraints for a general equilibrium economy with limited commitment in which the only consequence of default is losing the ability to borrow in future periods. First, we show that equilibrium debt limits must satisfy a simple condition that allows agents to exactly roll over existing debt period by period. Second, we provide an equivalence result, whereby the resulting set of equilibrium allocations with self-enforcing private debt is equivalent to the allocations that are sustained with unbacked public debt or rational bubbles. In contrast to the classic result by Bulow and Rogoff (1989a), positive levels of debt are sustainable in our environment because the interest rate is sufficiently low to provide repayment incentives. KEYWORDS: Self-enforcing debt, risk sharing, rational bubbles, sovereign debt.

1. INTRODUCTION CONSIDER A WORLD with integrated capital markets in which countries can borrow and lend at the world interest rate. Suppose that the only punishment for a country that defaults on its debt is the denial of credit in all future periods. What is the maximum amount of borrowing that can be sustained in equilibrium? In this paper, we address this question in the context of a general equilibrium model with complete securities markets and endogenous debt constraints. Following Alvarez and Jermann (2000), debt limits are set at the largest possible levels such that repayment is always individually rational. We show that, under some conditions, positive levels of debt are sustainable in equilibrium, and we characterize the joint behavior of intertemporal prices and debt limits. In the process, we identify a novel connection between the sustainability of debt by reputation and the sustainability of rational asset price bubbles. The key observation for our sustainability result is that the incentives for debt repayment rely not only on the amount of credit to which agents have access, but also on the interest rate at which borrowing and lending takes place. After default, agents are excluded from borrowing, but are allowed to save. 1 We thank for useful comments a co-editor, three anonymous referees, Marios Angeletos, Andy Atkeson, V. V. Chari, Hal Cole, Veronica Guerrieri, Nobu Kiyotaki, Narayana Kocherlakota, John Moore, Fabrizio Perri, Balázs Szentes, Aleh Tsyvinski, Iván Werning, Mark Wright, and seminar audiences at the Columbia, European University Institute (Florence), Mannheim, Maryland, Max Planck Institute (Bonn), NYU, Notre Dame, Penn State, Pompeu Fabra, Stanford, Texas (Austin), UCLA, UCSB, the Federal Reserve Banks of Chicago, Dallas, and Minneapolis, the 2002 SED Meetings (New York), the 2002 Stanford Summer Institute in Theoretical Economics, and the 2003 Bundesbank/CFS/FIC Conference on Liquidity and Financial Instability (Eltville, Germany) for feedback. Lorenzoni thanks the Research Department of the Federal Reserve Bank of Minneapolis for hospitality during part of this research. Hellwig gratefully acknowledges financial support through the ESRC. All remaining errors and omissions are our own.


DOI: 10.3982/ECTA6754

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Lower interest rates make both borrowing more appealing and saving after default less appealing. We show that sustaining positive levels of debt requires “low interest rates,” that is, intertemporal prices such that the present value of the agents’ endowments is infinite.2 Our result stands in contrast to the classic result of Bulow and Rogoff (1989a; henceforth BR), who analyzed the repayment incentives of a small open economy borrowing at a fixed world interest rate and showed that no debt can be sustained in equilibrium if the only punishment for default is the denial of future credit.3 The crucial difference is that BR took the world interest rate as given and assumed that the net present value of the borrower’s endowment, evaluated at the world interest rate, is finite. This rules out precisely the intertemporal prices that emerge in our general equilibrium setup. The difference between BR and this paper can also be interpreted in terms of unilateral versus multilateral lack of commitment. BR assumed that, after default, the borrowing country can save by entering a “cash-in-advance” contract with some external agent (some other country or some international bank), paying up front in exchange for future state-contingent payments. BR implicitly assumed that this agent has full commitment power for exogenous reasons, so he will never default on the cash-in-advance contract.4 In our model, instead, there is multilateral lack of commitment: no agent can commit to future repayments. Therefore, one country’s ability to save after default, and hence its default incentives, rests on the other countries’ repayment incentives.5 To illustrate our results, we first present a simple stationary example where, in equilibrium, positive borrowing arises and the interest rate is zero. More generally, debt is self-enforcing as long as the real interest rate is less than or equal to the growth rate of debt limits, which in steady state is equal to the growth rate of the aggregate endowment. As is well known, this is the same condition that makes bubbles sustainable in equilibrium (Tirole (1985)). 2

This terminology follows Alvarez and Jermann (2000). This result has sparked a vast literature that considers alternative ways to sustain positive levels of debt. A nonexhaustive list of contributions includes Eaton and Gersovitz (1981), Bulow and Rogoff (1989b), Cole and Kehoe (1995, 1998), Kletzer and Wright (2000), Kehoe and Perri (2002), Amador (2003), and Gul and Pesendorfer (2004). 4 The role of this assumption in BR was first pointed out by Eaton (1990) and Chari and Kehoe (1993a). 5 Two papers that share with BR the assumptions of one-sided commitment and credit exclusion after default are Chari and Kehoe (1993b) and Krueger and Uhlig (2006), the first in the context of a model of government debt with distortive taxes and lack of government commitment; the second in the context of competitive insurance markets with one-sided commitment by insurers but not households. Krueger and Uhlig (2006) showed that BR’s assumption about credit exclusion emerges naturally from competition among insurers with one-sided commitment. Two papers which, instead, assume multilateral lack of commitment are Kletzer and Wright (2000) and Kehoe and Perri (2002). In both papers, borrowing can arise in equilibrium, but the mechanism is quite different from ours: all transfers are publicly observable, and risk sharing and debt are sustained by a folk-theorem-type argument. 3

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In the rest of the paper, we give a complete characterization of the conditions under which debt is sustainable. Our first general result (Theorem 1) states that debt limits and intertemporal prices are consistent with self-enforcement if and only if all agents are able to exactly refinance outstanding obligations by issuing new claims. The result relies on arbitrage arguments that compare the feasible consumption sequences with and without a default (as in BR), as well as weak restrictions on preferences that guarantee the monotonicity of the agent’s optimal asset holdings in initial wealth. Our second result then establishes conditions for the existence of an equilibrium with self-enforcing debt and gives a characterization of sustainable equilibrium allocations by means of an equivalence result. Consider an alternative environment with no private debt, but where the government issues statecontingent debt that is not backed by any fiscal revenue, that is, the government must finance all existing claims by issuing new debt. If this unbacked public debt is valued in equilibrium, it is a rational bubble.6 Theorem 2 shows that any equilibrium allocation of the economy with self-enforcing private debt can also be sustained as an equilibrium allocation of the economy with unbacked public debt, and vice versa. Therefore, conditions that ensure the existence of valued unbacked public debt or, more generally, the existence of rational bubbles, also ensure the sustainability of positive levels of private debt in our economy. The possibility of rational bubbles in models with exogenous borrowing constraints à la Bewley (1980) has been recognized in Scheinkman and Weiss (1986), Kocherlakota (1992), and Santos and Woodford (1997). Our equivalence result shows that self-enforcing private debt can play the same role as a bubble and can take its place in facilitating intertemporal exchange. Our model is closely related to models with limited commitment and endogenous borrowing constraints (Kehoe and Levine (1993), Kocherlakota (1996), Alvarez and Jermann (2000)). The central difference is our assumption about default consequences: while previous authors assumed that agents are completely excluded from financial markets after default, our exclusion only rules out future credit. From a methodological point of view, the major difference is that under our punishment, the equilibrium utility after default is endogenous and depends on equilibrium prices. This prevents us from starting from a social planner’s problem with exogenous participation constraints and then decentralizing its solution, as in Alvarez and Jermann (2000). In Section 2, we describe our general model and define competitive equilibria with self-enforcing private debt and unbacked public debt. In Section 3, we illustrate our main results with a simple example. In Section 4, we study repayment incentives for individual agents and characterize debt limits consistent with no default (Theorem 1). In Section 5, we study the resulting general equilibrium implications (Theorem 2). The online Supplemental Material (Hellwig 6

In a deterministic environment, unbacked public debt is equivalent to fiat money.

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and Lorenzoni (2009)) contains additional results and extensions for the example in Section 3. 2. THE MODEL Uncertainty, Preferences and Endowments Consider an infinite-horizon endowment economy with a single nonstorable consumption good at each date t ∈ {0 1 2 }. For each t, there is a positive finite set S t of date-t events st . Each st has a unique predecessor σ(st ) ∈ S t−1 and a positive, finite number of successors st+1 ∈ S t+1 for which σ(st+1 ) = st . There exists a unique initial date-0 event s0 . Event st+τ is said to follow event st (denoted st+τ st ) if σ (τ) (st+τ ) = st . The set S (st ) = {st+τ : st+τ st } ∪ {st } denotes the subtree of all events starting from st and S ≡ S (s0 ) denotes the complete event tree. At date 0, nature draws a sequence {s0 s1 }, such that st−1 = σ(st ) for all t. At date t, st is then publicly revealed. The unconditional probability of st is denoted by π(st ), where π(st ) > 0 for all st ∈ S . For st+τ ∈ S (st ), π(st+τ |st ) = π(st+τ )/π(st ) denotes the conditional probability of st+τ given st . There is a finite number J of consumer types, each represented by a unit measure of agents and indexed by j. Each consumer type is characterized by a sequence of endowments of the consumption good, Y j ≡ {y j (st )}st ∈S .7 Preferences over consumption sequences C ≡ {c(st )}st ∈S are represented by the lifetime expected utility functional (1) βt π(st )u(c(st )) U(C) = st ∈S

where β ∈ (0 1), and u(·) is strictly increasing, concave, bounded, twice differentiable, and satisfies standard Inada conditions. Markets t

At each date s , agents can issue and trade a complete set of contingent securities, which promise to pay one unit of period t + 1 consumption, contingent on the realization of event st+1 st , in exchange for current consumption. If no agent ever defaults (as will be the case in equilibrium), securities issued by different agents are perfect substitutes and trade at a common price. If agents had the ability to fully commit to their promises, they would be able to smooth all type-specific endowment fluctuations. In our model, however, agents cannot commit: at any date st , they can refuse to honor the securities they have issued and default. Any default becomes common knowledge and 7 Throughout the paper, for any variable x, x(st ) denotes the realization of x at event st , X denotes the sequence {x(st )}st ∈S , and X(st ) denotes the subsequence {x(st+τ )}st+τ ∈S (st ) .


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the defaulting agent loses the ability to issue claims in all future periods. Creditors can seize the financial assets he holds at the moment of default (i.e., his holdings of claims issued by other agents), but they are unable to seize any of his current or future endowments or any of his future asset holdings. In sum, after a default, an agent loses the ability to issue debt and starts with a net financial position of 0, but he retains the ability to purchase assets.8 This form of punishment follows the assumptions of BR. It captures the idea that it is much easier for market participants to coordinate on not accepting the claims issued by a given borrower, than to enforce an outright ban from financial markets. As the future denial of credit eliminates the incentive to repay, a potential lender will assign zero value to the claims issued by a borrower who has defaulted in the past. Enforcing an outright ban from financial markets, on the other hand, requires that potential borrowers are dissuaded from accepting loans at market prices from agents who have defaulted in the past. The denial of future credit thus only requires the issuer of each security to be known, while a ban from financial markets requires that the identity of buyers and sellers in all financial transactions be observable, so that agents can be punished for dealing with others who have defaulted in the past. Let q(st ) denote the price of an st -contingent bond at the preceding event σ(st ). The date-0 price of consumption at st , p(st ), is defined recursively by p(st ) = q(st ) · p(σ(st )) for all st ∈ S . Let aj (st ) denote the agent’s net financial position at st , that is, the amount of st -contingent securities he holds net of the amount of st -contingent securities he has issued. An agent chooses a profile of consumption and asset holdings C j ≡ {c j (st )}st ∈S and Aj ≡ {aj (st )}st ∈S subject to the sequence of flow budget constraints (2) q(st+1 )aj (st+1 ) for each st ∈ S c j (st ) ≤ y j (st ) + aj (st ) − st+1 st

The amount of securities an agent issues is observable and subject to a statecontingent upper bound −φj (st+1 ), which then determines a lower bound on his net financial position at st+1 : (3)

aj (st+1 ) ≥ φj (st+1 ) for all st+1 st st ∈ S

Given the initial asset position aj (s0 ), the optimal consumption and asset profile for an agent who never defaults maximizes (1), subject to the constraints (2) and (3). 8

The assumption that any positive holdings of other agents’ claims are confiscated in case of default implies that agents can default only on their net financial position. This assumption is made only for analytic and expositional purposes: as we will show later, it can be relaxed without changing our results. Therefore, the only disciplining element that may prevent agents from defaulting is losing the privilege to borrow in future periods.

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Self-Enforcing Private Debt Since several arguments in the paper require the manipulation of budget sets, it is convenient to denote by C j (a Φj (st ); st ) the set of feasible consumption profiles C j (st ) for a type-j agent starting at event st with an asset position a ∈ R, facing the sequence of debt limits Φj (st ), that is, the set of profiles C j (st ) which satisfy (2) and (3) at all events in S (st ) for some asset holdings profile Aj (st ) with aj (st ) = a. If the agent chooses never to default, his lifetime expected utility is given by (4)

V j (a Φj (st ); st ) ≡

max

C(st )∈C j (aΦj (st );st ) t+τ s ∈S (st )

βτ π(st+τ |st )u(c(st+τ )) j

The lifetime utility of a consumer who has defaulted in the past is VD (a; st ) ≡ j V (a O(st ); st ), where O(st ) denotes the sequence of borrowing constraints equal to zero at every st+τ ∈ S (st ). Notice that V j (a Φj (st ); st ) is increasing j in a. Therefore, if φj (st ) is such that V j (φj (st ) Φj (st ); st ) = VD (0; st ), then an agent is exactly indifferent between default and no default if a = φj (st ), no default is strictly preferred if a > φj (st ), and default is strictly preferred if a < φj (st ). This leads to the following definition of debt limits that are “not too tight,” following the terminology in Alvarez and Jermann (2000). DEFINITION 1: The debt limits Φj ≡ {φj (st )}st ∈S are not too tight if and only if (5)

j

V j (φj (st ) Φj (st ); st ) = VD (0; st )

for all st ∈ S

Debt limits that are not too tight allow for the maximum amount of credit that is compatible with repayment incentives at all histories. In principle, any j set of debt limits for which V j (φj (st ) Φj (st ); st ) ≥ VD (0; st ) for all st ∈ S eliminates default incentives. However, if this was a strict inequality for some st , an agent facing a binding debt constraint at φj (st ) would be willing to borrow at a rate slightly higher than the market interest rate and market participants would not be willing to refuse him credit. Our debt limits are thus set so that (i) no borrower has an incentive to default and (ii) no lender has an incentive to extend credit beyond a borrower’s debt limit.9 A competitive equilibrium with self-enforcing private debt is then defined as follows: 9 Moreover, since V j (· Φj (st ); st ) is increasing in Φj (st ), the maximum extension in later periods can only help increase the sustainability of credit in earlier periods. In this sense, our notion of debt limits not being too tight really expands the provision of credit to the maximum sustainable level at all horizons.


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J DEFINITION 2: For given {aj (s0 )}j=1J with j=1 aj (s0 ) = 0, a competitive equilibrium with self-enforcing private debt consists of consumption profiles, asset holdings, and debt limits {C j Aj Φj }j=1J , and state-contingent bond prices Q such that (i) for each j, {C j Aj } maximize (1), subject to (2) and (3), for given aj (s0 ), (ii) the debt limits Φj are not too tight for each j, and J J J (iii) markets clear: j=1 c j (st ) = j=1 y j (st ) and j=1 aj (st ) = 0 for all st ∈ S . Our equilibrium definition follows exactly Alvarez and Jermann (2000) except for the default punishment, which only allows for the denial of future credit, instead of complete autarky. Conceptually, the debt limits are similar to prices in Walrasian markets, in that individuals optimize taking prices and debt limits as given, but both are endogenously determined by the market equilibrium to satisfy the market-clearing and self-enforcement conditions. Unbacked Public Debt For our equivalence result, we consider an alternative economy with unbacked public securities. As before, there are sequential markets with complete contingent securities. However, unlike before, agents can no longer issue these claims. Claims are only supplied by a government, which rolls over a fixed initial stock of claims d(s0 ) period by period by issuing new securities. The government must satisfy its budget constraint d(st ) ≤ st+1 st q(st+1 )d(st+1 ), for all st ∈ S , that is, at st the amount of resources raised by issuing new claims for all st+1 st must be sufficient to honor the previous period’s commitments. This budget constraint captures the notion that these securities are not backed by any tax revenues or other government income. We assume that the government’s budget constraint is satisfied with equality each period: (6) q(st+1 )d(st+1 ) for all st ∈ S d(st ) = st+1 st

Given initial asset holdings aj (s0 ) ≥ 0, optimal consumption allocations and asset holdings maximize (1), subject to (2) and the nonnegativity constraint aj (st ) ≥ 0 for all st ∈ S . A competitive equilibrium with unbacked public debt is then defined as follows: DEFINITION 3: For given initial asset positions aj (s0 ) ≥ 0 for all j and debt J supply d(s0 ) = j=1 aj (s0 ), a competitive equilibrium with unbacked public debt consists of consumption and asset profiles {C j Aj }j=1J , a debt supply profile D, and bond prices Q such that (i) for each j, {C j Aj } are optimal given J J aj (s0 ), (ii) D satisfies (6), and (iii) markets clear: j=1 c j (st ) = j=1 y j (st ) and J j t t t j=1 a (s ) = d(s ) for all s ∈ S .

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3. AN EXAMPLE In this section, we illustrate the main results of our paper by means of a simple example with two types of consumers. In each period, one type receives the high endowment e and the other receives the low endowment e, with e + e = 1. The types switch endowment with probability α from one period to the next. Formally, uncertainty is captured by the Markov process st , with state space S = {s1 s2 } and symmetric transition probabilities Pr[st+1 = s1 |st = s2 ] = Pr[st+1 = s2 |st = s1 ] = α. The event s t corresponds here to the sequence {s0 st } and the endowments y j (st ) only depend on the current realization of st , with y j (st ) = e if st = sj and = e if st = sj . We construct a symmetric Markov equilibrium, in which consumption allocations, asset holdings, debt limits, and prices of state-contingent bonds depend only on the current state st , consumption allocations and asset holdings are symmetric across types and states, and the debt limit is binding in the high-endowment state for each agent. To focus on a stationary equilibrium, assume that the economy begins in state s0 = s1 and the initial asset positions are a1 (s0 ) = −ω and a2 (s0 ) = ω, where −ω is the debt limit for both agents. Proposition 1 shows that equilibria with positive debt levels can exist. PROPOSITION 1: Let c be defined by 1 − β(1 − α) = βαu (1 − c)/u (c). If c < e, there exists a stationary equilibrium with self-enforcing private debt in which the following conditions are observed: (i) State-contingent bond prices are q(st+1 ) = qc = 1 − β(1 − α) if st+1 = st and q(st+1 ) = qnc = β(1 − α) if st+1 = st .10 (ii) Consumption allocations are c j (st ) = c if st = sj and = c if st = sj , where c = 1 − c. (iii) Asset holdings are aj (st ) = −ω if st = sj and = ω if st = sj , where ω = (e − c)/(2qc ). (iv) Debt limits are φj (st ) = −ω for all st ∈ S . PROOF: Given the conjectured equilibrium prices, the proposed allocations satisfy the consumer’s first-order conditions qc u (c) = βαu (c), qnc = β(1 − α), and qc u (c) ≥ βαu (c). In addition, budget constraints and market-clearing conditions are satisfied by construction, given our definition of ω. We therefore only need to check that debt limits are not too tight. Theorem 1 below shows that a sequence of debt limits Φ satisfies (5) if and only if φ(st ) = t+1 )φ(st+1 ) for all st ∈ S . With constant debt limit φj (st ) = −ω, st+1 st q(s this reduces to 1 = qc + qnc , which the equilibrium prices satisfy by construction. Q.E.D. Notice that c is well defined and unique, given Inada conditions and strict concavity. Moreover, c is defined independently of e, so the condition c < e is 10

The subscripts c and nc stand for “change” and “no change.”


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satisfied for an open set of parameters, under any smooth parametrization of u(·). Proposition 1 illustrates our first general result: positive levels of debt are sustainable in equilibrium if interest rates are sufficiently low. In particular, the no-default condition requires that qc + qnc = 1, tying the risk-free interest rate to zero. To explain where this result comes from and why it relates the incentives for repayment to bond prices, let us suppose the prices qc and qnc = β(1 − α) are stationary, and restrict agents to stationary consumption allocations of ch in high-endowment periods and cl in low-endowment periods.11 For a consumer in the high-endowment state, the expected lifetime utility associated to (ch cl ) is v(ch cl ) =

1 (1 − β(1 − α))u(ch ) + βαu(cl ) 1 − β + 2βα

A consumer who chooses constant asset positions of −ω in high-endowment states and a ≥ −ω in low-endowment states faces the budget constraints ch = e − ω + qnc ω − qc a and cl = e + a − qnc a + qc ω. Substituting for a gives the intertemporal budget constraint (7)

(1 − qnc )ch + qc cl = (1 − qnc )e + qc e + [qc2 − (1 − qnc )2 ]ω

If the consumer never defaults, his optimal allocation (ch cl ) maximizes v(ch cl ) subject to (7). After defaulting in a high-endowment period, the agent maximizes the same objective function subject to the same constraint (7), but with ω = 0. A default thus represents a parallel shift of the intertemporal budget constraint and is optimal as long as the shift is positive, which is the case whenever 1 − qnc > qc . On the other hand, not defaulting is strictly preferred when 1 − qnc < qc , and the agent is exactly indifferent when qc = 1 − qnc . This illustrates how repayment incentives depend on the interest rate: the higher is the interest rate, the less appealing is the opportunity to borrow and the more appealing is the option to lend after default. Figure 1 illustrates this argument graphically and shows how the equilibrium allocation is determined. The figure depicts the space of stationary allocations (ch cl ), along with the indifference curves for the function v(ch cl ). The solid straight line going through (e e), with slope −1, depicts the aggregate resource constraint ch + cl = e + e = 1. Our equilibrium allocation (c c) corresponds to the allocation that maximizes v(ch cl ), subject to the resource constraint. Consider now an allocation along the resource constraint to the upper left of (c c), for example, (c a c a ). The intertemporal budget constraint that supports this allocation as an equilibrium (the dashed line through (c a c a )) must 11 Since stationary allocations are optimal whenever state prices are stationary and satisfy qnc = β(1 − α), these restrictions are without loss of generality.

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FIGURE 1.—Feasible stationary allocations and default incentives.

be tangent to the indifference curve going through (c a c a ); its slope in turn equals the ratio of state prices (1 − qnc )/qc . Since by construction the slope of the indifference curve at (c c) is −1, the state prices supporting (c a c a ) satisfy (1 − qnc )/qc > 1. However, our previous argument implies that the no-default condition is violated. In the figure, the consumer can default in the highendowment state, trade along the post-default budget constraint (the dashed line through (e e)), and reach the consumption bundle (chd cld ), which gives strictly higher utility than (c a c a ). The same argument applies to all allocations to the upper left of (c c), but notice that as the allocation moves closer to (c c), the two intertemporal budget constraints become flatter and the gap between them, and hence the benefit from defaulting, becomes smaller. At the allocation (c c), at which the supporting state prices satisfy (1 − qn )/qc = 1, the two intertemporal budget constraints exactly coincide with each other and with the aggregate resource constraint, implying that agents are indifferent between defaulting and not defaulting. This corresponds to our equilibrium allocation with self-enforcing private debt. If we move to the lower right of (c c), the allocations between (c c) and (e e) along the feasibility constraint require supporting state prices that satisfy (1 − qnc )/qc < 1. At these prices, the intertemporal budget constraint with no default is to the right of the one with default, and hence agents would strictly prefer no default to default. Finally, the default and no-default budget constraints converge again at (e e), which corresponds to the autarkic equilibrium with zero borrowing and prices equal to qcaut = βαu (e)/u (e) and aut qnc = β(1 − α). Moreover, all allocations to the right of (e e) or to the left of (c a c a ) can be immediately ruled out since they are dominated by the autarky allocation, which is always within the agents’ budget sets. Our example allows for a simple comparison between our results and those obtained when default is punished by complete exclusion from financial mar-


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kets. In particular, the environment in our example is the same as the one in Krueger and Perri (2006), with the exception that they considered autarky as the consequence of default. We have chosen the point (c a c a ) in Figure 1 so that it also represents the stationary equilibrium allocation under the autarky punishment. To see this, notice that (c a c a ) is on the same indifference curve as the autarky allocation (e e).12 If agents are not allowed to save after default, when they make their default decision they only need to compare their expected utility at (c a c a ) and (e e). The interest rate plays no role in this comparison. As the figure shows, in our equilibrium there is strictly less risk sharing than in the economy with the autarky punishment, that is, c < c a . This is not surprising, given that our punishment is weaker. Somewhat more surprisingly and contrary to what one would expect from BR’s argument, the conditions for the existence of equilibria with positive debt are identical in the two environments: in both cases, a nonautarkic equilibrium exists only if e > c. The figure shows why this condition is necessary for sustaining positive levels of risk sharing: if e ≤ c, any departure from autarky towards more risk sharing would strictly lower the expected utility of a high-endowment consumer and would not be incentive compatible under either punishment. The condition e > c is equivaut ) < 1. In the terminology of Alvarez and alent to the condition 1/(qcaut + qnc Jermann (2000), a nonautarkic equilibrium exists if the autarky allocation displays “low implied interest rates,” that is, interest rates such that the present value of the aggregate endowment is infinite.13 However, the two forms of punishment have different implications for interest rates and debt levels in equilibrium: Under the autarky punishment, if positive levels of debt are sustainable, the equilibrium allocation displays “high implied interest rates,” that is, interest rates such that the present value of the aggregate endowment is finite.14 Under the no-borrowing punishment, instead, the equilibrium allocation displays low implied interest rates. The difference comes from the fact that, in our environment, utility after default is endogenous and depends on market interest rates. As long as interest rates are high, the BR result applies and the incentives for repayment disappear. The condition e > c also ensures that this economy admits a stationary equilibrium with valued unbacked public debt, with consumption allocations and bond prices that are identical to the ones in the equilibrium with self-enforcing private debt. 12 Notice that in the figure (c a c a ) lies to the lower right of the 45 degree line. If that was not the case, the stationary equilibrium under autarky punishment would feature perfect risk sharing. 13 aut When 1/(qcaut +qnc ) ≥ 1, autarky is the unique stationary equilibrium in both regimes. Proposition 4.8 in Alvarez and Jermann (2000) proves this result in general, under the autarky punishment. 14 See Proposition 4.10 in Alvarez and Jermann (2000) for a general statement of this result.

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PROPOSITION 2: If e > c, there exists an equilibrium with unbacked public debt, in which consumption allocations and prices are the same as in Proposition 1, asset holdings are aj (st ) = 0 if st = sj and = 2ω if st = sj , and the government’s supply of debt is d(st ) = 2ω for all st ∈ S . PROOF: Optimality of the proposed consumption allocations and asset holdings, as well as market-clearing in goods and asset markets, follows by construction. Moreover, since d(st ) is constant for all st ∈ S and qc + qnc = 1, the government’s rollover condition is also satisfied. Q.E.D. Proposition 2 illustrates our second general result: equilibrium allocations in an economy with self-enforcing private debt are equivalent to equilibrium allocations in an economy with unbacked public debt. This is proved in full generality in Theorem 2 and allows us to use equilibrium characterizations that apply in known environments with unbacked public debt to establish the existence and characterization of equilibria with positive levels of self-enforcing private debt. In the online Supplemental Material, we extend the analysis of this example in several dimensions. First, we augment the example to include aggregate endowment growth, showing that the equilibrium interest rate must equal the economy’s growth rate, which must equal the growth rate of the aggregate debt supply and the individual debt limits in steady state. We also show the existence of nonstationary equilibria in which there is a self-fulfilling collapse in the real value of debt (analogous to hyperinflations in economies with fiat money). Finally, we show how transitional dynamics depend on type-specific debt limits and initial asset holdings. 4. CHARACTERIZING REPAYMENT INCENTIVES In this section, we characterize the repayment incentives of an individual borrower. The main result in this section (Theorem 1) is that debt limits are not too tight if and only if they allow for “exact rollover,” that is, if and only if at each history, the agent is able to exactly repay his maximum outstanding debt −φ(st ) by issuing new debt, up to the limit −φ(st+1 ) for each st+1 st . Since we are exclusively concerned with the single-agent problem, we simplify notation throughout this section by dropping the superscript j. THEOREM 1: The debt limits Φ are not too tight if and only if they allow for exact rollover: (8) q(st+1 )φ(st+1 ) for all st ∈ S φ(st ) = st+1 st

Moreover, if the debt limits Φ are not too tight, C (a Φ(st ); st ) = C (a − φ(st ) O(st ); st ) for all st ∈ S .


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This theorem also shows that the budget set of an agent facing debt limits Φ which are not too tight is identical to that of an agent facing zero debt limits who starts with a higher initial asset position. Hence, optimal consumption and asset profiles are the same for the two agents. This simplifies the equilibrium characterization, since, rather than computing the fixed point between the consumer’s optimization problem and the self-enforcement condition (8), we only need to compute optimal consumption allocations for agents with zero debt limits, and these are identical to the optimal consumption allocations without default. Equilibrium debt limits are then constructed so as to satisfy (8) and market clearing in the asset market. Condition (8) states that debt can only be sustained if, instead of repaying, the borrower is able to infinitely roll over outstanding debt. This leads to a simple comparison with BR’s no-lending result, which follows almost immediately from Theorem 1. PROPOSITION 3—Bulow and Rogoff: Suppose prices and endowments are such that y(st+τ )p(st+τ )/p(st ) < ∞ for all st w(st ) ≡ st+τ ∈S (st )

Suppose the debt limits Φ are not too tight and satisfy φ(st ) ≥ −w(st ) for all st . Then φ(st ) = 0 for all st . PROOF: If w(st ) < ∞, it must be the case that limT →∞ st+T st p(st+T ) × w(st+T )/p(st ) = 0. Now, using the exact rollover condition and the condition φ(st ) ≥ −w(st ), we have φ(st ) = s

p(st+T ) p(st+T ) φ(st+T ) ≥ − w(st+T ) t t) p(s ) p(s t+T t t+T t s

s

s

for all T , which implies φ(st ) ≥ − lim

T →∞

s

p(st+T ) w(st+T ) = 0 t) p(s t+T t s

Q.E.D.

BR showed that positive levels of self-enforcing debt are ruled out if two conditions hold: (i) endowments are finite-valued at the prevailing state prices, that is, interest rates are high, and (ii) the agent’s debt is bounded by the present value of his future endowments, the “natural debt limit.” Condition (i) is imposed as an exogenous restriction on state prices. Condition (ii) on the other hand is a standard restriction which is usually imposed to rule out Ponzi

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games.15 With high interest rates, if debt limits are not zero and are not too tight, they are eventually inconsistent with the natural debt limits. With low interest rates, however, natural debt limits are infinite and impose no restriction on debt levels. While BR only focused on the first scenario, our characterization in Theorem 1 is sufficiently general to encompass both.16 To sustain positive debt levels, we must abandon condition (i) or (ii). From a partial equilibrium point of view, relaxing either one can lead to self-enforcing debt. However, a general equilibrium argument shows that the interesting case arises when we dispose of condition (i). If we relax condition (ii), but maintain high interest rates, Theorem 1 implies that if there are positive levels of debt, the aggregate stock of debt, and thus the savings of some lender, will eventually exceed the value of aggregate endowments. This clearly cannot happen in general equilibrium. On the other hand, it is possible to construct economies where, in general equilibrium, condition (i) fails to hold, as shown in the example in Section 3 and, more generally, in Section 5 below. Self-Enforcement and Exact Rollover Before we proceed to the proof of Theorem 1, we illustrate the relation between self-enforcement and exact rollover through a series of figures. For this, we assume that endowment fluctuations are deterministic and that agents trade a single uncontingent bond.17 The agents’ budget constraint can then be rewritten as ct = yt + (pt at − pt+1 at+1 )/pt . For a given sequence of prices {pt }, we can thus compare the consumption profiles resulting from different asset plans simply by comparing the period-by-period changes in the present value of asset holdings, pt at − pt+1 at+1 . In the following figures, we plot the time paths {pt at } of the present values of asset profiles with and without defaults to evaluate repayment incentives. Figure 2 considers repayment incentives when debt limits allow for exact rollover. In a deterministic environment, this requires that pt φt is constant over time; such debt limits are represented by the dotted line A. Line B represents an arbitrary asset profile that is feasible for an agent who defaults at some date t. Line C represents a parallel downward shift of the asset profile B to an initial asset position of φt . Notice that C generates the same consumption sequence as B. Moreover, since profile B remains nonnegative, C always 15 In the working paper version, Bulow and Rogoff (1988, p. 5) hinted at the idea that relaxing this condition may lead to positive debt when they remarked that this assumption rules out “Ponzi-type reputational contracts.” 16 Our formulation of the BR result is slightly different from the original version of the nolending result, which stated that any asset sequence that gives an agent no incentive to ever default must always remain nonnegative. The formal equivalence between the two statements follows from the additional (almost immediate) observation that any sustainable asset position can be bounded below by a sequence of debt limits that are not too tight. 17 We thus replace the dependence on st by a time subscript to simplify notation.


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FIGURE 2.—Debt limits satisfying exact rollover.

remains above A, and is therefore feasible for an agent who starts with an asset position of φt and does not default. Hence, this agent must be weakly better off not defaulting at date t. On the other hand, for any asset profile C that is feasible without default starting from an asset position of φt , there is some asset profile B that is feasible starting from a default at date t and gives the agent the same consumption profile as C. If debt limits allow for exact rollover, the agent must therefore be exactly indifferent between defaulting on an asset position of φt and not defaulting, that is, the self-enforcement condition is satisfied with equality. Along similar lines, we can illustrate how repayment incentives are violated when the present value of debt limits is shrinking over time. Figure 3 plots the case of BR, in which the natural debt limits are finite and act as a lower bound on the agent’s asset profile. In this case, their present value falls over time (line A). Then, given any asset profile B that is consistent with these debt limits and admits positive debt at some date t, there exists a date t ∗ ≥ t at which the present value of debt reaches a maximum. At that point, the agent can default and replicate the same consumption profile as B, just using positive asset holdings (line C), and can even improve upon the no-default profile by strictly increasing consumption at date t ∗ (line D).

FIGURE 3.—Shrinking debt limits.

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FIGURE 4.—Expanding debt limits.

Likewise, if the present value of debt limits is expanding over time (Figure 4), for every profile B that is feasible after a default at some date t, there exists a profile C that is feasible without default starting from asset position φt and implements the same consumption. Moreover, profile D remains feasible without default starting from asset position φt , but delivers strictly higher consumption at date t ∗ , where the debt limits are expanding. Hence, agents strictly prefer not to default when the present value of debt limits is expanding over time. PROOF OF THEOREM 1: Proposition 4 establishes the sufficiency part of Theorem 1 by generalizing the graphical argument of Figure 2. PROPOSITION 4: Suppose that the debt limits Φ allow for exact rollover. Then (9)

V (a Φ(st ); st ) = VD (a − φ(st ); st )

for all st ∈ S and any a ≥ φ(st )

PROOF: Starting from an arbitrary st , consider asset profiles {a(st+τ )}st+τ ∈S (st ) ˆ t+τ )}st+τ ∈S (st ) that satisfy a(s ˆ t+τ ) = a(st+τ ) − φ(st+τ ) for all st+τ ∈ S (st ), and {a(s t t t ˆ ) = a − φ(st ). By construction, {a(st+τ )}st+τ ∈S (st ) with a(s ) = a ≥ φ(s ) and a(s ˆ t+τ )}st+τ ∈S (st ) is feasible after deis feasible under no default if and only if {a(s t faulting at s . Moreover, the exact rollover (8) implies a(st+τ ) − condition t+τ+1 t+τ+1 t+τ+1 t+τ ˆ ˆ t+τ+1 ) for all q(s )a(s ) = a(s ) − q(s )a(s st+τ+1 st+τ st+τ+1 st+τ t+τ t t s ∈ S (s ) and, therefore, starting from a(s ) = a, asset plan {a(st+τ )}st+τ ∈S (st ) implements the same consumption allocation {c(st+τ )}st+τ ∈S (st ) as asset plan ˆ t ) = a − φj (st ). But then C (a Φ(st ); st ) = ˆ t+τ )}st+τ ∈S (st ) starting from a(s {a(s t t t C (a − φ(s ) O(s ); s ) and V (a Φ(st ); st ) = VD (a − φ(st ); st ). Q.E.D. The self-enforcement condition (5) follows from setting a = φ(st ) in (9). Condition (9) further implies that an agent who defaults on his maximum gross amount of debt −φ(st ), but keeps his own asset holdings a − φ(st ) after a default, is always exactly indifferent between defaulting and not defaulting. The assumption that agents start with a net financial position of zero after default can therefore be relaxed without weakening repayment incentives.


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Proposition 5 establishes the necessity part of Theorem 1 by showing that debt limits are not too tight only if they allow for exact rollover. This was already suggested by the graphical arguments in Figures 2–4. However, this graphical intuition is incomplete, since it only applies to sequences of debt limits whose present values are monotone increasing or decreasing. The argument for arbitrary nonmonotone sequences of debt limits turns out to be considerably more involved. PROPOSITION 5: If the debt limits Φ are not too tight, they allow for exact rollover. The proof of this proposition is given in the Appendix: here we sketch the key steps. Consider a sequence of debt limits Φ which are not too tight. Starting from some arbitrary event st , we first construct a sequence of auxiliary debt ˜ t ), as follows: limits Φ(s ⎧ φ(st+τ ) if a∗ (st+τ ) = φ(st+τ ) ⎪ ⎪ ⎨ ˜ t+τ+1 )} q(st+τ+1 ) min{φ(st+τ+1 ) φ(s ˜ t+τ ) = (10) φ(s ⎪ ⎪ ⎩ st+τ+1 st+τ otherwise where A∗ (st ) denotes the optimal no-default asset profile of an agent starting from st with asset position a∗ (st ) = φ(st ). The first step of the proof (and the major technical hurdle) consists of showing that this sequence is well defined and finite-valued. This is complicated by the fact that present discounted values need not be well defined in our environment since we cannot rely on an assumption of high interest rates. The characterization in turn makes use of the time separability, concavity, and boundedness of u(·). ˜ t+τ ) ≤ φ(st+τ ) and Next, using arbitrage arguments, we show that (i) φ(s t+τ t+τ ∗ t+τ t+τ ˜ ˜ t+τ ) ≥ φ(s ) = φ(s ) whenever a (s ) = φ(s ), and (ii) φ(s t+τ+1 ˜ )φ(st+τ+1 ) for every st+τ ∈ S (st ). The first property states st+τ+1 st+τ q(s that Φ˜ is a lower bound of Φ and is equal to Φ whenever the actual debt limit is binding. The second property states that Φ˜ satisfies (ER) with a weak inequal˜ at any event, the maximum outstanding debt obligations ity, so that under Φ, are weakly less than the funds that can be raised by exhausting debt limits on the continuation events. Property (i), together with the concavity of u(·) then implies that the value of the no-default problem starting from asset holdings of a(st ) = φ(st ) at st is the same under the original debt limits, Φ(st ), as under the auxiliary debt limits, ˜ t ) (since the latter only relaxes nonbinding debt limits). Since the origiΦ(s ˜ t ), we thus obtain VD (0; st ) = nal debt limits are not too tight and φ(st ) ≥ φ(s t t t t t t ˜ ) s ) ≥ V (φ(s ˜ t ) st ). On the other ˜ t ); Φ(s V (φ(s ); Φ(s ) s ) = V (φ(s ); Φ(s hand, using the same arbitrage argument as Proposition 4, property (ii) implies

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˜ t ) st ) and V (φ(s ˜ t ) st ) ≥ VD (0; st ) ˜ t ) Φ(s ˜ t ); Φ(s that C (0 O(st ); st ) ⊆ C (φ(s t+τ t+τ+1 ˜ t+τ+1 ˜ )φ(s ) for some st+τ ∈ with strict inequality if φ(s ) > st+τ+1 st+τ q(s t S (s ). Therefore, both inequalities must hold with equality, which requires ˜ t ) satisfies the exact rollover condition as an equality ˜ t ) and that Φ(s φ(st ) = φ(s t+τ t for all s ∈ S (s ). ˜ t+1 ) = φ(st+1 ) for all st+1 st . ReTo complete the proof, we show that φ(s peating the same steps as above, we construct additional sequences of auxˆ t+1 ), together with optimal asset holdings A(s ˆ t+1 ), for iliary debt limits Φ(s ˆ t+1 ) satisfies (ER) and φ(st+1 ) = φ(s ˆ t+1 ). Moreeach st+1 st . Clearly, Φ(s over, due to concavity and additive separability of U, optimal asset profiles ˆ t+τ ) ˆ t+τ ) = φ(st+τ ) = φ(s are monotone in initial asset holdings, so that a(s ∗ t+τ t+τ t+τ ˜ ). Together with (10), this implies that whenever a (s ) = φ(s ) = φ(s t+τ t+τ t+τ t+1 ˆ ˜ t+1 ) = φ(s ˆ t+1 ) = φ(st+1 ), ˜ φ(s ) = φ(s ) for all s ∈ S (s ) and hence φ(s which completes our proof. Q.E.D. REMARK 1: Proposition 5 is the only result where we use the assumptions of additive time separability, concavity, and boundedness of u(·). All other results rely purely on arbitrage arguments and therefore require only strict monotonicity. The boundedness assumption is a strong restriction, but it is required only for a partial equilibrium characterization. If one restricts attention to debt limits Φ such that optimal consumption allocations are bounded above by aggregate endowments (a condition that must hold in general equilibrium), Proposition 5 holds under the following weaker restriction. ASSUMPTION 1: For all C such that U(C) ≥ minj U(Y j ) and c(st ) ∈ [0 J j t t t t t

t st ∈S β π(s )c(s )u (c(s )) < ∞. j=1 y (s )] for all s ∈ S , This regularity condition bounds the rate at which individual and aggregate endowments can grow or decline, relative to the discount factor β. When relaJ tive risk aversion is bounded, this assumption holds whenever U( j=1 Y j ) and minj U(Y j ) are both finite. 5. GENERAL EQUILIBRIUM CHARACTERIZATION We now turn to the question of whether there exist equilibria with positive levels of self-enforcing debt and how they can be characterized. Theorem 2 shows that a given consumption allocation and price vector constitute a competitive equilibrium with self-enforcing private debt if and only if the same allocation and prices are an equilibrium of the corresponding economy with unbacked public debt. For the latter economy, there are known existence and characterization results (e.g., Santos and Woodford (1997)), which then extend immediately to the economy with self-enforcing private debt.


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THEOREM 2: An allocation {C j }j=1J and prices Q are sustainable as a competitive equilibrium (CE) with self-enforcing private debt if and only if {C j }j=1J and Q are also sustainable as a competitive equilibrium with unbacked public debt. j

PROOF: Step 1. If {C j Aj Φ Q} is a CE with self-enforcing private debt for j initial asset positions {aj (s0 )}j=1J , then {C j Aj − Φ } is optimal given initial J j asset holdings of aj (s0 ) − φj (s0 ), state prices Q, and zero debt limits, − j=1 Φ J j j J satisfies (6), and j=1 (Aj − Φ ) = − j=1 Φ , so that market clearing is sat j j J isfied. {C j Aj − Φ − j=1 Φ Q} therefore constitutes a CE with unbacked public debt for initial asset holdings {aj (s0 ) − φj (s0 )}j=1J . Step 2. If {C j Aˆ j D Q}j=1J constitutes a CE with unbacked public debt, starting from initial asset positions {aˆ j (s0 )}j=1J , we can construct debt limits Φ˜ j = −aˆ j (s0 )/d(s0 ) · D and asset holdings A˜ j = Aˆ j + Φ˜ j . By construction, debt limits Φ˜ j are not too tight, and the allocations {C j A˜ j } are optimal given initial asset holdings of a˜ j (s0 ) = 0 and debt limits holdings of Φ˜ j . Moreover, since J ˆ j J ˜ j J ˆj ˜j j=1 A = j=1 (A + Φ ) = j=1 A − D = 0, these allocations clear markets in the private debt economy. Therefore {C j a˜ j Φ˜ j p}j=1J is a CE with selfenforcing private debt, for initial asset positions of zero. Q.E.D. The proof of Theorem 2 makes repeated use of the exact rollover property. First, with exact rollover, the feasible, and hence the optimal, consumption allocations in the default and no-default problems exactly coincide for suitably chosen initial values of asset holdings. Second, the exact rollover property implies a mapping from debt limits Φj which are not too tight to a sequence of public debt supply that satisfies the government rollover constraint (6), and vice versa. Given this mapping from debt limits to public debt circulation and the equivalence of optimal consumption allocations, market-clearing conditions in the two economies turn out to be exactly equivalent. Theorem 2 formally establishes the connection between sustaining repayment incentives for private debt and sustaining rational bubbles. In the private debt economy, the agents’ commitment and enforcement power is so limited that any contract that, at some date, requires a positive transfer of resources in net present value is not sustainable. Likewise, in the economy with unbacked public debt, the government does not have the power to use taxation to guarantee its debt holders a positive net transfer of resources. In both cases, the result is that the only sustainable allocations roll over existing debt forever. The equivalence arises because neither side can credibly commit to future transfers, either via contract enforcement or via taxation.

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6. CONCLUDING REMARKS In this paper, we have studied a general equilibrium economy with selfenforcing private debt, in which borrowers, after default, are excluded from future credit, but retain the ability to save at market interest rates. For a partial equilibrium version of this model, in which a small open economy borrows internationally at fixed, positive interest rates, BR showed that debt cannot be sustained by reputational mechanisms only: eventually, the country always has an incentive to default. The BR result can be interpreted as follows: if there are some agents who are able to commit to intertemporal transfers at “high interest rates,” then the remaining agents, who are unable to commit, will accumulate and decumulate the securities issued by the committed agents, but will never become net borrowers. Krueger and Uhlig (2006) provided the analytical foundations for this interpretation. In contrast, we show that positive levels of debt can be sustained when no party has commitment power. The key to our result is that interest rates adjust downward to provide repayment incentives to all the potential borrowing parties. As a result, “low interest rates” emerge in equilibrium. These results helps to clarify the BR result in two directions. From a formal point of view, they highlight the role played by the interest rate in the BR argument. From a more substantive point of view, they clarify the role of multilateral versus unilateral lack of commitment and the role of credit exclusion versus stronger forms of punishment for the sustainability of debt in general equilibrium. This analysis helps to bridge the gap between the negative result of BR and the positive results obtained with stronger forms of punishment, in particular those in Kehoe and Levine (1993) and Alvarez and Jermann (2000).18 From an empirical point of view, the central implication of our model is that equilibrium borrowing requires low interest rates. In principle, this prediction could be used to test the hypothesis that international debt is sustained by our reputational mechanism. In practice, this is tantamount to testing for rational bubbles in international debt.19 One standard approach to such a test would be to compare the rate of return on international debt with the growth rate of the aggregate stock of debt in circulation. An alternative approach focuses 18 See the discussion in Section 3. Our analysis also relates to the model of private international capital flows of Jeske (2006) and Wright (2006). In their model, the ability of agents to participate in domestic capital markets after defaulting on external debt has the same effects as the ability to save in our model. See the discussion in Wright (2006) for formal details. 19 See Ventura (2004) and Caballero and Krishnamurthy (2006) for related work on bubbles in international capital flows. Notice that rational bubbles are often ruled out on theoretical grounds if there is a tradable asset that pays an infinite stream of dividends. This argument does not necessarily apply in a world with sovereign borrowers and multilateral lack of commitment. Real assets are typically tied to a physical location where the dividend is generated. The country controlling that location may choose to expropriate foreign nationals and bar them from accessing the asset’s dividends in the event of a default.


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on net flows, taking a sample of countries who borrow on the international capital markets and evaluating whether their debtor position is associated to a net outflow or to a net inflow of resources. This is analogous to the approach in Abel, Mankiw, Summers, and Zeckhauser (1989), who focused on the net flows between the corporate sector and consumers in a closed economy. Empirical work in either direction will face two significant challenges. First, as emphasized in Gourinchas and Rey (2005), net financial flows between countries are the outcome of gross flows that include instruments with very different rates of return and risk profiles (debt, equity, direct investment, and so on). Second, especially when looking at emerging economies, default episodes do take place regularly in international financial markets.20 In this paper, we allow for fully state-contingent securities, so we do not treat explicitly gross holdings of different assets and do not take a stand on how to interpret actual defaults within the context of our model. To map the model’s predictions to observed data on financial flows that include defaults, it would probably be necessary to extend the analysis by modelling explicitly gross positions in various financial instruments and to formulate an explicit interpretation of default episodes. Finally, our paper also has implications for the literature on inside and outside money. In particular, in our setup unbacked public debt and self-enforcing private debt are analogous, respectively, to outside (fiat) money and inside money. The existing monetary literature discusses the circulation of fiat money and inside money largely separately from each other. The circulation of fiat money requires that an intrinsically useless asset (a rational bubble) is traded at a positive price. The circulation of inside money instead relies on having the proper reputational mechanisms in place to guarantee that outstanding claims are honored. Although on the surface these seem to be conceptually distinct problems, our analysis shows that they are closely related.21

20 After disentangling the various gross positions and estimating rates of returns on the various instruments, Gourinchas and Rey (2005) found that the United States after 1988 became a net debtor and, in spite of that, has been typically receiving a net positive inflow of resources from the rest of the world. This finding is consistent with our story. Klingen, Weder, and Zettelmeyer (2004) looked at ex post realized returns for a broad sample of emerging markets over the last 30 years and argued that returns were very close to the returns on 10-year U.S. treasuries—a level which could very well be consistent with our story. Eichengreen and Portes (1989) and Lindert and Morton (1989) found similar results based on historical evidence. 21 See Cavalcanti, Erosa, and Temzelides (1999) and Berentsen, Camera, and Waller (2007) for matching models of private debt circulation under limited commitment. In Cavalcanti, Erosa, and Temzelides (1999), a fixed subset of agents is allowed to issue notes, which are sustained by the loss of a noncompetitive note-issuing rent if outstanding notes are not redeemed on demand. Berentsen, Camera, and Waller (2007) studied a matching model with money and competitive supply of bank credit, and showed among other things that with lack of commitment, such credit is sustainable only if the inflation rate is nonnegative.

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APPENDIX: PROOF OF PROPOSITION 5 Suppose that V (φ(st ) Φ(st ); st ) = VD (0; st ) for all st ∈ S , and that φ(st ) < 0 for some st (otherwise, the proposition holds trivially). The budget set C (a Φ(st ); st )is nonempty, and hence the problem is well defined, only if t+1 a ≥ −y(st ) + st+1 st q(s )φ(st+1 ), from which it follows that Φ(st ) must sat isfy φ(st ) ≥ −y(st ) + st+1 st q(st+1 )φ(st+1 ). Standard properties of the consumer problem (4) imply that V (a Φ(st ); st ) is strictly increasing in a, and the sequences C ∗ (st ) and A∗ (st ) of optimal consumption and asset holdings, starting from an initial asset position a∗ (st ) = φ(st ) at history st , are nondecreasing in φ(st ). Now, let N (st ) denote the subtree of events starting from st for which the debt limits are nonbinding: Starting from N0 (st ) ≡ {st }, define

Nτ (st ) = {st+τ st : a∗ (st+τ ) > φ(st+τ ) and σ(st+τ ) ∈ Nτ−1 (st )} Bτ (st ) = {st+τ st : a∗ (st+τ ) = φ(st+τ ) and σ(st+τ ) ∈ Nτ−1 (st )} N (s ) = t

∞

Nτ (s ) t

B (s ) = t

τ=0

N (s B (s

t+τ

t+τ

Bτ (st )

τ=0

; s ) = N (s ) ∩ S (s t

t

; s ) = B (s ) ∩ S (s t

∞

t

t+τ

t+τ

)

) for all st+τ ∈ N (st )

Nτ (st ) denotes the set of histories st+τ along which the debt limit was never binding between events st and st+τ , and N (st ) denotes the union of all such sets, including st . Bτ (st ) denotes the set of histories st+τ at which the debt limit is binding for the first time after st and B (st ) denotes the union of all such sets. N (st+τ ; st ) defines the “subtree” of N (st ), which starts at st+τ , and B (st+τ ; st ) denotes the set of events at which the debt limit is binding for the first time after st+τ . Finally, we define ˆ t+τ ; st ) = φ(s

st+τ+k ∈B(st+τ ;st )

p(st+τ+k ) φ(st+τ+k ) p(st+τ )

and w(st+τ ; st ) ≡

st+τ+k ∈N (st+τ ;st )

p(st+τ+k ) t+τ+k y(s ) p(st+τ )

as the present value of the first binding debt limits and of the endowments over N (st ). The proof of Proposition 5 then proceeds in five steps, which are stated as separate lemmas. Lemma 1 establishes that under mild regularity conditions,


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ˆ t+τ ; st ) are both finite-valued. Lemma 2 establishes the exw(st+τ ; st ) and φ(s ˜ t ), istence of a well defined, finite-valued sequence of auxiliary debt limits Φ(s which satisfies the recursive characterization: ⎧ ˜ t+τ+1 ; st )} ⎨ st+τ+1 st+τ q(st+τ+1 ) min{φ(st+τ+1 ) φ(s t+τ t ˜ ;s ) = (11) φ(s if a∗ (st+τ ) > φ(st+τ ) ⎩ t+τ φ(s ) if a∗ (st+τ ) = φ(st+τ ) ˜ t ) bounds Φ(st ) from below for all st+τ ∈ S (st ). Lemma 3 establishes that Φ(s and satisfies the (ER) condition as a weak inequality. Lemma 4 establishes that ˜ t ) satisfies (ER) as an equality and that φ(s ˜ t ; st ) = φ(st ). Finally, Lemma 5 Φ(s ˜ t+1 ; st ) = φ(st+1 ) for all st+1 st . shows that φ(s J LEMMA 1: Suppose either that u(·) is bounded or that c ∗ (st+τ ) ≤ j=1 y j (st+τ ) for all st+τ ∈ S (st ), and that Assumption 1 holds. Then w(st+τ ; st ) < ∞ and ˆ t+τ ; st ) > −∞. φ(st+τ ) + w(st+τ ; st ) > φ(s PROOF: Summing the agent’s budget constraint over st+τ+k ∈ N (st+τ ; st ), we get st+τ+k ∈N (st+τ ;st )

p(st+τ+k ) ∗ t+τ+k c (s ) p(st+τ )

ˆ t+τ ; st ) = a∗ (st+τ ) + w(st+τ ; st ) − φ(s p(st+τ+K ) ∗ t+τ+K a (s ) − lim K→∞ p(st+τ ) t+τ+K t+τ t s

∈N (s

;s )

Now, using the agent’s first-order condition for st+τ+k ∈ N (st+τ ; st ), we have st+τ+k ∈N (st+τ ;st )

=

p(st+τ+k ) ∗ t+τ+k c (s ) p(st+τ )

st+τ+k ∈N (st+τ ;st )

βk

π(st+τ+k ) u (c ∗ (st+τ+k )) ∗ t+τ+k c (s ) < ∞ π(st+τ ) u (c ∗ (st+τ ))

u(c) − u(0) ≤ U¯ if u(·) is bounded or because of Aseither because u (c)c ≤ J ∗ t+τ sumption 1 if c (s ) ≤ j=1 y j (st+τ ) holds for all st+τ ∈ S (st ). In addition, the agent’s first-order conditions over st+τ+k ∈ N (st+τ ; st ), along with the transversality condition, imply lim

K→∞

st+τ+K ∈N (st+τ ;st )

p(st+τ+K ) ∗ t+τ+K a (s ) = 0 p(st+τ )

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ˆ t+τ ; st ) > −∞. But then it follows immediatelythat w(st+τ ; st ) < ∞ and φ(s t t t+1 t+1 Finally, if a(s ) = −y(s ) + st+1 st q(s )φ(s ), the only feasible allocation without default yields c(st ) = 0 and a(st+1 ) = φ(st+1 ) for all st+1 st , which yields a lifetime expected utility of u(0) + β st+1 st π(st+1 |st )V (φ(st+1 ) t+1 Φ(st+1 ) st+1 ). If instead the agent defaults and 0 for all st+1 st , sets d(st+1 ) = t t his lifetime expected utility is u(y(s )) + β st+1 st π(s |s )VD (0 st+1 ), which must be strictly preferred to no default if Φ(st ) satisfies (SE). But then φ(st ) > t −y(s ) + st+1 st q(st+1 )φ(st+1 ) for all st ∈ S , and summing this inequality over ˆ t+τ ; st ) > 0. Q.E.D. st+τ+k ∈ N (st+τ ; st ), we find w(st+τ ; st ) + φ(st+τ ) − φ(s Lemma 2 establishes the existence of a finite-valued sequence of “auxiliary” ˜ t ). debt limits Φ(s ˜ t ) which satisfies (11), LEMMA 2: There exists a finite-valued sequence Φ(s t+τ t ˜ t+τ+K ; for all s ∈ S (s ) and for which limK→∞ st+τ+K ∈N (st+τ ;st ) p(st+τ+K )φ(s t s ) = 0. PROOF: Step 1 establishes the existence of such a solution for st+τ ∈ N (st ), together with the limit property. Step 2 extends the construction to S (st ). ˆ t+τ ; st ) − Y (st+τ ; st ) and define Step 1. Let Φ(0) be defined by φ(0) (st+τ ) = φ(s Φ(K) (st ) recursively by Φ(K) = T Φ(K−1) , where the operator T on sequences t B ∈ RN (s ) is defined by q(st+τ+1 ) min{φ(st+τ+1 ) b(st+τ+1 )} (T B)(st+τ ) = st+τ+1 ∈N (st+τ ;st )

+

q(st+τ+1 )φ(st+τ+1 )

st+τ+1 ∈B(st+τ ;st )

ˆ t+τ ; st ) − w(st+τ ; st ) = φ(0) (st+τ ), we have Since φ(st+τ ) ≥ φ(s (1) t+τ (0) t+τ φ (s ) ≥ φ (s ) for all st+τ ∈ N (st ) and, therefore, Φ(1) ≥ T Φ(0) . But T is a monotone operator and, therefore, {Φ(K) } is a nondecreasing sequence. Moreover, since φ(K) (st+τ ) ≤ 0 for all K and st+τ ∈ N (st ), {Φ(K) } must converge ˜ t ) which satisfies (11). The limit property then follows imto a finite limit Φ(s ˆ t+τ+K ; st ) − w(st+τ+K ; st ) ˜ t+τ+K ; st ) ≥ φ(0) (st+τ+K ) = φ(s mediately from 0 ≥ φ(s t+τ+K ˆ t+τ+K ; st ) − w(st+τ+K ; and the fact that limK→∞ st+τ+K ∈N (st+τ ;st ) p(s )[φ(s t s )] = 0. Step 2. Define B (1) (st ) = B (st ) and let

B (s ) = (k)

t

∞

τ=k

st+τ st : a∗ (st+τ ) = φ(st+τ ) and

σ(st+τ ) ∈ N st+τ for some st+τ ∈ B (k−1) (st )


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denote the subset of histories in S (st ), at which the debt limit is binding for the kth time after st . Since A∗ (st ) solves the consumer problem starting from st with asset position φ(st ) and since a∗ (st+τ ) = φ(st+τ ) for all st+τ ∈

∞ (k) t (s ), {a∗ (st+τ )}st+τ ∈S (st+τ ) solves the consumer problem starting from k=1 B

∞ t+τ any s ∈ k=1 B (k) (st ) with asset position φ(st+τ ). Since st was chosen arbi∞ trarily, we can replicate the same arguments as above for all st+τ ∈ k=1 B (k) (st ) ˜ t ) to (11) for all st+τ ∈ S (st ). Q.E.D. to construct a solution Φ(s ˜ t) ˜ t ) bounds Φ(st ) from below and (ii) Φ(s Lemma 3 establishes that (i) Φ(s satisfies the exact rollover property as a weak inequality. We prove these properties for N (st ) and B (st ); they can immediately be extended to S (st ) using the construction of the previous proof. ˜ t+τ ; st ) = φ(s ˆ t+τ ; st ) and LEMMA 3: (i) For all st+τ ∈ N (st ), φ(st+τ ) ≥ φ(s t+τ t t+τ+1 ˜ t+τ+1 t ˜ )φ(s ; s ). φ(s ; s ) = s+τ+1s+τ q(s t+τ t t+τ t ˜ ˜ t+τ+1 ; st ). (ii) For all s ∈ B (s ), φ(s ; s ) ≥ s+τ+1s+τ q(st+τ+1 )φ(s ˜ t+τ ; st ) for some PROOF: (i) Suppose to the contrary that φ(st+τ ) < φ(s t t+τ+k ˜ )}st+τ+k ∈S (st )\{st } denote the optimal asset profile s ∈ N (s ), and let {a(s without default, starting from a position of φ(st+τ ) at st+τ . Consider an agent who defaults and sets ⎧ ˜ t+τ+k ; st )} ˜ t+τ+k ) − min{φ(st+τ+k ) φ(s ⎨ a(s t+τ+k ˆ )= a(s for all st+τ+k ∈ N (st+τ ; st ) ⎩ 0 for all st+τ+k ∈ B (st+τ ; st ) t+τ

which is feasible after a default. From the monotonicity of optimal asset ˜ t+τ+k ) = a∗ (st+τ+k ) = φ(st+τ+k ) when˜ t+τ+k ) ≤ a∗ (st+τ+k ) and a(s holdings, a(s t+τ+k t+τ t ∈ B (s ; s ). Moreover, because of (SE), V (φ(st+τ+k ) Φ(st+τ+k ) ever s t+τ+k s ) = VD (0 st+τ+k ), so that the default profile provides the same lifetime utility going forward from any st+τ+k ∈ B (st+τ ; st ) as the nondefault profile. For st+τ+k ∈ N (st+τ ; st ), the difference in consumption between default and no default { c(st+τ+k )} is c(st+τ+k ) ˜ t+τ+k ; st )} = − min{φ(st+τ+k ) φ(s ˜ t+τ+k+1 ; st )} + q(st+τ+k+1 ) min{φ(st+τ+k+1 ) φ(s st+τ+k+1 st+τ+k

˜ t+τ+k ; st ) 0} ≥ 0 = − min{φ(st+τ+k ) − φ(s where the inequality is strict at st+τ . Therefore, consumption is weakly higher after a default for all st+τ+k ∈ N (st+τ ; st ) and is strictly higher at st+τ , imply˜ t+τ+1 ; st ) for all ing that default must be optimal. Now, using φ(st+τ+1 ) ≥ φ(s

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˜ t+τ ; st+τ ) for all st+τ+1 ∈ B (st+τ ; st ) in st+τ+1 ∈ N (st+τ ; st ) and φ(st+τ+1 ) = φ(s t+τ (11) then implies (ER) for all s ∈ N (st ). Solving (11) forward and using the ˜ t+τ ; st ) = φ(s ˆ t+τ ; st ). limit property from Lemma 2, we find φ(s (ii) Applying the same arbitrage argument as in part (i) at st+τ ∈ B (st ), we t+τ+1 ˜ ˜ t+τ ; st+τ ) and φ(s ˜ t+τ ; st+τ ) = have φ(st+τ ) ≥ φ(s )φ(st+τ+1 ; s+τ+1s+τ q(s t+τ t+τ t t+τ t+τ+1 ˜ ˜ s ). Moreover, by construction, φ(s ; s ) = φ(s ) and φ(s ; st ) = ˜ t+τ+1 ; st+τ ) for all st+τ+1 st+τ , from which the result follows immediately. φ(s Q.E.D. ˜ t ; st ) and that Lemma 4 uses these two properties to show that φ(st ) = φ(s t t+τ t ˜ ) satisfies (ER) with equality for all s ∈ S (s ). Φ(s ˜ t ) satisfies (ER) with equal˜ t ; st ) and Φ(s LEMMA 4: For all st ∈ S , φ(st ) = φ(s ity. PROOF: Consider the consumer problem with borrowing constraints equal ˜ t ). Since the objective is strictly concave, φ(st+τ ) ≥ φ(s ˜ t+τ ; st ) for all to Φ(s t+τ t t+τ t+τ t ∗ t+τ t+τ ˜ t ) relaxes ˜ s ∈ S (s ), and φ(s ) > φ(s ; s ) only if a (s ) > φ(s ), Φ(s ∗ t only nonbinding constraints and, hence, A (s ) is also optimal for the relaxed ˜ t ) st ) = ˜ t ), implying V (φ(st ) Φ(s problem with borrowing constraints Φ(s t t t V (φ(s ) Φ(s ) s ). Using the self-enforcement hypothesis and the monotonic˜ t ) st ) in a and φ(st ) ≥ φ(s ˜ t ; st ), this implies VD (0 st ) = ity of V (a Φ(s ˜ t ) st ) ≥ V (φ(s ˜ t ) st ). On the other ˜ t ; st ) Φ(s V (φ(st ) Φ(st ) st ) = V (φ(st ) Φ(s t ˜ hand, since Φ(s ) satisfies the exact rollover property as a weak inequality, the ˜ t ) st ) ≥ VD (0 st ), ˜ t ; st ) Φ(s same argument as Proposition 4 implies that V (φ(s t+τ+1 ˜ t+τ ; st ) > where the inequality is strict whenever φ(s )× st+τ+1 st+τ q(s t+τ+1 t t+τ t ˜ φ(s ; s ) for some s ∈ S (s ). Together these inequalities can hold ˜ t ; st ), and φ(s ˜ t+τ ; st ) = only as equalities, which requires that φ(st ) = φ(s t+τ+1 ˜ t+τ+1 t t+τ t )φ(s ; s ) for all s ∈ S (s ). Q.E.D. st+τ+1 st+τ q(s To complete the proof that Φ(st ) satisfies (ER), we thus need to show that ˜ t+1 ; st ) for all st+1 ∈ S (st ). Whenever st+1 ∈ B1 (st ), that is, whenφ(st+1 ) = φ(s ever the debt limit is binding at st+1 , this is true by construction. Our final lemma shows that this is also true whenever the debt limit is not binding. ˜ t+1 ; st ). LEMMA 5: For all st+1 ∈ N1 (st ), φ(st+1 ) = φ(s PROOF: Since st was chosen arbitrarily, applying all the preceding argu˜ t+1 ; st+1 ), so it suffices to show ments to st+1 ∈ N1 (st ) implies that φ(st+1 ) = φ(s t+1 t t+1 t+1 ˜ ˜ that φ(s ; s ) = φ(s ; s ). Now, from the monotonicity of {a∗ (st+τ )} with respect to the initial asset holdings, it follows that N (st+1 ) ⊆ N (st+1 ; st ) and


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B (st+1 ) ⊆ B (st+1 ; st ) ∪ N (st+1 ; st ), that is, debt limits must be binding for an agent starting from st+1 with assets φ(st+1 ) whenever they are binding for an agent starting from st+1 with assets a∗ (st+1 ) > φ(st+1 ). But then, by the defini˜ t) ˜ t+τ ; st+1 ) = φ(s ˜ t+τ ; st ) for all st+τ ∈ B (st+1 ; st ), and both Φ(s tion of (11), φ(s ˜ t+1 ) satisfy (ER) for all st+τ ∈ N (st+1 ; st ), from which it follows imand Φ(s ˜ t+τ ; st+1 ) = φ(s ˜ t+τ ; st ) for all st+τ ∈ N (st+1 ; st ) and, hence, mediately that φ(s ˜ t+1 ; st+1 ). ˜ t+1 ; st ) = φ(s Q.E.D. φ(s REFERENCES ABEL, A., G. MANKIW, L. H. SUMMERS, AND R. J. ZECKHAUSER (1989): “Assessing Dynamic Efficiency: Theory and Evidence,” Review of Economic Studies, 56, 1–20. [1157] ALVAREZ, F., AND U. JERMANN (2000): “Efficiency, Equilibrium, and Asset Pricing With Risk of Default,” Econometrica, 68, 775–797. [1137-1139,1142,1143,1147,1156] AMADOR, M. (2003): “A Political Model of Sovereign Debt Repayment,” Working Paper, Stanford University. [1138] BERENTSEN, A., G. CAMERA, AND C. WALLER (2007): “Money, Credit and Banking,” Journal of Economic Theory, 135, 171–195. [1157] BEWLEY, T. (1980): “The Optimum Quantity of Money,” in Models of Monetary Economies, ed. by J. H. Kareken and N. Wallace. Minneapolis: Federal Reserve Bank. [1139] BULOW, J., AND K. ROGOFF (1988): “Sovereign Debt: Is to Forgive to Forget?” Working Paper 2623, National Bureau of Economic Research. [1150] (1989a): “Sovereign Debt: Is to Forgive to Forget?” American Economic Review, 79, 43–50. [1138] (1989b): “A Constant Recontracting Model of Sovereign Debt,” Journal of Political Economy, 79, 43–50. [1138] CABALLERO, R., AND A. KRISHNAMURTHY (2006): “Bubbles and Capital Flow Volatility: Causes and Risk Management,” Journal of Monetary Economics, 53, 35–53. [1156] CAVALCANTI, R., A. EROSA, AND T. TEMZELIDES (1999): “Private Money and Reserve Management in a Matching Model,” Journal of Political Economy, 107, 929–945. [1157] CHARI, V. V., AND P. KEHOE (1993a): “Sustainable Plans and Mutual Default,” Review of Economic Studies, 60, 175–195. [1138] (1993b): “Sustainable Plans and Debt,” Journal of Economic Theory, 61, 230–261. [1138] COLE, H., AND P. KEHOE (1995): “The Role of Institutions in Reputation Models of Sovereign Debt,” Journal of Monetary Economics, 35, 45–64. [1138] (1998): “Models of Sovereign Debt: Partial versus General Reputations,” International Economic Review, 39, 55–70. [1138] EATON, J. (1990): “Sovereign Debt, Reputation and Credit Terms,” Working Paper 3424, National Bureau of Economic Research. [1138] EATON, J., AND M. GERSOVITZ (1981): “Debt With Potential Repudiation—Theoretical and Empirical Analysis,” Review of Economic Studies, 48, 289–309. [1138] EICHENGREEN, B., AND R. PORTES (1989): “After the Deluge: Default, Negotiation, and Readjustment During the Interwar Years,” in The International Debt Crisis in Historical Perspective, ed. by B. Eichengreen and P. H. Lindert. Cambridge, MA: MIT Press. [1157] GOURINCHAS, P., AND H. REY (2005): “From World Banker to World Venture Capitalist: US External Adjustment and the Exorbitant Privilege,” in G7 Current Account Imbalances: Sustainability and Adjustment, ed. by R. Clarida. Chicago: The University of Chicago Press. [1157] GUL, F., AND W. PESENDORFER (2004): “Self-Control and the Theory of Consumption,” Econometrica, 72, 119–158. [1138]

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HELLWIG, C., AND G. LORENZONI (2009): “Supplement to ‘Bubbles and Self-Enforcing Debt’,” Econometrica Supplemental Material, 77, http://www.econometricsociety.org/ecta/ Supmat/6754_extensions.pdf. [1140] JESKE, K. (2006): “Private International Debt With Risk of Repudiation,” Journal of Political Economy, 114, 576–593. [1156] KEHOE, P., AND F. PERRI (2002): “International Business Cycles With Endogenous Incomplete Markets,” Econometrica, 70, 907–928. [1138] KEHOE, T., AND D. LEVINE (1993): “Debt-Constrained Asset Markets,” Review of Economic Studies, 60, 865–888. [1139,1156] KLETZER, K., AND B. WRIGHT (2000): “Sovereign Debt as Intertemporal Barter,” American Economic Review, 90, 621–639. [1138] KLINGEN, C. A., B. WEDER, AND J. ZETTELMEYER (2004): “How Private Creditors Fared in Emerging Debt Markets, 1970–2000,” Discussion Paper 4374, CEPR. [1157] KOCHERLAKOTA, N. (1992): “Bubbles and Constraints on Debt Accumulation,” Journal of Economic Theory, 57, 245–256. [1139] (1996): “Implications of Efficient Risk-Sharing Without Commitment,” Review of Economic Studies, 63, 595–609. [1139] KRUEGER, D., AND F. PERRI (2006): “Does Income Inequality Lead to Consumption Inequality? Evidence and Theory,” Review of Economic Studies, 73, 163–193. [1147] KRUEGER, D., AND H. UHLIG (2006): “Competitive Risk-Sharing Contracts With One-Sided Commitment,” Journal of Monetary Economics, 53, 1661–1691. [1138,1156] LINDERT, P. H., AND P. J. MORTON (1989): “How Sovereign Debt Has Worked,” in Developing Country Debt and Economic Performance, Vol. 1, ed. by J. D. Sachs. Chicago: University of Chicago Press, 39–106. [1157] SANTOS, M., AND M. WOODFORD (1997): “Rational Asset Pricing Bubbles,” Econometrica, 65, 19–57. [1139,1154] SCHEINKMAN, J., AND L. WEISS (1986): “Borrowing Constraints and Aggregate Economic Activity,” Econometrica, 54, 23–46. [1139] TIROLE, J. (1985): “Asset Bubbles and Overlapping Generations,” Econometrica, 53, 1499–1528. [1138] VENTURA, J. (2004): “Bubbles and Capital Flows,” Working Paper, CREI, Universitat Pompeu Fabra. [1156] WRIGHT, M. (2006): “Private Capital Flows, Capital Controls and Default Risk,” Journal of International Economics, 69, 120–149. [1156]

Dept. of Economics, University of California at Los Angeles, Box 951477, Los Angeles, CA 90095-1477, U.S.A. and Centre for Economic Policy Research, London EC1V 0DG, U.K.; [email protected] and Dept. of Economics, Massachusetts Institute of Technology, E52-251C, 50 Memorial Drive, Cambridge, MA 02142, U.S.A. and NBER; [email protected]. Manuscript received October, 2006; final revision received December, 2008.


AN ELEMENTARY THEORY OF COMPARATIVE ADVANTAGE BY ARNAUD COSTINOT1 Comparative advantage, whether driven by technology or factor endowment, is at the core of neoclassical trade theory. Using tools from the mathematics of complementarity, this paper offers a simple yet unifying perspective on the fundamental forces that shape comparative advantage. The main results characterize sufficient conditions on factor productivity and factor supply to predict patterns of international specialization in a multifactor generalization of the Ricardian model which we refer to as an “elementary neoclassical economy.” These conditions, which hold for an arbitrarily large number of countries, goods, and factors, generalize and extend many results from the previous trade literature. They also offer new insights about the joint effects of technology and factor endowments on international specialization. KEYWORDS: Comparative advantage, neoclassical trade theory, log-supermodularity.

1. INTRODUCTION COMPARATIVE ADVANTAGE, whether driven by technology or factor endowment, is at the core of neoclassical trade theory. Using tools from the mathematics of complementarity, this paper offers a simple yet unifying perspective on the fundamental forces that shape comparative advantage in economies with an arbitrarily large number of countries, goods, and factors. Section 2 offers a review of some basic definitions and results in the mathematics of complementarity. Our analysis emphasizes one key property: logsupermodularity. Broadly speaking, the log-supermodularity of a multivariate function captures the idea that increasing one variable is relatively more important when the other variables are high. To fix ideas, consider the following statement. Countries with better financial systems produce relatively more in sectors with higher financial requirements. The formal counterpart to this statement is that aggregate output is log-supermodular in the quality of countries’ financial systems and the level of sectors’ financial requirements. In a trade context, log-supermodularity provides a powerful way to conceptualize the relationship between technology, factor endowment, and international specialization, as we will soon demonstrate. Section 3 describes our theoretical framework. We develop a multifactor generalization of the Ricardian model with an arbitrary number of countries 1 I thank Pol Antras, Vince Crawford, Gene Grossman, Gordon Hanson, Navin Kartik, Giovanni Maggi, Marc Melitz, David Miller, Marc Muendler, Jim Rauch, Esteban Rossi-Hansberg, Jon Vogel, and seminar participants at many institutions for helpful comments and discussions. I have also benefited from useful comments by a co-editor, the editor, and three anonymous referees that greatly improved the paper. This project was initiated at the department of economics at UCSD and continued at the Princeton International Economics Section, which I thank for their support. A previous version of this paper was circulated under the title “Heterogeneity and Trade.”


DOI: 10.3982/ECTA7636

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and sectors which we refer to as an “elementary neoclassical economy.” Factors of production are immobile across countries and perfectly mobile across sectors. Each country, sector, and factor is associated with a distinct characteristic denoted γ, σ, and ω, respectively. For instance, γ may capture the quality of a country’s educational system, σ may capture the skill intensity of a sector, and ω may capture the number of years of education of a worker. The two primitives of our model are (i) factor productivity, q(ω σ γ), which may vary across countries and sectors, and (ii) factor supply, f (ω γ), which may vary across countries. They reflect the two sources of comparative advantage in a neoclassical environment: technology and factor endowment. In this paper, we derive three sets of results on the pattern of international specialization. Section 4 focuses on the case in which only technological differences are a source of comparative advantage. Formally, we assume that q(ω σ γ) ≡ h(ω)a(σ γ). Under this restriction, our general model reduces to a standard Ricardian model. In this environment, we show that if a(σ γ) is log-supermodular, then aggregate output Q(σ γ) is log-supermodular as well. Economically speaking, if high-γ countries are relatively more productive in high-σ sectors, then they should produce relatively more in these sectors. This first result has played an important, albeit implicit, role in many applications and extensions of the Ricardian model. It is at the heart, for example, of the recent literature on institutions and trade; see, for example, Acemoglu, Antras, and Helpman (2007), Costinot (2007), Cuñat and Melitz (2006), Levchenko (2007), Matsuyama (2005), Nunn (2007), and Vogel (2007). At a formal level, these papers all share the same fundamental objective: providing microtheoretical foundations for the log-supermodularity of factor productivity with respect to countries’ “institutional quality” and sectors’ “institutional dependence,” whatever those characteristics may be. Section 5 analyzes the polar case in which factor productivity varies across countries in a Hicks-neutral way, q(ω σ γ) ≡ a(γ)h(ω σ). Hence, only factor endowment differences are a source of comparative advantage. This particular version of our model is a simple generalization of Ruffin (1988). In this environment, we show that if f (ω γ) and h(ω σ) are log-supermodular, then aggregate output Q(σ γ) also is log-supermodular. The basic logic is intuitive. On the one hand, high-γ countries have relatively more high-ω factors. On the other hand, high-ω factors are more likely to be employed in high-σ sectors because they are relatively more productive in these sectors. This explains why high-γ countries should produce relatively more in high-σ sectors. As in the Ricardian case, log-supermodularity provides the mathematical apparatus to make these “relatively more” statements precise. As we later discuss, this second set of results can be used to establish the robustness of many qualitative insights from the literature on heterogeneity and trade. Whether they focus on worker heterogeneity or firm heterogeneity à la Melitz (2003), previous insights typically rely on strong functional forms which guarantee explicit closed form solutions. For example, Ohnsorge and

THEORY OF COMPARATIVE ADVANTAGE

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Trefler (2004) assumed that distributions of worker skills are log-normal, while Helpman, Melitz, and Yeaple (2004) and Antras and Helpman (2004) assumed that distributions of firm productivity are Pareto. Our results formally show that assuming the log-supermodularity of f (ω γ) is critical for many of their results, whereas assuming log-normal and Pareto distributions is not. Section 6 considers elementary neoclassical economies in which both factor endowment and technological differences are sources of comparative advantage. In these economies, we show that unless strong functional form restrictions are imposed, robust predictions about international specialization can only be derived in the two most extreme sectors. In general, the log-supermodularity of f (ω γ) and q(ω σ γ) is not sufficient to derive the log-supermodularity of aggregate output. In the presence of complementarities between factor and sector characteristics, which are necessary for factor endowments to affect comparative advantage, the indirect impact of Ricardian technological differences on the assignment of factors to sectors may dominate its direct impact on factor productivity. This is an important observation that highlights the potential caveats of combining insights from distinct models without a generalizing framework. Although we are, to the best of our knowledge, the first ones to emphasize the role of log-supermodularity in a trade context, this property has been used previously in many areas of economics, including auction theory (Milgrom and Weber (1982)), monotone comparative statics under uncertainty (Jewitt (1987) and Athey (2002)), and matching (Shimer and Smith (2000)). From a mathematical standpoint, Jewitt’s (1987) and Athey’s (2002) works are most closely related to our paper. In particular, the fact that log-supermodularity is preserved by multiplication and integration is, like in Jewitt (1987) and Athey (2002), at the core of our analysis.2 In this respect, our contribution is to show that this mathematical property also has natural and useful applications for international trade. The theory of comparative advantage presented in this paper is attractive for two reasons. The first one is that it allows us to consider both sources of comparative advantage, technology and factor endowment—within a unifying yet highly tractable framework. This is important not only for generalizing results from the previous literature, but also because factor endowment in practice coexists with technology and institutional differences. Indeed, they often have the same causes (see, e.g., Acemoglu (1998)). The second reason is dimensionality. For pedagogical purposes, neoclassical trade theory is usually taught using simple models with a small number of countries, goods, and factors. The two most 2

This close mathematical connection notwithstanding, our results are not about monotone comparative statics. In this paper, we are interested in the cross-sectional variation of aggregate output within a given equilibrium, not changes in aggregate output across equilibria. In particular, the fact that all countries face the same prices within a given free trade equilibrium is crucial for our results.

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celebrated examples are the Ricardian model—with one factor, two goods, and two countries—and the Heckscher–Ohlin model—with two factors, two goods, and two countries.3 In these simple models, differences in either technology or factor endowments have strong implications for the pattern of international specialization. Unfortunately, strong results do not generally survive in environments with higher dimensionality (see, e.g., Ethier (1984) and Deardorff (2007)). By contrast, our predictions hold for an arbitrarily large number of countries, goods, and factors. In this respect, our paper is closely related to Deardorff (1980). Compared to Deardorff’s (1980) law of comparative advantage, our main results are less general in that we restrict ourselves to a multifactor generalization of the Ricardian model under free trade, but they are stronger in that they apply to any pair of goods and derive from restrictions on the model’s primitives—factor productivity and factor supply—rather than autarky prices. Finally, we believe that our general approach could also be useful outside international trade. The basic structure of our model is central to many models with agent heterogeneity. At the core of these models, there are populations of agents sorting across occupations. As we argue in our concluding remarks, whatever these categories may refer to in practice, they often are the formal counterparts to countries, factors, and sectors in our theory. 2. LOG-SUPERMODULARITY Our analysis emphasizes one particular form of complementarity: logsupermodularity.4 Since this concept is not widely used in the trade literature, we begin with a review of some basic definitions and results. Topkis (1998) and Athey (2002) offer an excellent overview and additional references.

n

2.1. Definition

Let X = i=1 Xi , where each Xi is totally ordered. For any x x ∈ X, we say that x ≥ x if xi ≥ xi for all i = 1 n. We let max(x x ) be the vector of X whose ith component is max(xi xi ) and let min(x x ) be the vector whose ith component is min(xi xi ). Finally, we denote by x−i the vector x with the ith component removed. With the previous notations, log-supermodularity can be defined as follows. DEFINITION 1: A function g : X → R+ is log-supermodular if for all x x ∈ X, g(max(x x )) · g(min(x x )) ≥ g(x) · g(x ).

3 Davis (1995) offered a simple combination of both models with three goods, two factors, and two countries. 4 In the statistics literature, Karlin (1968) referred to log-supermodularity as total positivity of order 2.


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If g is strictly positive, then g is log-supermodular if and only if ln g is supermodular. This means that if g also is twice differentiable, then g is logsupermodular in (xi xj ) if and only if ∂2 ln g/∂xi ∂xj ≥ 0. To get more intuition about the form of complementarities that log-supermodularity captures, consider g : X1 × X2 → R+ . For every x1 ≥ x1 , x2 ≥ x2 , the log-supermodularity of g in (x1 x2 ) implies that g(x1 x2 ) · g(x1 x2 ) ≥ g(x1 x2 ) · g(x1 x2 ) If g is strictly positive, this can be rearranged as g(x1 x2 )/g(x1 x2 ) ≥ g(x1 x2 )/g(x1 x2 ) Thus, the relative returns to increasing the first variable, x1 , are increasing in the second variable, x2 . This is equivalent to the monotone likelihood ratio property; see Milgrom (1981). In a trade context, this property may capture the fact that high-x1 countries are relatively more productive in high-x2 sectors, as in the Ricardian model, or that high-x1 countries are relatively more abundant in high-x2 factors, as in the Heckscher–Ohlin model. 2.2. Results Most of our analysis builds on the following two results: LEMMA 1: If g h : X → R+ are log-supermodular, then gh is log-supermodular. + LEMMA 2: Let μi be a σ-finite measure on Xi . If g : X → R is logsupermodular and integrable, then G(x−i ) = Xi g(x) dμi (xi ) is log-supermodular.

In other words, log-supermodularity is preserved by multiplication and integration. Lemma 1 directly derives from Definition 1. Proofs of Lemma 2 can be found in Lehmann (1955) for the bivariate case, and Ahlswede and Daykin (1978) and Karlin and Rinott (1980) for the multivariate case. In the rest of this paper, we assume that whenever integrals appear, requirements of integrability and measurability are met. 3. THEORETICAL FRAMEWORK We consider a world economy comprising c = 1 C countries with characteristics γ c ∈ Γ , s = 1 S goods or sectors with characteristics σ s ∈ Σ, and multiple factors of production indexed by their characteristics ω ∈ Ω, where Γ , Σ, and Ω are totally ordered sets.5 We let μ be a σ-finite measure on Ω. 5 We could allow for the existence of countries and sectors whose characteristics cannot be ordered. In this case, our results would simply apply to the subset of countries and sectors with

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The number of factors in Ω may be continuous or discrete. Factors of production are immobile across countries and perfectly mobile across sectors. f (ω γ c ) ≥ 0 denotes the inelastic supply of factor ω in country c. Factors of production are perfect substitutes within each country and sector, but vary in their productivity q(ω σ s γ c ) ≥ 0. In country c and sector s, aggregate output is given by s c Q(σ γ ) = q(ω σ s γ c )l(ω σ s γ c ) dμ(ω) (1) Ω

where l(ω σ s γ c ) is the quantity of factor ω allocated to sector s in country c. At this point, we wish to be clear that our theoretical framework is more general than a Ricardian model in that it allows multiple factors of production, but is less general than a standard neoclassical model in that it rules out imperfect substitutability between these factors within each sector. We come back briefly to the relationship between our model and the Heckscher–Ohlin model in Section 5. Throughout this paper, we focus on the supply side of this economy under free trade. Our goal is to determine how the cross-sectional variation of our two primitives, q(ω σ s γ c ) and f (ω γ c ), affects the cross-sectional variation of aggregate output, Q(σ s γ c ), taking world prices p(σ s ) > 0 as given. To this end, we follow the dual approach of Dixit and Norman (1980). DEFINITION 2: l(· · ·) is an efficient allocation if, for all c = 1 C, it solves the revenue maximization problem (2)

maxc

l(··γ )

S

p(σ s )Q(σ s γ c )

s=1

subject to

S

l(ω σ s γ c ) ≤ f (ω γ c )

for μ-almost all ω ∈ Ω

s=1

According to Definition 2, l(· · ·) is an efficient allocation if it is feasible and it maximizes the value of national output at given prices in all countries. Since there are constant returns to scale, a competitive equilibrium with a large number of profit-maximizing firms would lead to an efficient allocation. Because of the linearity of aggregate output, efficient allocations are easy to characterize. Unlike more general neoclassical models, the marginal return r(ω σ s γ c ) of factor ω in sector s and country c is independent of the allocaordered characteristics. By contrast, our analysis crucially relies on the fact that Ω is totally ordered.


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tion of factors in that sector: r(ω σ s γ c ) = p(σ s )q(ω σ s γ c ).6 As a result, we can solve problem (2) factor-by-factor the same way we would solve the revenue maximization problem in a simple Ricardian model. In any country c, almost all factors ω should be employed in the sector(s) where p(σ s )q(ω σ s γ c ) is maximum. In the rest of this paper, we restrict ourselves to environments where problem (2) admits a unique solution. ASSUMPTION 0: The solution to the revenue maximization problem (2) is unique for all c = 1 C and μ-almost all ω ∈ Ω. By our previous discussion, Assumption 0 requires p(σ s )q(ω σ s γ c ) to be maximized in a single sector for almost all factors and all countries. Since Assumption 0 plays a crucial role in our analysis, it is important to understand why and in which circumstances it is more likely to be satisfied. At a formal level, Assumption 0 is an implicit restriction on the demand side of the world economy, which requires world consumption to be at a vertex of the world production possibility frontier. This is illustrated in Figure 1 in the case of an economy with one factor, two goods, and two countries or, equivalently, two factors, two goods, and one country. Ceteris paribus, the more vertices there are on the world production possibility frontier, the milder that restriction on preferences becomes. From an economic standpoint, this means

FIGURE 1.—Uniqueness of the efficient allocation. 6

This would no longer be true, for example, if aggregate production functions were Cobb–Douglas, Q(σ s γ c ) = exp{ Ω α(ω) ln l(ω σ s γ c ) dμ(ω)}. Under this assumption, marginal returns would be given by r(ω σ s γ c ) = p(σ s )α(ω)Q(σ s γ c )/ l(ω σ s γ c ), which would depend on the price, p(σ s ), and exogenous technological characteristics, α(ω), but also Q(σ s γ c )/ l(ω σ s γ c ).

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that Assumption 0 is more likely to be satisfied in economies with the following attributes: (i) A large number of countries, as in the Ricardian models developed by Becker (1952), Matsuyama (1996), and Yanagawa (1996). (ii) A large number of factors, as in the trade models with worker heterogeneity developed by Grossman and Maggi (2000), Grossman (2004), and Ohnsorge and Trefler (2004). In particular, if there is a continuum of distinct factors in the economy, then Assumption 0 is generically true. Although prices are endogenous objects which may adjust to equalize marginal returns across sectors, a finite number of prices cannot, in general, equalize the returns of an infinite numbers of factors.7 Throughout this paper, we maintain Assumption 0, which allows us to express aggregate output under an efficient allocation as follows. PROPOSITION 1: Suppose that Assumption 0 holds. Then, for all c = 1 C and s = 1 S, aggregate output under an efficient allocation is given by s c Q(σ γ ) = (3) q(ω σ s γ c )f (ω γ c ) dμ(ω) Ω(σ s γ c )

where Ω(σ s γ c ) is the set of factors allocated to sector s in country c:

s c r ω σ γ Ω(σ s γ c ) = ω ∈ Ωr(ω σ s γ c ) > max (4) s =s

From now on, we refer to a world economy where Equations (3) and (4) hold as an elementary neoclassical economy. The rest of our paper offers sufficient conditions to make predictions on the pattern of international specialization in this environment. 4. SOURCE OF COMPARATIVE ADVANTAGE (I): TECHNOLOGY 4.1. Definition We first consider economies in which factor productivity satisfies (5)

q(ω σ γ) ≡ h(ω)a(σ γ)

with h(ω) > 0 and a(σ γ) ≥ 0. Equation (5) allows only Ricardian technological differences across countries. Since a is a function of σ and γ, some countries may be relatively more productive in some sectors than others. By contrast, factors may not be relatively more productive in some sectors than 7 Finally, note that Assumption 0 also is trivially satisfied in Ricardian models with Armington preferences; see, for example, Acemoglu and Ventura (2002). In those models, since q(ω σ s γ c ) is strictly positive in a single sector, p(σ s )q(ω σ s γ c ) is maximized in a single sector as well.


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others: if factor ω is twice as productive as factor ω in a given sector, then it is twice as productive in all of them. DEFINITION 3: An elementary neoclassical economy is a R-economy if Equation (5) holds. In a R-economy, there are no “real” differences across factors of produc tion. If there exists ω ∈ Ω such that r(ω σ s γ c ) > maxs =s r(ω σ s γ c ), then r(ω σ s γ c ) > maxs =s r(ω σ s γ c ) for all ω ∈ Ω. Hence, a R-economy is isomorphic to a standard Ricardian model. In this environment, Assumption 0 directly implies Ω(σ s γ c ) = Ω or ∅ Since there are no real differences across factors of production in a Reconomy, their marginal returns are maximized in the same sector. As a result, countries only produce one good.8 Given this restriction, our analysis of the Ricardian model is similar in terms of scope to the analysis of Jones (1961).9 4.2. Assumption To make predictions on the pattern of international specialization in a R-economy, we make the following assumption: ASSUMPTION 1: a(σ γ) is log-supermodular. Assumption 1 states that high-γ countries are relatively more productive in high-σ sectors. For any pair of countries, c1 and c2 , and goods, s1 and s2 , such that γ c1 ≥ γ c2 , σ s1 ≥ σ s2 , a(σ s1 γ c2 ) = 0, and a(σ s2 γ c2 ) = 0, Assumption 1 implies a(σ s1 γ c1 ) a(σ s2 γ c1 ) ≥ a(σ s1 γ c2 ) a(σ s2 γ c2 ) This is the standard inequality at the heart of the Ricardian model. The logsupermodularity of a simply requires that it holds for any ordered pairs of country and sector characteristics. 8 Of course, this stark implication of Assumption 0 will no longer be true in elementary neoclassical economies with more than one factor of production. 9 Section 4.4 briefly discusses how our results generalize to Ricardian environments where countries produce more than one good.

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4.3. Predictions The main result of this section can be stated as follows. THEOREM 1: In a R-economy, Assumption 1 implies Q(σ γ) log-supermodular. The formal proof as well as all subsequent proofs can be found in the Appendix. The argument is simple. If Q(σ γ) were not log-supermodular, then one could find a pair of countries and sectors such that the marginal returns of factors of production in the low-σ sector would be (i) strictly higher in the high-γ country and (ii) strictly lower in the low-γ country. Under free trade, this is precisely what the log-supermodularity of a precludes. Theorem 1 imposes strong restrictions on the pattern of international specialization. If a country with characteristic γ1 specializes in a sector with characteristic σ1 , then a country with characteristic γ2 < γ1 cannot specialize in a sector with characteristic σ2 > σ1 . In other words, there must be a ladder of countries such that higher-γ countries produce higher-σ goods. COROLLARY 1: Suppose that Assumption 1 holds in a R-economy. Then high-γ countries specialize in high-σ sectors. So far, we have shown that Assumption 1 is sufficient to make predictions on the pattern of international specialization in a R-economy. Conversely, we can show that Assumption 1 cannot be dispensed with if the logsupermodularity of Q is to hold in all R-economies. To see this, consider a two-sector R-economy. In this environment, if a were not log-supermodular, then one could find a high-γ country in which the marginal returns of factors of production would be strictly higher in the low-σ sector and a low-γ country in which the marginal returns of factors of production would be strictly higher in the high-σ sector. Therefore, the high-γ country would specialize in the low-σ sector and the low-γ country in the high-σ sector, which would contradict the log-supermodularity of Q. 4.4. Relation to the Previous Literature Making predictions on the pattern of international specialization in a Ricardian model with a large number of goods and countries is knowingly difficult. Deardorff (2007) noted that “Jones (1961) seems to have done about as well as one can, showing that an efficient assignment of countries to goods will minimize the product of their unit labor requirements.” We have just shown that by imposing the log-supermodularity of factor productivity across countries and sectors, one can generate much stronger predictions. The reason is simple.


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Using our notations and taking logs, Jones (1961) stated that an efficient assignment of countries to goods must solve max

ln a(σ γ)

Corollary 1 merely points out that the solution to this assignment problem exhibits positive assortative matching if ln a(σ γ) is supermodular (see, e.g., Becker (1973), Kremer (1993), and Legros and Newman (2002)). Though we have restricted ourselves in this section to the case where each country only produces one good, the formal connection between the Ricardian model and assignment models holds more generally. In Dornbusch, Fischer, and Samuelson (1977), for example, both countries produce a continuum of goods, but the pattern of international specialization still reflects the optimal assignment of goods to countries. Formally, let Σ(γ) ≡ {σ ∈ Σ|l(ω σ γ) > 0 for some ω ∈ Ω} be the set of goods produced in a country with characteristic γ. Without Assumption 0, this set may not be a singleton. However, using the same logic as in Theorem 1, it is easy to show that if a(σ γ) is strictly log-supermodular, then Σ(γ) must be increasing in the strong set order. Put simply, high-γ countries must specialize in high-σ sectors, as previously stated in Corollary 1. In light of this discussion, it should not be surprising that log-supermodularity has played an important, albeit implicit, role in many applications and extensions of the Ricardian model. In his “technology gap” model of international trade, Krugman (1986) assumed, using our notation, that labor productivity in country c and sector s is given by a(σ s γ c ) ≡ exp(σ s γ c ), where σ s is an index of good s’s technological intensity and γ c is a measure of country c’s closeness to the world technological frontier. Since ∂2 ln a/∂σ ∂γ > 0, this functional form satisfies Assumption 1, which is the critical sufficient condition for Krugman’s (1986) results to hold. Log-supermodularity also is at the heart of the recent literature on institutions and trade; see, for example, Acemoglu, Antras, and Helpman (2007), Costinot (2007), Cuñat and Melitz (2006), Levchenko (2007), Matsuyama (2005), Nunn (2007), and Vogel (2007). These papers have shown that crosscountry differences in institutions may give rise to a pattern of comparative advantage, even in the absence of true technological differences. Though the aforementioned papers differ in terms of the institutional characteristics they focus on—from credit market imperfections to rigidities in the labor market— they share the same fundamental objective: providing microtheoretical foundations for the log-supermodularity of factor productivity with respect to countries’ “institutional quality” and sectors’ “institutional dependence,” whatever those characteristics may be.

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5. SOURCE OF COMPARATIVE ADVANTAGE (II): FACTOR ENDOWMENT 5.1. Definition We now turn to the case where q satisfies (6)

q(ω σ γ) ≡ a(γ)h(ω σ)

with a(γ) > 0 and h(ω σ) ≥ 0. Equation (6) allows factor productivity to vary across countries, but only in a Hicks-neutral way.10 Therefore, there are no Ricardian technological differences in this environment. DEFINITION 4: An elementary neoclassical economy is a F -economy if Equation (6) holds. The key feature of a F -economy is that the set of factors allocated to a given sector is the same in all countries. Because of free trade and Hicks-neutrality, we have (7)

s) Ω(σ s γ c ) ≡ Ω(σ

s s h ω σ p σ = ω ∈ Ωp(σ s )h(ω σ s ) > max s =s

In a F -economy, the assignment function Ω(σ s γ c ) does not vary across countries. Hence, patterns of international specialization may only arise from crosscountry differences in factor endowments, f (ω γ c ). 5.2. Assumptions To make predictions on the pattern of international specialization in a F -economy, we make two assumptions: ASSUMPTION 2: f (ω γ) is log-supermodular. Assumption 2 states that high-γ countries are relatively more abundant in high-ω factors. For any pair of countries, c1 and c2 , and factors, ω1 and ω2 , such that γ c1 ≥ γ c2 , ω1 ≥ ω2 , and f (ω2 γ c1 ), f (ω2 γ c2 ) = 0, Assumption 2 implies f (ω1 γ c1 ) f (ω1 γ c2 ) ≥ f (ω2 γ c1 ) f (ω2 γ c2 ) As previously mentioned, this is equivalent to the assumption that the densities of countries’ factor endowments, fc (ω) ≡ f (ω γc ), can be ranked in terms of 10 One could easily extend the analysis of this section to the case where q(ω σ γ) ≡ a(ω γ)h(ω σ). Here, changes in a(ω γ) are isomorphic to changes in f (ω γ).


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monotone likelihood ratio dominance. Milgrom (1981) offered many examples of density functions that satisfy this assumption, including the normal (with mean γ) and the uniform (on [0 γ]). ASSUMPTION 3: h(ω σ) is log-supermodular. Assumption 3 states that high-ω factors are relatively more productive in high-σ sectors, irrespective of the country where they are located. In our model, Assumptions 2 and 3 play the same role as the ordinal assumptions on factor abundance and factor intensity, respectively, in the 2 × 2 × 2 Heckscher– Ohlin model. 5.3. Predictions The main result of this section can be stated as follows. THEOREM 2: In a F -economy, Assumptions 2 and 3 imply Q(σ γ) logsupermodular. The proof relies on Lemmas 1 and 2, but the broad logic is intuitive. If h(ω σ) satisfies Assumption 3, then high-ω factors are assigned to high-σ sectors under an efficient allocation. If, in addition, f (ω γ) satisfies Assumption 2, then a high value of γ raises the likelihood of high values of ω relative to low values of ω. This increases the likelihood that a given factor is allocated to high-σ sectors and, in turn, increases the relative output of these sectors. This, in a nutshell, explains why Q(σ γ) is log-supermodular. Now consider a pair of countries, c1 and c2 , producing a pair of goods, s1 and s2 , with γ c1 ≥ γ c2 and σ s1 ≥ σ s2 . Theorem 2 implies Qs1 c1 /Qs1 c2 ≥ Qs2 c1 /Qs2 c2 , where Qsc ≡ Q(σ s γ c ). Still considering the pair of countries, c1 and c2 , and applying Theorem 1 to an arbitrary subset of J goods, we obtain the following corollary. COROLLARY 2: In a F -economy where Assumptions 2 and 3 hold, if two countries produce J goods, with γ c1 ≥ γ c2 and σ s1 ≥ · · · ≥ σ sJ , then the high-γ country tends to specialize in the high-σ sectors: Q sJ c 1 Q s1 c 1 ≥ ··· ≥ s c s c Q12 QJ 2 s ) is the the same in all counIn a F -economy, the assignment function Ω(σ tries. As a result, cross-country differences in factor endowments are mechanically reflected in their patterns of specialization. With Hicks-neutral technological differences around the world, a country produces relatively more— compared to other countries—in sectors in which a relatively higher share of its

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factors selects. Corollary 2 operationalizes that idea by showing that Assumptions 2 and 3 are sufficient to characterize the “sectors in which a relatively higher share of [a country’s] factors selects” and, in turn, the pattern of international specialization. Compared to standard Heckscher–Ohlin predictions, Corollary 2 is a strong result. Even with an arbitrarily large number of goods and factors, it offers predictions on the cross-sectional variation of aggregate output rather than the factor content of trade. To get a better sense of where this strong result comes from, consider a Heckscher–Ohlin model with two factors, K and L, and multiple sectors with different factor intensities, (K/L)s . Because of constant returns to scale, such a model can always be rearranged as an elementary neoclassical economy in which ω ≡ (K/L)s . The key difference between this model and ours, however, is that its restrictions are about aggregate endowments of capital and labor, not the full distribution of (K/L)s . If there only are two sectors, the two sets of restrictions are equivalent,11 but with more than two sectors, restrictions on the full distribution are stronger and, hence, lead to stronger predictions. Corollary 2 also shows that imperfect competition and product differentiation, which are common assumptions in the empirical trade literature, are not necessary to derive “smooth” predictions about aggregate output in environments with multiple countries, goods, and factors.12 In Romalis (2004), for example, similar predictions are derived in an economy with two factors, two countries, and a continuum of goods because of monopolistic competition and non-factor-price equalization. In a F -economy, markets are perfectly competitive and factor price equalization holds. Yet, the log-supermodularity of h in (ω σ) creates a strong enough connection between factor and sector characteristics—namely, positive assortative matching—to guarantee that countries should produce relatively more in the sectors that use their abundant factors intensively. Finally, it is worth pointing out that Assumption 3 only matters indirectly s ). In the proof of Theorem 2, once we have esthrough its impact on Ω(σ tablished that high-ω factors are assigned to high-σ sectors, restrictions on 11 In the two-sector case, the supply of the two factors, ω1 ≡ (K/L)1 and ω2 ≡ (K/L)2 , must be given by

V (ω1 ) =

K − L(K/L)2 (K/L)1 − (K/L)2

and V (ω2 ) =

K − L(K/L)1 (K/L)2 − (K/L)1

respectively. Hence there is a one-to-one mapping between K and L, on the one hand, and V (ω1 ) and V (ω2 ), on the other hand. 12 Eaton and Kortum (2002) made a related point in a Ricardian environment, showing that a gravity equation can be derived under perfect competition in a multicountry–multisector economy. The mechanism emphasized by our model, however, is very different. Unlike Eaton and Kortum (2002), it relies on the efficient assignment of heterogeneous factors across sectors rather than random productivity shocks.


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h(ω σ) are irrelevant. Therefore, Assumptions 2 and 3 also imply that agc gregate employment, L(σ s γ c ) ≡ Ω(σ s ) f (ω γ ) dω, and aggregate revenues, s c s c R(σ γ ) ≡ Ω(σ s ) r(ω σ )f (ω γ ) dω, are log-supermodular in a F -economy. COROLLARY 3: In a F -economy where Assumptions 2 and 3 hold, if two countries produce J goods, with γ c1 ≥ γ c2 and σ s1 ≥ · · · ≥ σ sJ , then aggregate employment and aggregate revenues follow the same pattern of specialization as aggregate output: Ls1 c1 LsJ c1 ≥ · · · ≥ Ls1 c2 LsJ c2

and

Rs1 c1 RsJ c1 ≥ · · · ≥ Rs1 c2 RsJ c2

Corollary 3 is attractive from an empirical standpoint. To test such predictions on the pattern of international specialization, one is free to use aggregate data on either output, employment, or revenues. Moreover, our predictions all are ordinal in nature. This means that one does not need to observe the true country and sector characteristics to confront them with the data; any monotonic transformation of γ and σ will do. 5.4. Minimal Sufficient Conditions We have just shown that Assumptions 2 and 3 are sufficient conditions to predict the pattern of international specialization in a F -economy. This raises one obvious question: Are there weaker properties on f (ω γ c ) and h(ω σ s ) that may also lead to the log-supermodularity of Q(σ s γ c )? The short answer is, like in Section 4, that Assumptions 2 and 3 cannot be dispensed with if one wants to make predictions in all F -economies. To address that question formally, we follow the strategy of Athey (2002) and make the following statement: DEFINITION 5: H1 and H2 are a minimal pair of sufficient conditions for a given conclusion C if (i) C holds whenever H2 does, if and only if H1 holds and (ii) C holds whenever H1 does, if and only if H2 holds. Definition 5 states that if H1 and H2 are a minimal pair of sufficient conditions, then one cannot weaken either H1 or H2 without imposing further assumptions on the model. Note that this does not mean that a given conclusion C holds if and only if H1 and H2 are satisfied. It simply means that without one or the other, the conclusion C may not hold in all environments. In the next theorem, we show that the log-supermodularity of f (ω γ c ) and h(ω σ s ) is a minimal pair of sufficient conditions to predict the log-supermodularity of Q(σ s γ c ) in all F -economies. THEOREM 3: In a F -economy, Assumptions 2 and 3 are a minimal pair of sufficient conditions for Q(σ s γ c ) to be log-supermodular.

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From Theorem 2, we already know that Assumptions 2 and 3 are a pair of sufficient conditions. To establish that this pair is minimal, we need to prove the existence of F -economies in which Assumption 3 (resp. Assumption 2) and the log-supermodularity of Q(σ s γ c ) imply Assumption 2 (resp. Assumption 3). First, we construct a F -economy with sector-specific factors, that is, an economy where high-ω factors can only produce in one high-σ sector. In this environment, if Q(σ s γ c ) is log-supermodular, then f (ω γ c ) automatically is log-supermodular. Second, we construct a F -economy with country-specific factors, that is, an economy where high-ω factors can only be found in one high-γ country. Using the same logic, we show that the log-supermodularity of Q(σ s γ c ) implies the log-supermodularity of h(ω σ s ). 5.5. Relation to the Previous Literature The model presented in this section is a generalization of the r × n Ricardian model of Ruffin (1988) that allows for a continuum of factors and Hicksneutral technological differences. Whereas Ruffin (1988) offered analytical results in the two-good–two-factor case and the two-good–three-factor case, our results hold for an arbitrarily large number of goods and factors. Like in the Ricardian case, imposing the log-supermodularity of the primitives of our model is crucial to predict the pattern of international specialization in economies with higher dimensionality. Ohnsorge and Trefler (2004) have developed an elegant variation of Ruffin’s (1988) model with a continuum of workers.13 To derive closed form solutions, they assumed that workers’ endowments of human capital are normally distributed and that worker productivity takes an exponential form. In the working paper version of this paper (Costinot (2007)) we demonstrated that their model can be described as a F -economy. Once the signs of the crossderivatives of ln f and ln h have been computed using their functional forms, their results on the pattern of international specialization (Ohnsorge and Trefler (2004, Theorem 1, p. 15, Theorem 3, p. 19, and Theorem 5, p. 27)) are implications of Corollary 2. By identifying log-supermodularity as the critical sufficient condition for their results to hold, our analysis demonstrates not only the possibility of their results, but also their robustness. Interestingly, the forces that shape the pattern of international specialization in a F -economy also play a central role in determining the prevalence of organizational forms in trade models with firm-level heterogeneity à la Melitz (2003), most notably Helpman, Melitz, and Yeaple (2004) and Antras and Helpman (2004). The formal relationship between these papers and ours is discussed in detail in Costinot (2007). To map these models into our general framework, one needs to reinterpret each factor as a firm with productivity ω, 13

Ohnsorge and Trefler (2007) considered the case of a continuum of goods and workers.


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each country as an industry with characteristic γ, and each sector as an organization with characteristic σ. Then total sales Q(σ γ) by firms with a σ organih(ω σ)f (ω γ) dω, zation in a γ industry can be expressed as Q(σ γ) = Ω(σ) where Ω(σ) = {ω ∈ Ω|r(ω σ) > maxσ =σ r(ω σ )}. Under this interpretation, h(ω σ) and r(ω σ) are the sales and profits of a firm with productivity ω and organization σ, and f (ω γ) is density of productivity levels in sector γ.14 Like in Theorem 2, the assignment function Ω(σ) is the same across industries. As a result, differences in the distribution of firm productivity are mechanically reflected in the prevalence of various organizational forms. This explains why industries with relatively more productive firms have relatively more sales associated with the organization that more productive firms select, whether it is FDI or vertical integration.15 6. MULTIPLE SOURCES OF COMPARATIVE ADVANTAGE 6.1. A Simple Generalization With Factor-Augmenting Technological Differences Before offering a general analysis of economies with factor endowment and Ricardian technological differences, we present a simple generalization of a F -economy that allows for factor-augmenting technological differences. Formally, we assume that q satisfies (8)

q(ω σ γ) = a(γ)h[ω + t(γ) σ]

where ω + t(γ) ∈ Ω for all ω ∈ Ω and γ ∈ Γ . For example, one can think of ω as the log of workers’ number of years of education and of t(γ) as a measure of the quality of the educational system in a given country. If t(γ) ≡ 0, then we are back to the Hicks-neutral case, but if t(γ) = 0, Equation (8) introduces the possibility of technological complementarities between country and sector characteristics like in a R-economy. DEFINITION 6: An elementary neoclassical economy is a Fa -economy if Equation (8) holds. 14 The previous models and ours only differ in terms of market structure. In the previous models, monopolistic competition implies r(ω σ) = p(σ)q(ω σ γ). In Costinot (2007), we showed that our analysis carries over to that environment if r(ω σ) satisfies a single crossing property. In a F -economy, the only role of Assumption 3 is to guarantee that p(σ)q(ω σ γ) satisfies a single crossing property for all p(σ). 15 Antras and Helpman (2004, footnote 10, p. 571) also recognized the existence of a connection between the mechanism at work in their model and Melitz (2003) and Helpman, Melitz, and Yeaple (2004). However, they did not discuss the critical assumptions on which this logic depends. Our analysis shows, for example, that assuming the log-supermodularity of f (ω γ) is critical, whereas assuming Pareto distributions is not.

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Fa -economies are interesting because they offer a simple way to combine factor endowment and Ricardian technological differences. Formally, Fa economies are equivalent to F -economies up to a change of variable, ω ≡ ω + t(γ). Once endowments have been expressed in the same efficiency units across countries, aggregate output is equal to s c (9) a(γ c )h(ω σ s )f(ω γ c ) dμ(ω) Q(σ γ ) = s) Ω(σ

s ) is given by Equation (7). Therewhere f(ω γ c ) ≡ f [ω − t(γ c ) γ c ] and Ω(σ fore, if we can show that f(ω γ c ) is log-supermodular, then we can still use Theorem 2 to predict the pattern of international specialization. The next theorem offers sufficient conditions on f and t that allow us to do so. THEOREM 4: Consider a Fa -economy where Assumptions 2 and 3 hold. If t is increasing in γ and f is strictly positive and log-concave in ω, then Q(σ γ) is log-supermodular. Like log-supermodularity, log-concavity is satisfied by many standard distributions including the uniform, normal, and extreme value distributions (see Bagnoli and Bergstrom (2005)). The monotonicity of t and the log-concavity of f guarantee that technological differences reinforce the pattern of international specialization driven by factor endowment differences, Assumptions 2 and 3. Even in the absence of true cross-country differences in factor supplies, they imply that high-γ countries are relatively more abundant, now in efficiency units, in the high-ω factors.16 The pattern of international specialization in a Fa -economy follows. Finally, note that the previous results are stronger than standard Heckscher– Ohlin predictions with factor-augmenting productivity differences (see, e.g., Trefler (1993)). Like in a F -economy, our approach leads to predictions on the cross-sectional variation of aggregate output rather than the factor content of trade. 6.2. General Technological Differences We have just shown that the predictions of Sections 4 and 5 may extend to an environment with both Ricardian and factor-endowment sources of comparative advantage, albeit under strong functional form restrictions. We now turn to the case in which q satisfies the following assumption. 16 It is worth emphasizing that Theorem 4 crucially relies on the linear relationship between ω and t(γ). We could, in principle, generalize the nature of factor-augmenting technological differences by assuming that q(ω σ s γ c ) = a(γ c )h[t(ω γ c ) σ s ]. This generalized version of a Fa -economy would still be equivalent to a F -economy up to a change of variable, but predicting the log-supermodularity of Q would then require strong regularity conditions on f —as strong as the restrictions on t are weak .


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ASSUMPTION 4: q(ω σ γ) is log-supermodular. Assumption 4 is a strict generalization of the assumptions used in R- and F -economies.17 Compared to a R-economy, it allows complementarities between ω and σ; compared to a F -economy, it allows complementarities between σ and γ. Hence, both factor endowment and technological differences can, in principle, determine the pattern of international specialization. The question we want to ask is, “Does Assumption 4 put enough structure on the nature of technological differences to predict the pattern of international specialization across all countries and sectors of an elementary neoclassical economy?” Perhaps surprisingly, the answer is no. Under Assumption 4, robust predictions about international specialization can only be derived in the two most extreme sectors, as we now demonstrate. 6.2.1. A Counterexample We start by offering a counterexample in which factor supply satisfies Assumption 2, factor productivity satisfies Assumption 4, and yet aggregate output is not log-supermodular. Consider an elementary neoclassical economy comprising two countries with characteristics γ 2 > γ 1 > 0, S > 2 sectors with characteristics σ S ≡ (σ1S σ2S ) > · · · > σ 1 ≡ (σ11 σ21 ) > 0, and a continuum of factors ω ∈ [0 1]. Factor productivity and factor supply are given by (10)

q(ω σ γ) ≡ ωσ1 γ σ2

(11)

f (ω γ) ≡ 1

By Equations (10) and (11), Assumptions 4 and 2 are trivially satisfied. For any c = 1 2 and s = 1 S − 1, we denote by ω∗cs the factor whose marginal return is equalized between sector s and sector s + 1 in country c. By Equation (10), we have

(12)

∗ cs

ω

c −λs

= (γ )

p(σ s ) · p(σ s+1 )

1/(σ1s+1 −σ1s )

where λs ≡ (σ2s+1 − σ2s )/(σ1s+1 − σ1s ). For any γ 2 > γ 1 > 0 and σ S > · · · > σ 1 > 0, there exist p(σ S ) > · · · > p(σ 1 ) > 0 such that 1 > ω∗cS > · · · > ω∗c1 > 0 for c = 1 2. Assuming that the previous series of inequalities holds, we can write 17 Strictly speaking, Assumption 4 is not a generalization of the assumptions used in a Fa economy. According to Equation (8), if h is log-supermodular and t is increasing, then q is logsupermodular in (ω σ) and (σ γ), but not necessarily log-supermodular in (ω γ). However, as mentioned in Section 6.1, there exists a change of variable ω ≡ ω + t(γ) such that a Fa -economy reduces to a F -economy.

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Ω(σ s γ c ) = (ω∗cs−1 ω∗cs ), with the convention that ω∗c−1 = 0 and ω∗cS+1 = 1. Now combining this observation with Equations (3), (4), (10), and (11), we obtain

∗ σ s +1 s (ωcs ) 1 − (ω∗cs−1 )σ1 +1 s (13) Q(σ s γ c ) = (γ c )σ2 · σ1s + 1 Now take 1 < s1 < s2 < S such that λs1 = λs1 −1 = λs2 = λs2 −1 = 1. Equations (12) and (13) imply Q(σ s1 γ 1 )Q(σ s2 γ 2 ) ≥ Q(σ s1 γ 2 )Q(σ s2 γ 1 ) ⇔ s2 s1 s2 s1 (γ 2 /γ 1 )(σ2 −σ2 )−(σ1 −σ1 ) ≥ 1. Since γ 2 > γ 1 > 0, Q is not log-supermodular for s2 s1 s2 s1 18 σ2 − σ2 < σ1 − σ1 . What went wrong? The problem comes from the restrictions (or lack thereof) that Assumption 4 imposes on the variations of the assignment function, Ω(σ γ). In a R-economy, Equation (5) guarantees that Ω(σ γ) is either Ω or ∅. In a F -economy, Equation (6) guarantees that Ω(σ γ) is invariant across countries. Here, Equation (10) merely requires that (i) 1 > ω∗cS > · · · > ω∗c1 > 0 for c = 1 2, by the log-supermodularity of q in (ω σ), and that (ii) ω∗1s > ω∗2s for s = 1 S − 1, by the log-supermodularity of q in (σ γ). The latter condition opens up the possibility for country 2 to produce relatively less in the high-σ sectors. Though the log-supermodularity of q in (σ γ) makes identical factors relatively more productive in the high-σ sectors in country 2, it also leads to lower-ω factors to be assigned in these sectors under an efficient allocation. Since lower-ω factors are relatively less productive in the high-σ sectors by the log-supermodularity of q in (ω σ), this indirect effect tends to reduce the relative output of high-σ sectors in country 2. Our counterexample simply offers conditions under which the indirect effect of Ricardian technological differences on factor assignment dominates their direct effect on factor s s s s productivity: σ12 − σ11 > σ22 − σ21 . At a formal level, our complementarity approach breaks down because it requires conditions that are too strong. To predict patterns of specialization using Lemmas 1 and 2, we now need 1Ω(σγ) (ω) to be log-supermodular. Assuming that q is log-supermodular in (σ γ) and (ω σ) guarantees that 1Ω(σγ) (ω) is log-supermodular in (σ γ) and (ω σ), as in R- and F -economies, respectively. Both results are the counterparts of predictions about the monotonicity of solutions of assignment problems in the monotone comparative statics literature (see, e.g., Topkis (1998)). Unfortunately, there is no underlying assignment problem such that the log-supermodularity of q in (ω γ) implies the log-supermodularity of 1Ω(σγ) (ω) in (ω γ). On the contrary, if q is logsupermodular in (σ γ) and (ω σ), then 1Ω(σγ) (ω) must be log-submodular in s

s

s

s

The conditions σ S > · · · > σ 1 > 0, λs1 = λs1 −1 = λs2 = λs2 −1 = 1, and σ22 − σ21 < σ12 − σ11 all are satisfied, for example, if S = 6, s1 = 2, s2 = 5, (σ11 σ16 ) = (1 2 3 5 6 7), and (σ21 σ26 ) = (1 2 3 4 5 6). 18


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(ω γ). This is what we refer to as the “indirect effect of Ricardian technological differences on factor assignment.” As a careful reader may have already noticed, our counterexample is based purely on technological considerations. Although the fact that factor endowment differences can be a source of comparative advantage is crucial, our argument does not rely on factor-endowment differences per se: f (ω γ 1 ) = f (ω γ 2 ) = 1. Instead, it relies on the fact that for factor endowments to be a source of comparative advantage, sectors must vary in their factor intensity, that is, q must be log-supermodular in (ω σ). In our counterexample, the logsupermodularity of q in both (σ γ) and (ω σ) is the only issue. 6.2.2. Robust Predictions When only one source of comparative advantage is present, Theorems 1 and 2 show that the log-supermodularity of the primitives of our model implies the log-supermodularity of aggregate output. These are strong results that apply to all countries and all sectors. Theorem 4 offers similar predictions in the presence of multiple sources of comparative advantage if additional functional form restrictions are imposed. The previous counterexample shows, unfortunately, that these strong results do not hold generally. This is an important observation that highlights the potential caveats of combining insights from distinct models without a generalizing framework. In a R-economy, if high-γ countries are relatively more productive in high-σ sectors, then they produce relatively more in these sectors. In a F -economy, if high-γ countries are relatively more abundant in high-ω factors and high-ω factors are relatively more productive in high-σ sectors, then the same pattern of international specialization arises. In our counterexample, these assumptions on technology and factor endowments all are satisfied, yet high-γ countries do not produce relatively more in all high-σ sectors.19 Our last theorem offers weaker, but robust predictions on the cross-sectional variation of aggregate output in this general environment. THEOREM 5: Let σ ≡ min1≤s≤S σ s and σ ≡ max1≤s≤S σ s . In an elementary neoclassical economy, Assumptions 2 and 4 imply Q(σ γ c1 )Q(σ γ c2 ) ≥ Q(σ γ c2 )Q(σ γ c1 ) for any pair of countries such that γ c1 ≤ γ c2 . According to Theorem 5, if both factor endowment and technological differences are sources of comparative advantage, the pattern of international specialization is unambiguous only in the two most extreme sectors. The formal argument combines the main ideas of Theorems 1 and 2. Holding the assignment function Ω(σ γ) constant across countries, the log-supermodularity 19 In addition, the previous counterexample shows that assuming q(ω σ γ) ≡ h(ω σ)a(σ γ) with h and a log-supermodular is not sufficient to derive the log-supermodularity of Q.

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of f and q implies that high-γ countries produce relatively more in the highσ sectors, as in a F -economy. Since q is log-supermodular in σ and γ, the cross-country variation in Ω(σ γ) then reinforces this effect in the two most extreme sectors. As in a R-economy, a given factor ω is more likely to be found in high-σ sectors in high-γ countries. This implies that a given sector σ is more likely to be assigned low-ω factors in high-γ countries, and, in turn, that Ω(σ γ c2 ) ⊆ Ω(σ γ c1 ) and Ω(σ γ c1 ) ⊆ Ω(σ γ c2 ). In other words, Ricardian technological differences necessarily lead to more factors in the highest-σ sector in high-γ countries and more factors in the lowest-σ sector in low-γ countries, thereby strengthening the pattern of international specialization driven by factor endowment differences. Our final result is reminiscent of the Rybczynski results derived by Jones and Scheinkman (1977) in a standard neoclassical model with an arbitrary number of goods and factors. They showed, among other things, that if factor prices are constant across countries, as in our model, then an increase in the endowment of one factor must decrease output in one sector and increase output more than proportionally in another sector. An important difference between this result and ours is that Theorem 5 clearly identifies the two most extreme sectors as those being affected by changes in factor endowments.20 7. CONCLUDING REMARKS The present paper has developed an elementary theory of comparative advantage. Our theory emphasizes an intimate relationship between logsupermodularity and comparative advantage, whether driven by technology or factor endowment. If factor productivity and/or factor supply are logsupermodular, then many sharp predictions on the pattern of international specialization can be derived in economies with an arbitrarily large number of countries, goods, and factors. While we have focused on the determinants of international specialization, we believe that our general results could also be useful outside international trade. The basic structure of our model—Equations (3) and (4)—is central to many models with agent heterogeneity. These models may focus on different aggregate variables and different market structures, but at the core, there are “agents” with characteristics ω, these agents belong to “populations” with density f (ω γ) and they sort into “occupations” with characteristics σ based on their returns r(ω σ γ). In public finance, agents may be taxpayers with different income sorting across different cities; in corporate finance, agents may 20

Theorem 5 also is related, though less closely, to recent work by Zhu and Trefler (2005), Costinot and Komunjer (2007), Chor (2008), and Morrow (2008). For empirical purposes, the previous papers derived predictions on the pattern of international specialization in economies with Ricardian differences and multiple factors of productions. These predictions, however, are based on assumptions on factor prices, rather than primitive assumptions on factor supply.


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be firms with different productivity choosing different financial instruments; in development economics, agents may be individuals with different wealth choosing whether or not to become entrepreneurs; in labor economics, agents may be migrants with different levels of education sorting across destinations; and in organization economics, agents may be workers with different skills sorting across different layers of a hierarchy.21 In any of these circumstances, our analysis demonstrates that log-supermodularity potentially has important implications for the cross-sectional variation of aggregate variables across populations and occupations. APPENDIX: PROOFS PROOF OF THEOREM 1: We proceed by contradiction. Consider σ ≥ σ and γ ≥ γ . Suppose that Q(σ γ)Q(σ γ ) > Q(σ γ)Q(σ γ ). This implies Q(σ γ) > 0 and Q(σ γ ) > 0 with σ = σ and γ = γ . Since Q(σ γ) > 0, Equations (3) and (4) further imply the existence of ω ∈ Ω such that r(ω σ γ) > r(ω σ γ). Using Equation (5), we can simplify the previous inequality into p(σ )a(σ γ) > p(σ)a(σ γ). Since Q(σ γ ) > 0, similar reasoning implies p(σ)a(σ γ ) > p(σ )a(σ γ ). Combining the previous inequalities, we obtain a(σ γ)a(σ γ ) > a(σ γ)a(σ γ ), which contradicts Assumption 1. Q.E.D. h(ω σ) ≡ 1Ω(σ) (ω) · h(ω σ). We first show PROOF OF THEOREM 2: Let that h(ω σ) is log-supermodular. We proceed by contradiction. Consider h(ω σ ) h(ω σ) > h(ω σ) h(ω σ ). This ω ≥ ω and σ ≥ σ . Suppose that implies ω ∈ Ω(σ ) and ω ∈ Ω(σ) with ω = ω and σ = σ . Using Equation (7), we then get p(σ )h(ω σ ) > p(σ)h(ω σ) and p(σ)h(ω σ) > p(σ )h(ω σ ). The two previous inequalities imply h(ω σ )h(ω σ) > Assumption 3. By Equations (3), (6), h(ω σ)h(ω σ ), which contradicts and (7), we have Q(σ γ) = h(ω σ)a(γ)f (ω γ) dω. We have just shown that h(ω σ) is log-supermodular. a(γ) is trivially log-supermodular. By Assumption 2, we know that f (ω γ) is log-supermodular. Theorem 2 derives from these three observations and the fact that log-supermodularity is preserved by multiplication and integration, by Lemmas 1 and 2. Q.E.D. PROOF OF THEOREM 3: Theorem 2 shows that Assumptions 2 and 3 are sufficient conditions. That they are a minimal pair is proved by two lemmas. 21 See, for example, Epple and Romer (1991), Champonnois (2007), Banerjee and Newman (1993), Grogger and Hanson (2008), and Garicano and Rossi-Hansberg (2006), respectively.

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LEMMA 3: If Q(σ γ) is log-supermodular in any F -economy where Assumption 3 holds, then Assumption 2 holds. PROOF: We proceed by contradiction. Suppose that there exist ω ≥ ω and γ ≥ γ such that (14)

f (ω γ )f (ω γ) > f (ω γ)f (ω γ )

Inequality (14) implies ω = ω . Now consider a F -economy with two factors, ω and ω , two countries with characteristics γ and γ , and two sectors with characteristics σ > σ , such that (15)

a(γ) = a(γ ) = 1

(16)

h(ω σ) = h(ω σ ) = 1

(17)

h(ω σ ) = h(ω σ) = 0

which is possible since ω = ω and σ = σ . Combining Equations (15), (16), and (17) with Equations (3), (6), and (7) and inequality (14), we get Q(σ γ )Q(σ γ) > Q(σ γ)Q(σ γ ). By Equations (16) and (17), Assumption 3 is satisfied—a contradiction. Q.E.D. LEMMA 4: If Q(σ γ) is log-supermodular in any F -economy where Assumption 2 holds, then Assumption 3 holds. PROOF: We proceed by contradiction. Suppose that there exist ω ≥ ω and σ ≥ σ such that (18)

h(ω σ )h(ω σ) > h(ω σ)h(ω σ )

Inequality (18) implies ω = ω , h(ω σ ) > 0, and h(ω σ) > 0. This further implies the existence of p(σ) > 0 and p(σ ) > 0 such that (19)

h(ω σ) p(σ ) h(ω σ) > > h(ω σ ) p(σ) h(ω σ )

h(ω σ) where h(ω σ ) = +∞ if h(ω σ ) = 0. Now consider a F -economy with two factors, ω and ω , two countries with characteristics γ > γ , and two sectors with characteristics σ and σ , such that prices satisfy inequality (19) and

(20)

a(γ) = a(γ ) = 1

(21)

f (ω γ) = f (ω γ ) = 1

(22)

f (ω γ ) = f (ω γ) = 0


1189

which is possible since ω = ω and γ = γ . Inequality (19) implies p(σ ) × h(ω σ ) > p(σ)h(ω σ) and p(σ)h(ω σ) > p(σ )h(ω σ ). Combining these two inequalities with Equations (3), (6), (7), (20), (21), and (22), we get Q(σ γ )Q(σ γ) > Q(σ γ)Q(σ γ ). By Equations (21) and (22), Assumption 2 is satisfied—a contradiction. Q.E.D. PROOF OF THEOREM 4: We first show that if f (ω γ) is log-supermodular, strictly positive, and log-concave in ω, and t is increasing in γ, then f(ω γ) ≡ f [ω − t(γ) γ] is log-supermodular. We proceed by contradiction. Suppose that there exist ω ≥ ω and γ ≥ γ such that (23)

f [ω − t(γ) γ]f [ω − t(γ ) γ ] > f [ω − t(γ) γ]f [ω − t(γ ) γ ]

Since f is log-supermodular in (ω γ), we know that (24)

f [ω − t(γ ) γ]f [ω − t(γ ) γ ] ≥ f [ω − t(γ ) γ]f [ω − t(γ ) γ ]

Combining inequalities (23) and (24) and the fact that f > 0, we get (25)

f [ω − t(γ) γ]f [ω − t(γ ) γ] > f [ω − t(γ) γ]f [ω − t(γ ) γ]

Since t is increasing in γ, inequality (25) implies that f (· γ) cannot be of Polya frequency of order 2, which contradicts f log-concave in ω; see An (1998). At this point, we know that aggregate output is given by Equations (7) and (9), that f(ω γ) is log-supermodular, and that h(ω σ) is log-supermodular by Assumption 2. Thus, we can invoke Theorem 2, which implies Q(σ γ) is logsupermodular. Q.E.D. PROOF OF THEOREM 5: We use the following lemma. LEMMA 5: If q(ω σ γ) satisfies Assumption 4, then Ω(σ γ c2 ) ⊆ Ω(σ γ c1 ) and Ω(σ γ c1 ) ⊆ Ω(σ γ c2 ) for any γ c2 ≥ γ c1 . PROOF: Without loss of generality, suppose that σ = σ 1 . We only show that Assumption 4 implies Ω(σ γ c2 ) ⊆ Ω(σ γ c1 ) for any γ c2 ≥ γ c1 . The argument for Ω(σ γ c1 ) ⊆ Ω(σ γ c2 ) is similar. We proceed by contradiction. Take ω ∈ Ω(σ γ c2 ). By Equation (4), we have p(σ)q(ω σ γ c2 ) > / Ω(σ γ c1 ). By Equation (4), maxs=1 p(σ s )q(ω σ s γ c2 ). Now suppose that ω ∈ s s c1 we also have maxs=1 p(σ )q(ω σ γ ) > p(σ)q(ω σ γ c1 ). Let s∗ ≡ arg maxs=1 p(σ s )q(ω σ s γ c1 ). Combining the two previous inequalities, we ∗ ∗ ∗ get q(ω σ γ c2 )q(ω σ s γ c1 ) > q(ω σ s γ c2 )q(ω σ γ c1 ). Since σ s ≥ σ and Q.E.D. γ c2 ≥ γ c1 , this contradicts Assumption 4. PROOF that (26)

OF

γ) such THEOREM 5—Continued: Fix γ c1 ∈ Γ and define Q(σ

γ) ≡ Q(σ

q(ω σ γ)f (ω γ) dμ(ω) Ω(σγ c1 )

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for all σ ∈ Σ and γ ∈ Γ . We can use the same reasoning as in Theorem 2 γ) is log-supermodular. By Assumption 4, 1Ω(σγc1 ) (ω) is to show that Q(σ log-supermodular in (ω σ). By Assumptions 2 and 4, q(ω σ γ) and f (ω γ) γ) is log-supermodular by Lemmas 1 also are log-supermodular. Hence, Q(σ and 2, which implies (27)

γ c1 )Q(σ γ c2 ) ≥ Q(σ γ c2 )Q(σ γ c1 ) Q(σ

for any γ c2 ≥ γ c1 . By Equation (26), we have Q(σ γ c1 ) = Q(σ γ c1 ) and c c γ 1 ) = Q(σ γ 1 ). Since q(ω σ γ) ≥ 0 and f (ω γ) ≥ 0, we also have Q(σ γ c2 ) by Lemma 5. Combining Q(σ γ c2 ) ≥ Q(σ γ c2 ) and Q(σ γ c2 ) ≥ Q(σ the previous conditions with inequality (27), we get Q(σ γ c1 )Q(σ γ c2 ) ≥ Q.E.D. Q(σ γ c2 )Q(σ γ c1 ) for any γ c2 ≥ γ c1 . REFERENCES ACEMOGLU, A. (1998): “Why Do New Technologies Complement Skills? Directed Technical Change and Wage Inequality,” Quarterly Journal of Economics, 113, 1055–1089. [1167] ACEMOGLU, A., P. ANTRAS, AND E. HELPMAN (2007): “Contracts and Technology Adoption,” American Economic Review, 97, 916–943. [1166,1175] ACEMOGLU, D., AND J. VENTURA (2002): “The World Income Distribution,” Quarterly Journal of Economics, 117, 659–694. [1172] AHLSWEDE, R., AND E. DAYKIN (1978): “An Inequality for Weights of Two Families of Sets, Their Unions and Intersections,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 43, 183–185. [1169] AN, M. (1998): “Logconcavity versus Logconvexity: A Complete Characterization,” Journal of Economic Theory, 80, 350–369. [1189] ANTRAS, P., AND E. HELPMAN (2004): “Global Sourcing,” Journal of Political Economy, 112, 552–580. [1167,1180,1181] ATHEY, S. (2002): “Monotone Comparative Statics Under Uncertainty,” Quarterly Journal of Economics, 117, 187–223. [1167,1168,1179] BAGNOLI, M., AND T. BERGSTROM (2005): “Log-Concave Probability and Its Applications,” Economic Theory, 26, 445–469. [1182] BANERJEE, A., AND A. NEWMAN (1993): “Occupational Choice and the Process of Development,” Journal of Political Economy, 101, 274–298. [1187] BECKER, G. (1952): “A Note on Multi-Country Trade,” American Economic Review, 42, 558–568. [1172] (1973): “A Theory of Marriage: Part I,” Journal of Political Economy, 81, 813–846. [1175] CHAMPONNOIS, S. (2007): “Comparing Financial Systems: A Structural Analysis,” Mimeo, UCSD. [1187] CHOR, D. (2008): “Unpacking Sources of Comparative Advantage: A Quantitative Approach,” Mimeo, Singapore Management University. [1186] COSTINOT, A. (2007): “On the Origins of Comparative Advantage,” Mimeo, UCSD. [1166,1175, 1180,1181] COSTINOT, A., AND I. KOMUNJER (2007): “What Goods Do Countries Trade? New Ricardian Predictions,” Working Paper, NBER. [1186] CUÑAT, A., AND M. MELITZ (2006): “Labor Market Flexibility and Comparative Advantage,” Mimeo, Harvard University. [1166,1175] DAVIS, D. (1995): “Intra-Industry Trade: A Heckscher–Ohlin–Ricardo Approach,” Journal of International Economics, 39, 201–226. [1168]


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DEARDORFF, A. (1980): “The General Validity of the Law of Comparative Advantage,” Journal of Political Economy, 88, 941–957. [1168] (2007): “The Ricardian Model,” Mimeo, University of Michigan. [1168,1174] DIXIT, A. K., AND V. NORMAN (1980): Theory of International Trade. Cambridge, U.K.: Cambridge University Press. [1170] DORNBUSCH, R., S. FISCHER, AND P. SAMUELSON (1977): “Comparative Advantage, Trade, and Payments in a Ricardian Model With a Continuum of Goods,” American Economic Review, 67, 823–839. [1175] EATON, J., AND S. KORTUM (2002): “Technology, Geography, and Trade,” Econometrica, 70, 1741–1779. [1178] EPPLE, D., AND T. ROMER (1991): “Mobility and Redistribution,” Journal of Political Economy, 99, 828–858. [1187] ETHIER, W. (1984): “Higher Dimensional Issues in Trade Theory,” in Handbook of International Economics, Vol. 1, ed. by R. W. Jones and P. B. Kenen. New York: Elsevier Science, 131–184. [1168] GARICANO, L., AND E. ROSSI-HANSBERG (2006): “Organization and Inequality in a Knowledge Economy,” Quarterly Journal of Economics, 121, 1383–1435. [1187] GROGGER, J., AND G. HANSON (2008): “Income Maximization and the Selection and Sorting of International Migrants,” Working Paper, NBER. [1187] GROSSMAN (2004): “The Distribution of Talent and the Pattern and Consequences of International Trade,” Journal of Political Economy, 112, 209–239. [1172] GROSSMAN, AND MAGGI (2000): “Diversity and Trade,” American Economic Review, 90, 1255–1275. [1172] HELPMAN, E., M. MELITZ, AND S. YEAPLE (2004): “Export versus FDI,” American Economic Review, 94, 300–316. [1167,1180,1181] JEWITT, I. (1987): “Risk Aversion and the Choice Between Risky Prospects: The Preservation of Comparative Statics Results,” Review of Economic Studies, 54, 73–85. [1167] JONES, R. W. (1961): “Comparative Advantage and the Theory of Tariffs: A Multi-Country, Multi-Commodity Model,” Review of Economic Studies, 28, 161–175. [1173-1175] JONES, R. W., AND J. SCHEINKMAN (1977): “The Relevance of the Two-Sector Production Model in Trade Theory,” Journal of Political Economy, 85, 909–935. [1186] KARLIN, S. (1968): Total Positivity, Vol. I. Stanford, CA: Stanford University Press. [1168] KARLIN, S., AND Y. RINOTT (1980): “Classes of Orderings of Measures and Related Correlation Inequalities I. Multivariate Totally Positive Distributions,” Journal of Multivariate Analysis, 10, 467–498. [1169] KREMER, M. (1993): “The O-Ring Theory of Economic Development,” Quarterly Journal of Economics, 108, 551–575. [1175] KRUGMAN, P. (1986): “A ‘Technology Gap’ Model of International Trade,” in Structural Adjustment in Advanced Economies, ed. by K. Jungenfelt and D. Hague. London: MacMillan. [1175] LEGROS, P., AND A. NEWMAN (2002): “Monotone Matching in Perfect and Imperfect Worlds,” Review of Economic Studies, 69, 925–942. [1175] LEHMANN, E. (1955): “Ordered Families of Distributions,” Annals of Mathematical Statistics, 26, 399–419. [1169] LEVCHENKO, A. (2007): “Institutional Quality and International Trade,” Review of Economic Studies, 74, 791–819. [1166,1175] MATSUYAMA, K. (1996): “Why Are There Rich and Poor Countries?: Symmetry-Breaking in the World Economy,” Journal of the Japanese and International Economies, 10, 419–439. [1172] (2005): “Credit Market Imperfections and Patterns of International Trade and Capital Flows,” Journal of the European Economic Association, 3, 714–723. [1166,1175] MELITZ, M. (2003): “The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,” Econometrica, 71, 1695–1725. [1166,1180,1181] MILGROM, P. (1981): “Good News and Bad News: Representation Theorems and Applications,” The Bell Journal of Economics, 12, 380–391. [1169,1177]

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MILGROM, P., AND R. WEBER (1982): “A Theory of Auctions and Competitive Bidding,” Econometrica, 50, 1089–1122. [1167] MORROW, P. (2008): “East Is East and West Is West: A Ricardian–Heckscher–Ohlin Model of Comparative Advantage,” Mimeo, University of Toronto. [1186] NUNN, N. (2007): “Relationship-Specificity, Incomplete Contracts and the Pattern of Trade,” Quarterly Journal of Economics, 122, 569–600. [1166,1175] OHNSORGE, F., AND D. TREFLER (2004): “Sorting It Out: International Trade and Protection With Heterogeneous Workers,” Working Paper, NBER. [1167,1172,1180] (2007): “Sorting It Out: International Trade and Protection With Heterogeneous Workers,” Journal of Political Economy, 115, 868–892. [1180] ROMALIS, J. (2004): “Factor Proportions and the Structure of Commodity Trade,” American Economic Review, 14, 67–97. [1178] RUFFIN, R. (1988): “The Missing Link: The Ricardian Approach to the Factor Endowment Theory of Trade,” American Economic Review, 78, 759–772. [1166,1180] SHIMER, R., AND L. SMITH (2000): “Assortative Matching and Search,” Econometrica, 68, 343–370. [1167] TOPKIS, D. (1998): Supermodularity and Complementarity. Frontiers of Economic Research. Princeton, NJ: Princeton University Press. [1168,1184] TREFLER, D. (1993): “International Factor Price Differences: Leontief Was Right!” Journal of Political Economy, 101, 961–987. [1182] VOGEL, J. (2007): “Institutions and Moral Hazard in Open Economies,” Journal of International Economics, 71, 495–514. [1166,1175] YANAGAWA, N. (1996): “Economic Development in a Ricardian World With Many Countries,” Journal of Development Economics, 49, 271–288. [1172] ZHU, S., AND D. TREFLER (2005): “Trade and Inequality in Developing Countries: A General Equilibrium Analysis,” Journal of International Economics, 66, 21–48. [1186]

Dept. of Economics, Massachusetts Institute of Technology, 50 Memorial Drive, E52-391, Cambridge, MA 02142, U.S.A. and NBER; [email protected]. Manuscript received December, 2007; final revision received January, 2009.


NONPARAMETRIC IDENTIFICATION OF RISK AVERSION IN FIRST-PRICE AUCTIONS UNDER EXCLUSION RESTRICTIONS BY EMMANUEL GUERRE, ISABELLE PERRIGNE, AND QUANG VUONG1 This paper studies the nonparametric identification of the first-price auction model with risk averse bidders within the private value paradigm. First, we show that the benchmark model is nonindentified from observed bids. We also derive the restrictions imposed by the model on observables and show that these restrictions are weak. Second, we establish the nonparametric identification of the bidders’ utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders’ participation, leading to a latent distribution of private values that is independent of the number of bidders. The key idea is to exploit the property that the bid distribution varies with the number of bidders while the private value distribution does not. We then extend these results to endogenous bidders’ participation when the exclusion restriction takes the form of instruments that do not affect the bidders’ private value distribution. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values, and asymmetric bidders. Last, possible estimation methods are proposed. KEYWORDS: Risk aversion, private value, nonparametric identification, exclusion restrictions.

1. INTRODUCTION THE EMPIRICAL AND EXPERIMENTAL LITERATURE indicates that risk aversion is an important component of bidders’ behavior in auctions. For instance, Baldwin (1995) and Athey and Levin (2001) found that bidders diversify risk when bidding on species in U.S. Forest Service auctions in which payments are based on ex post harvested quantities. Adopting a structural approach, Campo, Guerre, Perrigne, and Vuong (2007) and Lu and Perrigne (2008) also found significant risk aversion in U.S. Forest Service auctions. Following Li and Tan’s (2000) theoretical results, Perrigne (2003) showed that a random reserve price leads empirically to larger revenues when bidders are risk averse, thereby justifying such an auction design. In eBay auctions, Ackerberg, Hirano, and Shahriar (2006) rationalized the popularity of the buy-it-now option by bidders’ risk aversion. In experiments, bidders’ risk aversion is used to explain the frequently observed overbidding over the neutral Bayesian Nash equilibrium strategy as in Cox, Smith, and Walker (1988), among others. Relying on a different equilibrium concept, Goeree, Holt, and Palfrey (2002) also found signif1 Financial support from the National Science Foundation Grant SES-0452154 is gratefully acknowledged by the last two authors. Versions of this paper were presented at the 2005 CIRANO Conference on Auctions in Montreal, the 2006 International Industrial Organization Conference in Boston, and at seminars at University of California Los Angeles, University of Arizona, University of California San Diego, University of British Columbia, and Yale University. We thank X. Chen, M. Peters, and P. C. B. Phillips as well as participants at conferences and seminars for their comments. We are grateful to a co-editor and two referees for thoughtful suggestions.


DOI: 10.3982/ECTA7028

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icant bidder risk aversion. Last, using recent structural econometric methods, Bajari and Hortacsu (2005) showed that risk aversion provides the best fit to some experimental data among a set of competing models, including learning models. With the exception of Lu and Perrigne (2008), who exploited additional bidding data, the previous papers all rely on a parameterization of the bidders’ utility function, yet various concepts of risk aversion have different implications on agents’ behavior. There is, however, no general agreement on which concept of risk aversion is the most appropriate to explain observed phenomena such as in finance through the diversification of portfolios, insurance when low risk car drivers tend to buy more insurance than needed, or in auctions through overbidding.2 A typical controversy is whether risk aversion is absolute or relative to economic agents’ wealth. As a matter of fact, little is known about the shape of agents’ utility functions. Several families of utility functions have been developed to embody some economic properties related to risk aversion; see Gollier (2001) for an extensive survey on risk aversion. Which family is relevant is an empirical question. Given the importance of risk aversion in auctions and our ignorance about bidders’ utility functions, we address the nonparametric identification of the latter in this paper. First, we show that the general model is not identified from observed bids. We then derive the theoretical restrictions imposed by the model on observables and show that these restrictions are weak. In particular, we show that any smooth bid distribution can be rationalized by a model with general risk aversion. Such a striking result implies that risk aversion does not impose testable restrictions on bids. This can explain why risk aversion is difficult to detect in observed bids. Second, we show that the bidders’ utility function is nonparametrically identified under some exclusion restrictions. To clarify ideas, our primary exclusion restriction takes the form of an exogenous bidders’ participation leading to a latent distribution of private values that is independent of the number of bidders. Such an assumption is made in theoretical auction papers; see Krishna (2002). It is also a standard assumption made in the empirical auction literature. Exclusion restrictions are widely used in econometrics. A typical example is the use of instrumental variables in labor economics to address the endogeneity of education in the estimation of the wage equation. Other examples can be found in Matzkin (1994) in a nonparametric context. Exclusion restrictions have also been used in the structural auction literature. Athey and Haile (2002) and Haile, Hong, and Shum (2003) exploited exogenous participation to detect the winner’s curse and hence to test for common values in first-price sealed-bid auctions. In a different framework, Bajari and Hortacsu (2005) used exogenous participation to estimate an auction model with con2

For an empirical analysis of risk aversion in car insurance, see Cohen and Einav (2007).

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stant relative risk aversion from experimental data.3 We then extend our results to a model with endogenous bidder participation, where the exclusion restriction takes the form of instruments that do not affect the bidders’ private value distribution but affect bidders’ participation.4 Though we consider first a benchmark model with symmetric bidders, independent private values, and no reserve price, our results extend to a binding reserve price, affiliated private values, and asymmetric bidders, whether asymmetry arises from private values and/or heterogenous preferences. As such, our results apply to a large class of auction models. The paper is organized as follows. Section 2 presents the benchmark model and establishes its nonparametric identification from observed bids. In view of this, Section 3 exploits exclusion restrictions to achieve nonparametric identification of the bidders’ utility function and private value distribution. We then characterize the theoretical restrictions that observed bids need to satisfy. Section 4 extends our nonidentification and identification results to a binding reserve price, affiliated private values, and asymmetric bidders. Section 5 concludes with a discussion of possible estimation methods. The Appendix collects the proofs. 2. MODEL AND NONIDENTIFICATION The first subsection presents the independent private value (IPV) first-price sealed-bid auction model with risk averse bidders and details properties of its equilibrium strategy. The second subsection establishes the nonparametric nonidentification of the model. 2.1. The Benchmark Model A single and indivisible object is sold through a first-price sealed-bid auction. Within the IPV paradigm, each bidder knows his own private value vi but not other bidders’ private values. There are I potential bidders with I ∈ I a finite subset of {2 3 4 }. Private values are drawn independently from a distribution F(·|I), which is absolutely continuous with density f (·|I) on a support [v(I) v(I)] ⊂ R+ . The distribution F(·|I) and the number of potential bidders I are common knowledge.5 Let U(·) be the bidders’ von Neumann– Morgenstern (vNM) utility function with U(0) = 0, U (·) > 0, and U (·) ≤ 0. 3 Exogenous participation is not necessary to estimate the model in their paper. Such a restriction avoids the use of a conditional quantile restriction as in Campo, Guerre, Perrigne, and Vuong (2007). 4 Haile, Hong, and Shum (2003) also introduced such instruments to deal with an endogenous number of bidders. 5 In the theoretical auction literature, F(·) is independent of I, which is the exclusion restriction considered in Section 3.1. In auction data, however, one may observe some correlation pattern between the number of bidders I and the value of the auctioned object, thereby justifying the dependence of F(·) on I. See also Section 3.2.

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All bidders are thus identical ex ante and the game is symmetric. Bidder i maximizes his expected utility (1)

EΠi = U(vi − bi ) Pr(bi ≥ bj j = i)

with respect to his bid bi , where bj is the jth player’s bid; see Maskin and Riley (1984, Case 1) and Krishna (2002). Because the scale is irrelevant, we impose the normalization U(1) = 1. The risk neutral case is obtained when U(·) is the identity function.6 From Maskin and Riley (1984), if a symmetric Bayesian Nash equilibrium strategy s(·) = s(· U F I) exists, then it is strictly increasing and continuous on [v(I) v(I)] and differentiable on (v(I) v(I)].7 Thus (1) becomes EΠi = U(vi − bi )F I−1 (s−1 (bi )|I), where s−1 (·) denotes the inverse of s(·). Hence, imposing bidder i’s optimal bid bi to be s(vi ) gives the differential equation (2)

s (vi ) = (I − 1)

f (vi |I) λ(vi − bi ) F(vi |I)

for all vi ∈ (v(I) v(I)]

where λ(·) = U(·)/U (·). From Maskin and Riley (1984), the boundary condition is s(v(I)) = v(I) because U(0) = 0. Moreover, the second-order conditions are satisfied. When the reserve price is nonbinding, the existence of a pure equilibrium strategy follows from Maskin and Riley (2000) and Athey (2001), while its uniqueness is established by Maskin and Riley (2003) using an argument similar to Lebrun (1999). The main contribution of Theorem 1 below is to derive the smoothness of the equilibrium strategy, which is used in the next subsection. Determining the smoothness of the equilibrium strategy is difficult when the differential equation (2) does not have an explicit solution, which is the case for general utility functions U(·). We assume that U(·) and F(·|I) belong to UR and FR defined as follows, respectively. DEFINITION 1: For R ≥ 1, let UR be the set of utility functions U(·) that satisfy the following conditions: (i) U : [0 +∞) → [0 +∞), U(0) = 0, and U(1) = 1. (ii) U(·) is continuous on [0 +∞) and admits R + 2 continuous derivatives on (0 +∞) with U (·) > 0 and U (·) ≤ 0 on (0 +∞). (iii) limx↓0 λ(r) (x) is finite for 1 ≤ r ≤ R + 1, where λ(r) (·) is the rth derivative of λ(·). 6 With bidders’ wealth w, the expected profit becomes [U(w + vi − bi ) − U(w)] Pr(bi ≥ bj j = i) + U(w). Different wealths wi lead to an asymmetric game if the wi ’s are common knowledge and to a multisignal game if the wi ’s are private information. See Che and Gale (1998) for a multisignal auction model. 7 From Maskin and Riley (1984, Remark 2.3), the only equilibria are symmetric when F(·) has bounded support.


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Conditions (i) and (ii) have been discussed previously. Note that limx↓0 λ(x) = 0 since U(0) = 0 and U (·) is nonincreasing. Thus, from (ii) and (iii) it follows that λ(·) admits R + 1 continuous derivatives on [0 +∞). These regularity assumptions are weak because they are satisfied by many vNM utility functions. DEFINITION 2: For R ≥ 1, let FR be the set of distributions F(·|I), I ∈ I , that satisfy the following conditions: (i) F(·|I) is a cumulative distribution function (c.d.f.) with support of the form [v(I) v(I)], where 0 ≤ v(I) < v(I) < +∞. (ii) F(·|I) admits R + 1 continuous derivatives on [v(I) v(I)]. (iii) f (·|I) > 0 on [v(I) v(I)]. Altogether (i)–(iii) imply that f (·|I) is bounded away from zero on [v(I) v(I)]. THEOREM 1: Let R ≥ 1. Suppose that [U F] ∈ UR × FR . Then for each I ∈ I , there exists a unique (symmetric) equilibrium strategy s(·). Moreover, this strategy satisfies the following statements: (i) ∀v ∈ (v(I) v(I)], s(v) < v with s(v(I)) = v(I). (ii) ∀v ∈ [v(I) v(I)], s (v) > 0 with s (v) = (I − 1)λ (0)/[(I − 1)λ (0) + 1] < 1. (iii) s(·) admits R + 1 continuous derivatives on [v(I) v(I)]. The proof of Theorem 1 can be found in the Supplemental Material (Guerre, Perrigne, and Vuong (2009)). 2.2. Nonidentification Results We study identification of the structure [U F] from observables. We assume that the number I of bidders is observed, as in a first-price sealed-bid auction with a nonbinding reserve price. We also assume that the distribution G(·|I) of an equilibrium bid is known. Thus the identification problem reduces to whether the structure [U F] can be recovered uniquely from the knowledge of (I G). A related issue is whether any bid distribution G(·|I) can be rationalized by a structure [U F]. Such a question relates to the possibility of testing the validity of the auction model. Following Guerre, Perrigne, and Vuong (2000), we express (2) using the distribution G(·|I) of an equilibrium bid. For every b ∈ [b(I) b(I)] = [v(I) s(v(I))], we have G(b|I) = F(s−1 (b)|I) = F(v|I) with density g(b|I) = f (v|I)/s (v), where v = s−1 (b). Thus (2) can be written as (3)

1 = (I − 1)

g(bi |I) λ(vi − bi ) for all bi ∈ (b(I) b(I)] G(bi |I)

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Since U(·) ≥ 0 and U (·) ≤ 0, we have λ (·) = 1 − U(·)U (·)/U (·)2 ≥ 1. Thus λ(·) is strictly increasing. Solving (3) for vi and using b(I) = v(I) with λ−1 (0) = 0 give 1 G(bi |I) −1 (4) vi = bi + λ I − 1 g(bi |I) ≡ ξ(bi U G I) for all bi ∈ [b(I) b(I)] where λ−1 (·) denotes the inverse of λ(·). This equation expresses each bidder’s private value as a function of his corresponding bid, the bid distribution, the number of bidders, and the utility function. Note that ξ(·) is the inverse of the bidding strategy s(·). The equilibrium bid distribution G(·|I) satisfies some regularity properties implied by the smoothness of s(·) given in Theorem 1 and the regularity assumptions on [U F]. DEFINITION 3: For R ≥ 1, let GR be the set of distributions G(·|I), I ∈ I , that satisfy the following conditions: (i) G(·|I) is a c.d.f. with support of the form [b(I) b(I)], where 0 ≤ b(I) < b(I) < +∞. (ii) G(·|I) admits R + 1 continuous derivatives on [b(I) b(I)]. (iii) g(·|I) > 0 on [b(I) b(I)]. (iv) g(·|I) admits R + 1 continuous derivatives on (b(I) b(I)]. (v) limb↓b(I) d r [G(b|I)/g(b|I)]/dbr exists and is finite for r = 1 R + 1. The regularity properties (i)–(iii) are similar to those of Definition 2 for F(·|I). They imply that g(·|I) is bounded away from zero on [b(I) b(I)] and that limb↓b G(b|I)/g(b|I) = 0 so that limb↓b(I) ξ(b U G I) = b(I). Properties (iv) and (v) are specific to the auction model. In particular, (iv) says that g(·|I) is smoother than f (·|I), extending a similar property noted by Guerre, Perrigne, and Vuong (2000) for the risk neutral model. Combined with (iii) and (iv), (v) implies that G(·|I)/g(·|I) is R + 1 continuously differentiable on [b(I) b(I)]. The following lemma provides necessary and sufficient conditions for rationalizing a distribution of observed bids by an IPV auction model with risk aversion. Hereafter, we say that a distribution is rationalized by an auction model with risk aversion if there exists a structure [U F] whose equilibrium bid distribution is identical to the given distribution. LEMMA 1: Let R ≥ 1 and let G(·|I) be the joint distribution of (b1 bI ) conditional on I ∈ I . There exists an IPV auction model with risk aversion [U F] ∈ UR × FR that rationalizes G(·|·) if and only if the following conditions hold:


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I (i) G(b1 bI |I) = i=1 G(bi |I) with G(·|·) ∈ GR . (ii) ∃λ : R+ → R+ with R + 1 continuous derivatives on [0 +∞), λ(0) = 0 and λ (·) ≥ 1 such that ξ (·) > 0 on [b(I) b(I)], where ξ(b U G I) = b + λ−1 [G(b|I)/((I − 1)g(b|I))]. Condition (i) is related to the IPV paradigm and requires that bids be independent and identically distributed (i.i.d.), where G(·|·) satisfies the regularity properties of Definition 3. Condition (ii) arises from ξ(· U G I) being the inverse of the equilibrium strategy, which is strictly increasing for each I ∈ I .8 The next proposition shows that any distribution G(·|·) ∈ GR can be rationalized by an IPV auction model with a utility function displaying risk aversion. PROPOSITION 1: Let R ≥ 1. A bid distribution G(·|·) can be rationalized by a risk aversion structure [U F] ∈ UR × FR if and only if G(·|·) ∈ GR . Proposition 1 is striking because it says that any bid distribution in GR can be rationalized by risk aversion. It implies that the restriction (ii) in Lemma 1 is redundant. Specifically, our proof indicates that we can always find a function λ(·) that corresponds to a utility function U(·) ∈ UR so that (ii) is satisfied whenever G(·|·) ∈ GR . Alternatively, the IPV auction model with general risk aversion does not impose any restriction on observed bids beyond their independence and the smoothness conditions embodied in GR . Thus, by allowing for risk aversion, one does enlarge the set of rationalizable bid distributions relative to the risk neutral case studied in Guerre, Perrigne, and Vuong (2000).9 A model is a set of structures [U F]. Hereafter, a structure [U F] is non˜ F] ˜ within the model that leads to identified if there exists another structure [U ˜ F] ˜ exists for any the same equilibrium bid distribution. If no such structure [U [U F], the model is (globally) identified. Given the weakness of the restrictions imposed by the model, it is not surprising that the model with risk aversion is not identified. PROPOSITION 2: Let R ≥ 1. Any structure [U F] ∈ UR × FR is not identified. Proposition 2 says that the risk aversion model is nonparametrically nonidentified. This contrasts with Guerre, Perrigne, and Vuong (2000), who 8 As shown in the proof of Lemma 1, if (ii) is satisfied, then G(·|I) is rationalized by the x structure [U F], where U(x) = exp 1 (1/λ(t)) dt and F(·|I) is the distribution of ξ(b U G I) 0 with b ∼ G(·|I). Because λ(x) ≈ λ (0)x in the neighborhood of 0, 1 (1/λ(t)) dt = −∞ so that U(0) = 0, as required. As a matter of fact, Lemma 1 can be extended to the rationalization of structures with some risk loving, that is, U (·) ≥ 0 as long as λ (·) > 0 so that λ(·) remains invertible. 9 Campo, Guerre, Perrigne, and Vuong (2007) showed another interesting result: Any distribution G(·|·) ∈ GR can be rationalized by some constant relative or absolute risk aversion model.

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showed that the risk neutral model is nonparametrically identified. Thus nonidentification arises from the unknown utility function U(·), which is restricted to the identity function under risk neutrality. In view of this result, one can entertain several strategies to identify risk aversion in first-price auctions. A first natural strategy is to require more data. For instance, the availability of ascending auction data allows one to identify [U F] nonparametrically as shown in Lu and Perrigne (2008). In particular, since risk aversion does not affect bidding in ascending auctions, ascending auction data allow one to identify F(·|I) as in Athey and Haile (2002), from which U(·) is identified using sealed-bid auction data. A second strategy is to impose minimal parametric restrictions on the structure [U F] through a parameterization of U(·) and/or F(·|·). This was pursued in Campo, Guerre, Perrigne, and Vuong (2007), where U(·) and one quantile of F(·|·) are parameterized to identify the model semiparametrically. In the next section, we explore another identifying strategy based on exclusion restrictions to identify the risk aversion model nonparametrically. 3. IDENTIFICATION UNDER EXCLUSION RESTRICTIONS To clarify ideas, we first consider the case of an exogenous participation in which the latent private value distribution is independent of the number of bidders. In a second subsection, we relax this assumption by considering the case of an endogenous participation with instruments that do not affect the underlying private value distribution. 3.1. Exogenous Participation We assume that F(·|·) does not depend on the number I of bidders, which corresponds to the exclusion restriction F(·|I) = F(·). Because bidders’ private values are independent of participation, the latter is exogenous. This exclusion restriction has been made implicitly in a majority of structural empirical studies involving various data sets such as Paarsch (1992) for timber sales, and Bajari and Hortacsu (2003) for eBay data. A notable exception is Athey, Levin, and Seira (2004) for U.S. Forest Service auctions. The functions U(·) and F(·) satisfy Definitions 1 and 2 with R ≥ 1. The previous definitions and equations defining the model need to be revised accordingly, with F(·) and f (·) defined on the support [v v], while the bid distribution G(·|I) still depends on I through s(· U F I), but its support is now [v s(v)], where the upper bound depends on I through s(·). The key idea of our nonparametric identification result is to exploit variations in the quantiles of the bid distribution with the number of bidders, while the corresponding quantiles of the private value distribution remain the same. Our result relies on a property of


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the equilibrium strategy, namely that s(·) is increasing in bidders’ participation. In simple terms, increased competition renders bidding more aggressive.10 Let I2 > I1 ≥ 2 be two different numbers of bidders. We use the index j = 1 2 to refer to the level of competition. Because the equilibrium strategy varies with the number of bidders, the bid distribution also varies with I, giving sj (·) and Gj (·) ≡ G(·|Ij ). Though the lower bound of the bid distribution remains the same because of the boundary condition, the upper bound bj varies with I. The next lemma gives some lower and upper bounds for each equilibrium strategy in terms of the other equilibrium strategy. In particular, it establishes that the equilibrium strategy strictly increases in the number of bidders. As far as we know, the latter property was obtained for the risk neutral case and the constant relative risk aversion case, but not with a general risk aversion U(·). LEMMA 2: Under the previous assumptions, we have I2 − I1 I2 − 1 I1 − I2 I1 − 1 s2 (v) + v < s1 (v) < s2 (v) < s1 (v) + v I2 − 1 I2 − 1 I1 − 1 I1 − 1 for any v ∈ (v v]. The preceding lemma provides some testable implications in terms of stochastic dominance between the observed equilibrium bid distributions as well as their quantiles.11 Let G1 (·) ≺b G2 (·) denote that the distribution G1 (·) is strictly (first-order) stochastically dominated by G2 (·) except at the common lower bound b. That is, G1 (b) > G2 (b) for any b ∈ (b b1 ], where the support of Gj (·) is [b bj ], which is a compact subset with nonempty interior of [0 ∞). For j = 1 2, let bj (α) denote the α-quantile of Gj (·), that is, Gj [bj (α)] = α for α ∈ [0 1]. Because bj = sj (v), where sj (·) is strictly increasing on [v v], bj (α) = sj [v(α)], where v(α) is the α-quantile of F(·). Hence, from Lemma 2 the quantiles of G1 (·) and G2 (·) satisfy (5)

I1 − 1 I2 − I1 I2 − 1 I1 − I2 b2 (α) + b < b1 (α) < b2 (α) < b1 (α) + b I2 − 1 I2 − 1 I1 − 1 I1 − 1

for any α ∈ (0 1]. Equivalently, let Gjk (·) denote the distribution of [(Ik − 1)bj + (Ij − Ik )b]/[Ij − 1], where j k = 1 2 and bj = sj (v).12 Thus, the lower bound of the support of Gjk (·) is b and we have G21 (·) ≺b G1 (·) ≺b G2 (·) ≺b G12 (·). 10 Identification of the bidders’ utility function when the equilibrium strategies are nonincreasing in competition is discussed in Section 4.4. 11 See Barrett and Donald (2003) for consistent tests of stochastic dominance. 12 When j = k, Gjk = Gj (·).

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When the number I of bidders can take more than two values, the previous results imply several testable stochastic dominance relations among the observed bid distributions. Several of them are actually redundant. For instance, suppose that I ∈ [I I] with 2 ≤ I < I < ∞. The above statement implies that there are 4[1 + 2 + · · · + (I − I)] = 2(I − I)(I − I + 1) stochastic dominance relations. The next corollary shows that there are at most 2(I − I + 1) relevant relations. COROLLARY 1: Suppose that I ∈ I ≡ [I I] with 2 ≤ I < I. Under the previous assumptions, the quantiles b(α I) of the equilibrium bid distribution G(·|I) satisfy I −1 1 max b(α I − 1) b(α I + 1) + b I I I −1 1 < b(α I) < min b(α I + 1) b(α I − 1) − b I −2 I −2 for any α ∈ (0 1] and any I ∈ [I I].13 Equivalently, let b(I) denote the equilibrium bid with I bidders. Let GI (·) denote the distribution of the maximum of b(I − 1) and [(I − 1)b(I + 1) + b]/I, and let GI (·) denote the distribution of the minimum of b(I + 1) and [(I − 1)b(I − 1) − b]/(I − 2). Hence, GI (·) ≺b G(·|I) ≺b GI (·) for any I ∈ [I I]. Given two numbers of bidders I2 > I1 , we now turn to the nonparametric x identification of [U(·) F(·)] or, equivalently, [λ(·) F(·)] as U(x) = exp 1 1/λ(t) dt using the normalization U(1) = 1. The key idea is to use the invariance of the quantile v(α) for two different numbers of bidders I1 and I2 . Specifically, using (4) leads to the compatibility conditions 1 1 α α −1 −1 b2 (α) + λ (6) = b1 (α) + λ I2 − 1 g2 (b2 (α)) I1 − 1 g1 (b1 (α)) for all α ∈ [0 1]. We then show that there exists a unique inverse function λ−1 (·) defined on the range R1 of the function R1 (α), where α ∈ [0 1] and (7)

Rj (α) =

α 1 Ij − 1 gj [bj (α)]

for j = 1 2. Note that the range Rj of Rj (·) is of the form [0 r j ] with 0 < r j < ∞ because gj (·) is bounded away from zero and is continuous on [0 bj ] by Definition 3 and Lemma 1. Moreover, Rj (α) = λ[v(α) − sj (v(α))] 13

Obviously, b(· I − 1) is dropped when I = I, while b(· I + 1) is dropped when I = I.


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from (4). Thus, identifying λ−1 (·) on Rj is equivalent to identifying λ(·) on the range of the markdown/rent v − sj (v), where v ∈ [v v]. Because s1 (·) < s2 (·) on (v v] by Lemma 2, we have R1 (·) > R2 (·) on (0 1]. Thus, r 2 < r 1 so that R2 ⊂ R1 . Hence, λ(·) is identified nonparametrically on the largest set of possible markdowns [0 maxv∈[vv] v − s1 (v)].14 The next proposition provides explicit expressions for λ(·) and F(·). PROPOSITION 3: Under the previous assumptions, λ−1 (·) is identified nonparametrically on R1 . Specifically, λ−1 (0) = 0 and for any u0 ∈ R1 \ {0}, λ−1 (·) is given by λ−1 (u0 ) =

+∞

b(αt )

t=0

where b(α) = b2 (α) − b1 (α) and the sequence {αt } is strictly decreasing with 0 < αt ≤ 1 satisfying the nonlinear recursive relation R1 (αt ) = R2 (αt−1 ) with initial condition R1 (α0 ) = u0 . Moreover, F(·) is identified nonparametrically on [v v] with F(·) = Gj [ξj−1 (·)] for j = 1 2. Our proof is constructive and shows that the sequence {αt } exists though it is not necessarily unique. When R1 (·) is strictly increasing, that is, when the markdown is strictly increasing in v, such a sequence is unique. When R1 (·) is not strictly increasing, the sequence {αt } may not be unique, but all ∞ such sequences must lead to the same value for t=0 b(αt ), which then de−1 fines λ (u0 ) uniquely. The construction of {αt } is illustrated in Figure 1, which displays the equilibrium strategies s1 (·) < s2 (·). For α0 ∈ (0 1], consider the α0 -quantile v(α0 ) of F(·). The markdown v(α0 ) − b1 (α0 ) is the sum of b(α0 ), which is known, and λ−1 [R2 (α0 )], which is unknown. The latter is equal to the markdown v(α1 ) − b1 (α1 ), which is also the sum of b(α1 ) and λ−1 [R2 (α1 )]. Continuing this construction gives the sequence {αt } and determines λ−1 [R2 (α0 )] as the infinite series of known differences in bid quantiles (αt ). An important related question is to characterize all the restrictions on the observed equilibrium bid distributions that arise from the auction model with exogenous participation. In particular, it is useful to assess whether the observed bid distributions, which vary with I, can be rationalized by a structure [U(·) F(·)] that is independent of I. In other words, these restrictions allow one to test the model and its assumptions. Violation of one of these restrictions leads to reject the model for explaining the observed bids. In particular, it could mean that the independence of F(·) from I is not justified. Lemma 3 provides such restrictions when I takes two different values I2 > I1 . 14 In general, maxv∈[vv] v − sj (v) = v − sj (v). On the other hand, if the markdown v − sj (v) is increasing in v, then maxv∈[vv] v − sj (v) = v − sj (v). Moreover, Rj (·) would be increasing in α and r j = Rj (1).

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FIGURE 1.—Identification with s(·) increasing in competition.

LEMMA 3: Let I = {I1 I2 } with I1 < I2 . Let Gj (· ·) be the joint distribution of (b1 bIj ), j = 1 2. The equilibrium bid distributions Gj (·), j = 1 2, are rationalized by a structure [U(·) F(·)] that is independent of I if and only if the following conditions hold: Ij (i) For each j = 1 2, Gj (b1 bIj ) = i=1 Gj (bi ), where Gj (·) = G(·|Ij ) with support of the form [b bj ] and {G(·|I); I ∈ I } ∈ GR . (ii) The α-quantiles of G1 (·) and G2 (·) satisfy b1 (α) < b2 (α) for α ∈ (0 1], that is, G1 (·) ≺b G2 (·). (iii) ∃λ(·) : R+ → R+ with R + 1 continuous derivatives on [0 +∞), λ(0) = 0, and λ (·) ≥ 1 such that (a) the compatibility condition (6) is satisfied for every α ∈ [0 1] and (b) for Ij ∈ I , ξj (·) > 0 on [b bj ], where ξj (b) = b + λ−1 [Gj (b)/((Ij − 1)gj (b))]. Unlike Lemma 2, which only provides some (testable) implications, Lemma 3 characterizes all the theoretical restrictions imposed by the model with an exogenous bidder’s participation. Relative to the general case of Section 2 in which F(·) can vary with I, the set of bid distributions that can be rationalized is much reduced because of the restrictions (ii) and (iii)(a). Together, Proposition 3 and Lemma 3 say that exogenous participation allows one to identify and test the model in sharp contrast to Propositions 1 and 2 for the


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model without the exclusion restriction. As in Corollary 1, Lemma 3 can be extended straightforwardly to the case where I ∈ I ≡ [I I]. Specifically, (i) and (iii)(b) hold, while (ii) and (iii)(a) still hold for all pairs (Ij Ik ) ∈ I , k = j. 3.2. Endogenous Participation From an empirical point of view, one can expect that private values depend on the number of bidders, as higher valued objects may attract more bidders. In this subsection, we introduce additional variables and possible unobserved ones that can explain auctioned objects’ heterogeneity and bidders’ participation. In particular, bidders’ participation may be endogenous and some of these additional variables can play the role of instrumental variables through exclusion restrictions. We first consider the case of unobserved heterogeneity in bidder’s participation. Let W be a vector of observed variables that characterizes heterogeneity across auctioned objects. These variables are assumed to affect both bidders’ private value distribution and bidders’ participation. Bidders’ participation is modeled as I = I(W ε), where ε can be interpreted as a term of unobserved heterogeneity or as a traditional error term. We assume v ⊥ ε|W , namely bidders’ private values are independent of ε given the object characteristics W . This assumption translates into the exclusion restriction F(v|W ε) = F(v|W ), while the observed bid distribution is G(b|I W ) since b = s(v U F I). This model is similar to the one in the previous subsection since bidders’ private values are independent of I given W . Thus the difference between the two models is the introduction of the vector of conditioning variables W . This exclusion restriction allows one to exploit variations in bidding behavior under two competitive environments, that is, I2 > I1 , at W given. Proposition 3 applies and the pair [U(·) F(·|·)] is nonparametrically identified, while the quantile becomes bj (α W ). Regarding Lemma 3, the bids are now conditionally independent given W in (i), while the rest extends straightforwardly. The term of unobserved heterogeneity ε may, however, affect private values because ε can capture some unobserved characteristics, such as quality, which also influence bidders’ participation. This leads to a model of endogenous participation. The introduction of additional variables or instruments Z combined with appropriate exclusion restrictions solves this problem. Specifically, bidders’ participation is modeled as I = I(W Z ε), while we assume v ⊥ Z|(W ε), namely the bidders’ private values are independent of Z given (W ε). This translates into the exclusion restriction F(v|W Z ε) = F(v|W ε). Hence, the variables Z can be viewed as instruments. This model corresponds to an endogenous number of bidders, as the unobserved heterogeneity ε affects both bidders’ private values and bidders’ participation.15 15 Haile, Hong, and Shum (2003) adopted a similar framework to test for common value in first-price sealed-bid auctions when I is endogenous.

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This model is reminiscent of entry models used recently in the empirical auction literature. Bajari and Hortacsu (2003) modeled bidders’ entry in eBay coin auctions within a common value framework. Their empirical results show that the variables that explain bidder entry are the appraisal value of the auctioned object (W ), the reserve price for the auctioned object (Z1 ), and the seller’s reputation (Z2 ), while the bidders’ private signal distribution depends on W only. Athey, Levin, and Seira (2004) modeled logger entry in U.S. Forest Service sealed-bid auctions and found that entry depends on road costs (Z1 ) and scale sale (Z2 ) in Northern sales, and on logging costs (Z1 ), road costs (Z2 ), density of timber (Z3 ), and timber volume (Z4 ) in California sales in addition to several variables (W ) that affect the private values. Krasnokutskaya and Seim (2005) modeled firms’ participation in road construction procurements and found that variables that affect participation but not firms’ costs are the distance of the firm to the project location (Z1 ) and the firms’ work load or backlog (Z2 ). As suggested in these papers, a reduced form analysis of the number of bidders and the observed bids can help determine the variables W and the instruments Z. The next result shows how the availability of instruments can help identify the model. COROLLARY 2: The structure [U F] with endogenous participation and unobserved heterogeneity is identified under the exclusion restriction F(·|W Z ε) = F(·|W ε) if either the following conditions hold: (i) ε = I − E[I|W Z]. (ii) Z = m(X ε), where m(· ·) is strictly increasing in ε with X ⊥ ε and X ⊆ W . The difficulty lies in that ε is unobserved, while it is needed to recover the structural distribution F(·|· ·). The two cases of Corollary 2 allow the unobserved heterogeneity ε to be continuous. Case (i) assumes that the unobserved heterogeneity ε is the residual in the nonparametric regression of I on (W Z), that is, I is a “sufficient” statistic for ε given (W Z) following Campo, Perrigne, and Vuong (2003) since F(·|Z W ε) = F(·|Z W I). The argument is then as follows. The observed bid distribution satisfies G(b|W Z ε) = G(b|I W ε) as s(·) = s(· I W , ε), where v ∼ F(v|W Z ε) = F(v|W ε) and I = I(W Z ε). The parallel with Proposition 3 appears, as we can exploit variations in bidding behavior under two competitive environments, while the latent distribution remains the same at (W ε) given, where ε can be recovered as the residual from the nonparametric regression of I on (W Z). Proposition 3 applies and [U F] is nonparametrically identified, while the quantile becomes bj (α W ε). Regarding Lemma 3, the bids are now conditionally independent given (W ε) in (i), while the rest extends straightforwardly.16 16 Endogenous entry can be used to solve the problem of unobserved heterogeneity. For instance, consider the risk neutral model with endogenous entry I = E(I|W )+ε, where the bidders’


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Case (ii) requires some additional variables X, as X cannot be a subset of W . Otherwise, I would be a function of (W ε) only and there would be no exclusion restriction that can be used to identify the model. The remaining assumptions follow Matzkin (2003) for the identification of the function m(· ·) with a nonadditive error term. Specifically, given X ⊥ Z and the normalization that ε is (say) uniformly distributed on [0 1], we have Pr(Z ≤ z|X = x) = Pr(ε ≤ m−1 (x z)) = m−1 (x z), where m−1 (x ·) is the inverse of m(x ·). In particular, ε is recovered as m−1 (X Z). A similar argument as in case (i) applies, showing that F(·|W ε) is identified. Examples of X include the number of the seller’s past transactions in eBay auctions when Z is the seller’s reputation (Bajari and Hortacsu (2003)) or the timber tract altitude and distance from a paved road when Z is the road cost or the logging cost (Athey, Levin, and Seira (2004)). 4. EXTENSIONS This section extends our results to a reserve price, affiliated private values, and asymmetric bidders in private values and/or preferences. Except for the first part of Proposition 7, we only sketch the proofs of Propositions 4–7 in the text. 4.1. Reserve Price A reserve price p0 > v introduces a truncation in the observed bid distribution as only the I ∗ bidders who have a value above p0 bid at the auction. Let G∗ (·|I) be the truncated bid distribution on [p0 b(I)]. We observe I ∗ the number of active bidders, I ∗ ≤ I I ∈ I . Because G∗ (b∗ |I) = [F(v|I) − F(p0 |I)]/[1 − F(p0 |I)] for b∗ ∈ [p0 b(I)], elementary algebra gives the inverse equilibrium strategy 1 G∗ (b∗ |I) 1 1 F(p0 |I) ∗ −1 (8) + v=b +λ I − 1 gj∗ (b|I) I − 1 g∗ (b∗ |I) 1 − F(p0 |I) ≡ ξ(b∗ U G∗ I F(p0 |I)) Definitions 1, 2, and 3 remain the same with the exception that p0 replaces b(I) in Definition 3. Moreover, because limb↓p0 g∗ (b|I) = +∞ as s (p0 ) = 0 from (2), derivatives and limits are infinite at p0 in Definition 3.17 Given that private value distribution is F(v|W ε). As above, ε can be recovered as ε = I − E(I|W ). Thus, F(·|W ε) is nonparametrically identified following Guerre, Perrigne, and Vuong (2000, Theorem 1). This method differs from Krasnokutskaya (2003), who identified the term of unobserved heterogeneity in a private value model F(v|W ε) with exogenous participation I = I(W ) using a multiplicative decomposition of private values and applying Li and Vuong’s (1998) results. 17 To avoid infinite derivatives/limits at p0 , we can consider the bid transformation used in Guerre, Perrigne, and Vuong (2000, Section 4), in which case rationalization and identification are based on the density of the transformed bids.

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I ∗ is binomially distributed with parameters [I 1 − F(p0 |I)], I and F(p0 |I) are identified under the assumption that I is constant across subsets of auctions. PROPOSITION 4: Any structure [U F] ∈ UR × FR with a reserve price is not identified. On the other hand, the structure [U F] with the exclusion restriction F(·|I) = F(·) is with U(·) identified on [0 maxv∈[vv] v − s1 (v)] and F(·) identified on [p0 v]. The joint bid distribution G∗ (· ·) is rationalized if and only if Lemma 1 is satisfied with GR and ξ(·) as defined above. From this rationalization result, any G∗ (·|I) ∈ GR I ∈ I can be rationalized by a risk aversion structure [U F] ∈ UR × FR . It is then straightforward to show that the structure [U F] ∈ UR × FR is not identified. Under the exogeneity of the number of bidders, we assume that we observe at least two levels of potential bidders I1 < I2 . Let G∗j (·) be the truncated bid distribution on [p0 bj ]. Lemma 2 still holds with a reserve price, as the latter simply reduces the shading relative to the case with no reserve price. In view of (8), Rj (α) becomes 1 F(p0 ) 1 α + Rj (α) = Ij − 1 gj∗ [b∗j (α)] 1 − F(p0 ) for j = 1 2. Note that Rj (α) differs from (6) by the additional term F(p0 )/(1 − F(p0 )). As before, F(p0 ) and the number of potential bidders Ij are identified from the binomial distribution of the number of actual bidders. The problem reduces to identifying λ−1 (·) and F(·) on [0 r 1 ] and [p0 v], respectively. A simple extension of Proposition 3 shows that [λ−1 (·) F(·)] is nonparametrically identified on these intervals using the quantiles b∗j (α) of G∗j (·). Similarly, Lemma 3 and Corollary 2 can be readily adapted. 4.2. Affiliated Private Values The vector (v1 vI ) is distributed as F(· ·|I), which is exchangeable in its I arguments, affiliated, and defined on [v(I) v(I)]I . We follow the notations of Li, Perrigne, and Vuong (2002). Let GBi |bi (bi |bi I) be the probability that i has a bid larger than all his opponents conditional on his bid bi with Bi = maxk=i bk and bi = s(vi ). Without loss of generality, we can consider GB1 |b1 (·|· I), as all bidders are symmetric. To simplify the notation, we omit the index 1. The inverse equilibrium strategy becomes GB|b (b|b I) ≡ ξ(b U G I) for all b ∈ [b(I) b(I)] (9) v = b + λ−1 gB|b (b|b I) with the joint bid distribution G(· ·|I). Definitions 1 and 2 remain the same except that F(· ·|I) is R + I continuously differentiable following


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Li, Perrigne, and Vuong (2000, 2002). Note that GB|b (·| · |I)/gB|b (·| · |I) = GB×b (· ·|I)/gBb (· ·|I), where GB×b (· ·|I) ≡ ∂GBb (· ·|I)/∂b and gBb (· ·|I) are the b-derivative of the joint c.d.f. and the joint density of (B b), respectively. Let GR be the set of exchangeable and affiliated distributions {G(· ·|I) I ∈ I } with R continuously differentiable densities such that GB×b (b b|I)/gBb (b b|I) is R + 1 continuously differentiable in b ∈ [b(I) b(I)] and strictly positive on (b(I) b(I)]. PROPOSITION 5: Any structure [U F] ∈ UR × FR with affiliated values is not identified. On the other hand, the structure [U F] with the exclusion restriction F(· ·|I) = F(· ·) is with U(·) identified on [0 maxv∈[vv] v − s1 (v)] and F(· ·) identified on [v v]I . The joint bid distribution G(· ·|·) is rationalized if and only Lemma 1 is satisfied with GR and ξ(·) as defined above. From this rationalization result, any G(· ·|·) ∈ GR can be rationalized by some risk aversion structure [U F] ∈ UR × FR . We can then show that the structure [U F] ∈ UR × FR is not identified by using a similar argument as in Proposition 2, where G(·|I)/[(I − 1)g(·|I)] is replaced by GB×b (· ·|I)/gBb (· ·|I) in view of (4) and (9). Under two competitive environments I1 < I2 and exogenous bidder participation, the vector (v1 vIj ) is distributed as Fj (· ·), which is exchangeable and affiliated. Proposition 5 uses the property that F1 (· ·) and v v F2 (· ·) are related by f1 (v1 vI1 ) = v · · · v f2 (v1 vI1 vI1 +1 vI2 ) dvI1 +1 · · · dvI2 , that is, f1 (· ·) is the marginal of f2 (· ·). Hence, Fj (· ·) has support [v v]j . Assume that the structures [U Fj ], j = 1 2, satisfy s1 (v) < s2 (v), that is, competition renders bidding more aggressive.18 Here again, the exogeneity of the number of bidders allows one to identify λ−1 (·) on [0 r 1 ] by exploiting variations in the number of bidders, where Rj (·) = j j GB×b (bj (α) bj (α))/gBb (bj (α) bj (α)) with bj (α) the α-quantile of the marginal bid density gj (·) associated with Ij bidders. Specifically, Proposition 3 and Lemma 3 extend to this case, while Corollary 2 requires that f1 (· ·|W ε) be the marginal of f2 (· ·|W ε). 4.3. Asymmetric Bidders Asymmetry among bidders, which is known ex ante to all participants, can arise from two different sources, namely from (i) different distributions of private values and/or (ii) different utility functions. We consider these cases separately. Hereafter, it is assumed that bidders’ identities are known. 18

Section 4.4 relaxes this assumption. The competition effect is unclear with affiliated private values, as some distributions F(· ·) may lead to bidding strategies decreasing in the number of bidders as shown by Pinkse and Tan (2005).

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Asymmetry in Private Values

Given I ∈ I , the joint private value distribution is F(· ·|I) = i Fi (·|I) with each Fi (·|I) satisfying Definition 2 on the support [v(I) v(I)]. To simplify, we assume that all Fi (·|I) have the same support. Let FR be the set of such distributions F(· ·|I) when I ∈ I . Because of the boundary conditions si (v(I)) = v(I) and si (v(I)) = sj (v(I)), j = i, bidder i’s distribution Gi (·|I) is defined on [b(I) b(I)] for all i = 1 I. Following Campo, Perrigne, and Vuong (2003), we have 1 −1 (10) vi = bi + λ Hi (bi |I) gk (·|I) ≡ ξi (bi U G I) where Hi (·|I) = Gk (·|I) k=i for i = 1 I. Let GR be the set of distributions G(· ·|·) such that each marginal distribution Gi (·|·) satisfies Definition 3 with G(b|I)/g(b|I) replaced by 1/Hi (b|I) in (v). PROPOSITION 6: Any structure [U F] ∈ UR × FR with asymmetry in private values is not identified. On the other hand, the structure [U F] with the exclusion restriction Fi (·|I) = Fi (·), I ∈ I , is partially identified with U(·) identified on [0 maxv∈[vv]i=1I0 v −si (v)] and Fi (·) i = 1 I0 , identified on [v v], where I0 is the number of bidders participating in both auctions. For the remaining bidders, Fi (·) is identified for the quantiles satisfying Ri1 (α) ∈ [0 maxk=1I0 r k1 ], where Ri1 (α) is defined below. The bid distribution G(· ·|·) is rationalized if and only if Lemma 1 is satisfied with GR and ξi (·), i = 1 I, I ∈ I , as defined above. Hence, any G(· ·|·) ∈ GR can be rationalized by a risk aversion structure [U F] ∈ UR × FR . It is then straightforward to show that any structure [U F] ∈ UR × FR is not identified. Under two competitive environments I2 > I1 ≥ 2 and exogenous bidders’ participation, each bidder i has a private value distribution independent of I. Thus, all Fi (·)’s are defined on the same support [v v]. Because our results under exclusion restrictions exploit the difference in bidding behavior under two competitive environments, it is crucial that at least one bidder participates in both auctions.19 For instance, when I1 = 2 and I2 = 3, at least one of the bidders in the two bidder auction must participate in the three bidder auction. In the case of asymmetry, because of the complexity of the system of differential 19 In empirical studies, a few bidders’ types are entertained as in Campo, Perrigne, and Vuong (2003), Athey, Levin, and Seira (2004), and Flambard and Perrigne (2006). In this case, it is crucial that at least one type is represented in both auctions.


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equations defining the equilibrium strategies, it is difficult if not impossible to prove that equilibrium strategies are increasing in competition. Nevertheless, because of the independence of private values, we postulate that equilibrium strategies are increasing in I due to the competition effect. Let sij (·) denote the equilibrium strategy for bidder i = 1 Ij , when the number of bidders is Ij , j = 1 2. The boundary conditions are si1 (v) = si 2 (v) = v for i = 1 I1 , and i = 1 I2 and sij (v) = si j (v) for i i = 1 Ij , j = 1 2. Bidders 1 I0 participate in both auctions, where 1 ≤ I0 ≤ I1 . Let vi (α) and bij (α) be the α-quantiles of Fi (·) and Gij (·) = Gi (·|Ij ), respectively. Instead of (4), we now have at the α-quantile (11)

vi (α) = bij (α) + λ−1 (Rij (α))

(i = 1 Ij , j = 1 2)

for α ∈ [0 1], where Rij (α) = 1/Hij (bij (α)) takes values in the range Rij = [0 r ij ] with Hij (·) = k=i gkj (·)/Gkj (·). A straightforward extension of Proposition 3 shows that λ−1 (·) is identified on [0 maxk=1I0 r k1 ], while [F1 FI0 ] are identified on [v v]. Because λ−1 (·) is identified on [0 maxk=1I0 r k1 ], which may be a strict subset of [0 r i1 ], where i refers to a remaining bidder, his private value distribution may not be identified everywhere, justifying the partial identification result of Proposition 6. Lemma 3 also extends where the compatibility conditions (6) now hold for each of the I0 bidders. Moreover, Corollary 2 applies provided the IV exclusion restrictions Fi (·|W Z ε) = Fi (·|W ε) for i = 1 I0 hold. Asymmetry in Preferences We consider structures of the form [U F] ∈ URI × FR with U = {(U1

I UI ) ∈ URI ≡ i=1 UR I ∈ I } ∈ URI . Given I ∈ I , we obtain for i = 1 I, 1 −1 ≡ ξi (bi Ui G I) (12) vi = bi + λi Hi (bi |I) where λi (·) = Ui (·)/Ui (·) and Hi (·|I) = k=i gk (·|I)/Gk (·|I). For each I, the boundary conditions s1 (v) = · · · = sI (v) = v and s1 (v) = · · · = sI (v) give a common support [b(I) b(I)] for the bid distributions across bidders. Let GR be the set of distributions G(· ·|·) such that each marginal distribution Gi (·|·) satisfies Definition 3 with G(b|I)/g(b|I) replaced by 1/Hi (b|I) in (v). PROPOSITION 7: Any structure [U F] ∈ URI × FR with asymmetry in preferences satisfying Hi (·|I) < 0, i = 1 I, I ∈ I , is not identified.20 On the other 20 The assumption Hi (·|I) < 0 corresponds to an increasing markup viα − biα in α from (12). If all the bid distributions G1 GI are log-concave, this assumption is automatically satisfied. Our requirement is weaker, as some bid distributions may not be log-concave. Log-concavity is usually verified on data.

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hand, the structure [U F] with the exclusion restriction F(·|I) = F(·) is with Ui (·) identified on [0 maxv∈[vv] v − si (v)] for i = 1 I and F(·) identified on [v v]. Because the α-quantiles (b1 (α) bI (α)) all correspond to the same αquantile v(α), (12) gives for an arbitrary pair (i k) of bidders the compatibility conditions 1 1 −1 −1 bk (α) + λk (13) = bi (α) + λi Hk (bk (α)|I) Hi (bi (α)|I) for all α ∈ [0 1]. These conditions reduce the set of bid distributions that can be rationalized relative to the symmetric case and may help in identification. Specifically, the bid distribution G(· ·|·) is rationalized if and only if (i) Lemma 1 is satisfied with GR and ξi (·), i = 1 I, as defined above and (ii) the compatibility conditions (13) are satisfied for any pair (i k) of bidders.21 Despite these compatibility conditions, the nonparametric model is still not identified. Because it is more involved than in previous cases, we provide a proof of such a result in the Appendix.22 Under exogenous bidders’ participation, we assume again that I0 bidders participate in both auctions, where I0 ≥ 1, and that equilibrium strategies are increasing in competition. Equation (11) takes a similar form with v(α) and −1 λ−1 i (·) replacing vi (α) and λ (·). A similar argument as in Proposition 3 ap−1 plies for identifying λi (·) nonparametrically on [0 r i1 ] for i = 1 I0 from which we can identify F(·) on [v v]. Since F(·) is identified everywhere, following a similar argument as in Lu and Perrigne (2008), the λ−1 j (·)’s for the remaining bidders are identified from (12). Specifically, because the v(α)’s are identified, we can recover the remaining λ−1 j (·) on [0 r j ]. Again Lemma 3 extends with the compatibility conditions (6) holding for each of the I0 bidders −1 and λ−1 i (·) replacing λ (·), to which we need to add the compatibility conditions (13) that must hold for any pair of bidders in each auction. Moreover, Corollary 2 applies. Asymmetry in Both Preferences and Private Values This third case involves asymmetry in both private value distributions and preferences. Given the above results, the model is not identified. We then consider exogenous bidders’ participation. Thus, (11) takes a similar form with −1 λ−1 i (·) replacing λ (·). Despite the complexity of this case, which has not been 21 Though similar in spirit, the compatibility conditions (13) apply within each auction, while the compatibility conditions (6) apply across auctions. 22 On the other hand, if bidder 1 participates in all auctions and his utility U1 (·) is known, the nonparametric model URI × FR becomes identified, as (12) for i = 1 allows us to identify F(·|·). Thus, evaluated at the α-quantile, (12) for i = 1 allows us to identify λi (·) on [0 maxα (vα − biα )]. This result is useful when bidders differ by their sizes: “large” bidders, who participate in all auctions, are risk neutral.


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considered to our knowledge, it can be shown that the structure [λi Fi ] is nonparametrically identified for the I0 ≥ 1 bidders who participate in both auctions. Specifically, we can apply Proposition 3 to these bidders to identify λ−1 i (·) and Fi (·) nonparametrically on [0 r i1 ] and [v v], respectively. On the other hand, we cannot identify the pair [λ−1 i (·) Fi (·)] for the other bidders. Again Lemma 3 extends with the compatibility conditions (6) holding for each of the −1 common bidders, where λ−1 i (·) replaces λ (·). Corollary 2 also applies. 4.4. Bidding Strategies Nonincreasing in Competition So far we have assumed that equilibrium strategies are increasing in the number of bidders. This may not be always the case. For instance, affiliated private values may lead to equilibrium strategies that are decreasing in competition for some F(· ·|·). In this section, we discuss how our results extend when the equilibrium strategies are nonincreasing in competition. As before, let I1 < I2 . We assume that for a bidder participating in both auctions, his equilibrium strategies s1 (·) and s2 (·) intersect a finite number of times at most. This excludes the case where these strategies are identical on some open interval of private values. The nonparametric identification of the model is established through the following steps: STEP 1: From the knowledge of G1 (·) and G2 (·), we can identify the positive values 0 < α∗1 < · · · < α∗K ≤ 1 at which the bid distributions and hence the bidder’s equilibrium strategies s1 (·) and s2 (·) intersect, that is, b1 (α∗k ) = b2 (α∗k ). STEP 2: Let sj (·) < sj (·) on (v v(α∗1 )), j j = 1 2. By Proposition 3, for any +∞ α0 ∈ (0 α∗1 ) , we can identify λ−1 (u0 ) as t=0 |b(αt )|, where u0 = Rj (α0 ). By −1 continuity of λ (·), we can also identify λ−1 (Rj (α∗1 )) which is also equal to λ−1 (Rj (α∗1 )) by the compatibility condition (6). Hence, λ−1 (·) is identified on [0 maxα∈[0α∗1 ] Rj (α)] = [0 max{maxα∈[0α∗1 ] Rj (α) maxα∈[0α∗1 ] Rj (α)}]. STEP 3: We have sj (·) < sj (·) on (v(α∗1 ) v(α∗2 )). For any α0 ∈ (α∗1 α∗2 ), we let u0 = Rj (α0 ) and by Proposition 3 we construct αt+1 recursively from the equation Rj (αt+1 ) = Rj (αt ) subject to αt+1 < αt . There are two possibilities: (i) If αt+1 ∈ [0 α∗1 ], we stop this sequence, as we switch t to values for which sj (·) ≤ sj (·). We have λ−1 (u0 ) = λ−1 (Rj (αt+1 )) + s=0 |b(αs )|. But λ−1 (Rj (αt+1 )) = λ−1 (Rj (αt+1 )) + |b(αt+1 )|, where λ−1 (Rj (αt+1 )) is identified +∞ from Step 1 and hence equal to r=0 |b(αr )| for some decreasing sequence +∞ t {αr } with α0 = αt+1 . Thus λ−1 (u0 ) = r=1 |b(αr )| + s=0 |b(αs )|, which gives +∞ λ−1 (u0 ) = t=0 |b(αt )| by letting αr ≡ αt+r . Figure 2 illustrates this case, where j = 1, j = 2, and t + 1 = 2. (ii) If αt+1 remains in (α∗1 α∗2 ) for all t, the sequence {αt } will converge +∞ to α∗1 . Taking the limit gives λ−1 (u0 ) = λ−1 (Rj (α∗1 )) + s=0 |b(αs )|. But −1 ∗ −1 ∗ λ (Rj (α1 )) = λ (Rj (α1 )), where the latter is identified from Step 2. Figure 3 illustrates this case with j = 1 and j = 2.

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FIGURE 2.—Identification with s(·) nonincreasing in competition: case (i).

By continuity of λ−1 (·), we can also identify λ−1 (Rj (α∗2 )) which is also equal to λ−1 (Rj (α∗2 )) by the compatibility condition (6). Hence, at the end of Step 3, λ−1 (·) is identified on [min{minα∈[α∗1 α∗2 ] Rj (α) minα∈[α∗1 α∗2 ] Rj (α)} max{maxα∈[α∗1 α∗2 ] Rj (α) maxα∈[α∗1 α∗2 ] Rj (α)}]. By combining Steps 2 and 3, λ−1 (·) is identified on [0 max{maxα∈[0α∗2 ] Rj (α) maxα∈[0α∗2 ] Rj (α)}]. STEP 4: For any k ≥ 2 and α0 ∈ (α∗k α∗k+1 ), we repeat Step 3 and its two ∗ possibilities. If αt+1 sequence and we have λ−1 (u0 ) = t ∈ [0 αk ], we stop the −1 −1 λ (Rj (αt+1 )) + s=0 |b(αs )|, where λ (Rj (αt+1 )) is identified from pre +∞ vious steps. Thus λ−1 (u0 ) = t=0 |bαt | as in Step 3(i). If αt+1 remains in (α∗k α∗k+1 ), Step 3(ii) applies and λ−1 (u0 ) is identified. As before, by continuity, λ−1 [Rj (α∗k+1 )] = λ−1 [Rj (α∗k+1 )] is identified. Applying a similar argument, combination of the various steps allows one to identify λ−1 (·) on [0 max{maxα∈[0α∗k+1 ] Rj (α) maxα∈[0α∗k+1 ] Rj (α)}] and hence on [0 max{maxα∈[01] Rj (α) maxα∈[01] Rj (α)}] when k + 1 = K.


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FIGURE 3.—Identification with s(·) nonincreasing in competition: case (ii).

The identification of the bidder’s private value distribution F(·) follows as in Proposition 3. The above procedure shows that assuming bidding strategies increasing in competition is not necessary to identify the model nonparametrically under exclusion restrictions. The construction of the sequence {αt } is more involved. 5. CONCLUSION This paper addresses the problem of nonparametric identification of bidders’ utility function(s). We show that the auction model with risk aversion is not identified and that it imposes weak restrictions on observables. This implies that the auction model with risk averse biddders is not testable in view of bids only. In view of these results, we exploit exclusion restrictions to identify the bidders’ utility function and their private value distribution nonparametrically. The exclusion restriction takes the form of an exogenous bidders’ participation as commonly assumed in theory and the empirical literature. When bidders’ participation is endogenous, we rely on the existence of instruments that affect the bidders’ participation but not their private values whether the unobserved heterogeneity appears additively or nonadditively in the equation for the number of bidders. The results are general, as they extend to a reserve

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price, affiliated private values, and asymmetric bidders. We also provide some conditions that must be verified by the data and can be used for model testing under exclusion restrictions. More generally, identifying risk aversion is an important issue in the analysis of microeconomic data. In this respect, it would be interesting to investigate how similar ideas can be exploited to identify agents’ utility function nonparametrically in other economic contexts such as in insurance. Based on our results, d’Haultfoeuille and Février (2007) applied a similar recursive construction of a quantile sequence to identify the principal’s objective function, the agent’s cost function, and the type distribution under exclusion restrictions in a principal–agent model. A nonparametric estimation method for [U F] or, equivalently, [λ−1 F] clearly needs to be developed. To clarify ideas, we consider the base case of Section 3.1. Consider first the estimation of λ−1 (·). A strategy could rely on the sequence of quantiles {αt } in Proposition 3. Specifically, we define Rˆ j (α) as the estimate of Rj (α) in (7), where gj (bj (α)) is replaced by its estimate gˆ j (bˆ j (α)) with gˆ j (·) a nonparametric estimate of the bid density gj (·) and bˆ j (α) a nonparametric estimate of the bid α-quantile bj (α). For any u0 ∈ R1 \ {0}, we estimate (αˆ 0 αˆ t ) by solving Rˆ 1 (α0 ) = u0 and Rˆ 1 (α1 ) = Rˆ 2 (αˆ 0 ) Rˆ 1 (αˆ t ) = Rˆ 2 (αˆ t−1 ) recursively. An estimator of λ−1 (u0 ) is λˆ −1 (u0 ) =

tL

ˆ αˆ t ) b(

t=0

where tL increases appropriately with the number L of auctions in the sample. The asymptotic properties of such an estimator are unknown. The difficulty relies on the serial correlation of the sequence {αˆ t } and on the accumulation of the errors in the estimation of λˆ −1 (u0 ). In addition to being computationally demanding, the rate of convergence of λ−1 (·) might be slow. A second strategy could exploit the compatibility conditions (6) in Lemma 3 to estimate λ−1 (·) nonparamerically. Given some quantiles α1 αnL with nL increasing appropriately with L, we estimate the quantiles bˆ j (αn ) and the bid ˆ n )) for j = 1 2 nonparametrically. We can densities at those quantiles gˆ j (b(α then use the method of sieves to estimate λ−1 (·) by minimizing the objective function nL 1 αn bˆ 2 (αn ) + λ−1 I2 − 1 gˆ 2 (bˆ 2 (αn )) n=1 2 αn 1 −1 ˆ − b1 (αn ) − λ I1 − 1 gˆ 1 (bˆ 1 (αn )) subject to λ−1 (·) ∈ L−1 , where L−1 is an appropriate set of increasing and smooth functions. See Chen (2007) for a survey on sieve estimation. Estab-


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lishing the asymptotic properties of such an estimation procedure is left for future research. In addition to having nonparametric estimates in the objective function, the main difficulty relies on controlling the increasing rate of the number nL of prescribed quantiles. From a practical point of view, this estimator is computationally convenient when L−1 is a finite dimensional linear space, in which case the above optimization problem can be solved by least squares. Moreover, such a method could be used in other auction models in which identification is based on similar compatibility conditions such as in the case of a procurement model with ex post cost uncertainty studied by Perrigne and Vuong (2008) in which λ−1 (·) is finitely parameterized. Regarding estimation of F(·), the private value density f (·) can be estimated from the estimated private values using (4) as in Guerre, Perrigne, and Vuong (2000) with λ−1 (·) replaced by its estimate λˆ −1 (·). The rate of convergence of fˆ(·) needs to be determined. APPENDIX This appendix contains the proofs of Lemmas 1–3, Propositions 1–3, the first part of Proposition 7 and Corollary 1. PROOF OF LEMMA 1: First, we prove that (i) and (ii) are necessary. Because bi = s(vi U F I) and the vi ’s are i.i.d., the bi ’s are also i.i.d. The fact that G(·|·) ∈ GR follows from Theorem 1, Definitions 1 and 2, and (3). To prove that (ii) is also necessary, consider (4), where λ(·) ≡ U(·)/U (·). Thus λ(·) is defined from R+ to R+ because λ(0) = limx↓0 λ(x) = 0, as noted after Definition 1. As U(·) admits R + 2 continuous derivatives on (0 +∞) with U (·) > 0 and as limx↓0 λ(r) is finite for r = 1 R + 1, then λ(·) has R + 1 continuous derivatives on [0 +∞). As λ (·) = 1 − λ(·)U (·)/U (·), we have λ (·) ≥ 1 because λ(·) ≥ 0, U (·) > 0, and U (·) ≤ 0. It remains to show that ξ (·) > 0. The equilibrium strategy must solve the differential equation (2). As (4) follows from (2), s(·) must satisfy ξ[s(v) U G I] = v for all v ∈ [v v]. We then obtain ξ(b U G I) = s−1 (b U F I). This implies ξ (·) = [s−1 (·)] > 0 using Theorem 1. Second, we show that (i) and (ii) are together sufficient. First, we construct a pair [U F] ∈ UR × FR . Let U(·) be such that λ(·) = U(·)/U (·) or U (·)/U(·) x = 1/λ(·). Integrating with the normalization U(1) = 1 gives U(x) = exp 1 1/λ(t) dt. We verify that U(·) ∈ UR . Because λ(·) admits R + 1 continuous derivatives on [0 +∞), then Definition 1(iii) is clearly satisfied. Moreover, in the neighborhood of zero, λ(t) ∼ λ (0)t with 1 ≤ λ (0) < ∞. 1 Thus x 1/λ(t) dt diverges to infinity, which implies that U(x) tends to zero as x ↓ 0. Define U(0) x = 0 so that U(·) is continuous on [0 +∞). Because U (x) = exp 1 1/λ(t) dt/λ(x), where λ(·) > 0 on (0 +∞), we have The second-order derivative gives U (x) = [−λ (x) + U (·) > x0 on (0 +∞). 2 1] exp 1 1/λ(t) dt/λ (x). Since λ (x) ≥ 1, U (·) ≤ 0 on (0 +∞). It remains to

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show that U(·) admits R + 2 continuous derivatives on (0 +∞). By assumption, λ(·) has R + 1 continuous derivatives on [0 +∞). It follows that U(·) admits R + 2 continuous derivatives on (0 +∞). Let F(·|I) be the distribution of X = b + λ−1 [G(b|I)/(I − 1)g(b|I)] conditional on I, where b ∼ G(·|I). We verify that F(·|·) ∈ FR . We have F(x|I) = Pr(X ≤ x|I) = Pr[ξ(b) ≤ x|I] = Pr[b ≤ ξ−1 (x)|I] = G[ξ−1 (x)|I], because ξ (·) > 0 by assumption. This implies F(·|I) = G[ξ−1 (·)|I] on [v(I) v(I)], where v(I) ≡ ξ(b(I)) = b(I) and v(I) ≡ ξ(b(I)) < ∞ by continuity of ξ(·). Because ξ(·) and G(·|I) are strictly increasing, F(·|I) is strictly increasing on its support [v(I) v(I)]. Moreover, ξ(·) is R + 1 continuously differentiable on [b(I) b(I)]. This follows from the definition of ξ(·), the R + 1 continuous differentiability of λ−1 (·) on [0 +∞), and the R + 1 continuous differentiability of G(·|I)/g(·|I) on [b(I) b(I)], which follows from Definition 3(iv) and (v). Thus F(·|I) = G[ξ−1 (·)|I] admits R + 1 continuous derivatives on [v(I) v(I)] because G(·|I) has R + 1 continuous derivatives on [b(I) b(I)]. It remains to show that f (·|I) > 0 on [v(I) v(I)]. This follows from f (·|I) = g[ξ−1 (·)|I]/ξ [ξ−1 (·)], where g(·|I) > 0 from Definition 3 and ξ (·) is finite on [b(I) b(I)]. Last, we show that the pair [U F] rationalizes G(·|·), that is, that G(·|I) = F[s−1 (· U F I)|I] on [b(I) b(I)], where s(· U F I) solves (2) with the boundary condition s(v(I) U F I) = v(I). By construction of F(·|I), G(·|I) = F[ξ(·)|I]. Thus, it suffices to show that ξ−1 (·) solves (2) with the boundary condition ξ−1 (v(I)) = v(I). The boundary condition is straightforward as ξ(b(I)) = b(I) = v(I). By construction of F(·|I), f (·|I)/F(·|I) = [ξ−1 (·)] × g[ξ−1 (·)|I]/G[ξ−1 (·)|I]. Thus ξ−1 (·) solves (2) if 1 = {(I − 1)g[ξ−1 (v)|I]λ[v − ξ−1 (v)]}/G[ξ−1 (v)|I] for all v ∈ [v(I) v(I)]. Making the change of variable v = ξ(b) and noting that ξ(b) − b = λ−1 [G(b|I)/(I − 1)g(b|I)] from the definition of ξ(·), it follows that ξ−1 (·) solves (2) with boundary condition ξ−1 (v(I)) = v(I). Q.E.D. PROOF OF PROPOSITION 1: In view of Lemma 1, it suffices to prove sufficiency. Specifically, it suffices to find a function λ(·) satisfying Lemma 1(ii), where G(·|·) ∈ GR . Denote G(b|I)/[(I − 1)g(b|I)] by ψ(b). Let minb∈[b(I)b(I)] ψ (b) = ψ , which is finite from Definition 3. If ψ > 0, any strictly increasing function λ(·) will satisfy ξ (·) ≥ 0, where ξ(b) = b + λ−1 (ψ(b)). It then suffices to take a λ(·) function that admits R + 1 continuous derivatives on [0 +∞) with λ(0) = 0 and λ (·) ≥ 1. If ψ < 0, we must find a strictly increasing and differentiable function λ(·) such that minb∈[b(I)b(I)] {(1/λ [λ−1 (ψ(b))]) × ψ (b)} > −1 to satisfy ξ (·) > 0 on [b(I) b(I)]. To satisfy the latter inequality, it suffices that the function λ(·) satisfies λ (·) ≥ 1 and ψ maxb∈[b(I)b(I)] 1/λ [λ−1 (ψ(b))] > −1, where the latter inequality is equivalent to ψ > −1/[maxb∈[b(I)b(I)] 1/λ [λ−1 (ψ(b))]]. But maxb∈[b(I)b(I)] 1/λ [λ−1 (ψ(b))] = maxx∈[0x] 1/λ (x) because x ≡ λ−1 (ψ(b)) takes


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its value between λ−1 [ψ(b(I))] = 0 and x = λ−1 [maxb∈[b(I)b(I)] ψ(b(I))] < +∞. Moreover, maxx∈[0x] 1/λ (x) = 1/ minx∈[0x] λ (x) ≡ 1/λ , where λ ≥ 1. Hence ψ > −λ , that is, 0 < −ψ < λ . Thus, λ(·) must have a sufficiently steep slope. To complete the proof, it suffices to take a λ(·) function that admits R + 1 continuous derivatives on [0 +∞) with λ(0) = 0. Q.E.D. PROOF OF PROPOSITION 2: Let [U F] ∈ UR × FR with bid distribution ˜ F] ˜ be such that U(·) ˜ = [U(·/δ)/U(1/δ)]δ G(·|·) ∈ GR by Lemma 1. Let [U ˜ with δ ∈ (0 1) and let F(·|I) be the conditional distribution given I of 1 G(b|I) ˜ ˜ G I) = b + λ˜ −1 ξ(b U I − 1 g(b|I) 1 G(b|I) = b + δλ−1 I − 1 g(b|I) = (1 − δ)b + δξ(b U G I) ˜ F] ˜ ∈ UR × FR . Because ξ(·) ˜ where b ∼ G(·|I). It is easy to check that [U is ˜ the weighted sum of two strictly increasing functions in b, then ξ(·) is strictly ˜ F] ˜ are obserincreasing. Hence, from Lemma 1, the structures [U F] and [U vationally equivalent, and [U F] is not identified. Q.E.D. PROOF OF LEMMA 2: (i) We first prove that s1 (v) < s2 (v) for any v ∈ (v v]. We have s2 (v) = s1 (v) = v. Moreover, from Theorem 1(ii) we have (A.1)

0 < sj (v) =

(Ij − 1)λ (0) 1 =1− < 1 (Ij − 1)λ (0) + 1 (Ij − 1)λ (0) + 1

where λ (0) ≥ 1. Thus 0 < s1 (v) < s2 (v) < 1. In particular, by continuity of sj (·) it follows that v < s1 (v) < s2 (v) for any v ∈ (v ε) for some ε satisfying v < ε ≤ v. The proof is now by contradiction. Suppose that s1 (v) ≥ s2 (v) for some v ∈ [ε v]. Because s1 (v) < s2 (v) for any v ∈ (v ε), the continuity of sj (·) would imply the existence of some v0 ∈ [ε v] such that s1 (v0 ) = s2 (v0 ). Moreover, for (at least) one of such v0 denoted v0∗ , the strategy s1 (·) must intersect the strategy s2 (·) from below, that is, s1 (vo∗ ) ≥ s2 (vo∗ ). From (2), we have sj (v0∗ ) = (Ij − 1)

f (v0∗ ) λ(v0∗ − sj (v0∗ )) F(v0∗ )

for j = 1 2, where f (v0∗ ) > 0 and F(v0∗ ) > 0 since v0∗ ∈ (v v], while λ(v0∗ − sj (v0∗ )) > 0 since v0∗ > sj (v0∗ ) by Theorem 1(i) and λ(·) > 0 on (0 +∞). By construction, s1 (v0∗ ) = s2 (v0∗ ). The previous equation then implies s1 (v0∗ ) < s2 (v0∗ ), contradicting s1 (v0∗ ) ≥ s2 (v0∗ ).

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(ii) Next, we prove the first inequality, which implies the third inequality after immediate algebra. For each j = 1 2, (2) gives (A.2)

sj (v) Ij − 1

=

f (v) λ(v − sj (v)) F(v)

for any v ∈ [v v]. From (i), v − s2 (v) < v − s1 (v) for any v ∈ (v v]. Because 0 < v − s2 (v) for any v ∈ (v v] by Theorem 1(i) and because λ(·) is strictly increasing with λ(·) > 0 on (0 +∞), then 0 < λ(v − s2 (v)) < λ(v − s1 (v)) for any v ∈ (v v]. Hence, because f (·) > 0 and F(·) > 0 on (v v], it follows from (A.2) that (A.3)

s (v) s2 (v) < 1 I2 − 1 I1 − 1

for any v ∈ (v v].23 Integrating (A.3) from v to v > v and using sj (v) = b give (A.4)

s2 (v) − b s1 (v) − b < I2 − 1 I1 − 1

for any v ∈ (v v]. The desired result follows after immediate algebra. Q.E.D. PROOF OF COROLLARY 1: The desired result holds when I = I and I = I by Lemma 2. Let I ∈ (I I). With I = I1 < I2 in (5), the first inequality in (5) gives (I − 1)

b2 (α) − b + b < bI (α) I2 − 1

for any α ∈ (0 1] and any I2 > I. Equation (A.4) applied to an arbitrary pair (I2 I2 ) with I2 < I2 shows that the left-hand side in the above inequality is strictly decreasing in I2 . Hence, the most stringent inequality is obtained when I2 is the smallest, that is, when I2 = I + 1. Similarly, with I = I2 > I1 in (5), the third inequality in (5) gives bI (α) < (I − 1)

b1 (α) − b +b I1 − 1

for any α ∈ (0 1] and any I1 < I. The right-hand side in the above inequality is strictly decreasing in I1 from (A.4). Hence, the most stringent inequality is obtained when I1 is the largest, that is, when I1 = I − 1. Combining these two results gives (A.5) 23

I −1 1 I −1 1 bI+1 (α) + b < bI (α) < bI−1 (α) − b I I I −2 I −2

Equation (A.1) shows that s2 (v)/(I2 − 1) < s1 (v)/(I1 − 1) also holds at v = v.


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for any α ∈ (0 1]. On the other hand, the middle inequality of (5) gives (A.6)

bI−1 (α) < bI (α) < bI+1 (α)

for any α ∈ (0 1] and any I ∈ [I I]. The desired result follows by combining (A.5) and (A.6). The second part of the corollary follows by noting that bI is a strictly increasing function of v, namely bI = sI (v) for each I. Hence, the random variables max{bI−1 [(I − 1)bI+1 + b]/I} and min{bI+1 [(I − 1)bI−1 − b]/(I − 2)} are also strictly increasing functions of v. It follows that the α-quantiles of their corresponding distributions GI (·) and GI (·) are equal to these functions evaluated at v(α). Thus, they are equal to the first term and third term of the two inequalities displayed in Corollary 1, respectively since bI (α) = sI [v(α)]. The stochastic dominance assertion then follows from these two inequalities. Q.E.D. PROOF OF PROPOSITION 3: From bj (α) = sj [v(α)] and (4) evaluated at v = v(α), we obtain the crucial relation α 1 (A.7) v(α) = bj (α) + λ−1 Ij − 1 gj [bj (α)] for j = 1 2 and any α ∈ [0 1]. Hence, using (6) we obtain the nonlinear relation (A.8)

λ−1 [R1 (α)] = λ−1 [R2 (α)] + b(α)

for any α ∈ [0 1]. For future use, we note that b(0) = 0 as s1 (v) = s2 (v) = b. Moreover, from Lemma 2, we know that s1 (v) < s2 (v) for any v ∈ (v v], which implies b1 (α) < b2 (α) for any α ∈ (0 1]. Hence, b(α) > 0 for any α ∈ (0 1]. Because λ−1 (·) is strictly increasing and Rj (α) > 0 for any α ∈ (0 1], it follows from (A.8) that R1 (α) > R2 (α) > 0 for any α ∈ (0 1]. In particular, because Rj (·) is continuous on [0 1] and Rj (0) = 0 for j = 1 2, the range Rj of Rj (·) must be of the form [0 r j ] with 0 < r j < ∞ and r 1 > r 2 , as claimed in the text. Now, by assumption, u0 belongs to R1 = [0 r 1 ]. If u0 = 0, then λ−1 (0) = 0. Next, consider the general case u0 ∈ (0 r 1 ]. Thus, there exists some α0 ∈ (0 1] such that u0 = R1 (α0 ). In particular, we have u0 = R1 (α0 ) > R2 (α0 ) > 0 = R1 (0) because R1 (·) > R2 (·) > 0 on (0 1]. Moreover, because R1 (·) is continuous on [0 1], there exists some α1 satisfying α0 > α1 > 0 and R1 (α1 ) = R2 (α0 ). Continuing such a construction, we have R1 (α1 ) > R2 (α1 ) > 0 = R1 (0), which implies that there exists some α2 satisfying α1 > α2 > 0 and R1 (α2 ) = R2 (α1 ). Thus, we have constructed a sequence, which is not necessarily unique, such that 1 ≥ α0 > α1 > · · · > αt > · · · > 0 with u0 = R1 (α0 ) > R2 (α0 ) = R1 (α1 ) > R2 (α1 ) = R1 (α2 ) > · · · > R2 (αt−1 ) = R1 (αt ) > · · · > 0, as indicated in the text.24 24 When R1 (·) is strictly increasing or, equivalently by (4), when the bidder’s rent is strictly −1 t increasing in v, then α0 = R−1 1 (u0 ) and αt = [R1 ◦ R2 ] (α0 ) for t = 1 2 where ◦ denotes the

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Because the sequence {αt } is strictly decreasing and is in (0 1], it must converge to some finite limit α∞ ∈ [0 1]. Because Rj (·) is continuous on [0 1], then limt→+∞ Rj (αt ) = Rj (α∞ ) for j = 1 2. But R2 (αt−1 ) = R1 (αt ) by construction, implying that R2 (α∞ ) = R1 (α∞ ). Because R2 (α) = R1 (α) only for α = 0, this implies that α∞ = 0 and, consequently, limt→+∞ Rj (αt ) = 0 for j = 1 2. We now iterate (A.8). Specifically, for any u0 ∈ R1 \ {0} and any corresponding sequence {αt } as constructed above, we must have the nonlinear dynamic relation λ−1 (u0 ) = λ−1 [R2 (α0 )] + b(α0 ) = λ−1 [R1 (α1 )] + b(α0 ) = λ−1 [R2 (α1 )] + b(α0 ) + b(α1 ) = λ−1 [R2 (αt )] + b(α0 ) + · · · + b(αt ) See Figure 1 for an illustration. Because λ−1 (·) is continuous on [0 +∞) with λ−1 (0) = 0 and limt→+∞ R2 (αt ) = 0, as shown above, then limt→+∞ λ−1 [R2 (αt )] = 0. Because λ−1 (u0 ) is finite, it follows from the above t equation that limt→+∞ τ=0 b(ατ ) must exist and that it is equal to λ−1 (u0 ), +∞ −1 that is, λ (u0 ) = τ=0 b(ατ ) as desired. Note that this must be so irrespective of the sequence {αt } and whether such a sequence is unique. Moreover, because b(ατ ) depends only on bj (·) and Rj (·), which depend only on the distributions Gj (·), the latter equality shows that λ−1 (·) is identified nonparametrically on R1 from observed equilibrium bids. The nonparametric identification of F(·) follows immediately from F(·) = Gj [ξj−1 (·)]. The nonparametric identification of λ−1 (·) on R1 and hence on R2 ⊂ R1 implies the nonparametric identification of ξj (·) on [b bj ] for j = 1 2 by (4) and (6). The latter implies the nonparametric identification of ξj−1 (·) = sj (·) on [v v]. Alternatively, pick an arbitrary α0 ∈ [0 1]. From (6) and (A.7) for (say) j = 1, we have v(α0 ) = b1 (α0 ) + λ−1 [R1 (α0 )]. Thus, the above explicit expression for λ−1 (u0 ) with u0 = R1 (α0 ) gives (A.9)

v(α0 ) = b1 (α0 ) +

+∞

b(αt )

t=0

showing that the α0 -quantile of F(·) is identified. Because α0 is arbitrary on Q.E.D. [0 1], it follows that F(·) is identified on [v v]. −1 t composition of two functions and [R−1 1 ◦ R2 ] denotes the t-composition of R1 ◦ R2 . Thus the sequence {αt } is unique.


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PROOF OF LEMMA 3: First, we prove that (i), (ii), and (iii) are necessary. We use a double index (i j) with i indexing bidder i among the Ij bidders and j = 1 2 indicating the level of competition. Because bij = sj (vi U F Ij ) and the vij ’s are i.i.d., the bij ’s are also i.i.d. given Ij j = 1 2. The fact that Gj (·) ∈ GR , j = 1 2, follows from Lemma 1. This establishes (i). Because s1 (v) < s2 (v) for any v ∈ (v v] from Lemma 2 and noting that bj = sj (v) with sj (·) strictly increasing, it follows that bj (α) = sj [v(α)]. Hence, b1 (α) < b2 (α) for any α ∈ (0 1] or, equivalently, G1 (·) ≺b G2 (·). This establishes (ii). Last, because λ(·) = U(·)/U (·) and U(·) satisfies Definition 1, then λ(·) is defined from R+ to R+ with λ(0) = 0 and λ (·) ≥ 1, and λ(·) is continuously differentiable on [0 ∞). Because F(·) is invariant in I, its quantiles are also invariant in I. Thus, considering (4) for two values I1 and I2 at any α-quantile with α ∈ [0 1] and I1 = I2 leads to (7). It remains to show that ξj (·) > 0, j = 1 2. The equilibrium strategy sj (·) must satisfy ξj [sj (v) U G Ij ] = v for any v ∈ [v v] and j = 1 2. We then obtain ξj (b U G Ij ) = sj−1 (b U F I). This implies ξj (·) = [s−1 (·)] > 0. This establishes (iii). Conversely, we show that (i), (ii), and (iii) are together sufficient. We construct a pair [U F] that satisfies Definitions 1 and 2, and is independent of I. Let U(·) be such that λ(·) = U(·)/U (·) or 1/λ(·) = U (·)/U(·). x Integration of the latter with the normalization U(1) = 1 gives U(x) = exp 1 1/λ(t) dt. We need to verify that U(·) satisfies Definition 1. This follows from the proof of Lemma 1. Let Fj (·) be the distribution of Xj = b + λ−1 [Gj (b)/((Ij − 1)gj (b))] given Ij , where b ∼ Gj (·), j = 1 2. Note that Fj (·) satisfies Definition 2 by the proof of Lemma 1. Moreover, because the compatibility condition is satisfied for any α ∈ [0 1], it implies that the corresponding α-quantile of Fj (·) does not depend on Ij . Hence F1 (·) = F2 (·) ≡ F(·), which thereby satisfies Definition 2. Last, we show that the pair [U F] can be rationalized by G1 (·) and G2 (·) with I2 > I1 , that is, that Gj (·) = F[sj−1 (· U F Ij )], j = 1 2, where sj (· U F Ij ) solves the first-order differential equation defining the equilibrium strategy with the boundary condition sj (v U F Ij ) = v. By construction, Gj (·) = F[ξj (·)]. Thus it suffices to show that ξj−1 (·), j = 1 2, solves the differential equation (2). This proof can be found in Lemma 1, which shows that ξj−1 (·), j = 1 2, solves the differential equation with I = Ij under the boundary Q.E.D. condition ξj−1 (v) = v. PROOF OF PROPOSITION 7: We prove the first part only. Let [U F] ∈ URI × FR . This structure generates G(· ·|·) ∈ GR whose marginal distributions satisfy Definition 3 and the compatibility condition (13). We show that there ˜ ∈ U I × FR rationalizing G(· ·|·). The proof ˜ F] exists another structure [U R is in four steps and is done for every fixed I ∈ I . ˜ Let U˜ 1 (·) = [U1 (·/δ)/ STEP 1—Construction of [U˜ 1 U˜ I F(·|I)]: −1 δ U1 (1/δ)] with δ ∈ (0 1). Thus, λ˜ 1 (·) = λ1 (·/δ) and λ˜ −1 1 (·) = δλ1 (·). For i =

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x 2 I, let U˜ i (x) = exp[ 1 1/λ˜ i (t) dt] so that λ˜ i (·) = U˜ i (·)/U˜ i (·), where λ˜ i (·) ˜ −1 is such that λ˜ −1 i [1/Hi (biα |I)] = λ1 [1/H1 (b1α |I)] + b1α − biα , for all α ∈ [0 1]. The latter defines λ˜ −1 i (·) well because 1/Hi (biα |I) strictly increases as α increases given Hi (·|I) < 0 by assumption. Moreover, λ˜ i (·) is strictly increasing as shown in Step 3. Note that the compatibility condition (13) is satisfied ˜ by construction. We then let F(·|I) be the conditional distribution given I of −1 ˜ ˜ v˜ i ≡ bi + λi [1/Hi (bi |I)] ≡ ξi (bi ) for an arbitrary i, where bi ∼ Gi (·|I). Using −1 −1 ˜ −1 λ˜ −1 1 (·) = δλ1 (·), we obtain λi [1/Hi (biα |I)] = δλ1 [1/H1 (b1α |I)] + b1α − biα . Thus, (13) with j = 1 gives (A.10)

λ˜ −1 i

1 1 −1 = δλi + (1 − δ)(b1α − biα ) Hi (biα |I) Hi (biα |I)

−1 −1 Equivalently, λ˜ −1 i [1/Hi (biα |I)] = λi [1/Hi (biα |I)] − (1 − δ)λ1 [1/H1 (b1α |I)]. ˜ ˜ In particular, since λ−1 i (·) is bidder’s i shading, the shading under [U1 UI ˜ is smaller than under [U1 UI F], that is, bidders bid more aggressively F] under the former than under the latter.

STEP 2 —λ˜ i (0) = 0 and ξ˜ i (·) > 0 on [b b]: Because [U F] ∈ URI × FR so that G(· ·|·) ∈ GR , we have λ−1 i (0) = 0 and limb↓b 1/Hi (b|I) = 0 for I ∈ I . Thus, (A.10) with the boundary conditions b1 = · · · = bI ≡ b = v im˜ ˜ ˜ ply λ˜ −1 i (0) = 0 and hence λi (0) = 0. Regarding ξi (·) > 0, we note that ξi (biα ) = −1 (1 − δ)b1α + δξi (biα ) from (A.10) and (12). Noting that b1α = G1 [Gi (biα )] ≡ Bi (biα ) and letting biα = b, we obtain ξ˜ i (b) = (1 − δ)Bi (b) + δξi (b), where Bi (b) = gi (b)/g1 [B(b)]. Hence, ξ˜ i (b) > 0 since Bi (b) > 0 and ξi (b) > 0. STEP 3—λ˜ i (·) ≥ 1: From (A.10) and (12), λ˜ −1 i [1/Hi (biα |I)] = δξi (biα ) + (1 − δ)b1α − biα , that is, 1/Hi (biα |I) = λ˜ i [δξi (biα ) + (1 − δ)b1α − biα ]. From the structure [U F], we have 1/Hi (biα |I) = λi [ξi (biα ) − biα ]. Thus, λi [ξi (biα ) − biα ] = λ˜ i [δξi (biα ) + (1 − δ)b1α − biα ]. Differentiating with respect to b = biα and noting that b1α = G−1 1 [Gi (biα )] ≡ Bi (biα ) gives (A.11)

λ˜ i (∗∗) =

ξi (b) − 1 λ (∗) ≡ Ri (b)λi (∗) δξi (b) + (1 − δ)Bi (b) − 1 i

where the different arguments of λi (·) and λ˜ i (·) are indicated by ∗ and ∗∗, respectively. Thus, it suffices to show that Ri (·) ≥ 1 since λi (·) ≥ 1. We note that ξ1 (b1α ) = ξi (biα ) = vα for all α ∈ [0 1] from the compatibility condition. Using b1α = Bi (biα ) gives ξ1 [Bi (b)] = ξi (b) for all b ∈ [b b]. Differentiating


1225

gives ξ1 [Bi (b)]Bi (b) = ξi (b), that is, Bi (b) = ξi (b)/ξ1 [Bi (b)]. Hence, (A.12)

ξi (b) − 1

Ri (b) =

δξi (b) − 1 + (1 − δ) =1+

ξi (b) ξ [Bi (b)] 1

(1 − δ)ξi (b){ξ1 [Bi (b)] − 1} δξ (b){ξ1 [Bi (b)] − 1} − {ξ1 [Bi (b)] − ξi (b)} i

for b ∈ [b b]. Note that ξi (·) > 1 on (b b] for every i = 1 I since dif 2 ferentiating (12) gives ξi (b) = 1 − λ−1 i [1/Hi (b|I)][Hi (b|I)/Hi (b|I)], where λ−1 i (·) > 0 and Hi (·|I) < 0 by assumption. Hence, ξi (·) ≥ 1 on [b b] by conti nuity. Since 1 − δ > 0 and ξi (·) > 0, it suffices to show that the denominator Di (b) (say) in the right-hand side is strictly positive for all b ∈ [b b] and some δ ∈ [δ∗ 1]. To study the sign of Di (·) on [b b], we note that gj (b) gj (b) + o(1) = Gj (b) gj (b)(b − b) + o(b − b) 1 1 gj (b) + o(1) = (1 + o(1)) b − b gj (b) + o(1) b − b Thus, a Taylor expansion of 1 = λi [ξi (b)−b] j=i [gj (b)/Gj (b)] from (12) gives =

I −1

(1 + o(1)) 1 = λi (0)[ξi (b) − 1](b − b) + o(b − b) b−b

= λi (0)[ξi (b) − 1](I − 1) + o(1) Hence, ξi (b) = 1 + {1/[(I − 1)λi (0)]} > 1 as λi (·) ≥ 1. Thus, because ξi (·) > 1 on [b b], then Di (·) > 0 on [b b] if and only if δ > δ∗ ≡ maxb∈[bb] δ(b), where δ(·) is continuous on [b b] with 1 ξi (b) − 1 ξ1 [Bi (b)] − ξi (b) = 1− δ(b) ≡ ξi (b){ξ1 [Bi (b)] − 1} ξi (b) ξ1 [Bi (b)] − 1 It remains to show that δ(·) < 1 on [b b] so that δ∗ < 1. Clearly, δ(·) < 1 on (b b], as ξi (·) > 1 on (b b]. Moreover, at b = b, we have δ(b) = [1/ξi (b)]{1 − [λi (0)/λ1 (0)]} < 1. ˜ ∈ U I × FR : From the previous steps and the rationalization ˜ F] STEP 4—[U R ˜ F] rationalizes G(· ·|·). It remains result given after (13), it follows that [U I ˜ ∈ U × FR . From the proof of Lemma 1, it suffices to show ˜ F] to show that [U R

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that λ˜ i (·) is R + 1 continuously differentiable on [0 ∞) for i = 1 I. This follows from (A.11) and (A.12), and the R + 1 continuous differentiablity of Q.E.D. λi (·) and ξi (·) as G(· ·|·) ∈ GR . REFERENCES ACKERBERG, D., K. HIRANO, AND Q. SHAHRIAR (2006): “The Buy-It-Now Option, Risk Aversion and Impatience in an Empirical Model of eBay Bidding,” Unpublished Manuscript, University of Arizona. [1193] ATHEY, S. (2001): “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,” Econometrica, 69, 861–889. [1196] ATHEY, S., AND P. HAILE (2002): “Identification of Standard Auction Models,” Econometrica, 70, 2107–2140. [1194,1200] ATHEY, S., AND J. LEVIN (2001): “Information and Competition in U.S. Forest Service Timber Auctions,” Journal of Political Economy, 109, 375–417. [1193] ATHEY, S., J. LEVIN, AND E. SEIRA (2004): “Comparing Open and Sealed Bid Auctions: Theory and Evidence From Timber Auctions,” Unpublished Manuscript, Stanford University. [1200, 1206,1207,1210] BAJARI, P., AND A. HORTACSU (2003): “The Winner’s Curse, Reserve Prices and Endogenous Entry: Empirical Insights From eBay Auctions,” Rand Journal of Economics, 34, 329–355. [1200,1206,1207] (2005): “Are Structural Estimates of Auction Models Reasonable? Evidence From Experimental Data,” Journal of Political Economy, 113, 703–741. [1194] BALDWIN, L. (1995): “Risk Averse Bidders at Oral Ascending-Bid Forest Service Timber Sales,” Unpublished Manuscript, RAND Corporation. [1193] BARRETT, G., AND S. DONALD (2003): “Test of Stochastic Dominance,” Econometrica, 71, 71–103. [1201] CAMPO, S., E. GUERRE, I. PERRIGNE, AND Q. VUONG (2007): “Semiparametric Estimation of First-Price Auctions With Risk Averse Bidders,” Unpublished Manuscript, Pennsylvania State University. [1193,1195,1199,1200] CAMPO, S., I. PERRIGNE, AND Q. VUONG (2003): “Asymmetry in First-Price Auctions With Affiliated Private Values,” Journal of Applied Econometrics, 18, 179–207. [1206,1210] CHE, Y. K., AND I. GALE (1998): “Standard Auctions With Financially Constrained Bidders,” Review of Economic Studies, 65, 1–21. [1196] CHEN, X. (2007): “Large Sample Sieve Estimation of Semi-Nonparametric Models,” in Handbook of Econometrics, Vol. VIB, ed. by J. Heckman and E. Leamer. Amsterdam: NorthHolland, 5549–5632. [1216] COHEN, A., AND L. EINAV (2007): “Estimating Risk Preferences From Deductible Choice,” American Economic Review, 97, 745–788. [1194] COX, J., V. SMITH, AND J. WALKER (1988): “Theory and Individual Behavior of First-Price Auctions,” Journal of Risk and Uncertainty, 1, 61–99. [1193] D’HAULTFOEUILLE, X., AND P. FÉVRIER (2007): “Identification and Estimation of Incentive Problems: Adverse Selection,” Unpublished Manuscript, CREST-INSEE. [1216] FLAMBARD, V., AND I. PERRIGNE (2006): “Asymmetry in Procurement Auctions: Evidence From Snow Removal Contracts,” The Economic Journal, 116, 1014–1036. [1210] GOEREE, J., C. HOLT, AND T. PALFREY (2002): “Quantal Response Equilibrium and Overbidding in Private-Value Auctions,” Journal of Economic Theory, 104, 247–272. [1193] GOLLIER, C. (2001): The Economics of Risk and Time. Cambridge, MA: MIT Press. [1194] GUERRE, E., I. PERRIGNE, AND Q. VUONG (2000): “Optimal Nonparametric Estimation of FirstPrice Auctions,” Econometrica, 68, 525–574. [1197-1199,1207,1217] (2009): “Supplement to ‘Nonparametric Identification of Risk Aversion in First-Price Auctions Under Exclusion Restrictions’,” Econometrica Supplemental Material, 77, http://www. econometricsociety.org/ecta/Supmat/7028_proofs.pdf. [1197]


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HAILE, P., H. HONG, AND M. SHUM (2003): “Nonparametric Tests for Common Values in FirstPrice Sealed-Bid Auctions,” Unpublished Manuscript, Yale University. [1194,1195,1205] KRASNOKUTSKAYA, E. (2003): “Auction Models With Unobserved Heterogeneity: Application to Michigan Highway Procurement Auctions,” Unpublished Manuscript, University of Pennsylvania. [1207] KRASNOKUTSKAYA, E., AND K. SEIM (2005): “Bid Preference Programs and Participation in Highway Procurement Auctions,” Unpublished Manuscript, University of Pennsylvania. [1206] KRISHNA, V. (2002): Auction Theory. New York: Academic Press. [1194,1196] LEBRUN, B. (1999): “First-Price Auctions and the Asymmetric N Bidder Case,” International Economic Review, 40, 135–142. [1196] LI, H., AND G. TAN (2000): “Hidden Reserve Prices With Risk Averse Bidders,” Unpublished Manuscript, University of British Columbia. [1193] LI, T., AND Q. VUONG (1998): “Nonparametric Estimation of the Measurement Error Model Using Multiple Indicators,” Journal of Multivariate Analysis, 65, 139–165. [1207] LI, T., I. PERRIGNE, AND Q. VUONG (2000): “Conditionally Independent Private Information in OCS Wildcat Auctions,” Journal of Econometrics, 98, 129–161. [1209] (2002): “Structural Estimation of the Affiliated Private Value Auction Model,” Rand Journal of Economics, 33, 171–193. [1208,1209] LU, J., AND I. PERRIGNE (2008): “Estimating Risk Aversion From Ascending and Sealed-Bid Auctions: The Case of Timber Auction Data,” Journal of Applied Econometrics, 23, 871–896. [1193,1194,1200,1212] MASKIN, E., AND J. RILEY (1984): “Optimal Auctions With Risk Averse Buyers,” Econometrica, 52, 1473–1518. [1196] (2000): “Equilibrium in Sealed High Bid Auctions,” Review of Economic Studies, 67, 439–454. [1196] (2003): “Uniqueness in Sealed High Bid Auctions,” Games and Economic Behavior, 45, 395–409. [1196] MATZKIN, R. (1994): “Restrictions of Economic Theory in Nonparametric Methods,” in Handbook of Econometrics, Vol. IV, ed. by R. Engle and D. McFadden. Amsterdam: North-Holland, 2523–2558. [1194] (2003): “Nonparametric Estimation of Nonadditive Random Functions,” Econometrica, 71, 1339–1375. [1207] PAARSCH, H. J. (1992): “Deciding Between the Common and Private Value Paradigms in Empirical Models of Auctions,” Journal of Econometrics, 51, 191–215. [1200] PERRIGNE, I. (2003): “Random Reserve Prices and Risk Aversion in Timber Auctions,” Unpublished Manuscript, Pennsylvania State University. [1193] PERRIGNE, I., AND Q. VUONG (2008): “Cost Uncertainty in Procurements,” Unpublished Manuscript, Pennsylvania State University. [1217] PINKSE, J., AND G. TAN (2005): “The Affiliation Effect in First-Price Auctions,” Econometrica, 73, 263–277. [1209]

Dept. of Economics, Queen Mary, University of London, Mile End Road, London E1 4NS, U.K.; [email protected], Dept. of Economics, Pennsylvania State University, 608 Kern Graduate Building, University Park, PA 16802-3306, U.S.A.; [email protected], and Dept. of Economics, Pennsylvania State University, 608 Kern Graduate Building, University Park, PA 16802-3306, U.S.A.; [email protected]. Manuscript received March, 2007; final revision received November, 2008.


PANEL DATA MODELS WITH INTERACTIVE FIXED EFFECTS BY JUSHAN BAI1 This paper considers large N and large T panel data models with unobservable multiple interactive effects, which are correlated with the regressors. In earnings studies, for example, workers’ motivation, persistence, and diligence combined to influence the earnings in addition to the usual argument of innate ability. In macroeconomics, interactive effects represent unobservable common shocks and their heterogeneous impacts on cross sections. We consider identification, consistency, and the limiting distribution of the interactive-effects estimator. Under both large N and large T , the estimator is √ shown to be NT consistent, which is valid in the presence of correlations and heteroskedasticities of unknown form in both dimensions. We also derive the constrained estimator and its limiting distribution, imposing additivity coupled with interactive effects. The problem of testing additive versus interactive effects is also studied. In addition, we consider identification and estimation of models in the presence of a grand mean, time-invariant regressors, and common regressors. Given identification, the rate of convergence and limiting results continue to hold. KEYWORDS: Additive effects, interactive effects, factor error structure, bias-corrected estimator, Hausman tests, time-invariant regressors, common regressors.

1. INTRODUCTION WE CONSIDER THE FOLLOWING units and T time periods (1)

PANEL DATA MODEL

with N cross-sectional

Yit = Xit β + uit

and uit = λi Ft + εit

(i = 1 2 N, t = 1 2 T ),

where Xit is a p × 1 vector of observable regressors, β is a p × 1 vector of unknown coefficients, uit has a factor structure, λi (r × 1) is a vector of factor loadings, and Ft (r × 1) is a vector of common factors so that λi Ft = λi1 F1t + · · · + λir Frt ; εit are idiosyncratic errors; λi , Ft , and εit are all unobserved. Our interest is centered on the inference for the slope coefficient β, although inference for λi and Ft will also be discussed. 1 I am grateful to a co-editor and four anonymous referees for their constructive comments, which led to a much improved presentation. I am also grateful for comments and suggestions from seminar participants at the University of Pennsylvania, Rice, MIT/Harvard, Columbia Econometrics Colloquium, New York Econometrics Camp (Saratoga Springs), Syracuse, Malinvaud Seminar (Paris), European Central Bank/Center for Financial Studies Joint Workshop, Cambridge University, London School of Economics, Econometric Society European Summer Meetings (Vienna), Quantitative Finance and Econometrics at Stern, and the Federal Reserve Bank of Atlanta. This work is supported in part by NSF Grants SES-0551275 and SES-0424540.


DOI: 10.3982/ECTA6135

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The preceding set of equations constitutes the interactive-effects model in light of the interaction between λi and Ft . The usual fixed-effects model takes the form (2)

Yit = Xit β + αi + ξt + εit

where the individual effects αi and the time effects ξt enter the model additively instead of interactively; accordingly, it will be called the additive-effects model for comparison and reference. It is noted that multiple interactive effects include additive effects as special cases. For r = 2, consider the special factor and factor loading such that, for all i and all t, 1 αi and λi = Ft = 1 ξt Then λi Ft = αi + ξt The case of r = 1 has been studied by Holtz-Eakin, Newey, and Rosen (1988) and Ahn, Lee, and Schmidt (2001), among others. Owing to potential correlations between the unobservable effects and the regressors, we treat λi and Ft as fixed-effects parameters to be estimated. This is a basic approach to controlling the unobserved heterogeneity; see Chamberlain (1984) and Arellano and Honore (2001). We allow the observable Xit to be written as (3)

Xit = τi + θt +

r k=1

ak λik +

r k=1

bk Fkt +

r

ck λik Fkt + πi Gt + ηit

k=1

where ak , bk , and ck are scalar constants (or vectors when Xit is a vector), and Gt is another set of common factors that do not enter the Yit equation. So Xit can be correlated with λi alone or with Ft alone, or can be simultaneously correlated with λi and Ft . In fact, Xit can be a nonlinear function of λi and Ft . We make no assumption on whether Ft has a zero mean or whether Ft is independent over time: it can be a dynamic process without zero mean. The same is true for λi . We directly estimate λi and Ft , together with β subject to some identifying restrictions. We consider the least squares method, which is detailed in Section 3. While additive effects can be removed by the within-group transformation (least squares dummy variables), the scheme fails to purge interactive effects. For example, consider r = 1, Yit = Xit β + λi Ft + εit . Then Yit − Y¯ i· = ¯ + εit − ε¯ i· where Y¯ i·, X¯ i·, and ε¯ i· are averages (Xit − X¯ i·) β + λi (Ft − F) over time. Because Ft ≡ F¯ , the within-group transformation with cross-section dummy variable is unable to remove the interactive effects. Similarly, the interactive effects cannot be removed with time dummy variable. Thus the within-

PANEL MODELS WITH FIXED EFFECTS

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group estimator is inconsistent since the unobservables are correlated with the regressors. However, the interactive effects can be eliminated by the quasidifferencing method, as in Holtz-Eakin, Newey, and Rosen (1988). Further details are provided in Section 3.3. Recently, Pesaran (2006) proposed a new estimator that allows for multiple factor error structure under large N and large T . His method augments the model with additional regressors, which are the cross-sectional averages of the dependent and independent variables, in an attempt to control for Ft . His estimator requires a certain rank condition, which is not guaranteed to be met, √ that depends on data generating processes. Peseran showed N consistency irrespective of the rank condition, and a possible faster rate of convergence when the rank condition does hold. Coakley, Fuertes, and Smith (2002) proposed a two-step estimator, but this estimator was found to be inconsistent by Pesaran. The two-step estimator, while related, is not the least squares estimator. The latter is an iterated solution. Ahn, Lee, and Schmidt (2001) considered the situation of fixed T and noted that the least squares method does not give a consistent estimator if serial correlation or heteroskedasticity is present in εit . Then they explored the consistent generalized method of moments (GMM) estimators and showed that a GMM method that incorporates moments of zero correlation and homoskedasticity is more efficient than least squares under fixed T . The fixed T framework was also studied earlier by Kiefer (1980) and Lee (1991). Goldberger (1972) and Jöreskog and Goldberger (1975) are among the earlier advocates for factor models in econometrics, but they did not consider correlations between the factor errors and the regressors. Similar studies include MaCurdy (1982), who considered random effects type of generalized least squares (GLS) estimation for fixed T , and Phillips and Sul (2003), who considered SUR-GLS (seemingly unrelated regressions) estimation for fixed N. Panel unit root tests with factor errors were studied by Moon and Perron (2004). Kneip, Sickles, and Song (2005) assumed Ft is a smooth function of t and estimated Ft by smoothing spline. Given the spline basis, the estimation problem becomes that of ridge regression. The regressors Xit are assumed to be independent of the effects. In this paper, we provide a large N and large T perspective on panel data models with interactive effects, permitting the regressor Xit to be correlated with either λi or Ft , or both. Compared with the fixed T analysis, the large T perspective has its own challenges. For example, an incidental parameter problem is now present in both dimensions. Consequently, a different argument is called for. On the other hand, the large T setup also presents new opportunities. We show that √ if T is large, comparable with N, then the least squares estimator for β is NT consistent, despite serial or cross-sectional correlations and heteroskedasticities of unknown form in εit . This presents a contrast to the fixed T framework, in which serial correlation implies inconsistency. Earlier fixed T studies assume independent and identically distributed (i.i.d.)

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Xit over i, disallowing Xit to contain common factors, but permitting Xit to be correlated with λi . Earlier studies also assume εit are i.i.d. over i. We allow εit to be weakly correlated across i and over t, thus, uit has the approximate factor structure of Chamberlain and Rothschild (1983). Additionally, heteroskedasticity is allowed in both dimensions. Controlling fixed effects by directly estimating them, while often an effective approach, is not without difficulty—known as the incidental parameter problem, which manifests itself in bias and inconsistency at least under fixed T , as documented by Neyman and Scott (1948), Chamberlain (1980), and Nickell (1981). Even for large T , asymptotic bias can persist in dynamic or nonlinear panel data models with fixed effects.2 We show that asymptotic bias arises under interactive effects, leading to nonzero centered limiting distributions. We also show that bias-corrected estimators can be constructed in a way similar to Hahn and Kuersteiner (2002) and Hahn and Newey (2004), who argued that bias-corrected estimators may have desirable properties relative to instrumental variable estimators. Because additive effects are special cases of interactive effects, the interactive-effects estimator is consistent when the effects are, in fact, additive, but the estimator is less efficient than the one with additivity imposed. In this paper, we derive the constrained estimator together with its limiting distribution when additive and interactive effects are jointly present. We also consider the problem of testing additive effects versus interactive effects. In Section 2, we explain why incorporating interactive effects can be a useful modelling paradigm. Section 3 outlines the estimation method and Section 4 discusses the underlying assumptions that lead to consistent estimator. Section 5 derives the asymptotic representation and the asymptotic distribution of the estimator. Section 6 provides an interpretation of the estimator as a generalized within-group estimator. Section 7 derives the bias-corrected estimators. Section 8 considers estimators with additivity restrictions and their limiting distributions. Section 9 studies Hausman tests for additive effects versus interactive effects. Section 10 is devoted to time-invariant regressors and regressors that are common to each cross-sectional unit. Monte Carlo simulations are given in Section 11. All proofs are provided either in the Appendix or in the Supplemental Material (Bai (2009)). 2. SOME EXAMPLES Macroeconometrics Here Yit is the output (or growth rate) for country i in period t, Xit is the input such as labor and capital, Ft represents common shocks (e.g., technological shocks and financial crises), λi represents the heterogeneous impact of 2 See Nickell (1981), Anderson and Hsiao (1982), Kiviet (1995), Hsiao (2003, pp. 71–74), and Alvarez and Arellano (2003) for dynamic panel data models; see Hahn and Newey (2004) for nonlinear panel models.


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common shocks on country i, and, finally, εit is the country-specific error term of output (or growth rate). In general, common shocks not only affect the output directly (through the total factor productivity or Solow resdidual), but also affect the amount of input in the production process (through investment decisions). When common shocks have homogeneous effects on the output, that is, λi = λ for all i, the model collapses to the usual time effect by letting δt = λ Ft , where δt is a scalar. It is the heterogeneity that gives rise to a factor structure. Recently, Giannone and Lenza (2005) provided an explanation for the Feldstein–Horioka (1980) puzzle, one of the six puzzles in international macroeconomics (Obstfeld and Rogoff (2000)). The puzzle refers to the excessively high correlation between domestic savings and domestic investments in open economies. In their model, Yit is the investment and Xit is the savings for country i, Ft is the common shock that affects both investment and savings decisions. Giannone and Lenza found that the high correlation is a consequence of the strong assumption that shocks have homogeneous effects across countries (additive effects); it disappears when shocks are allowed to have heterogeneous impacts (interactive effects). Microeconometrics In earnings studies, Yit represents the wage rate for individual i with age (or age cohort) t and Xit is a vector of observable characteristics, such as education, experience, gender, and race. Here λi represents a vector of unobservable characteristics or unmeasured skills, such as innate ability, perseverance, motivation, and industriousness, and Ft is a vector of prices for the unmeasured skills. The model assumes that the price vector for the unmeasured skills is time-varying. If Ft = f for all t, the standard fixed-effects model is obtained by letting αi = λi f . In this example, t is not necessarily the calendar time, but age or age cohort. Applications in this area were given by Cawley, Connelly, Heckman, and Vytlacil (1997) and Carneiro, Hansen, and Heckman (2003). As explained in a previous version, the model of Abowd, Kramarz, and Margolis (1999) can be extended to disentangle the worker and the firm effects, while incorporating interactive effects. Ahn, Lee, and Schmidt (2001) provided a theoretical motivation for a single factor model based on the work of Altug and Miller (1990) and Townsend (1994). In the setup of Holtz-Eakin, Newey, and Rosen (1988), the slope coefficient β is also time-varying. Their model can be considered as a projection of Yit on {Xit λi }; see Chamberlain (1984). Pesaran (2006) allowed β to be heterogeneous over i such that βi = β + vi with vi being i.i.d.. In this regard, the constant slope coefficient is restrictive. To partially alleviate the restriction, it would be useful to allow additional individual and time effects as (4)

Yit = Xit β + αi + δt + λi Ft + εit

Model (4) will be considered in Section 8.

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Finance Here Yit is the excess return of asset i in period t; Xit is a vector of observable factors such as dividend yields, dividend payout ratio, and consumption gap as in Lettau and Ludvigson (2001) or book and size factors as in Fama and French (1993); Ft is a vector of unobservable factor returns; λi is the factor loading; εit is the idiosyncratic return. The arbitrage pricing theory of Ross (1976) is built upon a factor model for asset returns. Campbell, Lo, and MacKinlay (1997) provided many applications of factor models in finance. Cross-Section Correlation Interactive-effects models provide a tractable way to model cross-section correlations. In the error term uit = λi Ft + εit , each cross section shares the same Ft , causing cross-correlation. If λi = 1 for all i, and εit are i.i.d. over i and t, an equal correlation model is obtained. In a recent paper, Andrews (2005) showed that cross-section correlation induced by common shocks can be problematic for inference. Andrews’ analysis is confined within the framework of a single cross-section unit. In the panel data context, as shown here, consistency and proper inference can be obtained. 3. ESTIMATION 3.1. Issues of Identification Even in the absence of regressors Xit , the lack of identification for factor models is well known; see Anderson and Rubin (1956) and Lawley and Maxell (1971). The current setting differs from classical factor identification in two aspects. First, both factor loadings and factors are treated as parameters, as opposed to factor loadings only. Second, the number of individuals N is assumed to grow without bound instead of being fixed, and it can be much larger than the number of observations T . Write the model as Yi = Xi β + Fλi + εi where ⎤ Yi1 ⎢ Yi2 ⎥ ⎥ Yi = ⎢ ⎣ ⎦

⎡ X ⎤

⎡

YiT

i1 i2

⎢X ⎥ ⎥ Xi = ⎢ ⎣ ⎦ XiT

⎡ F ⎤ 1 2

⎢F ⎥ ⎥ F =⎢ ⎣ ⎦ FT

⎤ εi1 ⎢ εi2 ⎥ ⎥ εi = ⎢ ⎣ ⎦ ⎡

εiT

Similarly, define Λ = (λ1 λ2 λN ) , an N × r matrix. In matrix notation, (5)

Y = Xβ + FΛ + ε


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where Y = (Y1 YN ) is T × N and X is a three-dimensional matrix with p sheets (T × N × p), the th sheet of which is associated with the th element of β ( = 1 2 p). The product Xβ is T × N and ε = (ε1 εN ) is T × N. In view of FΛ = FAA−1 Λ for an arbitrary r × r invertible A, identification is not possible without restrictions. Because an arbitrary r × r invertible matrix has r 2 free elements, the number of restrictions needed is r 2 . The normalization3 (6)

F F/T = Ir

yields r(r + 1)/2 restrictions. This is a commonly used normalization; see, for example, Connor and Korajzcyk (1986), Stock and Watson (2002), and Bai and Ng (2002). Additional r(r − 1)/2 restrictions can be obtained by requiring (7)

Λ Λ = diagonal

These two sets of restrictions uniquely determine Λ and F , given the product FΛ .4 The least squares estimators for F and Λ derived below satisfy these restrictions. With either fixed N or fixed T , factor analysis would require additional restrictions. For example, the covariance matrix of εi is diagonal or the covariance matrix depends on a small number of parameters via parameterization. Under large N and large T , the cross-sectional covariance matrix of εit or the time series covariance matrix can be of an unknown form. In particular, none of the elements is required to be zero. However, the correlation—either cross sectional or serial—must be weak, which we assume to hold. This is known as the approximate factor model of Chamberlain and Rothschild (1983). The usual identification condiSufficient variation in Xit is also Nneeded. 1 X M X tion for β is that the matrix NT F i is of full rank, where MF = i i=1 −1 IT − F(F F) F . Because F is not observable and is estimated, a stronger condition is required. See Section 4 for details. 3.2. Estimation The least squares objective function is defined as (8)

SSR(β F Λ) =

N (Yi − Xi β − Fλi ) (Yi − Xi β − Fλi ) i=1

3 The normalization still leaves rotation indeterminacy. For example, let G be an r × r orthogonal matrix, and let F ∗ = FG and Λ∗ = ΛG. Then FΛ = F ∗ Λ∗ and F ∗ F ∗ /T = F F/T = I. To remove this indeterminacy, we fix G to make Λ∗ Λ∗ = G Λ ΛG a diagonal matrix. This is the reason for restriction (7). 4 Uniqueness is up to a columnwise sign change. For example, −F and −Λ also satisfy the restrictions.

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subject to the constraint F F/T = Ir and Λ Λ being diagonal. Define the projection matrix MF = IT − F(F F)−1 F = IT − FF /T The least squares estimator for β for each given F is simply N

−1 N ˆ β(F) = Xi MF Xi Xi MF Yi i=1

i=1

Given β, the variable Wi = Yi − Xi β has a pure factor structure such that (9)

Wi = Fλi + εi

Define W = (W1 W2 WN ), a T × N matrix. The least squares objective function is tr[(W − FΛ )(W − FΛ ) ] From the analysis of pure factor models estimated by the method of least squares (i.e., principal components; see Connor and Korajzcyk (1986) and Stock and Watson (2002)), by concentrating out Λ = W F(F F)−1 = W F/T , the objective function becomes (10)

tr(W MF W ) = tr(W W ) − tr(F W W F)/T

Therefore, minimizing with respect to F is equivalent to maximizing tr[F (W × equal to the first r eigenW )F]. The estimator for √ F (see Anderson (1984)), is vectors (multiplied by T due to the restriction F F/T = I) associated with the first r largest eigenvalues of the matrix WW=

N i=1

Wi Wi =

N (Yi − Xi β)(Yi − Xi β) i=1

Therefore, given F , we can estimate β, and given β, we can estimate F . The ˆ is the solution of the set of nonlinear equaˆ F) final least squares estimator (β tions

−1 N N (11) Xi MFˆ Xi Xi MFˆ Yi βˆ = i=1

and (12)

i=1

N 1 ˆ ˆ ˆ ˆ (Yi − Xi β)(Y i − Xi β) F = FVNT NT i=1


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where VNT is a diagonal matrix that consists of the r largest eigenvalues of the above matrix5 in the brackets, arranged in decreasing order. The solution ˆ can be simply obtained by iteration. Finally, from Λ = W F/T , Λˆ is ˆ F) (β ˆ such that ˆ F) expressed as a function of (β ˆ Fˆ (YN − XN β)] ˆ Λˆ = (λˆ 1 λˆ 2 λˆ N ) = T −1 [Fˆ (Y1 − X1 β) We may also write ˆ Λˆ = T −1 Fˆ (Y − X β) where Y is T × N and X is T × N × p, a three-dimensional matrix. ˆ jointly minimizes the objective function (8). The pair ˆ Λ) ˆ F The triplet (β ˆ ˆ (β F) jointly minimizes the concentrated objective function (10), which, when substituting Yi − Xi β for Wi , is equal to (13)

tr(W MF W ) =

N

Wi MF Wi =

i=1

N

(Yi − Xi β) MF (Yi − Xi β)

i=1

This is also the objective function considered by Ahn, Lee, and Schmidt (2001), although a different normalization is used. They as well as Kiefer (1980) discussed an iteration procedure for estimation. Interestingly, convergence to a local optimum for such an iterated estimator was proved by Sargan (1964). Here we suggest a more robust iteration scheme (having a much better convergence property from Monte Carlo evidence) than the one implied by (11) and (12). Given F and Λ, we compute N

−1 N ˆ β(F Λ) = Xi Xi Xi (Yi − Fλi ) i=1

i=1

and given β, we compute F and Λ from the pure factor model Wi = Fλi + ei with Wi = Yi − Xi β. This iteration scheme only requires a single matrix inverse N ( i=1 Xi Xi )−1 , with no need to update during iteration. Our simulation results are based on this scheme. REMARK 1: The common factor F is obtained by the principal components method from the matrix W W /N. Under large N but a fixed T , W W /N → ΣW = FΣΛ F + Ω, where ΣΛ is the limit of Λ Λ/N—an r × r matrix—and Ω is a T × T matrix of the covariance matrix of εi ; see (9). Unless Ω is a scalar multiple of IT (identity matrix), that is, no serial correlation and heteroskedasticity, the first r eigenvectors of ΣW are not a rotation of F , implying that the 5

We divide this matrix by NT so that VNT will have a proper limit. The scaling does not affect Fˆ .

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first r eigenvectors of W W /N are not consistent for F (a rotation of F to be more precise). This leads to inconsistent estimation of the product FΛ , and, thus, inconsistent estimation of β. However, under large T , Ω does not have to be a scalar multiple of an identity matrix and the principal components estimator for F is consistent. This is the essence of the approximate factor model of Chamberlain and Rothschild (1983). REMARK 2: Instead of estimating F from (9) by the method of principal components, one can directly use factor analysis. Factor analysis such as the maximum likelihood method allows Ω to be heteroskedastic and to have nonzero off-diagonal elements. The off-diagonal elements (due to serial correlation) must be parametrized to avoid too many free parameters. Serial correlation can also be removed by adding lagged dependent variables as regressors, leaving a diagonal Ω. In contrast, the principal components method is designed for large N and large T . In this setting, there is no need to assume a parametric form for serial correlations. The principal components method is a quick and effective approach to extracting common factors. Note that lagged dependent variables lead to bias, just as serial correlation does; see Section 7. Small T models can also be estimated by the quasi-differencing approach (Section 3.3). This latter approach also parametrizes serial correlations by including lagged dependent variables as regressors so that εit has no serial correlation. The reason to parametrize serial correlation under quasi-differencing is that, with unrestricted serial correlation, lagged dependent variables will not be valid instruments. REMARK 3: The estimation procedure can be modified to handle unbalanced data. The procedure is elaborated in the Supplemental material; the details are omitted here. 3.3. Alternative Estimation Methods While the analysis is focused on the method of least squares, we discuss several alternative estimation strategies. METHOD 1: The quasi-differencing method in Holtz-Eakin, Newey, and Rosen (1988) can be adapted for multiple factors. Consider the case of two factors: yit = xit β + λi1 ft1 + λi2 ft2 + εit Multiplying the equation yit−1 by φt = ft1 /ft−11 and then subtracting it from the equation yit , we obtain yit = φt yit−1 + xit β − xit−1 βφt + λi2 δt + εit∗ where δt = ft2 − ft−12 φt and εit∗ = εit − φt εit−1 . The resulting model has a single factor. If we apply the quasi-differencing method one more times to the


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resulting equation, then the factor error will be eliminated. This approach was used by Ahn, Lee, and Schmidt (2006). The GMM method as in Holtz-Eakin, Newey, and Rosen (1988) and Ahn, Lee, and Schmidt (2001) can be used to estimate the model parameters consistently under some identification conditions. For the case of r = 1, GMM was also discussed by Arellano (2003) and Baltagi (2005). While not always necessary, there may be a need to recover the original model parameters; see Holtz-Eakin, Newey, and Rosen (1988) for details. This estimator is consistent under fixed T despite serial correlation and heteroskedasticity in εit . In contrast, the least squares estimator will be inconsistent under this setting. On the other hand, as T increases, due to the manyparameter and many-instrument problem, the GMM method tends to yield bias, a known issue from the existing literature (e.g., Newey and Smith (2004)). The least squares estimator is consistent under large N and large T with unknown form of correlation and heteroskedasticity, and the bias is decreasing in N and T . Furthermore, the least squares method directly estimates all parameters, including the factor processes Ft and the factor loadings λi , so there is no need to recover the original parameters. In many applications, the estimated factor processes are used as inputs for further analysis, for example, the diffusion index forecasting of Stock and Watson (2002) and factor-augmented vector autoregression of Bernanke, Boivin, and Eliasz (2005). The estimated loadings are useful objects in finance; see Connor and Korajzcyk (1986). Recovering those original parameters becomes more involved under multiple factors with the quasi-differencing approach. The least squares method is simple and effective in handling multiple factors. Furthermore, computation of the least squares method under large N and large T is quite fast. In summary, with small T and with potential serial correlation and time series heteroskedasticity, the quasi-differencing method is recommended in view of its consistency properties. With large T , the least squares method is a viable alternative. Remark 2 suggests another alternative under small T . METHOD 2: We can extend the argument of Mundlak (1978) and Chamberlain (1984) to models with interactive effects. When λi is correlated with the regressors, it can be projected onto the regressors such that λi = AX¯ i· + ηi , where X¯ i· is the time average of Xit and A is r × p, so that model (1) can be rewritten as Yit = Xit β + X¯ i· δt + ηi Ft + εit where δt = A Ft . The above model still has a factor error structure. However, when Ft is assumed to be uncorrelated with the regressors, the aggregated error ηi Ft + εit is now uncorrelated with the regressors, so we can use a random-

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effects GLS to estimate (β δ1 δT ). Similarly, when Ft is correlated with the regressors, but λi is not, one can project Ft onto the cross-sectional averages such that Ft = BX¯ ·t + ξt to obtain Yit = Xit β + X¯ ·t ρi + λi ξt + εit with ρi = Bλi . Again, a random-effects GLS can be used. When both λi and Ft are correlated with regressors, we apply both projections and augment the model with cross-products of X¯ i· and X¯ ·t , in addition to X¯ i· and X¯ ·t so that, with ρi = B ηi and δt = A ξt , (14)

Yit = Xit β + X¯ i· δt + X¯ ·t ρi + X¯ i· C X¯ ·t + ηi ξt + εit

where C is a matrix. The above can be estimated by the random-effects GLS. METHOD 3: The method of Pesaran (2006) augments the model with regressors (Y¯ ·t X¯ ·t ) under the assumption of Ft being correlated with regressors, where Y¯ ·t and X¯ ·t attempt to estimate Ft , similar to the projection argument of Mundlak. But in the Mundlak argument, the projection residual ξt is assumed to have a fixed variance. In contrast, the variance of ξt is assumed to converge to zero as N → ∞ in Pesaran (2006), who assumed Xit is N of the form Xit = Bi Ft + eit so that X¯ ·t = BFt + ξt with B = N −1 i=1 Bi and N −1 ξt = N −1 i=1 eit . The variance of ξt is of O(N ). Thus the factor error λ√i ξt is √ negligible under large N. He established N consistency and possible NT consistency for some special cases. It appears that when λi is correlated with the regressors, additional regressors Y¯ i· and X¯ i· should also be added to achieve consistency. 4. ASSUMPTIONS In this section, we state assumptions needed for consistent estimation and explain the meaning of each assumption prior to or after its introduction. Throughout, for a vector or matrix A, its norm is defined as A = (tr(A A))1/2 . The p × p matrix N N N 1 1 1 X MF Xi − X MF Xk aik D(F) = NT i=1 i T N 2 i=1 k=1 i where aik = λi (Λ Λ/N)−1 λk , plays an important role in this paper. Note that aik = aki since it is a scalar. The identifying condition for β is that D(F) is positive definite. If F were observable, the identification condition for β would be that the first term of D(F) on the right-hand side is positive definite. The pres-


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ence of the second term is because of unobservable F and Λ. The reason for this particular form is the nonlinearity of the interactive effects. Define a T × p vector Zi = MF Xi −

N 1 MF Xk aik N k=1

so that Zi is equal to the deviation of MF Xi from its mean, but here the mean is a weighted average. Write Zi = (Zi1 Zi2 ZiT ) . Then

N N T 1 1 1 Z Zi = Zit Zit D(F) = NT i=1 i N i=1 T t=1 N The first equality follows from aik = aki and N −1 i=1 aik aij = akj , and the second equality is by definition. Thus D(F) is at least semipositive definite. Since each Zit Zit is a rank 1 semidefinite matrix, summation of NT such semidefinite matrices should lead to a positive definite matrix, given enough variations in Zit over i and t. Our first condition assumes D(F) is positive definite in the p limit. Suppose that as N T → ∞, D(F) −→ D > 0. If εit are i.i.d. (0 σ 2 ), then the limiting distribution of βˆ can be shown to be √ NT (βˆ − β) → N(0 σ 2 D−1 ) This shows the need for D(F) to be positive definite. Since F is to be estimated, the identification condition for β is assumed as follows: ASSUMPTION A—E Xit 4 ≤ M: Let F = {F : F F/T = I}. We assume inf D(F) > 0

F∈F

The matrix F in this assumption is T × r, either deterministic or random. This assumption rules out time-invariant regressors and common regressors. Suppose Xi = xi ιT , where xi is a scalar and ιT = (1 1 1) . For ιT ∈ F and D(ιT ) = 0, it follows that infF D(F) = 0. A common regressor does not vary with i. Suppose all regressors are common such that Xi = W . For F = W (W W )−1/2 ∈ F , D(F) = 0. The analysis of time-invariant regressors and common regressors is postponed to Section 10, where we show that it is sufficient to have D(F) > 0, when evaluated at the true factor process F . For now, it is not difficult to show that if Xit is characterized by (3), where ηit have sufficient variations such as i.i.d. with positive variance, then Assumption A is satisfied.

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ASSUMPTION B: T p (i) E Ft 4 ≤ M and T1 t=1 Ft Ft −→ ΣF > 0 for some r × r matrix ΣF , as T → ∞. p (ii) E λi 4 ≤ M and Λ Λ/N −→ ΣΛ > 0 for some r ×r matrix ΣΛ , as N → ∞. This assumption implies the existence of r factors. Note that whether Ft or λt has zero mean is of no issue since they are treated as parameters to be estimated; for example, it can be a linear trend (Ft = t/T ). But if it is known that Ft is a linear trend, imposing this fact gives more efficient ∞ estimation. Moreover, Ft itself can be a dynamic process such that Ft = i=1 Ci et−i , where et are i.i.d. zero mean process. Similarly, λi can be cross-sectionally correlated. ASSUMPTION C—Serial and Cross-Sectional Weak Dependence and Heteroskedasticity: (i) E(εit ) = 0 and E|εit |8 ≤ M. (ii) E(εit εjs ) = σijts , |σijts | ≤ σ¯ ij for all (t s) and |σijts | ≤ τts for all (i j) such that N 1 σ¯ ij ≤ M N ij=1

T 1 τts ≤ M T ts=1

1 |σijts | ≤ M NT ijts=1

The largest eigenvalue of Ωi = E(εi εi ) (T × T ) is bounded uniformly in i and T . N (iii) For every (t s) E|N −1/2 i=1 [εis εit − E(εis εit )]|4 ≤ M. (iv) Moreover | cov(εit εis εju εjv )| ≤ M T −2 N −1 tsuv ij

T −1 N −2

| cov(εit εjt εks εs )| ≤ M

ts ijk

Assumption C is about weak serial and cross-sectional correlation. Heteroskedasticity is allowed, but εit is assumed to have a uniformly bounded eighth moment. The first three conditions are relatively easy to understand and are assumed in Bai (2003). We explain the meaning of C(iv). Let ηi = T T (T −1/2 t=1 εit )2 − E(T −1/2 t=1 εit )2 . Then E(ηi ) = 0 and E(η2i ) is bounded. N The expected value (N −1/2 i=1 ηi )2 is equal to T −2 N −1 tsuv ij cov(εit εis εju εjv ), that is, the left-hand side of the first inequality without the absolute sign. So the first part of C(iv) is slightly stronger than the assumption that the N second moment of N −1/2 i=1 ηi is bounded. The meaning of the second part is similar. It can be easily shown that if εit are independent over i and t with Eεit4 ≤ M for all i and t, then C(iv) is true. If εit are i.i.d. with zero mean and Eεit8 ≤ M, then Assumption C holds.


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ASSUMPTION D: εit is independent of Xjs , λj , and Fs for all i t j, and s. This assumption rules out dynamic panel data models and is given for the purpose of simplifying the proofs. The procedure works well even with lagged dependent variables; see Table V in the Supplemental Material. We do allow Xit , Ft , and εit to be dynamic processes. If lagged dependent variables are included in Xit , then εit cannot be serially correlated. Also note that Xit , λi , and εit are allowed to be cross-sectionally correlated. 5. LIMITING THEORY 0

0

We use (β F ) to denote the true parameters, and we still use λi without the superscript 0 as it is not directly estimated and thus not necessary. Here F 0 denotes the true data generating process for F that satisfies Assumption B. This F 0 in general has economic interpretations (e.g., supply shocks and demand shocks). The estimator Fˆ below estimates a rotation of F 0 .6 Define SNT (β F) as the concentrated objective function in (13) divided by NT together with centering, that is, SNT (β F) =

N N 1 1 (Yi − Xi β) MF (Yi − Xi β) − ε MF 0 εi NT i=1 NT i=1 i

The second term does not depend on β and F , and is for the purpose of centering, where MF = I − PF = I − FF /T with F F/T = I. We estimate β0 and F 0 by ˆ = arg min SNT (β F) ˆ F) (β βF

ˆ satisfies ˆ F) As explained in the previous section, (β N

−1 N βˆ = Xi MFˆ Xi Xi MFˆ Yi

i=1

i=1

1 ˆ ˆ ˆ ˆ (Yi − Xi β)(Y i − Xi β) F = FVNT NT i=1 N

where Fˆ is the the matrix that consists of the first r eigenvectors (multiplied by √ N 1 ˆ ˆ T ) of the matrix NT i=1 (Yi − Xi β)(Yi − Xi β) and where VNT is a diagonal If (6) and (7) hold for the data generating processes (i.e., F 0 F 0 /T = I and Λ0 Λ0 is diagonal) rather than being viewed as estimation restrictions, then Fˆ estimates F 0 itself instead of a rotation of F 0 . 6

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JUSHAN BAI

matrix that consists of the first r largest eigenvalues of this matrix. Denote PA = A(A A)−1 A for a matrix A. PROPOSITION 1—Consistency: Under Assumptions A–D, as N T → ∞, the following statements hold: p (i) The estimator βˆ is consistent such that βˆ − β0 −→ 0. p ˆ (ii) The matrix F 0 F/T is invertible and PFˆ − PF 0 −→ 0. The usual argument of consistency for extreme estimators would involve p showing SNT (β F) −→ S(β F) uniformly on some bounded set of β and F , and then showing that S(β F) has a unique minimum at β0 and F 0 ; see Newey and McFadden (1994). This argument needs to be modified to take into account the growing dimension of F . As F is a T × r vector, the limit S would involve an infinite number of parameters as N T going to infinity so the limit as a function of F is not well defined. Furthermore, the concept of bounded F is not well defined either. In this paper, we only require F F/T = I. The modification is similar to Bai (1994), where the parameter space (the break point) increases with the sample size. We show there exists a function SÑT (β F), depending on (N T ) and generally still a random function, such that SÑT (β F) has a unique minimum at β0 and F 0 . In addition, we show the difference is uniformly small, SNT (β F) − SÑT (β F) = op (1) where op (1) is uniform. This implies the consistency of βˆ for β0 . However, we cannot claim the consistency of Fˆ for F 0 (or a rotation of F 0 ) owing to its growing dimension. Part (ii) claims that the spaces spanned by Fˆ and F 0 are asymptotically the same. Alternative consistency concepts, including componentwise consistency or average norm consistency, are provided in the Appendix, as these consistency concepts are also needed. Given consistency, we can further establish the rate of convergence. THEOREM 1—Rate of Convergence: Assume Assumptions A–D hold. For √ comparable N and T such that T/N → ρ > 0, then NT (βˆ − β0 ) = Op (1). The theorem allows cross-section and serial correlations, as well as heteroskedasticities in both dimensions. This is important for applications in macroeconomics, say cross-country studies, or in finance, where the factors may not fully capture the cross-section correlations, and therefore the approximate factor model of Chamberlain and Rothschild (1983) is relevant. For microeconomic data, cross-section √ heteroskedasticity is likely to be present. Although the estimator is NT consistent, the underlying limiting distribution will not be centered at zero; asymptotic biases exist. The next two theorems provide the limiting behavior of the estimator. The first theorem deals

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with some special cases in which asymptotic bias is absent. This is obtained by requiring stronger assumptions: the absence of either cross-correlation or serial correlation and heteroskedasticity. The second theorem deals with the most general case that allows for correlation and heteroskedasticity in both dimensions. Introduce Zi = MF 0 Xi −

N 1 aik MF 0 Xk N k=1

Then in the absence of serial correlation and heteroskedasticity in one of the dimensions, and given an appropriate relative rate for T and N, it is shown in the Appendix that the estimator has the representation (15)

√ NT (βˆ − β0 ) =

N 1 Z Zi NT i=1 i

−1

N 1 Zi εi + op (1) √ NT i=1

If correlation and heteroskedasticity are present in both dimensions, there will be an Op (1) bias term in the above representation; see (21) in SecN tion 7. In all cases, we need the central limit theorem for (NT )−1/2 i=1 Zi εi = N T (NT )−1/2 i=1 t=1 Zit εit . Assuming correlation and heteroskedasticity in both dimensions, its variance is given by var √

N

1 NT

Zi εi =

i=1

N N T T 1 σijts E(Zit Zjs ) NT i=1 j=1 t=1 s=1

where σijts = E(εit εjs ). This variance is O(1) because assumption.

1 NT

ijts

|σijts | ≤ M by

ASSUMPTION E: For some nonrandom positive definite matrix DZ , (16)

plim

√

N T T N 1 σijts Zit Zjs = DZ NT i=1 j=1 t=1 s=1

1 NT

N

d

Zi εi −→ N(0 DZ )

i=1

In the absence of serial correlation and heteroskedasticity, we let σij = σijtt = E(εit εjt ) since it does not depend on t, and we denote DZ by D1 . Likewise, with no cross-section correlation and heteroskedasticity, we let ωts =

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JUSHAN BAI

σiits = E(εit εis ) since it does not depend on i, and we denote DZ by D2 . That is, D1 and D2 are the probability limits of (17)

plim

N N T 1 σij Zit Zjt = D1 NT i=1 j=1 t=1

plim

T N T 1 ωts Zit Zis = D2 NT t=1 s=1 i=1

The corresponding central limit theorem will be denoted by N d 1 N(0 D1 ) and √NT i=1 Zi εi −→ N(0 D2 ), respectively.

√1 NT

N i=1

d

Zi εi −→

THEOREM 2: Assume Assumptions A–E hold. As T N → ∞, the following statements hold: (i) In the absence of serial correlation and heteroskedasticity and with T/N → 0, √ d −1 NT (βˆ − β0 ) −→ N(0 D−1 0 D1 D0 ) (ii) In the absence of cross-section correlation and heteroskedasticity and with N/T → 0, √ d −1 NT (βˆ − β0 ) −→ N(0 D−1 0 D2 D0 ) 1 where D0 = plim D(F 0 ) = plim NT

N i=1

Zi Zi .

Noting that D1 = D2 = σ 2 D0 under i.i.d. assumption of εit , the following statement holds: COROLLARY 1: Under the assumptions of Theorem 1, if εit are i.i.d. over t √ d and i, with zero mean and variance σ 2 , then NT (βˆ − β0 ) −→ N(0 σ 2 D−1 0 ). It is conjectured that βˆ is asymptotically efficient if εit are i.i.d. N(0 σ 2 ), based on the argument of Hahn and Kuersteiner (2002). Part (i) of Theorem 1 still permits cross-section correlation and heteroskedasticity, and part (ii) still permits serial correlation and heteroskedasticity. The theorem also requires an appropriate rate for N and T . If T/N converges to a constant, there will be a bias term due to correlation and heteroskedasticity. The next theorem is concerned with this bias. We shall deal with the more general case in which correlation and heteroskedasticity exist in both dimensions.

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THEOREM 3: Assume Assumptions A–E hold and T/N → ρ > 0. Then √ d −1 NT (βˆ − β0 ) −→ N ρ1/2 B0 + ρ−1/2 C0 D−1 0 DZ D0 where B0 is the probability limit of B with (18)

−1 N N 1 (Xi − Vi ) F 0 F 0 F 0 N i=1 k=1 T T

−1 T 1 ΛΛ λk σiktt × N T t=1

B = −D(F 0 )−1

and C0 is the probability limit of C with (19) and Vi = E(εk εk ).

0 0 −1 −1 N ΛΛ 1 0 F F Xi MF 0 ΩF λi C = −D(F ) NT i=1 T N 0 −1

1 N

N j=1

aij Xj , aij = λi (Λ Λ/N)−1 λj , and Ω =

1 N

N k=1

Ωk with Ωk =

There will be no biases in the absence of correlations and heteroskedasticities. In particular, bias B0 = 0 when cross-sectional correlation and heteroskedasticity are absent, and similarly C0 = 0 when serial correlation and heteroskedasticity are absent. To see this, consider C in (19). The absence of serial and heteroskedasticity implies Ωk = σk2 IT ; thus, MF 0 ΩF 0 = correlation 2 0 ( k σk )MF 0 F = 0. It follows that C = 0 and hence C0 = 0. The parametric form of serial correlations is usually removed by adding lagged dependent variables. However, lagged dependent variables lead to bias with fixed-effect estimators; see Hahn and Kuersteiner (2002) in a different context. The bias will take a different form and is not studied here. The argument for B = 0 is not so obvious and is provided in the proof of Theorem 2(ii). When εit are i.i.d. over t and over i, then both B0 and C0 become zero, and the result specializes to Corollary 1. These results assume no lagged dependent variables, whose presence will lead to additional bias, which is not studied in this paper. REMARK 4: Suppose that k factors √ are allowed in the estimation, with k fixed but k ≥ r. Then βˆ remains NT consistent, albeit less efficient than k = r. Consistency relies on controlling the space spanned by Λ and that of F , which is achieved when k ≥ r. √ ˆ estimation of β does not affect REMARK 5: Due to NT consistency for β, the rates of convergence and the limiting distributions of Fˆt and λˆ i . That is, they are the same as that of the pure factor model of Bai (2003). This follows

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JUSHAN BAI

from Yit − Xit βˆ = λi Ft + eit + Xit (βˆ − β), which is a pure factor model with an added error Xit (βˆ − β) = (NT )−1/2 Op (1). An error of this order of magnitude does not affect the analysis. 6. INTERPRETATIONS OF THE ESTIMATOR The Meaning of D(F) and the Within-Group Interpretation Like the least squares dummy-variable (LSDV) estimator, the interactiveeffects estimator βˆ is a result of least squares with the effects being estimated. In this sense, it is a within estimator. It is more instructive, however, to compare the mathematical expressions of the two estimators. Write the additive-effects model (2) in matrix form: (20)

Y = β1 X 1 + β2 X 2 + · · · + βp X p + ιT α + ξιN + ε

where Y and X k (k = 1 2 p) are matrices of T × N, with X k being the regressor matrix associated with parameter βk (a scalar); ιT is a T × 1 vector with all elements being 1 and similarly for ιN ; α = (α1 αN ) and ξ = (ξ1 ξT ) . Define MT = IT − ιT ιT /T

MN = IN − ιN ιN /N

Multiplying equation (20) by MT from the left and by MN from the right yields MT Y MN = β1 (MT X 1 MN ) + · · · + βp (MT X p MN ) + MT εMN The least squares dummy-variable estimator is simply the least squares applied to the above transformed variables. The interactive-effects estimator has a similar interpretation. Rewrite the interactive-effects model (5) as Y = β1 X 1 + · · · + βp X p + FΛ + ε Then left multiply MF and right multiply MΛ to obtain MF Y MΛ = β1 (MF X 1 MΛ ) + · · · + βp (MF X p MΛ ) + MF εMΛ Let βˆ Asy be the least squares estimator obtained from the above transformed variables, treating F and Λ as known. That is, ⎤ ⎡ tr[MΛ X 1 MF X 1 ] · · · tr[MΛ X 1 MF X p ] −1 ⎦ βˆ Asy = ⎣ tr[MΛ X p MF X 1 ] · · · tr[MΛ X p MF X p ] ⎤ ⎡ tr[MΛ X 1 MF Y ] ⎦ ×⎣ tr[MΛ X p MF Y ]


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The square matrix on the right without inverse is equal to D(F) up to a scaling constant, that is, N 1 Z Zi D(F) = T N i=1 i ⎡ tr[MΛ X 1 MF X 1 ] 1 ⎣ = TN tr[MΛ X p MF X 1 ]

···

⎤ tr[MΛ X 1 MF X p ] ⎦

· · · tr[MΛ X p MF X p ]

This can be verified by some calculations. The estimator βˆ Asy can be rewritten as βˆ Asy =

N i=1

−1 i

Z Zi

N

Zi Yi

i=1

√ √ It follows from (15) that NT (βˆ − β) = NT (βˆ Asy − β) + op (1). To purge the fixed effects, the LSDV estimator uses MT and MN to transform the variables, whereas the interactive-effects estimator uses MF and MΛ to transform the variables. 7. BIAS-CORRECTED ESTIMATOR The interactive-effect estimator is shown to have the representation (see Proposition A.3 in the Appendix) (21)

1/2 N √ 1 T 0 0 −1 ˆ NT (β − β ) = D(F ) √ Zi εi + B N NT i=1 1/2 N + C + op (1) T

where B and C are given by (18) and (19), respectively, and they give rise to the biases. Their presence arises from correlations and heteroskedasticities in εit . We show that B and C can be consistently estimated so that a bias-corrected estimator can be constructed, as in the framework of Hahn and Kuersteiner (2002) and Hahn and Newey (2004). Attention is paid to heteroskedasticities in both dimensions, assuming no correlation in either dimension to simplify the presentation. We do point out how to estimate the biases consistently and outline the idea of the proof when correlation exists in either dimension.

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Under the assumption of E(εit2 ) = σit2 and E(εit εjs ) = 0 for i = j or t = s, term B becomes −1 −1 N ΛΛ 1 (Xi − Vi ) F 0 F 0 F 0 B = −D(F 0 )−1 (22) λi σ¯ i2 N i=1 T T N T where σ¯ i2 = T1 t=1 σit2 . The bias can be estimated by replacing F 0 by Fˆ , λi by T 2 ˆ λˆ i , and σ¯ i2 by σ¯î = T1 t=1 εˆ it2 . This gives, in view of Fˆ F/T = Ir , (23)

−1 N 2 (Xi − Vî ) Fˆ Λˆ Λˆ −1 1 ˆ ˆ λˆ i σ¯ˆ i B = −D0 N i=1 T N

The expression C is still given by (19), but Ω now becomes a diagonal matrix N N 2 2 N1 k=1 σkT ). Let Ωˆ = under no correlation, that is, Ω = diag( N1 k=1 σk1 N N 1 1 2 2 diag( N k=1 εˆ k1 N k=1 εˆ kT ) be an estimator for Ω. We estimate C by (24)

ˆ ˆ −1 N ΛΛ −1 1 ˆ ˆ ˆ ˆ Xi MFˆ ΩF λˆ i C = −D0 NT i=1 N

In the Appendix we prove (T/N)1/2 (Bˆ − B) = op (1) and (N/T )1/2 (Cˆ − C) = op (1). Define 1 1 ˆ βˆ † = βˆ − Bˆ − C N T THEOREM 4: Assume Assumptions A–E hold. In addition, E(εit2 ) = σit2 and E(εit εjs ) = 0 for i = j or t = s. If T/N 2 → 0 and N/T 2 → 0, then √ d −1 NT (βˆ † − β0 ) −→ N(0 D−1 0 D3 D0 ) 1 where D3 = plim NT

N T i=1

t=1

Zit Zit σit2 .

The limiting variance D3 is a special case of DZ due to the no correlation assumption. Bias correction does not contribute to the limiting variance. Also note that conditions N/T 2 → 0 and T/N 2 → 0 are added. Clearly, these conditions are less restrictive than T/N converging to a positive constant. There exist other bias correction procedures (e.g., panel jackknife) that could be used; see Arellano and Hahn (2005) and Hahn and Newey (2004). An alternative to bias correction in the case of T/N → ρ > 0 is to use the Bekker (1994) standard errors to improve inference accuracy. This strategy was studied by Hansen, Hausman, and Newey (2005) in the context of many instruments.


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REMARK 6: Consider estimating C in the presence of serial correlation. We need consistent estimators for T −1 Xi Ωk F 0 and T −1 F 0 Ωk F 0 , where Ωk = Eεk εk (T × T ), and then we take (weighted) averages over i and over k. Thus consider estimating them for each given (i k). These terms are standard expressions in the usual heteroskedasticity and autocorrelation (HAC) robust limiting covariance. To see this, let Wi = (Xi F 0 ) which is T × (p + r). Then the T long-run variance of T −1/2 Wi εk = T −1/2 t=1 Wit εkt is the limit of T1 Wi Ωk Wi , which contains T1 Xi Ωk F 0 and T1 F 0 Ωk F 0 as subblocks. A consistent estimator for T −1 Wi ΩWi can be constructed by the truncated kernel method of Newey and West (1987) based on the sequence Wˆ it εˆ kt (t = 1 T ). Similar argument has been made in Bai (2003). REMARK 7: While estimating B in the presence of cross-section correlation is not difficult, the underlying theory for consistency requires a different argument. In the time series dimension, the data are naturally ordered and distant observations have less correlations. The kernel method puts small weights for autocovariances with large lags, leading to consistent estimation. In the crosssection dimension, such an ordering of data is not available, unless an economic distance can be constructed so that the data can be ordered. In general, large |i − j| does not mean smaller correlation between εit and εjt . Bai and Ng (2006) studied the estimation of an object similar to B. They showed that if the whole cross-sample is used in the estimation, the estimator is inconsistent. A partial sample estimator, with N being replaced by n such that n/N → 0 and n/T → 0, is consistent. Thus, B can be estimated by (25)

ˆ −1 1 Bˆ = −D 0 n

−1 n n (Xi − Vî ) Fˆ Λˆ Λˆ i=1 k=1

T

N

T ˆλk 1 εˆ it εˆ kt T t=1

where n/N → 0 and n/T → 0. The argument of Bai and Ng (2006) can be adapted to show that Bˆ is consistent for B. Estimating the Covariance Matrices To estimate D0 , we define T N ˆ0 = 1 Zˆ it Zˆ it D NT i=1 t=1

ˆ rewhere Zˆ it is equal to Zit with F 0 , λi , and Λ replaced with Fˆ , λˆ i , and Λ, spectively. Next consider estimating Dj , j = 1 2 3. For all cases, we limit our attention to the presence of heteroskedasticity, but no correlation. Thus Dj

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JUSHAN BAI

(j = 1 2 3) are covariance matrices when heteroskedasticity exists in the crosssection dimension only, in the time dimension only, and in both dimensions, respectively. Thus we define

N T 1 1 ˆ1 = D σˆ 2 Zˆ it Zˆ it N i=1 i T t=1

T N 1 1 ˆ2 = D ω ˆ2 Zˆ it Zˆ it T t=1 t N i=1 T N ˆ3 = 1 D Zˆ it Zˆ it εˆ it2 NT i=1 t=1

where σˆ i2 =

1 T

T t=1

εˆ it2 , ω ˆ 2t =

1 N

N i=1

εˆ it2 , and Zˆ it was defined previously.

PROPOSITION 2: Assume Assumptions A–E hold. Then as N T → ∞, p ˆ D0 −→ D0 . In addition, in the absence of serial and cross-section correlations, p ˆ j −→ Dj , where D1 and D2 are defined in Theorem 2 with no correlation, and D D3 is defined in Theorem 4. REMARK 8: When cross-section correlation exists, we estimate D1 in (17) by n n T 1 1 ˆ ˆ ˆ D1 = Zit Zjt εˆ it εˆ jt n i=1 j=1 T t=1

where n satisfies n/N → 0 and n/T → 0; see Remark 7. It can be shown ˆ 1 is consistent for D1 . When serial correlation exists, we estimate D2 that D of (17) by estimating the long-run variance of the sequence {Zˆ it εˆ it } using the truncated kernel of Newey and West (1987); see Remark 6. It can be ˆ 2 is consistent for D2 . For estimating DZ —the covariance mashown that D trix when correlation exists in both dimensions—we need to use the partial sample method together with the Newey–West procedure. More specifically, n let ξˆ t = n−1/2 i=1 Zˆ it εˆ it , where n is chosen as before. The estimated long-run variance (e.g., truncated kernel) for the sequence ξˆ t is an estimator for DZ . While we conjecture the estimator is consistent, a formal proof remains to be explored. 8. MODELS WITH BOTH ADDITIVE AND INTERACTIVE EFFECTS Although interactive-effects models include the additive models as special cases, additivity has not been imposed so far, even when it is true. When addi-

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tivity holds but is ignored, the resulting estimator is less efficient. In this section, we consider the joint presence of additive and interactive effects, and show how to estimate the model by imposing additivity and derive the limiting distribution of the resulting estimator. Consider (26)

Yit = Xit β + μ + αi + ξt + λi Ft + εit

where μ is the grand mean, αi is the usual fixed effect, ξt is the time effect, and λi Ft is the interactive effect. Restrictions are required to identify the model. Even in the absence of the interactive effect, the restrictions (27)

N

αi = 0

i=1

T

ξt = 0

t=1

are needed; see Greene (2000, p. 565). The following restrictions are maintained: (28)

F F/T = Ir

Λ Λ = diagonal

Further restrictions are needed to separate the additive and interactive effects. They are (29)

N i=1

λi = 0

T

Ft = 0

t=1

N To see this, suppose that λ¯ = N1 i=1 λi = 0 or F¯ = not zero. Let λ†i = λi − 2λ¯ and Ft† = Ft − 2F¯ . Then

1 T

T t=1

Ft = 0, or both are

Yit = Xit β + μ + α†i + ξt† + λ†i Ft† + εit where α†i = αi + 2F¯ λi − 2λ¯ F¯ and ξt† = ξt + 2λ¯ Ft − 2λ¯ F¯ . It is easy to verify that F † F † /T = F F/T = Ir and Λ† Λ† = Λ Λ is diagonal, and at the same time, N † T † i=1 αi = 0 and t=1 ξt = 0 Thus the new model is observationally equivalent to (26) if (29) is not imposed. To estimate the general model under the given restrictions, we introduce some standard notation. For any variable φit , define N 1 φit φ¯ ·t = N i=1

T 1 φ¯ i· = φit T t=1

T N 1 φ¯ ·· = φit NT i=1 t=1

φ˙ it = φit − φ¯ i· − φ¯ ·t + φ¯ ·· and its vector form φ˙ i = φi − ιT φ¯ i· − φ¯ + ιT φ¯ ··, where φ¯ = (φ¯ ·1 φ¯ ·T ) .

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JUSHAN BAI

The least squares estimators are ˆ μˆ = Y¯ ·· − X¯ ·· β αˆ i = Y¯ i· − X¯ i· βˆ − μ ˆ ˆ ξˆ t = Y¯ ·t − X¯ ·t βˆ − μ −1 N N ˙ ˙ ˆ β= X MFˆ Xi X˙ MFˆ Y˙ i i

i=1

i

i=1

ˆ and by √ F is the T × r matrix consisting of the first r eigenvectors (multiplied N 1 ˙ T ) associated with the first r largest eigenvalues of the matrix NT i=1 (Yi − ˙ ˆ such that ˆ F) ˆ Y˙ i − X˙ i β) ˆ . Finally, Λˆ is expressed as a function of (β Xi β)( ˆ Fˆ (Y˙ N − X˙ N β)] ˆ Λˆ = (λˆ 1 λˆ 2 λˆ N ) = T −1 [Fˆ (Y˙ 1 − X˙ 1 β) ˆ αˆ i , Iterations are required to obtain βˆ and Fˆ . The remaining parameters u, ˆξt , and Λˆ require no iteration, and they can be computed once βˆ and Fˆ are obtained. The solutions for μ ˆ αˆ i , and ξˆ t have the same form as the usual fixedeffects model; see Greene (2000, p. 565). ˆ are indeed the least squares estiˆ Λ) ˆ F We shall argue that (μ ˆ {αˆ i } {ξˆ t } β mators from minimization of the objective function T N (Yit − Xit β − μ − αi − ξt − λi Ft )2 i=1 t=1

subject to the restrictions (27)–(29). Concentrating out (μ, {αi }, {ξt }) is equivalent to using (Y˙ it X˙ it ) to estimate the remaining parameters. So the concentrated objective function is T N (Y˙ it − X˙ it β − λi Ft )2 i=1 t=1

The dotted variable for λi Ft is itself, that is, c˙it = cit , where cit = λi Ft due to restriction (29). This objective function is the same as (8), except Yit and Xit are replaced by their dotted versions. From the analysis in Section 3, the least squares estimators for β, F , and Λ are as prescribed above. Given these estimates, the least squares estimators for (μ {αi } {ξt }) are also immediately obtained as prescribed.


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N We next argue that all restrictions are satisfied. For example, N1 i=1 αˆ i = T Y¯ ·· − X¯ ··βˆ − μˆ = μˆ − μˆ = 0. Similarly, t=1 ξˆ t = 0. It requires an extra argument T to show t=1 Fˆt = 0. By definition, N 1 ˆ NT = ˆ ˆ Y˙ i − X˙ i β) ˆ F FV (Y˙ i − X˙ i β)( NT i=1 Multiplying ιT = (1 1) on each side yields N 1 ˙ ˆ ˆ ˆ Y˙ i − X˙ i β) ˆ F ιT FVNT = ι (Yi − X˙ i β)( NT i=1 T T but ιT Y˙ i = t=1 Y˙ it = 0 and, similarly, ιT X˙ i = 0. Thus the right-hand side is N zero, implying ιT Fˆ = 0. The same argument leads to i=1 λˆ i = 0. ˆ we define To derive the asymptotic distribution for β, N 1 aik MF X˙ k Z˙ i (F) = MF X˙ i − N k=1

and

N 1 ˙ ˙ D(F) = Zi (F) Z˙ i (F) NT i=1

where aik = λi (Λ Λ/N)−1 λk . We assume (30)

˙ inf D(F) > 0 F

Let Z˙ i = Z˙ i (F 0 ). Notice that Y˙ it = X˙ it β + λi Ft + ε˙ it The entire analysis of Section 4 can be restated here. In particular, under the conditions of Theorem 2, we have the asymptotic representation √ NT (βˆ − β0 ) =

N 1 ˙ ˙ Z Zi NT i=1 i

−1 √

1 NT

N

Z˙ i ε˙ i + op (1)

i=1

In the Supplemental Material, we show the identity (see Lemma A.13) N ˙ ˙ i ≡ N Z˙ εi . That is, ε˙ i can be replaced by εi . It follows that if i=1 Zi ε i=1 i N 1 ˙ normality is assumed for √NT i=1 Zi εi , asymptotic normality also holds for √ NT (βˆ − β).

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N 1 ˙ 0 > 0; (ii) ASSUMPTION F: (i) plim NT Z˙ Z˙ i = D i=1 i 1 ˙ Z ), where D ˙ Z = plim ˙ ˙ N(0 D ijts σijts Zit Zjs . NT

√1 NT

N i=1

d Z˙ i εi −→

THEOREM 5: Assume Assumptions A–F hold. Then as T N → ∞, the following statements hold: (i) Under the assumptions of Theorem 2(i), √ d ˙ −1 D ˙ 1D ˙ −1 ) NT (βˆ − β0 ) −→ N(0 D 0 0 (ii) Under the assumptions of Theorem 2(ii), √ d ˙ −1 D ˙ 2D ˙ −1 ) NT (βˆ − β0 ) −→ N(0 D 0 0 ˙ 2 are special cases of D ˙ Z. ˙ 1 and D where D An analogous result to Theorem 3 also holds, and bias-corrected estimators can also be considered. Since the analysis holds with Xi replaced by X˙ i , details are omitted. 9. TESTING ADDITIVE VERSUS INTERACTIVE EFFECTS There exist two methods to evaluate which specification—fixed effects or interactive effects—gives a better description of the data. The first method is that of the Hausman test statistic (Hausman (1978)) and the second is based on the number of factors. We detail the Hausman test method, delegating the number-of-factors method to the Supplemental Material. Throughout this section, for simplicity, we assume εit are i.i.d. over i and t, and that E(εit2 ) = σ 2 . The null hypothesis is an additive-effects model Yit = Xit β + αi + ξt + μ + εit T N with restrictions i=1 αi = 0 and t=1 ξt = 0 due to the grand mean parameter μ. The alternative hypothesis—more precisely, the encompassing general model—is (31)

(32)

Yit = Xit β + λi Ft + εit

The null model is nested in the general model with λi = (αi 1) and Ft = (1 ξt + μ) . The interactive-effects estimator for β is consistent under both models (31) and (32), but is less efficient than the least squares dummy-variable estimator for model (31), as the latter imposes restrictions on factors and factor loadings. But the fixed-effects estimator is inconsistent under model (32). The principle of the Hausman test is applicable here.


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The within-group estimator of β in (31) is √ NT (βˆ FE − β) =

N 1 ˙ ˙ X Xi NT i=1 i

−1

N 1 ˙ X i εi √ NT i=1

where X˙ i = Xi − ιT X¯ i· − X¯ + ιT X¯ ··. Rewrite the fixed-effects estimator more compactly as √ NT (βˆ FE − β) = C −1 ψ N N 1 ˙ ˙ ˙ √1 where C = ( NT i=1 Xi Xi ) and ψ = NT i=1 Xi εi . The interactive-effects estimator can be written as (see Proposition A.3) √ NT (βˆ IE − β) = D(F 0 )−1 (η − ξ) + op (1) where (33)

η= √

1 NT

N

i

X MF 0 εi

i=1

N N 1 ξ= √ aik Xk MF 0 εi NT i=1 N k=1

The variances of the two estimators are √ var( NT (βˆ FE − β)) = σ 2 C −1

1

√ var( NT (βˆ IE − β)) = σ 2 D(F 0 )−1

In the accompanying document, we show, under the null hypothesis of additivity, (34)

E[(η − ξ)ψ ] = σ 2 D(F 0 )

This implies var(βˆ IE − βˆ FE ) = var(βˆ IE ) − var(βˆ FE ). Thus the Hausman test takes the form d J = NT σ 2 (βˆ IE − βˆ FE ) [D(F 0 )−1 − C −1 ]−1 (βˆ IE − βˆ FE ) −→ χ2p

Replacing D(F 0 ) and σ 2 by their consistent estimators, the above is still ˆ 0 . Let true. Proposition 2 shows that D(F 0 ) is consistently estimated by D p N T 1 σˆ 2 = L i=1 t=1 εˆ it2 , where L = NT − (N + T )r − p. Then σˆ 2 −→ σ 2 . REMARK 9: The Hausman test is also applicable when there are no time effects but only individual effects (i.e., ξt = 0). Then it is testing whether the individual effects are time-varying. Similarly, the Hausman test is applicable when αi = 0 in (31) but ξt = 0. Then it is testing whether the common shocks have heterogeneous effects on individuals. Details are given in the Supplemental Materials.

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JUSHAN BAI

10. TIME-INVARIANT AND COMMON REGRESSORS In earnings studies, time-invariant regressors include education, gender, race, and so forth; common variables are those that represent trends or policies. In consumption studies, common regressors include price variables, which are the same for each individual. Those variables are removed by the withingroup transformation. As a result, identification and estimation must rely on other means such as the instrumental variable approach of Hausman and Taylor (1981). This section considers similar problems under interactive effects. Under some reasonable and intuitive conditions, the parameters of the timeinvariant and common regressors are shown to be identifiable and can be consistently estimated. In effect, those regressors act as their own instruments; additional instruments, either within or outside the system, are not necessary. Ahn, Lee, and Schmidt (2001) allowed for time-invariant regressors, although they did not consider the joint presence of common regressors. Their identification condition relies on nonzero correlation between factor loadings and regressors. A general model can be written as (35)

Yit = Xit ϕ + xi γ + wt δ + λi Ft + εit

where (Xit xi wt ) is a vector of observable regressors, xi is time invariant, and wt is cross-sectionally invariant (common). The dimensions of regressors are such that Xit is p × 1, xi is q × 1, wt is × 1, and Ft is r × 1. Introduce ⎡ ⎤ Xi1 xi w1 ⎡ ⎤ ϕ ⎢ X x w ⎥ i 2 ⎥ ⎢ i2 ⎣ Xi = ⎢ ⎥ β = γ ⎦ ⎣ ⎦ δ XiT xi wT ⎡ w ⎤ ⎡ x ⎤ 1

⎢ x2 ⎥ ⎥ x=⎢ ⎣ ⎦ xN

1

⎢ w2 ⎥ ⎥ W =⎢ ⎣ ⎦ wT

Then the model can be rewritten as Yi = Xi β + Fλi + εi Let (β0 F 0 Λ) denote the true parameters (superscript 0 is not used for Λ). To identify β0 , it was assumed in Section 4 that the matrix −1 N N N 1 1 1 ΛΛ X MF Xi − X MF Xk λi λk D(F) = NT i=1 i T N 2 i=1 k=1 i N


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is positive definite for all possible F . This assumption fails when time-invariant regressors and common regressors exist. This is because D(ιT ) and D(W ) are not full rank matrices. However, the positive definiteness of D(F) is not a necessary condition. In fact, all that is needed is the identification condition D(F 0 ) > 0 That is, the matrix D(F) is positive definite when evaluated at the true F 0 , a much weaker condition than Assumption A. In the Supplemental Material, we show that the above condition can be decomposed into some intuitive assumptions. First, this means that the interactive effects are genuine (not additive effects); otherwise, we are back to the environment of Hausman and Taylor, and instrumental variables must be used to identify β. Second, there should be no multicollinearity between W and F 0 , and no multicollinearity between x and Λ. Finally, W and x cannot both contain the constant regressor (only one grand mean parameter). It remains to argue that D(F 0 ) > 0 (or equivalently, the four conditions above) implies consistent estimation. We state this result as a proposition. p PROPOSITION 3: Assume Assumptions B–D hold. If D(F 0 ) > 0, then βˆ → β0 .

The proof of this proposition is nontrivial and is provided in the Supplemental Material. The proposition implies that D(F 0 ) > 0 is a sufficient condition for consistent estimation. Given consistency, the rest of the argument for rate of convergence does not hinge on any particular structure of the regressors. Therefore, the rate of convergence of βˆ and the limiting distribution are still valid in the presence of the grand mean, time-invariant regressors, and common regressors. More specifically, all results up to Section 7 (inclusive) are valid. The result of Section 8 is valid for regressors with variations in both dimensions. Similarly, hypothesis testing in Section 9 can only rely on the subset of coefficients whose regressors have variations in both dimensions. Discussion When additive effects are also present, (35) becomes Yit = Xit ϕ + μ + αi + ξt + xi γ + wt δ + λi Ft + εit where μ is the grand mean (explicitly written out), and αi and ξt are, respectively, the individual and the time effects. The parameters γ and δ are no longer directly estimable. Under the restrictions of (27) and (29), the withingroup transformation implies Y˙ it = X˙ it φ + λi Ft + ε˙ it . The parameters φ and

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JUSHAN BAI

λi Ft are estimable by the interactive-effects estimator, so they can be treated as known in terms of identification. Letting Yit∗ = Yit − Xit φ − λi Ft , we have (36)

Yit∗ = μ + αi + ξt + xi γ + wt δ + εit

which is a standard model. As in Hausman and Taylor (1981), if we assume a subset of Xit ’s whose time averages are uncorrelated with αi but correlated with xi , time averages can be used as instruments for xi . We can assume a similar instrument for wt to estimate both γ and δ. This is a direct extension of the Hausman and Taylor framework for interactive-effects models. A more interesting setup is to allow time-dependent coefficients for the time-invariant regressors and, similarly, allow individual-dependent coefficients for the common regressors; namely (37)

Yit = Xit φ + μ + αi + ξt + xi γt + wt δi + λi Ft + εit

The observable variables are Yit , Xit , xi , and wt . Again the levels of γt and δi are not directly estimable due to αi + ξt . For example, αi + xi γt = (αi + xi c) + xi (γt − c) = α∗i + xi γt∗ . However, the deviations of γt from its time average γt − E(γt ) and the deviations of δi from its individual average δi − E(δi ) are directly estimable. In practice, these deviations or contrasts may be of more importance than the levels, since they reveal the patterns across individuals or changes over time, just like coefficients on dummy variables. Restrictions are needed to estimate (φ μ γt δi ). For example, we need to impose E(γt ) = 0 or t γt = 0, and we also need similar restrictions for δi , αi , ξt , λi , and Ft , together with some normalization and multicollinearity restrictions. Unreported simulations show that these deviations can be well estimated. To estimate the levels E(γt ) and E(δi ), the Hausman–Taylor approach appears to be applicable as well. In this case, Yit∗ in (36) is replaced by Yit∗ = Yit − Xit φ − xi γt∗ − wt δ∗i − λi Ft , where γt∗ = γt − E(γt ) and δ∗i = δi − E(δi ) are the deviations. The large sample theory of this model warrants a separate study. 11. FINITE SAMPLE PROPERTIES VIA SIMULATIONS We assess the performance of the estimator by Monte Carlo simulations. A general model with common regressors and time-invariant regressors is considered: Yit = Xit1 β1 + Xit2 β2 + μ + xi γ + wt δ + λi Ft + εit ((β1 β2 μ γ δ) = (1 3 5 2 4)) where λi = (λi1 λi2 ) and Ft = (Ft1 Ft2 ) . The regressors are generated according to Xit1 = μ1 + c1 λi Ft + ι λi + ι Ft + ηit1 Xit2 = μ2 + c2 λi Ft + ι λi + ι Ft + ηit2

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PANEL MODELS WITH FIXED EFFECTS TABLE I

MODELS WITH GRAND MEAN, TIME-INVARIANT REGRESSORS AND COMMON REGRESSORS (TWO FACTORS, r = 2)

T

Mean β1 = 1

SD

Mean β2 = 3

100 100 100 100 10 20 50

10 20 50 100 100 100 100

1.003 1.001 1.000 1.000 0.998 1.000 1.000

0.061 0.039 0.025 0.017 0.056 0.039 0.024

2.999 2.998 3.002 3.000 3.002 2.998 3.001

100 100 100 100 10 20 50

10 20 50 100 100 100 100

1.104 1.038 1.010 1.006 1.105 1.038 1.009

0.135 0.083 0.036 0.032 0.133 0.083 0.035

Interactive-Effects Estimator 3.103 0.138 4.611 0.925 3.036 0.084 4.856 0.524 3.012 0.037 4.981 0.156 3.006 0.033 4.992 0.115 3.108 0.135 4.556 0.962 3.037 0.084 4.859 0.479 3.010 0.037 4.974 0.081

N

SD

Mean μ=5

SD

Infeasible Estimator 0.061 4.994 0.103 0.041 5.002 0.065 0.024 5.000 0.039 0.017 5.000 0.029 0.055 4.998 0.098 0.039 5.000 0.064 0.025 4.999 0.040

Mean γ=2

SD

Mean δ=4

SD

1.998 2.000 1.999 1.999 2.002 2.002 2.001

0.060 0.040 0.024 0.017 0.066 0.040 0.025

4.003 4.000 4.000 3.999 4.001 3.999 4.000

0.087 0.054 0.030 0.020 0.063 0.046 0.029

1.952 1.996 1.995 1.996 1.939 1.991 2.000

0.242 0.104 0.098 0.066 0.240 0.109 0.041

3.939 3.989 3.999 3.997 3.949 3.996 4.000

0.250 0.114 0.058 0.061 0.259 0.082 0.033

with ι = (1 1). The regressors are correlated with λi , Ft , and the product λi Ft . The variables λij Ftj , and ηitj are all i.i.d. N(0 1) and the regression error εit is i.i.d. N(0 4). We set μ1 = μ2 = c1 = c2 = 1. Further, xi ∼ ι λi + ei and wt = ι Ft + ηi , with ei and ηi being i.i.d. N(0 1), so that xi is correlated with λi and wt is correlated with Ft . Simulation results are reported in Table I (based on 1000 repetitions). The infeasible estimator in this table assumes observable Ft . Both the infeasible and interactive-effects estimators are consistent, but the latter is less efficient than the former, as expected. The coefficients for the common regressors and time-invariant regressors are estimated well. The within-group estimator can only estimate β1 and β2 and is not reported. We next investigate what happens when interactive-effects estimator is used when the underlying effects are additive. That is, λi = (αi 1) and Ft = (1 ξt ) so that λi Ft = αi + δt . With regressors Xit1 and Xit2 generated with the earlier formula, the model is Yit = Xit1 β1 + Xit2 β2 + αi + ξt + εit We consider three estimators: (i) the within-group estimator, (ii) the infeasible estimator, and (iii) the interactive-effects estimator. All three are consistent. The results are reported in Table II. The interactive-effects estimator remains valid under additive effects, but is less efficient than the within-group estimator, as expected.

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JUSHAN BAI TABLE II MODELS OF ADDITIVE EFFECTS

Within-Group Estimator N

T

100 3 100 5 100 10 100 20 100 50 100 100 3 5 10 20 50

100 100 100 100 100

Interactive-Effects Estimator

Infeasible Estimator

Mean β1 = 1

SD

Mean β2 = 3

SD

Mean β1

SD

Mean β2

SD

Mean β1

SD

Mean β2

SD

1.002 1.001 1.000 0.999 1.001 0.999

0.146 0.099 0.068 0.048 0.029 0.021

2.997 3.002 2.996 2.999 2.999 3.000

0.144 0.100 0.066 0.046 0.029 0.021

1.001 1.001 1.000 0.998 1.001 0.999

0.208 0.114 0.072 0.048 0.029 0.021

2.998 3.003 2.995 2.998 2.999 3.000

0.206 0.118 0.072 0.047 0.029 0.021

1.155 1.189 1.110 1.017 1.003 1.000

0.253 0.194 0.167 0.083 0.029 0.021

3.164 3.190 3.106 3.016 3.000 3.001

0.259 0.186 0.167 0.080 0.029 0.021

1.001 1.000 1.000 1.001 0.998

0.142 0.102 0.069 0.047 0.030

2.995 3.005 2.999 3.000 3.002

0.143 0.100 0.069 0.047 0.029

1.002 1.000 1.001 1.001 0.998

0.113 0.093 0.066 0.045 0.030

2.996 3.006 2.999 3.000 3.002

0.116 0.092 0.065 0.046 0.028

1.163 1.179 1.106 1.018 1.000

0.240 0.190 0.167 0.080 0.030

3.165 3.180 3.106 3.017 3.004

0.251 0.189 0.164 0.080 0.029

Additional simulations are reported in the Supplemental Material, where we consider cross-sectionally correlated eit . Under cross-section correlation in eit and with a fixed N, the interactive-effects estimator is inconsistent. The estimator becomes consistent as N going to infinity. These theoretical results are confirmed by the simulations. A primary use of the factor model in practice is to account for cross-sectional correlations. With a sufficient number of factors included, much of the correlation in the error terms will either be removed or be reduced, making the correlation a less critical issue. We also report results that include lagged dependent variables as regressors. The idea is to parametrize and to control for serial correlation. The parameters are well estimated in the simulation. The interactive-effects estimator is effective under large N and large T . The computation is fast and the bias is decreasing with N and T , as shown in the theory and confirmed in the simulation. 12. CONCLUDING REMARKS In this paper, we have examined issues related to identification and inference for panel data models with interactive effects. In earnings studies, the interactive effects are a result of changing prices for a vector of unmeasured skills. The model can also be motivated from an optimal choice of consumption and labor supply for heterogeneous agents under a competitive economy with complete markets. In macroeconomics, interactive effects represent common shocks and heterogeneous impacts on the cross-units. In finance, the common factors represent marketwide risks and the loadings reflect assets’ exposure to the risks. A factor model is also a useful approach to controlling cross-section


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correlations. This paper focuses on some of the underlying econometric issues. We √ showed that the convergence rate for the interactive-effects estimator is NT , and this rate holds in spite of correlations and heteroskedasticity in both dimensions. We also derived bias-corrected estimator and estimators under additivity restrictions and their limiting distributions. We further studied the problem of testing additive effects against interactive effects. The interactiveeffects estimator is easy to compute, and both the factor process Ft and the factor loadings λi can also be consistently estimated up to a rotation. Under interactive effects, we showed that the grand mean, the coefficients of timeinvariant regressors, and the coefficients of common regressors are identifiable and can be consistently estimated. A useful extension is the large N and large T dynamic panel data model with multiple interactive effects. The argument for consistency and rate of convergence remains the same, but the asymptotic bias will take a different form. Another broad extension is nonstationary panel data analysis, particularly panel data cointegration, a subject that recently attracted considerable attention. In this setup, Xit is a vector of an integrated variable and Ft can be either integrated or stationary. When Ft is integrated, then Yit , Xit , and Ft are cointegrated. Neglecting Ft is equivalent to spurious regression and the estimation of β will not be consistent. However, the interactive-effect approach can be applied by jointly estimating the unobserved common stochastic trends Ft and the model coefficients, leading to consistent estimation. Finally, the models introduced in Section 10 (see Discussion) warrant further investigation. APPENDIX A: PROOFS

T We use the following facts throughout: T −1 Xi 2 = T −1 t=1 Xit 2 = Op (1) N or T −1/2 Xi = Op (1). Averaging over i, (T N)−1 i=1 Xi 2 = Op (1). Simi√ ˆ 2 = r, T −1/2 F ˆ = r, T −1 X F 0 = Op (1), larly, T −1/2 F 0 = Op (1), T −1 F i √ √ and so forth. Throughout, we define δNT = min[ N T ] so that δ2NT = min[N T ]. The proofs of the lemmas are given in the Supplemental material. LEMMA A.1: Under Assumptions A–D, N 1 Xi MF εi = op (1) sup NT F i=1 N 1 0 sup λi F MF εi = op (1) NT F i=1 N 1 sup εi PF εi = op (1) F NT i=1

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JUSHAN BAI

where the sup is taken with respect to F such that F F/T = I. PROOF OF PROPOSITION 1: Without loss of generality, assume β0 = 0 (purely for notational simplicity). From Yi = Xi β0 + F 0 λi + εi = F 0 λi + εi , expanding SNT (β F), we obtain N N 1 1 0 SNT (β F) = SÑT (β F) + 2β Xi MF εi + 2 λ F MF εi NT i=1 NT i=1 i N 1 + ε (PF − PF 0 )εi NT i=1 i

where (38)

SÑT (β F) = β

0 N 1 F MF F 0 ΛΛ Xi MF Xi β + tr NT i=1 T N

+ 2β

N 1 X MF F 0 λi NT i=1 i

By Lemma A.1, (39)

SNT (β F) = SÑT (β F) + op (1)

uniformly over bounded β and over F such that F F/T = I. Bounded β is in fact not necessary because the objective function is quadratic in β (that is, it is easy to argue that the objective function cannot achieve its minimum for very large β). Clearly, SÑT (β0 F 0 H) = 0 for any r × r invertible H, because MF 0 H = MF 0 and MF 0 F 0 = 0. The identification restrictions implicitly fix an H. We next show that for any (β F) = (β0 F 0 H), SÑT (β F) > 0; thus, SÑT (β F) attains its unique minimum value 0 at (β0 F 0 H) = (0 F 0 H). Define

A=

N 1 X MF Xi NT i=1 i

C=

N 1 (λ ⊗ MF Xi ) NT i=1 i

B=

Λ Λ ⊗ IT N

and let η = vec(MF F 0 ). Then S˜ NT (β F) = β Aβ + η Bη + 2β C η


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Completing the square, we have S˜ NT (β F) = β (A − C B−1 C)β + (η + β CB−1 )B(η + B−1 Cβ) = β D(F)β + θ Bθ where θ = (η + B−1 Cβ). By Assumption A, D(F) is positive definite and B is also positive definite, so SÑT (β F) ≥ 0. In addition, if either β = β0 = 0 or F = F 0 H, then SÑT (β F) > 0. Thus, SÑT (θ F) achieves its unique minimum at (β0 F 0 H). Further, for β ≥ c > 0, SÑT (β F) ≥ ρmin c 2 > 0, where ρmin is the minimum eigenvalue of the positive definite matrix infF D(F). This implies that βˆ is consistent for β0 = 0. However, we cannot deduce that Fˆ is consistent for F 0 H. This is because F 0 is T × r and as T → ∞, the number of elements goes to infinity, so the usual consistency is not well defined. Other notions of consistency will be examined. To prove part (ii), note that the centered objective function satisfies ˆ ≤ 0. Therefore, in view of (39), ˆ F) SNT (β0 F 0 ) = 0 and, by definition, SNT (β ˆ = SÑT (β ˆ + op (1) ˆ F) ˆ F) 0 ≥ SNT (β ˆ ≥ 0, it must be true that ˆ F) Combined with SÑT (β ˆ = op (1) ˆ F) S˜ NT (β From βˆ −→ β0 = 0 and (38), it follows that the above implies 0 F MFˆ F 0 Λ Λ = op (1) tr T N p

Because Λ Λ/N > 0 and (F 0 MFˆ F 0 )/T ≥ 0, the above implies the latter matrix is op (1), that is, (40)

F 0 MFˆ F 0 F 0 F 0 F 0 Fˆ Fˆ F 0 = − = op (1) T T T T

ˆ is invertible. By Assumption B, F 0 F 0 /T is invertible, so it follows that F 0 F/T Next, ˆ ) PFˆ − PF 0 2 = tr[(PFˆ − PF 0 )2 ] = 2 tr(Ir − Fˆ PF 0 F/T p p ˆ −→ Ir , which is equivalent to PFˆ − PF 0 −→ 0. But (40) implies Fˆ PF 0 F/T Q.E.D.

Note that for any positive definite matrices, A and B, the eigenvalues of AB are the same as those of BA, A1/2 BA1/2 , and so forth; therefore, all eigenvalues

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are positive. In all remaining proofs, β and β0 are used interchangeably, and so are F and F 0 . PROPOSITION A.1: Under Assumptions A–D, we can make the following statements: p (i) VNT is invertible and VNT −→ V , where V (r × r) is a diagonal matrix consisting of the eigenvalues of ΣΛ ΣF ; VNT is defined in (12). −1 ˆ )VNT (ii) Let H = (Λ Λ/N)(F 0 F/T . Then H is an r × r invertible matrix and T 1 ˆ 1 ˆ 0 2 F − F H = Ft − H Ft0 2 T T t=1

= Op ( βˆ − β 2 ) + Op

1 min[N T ]

PROOF: From N 1 ˆ ˆ ˆ ˆ (Yi − Xi β)(Y i − Xi β) F = FVNT NT i=1 ˆ + F 0 λi + εi , by expanding terms, we obtain and Yi − Xi βˆ = Xi (β − β) N N ˆ NT = 1 ˆ ˆ Xi Fˆ + 1 ˆ i F 0 Fˆ Xi (β − β)(β − β) Xi (β − β)λ FV NT i=1 NT i=1

+

N N 1 ˆ i Fˆ + 1 ˆ Xi Fˆ Xi (β − β)ε F 0 λi (β − β) NT i=1 NT i=1

N 1 ˆ Xi Fˆ εi (β − β) + NT i=1 N N N 1 0 ˆ 1 1 ˆ 0 ˆ F λi εi F + εi λi F F + ε i εi F + NT i=1 NT i=1 NT i=1

+

N 1 0 0 ˆ F λi λi F F NT i=1

= I1 + · · · + I9 ˆ ). Letting I1 I8 The last term on the right is equal to F 0 (Λ Λ/N)(F 0 F/T denote the eight terms on the right, the above can be rewritten as (41)

ˆ ) = I1 + · · · + I8 ˆ NT − F 0 (Λ Λ/N)(F 0 F/T FV


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ˆ )−1 (Λ Λ/N)−1 on each side of (41), we obtain Multiplying (F 0 F/T (42)

ˆ NT (F 0 F/T ˆ )−1 (Λ Λ/N)−1 ] − F 0 F[V ˆ )−1 (Λ Λ/N)−1 = (I1 + · · · + I8)(F 0 F/T

ˆ )−1 (Λ Λ/N)−1 is equal to H −1 , but the invertNote that the matrix VNT (F 0 F/T ibility of VNT is not proved yet. We have ˆ NT (F 0 F/T ˆ )−1 (Λ Λ/N)−1 ] − F 0 T −1/2 F[V ˆ )−1 (Λ Λ/N)−1 ≤ T −1/2 ( I1 + · · · + I8 ) · (F 0 F/T ˆ = Consider each term on the right. For the first term, note that T −1/2 F and N √ 1 Xi 2 −1/2 βˆ − β 2 r T I1 ≤ N i=1 T

√ r

= Op ( βˆ − β 2 ) = op ( βˆ − β ) because βˆ − β = op (1). Using the same argument, it is easy to prove that next four terms (I2–I5) are each Op (βˆ − β). The last three terms do not explicitly depend on βˆ − β and they have the same as those in Bai √ √ expressions and Ng (2002). Each of these terms is Op (1/ min[ N T ]), which was proved in Bai and Ng (2002, Theorem 1). The proof there only uses the property that ˆ Fˆ F/T = I and the assumptions on εi ; thus the proof needs no modification. In summary, we have (43)

ˆ NT (F 0 F/T ˆ )−1 (Λ Λ/N)−1 − F 0 T −1/2 FV √ √ = Op ( βˆ − β ) + Op (1/ min[ N T ])

(i) Left multiplying (41) by Fˆ and using Fˆ Fˆ = T , we have ˆ ) = T −1 Fˆ (I1 + · · · + I8) = op (1) VNT − (Fˆ F 0 /T )(Λ Λ/N)(F 0 F/T ˆ = because T −1/2 F

√ r and T −1/2 (I1 + · · · + I8) = op (1). Thus

ˆ ) + op (1) VNT = (Fˆ F 0 /T )(Λ Λ/N)(F 0 F/T Proposition 1 shows that Fˆ F 0 /T is invertible; thus VNT is invertible. To obtain the limit of VNT , left multiply (41) by F 0 and then divide by T to yield ˆ ) + op (1) = (F 0 F/T ˆ )VNT (F 0 F 0 /T )(Λ Λ/N)(F 0 F/T

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because T −1 F 0 (I1 + · · · + I8) = op (1). The above equality shows that the ˆ columns of F 0 F/T are the (nonnormalized) eigenvectors of the matrix (F 0 F 0 /T )(Λ Λ/N), and VNT consists of the eigenvalues of the same matrix p (in the limit). Thus VNT −→ V , where V is r × r, consisting of the r eigenvalues of the matrix ΣF ΣΛ . (ii) Since VNT is invertible, the left-hand side of (43) can be written as ˆ −1 − F 0 ; thus (43) is equivalent to T −1/2 FH √ √ T −1/2 Fˆ − F 0 H = Op ( βˆ − β ) + Op (1/ min[ N T ]) Taking squares on each side gives part (ii). Note that the cross-product term from expanding the square has the same bound. Q.E.D. The proofs for the next four lemmas are given in the Supplemental Material. LEMMA A.2: Under Assumptions A–C, there exists an M < ∞, such that statements (i) and (ii) hold: (i) We have 2 T T N −1/2 1 E N Fs Ft [εkt εks − E(εkt εks )] ≤ M T k=1

t=1 s=1

(ii) For all i = 1 2 N and h = 1 2 r, we have 2 T T N −1/2 1 E N Xit [εkt εks − E(εkt εks )]Fhs ≤ M T k=1

t=1 s=1

LEMMA A.3: Under Assumptions A–D, we have four equalities: (i) T −1 F 0 (Fˆ − F 0 H) = Op (βˆ − β) + Op (δ−2 NT ). (ii) T −1 Fˆ (Fˆ − F 0 H) = Op (βˆ − β) + Op (δ−2 NT ). ˆ −1 0 ˆ (iii) T Xk (F − F H) = Op (β − β) + Op (δ−2 NT ) for each k = 1 2 N. N 1 0 ˆ ˆ (iv) NT X M ( F − F H) = O ( β − β) + Op (δ−2 p Fˆ NT ). i i=1 LEMMA A.4: Under Assumptions A–D, we also have four equalities: (i) T −1 εk (Fˆ − F 0 H) = T −1/2 Op (βˆ − β) + Op (δ−2 NT ) for each k. N (ii) T √1 N k=1 εk (Fˆ − F 0 H) = T −1/2 Op (βˆ − β) + N −1/2 Op (βˆ − β) + ). Op (N −1/2 ) + Op (δ−2 N NT −1 1 ˆ (iii) NT k=1 λk (FH − F 0 ) εk = (NT )−1/2 Op (βˆ − β) + Op (N −1 ) + N −1/2 × Op (δ−2 NT ).


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N N 1 0 0 0 ˆ −1 −F 0 ) εk = (1/N 2 ) N (iv) NT k=1 (Xk F /T )(F F /T )(FH i=1 k=1 (Xk × T F 0 /T )(F 0 F 0 /T )(Λ Λ/N)−1 λi ( T1 t=1 εit εkt ) + (NT )−1/2 Op (βˆ − β) + N −1/2 × Op (δ−2 NT ) ˆ )−1 (Λ Λ/N)−1 . Under Assumptions A–D, we LEMMA A.5: Let G = (F 0 F/T have N N 1 ˆ X M ˆ (εk εk − Ωk )FGλ i N 2 T 2 i=1 k=1 i F 1 = Op + (NT )−1/2 [Op (βˆ − β) + Op (δ−1 √ NT )] T N 1 1 + √ Op ( βˆ − β 2 ) + √ Op (δ−2 NT ) N N

PROPOSITION A.2: Assume Assumptions A–D hold. If T/N 2 → 0, then N N √ 1 1 0 −1 ˆ √ Xi MFˆ − NT (βˆ − β ) = D(F) aik Xk MFˆ εi N k=1 NT i=1 N ζNT + op (1) + T where aik = λi (Λ Λ/N)λk and (44)

ˆ −1 ζNT = −D(F)

with Ω =

1 N

N k=1

0 ˆ −1 −1 N ΛΛ 1 F F Xi MFˆ ΩFˆ λi NT i=1 T N

Ωk and Ωk = E(εk εk ).

PROOF: From Yi = Xi β0 + F 0 λi + εi ,

−1 N N Xi MFˆ Xi Xi MFˆ F 0 λi βˆ − β0 = i=1

+

i=1

N i=1

or (45)

−1 Xi MFˆ Xi

N

Xi MFˆ εi

i=1

N N N 1 1 1 Xi MFˆ Xi (βˆ − β) = Xi MFˆ F 0 λi + X M ˆ εi NT i=1 NT i=1 NT i=1 i F

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ˆ In view of MFˆ Fˆ = 0, we have MFˆ F 0 = MFˆ (F 0 − FA) for any A. Choosing −1 A = H , from (42), we get ˆ −1 = −[I1 + · · · + I8](F 0 F/T ˆ )−1 (Λ Λ/N)−1 F 0 − FH It follows that N 1 X M ˆ F 0 λi NT i=1 i F

=−

0 ˆ −1 −1 N ΛΛ 1 F F Xi MFˆ [I1 + · · · + I8] λi NT i=1 T N

= J1 + · · · + J8 where J1–J8 are implicitly defined vis-à-vis I1–I8. For example, 0 ˆ −1 −1 N 1 ΛΛ F F J1 = − Xi MFˆ (I1) λi NT i=1 T N Term J1 is bounded in norm by Op (1) βˆ − β 2 and thus J1 = op (1)(βˆ − β). Consider N −1 N 1 ˆ k ΛΛ λi X Mˆ Xk (β − β)λ J2 = − 2 N T i=1 i F k=1 N −1 N N ΛΛ 1 (X M X ) λk λi (βˆ − β) k Fˆ i 2 N T i=1 k=1 N N N 1 1 1 X M ˆ Xk aik (βˆ − β) = T N N i=1 k=1 i F =

where aik = λi (Λ Λ/N)−1 λk is a scalar and thus commutable with βˆ − β. Now consider J3 =

ˆ ˆ 0 −1 −1 N N FF εk F ΛΛ 1 X M X λi (βˆ − β) ˆ k N 2 T i=1 k=1 i F T T N

ˆ Writing εk F/T = ε F 0 H/T + εk (Fˆ − F 0 H)/T = Op (T −1/2 ) + Op (βˆ − β) + √ √ k Op (1/ min[ N T ]), by Lemma A.4, it is easy to see that J3 = op (1)(βˆ − β).


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Next ˆ N N 1 0 Xk F ˆ X M ˆ F λk (β − β) J4 = − 2 N T i=1 k=1 i F T ˆ 0 −1 −1 FF ΛΛ λi × T N

ˆ −1 ) and using that T −1/2 F 0 − FH ˆ −1 is small, Writing MFˆ F 0 = MFˆ (F 0 − FH then J4 is equal to op (1)(βˆ − β). It is easy to show J5 = op (1)(βˆ − β) and thus it is omitted. The last three terms J6–J8 do not explicitly depend on βˆ − β. Only term J7 contributes to the limiting distribution of βˆ − β; the other two terms are op ((NT )−1/2 ) plus op (βˆ − β). We shall establish these claims. Consider J6 = −

ˆ ˆ 0 −1 −1 N N FF εk F ΛΛ 1 0 X M F λ λi ˆ k N 2 T i=1 k=1 i F T T N

Denote G = (Fˆ F 0 /T )−1 (Λ Λ/N)−1 for the moment: it is a matrix of fixed diˆ −1 ), we can mension and does not vary with i. Using MFˆ F 0 = MFˆ (F 0 − FH write ˆ N N ε F 1 1 ˆ −1 ) Gλi Xi MFˆ (F 0 − FH λk k J6 = − NT i=1 N k=1 T Now N N N 1 1 1 λk εk Fˆ = λk εk F 0 H + λk εk (Fˆ − F 0 H) NT k=1 NT k=1 NT k=1 1 = Op √ + (NT )−1/2 Op (βˆ − β) NT

+ Op (N −1 ) + N −1/2 Op (δ−2 NT ) 1 = Op √ + Op (N −1 ) + N −1/2 Op (δ−2 NT ) NT by Lemma A.4(iii). The last equality is because (NT )−1/2 dominates (NT )−1/2 × N 1 0 ˆ ˆ (βˆ − β). Furthermore, by Lemma A.3, NT i=1 Xi MFˆ (F − F H)λi = Op (β − −2 β) + Op (δNT ) for = 1 2 r, and noting G does not depend on i and

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G = Op (1), we have J6 = [Op (βˆ − β) + Op (δ−2 NT )] 1 −1 −1/2 −2 × Op √ Op (δNT ) + Op (N ) + N NT 1 −1 ˆ + Op (δ−2 = op (β − β) + op √ NT )N NT + N −1/2 Op (δ−4 NT ) The term J7 is simply N −1 N ΛΛ 1 λi Xi MFˆ εk λk J7 = − 2 N T i=1 N k=1 N N 1 =− 2 aik Xi MFˆ εk N T i=1 k=1

Next consider J8, which has the expression 0 ˆ −1 −1 N N 1 ΛΛ ˆ F F J8 = − 2 2 X M ˆ ε k εk F λi N T i=1 k=1 i F T N ˆ )−1 (Λ Λ/N)−1 and G = Let E(εk εk ) = Ωk (T × T ). Denoting G = (F 0 F/T Op (1), and rewriting gives (46)

J8 = −

N N 1 ˆ X M ˆ Ωk FGλ i N 2 T 2 i=1 k=1 i F

−

N N 1 ˆ X M ˆ (εk εk − Ωk )FGλ i N 2 T 2 i=1 k=1 i F

Denote the first term on the right by ANT . By Lemma A.5, we have

1 √

1

[Op (βˆ − β) + Op (δ−1 NT )] NT 1 1 − √ Op ( βˆ − β 2 ) + √ Op (δ−2 NT ) N N

J8 = ANT + Op

T N

+√


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Collecting terms from J1 to J8 with dominated terms ignored gives N 1 Xi MFˆ F 0 λi = J2 + J7 + ANT + op (βˆ − β) + op (NT )−1/2 NT i=1 1 + Op + N −1/2 Op (δ−2 √ NT ) T N

Thus,

N 1 X M ˆ Xi + op (1) (βˆ − β) − J2 NT i=1 i F

N 1 Xi MFˆ εi + J7 + ANT NT i=1 1 −1/2 + op (NT ) + N −1/2 Op (δ−2 + Op √ NT ) T N √ Combining terms and multiplying by NT yields √ ˆ + op (1)] NT (βˆ − β) [D(F) N N √ 1 1 Xi MFˆ − aik Xk MFˆ εi + NT ANT =√ N k=1 NT i=1 −1/2 + op (1) + Op T + T 1/2 Op (δ−2 NT )

=

ˆ −1 on each Thus, if T/N 2 → 0, the last term is also op (1). Multiply D(F) √ −1 ˆ side of the above and note that D(F) NT ANT = N/T ζNT . Finally, −1 −1 ˆ ˆ D(F) [D(F) + op (1)] = I + op (1), so we have proved the proposition. Q.E.D. LEMMA A.6: Under Assumptions A–D, ζNT = Op (1), where ζNT is given in Proposition A.2. LEMMA A.7: Under Assumptions A–D, we have the following equalities: (i) HH = (F 0 F 0 /T )−1 + Op ( βˆ − β ) + Op (δ−2 NT ). (ii) PFˆ − PF 0 2 = Op ( βˆ − β ) + Op (δ−2 ). NT Proposition A.2 still involves estimated F . To replace Fˆ by F 0 , we need some preliminary results.

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LEMMA A.8: Under Assumptions A–D, N N 1 1 Xi MFˆ − aik Xk MFˆ εi √ N k=1 NT i=1 N N 1 1 Xi MF 0 − =√ aik Xk MF 0 εi N k=1 NT i=1 √ T † ξNT + + T Op ( βˆ − β0 2 ) N √ + Op ( βˆ − β0 ) + T Op (δ−2 NT ) where (47)

† NT

ξ

−1 N N 1 (Xi − Vi ) F 0 F 0 F 0 =− N i=1 k=1 T T

−1 T 1 ΛΛ λk εit εkt = Op (1) × N T t=1

√ Combining Proposition A.2 and Lemma A.8 and noting that T Op ( βˆ − √ √ β0 2 ) + Op ( βˆ − β0 ) is dominated by NT (βˆ − β0 ) and T Op (δ−2 NT ) = op (1) if T/N 2 → 0, we have an additional statement: COROLLARY A.1: Under Assumptions A–D and as T/N 2 → 0, N N √ 1 1 0 −1 ˆ √ Xi MF 0 − NT (βˆ − β ) = D(F) aik Xk MF 0 εi N k=1 NT i=1 T N ξNT + ζNT + op (1) + N T † † ˆ −1 ξNT where ξNT = D(F) , ξNT is defined in (47) and ζNT is given in (44).

PROOF OF THEOREM 1: The assumption implies that both T/N and N/T † and hence are O(1). Furthermore, ζNT = Op (1) by Lemma A.6, and ξNT ξNT are Op (1) by Lemma A.8. The theorem follows from the expression for √ NT (βˆ − β) given in Corollary 1. Q.E.D. LEMMA A.9: Under Assumptions A–D, the following equalities hold: ˆ −1 − D(F 0 )−1 = op (1). (i) D(F)


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ˆ −1 − D(F 0 )−1 ] = op (1) if T/N 2 → 0. (ii) T/N[D(F) ˆ −1 − D(F 0 )−1 ] = op (1) if N/T 2 → 0. (iii) N/T [D(F) (iv) T/N(ξNT − B) = op (1) if T/N 2 → 0, where B is given in (18). (v) N/T (ζNT − C) = op (1) if N/T 2 → 0, where C is given in (19). PROPOSITION A.3: Under Assumptions A–D, if T/N 2 → 0 and N/T 2 → 0, then N N √ 1 1 0 0 −1 Xi MF 0 − NT (βˆ − β ) = D(F ) √ aik Xk MF 0 εi N k=1 NT i=1 T N B+ C + op (1) + N T where B and C are in (18) and (19), respectively. In this representation, the left-hand side involves no estimated quantities. ˆ −1 = D(F 0 )−1 + PROOF OF PROPOSITION A.3: From Lemma A.9(i), D(F) ˆ −1 in Corollary 1 can be replaced by D(F 0 ) since op (1), the matrix D(F) N N 1 1 √ i=1 [Xi MF 0 − N k=1 aik Xk MF 0 ]εi = Op (1). Parts (iv) and (v) of NT Lemma A.9 together with Corollary 1 again immediately lead to the proposition. Q.E.D. PROOF OF THEOREM 2: (i) We use the representation in Proposition A.3. Without serial correlation or heteroskedasticity, C in Proposition A.3 is zero. N This follows from Ωk = σk2 IT and MF 0 ΩF 0 = ( i=1 σk2 )MF 0 F 0 = 0. Further p more, T/NB −→ 0 since T/N → 0. Thus, by Proposition A.3, (48)

√

N 1 NT (βˆ − β0 ) = D(F 0 )−1 √ Zi εi + op (1) NT i=1

The limiting distribution now follows from Assumption E. (ii) The proof again uses the representation in Proposition A.3. From N/T → 0, we have N/T C → 0. We next argue that B = 0 when the cross-section correlation and heteroskedasticity are absent. Recall that σiktt = E(εit εkt ). By assumption, σijtt = σt2 for i = j and σijtt = 0 for i = j. Thus B in (18) is simplified as −1 −1 N ΛΛ 1 (Xi − Vi ) F 0 F 0 F 0 B = −D(F ) λi σ¯ 2 N i=1 T T N 0 −1

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T N where σ¯ 2 = T −1 t=1 σt2 . From Vi = N1 k=1 Xk aik and that aik is a scalar, thus N N commutable with all matrices, we have N1 i=1 λi aik = N1 i=1 λi λi (Λ Λ/N)−1 × λk = λk . This means that −1 −1 N 1 Vi F 0 F 0 F 0 ΛΛ λi N i=1 T T N =

−1 −1 N ΛΛ 1 Xk F 0 F 0 F 0 λk N k=1 T T N

which implies B = 0. Thus (48) holds and the limiting distribution once again follows from Assumption E. Q.E.D. PROOF OF THEOREM 3: This again follows from the representation of p Proposition A.3. Under the assumption that T/N → ρ, T/NB −→ ρ1/2 B0 , p where B0 is the probability limit of B, and N/T C −→ ρ−1/2 C0 , where C0 is the probability limit of C. Combining with Assumption E, we obtain the theorem. Q.E.D. Bias Correction LEMMA A.10: Under Assumptions A–D, the following equalities hold: N (i) N1 Λˆ − H −1 Λ 2 = N1 i=1 λˆ i − H −1 λi 2 = Op ( βˆ − β 2 ) + Op (δ−2 NT ) −2 −1 ˆ −1 ˆ (ii) N (Λ − H Λ )Λ = Op ( β − β ) + Op (δNT ). ˆ − H −1 (Λ Λ/N)H −1 = Op ( βˆ − β ) + Op (δ−2 (iii) Λˆ Λ/N NT ). ˆ −1 −1 ˆ ˆ (iv) (Λ Λ/N) − H (Λ Λ/N) H = Op ( β − β ) + Op (δ−2 NT ). N ˆ (v) N1 i=1 λˆ i − H −1 λi = Op (δ−1 ) + O ( β − β ). p NT N ˆ (vi) N1 k=1 T −1/2 Xi λˆ i − H −1 λi = Op (δ−1 NT ) + Op ( β − β ). LEMMA A.11: Under the assumptions of Theorem 4, LEMMA A.12: Under the assumptions of Theorem 4,

T/N(Bˆ − B) = op (1)

N/T (Cˆ − C) = op (1)

The proof of Theorem 4 follows from Proposition A.3, Lemma A.11, and Lemma A.12. For the proof of Proposition 2, see the Supplemental Material. REFERENCES ABOWD, J. M., F. KRAMARZ, AND D. N. MARGOLIS (1999): “High Wage Workers and High Wage Firms,” Econometrica, 67, 251–333. [1233]


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AHN, S. G., Y. H. LEE, AND P. SCHMIDT (2001): “GMM Estimation of Linear Panel Data Models With Time-Varying Individual Effects,” Journal of Econometrics, 101, 219–255. [1230,1231,1233, 1237,1239,1258] (2006): “Panel Data Models With Multiple Time-Varying Effects,” Mimeo, Arizona State University. [1239] ALTUG, S., AND R. A. MILLER (1990): “Household Choices in Equilibrium,” Econometrica, 58, 543–570. [1233] ALVAREZ, J., AND M. ARELLANO (2003): “The Time Series and Cross-Section Asymptotics of Dynamic Panel Data Estimators,” Econometrica, 71, 1121–1159. [1232] ANDERSON, T. W. (1984): An Introduction to Multivariate Statistical Analysis. New York: Wiley. [1236] ANDERSON, T. W., AND C. HSIAO (1982): “Formulation and Estimation of Dynamic Models With Error Components,” Journal of Econometrics, 76, 598–606. [1232] ANDERSON, T. W., AND H. RUBIN (1956): “Statistical Inference in Factor Analysis,” in Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 5, ed. by J. Neyman. Berkeley: University of California Press, 111–150. [1234] ANDREWS, D. W. K. (2005): “Cross-Section Regression With Common Shocks,” Econometrica, 73, 1551–1585. [1234] ARELLANO, M. (2003): Panel Data Econometrics. Oxford: Oxford University Press. [1239] ARELLANO, M., AND J. HAHN (2005): “Understanding Bias in Nonlinear Panel Models: Some Recent Developments,” Unpublished Manuscript, CEMFI. [1250] ARELLANO, M., AND B. HONORE (2001): “Panel Data Models: Some Recent Developments,” in Handbook of Econometrics, Vol. 5, ed. by J. J. Heckman and E. Leamer. Amsterdam: NorthHolland. [1230] BAI, J. (1994): “Least Squares Estimation of Shift in Linear Processes,” Journal of Time Series Analysis, 15, 453–472. [1244] (2003): “Inferential Theory for Factor Models of Large Dinensions,” Econometrica, 71, 135–173. [1242,1247,1251] (2009): “Supplement to ‘Panel Data Models With Interactive Fixed Effects’,” Econometrica Supplemental Material, 77, http://www.econometricsociety.org/ecta/Supmat/6135_proofs. pdf. [1232] BAI, J., AND S. NG (2002): “Determining the Number of Factors in Approximate Factor Models,” Econometrica, 70, 191–221. [1235,1267] (2006): “Confidence Intervals for Diffusion Index Forecasts and Inference for FactorAugment Regressions,” Econometrica, 74, 1133–1150. [1251] BALTAGI, B. H. (2005): Econometric Analysis of Panel Data. Chichester: Wiley. [1239] BEKKER, P. A. (1994): “Alternative Approximations to the Distributions of Instrumental Variable Estimators,” Econometrica, 62, 657–681. [1250] BERNANKE, B., J. BOIVIN, AND P. ELIASZ (2005): “Factor Augmented Vector Autoregression and the Analysis of Monetray Policy,” Quarterly Journal of Economics, 120, 387–422. [1239] CAMPBELL, J. Y., A. W. LO, AND A. C. MACKINLAY (1997): The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. [1234] CARNEIRO, P., K. T. HANSEN, AND J. J. HECKMAN (2003): “Estimating Distributions of Treatment Effects With an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College Choice,” Working Paper 9546, NBER; International Economic Review, 44, 362–422. [1233] CAWLEY, J., K. CONNELLY, J. HECKMAN, AND E. VYTLACIL (1997): “Cognitive Ability, Wages, and Meritocracy,” in Intelligence Genes, and Success: Scientists Respond to the Bell Curve, ed. by B. Devlin, S. E. Feinberg, D. Resnick, and K. Roeder. Berlin: Springer-Verlag, 179–192. [1233] CHAMBERLAIN, G. (1980): “Analysis of Covariance With Qualitative Data,” Review of Economic Studies, 47, 225–238. [1232] (1984): “Panel Data,” in Handbook of Econometrics, Vol. 2, ed. by Z. Griliches and M. Intriligator. Amsterdam: North-Holland. [1230,1233,1239]

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CHAMBERLAIN, G., AND M. ROTHSCHILD (1983): “Arbitrage, Factor Structure and MeanVariance Analysis in Large Asset Markets,” Econometrica, 51, 1281–1304. [1232,1235,1238, 1244] COAKLEY, J., A. FUERTES, AND R. P. SMITH (2002): “A Principal Components Approach to CrossSection Dependence in Panels,” Mimeo, Birkbeck College, University of London. [1231] CONNOR, G., AND R. KORAJZCYK (1986): “Performance Measurement With the Arbitrage Pricing Theory: A New Framework for Analysis,” Journal of Financial Economics, 15, 373–394. [1235,1236,1239] FAMA, E., AND K. FRENCH (1993): “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, 33, 3–56. [1234] FELDSTEIN, M., AND C. HORIOKA (1980): “Domestic Savings and International Capital Flows,” Economic Journal, 90, 314–329. [1233] GIANNONE, D., AND M. LENZA (2005): “The Feldstein Horioka Fact,” Unpublished Manuscript, European Central Bank. [1233] GOLDBERGER, A. S. (1972): “Structural Equations Methods in the Social Sciences,” Econometrica, 40, 979–1001. [1231] GREENE, W. (2000): Econometric Analysis (Fourth Ed.). Englewood Cliffs, NJ: Prentice-Hall. [1253,1254] HAHN, J., AND G. KUERSTEINER (2002): “Asymptotically Unbiased Inference for a Dynamic Panel Model With Fixed Effects When Both n and T Are Large,” Econometrica, 70, 1639–1657. [1232,1246,1247,1249] HAHN, J., AND W. NEWEY (2004): “Jackknife and Analytical Bias Reduction for Nonlinear Panel Models,” Econometrica, 72, 1295–1319. [1232,1249,1250] HANSEN, C., J. HAUSMAN, AND W. NEWEY (2005): “Estimation With Many Instruments,” Unpublished Manuscript, Department of Economics, MIT. [1250] HAUSMAN, J. (1978): “Specification Tests in Econometrics,” Econometrica, 46, 1251–1271. [1256] HAUSMAN, J. A., AND W. E. TAYLOR (1981): “Panel Data and Unobservable Individual Effects,” Econometrica, 49, 1377–1398. [1258,1260] HOLTZ-EAKIN, D., W. NEWEY, AND H. ROSEN (1988): “Estimating Vector Autoregressions With Panel Data,” Econometrica, 56, 1371–1395. [1230,1231,1233,1238,1239] HSIAO, C. (2003): Analysis of Panel Data. New York: Cambridge University Press. [1232] JÖRESKOG, K. G., AND A. S. GOLDBERGER (1975): “Estimation of a Model With Multiple Indicators and Multiple Causes of a Single Latent Variable,” Journal of the American Statistical Association, 70, 631–639. [1231] KIEFER, N. M. (1980): “A Time Series-Cross Section Model With Fixed Effects With an Intertemporal Factor Structure,” Unpublished Manuscript, Department of Economics, Cornell University. [1231,1237] KIVIET, J. (1995): “On Bias, Inconsistency, and Efficiency of Various Estimators in Dynamic Panel Data Models,” Journal of Econometrics, 68, 53–78. [1232] KNEIP, A., R. SICKLES, AND W. SONG (2005): “A New Panel Data Treatment for Heterogeneity in Time Trends,” Unpublished Manuscript, Department of Economics, Rice University. [1231] LAWLEY, D. N., AND A. E. MAXWELL (1971): Factor Analysis as a Statistical Method. London: Butterworth. [1234] LEE, Y. H. (1991): “Panel Data Models With Multiplicative Individual and Time Effects: Application to Compensation and Frontier Production Functions,” Unpublished Ph.D. Dissertation, Michigan State University. [1231] LETTAU, M., AND S. LUDVIGSON (2001): “Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time Varying,” Journal of Political Economy, 109, 1238–1287. [1234] MACURDY, T. (1982): “The Use of Time Series Processes to Model the Error Structure of Earnings in a Longitudinal Data Analysis,” Journal of Econometrics, 18, 83–114. [1231] MOON, R., AND B. PERRON (2004): “Testing for a Unit Root in Panels With Dynamic Factors,” Journal of Econometrics, 122, 81–126. [1231]


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MUNDLAK, Y. (1978): “On the Pooling of Time Series and Cross Section Data,” Econometrica, 46, 69–85. [1239] NEWEY, W., AND D. MCFADDEN (1994): “Large Sample Estimation and Hypothesis Testing,” in Handbook of Econometrics, ed. by R. F. Engle and D. McFadden. Amsterdam: North Holland, 2111–2245. [1244] NEWEY, W., AND R. SMITH (2004): “Higher Order Properties of GMM and Generalized Empirical Likeliood Estimators,” Econometrica, 72, 219–255. [1239] NEWEY, W. K., AND K. D. WEST (1987): “A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55, 703–708. [1251,1252] NEYMAN, J., AND E. L. SCOTT (1948): “Consistent Estimates Based on Partially Consistent Observations,” Econometrica, 16, 1–32. [1232] NICKELL, S. (1981): “Biases in Dynamic Models With Fixed Effects,” Econometrica, 49, 1417–1426. [1232] OBSTFELD, M., AND K. ROGOFF (2000): “The Six Major Puzzles in International Macroeconomics: Is There a Common Cause?” NBER Macroeconomic Annual, 15, 339–390. [1233] PESARAN, M. H. (2006): “Estimation and Inference in Large Heterogeneous Panels With a Multifactor Error Structure,” Econometrica, 74, 967–1012. [1231,1233,1240] PHILLIPS, P. C. B., AND D. SUL (2003): “Dynamic Panel Estimation and Homogeneity Testing Under Cross Section Dependence,” The Econometrics Journal, 6, 217–259. [1231] ROSS, S. (1976): “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 13, 341–360. [1234] SARGAN, J. D. (1964): “Wages and Prices in the United Kingdom: A Study in Ecnometrics Methodology,” in Econometric Analysis of National Economic Planning, ed. by P. G. Hart, G. Mill, and J. K. Whitaker. London: Butterworths, 25–54. [1237] STOCK, J. H., AND M. W. WATSON (2002): “Forecasting Using Principal Components From a Large Number of Predictors,” Journal of the American Statistical Association, 97, 1167–1179. [1235,1236,1239] TOWNSEND, R. (1994): “Risk and Insurance in Village India,” Econometrica, 62, 539–592. [1233]

Dept. Economics, New York University, 19 West 4th Street, New York, NY 10012, U.S.A., SEM, Tsinghua University, and CEMA, Central University of Finance and Economics, Beijing, China; [email protected]. Manuscript received October, 2005; final revision received January, 2009.


NOTES AND COMMENTS TESTING MODELS WITH MULTIPLE EQUILIBRIA BY QUANTILE METHODS BY FEDERICO ECHENIQUE AND IVANA KOMUNJER1 This paper proposes a method for testing complementarities between explanatory and dependent variables in a large class of economic models. The proposed test is based on the monotone comparative statics (MCS) property of equilibria. Our main result is that MCS produces testable implications on the (small and large) quantiles of the dependent variable, despite the presence of multiple equilibria. The key features of our approach are that (i) we work with a nonparametric structural model of a continuous dependent variable in which the unobservable is allowed to be correlated with the explanatory variable in a reasonably general way; (ii) we do not require the structural function to be known or estimable; (iii) we remain fairly agnostic on how an equilibrium is selected. We illustrate the usefulness of our result for policy evaluation within Berry, Levinsohn, and Pakes’s (1999) model. KEYWORDS: Monotone comparative statics, structure, multiple equilibria, complementarities, conditional quantiles.

1. INTRODUCTION IN MANY CONVENTIONAL ECONOMIC MODELS, equilibrium uniqueness comes at a cost of strong and often untenable assumptions. Consider, for example, general equilibrium models: the uniqueness conditions with some natural economic meaning imply the strong weak axiom, which in turn cannot be expected to hold beyond single-agent economies (Arrow and Hahn (1971)). Therefore it is not surprising to find equilibrium multiplicity present in a variety of contexts, ranging from general equilibrium models in microeconomics, to oligopoly models and network externalities in industrial organization, and nonconvex growth models in macroeconomics or models of statistical discrimination in labor economics. Performing comparative statics with multiple equilibria is a challenge. How changes in explanatory variables affect dependent variables depends on the way a particular equilibrium is selected. Unfortunately, the theoretical liter1 Many thanks to Rabah Amir, Kate Antonovics, Andrés Aradillas-López, Patrick Bajari, Peter Bossaerts, Colin Camerer, Chris Chambers, Xiaohong Chen, Valentino Dardanoni, Han Hong, Joel Horowitz, Matt Jackson, Michael Jansson, Chuck Manski, Andrea Mattozzi, Rosa Matzkin, Tom Palfrey, Jim Powell, Andrea Rotnitzky, Paul Ruud, Max Stinchcombe, Andrew Sweeting, Elie Tamer, Xavier Vives, Quang Vuong, Hal White, and Bill Zame, as well as participants at presentations given at Northwestern University, UC Berkeley, UC Davis, and the North American Econometric Society summer meeting, the EEE2005 conference, and the 2005 ESSET workshop. Echenique thanks the NSF for its support through award SES-0751980. Anonymous referees and a co-editor provided excellent suggestions and comments that greatly improved an earlier version of the paper. All remaining errors are ours.


DOI: 10.3982/ECTA6223

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ature offers little guidance on equilibrium selection.2 As a consequence, policy analysis seems impossible as policy effects may well vary across different equilibria. More to the point, without equilibrium selection, it is hard to identify the structure underlying economic models when multiple equilibria are present. And with no knowledge of the structure, we can say little about general comparative statics effects. We should emphasize that we are concerned with testing for the existence of a comparative statics effect; the counterfactual prediction of the effects of policies remains virtually impossible without substantial information about equilibrium selection. In this paper, we restrict our attention to economic models that exhibit complementarities between explanatory and dependent variables. In such models, despite the possible presence of multiple equilibria, a monotone comparative statics (MCS) prediction holds: there is a smallest and a largest equilibrium, and these change monotonically with explanatory variables (Milgrom and Roberts (1994), Villas-Boas (1997)). The paper’s main contribution is to show how MCS arguments translate into observable restrictions on the conditional quantiles of the dependent variable. Our framework is as follows: similar to Jovanovic (1989), we start with an underlying economic model relating dependent and explanatory variables. We disturb the model by adding an unobservable disturbance term that captures individual heterogeneity or other unaccounted random features. The assumptions we impose on the resulting structure are fairly weak: we allow for unknown structural function, unknown equilibrium selection, and reasonably general correlation between the disturbance and the explanatory variable. Our main result is that MCS produces testable implications on the (small and large) quantiles of the dependent variable.3 The result does not assume, or require estimating, an equilibrium selection procedure.4 The intuition behind our approach is fairly simple. Consider a model in which there are complementarities between explanatory and dependent variables. When the generated equilibrium is unique, then the model can be globally implicitly solved and the resulting reduced form is such that the dependent variable increases in the explanatory variable. This property translates into first-order stochastic dominance among distributions: all conditional quantiles of the dependent variable are increasing functions of the explanatory variable. When the model generates multiple equilibria, the above implicit function arguments fail to hold globally; it remains, however, the MCS property of the 2 Consider Kreps (1990), for example: “There are . . . lots of Nash equilibria to this game. Which one is the ‘solution’? I have no idea and, more to the point, game theory isn’t any help. Some (important) sorts of games have many equilibria, and the theory is of no help in sorting out whether any one is the ‘solution’ and, if one is, which one is.” 3 An early test for MCS can be found in Athey and Stern (1998) in the context of firms’ choice of organizational form. This prior work, however, does not address equilibrium problems. 4 Understandably, estimating the structural parameters requires additional parametric assumptions on the equilibrium selection.

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extremal equilibria. By focusing on the regions in which the monotonicity of equilibria holds, we still obtain that tail (small and large) conditional quantiles of the dependent variable increase in the explanatory variable. Testing for complementarities is thus possible by examining the behavior of the extreme conditional tails of the dependent variable. Our method applies to a large class of economic models with continuous dependent variables. These are models of individual decision making in which the equilibrium values are the solutions of an extremum problem and onedimensional equilibrium models where equilibria are fixed points. Since the dependent variable is continuous, our findings complement those developed by the growing literature on discrete games with multiple equilibria (Bresnahan and Reiss (1990, 1991), Berry (1992), Tamer (2003), Aguirregabiria and Mira (2007), Ciliberto and Tamer (2007)).5 The next section discusses equilibrium multiplicity in Berry, Levinsohn, and Pakes’s (1999) influential model of price-setting with differentiated products. In Section 3, we introduce the setup and present our results. We conclude in Section 4 with a discussion and possible extensions of our approach. Omitted proofs are contained in the Supplemental Material (Echenique and Komunjer (2009)). 2. EXAMPLE We now present a simplified version of Berry, Levinsohn, and Pakes’s (1995) model of price competition with differentiated products. We use this model for two purposes: first, to illustrate the challenges posed by equilibrium multiplicity, even in popular and well behaved economic models; second, to argue that our methods provide useful tools for policy analysis in these models. Concretely, we discuss the analysis of the Japanese “Voluntary Export Restraint” (VER) policy for automobile exports published in Berry, Levinsohn, and Pakes (1999; BLP hereafter). In our version of the BLP model, there are two firms, each producing one good. Firm 1 is foreign and firm 2 is a home firm. Following BLP, we model the VERs as (tax) increases in firms’ marginal costs. Firm i sets the price pi of its product and obtains profits Πi (pi p−i VER) = (pi − ci − λVERi )Di (pi p−i ), where Di (pi p−i ) is the demand for firm i’s good when its competitor sets a price p−i , ci is i’s marginal cost, VERi is a dummy variable for the VER, and λ is the corresponding tax per unit of i’s production. The firms are assumed to ¯ which—given the price p−i set by their competitor— choose prices p∗i ∈ [0 p], maximize their profits: p∗i = arg maxpi ∈[0p]¯ Πi (pi p−i VERi ). Only firm 1 (the foreign firm) is subject to the VER and we denote by VER = VER1 the VER dummy. 5 Unlike in these papers, however, our methods can only be applied to discuss comparative statics effects and are silent about other structural features of the model.

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Let p∗1 = βc1 (p2 VER) and p∗2 = βc2 (p1 0) denote the best responses (reactions) of firms 1 and 2 as determined by profit maximization. In our application, the existence of maximizers p∗i directly follows from the continuity of Πi ¯ in general, however, the maximizers need not be and the compactness of [0 p]; unique, so we allow the best responses βci to be correspondences (or set-valued functions). Hereafter, we denote by βi the best response function that firm i selects from βci ; by the maximum theorem, we can take βi to be continuous. 2.1. BLP Model With Multiple Equilibria We focus our analysis on the composed best response function for the foreign firm (firm 1): β1 (β2 (p1 0) VER). The equilibrium price p = p∗1 set by firm 1 is determined by the fixed-point condition: (1)

β1 (β2 (p 0) VER) − p = 0

Without additional restrictions on the demand functions, a solution p to the above equilibrium condition need not be unique. Not only are the known conditions for uniqueness very strong (Gabay and Moulin (1980), Caplin and Nalebuff (1991)), but there is a sense in which games generally tend to have large numbers of equilibria. In a model of randomly generated games, McLennan (2005) showed that the mean number of equilibria grows exponentially with the number of strategies. Games of strategic complements, which are especially relevant for our paper, tend to have particularly large numbers of equilibria (Takahashi (2005)). As pointed out by Berry, Levinsohn, and Pakes (1995, 1999) the BLP model in particular is not guaranteed to have a unique equilibrium. For example, Milgrom and Roberts (1990) established uniqueness for the linear demand, constant elasticity of substitution, logit, or translog models (under additional parameter constraints). Simple departures from these models result in multiple equilibria for p. Echenique and Komunjer (2007b) made this point in the context of a logit model with conditional heteroskedasticity. We now argue that, despite equilibrium multiplicity, it is still possible to evaluate the impact of the VER on the prices in Equation (1). If VER and prices are complements, then any selection from βc1 is monotone increasing in VER. To see this, note that ∂2 Π1 (p1 p2 VER)/∂p1 ∂VER = −λ ∂D1 (p1 p2 )/∂p1 has the same sign as λ provided the demand function D1 is everywhere decreasing in p1 , that is, ∂D1 (p1 p2 )/∂p1 < 0. When λ > 0, Π1 has strictly increasing differences (is strictly supermodular) in (p1 VER). Every selection from the best response p∗1 = βc1 (p2 VER) is then monotone increasing in VER (by Milgrom and Shannon’s (1994) monotone selection theorem). Hence, the comparative statics effects take the form of a monotone comparative statics (MCS) prediction: if the implicit tax on exports λ is positive, then the VER will cause the extremal price equilibria to increase. Figure 1 illustrates this effect. Evaluating the price impact of VER in the BLP framework is then equivalent to testing for the presence of MCS.


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FIGURE 1.—Monotone comparative statics of equilibria.

There are two difficulties in testing for MCS. First, we need to work with a stochastic version of the model (1). In the presence of unobservables, the observed prices will no longer be discretely distributed over equilibrium sets. Instead, they will have mixture distributions (in Section 3 we show how such mixtures arise). Since we allow for unknown equilibrium selection, the exact probabilities in the mixture remain unknown. This is the source of the second difficulty: certain equilibrium selections may induce smaller observations with the VER than without it, thus working against the positive effect of the VER. For example, assume that the price of firm 1’s product is like the one in Figure 2. Here, the average foreign car price decreases in the presence of the VER. This simple example shows that MCS has no general implications on the conditional mean of prices given VER. All we have to work with is the MCS property: that in the presence of the VER, the smallest equilibrium and the largest equilibrium have increased. 2.2. Observable Implications of MCS Write y ≡ ln p and x ≡ VER, and let r(y x) ≡ ln(β1 (β2 (exp y 0) x)) − y. Then the equilibrium values for y solve the equation (2)

r(y x) = 0

We first obtain a stochastic version of the model based on the equilibrium condition (2). Let Y be a scalar random variable whose realizations correspond to

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(2) (3) FIGURE 2.—Firm 1 equilibria are {p(1) p(2) p(3) } before and {p(1) VER pVER pVER } after VER.

the log prices y = ln p and let X be the VER dummy with realizations x. Consider the structural model r(Y X) = U, in which U is an unobservable scalar disturbance term. Different realizations of U induce values of Y that deviate from the exact equilibrium condition. The continuity and limit behavior of r (which we discuss in Section 3.1) guarantee that the disturbed equilibrium condition r(y x) = u always has at least one solution. We now study the comparative statics effects on the log prices Y following the introduction of the VER based on the model r(Y X) = U. For this, first note that since β1 is increasing in x, so is r. This property (which we later label Assumption S2) states that X and Y are complements. The existence of complementarities between the VER and the log prices is the starting point of our comparative statics analysis. Our main result provides simple conditions under which the effect of extremal equilibria will prevail for large (and small) values of the dependent variable. The conditions are simple and do not restrict the equilibrium selection rule. When they are satisfied, the effect of VER on extremal equilibria translates into testable implications on some large (and small) enough quantile of the distribution of log prices. The conditions (found in Assumptions S3 and S3 ) restrict both the dependence of U on X and the tail behavior of U. The first of these restrictions is an identification condition: we use it to prevent the variations in U from exactly canceling out with an increase in X. This condition is easily satisfied if U and X are independent, for example. Independence is, however, stronger than needed; Section 4.1.2 contains examples in which U and X are correlated in a reasonably general way. The role of the second condition is to ensure that any increase in X eventually translates into an increase of large enough conditional quantiles of Y given X. Its key feature is to only restrict the distribution of U given X without placing assumptions on the equilibrium selection procedure. Broadly speaking, this condition rules out heavy-tailed distributions; Section 3.2 contains a detailed discussion of the distributions that we allow for.


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3. STRUCTURAL MODEL AND RESULTS 3.1. Structure We consider a structural equation given by (3)

r(Y X) = U

where r : R × X → R is specified by economic theory.6 The variables that enter the structural model in (3) are: a dependent variable Y ∈ R, an explanatory variable X ∈ X ⊆ R, and a disturbance to the system U ∈ R. The exact form of the structural function r in Equation (3) need not be known. We assume that X and Y are observable, but U is not; U can be thought of as unaccounted heterogeneity in the model. We have in mind the structural equations derived from two classes of economic models. One class predicts equilibrium values y based on a first-order condition r(y x) = 0; these are single-person decision models, such as models solved by a social planner. A second class predicts equilibrium values y based on a fixed-point condition r(y x) = ρ(y x) − y = 0, as in the BLP model in Section 2. Given the function r, the structural econometric model is built by introducing the disturbance term U in the underlying economic model. Different realizations of U induce values of Y that deviate from the equilibria predicted by the economic model. The disturbance U has a clear interpretation as the extent to which a realized Y violates the exact (undisturbed) equilibrium condition. When Equation (3) determines Y as a function Y = m(X U), the distribution of the disturbance U conditional on the explanatory variable X, denoted by FU|X , determines unambiguously the conditional distribution of Y , denoted by FY |X . We say that FY |X is generated by the structure S = (r FU|X ). On the other hand, when Equation (3) has multiple solutions, a complete specification of the structure must include a rule that selects a particular realization y from the set of solutions. Such an equilibrium-selection rule can depend on the realized values of X and U. We assume that for any x in the support X of X, FU|X=x has a strictly positive density fU|X=x on R. The variable X can be discrete or continuous. The equilibrium set is the set of solutions to (3) when X = x and U = u, that is, Exu = {y ∈ R : r(y x) = u} when (x u) ∈ X × R. We work with the following assumption. ASSUMPTION S1: (i) The function r : R × X → R is continuous; (ii) for any x ∈ X , limy→−∞ r(y x) = +∞ and limy→+∞ r(y x) = −∞; (iii) for any (x u) ∈ 6

As in Matzkin (1994, 2008), we consider structural equations in which U is additively separable. The methods developed here are not suited for the nonseparable problem r˜(Y X U) = 0. In such cases, the framework in Echenique and Komunjer (2007a) may still be applied.

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X × R, Exu is a finite set. We write Exu = {ξ1xu ξnx xu } (ξ1xu ≤ · · · ≤ ξnx xu ) with nx = Card(Exu ). Assumption S1(i) and (ii) are standard. S1(ii) is akin to an Inada condition; in particular, S1(i) and S1(ii) imply, by the intermediate value theorem, that a solution to Equation (3) always exists. Assumption S1(iii) requires r not to be constant over any subintervals. By using suitable arguments from differential topology, S1(iii) can be shown to hold generically (see Mas-Colell, Whinston, and Green (1995) for examples of these arguments). That the number of equilibria only depends on the explanatory variable X is not a serious restriction; it can simply be satisfied by duplicating elements of the equilibrium set until its cardinality no longer depends on U. The BLP model in Section 2, for example, satisfies Assumption S1. Given that the best-response functions βi are continuous, so is r (Assumption S1(i)). That the demand functions Di are everywhere positive implies the limit conditions on r in S1(ii).7 Finally, when the distribution of U is absolutely continuous, the set of solutions to r(y x) = u will be finite with probability 1 (Assumption S1(iii)). We specify the selection rule as follows: let Pxu be a probability distribution over Exu , which assigns probabilities {π1x πnx x } to outcomes {ξ1xu ξnx xu }, such that π1x > 0 and πnx x > 0. For a given x, different realizations u can affect the support of Pxu , but not the probabilities assigned to different outcomes in the support. For example, Pxu might assign equal probabilities across all elements of Exu . The conditional distribution of Y is then obtained as follows. PROPOSITION 1: Assume S1 holds and fix a selection +∞ rule PXU . Then, for any x ∈ X , there are distribution functions FiY |X=x (y) = −∞ 1(ξixu ≤ y)fU|X=x (u) du for 1 ≤ i ≤ nx , such that j ≥ i implies that FjY |X=x nx first-order stochastically domiπix FiY |X=x (y). nates FiY |X=x and, for any y ∈ R, FY |X=x (y) = i=1 The proof of Proposition 1 is given in the Supplement Material. When multiple equilibria exist, FY |X is generated by the structure S = (r FU|X PXU ), which now includes the additional element PXU . Proposition 1 shows that under S, the conditional distribution of the dependent variable has a mixture form. When equilibrium is unique, that is, when the structural function r is monotone decreasing, the results of Proposition 1 reduce to the usual expression of the image distribution FY |X of Y given X : F¯Y |X=x (y) = FU|X=x (r(y x)), where we use F¯Y |X ≡ 1 − FY |X to denote the conditional distribution tail of Y given X. 7 Note that if Di (pi p) > 0 for all (pi p−i ), then pˆ i < ci + λVERi implies that Πi (pˆ i p−i VER) < 0 = Πi (ci + λVERi p−i VER), so βi (p−i VERi ) ≥ ci + λVERi for all values of p−i . As y → −∞, β1 (β2 (exp y 0) VER) → β1 (β2 (0 0) VER) ≥ c1 and so r(y x) → ¯ so β1 (β2 (exp y 0) VER) remains +∞. Moreover, for any value of p−i , βi (p−i VERi ) ≤ p, bounded as y → +∞ and r(y x) → −∞.


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In general, the structure S may not be known. We work with a class of structures that share a qualitative feature: they exhibit complementarities between the explanatory variable X and the dependent variable Y . ASSUMPTION S2: r(y x) is monotone increasing in x on R. Assumption S2 says that X and Y are complements. Such complementarity usually follows from a supermodularity property of the primitive model. An example is the BLP model in Section 2, in which Assumption S2 follows from the strict supermodularity of firm 1’s profits in its own price and the VER. The key feature of S2 is that it implies the MCS property: the extremal equilibria of r(y x) = 0 increase with x (Figure 1). We now review briefly some of the many economic models that fall into our framework. 3.1.1. Individual Decision Maker Consider models of individual decision making in which the dependent variable is one dimensional and determined through the first-order condition of an optimization problem. An important class of such models comprises those solved by a social-planning problem, such as growth and macroeconomic models in Barro and Sala-I-Martin (2003) and Ljungqvist and Sargent (2004). Other examples include models of firms’ investment choices used for testing whether investment is sensitive to Tobin’s q (Hayashi (1982), Hayashi and Inoue (1991)). 3.1.2. One-Dimensional Equilibrium Consider one-dimensional equilibrium models where equilibria are fixed points. For example, in a two-player game, one can compose the two players’ best-response functions similarly to how we dealt with BLP’s model in Section 2. As a consequence, duopoly models generally have the structure we need. Cournot n-firm oligopoly models also reduce to one-dimensional equilibrium models by an aggregation procedure as described by Amir (1996). One can thus examine whether entry of additional firms to a market causes a decrease in prices as in Amir and Lambson (2000). Additional examples can be found in overlapping-generations models and two-good general equilibrium models. 3.2. Main Result The presence of complementarities between X and Y is the basis of our main result: we show that an increase in x implies an increase in sufficiently large (and small) quantiles of FY |X=x . The result will follow from combining Assumption S2 with restrictions on U.

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Consider x1 and x2 in X with x1 < x2 . Let n1 = nx1 and n2 = nx2 . How does the MCS property translate into observable implications on F¯Y |X=x1 and F¯Y |X=x2 ? Recall that F¯Y |X = 1 − FY |X is the conditional distribution tail of Y . Let π1i = πix1 and π2j = πjx2 . Using the mixture result in Proposition 1 and focusing on the largest equilibria, we then have n1

(4)

F¯Y |X=x1 (y) F¯n1 Y |X=x1 (y) i=1 = n2 F¯Y |X=x2 (y) F¯n2 Y |X=x2 (y)

π1i [F¯iY |X=x1 (y)/F¯n1 Y |X=x1 (y)] π2j [F¯jY |X=x2 (y)/F¯n2 Y |X=x2 (y)]

j=1

≤

F¯n1 Y |X=x1 (y) 1 F¯n Y |X=x (y) π2n2 2

2

where the second inequality follows because π2n2 > 0, because F¯jY |X=x2 (y)/ F¯n2 Y |X=x2 (y) > 0, and because stochastic dominance implies F¯iY |X=x1 (y) ≤ F¯n1 Y |X=x1 (y). The upper bound in Equation (4) involves the probability of the largest equilibrium π2n2 —on which we place no restrictions other than being positive—as well as the ratio of the distributions F¯n1 Y |X=x1 and F¯n2 Y |X=x2 . These distributions are unknown and depend on the locations of the largest equilibria; hence they are difficult to control. A careful change of variables, however, transforms the problem so that (in the limit) the behavior of their ratio depends solely on the properties of r and FU|X . LEMMA 1: Under S1 and S2, and given (y0 x) ∈ R × X , we have 1(ξnx xu ≤ y) = 1(u ≤ r e (y x)) for any y ≥ y0 , where r e (y x) is the nonincreasing envelope of r(y x) on [y0 +∞), that is, r e (y x) = inf{q(y) : q is nonincreasing on [y0 +∞) and q(y) ≥ r(y x) for all y ∈ [y0 +∞)}. The proof is given in the Supplemental Material. The idea in Lemma 1 is to consider a nonincreasing transformation r e which coincides with r around the largest equilibrium (see Figure 3). For y ≥ y0 , then (5)

r e (yx1 )

fU|X=x1 (u) du FU|X=x1 (r e (y x1 )) F¯n1 Y |X=x1 (y) −∞ = = r e (yx ) 2 FU|X=x2 (r e (y x2 )) F¯n2 Y |X=x2 (y) fU|X=x2 (u) du −∞

Now, how the increase in x translates into F¯n1 Y |X=x1 and F¯n2 Y |X=x2 , depends on two factors: (i) the limit behavior or r e (y x1 ) and r e (y x2 ) as y grows,


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FIGURE 3.—Plots of y → r(y x) (dashed line) and y → r e (y x) (solid line).

and (ii) the limit behavior of the distribution FU|X . On (i), recall that, by S2, r(y x) is monotone increasing in x. Hence r(y x1 ) ≤ r(y x2 ), which given the continuity and limit conditions in S1 implies r e (y x1 ) ≤ r e (y x2 ) for all y ∈ [y0 +∞). We allow for two cases. Each case requires an assumption on FU|X . ASSUMPTION S3: (i) limy→+∞ [r(y x1 )/r(y x2 )] = λ with λ > 1; (ii) for any λ > 1, limu→−∞ FU|X=x1 (λu)/FU|X=x1 (u) = 0; (iii) FU|X=x1 (u)/FU|X=x2 (u) is bounded as u → −∞. ASSUMPTION S3 : (i) limy→+∞ [r(y x1 ) − r(y x2 )] = δ with δ < 0; (ii) for any δ < 0, limu→−∞ FU|X=x1 (u + δ)/FU|X=x1 (u) = 0; (iii) FU|X=x1 (u)/FU|X=x2 (u) is bounded as u → −∞. In S3(i), we control the limit ratio of r(y x1 ) to r(y x2 ); in S3 (i), we control the difference between r(y x1 ) and r(y x2 ), as in the BLP example in Section 2.8 Assumption S3(ii) prevents the left tail of the distribution FU|X=x1 from being too heavy. Letting V ≡ −U, S3(ii) can be restated as limv→+∞ F¯V |X=x1 (λv)/F¯V |X=x1 (v) = 0 for λ > 1, where F¯V |X ≡ 1 − FV |X denotes the tail of the conditional distribution of V given X. Put in words, Assumption S3(ii) requires that the distribution of −U be rapidly varying at +∞. 8 We have limy→+∞ [r(y 0) − r(y VER)] = limp1 →+∞ ln(β1 (β2 (p1 0) 0)/β1 (β2 (p1 0) ¯ 0) 0)/β1 (β2 (p ¯ 0) VER)) ≡ δ < 0, since β1 is increasing in VER for any VER)) = ln(β1 (β2 (p value of p2 .

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Rapid variation is a well known condition in the statistics of extreme values and is satisfied by a large variety of distributions whose tail behavior ranges from moderately heavy (log-normal, heavy-tailed Weibull) to light (exponential, Gamma, normal). Assumption S3 (ii) is more restrictive; it prevents the left tail of FU|X=x1 from decaying at a rate slower than (or equal to) that of an exponential. Assumption S3 (ii) is satisfied in distributions such as the normal or light-tailed Weibull. Both S3(ii) and S3 (ii) exclude the distributions with powerlike decaying tails (Student-t, Pareto). Finally, Assumption S3(iii) (S3 (iii)) ensures that FU|X=x2 does not decrease toward 0 faster than FU|X=x1 . This property is trivially satisfied when U is independent of X, and accommodates some interesting cases where U and X are dependent (see Section 4.1.2). Assumption S3 implies that the last term in Equation (5) converges to 0 as y grows. Indeed, let λ1 ∈ (1 λ): then by S3(i) and S1(ii) there is y1 ∈ R such that for every y ≥ y1 , r(y x1 ) ≤ λ1 r(y x2 ) and hence r e (y x1 ) ≤ λ1 r e (y x2 ). As FU|X is increasing, (6)

FU|X=x1 (r e (y x1 )) y→+∞ FU|X=x (r e (y x2 )) 2 lim

FU|X=x1 (λ1 r e (y x2 )) y→+∞ FU|X=x (r e (y x2 )) 2 FU|X=x1 (λ1 r e (y x2 )) FU|X=x1 (r e (y x2 )) = lim y→+∞ FU|X=x1 (r e (y x2 )) FU|X=x2 (r e (y x2 ))

≤ lim

B

A

By S1(ii), r goes to −∞ as y gets large and so does its envelope r e ; hence the term B in (6) remains bounded. Using the rapid variation in S3(ii), the term A goes to 0 as y increases and so does the product A × B. As a result, (7)

FU|X=x1 (r e (y x1 )) = 0 y→+∞ FU|X=x (r e (y x2 )) 2 lim

Similarly, under Assumption S3 (i), d(y) = r(y x1 ) − r(y x2 ) converges to δ < 0. Let δ1 ∈ (δ 0) and y1 ∈ R be such that, for y ≥ y1 , we have d(y) < δ1 . Hence, d e (y) ≤ δ1 , where d e (y) = r e (y x1 ) − r e (y x2 ). Noting that FU|X=x1 (r e (y x1 )) y→+∞ FU|X=x (r e (y x2 )) 2 FU|X=x1 (r e (y x2 ) + δ1 ) FU|X=x1 (r e (y x2 )) ≤ lim y→+∞ FU|X=x1 (r e (y x2 )) FU|X=x2 (r e (y x2 ))

lim

A

B


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and using the same reasoning as previously, we again get the limit result in Equation (7). Combining the latter with Equations (4) and (5) then shows that (8)

F¯Y |X=x1 (y) = 0 ¯Y |X=x (y) y→+∞ F lim

2

The statement in (8) is crucial. It says that for large enough values of y, x1 ≤ x2 implies that the corresponding conditional distributions are ordered. This ordering of large enough conditional quantiles of Y given X holds under very weak restrictions on the equilibrium selection; recall that we only needed the probability of the largest equilibrium to be positive. We have thus shown the following theorem. THEOREM 1: Assume S1, S2, and either S3 or S3 hold. Fix a selection rule PXU . Let (x1 xN ) ∈ X N be such that x1 ≤ · · · ≤ xN . Then there exists y¯N ∈ R such that for all y ≥ y¯N , F¯Y |X=x1 (y) ≤ · · · ≤ F¯Y |X=xN (y). Equivalently, there exists α¯ N ∈ (0 1) such that for all α ∈ [α¯ N 1), FY−1|X=x1 (α) ≤ · · · ≤ FY−1|X=xN (α). The previous analysis has focused on quantiles with probabilities close to 1, but an analogous result continues to hold for probabilities close to 0. 4. DISCUSSION Theorem 1 derives observable implications of models with complementarities between the dependent variable Y and the explanatory variable X that are valid despite the possible presence of multiple equilibria. These implications come in the form of order restrictions on the extreme (high and low) quantiles of Y conditional on X. We now discuss important features and possible limitations of the results of Theorem 1 when used for testing the presence of MCS. 4.1. Main Features We first discuss the applicability of our results. 4.1.1. Robustness to Identification Failures We have shown that the MCS property has implications for the conditional quantiles of Y given X. Given a sample of observations on the dependent and explanatory variables, these quantiles are by definition identified and consistently estimable using standard nonparametric methods. In particular, no additional restrictions on the structural function r are needed for estimation. Consequently, the results of Theorem 1 can be used to test for complementarities whether or not the structural function r is identified.9 9 Primitive conditions for identification of the structural function r are discussed in Chesher (2003), Newey and Powell (2003), and Matzkin (2008), for example.

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4.1.2. Unobserved Heterogeneity Given that Theorem 1 does not require the structural function r to be identified or estimable, its results are fairly robust to departures from independence or mean independence conditions between the latent disturbance U and the explanatory variable X. As a consequence, our testable implications apply even in models in which X is endogenous. In particular, under the assumptions of Theorem 1, the individual heterogeneity U can be correlated with the explanatory variable X in a reasonably general way. Say that conditional on X, U is normally distributed with mean μ(X) and variance σ 2 . When μ(x) is nonincreasing in x, a simple application of l’Hôpital’s rule shows that Assumptions S3(ii) and S3(iii) hold. A simple example would be when X and U are jointly normally distributed with a nonpositive correlation coefficient. A positive correlation between X and U, under which Assumption S3(iii) fails, prevents the econometrician from learning anything about the MCS property. The intuition behind this is simple: following an increase in X, U can in those cases increase so as to decrease the extremal equilibria. In addition to being correlated with the explanatory variable, we allow U to be heteroskedastic conditional on X. Say that given X, U is normally distributed with mean 0 and variance σ 2 (X). If σ 2 (x) is nondecreasing in x, then Assumption S3(iii) holds. Therefore, a normal disturbance whose conditional variance increases with the equilibrium level satisfies our Assumption S3. 4.2. Limitations We now caution about possible limitations of our approach. 4.2.1. Tail Observations and Robustness to Outliers Theorem 1 suggests that one can use observations from the extreme (high and low) quantiles of Y conditional on X to test for the presence of MCS. Such a test shall obviously be affected by the presence of outliers. When the latter are caused by mismeasurements, methods proposed in Chen, Hong, and Tamer (2005), for example, can be used to filter the errors prior to applying the test. Unless outliers are easy to detect, one should be careful when considering very large (or small) quantiles of the dependent variable. In particular, the results of Theorem 1 lend themselves to the study of cases where X can take some relatively small number of values for which large numbers of observations of Y are available. Evaluations of policy effects, such as those following the VER, are one such example: typically X then takes on two values. 4.2.2. Continuous Explanatory Variable The cutoff level y¯N in Theorem 1 is conditional on a realization of the sample of explanatory variables, (x1 xN ) ∈ X N . This is not a problem in applications in which the explanatory variables are treated as given. In some situations,


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however, an unconditional version of Theorem 1 is needed. The latter follows easily when the explanatory variables are discrete: it suffices to apply the reasoning in Section 3.2 to all the points in X . When the explanatory variables are continuous, we need to include an extra step which will ensure that x’s do not get too close: given a random sample (X1 XN ) drawn from FX , consider the joint distribution of the N − 1 spacings between the consecutive order statistics (X1N XNN ). Fix any ε > 0 and let δN > 0 be such that the probability of all spacings being greater than δN is greater than or equal to 1 − ε. Applying the reasoning in Section 3.2 to x and x + δN , we get the following corollary to Theorem 1: COROLLARY 2: Assume S1, S2, and either S3 or S3 hold. Fix a selection rule PXU . Given ε > 0, there exists y¯N ∈ R such that for all y ≥ y¯N , Pr{F¯Y |X1N (y) ≤ · · · ≤ F¯Y |XNN (y)} ≥ 1 − ε. Equivalently, there exists α¯ N ∈ (0 1) such that for all α ∈ [α¯ N 1), Pr{FY−1|X N (α) ≤ · · · ≤ FY−1|X N (α)} ≥ 1 − ε. 1

N

In a sense, Corollary 2 gives a stochastic version of the orderings in Theorem 1. 4.2.3. Test Implementation Finally, the conditional distributions (and quantiles) of the dependent variable are typically unknown and need to be estimated from the data. A statistical test of the orderings in Theorem 1 and Corollary 2 can then be constructed by deriving the asymptotic distribution of the conditional quantile estimators—the key is to derive the latter by imposing assumptions on the distributions FU|X while maintaining our agnosticism about the equilibrium selection PXU . When using the asymptotics, however, one needs to control the speed at which the probability α¯ N increases (or decreases) relative to the sample size N. See Echenique and Komunjer (2007a) for results, albeit in a somewhat different framework. REFERENCES AGUIRREGABIRIA, V., AND P. MIRA (2007): “Sequential Estimation of Dynamic Discrete Games,” Econometrica, 75, 1–53. [1283] AMIR, R. (1996): “Cournot Oligopoly and the Theory of Supermodular Games,” Games and Economic Behavior, 15, 132–148. [1289] AMIR, R., AND V. E. LAMBSON (2000): “On the Effects of Entry in Cournot Markets,” Review of Economic Studies, 67, 235–254. [1289] ARROW, K. J., AND F. H. HAHN (1971): General Competitive Analysis. San Francisco: Holden-Day. [1281] ATHEY, S., AND S. STERN (1998): “An Empirical Framework for Testing Theories About Complementarity in Organizational Design,” Working Paper 6600, NBER. [1282] BARRO, R. J., AND X. SALA-I-MARTIN (2003): Economic Growth. Cambridge, MA: MIT Press. [1289]

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BERRY, S. T. (1992): “Estimation of a Model of Entry in the Airline Industry,” Econometrica, 60, 889–917. [1283] BERRY, S. T., J. LEVINSOHN, AND A. PAKES (1995): “Automobile Prices in Market Equilibrium,” Econometrica, 63, 841–890. [1283,1284] (1999): “Voluntary Export Restraints on Automobiles: Evaluating a Trade Policy,” American Economic Review, 89, 400–430. [1283,1284] BRESNAHAN, T. F., AND P. C. REISS (1990): “Entry in Monopoly Markets,” Review of Economic Studies, 57, 531–553. [1283] (1991): “Empirical Models of Discrete Games,” Journal of Econometrics, 48, 57–81. [1283] CAPLIN, A., AND B. NALEBUFF (1991): “Aggregation and Imperfect Competition: On the Existence of Equilibrium,” Econometrica, 59, 25–59. [1284] CHEN, X., H. HONG, AND E. TAMER (2005): “Measurement Error Models With Auxiliary Data,” Review of Economic Studies, 72, 343–366. [1294] CHESHER, A. (2003): “Identification in Nonseparable Models,” Econometrica, 71, 1405–1441. [1293] CILIBERTO, F., AND E. TAMER (2007): “Market Structure and Multiple Equilibria in the Airline Industry,” Mimeo, University of Virginia and Northwestern University. [1283] ECHENIQUE, F., AND I. KOMUNJER (2007a): “A Test for Monotone Comparative Statics,” Working Paper 1278, Division of Humanities and Social Sciences, Caltech. [1287,1295] (2007b): “Testing Models With Multiple Equilibria by Quantile Methods,” Working Paper 1244R, Division of Humanities and Social Sciences, Caltech. [1284] (2009): “Supplement to ‘Testing Models With Multiple Equilibria by Quantile Methods’,” Econometrica Supplemental Material, 77, http://www.econometricsociety.org/ecta/ Supmat/6223_proofs.pdf. [1283] GABAY, D., AND H. MOULIN (1980): “On the Uniqueness and Stability of Nash-Equilibria in Noncooperative Games,” in Applied Stochastic Control in Econometrics and Management Science, ed. by P. K. A. Bensoussan and C. Tapiero. Amsterdam: North-Holland, 271–293. [1284] HAYASHI, F. (1982): “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econometrica, 50, 213–224. [1289] HAYASHI, F., AND T. INOUE (1991): “The Relation Between Firm Growth and Q With Multiple Capital Goods: Theory and Evidence From Panel Data on Japanese Firms,” Econometrica, 59, 731–753. [1289] JOVANOVIC, B. (1989): “Observable Implications of Models With Multiple Equilibria,” Econometrica, 57, 1431–1437. [1282] KREPS, D. M. (1990): Game Theory and Economic Modeling. London: Oxford University Press. [1282] LJUNGQVIST, L., AND T. J. SARGENT (2004): Recursive Macroeconomic Theory (Second Ed.). Cambridge, MA: MIT Press. [1289] MAS-COLELL, A., M. D. WHINSTON, AND J. R. GREEN (1995): Microeconomic Theory. London: Oxford University Press. [1288] MATZKIN, R. L. (1994): “Restrictions of Economic Theory in Nonparametric Methods,” in Handbook of Econometrics, Vol. IV, ed. by R. F. Engle and D. L. McFadden. New York: Elsevier Science, Chap. 42, 2523–2558. [1287] (2008): “Identification in Nonparametric Simultaneous Equations,” Econometrica, 76, 945–978. [1287,1293] MCLENNAN, A. (2005): “The Expected Number of Nash Equilibria of a Normal Form Game,” Econometrica, 73, 141–174. [1284] MILGROM, P., AND J. ROBERTS (1990): “Rationalizability, Learning and Equilibrium in Games With Strategic Complementarities,” Econometrica, 58, 1255–1277. [1284] (1994): “Comparing Equilibria,” American Economic Review, 84, 441–459. [1282] MILGROM, P., AND C. SHANNON (1994): “Monotone Comparative Statics,” Econometrica, 62, 157–180. [1284]


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NEWEY, W. K., AND J. L. POWELL (2003): “Instrumental Variable Estimation of Nonparametric Models,” Econometrica, 71, 1565–1578. [1293] TAKAHASHI, S. (2005): “The Number of Pure Nash Equilibria in a Random Game With Nondecreasing Best Responses,” Mimeo, Harvard University. [1284] TAMER, E. (2003): “Incomplete Simultaneous Discrete Response Model With Multiple Equilibria,” Review of Economic Studies, 70, 147–165. [1283] VILLAS-BOAS, J. M. (1997): “Comparative Statics of Fixed Points,” Journal of Economic Theory, 73, 183–198. [1282]

Division of Humanities and Social Sciences, California Institute of Technology, 311 Baxter Hall, Pasadena, CA 91125, U.S.A.; [email protected] and Dept. of Economics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508, U.S.A.; [email protected]. Manuscript received December, 2005; final revision received November, 2008.


MORE ON CONFIDENCE INTERVALS FOR PARTIALLY IDENTIFIED PARAMETERS BY JÖRG STOYE1 This paper extends Imbens and Manski’s (2004) analysis of confidence intervals for interval identified parameters. The extension is motivated by the discovery that for their final result, Imbens and Manski implicitly assumed locally superefficient estimation of a nuisance parameter. I reanalyze the problem both with assumptions that merely weaken this superefficiency condition and with assumptions that remove it altogether. Imbens and Manski’s confidence region is valid under weaker assumptions than theirs, yet superefficiency is required. I also provide a confidence interval that is valid under superefficiency, but can be adapted to the general case. A methodological contribution is to observe that the difficulty of inference comes from a preestimation problem regarding a nuisance parameter, clarifying the connection to other work on partial identification. KEYWORDS: Bounds, identification regions, confidence intervals, uniform convergence, superefficiency.

1. INTRODUCTION ANALYSIS OF PARTIAL IDENTIFICATION, that is, of models where only bounds on parameters are identified, has become an active field of econometrics.2 Within this field, attention recently turned to general treatments of estimation and inference. An important contribution in this direction is by Imbens and Manski (2004, IM henceforth), who pointed out that one might be interested in confidence regions for partially identified parameters themselves rather than “identified sets.” The intuitively most obvious, and previously used, confidence regions are of the latter type, meaning that they are conservative for the parameters. IM proposed a number of confidence regions for real-valued parameters that one can asymptotically conclude lie in an interval. This paper refines and extends IM’s main technical result—a confidence interval that exhibits uniform coverage of partially identified parameters if the length of the identified interval is a nuisance parameter. IM relied on a highlevel assumption that turns out to imply locally superefficient estimation of this nuisance parameter and that will fail in many applications. I take this discovery 1

I am indebted to Adam Rosen, whose insightful comments helped to much improve this paper. I also received helpful comments from an associate editor and three anonymous referees, Don Andrews, Xiaohong Chen, Aureo de Paula, Yanqin Fan, Guido Imbens, Nick Kiefer, Thierry Magnac, Chuck Manski, Francesca Molinari, Hyungsik Roger Moon, and Sang Soo Park, seminar audiences at Brown, Cologne, Cornell, Mannheim, NYU, Penn, Penn State, Vanderbilt, and Yale, and conference audiences in Bonn, Budapest, Durham (U.S.A.), London, Montréal, Munich, and Philadelphia. Financial support from a University Research Challenge Fund, New York University, is gratefully acknowledged. Of course, any and all errors are mine. 2 See Manski (2003) for a survey and many references. © 2009 The Econometric Society

DOI: 10.3982/ECTA7347

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JÖRG STOYE

as a point of departure for a new analysis of the problem, providing different confidence intervals that are valid both with and without superefficiency. A brief summary and overview of results goes as follows. In Section 2, I set up the inference problem, briefly summarize IM’s contribution, and explain the aforementioned issue. Section 3 contains the reanalysis. It reconstructs IM’s results from weaker and, as will be shown, more generic assumptions, but also proposes a different confidence region that is easily adapted to the case of no superefficiency. Section 4 concludes and the Appendix contains all proofs. 2. BACKGROUND Following Woutersen (2006), I consider a generalization of IM’s setup that removes some nuisance parameters. The object of interest is the real-valued parameter θ0 (P) of a probability distribution P(X); P must lie in a set P that is characterized by ex ante constraints (maintained assumptions). The random variable X is not completely observable, so that θ0 may not be identified. Assume, however, that the observable aspects of P(X) identify bounds θl (P) and θu (P) such that θ0 ∈ [θl θu ] almost surely. The interval Θ0 ≡ [θl θu ] will also be called an identified set. Let (P) ≡ θu − θl denote its length. Assume that estimators θl , θu , and exist and are connected by the identity ≡ θu − θl . Confidence regions for identified sets of this type are conventionally formed as σl σu cα cα CIα = θl − √ θu + √ N N σu ] is a standard error for θl [ θu ] and where cα is chosen such that where σl [ (1)

(cα ) − (−cα ) = 1 − α

For example, cα = −1 (0 975) ≈ 1 96 for a 95% confidence interval. Under regularity conditions, a simple Bonferroni argument establishes that limN→∞ Pr(Θ0 ⊆ CIα ) ≥ 1 − α. IM’s contribution is motivated by the observations that (i) one might be interested in coverage of θ0 rather than Θ0 , (ii) whenever > 0, then limN→∞ Pr(θ0 ∈ CIα ) ≥ 1 − α/2. In words, a 90% confidence interval (CI) for Θ0 is a 95% CI for θ0 . The reason is that asymptotically, is large relative to sampling error, so that noncoverage risk is effectively onesided at {θl θu } and vanishes otherwise. One would, therefore, be tempted to construct a level α CI for θ as CI2α .3 Unfortunately, this intuition works pointwise but not uniformly over interesting specifications of P . Specifically, limN→∞ Pr(θ0 ∈ CIα ) < 1 − α/2 along any local parameter sequence where N ≤ O(N −1/2 ), that is, whenever fails 3

To avoid uninstructive complications, I presume α ≤ 0 5 throughout.

PARTIALLY IDENTIFIED PARAMETERS

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to diverge relative to sampling error. While uniformity failures are standard in econometrics, this one is unpalatable because it concerns a very salient region of the parameter space; were it neglected, one would be led to construct confidence intervals that shrink as a parameter moves from point identification to slight underidentification.4 IM therefore proposed the confidence region σl σu c1 c1 θl − √α CI1α ≡ θu + √α N N where cα1 solves (2)

cα1 +

√ N − (−cα1 ) = 1 − α max{ σl σu }

Comparison of (2) with (1) reveals that (2) takes into account the estimated length of the identified set. For a 95% confidence set, cα1 will be −1 (0 975) ≈ 1 96 if = 0, that is, if point identification must be presumed, and will ap grows large relative to sampling error. IM proach −1 (0 95) ≈ 1 64 as 1 showed uniform validity of CIα under the following assumption. θl and θu that satisfy ASSUMPTION 1: (i) There exist estimators √ θ − θl d 0 σl2 ρσl σu N l →N θu − θu ρσl σu σu2 0 uniformly in P ∈ P , and there are estimators ( σl2 σl2 ρ) that converge to their population values uniformly in P ∈ P . (ii) For all P ∈ P , σ 2 ≤ σl2 ≤ σ 2 and σ 2 ≤ σu2 ≤ σ 2 for some positive and finite 2 σ and σ 2 , and θu − θl ≤ < ∞. (iii) √ For all > 0,v there are v > 0, K, and N0 such that N ≥ N0 implies − | > K ) < uniformly in P ∈ P . Pr( N| While it is clear that uniformity can obtain only under restrictions on P , it is important to note that is not bounded from below and thus the specific 4 In fact, CIα is not uniformly valid (at level 1 − α) for the identified set either: problems again occur if N is local to zero. This is not the focus of this paper however. All of these problem could be avoided by bounding away from 0, but such a restriction will frequently be inappropriate. For example, one cannot a priori bound from below the degree of item nonresponse in a survey or of attrition in a panel. Also, even when is known a priori (e.g., with interval data), the problem arguably disappears only in a superficial sense. Were it ignored, one would construct confidence intervals that work uniformly given any model but whose performance deteriorates across models as point identification is approached.

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uniformity problem that arises near point identification is not assumed away. Having said that, conditions (i) and (ii) are fairly standard, but (iii) deserves some explanation. It implies that approaches its population counterpart in a specific way. If = 0, then = 0 with probability approaching 1 in finite samples; that is, if point identification obtains, then this will be learned exactly and the limiting distribution of must be degenerate. What is more, degenerate limiting distributions occur along any local parameter sequence that converges to zero, as is formally stated in the following lemma.5 √ p LEMMA 1: Assumption 1(iii) implies that N| − N | → 0 for all sequences of distributions {PN } ⊆ P such that N ≡ (PN ) → 0. In words, Assumption 1(iii) requires to be superefficient at zero. It seems that this was not previously recognized, and it is certainly a nonstandard restriction. To be sure, there are relevant cases where superefficiency obtains naturally. One important example is that sample probabilities are superefficient estimators of population probabilities near zero; this is why Imbens and Manski (2004) were able to verify their assumptions for the mean with missing data. A new and less obvious example will be presented in this paper. Also, superefficiency has a somewhat dubious reputation due to pitfalls encountered when it is induced artificially. Such constructions incur large local risk that is, furthermore, easily overlooked in inappropriate asymptotic frameworks; see Leeb and Pötscher (2005) for some cautionary tales. These problems need not arise when superefficiency is assumed and they do not affect CI1α . Having said that, imposing superefficiency by assumption raises questions about generality. It fails when θl and θu come from individual moment conditions (as in the ATM example of Pakes, Porter, Ho, and Ishii (2006)) or when worst-case bounds are tightened by means of testable assumptions (i.e., the refined bounds may overlap). More generally, one might wonder whether CI1α applies in cases that interestingly generalize the mean with missing data. I now turn to answering these questions. Regarding the range of applications of CI1α , the superefficiency condition can be weakened and is then implied θl by construction. Regarding generality, CI1α is not valid withwhenever θu ≥ out superefficiency, but one can construct an alternative interval that is easily adapted to settings with and without superefficiency. 3. REANALYSIS OF THE INFERENCE PROBLEM It is instructive to begin by assuming that N is known. 5 This paper makes heavy use of local parameters, and to minimize confusion, I reserve the subscript (·)N for deterministic functions of N, including local parameters; hence the use of cα throughout. where IM used CN . Estimators are denoted by (·)


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θl that satisfies ASSUMPTION 2: (i) There exists an estimator √ d N[ θl − θl ] → N(0 σl2 ) σl2 that converges to σl2 uniformly uniformly in P ∈ P , and there is an estimator in P ∈ P . (ii) N ≥ 0 is known. (iii) For all P ∈ P , σ 2 ≤ σl2 ≤ σ 2 for some positive and finite σ 2 and σ 2 . Applications of this scenario include inference about the mean from interval data, where the length of intervals (e.g., income brackets) does not vary on the support of θ, as well as worst-case bounds on the average treatment effect when potential outcomes are supported on [0 1]. Define σl σl cα cα CIα = θl − √ θu + √ N N where (3)

√ NN − (− cα ) = 1 − α cα + σl

Lemma 2 establishes that this confidence interval is uniformly valid. LEMMA 2: Let Assumption 2 hold. Then lim inf

inf

N→∞ θ∈Θ P:θ0 (P)=θ

α ) = 1 − α Pr(θ0 ∈ CI

This result generalizes IM’s Lemma 3 from means with missing data to the present setting. It has the same proof idea: The normal approximation to α ) is concave in θ0 and equals (1 − α) if θ0 ∈ {θl θu }. The lemma’s Pr(θ0 ∈ CI α is not main purpose is as a backdrop for the case with unknown N , when CI feasible; preestimation of N will turn out to be the root cause of most problems. I turn to this case now. Thus, impose Assumption 1(i) and (ii), and drop Assumption 1(iii), but consider the following. √ ASSUMPTION 3: There exists a sequence {aN } such that aN → 0, aN N → ∞, √ p and N| − N | → 0 for all sequences of distributions {PN } ⊆ P with N ≤ aN .

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Assumption 3 is again a superefficiency condition, although it is more transparent than Assumption 1(iii). More importantly, it is weaker: By Lemma 1, Assumption 1(iii) implies that the above holds with the quantifier “for all sequences {aN }.” One consequence of this weakening is the availability of a simple sufficient condition for superefficiency.6 LEMMA 3: Let Assumption 1(i) and (ii) hold, and assume that Pr( θu ≥ θl ) = 1 for all P . Then Assumption 3 is implied. θu ) are jointly asymptotically norThus, Assumption 3 obtains whenever ( θl mal and are almost surely ordered, in particular, when they are ordered by construction. This immediately verifies superefficiency for a good number of applications including worst-case bounds on smooth functions of population moments or (under regularity conditions) quantiles, where in either case, partial identification could be due either to missing data or to interval observations with (a priori) unknown length of intervals. Assumption 1(iii) is not similarly implied and accordingly is harder to verify (although ultimately fulfilled in the examples just given). Nonetheless, Assumption 3 suffices for CI1α to be valid. PROPOSITION 1: Let Assumptions 1(i) and (ii) and 3 hold. Then lim inf

inf

N→∞ θ∈Θ P:θ0 (P)=θ

Pr(θ0 ∈ CI1α ) = 1 − α

Proposition 1 strengthens IM’s main result; furthermore, a clear understanding of its reliance on superefficiency informs a concise and intuitive proof. α , with c 1 being an estimator of cα . Think of CI1α as a feasible version of CI α 1 Validity of CIα would easily follow from consistency of cα1 , but such consistency √ √ seems to fail: ( − ) is usually of order O(N −1/2 ), so that ( N − N) does not vanish. This is where superefficiency comes into play. Think in terms of local parameters N , and distinguish between sequences where N vanishes fast enough for 3 to apply and sequences where this fails. In the √ √Assumption − N) does vanish, so the asymptotics are as if were former case, ( N known. In the latter case, N grows large relative to sampling error, so that the uniformity problem does not arise to begin with. The intuition highlights one channel through which superefficiency simplifies the analysis, namely the vanishing of sampling variation in N . There is θl + N , meaning that the a second, more subtle channel: Asymptotically, θu = limiting sampling distribution is univariate. This simplification allowed IM to calibrate cα through a single equation (2), even though the estimation problem 6

I thank Thierry Magnac for suggesting this result.


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is generally bivariate. One can avoid reliance on this simplification as follows: Let (cl2 cu2 ) minimize ( σl cl + σu cu ) subject to the constraint that √ N

2 (4) ρz1 ≤ cu + + 1−ρ z2 ≥ 1 − α Pr −cl ≤ z1 σu √ N

2 (5) − 1− ρ z2 ≤ ρz1 z1 ≤ cu ≥ 1 − α Pr −cl − σl where z1 and z2 are independent standard normal random variables.7 In typical cases, (cl2 cu2 ) will be uniquely characterized by the fact that both (4) and (5) hold with equality, but it is conceivable that one condition is slack at the solution. Let σl cl2 σu cu2 2 CIα ≡ θl − √ θu + √ N N Then the following statement holds. PROPOSITION 2: Let Assumptions 1(i) and (ii) and 3 hold. Then lim inf

inf

N→∞ θ∈Θ P:θ0 (P)=θ

Pr(θ0 ∈ CI2α ) = 1 − α

Calibration of (cl2 cu2 ) takes into account that the underlying estimation problem is bivariate. Expression (4) ensures that the nominal size of CI2α equals at least (1 − α) if θ0 = θl ; equation (5) does the same if θ0 = θu , where equality obtains if the according condition binds.8 In contrast, expression (2) enforces cl = cu and properly calibrates both only if σl = σu and ρ = 1. In general, the nominal size of CI1α for any finite N exceeds (1 − α) at one end and falls short of it at the other end, so that the interval appears invalid. This does not affect first-order asymptotics because for large N , the testing problem is asymptotically one-sided, whereas for small N , superefficiency ensures that σl = σu and ρ = 1 in the limit. Nonetheless, the nominal size of CI1α really converges to (1 − α) rather than equalling it, and CI1α corresponds to a biased hypothesis test, that is, Θ0 is not an upper contour set of nominal coverage probability. The construction of CI2α avoids these issues; indeed, CI2α is literally defined as the shortest confidence interval with correct nominal size. 7 Appendix B exhibits closed-form expressions for (4) and (5), illustrating that they can be evaluated without simulation. 8 By the nominal size of CI at θl , say, I mean CI 1/ σl φ((x − θˆ l )/σˆ l ) dx, that is, its size at θl as predicted from sample data. This size would be exact in finite samples if normal approximations were perfect and variances were known. Confidence regions are typically constructed by setting it equal to 1 − α.

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The improvement may be mostly conceptual if superefficiency obtains, because CI1α and CI2α are then asymptotically equivalent. It becomes more serious when superefficiency fails. The reason is that validity of CI2α depends on Assumption 3 only through the first channel mentioned above. This dependency can be eliminated by a straightforward modification: Consider a shrinkage estimator ∗ ≡ > bN , 0 otherwise, √ where bN is some preassigned sequence such that bN → 0 and bN N → ∞. By replacing with ∗ in the calibration of (cl2 cu2 ), one can artificially restore superefficiency where it is needed. Of course, this comes at the aforementioned cost of inducing superefficiency, which is discussed further below. σl c l + σu cu ) subject to the constraint that Let (cl3 cu3 ) minimize ( √ N∗

2 Pr −cl ≤ z1 ρz1 ≤ cu + + 1− ρ z2 ≥ 1 − α σu √ N∗

+ 1− ρ2 z2 ≤ ρz1 z1 ≤ cu ≥ 1 − α Pr −cl − σl where z1 and z2 are as before, and define ⎧ σl cl3 σu cu3 σl c 3 σu c 3 ⎨ θl − √ θu + √ θl − √ l ≤ θu + √ u , 3 CIα ≡ N N N N ⎩ ∅ otherwise Before discussing CI3α further, I state this paper’s final result. PROPOSITION 3: Let Assumption 1(i) and (ii) hold. Then lim inf

inf

N→∞ θ∈Θ P:θ0 (P)=θ

Pr(θ0 ∈ CI3α ) = 1 − α

The definition of CI3α reveals an additional modification: If θu is too far below θl , the interval is empty, which can be interpreted as rejection of the maintained assumption that θu ≥ θl . In other words, CI3α embeds a specification test. IM did not consider such a test, presumably for two reasons: It does not arise in their leading application, and it is trivial under their assumptions or whenθl are ordered by construction. But it is interesting in applications ever θu and such as moment inequalities, where < 0 is a generic possibility and samples θl might raise doubts whether θu ≥ θl . In other cases, the with θu much below possibility of CI3α being empty may be unattractive. It can be avoided by arbitrarily replacing ∅ in the above definition. One plausible alternative would be


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a Wald confidence region based on imposing θu = θl , that is, to estimate both θl and θu by a variance-weighted average θ of θl and θu , and to construct a −1 confidence region by adding and subtracting (1 − α/2) standard errors of θ. Any such enlargement of CI3α will make it potentially conservative if = 0; adjusting (cl3 cu3 ) to achieve exact size would cause complications. Whichever way one resolves this issue, an intriguing aspect of CI3α√is that it is analogous to CI2α except for the use of ∗ . (The event that θl − σl cl3 / N > θu + √ θu ≥ σu cu3 / N uniformly vanishes under superefficiency and is impossible if θl .) Together, CI2α and CI3α therefore provide a unified approach to inference for interval identified parameters. In contrast, CI1α does not become valid upon replacing with ∗ in (3). Some further remarks on these results are in order. First, inspection of proofs reveals that Propositions 1 and 2 do not really require joint (as opposed to marginal) normality of estimators. Hence IM’s result can be strengthened even further, although the additional gain in generality may be marginal. Furthermore, joint normality cannot be dropped from Lemma 3, so it is still needed if one wishes to invoke the simple sufficient condition. Joint normality is also required to claim that CI2α has exact nominal size and corresponds to a nominally unbiased test. Second, two ways in which ∗ can be modified are as follows. I defined a soft thresholding estimator for simplicity, but making ∗ a smooth function of would also ensure validity and might improve performance for close to bN . Also, the sequence bN is left to adjustment by the user. While the law of the iterated logarithm makes bN = (log log N)1/2 a salient choice, such adjustment is generally subject to the following trade-off: The slower bN vanishes, the less distortion is caused by shrinkage, but the quality of uniform normal approximations deteriorates, and they break down for bN = O(N −1/2 ). I do not expand on these possibilities to avoid redundancy with independent work by Andrews and Soares (2007). A modification of CI3α that is not recommended is to let cl3 = −1 (1 − α) and cu3 = −1 (1 − α), implying that CI3α = CI2α , whenever ∗ > 0. This would make sufCI3α shorter without affecting first-order asymptotics, because shrinking fices to ensure uniformity. But the new interval ignores the two-sided nature of noncoverage risk and hence has nominal size below 1 − α (although approaching 1 − α as N grows large). The improvement in length is, therefore, spurious. Third, ∗ = 0 can be interpreted as failure of a pretest to reject H0 : θu = θl , where the size of the pretest approaches 1 as N → ∞. Thus, shrinking resembles the “conservative pretest” solution to the parameter-on-the-boundary problem given by Andrews (2000, Section 4). Nonetheless, one should not interpret CI3α as being based on model selection. If θu = θl were known, the aforementioned Wald confidence region would be efficient, and a post-modelselection confidence region would use it, but it would be invalid if N =

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O(N −1/2 ). CI3α avoids the trap by employing a shrinkage estimator of in the pretest but not in the subsequent construction of the interval. Having said that, there is a tight connection between problems encountered here and known issues with post-model-selection estimators (Leeb and Pötscher (2005)), the underlying problem being discontinuity of pointwise limit distributions. In particular, now that superefficiency of ∗ is induced rather than assumed, it does follow that ∗ estimates N with large local risk near zero. Uniform validity of CI3α accordingly comes at the price of asymptotic dissimilarity, that is, the interval is conservative along certain local parameter sequences. This feature cannot be avoided by any of the modifications mentioned above (or in Andrews (2000) or Andrews and Soares (2007)). Finally, this paper’s assumptions improve on preceding work but still entail substantial loss of generality. Uniform validity of normal approximations could√be replaced by uniform validity of the bootstrap after minor adjustments, and N consistency plays no special role in and of itself. But these modifications will not help if upper and lower bounds are characterized as minima, respectively maxima, over a number of moment conditions (as in the hospitals example of Pakes et al. (2006); but see subsequent work by Chernozhukov, Lee, and Rosen (2008)). Also, it is not obvious how to generalize CI3α to multivariate moment conditions (but see subsequent work by Fan and Park (2007)). Other, independently developed methods (Andrews and Guggenberger (2007), Andrews and Soares (2007)) will work more generally. When CI3α applies, however, it is attractive for numerous reasons: It is trivial to compute, it is by construction the shortest interval whose nominal size is exact at both ends, and it is easily adjusted to the presence or absence of superefficiency. 4. CONCLUSION This paper extended Imbens and Manski’s (2004) analysis of confidence regions for partially identified parameters. Core findings are as follows. I establish that one assumption used for IM’s final result boils down to locally superefficient estimation of a nuisance parameter. This fact appears to have gone unnoticed before. The inference problem is then reanalyzed with and without superefficiency. IM’s confidence region is found to be valid under weaker conditions than theirs, a sufficient condition being that estimators of bounds are jointly asymptotically normal and ordered by construction. Furthermore, valid inference can be achieved by a confidence region that corresponds to a nominally unbiased hypothesis test, is easily adapted to the case without superefficiency, and embeds a specification test. A conceptual contribution is to recognize that much of the inference problem stems from preestimation of . This insight allows for brief and transparent proofs, and clarifies the connection to related work. For example,


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once the boundary problem is recognized, analogy to Andrews (2000) suggests that a straightforward normal approximation, as well as the bootstrap, will fail, whereas subsampling might work. Indeed, carefully specified subsampling techniques are known to yield valid inference for parameters identified by moment inequalities, of which the present scenario is a special case (Andrews and Guggenberger (2007), Chernozhukov, Hong, and Tamer (2007), Romano and Shaikh (2008)). The bootstrap, on the other hand, does not work in the same setting, unless it is modified in several ways, one of which is analogous to shrinking (Bugni (2007)). Against the backdrop of these (subsequent to IM) results, the validity of simple normal approximations in IM appears to be a puzzle that is now resolved. At the same time, the updated version of these normal approximations has practical value because it provides closed-form and otherwise attractive inference for important, if relatively simple, applications. APPENDIX A: PROOFS Preliminaries Most proofs consider sequences {PN } that will be identified with the implied sequences {N θN } ≡ {(PN ) θ0 (PN )}. For ease of notation, I suppress the N subscript on (θl σl σu ) and on estimators. Some algebraic steps treat (θl σl σu ) as constant; this is without loss of generality because by compactness implied in Assumption 1(ii), any sequence {PN } induces a sequence of values (θl σl σu ) with finitely many accumulation points, and the argument can be conducted separately for the according subsequences. Proofs of Lemma 2 and the propositions establish that lim Pr(θN ∈ CIiα ) ≥ 1 − α

N→∞

i = 1 2 3

These are pointwise limits, but they imply the claims because they are taken over sequences, in particular, they apply along least favorable sequences. Proofs present two arguments: one for the case that {N } is “small” and one for the case that it is “large” in a sense that will be delimited. Any sequence {PN } can be decomposed into one large and one small subsequence. PROOF OF LEMMA 1: The aim is to show that if N → 0, then √ − N | > δ) < ε ∀δ ε > 0 ∃N ∗ : N ≥ N ∗ ⇒ Pr( N| Fix δ and ε. By Assumption 1(iii), there exist N0 , v > 0, and K such that √ − N | > KvN ) < ε N ≥ N0 ⇒ Pr( N| uniformly over P (and hence N ). As N → 0, one can choose N1 such that N ≥ N1 ⇒ N ≤ δ1/v K −1/v ⇒ KvN ≤ δ. Combining these and choosing N ∗ ≡

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max{N0 N1 } yields N ≥ N∗

⇒

√ − N | > KvN ) ε > Pr( N| √ ≥ Pr( N| − N | > δ) Q.E.D.

as required.

PROOF OF LEMMA 2: Parameterize θ0 as θ0 = θl + aN for some a ∈ [0 1]. Then algebra reveals that α ) Pr(θ0 ∈ CI √ √ √ σl N N(θl − θl ) N σl aN ≤ ≤ (1 − a)N + cα = Pr − cα − σl σl σl σl σl √ √ N N cα − (1 − a)N − − aN → cα + σl σl θl , this converuniformly over P . Besides uniform asymptotic normality of gence statement uses that by uniform consistency of σl in conjunction with the lower bound on σl , σl /σl → 1 uniformly, and also that the derivative of the standard normal cumulative distribution function (c.d.f.) is uniformly bounded. Evaluation of derivatives establishes that the last expression in the preceding display is strictly concave in a, hence it is minimized at a ∈ {0 1}. But in those cases, the algebra simplifies to √ N α ) → Pr(θl ∈ CI cα ) = 1 − α N + cα − (− σl Q.E.D.

and similarly for θu . √

− N ) → N(0 σ2 ) uniformly PROOF OF LEMMA 3: By Assumption 3, N( 2 2 2 in P , where σ ≡ σl√ +σu −2ρσl σu . Hence, one can fix a sequence εN → 0 such − N ) ≤ d) − (d/σ2 )| ≤ εN for all N. (The lemma that supP∈P d∈R | Pr( N( 2 restates Lemma 2 if σ = 0, so this case can be ignored.) Fix a nonpositive sequence δN → −∞ such that (γδN ) > O(εN ) for any fixed γ ≥ 0. This is possible because of well known uniform bounds on the standard normal c.d.f., (e.g., (γδN ) > −(γδN )−1 (2π)−1/2 exp(−(γδN )2 /2) as δN → −∞); using this bound, one can verify that δN = −(log(− log εN ))1/2 will do. Fix any sequence PN such that N ≤ aN ≡ −δN N −1/2 . I will show that σ2 → 0, implying Assumption 3. Assume this fails. Then σ2 must have an accumulation 2 2 point σ∞ > 0. Along any subsequence converging to σ∞ , one would have √ √ Pr( θu ≤ θl ) = Pr( ≤ 0) = Pr( N( − N ) ≤ − NN ) √ ≥ (− NN /σ∞ ) − εN ≥ (δN /σ∞ ) − εN > 0


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for N large enough, a contradiction. Note how the conclusion of Lemma 2 (and hence, Assumption 1(iii)) is not implied because for N > aN , the second inequality above might fail. Q.E.D. √ p PROOF OF PROPOSITION 1: Let N ≤ aN . Then N| − N | → 0 by Assumption 3; thus √ √ p √ N( θu − θu ) = N( θl + − θl − N ) → N( θl − θl ) In conjunction with conditions (i) and (ii), this implies σu = σl , hence σu − p σl → 0. Also using Assumption 3 again, it follows that √ √ N p 1 1 → cα + N cα + N max{ σl σu } σl and the argument can be √ completed as in Lemma 2. √ aN . Then NN → ∞, hence lim supN→∞ N(θN − θl ) = ∞ or Let N >√ lim supN→∞ N(θu − θN ) = ∞ or both. Some algebra reveals that Pr(θN ∈ CI1α ) √ √ √ = Pr −cα1 σl ≤ N(θN − θl ) + N(θl − θl ) ≤ N + cα1 σu √ √ σl ≤ N(θN − θl ) + N(θl − θl ) = Pr −cα1 √ √ √ + cα1 θl ) > N σu − Pr N(θN − θl ) + N(θl − √ , divergence of Assume lim supN→∞ N(θN − θl ) < ∞. By√consistency of √ NN implies divergence in probability of N . Thus √ √ √ Pr N(θN − θl ) + N(θl − θl ) > N + cα1 σu √ √ √ θl ) > N − N(θN − θl ) → 0 ≤ Pr N(θl − √ where I used that cα1 σu ≥ 0 by construction and that N(θl − θl ) converges to a random variable by assumption. It follows that lim Pr(θN ∈ CI1α )

N→∞

√ √ σl ≤ N(θN − θl ) + N(θl − θl ) = lim Pr −cα1 N→∞ √ ≥ lim Pr −cα1 σl ≤ N(θl − θl ) = 1 − (cα1 ) ≥ 1 − α N→∞

√ the second inequality uses where the first inequality uses N(θN − θl ) ≥ 0, and √ σl and N(θl − θl )/σl . the definition of cα1 as well as convergence of

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√ For any subsequence of {PN } such that N(θN − θu ) fails to diverge, the argument is symmetric. If both diverge, coverage probability converges to 1. To see that a coverage probability of 1 − α can be attained, consider the case of = 0. Q.E.D. PROOF OF PROPOSITION 2: For a short proof that does not use joint normality, inspection of (4) and (5) reveals that CI2α is asymptotically equivalent to CI1α given local superefficiency. The longer argument below shows why CI2α generally has exact nominal size and will also be needed for Proposition 3. Let ( cl cu ) fulfil √ N

Pr − cl ≤ z1 ρz1 ≤ cu + + 1 − ρ2 z2 ≥ 1 − α σu √ N

2 + 1 − ρ z2 ≤ ρz1 z1 ≤ cu ≥ 1 − α Pr − cl − σl and write

(6) (7)

σl σu cl cu lim Pr θl ∈ θl − √ θu + √ N→∞ N N √ σl N = lim Pr − cl ≤ (θl − θl ) N→∞ σl σl √ √ N N σu (θu − θu ) ≤ + cu σl σl σl √ N cl ≤ (θl − θl ) = lim Pr − N→∞ σl √

σu √ N σu 1 2 ρ N(θl − θl ) − σu 1 − ρ z2 ≤ + cu σl σl σl σl √

N cl ≤ z1 ρz1 ≤ cu + + 1 − ρ2 z2 = lim Pr − N→∞ σu ≥1−α

uniformly. Here, the first step can be verified algebraically; the second step uses that by assumption, √ √ σu √ d ( N(θu − θu )| N(θl − θl )) → N ρ N(θl − θl ) σu2 (1 − ρ2 ) σl σu ) are uniformly consistent, and that uniformly, that ( σl √ σl σu ≥ σ, so that neither can vanish; the third step uses convergence of N/σl (θl − θl ). The


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argument for θu is similar. Note that in contrast to the very first step of Proposition 1, Assumption 3 was not used. As before, for any sequence {N } such that N < aN , superefficiency implies cl cu ) and CI2α therefore is valid at {θl θu }. Conthat (cl2 cu2 ) is consistent for ( vexity of coverage probability over [θl θu ] follows as before. For N ≥ aN , the argument entirely resembles Proposition 1. Finally, (7) will bind if (4) binds, implying that CI2α will then have exact nominal size at θl . A similar argument applies for θu . But (cl2 cu2 ) can minimize (cl σl + cu σu ) subject to (4) and (5) only if at least one of (4) and (5) binds, hence CI2α is nominally exact. Q.E.D. PROOF OF PROPOSITION 3: Let cN ≡ (N −1/2 bN )1/2 ; thus N 1/2 cN = (N 1/2 × bN )1/2 → ∞ and for parameter sequences such that N > cN , the proof is again as before. For the other case, consider ( cl cu ) as defined in the previous proof. ≤ By uniform convergence of estimators and uniform bounds on (σl σu ), Pr( bN ) is uniformly asymptotically bounded below by √ 1/2 /2σ → 1 ( N(bN − cN )/2σ) = N 1/2 bN − N 1/2 bN Hence, ∗ = 0 ≤ with probability approaching 1. Expression (6) is easily seen cu ), hence CI3α is valid (if potentially conservative) to increase in for every ( cl at θl . The argument for θu is similar. Now parameterize the true parameter value as θN ≡ aθl + (1 − a)θu for some a ∈ [0 1]. Then some algebra yields

σl σu cl3 cu3 Pr θN ∈ θl − √ θu + √ N N √ √ ) − N(1 − a) − cl3 σl = Pr N(1 − a)( − √ √ ≤ a N(θl − θl ) + (1 − a) N(θu − θu ) √ √ 3 − ) + Na + cu σu ≤ Na( Consider varying , holding (θl σl σu ρ a) constant. The cutoff values cl3 and cu3 depend on only through ∗ , but recall that Pr(∗ = 0) → 1. The estimators ( σl σu ) are uniformly consistent and the joint limiting distribution of all other random variables depends only on (σl σu ρ a). Hence, the preceding probability’s limit is minimized at = 0, in which case θN = θl and coverage was already shown. Q.E.D.

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APPENDIX B: CLOSED -FORM EXPRESSIONS FOR (cl cu ) This appendix provides a closed-form equivalent of (4) and (5). These expressions can be written as √ cl N ρ cu d(z) ≥ 1 − α

z+

+

1− ρ2 ρ2 1− ρ2 σu 1 − −∞ √ cu N ρ cl d(z) ≥ 1 − α

z+

+

1− ρ2 1− ρ2 σl 1 − ρ2 −∞ if −1 < ρ < 1;

√ N (cl ) − −cu − ≥ 1 − α σu √ N ≥1−α (cu ) − −cl − σl

if ρ = 1 (compare these expressions to (2)); and √ N min cl cu + ≥ 1 − α σu √ N ≥ 1 − α min cu cl + σl implying that (cl ) = (cu ) = 1 − α at the minimization problem’s solution, if = −1. There is no discontinuity at the limit because ρ √ √ N N cu ρ → I z ≥ [≤]cu + z+

+

σu 1− ρ2 1− ρ2 σu 1 − ρ2 as ρ → 1[−1]. REFERENCES ANDREWS, D. W. K. (2000): “Inconsistency of the Bootstrap When a Parameter Is on the Boundary of the Parameter Space,” Econometrica, 68, 399–405. [1307-1309] ANDREWS, D. W. K., AND P. GUGGENBERGER (2007): “Validity of Subsampling and ‘Plug-in Asymptotic’ Inference for Parameters Defined by Moment Inequalities,” Discussion Paper 1620, Cowles Foundation. [1308,1309] ANDREWS, D. W. K., AND G. SOARES (2007): “Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection,” Discussion Paper 1631, Cowles Foundation. [1307,1308] BUGNI, F. A. (2007): “Bootstrap Inference in Partially Identified Models,” Unpublished Manuscript, Northwestern University. [1309]


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CHERNOZHUKOV, V., H. HONG, AND E. T. TAMER (2007): “Parameter Set Inference in a Class of Econometric Models,” Econometrica, 75, 1243–1284. [1309] CHERNOZHUKOV, V., S. LEE, AND A. ROSEN (2008): “Intersection Bounds: Estimation and Inference,” Unpublished Mansucript, University College London. [1308] FAN, Y., AND S. S. PARK (2007): “Confidence Intervals for Some Partially Identified Parameters,” Unpublished Mansucript, Vanderbilt University. [1308] IMBENS, G., AND C. F. MANSKI (2004): “Confidence Intervals for Partially Identified Parameters,” Econometrica, 72, 1845–1857. [1299,1302,1308] LEEB, H., AND B. PÖTSCHER (2005): “Model Selection and Inference: Facts and Fiction,” Econometric Theory, 21, 21–59. [1302,1308] MANSKI, C. F. (2003): Partial Identification of Probability Distributions. New York: Springer Verlag. [1299] PAKES, A., J. PORTER, K. HO, AND J. ISHII (2006): “Moment Inequalities and Their Application,” Unpublished Mansucript, Harvard University. [1302,1308] ROMANO, J. P., AND A. M. SHAIKH (2008): “Inference for Identifiable Parameters in Partially Identified Econometric Models,” Journal of Statistical Planning and Inference, 138, 2786–2807. [1309] WOUTERSEN, T. (2006): “A Simple Way to Calculate Confidence Intervals for Partially Identified Parameters,” Unpublished Mansucript, Johns Hopkins University. [1300]

Dept. of Economics, New York University, 19 W. 4th Street, New York, NY 10012, U.S.A.; [email protected]. Manuscript received August, 2007; final revision received December, 2008.


INFORMATION INDEPENDENCE AND COMMON KNOWLEDGE BY OLIVIER GOSSNER, EHUD KALAI, AND ROBERT WEBER1 In Bayesian environments with private information, as described by the types of Harsanyi, how can types of agents be (statistically) disassociated from each other and how are such disassociations reflected in the agents’ knowledge structure? Conditions studied are (i) subjective independence (the opponents’ types are independent conditional on one’s own) and (ii) type disassociation under common knowledge (the agents’ types are independent, conditional on some common-knowledge variable). Subjective independence is motivated by its implications in Bayesian games and in studies of equilibrium concepts. We find that a variable that disassociates types is more informative than any common-knowledge variable. With three or more agents, conditions (i) and (ii) are equivalent. They also imply that any variable which is common knowledge to two agents is common knowledge to all, and imply the existence of a unique common-knowledge variable that disassociates types, which is the one defined by Aumann. KEYWORDS: Bayesian games, independent types, common knowledge.

1. INTRODUCTION THIS NOTE DEALS WITH the private information available to different agents in Bayesian environments, as described by the types of Harsanyi (1967/68), and with the notion of common knowledge, as described in Aumann (1976). In particular, we are interested in conditions of partial independence of agents’ types and their relationship to common-knowledge variables. Such conditions help explain the environment’s information structure: how types of agents are (statistically) associated with each other and what is common knowledge about such associations. A central condition of interest is subjective independence: the opponents’ types of every agent are (statistically) independent when the agent conditions on his own type. This condition is interesting for a variety of reasons. First, from an empirical point of view, when agents test whether their opponents’ types are independent, it is often done after they already know their own type. So in effect, they test conditions of subjective, and not full, independence. From an analytical point of view, subjective independence plays an important role when individual agents make use of laws of large numbers that require independence. In a strategic game, for example, a player who knows his own type can use laws of large numbers to make accurate predictions about the aggregate behavior of large groups of opponents. Kalai (2004) showed that in one-simultaneous-move Bayesian games with many semianonymous play1

Kalai’s research is partially supported by the National Science Foundation Grant SES0527656. We are grateful for excellent expositional and technical suggestions made by a co-editor and the anonymous referees. © 2009 The Econometric Society

DOI: 10.3982/ECTA7469

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ers, under subjective independence,2 the equilibria are robust to structural changes.3 Yet even if the number of players is small, a player who knows his own type and repeatedly observes his opponents’ choices can use laws of large numbers to predict the opponents’ future actions. Kalai and Lehrer (1993) studied the equilibria of Bayesian repeated games and showed that subjective independence implies that every Bayesian equilibrium converges to a Nash equilibrium of the repeated game, as if the privately known types were common knowledge. Gossner and Hörner (2006) studied repeated games with imperfect monitoring under subjective independence, where, conditional on each player’s signal (or on a garbled version of it), the opponents’ signals are independent. They showed that each player’s punishment level, defined as his min–max payoff in the repeated game (where the min is taken over the independent strategy profiles of the opponents), equals the player’s usual min–max payoff in mixed strategies of the stage game. Thus, in this context subjective independence allows for simple characterizations of individually rational payoffs, an important step in the analysis of folk theorems in repeated games with imperfect monitoring. In different contexts, which do not involve laws of large numbers, subjective independence is important in the epistemological analysis of game-theoretic solutions. For example, Bernheim (1986) used subjective independence to justify the use of Nash equilibrium (for the case of more than two agents), and, more recently, Brandenburger and Friedenberg (2008) used it to differentiate rationalizability from iterated dominance. With implications like those above, it is desirable to have a better understanding of the condition of subjective independence. How restrictive is it and in what types of situations does it arise? This note answers such questions. A second important condition in this paper is independence under common knowledge: conditional on some common-knowledge variable, the types of all the agents are independent. This condition is important for two reasons: (i) it enables an analyst to break down a complex strategic environment into individual components, each defined by the realized value of the common-knowledge variable and (ii) conditional on each such realized value, the analyst may assume type independence. For example, a Bayesian game can be broken down to smaller strategically separable Bayesian subgames, each with independent types. With three or more agents, this paper shows that subjective independence is equivalent to independence under common knowledge. Moreover, when there 2

While the formal statement there is stated with full independence, the proof only uses subjective independence. 3 The equilibria survive even if the simultaneous move assumption is relaxed to allow for sequential play, revisions of earlier choices, delegation, information transmission, communication, and more.

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is independence under a common-knowledge variable, there is a unique variable with this property, namely, the variable identified by Aumann (1976). This suggests useful algorithms for disassociating types in Bayesian environments. A direct proof of the equivalence of the two independence conditions above is long and tedious. The main body of the paper identifies general relationships between type independence and common knowledge. In addition to leading to a simpler proof of the equivalence above, these relationships offer better understanding of the structures of Bayesian environments. In particular, we show that, in general, any variable that disassociates the types must be more informative than any common-knowledge variable. Moreover, subjective independence implies (i) that there can be no coalitional secrets (any variable that is common knowledge to two agents is common knowledge to all) and (ii) that there is essentially a unique way to disassociate types. The next section offers examples to help with the underlying intuition. It is followed by a formal presentation that establishes the above results in a standard model where the agents hold a common-prior distribution over the possible environment’s states, as is customary in much of the epistemologic literature and applications. The concluding section offers two additional items: (i) an elaboration on how to disassociate types and (ii) a discussion on the role of the common-prior assumption, showing that without it the main results may fail. 1.1. Examples The following examples help illustrate the concepts. EXAMPLE 1—Sun and Moods: In an n-person Bayesian environment, the sun may either shine or not and, conditional on the state of the sun, σ, agent i’s mood, μi , is either happy or depressed. Each agent privately learns both whether it is sunny or not and his own mood (for example, “It is sunny and I am depressed”), and hence he may be one of four possible types. Let s(·) be the probability distribution over the states of the sun (with positive probability on each) and let mi (·|σ) be agent i’s probability distribution over moods, given the state of the sun. Assume that the likelihood of any vector of types (σ μ) is the product s(σ) i mi (μi |σ) and that all these distributions are commonly known to the agents. In the example above, the agents’ types are not independent. For example, knowledge of one’s type precludes possible types of the opponents. But the state of the sun disassociates the types in the sense that conditional on any state of the sun, the types are independent. Moreover, the state of the sun is common knowledge. Since in this situation a common-knowledge variable disassociates the types, the types are independent under common knowledge. One can also verify that in this example the agents’ types are subjectively independent.

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EXAMPLE 2—Sun, Ozone, and Moods: As in the previous example, the state of the sun, σ, can be shine or not shine, and each agent’s mood, μi , can be happy or depressed. In addition, the ozone level, ζ, may be good or bad. So each state of the environment is described by a triple: (σ ζ μi i=1n ). However, each agent knows only the weather and his own mood; he does not know the ozone level. So he can be one of the same four types as in the previous example. Pairs of sun–ozone combinations, (σ ζ), are determined by a full-support joint distribution k(σ ζ) and, conditional on any such pair, the n moods are independently and identically distributed with 0 < p(μi |σ ζ) < 1 denoting the probability that agent i’s mood is μi . Assume that for every fixed sun state σ, the probability of a happy mood is strictly greater when the ozone level is good than when it is bad. In the last example, the state of the sun is a common-knowledge variable. But it does not disassociate the moods (the moods are not independent conditional on it) and no common-knowledge variable disassociates the moods. To disassociate the moods, one needs to know both the state of the sun and the ozone level. Subjective independence fails too, since knowledge of the sun and one’s own mood is not sufficient to disassociate the opponents’ moods. 2. FORMAL PRESENTATION A set I = {1 2 n} (n ≥ 2) describes the agents in a Bayesian environment E = (Ω P T ), where T = TI = (Ti )i∈I is the profile of agents’ type functions, each being a random variable defined over the probability space (Ω P). For a group of players J ⊆ I, TJ = (Ti )i∈J denotes the profile of types of players in J. Implicitly, it is assumed that there is common knowledge of the three components of the environment E and that when a state ω ∈ Ω is drawn, each agent i is informed only of his own type ti = Ti (ω).4 However, through his knowledge of E , each agent makes inferences about the possible types of his opponents, the inferences that the opponents may make, and so forth. The space Ω may be finite or countable, and we assume without loss of generality that P(ω) > 0 for every ω ∈ Ω and that P(ti ) > 0 for every ti in the range of each Ti , that is, Ti is onto its range. We do not assume that P(t) > 0 for all profiles of types t, which would be a severe restriction. DEFINITION 1: A random variable Y is finer than (or refines) a random variable Z (alternatively Z is coarser than Y ) if for every pair of states, ω and ω , Z(ω) = Z(ω ) implies that Y (ω) = Y (ω ). 4 Readers who are used to epistemology models may prefer to think of the underlying partitions rather than the variables Ti . This would avoid the need for equivalence classes discussed below, but would be more cumbersome in notations and applications.


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Two variables, Y and Z, are (informationally) equivalent, Y ≈ Z, if they refine each other (Z(ω) = Z(ω ) if and only if Y (ω) = Y (ω )). Y strictly refines Z if Y refines Z and they are not equivalent. Clearly, “Y refines Z” is transitive and Y ≈ Z is an equivalence relation on the set of random variables. In the discussion that follows, we are mostly interested in the equivalence classes of variables [Z] instead of the variables Z themselves. It is easy to see that under the equivalence above and the “Y refines Z” relation, the variables form a (complete) lattice. The finest variable is represented by the function F(ω) = {ω} and the coarsest variable is represented by the function K(ω) = Ω. 2.1. Conditional Independence: Disassociating Types DEFINITION 2: Consider a subgroup of agents J ⊆ I. The variables TJ are independent if for every vector of values tJ , P(TJ = tJ ) = j∈J P(Tj = tj ). The variables TJ are independent conditional on a variable D if for all possible values tJ and d, P(TJ = tJ |D = d) = j∈J P(Tj = tj |D = d). When this is the case, we say that D disassociates TJ . When J is the group of all agents I, we may omit the subscript J, and simply say that the types are independent conditional on D and that D disassociates the types. In this case we may write P(T = t) =

d

P(Ti = ti |D = d)P(D = d)

i

Clearly, if D disassociates the types of T , it disassociates the types of any subgroup of agents, but the converse is not true. Moreover, disassociating types is always possible (for example, by conditioning on the finest variable F ) and often beneficial, but coarser disassociations are more desirable, since they help explain the relationships between the types with a smaller number of parameters. PROPOSITION 1: There exists a maximally coarse disassociation, that is, a disassociation D, such that any coarser disassociation is equivalent to it. PROOF: We apply Zorn’s lemma to the set of disassociations, partially ordered by the “is coarser than” relation. We only need to prove that for any totally ordered family (Dα )α∈D of disassociations, there exists a disassociation D that is coarser than any Dα . Define the equivalence relation ω ∼ ω to mean that there exists an α such that Dα (ω) = Dα (ω ) and define the variable D(ω)

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to be the equivalence class of ω, D(ω) = {ω ; ω ∼ ω }. D is clearly coarser than any Dα . For every vector of values tJ and every ω, P(TJ = tJ |Dα (ω)) =

P(TJ = tj Dα (ω)) P(Dα (ω))

Both P(TJ = tj Dα (ω)) and P(Dα (ω)) are increasing and converge, respectively, to P(TJ = tj D(ω)) and to P(D(ω)) on the chain defined by D . Thus, all the equalities defining each Dα as a disassociation can be taken to the limit, and D is also a disassociation. Q.E.D. DEFINITION 3: A type disassociation is coarsest if it is coarser than any other disassociation. Clearly, a coarsest disassociation must be unique (up to equivalence). Although there always exists a maximally coarse disassociation, it may not be unique and the coarsest disassociation may not exist. We defer further discussion about the negative implications of this phenomenon to the concluding section. 2.2. Common Knowledge DEFINITION 4: For a group of agents J ⊆ I, a variable C is J common knowledge if C is coarser than every Tj with j ∈ J. When C is I common knowledge, we simply say that it is common knowledge.5 Clearly, common-knowledge variables exist, since any constant variable is common knowledge for any group of agents. Also, common knowledge is coalitionally monotonic: if C is J common knowledge, it must also be L common knowledge for any L ⊆ J. Unlike disassociations, in the case of common knowledge it is desirable to identify the maximally refined (rather than the maximally coarse) variables, since they describe the most detailed information that is commonly known to the agents. DEFINITION 5: A common-knowledge variable C is finest if it refines every other common-knowledge variable. Clearly, a finest common-knowledge variable is unique (up to equivalence). Unlike disassociations of types, for which the coarsest one may not exist, the finest common-knowledge variable always exists. An explicit description of it is offered by a construction due to Aumann: 5 As we clarify below, this is equivalent to the standard definition of common knowledge described in Aumann (1976).


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For a fixed group of agents J ⊆ I , let ∼ be the equivalence relation defined by ω ∼ ω if there exists a chain ω0 = ω ω1 ωn = ω such that for every k = 1 n, there exists an agent j ∈ J who is the same type in states ωk−1 and ωk−1 : Tj (ωk−1 ) = Tj (ωk ). We define AumJ by the equivalence classes for this relation: AumJ (ω) = {ω : ω ∼ ω}. For notational simplicity we let Aum = AumI . Formally, as defined above, Aum is a random variable with a range in the set of all subsets of Ω, but it also describes the common-knowledge partition defined by Aumann (1976). The following lemma shows that Aum is common knowledge and it is the finest common-knowledge variable. LEMMA 1: Aum is the finest common-knowledge variable. PROOF: To see that Aum is common knowledge, note that under the construction above, Ti (ω) = Ti (ω ) implies Aum(ω) = Aum(ω ). Thus, Aum is refined by every Ti . To see that Aum is the finest common knowledge, consider any commonknowledge variable C. For any ω ω such that Aum(ω) = Aum(ω ), consider a chain ω0 = ω ω1 ωn = ω such that for every k = 1 n, there exists an i with Ti (ωk−1 ) = Ti (ωk ). For every k, Ti (ωk−1 ) = Ti (ωk ) implies C(ωk−1 ) = C(ωk ), so that C(ω) = C(ω ). Hence, Aum refines C. Finally, if both Aum and Aum are finest, Aum refines Aum and Aum refines Aum. Q.E.D. 2.3. Main Results THEOREM 1—Type Disassociations Refine Common Knowledge: Any variable D that disassociates the types is finer than any common-knowledge variable. PROOF: We prove a stronger conclusion, namely, that D refines any C which is common knowledge to any group of two agents. This is stronger because if C is common knowledge, it must be common knowledge to any group of two or more agents, yet the converse is not true. Notice, however, that if D disassociates the types of T , then it disassociates the types of any group of two agents. So it suffices to prove the theorem for the case of n = 2 agents. Since Aum refines all common-knowledge variables, it suffices to show that if D(ω1 ) = D(ω2 ), then Aum(ω1 ) = Aum(ω2 ). Let D(ω1 ) = D(ω2 ) = d, and let t1 = T1 (ω1 ) and t2 = T2 (ω2 ). Then P(T1 = t1 and T2 = t2 |D = d) = P(T1 = t1 |D = d)P(T2 = t2 |D = d) > 0. So there is a ω with T1 (ω) = t1 (= T1 (ω1 )) and T2 (ω) = t2 (= T2 (ω2 )). In other words, ω1 and ω2 are equivalent in the sense Q.E.D. of Aumann so that Aum(ω1 ) = Aum(ω2 ).

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COROLLARY 1: The only common-knowledge variable that may disassociate the types is Aum. DEFINITION 6: The types of T are subjectively independent if P(T−i = t−i |Ti = ti ) = j=i P(Tj = tj |Ti = ti ) for every agent i and every vector of types t (alternatively, Ti disassociates T−i for every i). Notice that the definition of subjective independence is global, rather than local, where one would require only subjective independence at individual states ω. Moreover, given the implicit common knowledge of P and T , if subjective indepence holds, then it is common knowledge: every agent can verify it for himself, for each one of his opponents, for their ability to verify it for their opponents, and so on. DEFINITION 7: The types of T are independent under common knowledge if there is a common-knowledge variable that disassociates them. THEOREM 2: For n ≥ 3 agents and any vector of types T , the following three conditions are equivalent: C1. The types are subjectively independent. C2. The types are independent under common knowledge. C3. Aum disassociates the types. Moreover, any of the conditions above implies the following statements: C4. The coarsest disassociating variable exists. C5. No Coalitional Secrets: Any variable Z is common knowledge to a group of agents J if and only if it is common knowledge to a group of agents K, where J and K are any two groups (overlapping or disjoint) with two or more agents each. PROOF: C2 ⇒ C1. The following lemma is useful. LEMMA 2: If X1 and (X2 X3 ) are independent conditional on X4 , then X1 and X2 are independent conditional on (X3 X4 ). PROOF: P(X1 |(X2 X3 ) X4 ) = P(X1 |X4 ) implies P(X1 |X3 X4 ) = P(X1 | X4 ), which implies the desired conclusion: P(X1 |X2 (X3 X4 )) = P(X1 |X3 Q.E.D. X4 ). Assume C2, that is, that Ti and (Tk )k=i are independent conditional on a common-knowledge variable Z. For any j = i, Lemma 2, with X1 = Ti , X2 = (Tk )k=ij , X3 = Tj , and X4 = Z, shows that Ti and (Tk )k=ij are independent conditional on (Tj Z). This proves that Ti and (Tk )k=ij are independent conditional on Tj , since (Tj Z) and Tj generate the same partition. This is true for every i j, establishing subjective independence.


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C1 ⇒ C5. Let J be any group of at least two agents and take i ∈ / J. Since the family (Tj )j∈J is independent conditional on Ti , Theorem 1 implies that Ti refines AumJ . Thus, AumJ∪{i} = AumJ . Since this is true for any J, an induction argument shows that for any group J of at least two agents, AumJ = Aum. C1 ⇒ C3. Let i, j, and k be three different agents. Since Tk and (Tl )l=ki are independent conditional on Ti , P(tk |(tl )l=ki ti ) = P(tk |(tl )l=ki ti ) if P((tl )l=ki ti )P((tl )l=ki ti ) > 0. Similarly, P((tl )l=kj tj )P((tl )l=kj tj ) > 0 implies P(tk |(tl )l=kj tj ) = P(tk |(tl )l=kj tj ). Thus for any a in the range of Aum{ij} , P(tk |(tl )l=k a) = P(tI |(tl )l=k a) holds whenever P((tl )l=k a)P((tl )l=k a) > 0. Hence, Tk and (Tl )l=k are independent conditional on Aum{ij} . Now using C1 and its implied C5, Tk and (Tl )l=k are independent conditional on Aum. Since this is true for any k, the family T is independent conditional on Aum. C3 ⇒ C2. This is immediate, since Aum is common knowledge. C3 ⇒ C4. This is immediate, since Aum disassociates the types and refines any variable that disassociates the types by Theorem 1. Q.E.D. 3. FURTHER ELABORATION The first example—sun and moods (no ozone)—illustrates a situation where subjective independence and all the other C1–C5 conditions hold. The state of the sun disassociates the moods of the agents and as the Aum variable, any other disassociation must simply be its refinement. In other words, other than the sun, there is no explanation of how the moods of the agents are related to each other (including among subsets of agents), and this sun disassociation is also the finest common knowledge. The second example—sun, ozone, and moods—illustrates a situation in which subjective independence and the equivalent conditions C1–C3 fail. The Aum variable is still the state of the sun, but it does not disassociate the moods. What can we say about disassociating the types in such situations? As it turns out, despite the failing of subjective independence, the main findings of this paper may still be helpful. 3.1. Disassociating Types Continuing with the sun, ozone, and moods example, we observe first that the two-dimensional variable that describes the values of the sun and the ozone, (S Z), does disassociate the types and that it is a maximally coarse variable with this property. To see that it is maximally coarse among all disassociations, assume to the contrary that there is a disassociation D that is strictly coarser than (S Z). By Theorem 1, D must be strictly finer than Aum. This leaves only two such possible disassociations: D1 , which discloses the ozone levels when the sun is shining, but does not do so when the sun is not shining, or, similarly, D2 , which discloses the ozone levels when the sun is not shining, but does not do so when the sun is shining.

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To show, for example, that D1 is not a disassociation, restrict yourself to the event “the sun is not shining,” which is common knowledge, and verify that conditional on this event the moods are not independent. Do the state of the sun and the ozone level, (S Z), offer the only way to disassociate the moods? No. For example, for every agent i, let N i = (S M−i ) describe the state of the sun and the vector of moods of all agents other than i. The reader may verify, using some of the main results described earlier, that every such N i is a maximally coarse variable that disassociates the types. Moreover, N i is not equivalent to (S Z). The nonexistence of the coarsest disassociation, or the existence of multiple maximally coarse disassociations, presents a conceptual difficulty. This is because unlike the example with the sun only, where there is a unique and full explanation of the mood relationships among the agents, now there are several competing such explanations. This means that an analyst, who wants to have one “natural” explanation of the mood relationships, has to introduce additional considerations. For example, Gossner, Laraki, and Tomala (2009) chose the decomposition that minimizes the entropy of the information not observed by the analyst. As the discussion above suggests, type disassociation may be attempted by the following series of steps: First, identify the variable Aum, a task that involves only unions and intersections, and disregards probabilities. Then check to see if the types are independent conditional on Aum. If they are, then Aum is the unique explanation of type relationships (and it is also common knowledge). If the types are not independent conditional on Aum, then there are no common-knowledge disassociations and all disassociations must be strict refinements of Aum. 3.2. Relaxing the Common-Prior Assumption In the model above, we (implicitly) assumed that the environment E = (Ω P T ) is common knowledge. Differential information was expressed only by the fact that when a state ω is realized, each agent i is informed only of the value of his own type Ti (ω). This formulation follows the original model of Aumann (1976) and most of the followup literature. What happens to subjective independence if E = (Ω P T ) is not common knowledge to all the agents? There are different ways to model such relaxations, while still retaining coherent concepts of common knowledge and disassociations. One simple relaxation is the following. Replace the triple E = (Ω P T ) by n triples E k = (Ωk P k T k ), k = 1 2 n, where every such triple represents the beliefs of agent k about the complete setting. For every such k, impose on E k the assumptions made in the paper on E . In other words, each agent has a different model of the world, but he believes that all the agents have the same model as his. It follows immediately that all the results hold in every agent’s mind.


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However, the above relaxation is extreme in opposite directions. On the one hand, the triples E k , modeling the different agents’ representations of the world, are completely unrelated to each other. One the other hand, within his own E k , every agent assumes full coincidence and common knowledge of the same environment. A model that relaxes the common-prior assumption in a minimal sense and is less extreme than the one above is a generalized Harsanyi model (GHM; see Harsanyi (1967/68)). While, in general, a minimal relaxation may not go far enough, it is sufficient for our purpose here. As we show below, such a minimal relaxation already overturns the main result of the paper. In this GHM there is common knowledge of the set of states Ω and of the vector of types T , but each agent k has a different prior probability P k over the states of the environment. Moreover, it is assumed that the different P k ’s are common knowledge. This means that each agent k is convinced that his prior P k is correct and that the priors of his opponents, P k ’s, are wrong. So in contrast to the model of Aumann (1976), they may agree to disagree, with disagreements that are common knowledge. The following is an example of such a GHM in which the main results of the paper fail.6 EXAMPLE 3 —Three Mood Setters: There are nine states described by triples μ = (μ1 μ2 μ3 ), where each μi , taking the values happy (H) or depressed (D), describes the mood of agent i. Each agent is informed only of his own mood. Moreover, based on his individual prior distribution over moods P i , he believes that he is the “mood setter.” For example, agent 1’s prior is (i) P 1 (μ1 = H) = P 1 (μ1 = D) = 050, and with each of X Y Z taking the values of H and D, he has (ii) P 1 (μ2 = X|μ1 = X) = P 1 (μ3 = X|μ1 = X) = 090 and (iii) P 1 (μ2 = Y and μ3 = Z|μ1 = X) = P 1 (μ2 = Y |μ1 = X)P 1 (μ3 = Z|μ1 = X). Condition (ii) represents the fact that agent 1 believes that he is a mood setter. Moreover, condition (iii) states his belief that his mood disassociates the moods of the others, which is subjective independence. In addition to the fact that each agent believes that he is the mood setter, as described by symmetric distributions P 2 and P 3 , there is implicit common knowledge that all three agents believe themselves to be the sole mood setter. In the example above the only common-knowledge variable (or the Aum variable) is the coarsest variable, represented by any constant function. Thus, the condition of independence under common knowledge translates to unconditional type independence. But under any P i , i = 1 2 3, no agent believes that the types are independent. (For example, P 1 (μj = H) = 050 for j = 1 2 3, yet P 1 (H H H) = 6 We thank an associate editor for suggesting a different (more complex) example in which the same phenomenon is observed.

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05 × (09)2 = (05)3 .) Given the common knowledge of the different priors, there is common knowledge that nobody believes the type independence, that is, independence under common knowledge fails. Yet condition (iii) above means that there is subjective independence, where each agent believes that for any type of himself, his opponents are independent of each other. Again, given the common knowledge of the individual priors, there is also common knowledge of subjective independence, that is, every agent knows that conditional on his types, the opponents’ types are independent, they all know that they all know that, and so forth. Clearly, one special case of the GHM is the case of a common prior studied in this paper, in which subjective independence and independence under common knowledge are equivalent. It is not clear whether, within the GHM, the equivalence can be obtained under weaker conditions than a common prior. REFERENCES AUMANN, R. J. (1976): “Agreeing to Disagree,” The Annals of Statistics, 4, 1236–1239. [1317,1319, 1322,1323,1326,1327] BERNHEIM, B. D. (1986): “Axiomatic Characterizations of Rational Choice in Strategic Environments,” The Scandinavian Journal of Economics, 88, 473–488. [1318] BRANDENBURGER, A., AND A. FRIEDENBERG (2008): “Intrinsic Correlation in Games,” Journal of Economic Theory, 141, 28–67. [1318] GOSSNER, O., AND J. HÖRNER (2006): “When Is the Individually Rational Payoff in a Repeated Game Equal to the Minmax Payoff?” Discussion Paper 1440, CMS-EMS. [1318] GOSSNER, O., T. LARAKI, AND R. TOMALA (2009): “Informationally Optimal Correlation,” Mathematical Programming B, 116, 147–172. [1326] HARSANYI, J. C. (1967/68): “Games With Incomplete Information Played by “Bayesian” Players, Parts I, II, and III,” Management Science, 14, 159–182, 320–334, 486–502. [1317,1327] KALAI, E. (2004): “Large Robust Games,” Econometrica, 72, 1631–1666. [1317] KALAI, E., AND E. LEHRER (1993): “Rational Learning Leads to Nash Equilibrium,” Econometrica, 61, 1019–1945. [1318]

Paris School of Economics, UMR CNRS-EHESS-ENPC-ENS 8545, 48 Boulevard Jourdan, 75014 Paris, France and London School of Economics, London, U.K.; [email protected], Dept. of Managerial Economics and Decision Science, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, U.S.A.; [email protected], and Dept. of Managerial Economics and Decision Science, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, U.S.A.; [email protected]. Manuscript received September, 2007; final revision received February, 2009.


ANNOUNCEMENTS 2009 FAR EAST AND SOUTH ASIA MEETING

THE 2009 FAR EAST AND SOUTH ASIA MEETING of the Econometric Society (FESAMES 2009) will be hosted by the Faculty of Economics of the University of Tokyo, on 3–5 August, 2009. The venue of the meeting is the Hongo campus of the University of Tokyo which is about ten minutes away from the Tokyo JR station. The program will consist of invited and contributed sessions in all fields of economics. We encourage you to register at your earliest convenience. Partial subsidy is available for travel and local expenses of young economists submitting a paper for a contributed session in the conference. Confirmed Invited Speakers: Dilip Abreu (Princeton University) Abhijit Banerjee (Massachusetts Institute of Technology) Markus K. Brunnermeier (Princeton University) Larry G. Epstein (Boston University) Faruk R. Gul (Princeton University) James Heckman (University of Chicago) Han Hong (Stanford University) Ali Hortacsu (University of Chicago) Michihiro Kandori (University of Tokyo) Dean Karlan (Yale University) Nobuhiro Kiyotaki (Princeton University) John List (University of Chicago) Charles Manski (Northwestern University) Daniel McFadden (University of California, Berkeley) Costas Meghir (University College London) Roger Myerson (University of Chicago) Sendhil Mullainathan (Harvard University) Ariel Pakes (Harvard University) Giorgio Primiceri (Northwestern University) Jean-Marc Robin (Paris School of Economics/UCL) Yuliy Sannikov (Princeton University) Hyun Song Shin (Princeton University) Christopher Sims (Princeton University) James Stock (Harvard University) David Weir (University of Michigan) Program Committee: Co-Chairs: Hidehiko Ichimura (University of Tokyo) Hitoshi Matsushima (University of Tokyo) © 2009 The Econometric Society

DOI: 10.3982/ECTA774ANN

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ANNOUNCEMENTS

Toni Braun, Macroeconomics (University of Tokyo) Xiaohong Chen, Econometric Theory (Yale University) Shinichi Fukuda, Monetary Economics, Macroeconomics, International Finance (University of Tokyo) Takeo Hoshi, Finance and Banking, Monetary Economics, Japanese Economy (UC San Diego) Toshihiro Ihori, Public Economics, Fiscal Policy (University of Tokyo) Hideshi Itoh, Contract Theory (Hitotsubashi University) Atsushi Kajii, Economic Theory, Information Economics, General Equilibrium, Game Theory (Kyoto University) Kazuya Kamiya, General Equilibrium, Decision Theory, Computational Economics (University of Tokyo) Yuichi Kitamura, Microeconometrics (Yale University) Kazuharu Kiyono, Trade, Industrial Economics, Applied Economics (Waseda University) Siu Fai Leung, Labor (Hong Kong University of Science and Technology) Akihiko Matsui, Game Theory (University of Tokyo) Tomoyuki Nakajima, Macroeconomic Theory (Kyoto University) Masao Ogaki, Macroeconometrics (Ohio State University) Hiroshi Ohashi, Industrial Organization, International Trade (University of Tokyo) Joon Y. Park, Time Series Econometrics (Texas A&M University) Tatsuyoshi Saijo, Experimental Economics, Mechanism Design (Osaka University) Makoto Saito, Asset Pricing, Consumption and Investment, Monetary Theory (Hitotsubashi University) Yasuyuki Sawada, Development, Applied Econometrics (University of Tokyo) Shigehiro Serizawa, Mechanism Design, Social Choice Theory (Osaka University) Akihisa Shibata, International Macroeconomics (Kyoto University) Takatoshi Tabuchi, Urban Economics, International Trade, Economic Geography (University of Tokyo) Noriyuki Yanagawa, Law and Economics, Financial Contract (University of Tokyo) Local Organizing Committee: Co-Chairs: Hidehiko Ichimura (University of Tokyo) Hitoshi Matsushima (University of Tokyo) Yoichi Arai (University of Tokyo) Yun Jeong Choi (University of Tokyo) Julen Esteban-Pretel (University of Tokyo)

ANNOUNCEMENTS

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Masahiro Fujiwara (University of Tokyo) Fumio Hayashi (University of Tokyo) Isao Ishida (University of Tokyo) Takatoshi Ito (University of Tokyo) Motoshige Itoh (University of Tokyo) Katsuhito Iwai (University of Tokyo) Yasushi Iwamoto (University of Tokyo) Yoshitsugu Kanemoto (University of Tokyo) Takashi Kano (University of Tokyo) Takao Kobayashi (University of Tokyo) Tatsuya Kubokawa (University of Tokyo) Naoto Kunitomo (University of Tokyo) Hisashi Nakamura (University of Tokyo) Tetsuji Okazaki (University of Tokyo) Yasuhiro Omori (University of Tokyo) Akihiko Takahashi (University of Tokyo) Yoshiro Miwa (University of Tokyo) Kazuo Ueda (University of Tokyo) Makoto Yano (Kyoto University) Jiro Yoshida (University of Tokyo) Hiroshi Yoshikawa (University of Tokyo) 2009 EUROPEAN MEETING

THE 2009 EUROPEAN MEETING of the Econometric Society (ESEM) will take place in Barcelona, Spain from 23 to 27 August, 2009. The Meeting is jointly organized by the Barcelona Graduate School of Economics and it will run in parallel with the Congress of the European Economic Association (EEA). The Program Committee Chairs are Prof. Juuso Välimäki (Helsinki School of Economics) for Theoretical and Applied Economics; Prof. Gerard J. van der Berg (Free University Amsterdam) for Econometrics and Empirical Economics. This year’s Fisher–Schultz Lecture will be given by Faruk Gul (Princeton University). The Laffont Lecture will be given by Guido Imbens (Harvard University). The Local Arrangements Committee: Albert Carreras, Chairman—Universitat Pompeu Fabra and Barcelona GSE Carmen Bevià—Universitat Autònomade Barcelona and Barcelona GSE

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ANNOUNCEMENTS

Jordi Brandts—Institute for Economic Analysis-CSIC and Barcelona GSE Eduard Vallory, Secretary—Barcelona GSE Director-General All details regarding the congress can be found on the website http:// eea-esem2009.barcelonagse.eu/. 2009 LATIN AMERICAN MEETING

THE 2009 LATIN AMERICAN MEETINGS will be held jointly with the Latin American and Caribbean Economic Association in Buenos Aires, Argentina, from October 1 to 3, 2009. The Meetings will be hosted by Universidad Torcuato Di Tella (UTDT). The Annual Meetings of these two academic associations will be run in parallel, under a single local organization. By registering for LAMES 2009, participants will be welcome to attend to all sessions of both meetings. Andrés Neumeyer (UTDT) is the conference chairman. The LAMES Program Committee is chaired by Emilio Espina (UTDT). The LACEA Program Committee is chaired by Sebastián Galiani (Washington University in St. Louis). Plenary Speakers: Ernesto Zedillo, Former President of Mexico, Yale University Roger Myerson, Nobel Laureate, University of Chicago, LAMES Presidential Address Mauricio Cardenas, Brookings Institution, LACEA Presidential Address Daron Acemoglu, MIT Guido Imbens, Harvard University John Moore, University of Edinburgh Invited Speakers: Fernando Alvarez, University of Chicago Jere Behrman, University of Pennsylvania Abhijit Banerjee, MIT Pablo Beker, University of Warwick Samuel Berlinski, University College of London Richard Blundell, University College of London Gustavo Bobonis, University of Toronto Michele Boldrin, Washington University in St. Louis Maristella Botticini, Boston University Francois Bourguignon, Paris School of Economics Francisco Buera, Northwestern University Guillermo Calvo, Columbia University

ANNOUNCEMENTS

Matias Cattaneo, University of Michigan V. V. Chari, University of Minnesota Satyajit Chatterjee, Philadelphia Federal Bank Lawrence Christiano, Northwestern University Ernesto Dal Bo, University of California, Berkeley David DeJong, University of Pittsburgh José de Gregorio, Banco Central de Chile Augusto de la Torre, The World Bank Rafael Di Tella, Harvard University Juan Dubra, Universidad de Montevideo Esther Duflo, MIT Jonathan Eaton, New York University Huberto Ennis, Universidad Carlos III de Madrid Martin Eichenbaum, Northwestern University Raquel Fernandez, New York University Sergio Firpo, Fundacao Getulio Vargas Sao Paulo Paul Gertler, University of California, Berkeley Edward Glaeser, Harvard University Ricardo Hausman, Harvard University Christian Hellwig, UCLA Bo Honoré, Princeton University Hugo Hopenhayn, UCLA Boyan Jovanovic, New York University Dean Karlan, Yale University Pat Kehoe, University of Minnesota Tim Kehoe, University of Minnesota Felix Kubler, Swiss Finance Institute Victor Lavy, Hebrew University David Levine, Washington University in St. Louis Santiago Levy, Inter American Development Bank Rodolofo Manuelli, Washington University Rosa Matzkin, UCLA Enrique Mendoza, University of Maryland Dilip Mookerjee, Boston University John Nye, George Mason University Rohini Pande, Harvard University Fabrizio Perri, University of Minnesota Andrew Postlewaite, University of Pennsylvania Martin Redrado, Banco Central de la Republica Argentina Carmen Reinhart, University of Maryland Rafael Repullo, CEMFI James Robinson, Harvard University Esteban Rossi-Hansberg, Princeton University Ernesto Schargrodsky, Universidad Di Tella

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ANNOUNCEMENTS

Karl Schmedders, Kellogg School of Management Northwestern University Paolo Siconolfi, Columbia University Michele Tertil, Stanford University Miguel Urquiola, Columbia University Martin Uribe, Columbia University Andres Velasco, Ministerio de Hacienda, Chile John Wallis, University of Maryland Chuck Whiteman, University of Iowa Stanley Zin, Carnegie Mellon University Further information can be found at the conference website at http://www. lacealames2009.utdt.edu or by email at [email protected]. 2010 NORTH AMERICAN WINTER MEETING

THE 2010 NORTH AMERICAN WINTER MEETING of the Econometric Society will be held in Atlanta, GA, from January 3 to 5, 2010, as part of the annual meeting of the Allied Social Science Associations. The program will consist of contributed and invited papers. The program committee will be chaired by Dirk Bergemann of Yale University. This year we are pleased to invite submissions of entire sessions in addition to individual papers. Program Committee: Dirk Bergemann, Yale University, Chair Marco Battaglini, Princeton University (Political Economy) Roland Benabou, Princeton University (Behavioral Economics) Markus Brunnermeier, Princeton University (Financial Economics) Xiahong Chen, Yale University (Theoretical Econometrics, Time Series) Liran Einav, Stanford University (Industrial Organization) Luis Garicano, University of Chicago (Organization, Law and Economics) John Geanakoplos, Yale University (General Equilibrium Theory, Mathematical Economics) Mike Golosov, MIT (Macroeconomics) Pierre Olivier Gourinchas, University of California (International Finance) Igal Hendel, Northwestern (Empirical Microeconomics) Johannes Hoerner, Yale University (Game Theory) Han Hong, Stanford University (Applied Econometrics) Wojcich Kopczuk, Columbia University (Public Economics) Martin Lettau, University of California, Berkeley (Finance) Enrico Moretti, University of California, Berkeley (Labor) Muriel Niederle, Stanford University (Experimental Game Theory, Market Design)

ANNOUNCEMENTS

Luigi Pistaferri, Stanford University (Labor) Esteban Rossi-Hansberg, Princeton University (International Trade) Marciano Siniscalchi, Northwestern University (Decision Theory) Robert Townsend, Massachusetts Institute of Technology (Development Economics) Oleg Tsyvinski, Yale University (Macroeconomics, Public Finance) Harald Uhlig, University of Chicago (Macroeconomics, Computational Finance) Ricky Vohra, Northwestern University (Auction, Mechanism Design)

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Econometrica, Vol. 77, No. 4 (July, 2009), 1336

FORTHCOMING PAPERS THE FOLLOWING MANUSCRIPTS, in addition to those listed in previous issues, have been accepted for publication in forthcoming issues of Econometrica. ABBRING, JAAP H., AND JEFFREY R. CAMPBELL: “Last-In First-Out Oligopoly Dynamics.” ANDREWS, DONALD W. K.: “Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection.” GANUZA, JUAN-JOSÉ, AND JOSE S. PENALVA ZUASTI: “Signal Orderings Based on Dispersion and the Supply of Private Information in Auctions.” GUERRIERI, VERONICA, AND GUIDO LORENZONI: “Liquidity and Trading Dynamics.” HAZAN, MOSHE: “Longevity and Lifetime Labor Supply: Evidence and Implications.” HECKMAN, JAMES, ROSA MATZKIN, AND LARS NEISHEIM: “Nonparametric Estimation of Nonadditive Hedonic Models.” IMAI, SUSUMU, NEELAM JAIN, AND ANDREW CHING: “Bayesian Estimation of Dynamic Discrete Choice Models.” NORETS, ANDRIY: “Inference in Dynamic Discrete Choice Models With Serially Correlated Unobserved State Variables.” QUAH, JOHN K.-H., AND BRUNO STRULOVICI: “Comparative Statics, Informativeness, and the Interval Dominance Order.” SWENSEN, ANDERS RYGH: “Corrigendum to ‘Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models’.”


DOI: 10.3982/ECTA774FORTH

Econometrica, Vol. 77, No. 4 (July, 2009), 1337

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 Andrew Chesher at andrew.chesher@ucl. ac.uk. 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 Andrew Chesher Department of Economics University College London Gower Street London WC1E 6BT, U.K. 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.


DOI: 10.3982/ECTA774SUM

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