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Asymmetry and risk aversion in first -price sealed bid auctions: Identification, estimation, and applications
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Asymmetry and risk aversion in first -price sealed bid auctions: Identification, estimation, and applications
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ASYMMETRY AND RISK AVERSION IN FIRST-PRICE SEALED BID AUCTIONS: IDENTIFICATION, ESTIMATION, AND APPLICATIONS. by Sandra Campo A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) August 2002 Copyright 2002 Sandra Campo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3094310 Copyright 2002 by Campo, Sandra All rights reserved. ® UMI UMI Microform 3094310 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA The G raduate School University Park LOS ANGELES, CALIFORNIA 900894695 This dissertation, w ritten b y c>-r\icl CjArf } ___________ U nder th e direction o f hsx... D issertation Com m ittee, an d approved b y a ll its m em bers, has been p resen ted to and accepted b y The Graduate School, in p a rtia l fulfillm en t o f requirem ents fo r th e degree o f DOCTOR OF PHILOSOPHY D ate A ugust 6, 2002 DISSER TA TION COMMITTEE Chairperson f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D edication To my Family who I love and cherish more than anyone else. They will never understand how much this dissertation is worth to me, but they will nonetheless be proud I have achieved my goals. Con todo mi cariho, vuestra Sandrih.a. To my Friends, who have gone through my bad tempers, my anguish, and so many unpleasant moments, but have stayed with me along this journey. They have made my experience at USC the source of extraordinary stories I will be enthusiastic, to share with my grand children. To all of you, thank you. To Carolina, who has been the best friend one can dream of, who has kept my spirits up, and made me a better person. Con todo mi corazon, muehos besos para ti, Carolina. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgem ents This dissertation was written under the guidance of my advisors Professor Quang Vuong and Professor Isabelle Perrigne. The first two chapters are based on my joint work with them and Emmanuel Guerre. Professor Quang Vuong’s high standards for research have made me reach higher. His patience and endurance have kept me from stopping to believe in my own research. From our meetings, I will always remember how his intuitions always triggered a new result and how, the most important of all, defeat was never an option. The most memorable feeling of all was the one that I experienced after each one of these meetings. Having my mentor share his knowledge with me made me feel like I was at the top of the world. I am forever endebted to him. Professor Isabelle Perrigne was an amazing source of new ideas. Her extensive knowledge of the literature help me build the pillars of this dissertation. She taught me to be a rigourous applied econometrician by finding the right balance between economic theory and data. I hope she will always be proud of me. I would like to thank two very important people, my dissertation committee mem bers: Professor Cheng Hsiao from the department of Economics at U.S.C. and iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Professor Thomas Gilligan from the Marshall business School at U.S.C. They have been as patient as one can wish for, and have always been supportive of my research. Thank You for your advices and your help. I am especially grateful to K. Hendricks and R. Porter for providing the data on the O.C.S. auctions analyzed in chapter 2, Susan Athey, Jason Cummins and Leigh Linden for providing the data set on the U.S. Forest Service timber auctions studied in chapter 3, and Robert Mancuso, Akio Takasue and James Zabala at the Los Angeles City Hall department of Engineering for giving me the opportunity to collect the data on the construction contracts procurement studied in chapter 4. I wish to thank the Haynes Foundation for their funding. Their fellowship has contributed to the completion of my last chapter, and has enable me to finish this dissertation. I admire their commitment to support and help research at the graduate level, and hope to see them pursue this goal within the U.S.C. Graduate School. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents D edication ii Acknowledgem ent s iii List of Tables viii L ist o f Figures ix A bstract x 1 Introduction 1 1.1 Introduction................................................................................................. 1 1.2 Models of asymmetry and risk av ersio n ................................................ 3 1.3 The reduced form analysis and the structural a p p ro a c h ................... 6 1.4 The empirical literature on asymmetry and risk a v e rs io n ................ 12 1.5 P u r p o s e ....................................................................................................... 15 2 A sy m m etry w ithin the Affiliated Private Value Paradigm: The exam ple of the OCS wildcat A uctions 18 2.1 Introduction................................................................................................. 18 2.2 The Model and the Structural A p p ro a c h ............................................. 22 2.2.1 The Asymmetric APV M o d e l.................................................... 22 2.2.2 Nonparametric Identification....................................................... 26 2.2.3 Structural Estimation ................................................................. 31 2.2.4 Practical Is s u e s .............................................................................. 36 2.3 An Application to Joint B idding............................................................. 38 2.3.1 The D a t a ........................................................................................ 39 2.3.2 Unobserved Heterogeneity and Common V alue....................... 43 2.3.3 Structural Estimation R e s u lts .................................................... 51 2.4 Conclusion.................................................................................................... 62 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Risk Aversion w ithin the Independent Private Value Paradigm: A n application to Tim ber A uctions 65 3.1 Introduction................................................................................................. 65 3.2 Model and Equilibrium S trateg y............................................................. 71 3.3 General Nonidentification R e s u lts .......................................................... 77 3.4 Semiparametric Id en tificatio n ................................................................ 84 3.5 Optimal Convergence R a te ....................................................................... 90 3.6 Semiparametric Estim ation....................................................................... 98 3.6.1 A Semiparametric Procedure......................................................... 98 3.6.2 Asymptotic P ro p e rtie s ..................................................................... 105 3.7 Data and Empirical R e s u lts ...................................................................... I l l 3.7.1 D a t a ......................................................................................................112 3.7.2 Estimation R e s u lts ............................................................................115 3.8 Conclusion.......................................................................................................122 4 A sym m etry and Risk Aversion w ithin the Independent Private Paradigm: The case o f the Construction Procurem ents. 124 4.1 Introduction....................................................................................................124 4.2 The m o d e l................................................................................................... 128 4.3 Identification ................................................................................................ 131 4.3.1 R ationalization........................................ 131 4.3.2 Identification result for CRRA utility functions ........................139 4.4 E stim ation.......................................................................................................143 4.4.1 The bid distributions........................................................................ 145 4.4.2 The risk aversion estim atio n ............................................................147 4.5 The private cost distribution...................................................................... 150 4.6 Application to construction procurem ents................................................151 4.6.1 D a t a ......................................................................................................152 4.6.2 Experience and risk aversion........................................................... 155 4.7 Conclusion.......................................................................................................163 References 165 Appendices 173 I Chapter 2, Appendix A ................................................................................ 173 II Chapter 3 .......................................................................................................179 Appendix 3.A .................................................................................... 179 Appendix 3 . B .................................................................................... 187 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Chapter 4 .................... 189 Appendix 4.A ....................................................................................189 Appendix 4 . B ....................................................................................190 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 2.1 Summary Statistics on B i d s ..................................................................... 41 2.2 Summary Statistics on Winner’s Informational Rents (% )................. 61 3.1 Some Summary Statistics ...........................................................................113 4.1 Some Summary Statistics ...........................................................................153 4.2 Firms Classification........................................................................................154 4.3 Firms E xperience...........................................................................................155 4.4 Summary Statistics on Informational R en ts..............................................161 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 2.1 Inverse Bidding S tra te g ie s ....................................................................... 53 2.2 Marginal Densities of Private Values ................................................... 54 2.3 Inverse Bidding S tra te g ie s ....................................................................... 57 2.4 Marginal Densities of Private Values ................................................... 58 2.5 Inverse Bidding Strategies of Joint B id d ers.......................................... 59 2.6 Inverse Bidding Strategies of Solo B id d e r s ......................................... 60 3.1 Conditional d e n sity .......................................................................................121 4.1 Bids and city’ s apparaisal value for the p ro je ct......................................155 4.2 Firms’ frequency per year of experience...................................................156 4.3 Percentage of specialized firm s...................................................................157 4.4 Bids versus ex p erien ce................................................................................158 4.5 The joint density g(b,z) .............................................................................159 4.6 The joint density f (c ,z ) ................................................................................163 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract This dissertation studies bidders’ asymmetry and risk aversion in first-price sealed bid auctions. It extends the structural empirical literature by considering asym metry within the affiliated private value paradigm, bidders’ risk aversion within the independent private value paradigm, and the asymmetry in risk aversion. The key issues are the identification and rationalization of these models. Differences in bidders’ characteristics may affect their private value distributions. In chapter two, I study the asymmetric affiliated auction model. Although the equilibrium does not have any closed form solution, the asymmetric distributions are identified. I propose a two-step nonparametric estimation procedure. The application on O.C.S. auction data reveals that asymmetry between firms, either solo firms or joint consortia is significant, and that the government extracts only 35% of the informational rent. The agents’ bidding behavior also depends on their attitude towards risk. They may fear to lose the object because there are not many alternatives for buying sim ilar objects, as in timber auctions. My third chapter studies symmetric risk averse bidders with independent private values. The model is not identified in general. x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Once I parametrize the agents’ risk aversion, I am able to derive semiparamet ric identification exploiting heterogeneity across auctions. I develop a multi-step semiparametric estimation procedure to recover the agents’ risk aversion parame ter and their private value distribution. In the U.S. Forest Service timber auctions, results show that bidders are fairly risk averse with a constant relative risk aversion coefficient equal to 0.61. Bidders’ wealth and experience can also affect differently their behavior towards risk, as for example in the construction industry. In Chapter four, I define a model of asymmetric risk averse bidders. As bidders share the same private value distribution, I achieve semiparametric identification without imposing any ad hoc condition. The identifying condition inspires the multi-step estimation procedure. Among the bidders competing for the Los Angeles City Hall construction contracts, I find that contractors with more experience are less risk averse. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.1 Introduction Auctions have become a popular method of sale. Governments use them to sell spectrum or oil drilling rights, to sell treasury bonds and used capital ( trucks, heavy machinery,...), as well as to attibute public contracts to the most competi tive firms (constniction contracts, school milk delivery,...) through procurements. The private sector uses auctions as well to sell art work, wine, and numerous other perishable products (fish, flowers,...). Households can even experience the bidding thrill on the web through the increasingly popular e-Bay website. The importance of the study of auctions resides in its financial stakes, the desire to attribute the object in the most efficient way, in particular in public procurements, and in its widespread use to organize markets, in particular for agricultural products. The literature in mechanism design studies the optimal auction design, which maximizes the auctionneer’s payoff and/or allocates the good to the bidder with the highest valuation for the object or the lowest cost to execute the contract in the case of procurement auctions. Rather inadvertedly, Vickrey (1961) first presents auctions as games. His agents maximize their expected utility assuming that their rivals behave as well. From 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. their optimization program, they find their optimal bidding strategies. This concept was later rigourously defined by Harsanyi (1967) as the Bayesian Nash Equilibrium. Bidders receive signals (their private value, or an estimate of the value of the object), have beliefs on their competitors participation (number of bidders) and their signals (the bidders’ valuations are either independent, affiliated or are estimates of the expected value of the object common to all, this last case is known as the common value paradigm). These beliefs are common knowledge. Their own signal is the only private information. They maximize their expected utility conditional upon their beliefs and derive their optimal bidding strategies. If an equilibrium does exist, it is then possible to consider the identification and the estimation of the primitives of the model (namely the utility functions and the value distribution). These primitives, unknown to the auctionneer, are of the utmost interest since they influence the choice of the optimal selling mechanism. For instance, if the agents are risk averse, a second-price auction is preferred to a first-price auction. Imposing a reserve price may raise the bids, and hence maximize the seller’s revenue. Such a price would depend on the distribution of the valuations as well as on the risk aversion of the agents. Thus identifying the source of asymmetries and risk aversion explaining the bidders’ behavior helps designing the optimal auction mechanism. The recovery of the model primitives is based on the simple concept that bids are the expression of the agents’ maximization program, and as such reflect the bidders’ characteristics. This idea is the starting point of the so-called structural approach. The purpose of this dissertation is to extend the empirical auction literature by adopting a structural approach to study the issues of asymmetry arising 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from different valuation distributions (chapter 2), to study bidders’ risk aversion (chapter 3), and heterogenous risk averse bidders (chapter 4) in first-price sealed bid auctions. I find the conditions for the rationalization of the model from the observables and establish the identification of the model. I then propose an estimation method to recover the primitives in each case. I illustrate each model by an application using the OCS oil drilling auction data in chapter 2, the U.S. Forest Service data in chapter 3, and the Los Angeles City Hall construction procurement auctions in chapter 4. In each example, I show the relevance of my assumptions and estimation results. I will first introduce a general game theoretic framework for analyzing asymmetry and risk aversion in first-price sealed bid auctions. I then present a short review of the empirical literature on these subjects, such as to put into perspective the contributions of my dissertation. 1.2 M odels o f asym m etry and risk aversion An indivisible item is offered for sale to I > 2 bidders, where I is the number of potential bidders. Each bidder has a private value for the object, which constitutes her only private information, and offers a bid bi for the object. Bidder Us utility function £/,;(•) is a function of her gain either — hi or bi — for procurement auctions, and her wealth ay. Note that bidder i’s utility depends on her own private value and not on her competitors valuations of the object. This defines the private value paradigm, which needs to be dissociated from the pure common value paradigm where bidders share the same value for the object, observe different signals about this value unknown at the time of the auction, and 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the bidder’s utility depends on the value of the object. Finally, the bidders’ private values (ux, • ■ • ,?>„) follow the distribution F(-, • • • , •) over [u,n]7. The bidders’ private value distribution, utility functions and wealths are all common knowledge. The Bayesian Nash equilibrium of this model is the vector of strategies F(-), {Ui(-)}j= 1, {Ui}{=1,1) = bi,-- - , «/(-, ••-,-) = hi) such that every bid der i, i = 1, • • • , I, maximizes her expected utility, namely max E(Ut) = U f a + vt - &i)Pr(&j > bj, i ± j, j = 1, ■ • • , /). (1.1) b i The resulting first-order condition provides a first-order differential equation in Si(-) with boundary conditions Sj(n) = v and Si(v) = Sj(v), for i ^ j — 1, • • • , I. It has no closed solution unless I define specific private value distribution F(-) and utility functions £/,;(•). I will study three special cases of this model: Case 1: The bidders are risk neutral, thus U i(u> i + — bi) = Ui + Vi — 6? ; . F(?q, • • • , vT) is affiliated and not exchangeable in its arguments. Lebrun (1999) and Bajari (1996) study a model of asymmetric bidders within the independent private value paradigm (IPV). They prove the existence of the equilibrium and list the conditions for its uniqueness. Maskin and Riley (2000a) and Athey (2001) prove the existence of a monotonic bidding strategy within the affiliated private value paradigm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Case 2: The bidders are risk averse. Bidder / “s utility function satisfies U'{-) > 0 and U"(-) < 0. The bidders’ utility is the same across agents and uy; = uj, for i = 1, • • • Their private values are i.i.d. drawn from the same distribution Fi(-) = F(-), bidders are ex-ante symmetric. This model was first defined by Harris and Raviv (1980) and Riley and Samuleson (1981). It is a symmetric game with risk averse bidders. Maskin and Riley (1984) prove the existence and the uniqueness of the symmetric equilibrium in the presence of a reserve price. Campo, Guerre, Perrigne and Vuong (2002) extend the proof to the model without reserve price. Case 3: Bidders exhibit a different behavior towards risk. Agent Vs utility function satisfies [//(•) > 0 and [/"(•) < 0, for i = 1, • • • all the £/*(•)s functions have the same functional form across bidders but bidder Vs measure of risk aversion is different from bidder j' s, for i ^ j. The bidders’ wealth is = u j , for i = 1, • • • , I. Athey (2001) shows the existence of monotonic bidding strategies at the equilibrium in models with asymmetric utility functions. Unless specific utility and private value distributions are defined, the equilibrium bidding strategies are solution of an intractable system of first-order differential equations. This explains the difficulty to prove any property of the equilibrium: existence, uniqueness, monotonicity, and the scarcity of theoretical papers on the subject. Two approaches have been adopted to deal with the complexity of the 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bidders’ bidding strategies and to confront these models to the data: the reduced form approach and the structural approach. 1.3 T he reduced form analysis and th e structural approach Both approaches assume that bidders are rational agents who maximize their espected utility EU;(-) with respect to their only strategic variable, their bid 6 ,:, i = 1, • • • The valuation distributions {Ti(-)}f=1, and the agents’ utility functions {U,;(•)}’ Lj are common knowledge. Thus the equilibrium concept is the Bayesian Nash equilibrium. The bidding strategies at the equilibrium, s,;(-, • • • , •), for i = 1 cannot be explicitly defined from the first order condition in general. The advocates of the reduced form analysis choose to design tests of auction theory from the the predictions of the model on the observables, the bids {6? : }f= 1 (cf. Porter (1995)). In the case of the U.S. Forest Service (USFS) timber auction, Athey and Levin (2001) cannot recover the functional form of the bidder’s strate gies in a model of first-price auctions with risk averse bidders. They nonetheless show that, if there is uncertainty on the value of the object, bids should be spread across the species composing the auctionned lot. Since their observations confirm the bid spread, they conclude that the agents may exhibit risk aversion in these particular auctions.1 Hendricks and Porter (1988) face a similar problem when 1 Athey and Levin aknowlcdge the fact that the bid spread across species could also be due to ex-ante asymmetry in the game. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. studying oil drilling auctions: the described asymmetric common value model does not have any closed form solution. Their interest lies in testing for the presence of asymmetry in joint bidding. Since the model predicts that joint bidding should influence positively the bids, they study the regression of the logarithm of bids, log hi for i = 1, • • • , I (where / is the number of participants to the auctions), on the exogenous variables of the problem: The object and sale characteristics Z, /, and the bidders’ characteristics Among the exogenous variables, the joint bidding dummy has a significant positive coefficient. They conclude that joint bidding is a source of asymmetry in the O.C.S. oil drilling rights auctions. The obvious weakness of such a test lies in the ad-hoc specification of a log-linear regression. The reduced form analysis of auction games has focused mainly on testing for the winner’ s curse phenomenon within the common value paradigm. Laffont, (1997) outline that these studies suffer from the same drawback stressed by Athey and Levin (2001): the tests are based on predictions which could be derived from different models of auction.3 The structural approach is based on a different philosophy. Since the agents’ bidding strategies are function of the unobservable characteristics of the bidders, namely their utility functions, and their private value distributions, there exists 2The variables exogeneity, such as the number of bidders, can be quest,ionned as bidders select themselves based on the object characteristics 3Hong (1997) show that bids are increasing with the number of bidders under the affiliated value paradigm as well as under the the common value paradigm in the presence of the winner’ s curse (sec also Pinkse and Tan (2001)). Unfortunately, empirical papers had already tested for the winner’s curse by observing whether bids were increasing with the number of bidders. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the possibility to recover these unobservables from the observables, the bids and exogenous variables of the auction. The challenge lies then in proving the exis tence and uniqueness of the utility and distribution functions explaining the actual observations. These issues are known as the bids rationalization and identification as illustrated in the following figure. The Prim itives Set The Observables Set {U, (■))<-1 W l i z I {k}£=l z * r where X* stands for the observed counterpart of variable X. The primitives set is made of all variables influencing the bidder’s optimization program: her private value ? ;* , the potential number of bidders in the auction I, the bidders’ private value distributions {F,(-)}f=1 and utility functions {f/j(-)}f=1, the bidders’ characteristics and the object characteristics Z. The observable set is made instead of all the observable variables and estimable functions of the auction: the number of participants, I*, the bids the bid distributions {GiYilii the observed bidders’ characteristics and the observed sale characteristics Z*. Note that the number of potential bidders I and the number of actual bidders I* may differ. If there is a reserve price for the auctionned object Po, it may or may not be oberved, i.e. po belongs to the set of variables Z but may not be in Z*. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rationalization is the study of the existence of inverse bidding strategies &(•) for i = 1, • • • , I such that from the observables, there exist primitives of the model which can explain these observations. Identification of the primitives is achieved if they are unique. Note that, for optimal design purpose, the recovery of the individual private values is not necessary as the rules of the auction are function of the agents’ utility function and valuation distribution. If the model is identified, restrictions imposed by the model on the observables may be unique to this model. Unfortunately models of auctions impose weak restrictions on the observations, as for example increasing inverse bidding strategies, which are common to general models of auction. It should be stressed that it is only within the structural approach that the agents’ primitives (valuation distribution and utility function) can be recovered, whereas the reduced-form analysis can only draw conclusions from the predictions of the model on the observables. In the case of the USFS timber auctions, I recover the bidders’ risk aversion parameter by adopting the structural approach, and test for risk neutrality. As mentionned earlier, Athey and Levin could only infer the presence of risk aversion from the study of the spread of observations across species. Identification and estimation of the structural parameters can be achieved without solving explicitly for the optimal bidding strategy. Two trends of the structural literature need here to be distinguished. The direct approach relies on the existence of either analytical solutions to the agents’ first order conditions or appromixations of the optimal bidding strategies. Bajari (1997) and Marshall, Richard, Stromquist and Meurer (194) design complex algorithms which approximate the true bidding strategies. Their computations are long and 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cumbersome. LafFont, Ossard and Vnong (1995) simulate the true strategies in a descending auction. Paarsch (1992), Donald and Paarsch (1996) define specific cost and utility distributions within the independent and common value paradigm. Within these specifications, the inverse bidding strategy can then be defined as a simple analytical function s_1(-) which defines the valuation distribution F(-) as F(s~l(bj)) — G(bi), where G(-) stands for the bid distribution. Defining such specific primitives provides the opportunity to either compute the exact bidding strategy or simulate it, but limits the number of applications as the observations may not satisfy these parametric assumptions. Guerre, Perrigne and Vuong (2000) introduce the indirect approach in a symmetric model with risk neutral bidders, which allows them to recover the primitives of the model without solving for the agents’ bidding strategies. They observe that, within the independent private value paradigm, the agents’ first, order condition (derived from the maximization of the expected utility in (1.1)) 1 = (/ - 1 )(ih - s (?;,;))|^ |jr h -y for all vt < G [v,v\, with boundary conditions s(v) — v can be rewritten as a function of the observables of the auction as 1 = (I — l)(vi — b i ) ^ ^ . From this expression, bidder Vs inverse bidding strategy £(•) can then be expressed as v, = ((k,G,I)=b,+ j F ^ l . ( 1,2) The inverse bidding strategy leads them to a two-step estimation procedure. First, they construct a vector of pseudo private values by replacing nonparametric 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. estimates of the bid distribution G(-) and density g(-) into equation (1.2). From this vector, they are then able, in a second step, to estimate nonparametrically the distribution of the private values, which is a consistent estimate of the private value distribution. The advantage of such an estimation procedure is obvious. They do not need to constrain the private value distribution function to a specific parametric form such as to solve for the agents’ bidding strategies, neither do they need to solve for these strategies numerically. As a m atter of fact, they do not need to solve for the first-order condition.4 Guerre et aVs method was developped under the assumptions that risk neutral agents shared the same private value distribution; my dissertation will illustrate how this method can be extended to more complicated model of auctions. I will introduce an affiliated distribution function with non exchangeable arguments in chapter 2, risk averse bidders in chapter 3, and heterogenous risk averse bidders in chapter 4 such as to study how asymmetry and risk aversion can be rationalized and identified in models of auction. 1.4 T he em pirical literature on asym m etry and risk aversion Bidders’ asymmetry in their private value distributions has been extensively studied. The literature on this subject aims to explain how some firms have always an advantage over their competitors in the attribution of some contracts 4For more details on these estimation methods, Perrigne and Vuong (1999) give an extensive survey of the econometric literature on auctions. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. even though the auction is designed to offer the same bidding opportunities to all the bidders. It also aims to correct these odds by changing the rules of the auctions as in the case of the Outer Continental Shelf (O.C.S.) auctions where the government was hoping to restore symmetry between big corporations and small firms by allowing the later to organize a consortium. If these consortia are. instead proven to be the source of asymmetry, the government would then choose to forbid their formation since asymmetry destroys the efficiency of the auc tion rules (the contract may be attributed to a firm who does not value it the most). Evidence of asymmetry has been found in numerous contexts. For different types of public contracts (Bajari (1997) for the construction industry, and Flambard and Perrigne (2001) for public snow removal contracts), show that firms have unequal private cost distributions due to their location or distance to the project. Jofre-Bonet and Pesendorfer (2001) outline another source of asymmetry as construction firms may have different production capacity constraints: if the firms have already assigned workers, machines and material to projects, they will suffer a corresponding cost disadvantage when bidding for a new contract. Firms may also decide to either submit a joint bid (Campo, Perrigne and Vuong (2002), Hendricks and Porter (1992)) or form a bidding cartel (Pesendorfer (2000), Porter and Zona (1993), Bajari(2001), Bajari and Ye (2001)). In the case of the O.C.S. auctions, Hendricks and Porter (1988) find evidence of a different source of asymmetry in firms’ access to information. They study oil drilling rights auctions where the auctionned fields are located next to fields already in exploitation. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The firm which drills on the latter has an obvious geological and technological informational advantage when bidding on the neighbour tracts. Even though all these papers rely on the assumption that agents bid according to the Bayesian Nash equilibrium, they differ in their methods of analysis and estimation. Porter and Zona (1993), Hendricks and Porter (1988,1992), Pesendorfer (2000) adopt the reduced form analysis, and test for asymmetry by testing for asymmetry in the bid distributions. The remaining papers adopt the structural approach, identify and estimate the private value or cost distribution. They eventually test for asymmetry by testing for asymmetry in the underlying private value distribution. The optimal auction design does not depend solely on the characterization of the agents’ private value distribution, but may also depend on the characterization of the agents’ utility functions, and more particularly on their risk aversion. Until very recently, the empirical literature has relied on the reduced form analysis to shed light on the agent’s risk aversion. Athey and Levin (2001) and Baldwin (1995) study the behavior of bidders in the U.S. Forest Service timber auctions. In these particular auctions, agents do not bid for a lot of timber but for each species of the auctionned lot. They find evidence of risk aversion in the bids spread across species as if bidders were managing a portfolio of assets to spread the risk of the auction due to the uncertainty on the volume of timber which can be harvested. They cannot give any estimate of the risk aversion measure, because they do not recover the agents’ utility functions. In chapter 3, I show the difficulty to imple ment the structural approach when agents exhibit risk aversion, even within the 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. independent private value paradigm, which explains why the analysis has stayed for so long in reduced form. Campo et al. (2002) and Perrigne (2001) eventually estimate the agents’ risk aversion parameter within the structural approach. This is the subject of my third chapter. They confirm the presence of risk aversion in the U.S. Forest Service timber auction, and give an actual estimate of the agents’ risk aversion when they exhibit a constant relative risk averse utility function. As found in chapter 3, the P ra tt’s measure of risk aversion is equal to 0.61, significantly different from 1, the risk neutral vahie of the coefficient. The experimental literature has been more extensive about the subject of risk aversion in auctions. Its purpose is to explain agents’ overbidding b}r the presence of risk aversion. As experimentalists define constant relative risk averse utility functions and uniform private value distributions within the independent private value paradigm, the optimal bidding strategies are simple functions of the parameter of interest, the risk aversion parameter. Note that in Goere et al. (2001) the estimate of the subjects’ risk aversion parameter is equal to 0.55, which is close to my own estimate, 0.61 (see chapter 3). A last issue lies in the possible asymmetry in the agent’s utility function and more particularly in their risk aversion. Bidders may exhibit different attitudes towards risk according to their wealth or assets, according to the spread of their activities or their experience, .... To my knowlegde this issue has only been studied in the experimental literature, cf. Cox, Smith and Walker (1982), and more recently, Palfrey and Pevnistkaya (2002). In chapter 4, I study heterogenous risk averse bidders within the structural approach. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.5 P urpose As mentionned earlier, bidders’ asymmetry and risk aversion influence the optimal auction design , which rewards the seller with the highest revenue and the bidder who has the highest evaluation with the object. Thus failing to recognize asymmetry and risk aversion in auction may lead to important revenue losses and/or inefficiency.5. To recover the bidders’ primitives (private value distributions, utility function) and their characteristics (asymmetry and/or risk aversion), I adopt the indirect approach as defined by Guerre et al (2000). Asymmetry has already been studied under the independent private value paradigm and the common value paradigm (see review of the literature). In a more realistic model, if agents highly value the auctionned object, there is a higher probability that their competitors will also have a high valuation for the object. This assumption is known as affiliation. In chapter 2, I study a model of asymmetric affiliated private value distributions. Even though, the first-order condition defining the equilibrium strategy does not have any closed form solution, I show nonparametric identification of the model. In the case of the O.C.S. wildcat auctions, firms are allowed to submit either joint bids or solo bids to win oil drilling rights. The government purpose was to restore symmetry between large and small companies by allowing the latter to gather and submit a joint bid. Nonetheless, the estimation shows that, within the asumptions of the model, there still exists some asymmetry 5Inefficiency occurs when, for example, a utility provider with a high production cost wins the contract in a procurement auction to the detriment of the lowest cost competitors. In this case, the contract may have been awarded to the higher cost provider because she was more risk averse than its competitors, and thus outbid the competition 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between joint and solo bidders as joint bidders are more likely to draw a larger private value for the tract than their competitors. Detecting risk aversion in the participants to an auction is a different matter. In chapter 3, I show that, within the independent private value paradigm, a symmetric model with risk averse bidders is unidentified even for particular utility functions as the Constant Relative Risk Averse (CRRA) utility functions. I am able to restore identification by exploiting the heterogeneity across auctions and imposing a constraint on the private value support. I eventually estimate the risk aversion parameter of a CRRA utility function using the semiparametric estimator developped by Campo et al. (2002). This estimator is optimal, though it converges at a slower rate than previously defined semiparametric estimators (see review of literature by Powell (1994), and Kyriazidou (1997)). In the case of the U.S. Forest Service (USFS) timber auctions, I show that the bidders’ risk aversion coefficient is significantly different from 1, the risk neutral value. Thus bidders are risk averse. Assuming that bidders share the same attitude towards risk is not realistic. Firms may have different assets, sizes, different activities or experiences. These factors may influence differently their utility and ultimately their bidding. In chapter 4, I study a model of heterogenous risk averse bidders within the independent private value paradigm. I provide a rationalization condition of the model and show that semiparametric identification can be achieved without imposing any ad hoc condition on the model, when agents exhibit CRRA utility functions. This result stands out when compared to the symmetric risk averse case identification result: idenfication was impossible within the latter model without imposing any 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. further restriction on the primitives. In the case of the Los Angeles construction contracts procurements, I am able to parametrically estimate the firms risk aversion coefficients. I not only show that firms are risk averse, but also that risk aversion is a function of the firm’s experience in the industry: as a firm grows older and gains more experience, it also bids less aggressively because it becomes less risk averse. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 A sym m etry within the Affiliated Private Value Paradigm: The exam ple of the OCS wildcat A uctions 2.1 Introduction Starting from Paarsch (1992), these past years have seen the development of the structural approach for analyzing auction data. Various estimation methods have been proposed mostly for the independent private value (IPV) paradigm. Donald and Paarsch (1993, 1996), and Laffont et al. (1995) have developed parametric estimation methods such as maximum likelihood and simulated nonlinear least squares, while Elyakime and al. (1994) and Guerre et al. (2000) have proposed some nonparametric ones. More recently, relying on the latter, the structural approach has been extended to other auction paradigms such as the more general affiliated private value (APV) by Li et al. (2000, 2002) and the pure common value (CV) paradigm by Haile et al. (2000), Li et al. (2000) and Hendricks et al. (2001). 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A common feature of this literature is the consideration of symmetric auction models, where all bidders are ex ante identical. Auction situations, however, provide many examples where the symmetry assumption is not tenable. This can arise because of bidders’ differences in size as noted in Laffont, et al. (1995), in geographic locations in Bajari (1999), Flambard and Perrigne (2001) and Hong and Shum (2002), and in capacity constraints in Jofre-Bonet and Pesendorfer (2000). Other examples include collusion among some bidders known as cartel (see Porter and Zona, 1993, Baldwin et al, 1997, Pesendorfer, 2000, Bayari and Ye, 2001), asymmetrically informed bidders in OCS drainage auctions (see Hendricks and Porter, 1988, and Hendricks et al., 1994) or joint bidding in OCS auctions (see Hendricks and Porter, 1992). These examples illustrate the necessity of developing general structural econometric methods in asymmetric auctions. This is the main contribution of the chapter for first-price sealed-bid auctions. Simultaneously, the theoretical economic literature has seen a renewal of interest in asymmetric auctions. Hong (1997) characterizes the equilibria in ascending auctions within the pure CV and a particular APV paradigms. Within the IPV paradigm, Lebrun (1999), Maskin and Riley (2000a, 2000b) and Bajari (2001) study the properties of the Bayesian Nash equilibrium in first-price sealed-bid auctions. In particular, these authors derive the system of differential equations characterizing the equilibrium strategies with two types of bidders. As is well known, however, a major difficulty is that no closed form solutions can be obtained in general unlike in the symmetric case first studied by Vickrey (1961). This 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. represents an important drawback for developing the structural approach since numerical methods are required to solve for the bidding strategies (see Marshall et al., 1994, and Bajari, 1999). As a result, only a few researchers (see Bajari, 1999, Crooke et al, 1997, Jofre-Bonet and Pesendorfer, 2000 and Flambard and Penigne, 2001) have tackled the structural empirical analysis of asymmetric first-price sealed-bid auctions within the IPV paradigm. In this chapter, I consider the asymmetric APV model for first-price sealed-bid auctions, which encompasses the IPV model as a special case. It is well known that the IPV model can be restrictive in practice as bidders may have private values related to each other. In contrast, the APV model allows for dependence among bidders’ private values while retaining a bidder’s private utility for the auctioned object. See Li et al (2000, 2002) for a discussion of the economic justification of the APV model in the symmetric case. 1 For the asymmetric APV model, I derive the differential equations that characterize the Bayesian Nash equilibrium strategies. For simplicity, I consider two types of bidders only but our method can be straightforwardly generalized to many types of bidders. I then establish the nonparametric identification of the model from observed bids and I propose a convenient two-step nonparametric procedure for estimating the underlying distribution of the model, namely the joint distribution of private values. My method extends Guerre et al (2000) to asymmetric auctions, and is especially convenient computationally as it circumvents the numerical resolution of the differential equations characterizing the equilibrium strategies. Moreover, because 1 Intuitively, affiliation means that when one bidder evaluates the object highly it is likely that others will evaluate; it highly too. The APV model is a special case of the general (symmetric) affiliated value model developed by Wilson (1977) and MilgTom and Weber (1982). This general model encompasses both the symmetric IPV and pure CV models as polar cases. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of its nonparametric nature, my method does not require a priori parametric specifications of the underlying private value distribution. As an illustration, I analyze the effects of joint bidding in OCS wildcat auctions, specifically whether joint bidding can be a source of asymmetry among bidders. Such a practice allows a firms’ consortium to submit one bid and was allowed to some extent by the federal government. The motivation was to encourage the participation of relatively small oil companies because of the large capital requirements for buying and developing OCS leases (see Hendricks and Porter, 1992). Two major issues then need to be addressed. First, it is likely that unobserved tract heterogeneity is present, affecting in particular the number of participants and the relative proportion of bidding consortia on any given tract. I show that my estimation method can deal with such unobserved heterogeneity under some reasonable assumptions. Second, I show that the asymmetric pure CV model is observationally equivalent to some asymmetric APV model while impos ing some additional testable restrictions. The latter are not supported by the data. The chapter is organized as follows. In Section 2.2, I introduce the asymmetric APV model with two types of bidders and derive the system of differential equa tions characterizing the equilibrium strategies. I then establish its nonparametric identifiability and characterize the restrictions on the bid distribution imposed by the model to assess its empirical validity. I also propose a two-step nonparamet ric procedure to estimate the underlying private values distributions of the model. Section 2.3 illustrates our estimation procedure and presents my empirical findings. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Unobserved tract heterogeneity and the pure CV assumption are discussed there. Section 2.4 collects some concluding remarks. Proofs are given in an appendix. 2.2 T he M odel and the Structural A pproach In this section, I first present the asymmetric APV model. We establish its nonparametric identification and I characterize the restrictions it imposes on the observed bid distribution. I then propose a computationally convenient two-step nonparametric procedure for estimating the latent distribution of the asymmetric APV model. 2.2.1 The Asym m etric A P V M odel A single and indivisible object is auctioned to n bidders who are assumed to be risk neutral. Though my results can be easily generalized to a larger number of types, for simplicity, I assume that there are only two types of bidders as in Maskin and Riley (2000a). For instance, subgroup G1 can be characterized by larger size bidders, better informed bidders, cartel of bidders or joint bidders as is the case in Section 2.3. This group is referred as the group of “strong” bidders. Subgroup GO gathers the other bidders, i.e the “weak” bidders. Type 1 contains m bidders while type 0 contains uq bidders, with n ,j + no = n and n > 2. Let vu, i = 1, ... ,ny, denote the strong bidders’ private values and va, i = 1, ... ,uq be the weak bidders’ private values. It is assumed that the vector (vn , .. . , Vyni, u0i , ... ,v0no) is the realization of a random vector whose n-dimensional cumulative distribution function is F(-). The 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. latter is assumed to belong to the set V of n-dimensional absolutely continuous (with respect to Lebesgue measure) distributions with hvpercube support that are affiliated and exchangeable (symmetric) in the first nx and last n0 arguments. Let [v, v\n denote the support of F(-), where v > 0.2 This probabilistic structure can be interpreted as follows. While there is symmetry within each subgroup as bidders in each group jointly draw their private values from an affiliated exchange able distribution, there is possible asymmetry between the two subgroups since their marginal distributions may differ across subgroups. Moreover, because of affiliation, there is general dependence among all private values. The distribution F(-) is assumed to be common knowledge and defines the asymmetric APV model. Each player i knows the value of his own signal, but does not know other players’ private signals. We focus below on the first-price sealed-bid auction. At the Bayesian Nash equilibrium, each bidder i of type 1 chooses his bid bX i to maximize his expected payoff E[(vX i — bX i)JI(B_i < 6 i,;)|u1 ? ;], where = m ax{si(y^),s 0 (|/oi)}, y * X i = maxjV ijG G 1 vX j and y0i = maxieG0 u0i, si(-) and s0(-) are the equilibrium strategies of bidders of types 1 and 0, respectively. The term E[-\vX i] denotes the expectation with respect to all random elements conditional upon vXi. As usual, I restrict myself to strictly increasing differentiable equilibrium strategies and, because of the symmetry within each group, I assume that bidders 2Sec Milgrom and Weber (1982) for a formal definition of affiliation. As in the theoretical literature on asymmetric auctions, I assume that all private values have the same finite support, namely [wJ], . See Maskin and Riley (2000a). 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the same subgroup adopt the same strategy.3 Hence the problem for any bidder i of type 1 can be written as max (vu - bu)Pr(yy < ](6 1 ? ;) and y0i < s0L(bh;)K?.), on where s j 1^) denotes the inverse of the equilibrium strategy = 0,1. The above probability can be written as Fy^yo\Vl(sil(bii), s0' 1(6 1j)|u1 , ;). Differentiating with respect to 6 1 ? ;, the equilibrium strategy si(-) for any strong bidder i , i = 1 , .. . , n,i satisfies the first-order differential equation Fy\Mvi(si so \hi)\vu) (^ii)5 so (^n)hhi) +(vii — bu) x dy*i s'i(si 1(bu)) dFy*,yo\vi(sl {bli),SQ ( f h ? : ) I ' D ? ) 1 dyo •X So( S 0 = 0 , (2 .1) 3In the asymmetric: IPV ease, Lebrun (1999) shows that the Bayesian Nash equilibrium exists and is unique. Moreover, it is in continuous strictly increasing pure strategies, which are fully characterized by the first-order and boundary conditions only. For the asymmetric APV case, Maskin and Riley (2000a) and Athey (2001) have established the existence of a monotonic pure strategy Nash equilibrium. Sec also Lizzeri and Pcrsico (2000) who consider a broad class of games with a common value component. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for all Vu G ju,u], where bu = Si(vu). Similarly, the equilibrium strategy s0(‘) for any weak bidder i,i = 1 ,.. . , v,q satisfies the first-order differential equation ^Vi,Vo\vo (S1 (b0i)i s 0 (b0i) l^O)) ^^yi,Vo\vo(Sl (b0i)) s 0 (^0i)N; 0i) dyi X dFyi,y*\V 0 (Sl (boi), s0 (hot)|uQ j) dy*o X i(si i boi)) 1 - 0 , (2 .2) for all vqi G [u, u], where b0i = s0 (u0 ? ;). The equilibrium strategies si(-) and so(-) are the solutions of the system of differ ential equations (2 .1 )-(2 .2 ) subject to the boundary conditions Si(u) = s0 (u) = v and s\(v) = so(v). This system is quite complex and intractable in general. When private values are mutually independent, it can be verified readily that this system reduces to the system of differential equations characterizing the Bayesian Nash equilibrium in the asymmetric IPV model studied by Maskin and Riley (2000a) . 4 Moreover, when rq = 0, or n 0 = 0, or F(-) is exchangeable in all its n arguments, it can be verified that the system (2 .1 )-(2 .2 ) reduces to the single differential equation characterizing the symmetric Bayesian Nash equilibrium in the symmetric APV model studied in Li et al. (2002). 4Bccausc of its intractability even in the IPV ease, Marshall et al. (1994) propose some numerical algorithms for solving the system when the latent distributions are uniform, while Bajari (1999) proposes some numerical procedures within a Bayesian estimation context. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.2 Nonparametric Identification The structural approach relies upon the hypothesis that observed bids are the equilibrium bids of the auction model under consideration. Depending on their type, bidders will bid according to the equilibrium strategies si(-) or s0(-) defined by the system (2.1)-(2.2) of differential equations. Formally, this means that, given an n-dimensional joint distribution F(-) of private values belonging to V, the structural econometric model is { h i = si(vu,F),i = 1 ,... ,71! (2.3) hi = s0(v0i,F),i = 1 ,... ,n 0, where the dependence on F(-) of both equilibrium strategies si(-) and sq(‘) is indicated. As private values are random, bids are naturally random with an n- dimensional joint distribution G(). The equilibrium bid distribution G(-) depends on the underlying distribution F(-) in two ways: (i) through the unobservables (u n ,... , v\ni, vqi,... , uono) which are jointly drawn from the distribution F(-), and (ii) through the equilibrium strategies si(-) and s0(-) that are both complex functions of F(-). This feature is common to auction models and complicates their identification and structural estimation. See Guerre et al. (2000) for a discussion. The structural element of the asymmetric APV model is the joint distribution F(-) of private values. A question of interest is to know whether this distribution is identified from observables, which are the bids (bn , ■ ■ ■ , h m , hi, ■ ■ ■ , & o n 0)- In particular, the number of bidders of each type as well as the bid and type of each bidder are assumed to be observed. A second issue is whether the structural econometric model (2.3) imposes some restriction(s) on the joint distribution of 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. observed bids. In other words, can any bid distribution G(-) be rationalized by an asymmetric APV model? Such a question is important for assessing the empirical validity of the asymmetric APV model. We introduce the conditional distribution GB*,B0\h of {B\, Bo) given 6 1, where B\ = maxj^i bu and B q = max,; & o ? ;- This gives GBhBolbl(X,X\x) = P v i B l ^ X ^ B o K X l h ^ x ) = P r(yl < s^(X ),yo < SqX (A:)|?;i = s^l (x)) - Fyl,yolvi(s^ (X ),s ^ (X )\s ^ (x )). It follows that dGB*,BQ \bi(X,X\x) dX dFyhyo]vl h l (X ),s^(X )\x) dy{ dy0 1 1 Using the last two equations and = s1 1(6 i), the first-order differential equation (2 .1 ) can be written as (2.4) 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A similar treatment applies to the first-order differential equation (2.2) by intro ducing the conditional distribution Gb1,b^ |& 0(’i 'I') and the derivative dGBuBS\b0(-,-\-)/dX. This gives , . GBl,B^b0(bo,bo\bQ ) _ vo - bo + — 7r~hlh \TJY = ^ °’ ( 5) aGBl,B*\b0{bo, oQ \bo)/dX where Bi — maxj&i* and B q = maxj^j bw. Hence, each private value in any of the two subgroups can be expressed as a function of the corresponding bid, an appropriate conditional bid distribution and its total derivative without solving the system of differential equations (2.1)-(2.2). As in Guerre et al. (2000) and Li et al. (2002), this is the key result that allows us to identify the asymmetric APV model as well as to estimate it easily without solving the mathematically intractable system of differential equations (2.1)-(2.2). In contrast to these previous papers, (2.4)-(2.5) involve in general a trivariate distribution and a total derivative in their denominators, which is not a density. The next proposition, whose proof is given in the appendix, shows that the asymmetric APV model is identified from observed bids. It also gives a necessary and sufficient condition on the joint distribution G(-) of observed bids for the existence of a latent distribution F(-) G V that can rationalize the bid dis tribution (?(■), i.e. for which G(-) is the corresponding equilibrium bid distribution. P ro p o sitio n 2.1: The asymmetric APV model is identified. Moreover, the joint distribution G(-) of observed bids can be rationalized by an asymmetric APV model with F(-) G V if and only if (i) G(-) belongs to V, 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and (ii) the functions (b,G) and £0(b,G) are strictly increasing in b 6 [b,h\ with £1 (b,G) = &(&, G), where [b, b]n is the support of G(-).5 The identification result of Proposition 2.1 is nonparametric in nature as it does not require any parametric specification of F(-). This contrasts with parametric identification results which can be achieved through misspecified parametric specifications. Second, it complements the few existing identification results for asymmetric models (see Laffont and Vuong, 1996, for the IPV model and a special CV model). Third, my result extends the identification result established by Li et al. (2002) for the symmetric APV model. Unlike the symmetric APV model, however, it is unknown whether the asymmetric APV model is the most general auction model identified from observed bids in the class of asymmetric auction models.6 Fourth, because V allows F(-) to be exchangeable in all its n arguments, my identification result implies that the asymmetric APV model can be distinguished from the symmetric one in view of observed bids. This is used to assess the presence of asymmetry in Section 2.3. Besides identification, Proposition 2.1 characterizes the game-theoretic restrictions imposed by the asymmetric APV model on the distribution of observed bids. In particular, the fact that bidders adopt the Bayesian Nash equilibrium strategies defined by the system of first-order differential equations (2 .1 )-(2 .2 ) imposes some 5I assume that the first-order conditions (2.1)~(2.2) with boundary conditions are sufficient for characterizing the equilibrium strategics, i.e. the second-order conditions an; autom atically satisfied. Sec; also footnote 3. sThis question is related to the problem of whether two asymmetric auction models can be discriminated from each other from observed bids as discussed in Laffont and Vuong (1996). 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. restrictions on the distribution of observed bids in the form of the monotonicity of the functions £1 (-,(?) and £o(',G). This monotonicity can be used as a basis of a formal test of the validity of the theoretical model. Rejection of such restrictions would imply that bidders do not adopt a Bayesian Nash equilibrium and/or that one or more of the underlying assumptions of the model are violated. These include situations with some common value, bidders’ risk aversion or an unknown number of participants. It should be noted that the rationalization conditions given in Proposition 2.1 and the identification of the APV model provide an empirical response to the existence and uniqueness of the Bayesian Nash equilibrium. In particular, If the observed bid distribution satisfies the rationalization conditions, then such a distribution can be viewed as the outcome of a (strictly increasing and differentiable) Bayesian Nash equilibrium of an APV model. Moreover, the identification property shows that such an APV model is unique whenever only strictly increasing and differentiable Bayesian Nash equilibria are considered. From equations (2.4) and (2.5), we note that, provided G\b*,£0|6i (■,-|-)> (•, -|-) and their total derivatives are known, one has neither to solve the complex system of differential equations (2 .1 )-(2 .2 ) for determining the equilibrium strate gies si(-) and sq(-). Specifically, knowledge of Gs «;B o|6l(-,-|-) and GBub*|60 (v |-) and hence of £i(-) and £o( 0 determines the private values rq and vq for any given bid hi and ho through (2.4) and (2.5), respectively. This provides a method for circumventing the extreme computational difficulties encountered in the structural 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analysis of auction data with asymmetric bidders. These equations form the basis upon which my proposed estimation procedure rests. 2.2.3 Structural Estim ation We focus hereafter on the bivariate densities •), •) and /io(-, •) of the pairs (vu, uy), (vQ i,v0j) and (vu ,v0j), respectively.7 If one knew GB*> B -|-) and GSiib*|(,0(-, -|* )> one could use (2.4) and (2.5) to compute the private values for all bidders and then estimate /(•) from the latter. This suggests the following two-step estimation procedure. In a first step, the conditional distributions GB*tBo\bi(', ’I') and Gb1ib»|60(-, -|-) and their total derivatives are estimated from observed bids. In a second step, private values estimated from (2.4) and (2.5) are used to estimate the aforementioned densities of private values. Though parametric methods could be used, in the continuation of our nonparametric identification result, I consider nonparametric techniques in each step. Specifically, the procedure is as follows. • Step 1. Construct a sample of pseudo private values based on (2.4) and (2.5) using nonparametric estimates of Gb*;B o|6i(u 'IOi dG\Bp£0|£ > i(‘, '\')/dX, GBl,B^\b0{-, -|-), and d,GBl,B*\bo{-,-\-)/dX from observed bids. • Step 2. Use the pseudo private values constructed in Step 1 to estimate nonparametrically the bivariate densities of interest. The analysis must be performed separately for each given pair (ni,n0) since the bid distributions and the inverse bidding strategies £i(-) and £0(-) estimated 7Knowiug these three bivariate densities, however, is not sufficient for recovering the joint distribution F(-). My results can be extended to the estimation of other multivariate densities. A vector of observed characteristics Z for the auctioned objects can be included as in Guerre et al. (2000). 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nonparametrically in the first step actually depend on (nx,n0). This is so because the bidding strategy of any bidder depends on the number and types of his opponents. If nx or n 0 is equal to zero, there is no asymmetry and the estimation procedure reduces to that presented in Li et al (2002). On the other hand, when r?4 and no are both strictly positive, the estimation procedure is in general more involved with a trivariate distribution and a total derivative. In particular, the analysis cannot be done by applying Li et al (2002) method separately on each group. Hereafter, I focus on the case where n ,Q and nx are both strictly positive. Let L be the number of corresponding auctions and let I index the Gth auction, £ = 1 ,... , L. In the first step, I note that the conditioning on bx or bo disappears from the ratios in (2.4) and (2.5) as for instance the ratio in (2.4) can be interpreted as _____________ P r(H | < bj, Bp < fei, hi = bj)______________ Pr(i?r = & i, Bq < b\, b\ — b\) + Pr(H* < bx, Bo = bx, b± — bi) Hence, using the observed bids {buf, i = 1 ,... , nx} and {b0if, i — 1 ,... , n,0} for £ = 1 ,.. . , L, I can estimate the numerators G B*jBoibx{bii bx, bi) and GBl,B£,b0{bo, b0, b0) nonparametrically by GB*tBoM { h M M ) and GBl,Bz,b0(bo,b0,b0), where nG\L ^ nx \ H gi 1 ^1 / fa = t— r ; c - : £ * ( i ^ < * ) * ( % < "GoB “ no “ \ nG o with #(■) the indicator function, Byf = m a x . b X jg, Bq ? = max,; bon, B ^ = maxj^i boje, Bu = max,; bug, some bandwidths hex and heo and a kernel iQ?(-). 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that I have used symmetry within each subgroup by averaging over nj and n0, respectively. Regarding the denominator of (3.6), I estimate it by the sum of £> 1 1(6 1, 6 1, 6 1) and £> 12(6 1, 6 1 , hi), where 1 ,A 1 ( x - B * ue\ ^ / z - bia with a bandwidth hgi and a kernel Kg{-). Similarly, I can estimate the denominator of (3.6) corresponding to type 0 using where hg0 is a bandwidth. Hence, estimates of the private values Vnt and vqu are c (1 \ , Gbi,b0m {huMit.Mn) . , via = ti(hue) = bm + ~ ----- :----- (2-7) ^n\Oue, °iH i °iif) + ^12[pm .-, One, One) t a, \ u , GBl,B^b0(bOi^hie,bon) voa - to(bw) - bm + j —- --- -----7-——^ — -----7-7— 7, (2-8) Mul&OiT, O qu, O qu) + Uo2\pOie, £ » 0 «£, for z = 1 ,... , n-i and z = 1 ,... , uq, respectively. Because of well-known boundary effects in nonparametric estimation, private values as defined in (2.7) and (2.8) may not be well estimated near the boundaries. This effect is corrected by introducing a trimming, which is explained in the next subsection. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The second step consists in the estimation of various densities of interest, such as / n (-, •), /oo(-,')> and /io('i ■ ), using the pseudo private values {viu,i = 1 ,... ,ni,v0i£,i = 1 ,... ,n0,£ = 1 ,... ,L}. To estimate f n (-, •), let /py)^-, •) be the bivariate density of the pair ( v u , v i j ) . This density can be estimated non parametrically by i t , 1 ^ f x - vue y - vijA where hfi is a bandwidth and Kf(-) is a bivariate kernel. Imposing symmetry, this gives = [/(ij),(.x,y) + f {idh(y,x)\/2. Thus, by considering all possible pairs a (symmetric) estimate of the bivariate density of private values for strong bidders is fn(*,y) = )Sx >y)- (2-°) '1' i<j Similarly, a (symmetric) estimate of the bivariate density of private values for weak bidders is foofay) = - 2 ^ T - - 2j-i ] T / J i)o (x, y), (2 -1 0 ) '° ' i< j where / ( fJ)o = [/pp)o(T y) + f(ij)0(y, -O]/2, and f(ij)0(*,y) = 1 T T Y / K .f(: x - 'I’M y - V Q j l h%L \ hfo hfo 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with hfo a bandwidth. To estimate the bivariate density /io(-, •)) we can proceed as follows. Let /,;lj0(-, •) be the estimate of the bivariate density of the pair (vuf ,v0 for any i = 1 ,... , n\ and j — 1 ,.. . , n 0, where i ( \ 1 ( x ~ ?,ia V ~~ i;oje fn d 0 ( - T y) = T 7 - 7 7 r 2 ^ K f — p — ’- j r - nfOnflL / = = 1 \ fl nf 0 and h!jl and h!f0 are two bandwidths. The bivariate density /io(-, •) can be esti mated by fio(x iV) — ~ J Z /mioOL V)- (2 -1 1 ) nin° ieGljeGO Using a similar argument as in Li et al. (2002), it can be shown that our two- step estimators are uniformly consistent using appropriate bandwidths. Moreover, simulations performed in that paper show the good behavior of the nonparametric estimator in small samples. The choice of bandwidths and kernel functions is dis cussed in the next subsection. Each bivariate density provides information on the degree of affiliation among bidders’ private values within the same type or across types. In particular, the shape of jio(-, •) tells whether there is some asymmetry between bidders of type 1 and 0. Because our estimation method is fully non parametric, such information is revealed by the data. A related advantage is that I do not have to parameterize the affiliation among private values, which can be linear or more complex. Lastly, a significant advantage of my method is its com putational simplicity. Indeed, my method does not require solving the differential equations (2 .1 )-(2 .2 ) and hence to compute the equilibrium strategies.8 8Under the assumption that the underlying private value distribution is independent of (r? 4 ,r),0), one can pool the pseudo private values (2.7)-(2.8) obtained for every pair (na,no) to improve estimates of the above bivariate densities. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.4 Practical Issues When working with real data, one frequently observes a highly skewed bid distribu tion with a large number of observations on the lower end. To minimize skewness effects, the data are transformed using a logarithmic function, which has been fre quently used in empirical work. Using the logarithmic transformation, (2.4) and (2.5) become ( i x /, . G D * D ^ d l (di,di\di) \ vi = ^xp(di) 1 + — — 1 ----- - 1 - n (d x) (2.12) \ dGoi,D0,di {di,di\di)/dX J , GDljD*ido(d0,d0\d0) \ = exp(do) I 1 + MD l.D lA<kM<k)Hx) - 1 s T ^ ' (2' 13) where d = log(l + b), Gd\,d04A'i "I") is th e conditional density of (D{,D0) = (maxj^i log(l + max* log(l + h0i)) given log(l + 6 X ), bx being chosen arbitraly among r ? ,x values, and dGD\,D04i (', '\')/dX is the appropriate total derivative. The notation in (2.13) is similarly defined with d0 = log(l + b0).9 Next, I adopt a trimming similar to that of Guerre et al. (2000). Let dmax = max{dm ax i , dm ax o}, where dmaxl and dm ax q are the maximum values of the log- transformed strong bids and weak bids, respectively. Thus dm ax estimates the common upper bound d of the supports of d\ and do. For j = 0,1, let fj(-) be the estimate of T j ( - ) using the previously defined nonparametric estimators for bid 9The transformation log(l + •) ensures that the support [d,d] is included in M+ and (»mpact whenever v < oo. In the application, I assume that v — 0 so that 6 = 0 and hence d = 0. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distributions with (doif.,duf) instead of {boa, hue). The pseudo private values v \ a and VQj£ are defined as if max{/?,ci, h g i } ^ d Xm ^ dm ax max{hGi, b g X } Too otherwise and { fo(dotf) if max{/?Go, ^go} < doa dmax - max{/?,G0, hgo} Too otherwise for t = 1 ,... , nx and f = 1 ,... , n0, respectively, and I — 1 ,... , L. Because the kernel Kf(-) in the second step has a compact support, the auctions that are relevant for estimating the densities of interest are those for which the values = 1, • • • ,nx and Vou,i — 1 ,... ,nQ are all finite, i.e. not trimmed. Let Lt denote the number of remaining auctions. It remains to discuss the choice of kernels and bandwidths. As is well known, the choice of kernels does not have much effect in practice. I choose the trightweight kernel, which satisfies the assumptions in Guerre et al. (2000). This kernel is of the form K(u) = (35/32)(1 — u2)3M(\u\ < 1). It is used for K g(-) and K G(-), while Kf(-, •) is the product of two univariate triweight kernels. In contrast, the choice of bandwidths requires more attention. I use band widths of the form hG X = cGX(niL)~1 /5, hgX = Cgi{nxL)~x/6, hG 0 = cG 0 (n0 L)_1/5, hgo — Cgo^ioL)^1 ^ for the first step, and hfX — CfX(2\(n,i — 2)\Lt / nx\)~l/lb, 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. hfo = c/o(2!(n0 - 2)\LT/n0\) ^ 15, h'fl = cn (nin0LT) 1/15, h'f0 = cf0(nin0LT) 1/15 for the second step, following Guerre et al. (2000) and Li eA al. (2002). The factors involving rq and n0 are due to the additional averaging of my estimators (see, e.g. (2.9), (2.10), and (2.11)). Regarding the constants cqi, cgj, cqo, cgo, c/i, and Cfo note that all the bandwidths except those in the second step correspond to the usual rate so that their constants can be obtained by the so-called rule of thumb. Hence, I use cc,\ = cgi = 2.978 x 1.06ddi and cqq — cg0 = 2.978 x 1.06<rrfo , where and are the standard deviations of the log ( 1 + bids) for each type, respectively. The factor 2.978 follows from the use of the triweight kernel instead of the Gaussian kernel (see Hardle, 1991). For the bandwidths in the second step, we use constants Cfi = 2.978 x 1.06oyi and Cfo = 2.978 x 1.06<5yo, where avi and avo are the standard deviations of the trimmed pseudo private values of strong bidders and weak bidders, respectively. 2.3 A n A pplication to Joint B idding This section illustrates my method with an empirical study of possible asymmetry arising from joint bidding in OCS wildcat auctions. I first present the data and motivate the problem. I then address two issues, namely unobserved heterogeneity and the adequacy of the pure CV model, that have been frequently raised in the empirical analysis of such data. Lastly, I apply my procedure and discuss my empirical results. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.1 The Data The US federal government began auctioning its mineral rights on oil and gas of offshore lands or Outer Continental Shelf (OCS) in 1954. My study focuses on the wildcat lease sales in the Mexico and Louisiana gulfs. Before each sale, the government announces that an area is available for exploration. This area is divided into a number of tracts, usually a block of 5000 or 5760 acres. Leases of drilling rights on these tracts are sold simultaneously through first-price sealed-bid auctions. The highest bidder wins the tract and pays his bid. 10 The highest bidder wins the tract and pays his bid. The participants to each auction are oil companies. A characteristic of interest in OCS auctions is the practice of joint bidding, which was allowed to some extent by the federal government. 11 Though joint bidding was allowed since 1954, its practice developed significantly in the 70s. Before 1970 joint bidding affected fewer than 20% of auctions, while this percentage rose to more than 80% after 1970. On average a joint bid involves two or three firms. A number of arguments for joint bidding have been given in the literature. For instance, joint bidding can weaken financial constraints, reduce costs by pooling cartel members’ information and capital through the joint venture and spread risks among firms. See e.g. DeBrok and Smith (1983), Millsaps and O tt (1985), Gilley et al. (1985) and Hendricks and Porter (1992). 10It can be considered that the reserve price at $15 per a,ere does not act as a screening device for participating to the auction, as recognized by many economists. See e.g. McAfee and Vincent (1992). 1 1 Until December 1975, any set of firms could organize a consortium so as to submit a so- called joint bid. Because of some concern about competition, the federal government restricted the practice of joint bidding by barring the eight largest firms from bidding jointly with each other. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As noted by many economists, however, joint bidding may have introduced some ex ante asymmetry among bidders. Because joint bidding is negligeable in the 50s-60s, our study focuses on auctions held between December 1972 and 1979.1 2 Because of data requirements explained subsequently, we consider auctions with two bidders who can be either joint or solo. This gives a total of 227 auctions from which 55 auctions have two solo bids, 60 auctions have two joint bids and 1 1 2 auctions have one solo bid and one joint bid. Among the latter, 63 auctions are won by the joint bidder. Using a normal approximation, the ratio 63/112 is greater than 1/2 at the 10% significance level in a one-sided test, where 1 / 2 would be the expected ratio if the two participants have equal chance of winning. 13 Thus joint bidding has increased the probability of winning suggesting some ex ante, aymmetry among participants. For each wildcat auction, we know the date, the acreage of the tract, the number of bidders, their bids in constant 1972 dollars and whether the bid is a solo or a joint bid. Table 2.1 gives some summary statistics in $ per acre for the 454 bids considered in our empirical study as well as on solo and joint bids separately, whether the opponent’s bid is of the same type or of a different type. 12I exclude auctions after 1979 since the rules of the auction mechanism have changed somewhat after this date. I also exclude the unique sale held in 1970 and the first sale in 1972 because the water depth of the tracts sold at these sales was much greater than usual. 13Hereafter, all tests are conducted at the 10% significance level. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1: Summary Statistics on Bids Variable # Obs Mean STD Min Max Within STD All bids 454 687.30 1,431.31 19.51 20,751.32 1,258.91 Joint Bids 232 837.32 1,717.54 21.46 20,751.32 — Solo Bids 2 2 2 532.53 1,033.20 19.51 11,019.08 Joint vs Joint 1 2 0 875.13 2,056.12 33.94 20,751.32 2,011.99 Joint vs Solo 1 1 2 796.83 1,266.32 21.46 6,377.94 — Solo vs Joint 1 1 2 603.28 1,226.61 19.51 11,019.08 — Solo vs Solo 1 1 0 456.45 788.19 20.80 7,009.10 747.43 A first feature revealed by the means displayed in Table 2.1 is that joint bids tend to be higher on average than solo bids, as a number of empirical studies have found. Moreover, joint bidders tend to bid higher when they face a joint bidder than when they face a solo bidder. Likewise, though their bids are lower than those of joint bidders, solo bidders tend to bid on average higher when they face a joint bidder than when they face another solo bidder. This suggests that the bidding strategy of each type of bidders depends on the type of their opponent. This could arise from bidders taking into account some possible asymmetry in their bidding strategies. For instance, a test of the equality of means for solo bids versus joint bids in the 1 1 2 auctions with one bidder of each type gives a f-statistic equal to 1 .6 6 , which (weakly) rejects their equality. It is also interesting to note that the within variability of solo versus solo bids is much smaller than the within variability of joint versus joint bids. This may again support the hypothesis of an asymmetry between joint and solo bidders. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A second feature of Table 2.1 is the large total variability of bids in all cases. This is confirmed by the wide range of bids. Such a large variability could arise from tract heterogeneity and/or bidders’ heterogeneity. Tract heterogeneity may also explain the differences in means noted above. In particular, higher value tracts could attract more likely joint bidders than solo ones, and conversely for lower value tracts. That is, the joint/solo structure could be endogenous in the sense of depending on tract characteristics. Moreover, the within standard deviation is important relative to the total standard deviation ranging from 70% to 98%.1 4 For instance, when considering auctions with two joint bidders, the within variability explains about 98% of the total variability of bids. This suggests that a large part of the differences across tracts can be explained by the joint/solo bidders’ composition. To further assess tract heterogeneity, we first regress the log of bids on a set of tract dummies and obtain a weak rejection of tract homogeneity. This may be due to the fact that the 227 tracts are sold through 16 sales spread over the 1972-1979 period. We then consider a regression with sale dummies and, in view of Table 2 .1 , three bidding structure dummies (whether the bid is the result of a joint consortium, whether the opponent in the auction is a joint bidder, and an inter action dummy between these wo dummies). The structure dummies coefficients are jointly significant while the equality of the 16 sale dummies is nearly accepted. Thus, controlling for bidding structure much decreases heterogeneity across tracts, while also controlling for changes in bidders’ types over time. It remains to discuss whether the oil crisis has increased bids after 1973 as well as whether the 1975 14The within variability for auctions with solo and joint bids is equal to 877.04. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ban 011 large firms bidding jointly has had some lowering effects on bids. Using dummies, none of these events turns out to be significant. 2.3.2 Unobserved H eterogeneity and Common Value In view of the previous analysis of bid data, we first discuss the heterogeneity issue within our econometric model presented in Section 2.2. Furthermore, the paradigm of the model in Section 2.2 is that of private values, while the pure CV model (also called the mineral right model) has been widely entertained in the empirical analysis of bidding in OCS auctions. This section derives some testable restrictions imposed by the (asymmetric) pure CV model, which allow us to bring some evidence on the debate private versus common value in such auctions. U n o b s e r v e d H e t e r o g e n e i t y From a broader perspective, heterogeneity in bid data can arise from differences across tracts and/or differences among bidders. Such tracts and bidders’ differ ences can be observed or unobserved by the analyst. Regarding bidders, their unobserved differences/heterogeneity are captured by their unobserved private val ues or information r? * , while the observed ones lead to an asymmetric game. For, ex ante differences known to all bidders such as their size or location define the asymmetries in the auction game. As indicated in Section 2.1, the main contribu tion of this chapter is to deal with observed heterogeneity among bidders despite generally intractable equilibrium strategies. Regarding tracts, their differences can be summarized by two vectors of characteristics, namely Z for the observed and W for the unobserved ones. The observed characteristics Z can be introduced by conditioning the latent distributions in the econometric model by Z as in Guerre et al. (2 0 0 0 ) among others. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As far as I know, the issue of unobserved heterogeneity has not been addressed formally in the previous literature. I provide here some assumptions, which allow us to deal with unobserved tract heterogeneity. Without such assumptions, major identification issues arise in the structural approach as it is impossible to disentangle the two types of unobserved heterogeneity coming from u,: and W. Specifically, for the asymmetric APV model of Section 2.2, I make the following assumptions. A ssum ptions: (% ) n\ = ni(Z, W), n0 = uq(Z, W) for som,e functions n0(-) and rci(-). (ii) The joint distribution of private values conditional upon the characteristics (Z, W ) of a tract is equal to the joint distribution of private values conditional upon (Z, ni, n0), i.e. F(-\Z,W) — F(-\Z, nu n0). Assumption (i) requires that the number of bidders from each type is a deter ministic function of the tract characteristics, observed and unobserved. This is a natural assumption satisfied by any entry model, which determines no and n,\ endogenously, whenever bidders decide about their participation prior to knowing their private information as in McAfee and McMillan (1987) and Levin and Smith (1994). Intuitively, Assumption (ii) says that, conditionally upon Z, unobserved tract heterogeneity is fully aggregated into (ni, no), i.e. the latter are sufficient statistics for W. It fully justifies my estimation procedure, which is conducted at (ni,n0) given. On empirical grounds, Assumption (i) is compatible with the general belief that higher value tracts are more likely to attract joint bidders as pointed out in 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Section 2.1. Likewise, Assumption (ii) is consistent with our previous findings that within variability is much important relative to total bid variability so that tract heterogeneity becomes negligeable when conditioning upon the structure (ni, no). Because no Z variable was found to be significant, tract heterogeneity is summarized by the structure (rq, n0) in my empirical analysis. Similarly, bidders’ observed heterogeneity is reduced to the joint or solo nature of the bidder. This approximation is justified as solo bids pertain mostly to large firms, while joint bids pertain to consortia composed mainly by a large and some fringe firms, as noted by Hendricks and Porter (1992). An important empirical issue is to assess whether differences in observed bids arise from structural asymmetry due to the solo or joint nature of the bidder or from unobserved tract heterogeneity. This issue can be answered under the preced ing assumptions. My estimation method delivers estimates of the joint densities /(-, -|ni, no) for (rq, no) equal to (2,0), (0,2) and (1,1). A comparison of these densities, however, does not provide a clear answer to this issue as the structure (n i, n0) is likely to depend on the unobserved tract characteristics W. On the other hand, under my assumptions, a comparison of the marginal densities for the joint and solo bidders in the ( 1 , 1 ) case provides a direct test of asymmetry among these two types of bidders. Moreover, the comparison of the marginal densities for the joint bidder in the (1 , 1 ) and (2 , 0 ) cases (for the solo bidder in the (1 , 1 ) and (0, 2) cases) can indicate the presence of unobserved tract heterogeneity. For, if these two marginal densities are different, then tracts attracting bidders of each type differ from tracts attracting two joint (solo) bidders because of unobserved tract heterogeneity. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e A s y m m e t r ic P u r e C o m m o n V a l u e M o d e l The pure CV model has been largely considered for explaining bidding behavior in OCS auctions. See Porter (1995) for a survey. Recently, Laffont, (1997) has raised serious concerns about the adequacy of the empirical evidence from reduced-form analyses supporting the CV paradigm in such auctions. This led to the recent debate of private versus common value in OCS auctions, while initiating a search for formal tests of either paradigm. Using OCS auctions prior to 1970, Hendricks et al. (2001) consider a symmetric pure CV model. Their tests rely on ex post returns of tracts and a different boundary condition of the bid distribution under the CV and PV paradigms with a binding reserve price. They conclude that bidding behavior appears to be more consistent with a pure symmetric CV model than a PV one, though recognizing that both components are probably present. Haile et al. (2000) also consider a symmetric game and develop an interesting nonparametric test for PV versus CV models. Their test requires that the underlying structure be independent of the number n of bidders, which precludes unobserved tract heterogeneity when n is endogenous. Moreover, their test requires that n varies across auctions, which is then exploited by estimating nonparametrically some functions for each number of bidders. As a result, this test is much demanding in auction data. In contrast, I study here asymmetric situations where unobserved heterogeneity may lead to latent distributions depending on the number of bidders because of unobserved tract heterogeneity (see Assumption (ii)). Moreover, my empirical analysis relies on auctions with two bidders. For, considering auctions with more than 2 bidders 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. leads to a greater number of (rq, no) situations to be entertained with fewer auctions in each case. 15 Thus, both previous tests do not apply to our situation. In this section, I study the asymmetric pure CV model with two players and derive some restrictions imposed by such a model. The latter are used to propose some simple tests of the pure CV model in the (1,1) case. Following Wilson (1969), the asymmetric pure CV model is defined as follows. Consider 2 bidders bidding for a tract of unknown but common value V with density f v(-) and support [u, v]. Each bidder i, i = 0,1 receives a signal or estimate e q for V. Conditional upon V the signals are independent but not necessarily identically distributed with a joint density /(•, • |V) = / 0 (-|V) x ,/j(-|V) on the support \a,a]2. Because / 0(-|-) and / i ( - 1 * ) need not be equal, the players have disparate information and hence, are asymmetric ex ante. Each bidder has a utility function (7(<x,, V) = V , which defines the pure CV model. The maximization for bidder 1 (say) is maxE[(V - &i)l(cr0 < Sq1(&i))|cr1 ] b l fSoHh) = max / V(cri,aro)fao\ffl((To\ai)dao - &ii? ir 0|0- 1(«o’1(*1)lcr1), 15With three bidders, I have 18 auctions for the (3,0) case, 19 auctions for the (0,3) case, and 48 and 60 auctions for the (2,1) and (1,2) cases, respectively. Moreover, the (2,1) and (1,2) cases involve the nonparamctrie estimation of trivariate distributions (see (2.4) and (2.5)). In contrast, the two bidders cases involve at most, bivariate distributions. See (2.18) and (2.19). 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where s0(-) is the strictly increasing and differentiable equilibrium strategy of bid der 0, F(tq \oi( 'I') is the conditional distribution of er0 given < j\ with density f ao\ai (-|-), and a0) = E[F|(7i, < t0]. The first-order condition gives for any o\ G [&,a] - i r<Toki(t S o1(^i)lcri) + sy1(&i)) - h\ - iL ^ ! ^ ” 0. (2.14) W o (b i )) Similarly, the first-order condition for bidder 0 gives for any a0 G [cr, < x ] — i4nicr0(«i 1(^o)|cro) + (sr 3(& o), M - j - = 0. (2.15) W i (4)) As in Section 2.2, we have Gbo\^ (60|&i) = T M M o 1 (b o )^ 1 (h)) and 9bo\bi (& o|& i) = faoki («0 1 (4 ) |^ r1 (& 1)) / 4 ( so 1 (f e o)) ■ So (14) and (15) become V'(C T l..S o 1(bi)) = 6, + A * A E l = 6 ( 6 . , G ) , (2.16) V(s11(ho),cro) = b0 - I 7 T irT = 4 ( 4 , G), (2.17) 9b1\b 0{bo\b0) for any < j? ; G [a, a], where = ^(cq) G [6, & ], * = 0,1. Equations (2.16) and (2.17) are directly comparable to (2.4) and (2.5) for the asymmetric APV model with two players. In particular, the right-hand sides of (2.4) and (2.16) (or (2.5) and (2.17)) are identical. On the other hand, the left-hand sides have a different, economic interpretation. In the pure CV model, it is V(a%, SqX (6i)) = E|V|<7i, So 1(& i))]> while it is bidder’s 1 private value Vi in the APV model. These equations are the key to the following proposition. A bivariate distribution Gb0,b i (■>') with support [ & , b]2 is said to be quasisym,metric if Pr[&i < & o|6i < M o < b ] = Pr[6i > bo\bi < M o < b ] for any b G [6, b]. In other words, a distribution 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In other words, a distribution is quasisymmetric if the likelihood for bid 1 being smaller than bid 0 is equal to one half given that both bids are less than any arbitrary value b. In particular, a distribution is quasisymmetric if it is exchangeable or symmetric. The converse is not true. P ro p o sitio n 2.2: Suppose that observed bids are the equilibrium bids of an asymmetric pure CV model with two players. Then their joint distribution G(-) must be quasisymmetric. The latter condition is equivalent to £i(b, G) = £o(b, G) for any b G [b, b]. Moreover, the asymmetric pure CV model is observationally equivalent to some asymmetric APV model with a quasisymmetric private value distribution and identical equilibrium strategies for both players. To my knowledge, this result has not appeared in the literature. In particular, by letting b = b, the quasisymmetry of the joint bid distribution implies that Pr(fci < bo) = Pr(&0 < bi) = 1/2, i.e. at equilibrium both players have equal probability of winning despite their disparate information. It is worthnoting that a related result holds in the extreme asymmetric pure CV model, where one bidder is perfectly informed about the value V of the object, while the other bidder is completely uninformed. As shown by Wilson (1967) in this case, the equilibrium is such that the uninformed bidder adopts a mixed strategy that is identical to the bid distribution of the informed bidder. Hence, both players have the same probability of winning. See Hendricks and Porter (1988) and Hendricks et al. (1994) for empirical applications and extensions. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Such a property provides a very simple test for the adequacy of the pure CV model in the two players case irrespective of symmetry. For, if players were symmetric ex ante, then their probabilities of winning will also be equal. Hence, the probability of winning for one bidder should be one half whether or not the game is symmetric in the pure CV paradigm with two bidders. As noted in Section 2.3.1, among the 112 auctions with one solo and one joint bidders, 63 were won by the joint bidder, suggesting that the joint bidder has a higher probability of winning. To investigate further this issue, it is interesting to split, the auctions in half and to test quasisymmetry on each half. Specifically, I consider the 56 auctions for which both bids are smaller than $460 per acre and the 56 auctions for which at least one bid is larger than $460. The joint bidder wins 28 times in the first subset and 35 times in the second subset. Quasisymmetry requires that the joint bidder’s probability of winning is one half in each subset.1 6 In the second subset, such an equality is rejected with a t-test statistic equal to 1.87. Thus, the pure CV model does not appear to be appropriate for these auctions. This does not, however, exclude the existence of a common value in a general model, where the utility of each player is of the form [/(cq, V), as in Wilson (1977). Such a model is known to be unidentified (see Laffont and Vuong, 1996). In my empirical analysis, I choose to consider the asymmetric APV model though we believe that most auction data contain both private and common values. Consideration of private values can be justified by important idiosyncratic differences among firms such as productive inefficiencies, capital constraints and opportunity costs. 16It is easy to see that P r[h < & o|b\ < h, ho < h ] = 1/2 for any b is equivalent to Pr[/>i < l> o \b i > b or b 0 > f t ] = 1/2 for any b . Testing simultaneously that this equality holds for every b is left for future research. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It should be noted that affiliation among private values may arise from an unknown common component (see Li et al, 2000). Regarding the last part of Proposition 2.2, note that the observationally equiva lent APV model is a particular asymmetric APV model since its latent distribution must be quasisymmetric and the bidders’ equilibrium strategies must be equal. My result extends the observationally equivalence of the symmetric pure CV model and the symmetric APV model established in Laffont and Vuong (1996) to the asymmetric case. It also extends the observationally equivalence of the extreme asymmetric pure CV model and the symmetric IPV model established in Laffont and Vuong (1996). For, Proposition 2.2 states that any asymmetric pure CV model is observationally equivalent to some quasisymmetric APV model. When considering the extreme asymmetric pure CV case, the two players’ bids are inde pendent. Such independence combined with quasisymmetry leads to a symmetric bid distribution and hence, a symmetric IPV model. 2.3.3 Structural Estim ation Results I have 227 auctions of which 60 are with two joint bids, 55 with two solo bids and 112 with one joint and one solo bids corresponding to (rq ,no) = (2,0), (rq,n-o) = (0,2) and (rq, n0) — (1,1), respectively. Following Section 2.2, esti mation is performed separately for each pair. Because n — 2, the fundamental equations (2.4)-(2.5) simplify greatly. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C a s e s (2,0) a n d (0,2) As only one type of bidder is involved, these cases correspond to the symmetric APV model studied by Li et al. (2002). I estimate both cases separately. For each case, I have pairs (bU) b2e) from the same type, dropping the 1/0 index for type. Thus, the ratio in (2.4) and (2.5) reduces to G '& 2|6 1(-|-)M 2|&i('I')- Because of the log transformation (see Section 2.2.4), I estimate the joint distribution Gd,d{•, •) from the pairs (du, d^) as well as its corresponding density Q d,d{-, •)• The former can be estimated by averaging the product of a counting process and a kernel, while the latter can be estimated by averaging the product of two kernels as shown in Li et al. (2002). For the (2,0) case, the bandwidths he and hg are equal to 1.41 and 1.65, while they are equal to 1.54 and 1.81 for the (0,2) case. Then, using an equation similar to (2.12), I obtain the pseudo private values for each case. Because of boundary effects, 3 auctions are trimmed out in the (2,0) case, while 6 auctions are trimmed out in the (0,2) case. Figure 2.1 displays the pairs (&«,?)«), i = 1, 2 I = 1 ,... , 57, which trace out the (inverse) equilibrium strategy £n(-), as well as the pairs (bu, vn), i = 1, 2, i — 1 ,... , 49 tracing out £oo(’)-1 ? The important feature is that both estimated functions are strictly increasing with £n(-) to the right of £oo(-)- Hence, for each case, the data do not reject the symmetric APV model in view of Proposition 2.1 of Li et al. (2002). It is interesting to compare the estimated £(■) functions. In particular, on the common interval where both functions can be estimated in Figure 2.1, a same private value leads to a higher bid when a joint bidder faces a joint bidder than 17Tho first index for the £(•) function refers to bidder’s type while the second index refers to his opponent’s type. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12000 ~ > • t 4000 ” J ' 0 0 500 1000 1500 2000 2500 3000 bids Figure 2.1: Inverse Bidding Strategies when a solo bidder faces a solo bidder. This is in agreement with Table 2.1. Turning to private values, I estimate the univariate marginal private value density for joint bidders using where L? = 57 and the bandwidth hf is equal to 2.978 x 1.06di(2Lj’)~1 //5 = 3, 702.08. Similarly, we estimate the univariate marginal private value density for solo bidders /o°’ 2)(-) with LT = 49 and hf equal to 1,476.48. Figure 2.2 displays the marginal density of joint versus joint in dashed line and the marginal density of solo versus solo in plain line. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.0004" 0.0003 " \ c \ u \ 0.0002" .-'" - \ 0.0001 - V \ 0.0000 " I ----------------------- i ------------------ — i— ........ L 0 4000 8000 12000 Private values Figure 2.2: Marginal Densities of Private Values The comparison of both marginal private value densities in Figure 2.2 shows some differences between joint and solo bidders. Specifically, the mean, mode and variance for solo bidders are smaller than the respective quanti ties for joint bidders. In fact, the estimated cumulative distribution func tion for joint bidders first-order stochastically dominates that for solo bidders.1 8 Hence, joint bidders are likely to draw larger private values than solo bidders with a relatively more important variability for the former. As pointed out in Sec tion 3.2, these differences can be explained by unobserved tract heterogeneity and differences between joint and solo bidders. This issue is further investigated below. 1 8 A one-sided Kolmogorov Smirnov test clearly rejects the equality in favor of stochastic domi nance for joint bidders. For the (2,0) case, the mean of the 114 trimmed private values is $2,195.14 per acre with a standard deviation equal to $3,024.10 and a range of [$33.94; $13,605.86]. For the (0,2) case, these numbers are $1,027.90, $1,170.15 and [$26.89; $5,031.39], respectively from the 98 trimmed private values. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As Figure 2.2 does not provide information on the affiliation between private values within the same auction whether they are both joint or solo, it is useful to test for their independence. I use the nonparametric test, proposed by Blum et al. (1961) (BKR hereafter), which is consistent and distribution free. For two variables X and Y, the test statistic is equal to (l/2)7r4J5, with B — N~4 — N 2(£)Ns(£))2, N the number of observations and Ni(£), A 2 CO, Ns(£), N ^i) the numbers of points lying respectively in the regions {(x,y)\x > Xi, y > Ye}. To impose symmetry among bidders of the same type, I duplicate the observations so that N = 2 x L. I find a test statistic equal to 6.57 using observed bids and equal to 4.69 using trimmed private values for joint bidders. For solo bidders, I obtained a test statistic equal to 9.52 using observed bids and equal to 6.92 using trimmed private values. The null hypothesis of independence is clearly rejected in all cases. C a s e ( n i ,n0) = ( 1, 1) The potentially asymmetric case is estimated using the 112 auctions with both types. Because there is only one bidder of each type, (2.4) and (2.5) simplify as B\ and Bq are void. In particular, their denominators reduce to the conditional densities gbo\h {hx\hi) and gbl\bo(b0\b0). Hence, (2.4) and (2.5) reduce to v i — £to(fri) — + G bo\bl(bi\hi) / g bo\bl(b\\h\), A ) — 6 i(M = h Q + G6l|6o(6o|bo)M1|6 0(& o|feo), (2.18) (2.19) 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. respectively. Consequently, (2.7)-(2.8) and the first step in our nonparametric estimation procedure simplify accordingly. For the first step, the bandwidths hoi-, hgi, h-G o and hgo are equal to 1.73, 2.03, 1.54 and 1.80, respectively. Figure 2.3 displays the estimated inverse bidding strategies £10(-) and |o i(-) traced out by the pairs (hu,vy) and (bw,v 0 ^), £ = 1 ,... ,91, for the joint and solo bidders, as 21 auctions are trimmed out. In particular, both functions are increasing indicating that the asymmetric APV model is not rejected by the data. Moreover, the inverse bidding strategy for solo bidders is to the right, of that for joint bidders. Given the same valuation, a solo bidder bids more aggressively than a joint bidder. Moreover, from Proposition 2.2, the comparison of £i0(-) and £oi(') indicates whether the joint bid distribution is quasisymmetric. Figure 2.3 suggests that this is not the case, especially for bids larger than $500 when the two £(•) curves start to diverge. This corroborates our findings of Section 2.3.2. Overall, this again agrees with the rejection of the pure CV model. The difference between the inverse equilibrium strategies constitutes a partial pic ture as bids also depend on bidders’ valuations and their distributions. The joint distribution is estimated using (2.11) and displays some asymmetry and correla tion.1 9 Figure 2.4 displays the marginal density of private values for each type, namely in dashed line and /q1 ,1 ^-) in plain line with bandwidths of the form 2.978 x 1.06(7^(LT)_ 1 /7 5 = 2,277.61 and 2.978 x 1.06ar^(LT)~1/s = 1,708.63 for joint and solo bidders, respectively 19The BKR test strongly rejects independence of both private values and bids with a test statistic equal to 6.45 for private values and 34.23 for bids. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S i cn 400 0 800 1200 b id s Figure 2.3: Inverse Bidding Strategies It appears that the density of solo private values has a slightly smaller mean, mode and variance than that of joint private values.2 0 Hence, solo bidders seem less likely to draw large private values than joint bidders are. This suggests some asymmetry though weak between joint and solo bidders as the empirical cumulative distribution functions slightly differ with a single crossing. 2 1 Such an asymmetry leads the solo bidders to shade less their private values than joint bidders, as found in Figure 2.3, so as to increase their probability of winning the auctions. See also Maskin and Riley (2000a) and Pesendorfer 20The moan of the trimmed pseudo private values for joint bidders is $1,208.73 per acre with a standard deviation equal to $1,622.34, while the mean of the trimmed pseudo private values for solo bidders is $1,161.62 per acre with a standard deviation equal to $1,334.21. 21A Kolmogorov Smirnov test docs not reject the equality of the c.d.f.s on either private! values or bids. Note, however, that the Kolmogorov Smirnov test is based on the independence! of the two samples. This is not the case as joint and solo private values (or bids) are affiliated, which decreases the power of the test. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.0004 0.0003 - -S 0.0002 0 .0 0 0 1 " 0.0000 0 2000 4000 6000 Private value Figure 2.4: Marginal Densities of Private Values (2000). However, the shading effect does not counterbalance fully the asymmetry in terms of valuation distributions, as indicated by the bid averages for joint vs solo and solo vs joint in Table 2.1 and the empirical probability of winning. As is well known, the aggressiveness of the weak bidder relative to the strong bidder may introduce some inefficiency in the auction in the sense that the winner of the auction has the lowest valuation. It turns out that this does not happen in our data set, which can be explained by the relatively weak asymmetry and the important variability of private values within each auction. It is interesting to compare these results to the first two cases where bidders are of the same type. Figure 2.5 displays the inverse bidding strategies for a joint bidder when facing a joint bidder (£n(-)) and when facing a solo bidder (6io(‘))> the former being to the right of the latter. Given a same tract valuation, a joint bidder will bid more aggressively when facing a joint bidder than when facing a solo bidder. For, the 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4000 ~ 0 500 1000 1500 2000 2500 3000 bids Figure 2.5: Inverse Bidding Strategies of Joint Bidders joint bidder faces less “competition” when facing a solo bidder who is more likely to draw a lower private value. Figure 2.6 displays the inverse bidding strategies for a solo bidder when facing a solo bidder (Coo(’)) and when facing a joint bidder (£oi(‘))> the former being to the left of the latter. Thus,a solo bidder will bid slightly more aggressively when facing a joint bidder than when facing a solo bidder to compensate for his lower private value. These results confirms the descriptive statistics of Table 2.1. Both figures indicate that bidders have integrated the type of their opponents in their bidding strategies. Lastly, as indicated in Section 2.3.2, the comparisons of / j 2,0^(-) and as well as of /o°’ 2)(-) and / q 1 ,1 ^-) provide some information about unobserved tract heterogeneity. A Kolmogorov Smirnov test gives a test statistic equal to 0.1917 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 400 800 1200 bids Figure 2.6: Inverse Bidding Strategies of Solo Bidders and 0.0965 for the former and latter comparisons, respectively. This represents a clear rejection of /{2 ,0 ^(-) = / P ’ 1 ^-). Under the assumptions of Section 2.3, this means that there is some unobserved tract heterogeneity as tracts attracting two joint bidders differ significantly from tracts attracting one joint bidder and one solo bidder. This is in agreement with the idea that higher value tracts are more likely to attract joint bidders. It also indicates that differences in bids observed in Table 2.1 are due to both asymmetry among solo and joint bidders and unobserved tract heterogeneity though the latter appear to be more important. In f o r m a t io n a l R e n t s An attractive feature of the structural approach is to estimate the underlying private value distribution as well as bidders’ private values. In particular, we can assess the informational rents left to the winners via {vw — bw)/vw, where vw is the estimate of the winner’s private value and bw is his bid. For those auctions that 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are kept after trimming, Table 2.2 gives the summary statistics on the winners’ informational rents in percentage decomposed as in Table 2.1. On average, the winner’ s informational rent is about 65%, indicating that the federal government is capturing 35% of the winner’s willingness-to-pay at the auction. Introducing a common value component in the model may lower the informational rents left to winning firms. Table 2.2 shows that the winner’s informational rent is the same on average irrespective of whether the winner is a solo or joint bidder and irrespective of the type of his opponent. This suggests that bidding bevahior and bidding structure correspond to an equilibrium as a large firm can choose to be either a solo bidder or constitute a consortium with fringe firms. Table 2.2: Summary Statistics on Winners’ Informational Rents (%) Variable # Obs Mean STD Minimum Maximum All Winners 197 65.07 9.39 31.82 81.51 Joint Winners 105 65.50 9.63 42.55 81.51 Solo Winners 92 64.59 9.14 31.82 79.00 Joint Winners vs Joint 57 65.86 9.56 42.55 78.52 Joint Winners vs Solo 48 65.08 9.80 47.83 81.51 Solo Winners vs Joint 43 63.85 9.87 31.82 79.00 Solo Winners vs Solo 49 65.24 8.50 43.17 78.55 Such a result indicates that our theoretical model is not rejected by the data. For, finding significantly different informational rents across structures would question bidders’ use of the Bayesian Nash equilibrium and the adequacy of the model and its assumptions. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Though informational rents in percentage are about the same, they differ in abso lute terms. For instance, the average informational rent vw — bw for a winning consortium is about $1,954.42 per acre compared to $1,151.40 per acre for a solo winner, i.e. 69.7% higher. This arises because the average valuation for a joint winner is $2,639.47 per acre, while it is only $1,608.34 per acre for a solo winner. Following our results, such a difference can be mainly explained by unobserved tract heterogeneity. The remainder is due to bidders’ asymmetry as pooling of capital contributes to a higher valuation for a consortium. Because asymmetry is relatively weak between joint and solo bidders, a ban on joint bidding would not have much impact on the federal government’s revenue provided participation remains the same. Previous empirical studies indicate that joint bidding occur mostly between a large firm and a number of fringe firms (see Hendricks and Porter, 1992), i.e. a fringe firm seldom bids solo. Some studies find that allowing for joint bidding has increased the level of competition through a larger partici pation of firms to the auctions. This combined with my results suggests that the federal government has increased its revenue by allowing for joint, bidding. 2.4 C onclusion The main contributions of this chapter are methodological as it pushes further the frontier of the new structural analysis of auction data (see Perrigne and Vuong, 1999, for a recent survey). While such an approach has been mostly confined to symmetric models, this chapter shows that the structural approach can be extended to the asymmetric case known to lead to intractable equilibrium strategies. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Moreover, I provide formal assumptions, which can deal with unobserved het erogeneity across auctioned objects. I also develop a simple test of the pure CV paradigm. Specifically, I consider the asymmetric APV model, which encompasses the IPV model. I establish its nonparametric identification from observed bids and I propose a two-step nonparametric procedure for estimating the underlying bidders’ private value distributions. A distinctive feature of our procedure is its computational convenience as it avoids the numerical integration of the system of differential equations characterizing the bidders’ equilibrium strategies. My method is fully nonparametric. This has the main advantage of leaving unspecified the underlying distribution and in particular the affiliation among private values, whose parameterization can be difficult. On the other hand, it requires a large number of data. Thus, parametric estimation methods need to be developed if more than two types of bidders are entertained and if some observed heterogeneity of the auctioned objects needs to be introduced. In this case, there would be as m any differential equations as types and one would need to know bidders’ identities to follow them across auctions. Lastly, my chapter illustrates the proposed methodology by analyzing joint bidding in wildcat OCS auctions after 1972. In particular, I find that the pure CV model does not seem to be supported by the data. I also find that asymmetry between joint and solo bidders is weak, while unobserved tract heterogeneity is important and well captured by the bidding solo/joint, structure. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. My findings indicate that the government could have benefited from allowing for joint bidding through mainly increased competition. My empirical analysis takes joint bidding as given. An important economic issue is the rationale of cartel formation prior to the auction. This constitutes a domain that needs further theoretical development. For a recent contribution, see Hen dricks et a ,I. (2000). 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Risk Aversion w ithin the Independent Private Value Paradigm: An application to Timber Auctions 3.1 Introduction Since the seminal unpublished work by Kenneth Arrow and its formalization by P ratt (1964), risk aversion has become a fundamental concept in economics when agents face situations under uncertainty. This is the case in auctions where bidders face various types of uncertainties relating to the value of the auctioned object, the strategies used by the other bidders and the characteristics of their opponents. In particular, a pervasive economic argument for justifying risk aversion of a bidder is that the value of the auctioned object is high relative to the assets of the bidder. Another argument is that a bidder does not have many alternatives for buying the object other than in the aiiction. On the other hand, many important results in the theory of auctions crucially depend on the assumption of risk neutrality. For instance, within the independent private value (IPV) paradigm, the revenue equivalence theorem established 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by Vickrey (1961) states that English, descending, first-price sealed-bid and second-price auctions lead to the same expected revenue for the seller provided bidders are risk neutral. Such an important result no longer holds when bid ders are risk averse. See Harris and Raviv (1980) and Riley and Samuelson (1981).1 Despite the importance of risk aversion in auction modeling and its likelihood in auction data, very few empirical studies have attempted to assess the extent of bidders’ risk aversion on field data. See Baldwin (1995) and Athey and Levin (2001) using US Forest Service timber auctions.2 A reason is that economic theory provides few implications of risk aversion that are difficult to test on bids data. An alternative approach is to consider that bids are precisely the outcomes of the Bayesian Nash equilibrium of the underlying auction game. This is known as the structural approach, which has been developed by Paarsch (1992) and Laffont, Ossard and Vuong (1995). For a recent survey, see Perrigne and Vuong (1999). This chapter adopts such an approach and focuses on identification and estimation under nonparametric assumptions in the spirit of Laffont and Vuong (1996) and Guerre, Perrigne and Vuong (2000). 1 Likewise, the optimal auction mechanism is quite involved under bidders’ risk aversion as it, requires complex transfers among bidders. See Maskin and Riley (1984) and Matthews (1987). For a recent survey of auction theory, see Klemperer (1999). 2 On the other and, the experimental literature has paid much attention to bidders’ risk aver sion as it can frequently explain observed overbidding (above the Bayesian Nash equilibrium) in experimental data. See Cox, Smith and Walker (1988) and Goere, Holt and Palfrey (2002) among others. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Throughout, I consider first-price sealed-bid auctions with risk averse bidders within the IPV paradigm.3 A first part presents briefly the model and discusses the existence, uniqueness and smoothness of the equilibrium strategy. A second part of this chapter is devoted to the identification of the auction model, i.e. whether its structural elements can be uniquely recovered from observed bids. In particular, general nonidentification results are given. The structural elements to be identified from observed bids are the bidders’ utility function and the bidders’ private values distribution. Unlike Donald and Paarsch (1996) who consider only a constant relative risk aversion (CRRA), we consider a general von Neuman Morgenstern (vNM) utility function exhibiting possibly risk aversion. First, I show that this general model is nonidentified from observed bids even when the utility function is restricted to belong to well known families of risk aversion. In particular, restricting bidders to have a constant relative risk aversion is not sufficient to achieve identification. Second, I show that any bid distribution can be rationalized by a CRRA model with zero wealth, a constant absolute risk aversion (CARA) model with zero wealth, and a fortiori a model with general risk aversion. Such a striking result implies that a CRRA model and a CARA model cannot be discriminated against each other. It also implies that the game theoretical model does not impose testable restrictions on bids. Furthermore, one 3Maskin and Riley (1984) show th at first-price auctions generate a larger revenue than ascend ing auctions with risk averse bidders. In particular, bidders’ risk aversion can provide a rationale for the use of first-price auctions. In many situations, empirical researchers have observed the exclusive use of a particular mechanism for a large variety of goods. For instance, ascending auctions are used for art and memorabilia, while first-price auctions are used for procurements and natural resources except for timber, which is sold through both mechanisms. Hence, the development of methods to assess the extent of risk aversion is especially im portant in first-price auctions . 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. can consider either CRRA or CARA utility functions with zero wealth without loss of power for explaining observed bids, despite either model not being identified. In view of the preceding results, a fourth part of this chapter seeks palatable restrictions that can be used to achieve identification of the auction model with risk averse bidders. Specifically, I exploit heterogeneity across auctioned objects combined with the assumption that private values distributions conditional upon the characteristics of the auctioned objects satisfy a parametric quantile restriction. Unlike previous work such as Guerre, Perrigne and Vuong (2000), Li, Perrigne and Vuong (2000, 2002), Campo, Perrigne and Vuong (2002), such an additional restriction was not necessary to identify the auction models under consideration. There are of course other possible restrictions such as requiring that some quantile be known, but the latter assumption is unattractive as the postulated value of some quantile directly affects the estimated degree of risk aversion. A second restriction relates to the parameterization of the bidders’ vNM utility function. Under these conditions, I show that the structural parameters and the conditional private values distributions of the model with risk adverse bidders are semiparametrically identified as no parametric assumption on the latent private values distributions is made. As a m atter of fact, I show that these two identifying conditions are jointly necessary as dropping either one of them will loose identification. In this sense, our semiparametric modeling is natural, while constituting a new direction for the literature on structural analysis of auction data. A fifth part of the chapter provides an upper boxmd for the convergence rate which can be attained by estimators of the parameter(s) of the utility function. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Given the semiparametric nature of our model, it is important to study the best (optimal) rate that an estimator of the risk aversion parameters can achieve. To do so, we rely on the minimax theory as developed by Ibragimov and Has’minskii (1981). As is well-known, estimation of the upper boundary of a distribution can be achieved at a faster rate than any other quantile. For this reason, I focus 011 a parametric restriction on the upper boundary of the private value distribution to achieve a faster convergence rate and a greater precision for the estimator. Specifically, when auctioned objects’ heterogeneity is characterized by d continuous variables and the underlying density is J?-times continuously differentiable, Guerre, Campo, Perrigne, Vuong (2002) show that the optimal rate for estimating the risk aversion parameter(s) is I\r(H+1)/(2R+3). Note that such a rate is independent of the dimension d of heterogeneity and is slower than \/N, which is unattainable given the assumed smoothness R. A sixth part of the chapter addresses the estimation of the structural elements, i.e. the parameter(s) of the vNM utility function and the underlying conditional density of bidders’ private values. I then develop a multistep semiparametric estimation procedure. A first step consists in estimating the conditional density of bids at its upper boundary. This involves nonparametric estimation of the upper boundary using Korostelev and Tsybakov (1993) theory of image reconstruction as well as the corresponding conditional densities at these upper bounds in the spirit of Cheng (1997) local polynomial estimators. In particular, the latter provides an automatic boundary correction. See Fan and Gijbels (1996).4 A second step 4Sce Heckman, Ichiinura, Smith and Todd (1998) and Hendricks, Pink.se and Porter (2000) for use of such a technique in other contexts. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. focuses on the estimation of the utility function parameter(s) relying upon the fundamental equation of the auction model. This leads to (possibly weighted) nonlinear least squares (NLLS) using the nonparametric estimates obtained in the first step. A third step allows me to recover the bidders’ private values and their underlying conditional density following Guerre, Perrigne and Vuong (2000) procedure given the estimation of the utility function parameter(s). Guerre, Campo, Perrigne and Vuong (2002) establish the asymptotic properties for this estimator. In particular, we show that our estimator of the utility function parameter (s) attains the optimal rate N(r+1W r+3\ This contrasts with most %/iV-consistent semiparametric estimators developed in the econometric literature, see Powell (1994) for a recent survey.5 As a m atter of fact, standard results on \f~N-consistent semiparametric estimators as given in Newey and McFadden (1994) do not apply. Another notable feature of my estimation problem is that it involves a nonlinear regression model with a bias and a variance that decrease and increase with the smoothing parameter, respectively. This diverging variance of the error term in the equation defining the utility function parameter(s) is the main reason why my semiparametric estimator does not achieve the standard parametric rate. Lastly in addition to providing an estimator converging at the optimal rate as well as not requiring a parametric specification of the bidders’ private values distributions, my method is computationally simple as it circumvents both the 5Notablo exceptions of semiparametric estim ators converging at a slower rate than y/N arc Manski (1985), Horowitz (1992), Kyriazidou (1997) and Honore and Kyriazidou (2000). 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. numerical determination and inversion of the equilibrium bidding strategy. This is especially convenient when there is no closed form solution to the differential equation defining the equilibrium strategy. This is the case when risk aversion is not of the simple CRRA form. I then illustrate our procedure on the US Forest Service timber auctions. In particular, a test of bidders’ risk neutrality is performed and bidders are found to be fairly risk averse. The chapter is organized as follows. Section 3.2 briefly presents the model and dis cusses the properties of the Bayesian Nash equilibrium strategy of the correspond ing auction game. Section 3.3 investigates the identification of the first-price auc tion model with risk averse bidders and provides general nonidentification results. Understanding of such results leads to additional restrictions used to achieve semi parametric identification of the model, which is the purpose of section 3.4. Section 3.5 provides an upper bound for the optimal (best) convergence rate, which can be attained by a semiparametric estimator of the utility function parameter(s). Sec tion 3.6 presents my semiparametric estimation procedure with its various steps and statistical properties. Section 3.7 is devoted to an illustration of our method to timber auction data. Section 3.8 concludes. Three appendices collect the proofs of our theoretical results. 3.2 M odel and E quilibrium S trategy This section presents the first-price sealed-bid auction model with risk averse bid ders within the IPV paradigm. It also establishes the existence, uniqueness and smoothness of the equilibrium bidding strategy. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e M o d e l A single and indivisible object is sold through a first-price sealed-bid auction. All sealed bids are collected simultaneously. The object is sold to the highest bidder who pays his bid to the seller. Within the IPV paradigm, each bidder is assumed to know his own private value for the auctioned object but not other bidders’ private values. The bidders’ private values are drawn independently from a common distribution F(-), which is absolutely continuous with density /(•) on a support [v,v] C M+. The distribution F(-) and the number of potential bidders I > 2 are assumed to be common knowledge. Moreover, each bidder is potentially risk averse. Let Uvn m {') be a von Neuman Morgensten utility function common to all bidders with U 'vNM{-) > 0. Because of potential risk aversion, the utility function is assumed to be weakly concave, i.e. U"NM(-) < 0. All bidders have the same initial wealth w > 0. This gives a utility function of the form Uvn m (w+'), where the argu ment refers to the monetary gain from the auction. All bidders are thus identical ex ante and the game is said to be symmetric.6 Bidder i then maximizes his expected utility Elf, = UvNM{w + - &j)Pr(&j > fy, j ^ i ) + UvNM{w)[ 1 - Pr(i>j > bj,j ^ *)] with respect to his bid where bj is the j th player’s bid. 6Relaxing the assumption of bidders’ common wealth, i.e. letting u> i be bidder *’s wealth, will lead to an asymmetric, game if the w,s are common knowledge and to a multisignals game if the t«jS are private information. Both cases are beyond the scope of this chapter. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bidder i problem is equivalent to maximizing '\UVN M { w + v i ~ b j ) — U VN M { w )]Pr(h > b j , j 7^ i). Let U(-) = Uvnm(w + -) — Uvnm(w)■ Note that U(-) is strictly increasing, weakly concave and satisfies U(0) = 0. Hereafter, we consider the maximization of EIL = U(vi - bj)Pr(bi > b j j ^ i), (3.1) where U(•) satisfies the preceding properties. This corresponds to the most studied case in the auction literature where the quality of the auctioned item is known and has equivalent monetary value. See Case 1 in Maskin and Riley (1984).7 In addition, because the scale is irrelevant, we impose the normalization U(1) = 1. The risk neutral case is obtained when U(-) is the identity function. It is useful to recall some basic properties of utility functions with risk aver sion. Let a and j 3 be arbitrary constants with a > 0. A common measure of absolute risk aversion is the ratio -U"NM{-)fU 'vNM{-), which can be constant or nonincreasing. This gives the set y CARA of constant absolute risk aversion utility functions and the set UDARA of decreasing absolute risk aversion utility functions. A well-known form for the former is given by Uvnm(x) = «(1 — exp(— ax)) + /?, with an absolute measure of risk aversion a > 0. Another common measure is the relative risk aversion, which is defined as — xU"NM (x)/U 'vNM (x). This quantity can be either constant or nonincreasing, which gives the set IJCRRA 0f constant, relative risk aversion utility functions and the set U DRRA of decreasing relative risk aversion utility functions. A well-known characterization for the former is given 7Maskin and Riley (1984) consider a more general model whore the utility of winning is of the form « (— Vi) and the utility of loosing is equal to w(-). Because I use a vNM utility function, u(—bi,Vi) = Uvn m (w + t> i — hi) and w(0) = U„n m (w)- Hence, the utility of loosing with no payment is equal to the utility of winning with payment equal to the bidder’s value. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by Uvn m(x ) — axl~c/(l - c) + (3 for c > 0 and 1 and Uvnm (x) = a log x + (3 for c = 1. Relative risk aversion is then measured by c. Note that if initial wealth w — 0, then 0 < c < 1 because the utility of loosing the auction would be unbounded otherwise. There exists other families of vNM utility functions exhibiting risk aversion, which have been considered in the literature. See Gollier (2001). Below I consider mostly the above four families, though our results also apply to other situations. E x i s t e n c e , U n iq u e n e s s a n d S m o o t h n e s s o f t h e E q u il ib r iu m S t r a t e g y From Maskin and Riley (1984), if a symmetric Bayesian Nash equilibrium strategy s(-,U,F,I) exists, then it is strictly increasing, continuous and differentiable.8 Thus (3.1) becomes EIR = f/(u, — bi)FI~1(s~1(bi)), where s_1(-) denotes the inverse of s(-). Hence, imposing bidder Fs optimal bid hj to be s(vj) gives the following differential equation ( 3 ' 2 ) for all L j 6 [v, v\, where A(-) = U(-)/U'(-). As shown by Maskin and Riley (1984), the boundary condition is U(v — s(v)) = 0, i.e. s(v) — v because [7(0) = 0. Moreover, the second-order conditions are satisfied. Maskin and Riley (1984, Theorem 2) prove the existence and uniqueness of s(-) by noting that the differential equation (3.2) with boundary condition has a unique solution when there is a binding reserve price, i.e. p0 > v. In our case, the reserve 8As noted by Maskin and Riley (1984, Remark 2.3), the only equilibria are symmetric when F(-) lias bounded support, which is assumed below. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. price is not binding. Consequently, there is a well-known singularity of (3.2) at the boundary v, which prevents the use of such an argument for establishing existence and uniqueness of s(-). This is the purpose of the next result, which provides in addition the regularity properties of s(-) used in the following section. We assume that [/(•) and F(-) belong to U r and T r defined as follows, respectively. D efinition 3.1: For R > 1, let Ur be the set of utility functions £/(•) satisfying (i) U : [0, Too) — > [0, Too), U(0) = 0 and U( 1) = 1. (ii) [/(•) is continuous on [0, Too), and admits R T 2 continuous derivatives on (0, Too) with U'(-) > 0 and [/"(•) < 0 on (0, Too). (iii) limX 4o (rr) is finite for 1 < r < i?T l, where A^(-) denotes the rth derivative. Conditions (i) and (ii) have been discussed previously. Note that limx to A(.r) - 0 since U(0) = 0 and U'(-) is nonincreasing. Thus, from (ii) and (iii) it follows that A(-) admits (R T 1) continuous derivatives on [0, Too). These regularity assumptions are weak. For instance, if U(x) = Uvnm(w + v) — Uvnm{w) with Uvnm(') a suitably normalized CRRA utility function, these conditions are satisfied for c > 1 when w > 0 and for 0 < c < 1 when w > 0, in which case R — oo. Similarly, with Uvnm(~) a suitable normalized CARA utility function, these conditions are satisfied for R = oo. D efinition 3.2: For R > 1, let T r be the set of distributions F(-) satisfying (i) F(-) is a c.d.f. with support of the form [u, v), where 0 < v < v < Too, (ii) F(-) admits R T 1 continuous derivatives on [u, u], (iii) /(■) > 0 on [u, u]. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These restrictions are weak with the exception of the finite upper bound 77 in (i). Relaxing (i) is p o s s ib le but would involve more technical aspects in the identification and e s tim a tio n sections. Altogether they imply that /(-) is bounded a w a y fro m z e ro o n [u, 77]. T h eo rem 3.0: Suppose that [U , F] € U r x !Fr for some R > 1, then there exists a u n iq u e (s y m m e tric ) e q u ilib r iu m a n d its equilibrium strategy s(-) satisfies: (i) Vu G (v, 77], s(v) < v , w h ile s(v) = v , (ii) Vu € i [u, 77], s'(u) > 0 w ith s'(v) < 1, (iii) s(-) admits R + l continuous derivatives on ju, 77]. When the reserve price is nonbinding, existence of a pure equilibrium strategy follows from Maskin and Riley (2000) and Athey (2001), while uniqueness has been established by Maskin and Riley (1996) using an argument similar to Lebrun (1999). The main contribution of Theorem 3.0, which is an immediate consequence of the more general Theorem B1 in Appendix B with no exogenous variables (corresponding to (Z, I) constant), is to derive the smoothness of the equilibrium strategy. Moreover, the proof of Theorem B1 differs significantly from previous work (e.g. Lebrun (1999), Lizzeri and Persico (2000)) and uses a continuation argument viewing s(-) as a zero of a regularized functional. An application of a Functional Implicit Function Theorem then allows us to obtain the smoothness of s(-), especially with respect to exogenous variables as needed in the estimation. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Except for some particular choices for U(-) and F(-), the equilibrium strategy does not have an explicit form. In practice, the integral form of the differential equation (3.2) can be useful: it is s(u)F f^1(w) = f (s(.x) + X(x — s(x))} dFI~1(x). Jy_ This form can be also written as s(v) = sn(v) + JJ{A(x— s(x)) — x+s(x)} dFI~1(x), where sjv(-) is the well-known equilibrium strategy in the risk neutral case derived by Riley and Samuelson (1981). Because A (u) > u for u > 0, it follows that the equilibrium bid with risk aversion will be strictly larger than the corresponding one under risk neutrality for v > v as noted by Milgrom and Weber (1982). 3.3 G eneral N onidentification R esu lts This section presents some general nonidentification results. I address the problem of identification of the structure [1 7 , F] from observables. Specifically, I assume that the number I of bidders is observed, as is typically the case in a first-price sealed-bid auction with a non binding reserve price. I also assume that the distribution G{ ) of an equilibrium bid is known. Knowledge of G(-) from observed bids is an estimation problem, which is addressed in Section 3.6. Thus the identification problem reduces to whether the structure [U, F] can be recovered uniquely from the knowledge of (/, G). A related issue is whether any distribution G(-) for an observed bid can be rationalized by a structure [U , F] given /. Such a question is important in itself as it is connected to the validity of the auction model under consideration. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Following Guerre, Perrigne and Vuong (2000) for the risk neutral case, I can express the differential equation (3.2) using the distribution G(-) of an equilibrium bid. For every b 6 [b, b] = [u, s(v)}, we have G(b) — F(s^1(b)) = F(v) with density g(b) = f(v)/s'(v). Thus the differential equation (3.2) can be written equivalently £ t S 1 = < /_ 1 ) ^ ) A(’’ i " ' ’ i)' (3-3) Since [/(•) > 0 and U"{-) < 0, we have A '(-) = 1 - U(-)U"(-)/U,2(-) > 1. Thus A(-) is strictly increasing. Solving (3.3) for u, gives Vi =l>i + v 1 ( j r y f j f y ) = «<6.. C G•!)’ (3.4) where A _ 1 (-) denotes the inverse of A (-). This equation gives each bidder’s private value as a function of its corresponding bid, the bid distribution, the number of bidders and the utility function. Note that £(•) is the inverse of the bidding strategy s(-). The equilibrium bid distribution G(-) satisfies some regularity properties, which are implied by the smoothness of s(-) stated in Theorem 3.0 and the reg ularity assumptions on [U , F]. It is convenient to introduce the following definition. Definition 3.3: For R > 1, let Qr be the set of distributions G(-) satisfying (i) G(-) is a c.d.f. with support of the form [b,b], where 0 < b < b < oo, (ii) G(-) admits R + l continuous derivatives on \b,b\, (iii) g(-) > 0 on [b,b], 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (iv) G(-) admits R + 2 continuous derivatives on (6, b], (v) limbihdr[G(b)/g(h)\/dhr exists and is finite for r = 1 ,... , R + 1. The regularity properties (i)— (iii) are similar to those of Definition 3.2 for F(-). They imply that g(-) is bounded away from zero on [b,b] and limhibG(b)/g(b) — 0 so that limh|t,((6, U, G, I) = b. Properties (iv) and (v) are specific to the auction model. In particular, (iv) says that G(-) is smoother than F(-), extending a similar property noted by Guerre, Perrigne and Vuong (2000) for the risk neutral model. Regarding (v), together with (iii) and (iv) this property implies that G(-)/g(-) is R + 1 continuously differentiable on [b, b]. The following lemma provides a necessary and sufficient condition for rationalizing a distribution of observed bids by an IPV auction model with risk aversion. Hereafter, we say that a distribution is rationalized by a risk averse auction model if there exists a structure [U, F] whose equilibrium bid distribution is identical to the given distribution. Lemma 3.1: Let I > 2, R > 1, and G(-) be the joint distribution of (&i,... , bj). There exists an IPV auction model with risk aversion [U , F] G Ur x T r that rationalizes G(-) if and only if (i) G(&1, ... , bi) = n[= i G(bi), (ii) G(-) £ Qr, 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (iii) 3A : JR+ — > ■ M + with R + 1 continuous derivatives on [0, + 0 0 ), A(0) = 0 and A'(-) > 1 such that £'(•) > 0 on [6, b], where Hi,U,G,I) = t + a-> ( y ^ g ) . Lemma 3.1 provides a necessary and sufficient condition for rationalizing an observed bid distribution. The first, condition is related to the use of the IPV paradigm and requires that bids be independently and identically distributed in agreement with private values. The second condition requires that the marginal observed bid distribution G(-) satisfies the regularity assumptions embodied in the set Qr of Definition 3.3. The third condition arises from the fact that £(•, U, G, I) is the inverse of the equilibrium strategy, which is strictly increasing. As shown in the proof of Lemma 3.1, if such a condition is satisfied, then G(-) is rationalized by the structure [U,F], where U(x) = exp f*(I/ \(t))dt and F(-) is the distribution of £(&,£/, G, I) with G(-). The next proposition shows that any distribution G(-) E Qr can be rationalized by an IPV auction model with a utility function displaying risk aversion. Proposition 3.1: Let I > 2 and R > 1. Any distribution G(-) E Qr can be rationalized by a CRRA structure with c E [0,1) as well as a CARA structure with both zero wealth and private value distributions in T r . Proposition 3.1 is striking. First, it implies that the restriction (iii) in Lemma 3.1 for rationalizing a bid distribution by an IPV auction model with risk averse bidders is redundant unlike Condition C2 in Theorem 1 in Guerre, Perrigne and 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vuong (2000) for the risk neutral case. Specifically, my proof indicates that we can always find a function A(-) corresponding to either a CRRA or CARA utility function so that (iii) is satisfied whenever G(-) £ Qr. Alternatively, the IPV auction model with general risk aversion does not impose any restrictions on observed bids beyond their independence and the weak regularity conditions embodied in Qr. Second, because a general risk aversion structure [U, F] £ Ur x T r leads to a bid distribution G(-) £ Qr by Lemma 3.1, Proposition 3.1 implies that there always exist a CARA structure and a CRRA structure with zero wealth that are observationally equivalent to [U, F], In other words, irrespective of initial wealth, the game theoretic auction model with arbitrary risk aversion does not provide enough restrictions on observed bids to discriminate it from a CRRA or a CARA model with zero wealth. Hence, without loss of power for explaining bids, an analyst could consider either a CRRA or a CARA model with zero wealth, provided these models are identified and can be estimated. Because a risk neutral model is a special case of a risk averse model, it follows from Proposition 3.1 that any risk neutral model is observationally equivalent to a risk averse model such as a CRRA or a CARA model. An interesting question is whether the reverse holds, i.e. whether any risk averse model is observationally equivalent to a risk neutral model. This is not true.9 Thus, by allowing for risk 9Thc following is an example with I = 2 of a CRRA structure that is not observationally equivalent to any risk neutral structure. Consider the distribution G(b) = kb for h G [0,1/2] r - | 3 /[8 (:C 2 — ) ] and G(b) — T-aa *3-% for b G [1/2,1], where x.i < x 2 are roots of the equation — & C 2 + 11 x — 2 = 0 and k such that G(-) is continuous at b = 1/2. The corresponding density g(-) is flat on [0,1/2] and sharply increasing on [1/2,1]. This distribution satisfies the regularity 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aversion, one does enlarge the set of rationalizable bid distributions relative to risk neutrality. As a matter of fact, Proposition 3.1 says that allowing for very simple forms of risk aversion such as CRRA or CARA rationalizes any bid distribution in Qr, A model is a set of structures [U, F]. For instance, the CARA model (with regularity R) is the set of structures [U,F] where [/(•) G U^ARA = { [/(•) = Uvn m (w + •) ~ UvNm (w ); w G JR+,U vn m (’) £ U CARA} n U r and F(-) G T r, The sets U^ARA, Uj{RRA and UrRRA are similarly defined. As suggested by Proposition 3.1, auction models with risk averse bidders are nonidentified, in general. Hereafter, we say that a structure [U, F] is nonidentified if there exists another structure [17, F] within the model under consideration that leads to the same equilibrium bid distribution. If no such [U, F] exists for any [U, F], I say that the model is (globally) identified. P ro p o sitio n 3.2: Let I > 2 and R > 1. Any structure [U , F] G Ur, x T r is not identified. Similarly, any structure [U, F] in UrARA x Tr, U^rra x Tr, UrARA x T r or UrRRA x T r is not identified. As shown by Guerre, Perrigne and Vuong (2000), the IPV auction model with risk neutral bidders is identified from observed bids. Thus the nonidentification of conditions of Definition 3.3 with R = 1 . Letting U(x) = T ~ c gives A(.r) = x/(l — c). The bid distribution G(-) can be rationalized by a CRRA structure where the inverse bidding strategy is c, G) = b+ (1 — c)G(h)/g(b) as soon as 2/5 < c < 1 by Lemma 3.l-(iii). On the other hand, as shown in Guerre, Perrigne and Vuong (2000) the distribution G(-) is rationalized by a risk neutral structure if and only if £(b,G) = b + G(b)/g(b) is strictly increasing. In our case, this function is £.(b,G) = 2b for 0 < b < 1/2 and £(b,G) = — §(!> — ^)(b — §) + 1 for 1/2 < b < 1. It can bo easily checked that this function is not strictly increasing. Hence there does not exist a risk neutral structure that is observationally equivalent to the above CRRA structure. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the general risk aversion model U r x T r arises from the unknown utility function [/(•), which is restricted to the identity function under risk neutrality. The second part of Proposition 3.2 indicates that restricting U(-) to be derived from the four well known families of utility functions is still insufficient for achieving identification even if this involves a simple parametric specification such as the CRRA model. Moreover, the proof of Proposition 3.2 shows that all these models remain nonidentified even if wealth was known to be zero. It is useful to understand the source of nonidentification by considering the CRRA model with zero wealth.1 0 Hence, U(x) = x 1“c with 0 < c < 1 and F(-) £ T r. Let G(-) be the corresponding equilibrium bid distribution. Consider the alternative CRRA structure [U, F] such that c = ac + 1 — a, with 0 < a < 1, while F(-) is the distribution of 1 — cG(h) c — c 1 — c ( 1 — cG(b)\ = b + 1 - 1 M = + T T v + 1 ” e w ) ’ where b ~ G(-). Because the above function is strictly increasing in b when c < c < 1, then G(-) can also be rationalized by [U,F]. Hence [t/, F] is observationally equivalent to [U, F]. This result sharply contrasts with Donald 10For the general risk aversion model, -where [U , F] € Ur X Tr with arbitrary wealth, lot [U , F] be such that U (■ ) = [U(-/a)/U(l/a)}a, with a € (0,1) and F(-) be the distribution of i(b ,U ,G ,I)= b + A- 1 = b + a \ - 1 > with b ~ G(-). The second equality follows from A(-) = U(-)/U'(-) = U(-/a)/U'{-/a) — X(-/a). It is easy to check that [U, F] e Ur x T r. Note that £(-) can be decomposed as the sum of (1 — a)b and <x(,(b) = a[b + X^1 (G(b) / (I — 1 )g(b))}, which are two strictly increasing functions in b. Hence, £(•) is strictly increasing. Thus, from Lemma 3.1 the structures [U,F] and [U, F] are observationally equivalent, i.e. lead to the same bid distribution G(-). 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and Paarsch (1996, Theorem 1), who state that the CRRA model is identified. In fact, because of their assumption 1, these authors restrict the distribution F(-) to have the same support as F(-). In contrast, our result shows that the CRRA model is not identified as F(-) and F(-) may have different supports. In particular, at 6 = 6 the above equation indicates that the support of F(-) must decrease, i.e. v < v, to compensate for the increase in the constant relative risk aversion parameter c > c. More generally, all the quantiles of F(-) are smaller than the corresponding ones for F(-) by the same argument. To summarize, considering risk aversion even under its simplest form such as CRRA much increases the explanatory power of the model relative to the risk neutral case since all bid distributions in Qr can now be rationalized. On the other hand, the validity of such a model can no longer be tested as the theory does not provide restrictions beyond the independence of bids and the regularity conditions of Qr. Moreover, risk averse models as simple as CRRA and CARA models are noniden tified from observed bids. In particular, parameterizing the utility function is not sufficient for achieving identification. Additional identifying restrictions are thus needed.1 1 3.4 Sem iparam etric Identification The purpose of this section is to exploit heterogeneity across auctioned objects combined with palatable identifying restrictions to achieve semiparametric 1 1 If v was known, Donald and Paarsch (1996) result would apply and the CRRA model would be identified. Assuming that v is known is, however, too strong as v directly affects the risk aversion parameter c because v — b + (1 — c)/[(J — 1 )g(b)]. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. identification of first-price auction models with risk averse bidders. Heterogeneity across objects is characterized by a vector of observed variables Z, which can be discrete or continuous with values in Z C W?. For instance, Z can include a dummy variable for the quality of the auctioned object or a continuous variable indicating the object’s appraisal value. As before, we assume that the number of bidders I E J is observed, which can be either constant or varying across auctions. Thus we consider hereafter that private values are drawn from the conditional distribution F(-\Z, I ).12 Our preceding nonidentification results of risk averse models then hold when the whole structure depends on (Z, I) namely [U , F) = {[U(-\z, I), F(-\z, I)), z E Z j E l } . A first natural restriction is to require that the utility function U(-) be independent of (Z, I). Hence risk aversion is independent of the characteristics of the auctioned objects and the number of bidders. This is justified in the case studied here as bidders do not face uncertainty about the quality and equivalent monetary value of the auctioned object. Restricting U(•) to be independent of (Z, I) is, however, insufficient for identifying the model as noted later. Thus I need to consider additional restrictions on both [/(•) and F(-1 -, •) to achieve identification. I impose the following ones. Assum ption A l: For R > 1, (i) £/(•) = £/(•; 9) E U r for every 9 E 0 , an open subset of JR P , 12Sudi a specification allows for unobserved heterogeneity provided / is a sufficient statistic for unobserved heterogeneity conditional upon Z. See Campo, Perrigne and Vuong (2002). Note that unobserved heterogeneity across bidders is allowed through differences in bidders’ private values. On the other hand, observed heterogeneity across bidders is ruled out as it will load to an asymmetric auction model. See Campo, Perrigne, and Vuong (2002) where bidders are ex ante different under risk neutrality. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (ii) F(-1 -, •) G T r ( Z x X) = •) : F ( - |x ,«) G J - r , V(z,i) G 2 x 1 } . The support of F(-|z, i) is denoted [r?(z, i), v(z, i)], (iii) For some a G (0,1], the a-quantile va(z,i) of F(-\z, i) satisfies va(z, i) — va(z, i; 7 ) for all (z,i) £ Z x X and 7 G F, an open subset of Mq, (iv) The function ij)a(z,i;0,^) = - ba(z,i);9) for (z,i) G Z x X determines uniquely (9, 7 ) G 0 x F, where ba(z,i) is the a-quantile of the equilibrium bid distribution G(-\z,i) generated by the structure [U , F], Condition (i) requires that U(-) belongs to a parametric family of utility functions that are smooth. Utility functions derived from CRRA and CARA vNM utility functions satisfy such a condition for R = 0 0 . It is also satisfied by many parametric families that allow for flexible patterns of risk aversion. Note that if initial wealth w is unknown, then w must be included in the parameter vector 0. Condition (ii) requires that the conditional distribution F(-\z,i) satisfies the regularity conditions of Definition 3.2 for every (2 , i) G Z x X. Condition (iii) is a parametric conditional quantile restriction on F(-\z, i), as frequently used in the semiparametric literature. See Powell (1994). For instance, va('-> Sh) can be chosen to be a constant or more generally a polynomial, where 7 is the vector of unknown coefficients. Note that a = 1 is allowed, in which case a parametric specification of the upper bound v(z, i) is considered. On the other hand, no assumption is made on the lower bound v(z,i) corresponding to a = 0 as v(z, i) is nonparametrically identified from the boundary condition v(z,i) = b(z, i). An alternative identifying assumption to (iii) would be to require that the difference va(z,i) — 1 is a parametric function of (z, i). 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is equivalent to imposing a restriction on the a-quantile as the lower boundary v(z,i) can be recovered from b(z, i). In particular when a = 1, vn(z, i) = v(z,i), in which case this alternative identifying assumption would be a parametric specification of the length v(z, i) — v(z, i) of the support of F(-\z, i). Condition (iv) is a standard identifying condition of the parameter vector (6, 7 ) from the knowledge of the function 7 ) on 2 x 1 . Note that it implies Card Z x X > p + q. Condition (iv) bears on [U, G ], where G implicitly depends on the structure [U , F). In particular, it can be easily verified. For instance, consider a CRRA model with zero wealth and a constant (unknown) a-quantile of F(-1 - , •), i.e. va(z,i) = 7 , in which case p = 1 and q = 1. Condition (iv) is then satisfied as soon as there are two a-quantiles ba(zi, q) and ba(z2,12) that differ as shown by Proposition 3.3. The next proposition establishes the semiparametric identification of the first-price auction model with risk averse bidders. Such a result relies upon the key equation (3.4) giving the inverse of the equilibrium strategy, taking into account the con ditioning variables (Z, I), the parameterization of the utility function U{-\0) and the a-quantile va(z,i) of F(-\z,i). Specifically, because the equilibrium strategy s(-,U,F,I) is strictly increasing, then ba(z,i) = s(va(z, i), U, F, i). Hence, (3.4) evaluated at the a-quantile ba(z,i) gives 1 a g(ba(z,i)\z,i) = t t t — 7— — 7 ---- 7 - 7 — 7 - 7 7 , (3.5) 1 - 1 \(va(z,i;^f) - ba(z,i);9) 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for any (z, i) G Z x I. This equation combined with condition (iv) in A1 suggests how the parameter vector (9, 7 ) can be identified given the knowledge of g(ba(z,i)\z,i), and the specified parametric forms for A(•;#) and va (■ ; 7)- P ro p o sitio n 3.3: Let I > 2. The semiparametric model defined as the set of structures [[/, F] satisfying Assumption Al is identified. In particular, if there exists (zj, ij) G Z x X, j = 1,2, such that 6a (A L> *1) 7^ then t h e CRRA model and th e CARA model with zero wealth, a constant conditional quantile restriction va = 7 and F ( - |- , •) G T r { Z x X) are semiparametrically identified. Proposition 3.3 provides a semiparametric identification result since [/(•) is parametrically identified through 9 while is nonparametrically identified subject to its parametric conditional quantile restriction. Morever, the proof shows that the parameter 7 is identified. Note that in the CRRA and CARA models with zero wealth and a constant quantile restriction, the additional requirement that the a-quantile ba(z, i) varies with (z, i) is readily verifiable. For instance, suppose that / does not vary, while Z is reduced to one dichotomous variable indicating e.g. the qixality of the auctioned object. The CRRA model is then identified if the a-quantiles of the conditional bid distributions corresponding to the two values of Z differ. It is worthnoting that parameterizing the utility function and the a-quantile of the distribution of private values arises naturally. In particular, dropping either one of these parameterizations would lead to a nonidentified model as the following examples indicate. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For instance, assume that the conditional quantile vn(z,i) is not parametrically specified, but a parametric specification for the utility function is retained. Specifically, consider the semiparametric auction model composed of the struc- identified. An example of this is the CRRA model with U(x) = xl~c for c 6 [0,1) and F(-|-, •) belonging to Tr{Z x X). The argument is similar to that given after Proposition 3.2 where G(-) and g(-) are replaced by G(-|-,-) and g(-1 -, •), respectively. Hence, restricting the utility function to be parametric does not [/(•) does not vary with (Z, I). Likewise, suppose that, the restriction to a parametric specification of the utility function is relaxed while the parametric conditional quantile restriction is retained. That is, consider the semiparametric model composed of structures [U , F ] satisfying Al-(ii) and (iii) with U(-) G U r . This model is not identified as soon as there is a constant term in the parametric conditional quantile specification va(z, i; 7 ). Specifically, let [U, F] be such a structure and consider the structure [[/, F], where tures [U, F] satisfying Al-(i) and (ii). Such a model would not be necessarily achieve by itself identification of the semiparametric model, despite the fact that ci[U(x/S)]6 for 0 < x < S2, C 2U{x + < 5 (1 — 6 )) for x > < 5 2, 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where 0 < < 5 < 1 , c\ — C 2\U{8)\1 5, and c2 = 1/U (1 + < 5 (1 — < §)).13 Let F(-\z,i) be The previous section has shown that the auction model with risk averse bidders is semiparametrically identified through a parameterization of the utility function and a parametric quantile restriction on the distribution of private values. This naturally leads to the search for estimators of [U , F], and in particular for semiparametric estimators of 6 as 6 parameterizes £/(•). From the semiparametric 13Notc that [7(0) = 0, and [/(•) has R + 2 continuous derivatives on (0,5) U (5, +oo). Thus [/(•) would belong to Ur if U(-) has R + 2 continuous derivatives at x — 52. In fact, [/(■) has only one continuous derivative at x — 52. Hence [/(•) should be smoothed out in the neighborhood of x = S2 so as to make it R + 2 continuously differentiable on (0, +oo), as required. Hereafter, I omit this smoothing requirement and use (/(•) directly. 14From Lemma 3.1, we need to show th at £'(-|; M ) > 0 f°r an.Y (-M) E - Z x l We have nalizcs the bid distribution G(-|-» •)• h remains to show th at F(-|-, •) satisfies Al-(iii). The a-quantile va(z,i) of F(-\z,i) satisfies va(z,i) = £(ba(z,i)\z,i). Consider G(ba(z,i)\z, *)/[(* - l)g(ba (z,i)\z,i)\ = «/[(* — 1 )9 (.ba {z,i)\z,i)]. Let 5 be sufficiently small so that 0 < A(5) < a/[{i — 1) sup(Zji)£Zxi 9 (ba(z,i)\z, ?')]. This is possible as A(-) is strictly increasing and A(0) = 0. Thus va{z,i) = t(l>a(z,i)\z,i) - 5(1 — 6) = va(z,i) - 5(1 - 5) > 0 for 5 sufficiently small. Thus va (z,i) satisfies Al-(iii) as soon va {z,v, 7 ) contains a constant term. the distribution of where b ~ G{-\z,i). It can be shown that [U, F] rationalizes G(-\-, •) and that F(-1 -, •) satisfies Al-(ii) and (iii) . 14 Hence, the parameterization of the conditional quantile of F{-\z,i) is not sufficient by itself for identification. 3.5 O ptim al C onvergence R ate Because £'(-|z,i) is strictly positive, £'(-\z,i) is strictly positive as required. Hence [17,F] ratio- 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. literature, it is known that semiparametric estimators can attain the parametric rate of convergence, while others converge at a slower rate. For the former, see Robinson (1988) and Newey and McFadden (1994) and Powell (1994) for surveys. For the latter, see Horowitz (1992), Manski (1995), Kyriazidou (1997) and Honore and Kyriazidou (2000). Given the nonstandard nature of our model, it is especially interesting to derive the best (optimal) convergence rate that can be attained by semiparametric estimators of 9. This is the primary purpose of this section. The optimal convergence rate for estimating the conditional density /(■) will follow from Guerre, Perrigne and Vuong (2000). I first need to strengthen our regularity assumptions on F(-\-, •) and £/(•;•) with respect to (z, i) and 9. Definition 3.4: For R > 1 and Z a rectangular compact of IR1 with nonempty interior, let Tr = ^Fr(Z x T) be the set of conditional distributions F(-\-, •) satis fying (i) V(z, i) 6 Z x X, 0 < v(z, i) = v < v(z, i) ~ v < oo, (ii) Vi < E X, F(-|-,z) admits R + 1 continuous derivatives on [u, u] x Z, (hi) Vi G X, inf{v!z)e^ v]xz f{v\z,i) > 0. Definition 3.4 follows Definition 3.2 taking into account the conditioning variables (X, I ) . 15 While conditions (ii) and (iii) are straightforward extensions of their counterparts in Definition 3.2, condition (i) needs further discussion. Because 15To simplify the; presentation, we c;xdude discrete exogenous variables by requiring Z to have a nonempty interior. Our results (Theorems 3.1 and 3.2) continue to hold with suitable modifications. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the singularity of the differential equation (3.2) at the lower boundary of the support of the private value distribution, assuming a constant lower boundary v(z,i) — v much simplifies the derivation of the smoothness of the equilibrium strategy s(-), which is needed to obtain the smoothness of the equilibrium bid distribution. On the other hand, such a restriction does not play a role in estimation as v(z,i) can be recovered from b(z,i). Regarding the upper boundary restriction, Section 3.4 indicates that a parame terization of a quantile of •) is necessary for achieving identification. The upper boundary is a particular quantile corresponding to a = 1. Our estimation procedure will rely on (3.5), which requires an estimate for ba(z,i). There is then an important difference between estimating a quantile corresponding to a. £ (0,1) and estimating the upper boundary. In particular, the convergence rate for estimating the latter is much faster than for estimating the former. This suggests that the optimal convergence rate for estimating 6 will be slower when considering a a-quantile restriction with a £ (0 , 1 ) than when considering the upper boundary. This will be discussed further below. Hereafter, we focus on the upper boundary restriction. For sake of simplicity, we consider a constant upper boundary restriction, in which case q = 1 . In view of the above, I then consider the semiparametric model composed of structures [U , F] satisfying the following assumption. A ssum ption A2: For R > 1, (i) In addition to Al-(i), U(•; ■ ) is thrice continuously differentiable on (0, Too) x 0 , 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (ii) F(-\;-)eT'x , (iii) The function tpi{z,i]9, v) = \{v — b(z,i)]6) for (z,i) £ Z x X determines uniquely (0,v) £ 0 x (0 , + 0 0 ), where b(z,i) is the upper boundary of the equilibrium bid distribution G(-\z, i) generated by the structure [U, F]. Conditions (i) and (ii) strenghthen conditions (i)--(iii) of Al. Condition (iii) simply expresses Al-(iv) at the upper boundary under a constant restriction. The next step is to derive the smoothness properties of the equilibrium bid distri bution G(-|-, •) corresponding to a structure [U , F] satisfying A2. Such properties are important as they relate to the implied statistical model for the observables, which are the bids, the number of bidders and the exogenous variables. To do so, I need to derive the smoothness properties of the equilibrium strategy s(-). Such properties are given in Theorem B1 for structures [U , F] £ U r x T*r . In addition to establishing the existence and uniqueness of the equilibrium strategy, Theorem B1 shows that such a strategy admits R + 1 continuous derivatives with respect to (v, z) in [v, u] x Z for every t £ X. The main difficulty in establishing Theorem B1 is to derive the smoothness of s(-) with respect to z because s(-) is defined through the differential equation (3.2), which does not have an explicit solution. The desire smoothness properties of G (-|-, •) then follows from Theorem Bl. L em m a 3.2: For every i £ X, the conditional distribution G(-\-,i) corresponding to a structure [U, F] £ U r x F*r satisfies (i) The upper boundary b(z, i) admits R + 1 continuous derivatives with respect to 2 £ Z and infzeZ(b(z,i) — b(z, i) > 0, 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (ii) For all (b, z) £ Sj{G) = {(b,z);z £ Z,b £ \b(z,i),b(z,i)]}, g(b\z,i) > cg > 0, (iii) G(-|-,i) admits R + 1 continuous partial dertivatives on 5)(G), (iv) c?(-1i) admits R + 1 continuous partial derivatives on Sll(G) = {(b, z);z £ Z ,b £ (b(z,i),b(z,i)]}. Lemma 3.2 parallels Proposition 1 in Guerre, Perrigne and Vuong (2000). In particular, properties (i)- (iii) for G(-|-, i) are similar to conditions (i)~ (iii) in Defi nition 3.4 for F(-\cdot,i). On the other hand, property (iv) implies that G(-\-,i) is smoother than F(-|-, i) on the interior of their respective supports. Such a property follows from (3.5) evaluated at a = 1, namely g(b(z,i)\z,i) = . i tu = m izi % $ )■ > (3.6) i - l \(v - b(z,i)-,8) for all (z, i) £ Z x I, where (3 = (9, v). It remains to specify the data generating process. For the fth auction, one observes all the bids Ba,i = 1 ,... ,If, the number of bidders If > 2 as well as the d-dimensional vector Zf characterizing the heterogeneity of the auctioned objects. This gives a total number N = °f 6 ids, where L is the number of auctions. Thus f(-\Zf, If) is the density of private values conditional upon (Zf, If) in auction I. Following the game theoretical model of Section 3.2, we make the following assumption on the data generating process and the specification of the semiparametric model composed of structures [U , F] satisfying A2. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ssum ption A3: (i) The variables (Zf, If), i = 1 ,2 ... are independently and identically distributed with support 2 x 1 , where J is a finite subset of {2 ,3 ,... Moreover, for every i E J , the probability P(i\z) of I = i given Z — z admits R + 1 continuous derivatives with respect to z G Z while the conditional density d>(z\i) of Z given I = i is bounded away from zero and infinity, (ii) For every t, the private values Vn,i = 1 ,... ,If are independently and identi cally distributed conditionally upon (Zf, If) as F0(-\Zf, If), (iii) The semiparametric model is correctly specified, i.e. the true utility function U0() and conditional distribution Fo(-|-, •) satisfy A2 . In particular, private values and hence bids are independent across auctions. 16 I am now in a position to establish the optimal rate at which (3 = (0, v) can be estimated. To do so, I follow Horowitz (1993) and invoke the minimax theory developed by Ibragimov and Has’minskii (1981). I consider the following norms W PW oo = max( max \6q\, |u|), ||/(-|-,-)||oo = sup sup \f(v\z,i)\ 1 <q<k ( z ,i) £ 2 x l v£[y_,v] 16Not, all of A3 is used to prove Theorem 3.1. Iu particular, (Z f,If ) need not ho independently and identically distributed, while < j> (-\) does not need to be bounded away from zero. Futhermore, A3-(i) can be weakened allowing Zf s not to be independently and identically distributed as Theorem 3.2 is derived conditionally upon {Z\, I i,... , Zf, If). 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and define the set of conditional densities for M > 0. As usual in studies of convergence rates, one considers a neighborhood of the true parameters (/3q, fo) in order to exclude superefficiency, i.e. where the indicator function restricts comparison of conditional densities on the intersection of their supports. Let P r^ j be the joint distribution of the V^s and the (Zf,If)s under {9, /, fzj), where fz,i is the joint, density of the (Zf,If)s. The next theorem establishes an upper bound for the optimal rate when estimating (3q. It crucially relies on Lemmas 3.1 and 3.2 and Proposition 3.3. Theorem 3.1: Under Assumptions A2 and A3, for any /?o € 0 x (0, +oo), any fo G T*r {M) and any deterministic sequence pN such that pNN~(R+1^(2R+3' > — > oo, we have ■ M Ve(Po,fo) for any t > 0, where the infinum infg is taken over all possible estimators f3 of (3 based upon {Bit., Zf, If), i = 1 ,... , If, I — 1 ,... , L. Vf (/3 o ,/o ) = { ( / ? , / ) G e X W (3 - /5o!U < e, l l ( / ( - h O - / o ( - K - ) ) W l v ) / o ( - K O > 0 ) | | o o < e } 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 3.1 reveals the nonparametric nature of the parameter p, which cannot be estimated at a faster rate than AKR+1)/(2/i+3)_ More precisely, for any estimator /?, Theorem 3.1 shows that pN{(3 — (3) diverges if pN grows faster than A T (i?+1 )/(2R +3)j converges at the rate jV '(fl:+1)/(2'R +3). Therefore, the optimal rate of convergence for estimating (3 q in the minimax sense is N^R+d/(2R+:i\ i.e. IV 2/ 5 when R = 1, which is independent of the dimension d of the exogenous variables Z. Insight of this result can be gained through the main idea of its proof, which introduces some perturbations of the true parameters (Pq, go). Specifically, assum ing R = 1 to simplify, we consider the parametric submodel {gN(b\z,i; /3)} approx imately defined by where T : IR~ — » IR is compactly supported with U/(0) = 1 , J ty(x)dx = 0 and k > 0. Using Proposition 3.3,1 first establish that each such density can be rationalized by an auction model with ((3, ■ ; 0)) E K(/?o,/o) for Pn sufficiently large. Computation of the corresponding Fisher information matrix at (3 = 0o gives hence also for p^ = y/N, i.e. the usual parametric rate. On the other hand, Theorem 3.2 in the next section will show that there exists an estimator (3 that gN(b\z, i ; (3) = go(b\z, i) + — V (Ky/p^(b - b0(z, i ))) ^ _ /g0) ? Pn Op I((30,z,i ) = / Jbr f bo(z,t) 1 dgN(b\z, i; pQ ) dgN(b\z, i; p0) b0(z,i) go{b\z,i) df3 dp' 1 dm,(z,z;P0) drnjz, r,P0) f r 2 ( « y ^ ( 6 - 6 pQ M ))) Aob,*) 9o(b\z,i) (z,i) 9o(b\z,i) f # 2(x) K/i f an an' J 1 dm{z , /: v i'ii j dm{z, r . f 1 ■dx K.pm(z, P #o) 1 1 dm.{z, i\ /3 0) dm,(z, i\ (30) j'p rn,(z, i] p0) dp dp1 1 1 / 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by a standard change of variable and using go(b0(z, i)\z, i) — m(z , i; 0O ). This gives the Fisher information matrix for the whole sample as t ta \ I V (x)dx h dm.(Zt, If, fo) dm(Ze, If, fo) t f A J AvfPo) — ------ 5/5— y , ~ T 7 — T o \ ------- n o --------------- ! Z q > -----------O p (N p kPn m (z e, If, Po) d(3 d(3' ■5/2, which is an O p ( N p ^ 2) and hence an op(l) whenever p^N ^2^ — > oo. Because the Fisher information matrix asymptotically vanishes, the model is asymptotically not identified and the parameter /3 cannot be estimated. 3.6 Sem iparam etric E stim ation This section proposes a semiparametric estimation procedure for estimating (i) the parameter ( 9 characterizing the bidders’ utility function U (•) and hence bid ders’ risk aversion, and (ii) the conditional latent density /(-|-) of bidders’ private values. Because we do not restrict /(-|-) to belong to a parametric family, the estimation problem is semiparametric in nature. A first subsection presents our semiparametric procedure and its different steps, while a second subsection estab lishes the asymptotic properties of our estimators of 8 and /(-|-). 3.6.1 A Semiparametric Procedure I first, strengthen slightly the regularity conditions satisfied by the semiparametric model [[/(•), F(-\-)] and maintain the identifying conditions of Section 3.2. Specifically, I assume Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ssum ption AO: (i) For R > 1, [/(•) G Ur{&) as defined in Definition 3.4 and F(-|-) G F r {Z) as defined in Definition 3.5. Moreover, [/(•;•) is thrice continuously differentiable on (0 , Too) x 0 . (ii) For any [U , F] G U r ( Q ) x T*r (Z), the set B = {b(z); z G Z} contains a set X satisfying Definition 3.4-(ii). (iii) The number of bidders is a deterministic function of Z, i.e. / = I(Z), whose range is a finite subset X of IN. Assumption AO-(iii) is a simplifying assumption. It includes the case when I is determined by an entry model (see also footnote 12). Following Guerre, Perrigne and Vuong (2000), extension to the case where I given Z is random is possible. 17 My semiparametric procedure follows closely the semiparametric identification result. If one knew the upper boundary b(-) and the density g(-\-), one could estimate the utility function parameter(s) 8 from (3.5) given the chosen paramet ric form for A (-;8). From the knowledge of G(-|-) and 8, one could then recover every bidder’s private value ? ; , • from (3.3) to estimate /(-|-). Unfortunately, b(-), G(-|-) and g(-|-) are unknown, but they can be estimated from observed bids. This suggests the following three steps procedure • Step 1 : From observed bids, estimate nonparametrically b(z) and g{b{z)\z) for the observed values 2 . 17In particular, the upper boundary b(z) becomes b(z, /). Similarly, conditioning on z becomes a conditioning on (z ,I ) in (3.6) and (3.7). 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Step 2: Using (3.5), where g(b{z)\z) and h(z) are replaced by their estimates obtained in the first step at the observed values z, estimate /? = (0,v) using NLLS. • Step 3: Using (3.3), where G(-|-) and p(-|-) are replaced by nonpararnetric estimators, recover pseudo private values vt for each bidder and estimate nonparametrically /(-|-). The next subsections detail each of these steps. Nonparam etric Boundary Estim ation This step consists in estimating the tipper boundary b(-) of the bids distribution and the conditional density g(-\-) at the upper boundary. I first discuss the estimation of the upper boundary b(-). Let R be the num ber of continuous derivatives of /(-|-) and d the dimension of Z. Under the assumptions of Definitions 3.4 and 3.5, the upper boundary b(-) is R + 1 con tinuously differentiable (see Appendix A, Lemma A l). Following Korostelev and Tsybakov (1993) theory of image reconstruction, one introduces a partition of Z into bins increasing in number. The boundary estimator for z in an arbitrary bin is obtained by minimizing the volume of the cylinder whose base is the bin and whose upper surface is defined by a polynomial of degree R in z 6 1 R ? ' subject to the constraints that the observations are contained in such a cylinder. The optimal polynomial evaluated at z gives the boundary estimate b(z). Under appropriate assumptions, the resulting piecewise polyno mial estimator converges to b(-) uniformly on Z at the rate (log Ar/Ar)( T+:0/( T+1+'d. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For instance, for R = 1 and d — 1, partition Z = [z, z] into fc/v bins {Zk;k — 1,... , kN} of equal length. A N cx (log IV/iV)1 /2. On each Zk — [zj., zf c ), the estimate of the upper boundary is the straight line dk + bk(z — zk), where (dk,bk) is obtained by solving rzk min / a,k + hk(z - zj^jdz = akA N + hk£k2N / 2. {(ak ,bk ):B u < a k+bk (Ze--z ^,i= \,... J e,Zt £ Z k } J ^ This estimator converges at the uniform rate (log N/N)2^, which is faster than \/N and sufficient in our case. Turning to the estimation of the density g{-1 -) at the upper boundary, it is well known that standard kernel density estimators suffer from higher bias at boundary points. This is known as boundary effects. Thus I consider a nonparametric esti mator in the spirit of Cheng (1997) local polynomial estimation. Specifically, for every I = 1,... , L, we consider the bins Bk({aN) = (b(Z() — (k + l)aN, b(Z() — kaN], where k — 0,... , KN, aN is a binwidth and = (Kn + 1 )«v is a bandwidth. To estimate g(b(Zf)\Zg), we use x Kn Y% t = — ' S ^ p k U{Bu G B k({aN )) a'N t^o = -TT— T< ------7~~----- < hjN V K m +1 h N K m +1 k — 0 N k JN ■ < 3 - 7 ) 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the Pks are some weights. The last equality shows that Yu, or more precisely Yt — ( l/ i » Y ? t = \ Yu, has a kernel type form with a one-sided step wise kernel with support [ - 1 , 0 ] taking the value (KN + 1 )pk on the interval (— (A ; + 1 )/(itTjv + 1), —kj(Kn + 1)], k = 0,... , K N.1S Thus, $ n (-) can be viewed as a constant piecewise approximation of a one-sided kernel with support [— 1 , 0] as in the binning approach discussed in Hardle and Linton (1994). The one-sided kernel $jv(‘) will have to satisfy some assumptions including being of order R + l (see Section 3.2). In particular, f ^ N(x)dx = 1 and J xr&N(x)dx — 0 for r = 1,... , R. The latter imply some conditions on the pks, namely J2k=o "P k — 1 and YLk=oPk{(k' + 1Y — ^r] = 0 for r = 2,... , R + I . 19 In practice, one does not know &(•). I thus define Yu and the bin Bi~t(aN) similarly to Yu and where b(-) is replaced by its estimator b(-) obtained previously. Sem iparam etric Estim ation of 9 Let T l be the sigma algebra generated by Zt, I = 1,... , L. I have the identity Yi = m,(Zi, Pq) + eu + cu, (3.8) 18Note, however, that I( remains bounded and hence doers not increase with N. Nevertheless, Y t and Yu arc asymptotically unbiased estim ators of g(b(Zt)\Zt) as vanishes. 19For density estimation, a common restriction is = o(ftjy), where the pks are chosen via a local polynomial regression, see Cheng (1997). See also Section 4.2 for another choice of the Pks when R — 1 . 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where e-u = E [^|JT L] — m(Zi,j30) and = Ye — Lemma B1 combined with (3.7) show that the bias term = 0(h^+i), while the variance of the error term eu is of order l/hN, namely Var[etf|J> J = — ^ + f $ 2 N{x)dx. (3.9) N J Hence, the Yus obey a parametric regression model with a vanishing bias and a variance of the error term diverging to infinity as hpj vanishes. To my knowledge, this last feature is original in econometrics and raises some technical difficulties when deriving the asymptotic properties of our estimator of 6. In particular, the diverging variance of the error term is the main reason why the rate of convergence of our estimator does not achieve the parametric rate y/N- Namely, its rate is (N/ log N Y r+1^(2R +3); which is optimal in the minimax sense. See Section 3.5. The previous model suggests to estimate /? by possibly weighted NLLS, i.e. by minimizing L It Qn (J3 ) = E E uj{Zs){Yu~-m{Zl-id)f (3.10) l i=i with respect to /3 £ Os, where the uj(Zf)s are strictly positive weights. The set 0,5 defines possible values of /?o, namely 0 ^ = {/? : 6 € 0 , maxz€zb(z) + 6 < v < TS U p} - 20 Because the upper boundary b(-) is unknown, the preceding estimator is 20The introduction of 6 > 0 is necessary to bound ami its derivatives as liui m(v — supZb(z)\6) = +oo when v — > nupZb(z) since A(0) = 0. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not, feasible. We thus replace b(-) by its estimator obtained in the first step. Thus our estimator of /? is obtained as (3N = Argmm/ 3eeArQiv(/3), where / h Q n (0) = EE u { Z e) { % - m { Z f J ) Y (3-11) l —\ i=l and m{Zt; (3) = 1 i ----, 0jv = {/? : 9 £ 0 , max 6 (Z/) + 6 < F < Fsup}. If- ~ 1 \(v -b(Z();0) l^ L Nonparam etric Estim ation of /(• |-) This step is similar to the second step in Guerre, Perrigne and Vuong (2000) with the difference that 6q in (3.6) is now estimated from step 2, while A(-) was known and equal to identity in that paper, which considers the risk neutral case. Specifically, to recover the pseudo private values Va for % = 1 ,... , Ip, L — 1,... , L from (3.6), we need an estimate of 0O , which is given by step 2, and an estimates of the ratio G(-|-)/g(-|-) evaluated at any pair (Bu, Zf). For an arbitrary pair (6 , z), G(b\z)/g(b\z) is estimated by . , f t f ' E f . T E b ^ < ( > ) i r c ( y a ) mb, z) = t-------------r— , h r ' S ' 1 E ' s ' 1* K ( b~ B it Z~ z t \ 1 If E a = i hg ’ kg J where Kg(-) and Kg(-) are kernels of order R + 1 with bounded supports, and h,Q and hg are bandwidths vanishing at the rates (N/ log N )1^ m+d+2' 1 and (N/ log N )1 /(2R+d+3) 5 respectively as in Assumption A4 in Guerre, Perrigne and Vuong (2000). 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The pseudo private values are defined as V i = B ie + - 7 ------ ;------------- ^ , (3.12) if ( ) + S(2hG) C S(G) and (Bu, Z() + S(2hg) C S(G). Otherwise, we let Vu be infinity, which corresponds to a trimming. The sets S(2hG) and S(2hg) are the supports of KG(-/ (2hG)) and Kg(-/(2hg)), respectively. The set S(G) is the estimated support of the conditional bids distribution G(-|-). Specifically, S(G) = {(b, z) : b G [b(z),b(z)], z G Z}, where b(-) is the boundary estimator discussed in Section 3.1.1 and & (•) is defined similarly. The N pseudo private values Vu hence obtained are used in a standard kernel estimation of /(-|-). Namely, for an arbitrary pair (v, z), f(v\z) is estimated by u ( v-Vit z-Zl f = M >, ■ ( h! e l , Kz (y f) where Kf(-) and Kz(-) are kernels of order R and R + 1 with bounded supports, and hf and hz are bandwidths vanishing at the rates (N/ log J /v)1 /(2jR +d+3) and (L/ logL) ( 2K+rf+2). The rate of convergence of /(-|-) is the same as achieved in Guerre, Perrigne and Vuong (2000), namely (N/ log j\f ')R/(2R+d+3) because the semiparametric estimator 0 converges at a faster rate. 3.6.2 A sy m p to tic P ro p erties In this section, I state the asymptotic properties of my estimators 6 and /(-|-) as derived in Campo, Guerre, Perrigne and Vuong (2002). In particular, we show that our semiparametric estimator 9 converges at the rate jsf(R+l)/(2R+ 'i) ^ which is 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. independent, of the dimension of Z, and that this rate is optimal in a minimax sense. Moreover, we establish the asymptotic distribution of 0, which can be used for testing, for instance, the presence of risk aversion in the CRRA model. Lastly, following Guerre, Perrigne and Vuong (2000), we show that the nonparametric estimator /(-|-) converges at the rate ]Sf(R+l)/(2R+d+s)^ which is also optimal in a minimax sense. Properties of 9 I make the following assumptions on ra(-;9), the weights cu(-), the kernel $jv(-), the bandwidth h^, the binwidth aN and the rate of uniform convergence a'N of the boundary estimator & (•). A ssum ption A2: (i) (3q = (9q, n0) belongs to the interior of 0$ = {(3 : 9 < G 0 , maxz€Z b(z) + 8 < v < !7 sup}. Moreover, the (k + 1) x (k + 1) matrix E[dm.(Z] f3 0)/d/3 • dm(Z; /3q)/ df3'] is nonsingular. (ii) The weight functions u(-) are uniformly bounded and bounded away from zero, i.e. supZ&zu(z) < oo and inizez^(z) > 0 . (iii) The kernel $jv(-) has a finite support [— 1,0] and satisfies f $ N(x)d:r, = 1, f xj$ N(x)dx = 0 for j = 1 ,R, sup^ s u p ^ ^ o ] |$v(-^)| < oo, and 0 < (x)dx < oo. (iv) hN = o(l), NhN — » oo and aN = 0(hN) (v) snpzeZ |b(z) - b(z) \ = 0 P(afN) with a'N = o(mm(a.Nh^+1, aNj\jN h N)). 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assumption A2-(i) is standard in parametric estimation. In particular, the second part strengthens the identification requirement of 0q from the parametric specifi cation m(-; 0). For instance, A2-(i) is satisfied under condition (ii) of Definition 3.4 whenever the matrix E[Om,(Z; 0Q ) / 00 ■ Om,(Z\ 00)/00'] has constant rank in the neighborhood of 0q (see, e.g. Rothenberg (1971, Theorem 1)). Assumption A2-(ii) combined with A2-(i) insures that the matrices A(0O ) and B(0O ) are nonsingular and finite, where Assumption A2-(iii) is standard in kernel estimation when using higher order function (3.8), this assumption translates into conditions on the weights pk-Namely, A2-(iii) is satisfied whenever YlkPk = J2kPk[(k + l)j — £4] = 0, j = 2,... , R + 1 , supf e \pk\ = 0(l/iFjv) and YhkPk, exactly of order 1/Km- The first part of Assumption A2 -(iv) is usual in kernel estimation, while the second part prevents K m from vanishing since hN = (Km + 1 )«w- Assumption A2 -(v) requires that the boundary estimator b(-) converges faster than the semiparametric estimator 9 (and hence than the parametric rate) so that estimation of the boundary does not affect the asymptotic distribution of 9. For instance, if R = 1 and d = 1, taking a'N = (log N/N)2^ is sufficient when K m = K > 0 and h-M — (log N/N)1^, which is the optimal vanishing rate in this case (see below). More generally, when d > 1 , K m is constant and Iim vanishes at the optimal rate (N/ log N)l^ 2RJrA\ A(/3) = E Iuj(z) dm(z ; 0 ) Om/z; 0 ) 00 00' B(0) = E Iuj2(z) kernels though the kernel is one-sided. In view of our choice for < & n (-) as the step 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assumption A2-(v) requires that the problem be sufficently smooth, namely R > d, for the boundary estimator & (-) of Section 3.6.1.1 to converge at the required rate.2 1 I introduce the following (k + 1)-square matrices familiar in the minimization of the NLLS objective function Q n ( P ) defined in (3.11) B„W = t , Q £\ I xR+1$ N(x)dx ^ dR+1g(b(Z()\Zt,) dm(Zf:, (3) M A / ) = ~ ( R T m ~ h l — ^ ----------- where the latter (k + l)-vector relates to the asymptotic bias of our estima tor. Let the matrices A N{(3) and BN((3) be defined as AN(p) and Bn((3) with m,{-]f3) replaced by m(-;{3). The next result establishes the consistency and asymptotic normality of (3. It also provides an estimate for its asymptotic variance. Theorem 3.3: Under Assumptions A0-A2 (i) f3 is a consistent estimator of (3$. (ii) f^ H x )d x r Y 1/2f ^ R+1 \ a h N Bn ((30) ) (AN(Po)0 - A )) - N hR+1bN(p0, f 0)) Af(0, Ik+1). 21 Specifically, wc obtain R > (d - 2 + (d2 + 4d) 1/2)/2, where the latter is strictly between d — 1 and d. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Moreover, (3 - fiQ = 0 P (hR+1 + 1 /y/N hN), and the best rate of convergence of (3 is Ar(K+1^ 2R+3\ which is achieved when the exact order of the bandwidth is p p —1/(2 /? + 3 ) (iii) If hN = o{N~lIVR+% ( J * ^ * )dXB n W ) ' An (i3)0 - ft) - i - jV(0, 4 +1), and P — /3q = Op(1 /\JNhjy). On technical grounds, the proof of Theorem 3.3-(i) is complicated by the fact that the variance (3.10) of the error term in the nonlinear model (3.9) diverges at the rate 1 / h^. In particular, omitting the estimation of the upper boundary £ > (•), which has no effect because of Assumption A2-(iv), (1 /N)Q^(p) = Op(l/hpf) because of the diverging variance. Hence, (1 /N)Qn((3) does not have a finite limit. This would lead to consider the normalization h,jsrQN((3)/N, but its limit is a constant independent of /?. To overcome such a difficulty, Campo, et al. (2002) show that (Qn{P) — Qn(Pq)~Q n (/3))/N vanishes asymptotically using a maximal inequality, where L QnW) = 'Ylu{Zi)[m{Zt\0) - m(Zf,p0)}2. £= 1 Consistency of (3 can then be established by standard arguments using the objective function Q^(P) (see, e.g. White (1994)). 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 3.3~(ii) gives t h e asymptotic distribution of ft — (30 and its rate of conver gence. In particular, (3 — /3q is approximately distributed as / f < 3 > 2 (r)dr \ l^2 A*1 (/3b) ( ' 0, Ik+i) + A -N \ p o)Nh%+lbN(0o, / 0), where A N(f30)/N, BN(f30)/N and 6jv(/?0, / 0) are Op(l) under our assumptions. This expansion corresponds to the usual bias/variance decomposition of nonpara- metric estimators (see e.g. Hardle and Linton (1994)). When = o(N~l^ 2R+3' > ), the leading term is the first term, otherwise it is the second term, i.e. the bias. Thus, the best rate of convergence of j3 is achieved when the variance and the bias have the same order, i.e. when hN is exactly of order 1V~1 /(2jR +3)_ in which case ft - A, = Op(7V~(k+1)/(2K +3)).2 2 The best rate jV '(fl+1)/(2 - R +3) of convergence of (3 is independent of the dimension d of Z and corresponds to the optimal rate for estimating an univariate density with R + 1 bounded derivatives. This seems surprising in view of the key relation (3.7), which suggests that fto is as difficult to estimate as the bivariate conditional density </(-|-)> while the latter cannot be estimated faster than js[(R+l)/(2R+3+d) from Stone (1982) given the (R + 1) bounded derivatives of <?(-|-)- The faster rate can be explained by noting t h a t (3.7) leads to th e moment conditions E[(g(b(Z)\Z) — m(Z; ft0))W(Z)] = 0 for any vector function W(-). Integrating then with respect to Z intuitively improves the rate of convergence by eliminating the d dimension. Note t h a t the previous moment conditions 22Wheu hft is as optimally chosen, the estim ator ft is asymptotically biased as for any non- parametric, estimator. See also Liu and Brown (1993). For a recent method for obtaining an asymptotically unbiased estim ator converging at the optimal rate, see Hcngartucr (1997). 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are similar to those considered by Newey and McFadden (1994) though the Assumptions (iii)-(iv) of their Theorem 8.1 is not satisfied in our case. In fact, because the variance (3.10) is diverging, the proof shows that the average gradient (l/N)dQN(Po)/d(3 — 0 P(hR+1 + 1 /s/Nh^), which is different from the usual O p ( 1 / V N ) . Hence, my estimator converges at a slower rate than ViV. Theorem 3.3-(iii) can be used in practice to make inference on (3 q as it gives an estimator of the variance of /3, namely ( f ^ jf(x)dx/h,N)A]v1(/3)BN0)A]v1(/3). Note that f3 depends on the weights n>(-), which can be chosen optimally to decrease the asymptotic variance of (3 as in weighted NLLS. From (3.10), the optimal weight function uT(-) is inversely proportional to the variance, i.e. ad(-) = l/m (-; (3o). This optimal weighted NLLS estimator /3 * can be implemented by a two-stage procedure, in which the optimal weights are estimated by l/m (-;/?), where [3 is obtained in the first step by ordinary NLLS. The estimator of the variance of (3* then reduces to ( J $ 2 N(x)dx/hN)A]A(f3*)- 3.7 D a ta and Em pirical R esu lts This section illustrates the previous methodology on timber auction sales from the US Forest Service. A first subsection briefly presents the data. A second subsection discusses the implementation of our estimation method for a CRRA utility specification and gives estimation results. In particular, risk aversion is found to be significative. I l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.7.1 Data The US Forest Service (USFS) timber auction data have been widely used in empirical studies on auctions. Comparing revenues generated from ascending and sealed-bid auctions, Hansen (1985) tests the revenue equivalence theorem. Adopting an independent private value framework, Baldwin, Marshall and Richard (1997) study collusion, while Baldwin (1995) attempts to test for the presence of risk aversion. More recently, Athey and Levin (2001) study the practice of skewed bidding when bidders bid on species and when payments are based on actual harvested value, while considering bidders’ risk aversion. Haile (2001) analyzes the bidding behavior under a common value framework when there are resale opportunities after the auctions. Each of these papers focuses on a particular economic issue relevant to these auctions. While suspecting bidders’ risk aversion in two of them, risk aversion has never been measured. The objective of this illustration is to shed some light on bidders’ risk aversion. For this reason, many characteristics of these auctions such as collusion, skewed bidding, resale markets, common value are let aside to focus on the issue of risk aversion. The West half of the United States has a large part of its forestry publicly owned and is an important provider of timber in the country.2 3 The Forest Service uses both oral ascending and sealed-bid auctions for selling its standing timber. I focus here on sealed-bid auctions for the year 1979. There are a total of 378 auctions held in 1979 in these regions involving a total of 1,400 sealed bids from sawmills. 23The data analyzed here (Mine from the Regions 1 to 6 , covering the states of Idaho, Montana, North and South Dakota, Nebraska, Kansas, Colorado, Wisconsin, Arizona, New Mexico, Nevada, Utah, California, Oregon and Washington. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The data contains a set of variables characterizing each timber lot on sale varying from the various species included in the lot, the estimated volume measured in mbf, the logging cost in dollars, the acreage of the lot, the term of the contract measured in months, the month during which the auction was held, the location of the lot, the total reserve price in dollars and the appraisal value in dollars.2 4 The latter is an estimated of the value of the lot taken into account the quality and quantity of timber. In addition to these variables, the data provides the number of bidders who have submitted a sealed bid as well as their bid in dollars and the identities of bidders. The following table gives some summary statistics on the bids per unit of volume mbf, the reserve price per unit of volume, the appraisal value per mbf, the volume in mbf, the density computed as the ratio of the volume per acre, the acreage and the number of bidders. Table 3.1: Some Summary Statistics Variable Mean STD Min Max Bids 97.28 71.51 1.05 1,149.28 Reserve Price 62.95 46.01 1.00 217.36 Appraisal Value 57.07 45.41 1.00 199.58 Volume 1,621.93 3,153,48 11.00 23,500.00 Density 2.05 5.17 0.002 46.43 Acreage 1,348.35 3,590.69 1.00 38,850 Number of Bidders 3.70 8.98 2 12 The auctioned lots display an important heterogeneity in terms of size and quality. When regressing the logarithm of bids per mbf on a complete set of variables characterizing the auctioned lot including region dummies and seasonality effects, 24 All data, arc; exprosced in current 1979 dollars. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. only two variables have a positive significant impact, namely the number of bidders and the appraisal value. A larger number of bidders increases the competition and therefore the bids, while bids are increasing in the lot value.2 5 The appraisal value seems to be the best candidate to capture heteregeneity across auctioned objects.2 6 The auctions are organized with a posted reserve price. It is well known that this reserve price does not act as a screening device to participating to this auction.2 7 To assess such a statement, we have estimated the probability a bid being close to the reserve price. Using a nonparametric estimator, I have estimated the conditional probability Pr(po < b < (1 + 8)po\Z), where po denotes the reserve price, b the bid variable, 8 an arbitrary value larger than 0 and Z an exogenous variable capturing heterogeneity, namely the appraisal value. Estimated at the average value Z — 57.07, we find this probability to be equal to 1.4% for 8 = 0.05, 4.5% for 8 = 0.10 and 9.5% for 8 = 0.20. This results clearly show that few bids are in the neighborhood of the reserve price and that the possible screening effect by the reserve price is negligeable. It is also interesting to assess whether there is no causality effect between the number of bidders and the value of each lot, such as a relation I(Z). No strong relation has been found using various regression models including regression models.2 8 25 A quadratic term has been included as well to capture some decreasing trend after some value for the number of bidders as expected with common value. It does not appear to Ik; significant. All regression results arc; available upon request. 26Sueh a feature has been already observed in previous empirical studies, such as in Haile (2 0 0 1 ), though using different data. 27Scc previous footnote. The; same comment applies. 28The number of bidders is very slightly decreasing in the appraisal value. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It remains to discuss the adequacy of the private values paradigm to the data. Previous papers, which have analyzed these data, have used both private and common value models. The fact that bids do not decrease after some value in the number of bidders is in favor of private values. Though no formal nonparametric test of conditional independence exists, some regressions could be performed such as considering the subset of auctions with two bidders and regress one bid on the other bid while controlling for heterogeneity. Nonparametric estimate of a conditional correlation between the two bids can also be performed. The latter confirms the assumption that bids are independent as the correlation estimate is not significant. 3.7.2 Estim ation Results The first step consists in estimating nonparametrically the upper boundary b{z?) and the bid density at this upper boundary g(b{zt)) for I — 1 ,... , L. As a matter fact, this step needs to be conducted for each value I = 1 ,... ,12 separately to take into account of the dependence of the observed bids on the number of bidders. In practice, the data provide enough auctions for two and three bidders. Above four bidders, the number of observed auctions is too small for implementing a nonparametric estimator. As bids increase in the number of bidders, it is expected that at z given the upper boundary also increases in the number of bidders. Such estimate of the boundary has been conducted separately using the 107 auctions with two bidders and the 109 auctions with three bidders. No significant increase has been found. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In view of this and to avoid curse of dimensionality problems, I have prefered to pool the data and to estimate an unique upper boundary for the 378 auctions. In practice, I have chosen a partition of the variable z over the interval [0,199.58] into 20 equal bins of approximative length 9.93. For each bin, I have esti mated the straight line defining for any Zf belonging to the bin the estimated upper boundary b(zf).29 I need to specify a one-sided kernel defined on [— 1,0] satisfying <fr(x)dx = 1 and being of order one, i.e. J^x<ft(x)dx — 0. The linear kernel 6x + 4 satisfies such assumptions. The kernel is then defined as < f> (x) = (6.x + 4)#( — 1 < x < 0). For the If bids corresponding to the value Zf, I have the same estimated upper boundary b(zf). To follow the notations of previous sections, Yu is estimated by (1 /hpf)4>((bu — g{z())/hjy), where h.^ is chosen following the rule of thumb proportional to 1,4001//5.3 0 The second step consists in estimating the parameter of risk aversion 6. Following previous studies in the experimental literature on auctions, I choose a CRRA specification, namely U(x) = x°, where the risk aversion parameter reduces to a single parameter at given wealth. In this case, m(zt\(3) takes a simple form, which gives the following equation to estimate 9 Yu = — — =-—- + eu + eu, (3.14) 7) — b{Z() 29Thc number of bins has been arbitrarely chosen. I have tried larger numbers such as 30 and 40. The obtained estimated upper boundaries are different but this docs not affect the estimation results for 0 in the second step. 30I could as well use an aggregated estim ator by averaging over the If estimates for each auction It gives equivalent results for the following step. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where eu is a vanishing bias and eif is an error term. Following Section 3.6, the estimator of (9,v) relying on weighted NLLS is obtained as 0,%) = Argmin(^ )GeQ(0,n), where L h ( 9 V QW'*) = Y , Y , * M \ Y a ------------------- 3-------- . (3.15) f c t b t V (I, - l)(v ~ b(z,)) J where the optimal weights are equal to (A — l)(v0 — b{zf)). This estimator can be implemented by a standard two-step procedure in which the optimal weights are first estimated by ordinary NLLS. Another possibility to implement this estimator is to solve for its first-order conditions associated to the maximization of Q with respect to 9 and v. The resulting estimator has the same asymptotic distribution as defined by Theo rem 3.3. When deriving the first-order conditions, the estimator is similar to an IV estimator of a linear model whose error term is rj( = b(z()(Ie — l)Yu—v(l£ — l)Yu+9. In particular, (A - 1 )b(Z,)Ya = - 9 + v(If, - 1 )YU + T]u- This equation is linear is the parameters (9, v). The error term rju, however, is correlated with the regressor requiring the use of an IV estimator. The instrumental variables are found to be equal to 1 /((It — l)(n — v(zg)) and — 1 /((If. — l)(u — v(zf))2. These instruments are the optimal ones as defined by Chamberlain (1987).3 1 In practice, this estimator can be implemented through a 31 Following Chamberlain (1987), the optimal instrum ents are defined as E(dp(Yu , ze,0Q,vo)/d(9,v\z() x (E(r?f|zeH ))~ \ where % = p(Yu , zf_,0o,vo) = b(zf) ( h - 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two-step procedure, where the first step involves a standard IV estimator with a vector of instruments (1 ,z,z 2,...) providing an estimate for v, which can be used for the optimal instruments in the second step. This estimator has one main advantage. It avoids to maximize the objective function Qn {8,v) while imposing the constraint on the set of parameters. In particular, constraining v to be larger than b(zg) for all £ can raise some problems when the data set contains some outlying bid observations. In the data set, the bid taking a value at 1,149 is clearly an outlier in the data set as the second and third highest bids are in the 700 range while the remaining 1,397 observed bids are below 500. As a result, there is an important sensitivity to outliers. Turning to the first-order conditions avoids such a problem. Moreover, it allows to determine the neighborhood of the true parameters. Theorem 3.3 derives the asymptotic distribution of the estimator for 0 and v. In practice, it suffices to compute the matrix Aj!1(0,v) and its inverse to derive the variance of the estimator. Using in the first step [1 z] as instruments, I find 0 = 0.3936 with a standard error equal to 0.2860, while v = 242.1988 with a standard error equal to 2.5198. I am interested in testing whether bidders show risk aversion, namely whether 0 ^ 1 . If 6 = 1, bidders are risk neutral. The test 0 = 1 leads us to reject the null hypothesis at 5% with a t-value equal to -2.12. Thus bidders are risk averse with a constant relative risk aversion coefficient of 0.6064.3 2 As an upper bound v independent of z can be restrictive in l)Yu + 0q — v0{I, — l)Ytf. If another specification for the utility function would be chosen, it would lead to a nonlinear IV estimator. 32This coefficient is close to the, one found in the experimental literature at about 0.5. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. practice, we have tried a linear parameterization of the upper boundary, namely v{z() = 7 q + = 1 ,... , 378. The same method as described above applies with some adjustments, in particular there are three parameters to be estimated instead of two. The estimate for 7 1 appears to be non singificant and the estimate for 6 is very similar. Such risk aversion implies that bidders bid more aggessively relative to the risk neutral case as they shade less their private values. Note also that such risk aversion coefficient in a CRRA model is equivalent of having more competition in the auctions. Namely, for an average auction with 4 bidders, a risk aversion parameter at 0.6064 is roughly equivalent of having about 8 bidders in an auction with risk neutrality. The third step can be then implemented. Using 6 — 0.3936, the private values can be recovered from (13). Applying the rule of thumb for the constants and the appropriate vanishing rates gives hg = 253.53 and he = 322.77. The estimated inverse equilibrium strategy is increasing in h satisfying the restriction imposed by the model on observables.33 I observe some boundary effects. As a result, some observations need to be trimmed out for the estimation of the underlying condi tional density of private values. The estimated conditional density is displayed in Figure 3.1 and has been obtained with a bandwidth h,f equal to 279.48. To assess the impact of risk aversion, I can compute winners’ gain in value and per centage for the auctions for which a good estimate of the private value is obtained. 33Sevcral graphs have been made for the different quartilcs of Z. They are available; upon request. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.1: Conditional density The results are given in the following table. As expected, risk aversion, which renders bidding more aggressive tends to reduce the gain for the winners. Table 3.2: Summary Statistics on the Winner’ s Gain Winner’s gain Mean STD Min Max in value 42.96 84.99 0.57 992.77 in % 19.83 12.18 3.11 75.73 It would be interesting to find an economic rationale for such risk aversion. Such risk aversion has been suspected by many authors. See Athey and Levin (2001) and Baldwin (1995). Their main argument is that bidders face uncertainty about the exact volume of each species in a lot leading bidders to split their bids across different species. The split of bids is then an indicator of bidder’s risk averison. Risk aversion has also been found in a different data set of timber auctions in which there is no bidding on species and bidders pay for their bids and not for harvested timber. See Perrigne (2001). A reason could arise from 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the uncertainty of the supply of timber and the limited outside opportunities regions of the US, the USES is a large supplier of timber. It is likely that given the uncertainty of the supply outside these auctions, it is important for bidders to win these auctions. Though more empirical studies need to be performed on other sectors, the experimental literature shows that overbidding is frequent though financial stakes are almost inexistent in experiments. It seeins that risk aversion would be a natural component of agent’s behavior when facing uncertainty. See the recent work by Goere, Holt and Palfrey (2001), where devi ations from the risk neutral Nash equilibrium are mainly explained by risk aversion. Measuring risk aversion is important for the seller when implementing the auc tion design. Though the optimal mechanism with risk averse bidders is especially difficult to implement as it involves some transfers (see Maskin and Riley (1984) and Matthews (1983)), an optimal posted reserve price can be set generated more revenue for the seller. It is can easily computed that for c ^ 1/J. the optimal reserve price is solution of When considering v = 0 equal to the appraisal value for a lot with average charac teristics in terms of value and number of bidders, I find p * 0 equal to approximatively $93. The same estimate conducted for an auction with risk neutrality following Riley and Samuelson (1981) result gives $132, which is significantly larger. The idea is that because bidders tend to bid more agressively with risk aversion, the precommitment effect does not need to be as important therefore reducing the besides the timber auctions organized by public institutions. In the western 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. level of the reserve price generating the maximum of profit for the seller. These figures show that assesing risk neutrality when implementing an optimal reserve price policy when risk aversion prevails may have dramatic affects on seller’s profit and revenue. 3.8 C onclusion This chapter is a step stone in the structural analysis of auctions as it solves the issue of risk aversion in symmetric games of auction. The difficulty to identify the structural elements of the game, namely the bidders’ utility function and private value distribution in a model with potentially risk averse bidders, explains why the subject was disregarded within the structural approach, until now. It shows that models of risk aversion are unidentified in general, and that the models stay unidentified, even for CARA or CRRA utility functions. Moreover, any model with either CARA or CRRA utility functions can explain the observations as the rationalization conditions turn out to be identical under either one of these specifications. Identifying the agents’ utility function and their private value distribution requires two additional restrictions on the primitives: the utility function should be parametrized, and the private value support, contrained. In the case of the U.S. Forest Service timber auction, I exploit the auctioned object heterogeneity by fixing an unknown but common upper bound on the private value distribution, across auctions. If bidders exhibit a CRRA utility function, the inverse bidding strategy evaluated at this upper bound is a function of the bid distribution, the risk aversion parameter and the private value upper bound. It suggests a multistep semiparametric procedure as the equation is a linear function of the unknowns: First, estimate the variables of the equation, Second, conduct 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the regression such as to recover estimates of the parameters of interest. Because the variables of the regression are nonparametrically estimated, the original estimator of the risk aversion parameter exhibits an optimal rate of convergence lower than a / / V (where N is the number of observations).3 4 The application conducted on the Forest Service timber auction data reveals that bidders are risk averse as the estimate of the P ratt’s measure of relative risk aversion is equal to 0.61, significantly different from 1, the risk neutral value. It implies that bidders enjoy lesser informational rents from the auctions as they bid more fiercely for the timber. This is may be explained by the small number of alternatives, offered to the bidders, to buy timber outside these public auctions. Building upon this methodology, I relax in the next chapter the assumption of homogeneous risk averse bidders. Surprisingly, the identification and estimation method are simplified by allowing heterogeneity across bidders. 3 4The properties of the estim ator arc detailed in Campo, Guerre, Perrigne and Vuong (2002). 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 A sym m etry and Risk Aversion w ithin the Independent Private Paradigm: The case of the Construction Procurem ents. 4.1 Introduction Vickrey’s (1961) founding paper of the auction literature studies a private value model with symmetric risk neutral agents. Within his model, agents vahie the object to be sold independently from one another. Because their private values are drawn from the same distribution, the agents are said to be symmetric. Their utility is equal to the net gain, private value minus bid: bidders are risk-neutral. Under these assumptions, Vickrey shows that first-price sealed bid auctions and oral auctions yield the same revenue to the seller. This result, known as the equivalence revenue theorem, is strongly dependent on the assumptions of the model: Independent private values, symmetry and risk-neutrality. The violation of any of these assumptions leads to important revenue loss/gain for the seller, depending on the choice of auction. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Empirical facts suggest, however, that bidders are asymmetric, and potentially risk-averse. Asymmetry lies in backlog capacities (Jofre-Bonet and Pesendorfer (2000)), location (Bajari (1997), Flambard and Perrigne (2000)), informational advantages (Hendricks and Porter (1993)), bidders’ sizes due to either joint ventures (Campo, Perrigne, Vuong (forthcoming)) or cartel formation and bid rigging (Porter and Zona (1993), Bajari and Summers (2001) survey). Bidders may also be risk averse because they fear to lose the auction, even more if the object value is important relative to their asset holdings. In experiments, Cox, Smith and Walker (1985), and more recently Goere, Holt, Palfrey (forthcoming) explain bidders overbidding, relative to the Nash risk neutral equilibrium, by the presence of risk aversion. In U.S. Forest Service timber auctions, where firms bid per species, Athey and Levin (2001) and Baldwin (1995) found evidence of risk aversion in the spread of bidding aceross different species in a same lot. Both papers adopt a reduced form analysis, which, as they outline themselves, cannot help estimate the bidder’ s risk aversion. For this purpose, Campo, Guerre, Perrigne and Vuong (2002), and Perrigne (2001) adopt the structural approach in models with independent private values to identify and estimate the agents’ risk aversion. Within this approach, the observed bids are the bidders’ Bayesian Nash equilibrium bids.1 The main issue, as raised in chapter II , is the difficulty to isolate and estimate the risk aversion parameter, which stays unidentified for any concave utility function. In chapter II, I achieve identification by imposing some restriction on the private value support. 1 Gocre, Holt, Palfrey (forthcoming) use the Quantal Response Equilibrium concept. The equi librium is derived from a logistic model which explains how deviations from the Nash equilibrium may arise due to errors. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Even though bidders share the same risk aversion parameter, identification is impossible without imposing some restrictions on the other primitives of the model. The major drawback of these last models lies in the assumption that bidders share the same behavior towards risk. Bidders’ characteristics such as their size, cash-flow, age, insurance coverage, experience should affect differently their behavior towards risk. Building on the results of existence derived by Athey (2001)2, I adopt the structural approach to study a model of auction with asymmetric utility functions. My purpose is to identify and estimate the bidders’ heterogenous behavior towards risk. In section 4.1, I define a model where agents are risk averse but exhibit different risk-aversion coefficients. Unlike previous papers which defined asymmetry in bidders’ different private value distributions, asymmetric bidders are here agents drawing private values from the same distribution but evaluating risk and gain differently: Agents have different risk-averse utility parameters. The model leads to appealing identification result and estimation procedure if the agents exhibit Constant Relative Risk Aversion (CRRA) utility functions.3 Identification and estimation of the model rely on Guerre, Perrigne, Vuong (2000) indirect approach. In section 4.2, I show that the private value distribution as well as the agents’ risk-aversion coefficients are semiparametrically identified when the bidders’ utility functions belong to the family of constant relative risk aversion utility functions. Note that identification is achieved without imposing any restriction, 2Athey shows the existence of a monotonic strategy in a game where agents exhibit heteroge nous utility or asymmetric private value distributions. 3 P ratt (1961) gives a detailed and complete definition of risk aversion in utility functions. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unlike in Campo, Guerre, Perrigne, Vuong (2002). This result follows from one simple characteristic of the model: different bidding strategies originate from the same private value distribution. In Campo et al. (2002) no such heterogeneity can be exploited from the primitives of the model as there exist a unique risk aversion coefficient and a unique private value distribution. The multi-step parametric estimation procedure directly relies on the identification result. Section 4.3 presents the estimation procedure . Because bids (and private values) are observed if only if they lie above a threshold, unknown to the analyst, the choice of the estimation method is non trivial. Paarsch’s extensive work (see Hendricks and Paarsch (1995) survey) focuses on the issue involved in estimating a distribution which support depends on the parameters to be estimated. Donald and Paarsch (1993) introduces the piecewise maximum likelihood estimator which properties hold if there is no exogenous variable. Bajari (1997) solves the problem by using priors on the boundaries. Instead I choose to apply the boundary estimation procedure introduced in Korostelev and Tsybakov (1993) to estimate the support of the bid distribution. Since this estimator converges faster than \/fV, one can then apply results from the censored model literature to the estimation of the bid distributions. Bajari (1997), Bajari et al. (2001), Porter and Zona (1993), Jofre-Bonet and Pesendorfer (2000) study the market for public construction contracts to highlight the existence of ex-ante asymmetry among firms: Firms may either have different capacity constraints or decide to form a bidding cartel ( also called bid rigging). Asymmety lies there in the firms’ private cost distribution. Instead, I assume 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that asymmetry arises through different attitudes towards risk, related to, for example, the firm’s experience. From interviews with professionals in this industry, contractors consider building and constructing as activities of high risk. Every new project brings its own challenge, its own risk, and may even lead to bankruptcy as argued extensively in Schleifer (1990). These two observations motivate my study of the construction industry to illustrate the firms’ potentially asymmetric risk aversion. In the case of the constuction contracts offered for bidding by the Los Angeles City Hall between 1994 and 2001, the firms’ bids suggest that the contractors’ capacity in risk management depends on their experience in the industry. Section 4.4 presents the estimates of the different risk aversion coefficients according to the firm’s experience. My estimations show a downward trend in risk aversion, consistent with the fact that older firms with potentially larger assets are able to diversify and control better for risk than younger firms with potentially less capital. 4.2 T he m odel A contract is offered to bidding through a public procurement auction. Each bidder submits a bid (prices), 6 * , i = 1, •••,/, in a sealed envelope. The public institution organizing the auction awards the contract to the lowest bidder who receives as payment her bid. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W ithin the independent private value paradigm, each bidder knows her own cost Cj, i.e. her cost of completion of the contract, but not her competitors’ costs.4 The bidders’ costs are drawn from the same distribution F(-), which is absolutely continuous with density /(•) on the support [c, c] C JR+. The distribution F(-), the agents’ endowments of wealth {uy}f=1, and the number of participants I are assumed to be common knowledge. Let [/,;(•) be agent Vs von Neuman Morgenstern (vNM) utility function. Agents are potentially risk-averse: U-(-) > 0, U"{-) < 0, i — 1 ,•••,!. Agents are asymmetrically risk-averse: [/*(■) differs from Uj(-), j ^ i such that the agents’ measures of risk-aversion are different, while [/;(•) and Uj(-) are assumed to belong to the same parametric family of utility functions.5 Assume I > 2, agent i chooses her bid hi : such as to maximize her expected profit from the auction: Ei(Ui) = Uiiui + bi~ c H ) Pv(bj < bj,Vj ^ i) + Ui(uji) Pr(6, > bh for some j ^ i) Uiiui + bt-d) [1 - F i s f i b ^ + U ^ ) where , s = 6 * is agent Vs equilibrium bidding strategy and her inverse strategy, for i = 1 The equilibrium strategy s*(-) is strictly increasing on [c,c] and expresses the equilibrium bid as a function of agent r s cost, the 4Costs include the base cost, namely the construction cost, and the overhead cost (manage ment cost). Because contractors are more or less efficient and bear different opportunity costs, costs can be considered idiosyncratic to each firm and unknown to the firm’s competitors. 5If agents i and j exhibit a CRRA utility, their coefficients of relative risk aversion < 9 * and 0 ,- should satisfy 0 , ; / 0 ,-. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cost distribution F(-), the agents’ endowments of wealth {o^T ;}f=1, the number of bidders J, and the bidders’ utility functions The Bayesian-Nash equilibrium strategy satisfies then the first-order differen tial equation6 U'M + bj - a) [ i - F(sjl(b,))] - [UM + h - a) - UM)] e , , , ,- v n t1 -%■(«)]=«. < 4 ^) J ' J \ *// k=l,kjtj,i with the initial conditions s,;(c) = c, V i = 1, • • • , I and s,:(c) = s? (c), V i ^ j, h i = 1, • ’ ' >^-7 By rearranging the terms in equation (4.1), the first-order condition appears more familiar as8 .. _ [Uj(wj + F — Cj) — Uj(o>j)] / ( gj (M )________ 1 /. + h - C i) 1 - F (sj \ bi)) ' 6I assume second-order conditions being automatically satisfied so th at the equilibrium is determined by the first-order conditions only. 7Not,iec th at by definition of the Bayesian Nash equilibrium, bidder i knows each one of his opponents’ inverse bidding strategy, «,(•), and his opponents’ risk averse utility functions. This assumption may seem unrealistic but can be motivated either in small industries or in industries where firms organize themselves into associations. In both cases, I have described industries where the competition is well known. 8In a first-price sealed bid auction with symmetric risk-averse bidders, we have f(ci) U (w,; + hi - Cj) - Ui(u>i) 1 1 = (I - 1) 1-F(c,;) U Uwi + bi-CH) s’iaY 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The above differential equation does not have any analytical solution. In a different framework, with risk neutral agents and asymmetric private value distributions, Marshall, Meurer and Richard (1994) solve numerically for the equilibrium bidding strategies. I choose to avoid these cumbersome computations and follow instead Guerre, Perrigne and Vuong (2000) indirect approach. Their method enables me to study the model rationalization and identification without solving explicitly for the equilibrium bidding strategies. 4.3 Identification This section states the conditions for rationalizing the model and the conditions for its semiparametric identification. Throughout this section, I assume that I observe the number of participants /, the agents’ bids hi, their bid distribution functions Gi(-), and wealth endowments Ui, V i = 1, • • • , J.9 4.3.1 R ation alization The observed bids are rationalized by the aforementionned auction model if there exists a structure [{f/,:(-)}f=1, F(-)] which can explain these observations. I will study the uniqueness of this structure only once its existence has been proven. The bidders’ first order condition in (4.2) defines the observables as a function of the unobservables { ( / * ( - ) , c7 ;}f=1 and F(-): bt = S j(c ,;, Ui, F, cp,;). As suggested by Guerre, Perrigne and Vuong (2000), I rewrite equation (4.2) as a 9Thc wealth endowments are observed and known to the agents, which creates ex-ante asym m etry in the game. If these endowments were unknown to the agents, the model would become a, double signal game, which is out of the scope of this chapter. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function of the observables and potentially estimable arguments only. For e M = Ms),4 G#i) = F(»jl(W ) and aft) = /(»7*(W)/*J(*7'(W). («4> becomes (4,3) where A ? ( a C / - bj Ci) UiiuJi -f- Cj) Ui (uJi ) U-(uii + hi - Ci) I define U^jibi — c,;) = £4(uy + 6 ,; — q ) — Ui(u>i). The change of notation is not trivial. Maskin and Riley (2000) define a model where the agents’ loss utility is equal to U(0) = 0. Instead I define a model where the agents’ outside options Ui{ui) (in the case of a loss) are different across agents, but where the function UUui{-) satisfies £4,^(0) = 0. The reparametrization brings back the analysis on the familiar ground defined by Maskin and Riley, without restricting the agents’ loss utility to be the same across bidders. The function A ,;(•) becomes From (4.3), I then recover the inverse bidding strategy £*(-, • • • , •), for i = 1, • • • , I, (4.4) 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where G stands for the vector of distributions (G\, • • • , Gj), and YJ(E = From iny previous papers, I borrow the familiar regularity conditions on {Ui(-)}j= 1 and F(-) gathered in Definition 4.1 and 4.2. These assumptions will appear to be necessary and sufficient for the model rationalization (cf. Lemma 4.1). D efinition 4.1: Let be the set of utility functions UW iti(-) satisfying (*) : [0, +oo) -+ 1R+, UU i,i(0) = 0 and UUiji{ 1) = l . 1 0 (ii) UW ij{ ) is continuous on [0, +oo), and admits 2 continuous derivatives on [0, Too) with E ,;,* (■ ) > 0 and E E ( ') - 0 on (°> +°°)- (Hi) limx_ > 0- is finite. Condition (i) is an important normalization on the agents’ utility space. I implic itly assume that an agent’s utility in case of a loss is equal to the utility she derives from a win where the agent pays her exact private cost. Her utility is in both cases equal to Ui(u>i).n The utility function is required to be monotonic with respect to its argument: U'(-) > 0, and potentially concave, U"(-) < 0. Conditions (in) is borrowed from Guerre, Perrigne and Vuong (2000). Along with (ii), condition (Hi) implies that _ ,;(•) admits a continuous derivative over [0, oo). This will help define the inverse bidding strategy properties as £*(•) is a function of A^,^-) (see equation (4.6)). Note that popular risk-averse utility functions, such as CRRA and CARA utility functions, satisfy these three conditions. 10Notc that by definition UUiti(x) = Ui(x + w ,:) — Ui(wi). 11In experimental economics, authors question this assumption: An agent may enjoy a positive utility from the only thrill of winning the auction. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The private cost distribution satisfies the following assumptions: D efinition 4.2: Let T C V be the set of distributions F(-) such that 1 2 (i) F(-) is a c.d.f. with support of the form [c, c], where 0 < c < c < oo. (ii) F(-) admits one continuous derivative on [c, c]. (Hi) /(•) > 0 on [c, c]. The proof of rationalization requires a continuous and everywhere positive den sity function, as assumed in points (ii) and (Hi).13 The private cost support is assumed to be compact. It implies that private costs cannot follow, among other distributions, any log-normal or exponential distribution. Because equations (4.2) and (4.3) are the same expression of a representative bid der’s first order condition, Definitions 4.1 and 4.2 imply in turn some restrictions on the bid distributions (?*(•), for i = 1, • • • , /. These conditions are gathered in Definition 4.3. D efinition 4.3: Let Q be the set of distributions (?,;(•) C V, for i = 1, • • • , I, such that (z) Gi(-) is a c.d.f. with support of the form [6, b], where 0 < b < b. (ii) Gi(-) admits one continuous first derivative on [6,6]. (Hi) gi(-) > 0 on [6,6]. r2V is defined as the set of probability distributions F(-) on M+ such that F = {F (-) is absolutely continuous with an interval support in 2R+ }. 13These conditions art; common to the proofs of existence in models with either asymmetric private costs distributions or symmetric risk averse bidders. To my knowledge, there does not exist any paper which studies the existence and unicitv of an equilibrium when bidders exhibit different utility functions. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (iv) Gj(-) admits two continuous derivatives on \b,b). (?;) lirnb _ ^ 5 d[l/Yi(bj\/db = 0. Condition (i) follows from the boundedness of the private cost distribution support: c G [c, c]. To ensure that Si(b) = c, I still need to impose condition (u). Finally the properties of differentiation of the bid distribution (?,;(■) in (ii) and (iv) are shown in appendix. They are derived from the simple observation that the agent Vs bid distribution is equal to the private cost distribution F(-) trough the appropriate change of variable = c,. Building on these three definitions, the following lemma states how the observed bid distributions are rationalized by an asymmetric risk-averse model if and only if there exists a structure F(-)] whose equilibrium bid distributions are identical to the given observed distributions. L em m a 4.1: Let I > 2 and Gi(-) be the distribution of 6 ,:, for i = 1, • • • , then is rationalized by an IPV model with asymmetric risk averse agents [{GW l,*(•)}[= i, F(-)} G Ul x T if and only if CO G(h,--- ,hj)^YlliGi(k), (ii) Gi(-) G Q, for i = = l,--- (Hi) such that G,:(6“ ) = Gj(bJ) = a,a G [0,1),3 { s a t i s f y i n g : (a) A ^O ) : M+ — > 1R+ with one continuous derivative on (0, Too], XWi,i(0) = 0) A C,,,;(•) > 1, such that d£/dbi > 0 for &(&“, Ui,Ui, G, I) = bf — Aj,p [1/> W )] and (b) [1/K(4f)l - [1/J5(«f)]. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These conditions are the necessary and sufficient conditions to the model rationalization for strictly increasing equilibrium strategies. Condition (i) is a direct consequence of the independent costs assumption: if the agents’ private costs are independent, their bids are also independent. In (ii), because the observed bids are the expression of the agents’ equilibrium strategies for a given cost, their distribution functions £?;(•) satisfies Definition 4.3 ( for the details of the proof see appendix C). Note that Definition 4.1 assumes a compact private cost support. I preserve such property by assuming that gi(-) is bounded away from 0 and limb ^ b _ (1 — G*(6))/gi(b) is equal to zero (conditions (Hi) and (iv)). The bid distributions G,;(-) satisfying Lemma 4.1 conditions (i), (ii) and (Hi)(a) can be rationalized by models with either asymmetric private cost distributions (Campo, Perrigne and Vuong (2002)) or symmetric risk averse agents (Campo, Perrigne, Guerre and Vuong (2002)). Instead a model with asymmetric risk averse bidders imposes a new restriction upon the bid distributions. Condition (m)(b) defines this distinctive restriction. It is derived as follows. Suppose that two agents, called i and j, draw the same private cost c then F(c) — a. Because agents share the same private cost distribution F(-). They submit bids bf and 6“ satisfying Gi(bi) = Gj(bf) = F(c) = a, because the agents’ equilibrium statregies Sj(c) = h and Sj(c) = bj are increasing functions of the cost. For different coefficients of relative risk aversion, these bids are different, i.e. 6, ^ bj. Thus I am able to recover two different bids which originate not only from the same cost c but also 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from the same cost distribution function F(-). Agents i and j inverse bidding strategies are c = U ,,Ui, G, /)=(,»- A -;, (1 /Y ,< m ) . and c = <i(d?,i7j , ^ , G , / ) = i . J - A - 1 i i ( l / y i (6J)). By equating these two inverse bidding strategies, I recover the condition (in), namely K - > > 1 = Kli [1/W ) 1 - K-J [1 / % “)] ' This lemma gives an alternative definition of the Bayesian Nash Equilibrium. The equilibrium is defined as the vector of increasing functions (£i(-), • • • ,£/(•)) such that J m , u , , u „ G , i ) = \ i f - f ? = for i = 1, • • • , I, i ^ j and a £ [0, l).1 4 Unlike equation (4.2) which defines the equilibrium as the solution to an unsolvable first-order differential equation, this system defines the equilibrium in terms of the inverse bidding strategies £»(•), i = 1, • • • , /, and highlights one crucial property of the model: agents i and j share the same cost distribution. 14I exclude a = 1 because 1 — (?*(•) appears in the denominator of Yi(-). 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In a model of symmetric or even asymmetric private value distributions, the equi librium is only defined by the first equation in system (4.5). This equation defines the agents’ inverse bidding strategies. It implies that the set of bid distributions rationalized by a model of either asymmetric private cost distributions or symmet ric risk averse bidders Q is larger than the set of distributions rationalized by a model of asymmetric risk averse agents Q if and only if the second equation of the system defines a non redundant restriction on the bid distributions. Suppose that two agents compete for the same contract. Agents 1 and 2 exhibit CRRA utility functions, Ui(x) — x 0i, for * = 1, 2, 9\ ^ 02, and have the following bid distributions: Gi(b) — 6 , for b G [0,1] and G2(b) = (5/8)6, for 6 G [0,4/5], G2(b) = - 3 /2 + 5/26 for 6 G [4/5,1], Let 6 j = 2/3 and 6 2 = 13/15, then G i(6 x) = G2(b2). Condition (m)(b) becomes 2________ 9\ 3 3 (7? ,] - 1) + f n 2 3 76\ 15 + 21(77,! - 1) + 1 5 7 7 ,2 and 02 equals [3(77,1 + 77,2 - 1)] [21(771 — 1) + 1577,2] + 7 X 15(77,1 + 7 7 ,2 - 1)#1 2 [2 l(7?,i - 1) + 1577,2] ' Since 77,1 and n 2 satisfy 77,1 + n2 > 2, the parameter 02 as defined above is greater than one, which contradicts 02 G (0,1]. It implies that there exist two distributions Gi(-) G G and G2(-) G Q such that, for at least one pair of bids (6 1, b2), condition (m )(b) is violated: Gi(-) and G(-) do not belong to Q. Thus Q C Q and condition (in) defines an original restriction on the bid distributions. 138 1 3 _______ 2 (9 2 15 15(77,2 + 7 7 - 1 — 1) ’ 202 15(77,1 + 77,2 - 1 ) ’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Condition (in) distinctive role in defining the asymmetric risk averse model and the condition to its rationalization explains its key role in the model identification and estimation procedure. 4.3.2 Identification result for CRRA utility functions In the previous section, I have defined the necessary conditions for the existence of a model [{U i}1 ^ , F(-)] explaining the observations i- In this section, I define the condition of its uniqueness for a given parametrization of the bidders’ utility functions £/*(•). Note that the vector of wealth endowments {uy}(=1 is set to zero, all agents have the same wealth.1 5 Suppose {/,;(•) belongs to the set of Constant Relative Risk Aversion utility func tions (CRRA) defined R S Ui(x) = x 0i, where 1 — is agent Vs coefficient of relative risk aversion. Agent i is risk neutral for 6i = 1. For i = 1, • • • , I, the risk aversion parameter ( 9 * is restricted to belong to the interval (0,1] because ny is zero. The bidding strategies are non decreasing in the bidders’ private cost at this only condition. An appealing characteristic of CRRA utility functions is that risk aversion is simply measured by the relative risk aversion coefficient 1 — 0, asymmetry in risk aversion is defined by 0 ,: ^ 0j. Where previous literature identifies asymmetric private cost distributions as the source of 15Sctting Wi = 0 is a strong theoretical assumption. It is necessary not only for the identification result but also for the application since wealth is an unobserved variable in my d ata set. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. asymmetric bidding, my model explains the observed asymmetric bidding by the existence of asymmetric utility functions as & i ^ Qj «t=t- Gi(-) ^ Gj(-). To estimate the agents’ risk aversion coefficients and private cost distribution, I first need to prove that there does not exist two structures {{UUui( )}{=1, T'(-)] and [{t/wj,?:(•)}> F(-)\ explaining the observed bids (or bid distributions). In the case of CRRA utility functions, the bidders’ utility functions are defined by the vector of risk aversion parameters I show that, without imposing any ad hoc assumptions, the vector of parameters explaining the observations is unique. Once the utility function identified, proving the existence of a unique private cost distribution F(-) explaining the observed bids is straightforward since I fall into the case described by Guerre et al. (2000) where the only unknown primitive is the private value (cost) distribution. Suppose I observe £?,;(•) ^ Gj(-). Agents are then characterized by a different 0; and a different inverse bidding strategy ^(-) defined as ( 4 -6 ) Under the CRRA specification, condition (m )(b) in Lemma 4.1 becomes fj n . ha _ ha — i__________ i — (A 7) • 1 w ) Yj(b?y for (bf, ¥■) such that Gi(bf) = Gj(bj) = a . 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The bidders’ costs have disappeared from equation (4.7), and the parameters O f and 6j are the only remaining unknowns. Since the equality holds for all pair of bids (bf, bf) and for any value a G [0,1), equation (4.7) defines a system of an infinite number of equations for a finite number of unknown parameters, (6fi, • • • , < 9j). The formulation of a CRRA utility function simplifies the Lemma 4.1 identification condition into the following full rank condition. L em m a 4.2: An asymmetric CRRA-IPV model [F(-), is identified if and only if there exists at least one pair of distributions (Gi(-), Gj(-)) such that Yj(hT) r m 2 ) Yj(bf) * Y,(bfY for = G # “‘ ) - a t , a £ [0,1), k = 1,2 a , ^ a2. The intuition of the result is rather simple. Suppose two agents share the same private cost c for a project, but have different relative risk aversion coefficients 9\ and 02- Assuming that there are only two types 1 and 2, the agents bid b\ and b2 and their respective inverse bidding strategies are | c = U K G l(-),G2(-)dl), I c = U h , G 2(-),G1(-),e2) 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I have a system of 2 equations in 3 unknowns (c, 9\, 62). Identification is not yet possible. Suppose now that the two bidders have another private cost c'. They bid respec tively b[ and b ' 2 and their inverse bidding strategies yield to By stacking the two systems, I finally recover a system of 4 equations in 4 unknowns (c, c', Oi, 62). Identification is then achievable as stated in Lemma 4.2 (for a complete proof see appendix B). Here is a surprising result: Although the number of parameters to be estimated in a model with asymmetric risk averse bidders is larger than in the case of symmetric risk averse bidders, identification is easier. In the case of symmetric risk averse bidders, Campo, Guerre, Perrigne and Vuong (2 0 0 2 ) achieve identification by imposing an additional restriction on the private value (cost) support. A quantile of the value distribution is the same across het erogenous goods. I avoid any new restriction in the asymmetric case because the model itself conveys enough information on the private cost distribution: Agents have different bid distributions but share the same private cost distribution. As a consequence, I can recover a unique F(-) from the observation of different bid distributions Gj(-). This property obviously does not hold if we observe a unique bid distribution as in the symmetric risk averse case. Note that I am not merely c c' 6 ( & k G 2(.), <?!(.), fc). 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. substituting one restriction on the private cost support by a restriction on the private value distribution. I can still relax the assumption that bidders share the same private cost distribution in a model of asymmetric risk aversion and keep my identification result, but I cannot relax the quantile constraint in a model with symmetric risk aversion and private value distribution F(-) without losing identification (see Campo (in progress)). 4.4 E stim ation I have considered so far that a homogenous contract was offered to bidding. The identification and rationalization results found in previous sections apply as well to heterogenous contracts characterized by a vector Z of variables, Z C H Z1 . The private cost density becomes the conditional density f(-\Z). Let X denote the agent specific variables and 7 the vector of parameters influencing the bidders’ utility function, such that £/,(•) = U(-\X = 7 = 7 *). The vector 7 encompasses among other parameters the agents’ CRRA coefficients (coefficient to be estimated). Note that Uj{-) does not depend on the sales characteristics Z. In particular, it implies that, within a CRRA specification, the bidders’ risk aversion parameters do not depend on the variables Z. The following section proposes a parametric estimation procedure for estimating (i) the parameter 0j characterizing the bidder i’s CRRA utility function [/,(•), for i = 1 , • • • , /, and defining the bidder’ risk aversion, and (ii) the conditional latent density f{-\Z) of the bidders’ private costs. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I consider auctions with at least 2 bidders. Let L be the number of such auctions, £ indexes the Gth auction. The identification result suggests a three-step estimation procedure: • F irst Step: From observed bids, estimate G{-\Z = z,Xi = . 1 7 , 7 = 7 , ;) = Gi(-\Z — z) for a given z, defining the characteristics of th e contract, where Xi are agent i’s characteristics, and 7 * , agent Vs utility parameters. Note t h a t G(-|-) is truncated since bi t € [ft, b ]. • Second Step: Recover the pair of bids bf(zp) and bj(ze) such that Gj(bf(zf) |Z = Zf) = Gj(b^(zt)\Z = Z() = a for some a G [0,1). I denote them by b^(zi) and b'j(zf). Estimate Yi(bf, z) by distribution estimates obtained in the first step. Replacing Fj(-), and bids bf by their estimates, the equation system (4.7) becomes where e is an error term associated with the use of the estimates instead of equation variables are function of hi and as such endogenous. To recover the estimates of 0 I then use the instrumental variable method. where gj(-\Z = z) and Gj{-\Z = z) are the conditional bid density and the exact values of the bids and the bid distribution. The right-hand side 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • T h ird Step: Using equation (4.6), recover the pseudo costs c * for each agent, and estimate nonparametrically their conditional density f(-\Z = z). The next subsections detail each estimation step. 4.4.1 The bid distributions The bid distributions may be estimated nonparametrically if there is a sufficient number of observations. Unfortunately, the bid distributions depend on contracts as well as agents’ characteristics. The estimation would then suffer from the dimensionality curse. The number of exogenous variables, which increases with the number of bidders I, restricts the choice of the estimation method to parametric estimators. I define Bu = log(6«), where £ is the index of the auction or contract, and i is the agent’s index. Suppose agent’s i bid follow a truncated lognormal distribution with paramters (/q, cq) on [b, b], i = 1, • - • , I .1 6 I denote by H(-1-), the conditional distribution of the logarithm of the bids and by h,(-1-), its respective density. They are truncated as Bn belongs to the support [B _ , B], where jB = log(h) and B = log(b). Let H(-\z) be the standardized normal distribution conditional on z, the charac teristics of the contract, the truncated distribution ]?,;(• \z) is then equal to H (Bh) = H([B - /..M I M ) - H((B(z) - ,k(z))l<T0 ) - /ij(2 ))M ) - H((B{z) - 16If the upper bound of the bid distribution support B is large, the upper bound truncature lias little effect, on the estimations when considering a log-normal distribution. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for B £ [B(z), B(z)]. I assume that the distribution mean /x,;(z) = d o ,: 4- d1 ? : z is linear in the characteristics of the contract z. The lower bound of the distribution H(-\Z) , J3(z), is estimated using the trapezoid maximization method introduced by Korostelev and Tsybakov (1993). Suppose that B is a function of the sales characteristics z £ 1R+. The estimates c0 and c\ satisfy max _ / C o + ci(z — z)dz — coA + ctA 2/2. {(c0lci):B,i>co+ci (zt-z),i=1 ,- ,It,ze£[z,z)} J ^ where A is the length of the support [z, z]. The solution to the maximization program provides the estimated lower bound B_(z) = co;+ci,;(z — z) = cq + Ci(z — z), where z = min^ Zf. Similarly, the upper bound is estimated as the function B{z) = dw + du(z — z) = do + d\(z — z) with (d0, d\) solution of min _ [ d0 + d\(z — z)dz = d0A + d\A2/2. {(do,di):J5i(<do+</i(z£-z),i=l,— J z The boundary estimators converge at the uniform rate (log N/N)2^ (N stands for the total number of bids), which is faster than the conventional parametric rate of convergence y/N. I can then estimate the distributions idj(-|z), for i = 1, • • • , I using the maximum likelihood method, considering the estimated truncatures as given. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 The risk aversion estim ation The estimation of the risk aversion coefficients relies entirely on the identification condition < ? (* ) - w E T T T ? = * ? (* ) e > holding for all the pairs of bids (bf(z),bj(z)) such that, for i ^ j, Gi(b?(zf)\Z = z() = Gj(b°-(z{)\Z = ze ) = a. By an adequate change of variable, the identification condition becomes a function of the log normal distributions of the bids as estimated in the previous section, namely <48) where Yni{Bf) stands for the L-vector whose element is h,j(Bfe \ Z = z) 1 1 — Hj(Bf(\Z — z) exp (B£) ’ and (Bf (z),B^(z)) satisfy Hi(Bf\z) = Hj(Bj\z). Unfortunately I do not observe the pairs (B?(z), Bj(z)) satisfying this equality. An elegant way to circumvent this problem is to construct pair of estimates (Bf(z)1Bf(z)), satisfying Hi(Bf{zt)\Z = zf) = Hj(B^{zt)\Z = z() = a, for every value of Z (characteristics of the contract) and any given a E [0,1). 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tliese pairs of estimates define the left-hand side variables in (4.8). In a second step, I estimate Ym(-), for i = 1, • • - , I. I denote by Ym (-) their estimates. They are the right-hand side variables in (4.8). This completes the estimation of (4.8). The parameters < 9 ; are the only unknowns left as the identification condition becomes exp Bf{zf) - exp B a A z t) = YHi{Bf(zi)) YHj(Bj(zg)) + € ? ;r - (4.9) An important feature is to be stressed here: All variables in equation (4.9) are estimated, and not observed. This justifies the introduction of an error term e. If these variables were instead observed, the equality would exactly hold (cf. Lemma 4.1). The introduction of the error term makes the parameters of interest appear natu rally as solution of a regression in and 6j. Since the equation holds for every pair of bidders, = ,1, i ^ j and any value a € [0,1), the estimator efficiency increases by pooling all bidders’ (1 through I) identification conditions into one equation, 1 '• 1 ta 2 6 ? - iff 1 • • C C N ha 3 ha - ° I - 1 1 Y m ( B f ) Ym (B§) \ 0 0 1 0 0 0 0 -1 -1 YH2(B$) YHi(B f ) 0 -1 0 0 1 V / m - o W U ) Ynr(B f) 81 9i +€,(4.10) where bf — 6“ expBf- - expB f and l/YHi{Bf) are Liyj x 1 vectors which 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. typical element are bf{z() — b^{z() and 1 jYfn{Bf{z()) respectively.1 7 The error term e is a ((/ — 1)L1 j2 + • • • + Lj,i- i) x 1 vector. As rnentionned earlier, the right-hand side variables in (4.10) are endogenous: E(e\YHi, Z = ze, X = .x - ,:) ^ 0 for i = 1, • • ■ , I. Since Z and X represent all the variables of the model, the choice of instrumental variables cannot be straightforward if these are also the only observed variables. Lewbel (1997) studies a similar problem: he does not observe any exogenous variable which could play the role of instrumental variable. Let Xi = l/Y f/,;(5f), Lewbel suggests the instrument variable, which typical Zth element is (Xu — Xi)(Yi — Y), where h = 1, ■ • • , (I — l)L i ,2 H h Y is the left-hand side variable in (4.10), and X and Y are the respective average value of the vectors. I will use these instruments to conduct the estimation of the CERA coefficients i — 1, ■ • • , I. The optimal regression would consider every value of a G [0,1), but for simplicity purpose, the estimation was conducted for one given value of a such that Gi(ht) = Gi(bj) = at. (t is the L x 1 unit vector).1 8 n Li j is the number of auctions where there is at least one agent of group i and one agent of group j. The vectors size varies with the number of auctions where agent i and j compete. 18Further developments of the chapter will consider the extension of the estimation procedure to multiple values of a. 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 T he private cost distribution From the estimates of the coefficients of relative risk aversion 0t, i = 1, • • • ,/ , I recover the estimated private cost of the agents for any given bid b^. The inverse bidding strategies equation provides the values of the socalled pseudo costs : Y i ( b u ) i — 1, • • • , If, £ — 1, • • • , L, where Yi,(he.) = Ym{Bu) was estimated in a previous section. From the vector of pseudo costs cif_ , i = 1, • • ■ ,1? i = 1, • • • ,L, I can either estimate the conditional cost distribution or its conditional density /(-|-). I use a kernel nonparametric estimator of the density, l V ' b iz (c-Cu z-zt \ L h,i+1 Y , e = i Ie Z v ,;= i hc , hz ) f(c\z) = ---- J ------------------- , (4.11) L h% i hz 1 where the numerator stands for the kernel estimator of the joint density /(c, z), the denominator stands for the kernel estimator of the density f(z), and d is the dimension of the exogenous variable space Z. Note that the observations have been trimmed at the boundaries (using the same rule than in previous chapters) to avoid the boundary effects. The functions Kcz{-) and Kz(-) are triweight kernels of order 1 and 2 with bounded supports satisfying the regularity assumptions defined in Guerre et al. (2000). 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The kernel function Kz(-) is the univariate kernel of the form K(u) = (35/32)(l — < 1), Kcz{-) is the product of two univariate kernels. The band widths hc and hz converge to zero at the rates (N/ log jV)1 ^ 5" 1 "^ and {Lj log L)V(5+d) respectively, where N stand for the number of observations, and L for the number of auctions. 4.6 A pplication to construction procurem ents The construction industry provides an illustration of the model. Contractors build their cost estimates without knowing their competitors’ computations. Thus costs can be reasonnably assumed to be independent from one another. They are also specific to each firm because firms differ in production and opportunity costs: Bidding for a given project and eventualy winning the contract does not only imply more revenue from this given project, but also less capacity to compete for other projects (i.e. higher opportunity costs). Ultimately, if the firm decides to participate to the auction, it still bears the risk of losing the project. Its risk aversion and hence its aggressiveness in bidding depends on its experience in the industry, its capital, etc .... To take into account these indiosyncratic differences, I model the construction industry as a sector where firms have different CRRA coefficients. The data suggest that a firm’ s risk aversion depends on its experience. I will adapt my model to capture this characteristic. The following sections present the data set, how the model captmres the relation between experience and risk aversion, and the estimation results. Results support the idea that as a firm becomes more experienced and expand its activities, it becomes less risk averse. 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.6.1 D ata The data set consists of 435 construction contracts offered to bidding by the Engineering Department of the Los Angeles City Hall Department of Public Works (DPW) between 1994 and 2001. There is a total of 2316 bids from 375 firms. The projects range from sewage and street repairs to library branches construction. The DPW Engineering Department advertises the list of projects out to bid on the web, such as the official deparment of Engineering web page and Dodge news, among others, and on paper, such as in the Metropolitan news, the construction market data, for 4 to 6 weeks before the auction. The project characteristics, terms and code number are specified as well as the city’ s estimate for the project, but the DPW does not specify any reserve price. Firms can then retrieve bid proposals from the department bureau, and submit their final itemized bids in person in a sealed envelope. The Engineering Department, after checking for conformity to construction and insurance regulation, gives its approval for the bid to compete in the auction. Less than 10% of the bids are rejected on average. After closing time, bids are made public during a City Hall DPW board meeting. The lowest bid wins the auction. The Los Angeles City Hall Engineering department is known not to renegotiate the terms of the contract (namely the prices) once it has attributed it. There is perfect commitment of the DPW. The data contains the bids in dollars, the bidder’s name, the number of participants (nb) and the city’s estimate (city) which statistics are reported below. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1: Summ ary S tatistics variable m ean stdv min m ax bids winning bids city nb .899915 .749338 .808378 5.51 1.085423 .966114 .992186 3.06 .017732 .017732 .019000 2 4.965415 4.391280 4.509276 17 Figures for bids and city’s estimate are expressed in millions of dollars, and deflated according to the construction cost index, base(1996)=100 provided by the U.S. Census Bureau. Figure 4.1 is a plot of the bid observations against the city’ s appraisal value for the project. The observed increasing relationship between the bids and the city’s estimate, denoted by Z in the remainder of this chapter, will be reflected in the bid distribution estimates. To complete such figures, the Californian state licence board’s web page (www. cslb.ca.gov) provides individual information on the participants to the auctions, namely their date of foundation and their classification. Firms are either classified as A, B, both A & B, either with or without any other specialization code. A stands for General Engineering contractor, B for General Building contractor, specialized classifications extend to electrical work, sewage repairs,.... Details and definitions are provided in the appendix. Figures 4.2 and 4.3 show the frequency of the firms per year of experience and the percentage of firms in the “other” category for every year of experience. Note that 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 - 4 - "2 IS 2 - 0 0 1 2 3 4 city's estimate Figure 4.1: Bids and city’s apparaisal value for the project a larger percentage of older firms are classified as specialized compared to younger firms. Aside from having a general contractor permit, they have added to the list of their permits specialized ones. In the table of statistics below, the “other” category is made of firms which have either both general and specialized permits, or only specialized ones but this case is marginal. The number of firms in each classification code are Table 4.2: Firm s Classification A B AB other 91 44 84 156 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Years o f experience Figure 4.2: Firms’ frequency per year of experience Table 4.3 gathers a few statistics on the years of experience: Table 4.3: Firm s Experience mean stdv min max 13.42 12.21 .00 72.00 I define a firm’s experience as the numbers of years in activity at the date of the auction. The oldest firm was founded in 1929, the younger ones in 2000. 4.6.2 Experience and risk aversion I conduct the nonparametric regression of the bids on the firms’ experience (exper) and the average value of city’s appraisal value Z = zm. The graph of the predicted bids against the bidders’ experience, E(hj\exper = experi, z = zm), appears in figure 4.4. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N 0.6 05 0.4 0.0 i 40 experience Figure 4.3: Percentage of specialized firms It shows that experience influence the firms’ bidding behavior as bids have a positive correlation with experience. The theory predicts that risk averse agents bid more aggressively than less risk averse bidders, because they fear to lose the object more than their competitors do. It suggests that experience may affect the agents’ bidding by influencing their behavior towards risk, i.e. their risk aversion parameter in a CRRA model. Suppose I model the risk aversion parameter as a function of the firm’s experience to capture this pattern. If I denote by e.rper,; firm Vs number of years of experience and define 0? ; as = (30 + /Ae.rper,-, I should expect to be positive. As a firm gains experience, its 0 ,; increases, or equivalently its risk aversion coefficient 1 — 0j decreases, i.e. she becomes less risk averse. To confirm such intuition, I need to adapt equation (4.10) to the new framework. Since a bidder’s risk aversion depends on her experience, I will estimate ( 9 ? ; for 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.50 ~ 0.45“ 0.40- tn -a x > 0.35- 0.30- 0.25- every given level of experience. The index i refers now to i — 1 years of experience. Firms are then grouped by years of experience assuming implicitly that a one year old firm shares the same behavior towards risk that another firm, the same age. 10 15 experience Figure 4.4: Bids versus experience Equation (4.10) becomes % - % 1 1 - 1 Yh 2(B%) Yh 2(B%) b« 1 1 ( i - 1 ) 0 ‘—i ) Y m (B . ? ) YHj(Bj)a Ym(B?) YHj(B?) 1 1 1 1 1 U - 2 ) ( I - 1) /Yjji(Bf) Yh ( , - i )(B?_Y YH, ( B f) J +e,(4.12) for i = 1, ■ • • ,24 corresponding up to 23 years of experience, there are enough observations such as to estimate group i bid distribution and function Ym (bj). Similarly, I obtain estimates of the bid distributions of groups 25, 26 and 27, made 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.5: The joint density g(b,z) of firms with 24 and 25 years of experience, 26 to 30 years of experience and 31 years and more years of experience. This regrouping was motivated by the insufficient number of observations for firms with 24 and plus years of experience. I then perform steps one and two of the estimation procedure.1 9 The nonparainateric estimation of the joint density g(b, z), as plotted in figure 4.5 (bids and appraisal value are expressed in millions of dollars), suggests that bids follow a lognormal distribution with its mean linear in log z. Note the unimodal distribution, which mode lies on the log-shaped curve. The lower and upper truncature estimates, as shown in figure 4.1, are B(z) = -4.03239435 + 0.87649452z and t (z ) = -0.17285272 + 0.36123635. 19On average a firm participates only 6 times to the auctions. W ith a higher participation rate, I would be able to estimate a relative coefficient of risk aversion for each of the 375 firms participating in these auctions. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equation (4.12) holds for pairs of bids (bf, bj) = (exp Bf, exp Bf) such that Hi(B®(z)\z) = Hj(Bj (z)\z) = a, for a E [0,1). The value of a for which the model is the best fit (with the highest pseudo-R2) is a. = 0.8. The instrumental variables estimates of /?o and (3\ in 9^ — (3 o + (3 texperi are2 0 §i = 0.16114 + 0.050209 exper (0.22316) (9.8177) where the t-values appear in parentheses. For groups i = 1, • • • ,17, the predicted fys belong to the interval (0,1], as assumed in section 3.1. They are significantly different from 1, the risk neutral value of the coefficients: Firms with 0 to 16 years of experience are risk averse. The estimates of the risk aversion parameter are greater than one for those firms with more than 16 years of experience. This result may be due to the assumption that 9 is linear in the number of years of experience, which may not be the best fit. One might expect to observe an asymptote in 9, as after a given number of years of experience, a firm’s behavior towards risk should not change. The estimates of 9i, for i = 17, • • • , 24, may also be greater than one because as uji = 0, for i = 1, • • • , /, the corresponding ( 9 ,: overestimates the true value to compensate for the absence of wealth in the model. This problem should be solved with the introduction of wealths uy ^ 0 such that uy + 6, — c * > 0. The assumption ujj = 0, for i = 1, • • - , I is needed for the identification of the model. In future papers, I will study how to relax this assumption and still achieve identification. 20See previous section on the choice of the set of instrumental variables. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The test of asymmetry is rather simple, f3\ is significant,: Firms are asymmetrically risk averse. The increasing trend confirms the idea that as a firm grows older, it becomes less risk averse (its coefficient of relative risk aversion, 1 — (9 , becomes smaller). As a firm gains experience, it attracts more investors and capital. Moreover, it can find and rely on other sources of revenue to support its existence, and even diversify into other activities. In other words it can spread risk across contracts. It can also reduce risk: An older firm is able to better manage risk and reduce its relative risk aversion parameter. Figure 4.3 supports the idea that an older firm is able to spread risk across different activities: The percentage of firms with specialty permits increases with age. Other characteristics of the firm may be correlated to its age or experience: its assets, its subcontractors network, its reputation, etc Although those are unobserved, their influence on a firm’s utility can be inferred through the estimation of the firm’s coefficient of relative risk aversion. For example, a firm’s reputation may help reduce its risk aversion. A firm, which survives in the construction industry is more likely to enjoy a good reputation. This will bring more opportunities for other contracts. It will also lessen the firm’s agressiveness when bidding for any particular project, because other activities exist. Results confirm this idea: the P ratt measure of relative risk aversion 1 — 0 ,; is decreasing with the number of years of experience. Wealth is another important unknown factor which influences directly the bidder’s utility, namely U^jXh — Ci) = Ui(uJi + — q). Wealth cannot be measured or estimated but its effect on the agent’s utility can be inferred from its rela tion to the agent’s risk aversion. As wealth is positively correlated to the firm’s age, 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the firm’s risk aversion is negatively correlated to its experience, I can conclude that the firm’ s coefficient of relative risk aversion is decreasing in the firm’s wealth. From equation (4.11) and the estimated , I recover the the pseudo private costs cn for every agent i which participated to auction £, for £ = 1, ■ • ■ , L and i — 1, • • • , If. The summary statistics of the costs and the informational rents (hw — cw) jbw (where bw and cw are the bid and the pseudo cost of the winning firm) are gathered in the following table Table 4.4: Summary Statisctics on Inform ational Rents variable mean std min max pseudo cost 0.73020 0.94926 0.00272 4.73695 rents 0.23982 0.47255 0.00138 4.08753 rents (%) 28.2209 23.0088 1.1137 97.0777 Figures in the table are expressed in millions of dollars. On average the City Hall fails to capture 28% of the informational rents of the procurements auctions. The costs can reach as high as $4 millions and as low as $2,720 due to the high variation on the size of the project (The contracts in the data set range from library construction to sewage repair). The nonparametric estimate of the conditional cost density given the appraisal value f(c\z) was conducted over 1788 pseudo costs after trimming. The density appears in figure 4.6. Note that the distribution is unimodal and suggest a log normal distribution with a linear mean in the city’s appraisal value. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.6: The joint density f(c,z) This is not surprising since the estimated bid distribution suggested that bids fol low a log normal distribution. Note that this distribution had a linear mean in log z instead of z. It raises the issue of how the assumptions on the bid distri bution may affect the cost distribution. This problem would be avoided if more observations were available such as to nonparametrically estimate the bid distri bution. Note also that the graph of the distribution suggests that the cost dis tribution may depend on other exogenous variables as the cost density is more dense for some values of z (dark shaded areas). In particular, the endogeneity of the number of bidders may explain this observation. I performed the regres sion of log(6) on, among other contract and bidders’ characteristics, the city’s appraisal value z, the number of bidders, the number of years of experience of the firm. As mentionned earlier, z was by far the most significant variable of the model. Nonetheless, the experience and the number of bidders were also sig nificant,. While the influence of experience on the bids can be explained by the 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. firms’ risk aversion parameters, the endogeneity of the number of bidders is an issue still to be resolved within the specification of my model. If I assume, as in chapter 2, that f(-\Z,n) = /(-|Z ), where n stands for the number of bidders. I do not deny the endogeneity of the number of bidders as n is a function of the characteristics of the sale Z, but I assume that entry depends only on these char acteristics, and not on the state of the economy, or the status of other contracts (these define the opportunity costs) for example. In a future paper, I will investi gate how these variables may influence the cost density. 4,7 C onclusion The empirical auction literature on construction contracts focuses its efforts on proving the existence of asymmetric private value distributions. It is implicity assumed that, only these heterogenous distributions can explain the observed asymmetric bidding behavior in an auction game. Bidders’ differences in pro duction processes, economies of scale or in their cartel decision affect their costs and eventually their bidding. W ithout undermining the importance of such a modelization in explaining the observed bidders’ behavior, this chapter shows that another possible source of asymmetry has been neglected: Agents may also be heterogenous in taste, because of their wealth, their portfolio, their experience, etc.... I adopt a model of asymmetric risk averse bidders where agents’ wealth and experience may affect differently the agents’ attitude towards risk. In the case of the L.A. city hall construction contracts, the data shows that contractors are not only risk averse but also have different risk aversion tolerance according to their experience. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This model leads to two unique and surprising results. First, the set of dis tributions rationalized by a model of asymmetric risk aversion is a subset of the set of distributions which can be rationalized by a model of symmetric risk aversion. For example, in the case of CRRA utility functions, more parameters (and hence more degrees of freedom) should help in explaining a larger set of bid distributions, but the heterogeneity which lies in the agents’ risk aversion parameters is actually constrained by the private cost distribution homogeneity. This constraint does not exist in a model where both the risk averse utility function and the private cost distribution are homogenous. Second, identifi cation in the asymmetric case is much simpler than in the symmetric case. Although, in the latter model, within a CRRA specification, risk aversion is defined by only one parameter, identification cannot be achieved without some restrictions on the private value support. I do not need to impose any ad hoc restriction to my model. I assume, however, that bidders share the same private value distribution, this is the key assumption around which all these results unfold. Imposing symmetry on the agents’ private value distribution might force asym metry upon the bidders’ risk averse utility functions, which would otherwise be homogenous across agents. The challenge lies in identifying the dominant source of asymmetry. To achieve this purpose, I will study a model of both heteroge nous private values distributions and heterogenous risk aversion coefficients such as to test for the existence and the importance of either one of those sources of asymmetry. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. References Ahtey, S. (2001): “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,” Econometrica, 69, 861-889. Athey, S. and J. Levin (2001): “Information and Competition in U.S. Forest Service Timber Auctions,” Journal of Political Economy, 375-417. Bajari, P. (1996):“ Properties of the First-Price Sealed Bid Auction with Asymmetric Bidders,” Working paper, Stanford University. Bajari, P. (1999): “Econometrics of the First-Price Auctions with Asymmetric Bidders,” Working Paper, Stanford University. Bajari, P. (2001): ”Comparing Competition and Collusion: A Numerical Approach,” Economic Theory, 18,187-205. Bajari, P. and G. Summers (2002): “Detecting Collusion in Procurement Auctions: A Selective Survey of Recent Research,” forthcoming in Antitrust Law Journal. Bajari, P. and L. Ye (2001):“ Competition Versus Collusion in Procurement Auctions: Identification and Testing,” Stanford University, Working Paper. Baldwin, L. (1995): “Risk Averse Bidders at Oral Ascending-Bid Forest Timber Sales,” Working Paper, RAND Corporation. Baldwin, L., R. M arshall and J.F. Richard (1997): “Bidder Collusion at Forest Service Timber sales,” Journal of Political Economy, 105, 657-69. Billingsley, P. (1968): Convergence of Probability Measures, Wiley. 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B lu m J.R ., J. K iefer, M . R o sen b latt (1961):“ Distribution Free Tests of Inde pendence Based on the Sample Distribution Function,” Annals of Mathematical Statistics, 32, 485-498. C am po, S. (2002): “Attitudes towards Risk and Asymmetric Bidding: Evidence from Construction Procurements,” Working Paper, University of North Carolina, Chapel Hill. Cam po, S., E. Guerre, I. Perrigne and Q. Vuong (2002):“Semiparametric Estimation of First-Price Auctions with Risk Averse Bidders,” University of Southern Califronia, Working Paper. Cam po, S., I. Perrigne and Q. Vuong (2002): “Asymmetry in First-Price Auctions with Affiliated Private Values,” Journal of Applied Econometrics, forthcoming. Chamberlain, G. (1987): “Asymptotic Efficiency in Estimation with Condi tional Moment Restrictions,” Journal of Econometrics, 34, 305-334. Cheng, M .Y . (1997): “A Bandwidth Selector for Local Linear Density Estima tors,” The Annals of Statistics, 25, 1001-1013. Cox, J., R obertson,B. and Sm ith, V . (1982): “Theory and Behavior of Single object Auctions,” Vernon L. Smith cd., Research in Experimental Economics, vol. 2, Greenwich, CT.JAI Press, 1-43. Cox, J., V. Sm ith and J. Walker (1982): “Auction Market Theory of heteroge nous Bidders,” Economics Letters, 9, 319-325. Cox, J., V. Sm ith and J. Walker (1988): “Theory and Individual Behavior of First-Price Auctions,” Journal of Risk and Uncertainty, 1, 61-99. Crooke, P., L. Froeb and S. Tschantz (1997): “Mergers in Sealed vs Oral Asymmetric Auctions,” Working Paper, Vanderbilt University. DeBrock L. and J. Sm ith (1983): “Joint Bidding, Information Pooling, and the Performance of Petroleum Lease Auctions,” The Bell Journal of Economics, 14, 395-404. Donald S. and H. Paarsch (1993): “Piecewise Maximum Likelihood Estimation in Empirical Models of Auctions,” International Economic Review, 34, 121-148. 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D onald, S. and H. Paarsch (1996): “Identification, Estimation, and Testing in Parametric Empirical Models of Auctions within the Independent Private Values Paradigm,” Econometric Theory, 12, 517-567. Elyakim e B., J.J. Laffont, P. Loisel, Q. Vuong (1994): “First-Price Sealed-Bid Auctions with Secret Reservation Prices,” Annales d ’ Economic. e,t de Statistique, 34, 115-141. Fan, J. and I. Gijbels (1996): Local Polynomial Modelling and its Applications, Monographs on Statistics and Applied Probability, 66, Chapman and Hall. Flam bard, V. and I. Perrigne (2001): “Asymmetry in Procurement Auctions: Evidence from Snow Removal Contracts,” Working paper, University of Southern California. G oere, J., C. Holt and T. Palfrey (2002): “Quantal response Equilibrium and Overbidding in Private Value Auctions,” Journal of Economic Theory, forthcoming. Gollier, C. (2001): The Economics of Risk and Time, MIT Press. G illey O., G. Karels, R. Lyon (1985): “Joint Ventures and Offshore Oil Lease Sales,” Economic Inquiry, XXIV, 321-339. Guerre, E., I. Perrigne and Q. Vuong (2000): “Optimal Nonparametric Estimation of First-Price Auctions,” Econom,etrica, 68, 525-574. Haile, P. (2001): “Auctions with Resale Markets: An Application to US Forest Service Timber Sales,” American Economic Review, 91, 399-427. Haile P., H. Hong, M. Shum (2000): “Nonparametric Tests for Common Values in First-Price Auctions,” Princeton University, Working Paper. Hansen, R. (1985): “Empirical Testing of Auction Theory,” American Economic Review, 75, 156-159. Hardle, W . (1991): Smoothing Techniques with Implementation in S., Springer- Verlag, New York, NY. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Hardle, W . and O. L inton (1994): “Applied Nonparametric Methods,” in R.F. Engle and D.L. McFadden, eds., Handbook of Econometrics, Volume IV, Amsterdam: North Holland. H arris, M . and A. Raviv (1981): “Allocation Mechanisms andthe Design of Auctions,” Econom.etrica, 49, 1477-1499. H arsanyi, J. (1967): “Games with Incomplete Information Played by Bayesian Players,” Management Science, 14, 159-182. Heckm an, J., H. Ichimura, J. Sm ith and P. Todd (1998): “Characterizing Selection Bias Using Experimental Data,” Econometrica, 66, 1017-1098. Hendricks, K., J. Pinkse and R. H. Porter (2001): “Empirical Implications of Equilibrium Bidding in First-Price, Symmetric, Common Value Auctions,” Working Paper, Northwestern University. Hendricks, K. and R. H. Porter (1988): “An Empirical Study of an Auction with Asymmetric Information,” American Economic Review, 78, 865-883. Hendricks, K. and R. H. Porter (1992): “Joint Bidding in Federal OCS Auctions,” American Economic Review, 82, 506-511. Hendricks K., R. H. Porter and G. Tan (2000): “ Joint Bidding in Federal Offshore Oil and Gas Lease Auctions,” University of British Columbia, Working Paper. Hendricks, K., R. H. Porter and C. W ilson (1994): “Auctions for Oil and Gas Leases with an Informed bidder and a random reserve Price,” Econometrica, 62, 1415-1444. Hengartner, N. W . (1997): “Asymptotic Unbiased Density Estimators,” Working Paper, Yale University. H ong H. (1997): “Characterization of Equilibria in Asymmetric Ascending Auctions,” Stanford University, Working Paper. Hong H. and M. Shum (2002): “Econometric Models of Asymmetric Ascending Auctions,” Princeton University, Working Paper. 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Honor e, B. E. and E. K yriazidou (2000): “Panel Data Discrete Choice Models with Lagged Dependent Variable,” Econometrica, 68, 839-874. H orowitz, J. L. (1992): “A Smoothed Maximum Score Estimator for the Binary Response Model,” Econometrica, 60, 505-531. Horowitz, J. L. (1993): “Optimal Rates of Convergence of Paprameter Esti mators in the Binary Response Model with Weak Distributional Assumptions,” Econometric Theory, 9, 1-18. Ibragim ov, I. A. and R . Z. H a s’m inskii (1981): Statistical Estimation. Asymptotic Theory, New York: Springer Verlag. Jofre-Bonet, M. and M. Pesendorfer (2000): “Bidding Bhavior in a Repeated Procurement Auction,” Working Paper, Yale University. Klem perer, P. (1999): “Auction Theory: A Guide to the Literature,” Journal of Economic Surveys, 13, 227-260. K orostelev, A. P. and A .B. Tsybakov (1993): Minimax Theory of Im.age Reconstruction, Lecture Notes in Statisics, 82, New York: Springer Verlag. Kyriazidou, E. (1997): “Estimation of a Panel Data Sample Selection Model,” Econometrica, 65, 1335-1364. Laffont, J.J. (1997): “Game Theory and Empirical Economics: The Case of Auction Data,” European Economic Review, 41, 1-35. Laffont, J. J., H. Ossard and Q. Vuong (1995): “Econometrics of First-Price Auctions,” Econometrica, 63, 953-980. Laffont, J.J. and Q. Vuong (1996): “Structural Analysis of Auction D ata,” American Economie Review, Papers and Proceedings, 86, 414-420. Lebrun, B. (1999): “First-Price Auctions in the Asymetric N Bidder Case,” Intern,ational Economic Review, 40, 125-142. Le Cam, L. and G. Yang (1990): Asymptotics in Statistics, Some Basic Concepts, Springer Verlag. 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Levin D. and J . L. Sm ith (1994): “Equilibrium in Auctions with Entry,” American Economic Review, 84, 585-599. Li, T., I. P errig n e and Q. Vuong (2000): “Conditionally Independent Private Information in OCS Wildcat Auctions,” Journal of Econometrics, 98(1), 129-161. Li, T., I. Perrigne and Q. Vuong (2002): “Structural Estimation of the Affili ated Private Value Auction Model,” Rand Journal of Economics, forthcoming. Liu, R. C. and L. D. Brown (1993): “Nonexistence of Informative Unbiased Estimators in Singular Problems,” Annals of Statistics, 21, 1-13. Lizzeri, A. and N. Persico (2000): “Uniqueness and Existence of Equilibrium in Auctions with a Reserve Price,” Games and Economic Behavior, 30, 83-114. M anski, C. F. (1985): “Semiparametric Analysis of Discrete Response, Asymp totic Properties of the Maximum Score Estimator,” Journal of Econometrics, 27, 205-228. M arshall R ., M. Meurer, J. F. Richard and W . Strom quist (1994): “Numerical Analysis of Asymmetric Sealed High Bid Auctions,” Games and Economic Behavior, 7, 193-220. M askin, E. and J. R iley (1984): “Optimal Auctions with Risk Averse Buyers,” Econometrica, 52, 1473-1518. M askin, E. and J. R iley (1996): “Uniqueness in Sealed High Bid Auctions,” Working Paper, UCLA. M askin, E. and J. R iley (2000a): “Asymmetric Auctions,” Review of Economic Studies, 67,413-438. M askin, E. and J. R iley (2000b): “Existence of Equilibrium in Sealed High Bid Auctions,” Review of Economic Studies, 67, 439-454. M atthew s, S. (1987): “Comparing Auctions for Risk Averse Buyers: A Buyer’s Point of View,” Econometrica, 633-646. M cAfee, P. and J. M cM illan (1987): “Auctions with Entry,” Economics Letters, 23, 343-347. 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M cA fee, P. and D. V incent (1992): “Updating the Reserve Price in Cornmon- Value Models,” Am,erican Economic Review, 82, 512-518. M ilgrom , P. and R. W eber (1982): “A Theory of Auctions and Competitive Bidding,” Econometrica, 50, 1089-1122. M illsaps, W . and M. Ott (1985): “Risk Aversion, Risk Sharing and Joint Bid ding: A Study of Outer Continental Shelf Petroleum Auctions,” Land Economics, 61, 372-386. N ewey, W . K. and D. L. M cFadden (1994): “Large Sample Estimation and Hypothesis Testing,” in R.F. Engle and D.L. McFadden, eds., Handbook of Econometrics, Volume IV, Amsterdam: North Holland. P aarsch , H. (1992): “Deciding between the Common and Private Value Paradigms in Empirical Models of Auctions,” Journal of Econometrics, 51, 191-215. Palfrey, T. and Pevnistkaya, S. (2002): “Endogenoi is Entry and Self-selection in Private Value Auctions: An Experimental Study,” Caltech working paper. Perrigne, I. (2001): “Random Reserve Prices and Risk Aversion in Timber Auctions,” Working Paper, University of Southern California. Perrigne, I. and Q. Vuong (1999): “Structural Econometrics of First-Price Auctions: A Survey of Methods,” Canadian Journal of Agricultural Economics, 47, 203-223. Pesendorfer, M. (2000): “A Study of Collusion in First-Price Auctions,” Review of Economic Studies, 67, 381-411. Pinkse, J. and G. Tan (2001): “The Affiliation Effect in First-Price Auctions,” Working paper, University of British Columbia. Porter, R. H. (1995): “The Role of Information in U.S. Offshore Oil and Gas Lease Auctions,” Econometrica, 63, 1-27. Porter, R. H. and J. Zona (1993): “Detection of Bid Rigging in Procurement Auctions,” Journal of Political Economy, 101, 518-538. 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pow ell, J. L. (1994): “Estimation of Semiparametric Models,” in R.F. Engle and D.L. McFadden, eds., Handbook of Econometrics, Volume IV, Amsterdam: North Holland. P r a tt, J. (1964): “Risk Aversion in the Small and in the Large,” Econometrica, 32, 122-136. Riley, J. and W . Samuelson (1981): “Optimal Auctions,” American Economic Review, 71, 381-392. R obinson, P. (1988): “Root-iV-Consistent Semiparametric Regression,” Econo- metrica, 56, 931-954. Rothenberg, T. J. (1971): “Identification in Parametric Models,” Econometrica, 39, 577-592. S tone, C. J. (1982): “Optimal Rates of Convergence for Nonparametric Regres sions,” The Annals of Statistics, 10, 1040-1053. van d er V aart, A. (1998): Asymptotic Statistics, Cambridge University Press. Vickrey, W . (1961): “Counterspeculation, Auctions, and Sealed tenders,” Journal of Finance, 16, 8-37. W hite, H. (1994): Estimation, Inference and Specification Aanlysis, Econometric Society Monograph, 22, Cambridge University Press. W ilson, R. (1967): “Competitive Bidding with Asymmetric Informa tion,” Management Science, 13, 816-820. W ilson, R. (1969): “Competitive Bidding with Disparate Informa tion,” Management Science, 15, 446-448. W ilson, R. (1977): “A Bidding Model of Perfect Competition,” Review of Economic Studies, 44, 511-518. Zeidler, E. (1985): Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems, Springer Verlag. 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendices I. C hapter 2 Proof of Proposition 2.1: The proof is in two parts. First, I prove that the asymmetric APV model is identified. Second, I prove the necessary and sufficient conditions under which a distribution G(-) can be rationalized by an asymmetric APV model. (1) Let G(-) be the joint distribution of the observed (equilibrium) bids with support [b , b]n in the asymmetric APV model. Suppose that there are two possible underlying distributions F(-) and F(-) of private values both leading to the same joint bid distri bution G(-) in the asymmetric APV model. By assumption, both distributions F(-) and F(-) belong to the set V of n-dimensional absolutely continuous distributions with hypercube supports that are affiliated and exchangeable in their first n\ and last no arguments. Let si(-, F), so(-, F ), si(-, F) and §i(-, F) be the strictly increasing Bayesian Nash equilibrium strategies corresponding to F(-) and F(-), respectively. Thus si(-,F), sq(-,F ), si(-, F ) and s%(-,F) satisfy the first-order differential equations (2.1) and (2.2), which can be written as in (2.4) and (2.5). Therefore F ( v i , v 0 ) = P r ( 6 ( b i , G ) < v l 5 C o ( b 0 ,G) < v 0 ) = G ^ r ^ v i , G), £o” V o , G)), F ( v i , v 0 ) - Pr(6(bi,G ) < v j ^ o ^ o , ^ ) < v 0 ) - G ( ^ i _ 1 ( v i , G ) , ^ 0 _ 1 ( v o , G)). 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It follows that F(-) = F(-) on their common support [v,v]n = [£o(£, G), £ o (& > G)]n — [£i(& , G), £1 (b,G)]n in the asymmetric APV model. Thus the asymmetric APV model is identified.2 1 (2) I first show necessity of (i)-(ii) in Proposition 2.1. Let si(-,F) and sq (-,F ) be the strictly increasing differentiable Bayesian Nash equilibrium strategies corresponding to F(-), which is affiliated and exchangeable in its first n\ and last no arguments and G(-) be the joint distribution of observed bids. Thus, G(t>i,t>o) = F (si_1(bi, F ), so^1 (bo, F )) for every bj € [b, b]ni = [u, Si(u, F)]”1 and bo € E [b,b]n° = [u, so(v, F)]"°. Because the strategies si(-,F) and sq(-,F ) are strictly increasing and F(-) € V , then G(-) belongs to V . Moreover, these strategies must solve the system of first-order differ ential equations (2.1) and (2 .2). Because the system composed by (2.4) and (2.5) is equivalent to the system of equations (2 .1 ) and (2 .2 ), then si(-,F) and sq(-,F ) must satisfy £i(si(vi, F), G) = v \ and £0 (3 0 (^0 1 F ), G) = vq for all v\ e [u, u] and v0 € [v, v \. Making the change of variables b\ = S ] (ni, F) and bo = sq(vq, F), we obtain £,i{b\,G) = s^1(6i,F) and £o(bo, G) — S q 1( t> o > F ) for every b\ and bo in [6 , b]. Thus (-,G) and £o(-, G) must be strictly increasing on [b, b] because sj)1(-,F) and S q 1(-, F) are strictly increasing on [6 , b). To prove sufficiency, let the joint distribution G(-) belong to V with support [b,b]n. This implies that G(-) is exchangeable in its first n\ and last no arguments and affiliated. We note that lim^i, £i (b, G) = b. This follows from (2.4) and the fact that (i) b is finite, (ii) linp,_>6 logGb*,B0\b i (b,b\b) = -oo, and 21 To simplify the notation, the vector £i(foi, G) denotes the vector (£i(?Ui, G), ... , G (bjni,G)), etc. 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (iii) dlogGBhBalbl(B,b\b)/dB = (dGBlBolbl (B, b\b)/dB)/GB*,Bolbl (B, 6|6), d logGB*tBo\bl(B, b\b)/db = (dGB*jBo^ bl(B,b\b)/ db)/GB^ Bo\bl (B, b\b), so that lim^b {dGB*Bo\bl (b, b\b)/db)/GB*jBo\bl (6, b\b) = + 0 0 . Using similar arguments in (2.5), I can show that lim & _ > b C o(b,G) = b. Next, I define a distribution F(-) as F(vi, vo) = G(Ci_1(vi, G), C o-1 (v0, G)), for all (vj, v q ) € [n,v]n, where v = £1 (6,G) = £o{b,G) = 6 and v — Ci(6,G) = ^o(b,G). Because Ci(-,G) and Co('>G) are strictly increasing on [6,6] by assumption, F(-) is a valid distribution. Since G(-) is strictly increasing on [6 , 6 ]” , then F(-) is strictly increasing on [v,v]n. Hence the support of F(-) is the hvpercube \v, v]n. Moreover, because ^1 ”1(-,G) and ^ x(-, G) are strictly increasing and G(-) € V , then F(-) belongs to V. It remains to show that that this distribution F(-) can rationalize G(-) in an asymmetric APV model, i.e., that G(-, •) = F (si-1(-, F ), So“ 1(-, F )) on [6,6]”, where Sj(-,F) and sq(-,F) solve (2.1) and (2.2) with the boundary conditions s i ( v , F ) = sq(v,F) = v and s\( v , F) = 6 ‘ o(u, F). By construction of F(-), we have G(-) = F(Ci(-, G), Co(‘? G)). Thus it suffices to show that Cf1(-, G) and C o "1 (•, G) solve the system of equations (2 .1 ) and (2.2) with the boundary conditions G) = ^q1(v,G) = v and ^ 1(u, G) = CcTHf G). It is easy to see that these boundary conditions are satisfied by construction and assumption, respectively. From the construction of F(-), note that F y* yo\Vl(-, ■ |t>i) = GJ gj) J Bo|&i(£r1(‘,G), C o T H -* G)|Cf1(ui, G)). Differentiating and taking the ratio, this gives f y ^ i - l ^ / F y b y o l v A - l v i ) = G ) ( 9 G Bj >Bo|6l( ^ l (-,G ),e 0 " 1( ', G ) |^ r 1(^1, G)) / d B D / G s ^ ^ ^ G l ^ H ^ G ^ ^ G ) ) + ^ 1 '(.,G)(9(?B, B o |,1 (er1 ('1 G), C o l(-, G ) \ ^ \ V1, G ) ) / d B 0) / G BhBolbl (CrX (-, G), Co'(-, G)|Cf1('ui’ G)). 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I can develop similar computations for F y i y *\iVl(-,-\vo) = G BlB* \bl(£j 1 (•,G),f;0 1 ( ; G ) \ ^ l (v0,G )). Thus ^ ( - .G ) and £^x(-, G) solve (2.1) and (2.2) if 1 = (n - £ x 1(vi,G)) ( d G B l^ \ bl{ ^ \ v 1, G ) , ^ \ v l , G ) \ ^ \ v x, G ) ) / d B l ) dGBl,Bo\bl { ii \ v i ,G U z \ v u G m \ v i ,G ) ) / d B Q )\ + G 'BIiBo|bl(^r1( ^ ,G ) ,^ 1(n1,G)|^r1(u1,G)) J ’ 1 = («o-C o1K G ! )) V GBl^ b M \ ^ G ) , ^ \ v o , G ) \ ^ l{vQ ,G)) d G BltB5lbo( ^ H v o , G ) , C o H v o , G M o \ v o , G ) ) / d B ^ ) \ G f B1 ,^|&o(?r1(t’ o,G),C(T1(? ;o, G ) |^ 1(n0,G)) J ’ holds for any (vx,vo) € [v,v]2- This clearly holds by definition of £i(-, G) and £o(-, G). Proof of Proposition 2.2: First, I prove that the joint (equilibrium) bid distribution G(-) in an asymmetric pure CV model with two players must satisfy £i(-, G) = £o(-> G) on [b,h]. As noted in the text, such a distribution must satisfy (16) and (17) for any £ [ai,cr], where bx = ^(cq) for i = 1,2, i.e. for any b, < E [b,b], where cq = s ^ l (bj). Replacing a,; by s ~ l (bi) in both left-hand sides of (16) and (17) shows that these two equations are equivalent to P (S l- » , S o » ) = Zi(b,G ), (2.A.1) V ( s ^ ( b ) , s ^ ( b ) ) = £o(b, G), (2.A.2) for any b G [b,b]. The desired result follows from the equality of the left-hand sides. 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next, I show that the condition £i(-, G) = £o(-,G) on [ 6, 6] is equivalent to the qua- sisvmmetry of G(-). By definition of £ ,•(•, G), the condition £i(-, G) = £o(-,G) on [ 6, 6] is equivalent to G bo\bl(b\b)/gho\bl(b\b) = G h \ho(b\b) / g bl\bo(b\b) for any b G [6 , 6 ], i.e. to can be interpreted as Pr[6o < 6, bi = 6 ] = Pr[6o = 6, b\ < 6 ] for any 6 G [6,6]. Hence, it is equivalent to the condition Pr[&o < hi < 6] — Pr[6i < bo < 6 ] for any 6 G [6,6], which can be proved formally by integrating the former and differentiating the latter with respect to 6 . It remains to note that the latter condition is equivalent to Pr[6i < 6o|6i < 6, 6 o < 6 ] = Pr[6i > 6o|6i < 6, 6 o < 6 ] for any b G [6,6]. i.e. to the quasisymmetry of (?(•).2 2 Lastly, to prove the third statement in Proposition 2.2, consider the bivariate private value distribution F(-) generated by (v%, vq), where (n x ,« o ) is given by (2.4) and (2.5), while (6i, 6o) is distributed as G(-), which is the bivariate equilibrium bid distribution in the pure CV model under consideration. In the two players case, it is easily seen that (2.4) and (2.5) reduce to (2.18) and (2.19), respectively. Now, by Theorem 2.3 in Milgrom and Weber (1982) note that G(-) is affiliated because (& i, 6q) = (si(oi), so^o)), o \ and c to are affiliated, and the equilibrium strategies si(-) and so(-) in the (asymmetric) pure CV model are restricted to be strictly increasing. Second, (2.A.1) and (2.A.2) imply that & (•,(?), i = 1 , 2 must be strictly increasing on [6,6] because s,r l (‘) * s strictly increasing, and because V(<ji,cto) is strictly increasing in (<xi,<ro) by Theorem 2.5 in Milgrom and Weber (1982). Hence, from Proposition 2.1 it follows that the bid 2 2 Notc that the equivalences in this paragraph do not depend on the fact that, G(-) is the equilibrium distribution. Hence, they apply to any bivariate distribution. for any 6 G [ 6, 6], where gbljbo(-) denotes the joint density of (6 1, 6 0). This equality 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distribution G(-) can be rationalized by an (asymmetric) APV model. Therefore, such an APV model is observationally equivalent to the (asymmetric) pure CV model under consideration. To complete the proof, it remains to show that the private value distribution F(-) of the observationally equivalent APV model is quasisymmetric. Because G(-) is the equilib rium bid distribution of the pure CV model, it must be quasisymmetric, as shown above. Moreover, F(v 1, 1* 2 ) = G(^1 “1(ni),£o"X (uo)) by definition, where &(•) is strictly increas ing, and Ci(-) = £o(0> as previously shown. It follows easily that F(-) is quasisymmetric. Moreover, both players adopt the same equilibrium strategy *(•) = ^ (•). 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. II. C hapter 3 Appendix 3.A Appendix A gathers proofs of Lemma 3.1 and Propositions 3.1-3.3 stated in Sections 3.3 and 3.4. P roof of Lemma 3.1: First, I prove that conditions (i), (ii) and (iii) are necessary. Because f > , ; = s(vt, U, F, I) and the V {S are i.i.d., it follows that the 6 ,;s are i.i.d. so that (i) must hold. Condition (ii) requires that G(-) must satisfy the regularity properties (i)-(v) of Defi nition 3.3. By the Implicit Function Theorem (see e.g. Lemma C l in Guerre, Perrigne and Vuong (2000)), note that Theorem 3.0 implies the following properties on (•): — s~'1(-) is continuous and strictly increasing on [6 , b], with b = v and b = s(v) < v, — s~1(-) admits R + \ continuous derivatives on [6,b], — [s_1(-)]' > 0 on [b,b] and [s_1(6)]' > 1. From the model, I have G(b) = F[s_1(6)] with s“1(-) continuous and increasing on [ & ,£ > ] . Thus, because /(•) > 0 and [s“1(-)]' > 0, the support of G(-) is [s(u), s(u)] = [b , b], where b = v > 0. Moreover, b = s(v) < v < oo. This proves (i). Next, s~1(-) admits R + 1 continuous derivatives on [b,b}, while F(-) admits R+ 1 continuous derivatives on [u, u]. Hence G(-) — A[s_1(-)] has i2 + l continuous derivatives on [b , b]. This proves (ii). I have g{b) = /[s— 1 (6)]/s/[s— 1 (£ » )]. Hence (iii) follows from Theorem 3.0 and Definition 3.2-(iii). Regarding (iv), (3.3) implies „(h\ - F [s~l{h)] 9W ( i- i ) \[ s ^ ( b ) - b y 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I know from Definitions 3.1 and 3.2 that A(-) and F(-) have R + l continuous derivatives on [0, Too) and [u,u], respectively, while s_1(-) has R + l continuous derivatives on [b , b]. Thus G(-) has R + 2 continuous derivatives on (b, b]. The fact that A(0) = 0 prevents the continuity of derivatives at b. It remains to show (v). From (3.3), I have G(b)/g(b) = (J — lJAfs"^^) — 6], Because A(-) and s_1(-) are R + l continuously differentiable on [0, +oo) and [b , b\, then G(-)/g(-) is R + 1 continuously differentiable on [b, b], and hence admits a finite limit as bib. To prove that condition (iii) is also necessary, consider (3.4), where the function A(-) is the ratio U(-)/Ur(-). Thus A(-) is defined from 1R+ to iR+ because A(0) = lim^o A (.x) = 0, as noted after Definition 3.1. Moreover, U(-) admits R + 2 continuous derivatives on (0, Too). As linxjjo A^ is finite for r = 1 ,... , R + 1, these imply that A (-) has R + l continuous derivatives on [0, Too). As A'(-) = 1 — \{-)U"{-)jU'{■ ), we have A'(-) > 1 because A(-) > 0, U'(-) > 0 and U"{-) < 0. It remains to show that the function £(•) is increasing. The equilibrium strategy must solve the differential equation (3.2). As (3.3) follows from (3.2), s(-) must satisfy £(s(r), 1 7 , G, I) = v for all v E [u, v\. I then obtain ^(6, U, G, I) = s^1(6, U, F, I). This implies £'(•) = [s^1(-)]/, which is strictly positive as required. Second, I have to show that conditions (i), (ii) and (iii) are together sufficient. Assume that bids are independently and identically distributed as G(-) E Q r and there exists a function A(-) satisfying the properties of Lemma 3.1. First, I construct a pair [1 7 , F] belonging to U r x F r . Let U(-) be such that A(-) = U{-)/U'{-) or U\-)/U{-) = 1/A(-). By integrating, we obtain U(x) = U(a) exp[ /a T 1 / X(tl)dt] for arbitrary a > 0. With the normalization 17(1) = 1, this gives U(x) = exp / * l/\(t)dt. I verify that such a utility function belongs to U r . Because A(-) admits R + l continuous derivatives on [0, +oo), then condition (iii) of Definition 3.1 is clearly satisfied. Moreover, in 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the neighborhood of zero, A(f) ~ A ;(0)f with 1 < m < oo. Thus the integrand J^l/X(t)dt diverges to infinity, which implies that U(x) tends to zero. Define U(0) = 0. The derivative U'{x) is equal to exp f* 1 / X(t,)dt/ X(x), where A(-) > 0 on (0, +oo) because A(0) = 0 and A'(-) > 1. This implies that U'{-) > 0 on (0, +oo). The second-order derivative gives U"(x) = (— A'(x) + 1) exp )* I / X(t)dt / X2 (x). We know that X'(x) > 1, which implies that U"(-) < 0 on (0, Too). It remains to show that [/(■) admits R + 2 continuous derivatives. By assumption, A(-) has R + 1 continuous derivatives on [0, +oo). It follows that U(-) admits R + 2 continuous derivatives on (0, +oo). Lastly, U(-) is continuous on (0, +oo) as limT 1 0 U (x) — U(0) — 0. Let F(-) be the distribution of X = h + X~1(G(b)/(I — 1 )g(b)), where b ~ G(-). I verify that such a distribution F(-) belongs to J-r. I have F(x) = Pr(X < x) = Pr(£(fc) < x). The latter can be written as Pr(6 < £_1(a:)), which is equal to G[£'~1(x)], because £(•) is strictly increasing by assumption. This implies that F(-) = G(£_1(-)) on [n,U], where v = £(6) = 6 and v = £(6) < oo by continuity of £(•). Because £(•) and G(-) are strictly increasing, then F(-) is strictly increasing on [u, v] and its support is [n,U], which is a finite interval of M+. From Definition 3.3, G(-) has R + 1 continuous derivatives on [6,6]. Moreover, £(•) is R + 1 continuously differentiable on [6 , b]. This follows from the definition of £(•), the R + l continuous differentiability of A~x(-) on [0,+oo), and the R + l continous differentiability of G(-)/g(-) on [6,6], which follows from Definition 3.3-(iv) and (v). Thus F(-) = G ( ^ 1(-)) admits R + l continuous derivatives on [n, v\. It remains to show that the corresponding density /(•) is strictly positive. I have /(•) = < 7 (£ _1('))/£/(£~1(‘))t where g(-) > 0 from Definition 3.3 and £'(•) is finite on [6 , 6]. Thus /(•) > 0 on [v, v\. Lastly, I have to show that the pair [1 7 , F] can rationalize G(-) in a first-price sealed- bid auction with risk averse bidders, i.e. that G(-) = F(s~l(-,U, F, I)) on [6,6], where 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s(-,U,F,I) solves (3.2) with the boundary condition s(v,U,F,I) = v. By construction of F(-), I have G(-) — F(g(-)). Thus, it suffices to show that £_1(-) solves (3 .2 ) with the boundary condition £_1(u) = v. The boundary condition is straightforward as £(b) = b — v. From the construction of F(-), we have f(-)/F{-) = [C _1(,)]/. I ?($_1('))/C;(C ” 1('))- Thus £~1 '(-) solves the differential equation (3.2) if for all v € [u, v]. Making the change of variable v = £(b) and noting that £(b) — b = A~l[G(h)/(I — 1 )g(b)] from the definition of £(•), it follows that £_1(-) solves the differential equation (3 .2 ) with boundary condition £_1(u) = v. Proof of Proposition 3.1: (i) Consider a bid distribution G(-) 6 Q r generated by a structure [ C 7 , F] € U r x T r . I have to show that there exists a structure [U, F], where U(x) = .t1-c, 0 < c < 1 and F € F r , that rationalizes the distribution G(-). In this case, A (.x) = .x/(l — c) so that A(0) = 0 and A'(-) > 1. From Lemma 3.1, it suffices to show that there exists a value c € [ 0 , 1) such that the function has a strictly positive derivative on [6 , b]. Differentiating, this is equivalent to [G(b)/ g(b)]r > —(I — 1)/(1 — c) for all b € [b , b]. The latter is true if inf_ b a \b ,b ] G(b) .9(P) Note that the left-hand side is finite because G(-)/g(-) is R+l continuously differentiable on [b, b], as noted after Definition 3.3. I consider two cases. If inf\,{G{b)/g{b)}' > 0, then we can choose any value c € (0,1) to satisfy (A.l). Second, if infb[G(b)/g(b)]' < 0, 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.A.1) can be written as c > 1 — (I — l) /( — inf;, [G(b)/g(b)}'), where the right-hand side term is less than one. Thus I can find many values for c in the interval (0,1). Therefore I can always find a value for c G (0,1) such that G(-) can be rationalized by a CRRA model. (ii) The proof for the CARA case is similar. Consider U € U ^ A R A . This gives the utility function U(x) = (1 — e~ax)/(l — e~“) with a > 0. Hence X(x) = (eax — I)/a and Note that limbib[G(b)/ g(b)\' = lim ^ 1 — G(b)g'(b)/g2(b) = 1 because R > 1 and g(b) > 0. Hence the preceding inequality holds at b for any c > 0. Thus the preceding inequality becomes This is satisfied for an infinity of values for a > 0 provided the supremum is not Too. I know that —(g(b)/G(b))[I — 1 T (G(b)/g(b))'} is R continuously dif ferentiable on (b,b\ and hence continuous on (b, b ] because R > 1. Moreover, lim & jft —(g(b)/G(b))[I — 1 T (G(b)/g(b))r] = — oo because g(b)/G(b) tends to T oo and [G(b)/ g(b)Y tends to 1. Thus, I can always find a value for a and hence a CARA model that can rationalize any bid distribution G(-). A 1(x) = (1/a) log(l T ax). This defines the following inverse bidding strategy I have to show that there exists a > 0 such that £'(6) > 0 on [b , b]. Differentiating gives the following inequality on a 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P ro o f of Proposition 3.2: (i) Nonidentification of the general model. Consider a structure [U , F] G U r x F r , which generates a bid distribution G(-) G Q r by Lemma 3.1. Suppose first that U(-) is not of the form .t1_c for any c, 0 < c < 1. From Proposition 3.1, it follows that there exists a CRRA structure [U,F] with zero wealth and F G F r that leads to the same equilibrium bid distribution G(-). Because a CRRA utility function with zero wealth belongs to U r , the original structure [U, F] is not identified. Suppose next that U(-) is of the form x } ~ c for some c, 0 < c < 1. From the second part of the proposition, which is proven below, there exists another CRRA structure with zero wealth, and hence another risk aversion structure [U, F] with F € F r that is observationally equivalent to [U, F]. Hence [U, F] is again not identified. (ii) Nonidentification of the CRRA, CARA, DRRA and DARA models. I show first that a CRRA model is not identified. Consider a structure [U , F] where U(-) is derived from a CRRA vNM utility function with wealth w > 0 and F € F r . This generates a bid distribution G(-) € Qr. Proposition 3.1 shows that there exist a CRRA utility function U(-) with zero wealth and 0 < c < 1 and a distribution F(-) G F r leading to the bid distribution G(-). Thus [U , F] is not identified. When w = 0, the proof of Proposition 3.1 shows that there exists an infinity of values for c, c < c < 1, generating the same distribution G(-). Thus the CRRA model is unidentified. I can use a similar argu ment to show that the CARA model is unidentified from the proof of Proposition 3. l-(ii). Next, consider a structure [U, F] G U ^ RRA x F r defining a DRRA model and generating a bid distribution G(-) € Qr . Note that U ^ RRA C U r RRA. If U(-) is generated from a vNM utility function with constant relative risk aversion, I know from above that there exists another CRRA structure that is observationally equivalent to [U, F]. On the other hand, if U(-) is generated from a vNM utility function with strictly decreasing relative risk aversion, I know from Proposition 3.1 that there exists an observationally 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equivalent CRRA structure with zero wealth. Thus the DRRA model is unidentified. A similar argument shows that the DARA model is unidentified. P roof of Proposition 3.3: I distinguish two parts. The first part concerns the identification of the general semiparametric model composed of structures [U, F] satisfying Al, while the second part concerns the identification of the CRRA and CARA models. P a r t 1. Let [U, F] satisfying Al with parameters (8 , 7 ) and G(-|-, •) be the corresponding equilibrium bid distribution given ( Z , I ) . Suppose that there exists another structure [U, F] satisfying Al with parameters (8 , 7 ) and leading to the same conditional bid distribution. I first, show that (8 , 7 ) is identified, i.e (8, 7 ) = (6 , 7 ). Writing (3.5) for each structure gives 1 C K = - ba (z, i); 8) = A(va ( z , i ; j ) - ha (z,i);8 ), (3.A.2) i - l g ( b a ( z ,i) \z ,i) for every (z, i) E Z x X. Hence Al-(iv) implies that ((9,7 ) = ( 8 , 7 ). From Al-(i), U(-) = [/(•; 8) = U(-; 8) = U(-), which establishes the identification of U(-). Moreover, from (3.4), I have v = b + A -l 1 G ( b \ z ,i ) _0 i - 1 g{b\z,i) ’ = v. for every b E [b(z, ?'), b(z, * )] and (z, i) E Z x T . This shows that F ( - 1 - , •) = F (-j -, -), i.e. that the latter is identified. 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PART 2. I have U(x) = x}~~c with 0 < c < 1 for the CRRA model and U (x ) = (1 — exp~aT)/(l — exp~“) with a > 0 for the CARA model. Conditions (i)— (iii) of Al are satisifed, where va(z, i) = 7 . Thus, it suffices to verify condition (iv). For the CRRA model, I have A ( 7 — ba(z,i);0) — ( 7 — ba(z,i))/0, where 0 = 1 — c. By assumption there exist two pairs (2 7,i{) and (z2,* 2 ) belonging to 2 x 1 such that ba(zh ii) ^ ba(z2,i2). Hence, ( 7 - ba(zx,ii))/9 ^ ( 7 - ha{z2, * 2))/^- Thus knowing the function A ( 7 — £ > < * ( • , •); 6) for every (z, i) £ 2 x 1 and hence for (2 7, i\) and (z2, i2) gives a system of two linear equations in two unknown parameters (0 , 7 ). Because the determinant of such a system is not equal to zero, there is a unique solution. For the CARA model, we have A(y — ba{z,i);9) = (exp^7”^ ^ ’ ’^ —l)/9, where 9 — a. By assumption there exist two pairs (z\,i\) and (z2,i2) belonging to 2 x 1 such that ba(zi,h) ba(z2, i2). Hence, Ai ^ A 2, where Xj = (exp0^ ^ ^ 77^ -1 )/# for j = 1,2. Rearranging terms, eliminating 7 and taking the logarithm give log ) + r x r - ~ - ba(z2,i2)), I + VAX where ba(z\,ii) > ba(z2,i2) without loss of generality and hence X2 > Ai > 0 . Differentiating twice with respect to 9 the left-hand side shows that the left-hand side is strictly increasing and concave in 9 on [0,+ 0 0 ). Because one root of the above equation is 0 = 0, there is at most one other strictly positive root. Thus, 0 is uniquely determined, which gives a unique 7 . 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix 3.B Appendix B gathers proofs of Lemma 3.2 Proofs of Theorem 3.1-3.2 stated in sections 3.5 and 3.6, and Proofs of Theorem B1 and Lemmas B1-B9 can be found in Campo, et al. (2002). B.l Smoothness of G(-|-,-) For R > 1, consider the (nonparametric) model defined by structures [U, F] € U r x where U r and are as defined in Definitions 3.1 and 4. For any such structure, the next result establishes the existence, uniqueness, and smoothness of the equilibrium strategy s(-;z,i). In addition to obtaining the smoothness of s(-;-,i) with respect to (v , z ), which is nontrivial because s(-; •, •) does not have an explicit form in general, its proof is interesting in its own right as it tackles directly the singularity at v of the differential equation characterizing s(-; -, •) in contrast to previous work (e.g. Maskin and Riley (1996), Lebrun (1999), Lizzeri and Persico (2000)). Theorem B l: Suppose that [U, F] € U r x for some R > 1, then there exists a unique (symmetric) equilibrium and its equilibrium strategy s(-; •, •) satisfies: (i) V(v, x, ?) € (v,v] x Z x T, s(v; z, i) < v, while s(v; z, i) = v, (ii) V(v,z,i) £ [v_,v\ x Z x X , s'(v;z,i) > 0 with s'(v;z,i) < 1, (iii) Vi £ l , s(-; •, i) admits R + l continuous derivatives on [n, v] x Z. Theorem Bl is crucial for establishing Lemma 3.2. Let £(•; z, i) = s_1(-; z, i). P roof of Lemma 3.2. (i)-(iii) by Theorem Bl b(z, i) = s(v; z, i) admits R + l continuous derivatives on Z. The other assertion of (i), (ii) and (iii) follow from Lemma C5. 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (iv) From (3.3) we have 1 G(b\z,i) g(b\z,i) = - i - 1 A (£{b;z,i) - b) with A (£(b;z,i) — b) > 0 because s(v; z, i) < v if v > v by Theorem Bl. Therefore, Lemma C5 and the composition rule for differentation give that g(b\z, i) admits up to R + 1 continuous partial derivatives on Sf(G), which establishes (iv). 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. III. C hapter 4 A P P E N D I X 4 .A D a ta C lassification Class ” A ” - General Engineering Contractor. The principal business is in connection with fixed works requiring specialized engineering knowledge and skill. Class ” B ” - General Building Contractor. The principal business is in connection with any structure built, being built, or to be built, requiring in its construction the use of at least two unrelated building trades or crafts; however, framing or carpentry projects may be performed without limitation. In some instances, a general building contractor may take a contract for projects involving one trade only if the general contractor holds the appropriate specialty license or subcontracts with an appropriately licensed specialty contractor to perform the work. Class ” C ” - Specialty Contractor. There are 39 separate ” C” license classifications for contractors whose construction work requires special skill and whose principal contract ing business involves the use of specialized building trades or crafts. Manufacturers are considered to be contractors if engaged in on-site construction, alteration, or repair. A few example of specializations are Boiler, Hot water Heating and Steam Fitting C - 4 Building Moving and Demolition C - 21 Carpentry, Cabinet and Millwork C — 5 Concrete c-s Drywall C - 9 Earthwork and Paving C - 12 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX 4 .B P r o o f o f D efin itio n 4.3: Conditions (i) through (in) are straigforward consequences of Definition 4.2 (i) and (ii). Condition (iv) is proven as follows. Note that by definition, for i = 1, • • • where gj(-) is once continuously differentiable since A^.^-) is once continuously differentiable, and Gj(-) admits then two continuous derivatives over (b, b ), for j = 1 ,•■•,/. Note that this support is not closed as (?,;(•) appears in the denominator of the expression here above. P r o o f o f Lemma 4.1: Necessary con d ition s: Suppose that the observations are rationalized by a model [{f7j}f=1(-), F(-)} such that F(-) € F and Ui(-) G for i = 1, • - • ,/. Let s,;(-, F) be strictly increasing differentiable Bayesian Nash equilibrium strategies corresponding to this model. Thus Along with condition (in) in Definition 4.1, it implies that A (-) admits one continous derivative over its support [0, +oo). Condition (iv) then follows from equation (4.2) k = Si(-,F, { U i } ^ ) and G,;(6j) = F ( s i 1 (& ,;)) for every h G [b,b] = [si(v, F, {Ui}(= l ),v] 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In turn, si“1(-, {Gi}(=1) satisfies the first order condition for s,” 1 = £,;(•, {Gi}] i=l). So there exist increasing inverse bidding strategies i = 1, • • • , I, satisfying equation (4.4) and consequently there exist the corresponding functions A W i y(&7 ;) = (G? ;}[=1). Since the bid observations are rationalized by the model (bidder’s i FOC is satisfied), A w , ;,«(•) is also equal to Ui(-)/U^(-), and A ^ ^(-) is greater or equal to 1. I have then proven that there exists a function A u > l}i('), for i — 1, • ■ • , which satisfies condition (in) in Lemma 4.1. With these preliminary statements, I can start to prove condition (i) through (Hi). To prove (i), consider the bid distribution evaluated at an arbitrary value (bw, • • • , 6/o) G(b\o, • • • , bio) = Pr(fei < f> io , • • • , bj < bio) = Pr(f>i < si(n io), • • • , bi < s i (vi0)) = Pr(sj“1(6i) < nio, , s jx(bi) < vJ0) = Pi'(6(Fl) < nio, • • • ,£/(£>/) < vio) = Pr(£i(&i) < n10 ) • • • Pr(£/(&/) < vI0) = F(vw) ■ ■ ■ F ( vjo) = G i(b i) ■ ■ ■ G i(b i) Bids are independently distributed, i.e. condition (i) is satisfied. Part (ii) follows from the analysis of the equality 9ji i] where /(•) is continuous over [sj1^), f e ] and positive everywhere and s'-(-) is continuous 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. over [Si l(b),si l(b)} (see equation (4.2)). It implies that ,g ? -(-) satisfies the same properties than /(•) over its support [b,b]. S ufficient conditions: Define a distribution F(-) as F(ci, ■ ■ ■ , c/) = G(^f1(ci, {Gi}j=1), ■ ■ ■ , £;Jl (ci, for C i G [c,c], where c = &(6, (G,;}[=1) and c = & (& , {G}f=1). Under the assumption that Gi(bi) satisfies part (i), I obtain F(c \ 7 ■ ■ ■ , cj) = i'i(ci) • • • F](cj). The agents’ private costs are independent draws. By construction, /.(= .)= is K d te , (4.b .i ) Since the partial derivative of the inverse bidding strategy £ » ;(•, •) is positive, and gi(-) is also positive over its support, /*(•) > 0. Thus the distribution F(-) satisfies then condition (in) in Definition 4.2. I still have to prove that T)(-) admits one continuous derivative over its support. Consider equation (4.B.1). By definition, < /* (•) and AJ^(-) admit one continuous derivative, for all i — 1, ••• As a consequence, F)(-) admits also one continuous derivative over its support, and thus satisfies condition (ii) in Definition 4.2. Finally the distribution functions Fi(-) = F(-), for i = 1, • • • , I. This result follows from condition (in). For (bf, £ > “) such that Gi(bf) = Gj(b‘ “) = a, £i(bf, ■ ■ ■ ) = €j(Wj, • ■ •) so that Gi(bf) = F(UK, ■••)) = •••))• To complete the proof, I need to find a utility function Ui(-) for i = 1 ,-••,/ such that the bid distribution G,;(-) = F (s^1(-, {Gf=1}) on [b,!)}1, where s*(-,G) solves for 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the differential equation (4.2) with boundary conditions Si(v,F) = b and Si(v) = b. To prove my point, I need to show that £*(-, {G}f=1). The function £i(-,G) satisfies condition (in) in Lemma 4.1. By the adequate change of variable, 1 — F(£i(., G)) = 1 — G,;(-), the first condition in (in) then becomes , ^ f m b i , 0 ) ^ , 0 ) l-F (tj(h ,G ) • This expression is equal to equation (4.2) for £ ,;(■ , {G}^=1) = x(-, {G}j=1). P r o o f of L em m a 4.2: By isolating two agents, arbitrarely called agent 1 and agent 2, one can solve for the pair of CRRA coefficients 9\ and 62 which satisfy condition (iii)(b). In matrix form, condition (iii)(h) becomes W ) - Y!(& ? * ) 0! (b ^ -b ^ y ^ y * 1 _ Y2(b?) -ri(6i)“2 ) 0 2 (b “2 - 6 “2)y«2y“2 which should be satistfied for Gi(6“1) = G2(& 2 1) = Qi and Gi(b“2) = G2 (b“2) = a 2- The system admits a unique solution in 9 if the A matrix is invertible. The full rank condition is yc*l y a 2 y a iy a 2 _ y«ayai ^ 0 which is equivalent to -d _ ^ ^L_ Y o Yo 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the solutions in (6> i, #2) are y a i - y r a 2 y r a 2 y a 2 y t t i y O ! 2 y C ( i y « 2 _ y - a zy -a i ^2 y a i y » 2 ^ 2 5 12 12 12 12 where Xi = (fef - bf) + (b*1 - )F2 ai / r 2 “2 and X2 = (6 f - 6?2) + (h*1 - ft?1 )F“1 / F “2. The parameters fys are uniquely recovered from the system of equations above, once at least one coefficient 9j is known since they can be expressed as a function of this parameter. Namely Yj(bT) «, = ? ) ( % ' - vjl) y « iy a2 y “2 = y e , , | Y f , Y i -_ ~ Yi ' W ~ b>a ^ for any oq € [0,1). 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Campo, Sandra
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Asymmetry and risk aversion in first -price sealed bid auctions: Identification, estimation, and applications
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Economics
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