1 Itroductio South Tyrol i Norther Italy is the most importat apple growig regio i Europe. Apple growers have the possibility to sell their products t

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1 Quality eforcemet as a public good (prelimiary ad icomplete) Rudolf Kerschbamer, Muriel Niederle ad Josef Perktold Uiversity of Viea, Harvard Uiversity, Uiversity of Chicago December 1999 Abstract A competitive market for a experiece good is cosidered where high quality is eforced by repeated game trigger strategies, as e.g. i Klei ad Leffler (1981). The good is demaded by two types of customers, log ru (LR) ad short ru (SR), the former buyig repeatedly, the latter oly oce. I this settig quality eforcemet has public good characteristics: SR buyers ca free ride o quality eforcemetbylrbuyers but, by doig so, they may prevet LR buyers from puishig firms for producig low quality. We characterize equilibria i differet market istitutios ad show that o-exclusivity has a egative impact o quality eforcemet whe the market istitutio provides some public iformatio. 1

2 1 Itroductio South Tyrol i Norther Italy is the most importat apple growig regio i Europe. Apple growers have the possibility to sell their products trough three differet chaels, through a cooperative, i auctios or directly to wholesale traders. Buyers are either log-term whole sale traders or traders that buy oly occasioally, e.g. traders from other regios that have to cover a shortfall i the harvest i their ow regio. A apple grower ca icrease his quatity o the market by mixig low quality apples together with high quality ad tryig to sell them off as high quality. The market istitutio differ i their quality cotrol. Cooperatives ispect ad categorize the quality of each shipmet by growers. I the auctios, quality is ot verified, but the grower is idetified by his membership umber. The cases are large (approximately 340 kg) ad create the cheatig possibility for growers. A grower that obtaied a reputatio for doig so, was oly able to sell at low prices i the auctio. He tried to sell uder a differet membership umber, but was ot allowed by the auctio orgaizers to do so. This market is characterized by three features that form the basis for our model, the possibility of moral hazard i the provisio of high quality, the existece of both log ru ad short ru buyers ad icomplete iformatio exchage about past qualities. Also, the market istitutios used differ i their treatmet of the quality eforcemet problem. This paper aalyses i a repeated game settig how the outcome ad efficiecy of differet market istitutios, especially auctios ad decetralized bargaiig markets, are affected by these features. We cosider a market for a experiece good with a fiite umber of buyers ad sellers where high quality is eforced by repeated game trigger strategies, as e.g. i Klei ad Leffler (1981). The good is demaded by two types of customers, log ru (LR) ad short ru (SR), the former buyig repeatedly, the latter oly oce. I the auctio settig the idetity or type of the seller caot directly ifluece the allocatio. All bids are treated i the same way ad short ru buyers caot be discrimiated agaist i favor of log ru buyers. If past trades, but ot the produced qualities are publicly observed, quality eforcemet obtais public good characteristics. Observig past tradig prices ad quatities makes it possible that SR buyers buy from firms with high price histories. SR buyers ca, therefore, free ride o quality eforcemet by LR buyers but, by doig so, they may prevet LR buyers from puishig firms for producig low quality. The presece of short ru buyers creates a additioal problem for eforcemet. Whe a firm starts to produce low quality, its LR customers migrate to other firms but puishmetisweakeed by ogoig sales to SR buyers who fill i for the reductio i demad.. With some probability all cheated LR customers get replaced by SR buyers who are ot able to sigal low quality to the ext geeratio. A partial market breakdow results ad oly few firms ca be supported as high quality producers. As a cosequece a partial market breakdow might occur. However, the outcome of auctio markets ca be improved by coordiatio amog LR buyers. Coordiatio strategies allows a better use of iformatio geerated by a market istitutio. It uses a sigal of the auctio istitutio ad 2

3 makes it to carry iformatio about cheatig, which it is ot able to without coordiated strategies. Auctio markets with a large iformatio flow, as e.g. whe all bids are published, produce high puishmet probabilities eve without coordiatio. That is, additioal iformatio geerated through coordiatio ca act as a substitute for iformatio iheret i a market istitutio. I decetralized markets, where oly match specific iformatio is available, some firms ca form a log term relatioship with log ru buyers. Those firms sell high quality tolrbuyers, but SR buyers receive oly low quality. Better outcomes ca be obtaied if firms are allowed to price discrimiate betwee LR ad SR buyers. With price discrimiatio, SR buyers reward the LR buyers for their eforcemet activity. Without discrimiatio or exclusivity, iformatio spill-overs from LR to SR ca reduce welfare, while iformatio flows amog LR ca oly icrease welfare. Depedig o the market istitutio, more public iformatio ca improve the outcome but ca also make itworse. To illustrate the role of exclusioary or preferetial treatmet i our repeat purchase settig cosider the followig hypothetical example. A restaurat has build up a customer base of repeat purchasers ad is sellig high quality to them. Now, the restaurat receives a very good review i a widely published restaurat guide. Assume, that the restaurat guide will ot be updated or oly after a log time. Through the restaurat review may short-term customers, e.g. tourists, lear about the high quality history of the restaurat ad will show up at its doors. Give the limited capacity of the restaurat, its ower has to decide whether to reserve seats for its regular customers or to accept short term customers who might be willig to pay more. I the latter case, the restaurat ca lower its quality ad maybe icrease its price ad sell to a sequece of short ru customers that are attracted by the review. The tourists will see a full restaurat ad high prices, but do ot lear from the experiece of previous tourists. This seems to be, therefore, a profit icreasig strategy for the firm. However, with the absece of log term customers, eforcemet is lost, ad the above sceario caot be a equilibrium, if tourists have ratioal beliefs. A equilibrium, where the restaurat attracts both log ru ad short ru customers, might require mixed strategies which result i reduced puishmet probabilities ad a partial market break dow. If, i cotrast, the restaurat gives preferetial treatmet to its log term customers ad restricts sales to tourists, eforcemet ca be maitaied ad the log term customers ad some short term customers beefit from high quality productio. If the restaurat guide is frequetly updated, the iformatio asymmetry is strogly reduced ad eforcemet is doe by both log ru ad short ru customers. Therefore, the iteractio of o-discrimiatio betwee LR ad SR buyers, ad the existece of some but limited iformatio spill-overs is the source of the eforcemet problem ad the potetial market breakdow. Past eforcemet obtais public good characteristics, whose rivalrous cosumptio by short ru buyers destroys it. Price discrimiatio, such asloyalty discouts, makes 3

4 eforcemet to a private good ad improves welfare. A evidet strad of literature related to our model is the literature o quality eforcemet. A semial cotributio is Klei ad Leffler (1981). I this model firms face the moral hazard problem of producig either high or low quality. Quality is a experiece good, either observable or verifiable. hece the oly way to eforce high quality is through repeat purchase. I Klei-Leffler's model all the buyers (as well as the firms) are ifiitely lived. Furthermore after oe buyer has bee cheated all other buyers get iformed ad together collectively puish the firm by ever tradig with her agai. This allows a very effective detectio of "deceptive productio" (or cheatig). This "shoutig" of every buyer who got deceived is however ot dealed with as a strategic variable. Buyers ca ot choose whether to use it, or are there ay costs ivolved i shoutig. If shoutig were made strategic it is however uclear whether i equilibrium oly cheated players would start shoutig ad whether buyers would ivest i a "shoutig" techology.. I our model we take oly the market istitutio as exogeous. We also do ot allow for a shoutig techology, though i our model buyers would ot ecessarily wat toivest i it. 1 Papers o Folk theorems with a public sigal by Fudeberg et al (1991), (1994) ad (1994) are i a similar, though much more geeral spirit. Though also there the public sigals remais a exogeous variable. The ext sectio presets the model ad aalyzes the repeated game. The followig sectios aalyze the auctio ad decetralized bargaiig markets i tur. 2 The Market Cosider a repeated market for a experiece good with a fiite umber of sellers ad buyers. Sellers are idetical, ifiitely lived ad ca produce at most q uits per period of either high quality at uit cost c, or of low quality at uit cost c, with c >c. The good is ot storable. Buyers differ i their valuatio ad i their time horizo. Cosumers wat to cosume at most oe uit per period durig their life. A cosumer with valuatio fi receives a utility offi p: The distributio of cosumer types remais costat i each period ad is commo kowledge. There are m k buyers with valuatio fi k, with k =1::K ad fi 0 <fi 1 < ::: < fi K. A fractio of cosumers of each valuatio type are ifiitely lived (log ru buyers LR). A fractio (1 ) live oly for oe period ad are replaced i the ext period by a ew geeratio (short ru buyers SR). Sellers ad log ru buyers have the same discout factor ffi: (Note: Discout factor of LR buyers is i most settigs ot relevat, relevat is oly their memory.) Cosumers caot observe the quality of the good at the time of purchase, but lear it durig cosumptio. High quality will be eforced by the threat of puishmet i the repeated game. With respect to the timig, we assume that firms have to produce at the begiig of the period, so that the firm does ot have ay iformatio 1 The poit ofitroducig short ru buyers i our model is to itroduce buyers who ca ot covey iformatio that they got cheated, eve if a shoutig techology were available. 4

5 about the cosumer that buys the product ad the market outcome whe decidig which quality to produce. This rules out, that the firm ca make its quality choice coditioal o the idetity of the buyer, if it lears this as a cosequece of the market process. Cosumer choice If cosumers kow which firms are supposed to produce high quality o the equilibrium path, their demad fuctios will be the same as uder full iformatio. Short ru buyers are ucocered about the future ad will therefore always play their static (oe period) best respose. Log ru buyers could choose a actio that does ot maximize their period payoff if the repeated game yields a higher preset value payoff for aother choice. I our market structure it is ot ecessary for log ru buyers to pick a actio that does ot maximize their period utility o the equilibrium path. Buyers with correct expectatios will demad high quality if fi μ μp fi p ad fi μ μp 0 low quality if fi μ μp» fi p ad fi p 0 There are two cases to cosider. First assume that p=p =. The high valuatio buyers with fi 2 p p = ;fi K Λ will buy high quality, medium valuatio buyers with fi 2 p= ; p p = Λ will buy low quality, ad low valuatio buyers with fi 2 fi0 ;p= Λ will ot buy, with possible empty sets. I the opposite case, where p=p < =, high valuatio buyers with fi 2 p= ; fi K Λ will buy high quality, ad other buyers (with valuatio fi 2 fi0 ; p= Λ ) will ot buy, agai with possible empty sets. We deote the demad for high ad low quality uits for give prices p; p by D p;p ad D p; p, with idices LR ad SR whe we refer to demad by log ru or short ru buyers respectively. I order to reduce the umber of possible cases for demad fuctios to cosider ad to rule out trivial equilibria we make the followig assumptios o the full iformatio Walrasia equilibrium ad the full puishmet equilibrium. results. These assumptios are ot ecessary for the qualitative Assumptio 1: = = c=c: This assumptio has two importat implicatios. First, that total productio is idepedet of the compositio of total productio betwee high ad low quality sice fi c 0, fi c 0; ad secodly, that producig low quality isever optimal i the first best (FB), (i.e., with verifiable quality) sice fi c fi c for all fi c= = c=. Assumptio 2: Free Etry of firms ad fi 0 <c= ; fi 1 = c= + ffl (for some small ") Together with Assumptio 1, Assumptio 2 implies that the umber of active firms i the FB P is determied by = K fi k m k =q: Note that the same umber of firms will also be active i our 1 5

6 secod best world (where quality is ot verifiable) sice free etry esures that p = c. Notice, that is ot ecessarily uique: Next otice that i our secod best world high quality ca oly be eforced by future "puishmet" that is harsh eough to iduce the firm to forego a curret gai of c c. I our model the "harshest" puishmet occurs if a deviatig firm (that is, a firm that sells low quality at the high price p) is immediately detected ad if this firm is uable to participate i the high quality market i ay subsequet period (ex post observable quality + grimm strategies by buyers). The harshest puishmet will iduce a firm to produce high quality if c c» (p p)=ffi. I the sequel we call the price p = c +(c c)=ffi, the miimum eforcemet price uder full puishmet ad deote it by p FP. The maximal umber of firms that ca sell high quality atthis price is deoted by FP. P This umber is determied by FP = D(p FP ;c) = K fi k m k =q, where k is the lowest k for which fi k p FP fi k c. Notice, that p FP is strictly higher tha the Walrasia equilibrium price for ay ffi<1. This implies that asymmetric iformatio is welfare reducig sice the umber of high quality firms i equilibrium Λ» FP <for ay ffi<1. k Assumptio 3: FP > 0: Assumptio 3 is a coditio o the fis ad o ffi. It requires that at the miimum eforcemet price uder full puishmet there is positive demad. This rules out cases where high quality ca ever be eforced. Uder Full iformatio, i a Walrasia equilibrium oly high quality uits are produced ad prices are equal to uit cost, i.e. μp = μc; p = c, μ = = D(1; μc) = D(1;c) ad = 0: I ay asymmetric iformatio equilibrium p Λ > c, p Λ = c ad Λ + Λ =. The welfare loss uder these assumptio comes from the compositio of productio, fewer high quality uits ad more low quality uits are produced tha i the first best. The aggregate welfare of differet market istitutios ca simply be raked by the umber of high quality uits produced. 2.1 The repeated Game I our model oly the cosumer of a firms product lears directly about the produced quality. Other buyers ad sellers lear oly from observed cosequeces i the market stage game. The situatio, therefore costitutes a repeated game with imperfect moitorig where players observe public ad also private sigals about the actios of other players. The equilibrium i this case is i geeral difficult to characterize. The case of public sigals, i.e. all players receive the same (imperfect) sigals, has bee thoroughly aalyzed. Fudeberg, Levie, Maski (1994) showed that uder some assumptio o the sigal a Folk theorem obtais. Fudeberg Levie (1994) derived sufficiet coditios whe short ru players are also preset. If players are symmetrically iformed about actios of others they agree o the eed to puish. If players receive coflictig sigals, oly those buyers that were cheated will avoid the cheatig firm, while other buyers will still buy from it. 6

7 Cosequetly, puishmet by some buyers might ot have ay immediate payoff cosequeces for the cheatig firm. Fudeberg ad Levie (1991) showed that a partial Folk Theorem is possible if players are either perfectly patiet or are oly required to play "-best strategies. Recetly Kadori ad Matsushima (1998) ad Compte (1998) show that a Folk theorem ca be obtaied whe players are allowed to sed messages. I our case, we are iterested i the iformatioal properties of market istitutios, therefore we do ot allow players to directly commuicate ad we restrict ourselves to strictly positive discout rates. We derive approximate bouds o puishmet probabilities ad size of the high quality market depedig o the market istitutio ad the amout of coordiatio amog buyers. Usig the special structure of our market game, we obtai strategies that are fuctios of a simple state space. The evolutio of the game has as a cosequece a Markov property ad the stadard tools of dyamic programmig ca be used. The market is divided ito two submarkets, oe desiged for high quality ad oe for low quality. Eforcemet of high quality productio requires that the access of firms to the high market is ratioed. As i Klei Leffler (1981), firms eed a positive expected excess profit stream that they will lose if they start to produce low quality. The required price premium reduces the demad ad oly a restricted umber of firms ca participate i the high quality market. Iitially, firms are arbitrarily selected for participatio i the high quality market. Firms that have bee detected cheatig will be replaced by ew firms from the pool of o producig ad low quality producig firms. This, however, does ot occur o the equilibrium path. Productio for the low quality market does ot have ay icetive problems. Therefore, i the followig aalysis we ca restrict attetio exclusively o the high quality market. High quality productio will be eforced with trigger strategies. Buyers that have bee cheated or leared from others that a firm is cheatig will belief that this firm will produce low quality for the rest of the game ad ever agai buy from this firm. This meas that a firm that cheats some LR buyers will lose their demad ad will lose all high quality demad oce the deviatio has bee publicly sigalled. If a short ru buyer has bee cheated, this iformatio disappears from the game. SR buy oly oce ad have o possibility to sigal to other players which quality they received. The history of the game with respect to each firm is i oe of four differet phases, the ocheatig phase o the equilibrium path, the cheatig phase, whe the firm produces low quality but it is ot yet publicly kow to do so, the detectio phase, whe the market sigals idicate that a firm is cheatig, ad a puishmet phase, whe the firm has left forever the high quality market. Players have the same iformatio ad beliefs i all phases except for the cheatig phase. I the cheatig phase, oly the firm ad the already cheated buyers kow this. Other players caot distiguish cheatig ad o-cheatig phase. A log ru buyer chooses his actios coditioal o beig cheated ad i some settigs also o the umber of periods that passed sice he was cheated. 7

8 Acheatig firm ad cheated log ru buyers have to form beliefs about the umber of LRs that have bee cheated. Therefore, the state durig a cheatig phase icludes the umber of cheated log ru players, the time that passed sice they were cheated ad the beliefs about this. We are mostly lookig for strategies i which oly the support of beliefs matters. This support is ecoded i the defiitio of the state. Actual beliefs eed ot be part of the state, which greatly simplifies the aalysis. We are therefore able to rak market istitutios ad equilibria i their efficiecy usig approximate bouds, without havig to derive a complete solutio. Our mai objective is to shed light o the qualitative differeces ad problems that various market micro structures have i order to eforce good performace with imperfect moitorig ad imperfect iformatio aggregatio. 2.2 The geeral repeated game The geeral model of the repeated game follows Fudeberg, Levie, however, i cotrast to their model we have ot oly imperfect iformatio about players actio but also icomplete iformatio about players types. The valuatio of idividual buyers for the good are private iformatio. We assume that the distributio of valuatios is commo kowledge. May markets istitutios operate uder aoymity of traders. Firms i our model always have ames ad ca be idetified. Whether ames or characteristics of buyers plays a role i the equilibrium depeds o two types of assumptios, aoymity ad discrimiatio. By aoymity we mea that the ame (ad the type) of the buyer is ot revealed, which implies that oly aggregate (statistical) iformatio for the actio of all buyers is observed. Discrimiatio deotes the property of the market istitutio ad thereby of the game structure that does ot allow players to coditio their behavior o the idetity of other players. For example i the auctio settig firms are ot able to accept oly offers by log ru buyers. Exclusio by ame or idetity is assumed to be illegal i this case. We assume that the idetity ofbuyers caot be observed i the auctio markets, however i the decetralized bargaiig markets firms ca recogize a log ru buyer that they have met i the past. With these iformatioal assumptios, buyers ad sellers have to form beliefs about the aggregate distributio of actios ad histories. For the followig let i deote players, player i 2 I is of type fl i 2V, with I ad V fiite. Firms are all of the same type. The type of a buyer is give by his valuatio fi ad his time horizo, either LR or SR. The distributio of types i the ecoomy is give by a coutig measure m(fl). The market game is played i each period. Each player chooses a stage game strategy or actio a i : Each actio profile a =(a 1 ; ::; a ) iduces a probability distributio over outcomes z =(z 1 ; ::; z ). The outcome reflects a stochastic elemet ο i the market istitutio ad possibly mixed strategies. At the ed of a period t, each player i observes a outcome z it which cotais a private sigal as well as a public compoet, ad the players actio. A players realized payoff depeds oly o his observed outcome. The history for player i at the begiig of period t is h it =(h it 1 ;z it 1 ). Let H deote the set of all possible histories. A strategy ff i of a player is a map from his set of 8

9 private histories H i to stage game strategies. Players form beliefs μ i (h it ) 2 M(I V H) about the distributio of histories of other players ad their types coditioal o his ow history. Beliefs of players have to be cosistet with the distributio of types ad with the strategy profile ff: Give a system of beliefs μ ad a history h t, the strategies ad the radom elemet iduce a distributio over actio ad histories of all players at all times. The expected payoff i a period for player i for give history is u i (ff; h it )= Z u i (a; ο)df (ο)dμ i (h it ) where a = (ff i (h it )) i2i is the actio profile give strategies ad histories at period t, ad the discouted average cotiuatio payoff of log ru players V it =(1 ffi)e t 1 X fi =1 ffi fi u i (ff;h ifi ) where the expectatio is take over possible cotiuatio histories coditioal o the history of player i at time t. APerfect Equilibrium requires that players form cosistet beliefs ad maximize their utility for every cotiuatio game. Short ru players will always play a best respose i the stage game give the public history. I the equilibria that we look at, it is ot ecessary for log ru buyers to play aactio o the equilibrium path that is ot a stage game best respose. Off the equilibrium path after a firm has started to cheat, the behavior of log ru buyers determies the probability ofpublic detectio. This probability ca possibly be icreased if log ru buyers are puished for ot sigallig. However, with imperfect iformatio about idividual purchase histories, this is a difficult problem. For most cases we cocetrate therefore o the eforcemet of behavior o the seller side. 2.3 A simplified repeated game The followig provides a simplified versio of the repeated game that makes it easier to uderstad the structure ad dyamics of the game (off-the equilibrium path). We cocetrate o the high quality market, simplify the payoff structure ad use detectio probabilities that are exogeous. I each period a simultaeous move stage game is played. Each firm chooses to produce either high (H) or low (L) quality, each buyer aouces a set of firms from which he is willig to buy. The payoff to a firm is p c; c 2 fc; μcg; if at least oe buyer aouces a demad for the firms products, ad zero otherwise. The expected payoff to a buyer is a icreasig fuctio of the average quality of all firms which are i his aouced set. I cotrast to the actual market game this abstracts from buyer iteractio ad market clearig off the equilibrium path. Each buyer is radomly assiged to a firm i the demaded set ad lears the quality that this firm has produced. If a log ru buyer lears that a firm has produced low quality, he will expect the firm to always produce low quality ad stop buyig from this firm. If low quality was sold to a short ru buyer, the o other buyers will lear about this. The probability that a firm is puished 9

10 depeds o the joit iformatio of all buyers. Cheatig a short ru buyer, therefore, does ot affect the joit iformatio of buyers i future periods. The iformatio o the buyer side ca be summarized by the umber of log ru buyers x that have bee cheated. For ow, we assume that a deviatig firm ca observe how may log ru buyers it has cheated. Log ru buyers that have bee cheated will avoid the cheatig firm. I the market game this results i actios that lead to publicly observable outcomes that sigal that the firm is producig low quality. Here, we assume that there is a exogeous probability of public detectio ß(x; D) which is icreasig i the umber of cheated log ru buyers x. With trigger strategies, public detectio of a firm's cheatig leads to puishmet by all buyers ad to zero payoff for the firm. Uder these assumptios, the iformatio of buyers with respect to a firm ca be summarized with the followig state variables: 2 State 0 : o log ru buyer has bee cheated, state x 2 X :umber of LR cheated but cheatig ot publicly sigalled, state detectio D : cheatig is publicly sigalled, ad state P :every buyer kows that firm has cheated ad is beig puished. Histories that map ito the same state iduce the same distributio of beliefs across buyers, i.e. are idistiguishable for buyers subject to permutatios of buyers' ames. Each firm i has a state space Y i with geeric elemet y i 2 X [f0;d;pg, the joit state is give by y =(y 1 ; :::; y ) 2 Y =X Y i. i The strategy of a buyer has to be measurable with respect to his iformatio. SR ad LR that have ot bee cheated ca distiguish the 3 types of histories ad states fdg; fp g ad X [f0g: Cheated LR buyers kow that the firm has produced low quality, i.e. y i 2 X; but ot the total umber of buyers x that kow this. However, give the curret simplificatios, this later iformatio is ot directly payoff relevat to the cheated buyers ad they do ot eed to coditio their behavior o beliefs about x. We cosider oly strategies that are idepedet of caleder time which implies that the cotiuatio of the game at a time t depeds oly o the state y. I the equilibrium that we are iterested i, traders follow the followig strategies ff i (y i )= ( H if y i = 0 L if y i 6= 0 for each firm i ff j ((y i ) i )= fi : suppμ j (y i )=f0gg for each buyer j Beliefs of buyers μ j (y i ) for all firms i ad buyers j are give by the above observability assumptios, i.e. 2 This is ot the miimal state space possible, but used to make expositio easier ad more compatible with later versio of game. (1) 10

11 suppμ j (y i )= 8 >< >: f0g X fdg fp g if y i 2 X [f0g ad j has ot bee cheated by firm i if y i 2 X ad j has bee cheated by firm i if y i 2 D if y i 2 P The strategy profile of all players iduces a probability distributio over the evolutio of the state. The evolutio of the state coditioal o strategies has a Markov property ad is therefore fully summarized by the trasitio matrix ad the iitial state. Iitially, all firms that are o the high quality market are i state 0: The trasitio matrix Π at a period t is a fuctio of the firms actio i this period, i.e. Π a where a 2 fl; Hg, coditioal o the strategies of buyers as give above. The probability ß(x; D) of beig detected give state x is idepedet of whether the chose actio was H or L. The trasitio matrix whe the firm produces low quality has the form Π L 0... x... x 0... D P 0 ß L (0; 0) ß L (0;x) ß L (0;x 0 ) ß L (0; ) x 0 ß L (x; x) ß L (x; x 0 ) ß L (x; ) ß(x; D) 0.. x ß L (x 0 ;x 0 )... ß L (x 0 ; ) ß(x 0 ;D) ß( ;D) ß( ;D) 0 D P ad whe the firm produces high quality it has the followig form Π H 0... x... x 0... D P x 0 1 ß(x 0 ;D) 0 0 ß(x; D) 0.. x ß(x 0 ;D)... 0 ß(x 0 ;D) ß( ; D) ß( ;D) 0 D P

12 is The probability of beig detected after exactly fi periods of cheatig whe the iitial state is y t ffi L;fi (y t )=ß fi L(y t ;D) where ß fi a(y; y 0 ) = Pr(y t+fi = y 0 jy t = y; a) is the fi-period trasitio probability from state y to y 0 if the firm always chooses actio a. The expected discout factor util detectio whe startig from state y t is b ffil (y t )= 1 P fi =1 ffi fi ffi L;fi (y t ). The optimal strategy of the firm is the solutio to the followig Bellma equatio V (y) = maxfv H (y);v L (y)g where V H (y) = V L (y) = 8 >< >: 8 >< >: Assumptios: (μp μc)(1 ffi)+ffiv (0) if y = 0 (μp μc)(1 ffi)+ffi [ß H (x; x)v (x)+ß(x; D)V (D)] if y = x 2 X (p μc + v D )(1 ffi)+ffiv (P ) if y = D (p μc)(1 ffi)+ffiv (P ) if y = P (μp c)(1 ffi)+ffi P ~x (μp c)(1 ffi)+ffi (A1) ß(x; D) is icreasig i x (A2) ß(x; D) is cocave i x» P ~x ß L (0; ~x)v (~x) if y = 0 ß L (x; ~x)v (~x)+ß(x; D)V (D) if y = x 2 X (p c + v D )(1 ffi)+ffiv (P ) if y = D (p c)(1 ffi)+ffiv (P ) if y = P P P (A3) ß L (x; x + 1)) L (x +1;x+ k) ß L (x; x + k)), for all x k 1 k 2(ß (A4) ß L (x +1;x+ k) ß L (x; x + k), for all k 2 ad for all x Propositio 1 Assume A1 to A4 hold, the prices p Λ = c; μp Λ = c +(μc c)=^ffi; where ^ffi = 1P fi =1 ffi fi ß fi L (0;D) ad quatities Λ = max D(p Λ ;c); Λ = Λ = D(p Λ ;c) are the outcome of the subgame perfect equilibrium with strategies give by (1). For give trasitio probabilities ß this is the equilibrium outcome with the lowest welfare loss. Proof: The followig lemmas show that if the gais to cheatig V L (x) V H (x) is positive for some state x the it is also positive for all higher states x + k. A firm for which itwas optimal to cheat a buyer will therefore always produce low quality. No buyer that has leared that a firm has produced low quality i the past will therefore demad from this firm. We show later that if the 12

13 price is at least μp Λ ; the the firm does ot have a icetive to start producig low quality after a good history. μp Λ is therefore the miimum eforcmet price ad Λ the largest umber of firms that ca be supported to produce high quality i ay SPE. Lemma 2 Assume V (D) is idepeded of x ad V (D) < (μp μc). If (A1) ß(x; D) is icreasig i x the V H (x) is decreasig i x: If (A2) ß(x; D) is cocave i x;the V H (x) is covex i x: The payoff for the firm for producig high quality at state x is give by V H (x) =(μp μc) 1 ffi + 1 ffi(1 ß(x;D)) The lemma ca be easily verified. ffiß(x;d) 1 ffi(1 ß(x;D)) V (D) Lemma 3 If V (x) is decreasig ad covex ad (A3) holds the the icetives to cheat are icreasig i the umber of cheated log ru buyers x: Proof: Recall that V L (x) =(μp c)(1 ffi)+ffi» P ~x ß L (x; ~x)v (~x)+ß(x; D)V (D) V H (x) =(μp μc)(1 ffi)+ffi [ß H (x; x)v (x)+ß(x; D)V (D)] the the payoff differece is (x) =V L (x) V H (x) =(μc c)(1 ffi)+ffi " X ~x ß L (x; ~x)v (~x) ß H (x; x)v (x) This differece is icreasig i x if the term i brackets ist icreasig i x. i.e. P P ß L (x; ~x)v (~x) ß H (x; x)v (x)» ß L (x +1; ~x)v (~x) ß H (x +1;x+1)V (x +1) ~x2x collectig terms, we have ~x2x [ß H (x; x) ß L (x; x)] V (x)+ X ~x6=x;~x6=x+1 [ß L (x +1; ~x) ß L (x; ~x)] V (~x) [ß P H (x+1;x+1) ß L (x+1;x+1) + ß L (x; x+1)] V (x+1) ote ß H (x; x) = ß L (x; ~x) ~x2x The sum of the coefficiets o the left ad right had sides are equal. Dividig by the expressio i brackets o the left had side, implies that the right had side is a covex combiatio as log as ß L (x +1; ~x) ß L (x; ~x):for all ~x x + 2 ad all x P ~x6=x ß L (x; x +1)+ ß L (x; ~x)) P ~x6=x+1 ß L (x +1; ~x) V (x)+ X ~x6=x;~x6=x+1 13 (ß L (x +1; ~x) P ß L (x; ~x)) V (~x) V (x+1) ß L (x; x +1)+ ß L (x +1; ~x) ~x6=x+1 # (2)

14 If V (x) is a decreasig, covex fuctio, the the above iequality (2) will hold as log as E(x) x + 1, where the expectatio is evaluated at the probability distributio defied by the left had side( 3 ). A sufficiet coditio for this to hold is that the first term is larger tha a half, or equivaletly X (ß L (x +1;x+ k) ß L (x; x + k)) X ß L (x; x + 1)) k 1 k 2 I the case q = 1 this simplifies to ß L (x; x + 1)) (ß L (x +1;x+ 2), i.e. the probability of cheatig a additioal log ru buyer ad ot beig dedected is a decreasig fuctio of the umber of already cheated log ru buyers x: Remark: Secod Order Stochastic Domiace caot be used, because it is ot satisfied i most relevat examples, e.g. ever if q =1. Lemma 4 Assume (A1,A2,A3) hold. If (x) 0; the (x 0 ) 0 for all x 0 >x. If (0) 0; the (x) 0 for all x 2 X. If a firm is willig to cheat after a good history, i.e. i state y = 0; the it is willig to always cheat. This implies that the relevat icetive costrait for the miimum eforcemet price compares ever cheatig ad always cheatig. It also implies that the strategy of the firm i the case whe the firm caot observe the umber of cheated log ru buyers depeds oly o the support of the beliefs ad ot o the exact distributio. Proof: We have prove that uder the above assumptios V H (x) is decreasig ad covex ad the gais to cheatig ad switchig back to high quality productio are icreasig i x. We did ot proof it for the optimal value fuctio V (x) ad optimal deviatio payoffs (x): However, the former is sufficiet for the proof of the lemma as the followig shows for the case q =1: Defie aa 0(x) =V La (x) V Ha (x) =(μc c)(1 ffi)+ffi [ß a 0(x; x +1)V a 0(x +1) ß a (x; x)v a (x)] (3) where a ad a 0 stad for the cotiuatio strategy ff(x) = a ad ff(x +1) = a 0. Note aa 0(x) is icreasig i V a (x +1) ad decreasig i V a (x). Let a = Λ ad a 0 = Λ represet the optimal cotiuatio strategies. The previous lemmas imply HH (x) is icreasig i x: Note the followig properties which follow from optimal choice ad equatio (3). HΛ (x) =0, LΛ (x) =0 HΛ (x) > 0, LΛ (x) > 0 aa 0(x) 0 ) aλ (x) 0 for all a; a 0 3 EV (x) V (E(x)) V (x + 1); where the first iequality follows from covexity for a extesio of the fuctio V to real umbers; the secod iequality follows from mootoicity, V (weakly) decreasig, ad from E(x) x + 1: 14

15 Assume ΛΛ (x) 0: The either (a) ΛΛ (x) = ah (x) 0 or (b) ΛΛ (x) = al (x) 0 for ay a. Case (a): HH (x) 0 implies HH (x +1) 0 ad by mootoicity ad optimality also all ΛΛ (x) 0;x x +1 Case (b): LL (x) 0 implies ΛΛ (x +1)= LΛ (x +1) 0 I both cases, if producig low quality is preferred at state x the it is also preferred at state x +1 ad by iductio at all higher states.the same is true for x = 0, i.e. y = 0. For geeral q the same argumet applies whe appropriate chages are made (cosider a 0 as a vector ad cosider all higher states x + k; k =1; 2; 3; :::). At the miimum eforcemet price, i.e the price such that (0) = 0 the icetive costrait of cheatig ad the switchig back to producig high quality is ot bidig sice ΛH (x)» ΛL (x) uder the assumptios of the lemma. Miimum Eforcemet Price I the followig we derive the miimum eforcmet price i a represetatio comparable to the full puishmet case. The icetive costrait is coditioal o the iformatio of the firm whe it starts to produce low quality for the first time. The ex ate detectio probabilities ffi fi = ß fi L (0;D) ad the profit v D i the detectio period summarize the essetial features of the market game for the firms icetive costrait. As i stadard repeated game aalysis it suffices to look oly at sigle deviatios from a equilibrium. The followig icetive costrait is derived uder the assumptio that a firm that fids it optimal to cheat oce will produce low quality products i all cotiuatio games, which, as show, is satisfied if assumptios A1 through A4 hold. v D = v D (p c) is used to correct for the differece betwee the payoffs i the detectio ad puishmet periods. The expected profit from cheatig is the give by V L = 1X Λ (μp c)(1 ffi fi )+(p c)ffi fi +(1 ffi)ffi fi E jfi v D ffifi +(μp c)ffi 1 fi =1 P where ffi 1 = (1 1 ffi fi ) is the probability of ever beig detected. The term i brackets is the t=1 average profit if detectio occurs fi periods after the firm starts to produce low quality. The average discouted profit from always producig high quality is μ V =μp μc: The icetive costrait μ V VL implies μp p + μc c ^ffi + (1 ffi) E [ffi fi v D ] ^ffi where ^ffi P = 1 ffi fi ffi fi is the expected discout factor util detectio. This is the same characterizatio fi =1 as i the stadard case with full puishmet i the period followig cheatig whe ^ffi is replaced by P ffi. If the hazard rate ff = ffi fi =(1 fi 1 ffi s ) is costat, the ^ffi = s=1 ffffi : 1 (1 ff)ffi The smallest price μp that satisfies this iequality is the miimum eforcemet price. The 15

16 miimum eforcemet price is lower with earlier expected detectio. Therefore, market istitutios ca be welfare-raked by their implied ^ffi. 2.4 Market equilibrium (icomplete) The outcome of the repeated game is a costat market equilibrium i each period characterized by prices (μp Λ, p Λ ) ad umber of firms μ Λ ; Λ producig high or low quality respectively. Total supply ad demad fuctio imply equilibrium price for low quality alog the equilibrium path p Λ = c The price ad umber of firms o the high quality market are determied by the miimum eforcemet price ad the demad curve. The detectio probabilities are the outcome of the iteractio betwee market micro structure ad stage game strategies of buyers ad are a (weakly) decreasig fuctio of the size of the high quality market ffi(μ): Together with the icetive costrait o the firm, this implies a miimum eforcemet price μp E (μ) that is icreasig i μ, which is the icetive compatible supply" curve. Ay price-quatity pair above it ca be eforced o the firm side. The itersectio with the demad fuctio defies the maximum size of the high quality market μ Λ that ca be supported by a equilibrium of a specific market game. The umber of uits that ca be produced is qμ Λ : The market equilibrium ca be calculated i the followig steps: ffl Market micro structure implies (ffi fi ) 1 fi =1, ad v D for give Λ ffl ffi fi ad v D ad icetive costrait for firms imply miimum eforcemet price μp Λ ffl Total supply ad demad fuctio imply equilibrium price for low quality alog the equilibrium path p Λ ffl miimum eforcemet price μp Λ,low quality price p Λ ad demad fuctio imply the demad for high quality uits ffl which i tur implies the umber of firms Λ that ca be supported as high quality producers, other firms ca oly produce low quality (ratioig effect because of profit premium for high quality productio) ffl fid fixed poit i Λ I the ext sectio, we derive the detectio ad puishmet probabilities if the market is orgaized i the form of sequetial auctios 16

17 3 Auctios We ow cosider a eviromet where the uits of high quality are sold at a auctio, whereas there remais a separate low quality market for which we do ot specify a particular market mechaism. I a first stage, after sellers produced their desired quality, they decide whether to participate i the auctio - i.e. eter the "high" quality market or remai i the low quality market. We will try to characterize the most efficiet equilibrium of particular auctios, i.e. the highest amout of high quality that ca be sustaied by the specific auctio mechaism. A importat determiat of the puishmet probability iaauctio market is the amout of iformatio available to the buyers (i) istataeously, i.e., durig the biddig process; ad (ii) ex post (to all buyers, or oly to LR?). The followig summarizes the mai possiblities of sigals i the auctio process that covey the iformatio that a firm has bee cheatig. Ad (i) the amout of istataeous iformatio is maily predetermied by the chose auctio format: I a sealed bid auctio, for istace, there is o istataeous iformatio, while mimickig (uiformed buyers try to imitate the biddig behavior of iformed oes) ad trickig (iformed LRs try to iduce uiformed buyers to buy low quality at the high price to avoid a icrease i p) possibilities are preset i a Eglish auctio. To keep thigs simple, we cocetrate o two sealed bid formats (first ad secod price). Ad (ii) The amout of ex post iformatio is a matter of assumptio. LRs kow, of course, the past prices ad quatities of the differet firms, ad their "raks". I the sequel we assume that this iformatio is also available to SRs. If it is ot, the puishmet probability islower, of course. I some real auctios additioal iformatio o the "order state" is made available. Araw idicator of the order state is the price-supplemet "EDI" (existece of extra demad at market price, "repartiert" i Germa) for "the demad for the uit(s) produced by a give firm was higher tha the supply (=1, sice we cocetrate o the special case q = 1 here)". We will discuss the impact of this iformatio o the puishmet probability below. For the sealed bid first price auctio we will also cosider the case where the umber of bids at the sellig-price is made public ex post. We call this the "demad overhag observed" versio. Whe will a firm be puished i a auctio? Puishmet (the firm is uable to sell its uit at the high price) will oly take place if some publicly available iformatio idicates that the firm is or has bee ( either with some probability or for sure) cheatig. The relevat sigal may be the quatity orthe price. A egative quatity sigal (a firm was uable to sell its uit) idetifies the cheater exactly. I the "EDI" ad i the "demad overhag observed" versio, buyers might also be able to observe positive quatity sigals (the demad for the uit sold o a earlier rak was "too high"). High demad ca, however, oly have sigallig value if the buyers coordiate 17

18 their bids (without coordiatio all buyers that have't got a uit yet will submit a bid for the uit sold by a give o-cheatig firm). A positive quatity sigal usually (except for rak 1) meas "set detectio": It idicates that some firm o a later rak is (or was) cheatig. Negative price sigals (the sellig-price is lower tha p) are ever observed i the auctios cosidered below sice firms are assumed to aouce reservatio prices. We allow firms to aouce reservatio prices to avoid the bilateral moopoly problem metioed earlier (ex post, oce high quality has bee produced, buyers have a icetive to bid less tha p; if firms take this ito accout, they will ever produce high quality). A positive price sigal (the sellig-price exceeds p) o a earlier rak is similar to a positive quatity sigal: It idicates that some firm o a later rak has cheated i the past. I the sealed bid first price format a sigle buyer that offers more tha p for a uit o a earlier rak suffices to geerate a positive price sigal. I the sealed bid secod price auctio at least two buyers that offer more tha p for the same uit are eeded to chage the sellig-price. 3.1 Sealed Bid Secod Price Auctio Format: The uits desigated for the high quality market are put up for sale i radom sequece. The auctio begis with the auctioeer aoucig the sequece of uits (= the "raks" of the firms) ad the firms' reservatio prices 4. The sealed bids for the first uit (= for the uit sold by the firm o rak 1) are solicited. The highest bid is accepted at a price equal to the 2d highest bid. The sealed bids for the uit produced by the firm o rak 2 are solicited. The process cotiues util the uit of the firm o rak is sold. We cosider two differet strategy-specificatios. I the first, LR-buyers bid p alog the equilibrium path. I the secod, they bid p Secod Price with p Bids Strategies: O the equilibrium path all buyers whose type fi k is greater or equal tha (p c) = bid p o each rak util they get their uit. Off the equilibrium path the behavior of the cheated LRs depeds o their beliefs o how may LRs have already bee cheated. These beliefs imply a critical rak. For a give period t we deote the critical rak for a LR buyer first cheated fi periods ago by br fi t. If the rak of the cheatig firm, deoted by r t, is strictly lower tha f brt fi the the LR buyer cheated fi periods ago bids p o all r 6= rf t. Otherwise he bids p o all raks strictly lower tha br fi t fi (provided br fi t fi 1; otherwise...). If he does't get a uit o oe of these raks he bids p + o rak br t fi fi. Pessimistic Beliefs: Cheated LR buyers assume that with strictly positive probability all other LRs have already bee cheated. Pessimistic beliefs are the worst beliefs regardig the support. 4 Here ad throughout the rest of the auctio-aalysis we assume that all high price firms aouce p as their reservatio price. 18

19 Thus, they give a upper boud for the puishmet probability. Pessimistic beliefs igore iitial periods ad are ot fully cosistet if the LR buyer uder cosideratio has bought high quality from the cheatig firm a short time before (provided "hit ad ru" cheatig is ot optimal). With pessimistic beliefs br fi t =(1 ) + 1 for all cheated LRs idepedetly of fi (ad t). (If (1 ) is ot a iteger the take the ext lower iteger.) Optimistic Beliefs: The LR buyer cheated fi periods ago believes that cheatig firm bega cheatig fi periods ago. Optimistic beliefs are best beliefs regardig the support. For a LR buyer with optimistic beliefs that has bee cheated fi periods ago br fi t = +1 fi if fi» ad br fi t = (1 ) + 1 otherwise. That is, br fi t = mi f +1 fi;(1 ) +1g : Results: Claim 3: With pessimistic beliefs the probability of (set) detectio is zero for at least (1 ) cheatig periods. That is, with pessimistic beliefs ffi L;s = 0 for s» (1 ). Similarly, with optimistic beliefs the probability of (set) detectio is zero for at least 1 cheatig periods. Proof: Cheatig is detected (set detectio) if more tha oe LR buyer bids p + for the same uit. With the described strategies this caot happe i the first... cheatig periods Secod Price with p + Bids Strategies: Alog the equilibrium path the SRs bid p from rak 1 o util they get their uit. LRs do the same i the first roud. From the o they bid p + for the firm from which they bought previously. Off the equilibrium path the behavior of the cheated LRs depeds agai o their beliefs which determie their critical rak br t fi. If the rak of the cheatig firm is strictly lower tha br fi t the the LR buyer cheated fi periods ago bids p o all raks r 6= r t f. Otherwise he bids p o all raks lower tha br fi t 2. If he does't get a uit o oe of these raks he bids p + o rak br t fi 1. From the o he bids p + for the firm he gets util he is cheated agai. Results: Results deped upo whether "EDI" is observed or ot. The followig claim holds for the ot-observed versio. Claim 4: With pessimistic beliefs ffi L;1 < 3 ; ad ffi L;fi =0for all fi > 1. Proof: Set detectio occurs if more tha oe LR bids p + ; that is, if (i) the cheatig firm had a LR buyer (probability ), (ii) the cheatig firm gets oe of the rear raks (probability ), (iii) the cheated LR does't get a uit before rak (1 ), ad (iv) the firm o rak (1 ) has a LR give that the cheatig firm has a LR ad the cheated LR has't got a uit before rak (1 ) (probability strictly lower tha ). 19

20 3.2 Sealed Bid First Price Auctio Format: The uits desigated for the high quality market are put up for sale i radom sequece. We assume that the firms ca set reservatio prices. 5 The auctio begis with the auctioeer aoucig the sequece of uits (= the "raks" of the firms). The sealed bids for the first uit (= for the uit sold by the firm o rak 1) are solicited. The highest bid is accepted at the stated price. The sealed bids for the uit produced by the firm o rak 2 are solicited. The process cotiues util the uit of the firm o rak is sold. We will cosider several cases, depedig o the amout of iformatio bidders get, apart from the trasactios No History ad reservatio prices observed Cosider the case where all uits, high ad low quality, are sold i a sigle auctio ad buyers caot observe the prices of uits sold at previous auctios or at the curret auctio of uits other tha their ow. I this case the first price auctio resembles i its iformatio trasmissio the decetralized bargaiig market. However the differece is that i the auctio the firms lose ay "exclusivity" right, i.e. they ca ot deter buyers who are makig the highest bid from wiig the auctio. This is mostly a hypothetical case that we cosider i order to distiguish the role of iformatio ad the role of exclusio, i.e. makig the offers coditioal o the type or idetity of the buyers. Cosider first the case where (1 + ffi)(c c) fi K < ffi( ) ER The followig strategies costitute a equilibrium. = fi ER SR Strategies The Firms: Firms that had a LR i the first period cotiue to produce high quality ad set a reservatio price p ER : Whe they ca ot sell their good, they produce ext period low quality ad set a reservatio price of p: Wheever they sell their low quality uit at a price p>p they believe that a LR bought their good ad ext period produce high quality ad set a reservatio price of p ER : If a firm sells her good oly at a price p; she produces i the ext period a low quality uit ad sets her reservatio price to p: The Log Ru Buyers: Each LR bids p ER for the firm from which he bought from last period. Beliefs out of equilibrium. LRs without a regular firm bid oly p + ", ad the ext period bid p ER for the uit of that same firm, sice they believe that the firm produced a high quality uit ad set a reservatio price of p ER. 5 Here ad throughout the rest of the auctio-aalysis we assume that all high price firms aouce p as their reservatio price. 20

21 The Short Ru Buyers: Each SR oly bids p. SRs believe that oly firms that produced low quality set a reservatio price below p ER. Propositio 1 (Exclusive Relatioship Equilibrium) I the case of fi K < μper p ( ) ER is a equilibrium where i each period after the first oe the umber of high quality uits equals the umber of log ru players with valuatio fi k > fi ER =(μc c)=ffi( ). Each Log Ru buyer with valuatio fi k > fi ER gets a high quality uit at a price p ER = c +(μc c)=ffi from his "regular" firm. LRs with valuatio fi k < fi ER ad Short Ru buyers get low quality uits from other firms at a price p = c. Hece, μ ER = P fi k >fi ER m k : there Proof: No firm i a relatio with a LR Buyer has a icetive to deviate: If the firm sets a reservatio price p>p ER she will ot be able to sell to her LR;who oly bids p ER : A firm that starts to produce low quality ca expect earigs of p ER c + ffi (p c) 1 ffi sice if she produces low quality oce, she will lose her LR forever. If she cotiues to produce high quality her expected earigs are Hece, we have the coditio p ER c + ffi 1 ffi (per c): p ER c + For p = c we have ffi 1 ffi (per c) > p ER c + ffi 1 ffi (p c), (μc c) μper > + p: ffi μp ER > c + μc c : ffi No firm producig low quality has a icetive to deviate: Will the firm try to set a reservatio price p > p? Alog the equilibrium path o LR will bid p > p either a SR, hece o icetive to do so. Furthermore, for the same reaso, the firm will ot start to produce high quality sice she wo't be able to get a price higher tha p. No LR Player has a icetive to deviate: Cosider a LR with fi > fi ER : Sice the firm sets a reservatio price of μp ER the LR will bid μp ER. No SR has a icetive to deviate: Whe will a high valuatio SR be willig to pay ^p >p ER? Whe the SR bids ^p >p ER he gets market average quality, hece is willig to do that oly if fi k ( μer μer +(1 ) ) bp > fi μ k p, bp 6 ER p + fi k ( ) 21