Durable Goods, Coasian Dynamics, and Uncertainty: Theory and Experiments

Transcription

1 Durable Goods, Coasian Dynamics, and Uncerainy: Theory and Experimens Timohy N. Cason Purdue Universiy Tridib Sharma Insiuo Tecnológico Auónomo de México This paper presens a model in which a durable goods monopolis sells a produc o wo buyers. Each buyer is privaely informed abou his own valuaion. Thus all players are imperfecly informed abou marke demand. We sudy he monopolis s pricing behavior as players uncerainy regarding demand vanishes in he limi. In he limi, players are perfecly informed abou he downward-sloping demand. We show ha in all games belonging o a fixed and open neighborhood of he limi game here exiss a generically unique equilibrium oucome ha exhibis Coasian dynamics and in which play lass for a mos wo periods. A laboraory experimen shows ha, consisen wih our heory, oucomes in he Cerain and Uncerain Demand reamens are he same. Median opening prices in boh reamens are roughly a he level prediced and considerably below he monopoly price. Consisen wih Coasian dynamics, hese prices are lower for higher discoun facors. Demand wihholding, however, leads o more rading periods han prediced. We are graeful o Larry Ausubel, Mark Bagnoli, Indranil Chakrabary, Oliver Har, Charlie Kahn, David Levine, Nicola Persico, San Reynolds, Jean Tirole, Ricard Torres, and hree anonymous referees for heir commens. Seminar paricipans a Insiuo Tecnológico Auónomo de México, he Universiy of Melbourne, he Midwes Economic Theory conference, he annual meeing of he Economic Science Associaion, he Tinbergen Insiuu (Erasmus Universiy), and he Lain American Meeing of he Economeric Sociey provided valuable feedback. Jorge Aburdene, Minerva Nava, Agusin Perez, and Erneso Revilla provided valuable assisance in organizing and conducing he experimens. Funding from Consejo Nacional de Ciencia y Tecnologia, Mexico, is graefully acknowledged. The usual disclaimer applies. [Journal of Poliical Economy, 2001, vol. 109, no. 6] 2001 by The Universiy of Chicago. All righs reserved /2001/ $

2 1312 journal of poliical economy I. Inroducion Theoreical predicions on he dynamic pricing behavior of a durable goods monopolis range from compeiive pricing o perfec price discriminaion over ime. This wide range of predicions suggess an imporan role for empirical research on his opic, bu unforunaely knowledge of empirical regulariies in such markes is limied. In his paper, we provide some new evidence on durable goods monopoly pricing in a simple laboraory marke. The marke is designed keeping in mind he remarkable sensiiviy of equilibria o changes in he exensive form game. Our resuls provide suppor for Coase s (1972) compeiive pricing conjecure in all reamens. Coase conjecured ha a durable goods monopolis, selling a good over ime, would price her producs a less han he saic monopoly price. Theoreical suppor for he Coase conjecure is by no means unanimous. Gul, Sonnenschein, and Wilson (1986) provide a jusificaion for he conjecure based on a model wih a coninuum of buyers. 1 Srikingly, Bagnoli, Salan, and Swierzbinski (1989) show ha for discree demand and for sufficienly high discoun facors, a monopolis who is perfecly informed abou demand is able o perfecly pricediscriminae. 2 Furhermore, here are no Coasian dynamics. 3 They do, however, poin ou ha oher equilibria could also exis. Tha equilibrium oucomes may change drasically for apparenly small changes in exensive form games has been known for quie some ime (see, e.g., Fudenberg, Levine, and Tirole 1985). Furhermore, no much is known abou acual srucures of exensive form games in realworld durable goods monopolies. As such, our abiliy o predic pricing paerns in such markes is severely consrained. To compound problems, empirical research is limied, excep for Suslow s (1986) analysis of Alcoa s aluminum pricing (moivaed by an imporan anirus case, Unied Saes v. Aluminum Company of America [148 F.2d 416, 1945]) and Reynolds (2000) recen laboraory sudy. Our basic objecive in his paper is o provide more evidence of durable goods monopoly pricing from simple laboraory markes. Though evidence from experimens has drawbacks because of poenial limiaions regarding parallelism wih he naurally occurring economy, ex- 1 See Fudenberg and Tirole (1991) for a more complee lis of references. 2 We use he erm perfecly no in a precise sense since he monopolis charges differen prices over ime. Hence her profis need o be correced for he discoun facor. 3 Coasian dynamics will refer o he rae a which he monopolis s (iniial) price approaches he perfecly compeiive level, as he ime period beween price offers approaches zero (capured by he discoun facor approaching one). Our usage of he erm Coasian dynamics differs from ha commonly employed in he lieraure. Auhors ofen use he phrase Coasian dynamics o refer o he siuaion in which high-valued buyers buy earlier han low-valued ones and here is a paern of monoonically decreasing prices over ime.

3 durable goods 1313 perimens have he advanage ha hey provide evidence from exogenously formulaed games. Therefore, predicions can be associaed wih specific exensive form games (or insiuions). Neverheless, a bi of cauion may be required in such exercises. In he following paragraph we explicily sae our concern. In many experimens, as in our case, he experimener may never be able o compleely implemen he game of his or her choice in acual laboraory environmens. Paricipans in he laboraory could be playing games close o he one srucured by he experimener. Also, he experimener may no know he naure of he nearby game ha is played by paricipans. In such siuaions, he equilibrium of he exogenously specified game (whose oucome is being esed) should be robus o various ypes of minor perurbaions. To be specific, an equilibrium of a game is said o be robus when he equilibrium oucome of some nearby game converges o he equilibrium oucome of he original game as he disance beween he games converges o zero. Or loosely speaking, he equilibrium oucome of he specified game should no change drasically wih (i.e., should no be sensiive o) small changes in he original game. In laboraory markes such as ours, he discree demand srucure of Bagnoli e al. is of paricular significance because wih a finie number of (human) buyers and a finie number of goods, i is no possible o consruc coninuous demand curves. Even hough one does no expec o see complee exracion of consumer surplus in experimens, we ake he Bagnoli e al. resul of perfec price discriminaion as a benchmark hypohesis. The reason is ha heir resul is robus o a leas wo kinds of perurbaions of heir original game. Levine and Pesendorfer (1995) consider a game in which he monopolis observes her sales wih noise, so here is uncerainy over realized demand. Though, in general, perfec price discriminaion canno be susained as an equilibrium oucome, i urns ou ha for a finie number of buyers and small levels of noise, he monopolis s profi is close o ha under perfec price discriminaion. Von der Fehr and Kühn (1995) consider a model in which, consisen wih laboraory seings, prices are no fully divisible. Under addiional condiions, as indivisibiliies are reduced heir equilibrium oucome converges o he Bagnoli e al. oucome. In experimenal seings, anoher perurbaion may be of significance. I is quie clear ha even if we provide he monopolis wih perfec informaion abou buyers moneary payoffs (as we do in he Cerain Demand reamen discussed below), i is impossible o provide perfec informaion abou uiliy over moneary payoffs. Furhermore, as discussed in Fehr, Kirchseiger, and Riedl (1998), ase parameers such as envy and spie migh lead subjecs o rejec prices ha provide hem wih lile or no surplus. The exensive lieraure on he ulimaum

4 1314 journal of poliical economy game provides ample evidence ha subjecs rejec posiive surplus offers when he proposed allocaion is highly unequal (see Roh [1995] for a survey). In experimenal seings, privae informaion over ase parameers is a norm raher han an excepion. This, in our durable goods (experimenal) seing, ranslaes o he fac ha he monopolis can never be cerain abou he demand curve (in an a priori sense). Is he perfecly discriminaing equilibrium of Bagnoli e al. robus o perurbaions in which he monopolis is uncerain abou demand? We address his quesion heoreically in Secion II and show ha here exiss a unique Nash equilibrium oucome in he Cerain Demand game ha is robus o uncerainy abou he demand curve. This oucome generaes Coasian pricing and dynamics, wih prices far below he price discriminaion equilibrium of Bagnoli e al. Hence, we use he oucome of Coasian dynamics as a second hypohesis for our experimens. These wo hypoheses correspond o he exreme ends of he prevailing heoreical predicions. We are aware ha careful consideraion of complicaions such as learning and misakes would generae addiional hypoheses for he experimen. We do no model such complicaions. So our resuls, hough poenially imporan for regulaory policies, should be inerpreed cauiously. In our experimen we implemen wo basic reamens. In he Cerain Demand reamen, buyers moneary valuaions are announced and each buyer is assigned one and only one valuaion. Announcing buyers moneary valuaions is no he same as announcing uiliy over hese valuaions, however, and he monopolis canno know for cerain ha buyers would pay prices equal o hese known valuaions. Thus here is inheren privae informaion even in he Cerain Demand reamen. In our second, Uncerain Demand, reamen, buyers know heir moneary valuaions wih cerainy bu sellers (as well as buyers) are informed ha any paricular buyer s valuaion is a random variable. The probabiliies assigned o hese variables are announced. The uncerainy (exogenously) induced in his second reamen is small. Our resuls show ha we can rarely rejec he hypohesis ha he oucomes are he same in hese wo reamens. Moreover, he oucomes in boh reamens are consisen wih Coasian price predicions raher han he high prices of he Bagnoli e al. equilibrium oucome. The experimen implemens hree differen discoun facors hrough differen game coninuaion probabiliies. As prediced by Coasian dynamics, and again conrary o he Bagnoli e al. perfec price discriminaion oucome, in he Cerain Demand reamen, prices shif down oward compeiive prices as he discoun facor increases. For readers who would like o skip Secion II and go direcly o secions on our experimen, here we provide a brief descripion of he heory presened in Secion II. We consider a marke wih wo buyers.

5 durable goods 1315 Call he buyers 1 and 2. In he Bagnoli e al. model, buyers 1 and 2 have high valuaions for he good wih respecive probabiliies of one and zero. The monopolis is uninformed abou he buyers ideniy, hough she knows ha here is one high-valued and one low-valued buyer. So he monopolis and he buyers know wih cerainy he downward-sloping marke demand curve. In our alernaive model, each buyer is privaely informed abou only his own valuaion, which can be eiher high or low. As in Bagnoli e al., he seller does no know he ideniy of he buyers. In paricular, in our model he seller as well as he buyers are uncerain abou marke demand. The buyers valuaions are drawn from independen disribuions, and hese disribuions along wih he seller s valuaion are known. 4 Le m 11 and m 21 be he prior probabiliies ha buyers 1 and 2, respecively, have a high valuaion. To keep he monopolis s uncerainy abou demand small, we consider all games in which m 11 and m 21 belong o he inervals (0, 1) and (0, a ), respecively. We explicily specify a small and posiive a. Thus we consider a se of games in which boh buyers are privaely and independenly informed abou heir own valuaions. These games belong o a fixed open neighborhood of he perfec informaion game of Bagnoli e al. Our main resul is ha all games wih parameers belonging o his neighborhood have a generically unique perfec Bayesian equilibrium oucome ha exhibis Coasian dynamics. This esablishes he fac ha he Bagnoli e al. resul is no robus o perurbaions of he kind ha we are considering. In oher words, consider all sequences of games in which buyers 1 and 2 have high valuaions wih probabiliies ha belong o (0, 1). All nonrivial sequences of such kinds will evenually converge o he Bagnoli e al. game hrough our specified neighborhood. Bu in his neighborhood, we show ha all equilibria exhibi Coasian oucomes. There is anoher srengh of our resul. In general, limis of equilibrium oucomes may depend on he speeds of convergence of various parameers. For example, depending on differen speeds of convergence in he Levine and Pesendorfer and von der Fehr and Kühn models, one can alernaely ge he Coasian and Bagnoli e al. oucomes as robus oucomes. By conras, in all he games (in he neighborhood) ha we consider, and for all discoun facors, i urns ou ha, along he equilibrium pah, play lass for a mos wo periods. This provides an upper bound on he equilibrium opening price, which is independen of he disribuion over valuaions. As a resul, our convergence resul is independen of he speeds of convergence of he discoun facor and uncerainy. 4 Since we assume ha he space of buyers valuaions is discree, he assumpion on he seller s valuaion is no necessary for our resuls.

6 1316 journal of poliical economy The inuiion behind our resul wih imperfec informaion is as follows. I is worhwhile for he monopolis o charge high prices only if buyers wih high valuaions buy wih a (required) posiive probabiliy. Along equilibrium, when a good is (supposed o be) bough a a high price, he monopolis updaes her beliefs a he beginning of he nex period, regardless of wheher a good is bough or no. By assumpion, here is a low probabiliy ha boh buyers have high valuaions. Hence, poserior beliefs aach a low weigh o he even ha he remaining buyer(s) has a high valuaion. As a resul, he monopolis lowers prices in he succeeding period. Off he equilibrium pah, if he seller were o charge a very high price in he firs period, hen each high-valued buyer would wai o buy he good in he succeeding period. Bu hen he monopolis would no be able o sell a he very high price charged in period 1. The remainder of he paper is organized as follows. Secion II presens he heoreical model and collecs he heoreical resuls. Secion III describes he experimenal design and procedures. Secion IV conains he experimenal resuls and relaes our findings o he experimenal lieraure. Secion V presens conclusions. II. The Model There are wo risk-neural buyers, 1 and 2, and a risk-neural monopolis whose consan marginal cos is normalized o zero. The monopolis canno idenify he buyers; ha is, faced wih a buyer, she canno say i wheher he is buyer 1 or 2. Le buyer i s valuaion be denoed by v, i i where v can ake values v 1 or v 2. For buyer i, v p v j wih probabiliy m ij, i, j {1, 2}. For example, buyer 2 has valuaion v 1 wih probabiliy m 21. Noe ha mi1 mi2 p 1. Each buyer is privaely informed abou his own valuaion and knows he disribuion from which his valuaion is drawn. 5 The disribuions are assumed o be independen and o be common knowledge. We assume ha v 1 1 v The assumpion v is quie weak because in he finie case a buyer would have a valuaion exacly equal o ha of he monopolis wih probabiliy zero and he monopolis will never rade wih hose wih valuaions below her cos. 6 There are infinie ime periods, and in each ime period, he monopolis ses a price in he firs sage. Consumers who have no bough previously decide o buy one uni of he good or no buy in ha period. Once a consumer buys a good, he leaves he marke. For any period 1 0, he game may no proceed o 1 wih a posiive probabiliy 5 Our resuls would no change if we were o assume ha buyers know heir valuaion bu no he disribuion from which heir valuaion is drawn. 6 The assumpion also induces a unique perfec Bayesian equilibrium oucome for m (0, 1). ij

7 durable goods d. All players are indifferen beween one uni of payoff a period and one uni a 1. In wha follows in his secion, we shall inerpre his o be a model in which all players have a common discoun facor of d and he game proceeds o he nex period wih probabiliy one. This change is made wihou loss of generaliy and is consisen wih he cied lieraure. A dae hisory for he monopolis is a sequence 1 of vecors {(p n, q n)} 0, where p n is he price charged in period n and q n is he number of goods sold hrough period n (inclusive); for he consumer, a dae hisory is a sequence {(p n 1, q n)} 0. For any paricular 1 hisory h h, le m ij denoe he belief of he monopolis ha buyer i has valuaion v j. Similarly, for a given hisory h kh, le m ij be buyer k s belief ha buyer i has valuaion v j. Le p 0 p and q 0 p 0. Noe ha for any period here are an infinie number of hisories. A pure sraegy of he monopolis is a price choice for every informaion se in a period for all periods. Similarly, for he consumers i is a decision o buy or no o buy for every hisory in a period for all periods. We sar wih he perfec informaion model of Bagnoli e al., where m11 p 1 and m 21 p 0. The concep of equilibrium is ha of a subgame- perfec Nash equilibrium. Bagnoli e al. show ha here exiss an equilibrium in which he monopolis is able o perfecly discriminae for high values of d. Consider he following consumer sraegies. For any period, any period hisory, and any price p charged by he monopolis, consumer i buys he good iff v p i (hence he consumer s sraegy is independen of hisory and myopic). The monopolis uses he following sraegy (Pacman sraegy): For any period, define v max() o be he highes valuaion of he se of consumers who have no purchased he good hrough period. For each period, he monopolis ses p p v max() and drops his price in fuure periods iff a leas one uni of he good is sold a his price (so he Pacman sraegy depends on pas sales). I is shown ha hese wo sraegies consiue an equilibrium when 1 1 d 1 1 [D/(v 1 v 2)], where D p v 1 v 2. Proposiion 1 is aken from Bagnoli e al., and hence we sae i wihou proof. Proposiion 1. The myopic sraegy is a sequenially bes response for a buyer playing agains he Pacman sraegy. If all buyers use he myopic sraegy, hen he Pacman sraegy is opimal whenever 1 1 d 1 1 [D/(v 1 v 2)]. Prices in he Pacman sraegy are independen of d (in he relevan range). This is sharply a odds wih he Coase conjecure. The driving force behind his sraegy is he assumpion ha m11 p 1 and m 21 p 0. If i were o be opimal for he monopolis o charge a price equal o v 1 in period 1 bu if buyer 1 were o deviae from his myopic sraegy and no buy, hen i is sill he case ha in period 2, m11 p 1 and m 21 p 0. As a resul, since i was opimal for he seller o charge v 1 in period 1, i is opimal for her o charge v 1 in period 2. The deviaing

8 1318 journal of poliical economy buyer is hus punished. Commimen o such punishmen arises because of he monopolis s saionary beliefs. Commimen hrough saionary beliefs is, however, no possible wih imperfec informaion. In wha follows we shall characerize equilibria for all games wih parameers 0! m 11! 1 and 0! m 21! a! 1. The erm a is a fixed number o be specified laer. The concep of equilibrium is perfec Bayesian. The games ha we analyze belong o a fixed neighborhood around he Bagnoli e al. game (where m11 p 1 and m 21 p 0). In all he games ha we sudy, in equilibrium he game ends in finie ime wih probabiliy one. The reason is ha he seller updaes her beliefs in favor of he low-valued ype when here is no sale in a given period. So, afer a finie number of periods, eiher all goods are sold or he monopolis is willing o sell o buyers wih he lowes ype insead of charging a higher price and waiing for high-valued buyers o buy. Fudenberg e al. (1985) formalize his inuiion in a one-buyer, oneseller environmen. This skimming propery of Fudenberg e al. applies almos direcly in our environmen. The nex lemma, which provides a formal saemen, is hence saed wihou proof. Lemma 1. In any perfec Bayesian equilibrium, for any period, if a buyer wih valuaion v is willing o accep an offer p, hen a buyer wih valuaion v 1 v acceps he offer wih probabiliy one. Furhermore, he game ends wih probabiliy one afer a finie number of periods. By lemma 1 he game ends in a finie number of periods. Hence, we can use backward inducion o compue equilibria in his game. The nex sep is o compue equilibria along differen kinds of coninuaion games ha may arise. Consider he coninuaion game ha sars afer one good is sold. Since in his game here is only one buyer in he marke, i is equivalen o he game sudied in Har (1989). To adap Har s resul o our framework we need some noaion. Le v in period be he monopolis s belief ha he surviving buyer is buyer 1. Le m be he monopolis s belief ha he surviving buyer has valuaion v 1. Then m p vm 11 (1 v )m 21, where mi1 is he probabiliy ha he ih buyer has a valuaion of v 1. Define a 1 { v 2/v 1. Har s resul is as follows. Lemma 2. In all coninuaion games saring from period, wih one surviving buyer, he game ends in a finie number of periods. The equilibrium oucome is generically unique. There exiss a sequence of numbers 0 p a 0! a 1! a 2! such ha, when m (a i, a i 1], he game lass for a mos i 1 periods. The price charged in period is p p (1 i i d )v The equilibrium price p 1 d v 2. converges o v 2 as d converges o one. Proof. See Har (1989). Lemma 2 sheds ligh on an ineresing aspec of he robusness of one-buyer, one-seller games. Oucomes of he perfec informaion ver-

9 durable goods 1319 sion may be similar o ha of a slighly perurbed version. A perfec informaion game in which he buyer has valuaion v 2, ha is, m p 0, yields he same oucome as he perurbed version in which he buyer can have valuaion v 1 ( 1 v 2) wih probabiliy m (0, a 1) and v 2 wih probabiliy 1 m. Thus one can say ha he oucome in Bagnoli e al. is robus o informaion perurbaions (of our kind) in a one-buyer, oneseller game. Bagnoli e al. (1992) ge a similar resul when all buyers have he same valuaion bu he monopolis is uninformed abou he valuaion. Since all buyers have he same valuaion, his case is similar o he case in which here is jus one buyer. The Bagnoli e al. resul is robus o oher perurbaions as well. Lemma 3 saes ha when a buyer is high-valued wih a high enough probabiliy (relaive o d), opening prices will be close o v 1. Lemma 3. Le m (0, 1). For any finie N, here exiss m such ha he game lass for a mos N periods. The price charged in period is N N p p (1 d )v As N converges o infiniy, p 1 d v 2. converges o v 1. Proof. I follows from Har (1989) ha he lowes upper bound of he sequence {a i} ip0 is one, and furhermore, ai 1 1 ai for all i. The proof follows from he previous lemma and aking he limi of p as N ends o infiniy. Q.E.D. So wha happens when here is more han one buyer? The quesion brings us o he sudy of anoher possible coninuaion game in which equilibrium oucomes may be similar o he Bagnoli e al. oucome (also see Bagnoli e al. 1992). Consider coninuaion games in which boh buyers are in he marke, bu wih cerainy one buyer is known o have he lower valuaion of v 2. Wihou loss of generaliy le m 21 p 0. Lemma 4 characerizes equilibria along all such coninuaion games. Since he proof is similar o ha of lemma 2, i is omied. Lemma 4. In all coninuaion games saring from period, wih wo surviving buyers and m 21 p 0, he game ends in a finie number of periods. The equilibrium oucome is generically unique. There exiss a sequence of numbers 0 p b 0! b 1! b 2! such ha, when m11 (b, i b i 1], he game lass for a mos i 1 periods; b1 p 2a1 and b 2! i i 2a 2. The price charged in period is p p (1 d )v 1 d v 2. The equi- librium price p converges o v 2 as d converges o one. The wo coninuaion games sudied above are quie similar. In boh cases, only one buyer has privae informaion. A lemma similar o lemma 3 can also be proved, and one could show ha when m 11 is high, ha is, he probabiliy ha buyer 1 has valuaion v 1 is high, hen he monopolis would charge an opening price close o v 1 and buyer 1 buys a his price wih posiive probabiliy. Following he sale of a good, he monopolis lowers prices o v 2, a which price buyer 2 buys wih prob-

10 1320 journal of poliical economy abiliy one. Thus wih posiive probabiliy he Bagnoli e al. oucome could occur. We now sudy he overall game in which m11 (0, 1) and m 21 ( 0 bu is small. The main difference beween he coninuaion games sudied above and he overall game is ha boh buyers have privae informaion. An ineresing resul for he overall game is ha coninuaion games, like he ones sudied above, do no belong o he equilibrium pah. Le 2 us define a p [2/(1 a 1)]a 1, where a! a1 as a 1! 1. We are now in a posiion o characerize he equilibrium oucomes of he overall game in which here are wo buyers, m11 (0, 1) and m 21 (0, a ). Our main resul is saed in he nex proposiion. Proposiion 2. Le m11 (0, 1) and m 21 (0, a ). The game lass for a mos wo more periods in equilibrium. All equilibria have a generically 1 unique oucome. Along he equilibrium pah, p p (1 d)v 1 dv 2and 2 1 p p v 2 when m11 m a1 and p p v 2 when m11 m 21 2a 1. Proof. The proof, which depends on lemmas 5 8, discussed below, is presened in he Appendix. Proposiion 2 ells us ha in he overall game, he price charged in period 1 is eiher (1 d)v 1 dv 2 (when m 11 is high) or v 2 (when m 11 is low). The equilibrium oucome is generically unique because here exiss only one paricular value of m 11 in which he price charged in period 1 can be eiher (1 d)v 1 dv 2 or v 2. Thus, for all m11 (0, 1), he game lass for a mos wo more periods. Hence, in conras o lemmas 2 and 4, no maer how close m 11 is o one, opening prices are always bounded above by (1 d)v 1 dv 2. I is also he case ha he price se in period 1 eiher is equal o v 2 or converges o v 2 as d converges o one. So for m 11 close o one and m 21 close o zero, ha is, when he monopolis is almos cerain ha she is facing a downward demand curve, he equilibrium induces he following unique oucome. In period 1 he monopolis ses a price equal o (1 d)v 1 dv 2. One or boh goods are bough a his price wih posiive probabiliy. Whaever he oucome, he monopolis reduces he price o v 2 in he nex period. The remaining buyers buy he good a his price and he game ends. To prove our resul we proceed as follows. Proposiion 2 claims ha, along equilibrium, he game lass for a mos wo periods. We shall prove proposiion 2 by showing ha he se of all equilibrium pahs in which he game lass for more han wo periods is empy. Recall from lemma 1 ha in equilibrium he game lass for a finie number of periods. So consider all finie equilibrium pahs along which he game lass for more han wo periods wih posiive probabiliy. If such pahs do exis, hen along any such pah here has o exis such ha he game goes o period 2 wih posiive probabiliy and ends in period 2 wih probabiliy one. Call he se of such pahs Y. Our objecive will be o show ha Y is empy. This would hen imply ha

11 durable goods 1321 he game lass for a mos wo periods. Consider a subse of Y, where m11 (0, 1) and m 21 ({ e ) (0, a ) and he game goes o period 2 wih posiive probabiliy and ends in period 2 wih probabiliy one. Call his se Y. Noe ha since m 21 (0, a ), Bayesian updaing implies ha e [0, a ). The se Y imposes a resricion on Y in he sense ha i does no allow e p 0. Before we go on o prove our resul, i migh be worhwhile o reierae he inuiion. Firs noe ha we require m 21 o be greaer han zero. Oherwise we are back o he siuaion characerized in lemma 4. So m is a necessary condiion. In oher words, for our resul i is nec- essary o have more han one high-valued buyer wih posiive probabiliy. In his scenario, i is worhwhile for he monopolis o charge high prices only if here exis, wih a sufficienly large probabiliy, high-valued buyers and a leas one buyer wih high valuaion buys wih a (required) posiive probabiliy. When a good is bough a a high price, he monopolis updaes her beliefs a he beginning of he nex period. By assumpion, m 21 (0, a ), and hence here is a low probabiliy ha boh buyers have high valuaions. So once a good is bough, poserior beliefs aach a low weigh o he even ha he remaining buyer has a high valuaion. As a resul he monopolis lowers prices. Buyers are now able o ake advanage of an exernaliy. A high-valued buyer wais o buy he good because he believes ha, wih posiive probabiliy, here is anoher buyer who would buy a he high price. Bu, as a consequence, all high-valued buyers wai, and he monopolis canno sell a a high price. Noe ha for buyers o be able o ake advanage of his exernaliy, we need a leas wo high-valued buyers wih posiive probabiliy. We prove proposiion 2 using a series of lemmas. The sequence of lemmas follows he sequence of logic in he inuiion provided in he previous paragraph. All proofs of he lemmas are in he Appendix. Here we provide brief skeches. Firs noe ha when he probabiliy ha boh buyers have valuaion v 1 is oo low, hen he game lass for one period. The reason is ha even if he monopolis were o be able o separae he wo buyers and commi o specific prices for each of hem, she would sill no commi o a price higher han v 2 since he probabiliy of execuing a sale a his price would be oo low. Hence he price charged by he monopolis would be v 2. Since he monopolis would never charge a lower price, i becomes worhwhile for all buyers o buy a his price. Lemma 5. In all coninuaion games saring from period, if m11 a1and m21 a 1, hen he game lass for one period. The monopolis charges p p v 2 and boh buyers buy a he price. As menioned above, i should be worhwhile for he monopolis o charge high prices only if a leas one buyer wih a high valuaion buys wih a (required) posiive probabiliy. We now proceed o formalize his

12 1322 journal of poliical economy noion. For all elemens in Y (recall ha in his se boh buyers have high valuaion wih posiive probabiliy), le consumers 1 and 2 (wih valuaion v 1 ) buy he good wih probabiliies x and y, respecively, in period. For all oher periods, we shall refer o hese probabiliies as x and y. Since e is small, consumer 1 has o buy wih a probabiliy (x) large enough such ha he monopolis finds i o her benefi no o charge a lower price and ends he game in 1 (recall ha in Y he game goes o 2 wih posiive probabiliy). Lemma 6 provides a lower bound on x, for all equilibrium pahs in Y. Lemma 6. For all elemens in Y, i has o be he case ha x (1 d) (d/m 11)(2a1 e ) 1 0. Along all pahs in Y, in period wih p 1 v 2, eiher boh goods are bough, one good is bough, or no good is bough. If boh goods are bough, hen he game ends. If exacly i good is bough, i {0, 1}, hen he probabiliy ha he surviving buyer is 1 and has a high valuaion is 1 1 vi m11 ; similarly, he probabiliy ha he surviving buyer is 2 and has a 1 1 high valuaion is (1 v i )e, where m 11(1 x)e y 1 1 v1 m11 p, (1) m x(1 e y) (1 m x)e y m11xe(1 y) 1 1 (1 v 1 )e p, (2) m11x(1 e y) (1 m11x)e y and m 11(1 x) 1 1 v0 m11 p, (3) 1 m11x e(1 y) 1 1 (1 v 0 )e p. (4) 1 e y Since all pahs in Y go o 2 wih posiive probabiliy, i canno be ha he monopolis finds i opimal o end he game in period 1 by charging v 2. Therefore, following lemma 5, we have o rule ou he case (case 0) in which v1 m 11! a1 and v0 m 11! a 1. So, along all pahs in Y, eiher v1 m11 a1or v0 m11 a1 (or boh). Now, here are wo 1 1 cases in which v1 m11 a 1, call hem cases 1 and 2, and one more case in which v1 m 11! a 1, call i case 3. Noe ha in case 3, v0 m11 a 1. Lemma 7. There exiss no elemen in Y under which case 1 or case 2 would hold. Since, by definiion of Y, case 0 is ruled ou and he previous lemma rules ou cases 1 and 2, i follows ha case 3 mus be rue for all pahs in Y. Now, under case 3, when no good is bough in period, he

13 durable goods 1323 probabiliy ha he surviving buyer is 1 and has valuaion v 1 has o be high; furhermore, i should be profiable for he monopolis o sell only o he high-valued buyer in 1. Bu, as i urns ou again, his is no possible given he requiremen on x. As a resul, case 3 also canno hold, and herefore he se Y, as defined, has o be empy. Lemma 8. The se Y p M. From he previous lemma we undersand he following. Wih wo buyers in he marke, if he game were o go on for hree more periods wih posiive probabiliy and end here wih probabiliy one, hen i mus be he case ha he monopolis knows ha buyer 2 has valuaion v 2. Tha is, m 21 p 0, and lemma 4 is hen applicable. Bu hen if buyer 2 were o be of he high ype, he mus have bough before a a price higher han p. Would he do so? He would no because of an exernaliy. Suppose, in equilibrium, ha he high-valued buyer 2 is expeced o buy in some period. Bu suppose ha he decides no o buy. Then he knows ha in 2 he price would fall o v 2 wih posiive probabiliy (since he high-valued buyer 1 would buy in 1 wih posiive prob- abiliy). So, o induce he high-valued buyer 2 o buy, he price charged in period has o be sufficienly low. I urns ou ha his price is so low ha he high-valued buyer 1 also finds i o his benefi o buy wih probabiliy one a his price. Since all high-valued buyers buy a, he buyers remaining in period 1 have o have valuaion v 2.Soiis opimal for he monopolis o charge v 2 in 1 and end he game. Thus he game can las for a mos wo periods. Proposiion 2 implies ha equilibrium oucomes exhibi Coasian dynamics when he monopolis knows ha here exiss a buyer (buyer 1) who almos surely has a valuaion of v 1 and anoher buyer (buyer 2) who almos surely has a valuaion of v 2. 7 In paricular, he resul holds for values of m 11 close o one and m 21 close o zero. Furhermore, our resul is invarian o values of m 11 and m 21 in (nonrivial) ranges around one and zero, respecively. This, coupled wih he fac ha laboraory environmens are inherenly noisy, leads o he conjecure ha (i) experimenal reamens wih perfec informaion should give us resuls similar o hose wih reamens ha exogenously impose small imperfec informaion, and (ii) he resuls should be consisen wih Coasian dynamics. The key for his conjecure o hold is ha he high-valued buyer 1 (and of course he seller) believes ha buyer 2 would pay more han his assigned valuaion of v 2 wih an arbirarily small (bu posiive) 7 Recall ha he monopolis does no know he ideniy of he buyers. Faced wih a buyer, he monopolis canno ell wheher he is buyer 1 or buyer 2.

14 1324 journal of poliical economy probabiliy. This belief is quie plausible given he nonmoneary payoffs ofen observed in laboraory markes. 8 III. Experimenal Design and Procedures In his durable goods seing he seller has power over price and buyers make only accep or rejec purchase decisions. Therefore, he seller poss prices each period as in he sandard posed offer rading insiuion. Each marke includes wo buyers and a single seller, and each buyer can purchase up o one uni per rading round. If buyer i wih j j redempion value vi buys a price p, she earns vi p ha round. Sellers always face zero marginal coss, and here are no fixed coss. Therefore, if a seller sells a uni a price p, she simply earns p on ha uni. 9 Since he seller faces wo buyers, she can sell a mos wo unis per round. From now on we shall use he erm valuaion insead of redempion value. Wihin each rading round he seller poss one price per period across a sequence of periods. The seller also specifies he number of unis she is willing o offer a his price, which mus equal he number of unis she has remaining in invenory for ha round (eiher one or wo). Buyers hen decide simulaneously wheher o execue a ransacion by acceping he seller s offer price. 10 In subsequen periods, sellers are free o submi new price offers. A rading round ends if eiher (a) boh buyers purchase heir single unis or (b) he round erminaes randomly. A he end of every offer period (afer buyer purchase decisions), if a leas one buyer has no purchased her uni, a die is rolled in full view of subjecs. This die roll implemens a coninuaion probabiliy ha we manipulaed a hree levels across sessions: d p.6,.75, and.9. For example, when d p.9, if he oucome of he roll of a 10-sided die is a zero, hen he round ends immediaely; oherwise he seller poss a new price and he round coninues o he nex offer period. This coninuaion probabiliy is fixed across periods wihin a round and wihin a session. Each session employed 12, 15, or 18 subjecs, wih wo buyers for each monopoly seller, organized ino one marke for each seller. Sixeen of he 25 sessions employed 18 subjecs. For example, wih 18 subjecs, six 8 For example, buyers in firs-price aucions ofen bid as hough hey have a (nonmoneary) uiliy of winning. Some small number of subjecs in he presen seing could receive a uiliy from successfully execuing a ransacion or could simply make misakes. 9 Noe ha redempion valuaion and profis are given in moneary erms. They may or may no be equivalen o uiliy payoffs. 10 Requiring he seller o offer boh unis when she has no ye sold any in he round ensures ha all buyers are able o purchase in any period. This permis simulaneous accepance decisions by he buyers raher han requiring buyers o make purchase decisions in a random order, as is frequenly done in posed offer experimens.

15 durable goods 1325 subjecs are sellers, 12 subjecs are buyers, and six markes are conduced simulaneously. A he end of each rading round, buyers and sellers are reorganized ino new markes wih differen buyers and sellers for he nex round a so-called srangers design. This random shuffling of subjecs each round essenially implemens a sequence of one-sho games and has been shown o successfully minimize any repeaed game incenives across rounds (e.g., Durham, Hirshleifer, and Smih 1998). Subjecs ideniies are always concealed, alhough he sellers and buyers sa on opposie sides of he laboraory, which was obvious o subjecs. Usually six rading rounds were conduced in each session. Occasionally only five rounds were possible wihin he session ime period when d p.9, and we were able o complee more han six rounds someimes when d p.6. The experimen was conduced using he Muliple Uni Double Aucion rading sofware developed a California Insiue of Technology (Plo 1991). Alhough designed iniially for conducing experimens using he coninuous double-aucion rading insiuion, i is easily adaped for posed offer rading. Sellers pos offers in differen markes shown on differen pages, which are shown on differen screens of he program, and buyers and sellers move o new markes in differen rounds using he page up or page down keys on heir keyboards. As menioned in he Inroducion, for each of he hree coninuaion probabiliies, we implemen wo experimenal reamens ha varied he composiion and informaion of demand. In he Cerain Demand reamen, one buyer in each marke has a valuaion of 54 pesos and he oher buyer has a valuaion of 18 pesos. This is explained o all subjecs in he insrucions. The insrucions also explain ha buyers valuaions can change beween 18 and 54 in differen rading rounds, bu each marke always conains one buyer wih each valuaion. Furhermore, hese valuaions remain consan across periods wihin he same rading round. The raio of he difference in valuaions (D p 36) o he sum of valuaions (72) is 0.5. If valuaions were o be equivalen o uiliy payoffs, hen one could inerpre he Cerain Demand reamen in he lieral sense. If so, hen wih any coninuaion probabiliy d 1.5, he Pacman sraegy is an equilibrium sraegy (proposiion 1). As summarized in able 1, if subjecs play he Pacman equilibrium, he game should las a mos wo periods, wih he opening price equal o 54 and he second price equal o 18. Noe ha in his equilibrium, offer prices are independen of d. In he Uncerain Demand reamen, each buyer knows his own valuaion wih cerainy, bu sellers know only he probabiliy disribuion underlying buyer valuaions. Buyer 1 has a valuaion of 54 pesos wih probabiliy.95 and a valuaion of 18 wih probabiliy.05. Buyer 2 has a valuaion of 18 pesos wih probabiliy.95 and a valuaion of 54 wih

16 d Treamen TABLE 1 Summary of Experimen Parameers and Theoreical Predicions A. Parameers Cerain Demand Treamen Uncerain Demand Treamen v 1 p54 v 2 p18 v 1 p54 v 2 p18 m 11 pm 22 p1 m 12 pm 21 p0 m 11 pm 22 p.95 m 12 pm 21 p.05 Number of Periods B. Equilibrium Predicions Cerain Demand Treamen Price p 1 (pv 1 ) (Pacman Equilibrium) Price p 2 (pv 2 ) Number of Periods Uncerain Demand Treamen Price p 1 (p[1 d]v 1 dv 2 ) Price p 2 (pv 2 ) dp dp dp Noe. Each session employed wo buyers for each monopoly seller. Sellers and buyers were randomly reassigned o new markes each rading round.

17 durable goods 1327 probabiliy.05. If one were o assume no divergence beween uiliy and valuaion, hen in erms of he noaion used in Secion II, m 11 p m 22 p.95 and m12 p m 21 p.05. Therefore, he seller knows ha wih probabiliy.905 she faces one buyer wih a value of 54 and one buyer wih a value of 18, as in he Cerain Demand reamen. She also knows ha wih probabiliy.0475 she faces wo buyers ha boh have a value of 54 and wih probabiliy.0475 she faces wo buyers ha boh have a value of 18. The experimeners deermined he random values prior o he experimen using a 20-sided die, and he same drawn sequence of random values was used in all Uncerain Demand sessions. The insrucions also explain ha buyers can change beween buyers 1 and 2 in differen rading periods and ha buyer valuaions remain consan across periods in he same rading round. Noe ha m 21 p.05, and since a1 p, we have m 21 p.05! a p. Furher, m11 m 21 p 1 1 ( p 2a1) So from proposiion 2, i follows ha he unique oucome is such ha he game should las wo periods, wih he opening price equal o (1 d)54 d18 and he second price equal o v 2 p 18. Table 1 summarizes hese parameers and price predicions. As discussed above, he Pacman equilibrium in he Cerain Demand reamen may no be robus o he endogenous (or inheren) incomplee informaion regarding buyer reservaion values. 11 Noe also ha he specific price predicions in he Uncerain Demand reamen are sensiive o d bu no o he level of uncerainy (by proposiion 2 he 1 oucome should no change for all m 21 (0, ) and m 11 such ha 9 2 m11 m 21 1 ). So if he endogenous uncerainy in he Cerain Demand 9 reamen is small and if adding furher noise in he Uncerain Demand reamen does no violae he condiion imposed in proposiion 2, hen here is reason o expec ha endogenous uncerainy in he Cerain Demand reamen may lead o prices ha are similar o prices in he Uncerain Demand reamen. We repor 13 sessions in he Cerain Demand reamen and 12 sessions in he Uncerain Demand reamen, employing a oal of 408 subjecs: 336 subjecs were inexperienced and 72 were experienced in a previous session of his series. Table 2 summarizes he number of monopoliss in each of he reamens. Experienced sessions were conduced in only he d p.9 and.6 reamens, where he larges difference in opening prices is expeced. Subjecs were recruied using in-class 11 Some readers have suggesed an alernaive robo buyers reamen o es he Pacman equilibrium. In such a reamen, he experimener would make purchase decisions for he buyers using an announced myopic sraegy (i.e., buy one uni a a price no higher han 54 and a second uni a a price no higher han 18). Alhough we considered such a reamen, we expec ha hese announced buyer sraegies would make he Pacman sraegy raher obvious, so lile would be learned in such a nonsraegic environmen. We herefore chose o devoe he scarce resources o pay subjecs o he sraegic environmen wih human buyers.

18 1328 journal of poliical economy TABLE 2 Number of Monopoliss in Each Treamen d Treamen Cerain Demand Uncerain Demand dp.6 27 monopoliss (6 experienced) 22 monopoliss (6 experienced) dp monopoliss 18 monopoliss dp.9 26 monopoliss (6 experienced) 24 monopoliss (6 experienced) Noe. d denoes he coninuaion probabiliy from one period o he nex. announcemens from he general undergraduae populaion a he Insiuo Tecnológico Auónomo de México. All experimens were conduced in Spanish. 12 Subjecs were paid 50 pesos as an appearance fee plus heir profi from rading. Salien earnings per subjec ranged from a minimum of 2 pesos o a maximum of 342 pesos, wih an average of 107 pesos. The approximae exchange rae when he experimen was conduced was 10 pesos equal o U.S.$1.00, bu in erms of local purchasing power, he average earnings considerably exceed he equivalen of U.S.$30. On average, a session required abou 105 minues o complee. IV. Resuls We organize he resuls using hypoheses derived from he heory presened in Secion II. Subsecions A and B presen resuls from he Cerain and Uncerain Demand reamens. Subsecion C compares he resuls in hese wo reamens, and Subsecion D compares prices for he hree d reamens. Subsecion E briefly compares he resuls o he relaed experimenal lieraure. A. Cerain Demand Hypohesis 1. Median opening offer prices equal 54 he highes buyer valuaion. Table 3 presens he opening offer prices in he inexperienced Cerain Demand sessions, separaely by he coninuaion probabiliy reamen (d). Nearly all offer prices fall well below 54, rejecing hypohesis 1. In some rounds he mean offer price exceeds he median offer price by a wide margin, because of a few oulying prices. However, over all, only 14 ou of 378 opening prices (4 percen) are greaer han or equal o 54. The boom of able 3 indicaes ha he median offer pooled across rounds is 35 pesos when d p.6, 30 pesos when d p.75, and 26 pesos 12 An English ranslaion of he insrucions is available from he auhors.

19 durable goods 1329 when d p.9. The medians are, respecively 35, 28, and 25 in round 6. As we discuss below, noe ha hese median offer prices are roughly consisen wih he equilibrium predicions of he incomplee informaion model (which are 32.4, 27, and 21.6, respecively). Figure 1 summarizes he opening prices across he six rading rounds, separaely for each d reamen. Opening prices end o fall, and he variance ends o decline across rounds. In every round and in every reamen, a nonparameric Wilcoxon sign rank es rejecs he null hypohesis ha he median prices equal 54 a he 1 percen significance level. 13 Figure 2 presens he opening offer prices across he six rading rounds for he experienced sessions. For boh d reamens, prices are subsanially below 54, again rejecing hypohesis 1. For d p.6, opening prices are remarkably close o he incomplee informaion model predicion. For d p.9, opening prices usually exceed he incomplee in- formaion model predicion. 14 Hypohesis 2. Median offer prices afer one uni has been sold equal 18 he second-highes buyer valuaion. When all rounds in he inexperienced sessions are pooled, he median offer price immediaely afer one uni has been sold is 20 pesos when d p.6, 16 pesos when d p.75, and 15 pesos when d p The medians are 18, 16, and 15, respecively, in he final round. According o he Wilcoxon es, median offers are no significanly differen from 18 a he 5 percen level in all bu wo individual rounds. This provides some suppor for hypohesis 2. In he experienced sessions, when all rounds are pooled, he median offer price immediaely afer one uni has been sold is 14 pesos when d p.6 and 15 pesos when d p.9. One inerpreaion of he resul ha opening prices are subsanially below 54 is ha subjecs as in mos experimens on he ulimaum game resis highly asymmeric disribuions of exchange surplus. The observed opening prices are ofen in he range of pesos, which if acceped lead he monopolis o earn roughly percen of he gains from rade. Ulimaum offers are in his same range, alhough hey ypically offer no more han one-half of he surplus o he respon- 13 Here and in wha follows we repor saisical ess only for he independen markes, separaely for each round. We canno pool observaions for his saisical es across rounds because he same seller makes muliple price offers in differen rounds, causing he daa o be nonindependen when all rounds are pooled. 14 Saisical ess for individual rounds of he experienced daa have low saisical power because only six sellers paricipaed in each reamen for he experienced daa. Therefore, we focus on saisical ess for he inexperienced daa and usually repor only summary saisics for he experienced daa o assess if a major shif in behavior occurs as subjecs gain experience. 15 Some of he offers exceed he low valuaion of 18, which may reflec misakes or confusion on he par of sellers when hey have already sold a uni a a price above 18 (mos probably o heir high-valuaion buyer). Some sellers, however, may believe (someimes correcly) ha hey sold heir firs uni o he low-valued buyer.

20 TABLE 3 Median and Mean Opening Period Prices in he Cerain Demand Treamen wih Inexperienced Subjecs 1330 Np21 per Round a dp.6 dp.75 dp.9 Pacman Equilibrium Uncerain Demand Equilibrium Np18 per Round Pacman Equilibrium Uncerain Demand Equilibrium Np20 per Round b Pacman Equilibrium Uncerain Demand Equilibrium Round 1: Median offer 45 54* 32.4* * * 21.6* Mean offer Sandard error Round 2: Median offer 40 54* 32.4* 30 54* 27* 30 54* 21.6* Mean offer Sandard error Round 3: Median offer 36 54* * 27* * 21.6* Mean offer Sandard error Round 4: Median offer 32 54* * 27* 25 54* 21.6 Mean offer Sandard error

21 Round 5: Median offer 33 54* * * 21.6 Mean offer Sandard error Round 6: Median offer 35 54* * * 21.6 Mean offer Sandard error All rounds pooled: Median offer Mean offer Sandard error a Round 6 has only 17 observaions. b Round 6 has only 15 observaions. * This equilibrium price predicion is rejeced a he 5 percen significance level using a wo-ailed Wilcoxon es. This es is no conduced on he pooled rounds daa. 1331

22 Fig. 1. Opening offer prices, by rading round, for inexperienced cerain demand. a, Coninuaion probabiliy d p.6; 21 observaions per round excep round 6 (17 obser- vaions). b, Coninuaion probabiliy d p.75; 18 observaions per round. c, Coninuaion probabiliy d p.9; 20 observaions per round excep round 6 (15 observaions). 1332

23 Fig. 2. Opening offer prices, by rading round, for experienced cerain demand. a, Coninuaion probabiliy d p.6; six observaions per round. b, Coninuaion probabiliy d p.9; six observaions per round. 1333