OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS

Transcription

1 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS RAHUL DEB Absrac. We model new experience goods in he conex of dynamic mechanism design. These are goods for which an agen is unsure of her valuaion bu can learn i hrough consumpion experience. We consider a dynamic environmen wih a single buyer and seller in which conracing occurs over T periods, where each ime he agen consumes he objec, she receives a signal which allows her o revise her valuaion. In his seing, experimenaion wih he produc is sraegic boh for he buyer and seller. We derive he efficien and seller opimal conracs and compare hem. We presen a simple wo period example which highlighs some of he key feaures of he model. Finally, he mehodology developed in he paper can be used o design efficien and opimal conracs in a muli-buyer seing wih learning, where each buyer has single uni demand and here is a single objec for sale in each period. Keywords: Dynamic mechanism design, new experience goods, bandi problems. 1. Inroducion Mechanism design is perhaps he mos prominen mehod used by economiss o design conracs and sudy price discriminaion in a variey of differen seings. However, he majoriy of he lieraure focuses on saic models in which conracing occurs only in a single period and assumes, ha he agens know heir valuaions a he ime a which hey agree o he conrac. Recenly, here have been exciing developmens which exend some sandard resuls of saic mechanism design heory o a general dynamic seing. In dynamic mechanism design, he agens valuaions follow a sochasic process, which implies ha hey have new privae informaion a each period. The conrac offered by he principal in a given period depends no only on he curren repor by an agen bu also on all previous repors as well. Ahey and Segal (007) and Bergemann and Välimäki (008) sudy he design of efficien mechanisms in such a dynamic conex. In conras, Pavan, Segal and Toikka (008) (henceforh referred o as PST) derive a seller opimal mechanism for muliperiod conracing, exending he resuls of Myerson (1981) o a dynamic seing. They sudy incenive compaibiliy in a very general environmen where agens ypes are allowed o evolve in an unresriced way, decisions affec he ype disribuion and he payoff o he agen need no be separable across ime. Deparmen of Economics, Yale Universiy, Box 0868, New Haven, CT address: rahul.deb@yale.edu. Dae: February, 009. I would like o hank Dirk Bergemann for invaluable guidance on his projec. I would also like o hank Johannes Hörner, Maher Said and Larry Samuelson for useful commens. Needless o say, all errors are mine. 1

2 RAHUL DEB This paper sudies a muli period conracing problem wih learning. We se up a model where an agen would like o conrac wih he principal for T periods. A each period he agen has single uni demand for he objec, he valuaion of which evolves over ime. If he agen consumes he objec a any period, she receives a signal from her consumpion experience which allows her o updae her valuaion for he objec. This is he key difference beween a model of learning and he sandard dynamic mechanism design problem. In sandard dynamic mechanism design he agen s valuaion evolves over ime wheher or no she consumes he objec. In our seing, valuaions are only updaed by consumpion and hence here is a poenial benefi o experimenaion for he buyer. Moreover, if he seller expecs he buyer o like he produc afer she ries i, she can exrac surplus from he buyer for allowing her o experimen and updae her valuaion. Hence, experimenaion affecs boh he buyer s and seller s incenives and hence social welfare and seller revenue. This model is relevan o a number of commonly occurring siuaions. Inroducory pricing is a well-known phenomenon where a seller offers a low price iniially o he buyer, so ha he buyer can learn he aribues of he good. Afer he inroducory pricing period, he seller ypically offers a differen price o he buyer who may or may no choose o coninue consuming he produc. An implici moivaion of such behavior is ha he buyer may be imperfecly informed or simply unaware abou her valuaion for he produc bu by allowing her o experimen for a low price, she can revise her valuaion hrough consumpion experience. Such siuaions are rife in he real world. We observe cable elevision companies offering discouns for he firs few monhs of service, gyms which allow one o ry he equipmen for free for he firs few weeks and sofware which offer a free iniial rial period following which a license mus be purchased o coninue using he sofware. Ineresingly, our model predics ha, in general, such pricing is no revenue maximizing. Specifically, we model a muli-period environmen wih a single buyer and seller. The buyer has an iniial privae valuaion θ 1 for he objec which is unobserved by he seller. If he buyer consumes he objec in any period, she receives a signal ξ from her experimenaion which allows her o reassess he value of he objec and hen decide wheher o consume i again in he subsequen periods. The signal ξ is unobserved by he seller. A each period he buyer repors a message o he seller if she has new privae informaion, i.e., a revised valuaion for he objec. The seller commis o a T period conrac a period 1. This conrac offers a menu of prices and probabiliies of receiving he objec a each period, which depends on no only he period message (if here is new privae informaion) bu also on all previous messages. Lasly, we assume ha he buyer can break he conrac and walk away a any period. We derive he socially efficien conrac and he revenue maximizing conrac for he seller. This problem bears resemblance o he muli-armed bandi problem. The muli-armed bandi problem is a saisical decision model of an agen rying o opimize her decisions while simulaneously improving her informaion. I can be hough of in erms of a gambler who sequenially chooses which of K differen arms of a slo machine o play so as o maximize her reward. Choosing an arm leads o an insananeous payoff bu he process which deermines he payoff evolves during he

3 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 3 course of play. The key feaure is ha he disribuion of reurns from an arm only change when i is chosen. Hence, here is a rade-off beween exploring differen arms o discover he opimal one and exploiing he arm which is known o give he bes payoff a he presen ime (see Bergemann and Välimäki (006) for a survey of he economics lieraure on bandi problems). In our seing, he buyer experimens by consuming he produc a a price. This experimenaion leads her o updae her value and hence her payoff. Of course, she always has a safe arm which offers a payoff of 0, which is he opion of breaking he conrac. The key difference, however, is ha in our seing, experimenaion is sraegic for boh he buyer and he seller and as a resul incenives need o be aligned on boh sides of he marke. A surprising resul is ha like a wo-armed bandi problem, he seller opimal conrac is a sopping rule. PST presen a special case of he learning model in his paper as an applicaion of heir resuls. They assume ha ypes evolve according o an AR(1) process and ha he seller canno offer a probabilisic conrac. Moreover, hey make qualiaive argumens regarding he opimal and efficien conracs and do no derive closed form soluions. By conras, his paper allows for more general evoluion of ypes and allows he seller o offer a probabilisic conrac. Our closed form soluion confirms heir inuiion in his general seing. Lasly, i should be poined ou ha he presenaion of his paper bears closes resemblance o Pavan (007) which is an older working paper version of PST. For compleeness, we would like o refer he reader o a few oher imporan papers in he mechanism design lieraure which are similar in spiri o our work- Baron and Besanko (1984), Baaglini (005), Coury and Li (000) and Ëso and Szenes (007), o name bu a few. The paper is organized as follows. Secion describes he framework of he model and some key assumpions. Secion 3 derives he socially efficien conrac. Secion 4 ses up he seller s revenue maximizaion problem and derives is closed form soluion. Secion 5 presens a simple period example which highlighs some of he imporan feaures of our resuls. Secion 6 discusses he assumpion of independen shocks made in his paper. Finally, secion 7 discusses some simple exensions of our model and provides concluding remarks. The appendix conains some of he proofs.. The Model The model consiss of a single buyer and a single seller. Where no explicily menioned he buyer is referred o using feminine pronouns and he seller using masculine pronouns. Conracing occurs over T ime periods where individual ime periods are denoed by 1,,..., T }. We assume ha boh he buyer and he seller are risk neural and ha he seller has a cos of zero (assuming zero seller cos is purely for ease of exposiion as he resuls in he paper generalize rivially o nonzero seller cos). The discoun facor is given by δ [0, 1. The valuaion of he buyer a ime is given by θ. A period 1, he buyer realizes an iniial valuaion which is unobserved by he seller. This value is denoed by θ 1 Θ where Θ is he bounded inerval [θ 1, θ 1 R. θ 1 is drawn from a disribuion F, which is assumed o be sricly increasing on he

4 4 RAHUL DEB suppor Θ. If he buyer consumes he objec a a ime she receives a signal from her consumpion experience which allows her o revise her valuaion. The signal ξ Ξ Ξ (, ) is drawn independenly from a disribuion G. 1 We will use Ξ and Ξ inerchangeably for noaional convenience, bu he meaning will remain clear. Since we make no assumpions abou he suppor of G, his assumpion of ξ being unbounded is wihou loss of generaliy. Le C = c 1,..., c C } denoe he se of ime periods up o and including period 1, or equivalenly, a he beginning of period, where he buyer consumes he objec. We denoe he se of all possible consumpion hisories C by C 1,..., 1}. For example, a he beginning of he fifh period, if he buyer had consumed he objec in periods 1, and 4, hen C 5 = c 5 1, c5, c5 4 } = 1,, 4}. For ease of noaion we define he vecors ξ C and ξ as follows ξ C = (ξ c 1 +1,..., ξ c C +1 ) ξ = (ξ,..., ξ ) ξ C denoes he vecor of signals he buyer receives up o period due o consumpion hisory C. In our example when C 5 = 1,, 4}, ξ C5 = (ξ, ξ 3, ξ 5 ). Values evolve according o flow valuaion funcions v, which depend on he iniial valuaion θ 1 and subsequen signals ξ. Hence if he buyer receives r 1 signals, v r is defined as v r : Θ Ξ r 1 R. We assume hese flow value funcions are Markov, ha is, for any θ 1, ξ r and θ 1, ξ r such ha v r (θ 1, ξ r ) = v r ( θ 1, ξ r ), i mus be he case ha for all s > r (A0) v s (θ 1, ξ r, ξ r+1,..., ξ s ) = v s ( θ 1, ξ r, ξ r+1,..., ξ s ) We can now define period flow valuaion θ in erms of he iniial value and he subsequen consumpion hisory C (wih r = C ) by using flow valuaion funcions v. θ = v C (θ 1, ξ C ) = v r (θ 1, ξ C ) This funcion defines how he buyer s flow valuaion evolves as she receives new signals from her experimenaion. I is assumed ha he seller knows he funcions v and he disribuions F and G. We will make he following key assumpions abou he disribuion of iniial value F and flow value funcions v. (A1) The densiy f of cdf F is log concave, implying he monoone hazard rae condiion. (A) v r is differeniable and sricly increasing in all argumens. (A3) For r s, lim ξr v s (θ 1, ξ s ) and lim ξr v s (θ 1, ξ s ). (A4) v s (θ 1, ξ s ) are concave in θ 1 and v s(θ 1,ξ s ) θ 1 ξ r 0 for any (θ 1, ξ s ), r s. (A5) For any r 3, any s r and any (θ 1, ξ r, ξ r+1,..., ξ s ) v s (θ 1, ξ r, ξ r+1,..., ξ s ) θ 1 ξ r 1 v r (θ 1, ξ r ) ξ r v s (θ 1, ξ r, ξ r+1,..., ξ s ) θ 1 ξ r v r (θ 1, ξ r ) ξ r 1 1 We discuss his assumpion in secions 6 and 7. The assumpion implies ha he mechanism mus be defined for repored values ha are poenially ouside he suppor of G. However, in equilibrium he only values ha he buyer will repor will lie in he suppor.

5 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 5 Similarly for r = and any (θ 1, ξ,... ξ s ), s v s (θ 1, ξ,..., ξ s ) v (θ 1, ξ ) θ1 v s (θ 1, ξ,..., ξ s ) v (θ 1, ξ ) ξ θ 1 ξ θ 1 These assumpions are idenical o hose required by Pavan (007). Assumpion A1 is he sandard monoone hazard rae assumpion used in much of he mechanism design lieraure. Assumpions A and A3 are naural properies of a value funcion and are hence innocuous. Assumpion A4 is an inuiive concaviy condiion, which one would expec a dynamic valuaion funcion o saisfy. Assumpion A5, admiedly has no obvious economic inerpreaion. I is purely a echnical assumpion required for our resuls. As a jusificaion, we observe ha a number of naural funcions saisfy our assumpions. Consider for example he following value funcions. (1) () (3) θ = v (θ 1, ξ ) = θ 1 + i=1 θ = v (θ 1, ξ ) = α1θ αk θ k + ξ θ = v (θ 1, ξ ) = θ 1 The flow value funcion (1) is he random walk case and is a special case of a general linear auoregressive process of order k (AR(k)), in which values are defined recursively by equaion (). Finally, (3) is an example of a value funcion where he shocks are muliplicaive. We now define he message space. A period 1, he se of messages is denoed by M 1. A any period > 1, if he buyer receives he objec a period 1, she receives a signal ξ abou her valuaion from her consumpion experience and she can repor a message from message space M. If she does no receive he objec a 1 hen she has no new privae informaion and hence does no make a repor. The seller commis o a mechanism consising of a series of funcions q, p, which depend on he curren ime period, he hisory of buyer consumpion C and he hisory of repored messages. q denoes he probabiliy of receiving he objec and p is he price. More formally i=1 q C : M 1 M c 1 +1 M c r +1 [0, 1 p C : M 1 M c 1 +1 M c r +1 R where r = C Reurning o he above 5 period example, where C 5 = 1,, 4}. A period 5, he mechanism offers a conrac (q5 C5, pc5 5 ) where qc5 5 : M 1 M M 3 M 5 [0, 1 and similarly p C5 5 : M 1 M M 3 M 5 R. In oher words since he buyer does no receive he objec in period 3, she has no new privae informaion, and hence does no repor in period 4. However, she ends up consuming he objec in period 4 and herefore once again makes a repor in period 5. Noe ha while we force he seller o commi o a conrac a period 1, we allow he buyer o walk away and end he conrac a any ime period. Because he conrac canno be renegoiaed, his does no give he buyer addiional bargaining power by hreaening o walk away. I merely ξ ξ

6 6 RAHUL DEB forces he conrac o be individually raional for he buyer a all ime periods. The iming and he conracing are made more explici below. The buyer learns her iniial value θ 1. The seller offers a mechanism given by (q 1, p 1 ), (q C, pc ),..., (qct T, pct T ) for all C C,..., C T C T. The buyer repors a message m 1 M 1. Having repored m 1, she is offered he objec wih probabiliy q 1 (m 1 ) wih corresponding ransfer p 1 (m 1 ). If she agrees hen he conrac is enaced, oherwise, if she refuses o paricipae hen boh she and he seller ge a payoff of 0 and he conrac erminaes. If she receives he objec (if q 1 (m 1 ) < 1, she migh no), she receives signal ξ hrough her consumpion experience. If she doesn receive he objec, she ges no furher informaion. If she ends up receiving he objec in period 1, she repors a message m M and is offered q C(m 1, m ), p C (m 1, m ) where C = 1}. If she agrees hen he conrac is enaced, oherwise, if she refuses o paricipae hen boh she and he seller ge a payoff of 0 in period and he conrac erminaes. If she does no end up receiving he objec in period 1, she is offered q C(m 1), p C (m 1) where C = φ. If she agrees hen he conrac is enaced, oherwise, if she refuses o paricipae hen boh she and he seller ge a payoff of 0 in period and he conrac erminaes. This process coninues for he duraion T or unil he buyer decides o break he conac. Finally, we observe ha since he seller commis o he full T period mechanism, he Revelaion Principle applies and we need only consider direc mechanisms. In oher words, we need only consider mechanisms where M 1 Θ and M Ξ. In all subsequen secions we analyze only he direc mechanisms q C : Θ Ξ c 1 +1 Ξ c r +1 [0, 1 p C : Θ Ξ c 1 +1 Ξ c r +1 R where r = C PST sudy a special case of his learning model. They assume ha he iniial valuaion and subsequen shocks are normally disribued. In heir seing, he period valuaion θ is a weighed average of value θ 1 and signal ξ. Clearly his is a special case of our seing. Moreover, hey also assume ha he seller offers a conrac q a each period where q 0, 1}. In oher words hey do no allow he seller o offer probabilisic conracs o he buyer. Lasly, hey do no derive closed form soluions for he efficien and seller opimal conrac which is in conras o his paper.

7 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 7 3. The Efficien Policy We sar off by deriving he efficien policy. Since he seller has no cos, he ex-ane efficien policy maximizes he following surplus funcion T θ 1 q 1 (θ 1 ) +... µ q (C )q C (θ 1, ξ C )v C (θ 1, ξ C )dg(ξ )... dg(ξ T ) Ξ Ξ T δ = C C Where µ q (C ) is he probabiliy ha hisory of consumpion C occurs as a resul of conrac q. Hence, µ q (C ) = s= w Ds s (θ 1, ξ Ds ) where D s = C,..., s} and w Ds s (θ 1, ξ Ds ) = qs Ds (θ 1, ξ Ds ) if s C 1 qs Ds (θ 1, ξ Ds ) if s / C We now derive he efficien conrac by backward inducion. A ime period T, for any hisory of consumpion C T, he efficien conrac mus be 1 if v qt CT (θ 1, ξ CT T CT ) = (θ 1, ξ CT ) 0 0 oherwise Similarly, a ime period T 1, for any hisory of consumpion C T 1, he efficien conrac will be CT 1 qt 1 (θ 1, ξ CT 1 ) = CT 1 1 if v δ Ξ T qt CT 0 oherwise T 1 (θ 1, ξ CT 1 )+ (θ 1, ξ CT 1, ξ T )v CT T (θ 1, ξ CT 1, ξ T )dg(ξ T ) 0 where C T = C T 1 T 1} and q T is defined above. We observe ha i is never efficien o no allocae he objec in period T 1 and hen allocae i in period T. This is because, from he definiion of q T, he second erm δ Ξ T qt CT (θ 1, ξ CT 1, ξ T )vt CT (θ 1, ξ CT 1, ξ T )dg(ξ T ) is always non-negaive. Hence, if he objec is no allocaed in period T 1, i mus be he case ha CT 1 vt 1 (θ 1, ξ CT 1 ) < 0 and hence he objec will no be allocaed in period T eiher. Conversely, noice ha he efficien policy could allocae he objec o he consumer even hough her period CT 1 T 1 flow valuaion vt 1 (θ 1, ξ CT 1 ) is negaive. This is because here could be posiive welfare gains from experimenaion. Lasly, we also observe ha boh q T 1 and q T never ake a value beween 0 and 1. When vt CT (θ 1, ξ CT ) = 0, qt CT could poenially ake a value beween 0 and 1 bu his will be oucome equivalen o qt CT CT 1 = 1. The same argumen can be applied o qt 1. This argumen can be backward induced o period 1. We denoe hisories S = 1,..., 1}. These are hisories a ime where he buyer ges he objec in every period up o. Hence, a hese hisories he value funcion v S is simply equal o v.

8 8 RAHUL DEB Proposiion 1. The efficien conrac is given by he following quaniies. q C (θ 1, ξ C ) = q CT T (θ 1, ξ CT ) = 1 if C T = S T = 1,..., T 1} and v T (θ 1, ξ ST ) 0 0 oherwise 1 if C = S = 1,..., 1} and v (θ 1, ξ S )+ T s=+1 Ξ Ξ T δ s ( s r=+1 qsr r (θ 1, ξ Sr ) ) v s (θ 1, ξ Ss )dg(ξ +1 )... dg(ξ T ) 0 0 oherwise The efficien conrac will allocae he objec o he buyer wih probabiliy 1 as long as he sum of heir value and heir expeced value from experimenaion is non-negaive. The efficien conrac induces efficien experimenaion, ha is, if he flow value v of he buyer falls below 0, she is allowed o experimen if her value is expeced o improve wih experimenaion. Also he efficien conrac is a sopping rule where if he buyer does no receive he objec in period she will no receive i in any subsequen period. 4. The Seller Opimal Mechanism In his secion we will derive he seller opimal conrac for periods. This will highligh he key seps of he proof wihou he reader having o deal wih complicaed noaion. We will sae he resul for he general T period case a he end of he secion and provide he proof of incenive compaibiliy in he appendix. We sar off by defining he buyer s uiliy a period. There are wo possible hisories which we denoe by C1 = 1} and C = φ. When he buyer receives he objec in period 1 having repored ˆθ 1 (wih rue value θ 1 ), she receives a signal given by ξ. Her uiliy in period is hen given in erms of her rue valuaions θ 1, ξ, repored valuaions ˆθ 1, ˆξ and hisory of consumpion C1 as follows Ũ C 1 (θ 1, ξ ; ˆθ 1, ˆξ ) = v (θ 1, ξ )q C 1 (ˆθ 1, ˆξ ) p C 1 (ˆθ 1, ˆξ ) The ilde superscrip is used for uiliy funcions where he buyer has new informaion o repor. The semicolon is used o separae he rue from he repored values. If she does no receive he objec her uiliy a period is simply given by Û C (θ 1; ˆθ 1 ) = θ 1 q C (ˆθ 1 ) p C (ˆθ 1 ) If he buyer received he objec a = 1 by reporing value ˆθ 1 and has realized rue value and signal θ 1, ξ, he highes uiliy she can ge a = is given by he soluion o he following opimizaion problem. } (4) Û C 1 (θ 1, ξ ; ˆθ 1 ) = max ˆξ v (θ 1, ξ )q C 1 (ˆθ 1, ˆξ ) p C 1 (ˆθ 1, ˆξ ) The ha superscrip on uiliy funcions is used denoe he maximum obainable uiliy for he buyer, coningen on boh he rue values of he signals and he previously repored values of he

9 signals. In oher words OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 9 Û C 1 (θ 1, ξ, ; ˆθ 1 ) = max ˆξ Ũ C 1 (θ 1, ξ ; ˆθ 1, ˆξ ) We now define incenive compaibiliy in his conex. Definiion 4.1 (Incenive Compaibiliy). A mechanism (q, p) is incenive compaible if ruhelling is opimal for he buyer a period 1 and ruhelling is opimal for all oher periods (where she has new privae informaion o repor) condiional on having repored ruhfully in he pas. This is clearly a weaker requiremen han forcing ruhelling o be opimal regardless of pas repors. Formally, if he buyer reveals her value ruhfully a = 1 and receives he objec hen ruh elling is incenive compaible a = if (5) Ũ C 1 (θ, ξ; θ, ξ) = Û C 1 (θ, ξ; θ) = U C 1 (θ, ξ) where U (θ, ξ) is concise noaion for uiliy from ruh elling. Clearly, when she does no receive he objec she has no new privae informaion and hence here are no incenive issues a period. We define he expeced uiliy of he buyer having realized value θ a = 1 and reporing opimally as follows (6) Û1(θ 1 ) = max θ 1 q 1 (ˆθ 1 ) p 1 (ˆθ 1 ) + δq 1 (ˆθ 1 ) ˆθ 1 Ξ } Û C 1 (θ 1, ξ ; ˆθ 1 )dg(ξ ) + δ(1 q 1 (ˆθ 1 ))Û C (θ 1; ˆθ 1 ) Now, we can define he uiliy from ruh elling in boh periods as follows. U 1 (θ 1 ) = θ 1 q 1 (θ 1 ) p 1 (θ 1 ) + δq 1 (θ 1 ) U C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))U C (θ 1) The revenue of he seller when he buyer repors ruhfully in boh periods is given by [ (7) p 1 (θ 1 ) + δq 1 (θ 1 ) p C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))p C (θ 1) df (θ 1 ) Θ Ξ Ξ A mechanism is said o be incenive compaible if ruh elling maximizes expeced uiliy for he buyer in period 1 and ruh elling is also opimal in period, condiional on having repored ruhfully in period 1. In oher words, U 1 (θ 1 ) = Û1(θ 1 ) and (5) should hold simulaneously. Individual raionaliy requires ha U 1 (θ 1 ) 0, U C 1 (θ 1, ξ ) 0 and U C (θ 1) 0 for all (θ 1, ξ ) Θ Ξ. The seller designs an opimal conrac by solving he following problem. [ p 1 (θ 1 ) + δq 1 (θ 1 ) max q 1 ( ),p 1 ( ),q (, ),p (, ) Θ p C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))p C (θ 1) Ξ subjec o U 1 (θ 1 ) = Û1(θ 1 ), U (θ 1, ξ ) = Û(θ 1, ξ ; θ 1 ), U 1 (θ 1 ) 0, U C 1 (θ 1, ξ ) 0, U C (θ 1) 0 df (θ 1 ) In order o solve his problem we use a similar reverse engineering process as he one in Ëso and Szenes (007). We firs derive necessary condiions for incenive compaibiliy and define a relaxed opimizaion problem in which he seller is maximizing his revenue subjec o hese necessary condiions.

10 10 RAHUL DEB We sar by examining he incenive consrain in period. Using he sandard echnique of Myerson (1981), equaion (4) implies ha a necessary and sufficien condiion for ruh elling a = given ruhful reporing a = 1 is (ENV) U C 1 (θ 1, ξ ) = U C 1 (θ 1, ξ ) + ξ v(θ 1, ξ ) ξ q C 1 (θ 1, ξ )d ξ where U C 1 (θ 1, ξ ) = lim ξ U C 1 (θ 1, ξ ). We can use he envelope heorem o differeniae 3 equaion (6) wih respec o θ 1 o ge (8) C1 du 1 (θ 1 ) Û = q 1 ( θ 1 ) + δq 1 ( θ 1 ) (θ 1, ξ ; θ 1 ) dθ 1 Ξ θ 1 dg(ξ ) + δ(1 q 1 (θ 1 )) Û C (θ 1; θ 1 ) θ 1 where θ 1 is assumed o be he maximizer of (6). We can use he envelope heorem again on (4) yielding Û C 1 (θ 1, ξ ; θ 1 ) θ 1 = v (θ 1, ξ ) θ 1 q C 1 ( θ 1, ξ ) where ξ is assumed o be he maximizer of (4). We can do he same for Û C. Plugging hese ino equaion (8) we arrive a a necessary condiion for ruh elling: (9) du 1 (θ 1 ) dθ 1 = q 1 (θ 1 ) + δq 1 (θ 1 ) θ 1 Ξ v (θ 1, ξ ) θ 1 q C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))q C (θ 1) While his condiion is necessary, i is no, in general, sufficien for ruh elling o be incenive compaible. This is in conras wih he saic model and his observaion has also been made by Ëso and Szenes (007) and Pavan (007). (9) inegraes o θ1 [ v(, ξ ) (ENV1) U 1 (θ 1 ) = U 1 (θ 1 ) + q 1 () + δq 1 () q C 1 Ξ (, ξ )dg(ξ ) + δ(1 q 1 ())q C d () We can now rewrie revenue equaion (7) in erms of he buyer s uiliy and define he seller s revenue as [ θ 1 q 1 (θ 1 ) + δq 1 (θ 1 ) v(θ 1, ξ )q C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))θ 1 q C (θ 1) U 1 (θ 1 ) df (θ 1 ) Θ Ξ We now define he relaxed problem for he seller. In his problem he maximizes his revenue subjec o he necessary condiions for incenive compaibiliy ha we have derived: [ θ 1 q 1 (θ 1 ) + δq 1 (θ 1 ) v (θ 1, ξ )q C 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))θq C (θ 1) Θ Ξ max q 1 ( ),U 1 ( ),q (, ),U C (, ) U 1 (θ 1 ) df (θ 1 ) subjec o (ENV1), (ENV), U 1 (θ 1 ) 0, U C 1 (θ 1, ξ ) 0, U C (θ 1) 0 3 We derive his condiion explicily in he appendix as here is no immediaely obvious reason o assume ha he buyer s uiliy is differeniable. This derivaion will also show why he envelope condiion is necessary bu no sufficien for incenive compaibiliy.

11 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 11 By subsiuing equaion (ENV1) ino he objecive funcion of he above relaxed problem and inegraing by pars, we can rewrie he objecive funcion as [ (θ1 1 F (θ 1) ) (10) q1 (θ 1 ) + Θ ( δq 1 (θ 1 ) v (θ 1, ξ ) v (θ 1, ξ ) 1 F (θ 1 )) C q 1 Ξ θ 1 (θ 1, ξ )dg(ξ ) + δ(1 q 1 (θ 1 ))(θ 1 1 F (θ 1) ) C q (θ 1) U 1 (θ 1 ) df (θ 1 ) I is hen immediae ha he soluion o he relaxed problem is independen of paricipaion and incenive consrains for period. These consrains are only relevan for deriving he prices. The following soluion of he relaxed problem hen follows Proposiion. The soluion o he wo period seller s relaxed problem is given by U 1 (θ 1 ) = 0 and he following allocaion rules: q C 1 1 if (θ v (θ 1, ξ ) 1 F (θ 1) v (θ 1,ξ ) 1, ξ ) = θ oherwise q C (θ 1) = 0 q 1 (θ 1 ) = 1 if [ θ 1 1 F (θ [ 1) +δ Ξ v (θ 1, ξ ) 1 F (θ 1) v (θ 1,ξ ) C θ 1 q 1 (θ 1, ξ )dg(ξ ) 0 0 oherwise These quaniies are derived by poinwise maximizaion of he seller revenue funcion (10). This can be seen as follows. The second erm of (10) is clearly maximized when q C 1 (θ 1, ξ ) is given as above. The hird erm is maximized when q C (θ 1) = 1 if θ1 1 F (θ 1) 0 0 oherwise Bu hen he opimal selecion of q 1 when will be given by 1 if [ θ 1 1 F (θ [ 1) +δ Ξ v (θ 1, ξ ) 1 F (θ 1) v (θ 1,ξ ) C θ 1 q 1 (θ 1, ξ )dg(ξ ) q 1 (θ 1 ) = δ [ θ 1 1 F (θ 1) C q (θ 1) 0 oherwise Bu from he derived value of q C 1, i is always he case ha [ θ1 1 F (θ 1) [ +δ v (θ 1, ξ ) Ξ 1 F (θ 1) v (θ 1, ξ ) q C 1 θ 1 (θ 1, ξ )dg(ξ )δ [ θ 1 1 F (θ 1)

12 1 RAHUL DEB as he second erm on he lef hand side is always non-negaive. Moreover, since 0 q C 1, i is always he case ha [ θ1 1 F (θ 1) = [ θ 1 1 F (θ 1) δ [ θ 1 1 F (θ 1) [ +δ v (θ 1, ξ ) Ξ 1 F (θ 1) [ +δ v (θ 1, ξ ) Ξ 1 F (θ 1) C q (θ 1) v (θ 1, ξ ) θ 1 q C 1 (θ 1, ξ )dg(ξ ) 0 v (θ 1, ξ ) θ 1 q C 1 (θ 1, ξ )dg(ξ ) Hence, whenever q 1 (θ 1 ) = 0 or hisory C occurs i mus be he case ha θ 1 1 F (θ 1) < 0 or q C (θ 1) = 0. Hence, i is oucome equivalen for he seller o define q C (θ 1) as idenically 0. This will yield he expression for q 1 in he proposiion. Noice, ha he seller opimal conrac like he efficien conrac, is a sopping rule. If i is no opimal for he seller o allocae he objec o he buyer in he firs period i will no be opimal o allocae i o her in he second period. Moreover, in he opimal conrac q s do no ake values beween 0 and 1. I is sraighforward o generalize he inuiion of he period resul o arbirary T periods. The T period opimal conrac will also be a sopping rule such ha if he buyer does no ge he objec in period, she will no ge he objec in any period s >. Once again, we denoe hisories S = 1,..., 1}. Recall, hese are hisories a ime where he buyer ges he objec in every period up o. Hence, a hese hisories he value funcion v S is simply equal o v. Proposiion 3. The soluion o he T period seller s relaxed problem is given by U 1 (θ) = 0 and he following allocaion rules: 1 if C T = S T = 1,..., T 1} and q C (θ 1, ξ C ) = q CT T (θ 1, ξ CT ) = 1 if C = S = 1,..., 1} and [ v (θ 1, ξ S ) v(θ 1,ξ S ) + v T (θ 1, ξ ST ) 1 F (θ 1) v T (θ 1,ξ ST ) θ oherwise 1 F (θ 1 ) θ 1 T s=+1 Ξ Ξ T δ s ( s dg(ξ +1 )... dg(ξ T ) 0 0 oherwise r=+1 qsr r (θ 1, ξ Sr ) )[ v s (θ 1, ξ Ss ) vs(θ 1,ξ S s ) 1 F (θ 1 ) θ 1 The proof of his resul uses he idenical backward inducion procedure of he period case presened in his secion and is hence omied. Ineresingly, noe ha he seller opimal conrac is idenical o ha of he efficien conrac wih he flow values replaced by he virual flow values. In he dynamic seing he virual flow values are given by v (θ 1, ξ S ) 1 F (θ 1) v (θ 1,ξ S ) θ 1. These values depend only on firs period disribuion F and no on G. This reflecs he fac ha he seller commis o he conrac a period 1. The disribuions G however do affec he opimal and efficien conrac as hey represen he value of experimenaion.

13 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 13 Finally, we need o show ha we can find prices so ha he soluion o he relaxed problem can be implemened in an incenive compaible way. This is summarized by he following proposiion he proof of which is in he appendix. Proposiion 4. The quaniy schedules of he incenive compaible, individually raional seller opimal conrac coincide wih ha of he relaxed problem saed in proposiion (3). In oher words, we can find prices o implemen he quaniy schedules of (3) such ha he resuling mechanism gives boh he buyer and he seller he same uiliy and is incenive compaible. 5. A Two Period Example In his secion we discuss a simple example of he above model. This example highlighs some of he key feaures of he model. We assume ha iniial valuaion θ 1 is drawn from he uniform U[0, 1 disribuion (which is log concave and hence saisfies he monoone hazard rae condiion). If he buyer receives he objec hen her signal ξ is drawn from he U[ 1, 1 disribuion. The value funcion v is assumed o be addiive and is given by v (θ 1, ξ ) = θ 1 + ξ. Thus he signal can boh decrease or increase he value in period ; in paricular, i can drive he buyer s valuaion below 0. Linear v clearly saisfies he assumpions we need for our resuls. We can now derive he opimal allocaions: q 1 (θ 1 ) = q (θ 1, ξ ) = 1 if θ 1 θ l 0 oherwise 1 if ξ 1 θ 1 0 oherwise and he corresponding opimal prices p 1 (θ 1 ) = p (θ 1, ξ ) = where θ l = θ l + δ [θ l + θ 1 if θ 1 θ l 0 oherwise 1 θ 1 if ξ 1 θ 1 0 oherwise 1 + δ 1 δ I is ineresing o noe ha he buyer wih he lowes value who receives he objec (he buyer wih value θ l ) pays more for he objec han her value. Hence, in he opimal mechanism he seller does no offer a low inroducory price in period 1 which allows everyone o revise heir valuaion for he produc. Raher, he does no offer he produc o buyers of lowes valuaions and insead levies a charge for he consumpion experience. Of course, i is possible o consruc a differen example of he model such ha θ l = θ, so ha he seller sells he objec o he enire marke. However, even in his case he buyer wih he lowes valuaion (who ges he objec) will pay a leas her value in period 1 o ry he objec. This behavior is individually raional because she has

14 14 RAHUL DEB he opion o erminae he conrac a period, hence, expecs a leas a weak increase in her valuaion in period and as a resul her expeced uiliy in period 1 is non-negaive. Noice, also ha he cuoff value θ l is decreasing in δ. When δ approaches 0, θ l approaches 1/ which is he cuoff value of he saic opimal aucion of Myerson (1981). This is because when δ approaches 0, neiher he buyer nor he seller (iniially) cares abou he nex period and he seller ries o exrac as much revenue as possible a = 1. As δ increases, he cuoff value θ l begins o fall. This reflecs he fac ha he seller values revenue from fuure paricipaion and also he fac ha he buyer benefis more from experimenaion. The price charged by he seller in period 1 is sricly increasing in he announced valuaion. The price in period is independen of he signal ξ and is decreasing in he θ 1 announced by he buyer in period 1. This mechanism is inuiively incenive compaible due o he following reason. For buyers wih high valuaion, here is a high probabiliy ha hey would like o purchase he produc in period afer observing ξ. Announcing a high valuaion guaranees hem a relaively low price omorrow as he price in period depends only on he θ 1 announced in period 1. Buyers wih low valuaions prefer o consume he produc in period 1 for a low price and as a resul face a relaively higher price in period. However, since hey feel i is unlikely ha hey will like he produc once hey ry i, hey are conen simply o ry i a a low price. The opimal conrac is clearly inefficien. In his seing an efficien conrac would allocae he objec o he buyer wih probabiliy 1 if her valuaion θ 1 is sricly greaer han 0 in period 1. In period, he efficien conrac would allocae he objec o he buyer wih probabiliy 1, if her revised valuaion v (θ 1, ξ ) is sricly greaer han 0. In conras, he opimal conrac does no allocae he objec he buyers wih valuaion below θ l in period 1, and o buyers wih valuaion v (θ 1, ξ ) less han 1 θ 1 in period. The inuiion for his is idenical o ha of he opimal aucion in a saic seing. 6. Independen Shocks In his secion we briefly address he assumpion of independen shocks. We assumed ha he disribuion G of ξ is independen of he disribuion F of θ 1. This assumpion is acually less resricive han i may iniially appear. We discuss he implicaion of his assumpion when T = as discussed in Ëso and Szenes (007). Again resricing he discussion o periods makes i easy o inerpre he naure of he assumpion. The inuiion in his secion has been generalized o T periods by PST. Le H(θ θ 1 ) be he disribuion of period values condiional on realized period 1 value and le h(θ θ 1 ) be he associaed pdf. The following is an insighful resul from Ëso and Szenes (007). Lemma 6.1 (Ëso and Szenes (007)). Suppose he disribuion H(θ θ 1 ) of θ condiional on θ 1 is coninuous and sricly increasing in θ. Then here exiss a real valued funcion v and a random

15 OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 15 variable ξ where ξ and θ 1 are independen such ha θ = v (θ 1, ξ ) The inuiion of his resul sems from he following observaion. We can define ξ = H(θ θ 1 ). Then ξ is disribued uniformly U[0, 1 and is independen of θ 1. The value funcion v can hen be defined as v (θ 1, ξ ) = H 1 θ 1 (ξ ) where H 1 θ 1 (ξ ) = θ such ha H(θ θ 1 ) = ξ. The funcion is inverible because we assumed H was coninuous and sricly increasing. So in general, his assumpion is no resricive a all. However, for our resuls we require he addiional assumpions A-A4 on he value funcion v (assumpion A5 is no required when T = ). Hence, we need addiional assumpions on he sochasic process H. The following assumpions also derived by Ëso and Szenes (007), on he sochasic process are sufficien for our resuls. A6 H(θ θ 1 ) is coninuous and sricly increasing in θ. A7 For θ 1 > θ 1, H(θ θ 1 ) firs order sochasically dominaes H(θ θ 1 ) or in oher words H(θ θ 1 )/ θ 1 < 0. A8 H(θ θ 1 )/ θ 1 h(θ θ 1 ) is increasing in θ. A9 H(θ θ 1 )/ θ 1 h(θ θ 1 ) is increasing in θ 1. PST show ha for arbirary T periods, he sufficien condiion o saisfy assumpions A A5, is firs order sochasic dominance and ha H(θ θ 1)/ θ 1 h (θ θ 1 ) is increasing in boh θ and θ 1, where H is he condiional disribuion of θ given θ 1. These assumpions are saisfied by many sochasic processes. Therefore, while he assumpion of independen shocks does resul in a loss of generaliy, we do no find i o be excessively resricive and hence, assuming independen shocks is no as severe a consrain as i may have iniially seemed. 7. Exensions The model presened in his paper can be exended in a fairly sraighforward way o he case of N buyers compeing for a single objec in each period, over he course of T periods. Only he buyer who wins he objec in a given period ges o consume he objec and hence updae her valuaion hrough her consumpion experience. A period, he opimal conrac will award he objec he buyer wih he highes expeced discouned sum of virual flow values. The closed form soluions hough are exremely complicaed as we need o consider a large se of possible hisories. I is possible for a buyer o win an objec in period, receive a bad signal and lose he objec in period + 1, only o win i again in period + afer he buyer who won in + 1 also receives a bad signal. This is in conras o he case of he single buyer which is simply a sopping rule. Sponsored search in paricular and adverising in general is an example of a real life siuaion of such an aucion model wih learning. Adverisemens, by heir very naure, are producs where he adveriser canno accuraely asses he value of adverising before placing he ad. Hence, he

16 16 RAHUL DEB expeced value of an ad o an adveriser could poenially differ drasically before and afer placing he ad. Sponsored search aucions are dynamic in naure, wih he compeing adverisers consanly updaing heir valuaions over ime. There is a curren and ongoing economics lieraure which sudies sponsored search (see Edelman e al. (007), Varian (007), Borgers e al. (007) and Nazerzadeh e al. (007) o name bu a few papers). Of course, while our model capures he learning aspec of adverising i ignores he fac ha adverisers ofen have more han one opion in each period. In sponsored search aucions, bidders who lose he aucion for he bes link posiion can poenially win he second bes posiion and so on. A model ha capures boh his aspec and learning is a subjec of fuure research. Finally, i is also possible o exend he model in such a way so ha he disribuion of shocks G is no he same in all periods. The analysis will go hrough idenically if we assume ha he firs ime he buyer consumes he produc her signal comes from disribuion G 1, he second ime from disribuion G and so on. Noice ha his is no he same as saying ha if she consumes he produc in period 1 she will receive a signal from disribuion G. If he disribuion of he signals were a funcion of he ime period as opposed o he number of imes he buyer consumes he objec hen he opimal conrac need no be a sopping rule. Consider he former case, where disribuion G has a very low mean and G +1 has a very high mean. A period 1, i may be opimal for he seller no o allocae he objec o he buyer as hen she will probably receive a low signal from G and insead allocae i o her in period allowing her o receive he high signal from G +1 and hence allowing he seller o exrac more surplus. Appendix A. Deriving he Envelope Condiion We now explicily derive he envelope condiion a period 1. Consider he uiliy o he buyer from reporing ruhfully a period 1. U 1 (θ 1 ) = θ 1 q 1 (θ 1 ) p 1 (θ 1 ) + (11) T [... µ q (C ) Ξ Ξ T δ = C C q C (θ 1, ξ C )v C (θ 1, ξ C ) p C (θ 1, ξ C ) dg(ξ )... dg(ξ T ) where µ q (C 1 ) is he probabiliy ha hisory of consumpion C 1 occurs as a resul of conrac q and reporing θ 1 in period 1 and ruhful reporing hereafer. If insead she repors θ 1 in period 1 and subsequenly repors ruhfully she ges θ 1 q 1 (θ 1) p 1 (θ 1) + T [... µ q(c ) Ξ Ξ T δ = C C q C (θ 1, ξ C )v C (θ 1, ξ C ) p C (θ 1, ξ C ) dg(ξ )... dg(ξ T ) where µ q(c ) is he probabiliy ha hisory of consumpion C occurs as a resul of conrac q and reporing θ 1 in period 1 and ruhful reporing hereafer. Noice ha in he above expression he buyer does no repor sraegically afer period 1. There is no reason o believe ha ruhelling is opimal afer he buyer has misrepored in he firs period. Incenive compaibiliy implies ha

17 he (11) (1). Rearranging (1) OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 17 U 1 (θ 1 ) U 1 (θ 1) (θ 1 θ 1)q 1 (θ 1) + T... Ξ δ = C C Exchanging he roles of θ 1 and θ 1 we ge (13) U 1 (θ 1 ) U 1 (θ 1) (θ 1 θ 1)q 1 (θ 1 ) + T... Ξ δ = C C µ q(c )q C Ξ T µ q (C )q C Ξ T (θ 1, ξ C ) [ v C (θ 1, ξ C ) v C (θ 1, ξ C ) dg(ξ )... dg(ξ T ) (θ 1, ξ C ) [ v C (θ 1, ξ C ) v C (θ 1, ξ C ) dg(ξ )... dg(ξ T ) Dividing boh sides of (1) and (13) by θ 1 θ 1 and aking he limi θ 1 θ 1 we ge he required envelope condiion. du 1 (θ 1 ) dθ 1 = q 1 (θ 1 ) + T δ = C C [... µ q (C ) Ξ Ξ T q C (θ 1, ξ C ) vc (θ 1, ξ C ) θ 1 dg(ξ )... dg(ξ T ) Appendix B. Proof of Incenive Compaibiliy Proof. The opimal conrac has q C = 0 for all C S. Clearly, i mus also be he case ha = 0 for all C S. Hence, revenues for he seller and uiliy for he buyer will only conain p C he erms q S, p S. In his proof we will refer o q S, p S simply as q, p. This makes he proof considerably more inuiive as we have o deal wih less noaion. We need o show ha here are prices which make he soluion o he relaxed problem incenive compaible. We firs need o check he following monooniciy condiion. This condiion is he following. (1) Quaniy schedules q are nondecreasing in each argumen. () For any and any s + 1 and any (θ 1, ξ 1, ξ, ξ +1 ) and (θ 1, ξ 1, ξ, ξ +1 ) such ha v +1 (θ 1, ξ 1, ξ, ξ +1 ) = v +1 (θ 1, ξ 1, ξ, ξ +1 ), if and only if ξ ( ) ξ and vice versa. q s (θ 1, ξ 1, ξ, ξ +1,..., ξ s ) ( )q s (θ 1, ξ 1, ξ, ξ +1,..., ξ s ) (3) Similarly, for any (θ 1, ξ ) and ( θ 1, ξ ) such ha v (θ 1, ξ ) = v ( θ 1, ξ ), if and only if θ 1 ( ) θ 1. q s (θ 1, ξ,..., ξ s ) ( )q s ( θ 1, ξ,..., ξ s )

18 18 RAHUL DEB I is sraighforward o observe ha A1-A4 imply ha he quaniy schedules are nondecreasing in each argumen. Hence, we only need o show monooniciy properies () and (3) as above. We will show propery () and (3) will follow similarly. We do his by backward inducion on he quaniy schedules. We will show he ξ ξ case and he ξ ξ case is analogous. Pavan (007) shows ha he assumpions A1-A5 imply ha v s (θ 1, ξ 1, ξ, ξ +1,..., ξ s ) v s(θ 1, ξ 1, ξ, ξ +1,..., ξ s ) 1 F (θ 1 ) θ 1 v s (θ 1, ξ 1, ξ, ξ +1,..., ξ s ) v s(θ 1, ξ 1, ξ, ξ +1,..., ξ s ) 1 F (θ 1 ) θ 1 whenever v +1 (θ 1, ξ 1, ξ, ξ +1 ) = v +1 (θ 1, ξ 1, ξ, ξ +1 ) and ξ ξ. In paricular his implies ha he above holds for s = T and hence q T saisfies he monooniciy condiion. Now since v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) 1 F (θ 1 ) θ 1 v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) 1 F (θ 1 ) θ 1 v T 1 (θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ) v T 1(θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ) 1 F (θ 1 ) θ 1 v T 1 (θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ) v T 1(θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ) 1 F (θ 1 ) θ 1 q T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) q T (θ 1, ξ 1, ξ, ξ +1,..., ξ T ) This implies ha q T 1 (θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ) q T 1 (θ 1, ξ 1, ξ, ξ +1,..., ξ T 1 ). This argumen can be furher induced backwards o show ha he monooniciy condiions are saisfied by all q. We are now in a posiion o derive prices so ha he resuling conrac is incenive compaible. From he definiion of buyer uiliies we can inducively derive he ransfers. In oher words, we use he following equaions and U T (θ 1, ξ T ) = v T (θ 1, ξ T )q T (θ 1, ξ T ) p T (θ 1, ξ T ) U (θ 1, ξ ) = v (θ 1, ξ )q (θ 1, ξ ) p (θ 1, ξ ) + δq (θ 1, ξ ) U +1 (θ 1, ξ, ξ +1 )dg(ξ +1 ) Ξ +1 We firs observe ha if he buyer repored ruhfully up o period T i is immediae ha she will repor ξ T ruhfully as well. This is because he envelope condiion for he final period is necessary and sufficien for incenive compaibiliy (see Myerson (1981)). However, as we have poined ou, for any < T, he envelope condiion is merely necessary for incenive compaibiliy. We proceed inducively by choosing an arbirary < T and assuming ha he conrac is incenive compaible for periods s >. This implies ha if he buyer repors ruhfully ill period, hen he bes she can do is o repor ruhfully for any subsequen period. Now we show ha i is opimal for he buyer o repor ruhfully a period having repored ruhfully ill period 1. If insead, and and

19 she repors ˆξ ξ in period, she ges OPTIMAL CONTRACTING OF NEW EXPERIENCE GOODS 19 Û (θ 1, ξ 1, ξ ; θ 1, ξ 1, ˆξ ) = v (θ 1, ξ )q (θ 1, ξ 1, ˆξ ) p (θ 1, ˆξ ) + δq (θ 1, ξ 1, ˆξ ) Û +1 (θ 1, ξ, ξ +1 ; θ, ξ 1, ˆξ )dg(ξ +1 ) Ξ +1 Now noe ha Û (θ 1, ξ 1, ξ ; θ 1, ξ 1, ˆξ ) = U (θ 1, ξ 1, ˆξ ) + [v (θ 1, ξ ) v (θ 1, ˆξ )q (θ 1, ξ 1, ˆξ ) + [ δq (θ 1, ξ 1, ˆξ ) Û +1 (θ 1, ξ, ξ +1 ; θ, ξ 1, ˆξ ) U +1 (θ 1, ξ 1, ˆξ, ξ +1 ) dg(ξ +1 ) Ξ +1 Hence we can rewrie he incenive consrain as (14) U (θ 1, ξ 1, ξ ) U (θ 1, ξ 1, ˆξ ) ξ [ v (θ 1, ξ 1, ξ ) ˆξ T ξ q (θ 1, ξ 1, ˆξ ) d ξ + δq (θ 1, ξ 1, ˆξ ξ [ Û+1(θ 1, ξ 1, ) ξ, ξ +1 ; θ, ξ 1, ˆξ ) Ξ +1 ξ dg(ξ +1 ) d ξ ˆξ T Taking he difference beween uiliies and using he envelope heorem a period yields (15) U (θ 1, ξ 1, ξ ) U (θ 1, ξ 1, ˆξ ) = ξ v (θ 1, ξ 1, ξ ) ˆξ ξ q (θ 1, ξ 1, ξ ) + [ q (θ 1, ξ 1, ξ )... δ v +1(θ 1, ξ 1, ξ, ξ +1 ) Ξ +1 ξ q +1 (θ 1, ξ 1, ξ, ξ +1 ) + + Ξ T δ T v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T )( ξ T s=+1 Similarly he envelope heorem a period + 1 yields (16) Û+1(θ 1, ξ 1, ξ, ξ +1 ; θ 1, ξ 1, ˆξ ) ξ = q s (θ 1, ξ 1, ξ,..., ξ s ) ) } dg(ξ +1 )... dg(ξ T ) d ξ v +1 (θ 1, ξ 1, ξ, ξ +1 ) ξ q +1 (θ 1, ξ 1, ˆξ, ˆξ +1( ξ )) + δq +1 (θ 1, ξ 1, ˆξ, ˆξ +1( ξ Û+(θ 1, )) ξ 1, ξ, ξ +1, ξ +; θ 1, ξ 1, ˆξ, ˆξ +1 ( ξ )) Ξ + ξ dg(ξ + ) where ˆξ +1 ( ξ ) is he opimal repor in period + 1 given ha she repored ˆξ in period. ˆξ +1 ( ξ ) is he implici soluion o he equaion v +1 (θ 1, ξ 1, ˆξ, ˆξ +1 ) = v +1(θ 1, ξ 1, ξ, ξ +1 ). The reason for his is he following. Assume ha he rue signals observed by he buyer up o period + 1 are θ 1, ξ 1, ˆξ, ˆξ +1 ( ξ ). Then if he buyer repors ruhfully up o period i mus be opimal for he buyer o ruhfully repor ˆξ +1 ( ξ ) in period +1 by he inducive hypohesis. Bu from he definiion of ˆξ +1 ( ξ ), he buyer s valuaion in period +1 is v +1 (θ 1, ξ 1, ˆξ, ˆξ +1 ( ξ )) = v +1 (θ 1, ξ 1, ξ, ξ +1 ).

20 0 RAHUL DEB Hence, when he buyer has his value i is opimal for her o repor ˆξ +1 ( ξ ) when she misrepors ˆξ in period. We can now plug (16) ino he righ hand side of (14) and we ge ξ [ v (θ 1, ξ 1, ξ ) ˆξ T ξ q (θ 1, ξ 1, ˆξ ) + q (θ 1, ξ 1, ˆξ )... δ v +1(θ 1, ξ 1, ξ, ξ +1 ) Ξ +1 ξ q +1 (θ 1, ξ 1, ˆξ, ˆξ +1( ξ )) + Ξ T δ v +(θ 1, ξ 1, ξ, ξ +1, ξ + ) ξ q +1 (θ 1, ξ 1, ˆξ, ˆξ +1( ξ ))q + (θ 1, ξ 1, ˆξ, ˆξ +1( ξ ), ξ + ) + + δ T v T (θ 1, ξ 1, ξ, ξ +1,..., ξ T )( ξ T s=+1 q s (θ 1, ξ 1, ˆξ, ˆξ +1( ξ ),..., ξ s ) )} dg(ξ +1 )... dg(ξ T ) d ξ We can compare he above expression wih (15). The monooniciy propery ensures ha he incenive compaibiliy condiion (14) is saisfied and his complees he proof. References [1 S. Ahey and I. Segal (007), An Efficien Dynamic Mechanism. Working Paper. [ D. Baron and D. Besanko (1984), Regulaion and Informaion in a Coninuing Relaionship. Informaion Economics and Policy. [3 M. Baaglini (005), Long-erm Conracing wih Markovian Consumers. American Economic Review, 95, [4 D. Bergemann and J. Valimaki (006), Bandi Problems. Cowles Foundaion Discussion Paper. [5 D. Bergemann and J. Valimaki (008), The Dynamic Pivo Mechanism. Cowles Foundaion Discussion Paper. [6 T. Borgers, I. Cox, M. Pesendorfer, and V. Pericek (007), Equilibrium Bids in Sponsored Search Aucions: Theory and Evidence. Working Paper. [7 P. Coury and H. Li (000), Sequenial Screening. Review of Economic Sudies. [8 B. Edelman, M. Osrovsky, and M. Schwarz (007), Inerne Adverising and he Generalized Second-price Aucion: Selling Billions of Dollars Worh of Keywords. American Economic Review, 97, [9 P. Eso and B. Szenes (007), Opimal Informaion Disclosure in Aucions and he Handicap Aucion. Review of Economic Sudies, 74, [10 R. Myerson (1981), Opimal Aucion Design. Mahemaics of Operaions Research. [11 H. Nazerzadeh, A. Saberi, and R. Vohra (007), Dynamic Cos-per-acion Mechanisms and Applicaions o Online Adverising. Working Paper. [1 A. Pavan (007), Long-erm Conracing in a Changing World. Working Paper. [13 A. Pavan, I. Segal, and J. Toikka (008), Dynamic Mechanism Design: Revenue Equivalence, Profi Maximizaion, and Informaion Disclosure. Working Paper. [14 H. Varian (007), Posiion Aucions. Inernaional Journal of Indusrial Organizaion, 5,