Dynamic Contracting: An Irrelevance Result

Transcription

1 Dynamic Conracing: An Irrelevance Resul Péer Eső and Balázs Szenes Sepember 5, 2013 Absrac his paper considers a general, dynamic conracing problem wih adverse selecion and moral hazard, in which he agen s ype sochasically evolves over ime. he agen s final payoff depends on he enire hisory of privae and public informaion, conracible decisions and he agen s hidden acions, and i is linear in he ransfer beween her and he principal. We ransform he model ino an equivalen one where he agen s subsequen informaion is independen in each period. Our main resul is ha for any fixed decision-acion rule implemened by a mechanism, he maximal expeced revenue ha he principal can obain is he same as if he principal could observe he agen s orhogonalized ypes afer he iniial period. In his sense, he dynamic naure of he relaionship is irrelevan: heagenonlyreceivesinformaionrensforheriniialprivae informaion. We also show ha any monoonic decision-acion rule can be implemened in a Markov environmen saisfying cerain regulariy condiions. Keywords: asymmeric informaion, dynamic conracing, mechanism design 1 Inroducion We analyze muliperiod principal-agen problems wih adverse selecion and moral hazard. he principal s per-period decisions and he moneary ransfers are governed by a conrac signed a he beginning of he relaionship, in he presence of some iniial informaional asymmery, and he agen s privae informaion sochasically evolves over ime. he agen s final payoff can depend, quie generally, on he enire hisory of privae and public informaion, conracible decisions and he agen s hidden acions, and i is linear in he ransfer beween her and he principal. We discuss he wide-ranging applicaions of such models in micro- and macroeconomic modeling below. Deparmen of Economics, Oxford Universiy, Oxford, UK. [email protected]. Deparmen of Economics, London School of Economics, London, UK. [email protected]. 1

2 Our main resul is an irrelevance heorem: In a mechanism implemening a given aciondecision rule, he maximal expeced revenue ha he principal can obain (and his maximal payoff if i is linear in he ransfers) is he same as if he could conrac on whaever new (orhogonal) informaion is observed by he agen in any fuure period. Noe ha in he hypoheical benchmark case where he agen s fuure orhogonalized ypes are observable and conracible he paries need no inerac beyond he iniial period, and he agen has no access o dynamic deviaion reporing sraegies. In his sense he dynamic naure of he adverse selecion problem is irrelevan. his irrelevance resul holds in a rich environmen, wih very lile assumed abou he agen s uiliy funcion (no single-crossing or monooniciy assumpions are made), he informaion srucure, and so on. We also show ha monoonic decision rules can be implemened in Markov environmens wih ime-separable payoffs, subjec o addiional regulariy condiions. he regulariy assumpions include familiar single-crossing condiions for he agen s uiliy funcion, and also assumpions concerning he availabiliy of a conracible public signal abou he agen s acion and ype. If he signal is informaive (however imperfecly) abou a summary saisic of he agen s hidden acion and ype, and is disribuion is generic, hen any monoonic decision rule coupled wih any monoonic acion rule is approximaely implemenable. 1 If he conracible signal is uninformaive abou he agen s acion (bu he oher regulariy condiions hold), hen monoonic decision rules coupled wih agen-opimal acions can be implemened. he significance of he implemenaion resuls is ha when hey apply, he dynamic conracing problem can indeed be reaed as a saic one and solved as follows. Consider he benchmark case in which he agen s only privae informaion is her iniial ype, and he principal can observe her orhogonalized fuure ypes. Solve his relaxed case, eiher by opimizing he acion rule as well or aking i o be he agen-opimal one, depending on wheher or no a public summary signal abou he agen s ype and acion is available. If he resuling rule is monoonic in he agen s ype profile hen i can be implemened in he original problem wih he same expeced paymens as in he benchmark, hence i is opimal in he original problem as well. 2 I is imporan o noe ha despie he validiy of his soluion mehod he original and he benchmark problems are no equivalen: he monooniciy requiremen on he decision rule is more sringen, hence he se of implemenable decision rules is smaller, in he benchmark. However, under he regulariy condiions he opimal soluion in he relaxed (benchmark) problem is implemenable in he more resricive original problem. Models in he class of dynamic conracing problems ha we analyze can be, and indeed 1 he genericiy condiion and he noion of approximae implemenabiliy will be precisely defined in Secion 4. 2 However, Baaglini and Lamba (2012) poin ou ha he regulariy condiions guaraneeing he monooniciy of he poinwise-opimal decision rule are quie srong. 2

3 have been, applied o a wide range of economic problems. 3 he roos of his lieraure reach back o Baron and Besanko (1984) who used a muli-period screening model o address he issue of regulaing a monopoly over ime. Coury and Li (2000) sudied opimal advance icke sales, Eso and Szenes (2007a) he opimal disclosure of privae informaion in aucions, Eso and Szenes (2007b) he sale of advice as an experience good. Farhi and Werning (2012), Golosov, roshkin and sivinsky (2011) and Kapička (2013) apply a similar approach o opimal axaion and fiscal policy design, respecively. Pavan, Segal and oikka (2012) apply heir (o dae, mos general) resuls on he muli-period pure adverse selecion problem o he aucion of experience goods (bandi aucions). Garre and Pavan (2012) use a dynamic conracing model wih boh adverse selecion and moral hazard o sudy opimal CEO compensaion. Such mixed, hidden acion hidden informaion models could also be applied in insurance problems. In his paper we also develop hree applicaions in order o illusrae our echniques and new resuls. he firs wo examples are dynamic monopoly problems in which he buyer s valuaion for he good (her ype) sochasically evolves over ime. In he second example he valuaion also depends on he buyer s hidden, cosly acion: e.g., she may privaely inves in learning how o beer enjoy he good. he monopolis canno observe any signal abou he buyer s ype and acion; all he can do is o offer a dynamic screening conrac. We derive he opimal conrac and show ha our dynamic irrelevance heorem holds: all disorions are due o he buyer s iniial privae informaion. he hird applicaion is a dynamic principal-agen problem wih adverse selecion and moral hazard, where he principal is an invesor and he agen an invesmen advisor. he conracible decision is he amoun of money invesed by he principal wih he agen. he agen s ype is her abiliy o generae higher expeced reurns, whereas her cosly acion is aimed a picking socks ha conform wih he principal s oher (e.g., ehical) consideraions. Here he principal (invesor) observes a summary signal abou he agen s (advisor s) ype and acion, in he form of he principal s flow payoff. We fully solve his problem as well and show ha he dynamic irrelevance heorem applies. In order o formulae he main, irrelevance resul of he paper we rely on an idea inroduced in our previous work (Eso and Szenes (2007a)): We ransform he model ino an equivalen orhogonal represenaion, in which he agen s privae informaion in each period is independen of ha obained in earlier periods. he irrelevance heorem obains by showing ha in he original problem (where he agen s orhogonalized fuure ypes and acions are no observable), in any incenive compaible mechanism, he agen s expeced payoff condiional on her iniial ype are fully deermined by her on-pah (in he fuure, ruhful) behavior. 3 Our review of applicaions is deliberaely incomplee; for a more in-deph survey his lieraure see Krähmer and Srausz (2012) or Pavan, Segal and oikka (2012). 3

4 herefore, he agen s expeced payoff (and paymens) coincide wih hose in he benchmark case, where he orhogonalized fuure ypes are publicly observable. he resuls on he implemenabiliy of monoonic decision rules are obained in Markovian environmens subjec o addiional regulariy condiions. Here, he key sep in he derivaion (also used in Eso and Szenes (2007a) in a simpler model) is o show ha if he agen is unruhful in a given period in an oherwise incenive compaible mechanism, she immediaely undoes her lie in he following period o make he principal s inference regarding her ype correc in all fuure periods. he explici characerizaion of ou-of-equilibrium behavior in regular, Markovian environmens enables us o pin down he ransfers ha implemen a given monoonic decision rule in a model wih adverse selecion. he resuls for models wih boh moral hazard and adverse selecion are obained by appropriaely reducing he general model o ones wih only adverse selecion, he exac way depending on he assumpions made regarding he observabiliy of a public signal on he agen s ype and acion. he echnical conribuions nowihsanding, we believe he mos imporan message of he paper is he dynamic irrelevance resul. he insigh ha he principal need no pay his agen rens for pos-conracual hidden informaion in a dynamic adverse selecion problem has been expressed in previous work (going back o Baron and Besanko (1984)). Our paper highlighs boh he deph and he limiaions of his insigh: Indeed he principal ha conracs he agen prior o her discovery of new informaion can limi he agen s rens o he same level as if he could observe he agen s orhogonalized fuure ypes; however, we also poin ou ha he wo problems are no equivalen. he paper is organized as follows. In Secion 2 we inroduce he model and describe he orhogonal ransformaion of he agen s informaion. In Secion 3 we derive necessary condiions of he implemenabiliy of a decision rule and our main, dynamic irrelevance resul. Secion 4 presens sufficien condiions for implemenaion in Markov environmens. Secion 5 presens he applicaions; Secion 6 concludes. Omied proofs are in he Appendix. 2 Model Environmen. here is a single principal and a single agen. ime is discree, indexed by =0, 1,...,. he agen s privae informaion in period is θ Θ, where Θ = θ, θ R. In period, he agen akes acion a A which is no observed by he principal. he se A is an open inerval of R. hen a conracible public signal is drawn, s S R. Afer he public signal is observed in period, a conracible decision is made, denoed by x X R n, which is observed by boh paries. Since x is conracible, i does no maer wheher i is aken by he agen or by he principal. he conrac beween he principal and he agen is 4

5 signed a =0, righ afer he agen has learned her iniial ype, θ 0. We denoe he hisory of a variable hrough period by superscrip ; for example x = (x 0,...,x ), and x 1 = { }. he random variable θ is disribued according o a c.d.f. G θ 1,a 1,x 1 suppored on Θ. he funcion G is coninuously differeniable in all of is argumen, and he densiy is denoed by g θ 1,a 1,x 1. he public signal s is disribued according o a coninuous c.d.f. H ( f (θ,a )), where f : Θ A R is coninuously differeniable. We may inerpre s as an imperfec public summary signal abou he agen s curren ype and acion; for example, in Applicaion 3 of Secion 5 i will be s = θ + a + ξ, where ξ is noise wih a known disribuion. In he general model we assume ha for all θ, θ and a here is a unique a such ha f (θ,a )=f ( θ, a ). 4 he agen s payoff is quasilinear in money, and is defined by u θ,a,s,x p, where p R denoes he agen s paymen o he principal, and u :Θ A S X R is coninuously differeniable in θ and a for all =0,...,. We do no specify he principal s payoff. In some applicaions (e.g., where he principal is a monopoly and he agen is cusomer) i could be he paymen iself, in ohers (e.g., where he principal is a social planner and he agen he represenaive consumer) i could be he agen s expeced payoff; in ye oher applicaions i could be somehing differen. A noaional convenion: We denoe parial derivaives wih a subscrip referring o he variable of differeniaion, e.g., ũ θ ũ/ θ, f θ f / θ, ec. Orhogonalizaion of Informaion. he model can be ransformed ino an equivalen one where he agen s privae informaion is represened by serially independen random variables. Suppose ha a each =0,...,, he agen observes ε = G θ θ 1,a 1,x 1 insead of θ. Clearly, ε can be inferred from θ,a 1,x 1. Conversely, θ can be compued from (ε,a 1,x 1 ), ha is, for all =0,..., here is ψ :[0, 1] A 1 X 1 Θ such ha ε = G ψ (ε,a 1,x 1 ) ψ 1 (ε 1,a 2,x 2 ),a 1,x 1, (1) where ψ (ε,a 1,x 1 ) denoes ψ 0 (ε 0 ),...,ψ (ε,a 1,x 1 ). In oher words, if he agen observes ε,a 1,x 1 a ime in he orhogonalized model, she can infer he ype hisory ψ (ε,a 1,x 1 ) in he original model. Of course, a model where he agen observes ε for all is sraegically equivalen o he one where she observes θ for all (provided ha in boh cases she observes x 1 and 4 his assumpion ensures ha he principal canno resolve he adverse selecion problem by requiring he agen o ake a cerain acion and using he public signal o deec he agen s ype. 5

6 recalls a 1 a ). By definiion, ε is uniformly disribued on he uni inerval 5 for all and all realizaions of θ 1, a 1 and x 1, hence he random variables {ε } 0 are independen across ime. here are many oher orhogonalized informaion srucures (e.g., ones obained by sricly monoonic ransformaions). In wha follows, o simplify noaion, we fix he orhogonalized informaion srucure as he one where ε is uniform on E =[0, 1]. he agen s gross payoff in he orhogonalized model, u : E A S X R, becomes u ε,a,s,x = u ψ ε,a 1,x 1,a,s,x. Revelaion Principle. A deerminisic mechanism is a four-uple Z, x, a,p, where Z is he agen s message space a ime, x : Z S X is he conracible decision rule a ime, a : Z S 1 A is a recommended acion a, andp : Z S R is he paymen rule. he agen s reporing sraegy a is a mapping from previous repors and informaion o a message. We refer o a sraegy ha maximizes he agen s payoff as an equilibrium sraegy and he payoff generaed by such a sraegy as equilibrium payoff. he sandard Revelaion Principle applies in his seing, so i is wihou loss of generaliy o assume ha Z = E for all, ando resric aenion o mechanisms where elling he ruh and aking he recommended acion (obedience) is an equilibrium sraegy. A direc mechanism is defined by a riple x, a, p, where x : E S X, a : E S 1 A and p : E S R. Direc mechanisms in which elling he ruh and obeying he principal s recommendaion is an equilibrium sraegy are called incenive compaible mechanisms. We call a decision-acion rule x, a implemenable if here exiss a paymen rule, p : E R such ha he direc mechanism x, a, p is incenive compaible. echnical Assumpions. We make hree echnical assumpions o ensure ha he equilibrium payoff funcion of he agen is Lipshiz coninuous in he orhogonalized model. Assumpion 0. (i) here exiss a K N such ha for all =1,..., and for all θ,a,s,x, u θ θ,a,s,x, u a θ,a,s,x <K. (ii) here exiss a K N such ha for all =1,...,, τ<, and for all θ,a 1,x 1, G θ θ θ 1,a 1,x 1, G θτ θ θ 1,a 1,x 1 <K. 5 o see his, noe ha since ε = G `θ θ 1,a 1,x 1, he probabiliy ha ε ε is Pr `G `θ θ 1,a 1,x 1 ε =Pr`θ G 1 ` ε θ 1,a 1,x 1 = G `G 1 ` ε θ 1,a 1,x 1 = ε. 6

7 (iii) here exiss a K N such ha for all =1,..., and for all θ,a, f θ (θ,a ) f a (θ,a ) <K. 3 he main resul We refer o he model in which he principal never observes he agen s ypes as he original model, whereas we call he model where ε 1,...,ε are observed by he principal he benchmark case. he conracing problem in he benchmark is a saic one in he sense ha he principal only ineracs wih he agen a =0, and he agen has no access o dynamic deviaion reporing sraegies. Our dynamic irrelevance resul is ha in any mechanism ha implemens a given decision-acion rule in he original model he principal s maximal expeced revenue is he same as i would be he benchmark case. Specifically, wha we show below is ha he expeced ransfer paymen of an agen wih a given iniial ype when he principal implemens decision-acion rule x, a in he original problem is he same (up o a ype-invarian consan) as i would be in he benchmark. his implies ha he principal s maximal expeced revenue (and his payoff, in case i is linear in he revenue) when implemening a decision-acion rule in he original problem is jus as high as i would be in he benchmark. his does no imply, however, ha he wo problems are equivalen: sufficien condiions of implemenabiliy (of a decision rule) are sronger in he original problem han hey are in he benchmark. We will urn o he quesion of implemenabiliy in Secion 4. In he nex subsecion we consider a decision-acion rule x, a and derive a necesarry condiion for he paymen rule p such ha x, a, p is incenive compaible. his condiion urns ou o be he same in he benchmark case and in he original model. We hen use his condiion o prove our main, irrelevance resul. 3.1 Paymen rules We fix an incenive compaible mechanism x, a, p and analyze he consequences of ime-0 incenive compaibiliy on he paymen rule, p, in boh he original model and he benchmark. We consider a paricular se of deviaion sraegies and explore he consequence of he non-profiabiliy of hese deviaions in each case. o his end, le us define his se as follows: If he agen wih iniial ype ε 0 repors ε 0 hen (i) she mus repor ε 1,...,ε ruhfully, and (ii) for all =0,...,, afer hisory ε,s 1, she mus ake acion a (ε, ε 0,s 1 ) such ha he disribuion of s is he same as if he hisory were ε 0,ε 0,s 1 and acion a (ε 0,ε 0,s 1 ) were aken, where ε 0 =(ε 1,...,ε ). Since he disribuion of s only depends on f (θ,a ), 7

8 he acion a (ε, ε 0,s 1 ) is defined by where f θ, a (ε 0,ε 0,s 1 ) = f θ, a (ε, ε 0,s 1 ), (2) θ = ψ ε0,ε 0, a 1 ε 0,ε 1 0,s 2, x 1 ε 0,ε 1 0,s 1, θ = ψ ε, a 1 ε 1, ε 0,s 2, x 1 ε 0,ε 1 0,s 1. In oher words, he deviaion sraegies we consider require he agen (i) o be ruhful in he fuure abou her orhogonalized ypes, and (ii) o ake acions ha mask her earlier lie so ha he principal could no deec her iniial deviaion based on he public signals, even in a saisical sense. 6 Noe ha in he benchmark case we only need o impose resricion (ii) since he principal observes ε 1,...,ε by assumpion. Also noe ha he sraegies saisfying resricions (i) and (ii) include he equilibrium sraegy in he original model because if ε 0 = ε 0 he wo resricions imply ruh-elling and obedience (adherence o he acion rule). We emphasize ha we do no claim by any means ha afer reporing ε 0 i is opimal for he agen o follow a coninuaion sraegy defined by resricions (i) and (ii). Neverheless, since he mechanism x, a, p is incenive compaible, none of hese deviaions are profiable for he agen. We show ha his observaion enables us o characerize he expeced paymen of he agen condiional on ε 0 up o a ype-invarian consan. Le Π 0 (ε 0 ) denoe he agen s expeced equilibrium payoff condiional on her iniial ype ε 0 in he incenive compaible mechanism x, a, p. ha is, Π 0 (ε 0 )=E u ε, a ε,s 1,s, x ε,s p ε ε 0, (3) where E denoes expecaion over ε and s. Proposiion 1 If he mechanism x, a, p is incenive compaible eiher in he original model or in he benchmark case, hen for all ε 0 E 0 : ˆ ε0 Π 0 (ε 0 )=Π 0 (0) + E u ε0 y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s dy ε 0 (4) 0 ˆ ε0 +E u a y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s a ε0 y, ε 0,y,s 1 dy ε 0, 0 =0 6 Similar ideas are used by Pavan, Segal and oikka (2012) in a dynamic conracing model wihou moral hazard and by Garre and Pavan (2012) in a more resricive environmen wih moral hazard. 8

9 where y, ε 0 =(y, ε1,...,ε ). Proposiion 1 esablishes ha in an incenive compaible mechanism ha implemens a paricular decision-acion rule he expeced payoff of he agen wih a given (iniial) ype does no depend on he ransfers. Analogous o he necessiy par of he Spence Mirrlees Lemma in saic mechanism design (or Myerson s Revenue Equivalence heorem), necessary condiions similar o (4) have been derived in dynamic environmens by Baron and Besanko (1984), Coury and Li (2000), Eso and Szenes (2007a), Pavan, Segal and oikka (2012), Garre and Pavan (2012), and ohers. In our environmen, which is no only dynamic bu incorporaes boh adverse selecion and moral hazard as well, he real significance of he resul is ha he same formula applies in he original problem and in he benchmark case. I may be insrucive o consider he special case where he principal has no access o a public signal, or equivalenly, he disribuion of s is independen of (θ,a ). Since he choice of a has no impac on x and p, he agen chooses a o maximize her uiliy. A necessary condiion of his maximizaion is E u a ε, a (ε,s 1 ),s, x (ε,s ) ε,s 1 =0for all. As a consequence he las erm of Π 0 (ε 0 ), i.e. he second line of equaion (4), vanishes. Proof of Proposiion 1 Firs we express he agen s reporing problem a =0in he benchmark case as well as in he original problem subjec o resricions (i) and (ii) discussed a he beginning of his subsecion. In order o do his define U (ε 0, ε 0 )=E u ε, a ε, ε 0,s 1,s, x ε 0,ε 0,s ε 0 and P (ε 0 )=E p ε 0,ε 0,s ε 0,a = a ε, ε 0,s 1,x = x ε 0,ε 0,s, (5) where a is defined by (2). Recall ha he acion a ε, ε 0,s 1 generaes he same disribuion of s as if he agen s rue ype hisory was ε 0,ε 0 and he agen had aken a ε0,ε 0,s 1. he significance of his is ha E p ε 0,ε 0,s ε 0,a = a ε, ε 0,s 1,x = x ε 0,ε 0,s = E p ε 0,ε 0,s ε 0,a = a ε 0,ε 0,s 1,x = x ε 0,ε 0,s, so he righ-hand side of (5) is indeed only a funcion of ε 0 bu no ha of ε 0. In he benchmark case, he payoff of he agen wih ε 0 who repors ε 0 and akes acion a ε, ε 0,s 1 a every is W (ε 0, ε 0 ) = U (ε 0, ε 0 ) P (ε 0 ).NoehaW (ε 0, ε 0 ) is also 9

10 he payoff of he agen in he original model if her ype is ε 0 a =0, repors ε 0 and her coninuaion sraegy is defined by resricions (i) and (ii) above, ha is, she repors ruhfully afererwards and akes acion a ε, ε 0,s 1 afer he hisory ε,s 1. he incenive compaibiliy of x, a, p implies ha ε 0 arg max bε0 E 0 W (ε 0, ε 0 ) boh in he benchmark case and in he original model. In addiion, Π 0 (ε 0 )=W (ε 0,ε 0 ) and, by Lemma 6 of he Appendix, Π 0 is Lipshiz coninuous. herefore, heorem 1 in Milgrom and Segal (2002) implies ha almos everywhere. Noe ha =0 dπ 0 (ε 0 ) = U (ε 0, ε 0 ), dε 0 ε 0 bε 0 =ε 0 U (ε 0, ε 0 ) = E u ε0 ε, a ε,s 1,s, x ε,s ε 0 ε 0 bε 0 =ε 0 +E u a ε, a ε,s 1,s, x ε,s a ε0 ε, ε 0,s 1 bε 0 =ε 0 ε 0. Since Π 0 is Lipshiz coninuous, i can be recovered from is derivaive, so he saemen of he proposiion follows. By Proposiion 1, for a given decision-acion rule, incenive compaibiliy consrains pin down he expeced paymens condiional on ε 0 uniquely up o a consan in boh he benchmark case and he original model. paymen condiional on ε 0 can be expressed as o see his, noe ha from (3) and (4) he expeced E p ε,s ε 0, a, x = E u ε, a ε,s 1,s, x ε ε 0 Π0 (0) ˆ ε0 E u ε0 y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s dy ε 0 0 ˆ ε0 E u a y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s a ε0 y, ε 0,y,s 1 ε 0. 0 =0 An immediae consequence of his observaion and Proposiion 1 is he following Remark 1 Suppose ha x, a, p and x, a, p are incenive compaible mechanisms in he original and in he benchmark case, respecively. hen E p ε,s ε 0, a, x E p ε,s ε 0, a, x = c, where c R. 10

11 3.2 Irrelevance of dynamic adverse selecion Now we show ha he principal s maximal revenue of implemening a decision rule is he same as if he were able o observe he orhogonalized ypes of he agen afer =0. ha is, whenever a decision rule is implemenable, he agen only receives informaion ren for her iniial privae informaion. his is our irrelevance resul. o sae his resul formally, suppose ha he agen has an ouside opion, which we normalize o be zero. his means ha any mechanism mus saisfy Π 0 (ε 0 ) 0 (6) for all ε 0 E 0. We call he maximum (supremum) of he expeced paymen of he agen across all he mechanisms ha implemen x, a and saisfy (6) he principal s maximal revenue from implemening his rule. 7 heorem 1 Suppose ha he decision rule x, a is implemenable in he original model. hen he principal s maximal revenue from implemening his rule is he same as in he benchmark case. Proof. Consider firs he benchmark case where he principal observes ε 1,...,ε and le p denoe he paymen rule in a revenue-maximizing mechanism. hen he revenue of implemening x, a is jus E p ε,s a, x. Of course, he principal s revenue in he benchmark case is an upper bound on his revenue in he original model. herefore, i is enough o show ha he principal can achieve E p ε,s a, x from implemening x even if he he does no observe ε 1,...,ε.Suppose ha he direc mechanism x, a, p is incenive compaible. hen, by Remark 1, E p ε,s ε0, a, x = E p ε,s ε0, a, x + c for some c R. Define p ε,s o be p ε,s c. Since adding a consan has no effec on incenives, he mechanism x, a, p is incenive compaible. In addiion, E p ε,s ε 0, a, x = E p ε,s ε 0, a, x, ha is, he principal s revenue is he same as in he benchmark case. Finally, noice ha he paricipaion consrain of he agen, (6), is also saisfied because 7 Requiring (6) for all ε 0 implies ha we resric aenion o mechanisms where he agen paricipaes irrespecive of her ype. his is wihou he loss of generaliy in many applicaions where here is a decision which generaes a uiliy of zero for boh he principal and he agen. Alernaively, we could have saed our heorem for problems where he paricipaing ypes in he opimal conrac of he benchmark case is an inerval. 11

12 he agen s expeced payoff condiional on her iniial ype, ε 0,ishesameashainhe benchmark case. he saemen of heorem 1 is abou he revenue of he principal. Noe ha if he payoff of he principal is also quasi-linear (affine in he paymen), hen he decision rule and he expeced paymen fully deermines his payoff. Hence, a consequence of heorem 1 is, Remark 2 Suppose ha he decision-acion rule x, a is implemenable in he original model and he principal s payoff is affine in he paymen. hen he principal s maximum (supremum) payoff from implemening x, a is he same as in he benchmark case. I is imporan o poin ou ha our dynamic irrelevance resul does no imply ha he original problem (unobservable ε 1,...,ε ) and he benchmark case (observable ε 1,...,ε ) are equivalen. heorem 1 only saes ha if an decision-acion rule is implemenable in he original model, hen i can be done so wihou revenue loss as compared o he benchmark case. his resul was obained under very mild condiions regarding he sochasic process governing he agen s ype, her payoff funcion and he srucure of he public signals. he obvious, nex quesion is wha ype of decision-acion rules can be implemened (under wha condiions) in he original problem. We find some answers o his quesion in he nex secion. 4 Implemenaion his secion esablishes resuls regarding he implemenabiliy of cerain decision rules. 8 We resric aenion o a Markov environmen wih ime-separable, regular (monoonic and single-crossing) payoff funcions, formally saed in Assumpions 1 and 2 below. Firs, we show ha in he pure adverse selecion model (where here are neiher unobservable acions nor public signals) any monoonic decision rule is implemenable. hen we urn our aenion o he general model wih moral hazard. here he se of implemenable decision rules depends on he informaion conen of he public signal. If he public signal has no informaional conen, ha is, he disribuion of s is independen of f (θ,a ), hen naurally he agen canno be given incenives o choose any acion oher han he one ha maximizes her flow uiliy in each period. In his case, we show ha any decision-acion rule can be implemened if x is monoonic and a is deermined by he agen s per period maximizaion problem. he mos ineresing (and permissive) implemenaion resul is obained in he general model wih adverse selecion and moral hazard in case he signal is informaive and is dis- 8 hroughou his secion we require a ype-invarian paricipaion consrain for he agen wih her ouside opion normalized o zero payoff, ha is, (6) o hold. 12

13 ribuion saisfies a genericiy condiion due o McAfee and Reny (1992). his condiion requires ha he disribuion of s condiional on any given y = f (θ,a ) is no he average of signal disribuions condiional on oher y = y s such ha y = f ( θ, a ). In his case, we show ha any monoonic decision-acion rule x, a can be approximaely implemened (o be formally defined below). 9 he resul is based on argumens similar o he Full Surplus Exracion heorem of McAfee and Reny (1992) and explois he propery of he model ha f is approximaely conracible and he agen is risk neural wih respec o moneary ransfers. he main resul of his secion is ha in our general model, in a regular Markovian environmen wih ransferable uiliy and generic public signals, he principal is able o implemen any monoonic decision-acion rule while no incurring any agency cos apar from he informaion ren due o he agen s iniial privae informaion. In order o sae he regulariy assumpions made hroughou he secion, we reurn o he model wihou orhogonalizaion. hroughou his secion, we assume ha he public signal does no direcly affec he agen s payoff direcly and we remove s from he argumens of u, ha is, u :Θ A X R. We make wo ses of assumpions regarding he environmen. he firs se concerns he ype disribuion, he second one he agen s payoff funcion. Assumpion 1. (ype Disribuion) (i) For all {0,...,}, herandomvariableθ is disribued according o a coninuous c.d.f. G ( θ 1 ) suppored on an inerval Θ = θ, θ. (ii) For all {1,...,}, G ( θ 1 ) G ( θ 1 ) whenever θ 1 θ 1. Par (i) of Assumpion 1 saes ha he agen s ype follows a Markov process, ha is, he ype disribuion a ime only depends on he ype a 1. Inaddiion,hesupporofθ only depends on, soanyypeonθ can be realized irrespecive of θ 1. Par (ii) saes ha he ype disribuions a ime are ordered according o firs-order socahasic dominance. he larger he agen s ype a ime 1, he more likely i is o be large a ime. Assumpion 2. (Payoff Funcion) (i) here exis {u } =0, u :Θ A X R coninuously differeniable, such ha u θ,a,x = u θ,a,x. =0 (ii) For all {0,...,}, u is sricly increasing in θ. (iii) For all {0,...,}, θ Θ,a A : u θ θ,a,x u θ θ,a, x whenever x x. 9 he approximaion can be dispensed wih if he public signal is he summary saisic f (θ,a ) iself. 13

14 Par (i) of Assumpion 2 says ha he agen s uiliy is addiively separable over ime, such ha her flow uiliy a ime only depends on θ and a (and no on any prior informaion and acion) besides all decisions aken a or before. Par (ii) requires he flow uiliy o be monoonic in he agen s ype. Par (iii) is he sandard single-crossing propery for he agen s ype and he conracible decision. We refer o he model as he one wih pure adverse selecion if u a 0 for all and he disribuion of s is independen of f. Nex, we sae our implemenaion resul for his case (Proposiion 2). hen, in Secions 4.1 and 4.2 we reurn o he general model wih moral hazard. In boh scenarios regarding he informaional conen of he public signal discussed above we reduce he problem of implemenaion o ha in an appropriaely-defined pure adverse selecion problem. Proposiion 2 Suppose ha Assumpions 0,1 and 2 hold in a pure adverse selecion model. hen a decision rule, x, x :Θ X,isimplemenableifx is increasing for all. By Corollary 2 of Pavan, Segal and oikka (2012), Assumpions 1-2 imply heir inegral monooniciy condiion; sligh differences in heir and our echnical assumpions nowihsanding, our Proposiion 2 appears o be an implicaion of heir heorem 2. We presen a proof of his resul in Secion 4.3 relying on he echniques used in Eso and Szenes (2007a). 4.1 Uninformaive public signal Suppose ha he public signal is uninformaive (i.e. s is independen of f ). We mainain he assumpion ha he payoff funcion of he agen is ime-separable and saisfies Assumpion 2, bu now he flow uiliy a ime is allowed o vary wih a. Recall ha he acion space of he agen a ime, A, was assumed o be an open inerval of R in Secion 2. We needed his assumpion because we posied ha for all θ, θ and a here is a unique a such ha f (θ,a )=f ( θ, a ). 10 Since here is no public signal in he case considered here, we can relax he requiremen ha A is open. In fac, in order o discuss he implemenabiliy of allocaion rules which may involve boundary acions, we assume ha A =[a, a ] is a compac inerval hroughou his subsecion. Assumpion 3. For all {0,...,}, for all θ Θ,a, a A,x, x X (i) u a 2 θ,a,x 0, (ii) u θ θ,a,x u θ θ, a, x whenever a a,and (iii) u a θ,a,x u a θ,a, x whenever x x. 10 If A was compac hen here would be a pair, (θ,a ), which maximizes f. herefore, if b θ / arg max θ [max a f (θ,a )], henherewasnoba such ha f (θ,a )=f ( b θ, ba ). 14

15 Par (i) of he assumpion saes ha he agen s payoff is a concave funcion of her acion. his is saisfied in applicaions where he acion of he agen is inerpreed as an effor, and he cos of exering effor is a convex funcion of he effor. Par (ii) saes ha he single-crossing assumpion is also saisfied for he acion. In he previous applicaion, his means ha he marginal cos of effor is decreasing in he agen s ype. Par (iii) requires he single-crossing propery o hold wih respec o acions and decisions. In wha follows, we urn he problem of implemenaion in his environmen wih adverse selecion and moral hazard ino one of pure adverse selecion. Since here is no publicly available informaion abou he agen s acion, her acion maximizes her payoff in each period and afer each hisory. ha is, if he agen has ype θ and he hisory of decisions is x, hen she akes an acion which maximizes u θ,a,x. Moivaed by his observaion, le us define he agen s new flow uiliy funcion a ime, v :Θ X R, obe v θ,x = max a u θ,a,x. We will apply our implemenaion resul for he pure adverse selecion case (Proposiion 2) o he seing where he flow uiliies of he agen are {v } =0 while keeping in mind ha he acion of he agen in each period maximizes u. o his end, le a θ,x denoe he generically unique arg max a u θ,a,x for all θ Θ and x X. By par (i) of Assumpion 3, if a θ,x is inerior, i is defined by he firs-order condiion u a θ, a (θ,x ),x =0. (7) he nex lemma saes ha he flow uiliies, {v } 0, saisfy he hypohesis of Proposiion 2. saisfy Assumpions 2 and 3. hen he func- Lemma 1 Suppose ha he funcions {u } =0 ions {v } =0 saisfy Assumpion 2. Suppose ha he decision-acion rule x, a is implemenable. hen, since he agen s acion maximizes her payoff in each period, a θ = a θ, x θ. In addiion, he decision rule x mus be implemenable in he pure adverse selecion model, where he agen s flow uiliy funcions are {v } =1. Hence, he following resul is a consequence of Proposiion 2 and Lemma 1. Proposiion 3 Suppose ha Assumpions 0-3 hold. hen a decision rule, x, a, x : Θ X and a :Θ A,isimplemenableifx is increasing and a θ = a θ, x θ for all {0,...,}. 15

16 Of course, he saemen of his proposiion is valid even if he public signal is informaive (s depends on f ) bu he principal ignores i and designs a mechanism which does no condiion on s. However, if s is informaive abou f (θ,a ) he principal can implemen more decision rules which is he subjec of he nex subsecion. 4.2 Informaive public signal We urn our aenion o he case where public signal is informaive. he nex condiion is due o McAfee and Reny (1992); i requires ha he disribuion of he public signal condiional of any given value of y 0 = f (θ,a ) is no he average of he disribuion of s condiional on oher values of f. his condiion is generic. Assumpion 4. Suppose ha for all θ Θ and a A, f (θ,a ) Y =[y, y ]. hen, for all µ [y, y ] and y 0 [y, y ], µ ({y 0 }) = 1implies h ( y 0 ) = 1 0 h ( y) µ (dy). Nex, we make furher assumpions on he agen s flow uiliy, u,andonheshapeofhe funcion f. Assumpion 5. For all {0,...,}, for all θ Θ,a A,x X (i) u a θ,a,x < 0, (ii) here exiss a K N such ha f a (θ,a ),f θ (θ,a ) > 1/K, (iii) f a 2 (θ,a ) f θ (θ,a ) f a (θ,a ) f aθ (θ,a ), and (iv) u θx τ θ,a,x f a (θ,a ) u ax τ θ,a,x f θ (θ,a ). Par (i) requires he agen s flow uiliy o be decreasing in her acion. his is saisfied in applicaions where, for example, he agen s unobservable acion is a cosly effor from which she does no benefi direcly. Par (ii) says ha he funcion f is increasing in boh he agen s acion and ype. In many applicaions, he disribuion of he public signal can be ordered according o firs-order sochasic dominance. In hese applicaions, par (ii) implies ha an increase in eiher he acion or he ype improves he disribuion of s in he sense of firs-order sochasic dominance. Par (iii) is a subsiuion assumpion regarding he agen s ype and hidden acion in he value of f. I means ha an increase in a,holdinghevalueof f consan, weakly decreases he marginal impac of a on f. 11 his assumpion is saisfied, for example, if f (θ,a )=θ + a, bu i is clearly more general. As will be explained laer par (iv) is a sreghening of he single crossing propery posied in par (iii) of Assumpion 2. I requires he marginal uiliy in ype o be increasing in he conracible decision while 11 o see his inerpreaion, noe ha he oal differenial of f a (he change in he marginal impac of a ) is f a 2 da + f a θ dθ. Keeping f consan (moving along an iso-value curve) means dθ =( f a /f θ )da. Subsiuing his ino he oal differenial of f a yields (f a 2 f a θ f a /f θ )da. his expression is nonposiive for da > 0 if par (iii) is saisfied. 16

17 holding he value of f fixed. his assumpion is saisfied, for example, if he effor cos of he agen is addiively separable in her flow uiliy. he key observaion is ha due o Assumpion 4, he value of f becomes an approximaely conracible objec in he following sense. For each value of f, y, he principal can design a ransfer scheme depending only on s ha punishes he agen for aking an acion which resuls a value of f which is differen from y. Perhaps more imporanly, he punishmen can be arbirarily large as a funcion of he disance beween y and he realized value of f.we use his observaion o esablish our implemenaion resul in wo seps. Firs, we rea f (for all ) as a conracible objec, ha is, we add anoher dimension o he conracible decisions in each period. Since, condiional on θ,hevalueoff is deermined by a, we can express he agen s flow uiliy as a funcion of f insead of a. hese new flow uiliies depend only on ypes and decisions, so we have a pure adverse selecion model. We hen show ha he new flow uiliies saisfy he requiremens of Proposiion 2 and hence, every monoonic rule is implemenable. he second sep is o consruc punishmen-ransfers menioned above and show ha even if f is no conracible, any monoonic decision rule can be approximaely implemenable. For each y {f (θ,a ):θ Θ,a A } and θ Θ, le a (θ,y ) denoe he soluion o f (θ,a )=y in a. For each =0,...,, we define he agen s flow uiliy as a funcion of y as follows: w θ,y,x = u θ, a (θ,y ),x. Nex, we show ha he funcions {w } =0 saisfy he hypohesis of Proposiion 2. Lemma 2 Suppose ha Assumpions 2-5 are saisfied. hen he funcions {w } =0 saisfy Assumpion 2. By his lemma and Proposiion 2 if he value of f was conracible for all, any increasing decision rule was implemenable. However, f is no conracible; neverheless we can sill implemen increasing decisions rules approximiaely in he sense ha by following he principal s recommendaion he agen s expeced uiliy is arbirarily close o her equilibrium payoff. Formally: Definiion 1 he decision rule x, a is approximaely implemenable if for all δ>0 here exiss a paymen rule p: Θ S R such ha for all θ 0 Θ 0, E s =0 u θ, a θ, x θ p θ,s θ 0 Π 0 (θ 0 ) δ, (8) where Π 0 (θ 0 ) denoes he agen s equilibrium payoff wih iniial ype θ 0. 17

18 We are ready o sae he implemenaion resul of his subsecion. Proposiion 4 Suppose ha Assumpions 0-5 are saisfied. hen a decision rule, x, a, x :Θ X and a :Θ A,isapproximaelyimplemenableifx and a are increasing for all {0,...,}. 4.3 he proof of Proposiion 2 Since in a pure adverse selecion model u a 0 for all, hroughou his secion, we remove a from he argumens of u,hais,u :Θ X R. We firs inspec he consequences of Assumpions 1 and 2 on he orhogonalized model. Noe ha since θ does no depend on x 1, he inference funcions defined in equaion (1) do no depend on he decisions eiher, so ψ : E Θ. he ime-separabiliy of he agen s payoff (par (i) of Assumpion 2) is preserved in he orhogonalized model, excep ha he flow uiliy a, u : E X R, now depends on he hisory of ypes up o and including ime : u ε,x = u ψ ε,x. (9) Par (iii) of Assumpion 1 implies ha he larger he ype hisory in he orhogonalized model up o ime, he larger is he corresponding period- ype in he original model. his, coupled wih par (ii) of Assumpion 2 implies ha u is weakly increasing in ε 1 and sricly in ε. Monoonicy in x as well as single-crossing (par (iii) of Assumpion 2) are also preserved in he orhogonalized model. We sae hese properies formally in he following Lemma (see he proof in he Appendix). Lemma 3 (i) For all {0,...,} and ε,ε E, ε ε ψ ε ψ ε, (10) and he inequaliy is sric whenever ε <ε. (ii) he flow uiliy, u defined by (9), is weakly increasing in ε 1 and x 1,andsricly increasing in x and ε. (iii) For all {0,...,}, u ε ε,x u ε ε, x whenever x x. Anoher imporan consequence of par (i) of Assumpion 1 is ha for all ε +1 and ε, here exiss a ype σ +1 (ε +1, ε ) E such ha, fixing he principal s pas and fuure decisions as well as he realizaions of he agen s ypes beyond period +1, he agen s uiliy flow from period +1on is he same wih ype hisory ε +1 as i is wih ε 1, ε,σ +1 (ε +1, ε ). We will show below ha σ +1, inerpreed in Eso and Szenes (2007) as he agen s correcion of 18

19 a lie, defines an opimal sraegy for he agen a ime +1 afer a deviaion from ruhelling in an incenive compaible direc mechanism a. his is formally saed in he following Lemma 4 For all {0,..., 1}, ε +1 E +1 and ε E, here exiss a unique σ +1 ε +1, ε E+1 such ha for all k = +1,...,,allε k E k and x k X k, u k (ε 1,ε,ε +1, ε +2,...,ε k, x k )=u k (ε 1, ε,σ +1, ε +2,...,ε k, x k ). (11) he funcion σ +1 is increasing in ε,sriclyincreasinginε +1 and decreasing in ε. he saemen of he lemma migh appear somewha complicaed a firs glance, bu is meaning and is inuiive proof are quie sraighforward. Par (i) of Assumpion 1 requires he suppor of θ o be independen of θ 1. herefore, if he ype of he agen is ψ ε 1, ε a ime, here is a chance ha he period-( + 1) ype will be ψ +1 ε +1. he ype σ +1 ε +1, ε denoes he orhogonalized informaion of he agen a +1 which induces he ransiion from ψ ε 1, ε o ψ+1 ε +1,hais, ψ +1 ε 1, ε,σ +1 ε +1, ε = ψ+1 ε +1. his means ha he inferred ype in he original model is he same afer he hisories ε 1, ε,σ +1 ε +1, ε and ε +1. Par (i) of Assumpion 1 and par (ii) of Assumpion 2 imply ha, given he decisions, he flow uiliies in he fuure only depend on curren ype which, in urn, imply (11). he decision rule in he orhogonalized model, x : E X, which corresponds o =0 {x } =0, is defined by x ε = x ψ ε for all and ε. Noe ha, by (10), if {x } =0 is increasing in ype (x is increasing in θ for all ) hen he corresponding decision rule {x } =0 in he orhogonalized model is also increasing in ype. 12 In fac, he monooniciy of {x } =0 implies a sronger monooniciy condiion on {x } =0. Consider he following wo ype hisories, ε k and ε 1,...,ε 1, ε,σ +1 ε +1, ε,ε+2,...,ε k. Noe ha he inferred ypes in he original model are exacly he same along hese hisories excep a ime. A ime, he inferred ype is smaller afer ε if and only if ε ε. Since x k is increasing in θ, he decision is smaller afer ε k if and only if ε ε.formally, Remark 3 If {x } =0 is increasing hen for all k =1,...,, <k, εk E k : x k (ε k ) x k ε 1, ε,σ +1 ε +1, ε,ε+2,...,ε k ε ε. (12) 12 o see his, noe ha if v bv hen x `bv = ex `ψ `bv ex `ψ `v = x `v, where he inequaliy follows from he monooniciy of {ex } 0 and (10). 19

20 o simplify he exposiion, we inroduce he following noaion for =0,...,, k : ζ k (ε k,y)= ε 1,y,ε +1,...,ε k, ρ k (ε k,y,ε )= ε 1, ε,σ +1 ε 1,y,ε +1, ε,ε+2,...,ε k. he vecors ζ k (ε k,y) and ρ k (ε k,y,ε ) are ype hisories up o period k, rue or repored, which are differen from ε k only a or a and +1.Fork = hese are appropriaely runcaed, e.g., ρ (ε,y,ε )= ε 1, ε. As we explained, he monooniciy of {x } =0 implies boh he monooniciy of {x } =0 and (12). herefore, in order o prove Proposiion 2, i is sufficien o show ha any increasing decision rule in he orhogonalized model which saisfies (12) can be implemened. In wha follows, fix a direc mechanism wih an increasing decision rule {x } =0 ha saisfies (12). Le Π (ε ε 1 ) denoe a ruhful agen s expeced payoff a condiional on ε. ha is, Π (ε ε 1 )=E k=0 u k ε k,x k (ε k ) p(ε ) Define he paymen funcion, p, such ha for all =0,..., and ε E, Π ε ε 1 =Π 0 ε 1 + E ˆ ε 0 k= ε u kε ζ k (ε k,y),x k ζ k (ε k,y) dy. (13) ε. (14) I is no hard o show ha he inegral on he righ-hand side of (14) exiss and is finie because of par (ii) of Assumpion 1, par (i) of Assumpion 2 and he monooniciy of x k.i should be clear ha i is possible o define p such ha (14) holds. In his mechanism, le π ε, ε ε 1 denoe he expeced payoff of he agen a ime whose ype hisory is ε and has repored ε 1, ε. his is he maximum payoff she can achieve from using any reporing sraegy from +1 condiional on he ype hisory ε and on he repors ε 1, ε. Π (ε ε 1 )=π (ε,ε ε 1 ). If he mechanism is incenive compaible hen, clearly, We call a mechanism IC afer ime if, condiional on elling he ruh before and a ime 1, i is an equilibrium sraegy for he agen o ell he ruh aferwards, ha is, from period on. By Lemma 4, he coninuaion uiliies of he agen wih ype ε +1 are he same as hose of he agen wih ype ε 1, ε,σ +1 ε +1, ε condiional on he repors and he realizaion of ypes afer +1. herefore, if a mechanism is IC afer +1, he agen whose ype hisory is ε +1 and repored ε 1, ε up o ime maximizes her coninuaion payoff by reporing σ +1 ε +1, ε a ime +1 and reporing ruhfully aferwards. If his were no 20

21 he case hen he agen wih ε 1, ε,σ +1 ε +1, ε would have a profiable deviaion afer ruhful repors up o and including, conradicing he assumpion ha he mechanism is IC afer +1. herefore, in a mechanism ha is IC afer +1,wehave π ε, ε ε 1 = u ε,x ε 1, ε u ε 1, ε,x ε 1, ε (15) ˆ + Π +1 σ+1 ε +1, ε ε 1, ε dε+1. We use (15) in he following Lemma o characerize he agen s coninuaion payoff who deviaes a in a mechanism ha is IC afer. Lemma 5 Suppose ha he mechanism is IC afer ime +1 and (14) is saisfied. hen, for all ε and ε, π ε, ε ε 1 π ε, ε ε 1 = ˆ ε E u kε ζ k (ε k,y),x k ρ k (ε k,y,ε ) dy ε. (16) bε k= his lemma is a direc generalizaion of Lemma 5 of Eso and Szenes (2007a); is proof can be found in he Appendix. We are now ready o prove Proposiion 2. Proof of Proposiion 2. In order o prove ha he ransfers defined by (14) implemen {x } =0, i is enough o prove ha he mechanism is IC afer all =0,..., 1. We prove his by inducion. For = 1 his follows from he sandard resul in saic mechanism design wih he observaion ha x is monoone and (14) is saisfied for. Suppose now ha he mechanism is IC afer +1. We show ha he mechanism is IC afer, hais,he agen has no incenive o lie a if she has old he ruh before. Consider an agen wih ype hisory ε and repor hisory ε 1 who is conemplaing o repor ε <ε.wehaveoshowhaπ ε, ε ε 1 π ε,ε ε 1 0 which can be wrien as π ε, ε ε 1 π ε, ε ε 1 + π ε, ε ε 1 π ε,ε ε 1 0. By (14) and (16), he previous inequaliy can be expressed as ˆ ε E u kε ζ k (ε k,y),x k ζ k (ε k,y) dy ε bε ˆ ε E u kε ζ k (ε k,y),x k ρ k (ε k,y,ε ) dy ε. bε k= k= (17) 21

22 In order o prove his inequaliy i is enough o show ha he inegrand on he lef-hand side is larger han he inegrand on he righ-hand side. By par (iii) of Lemma 3, in order o show his, we only need ha x k ρ k ε k,y,ε x k ζ k (ε k,y) on y [ε,ε ], which follows from Remark 3. An idenical argumen can be used o rule ou deviaion o ε >ε. From he proof of Proposiion 2 i is clear ha in he environmen saisfying Assumpions 1 and 2 (i.e., wih Markov ypes and a well-behaved agen payoff funcion), a decision rule { x } =0 is implemened by ransfers saisfying (14) if, and only if, condiion (17) holds in he orhogonalized model. 13 Bu (14) is also a necessary condiion of implemenaion (differeniae i in ε and compare ha wih (4) in Proposiion 1), herefore condiion (17) is indeed he necessary and sufficien condiion of implemenabiliy of a decision rule in he regular, Markov environmen. Formally, we sae, is imple- Remark 4 Suppose ha Assumpions 1 and 2 hold. hen a decision rule, {x } 0 menable if, and only if, (17) holds in he model wih orhogonalized informaion. Implemenabiliy in he Benchmark Case. Suppose ha he principal can observe ε 1,...,ε. hen, using argumens in sandard saic mechanism design, a decision rule {x } 0 can be implemened if, and only if, for all ε 0,ε 0 E 0, ε 0 ε 0, E ˆ ε0 u kε0 k=0 bε 0 y, ε k 0,x k y, ε k 0 dy ε 0 E ˆ ε0 u kε0 k=0 bε 0 y, ε k 0,x k ε 0,ε k 0 dy ε 0. his inequaliy is obviously a weaker condiion han (17), so he principal can implemen more allocaions in he benchmark case. 5 Applicaions We presen hree applicaions o illusrae how our echniques and resuls can be applied in subsanive economic problems. In each applicaion we firs solve he benchmark case, where he he principal can observe he agen s orhogonalized fuure ypes. (In he absence of a conracible summary signal abou he agen s ype and hidden acion he acion rule is aken o be he agen-opimal one; in he presence of such a signal he acion rule is also opimized.) hen we verify he appropriae monooniciy condiion regarding he decision-acion rule and conclude ha he soluion is implemenable, hence opimal, in he original problem as well. 13 Noe ha condiion (17) is a join resricion on {x } 0 and he marginal uiliy of he agen s ype, and i is implied by he monooniciy of he decision rule in he environmen of Assumpions

23 In all hree applicaions we assume ha he agen s ype follows he AR(1) process θ = λθ 1 +(1 λ)ε, =0,...,, where θ 1 =0and ε 0,...,ε are iid uniform on [0, 1]. he exac specificaion is adoped for he sake of obaining a simple orhogonal ransformaion of he informaion srucure: θ =(1 λ)λ k=0 λ k ε k, =0,...,. (18) he ype process is Markovian. Assumpion 1 is saisfied excep ha he suppor of θ depends on he realizaion of θ 1. However, i is easy o make he suppor of θ he uni inerval for all by mixing he disribuion of θ in (18) wih he uniform disribuion on [0, 1]; our specificaion obains in he limi as he weigh on he uniform disribuion vanishes. In all hree examples he agen s uiliy is ime-separable, and he flow uiliy, ũ (θ,a,x ), only depends on he agen s ype, hidden acion and he conracible decision. 14 Denoe he flow uiliy in he orhogonally ransformed model by u (ε,a,x ). By Proposiion 1, in any incenive compaible mechanism x, a,p he agen s equilibrium payoff can be wrien as ˆ ε0 Π 0 (ε 0 )=Π 0 (0) + E u ε0 y, ε 0,a (y, ε 0,s 1 ),x (y, ε 0,s ) dy 0 ε 0 =0 ˆ ε0 + E u a y, ε 0,a (y, ε 0,s 1 ),x (y, ε 0,s ) a ε0 (y, ε 0,y,s 1 ) dy ε 0, (19) 0 =0 where a (ε, ε 0,s 1 ), defined by equaion (2), is he period- acion of he Agen ha masks her iniial misrepor of ε 0 condiional on he hisory of ypes and public signals. Nex, we describe Applicaions 1 3, ordered according o increasing complexiy of he agen s payoff funcion. he firs applicaion is a pure adverse selecion model; he second one is a varian ha includes a hidden acion as well, bu no conracible signal abou he agen s ype and acion. he hird applicaion has boh adverse selecion and moral hazard, and he agen s ype and acion generae a conracible summary signal. Applicaion 1. he principal is he seller of an indivisible good; he agen is a buyer wih valuaion θ in period. he conracible acion, x [0, 1], is he probabiliy ha he buyer receives he good. he buyer has no hidden acion; her flow uiliy is simply ũ (θ,x )=θ x, or equivalenly, in he orhogonalized model, u (ε,x )=(1 λ)λ k=0 λ k ε k x. Noe ha Assumpion 2 holds and u ε0 =(1 λ)λ x. 14 Assumpion 0 is also saisfied due o he boundedness of all relevan domains and he coninuous differeniabiliy of all involved funcions. 23

24 Since he agen has no hidden acion he second line in equaion (19) is zero, and so Π 0 (ε 0 )=Π 0 (0) + E ˆ ε0 0 (1 λ)λ x (y, ε 0) dy ε 0. (20) =0 Suppose he buyer s paricipaion is guaraneed if she ges a non-negaive payoff; by (20) his is equivalen o Π 0 (0) 0. In order o compue E [Π 0 (ε 0 )] we noe ha by Fubini s heorem, ˆ 1 ˆ ε0 0 0 x (y, ε 0) dydε 0 = ˆ 1 ˆ 1 0 y x (y, ε 0) dε 0 dy = ˆ 1 0 (1 ε 0 )x (ε ) dε 0, herefore E [Π 0 (ε 0 )] = Π 0 (0) + E (1 λ)λ (1 ε 0 )x (ε ). (21) =0 Assume he seller (principal) maximizes his expeced revenue; here is no cos of producion. he expeced revenue equals he expeced social surplus generaed by he mechanism less he buyer s expeced payoff: E θ x (ε ) (1 λ)λ (1 ε 0 )x (ε ) Π 0 (0), =0 where θ is given by equaion (18). Solve he seller s problem by seing Π 0 (0) = 0 and poinwise maximizing he objecive in x (ε ): he soluion is found by seing x ε =1if and only if θ (1 λ)λ (1 ε 0 ) and x ε =0oherwise. Equivalenly, in he noaion of he original model, x (θ )=1 θ+λ θ 0 (1 λ)λ, where 1 is he indicaor funcion. his decision rule in monoone in θ ; herefore, by Proposiion 2, i is implemenable in he original problem as well as in he benchmark case. Hence i is he opimal soluion in boh. In his muli-period rading (single-buyer aucion) problem he firs-bes oucome would be o rade he good whenever θ 0. In conras, in he revenue-maximizing mechanism he good is sold whenever θ λ (1 λ θ 0 ). As in he one-period problem, his decision rule corresponds o seing a reservaion price in each period. he reservaion price is always nonnegaive because θ 0 1 λ by (18). Ineresingly, he reservaion prices and he disorion ha hey induce only depend on he buyer s iniial informaion (confirming our dynamic irrelevance resul) and disappear over ime as. 24

25 Applicaion 2. In his applicaion, as in he previous one, he principal is a seller and he agen a buyer wih period- valuaion θ. Assume he good is divisible, so x [0, 1] is inerpreed as he amoun bough by he buyer, and he seller has producion cos x 2 /2. he imporan difference in his applicaion (as compared o he previous one) is ha we assume he buyer akes a cosly, hidden acion inerpreed as invesmen in every period, which increases her valuaion. 15 he buyer s flow uiliy is ũ (θ,a,x )=(θ + a )x ψa 2 /2, or equivalenly, in he orhogonalized model, u (ε,a,x )= (1 λ)λ k=0 λ k ε k + a x 1 2 ψa2. Noe ha Assumpions 0 3 hold, and u ε0 =(1 λ)λ x (same as in Applicaion 1). Assume ha he seller canno observe any signal abou he buyer s valuaion and invesmen. Hence he second line in equaion (19) is zero, and so Π 0 (ε 0 ) is given by equaion (20), and E[Π 0 (ε 0 )] by equaion (21). he seller s (principal s) expeced profi is he expeced social surplus generaed by he mechanism less he buyer s (agen s) expeced payoff: E θ + a (ε ) x (ε ) 1 2 ψa (ε ) x (ε ) 2 (1 λ)λ (1 ε 0 )x (ε ) Π 0 (0). =0 Since he seller can make no inference abou a, moreover he buyer s fuure valuaions are no affeced by her curren invesmen eiher, a is se by he buyer o maximize her curren flow uiliy: a (ε ) x (ε )/ψ. Subsiuing his ino he seller s expeced payoff, he firs-order condiion of poinwise maximizaion of he seller s objecive in x (ε ) is θ + x (ε ) ψ x (ε ) (1 λ)λ (1 ε 0 )=0. (22) Assuming ha he buyer paricipaes wih non-negaive payoff i is opimal o se Π 0 (0) = 0. Using (18) in rearranging (22) yields, in erms of he original model, x (θ )= ψ θ + λ θ 0 (1 λ)λ. ψ 1 Assume ψ>1. hen x (θ ) is sricly increasing; by Proposiion 3 i is implemenable boh in he original problem and he benchmark when coupled wih invesmens ã (θ )= x (θ )/ψ. herefore his allocaion rule is he opimal second-bes soluion in boh problems. In his applicaion, in he firs bes (conracible θ, a ), he relaionship beween he 15 Inerpreing a as a cosly acion aken righ before θ is realized and shifing he disribuion of θ, his applicaion can be hough of as a muli-period generalizaion (of a specific example) of Bergemann and Välimäki (2002). Our focus is on he revenue-maximizing sales mechanism insead of he efficien one. 25

26 buyer s invesmen level and her anicipaed purchase (rade) would be he same, a FB x FB /ψ. However, he firs-bes level of rade would be x FB (θ )=ψθ /(ψ 1). he disorion, which maerializes in he decision rule in he form of less rade, and in he acion rule as less invesmen in comparison o he efficien levels, is again due o he buyer s (agen s) iniial privae informaion and i disappears over ime. Applicaion 3. he principal is an invesor and he agen is an invesmen advisor (banker); he conracible acion x is he amoun invesed. Assume ha he agen invess κx for herself and (1 κ)x for he principal. he proporion κ [0, 1] is fixed exogenously; κ>0 is realisic bu κ =0is an ineresing special case. he agen s ype θ represens her abiliy o achieve a greaer expeced reurn. Her effor (hidden acion a ) is direced a finding asses ha fi he principal s oher (e.g., ehical) invesmen goals; i generaes a payoff propoional o he invesed amoun for he principal bu imposes an up-fron cos on he agen. Le ũ = θ κx ψa 2 /2 be he agen s payoff and v =(θ + a + ξ )(1 κ)x rx 2 /2 he principal s; in he laer rx 2 /2 represens he principal s (convex) cos of raising funds for invesmen, and ξ is a noise erm (e.g., uncerainy in how he advisor s effor affecs he invesor s non-pecuniary reurn on invesmen), included for he sake of generaliy. Assume ha v (bu no θ nor a ) is conracible, and define s = θ +a +ξ as he conracible public signal. he paries payoffs are ransferable, i.e., hey may conrac on moneary ransfers as well. I is easy o check ha in his applicaion Assumpions 0 5 are all saisfied. 16 his is a parameric example of he model discussed in Secion 4.2. Garre and Pavan (2012) solve a relaed problem where, using he noaion of his example, κ =0and r =0;he decision x {0, 1} corresponds o wheher or no he principal employs he agen insead of a coninuous invesmen decision (which is more meaningful in our example). In he orhogonalized model u (ε,a,x )=(1 λ)λ k=0 λ k ε k κx 1 2 ψa2, hence u ε0 = κ(1 λ)λ x and u a = ψa. he period- acion of he agen ha masks her iniial misrepor of ε 0 condiional on he hisory of ypes and prior public signals is a (ε, ε 0,s 1 ), formally defined by θ + a (ε 0,ε 0,s 1 ) θ + a (ε, ε 0,s 1 ), 16 If κ =0hen Assumpion 2(ii) only holds weakly. However, we will show ha he opimal decision rule is coninuous in κ a κ =0,andhenceheapproximaeimplemenaionresulholdsinhelimi. 26

27 where θ =(1 λ)λ ε 0 +(1 λ) k=1 λ k ε k, hence a (ε, ε 0,s 1 )=a (ε 0,ε 0,s 1 )+(1 λ)λ (ε 0 ε 0 ). Noe ha a ε0 (ε, ε 0,s 1 )= (1 λ)λ. By equaion (19), he agen s expeced payoff wih iniial ype ε 0 is Π 0 (ε 0 )=Π 0 (0) + E ˆ ε0 0 (1 λ)λ κx (y, ε 0,s )+ψa (y, ε 0,s 1 ) dy ε 0. =0 Coninue o use Π 0 (0) 0 as he paricipaion consrain. Again, using Fubini s heorem as in he previous applicaions we ge E [Π 0 (ε 0 )] = Π 0 (0) + E (1 λ)λ (1 ε 0 ) κx (ε,s )+ψa (ε,s 1 ). =0 he principal s ex-ane expeced payoff is he difference beween he expeced social surplus generaed by he mechanism and he agen s expeced payoff: E (θ + a + ξ )(1 κ)x 1 2 rx2 + θ κx 1 2 ψa2 (1 λ)λ (1 ε 0 )(κx + ψa ) Π 0 (0), =0 where he argumens of a (θ,s 1 ) and x (θ,s ) are suppressed for breviy. If he public signal s conained no noise erm (i.e., in case ξ 0), hen he principal could infer a from he agen s ype repor and he realized signal and indirecly enforce any acion. (23) In his case, he firs-order condiion of (poinwise) maximizaion of (23) in a is (1 κ)x ψa (1 λ)λ (1 ε 0 )ψ = 0, whereas he same wih respec o x is θ +(1 κ)a rx (1 λ)λ (1 ε 0 )κ =0. Subsiue he former ino he laer and wrie θ 0 /(1 λ) for ε 0 o ge x (θ )= θ + λ θ 0 (1 λ)λ rψ (1 κ) 2. Assuming rψ > (1 κ) 2 he resuling x is sricly increasing in θ, and so is he corresponding opimal ã, which is is posiive affine ransformaion. herefore, by Proposiion 4, his decision-acion rule is approximaely implemenable in he original model as well as in he benchmark. I is easy o see ha he firs-bes decision rule would be x FB (θ )= θ / r (1 κ) 2 /ψ. Again, he disorion in x (θ ) is purely due o he agen s iniial privae informaion, illusraing our dynamic irrelevance heorem. 27

28 6 Conclusions In his paper we considered a dynamic principal-agen model wih adverse selecion and moral hazard and proved a dynamic irrelevance heorem: In any implemenable decision rule he principal s expeced revenue and he agen s payoff are he same as if he principal could observe he agen s fuure, orhogonalized ypes. We also provided resuls on he implemenabiliy of monoonic decision rules in regular, Markovian environmens. he implemenaion resuls imply a sraighforward mehod of solving a large class of dynamic principal-agen problems wih meaningful economic applicaions. he model considered in his paper could be exended in wo direcions wihou much difficuly, a he expense of addiional noaion and echnical assumpions. Firs, i would be possible o accommodae muliple agens in he principal-agen model by replacing he agen s incenive consrains wih an appropriae (Bayesian) equilibrium. Second, he model could be exended o have an infinie ime horizon. In his case our main heorem sill holds assuming ime-separable uiliy, discouning, and uniformly bounded feliciy funcions. References Baron, D. and D. Besanko (1984): Regulaion and Informaion in a Coninuing Relaionship, Informaion Economics and Policy 1, Bergemann, D. and J. Välimäki (2002): Informaion Acquisiion and Efficien Mechanism Design, Economerica 70(3), Coury, P. and H. Li (2000): Sequenial Screening, Review of Economic Sudies 67, Eső, P. and B. Szenes (2007a): Opimal Informaion Disclosure in Aucions and he Handicap Aucion, Review of Economic Sudies 74(3), Eső, P. and B. Szenes (2007b): he Price of Advice, RAND Journal of Economics 38(4), Farhi, E. and I. Werning (2012): Insurance and axaion over he Life Cycle, Review of Economic Sudies, 80(2), Garre, D. and A. Pavan (2012): Managerial urnover in a Changing World, Journal of Poliical Economy 120(5), Golosov, M. M. roshkin, and A. syvinski (2011): Opimal Dynamic axes, Mimeo, Yale Universiy. 28

29 Kapicka, M. (2013): Efficien Allocaions in Dynamic Privae Informaion Economies wih Persisen Shocks: A Firs-Order Approach, Review of Economic Sudies 80 (3): Krähmer, D. and R. Srausz (2012): Dynamics of Mechanism Design in: An Inroducion o Mechanism Design by ilman Börgers, manuscrip. McAfee, R. P. and P. Reny (1992): Correlaed Informaion and Mecanism Design, Economerica 60(2), Milgrom, P. and I. Segal (2002): Envelope heorems for Arbirary Choice Ses, Economerica 70(2), Pavan, A., I. Segal and J. oikka (2012): Dynamic Mechanism Design mimeo MI, Norhwesern Universiy, and Sanford Universiy. Appendix Lemma 6 If he mechanism x, a, p is incenive compaible, he equilibrium payoff funcion of he agen, Π 0,isLipshizconinuous. Proof. hroughou he proof, le K denoe an ineger such ha he inequaliies in Assumpion 0 are saisfied and, in addiion, for all =1,...,, τ<, and for all θ,a,x G θτ θ θ 1,a 1,x 1 g (θ θ 1,a 1,x 1 ) <K. Firs, we show ha here exiss a K N such ha ψε0 ε,a 1,x 1 < K. For =0, ψ 0ε0 (ε 0 )=G 1 0ε 0 (ε 0 )=1/g 0 G 1 0 (ε 0 ) <Kby par (ii) of Assumpion 0. We proceed by inducion and assume ha ψ τε0 ε τ,a τ 1,x τ 1 < K (τ) for τ =0,..., 1. hen ψ ε0 ε,a 1,x 1 = G 1 ε ε 0 ψ 1 ε 1,a 2,x 2,a 1,x 1 = 1 g (ψ (ε,a 1,x 1 ) ψ 1 (ε 1,a 2,x 2 ),a 1,x 1 ) τ 1 + G 1 ψ ε τ ψ 1 ε 1,a 2,x 2,a 1,x 1 ψ τε0 ε τ,a τ 1,x τ 1 τ=0 K + max K (τ) τ 1 G 1 τ ψ ε τ ψ 1 ε 1,a 2,x 2,a 1,x 1, τ=0 where he inequaliy follows from he inducive hypohesis and par (ii) of Assumpion 0. 29

30 However, K + max K (τ) τ 1 G 1 τ ψ ε τ ψ 1 ε 1,a 2,x 2,a 1,x 1 τ=0 = K + max K (τ) τ 1 G τ ψτ ψ ε,a 1,x 1 ψ 1 ε 1,a 2,x 2,a 1,x 1 τ=0 1 g (ψ (ε,a 1,x 1 ) ψ 1 (ε 1,a 2,x 2 ),a 1,x 1 ) K + max τ K (τ) K2, by Assumpion 2. So, we can conclude ha ψε0 ε,a 1,x 1 <K+ maxτ K (τ) K 2. We are ready o prove ha Π 0 is Lipshiz coninuous. Suppose ha Π 0 (ε 0 ) Π 0 (ε 0 ). Le π 0 (ε 0,ε 0 ) denoe he payoff of an agen whose iniial ype is ε 0, repors ε 0, hen repors ruhfully aferwards and akes acion a ε, ε 0,s 1 afer hisory ε,s 1. Since he mechanism x, a, p is incenive compaible, π 0 (ε 0,ε 0 ) < Π(ε 0 ) and hence, Π 0 (ε 0 ) Π 0 (ε 0 ) < Π 0 (ε 0 ) π 0 (ε 0,ε 0 ). So, i is enough o prove ha Π 0 (ε 0 ) π 0 (ε 0,ε 0 ) <K ε 0 ε 0. (24) In addiion, Π 0 (ε 0 ) π 0 (ε 0,ε 0 ) = E u ε, a ε,s 1,s, x ε ε 0 E u ε, a ε, ε 0,s 1,s, x ε 0,ε 0,s ε 0. In order o esablish (24) i is sufficien o show ha he absolue value of he difference beween he erms whose expecaions are aken on he righ-hand side of he previous equaion is smaller han K ε 0 ε 0.Noeha u ε, a ε,s 1,s,x u ε 0,ε 0, a ε 0,ε 0,s 1,s,x = ˆ ε0 u ε0 y, ε 0, a y, ε 0,s 1,s,x + bε 0 u a y, ε 0, a y, ε 0,s 1,s,x a bε0 ε,y,s 1 dy. =0 We will show ha boh erms on he righ-hand side of he previous equaion is bounded by 30

31 a consan imes ε 0 ε 0.Noeha = ˆ ε0 bε 0 ˆ ε0 bε 0 =0 u ε0 y, ε 0,a,s,x dy u θ ψ y, ε 0,a 1,x 1,a,s,x ψ ε0 y, ε 0,a 1,x 1 dy KK ε 0 ε, by par (i) of Assumpion 0 and since ψ ε0, ε < K, as shown above. In addiion, = ˆ ε0 bε 0 =0 ˆ ε0 bε 0 =0 u a y, ε 0, a y, ε 0,s 1,s,x a bε0 ε,y,s 1 dy (25) u a ψ y, ε 0,, a y, ε 0,s 1,s,x a bε0 ε,y,s 1 dy. By he Implici Funcion heorem, a bε0 ε,y,s 1 = f θ ψ y, ε 0,, a ε,y,s 1 f a ψ y, ε 0,, a (ε,y,s 1 ) ψ bε 0 ε 0,y,a 1,x 1, which does no exceed KK by par (iii) of Assumpion 0 and he argumen above showing ha ψ ε0 < K. Hence, (25) is smaller han KK ˆ ε0 bε 0 u a ψ y, ε 0,, a y, ε 0,s 1,s,x dy K 2 K ε 0 ε =0 by par (i) of Assumpion 0. Proof of Lemma 1. Par (i) of Assumpion 2 is saisfied by definiion. o see par (ii), noice ha if a θ,x is inerior hen v θ (θ,x ) = u θ θ, a θ,x,x + u a θ, a θ,x,x a θ,x (26) θ = u θ θ, a θ,x,x > 0, where he second equaliy follows from (7), and he inequaliy follows from par (ii) of Assumpion 2. If a θ,x is no inerior hen, generically, v θ (θ,x )=u θ θ, a θ,x,x > 0, (27) where he inequaliy again follows from par (ii) of Assumpion 2. 31

32 I remains o prove ha v saisfies par (iii) of Assumpion 2. o simplify noaion, we only prove his claim for he case when he conracible decision is unidimensional in each period, ha is, X R for all =0,...,.Supposefirshaa θ,x is inerior. Noe ha for all τ, v θxτ (θ,x ) = u θxτ θ, a θ,x,x + u θa θ, a θ,x,x a θ,x x τ = u θxτ θ, a θ,x,x u θa θ, a θ,x,x u a x θ, a θ,x,x u a 2 (θ, a (θ,x ),x ), where firs equaliy follows from (26) and he second one follows from (7) and he Implici Funcion heorem. Noe ha u θx τ, u θa and u ax are all non-negaive by par (iii) of Assumpion 2 and pars (ii) and (iii) of Assumpion 3. In addiion, u a 2 is negaive by par (i) of Assumpion 3. herefore, v θxτ (θ,x ) 0. Supposenowhaa θ,x is no inerior. hen, for all τ, generically, v θx τ (θ,x )=u θx τ θ, a θ,x,x 0, where he equaliy follows from (27) and he inequaliy follows from par (ii) of Assumpion 3. Proof of Lemma 2. Par (i) of Assumpion 2 is saisfied by definiion. o see par (ii), noice ha w θ θ,y,x = u θ θ, a (θ,y ),x + u a θ, a (θ,y ),x a (θ,y ) θ. (28) We apply he Implici Funcion heorem for he ideniy f (θ, a (θ,y )) y o ge a (θ,y ) θ = f θ (θ, a (θ,y )) f a (θ, a (θ,y )), which is negaive by par (ii) of Assumpion 5. Since u θ > 0 by par (ii) of Assumpion 2 and u a < 0 by par (i) of Assumpion 5, we conclude ha w is sricly increasing in θ. Nex, we prove ha w saisfies par (iii) of Assumpion 2. Firs, we esablish he singlecrossing propery wih respec o θ and y. By (28), w θy θ,y,x = u θa θ, a (θ,y ),x a (θ,y ) y + u a 2 θ, a (θ,y ),x a (θ,y ) y a (θ,y ) θ + u a θ, a (θ,y ),x 2 a (θ,y ) θ y. 32

33 In order o sign a / y and 2 a / θ y, we appeal o he Implici Funcion heorem once again, a (θ,y ) y = 1 f a (θ, a (θ,y )) and 2 a (θ,y ) = f (θ, a a2 (θ,y )) f θ (θ,a (θ,y )) f a (θ,a (θ f,y )) a θ (θ, a (θ,y )) θ y fa 2. (θ, a (θ,y )) herefore, w θy θ,y,x can be rewrien as u θ θ, a (θ,y ),x f a (θ, a (θ,y )) + u a 2 θ, a (θ,y ),x f θ (θ, a (θ,y )) f 2 a (θ, a (θ,y )) + u a θ, a (θ,y ),x f a 2 (θ, a (θ,y )) f θ(θ,a (θ,y)) f a (θ,a (θ f,y )) a θ (θ, a (θ,y )) fa 2. (θ, a (θ,y )) he firs erm is posiive by par (ii) of Assumpion 2 and par (ii) of Assumpion 5. he second erm is posiive by par (i) of Assumpion 3 and par (ii) of Assumpion 5. he hird erm is posiive by pars (i) and (iii) of Assumpion 5. herefore, we conclude ha w θy 0. I remains o show ha he single crossing propery in par (iii) of Assumpion 2 also holds wih respec o θ and x τ for all τ. o simplify noaion, we only prove his claim for he case when he conracible decision is unidimensional in each period, ha is, X R for all =0,...,. By (28), w θx τ θ,y,x = u θx τ θ, a (θ,y ),x + u ax τ θ, a (θ,y ),x a (θ,y ) θ = u θx τ θ, a (θ,y ),x u ax τ θ, a (θ,y ),x f θ (θ, a (θ,y )) f a (θ, a (θ,y )), which is posiive by par (iv) of Assumpion 5. Proof of Proposiion 4. Fix an increasing decision rule x, a and a δ>0. Below, we consruc a ransfer rule, p, suchha x, a, p saisfies (8). o his end, define he funcion y :Θ Y such ha y θ = f θ, a θ for all and θ. Since a is increasing in θ and f is sricly increasin in boh θ and a (see par (ii) of Assumpion 5), he funcion y is also increasing in θ. herefore, by Lemma 2 and Proposiion 2, he decision rule x, y is implemenable in a pure adverse selecion model where he agen flow uiliies are {w } =0. Le p :Θ R denoe a ransfer rule which implemens x, y. Fix a K N such ha u a <K and f a > 1/K. By par (i) of Assumpion 0 and par (ii) of Assumpion 5, such a K exiss. By heorem 2 of McAfee and Reny (1992), for each 33

34 =0,...,, exiss a funcion p : S Y R such ha E s (p (s,y ) f (θ,a )=y )=0and Le us now define p :Θ S R by E s p (s,y ) f (θ,a )=y K 2 y y δ +1. (29) p θ,s = p θ + p s, y θ. (30) Nex, we show ha he agen canno generae an excess payoff of δ by deviaing from ruhelling and obidience in he mechanism x, a, p. Firs, noe ha he agen canno benefi from making her sraegy a ime coningen on he hisory of public signals, s 1, because her coninuaion payoff does no depend on hese variables in he mechanism x, a, p. herefore, we resric aenion o sraegies which do no depend on pas realizaions of he public signal. =0 Any such sraegy induces a mapping from ype profile o repors and acions in each period. Le ρ θ and α θ denoe he agen s repor and acion a ime, respecively, condiional on her ype hisory θ. Le α θ denoe he soluion of f (θ,a )=f ρ θ, a ρ θ = y ρ θ (31) in a. In oher words, α θ would is he agen s acion which generaes he same value of f condiional on θ as if he agen s rue ype was ρ θ and she ook acion a ρ θ. hen he expeced payoff generaed by ρ,α, condiional on θ 0,is E θ,s u θ,α θ, x ρ (θ p ρ θ,s θ 0 =0 = E θ u θ, α θ, x ρ (θ p ρ θ θ 0 + =0 E θ,s u θ,α θ, x ρ (θ u θ, α θ, x ρ (θ p s, y θ θ 0, =0 where he equaliy follows from (30). (32) 34

35 We firs consider he firs erm on he righ-hand side of he previous equaliy. Noe ha E θ u θ, α θ, x ρ (θ p ρ θ θ 0 =0 = E θ w θ, y ρ θ, x ρ (θ p ρ θ θ 0 E θ =0 w θ, y θ, x θ p θ θ 0 = E θ u θ, a θ, x θ p θ θ 0, =0 where he inequaliy follows from he assumpion ha he ransfer rule p implemens x, y if he flow uiliies are w =0.Alsonoeha u θ,α θ, x ρ (θ u θ, α θ, x ρ (θ E s p s, y ρ θ θ,α θ K α θ α θ E s p s, y ρ θ θ,α θ K 2 f θ, α θ f θ,α θ E s p s, y ρ θ θ,α θ = K 2 y ρ θ f ρ θ,α θ E s p s, y ρ θ θ,α θ δ +1, where he firs and second inequaliies follow from u a <Kand f a > 1/K, he equaliy follows from (31), and he las inequaliy follows from (29). Summing up hese inequaliies for =0,..., and aking expacaion wih respec o θ, E θ,s u θ,α θ, x ρ (θ u θ, a ρ θ, x ρ (θ p s, y θ θ 0 δ. =0 herefore, plugging (33) and (34) ino (32) we ge ha E θ,s u θ,α θ, x ρ (θ p ρ θ,s θ 0 E θ =0 =0 u θ, a θ, x θ p θ θ 0 + δ =0 = E θ,s u θ, a θ, x θ p ρ θ,s θ 0 + δ, =0 where he equaliy follows from E s [p (s,y ) f(θ,a )=y ]=0. his implies ha he agen canno gain more han δ by deviaing from ruh-elling and obedience in he mechanism x, a, p. (34) (33) 35

36 Proof of Lemma 3. Par (i) ε = H 1 >0 is defined recursively by ψ ε = H 1 (θ θ 1 ), herefore ψ 0 (ε 0 )=H0 1 (ε 0 ) and ψ for ε ψ 1 ε 1. We prove he saemen of his par by inducion. For =0,wehaveH 1 0 (ε 0 ) H 1 0 (ε 0 ) whenever ε 0 ε 0 and he inequaliy is sric if ε 0 > ε 0. Suppose ha (10) holds for, ha is, ψ ε ψ ε whenever ε ε and he inequaliy is sric whenever ε < ε. Noe ha ψ +1 (ε +1 ) = H+1 1 ε+1 ψ ε and ψ +1 ε +1 = H+1 1 ε+1 ψ ε. Since ψ ε ψ ε by he inducive hypohesis, par (ii) of Assumpion 1 implies ha ψ +1 ε +1 ψ +1 ε +1.Inaddiion,ifε +1 > ε +1 hen H 1 +1 ε+1 ψ ε >H+1 1 ε +1 ψ ε. Par (ii): he funcion u is sricly increasing in x and weakly increasing x 1 because of par (ii) Assumpion 2 and (9). Equaliies (9) and (10) imply ha u is sricly increasing in ε and weakly increasing in ε 1. Par (iii): Fix a {0,...,} and noe ha by (9), u ε ε,x = u θ ψ ε,x ψ ε. ε he resul follows from (10) and par (iii) of Assumpion 2. Proof of Lemma 4. Fix a {0,..., 1}, ε +1 E +1 and ε E. Le σ +1 = H +1 ψ+1 ε +1 ψ ε 1, ε. (35) By he full suppor assumpion in par (i) of Assumpion 1, i follows ha ψ +1 ε +1 = ψ +1 ε 1, ε,σ +1, ha is, he compued ime-( + 1) ype of he original model is he same afer ε +1 and ε 1, ε,σ +1. herefore he inferred ype in he original model is also he same afer any fuure observaions, ha is, ψ k ε 1,ε,ε +1, ε +2,...,ε k = ψk (ε 1, ε,σ +1, ε +2,...,ε k ), for all k = +1,...,,allε k E k. his equaliy and (9) imply (11). Also noe ha σ +1 ε +1, ε, defined by (35), is increasing in ε, sricly increasing in ε +1 by par (i) of Lemma 3 and decreasing in ε by par (i) of Lemma 3 and par (iii) of Assumpion 1. I remains o show ha here does no exis any oher σ +1 which saisfies (11). his follows from par (ii) of Lemma 3, which saes ha u +1 is sricly increasing in ε +1, which implies ha (11) wih k = +1canno hold for wo differen σ +1 s. 36

37 Proof of Remark 3. Recall from he proof of Lemma 4 ha for all k = +1,...,, ψ k ε +1,ε +2..., ε k = ψk ε 1, ε,σ +1 ε +1, ε,ε+2,...,ε k. By (10), ψ ε ψ ε 1, ε if and only if ε ε. hen (12) follows from he monoiniciy of {x } 0 and he definiion of {x} 0. Proof of Lemma 5. Le γ k ε k, ε,y denoe ε 1, ε,y,ε +2,...,ε k for k = +1,...,. Suppose firs ha ε > ε. hen σ +1 ε +1, ε >ε+1,and π ε, ε ε 1 = u ε,x ε 1, ε u ε 1, ε,x ε 1, ε ˆ + Π +1 σ+1 ε +1, ε ε 1, ε dε+1 = u ε,x ε 1, ε u ε 1, ε,x ε 1, ε +Π ε ε 1 ˆ ˆ ˆ σ+1(ε +1,bε ) +... γ k ε k, ε,y,x k γ k ε k, ε,y dydε +1...dε k k=+1 ε +1 u k ε +1 = u ε,x ε 1, ε u ε 1, ε,x ε 1, ε + π ε, ε ε 1 ˆ ˆ ˆ σ+1(ε +1,bε ) +... u kε+1 γ k ε k, ε,y,x k γ k ε k, ε,y dydε +1...dε k k=+1 ε +1 where he firs equaliy is jus (15), he second one follows from (14), and he hird one from Π ε ε 1 = π ε, ε ε 1. So, in order o prove (16), we only need o show ha and u ε,x ε 1, ε u ε 1, ε,x ε 1, ε = ˆ ε bε u ε ε,y,x ρ (ε, ε ) dy. (36) k=+1 ˆ ˆ ˆ σ+1(ε +1,bε )... ε +1 = k=+1 ˆ u kε+1 γ k ε k, ε,y,x k γ k ε k, ε,y dydε +1...dε k ˆ ˆ ε... u ε ζ k (ε k,y),x k ρ k (ε k,y,ε ) dydε +1...dε k. (37) bε Equaion (36) direcly follows from he Fundamenal heorem of Calculus. We now urn our aenion o (37). By Lemma 4, σ +1 is coninuous and monoone. he image of σ +1 ε +1,y on y [ε,ε ] is ε +1,σ +1 ε +1, ε. Hence, afer changing he variables of inegraion, for 37

38 all k = +1,...,: ˆ σ+1(ε +1,bε ) ε ˆ +1 ε u kε+1 bε u kε+1 γ k ε k, ε,y,x k γ k ε k, ε,y dy = γ k ε k, ε,σ +1 ε 1,y,ε +1, ε,x k γ ε k k, ε,σ +1 ε 1,y,ε +1, ε σ +1 ε 1,y,ε +1, ε Recall ha by (11) he following is an ideniy in y: u k ε 1,y,ε +1,...,ε k,x k u k γ k ε k, ε,σ +1 ε 1,y,ε +1, ε,x k, so, by he Implici Funcion heorem, u kε ε 1,y,ε +1,...,ε k,x k y dy. (38) = u kε+1 ε 1, ε,σ +1 ε 1,y,ε +1, ε,...,εk,x k σ +1 ε 1,y,ε +1, ε. (39) y Plugging (39) ino (38) and noing ha γ k ε k, ε,σ +1 ε 1,y,ε +1, ε = ρ k ε k,y,ε yields (37). An idenical argumen can be used o deal wih he case where ε >ε. 38