Selling Options. Simon Board. This Version: November 17, 2004 First Version: June Abstract

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1 Sellng Optons Smon Board Ths Verson: November 17, 2004 Frst Verson: June 2001 Abstract Contracts often take the form of optons: deals can be reneged on, brdges may not be bult, and acqured frms mght go bankrupt. Ths paper consders the auctonng of a dynamc opton, where post aucton nformaton nteracts addtvely wth prvate nformaton but s allowed to take any stochastc form. The revenue maxmsng aucton conssts of an up front bd and a contngent payment (strke prce). The contngent payment equals the opportunty cost of exercsng the contract plus a rent tax that s tme and state nvarant and nversely related to the up front payment. The rent tax s desgned n a Pgouvan manner so that the agents choce of exercse decson maxmses the seller s revenue; t can also be nterpreted as a generalsed reservaton prce. The revenue maxmsng mechansm nduces a dynamc dstorton: the opton s exercsed later than under the comparable welfare maxmsng mechansm. 1 Introducton Ths paper derves the optmal method to sell an opton, a contract that when exercsed yelds a state dependent payoff. The desgner of such a sales procedure has a lot of flexblty. Payment can depend on whom the good s awarded to, when the opton s exercsed (f ever), and the nformaton revealed after the sale. Allowng for all these possbltes, the optmal aucton s derved wth only an addtvty restrcton on the post aucton nformaton. Moreover, ths optmal contract takes a very smple form. The model can be appled to any knd of opton, whether t s physcal or fnancal, real or surreal. After a tmber aucton, the wnner has the rght to harvest the trees at a date of ther Department of Economcs, Unversty of Toronto. I am grateful for comments from Davd Ahn, Alan Beggs, Jeremy Bulow, Peter DeMarzo, Paul Klemperer, John McMllan, Rob McMllan, Martn Osborne, José Quntero, Ilya Segal, Danel Sgro, Jean Trole and Bob Wlson. JEL classfcaton: C7, D4, D8, D9, G3. 1

2 choosng, dependng upon the market prce of lumber. 1 over may be lqudated n a suffcently bad state of nature. 2 A frm that has recently been taken Hollywood studos buy optons on books and make further contngent payments f they decde to make a flm. 3 Ol felds are often sold n auctons yet can be abandoned f the ol prce snks low enough. 4 Optons are also the canoncal example of a state dependent decson, provdng a base for more complcated decson problems. In order to fx deas, suppose N agents compete over the rght to buld houses on a dsused ndustral ste. Before the aucton, the agents have prvate nformaton about ther revenue from house sales, whch may have common value elements. After the aucton, the wnnng agent s costs evolve as nput prces change, and as they dscover more about the cost of clearng up the plot. At any tme after the aucton the wnnng agent can commence constructon, or they may choose to abandon the ste. Ths paper consders mechansms n whch the payment to the seller conssts of two components: an up front charge and a contngent fee pad when the houses are bult. The contngent payment s allowed to depend upon (1) who wns the good; (2) all the agents types; (3) when the opton s exercsed; and (4) any nformaton revealed after the aucton. The crucal dfference between up front and contngent payments s that the latter ntroduces a dstorton. Up front payments are sunk and do not affect when the houses are bult. In comparson, a postve contngent payment wll delay constructon (Proposton 1). Welfare s therefore maxmsed by settng contngent payments equal to zero and usng an up front scheme, such as an Englsh aucton (Theorem 1). In contrast, the revenue maxmsng aucton nvolves a postve contngent payment (Theorem 2). For a loose ntuton, notce that a hgh value agent s more lkely to exercse the opton than a low value agent, and therefore cares more about the sze of the contngent payment. Makng low types pay a contngent fee thus helps the seller separate agents, reducng the ncentve for a hgh value agent to copy a low value agent. More precsely, contngent fees have the effect of reducng nformaton rents. An agent s payoff s ndependent of ther prvate nformaton when they do not exercse the opton, so agents extract nformaton rents only when they choose to exercse. Snce rents are ncreasng n the probablty of exercsng the opton, rasng contngent payments makes the opton less 1 For a descrpton of ths market see Hale (2001). 2 In 1990 a bankng consortum, Old Bond Street Holdngs, bought Yardley, a strugglng Brtsh cosmetcs company. In 1999, after falng to rebrand ts mage, the owners called n the recevers. In a smlar story, ITV Dgtal pad 315 mllon for the rght to broadcast Football League games. In 2002 the owners, two other TV companes, put the frm nto admnstraton after subscrptons fell below expectatons. 3 These optons normally last for one or two years whle the producer arranges for a cast, crew and acceptable screenplay. If flmng goes ahead, the opton s usually exercsed on the frst day of prncpal photography; otherwse the property rghts revert back to the author (Ltwak (1999)). 4 Over , 22% of 5 year OCS wldcat leases expred wthout any wells beng drlled (Porter (1995)). 2

3 desrable, lowerng the probablty of exercse and reducng rents. Contngent payments thus lower welfare by dstortng the exercse decson, but reduce nformaton rents, gvng rse to a tradeoff. An agent chooses an exercse tme to maxmse ther valuaton mnus the contngent payment, whle revenue from ths agent equals the valuaton mnus the expected nformaton rent. Settng the contngent payment equal to ths nformaton rent thus algns ncentves n the style of a Pgouvan tax. The agent s choce of exercse tme then maxmses revenue an act of perfect delegaton. Moreover, ths revenue maxmsng contngent fee s (1) postve, (2) declnng n valuatons, (3) ndependent of when the opton s exercsed, and (4) ndependent of post aucton nformaton. Ths last property s partcularly attractve: t means the seller does not have to observe post aucton data n order to mplement the optmal mechansm. Moreover, f the seller can commt to release extra nformaton they wll always choose to do so, even f they cannot observe ts effect. Hence the seller of the ndustral ste should commt to release any reports on the levels of contamnaton. The revenue maxmsng aucton ntroduces neffcences. In addton to an excessve reservaton prce and a bas aganst larger agents, the postve contngent payment means the opton wll be exercsed later than s socally optmal (Proposton 2). Ths extends a result of Stokey (1979) n the context of durable goods monopoles. The paper consders two extensons of the basc model. The frst extenson s to allow the good to be allocated to dfferent agents dependng upon the state of the world. The welfare maxmsng mechansm can be mplemented by the welfare maxmsng aucton n Theorem 1 f the aucton s held at the rght tme. It can also be mplemented va a Vckrey mechansm when agents have prvate values. Smlar results apply to the revenue maxmsng aucton. The second extenson allows for non addtve valuatons. If the seller can observe the state of the world, the revenue maxmsng mechansm sets the contngent payment equal to the agent s nformaton rent. Unlke the addtve case, revenue wll generally be reduced f the seller cannot observe the state of the world. However, n specal cases, the mechansm wthout observablty can attan the maxmal revenue wth observablty. The paper s structured as follows: Secton 2 ntroduces the model. Secton 3 derves the welfare and revenue maxmsng auctons, and dscusses the nterpretaton. Secton 4 explores the two extensons, and Secton 5 concludes. Omtted proofs are contaned n Appendx A. 1.1 Lterature Mlgrom and Weber (1982) show revenue s ncreased when the sale prce s lnked to nformaton correlated wth agents prvate nformaton. Rley (1988) used ths lnkage prncple to prove royaltes can ncrease revenue n mneral rghts auctons. An extreme example of ths was gven by Hansen (1985) where the wnnng agent s valuaton was ex post observable, leadng to full- 3

4 extracton. In contrast to these models, post aucton nformaton wll tell the seller nothng about the agent s prvate nformaton; however, the act of exercsng the opton wll. The paper s most closely related to the optmal regulaton lterature. Baron and Myerson (1982) consder procurng from a suppler wth unknown cost. Ths s extended by Baron and Besanko (1984) who suppose the suppler s cost may change after the contract s sgned. The optmal procurement aucton s consdered by McAfee and McMllan (1987), Laffont and Trole (1987) and Rordan and Sappngton (1987) who show that the problem separates the contract should be gven to the best type, and then the optmal sngle agent regulaton contract should be mplemented. Analogous reasonng s appled to prce dscrmnaton by Mussa and Rosen (1978) and Courty and L (2000). Secton 3.8 consders these related deas n more detal. The problem of sellng an opton s dentcal to the problem of sellng a durable good where agents valuatons vary over tme. When there s one agent, the model of ths paper can thus be nterpreted as the optmal sales contract for a durable good monopolst wth elastc demand. To llustrate, consder a car manufacturer whose sales depend on the (uncertan) level of nterest rates. Rather than chargng a sequence of prces, Theorem 2 suggests the manufacturer may do better by askng for a down payment when a new model frst comes out n exchange for a reduced sale prce. Ths problem was frst consdered by Conlsk (1984). More recently, Laffont and Trole (1996) and Behl (2001) have derved the optmal rental contracts for such a durable goods monopolst n a two perod model. Other authors have consdered constraned mechansms. Stokey (1979) analyses a mult perod model, but assumes payments are only contngent,.e. the frm charges a sequence of prces. Hansen (1988), Board (2003), and DeMarzo, Kremer, and Skrzypacz (2004) also analyse contngent payment schemes, whle Waehrer (1995) supposes that the contngent payment s ndependent of types. Hale (2001, 2003) and McAfee, Takacs, and Vncent (1999) consder a two perod game where a good s sold n perod 1. New nformaton then becomes avalable and, n perod 2, agents may engage n resale. Ths s a hard problem snce the opportunty to purchase n the resale market affects bddng n the orgnal aucton. The optmal auctons are, however, more straghtforward. Welfare s maxmsed by sellng the good n the second perod, after the uncertanty has resolved. Smlarly, revenue s maxmsed by runnng an aucton n the second perod and also demandng the wnner make a contngent payment equal to ther nformaton rent (Eso and Szentes (2003) and Secton 4.1). However ths soluton s not wthout problems. Schwarz and Sonn (2001) observe that the seller cannot smply wat untl the second perod f an agent must make nvestments before they can harvest the tract. In addton, as ths paper hghlghts, the seller cannot smply wat f they do not know when the agents wll exercse ther opton. 4

5 2 Model Suppose each of N agents has a net valuaton consstng of two elements: an ex ante valuaton and an ex post cost. The ex ante valuaton of agent s determned by ther prvately observed type θ [θ, θ]. Agents types are mutually ndependent, where θ s dstrbuted accordng to F (θ ) wth densty f (θ ). Agent s ex ante valuaton s denoted v (θ, θ ), where θ := (θ 1,..., θ 1, θ +1,..., θ N ). The functon v ( ) has several standard propertes: t s contnuously dfferentable n all arguments, ts dervatve v / θ s strctly postve and bounded above, and v j / θ v / θ for j. Ths formulaton ncludes prvate values (v (θ) = θ ) and common values (v (θ) = j θ j) as specal cases, where θ := (θ 1,..., θ N ) The ex post cost evolves over tme, whch s dscrete and fnte, t {1,..., T }. The cost of each agent consttutes a sequence {c,t } of random varables, determned by the state of the world, ω Ω. These costs are unrestrcted n ther dependence across tme and across agents, although they are ndependent of the agents types. The nformaton about costs possessed by agent after the sale s descrbed by a fltered space (Ω, F, {F,t }, Q), where Ω are the states of the world, F the measurable sets, {F,t } the nformaton parttons, whch grow fner over tme, and Q the probablty measure. At each tme t, the agent knows ther current costs, {ω : c,t c} F,t for each c R. That s c,t s F,t measurable ( t), or more smply, c,t s F,t adapted. Let F t = F,t be the total nformaton avalable at tme t, and assume F t contans no more nformaton about s costs than F,t,.e. E[c,t F t ] = E[c,t F,t ] for t t ( ), where E denotes the expectaton over ex post costs. In terms of the example, mentoned n the Introducton, of a frm who has the rght to buld a housng estate on a pece of land, one can thnk of v (θ) as the revenue from the sale of the houses. The cost c,t can ether be nterpreted as (a) the materal cost of startng constructon at tme t, or (b) the tme t estmate of the ultmate clean up cost c, where c,t = E[c F,t]. Denote the common dscount factor by δ 1. If agent exercses the opton n perod t T ther net valuaton s (v (θ) c,t )δ t and f they never exercse the opton, they receve 0. Ths addtve formulaton s mportant for the smplcty of the revenue maxmsng aucton. Secton 4.2 examnes the robustness of the results to relaxng ths assumpton. Absent any payments, the problem of an agent, who knows ther valuaton v (θ), s to choose a stoppng tme (exercse tme) τ to solve the problem: max τ E[(v (θ) c,τ )δ τ ] 5

6 The stoppng tme depends upon the random sequence of costs, so s tself a random varable takng values n {1,..., T } { }, where τ = means that the opton s never exercsed. In addton, the decson to stop at tme t can depend only on nformaton avalable at tme t, {ω : τ t} F,t ( t). Abusng termnology slghtly, we wll say the such a stoppng tme τ s F,t adapted. Snce F,t s as nformatve about agent s costs as F t, the agent s choce of stoppng tme also maxmses utlty amongst the class of F t adapted stoppng rules. Now consder the seller s problem of desgnng a sales procedure. The seller observes the evoluton of costs, {F t }, and chooses a three part drect revelaton mechansm P, y, z consstng of an allocaton functon, an up front payment and a contngent payment. After the mechansm s determned, each agent of type θ reports θ, where the vector of reports s denoted θ. After the the opton s awarded to an agent and the up front fees are pad, the seller publcly releases the reports θ. 5 The wnnng agent then chooses whether and when to exercse the opton, makng a contngent payment when they do so. More formally, the mechansm s defned as follows: The allocaton functon P : [θ, θ] N [0, 1] s the probablty that agent wns the object, where P ( θ) 1. The up front payment functon y : [θ, θ] N R s a transfer to the seller made by agent mmedately after the aucton and s ndependent of post aucton costs. The contngent payment functon z,t : [θ, θ] N Ω R s a F t adapted transfer made by agent f they wn and exercse the opton n perod t. One should note that the contngent payment s allowed to depend upon: () the dentty of the agent, () all the agents reports, () the tme the opton s exercsed, and (v) the entre hstory of the stochastc process. Ths contngent payment can also be nterpreted as the strke prce. Let z = {z,t } t. An equvalent, and more standard, way to formulate the problem s to have the seller choose a drect revelaton mechansm P, x, τ consstng of an allocaton functon P : [θ, θ] N [0, 1] determnng whch agent obtans the good, a payment functon x : [θ, θ] N R, and a F t adapted stoppng tme τ : [θ, θ] N Ω {1,..., T } for the wnnng agent. The equvalence of these mechansms follows from the revelaton and taxaton prncples and proved n Appendx A.1. Ths latter mechansm, however, s harder to nterpret and mplement than the contngent payment mechansm, P, y, z. 5 Releasng ths nformaton wll always make the seller better off snce they can contract on θ. Any allocaton mplemented when θ s not released can be mplemented when θ s released by choosng approprate payments. When agents have prvate values (v (θ) = θ ) there s no need for the seller to release these reports. 6

7 The mechansm P, y, z contans one notable restrcton: the allocaton of the good does not depend upon post aucton costs. Ths restrcton may be serous f the wnner turns out to have substantally hgher costs than a losng agent. However, f the agents post aucton costs are smlar, reallocaton wll not be desrable. The ssue s further explored n Secton 4.1. If all agents report truthfully and the opton s allocated to agent, who exercses the opton usng stoppng rule τ, they obtan ex post utlty u (θ, z, τ ) := E[(v (θ) c,τ z,τ (θ))δ τ ] (2.1) The aucton mechansm can be summarsed by the followng steps: Tme t = 1. Each agent observes type θ. The seller chooses the mechansm P, y, z to maxmse welfare or revenue (as defned n Secton 3). Tme t = 0. Each agent reports ther type θ. The good s allocated to agent wth probablty P ( θ) and they make up front payment y ( θ). The seller then reveals θ to the wnnng agent. Tme t {1,..., T }. After the aucton, costs {c,t } are revealed, whle the agent observes F,t. Agent may then choose exercse at tme t and make contngent payment z,t ( θ), or they may choose never to exercse the opton. 2.1 Optmal Stoppng Problem Before analysng the mechansm desgn problem, t wll be useful to establsh some propertes of the wnnng agent s optmal exercse decson. In stage 3 of the game, the mechansm P, y, z has been determned, and the opton has been awarded to some agent. Ths agent must then choose a stoppng tme τ to maxmse ther ex post utlty (2.1). Denote the set of maxmsers by ˆτ. The set of stoppng rules forms a lattce, where τ H τ L f τ H (ω) τ L (ω) (a.e. ω Ω). Comparng two sets of stoppng rules, ˆτ H ˆτ L n strct set order f τ ˆτ H and τ ˆτ L mply that τ τ ˆτ H and τ τ ˆτ L. Proposton 1. The soluton ˆτ of agent s optmal stoppng problem (2.1) has the followng propertes: (a) ˆτ s a nonempty sublattce contanng a greatest and least element. (b) ˆτ s decreasng n θ n strct set order. (c) Fx z,t and consder chargng a contngent payment z,t + K ( t), for a constant K. Then ˆτ s ncreasng n K n strct set order. Proof. (a) Nonemptness follows from backwards nducton (e.g. Chow, Robbns, and Segmund (1971, Theorem 3.2)). Ths constructon also mples the set of maxmsers has a greatest and 7

8 least element. To fnd the least (greatest) element use the rule: stop when current payoffs are weakly (strctly) larger than the contnuaton utlty. Snce the set of stoppng tmes s a lattce and u (θ, z, τ ) s modular n τ, the set of maxmsers s a sublattce by Topks (1998, Theorem 2.7.1). (b) u (θ, z, τ ) satsfes decreasng dfferences n (θ, τ ), snce δ 1, and s modular n τ. Hence the optmal soluton s decreasng by Topks (1998, Theorem 2.8.1). (c) u (θ, z + K, τ ) satsfes ncreasng dfferences n (K, τ ), snce δ 1, and s modular n τ. Hence the optmal soluton s ncreasng by Topks (1998, Theorem 2.8.1). Proposton 1(b) says that agents wth hgh valuatons are more mpatent and choose to stop earler. 6 Smlarly, Proposton 1(c) says that when the contngent payment decreases the agent s effectve valuaton ncreases and they stop earler. Let τ be the least element from ˆτ. From Proposton 1, ths exsts, s decreasng n θ and ncreasng n K. 3 Optmal Auctons Ths secton characterses the mechansms P, y, z that maxmse revenue and welfare. 3.1 Informaton Rents Agent chooses ther stoppng tme τ and ther reported type θ to maxmse nterm utlty E θ [P ( θ, θ ) E [( v (θ, θ ) c,τ z,τ ( θ, θ ) ] ) δ τ y ( θ, θ ] ) where E θ s the expectaton over other agent s types. Truthful revelaton s a Bayesan Nash equlbrum f the drect revelaton mechansm satsfes ncentve compatblty and ndvdual ratonalty. From agent s perspectve, f the other agents report truthfully, ther nterm utlty gven type θ, report θ and stoppng tme τ s, U (θ, θ, τ ) = E θ [P ( θ, θ ) E [( ) ] ] v (θ) c,τ z,τ ( θ, θ ) δ τ y ( θ, θ ) Incentve compatblty then says U (θ, θ, τ ) U (θ, θ, τ ), whle ndvdual ratonalty states U (θ, θ, τ ) 0. (3.1) In order to examne how utlty s affected by a change n θ, we would lke to use an envelope theorem where the agent optmses over ( θ, τ ). The space of stoppng tmes s too complcated for the usual envelope theorem on R N to be appled, so we wll use the generalsed 6 There s emprcal support for Proposton 1(b). In OCS wldcat auctons, Porter (1995) observes that the frst tracts to be drlled were those wth hgh bds; these tracts also led to hgher ol revenues. 8

9 envelope theorem of Mlgrom and Segal (2002). 7 Interm utlty, when the agent chooses the optmal stoppng rule, τ, and reports ther type truthfully, θ = θ, can then be expressed as the ntegral equaton, U (θ, θ, τ ) = E θ [ θ θ ] P (s, θ )E[δ τ ] v (s, θ ) ds θ + U (θ, θ, τ ) (3.2) Equaton (3.2) mples that an agent s nformaton s useful only when they execute the opton, so ths s the only tme when they collect rents. Thus by delayng when the agent exercses, the seller can reduce the agent s utlty and potentally ncrease revenue. Incentve compatblty mples that nterm utlty can be expressed by the ntegral representaton (3.2). It also mples that nterm utlty U (θ, θ, τ ) s supermodular n (θ, θ ). Ths s the monotoncty condton and mples that [ ] E θ P ( θ, θ )E[δ τ ] v (θ) θ s ncreasng n θ. Together, (3.2) and (3.3) are necessary and suffcent for ncentve compatblty, as shown n Appendx A.2. Generally, the monotoncty condton mght be very complcated (e.g. f contngent payments dffer over tme), but s smple for the optmal mechansm. (3.3) Indvdual ratonalty states that nterm utlty s postve, U (θ, θ, τ ) 0 ( θ ). Snce transferrng money from the seller to the agents wll not mprove welfare or revenue, we henceforth assume that ncentve compatblty bnds for the lowest type, U (θ, θ, τ ) = 0. Takng expectatons wth respect to type θ and ntegratng by parts yelds ex ante utlty: 3.2 Seller s Payoffs [ E θ [U (θ, θ, τ )] = E θ P (θ)e[δ τ ] v (θ) 1 F ] (θ ) θ f (θ ) Consder sellng a tract of land to an ol company, a loggng frm, or a constructon company, that goes to waste f not used. The seller s payoff equals the sum of the up front and contngent payments when the opton s sold. When the opton s not sold the seller obtans v 0, whch may 7 The theorem requres three condtons be met: (1) U s dfferentable wth respect to θ, whch holds because v s dfferentable, (2) The dervatve U s bounded, whch holds because v (θ)/ θ s bounded, (3) The maxmum s attaned. Proposton 1(a) says the stoppng rule attans ts supremum for any gven θ whle, n a Bayesan Nash equlbrum, the agent s payoff s maxmsed at θ = θ by the revelaton prncple. 9

10 be stochastc, but s ndependent of θ. To summarse, the payoffs are: Agent Seller Not Awarded 0 v 0 Exercsed at t (v c,t z,t ) δ t y z,t δ t + y Not Exercsed y y Some examples may have dfferent payoff structures. Ths s dscussed n Secton Welfare Maxmsaton Defne welfare as the sum of the agents utltes and the seller s revenue. Ths equals the value of the opton: [ ] [( Welfare = E θ P (θ) E[(v (θ) c,τ )δ τ ] + E θ 1 ) ] P (θ) E[v 0 ] (3.4) The welfare maxmsaton problem s to maxmse (3.4) subject the to monotoncty condton (3.3) and τ maxmsng ex post utlty u (θ, z, τ ), as defned by (2.1). Notce that f there s no contngent payment then utlty concdes wth welfare and τ s welfare optmal. Hence: Theorem 1. Suppose ether: (a) agents have prvate values, or (b) the dstrbuton of costs s the same for all agents. Then welfare s maxmsed by a mechansm wth contngent payments z,t W (θ) = 0 ( θ) and allocaton P W (θ) = 1 f u (θ, z W, τ ) > u j(θ, zj W, τ j ) j and u (θ, z W, τ ) > E(v 0) 0 otherwse (3.5) Proof. See Appendx A.3. Corollary 1 (Symmetry). Suppose the valuaton functon v ( ) and the dstrbuton of costs are the same for all agents. Then welfare s maxmsed by awardng the opton to the agent wth the hghest type, θ, f the ex post utlty (2.1) from the opton exceeds the value to the seller. Proof. Under symmetry, θ θ j mples u (θ, z W, τ ) u j(θ, zj W, τ j ). Contngent payments dstort the wnnng agent s optmal stoppng problem, delayng the exercse tme. Hence the welfare maxmsng aucton sets contngent payments to zero and awards the object to the agent wth the hghest ex post utlty. 10

11 Wth prvate values the seller can use any mechansm that awards the good to the agent wth the hghest utlty, such as an Englsh or second prce aucton. If agents are symmetrc (as s Corollary 1) the seller can use any standard aucton that allocates the good to the agent wth the hghest type, such as a frst prce, second prce, or all pay, even s there are common value components. Wth asymmetres and common values one can use the scheme of Dasgupta and Maskn (2000). For the welfare maxmsng mechansm to satsfy the monotoncty condton (3.3) we requre that the probablty s awarded the good to ncrease n s reported type. Ths s equvalent to assumng u (θ, z W, τ ) u j(θ, z W, τj ) s quas ncreasng n θ ( θ ). 8 Whle the condtons n Theorem 1 are suffcent for monotoncty to be satsfed, the effcent aucton may be mpossble to mplement wth asymmetrc costs and common values. Example 1. Suppose a European opton, where T = 1, s auctoned to two agents, A and B, who have common values. Assume only agent A has any post aucton uncertanty. Payoffs are u A = E max{θ A + θ B c, 0} and u B = θ A + θ B where c may be negatve. Defne θ by E θa [E max{θ A + θ c, 0}] = E θa [θ A + θ ] (3.6) The welfare maxmsng aucton then sets contngent payments to zero and allocates the contract to A f θ B θ, and to B f θ B > θ. Ths allocaton functon s ndependent of A s type Revenue Maxmsaton Expected revenue equals welfare mnus agents total utlty. [ Revenue = E θ P (θ) E[(v (θ) c,τ )δ τ v 0 ] ] U (θ, θ, τ ) + E[v 0 ] [ ] = E θ P (θ) E[(MR (θ) c,τ )δ τ v 0 ] + E[v 0 ] (3.7) where we follow Bulow and Roberts (1989) n denotng s margnal revenue (or vrtual utlty) by MR (θ) := v (θ) 1 F (θ ) v (θ) (3.8) f (θ ) θ 8 A functon f : R R s quas ncreasng f f(x) 0 mples f(y) 0 for y x. Ths condton s relatvely common, e.g. Dasgupta and Maskn (2000). 9 See Appendx A.4 for a proof. 11

12 Equaton (3.7) yelds a revenue equvalence result: any mechansm that has the same allocaton functon P ( ), gves the lowest type no surplus and nduces the same optmal stoppng rule yelds the same revenue. Consequently the revenue maxmsng aucton wll not be able to pn down the up front payment scheme, just the expected up front payment. Wth ths n mnd, the seller s am s to pck P, z to maxmse revenue (3.7) subject to the monotoncty condton (3.3) and τ maxmsng ex post utlty (2.1). Assumpton (MON). The nformaton rent (or margnal consumer surplus) 1 F (θ ) v (θ) f (θ ) θ s dfferentable and decreasng n θ. If v (θ) s concave n θ ( θ ), (MON) s mpled by the usual monotone hazard condton. The wnnng agent s ex post utlty (2.1) equals ther net valuaton mnus ther contngent payment. Revenue (3.7) equals the wnnng agent s net valuaton mnus ther nformaton rent. Hence settng the contngent payment equal to the nformaton rent term nduces the agent to choose the optmal stoppng rule. Theorem 2. Suppose (MON) holds and ether: (a) agents have prvate values, or (b) the dstrbuton of costs s the same for all agents and MR (θ) MR j (θ) s quas ncreasng n θ (, j). Then revenue s maxmsed by a mechansm wth contngent payments z,t(θ) R = 1 F (θ ) v (θ) (3.9) f (θ ) θ and allocaton P R (θ) = 1 f u (θ, z R, τ ) > u j(θ, zj R, τ j ) j and u (θ, z R, τ ) > E(v 0) 0 otherwse (3.10) Proof. (a) Contngent payments. Settng contngent payments accordng to (3.9) mples τ arg max E[(v (θ) c,τ z,τ R (θ))δ τ ] [ ] arg max E θ P (θ)e[(mr (θ) c,τ )δ τ v 0 ] + E[v 0 ] so when agents choose τ to maxmse ex post utlty (2.1) they also maxmse revenue (3.7). (b) Allocaton rule. The opton should be awarded to the agent who nduces the hghest revenue, subject to ths value exceedng the seller s valuaton, yeldng equaton (3.10). 12

13 (c) Fnally, we need to check the monotoncty condton (3.3). Ths s done through three clams. Clam 1: E[δ τ ] s ncreasng n θ. Proof: An ncrease n θ reduces z,t R unformly across tme, by (MON). By Proposton 1(c), ths reduces τ and therefore ncreases E[δ τ ]. Clam 2: P R ( θ, θ ) s ncreasng n θ. Proof: In the revenue maxmsng aucton ex post utlty s Let us establsh two facts. u (θ, z R, τ ) = E [ ] (MR (θ) c,τ )δ τ (1) u (θ, z R, τ ) s ncreasng n θ. The envelope theorem mples θ u (θ, θ, z R, τ ) = u (θ, θ, z R, τ ) + θ θ MR (s, θ )E[δ τ (s,θ ) ] ds whch s ncreasng n θ snce margnal revenue MR (θ) s ncreasng n θ, by (MON). (2) u (θ, z R, τ ) u j(θ, z R j, τ j ) s quas ncreasng n θ. Applyng the envelope theorem to u j (θ, z R j, τ j ), θ u j (θ, θ, zj R, τj ) = u j (θ, θ, zj R, τj ) + θ θ MR j (s, θ )E[δ τ j (s,θ ) ] ds If agents have prvate values then u j (θ, z R j, τ j ) s constant n θ, so (1) mples (2). Next, suppose all types have the same dstrbuton of costs and u (θ, z R, τ ) = u j(θ, z R j, τ j ) > 0. Then MR (θ) = MR j (θ) and E[δ τ ] = E[δ τ j ]. The assumpton that MR (θ) MR j (θ) s quas ncreasng n θ means whch mples (2). θ [MR (θ) MR j (θ)] 0 so that Clam 3: U (θ, θ, τ ) s supermodular n (θ, θ ). Proof: The two clams then mply that θ U (θ, θ [, τ ) = E θ ncreasng n θ, as requred. θ [u (θ, z R, τ ) u j(θ, z R j, τ j )] 0 ] ( θ, θ )E[δ τ ] θ v (θ) s Corollary 2 (Symmetry). Suppose (MON) holds, the valuaton functon v ( ), the dstrbuton of types F (θ ) and the dstrbuton of costs are the same for all agents, and MR (θ) MR j (θ) s quas ncreasng n θ. Then revenue s maxmsed by awardng the opton to the agent wth the hghest type, f the ex post utlty (2.1) from the opton exceeds the value to the seller. P R Proof. Under symmetry, θ θ j mples u (θ, z R, τ ) u j(θ, z R j, τ j ). Theorem 2 states that the optmal contngent payment s set accordng to equaton (3.9), and the opton then allocated to the agent wth the hghest ex post utlty. There s more 13

14 flexblty over the up front payment whch can be determned n any number of ways so long as nterm utlty (3.1) equals the ntegral equaton, (3.2). If agents have prvate values the mechansm can be mplemented by a second prce aucton whch reveals the hghest agent s type and allocates the good to the agent wth the hghest ex post utlty. If agents are also symmetrc (as n Corollary 2), the seller can use any standard aucton that reveals the wnnng agent s type, such as a frst prce, second prce or all pay aucton (but not an Englsh aucton). The optmal contngent payment can then be deduced from the wnner s up front bd. Wth common values, the seller needs to use a mechansm that reveals all agents types. Under symmetry, the frst prce, second prce and all pay auctons also satsfy ths crteron. The optmal mechansm n Theorem 2 s not unque. For example, the usual mechansm desgn approach s to derve the optmal stoppng tme τ and to use a forcng contract, demandng the agent pay an nfnte amount f they choose anythng other than τ. Ths mechansm, however, s hard to nterpret, harder to mplement, and depends on post aucton costs. In contrast, the contngent payment n Theorem 2 s ndependent of post aucton costs. Ths means that the optmal mechansm can be mplemented when the seller cannot observe, or cannot contract on, cost data. Suppose the seller has some extra nformaton about costs {c,t } not possessed by the agents. That s, the seller knows F t F t. If the seller can contract on F t then they should automatcally release ths nformaton snce they can always mplement the outcome wthout revelaton by choosng approprate payments. After the nformaton s released, the revenue maxmsng mechansm s gven by Theorem 2, whch s ndependent of costs. Consequently the seller always wshes to release cost nformaton, even when they cannot observe the nformaton they are releasng. The revenue maxmsng aucton ntroduces two dstortons. Frst, the strke prce s too hgh. Second, the good may be allocated to the wrong agent (f agents are asymmetrc) or not awarded at all. Ths tells us about the type of dynamc dstorton that market power can nduce. Proposton 2. If the dstrbuton of costs s the same for all agents then the opton wll be exercsed earler under the welfare maxmsng aucton (Theorem 1) than under the revenue maxmsng aucton (Theorem 2). Proof. Suppose agent wns the welfare maxmsng aucton and let ˆτ W be the set of stoppng rules that maxmse ex post utlty (2.1). Smlarly, suppose agent j (who may be the same as ) wns the revenue maxmsng aucton and let ˆτ j R be the set of stoppng rules that maxmse ex post utlty (2.1). Snce the dstrbuton of costs s the same for all agents, v (θ) v j (θ) MR j (θ). Proposton 1(b) then mples that ˆτ j R ˆτ W n strct set order, and the least stoppng tme s greater under the revenue maxmsng mechansm. 14

15 Settng N = 1, Proposton 2 mples that a durable goods monopolst wth varyng demand (or varyng costs) always sells later than a perfectly compettve frm. Ths extends the result of Stokey (1979), who assumes the sequence of costs {c,t } s determnstc and gradually decreases. 3.5 Interpretaton Theorem 2 says that the problem separates. The opton s frst allocated to the agent wth the hghest ex post utlty (2.1). In the second stage, the contract mplements the optmal sngle agent contract, where the contngent payment (strke prce) s postve, declnng n the agent s type, ndependent of post aucton costs, and ndependent of the tme of executon. The revenue optmal mechansm works n a smple way. Informaton rents drve a wedge between welfare and revenue. When the agent chooses the stoppng tme they are nterested n maxmsng ther dscounted valuaton, whereas the seller s nterested n maxmsng ther dscounted margnal revenue. Insertng a Pgouvan tax equal to the expected nformaton rents means the agent s and seller s problems concde an act of perfect delegaton. As shown n Secton 3.1, agents collect rents only when they exercse the opton. Thus the Pgouvan tax, the rent tax, wll be charged only when an agent executes the opton, and can be fully captured by the contngent payment. Rents are smaller for hgher types, snce there are fewer agents who wsh to copy them, so agent s contngent payment s decreasng n θ. Consequently the hghest type makes no contngent payment (.e. no dstorton at the top). Rents are lnear n the dscount factor, so the contngent payment s ndependent of tme. And post aucton costs are addtve, so rents do not depend on the cost, nor does the contngent payment. 3.6 Some Numbers The contngent payment can be substantal. Fgure 1 shows the revenue maxmsng frst prce bddng locus for a European opton (T = 1) wth prvate values (v (θ) = θ ), where c, θ U[0, 1], v 0 = 0, and δ = 1. In ths example, the contngent payment s larger than the up front payment for many types of agents. Whle ths provdes some ndcaton about the sze of the optmal contngent payment, t says nothng about the welfare and revenue effects of runnng such a mechansm. For example, consder a European opton (T = 1) wth prvate values (v (θ) = θ ), where ln c N(0, 1), θ exp(1), v 0 = 0 and δ = 1. The followng table compares the welfare maxmsng aucton (Theorem 1) and the revenue maxmsng aucton (Theorem 2): These numbers are based on 10,000 smulatons. 15

16 Up Front Payment Bdders 5 Bdders 100 Bdders 10 Bdders Contngent Payment Fgure 1: Frst Prce Bddng Locus for a European Opton Revenue Welfare #of agents Revenue Max Welfare Max Change +12.8% +6.1% +1.1% 20.1% 5.7% 1.0% 3.7 Generalsng the Seller s Payoff In Secton 3.2 we assumed the seller s payoffs are fully captured by the transfer payments. However, ths s not always the case. If a ol company fals to drll wthn fve years of acqurng an OCS lease, the tract returns to the government. Smlarly, before a fnancal opton s exercsed the dvdends accrue to the seller. Instead, suppose payoffs are gven by: agent Seller Not Awarded 0 v 0 Exercsed n t (v c,t z,t ) δ t y a t + z,t δ t + y Not Exercsed y b + y 16

17 where b and a t may be random and are ndependent of agent s types. When the agent chooses to stop at tme t the seller loses b a t, so the welfare maxmsng aucton nserts a Pgouvan tax equal to the opportunty cost of the opton. z W,t (θ) = E[b a t F t ] The contngent payment may now depend on tme and the post aucton costs. Smlarly, the optmal contngent payment under revenue maxmsaton s z,t(θ) R = 1 F (θ ) v (θ) + E[b a t F t ] (3.11) f (θ ) θ Hence the revenue maxmsng strke prce equals the opportunty cost plus the rent tax. 3.8 Dscusson The dea of a rent tax can also be used to renterpret the solutons to a number of other classc economc problems. Frst, consder the textbook monopolst, where consumer has valuaton, θ, dstrbuted accordng to F (θ ). The proft maxmsng soluton s to sell to the agent f margnal revenue exceeds costs, MR(θ ) c. The usual way to mplement ths outcome s to charge a prce MR 1 (c). Alternatvely the frm could set a contngent payment z = c + (1 F (θ ))/f(θ ) n addton to an up front fee. Agents would then purchase f θ z 0,.e. f MR(θ ) c. Smlarly, one can use the rent tax to nterpret the reserve prce n a standard aucton as an opton contract. The seller should charge a contngent payment z = v 0 + (1 F (θ ))/f(θ ) and then award the good to the hghest agent wthout a reserve prce. Agents wll then only execute the opton f θ z 0,.e. f MR(θ ) v 0. Hence a reserve prce s a specal case of an opton. Moreover, ths mechansm s robust to addtve uncertanty. In addton, f a prce s more credble than an allocaton, ths method of mplementaton mght help overcome the Coase conjecture (McAfee and Vncent (1997)). 11 Theorem 2 can be used to derve the optmal contngent payment n a ol tract aucton. Suppose agent has estmate, θ F, of the extracton cost, k (θ), whch may have common value components. Ol prces, p t, are uncertan, whle the amount of ol n the ground s normalzed to 1. Gven contngent payments z,t (θ), suppose agent s ex post utlty from extractng the ol usng exercse rule τ s u (θ, z, τ ) := E[(p τ k (θ) z,τ (θ))δ τ ]. Ths equaton s just a relabelng of (2.1) and, correspondngly, the revenue maxmsng contngent 11 For example, sellng an opton s lkely to gve the wnnng agent a lot of barganng power. In comparson, not awardng the good keeps any barganng power n the hands of the seller. 17

18 payment s z,t(θ) R = F (θ ) k (θ) f (θ ) θ The wnnng agent chooses τ to maxmse u (θ, z R, τ ) = E[(p τ k (θ) z R,t (θ))δτ ]. In comparson, the OCS wldcat auctons use a contngent payment of z,t = p t /6 (Porter (1995)). The wnnng agent then chooses τ to maxmse u (θ, z R, τ ) E[(p τ (6/5)k (θ))δ τ ]. Whle the optmal exercse decsons may concde, there s no reason to beleve they wll do so. Next, consder a second degree prce dscrmnatng monopolst who can award dfferent qualty, q, to dfferent agents. Suppose that for agent the margnal beneft from qualty s θ F (θ ), and the cost of qualty s gven by the convex functon c(q). Mussa and Rosen (1978) show the prncpal equates margnal costs to margnal beneft mnus the nformaton rent: c (q) = θ 1 F (θ ) f(θ ) Ths outcome can be mplemented by chargng a prce, z (q) = c(q) + q(1 F (θ ))/f(θ ), equal to cost plus the rent tax, n addton to an up front payment. Once the rent tax has been mposed, the prncpal s problem has been perfectly delegated and the agent s choce of q maxmses the prncpal s revenue. Baron and Myerson (1982) consder a prncpal who s tryng to procure quantty q of an nput from an agent wth unknown cost. If V (q) s the prncpal s beneft and θ the margnal cost, ths s the reverse of the prce dscrmnaton story. The optmal soluton can then be mplemented by payng z (q) = V (q) qf (θ )/f(θ ) for quantty q, and settng an approprate up front payment. Ths s an adjusted verson of the sellng the frm strategy suggested by Loeb and Magat (1979) and nduces the agent to supply the optmal quantty. Snce a durable good s an opton, Theorem 2 derves the optmal sales contract for a sngle durable good wth stochastc valuatons. The optmal sales contract for a durable goods monopolst wth elastc supply can then be obtaned by settng N = 1, where v 0 s nterpreted as the margnal cost. Agent s awarded an opton f u (θ, z R, τ ) v 0. The contract then conssts of a contngent payment z,t R (θ) along wth an up front payment such that nterm utlty (3.1) equals the ntegral equaton, (3.2). Ths sales contract can be compared wth Laffont and Trole (1996) and Behl (2001) who analyse the optmal rental contract wth stochastc valuatons. Ther models allow for non addtve valuatons and consder only two perods, however, for the sake of comparson suppose the rental value each perod t s (1 δ)(θ c t ). Usng the rent tax approach, the optmal rental contract then charges a contngent payment z = (1 δ)(1 F (θ ))/f(θ ) every perod the good s rented, along wth an up front fee The choce between a rental and sale contract depends upon the crcumstances. Whle rentng wll generally be preferred to sellng, t may not be possble. Many goods are rreversble: once a brdge has been bult there 18

19 These examples show how the rent tax approach can be used n a varety of dfferent stuatons. In the smple examples, such as textbook monopoly prcng, there are smpler schemes. However, n more complex problems, rent taxaton seems appealng and has the advantage that t s ndependent of addtve shocks to valuatons. 4 Extensons Ths secton looks at two extensons of the basc model. Secton 4.1 consders the queston of aucton tmng and the optmal state dependent allocaton functon. Secton 4.2 analyses the payoffs wth non addtve valuatons. The secton pants a general pcture of these extensons, but wll only consder the relaxed problems, omttng the monotoncty condton. nterpretng the results, but should not affect the basc ntuton. 4.1 Aucton Tmng and State Dependent Allocaton Ths caveat should be born n mnd when One advantage of a mult perod model s that s allows us to analyse when the aucton should be held. Suppose the seller contracts at tme 0 (as n Secton 3), but chooses to hold the aucton at tme s, whch tself s an F t adapted stoppng tme, {ω : s t} F t. 13 Defne the nformaton avalable at the start of the aucton to be F s := {A : A {s = t} F t t}. The mechansm can be descrbed by the trple P, y, z, where P : [θ, θ] N [0, 1] s a F s measurable allocaton functon, y : [θ, θ] N R s the up front payment made at tme 0, and z,t : [θ, θ] N Ω R s a F t adapted contngent payment. As n Secton 3, the revenue maxmsaton problem s [ Revenue = E θ E P (θ)e [ (MR (θ) c,τ )δ τ v 0 F s ] ] + E[v 0 ] s.t. τ arg max τ s E[(v (θ) c,τ z,τ )δ τ ] where MR (θ) s defned by (3.8). Gven a startng tme s, revenue s maxmsed by the polcy s Theorem 2. The contngent payment s set equal to the nformaton rent (3.9), and the good gven to the agent wth the hghest tme s ex post utlty, E[(v (θ) c,τ z,τ (θ))δ τ F s ]. 14 Wth prvate values ths can be mplemented usng the handcap aucton of Eso and Szentes s lttle pont n takng t down n the off season. Other goods are one tme experences: watchng Ctzen Kane s a certanly durable event yet can only be rented f the agent suffers from severe amnesa. 13 The fact the contract s sgned at tme 0 s mportant and means that the agents do not gan rents from knowledge of post aucton costs. 14 Smlarly, welfare s maxmsed by the polcy n Theorem 1 when mplemented at tme s. 19

20 (2003). At tme 0 each agent s assgned contngent payment z,t R (θ) and makes up front payment ŷ (θ) chosen so that ncentve compatblty holds. At tme s the auctoneer than holds a second prce aucton that allocates the good to the agent wth the hghest tme s ex post utlty. When choosng the startng tme s, there are two man effects at work. Frst, the decson effect. By delayng the aucton the agent may have mssed good opportuntes to execute the opton, reducng revenue. Second, the allocaton effect. By delayng the aucton, nformaton s revealed and the seller ncreases revenue. In the case of a tmber auctons, contract terms are generally longer than the tme needed to harvest, so most agents wat untl the untl the end of ther contract (Hale (2001)). By delayng the aucton, a more effcent allocaton would be possble. However, delayng the aucton may also mean some tracts were harvested too late. The begs the queston: when s the best tme to hold the aucton? In order to answer ths, let us consder the optmal state dependent mechansm State Dependent Allocaton Defne the state dependent mechansm by a par σ, x. The exercse functon σ : [θ, θ] N Ω {1,..., T } { } descrbes when agent exercses the object, where {ω : σ ( θ) t} F t. If σ ( θ) = then agent never exercses, and 1 σ ( θ)< 1 snce only one agent can wn.15 Let the seller be agent 0. Payment s gven by x : [θ, θ] N R. In the Bayesan Nash equlbrum, agent chooses ther reported type θ to maxmse nterm utlty U (θ, θ ) = E θ E[(v (θ) c,σ ( θ,θ ) )δσ ( θ,θ ) x ( θ, θ )] Followng Secton 3, welfare s the sum of the seller s and agents utltes, Welfare = E θ E[(v (θ) c,σ (θ))δ σ (θ) ] Let σ W (θ) = {σ W mnus rents, (θ)} be the welfare optmal stoppng rule. 16 Smlarly, revenue equals welfare Revenue = E θ E[(MR (θ) c,σ (θ))δ σ (θ) ] 15 In Secton 3 the allocaton was determned by two functons: P ( ) says whch agent wns the opton, and τ says when the wnnng agent executes. The σ ( ) formulaton merges these two functons nto one. 16 Under prvate values the welfare maxmsng polcy satsfes the monotoncty constrant, that U (θ, θ ) s supermodular n (θ, θ ). Intutvely, when agent s type rses they are allocated the good more often and are allocated t earler (by Proposton 1)(b). Wth common value components, monotoncty wll requre other assumptons, as n Theorem 1. 20

21 where the seller s margnal revenue s MR 0 (θ) = v 0. Let σ R (θ) be the revenue optmal stoppng rule. The welfare maxmsng rule can be mplemented by the mechansm n Theorem 1 f the aucton s started at tme s = mn {σ W (θ)}. 17 Smlarly, the revenue maxmsng rule can be mplemented by the mechansm n Theorem 2 f the aucton s started at tme s = mn {σ R(θ)}. Intutvely, once the opton s n the hands of the rght agent they wll execute at the tme that maxmses the seller s objectve. The auctoneer thus delays the aucton untl they have enough nformaton to allocate the opton to the rght agents n the each state. Whle t s easer to run an aucton than drectly mplement the stoppng functon σ(θ), the seller stll requres a lot of knowledge. In partcular, they need to know cost nformaton, F t, n order start the aucton at the correct tme. The next secton consders how to mplement these mechansms when the seller cannot observe post aucton costs State Dependent Allocaton wth Unobservable State The frst task s to derve the welfare maxmsng mechansm when costs are not observable. Recall that each agent observes nformaton F,t, and assume agents have prvate values, v (θ) = θ. 18 The welfare maxmsng mechansm, σ W (θ), can then be mplemented through a Vckrey aucton. Let {σ j (θ )} j maxmse welfare n a world wthout agent. Suppose at tme 0 each agent reports ther type θ. In every subsequent perod, t {1,..., T }, agent then reports an nformaton partton F,t whch mples reported costs c,t. Eventually, f s awarded the good, they pay the externalty they exert on agents j, [ ] ( θ j c j,σ )δ σ j ( θ j c j j,σ W )δ σw j j j Truthful revelaton s a weakly domnant strategy n each perod. See Appendx A.6 for a proof. To mplement the the revenue maxmsng mechansm, we can use a verson of the handcap mechansm of Eso and Szentes (2003) whch s a natural complment to the Theorem 2. At tme 0 agents make a report of ther types θ, are assgned a contngent payment z R (θ ), equal to the rent tax (3.9), and make an up front payment ŷ (θ) chosen so that ncentve compatblty holds. Once the contngent payment has been ntroduced each agent acts as f ther valuaton s MR (θ ), so we can run the above Vckrey aucton where valuatons θ are replaced by margnal revenues MR (θ ). 17 See Appendx A.5 for a proof. 18 One may be able to extend the treatment to common values usng mechansms such as Dasgupta and Maskn (2000). 21

22 4.2 Optons Wthout Addtvty So far we have assumed prvate nformaton s addtvely separable from post aucton nformaton. Suppose nstead that agent s net valuaton n perod t s gven by the F,t -adapted functon v,t : [θ, θ] N Ω R +. The welfare maxmsng mechansm (Theorem 1) awards the opton to the agent wth the hghest ex post utlty and so s unaffected by droppng the addtvty assumpton. Smlarly, as shown n Secton 4.2.1, f the seller can contract on the state then the revenue maxmsng mechansm s essentally the same as Theorem 2. However, the optmal contngent payment may now be state dependent. Therefore, f the seller cannot observe the state, the optmum wth observablty can only be acheved n specal cases Observable State Frst suppose the total post aucton nformaton, F t = F,t, s observed by the seller. Agents choose ther stoppng tme to maxmse ex post utlty, u (θ, z, τ ) := E[(v,τ (θ) z,τ (θ))δ τ ] (4.1) The dervaton of revenue s the same as n Secton 3 and s gven by Revenue = E θ [ P (θ) E [( v,τ (θ) v,τ (θ) ) ] ] 1 F (θ ) δ τ v 0 + E[v 0 ] (4.2) θ f (θ ) where τ s chosen to maxmse (4.1). Comparng equatons (4.1) and (4.2) one can see that the agent s choce of stoppng rule concdes wth the seller s choce f the contngent payment equals nformaton rents, z,t(θ) O = v,t(θ) 1 F (θ ) θ f (θ ) where the O superscrpt stands for observable. So the optmal contngent payment s tme (state) ndependent f s rents are ndependent of the tme (state) Unobservable State Next, suppose the state of the world s not observed by the seller. If the seller can learn agent s state from agent j, then t s easy to mplement the optmum under observablty. To avod ths trval case, suppose the set of states s a product space, where ω = {ω 1,..., ω N } and F j,t s ndependent of the sgma algebra generated by ω for j. In ths stuaton revenue stll s gven by equaton (4.2); however, the contngent payment z,t can no longer depend upon the state. Consequently the optmum wth observablty may not be attanable. 22

23 Example 2. Suppose the opton s European (T = 1), agent wns the aucton, and ther state s represented by ω = (ω,1, ω,2 ) [0, 1] 2, whle θ U[0, 1]. Let v(θ, ω,1, ω,2 ) = θ ω,1 +ω,2 so that MR (θ, ω,1, ω,2 ) = (2θ 1)ω,1 +ω,2. The seller would then lke to execute f (2θ 1)ω,1 + ω,2 0, whle the agent executes f θ ω,1 + ω,2 z, gven some contngent payment z. Wth observablty the optmum can be mplemented by choosng z = (1 θ )ω,1 ; however, there s no state ndependent contngent [ payment z whch wll algn] these ncentves. Hence prncpal chooses z to maxmse E [(2θ 1)ω,1 + ω,2 ]1 {θ ω,1 +ω,2 z }. Under some crcumstances the maxmum revenue wth observablty can be attaned wthout observablty. For example, consder a European opton (T = 1) where ω [0, 1] and v (θ, ω ) s ncreasng n the state, ω. Defne the margnal revenue from agent n state ω to be MR (θ, ω ) := v (θ, ω ) v (θ, ω ) 1 F (θ ) θ f (θ ) If MR (θ, ω ) s quas ncreasng n ω then the states where the seller wshes to exercse, {ω : MR (θ, ω ) 0}, s an ncreasng set of the form {ω ω }. The revenue maxmsng exercse rule can then be mplemented under non observablty by settng the contngent payment z,t NO (θ) := v,t(θ, ω ) 1 F (θ ) θ f (θ ) That s, the seller ntroduces a Pgouvan tax equal to the nformaton rent n the margnal state. Snce v (θ, ω ) s ncreasng n ω, the wnnng agent wll choose the revenue maxmsng exercse decson n the nfra margnal states. 19 The crucal assumpton s that there s only one margnal state n the seller s and agent s exercse decsons. margnal states. Then the contngent payment can be chosen to algn ncentves n the Ths approach can also be extended to multple perods. 20 Suppose agent s state s gven by ω = (ω,1,..., ω,t ) [0, 1] T, and that payoffs are markov, so the decson to stop at tme t depends only on the state ω,t. Also suppose that the seller s and agents stoppng problems are monotone, so that {ω,t : τ = t} are ncreasng sets. Intutvely, ths means that payoffs dsplay mean reverson, so an ncrease n today s state ncreases today s payoff more than future payoffs. Under these assumptons, the maxmum revenue wth observablty can be mplemented wthout observablty by choosng the contngent payments to algn ncentves n these (unque) margnal states. 19 Ths one perod result s smlar to Baron and Besanko (1984), Laffont and Trole (1996) and Courty and L (2000). These authors work wth the nduced dstrbuton of valuatons, rather than specfyng the underlyng state space. 20 Monotone markov stoppng problems are further examned by Fredman and Johnson (1997). 23