Lotteries, Sunspots, and Incentive Constraints 1

Transcription

1 Lottere, Sunpot, and Incentve Contrant 1 Tmothy J. Kehoe Department of Economc, Unverty of Mnneota, Mnneapol, Mnneota Reearch Department, Federal Reerve Bank of Mnneapol, Mnneapol, Mnneota tkehoe@atla.occ.umn.edu Davd K. Levne Department of Economc, Unverty of Calforna Lo Anele, Lo Anele, CA dlevne@ucla.edu and Edward C. Precott Department of Economc, Unverty of Mnneota, Mnneapol, Mnneota Reearch Department, Federal Reerve Bank of Mnneapol, Mnneapol, Mnneota ecp@re.mpl.frb.fed.u Frt veron: September 11, 1997 Th veron: Aprl 8, 00 Abtract: We tudy a prototypcal cla of exchane econome wth prvate nformaton and ndvblte. We etablh an equvalence between lottery equlbra and unpot equlbra and how that the welfare and extence theorem hold. To etablh thee reult, we ntroduce the concept of the tand-n conumer economy, whch a tandard convex, fnte conumer, fnte ood, pure exchane economy. Wth decrean abolute rk averon and no ndvblte, we prove that no lottere are actually ued n equlbrum. We provde a mple numercal example wth ncrean abolute rk averon n whch lottere are necearly ued n equlbrum. We alo how how the equlbrum allocaton n th example can be mplemented n a unpot equlbrum. 1 We are rateful to the partcpant at the Cowle Semnar at Yale Unverty and at the Smpoo de Anála Económco. Natonal Scence Foundaton Grant SBER , , and , and the UCLA Academc Senate provded fnancal upport.

2 1 1 Introducton There conderanle emprcal evdence that, unlke n the tandard complete market model, ndvdual bear ubtantal doyncratc rk. See Kreuer [18] for a urvey and dcuon. Both ncomplete market model, uch a thoe of Geanakoplo [13], and model of ndvdual ratonalty contrant, uch a thoe of Kehoe and Levne [15, 16], Kocherlakota [17], and Alvare and Jermann [1], have been ued to tudy doyncratc rk bearn. None of thee model can explan a tron concentraton of ndvdual portfolo n a narrow rane of aet. Why, for example, doe Bll Gate hold larely Mcrooft equty, or doe a car dealer portfolo cont larely of the dealerhp nventory? Whle ndvdual wth uch undverfed portfolo are a mall fracton of the populaton, they hold a lare percentae of wealth. Moral haard an obvou explanaton for uch undverfed portfolo, and recently there ha been a reurence of nteret n ntroducn th feature nto eneral equlbrum theory. Bennardo [3], Bennardo and Chappor [4], and Bn and Guatol [5] have been uch effort. The pont of departure ha been Precott and Townend [19, 0], who ntroduce both the dea that ncentve contrant can be ntroduced nto eneral equlbrum theory n a enble way and the dea that lottere play a potentally mportant role n the reultn theory. Althouh ther theory ha been wdely ued to tudy ndvblte n the areate economy by Hanen [13], Roeron [], Cole and Precott [9], and other, untl recently lttle effort ha been made to tudy ncentve contrant from th pont of vew. The dea of un lottere to tudy aet market reman controveral. Th paper tude a prototypcal cla of ncentve contraned envronment n an effort to clarfy a number of ue. One ue how lottery equlbra are to be mplemented. In the ndvblty cae wth a fnte number of houehold, Shell and Wrht [3] how that there a cloe connecton between lottery equlbra and unpot equlbra, a connecton that made tht n Garratt et al. [1]. 3 Here we how that A recent excepton Precott and Townend [1] model of the frm. 3 Shell and Wrht [3] conder a model wth complete nformaton and a contnuum of conumer. They how that every lottery equlbrum allocaton can be decentraled a a unpot equlbrum. In a model wth complete nformaton and a fnte unpot tate pace, Garratt [11] how that, whle every lottery equlbrum allocaton can be decentraled a a unpot equlbrum, not every unpot equlbrum allocaton a lottery equlbrum allocaton. Garratt et al. [1] etablh the equvalence, n eneral

3 mlar reult hold n the cae of ncentve contraned econome wth a contnuum of houehold. In the mportant cae of decrean abolute rk averon and no ndvblte, we how that lottere are not actually needed n equlbrum. We provde a mple numercal example wth ncrean abolute rk averon n whch lottere are necearly ued n equlbrum. We alo how how the equlbrum allocaton n th example can be mplemented n a unpot equlbrum. The proof of theorem, the analy of the example, and the computaton of equlbra n thee ort of econome are reatly facltated by the noton of the tand-n conumer economy. Overall, we arue that the Precott and Townend framework repreent a enble and ueful framework for analyn moral haard and advere electon n eneral equlbrum theory. In recent related work, Cole and Kocherlakota [9] conder an envronment lke our wth prvate nformaton over endowment. They how that, f torae poble and unobervable by other houhold, then the equlbrum allocaton the ame a that n an economy wth an ncomplete market economy wth a nle aet that pay the ame n all tate. A Smple Inurance Problem There a contnuum [,] 0 1 of houehold who are ex ante dentcal. There are two ood j 1,. Let c j denote conumpton of ood j. Utlty ven by v1( c1) v( c), where each v () trctly concave and trctly ncrean. The endowment of ood 1 rky, whle ood ha a certan endowment. Each houehold ha an ndependent 50 percent chance of ben n one of two tate, {, b}. The endowment of ood 1 tate dependent and can take on one of the two value 1 and 1b, where 1 1b, whle the endowment of ood fxed at. Vewed n the areate, after the tate realed, half of the populaton ha the ood endowment, and half the bad endowment. After the tate realed, there are an from trade, a the bad endowment houehold want to purchae ood 1 and ell ood. Before the tate realed, there are addtonal an from trade nce houehold want to purchae nurance aant the bad tate. In fact, nce all houehold are ex ante dentcal and utlty trctly concave, the bet ymmetrc allocaton that n whch houehold complete nformaton econome, of the et of lottery equlbrum allocaton and the et of unpot equlbrum allocaton baed on a contnuou unpot randomaton devce.

4 3 conume ( )/ of ood 1, and of ood. Follown the mechanm den 1 1b lterature, we refer to th allocaton a the frt bet. Suppoe that the realaton of the doyncratc rk prvate nformaton known only to the ndvdual houehold. In th cae, the frt bet allocaton not ncentve compatble. In the frt bet allocaton, bad endowment houehold receve an nurance payment of ( )/ 1 1b, whle ood endowment houehold mut make a payment of the ame amount. Conequently, ood endowment houehold would mrepreent ther endowment n order to receve a payment rather than make one. One approach to modeln equlbrum to prohbt tradn n nurance contract, and conder only tradn that take place ex pot after the tate realed. Th an example of an ncomplete market model. The reultn compettve equlbrum lead to an equalaton of marnal rate of ubttuton between the two ood for the two type of houehold, but there are unrealed an from ex ante trade n nurance contract. A econd approach to modeln equlbrum to oberve that t poble to trade n nurance contract provded that no houehold buy a contract that would later lead t to mrepreent t tate. If endowment can be made publc, but only voluntarly, then the ood endowment houehold can mtate the bad endowment type, but not converely. Suppoe that a houehold attempt to purchae ( x1 b, x b) n tate b n exchane for ( x1, x ) n tate. In the ood endowment tate, utlty wll be c h ( ). In th cae, the ood endowment houehold may want to v x v x pretend that the tate actually the bad endowment tate, tate b. To avod detecton, t mut make the ame pot market purchae that a bad endowment houehold would make, ( x1b, x b). Th reult n utlty v x v x compatblty contrant c h c h. Therefore the ncentve 1 1 1b b c h c h c h c h. v x v x v x v x b b If th contrant atfed, the ood endowment houehold ha no ncentve to le about t prvate nformaton. We now etablh that, f tradn n nurance contract prohbted, there are ncentve compatble an to allown th trade. Let ( x,,, 1 x x1b x b) denote net trade by a houehold n an equlbrum n whch tradn n nurance contract prohbted.

5 4 Snce a bad endowment houehold cannot mtate a ood endowment houehold, t face no ncentve contrant. Snce the ood endowment houehold could have purchaed the net trade of the bad endowment houehold and had ncome left over, t trctly prefer t own net trade to that of the bad endowment houehold, c h c h. v ( x ) v ( x ) v x v x b b We already know that ( ~ x, ~ x, ~ x, ~ x ) ( ( )/ 0,,( )/ 0, ) would be 1 1b b 1 1b 1 1b the equlbrum trade of nurance f there were no ncentve contrant and would yeld trctly hher ex ante utlty than ( x,,, 1 x x1b x b) becaue utlty trctly concave. Conder the net trade ( x, x, x, x ) (( 1) x ~ x,( 1 ) x ~ x,( 1 ) x ~ x,( 1 ) x ~ x ). 1 1b b 1 1 1b 1b b b If mall enouh, then ood endowment houehold tll ha no ncentve to mrepreent, but ex ante utlty trctly hher. Therefore, there are addtonal ncentve compatble an to trade that are not realed when tradn n nurance contract prohbted. Suppoe, more enerally, that houehold trade ood contnent on announcement. No houehold wll ever delver a bundle that not ncentve compatble. Every houehold know th fact, and o only ncentve compatble bundle can be traded. Notce, however, that th troner arument doe not uarantee that all ncentve compatble bundle actually can be traded unle thee contract prohbt ex pot trade: If a ood endowment houehold can receve an nurance payment by clamn a bad endowment and then turn around and trade the nurance payment of ood 1 for addtonal unt of ood, t wll prefer th to admttn a ood endowment. The contract mut pecfcally prohbt houehold clamn to have a bad endowment from tradn ood 1 for ood. Contract that preclude other trade are often referred to a excluve contract. Contract of th type are common n nurance market. Often nurance contract pecfy that the nurance payment can be ued only for a pecfc purpoe, uch a replacn a tructure on a pecfc locaton. We conder only excluve contract n th paper. Let X denote the pace of all net trade that atfy the ncentve contrant. Our proram to retrct houehold to tradn plan n X and then do ordnary compettve

6 5 equlbrum theory. There are two complcaton wth th proram. Frt, fxn ( x1, x ), the et of ( x1b, x b) that atfe the ncentve contrant fal to be convex, o X not a convex et. Th mean that Pareto mprovement may be poble by un lottere. Second, we can ue lottere to weaken the ncentve contrant; that, contnent on t announcement, the houehold receve net trade ( x1, x) that are random. Conequently, the ncentve contrant need only hold n expected value. If we let E denote the expectaton condtonal on the announcement of tate the ncentve contrant become Eov1c 1 x1h vc xht Ebov1c 1 x 1bh vc xbht. For thee two reaon, once we ntroduce ncentve contrant nto eneral equlbrum, we alo ntroduce lottere. 4 3 The Envronment Houehold are of I type 1,, I. There a contnuum of ex ante dentcal houehold of each type. An ndvdual houehold denoted by hh [, 0 ], where 0 the e of the populaton of type houehold. A houehold type commonly known. There are J traded ood j 1,, J. There are alo two ource of uncertanty: a commonly oberved unpot and houehold pecfc doyncratc rk. A unpot a random varable unformly dtrbuted on [0,1]. Idoyncratc rk repreented by pecfyn that each houehold of type conume n one of a fnte number of tate S. Each tate ha probablty 0 where 1. Th probablty ha two nterpretaton: Frt, from the perpectve of the ndvdual houehold, t the probablty of ben n the tate. Second, a we explan below, from the perpectve of the entre populaton of houehold of type, t repreent the fracton of houehold n tate. We aume that houehold can contract for delvery of ood contnent on the unpot and the ndvdual tate of the houehold. 5 We wrte x ( h, ) for the net S j 4 In addton to Precott and Townend [1984a, 1984b], uch other author a Arnott and Stlt [] have remarked on the potental of lottere to mprove welfare. 5 Strctly peakn, we hould allow houehold to bae contract on the doyncratc tate of other houehold. In the type of equlbrum we wll conder contract baed on other houehold doyncratc tate do not erve any purpoe. We omt them to avod notatonal complcaton. Th pont dcued further n Secton 4.

7 6 amount of ood j delvered to houehold h of type when the doyncratc tate and the unpot tate. The dtrbuton of doyncratc hock and unpot are aumed to be ndependent. The doyncratc hock are uch that the areate net trade of all type houehold of ood j when the unpot ven by yj ( ) xj( h, ) dh, S whch the aumpton that the fracton of houehold of type n tate. There are everal jutfcaton for th aumpton. The eaet aumpton that doyncratc hock are ndependent acro houehold. It known that th ncontent wth areate net trade defned by Lebeue nteraton and a pace of conumer on the unt nterval; th dcued n Boylan [6]. Alternatvely, we could defne areate net trade by the Pett nteral, a n Uhl [4]. Or we could mply allow doyncratc tate to be correlated acro ndvdual. We prefer to avod thee techncal ue, however, and mply jutfy the defnton of areate conumpton ven a the lmt of areate net trade n fnte houehold econome wth hock ndependent acro houehold. Th, after all, the purpoe n ntroducn contnuum econome n the frt place. Tradn take place before any uncertanty realed. Then the doyncratc tate are realed and announcement of tate are made. Next, unpot are realed. Fnally, delvere are made, no further trade allowed, and conumpton take place. Notce that at th pont we do not allow ex ante unpot that are realed pror to the realaton and announcement of ndvdual tate. A more eneral model would allow both ex ante and ex pot unpot. Later, we how that equlbra n the more eneral model are equvalent to equlbra n the ex pot model we conder here. 6 Fx houehold h of type. For each announcement of the doyncratc tate and J the unpot th houehold reale a net trade x ( h, ). We aume that th net trade mut belon to the feable net trade et X. Notce that endowment are ncorporated drectly nto the feable net trade et and are not pecfed eparately; we allow X to depend on o that endowment may be doyncratc tate contnent. In a 6 In a model wth only ex ante unpot, equlbrum can be chaned by addn ex pot unpot, becaue lottere condtonal on prvate nformaton may be ued to eparate houehold wth dfferent rk preference. Th wa ornally ponted out by Cole [7].

8 7 tandard endowment economy, wth endowment, we have X ven by u : X. We ue the notaton x ( h, ) JS { x x }. Utlty for the et of net trade vector correpondn to dfferent doyncratc tate and for expected utlty. S u ( x ( h, )) u ( x ( h, )) We next conder ncentve contrant. Thee are derved from requrn that houehold not want to mrepreent prvate nformaton about ther own tate. Prvate nformaton about tate pecfed by et of feable report F S. Thee et repreent the report that a houehold can make about h tate when h true tate wthout ben contradcted by ether publc nformaton or phycal evdence. Conequently, a et of feable report mut atfy two aumpton: Feable Truthtelln: For all S, F. Feable Mrepreentaton: If ' F, then X ' X. The econd aumpton requre that t not poble to report a net trade et that nfeable wth repect to the true net trade et. Th aumpton rule out more complcated poblte, uch a tuaton where the feablty of tradn plan can only be dcovered ex pot and punhment mpoed for volatn contract. In uch a cae, feable report would depend on the partcular contract offered. The noton of feable report that atfy thee two aumpton leave ubtantal flexblty n buldn model econome. We provde two example of clae of econome that atfy our aumpton: Publc Endowment: The endowment not. Then F m' ' r. are publcly oberved, but preference u are Voluntary Publc Endowment: A houehold wth endowment may publcly dplay any porton of t endowment. Preference u endowment be dplayed n order to be reported, we obtan F are prvate. By requrn that an m ' ' r. In other word, houehold clamn a partcular endowment can be requred to dplay the

9 8 clamed endowment, preventn houehold wth maller endowment from mrepreentn that they have larer endowment. Notce that we treat et of feable report a data. Th avod the deeper and more dffcult queton of how et of feable report are enerated from underlyn fundamental. Notce n partcular that we could ue the et F m' ' r even n the cae of publc endowment. It apparent that th would lead to maller et of feable allocaton than makn full ue of the publc nformaton and takn F m' ' r. A caveat to the reult reported n th paper that effcency condtonal on a partcular et of feable report. A unpot contnent tradn plan x ( h, ) called ncentve compatble f for all ' F ( ) u( x( h, )) d u( x' ( h, )) d 0. We do not aume that X convex or that u concave or non-decrean. We do aume: Cloed and Bounded Trade: X cloed and bounded below. Voluntary Trade: 0 X. Cheaper Pont: For every and ome S and x 0, x Convex Hull{ X }. Contnuty: u contnuou. Non-ataton: From ome tate and all x u ( x ~ ) u ( x ). X there ext ~ x X uch that Boundary: If x, then lm u ( x ) / x 0. Wth the excepton of the boundary condton, thee aumpton are elf-explanatory. The boundary condton requre marnal utlty to aymptote to ero; t ay that eventually utlty ncreae lower than any lnear functon. 7 7 Notce that nothn n thee aumpton rule out ndvblte, nor t neceary to, a the lterature on lottere wth ndvblte dcued n the ntroducton how.

10 9 An mportant fact about the non-ataton condton that t mple non-ataton for ncentve compatble net trade. Lemma 3.1: If x X S ncentve compatble, then there an ncentve compatble ~x X uch that u ( x ~ ) u ( x ). S S S Proof: By the non-ataton aumpton there a tate and ~ x wth u( x ~ ) u( x). In other tate ', f ~ x X ' or f ~ x X' and u' ( ~ x ) u' ( x), take ~ x' x' ; otherwe, take ~ x ~ ' x. Clearly, u( x ~ ) S S u( x) becaue 0. We now arue that ~ x ncentve compatble. Suppoe " F '. Snce u' ( x' ) u' ( x" ), t follow that u' ( ~ x' ) u' ( ~ x" ) hold f ~ x" x". If ~ x" x" then ~ x ~ " x. If u' ( x' ) u' ( ~ x" ), nce u' ( ~ x' ) u' ( x' ), the ncentve contrant hold. If u' ( x' ) u' ( ~ x" ), we have u' ( x' ) u' ( ~ x" ) u' ( ~ x ), mplyn that ~ x ~ ' x. By our contructon of ~ x ', aan the ncentve contrant hold. Three pont to emphae about the model are Type are commonly known; the doyncratc tate may or may not be prvate nformaton. It mportant that contractn take place pror to learnn any prvate nformaton. If contractn poble only after learnn prvate nformaton, or, what amount to the ame thn, f type are prvate nformaton, then ncentve to mrepreent nformaton wll depend on the net trade of rval houehold. Th repreent an externalty that may nvaldate the welfare theorem. 8 Houehold do not care about the prvate nformaton of rval houehold. Th aumpton could be relaxed, but t would then be neceary to allow contractn baed upon the announcement of the relevant rval. We have mplctly aumed that contract are excluve that, that tradn not poble after delvere are made. A we noted n the example, equlbrum wth nonexcluve tradn qute dfferent than wth excluvty. A ponted out by Precott and Townend [19], the welfare theorem can fal wthout excluvty. 8 Precott and Townend [1984b] ve an example n whch the frt welfare theorem hold, but the econd fal. In other example, uch a thoe n Precott and Townend [1984a], both welfare theorem fal and equlbrum may not ext.

11 10 We conclude th ecton by llutratn how the example of the prevou ecton ft nto th framework. In the example I 1, J, 1, S {, b}, b 1/. The net trade et are thoe net trade that exceed the neatve of the endowment X {( x, x ) x, x } The utlty functon for net trade derved from the utlty of conumpton accordn to u( x1, x) v1( 1 x1) v( x) u ( x1, x, x1 b, xb) u( x1, x ) ub( x1 b, xb). The feable reportn et reflect the fact that X X, 1 1 b 1 1 F {}, b F {, b}. b There one ncentve contrant, correpondn to a ood endowment tate reportn a bad endowment: u( x( h, )) d u( xb( h, )) d 0. 4 Equlbrum wth Sunpot A unpot allocaton a meaurable map for each type from houehold to ndvdual tradn plan; that x ( h, ) X. An allocaton ocally feable f for each unpot realaton I x ( h, ) dh 0. 1 S Notce that th defnton ncorporate publc free dpoal; we do not aume ndvdual can ecretly dpoe of ood. We ay that an allocaton ha equal utlty f for each type u ( x ( h, )) d u ( x ( h', )) d for almot all h, h'. Let e denote the doyncratc tate of all houehold of all type; e(, h) the doyncratc tate of houehold h of type. Then the Arrow-Debreu commodty a delvery of j ood contnent on ( e, ). Arrow-Debreu prce are q J ( e, ). Becaue there no areate rk, we retrct attenton to Arrow-Debreu equlbra n whch prce are ndependent of e; that, q( e, ) q( ). Wth thee prce, the cot to a houehold h of type to purchae x at q x de e q( ) x de q( ) x. { (,) ee h } (, ) { (,) ee h }

12 11 J We refer to a non-ero meaurable functon q( ) a a prce functon. A unpot equlbrum wth tranfer cont of a ocally feable unpot allocaton x toether wth a prce functon q. For all type and almot all h [ 0, ], x ( h,) mut maxme u ( x ( h, )) d over unpot contnent tradn plan x ( h,) atfyn the unpot budet contrant q x S h d S q x h d ( ) (, ) ( ) (, ), and the ncentve contrant u( x( h, )) u( x ( h, )) d ' 0, ' F ( ). The tranfer themelve mut atfy the equal treatment condton that they depend only on type: S S q( ) x ( h, ) d q( ) x ( h', ) d for almot all h, h'. A unpot equlbrum a unpot equlbrum wth tranfer n whch the tranfer are ero: q x ( ) S ( h, ) d 0 for almot all h. Fnally, a unpot allocaton Pareto effcent f there no alternatve ocally feable allocaton atfyn the ncentve contrant n whch almot all houehold have no le utlty and a potve meaure of houehold have trctly more utlty. An mmedate conequence of the fact that the tranfer atfy the equal treatment condton the concluon that the equlbrum allocaton mut be an equal utlty allocaton. If t were not, then a potve meaure of type could ncreae ther utlty by wtchn to a conumpton plan ued by other of the ame type. Lemma 4.1 A unpot equlbrum allocaton wth tranfer an equal utlty allocaton. Our man oal to etablh the man theorem of compettve eneral equlbrum theory for the unpot economy Theorem 4. (Frt Welfare Theorem) Every unpot equlbrum allocaton wth tranfer Pareto effcent. Theorem 4.3 (Second Welfare Theorem) For every Pareto effcent allocaton x wth equal utlty there are prce q uch that ( x, q ) are a unpot equlbrum wth tranfer. Theorem 4.4 (Extence Theorem) There at leat one unpot equlbrum.

13 1 The frt welfare theorem a relatvely drect conequence of the non-ataton aumpton and the tandard proof of the frt welfare theorem. The remann reult follow from equvalence theorem below. 5 Equlbrum wth Lottere A probablty dtrbuton over X u ( ) u ( x ) d ( x ) referred to a a lottery. We defne a the expected utlty from the lottery. From the pont of vew of ndvdual utlty, all tradn plan that nduce the ame et of lottere m r yeld the ame utlty, and the ncentve contrant can alo be computed drectly from S the lottery. The areate reource ued by a et of lottere are S y x d ( x ). Notce that wth a contnuum of houehold we need not dtnuh between realed and expected net trade. Th dtncton mportant n decentraln lottere n the ndvble cae wth a fnte number of houehold, a can be een n the work of Garratt [11] and Garratt et al. [1]. To llutrate our notaton we apply t our nurance example. In the example, there are two tate S 1 {, b}, and two net tradn et X 1 contn of net trade that are at leat a reat a the neatve of the endowment. The et of lottere ha one lottery correpondn to each tate 1 1. The lottere are each non-neatve meaure that atfy d 1 ( x 1 1 ). We now conder the Precott and Townend perpectve, n whch houehold trade drectly n lottere. Our oal to how that th formulaton equvalent to the unpot formulaton.

14 13 1 I A lottery allocaton a vector of et of lottere, (,..., ), where the et of lottere an lottere to type n each doyncratc tate. 9 Notce that th requre all houehold of a ven type to purchae the ame lottery. Becaue preference are convex over lottere, t make ene to mpoe th retrcton and we demontrate n the next ecton that there no lo of eneralty n th. A et of lottere ocally feable f I 1 xd ( x) 0. S Th ay that n the areate the expected net trade ued by the lottery allocaton non-potve. A lottery allocaton Pareto effcent f no ocally feable, ncentve compatble Pareto mprovement poble. A lottery equlbrum wth tranfer cont of a ocally J feable lottery allocaton toether wth non-ero prce vector p. For all type, mut maxme u ( ) S over lottere atfyn the lottery budet contrant S S p x d ( x ) p x d ( x ), and the ncentve contrant u( ) u( ' ) 0, ' F ( ). A lottery equlbrum a lottery equlbrum wth tranfer n whch the tranfer are ero p x d ( x ) S 0. Notce that n th formulaton, lottere are prced accordn to the areate reource they ue. Th a no-arbtrae condton: two lottere that ue the ame areate reource mut have the ame prce. If one lottery ue areate reource y and another y, and f the cot of buyn y and y eparately exceed the cot of buyn y y, t would be proftable to buy the jont lottery y y and ell the pece, whle n the oppote cae, the pece hould be bouht eparately, then packaed and old. Only 9 1 I We could have equally well followed the formalm of defnn a trade vector (,..., ), and retrctn houehold of type to the trade et n whch the lottery vector for all other type put probablty 0 on all trade. The notaton followed here ha the advantae of ben le cumberome, but ha the dadvantae of mplctly havn dfferent trade pace, and underlyn commodty pace, for dfferent houehold type.

15 14 lnear prcn n the areate reource uarantee that there are no arbtrae opportunte. 10 Let k (# F 1) denote the number of ncentve contrant. There are four S bac feature of lottere that are worth emphan: Lemma 5.1 (a) A convex combnaton of ncentve compatble et of lottere ncentve compatble. (b) Let y, ~ y be the reource ued by the et of lottere, ~ and let 0 1. Then the et of lottere ( 1 ) ~ ue areate reource y ( 1 ) ~ y. (c) For any ncentve compatble et of lottere there another ncentve compatble et of lottere ~ un the ame areate reource, yeldn the ame utlty, and each lottery ~ havn upport on k J pont. (d) Let y be the reource ued by the ncentve compatble et of lottere, and uppoe ~ y y. Then there an ncentve compatble et of lottere ~ un no more reource than ~ y that yeld trctly more utlty than. Proof: (a) and (b) are mmedate. The proof of (c) larely mathematcal, and provded n the Appendx. To prove (d), frt apply (c) to fnd an ncentve compatble lottery ~ wth fnte upport yeldn the ame utlty a. Becaue th lottery ha fnte upport, t follow from Lemma 3.1 that there an ncentve compatble net trade S S S. Let ~ x wth u ( x ) u ( ~ ) u ( ) be the deenerate lottery wth pont ma on x. Then for all 0 1, the et of lottere ( 1) ~ ncentve compatble by (a) and yeld trctly more utlty than. A approache 0, however, the areate reource ued by th et of lottere approach y and, therefore, for uffcently mall, are le than ~ y. 10 Much of the lterature on lottere tude producton econome n whch frm can repackae lottere nto dfferent lottere un the ame reource; what we refer to a a no-arbtrae condton follow n that ettn from proft maxmaton by frm. Our approach follow Hanen [1985].

16 15 We wll etablh the man theorem of compettve eneral equlbrum theory for the lottery economy, a well a the unpot economy. Theorem 5. (Frt Welfare Theorem) Every lottery equlbrum allocaton wth tranfer Pareto effcent. Theorem 5.3 (Second Welfare Theorem) For every Pareto effcent allocaton there are prce formn a lottery equlbrum wth tranfer. Theorem 5.4 (Extence Theorem) There at leat one lottery equlbrum. In Precott and Townend [0], thee theorem are proved drectly; we ve alternatve proof below. Our reult on unpot equlbra then follow from hown that lottery and unpot allocaton are equvalent. 6 Sunpot Equlbrum veru Lottery Equlbrum Sunpot allocaton and lottery allocaton are dfferent decrpton of randomaton. For example, uppoe that there are two dentcal type, and one ood, automoble, for whch the conumpton vector ether one automoble or ero. Suppoe moreover, that each type endowed wth one half an automoble per capta. From the lottery perpectve, the tuaton mple: there can be no trade between the two type, o each houehold hould receve an automoble wth probablty 1/. In other word, n equlbrum, each houehold of each type purchae a lottery wth a 50 percent chance of 1 automoble, and a 50 percent chance of 0 automoble. In our notaton, ( 1/ ) 1/ and ( 1/ ) 1/ where 1/ and x equal to 1 or 0 for both houehold type 1,. Th lottery can be mplemented n many way by mean of unpot. For example, we could mane that the ndvdual lottere are ndependent, 11 and that n the areate the tron law of lare number lead to ocal feablty. An alternatve formulaton would be to have a mple unpot allocaton n whch when the unpot varable atfe 1/, the frt type receve all the car and, when 1/, the econd type receve all the car. From an ndvdual pont of vew t make no dfference whch of thee method ued to allocate car. 11 Subject to uual caveat about a contnuum of ndependent random varable; ee the dcuon above.

17 16 A unpot allocaton may nduce dfferent lottere for dfferent houehold. To et a nle et of lottere for each type, a requred for a lottery allocaton, we averae toether the houehold pecfc lottere. Ben wth a unpot allocaton ~ xh (, ). For each houehold, there correpond lottere ( x ~ ( h,)). We can then averae thee lottere over houehold to et a mean lottery for the entre type ~ ( ~ x ( h,)) dh/. Notce that the reource ued by th lottery are equal to the expected reource ued by the unpot allocaton; that, x d ~ ( x ) ~ x ( h, ) dhd /. Moreover, by defnton, n an equal utlty unpot allocaton houehold of type mut be ndfferent between the allocaton ~ x ( h,) and ~ x ( h',) for almot all h, h'. Snce ther utlty lnear n probablte, th mean they mut be ndfferent between ~ x ( h,) and the mean et of lottere ~ for almot all h. In a mlar ven, nce the ncentve contrant hold for almot all ndvdual et of lottere and are alo lnear n probablte, the mean et of lottere mut atfy the ncentve contrant. Conequently, the mean et of lottere correpondn to a unpot equlbrum allocaton a natural canddate to be an equlbrum of the lottery economy. We provde an example of averan of lottere n Secton 10. If q ( ) a prce functon n the unpot economy, we can n a mlar way defne the mean prce q q( ) d. Althouh t not obvou, we wll how below that the mean prce n fact a correct way to prce the mean lottery n the lottery economy. 1 To apprecate the poblty of q( ) not ben contant, conder a varant of our automoble example. Aan there are two dentcal type, but now there are two ood, clothe waher and dryer. Houehold of each type are endowed wth 1/ unt each of waher and dryer and can conume only 1 or 0 of each of thee ood. Furthermore, thee ood are optmally be conumed n fxed proporton: u ( x, x ) mn[ x, x ] Alternatvely, we could mply requre that n a unpot allocaton each houehold of a type have the ame lottery. Whle th retrcton eem natural n the context of the lottery model wth a repreentatve houehold, t doe eem a natural n the unpot model. Fortunately, we can how that only the mean lottery and mean prce matter.

18 17 It eay to check that one equlbrum where q( ) q ( 10, ) and RST ( x 1 h 1 /, 1 / ) f (, ) 1 / ( 1/, 1/ ) f 1/, wth the oppote allocaton to houehold of type. Another equlbrum ha the ame allocaton but the prce vector q( ) q ( 01., ) Yet another equlbrum would have (, ) / / q( ) RST 10 f 1 4 or 3 4 ( 01, ) f 1/ 4 3/ 4. The poblty of prce q( ) that vary wth the unpot are becaue there more that one contant prce vector that can upport an allocaton. Notce, n our example, that the mean prce q ( 1/, 1/ ) alo an equlbrum prce. In Theorem 6., we prove that th true n eneral. In our example, n whch the role for unpot are becaue of ndvblte, the poblty of more than one contant prce vector upportn an allocaton deenerate n that t dappear f we perturb the endowment. In econome where the role for unpot are becaue of ncentve contrant, however, there no need for equlbra wth more than one upportn prce to be deenerate. We defne a unpot allocaton to be equvalent to a lottery allocaton f for each type the mean et of lottere of the unpot allocaton equal to the correpondn et of lottere n the lottery allocaton. We defne unpot prce to be equvalent to a lottery prce f the mean prce of the unpot prce equal to the lottery prce. By defnton, there only one lottery allocaton and prce that equvalent to a ven unpot allocaton and prce functon. A we have already noted, however, there not a unque way to contruct a unpot allocaton (or prce) from a lottery allocaton. Neverthele, there one mportant contructon that play a key theoretcal role n movn from lottery econome to unpot econome. For a ven lottery prce p we defne the contant functon q( ) p to be the canoncal unpot prce functon 13. For a ven lottery allocaton we defne the canoncal unpot allocaton to be a partcular allocaton n whch the areate reource ued by each type are ndependent of the unpot tate. Specfcally, correpondn to the lottery a random varable ~ x ( ). Recall that 13 Garratt et al. [1] call thee prce contant probablty adjuted prce. They how that n econome wth complete nformaton all unpot equlbrum allocaton can be upported by prce that are collnear wth probablte f the unpot randomaton devce contnuou.

19 18 amod b the remander of a dvded by b. We defne the canoncal unpot allocaton a x( h, ) ~ x (( h ) mod ). 14 Notce that at th canoncal allocaton, the areate net trade by all houehold of a type ndependent of the realaton of the unpot. Thee mple contructon how that for every lottery allocaton and prce there at leat one equvalent unpot allocaton and prce. Becaue the contructon of the lottery allocaton preerve utlty, ocal feablty and the ncentve contrant, we can draw an mmedate concluon about Pareto effcency. Theorem 6.1 An equal utlty allocaton Pareto effcent n the unpot economy f and only f any (or all) equvalent allocaton n the lottery economy are Pareto effcent. Moreover, the ocally feable, ncentve compatble equal utlty et n the unpot economy the ame a the ocally feable ncentve compatble utlty et n the lottery economy. Le mmedately obvou the equvalence of equlbra n the two econome. Theorem 6. An allocaton and prce are an equlbrum wth tranfer n the lottery economy f any (or all) equvalent allocaton and prce functon are an equlbrum wth tranfer n the unpot economy. An allocaton and contant prce functon are an equlbrum wth tranfer n the unpot economy f the equvalent allocaton and prce functon are an equlbrum wth tranfer n the lottery economy. In both cae the e of the tranfer the ame n the two econome. Proof: Conder a unpot allocaton x and prce functon q and an equvalent lottery allocaton and prce p. Suppoe frt that q contant (n partcular, that q( ) p ) and that, p are an equlbrum wth tranfer n the lottery economy. Snce houehold care only about ther ndvdual lottery and nce q( ) p, x, q are an equlbrum wth tranfer n the unpot economy. Snce n both cae each type pay only for the areate reource ued, whch the ame n both econome, the tranfer mut be the ame n both cae. Now uppoe ntead that x, q are an equlbrum wth tranfer n the unpot economy, and that pobly q not contant. We mut how that, p are a lottery 14 There are many way of mappn a lottery allocaton nto a unpot allocaton nvolvn dfferent way of correlatn outcome acro ndvdual. See Shell and Wrht [3] and Garratt et al. [1] for dcuon.

20 19 equlbrum wth tranfer. To how th we mut how that for each type any et of lottere that yeld more utlty than cannot be afforded at the prce p and that can be purchaed at thoe prce. Frt, we how that for each type, any et of lottere that yeld more utlty than cannot be afforded at the prce p. Suppoe that n fact affordable and yeld more utlty than. Notce that nce they are equvalent, the utlty from the ame utlty x ve almot all houehold. We ue to contruct a unpot plan that affordable at prce q yeldn the ame utlty a ; th wll be the dered contradcton. Conder the canoncal unpot allocaton x correpondn to. Th ve every houehold n more utlty than x. It alo contructed o that x ( h, ) dh ndependent of. By contructon t affordable at prce q( ) p ; becaue p the averae of q( ) and x ( h, ) dh ndependent of, t therefore affordable at prce q( ). It follow that for a potve meaure et of houehold h, x ( h,) alo affordable at prce q( ). Th ve the dered contradcton, nce x ( h,) ve the ame utlty a for all h. To conclude the proof, we how that can be purchaed at prce p. Suppoe for ome th not the cae, that cot more than the tranfer to that type. Then nce ocally feable, for ome other type ' expendture on ' mut be le than the tranfer payment. But by Lemma 5.1, uch a type could ue the extra ncome to purchae a better lottery than ', whch we have hown cannot happen. 7 The Stand-n Conumer Economy We now prove the welfare theorem and the extence of an equlbrum. From the equvalence of the unpot and lottery equlbra, t uffcent to do o n ether of the two type of econome. Each approach, however, poe t own complcaton. The unpot economy ha a net trade et that complcated and non-convex. The lottery economy ha a net trade et that convex but nfnte dmenonal. One approach that of Precott and Townend [0], whch to work drectly wth theorem for nfnte dmenonal econome. The alternatve purued here lead to fnte dmenonal and mathematcally mpler proof by obervn that the houehold problem of maxmn utlty ubject to a budet contrant can be broken n two part. The frt part, nce the

21 0 cot of a et of lottere mply the cot of the expected net trade t ue, to thnk of the houehold a purchan an expected net trade vector. The econd part to thnk of the houehold a choon the et of lottere that maxme utlty ubject to th expected net trade contrant. Th utlty depend only on the expected net trade vector, whch fnte dmenonal, o n effect reduce the economy to a fnte one. Specfcally, we conder net trade vector y J. The et of nteret are net trade vector that are content wth feable tradn plan of type houehold: J Y y x X Cloure(ConvexHull{, y x S }). Gven that a bundle y Y ha been purchaed, how much utlty can a type houehold et? The anwer ven by U ( y ) up u x d x S ( ) ( ) ubject to upport X, xd ( x) y, u ( ) u ( ) 0, ' F ( ). S Th contruct wll be mot ueful f we can replace the up wth a max, o that there at leat one lottery that actually yeld the utlty U ( y ). ' Lemma 7.1 If the boundary condton hold, then U ( y ) max u x d x S ( ) ( ) ubject to upport u ( ) u ( ) 0, ' F ( ). ' X, xd ( x) y, S Proof: By Lemma 5.1 we can aume that there a equence of et of lottere wth each lottery havn upport at k J pont convern to the up. Let x, be the pont and probablte n th equence. Th ha a converent ubequence on the extended real lne. Becaue X bounded below, any component of x that convere to ha correpondn probablty convern to ero. By the boundary condton the lmt of expected utlty for uch a pont alo ero. So the lmt et of lottere place weht only on fntely many pont, and ve the ame utlty and atfe the feablty and ncentve condton. It the optmal et of lottere. We now tudy trade n the economy, by condern I conumer wth utlty functon U and conumpton et Y. We refer to conumer a the tand-n conumer,

22 1 a he repreent all houehold of type. The tand-n conumer make purchae on behalf of the ex ante dentcal houehold he repreent, then allocate the purchae to ndvdual houehold by mean of an optmal lottery. Notce the role played here by the aumpton that all houehold of a ven type are ex ante dentcal: there no ambuty about how a lottery hould be choen to allocate reource amon ndvdual houehold. In the tand-n conumer economy, an allocaton y a vector y Y for each type. The allocaton ocally feable f y 0. A tand-n conumer equlbrum J wth tranfer cont of a non-ero prce vector p, and a ocally feable allocaton y. For each type, y hould maxme U ( y ) ubject to py p y, y Y. An endowment equlbrum and Pareto effcency are defned n the obvou way. Notce that equlbra n the tand-n conumer economy are equvalent to equlbra n the lottery economy n a drect and mple way. Gven a lottery equlbrum, p, we can compute the expected reource ued by the equlbrum lottery y x d ( x ). Clearly y, p are a tand-n conumer equlbrum. S Converely, ven a tand-n conumer equlbrum y, p, we can ue Lemma 7.1 to fnd for each tand-n conumer an optmal et of lottere, and t clear that, p are a lottery equlbrum. To prove the welfare and extence theorem for the unpot economy and lottery economy, t uffce to prove them for the tand-n conumer economy. A th a fnte dmenonal pure exchane economy, th follow from verfyn tandard properte of utlty functon and conumpton et. Lemma 7. Utlty U contnuou, concave, and, f non-ataton hold, trctly ncrean. The net trade et ha 0 Y and cloed, convex and bounded below. If the cheaper pont aumpton hold, then there a pont 0 y Y. 8 Excluvty and Incentve Contrant We have already ponted out that ncentve contrant demand excluvty of contract: althouh houehold of a partcular type are ex ante dentcal, ex pot they reale dfferent value of the doyncratc hock, and would want to trade wth one another. The ue of unpot or lottere ntroduce another dmenon n whch houehold are ex pot dfferent: even houehold who reale the ame doyncratc

23 hock wll have dfferent ex pot net trade, a ome wn and ome loe n the lottery. Th rae the queton of whether even houehold wth the ame doyncratc tate wll want to trade n equlbrum. The anwer that, n the abence of ncentve contrant, for example, when there are ndvblte, houehold do not want to trade. Conequently, t only n econome wth ncentve contrant that we requre excluvty. For mplcty, we lmt attenton to lottere that have countable upport. Th wll be the cae f the conumpton et are dcrete, a they may be wth ndvblte. From the proof of Lemma 7., we alo know that for any lottery equlbrum, there another lottery equlbrum yeldn exactly the ame utlty and wth each type conumn the ame areate reource, n whch the upport of the lottery fnte. The reult we prove hold more enerally, but the proof of the mot eneral cae more techncal. Lemma 8.1: Aume that for all S, F {}, o that there are no ncentve contrant. Suppoe that p upport the upper contour et of U at y, upport S xd ( x) y and, u ( x ) d ( x ) U ( y ), and that upport countable. Let x u at x. S X, X be uch that ( x ) 0. Then p upport the upper contour et of Proof: Suppoe converely to the Lemma that there ~ x p( ~ x x ) 0. Conder ~ defned by R X wth u ( x ~ ) u ( x ), and ( ), ~ x x ~ ( ) S 0 ( ~ x ) ( ) ~ T x x x For ' take ~ ' '. Set w x d x S ~ ( ), where x the dummy varable of nteraton. Then p( w y ) 0 and u ( w ) u ( y ), a contradcton. 9 Rk Averon and Lottere Whle n prncple lottere may be ueful when there are ncentve contrant, n many practcal example, equlbrum lottere are deenerate. Th not a neceary concluon: Cole [7] ve a robut example n whch lottere are ued to ort hh

24 3 marnal utlty from low marnal utlty tate. Cole example ha the odd feature, however, that the hh marnal utlty houehold, who we would enerally thnk of a havn low endowment, are le rk avere than low marnal utlty houehold. In th ecton, we how that n the more plauble cae of decrean abolute rk averon, equlbrum lottere are n fact deenerate. Th the cae n our ntal example. We now pecale to the cae of an economy n whch there are no ndvblte. We aume that each houehold of type n tate ha an endowment of J, and a utlty functon for conumpton v( c) that trctly ncrean and concave. The net trade et X mx x r allow no ndvblte, and the utlty functon u ( x ) v ( x ). In th ettn, ven a lottery over net trade, we J defne the certanty equvalent c ( ) to be the fracton of the expected conumpton from the lottery that equvalent to the expected utlty from the lottery, c ( ) xd( x ) j, where e v ( x d ( x ) ) v ( x ) d ( x ) e j and, nce utlty aumed concave, 0 1. State Independent Preference: v ( c ) v ( c ) Decrean Abolute Rk Averon: If 0, then v ( c ( )) v ( x ) d ( x). Th ay that the certanty equvalent an ncrean functon of conumpton, or equvalently, that the rk premum declnn. It trahtforward to check that n the cae of a nle ood, th equvalent to the uual defnton. We wll how that, f (for all type) preference are tate ndependent and exhbt non-ncrean abolute rk averon, then there alway an equlbrum wth deenerate lottere. It convenent to prove th un a weaker condton that doe not requre tate ndependent utlte. Generaled Decrean Rk Averon: If ' F then v( c' ( ' ) ' ) v( x) d' ( x )

25 4 From the aumpton of feable mrepreentaton th defnton make ene, nce X X ' mean that ' n fact a lottery n X ; th aumpton combned wth tate ndependent decrean abolute rk averon alo mple eneraled decrean rk averon. Bacally, th aumpton ay that a houehold that actually ha tate ' more rk avere than a houehold that maqueradn a tate '. Theorem 9.1 Wth eneraled decrean abolute rk averon (and no ndvblte), every oluton to the tand-n conumer problem a pont ma on a nle pont for each. Proof: Let be lottere that olve the tand-n conumer problem. Conder the alternatve deenerate lottery n that put ma one on c ( ). Th lottery conume no more reource than nce v concave, and, f houehold tell the truth, they yeld exactly the ame utlty. Moreover, the eneraled decrean rk averon condton mean that any ncentve contrant atfed under atfed a well under the new plan. Fnally, f any of the are non-deenerate, then, nce the rk premum aumed trctly potve, trctly fewer reource are conumed by the deenerate alternatve, whch contradct Lemma An Example Havn hown that decrean abolute rk averon lead to deenerate lottere, we turn now to an example n whch ncrean abolute rk averon lead to nondeenerate example. A n Cole [7], we focu on the cae where there are two tate, a nle ood and a nle type. For notatonal convenence, we omt the upercrpt. Wth two tate, whch we denote, b, and a nle ood, we denote the endowment a b. In addton, we aume voluntary publc endowment. Th mean that houehold can optonally reveal ther endowment. Conequently, the et of feable report are F {, b}, F b b {}. Let { x } be a fnte et of pont on whch put weht. Stand-n conumer utlty then ven by max ( ) ( ) v x S x

26 5 ubject to S x x ( ) 0 v( x ) x v x x ( ) ( ) b ( ) 0 b ( x ) 1 ( x ) 0. Th a lnear proram that can be olved on any rd { x }. Lemma 5.1(c) ay that there wll be a oluton that place weht on at mot 4 dfferent pont for each. A the rd refned, the et of approxmate oluton wll approach the et of exact oluton to the problem; f the ornal rd carefully choen, t be poble to fnd an exact oluton on the rd. Notce that th the eneral ort of lnear prorammn problem that we need to olve to fnd the optmal lottery allocaton for a houehold type n the tand-n conumer economy. In the eneral cae, there are J reource contrant, whch replace the 0 on the rht-hand de wth y j, and k ncentve contrant. To have non-deenerate lottere requre ncrean abolute rk averon. A convenent famly wth th property that of quadratc utlty functon. Conder the quadratc utlty functon vc () 78c c wth endowment 30, b 10 and probablte b 1/. It can be verfed that a oluton to the lnear prorammn problem defnn the tand-n conumer utlty a deenerate lottery n the ood tate wth ( 1) 1, and n the bad tate the non-deenerate lottery b( 7) 1/, b( 5) 1/. The mean tranfer n the ood tate 1, wth the mean tranfer n the bad tate of +1. The tranfer n the bad tate nvolve a lare amble between +7 and 5, however, and the well-endowed houehold prefer to avod th rk. There are multple oluton to th example. For example, ( 1) 1, b() 1 7 / 16, b( 9) 9 / 3, b( 7) 9 / 3 alo a oluton. To verfy that all oluton to our example nvolve non-deenerate lottere, we provde a uffcent condton under whch a non-deenerate lottery mprove welfare, o that the oluton to the tand-n conumer problem wll necearly be non-deenerate. If there only one ood, then the only deenerate lottery that atfe the ncentve contrant and doe not lower welfare autarky: ( 0) 1, b( 0) 1. We now earch for condton under whch a mall lottery can mprove welfare whle atfyn the ncentve

27 6 contrant. Suppoe that we replace b ( 0) 1 wth a mall lottery x wth mean x and varance x. To be pecfc, let b ( x' ) be a lottery wth mean 0 and varance 1 (for example, b () 1 1/ and b ( 1) 1/ ), and let x x x x'. We et ( x ) 1, where, to mantan ocal feablty, we requre that x b x. The ncentve contrant holdn exactly can be wrtten b v( x) Ev( x). To econd order, th contrant can be wrtten approxmately a b b v( ) v'( ) 1 x v''( ) ( ) x 1 v( ) v'( ) x v''( )( x x ). Un th equaton we can now olve, at leat to econd order, for the varance of the lottery for whch the ncentve contrant atfed a a functon of the mean of the lottery: v'( ) ( x) x v''( ) x b x. A econd order Taylor ere expanon allow u to approxmate the ex ante utlty of a mall lottery that atfe the ncentve contrant a a functon of t mean: b b V( x) [ v( ) v'( ) 1 x v''( )( ) x ] 1 b[ v( b) v'( b) x v''( b)( x( x) x )]. Dfferentatn wth repect to x, we fnd V'( 0) b( v'( b) v'( )) b v'( ) v''( b). v''( ) If th expreon trctly potve, a t n our numercal example, then ntroducn a mall lottery wth a potve mean and a varance jut lare enouh to make the ncentve

28 7 contrant hold ncreae welfare. Conequently, the deenerate lottery cannot be the oluton to the tand-n conumer problem, and a non-deenerate lottery mut be ued n equlbrum. It worth pontn out two feature of th example and our calculaton. Frt, notce that, f utlty exhbt decrean abolute rk averon, then v''( v b ) ''( ) v'( ) v'( ) b and t mpoble for V'( 0 ) to be potve at a mall non-deenerate lottery. Even f there ncrean abolute rk averon, however, there may not be non-deenerate lottere: for th, ncrean abolute rk averon neceary but not uffcent. Second, notce that, f utlty quadratc, then the formula we obtan for V( x) exact, and not jut a ood approxmaton for mall lottere. In fact, t ha been by maxmn th functon that we have obtaned the numercal example. The ntuton for the preence of multple equlbra n our numercal example mple: The functon V( x) maxmed by a lottery n the bad tate wth mean 1 and varance 36. Th pn down expected utlty n both tate, but there an nfnte number of lottere n the bad tate wth th mean and varance, whch all that matter for a quadratc utlty functon. The trck to make the conumer n the ood tate ndfferent between reportn the ood tate and makn a tranfer of 1 and reportn the bad tate and recevn the tranfer. We can ue our numercal example to llutrate ome ue related to unpot allocaton. One unpot allocaton, equvalent to the lottery allocaton ( 1) 1 and b( 7) 1/, b( 5) 1/ x ( h, ) 1 and R ( h)mod / xb ( h, ) S T 7 f f ( h)mod 11/. Th allocaton far from unque, however, nce any way of relatn the unpot varable to the ndex of the houehold h that reult n each houehold recevn +7 wth probablty 1/ and 5 wth probablty 1/ work jut a well. Yet another optmal unpot allocaton