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.fr.fed.u Frt veron: Septemer 11, 1997 Th veron: Auut 1, 000 Atract: We tudy a prototypcal cla of exchane econome wth prvate nformaton and ndvlte. We etalh an equvalence etween lottery equlra and unpot equlra and how that the welfare and extence theorem hold. To etalh 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 aolute rk averon and no ndvlte, we prove that no lottere are actually ued n equlrum. We provde a mple numercal example wth ncrean aolute rk averon n whch lottere are necearly ued n equlrum. We alo how how the equlrum allocaton n th example can e mplemented n a unpot equlrum. 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 ear utantal 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 Alvarez and Jermann [1], have een ued to tudy doyncratc rk earn. 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 hazard an ovou explanaton for uch undverfed portfolo, and recently there ha een a reurence of nteret n ntroducn th feature nto eneral equlrum theory. Bennardo [3], Bennardo and Chappor [4], and Bn and Guatol [5] have een uch effort. The pont of departure ha een Precott and Townend [19, 0], who ntroduce oth the dea that ncentve contrant can e ntroduced nto eneral equlrum theory n a enle way and the dea that lottere play a potentally mportant role n the reultn theory. Althouh ther theory ha een wdely ued to tudy ndvlte n the areate economy y Hanen [13], Roeron [], Cole and Precott [9], and other, untl recently lttle effort ha een 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 numer of ue. One ue how lottery equlra are to e mplemented. In the ndvlty cae wth a fnte numer of houehold, Shell and Wrht [3] how that there a cloe connecton etween lottery equlra and unpot equlra, 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 equlrum allocaton can e decentralzed a a unpot equlrum. In a model wth complete nformaton and a fnte unpot tate pace, Garratt [11] how that, whle every lottery equlrum allocaton can e decentralzed a a unpot equlrum, not every unpot equlrum allocaton a lottery equlrum allocaton. Garratt et al. [1] etalh 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 aolute rk averon and no ndvlte, we how that lottere are not actually needed n equlrum. We provde a mple numercal example wth ncrean aolute rk averon n whch lottere are necearly ued n equlrum. We alo how how the equlrum allocaton n th example can e mplemented n a unpot equlrum. The proof of theorem, the analy of the example, and the computaton of equlra n thee ort of econome are reatly facltated y the noton of the tand-n conumer economy. Overall, we arue that the Precott and Townend framework repreent a enle and ueful framework for analyzn moral hazard and advere electon n eneral equlrum theory. In recent related work, Cole and Kocherlakota [9] conder an envronment lke our wth prvate nformaton over endowment. They how that, f torae pole and unoervale y other houhold, then the equlrum 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 Prolem There a contnuum [,] 01 of houehold who are ex ante dentcal. There are two ood j = 1,. Let c j denote conumpton of ood j. Utlty ven y 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 en n one of two tate, {, }. The endowment of ood 1 tate dependent and can take on one of the two value ω 1 and ω 1, where ω 1 > ω 1, whle the endowment of ood fxed at. Vewed n the areate, after the tate realzed, half of the populaton ha the ood endowment, and half the ad endowment. After the tate realzed, there are an from trade, a the ad endowment houehold want to purchae ood 1 and ell ood. Before the tate realzed, there are addtonal an from trade nce houehold want to purchae nurance aant the ad tate. In fact, nce all houehold are ex ante dentcal and utlty trctly concave, the et ymmetrc allocaton that n whch houehold complete nformaton econome, of the et of lottery equlrum allocaton and the et of unpot equlrum allocaton aed on a contnuou unpot randomzaton devce.

4 3 conume ( ω + ω )/ of ood 1, and of ood. Follown the mechanm den 1 1 lterature, we refer to th allocaton a the frt et. Suppoe that the realzaton of the doyncratc rk prvate nformaton known only to the ndvdual houehold. In th cae, the frt et allocaton not ncentve compatle. In the frt et allocaton, ad endowment houehold receve an nurance payment of ( ω ω )/ 1 1, 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 equlrum to proht tradn n nurance contract, and conder only tradn that take place ex pot after the tate realzed. Th an example of an ncomplete market model. The reultn compettve equlrum lead to an equalzaton of marnal rate of uttuton etween the two ood for the two type of houehold, ut there are unrealzed an from ex ante trade n nurance contract. A econd approach to modeln equlrum to oerve that t pole to trade n nurance contract provded that no houehold uy a contract that would later lead t to mrepreent t tate. If endowment can e made pulc, ut only voluntarly, then the ood endowment houehold can mtate the ad endowment type, ut not converely. Suppoe that a houehold attempt to purchae ( x1, x ) n tate n exchane for ( x1, x ) n tate. In the ood endowment tate, utlty wll e ( ). In th cae, the ood endowment houehold may want to v ω + x + v ω + x pretend that the tate actually the ad endowment tate, tate. To avod detecton, t mut make the ame pot market purchae that a ad endowment houehold would make, ( x1, x ). Th reult n utlty v ω + x + v ω + x compatlty contrant. Therefore the ncentve v ω + x + v ω + x v ω + x + v ω + x If th contrant atfed, the ood endowment houehold ha no ncentve to le aout t prvate nformaton. We now etalh that, f tradn n nurance contract prohted, there are ncentve compatle an to allown th trade. Let ( x,,, 1 x x1 x ) denote net trade y a houehold n an equlrum n whch tradn n nurance contract prohted.

5 4 Snce a ad 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 ad endowment houehold and had ncome left over, t trctly prefer t own net trade to that of the ad endowment houehold,. v ( ω + x ) + v ( ω + x ) > v ω + x + v ω + x We already know that ( ~ x, x ~, ~ x, x ~ ) = ( ( ω ω )/ 0,,( ω ω )/ 0, ) would e the equlrum trade of nurance f there were no ncentve contrant and would yeld trctly hher ex ante utlty than ( x,,, 1 x x1 x ) ecaue utlty trctly concave. Conder the net trade ( x, x, x, x ) = (( 1 θ) x + θ x ~,( 1 θ ) x + θ x ~,( 1 θ ) x + θ x ~,( 1 θ ) x + θ x ~ ) If θ mall enouh, then ood endowment houehold tll ha no ncentve to mrepreent, ut ex ante utlty trctly hher. Therefore, there are addtonal ncentve compatle an to trade that are not realzed when tradn n nurance contract prohted. Suppoe, more enerally, that houehold trade ood contnent on announcement. No houehold wll ever delver a undle that not ncentve compatle. Every houehold know th fact, and o only ncentve compatle undle can e traded. Notce, however, that th troner arument doe not uarantee that all ncentve compatle undle actually can e traded unle thee contract proht ex pot trade: If a ood endowment houehold can receve an nurance payment y clamn a ad 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 proht houehold clamn to have a ad 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 e 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 equlrum theory. There are two complcaton wth th proram. Frt, fxn ( x1, x ), the et of ( x1, x ) that atfe the ncentve contrant fal to e convex, o X not a convex et. Th mean that Pareto mprovement may e pole y 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 ecome E v ω + x + v ω + x E v ω + x + v ω + x For thee two reaon, once we ntroduce ncentve contrant nto eneral equlrum, 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 y h H =[ 0, λ ], where λ > 0 the ze 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 oerved unpot and houehold pecfc doyncratc rk. A unpot a random varale σ unformly dtruted on [0,1]. Idoyncratc rk repreented y pecfyn that each houehold of type conume n one of a fnte numer of tate S. Each tate ha proalty π > 0 where π 1. Th proalty ha two = S nterpretaton: Frt, from the perpectve of the ndvdual houehold, t the proalty of en n the tate. Second, a we explan elow, 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, σ ) R for the net j 4 In addton to Precott and Townend [1984a, 1984], uch other author a Arnott and Stltz [] have remarked on the potental of lottere to mprove welfare. 5 Strctly peakn, we hould allow houehold to ae contract on the doyncratc tate of other houehold. In the type of equlrum we wll conder contract aed 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 σ. The dtruton of doyncratc hock and unpot are aumed to e ndependent. The doyncratc hock are uch that the areate net trade of all type houehold of ood j when the unpot σ ven y yj ( σ) = x h dh π S j(, σ), whch the aumpton that π 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 y Leeue nteraton and a pace of conumer on the unt nterval; th dcued n Boylan [6]. Alternatvely, we could defne areate net trade y the Pett nteral, a n Uhl [4]. Or we could mply allow doyncratc tate to e 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 efore any uncertanty realzed. Then the doyncratc tate are realzed and announcement of tate are made. Next, unpot are realzed. 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 realzed pror to the realzaton and announcement of ndvdual tate. A more eneral model would allow oth ex ante and ex pot unpot. Later, we how that equlra n the more eneral model are equvalent to equlra 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 realze a net trade x ( h, σ ) R. We aume that th net trade mut elon to the feale net trade et X. Notce that endowment are ncorporated drectly nto the feale net trade et and are not pecfed eparately; we allow X to depend on o that endowment may e doyncratc tate contnent. In a 6 In a model wth only ex ante unpot, equlrum can e chaned y addn ex pot unpot, ecaue lottere condtonal on prvate nformaton may e ued to eparate houehold wth dfferent rk preference. Th wa ornally ponted out y Cole [7].

8 7 tandard endowment economy, wth endowment ω, we have X ven y u : X R. We ue the notaton x ( h, σ ) R JS = { x x ω }. Utlty for the et of net trade vector correpondn to dfferent doyncratc tate and for expected utlty. S u ( x ( h, σ)) = π u ( x ( h, σ)) We next conder ncentve contrant. Thee are derved from requrn that houehold not want to mrepreent prvate nformaton aout ther own tate. Prvate nformaton aout tate pecfed y et of feale report F S. Thee et repreent the report that a houehold can make aout h tate when h true tate wthout en contradcted y ether pulc nformaton or phycal evdence. Conequently, a et of feale report mut atfy two aumpton: Feale Truthtelln: For all S, F. Feale Mrepreentaton: If F, then X X. The econd aumpton requre that t not pole to report a net trade et that nfeale wth repect to the true net trade et. Th aumpton rule out more complcated polte, uch a tuaton where the fealty of tradn plan can only e dcovered ex pot and punhment mpoed for volatn contract. In uch a cae, feale report would depend on the partcular contract offered. The noton of feale report that atfy thee two aumpton leave utantal flexlty n uldn model econome. We provde two example of clae of econome that atfy our aumpton: Pulc Endowment: The endowment ω not. Then F = ω = ω. are pulcly oerved, ut preference u are Voluntary Pulc Endowment: A houehold wth endowment ω may pulcly dplay any porton of t endowment. Preference u endowment e dplayed n order to e reported, we otan F are prvate. By requrn that an = ω ω. In other word, houehold clamn a partcular endowment can e 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 feale report a data. Th avod the deeper and more dffcult queton of how et of feale report are enerated from underlyn fundamental. Notce n partcular that we could ue the et F = ω ω even n the cae of pulc endowment. It apparent that th would lead to maller et of feale allocaton than makn full ue of the pulc nformaton and takn F = ω = ω. A caveat to the reult reported n th paper that effcency condtonal on a partcular et of feale report. A unpot contnent tradn plan x ( h, σ ) called ncentve compatle f for all F ( ) u( x( h, σ)) dσ u( x ( h, σ)) dσ 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 ounded elow. 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 oundary condton, thee aumpton are elf-explanatory. The oundary condton requre marnal utlty to aymptote to zero; t ay that eventually utlty ncreae lower than any lnear functon. 7 7 Notce that nothn n thee aumpton rule out ndvlte, nor t neceary to, a the lterature on lottere wth ndvlte dcued n the ntroducton how.

10 9 An mportant fact aout the non-ataton condton that t mple non-ataton for ncentve compatle net trade. Lemma 3.1: If x X S ncentve compatle, then there an ncentve compatle ~x X uch that π u ( ~ x ) > π 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, π u( ~ x ) > π S S u( x ) ecaue π > 0. We now arue that ~ x ncentve compatle. 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 emphaze aout the model are Type are commonly known; the doyncratc tate may or may not e prvate nformaton. It mportant that contractn take place pror to learnn any prvate nformaton. If contractn pole 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 aout the prvate nformaton of rval houehold. Th aumpton could e relaxed, ut t would then e neceary to allow contractn aed upon the announcement of the relevant rval. We have mplctly aumed that contract are excluve that, that tradn not pole after delvere are made. A we noted n the example, equlrum wth nonexcluve tradn qute dfferent than wth excluvty. A ponted out y Precott and Townend [19], the welfare theorem can fal wthout excluvty. 8 Precott and Townend [1984] ve an example n whch the frt welfare theorem hold, ut the econd fal. In other example, uch a thoe n Precott and Townend [1984a], oth welfare theorem fal and equlrum may not ext.

11 10 We conclude th ecton y llutratn how the example of the prevou ecton ft nto th framework. In the example I = 1, J =, λ = 1, S = {, }, π = π = 1/. The net trade et are thoe net trade that exceed the neatve of the endowment X = {( x, x ) x ω, x ω } The utlty functon for net trade derved from the utlty of conumpton accordn to u( x1, x) = v1( ω1 + x1) + v( ω + x) u ( x1, x, x1, x) = u( x1, x ) + u( x1, x). The feale reportn et reflect the fact that X X, F = {}, F = {, }. There one ncentve contrant, correpondn to a ood endowment tate reportn a ad endowment: u( x( h, σ)) dσ u( x( h, σ)) dσ 0. 4 Equlrum wth Sunpot A unpot allocaton a meaurale map for each type from houehold to ndvdual tradn plan; that x ( h, σ ) X. An allocaton ocally feale f for each unpot realzaton σ I λ π x( h, σ) dh 0. = 1 S Notce that th defnton ncorporate pulc 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, σ)) dσ = u ( x ( h, σ)) dσ 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-Dereu commodty a delvery of j ood contnent on (, e σ ). Arrow-Dereu prce are q R J. Becaue ( e, σ ) + there no areate rk, we retrct attenton to Arrow-Dereu equlra n whch prce are ndependent of e; that, q( e, σ ) = q( σ ). Wth thee prce, the cot to a houehold h of type to purchae x at σ q e σ x de = q( σ) x de = π q( σ) x. { (,) ee h = } (, ) { (,) ee h = }

12 11 J We refer to a non-zero meaurale functon q( σ ) R + a a prce functon. A unpot equlrum wth tranfer cont of a ocally feale unpot allocaton x toether wth a prce functon q. For all type and almot all h [ 0, λ ], x ( h,) mut maxmze u ( x ( h, σ)) dσ over unpot contnent tradn plan x ( h,) atfyn the unpot udet contrant π q σ x σ σ π σ σ σ S h d S q x h d ( ) (, ) ( ) (, ), and the ncentve contrant u( x( h, σ )) u( x ( h, σ )) d σ 0, F ( ). The tranfer themelve mut atfy the equal treatment condton that they depend only on type: S S π q( σ) x ( h, σ) dσ = π q( σ) x ( h, σ) dσ for almot all h, h. A unpot equlrum a unpot equlrum wth tranfer n whch the tranfer are zero: π q σ x σ σ ( ) S ( h, ) d = 0 for almot all h. Fnally, a unpot allocaton Pareto effcent f there no alternatve ocally feale 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 equlrum allocaton mut e an equal utlty allocaton. If t were not, then a potve meaure of type could ncreae ther utlty y wtchn to a conumpton plan ued y other of the ame type. Lemma 4.1 A unpot equlrum allocaton wth tranfer an equal utlty allocaton. Our man oal to etalh the man theorem of compettve eneral equlrum theory for the unpot economy Theorem 4. (Frt Welfare Theorem) Every unpot equlrum 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 equlrum wth tranfer. Theorem 4.4 (Extence Theorem) There at leat one unpot equlrum.

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 elow. 5 Equlrum wth Lottere A proalty dtruton µ over X u ( µ ) u ( x ) dµ ( 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 µ = µ S yeld the ame utlty, and the ncentve contrant can alo e computed drectly from the lottery. The areate reource ued y a et of lottere are y = λ xd x π S µ ( ). Notce that wth a contnuum of houehold we need not dtnuh etween realzed and expected net trade. Th dtncton mportant n decentralzn lottere n the ndvle cae wth a fnte numer of houehold, a can e 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 = {, }, 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 µ 1 1. The lottere µ are each non-neatve meaure that atfy d µ 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, µ = ( µ,..., µ ), where the et of lottere µ an lottere µ 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 feale f = I 1 λ π xdµ ( x) 0. S Th ay that n the areate the expected net trade ued y the lottery allocaton non-potve. A lottery allocaton Pareto effcent f no ocally feale, ncentve compatle Pareto mprovement pole. A lottery equlrum wth tranfer cont of a ocally J feale lottery allocaton µ toether wth non-zero prce vector p R +. For all type, µ mut maxmze π µ u ( ) S over lottere µ atfyn the lottery udet contrant S S p x d ( x ) p x d π µ π µ ( x ), and the ncentve contrant u( µ ) u( µ ) 0, F ( ). A lottery equlrum a lottery equlrum wth tranfer n whch the tranfer are zero p x d ( x ) π S µ = 0. Notce that n th formulaton, lottere are prced accordn to the areate reource they ue. Th a no-artrae 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 uyn y and y eparately exceed the cot of uyn y y, t would e proftale to uy the jont lottery y y and ell the pece, whle n the oppote cae, the pece hould e ouht eparately, then packaed and old. Only 9 1 I We could have equally well followed the formalm of defnn a trade vector µ = ( µ,..., µ ), and retrctn houehold of type to the trade et n whch the lottery vector for all other type put proalty 0 on all trade. The notaton followed here ha the advantae of en le cumerome, ut 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 artrae opportunte. 10 Let k = (# F 1) denote the numer of ncentve contrant. There are four S ac feature of lottere that are worth emphazn: Lemma 5.1 (a) A convex comnaton of ncentve compatle et of lottere ncentve compatle. () Let y, ~ y e the reource ued y the et of lottere µ, ~ µ and let 0 α 1. Then the et of lottere αµ + ( α ) ~ 1 µ ue areate reource αy + ( 1 α) ~ y. (c) For any ncentve compatle et of lottere µ there another ncentve compatle et of lottere ~ µ un the ame areate reource, yeldn the ame utlty, and each lottery ~ µ havn upport on k + J + pont. (d) Let y e the reource ued y the ncentve compatle et of lottere µ, and uppoe ~ y > y. Then there an ncentve compatle et of lottere ~ µ un no more reource than ~ y that yeld trctly more utlty than µ. Proof: (a) and () are mmedate. The proof of (c) larely mathematcal, and provded n the Appendx. To prove (d), frt apply (c) to fnd an ncentve compatle lottery ~ µ wth fnte upport yeldn the ame utlty a µ. Becaue th lottery ha fnte upport, t follow from Lemma 3.1 that there an ncentve compatle net trade x wth π u( x) π µ π µ S S u( ~ S u > ) = ( ). Let ~ µ e the deenerate lottery wth pont ma on x. Then for all 0 < α 1, the et of lottere ( 1 αµ ) + αµ ~ ncentve compatle y (a) and yeld trctly more utlty than µ. A α approache 0, however, the areate reource ued y th et of lottere approach y and, therefore, for α 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-artrae condton follow n that ettn from proft maxmzaton y frm. Our approach follow Hanen [1985].

16 15 We wll etalh the man theorem of compettve eneral equlrum theory for the lottery economy, a well a the unpot economy. Theorem 5. (Frt Welfare Theorem) Every lottery equlrum allocaton wth tranfer Pareto effcent. Theorem 5.3 (Second Welfare Theorem) For every Pareto effcent allocaton there are prce formn a lottery equlrum wth tranfer. Theorem 5.4 (Extence Theorem) There at leat one lottery equlrum. In Precott and Townend [0], thee theorem are proved drectly; we ve alternatve proof elow. Our reult on unpot equlra then follow from hown that lottery and unpot allocaton are equvalent. 6 Sunpot Equlrum veru Lottery Equlrum Sunpot allocaton and lottery allocaton are dfferent decrpton of randomzaton. For example, uppoe that there are two dentcal type, and one ood, automole, for whch the conumpton vector ether one automole or zero. Suppoe moreover, that each type endowed wth one half an automole per capta. From the lottery perpectve, the tuaton mple: there can e no trade etween the two type, o each houehold hould receve an automole wth proalty 1/. In other word, n equlrum, each houehold of each type purchae a lottery wth a 50 percent chance of 1 automole, and a 50 percent chance of 0 automole. In our notaton, µ ( 1/ ) = 1/ and µ ( 1/ ) = 1/ where ω = 1/ and ω + x equal to 1 or 0 for oth houehold type = 1,. Th lottery can e mplemented n many way y 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 numer lead to ocal fealty. An alternatve formulaton would e to have a mple unpot allocaton n whch when the unpot varale atfe σ 1/, the frt type receve all the car and, when σ > 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 Suject to uual caveat aout a contnuum of ndependent random varale; ee the dcuon aove.

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 ~ xhσ (, ). For each houehold, there correpond lottere µ ( x ~ ( h,)). We can then averae thee lottere over houehold to et a mean lottery for the entre type ~ ( ~ µ = µ x ( h,)) dh/ λ. Notce that the reource ued y th lottery are equal to the expected reource ued y the unpot allocaton; that, x d~ µ ( x ) = ~ x ( h, σ) dhd σ / λ. Moreover, y defnton, n an equal utlty unpot allocaton houehold of type mut e ndfferent etween the allocaton ~ x ( h,) and ~ x ( h,) for almot all h, h. Snce ther utlty lnear n proalte, th mean they mut e ndfferent etween ~ x ( h,) and the mean et of lottere ~ µ 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 proalte, the mean et of lottere mut atfy the ncentve contrant. Conequently, the mean et of lottere correpondn to a unpot equlrum allocaton a natural canddate to e an equlrum of the lottery economy. We provde an example of averan of lottere n Secton 10. If q ( σ ) a prce functon n the unpot economy, we can n a mlar way defne the mean prce q = q( σ ) dσ. Althouh t not ovou, we wll how elow that the mean prce n fact a correct way to prce the mean lottery n the lottery economy. 1 To apprecate the polty of q( σ ) not en contant, conder a varant of our automole example. Aan there are two dentcal type, ut 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 e conumed n fxed proporton: u ( x, x ) = mn[ ω + x, ω + 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 equlrum where q( σ ) = q = ( 10, ) and ( x 1 h 1 /, 1 / ) f σ (, ) 1 / σ = ( 1/, 1/ ) f σ > 1/, wth the oppote allocaton to houehold of type. Another equlrum ha the ame allocaton ut the prce vector q( σ ) = q = ( 01., ) Yet another equlrum would have (, ) σ / σ > / q( σ ) = 10 f 1 4 or 3 4 ( 01, ) f 1/ 4 < σ 3/ 4. The polty of prce q( σ ) that vary wth the unpot σ are ecaue 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 equlrum prce. In Theorem 6., we prove that th true n eneral. In our example, n whch the role for unpot are ecaue of ndvlte, the polty of more than one contant prce vector upportn an allocaton deenerate n that t dappear f we pertur the endowment. In econome where the role for unpot are ecaue of ncentve contrant, however, there no need for equlra wth more than one upportn prce to e deenerate. We defne a unpot allocaton to e 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 e 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( σ ) = p to e the canoncal unpot prce functon 13. For a ven lottery allocaton µ we defne the canoncal unpot allocaton to e a partcular allocaton n whch the areate reource ued y each type are ndependent of the unpot tate. Specfcally, correpondn to the lottery µ a random varale ~ x ( σ ). Recall that 13 Garratt et al. [1] call thee prce contant proalty adjuted prce. They how that n econome wth complete nformaton all unpot equlrum allocaton can e upported y prce that are collnear wth proalte f the unpot randomzaton devce contnuou.

19 18 amod the remander of a dvded y. We defne the canoncal unpot allocaton a x( h, σ ) = ~ x (( σ + h ) mod λ ). 14 Notce that at th canoncal allocaton, the areate net trade y all houehold of a type ndependent of the realzaton 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 fealty and the ncentve contrant, we can draw an mmedate concluon aout 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 feale, ncentve compatle equal utlty et n the unpot economy the ame a the ocally feale ncentve compatle utlty et n the lottery economy. Le mmedately ovou the equvalence of equlra n the two econome. Theorem 6. An allocaton and prce are an equlrum wth tranfer n the lottery economy f any (or all) equvalent allocaton and prce functon are an equlrum wth tranfer n the unpot economy. An allocaton and contant prce functon are an equlrum wth tranfer n the unpot economy f the equvalent allocaton and prce functon are an equlrum wth tranfer n the lottery economy. In oth cae the ze of the tranfer the ame n the two econome. Proof: Conder a unpot allocaton x and prce functon q and an equvalent lottery allocaton µ and prce p. Suppoe frt that q contant (n partcular, that q( σ ) = p ) and that µ, p are an equlrum wth tranfer n the lottery economy. Snce houehold care only aout ther ndvdual lottery and nce q( σ ) = p, x, q are an equlrum wth tranfer n the unpot economy. Snce n oth cae each type pay only for the areate reource ued, whch the ame n oth econome, the tranfer mut e the ame n oth cae. Now uppoe ntead that x, q are an equlrum wth tranfer n the unpot economy, and that poly q not contant. We mut how that µ, 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 equlrum wth tranfer. To how th we mut how that for each type any et of lottere µ that yeld more utlty than µ cannot e afforded at the prce p and that µ can e purchaed at thoe prce. Frt, we how that for each type, any et of lottere µ that yeld more utlty than µ cannot e afforded at the prce p. Suppoe that µ n fact affordale and yeld more utlty than µ. Notce that nce they are equvalent, the utlty from µ the ame utlty x ve almot all houehold. We ue µ to contruct a unpot plan that affordale at prce q yeldn the ame utlty a µ ; th wll e the dered contradcton. Conder the canoncal unpot allocaton x correpondn to µ. Th ve every houehold n more utlty than x. It alo contructed o that x ( h, σ ) dh ndependent of σ. By contructon t affordale at prce q( σ ) = p ; ecaue p the averae of q( σ ) and x ( h, σ ) dh ndependent of σ, t therefore affordale at prce q( σ ). It follow that for a potve meaure et of houehold h, x ( h,) alo affordale at prce q( σ ). Th ve the dered contradcton, nce x ( h,) ve the ame utlty a µ for all h. To conclude the proof, we how that µ can e purchaed at prce p. Suppoe for ome th not the cae, that µ cot more than the tranfer to that type. Then nce µ ocally feale, for ome other type expendture on µ mut e le than the tranfer payment. But y Lemma 5.1, uch a type could ue the extra ncome to purchae a etter lottery than µ, whch we have hown cannot happen. 7 The Stand-n Conumer Economy We now prove the welfare theorem and the extence of an equlrum. From the equvalence of the unpot and lottery equlra, 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 ut 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 y oervn that the houehold prolem of maxmzn utlty uject to a udet contrant can e roken 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 maxmze utlty uject 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 feale tradn plan of type houehold: J Y y x X Cloure(ConvexHull{ R, y = π x }). Gven that a undle y Y ha een purchaed, how much utlty can a type houehold et? The anwer ven y U ( y ) = up π u x d x S ( ) µ ( ) uject to upport µ X, λ π xdµ ( x) y, u ( µ ) u ( µ ) 0, F ( ). S Th contruct wll e 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 ). S Lemma 7.1 If the oundary condton hold, then U ( y ) = max π u x d x S ( ) µ ( ) uject to upport µ u ( µ ) u ( µ ) 0, F ( ). X, π xdµ ( 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 + l l J + pont convern to the up. Let x, µ e the pont and proalte n th equence. Th ha a converent uequence on the extended l real lne. Becaue X ounded elow, any component of x that convere to ha correpondn proalty convern to zero. By the oundary condton the lmt of expected utlty for uch a pont alo zero. So the lmt et of lottere place weht only on fntely many pont, and ve the ame utlty and atfe the fealty and ncentve condton. It the optmal et of lottere. We now tudy trade n the economy, y 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 ehalf of the ex ante dentcal houehold he repreent, then allocate the purchae to ndvdual houehold y mean of an optmal lottery. Notce the role played here y the aumpton that all houehold of a ven type are ex ante dentcal: there no amuty aout how a lottery hould e 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 feale f y 0. A tand-n conumer equlrum J wth tranfer cont of a non-zero prce vector p R +, and a ocally feale allocaton y. For each type, y hould maxmze U ( y ) uject to p y p y, y Y. An endowment equlrum and Pareto effcency are defned n the ovou way. Notce that equlra n the tand-n conumer economy are equvalent to equlra n the lottery economy n a drect and mple way. Gven a lottery equlrum µ, p, we can compute the expected reource ued y the equlrum lottery y λ π x dµ ( x ). Clearly y, p are a tand-n conumer equlrum. = S Converely, ven a tand-n conumer equlrum y, p, we can ue Lemma 7.1 to fnd for each tand-n conumer an optmal et of lottere µ, and t clear that µ, p are a lottery equlrum. 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 ounded elow. 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 realze 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 realze 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 equlrum. The anwer that, n the aence of ncentve contrant, for example, when there are ndvlte, 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 countale upport. Th wll e the cae f the conumpton et are dcrete, a they may e wth ndvlte. From the proof of Lemma 7., we alo know that for any lottery equlrum, there another lottery equlrum 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, ut 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 µ X, π xdµ S ( x ) y and, u x d x U y ( S ) µ ( ) = ( ), and that upport µ countale. Let x X e uch that µ ( x )> 0. Then p upport the upper contour et of u at x. Proof: Suppoe converely to the Lemma that there ~ x p ( ~ x x ) 0. Conder ~ µ defned y X wth u ( ~ x ) > u ( x ), and ( ), ~ µ z z x x ~ µ ( ) z = 0 z µ ( ~ = x ) µ ( ) ~ x + x z = x For take ~ µ = µ. Set w = π x d ~ x S µ ( ), where x the dummy varale of nteraton. Then p ( w y ) 0 and u ( w ) > u ( y ), a contradcton. 9 Rk Averon and Lottere Whle n prncple lottere may e ueful when there are ncentve contrant, n many practcal example, equlrum lottere are deenerate. Th not a neceary concluon: Cole [7] ve a rout 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 plaule cae of decrean aolute rk averon, equlrum lottere are n fact deenerate. Th the cae n our ntal example. We now pecalze to the cae of an economy n whch there are no ndvlte. We aume that each houehold of type n tate ha an endowment of J ω R ++, and a utlty functon for conumpton v( c) that trctly ncrean and concave. The net trade et X = x x ω allow no ndvlte, and the utlty functon u ( x ) = v ( ω + x ). In th ettn, ven a lottery µ over net trade, we J defne the certanty equvalent c ( µ ) R ++ to e the fracton of the expected conumpton from the lottery that equvalent to the expected utlty from the lottery, c ( µ ) θ ω xdµ ( x ), where + = v ( θ ω x dµ ( x ) ) v ( ω x ) dµ ( x ) + = + and, nce utlty aumed concave, 0< θ 1. State Independent Preference: v ( c ) = v ( c ) Decrean Aolute Rk Averon: If z > 0, then v ( z + c ( µ )) < v ( z + ω + x ) dµ ( 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 exht non-ncrean aolute rk averon, then there alway an equlrum wth deenerate lottere. It convenent to prove th un a weaker condton that doe not requre tate ndependent utlte. Generalzed Decrean Rk Averon: If F then v( ω + c ( µ ) ω ) < v( ω + x) dµ ( x )

25 4 From the aumpton of feale mrepreentaton th defnton make ene, nce X X mean that µ n fact a lottery n X ; th aumpton comned wth tate ndependent decrean aolute rk averon alo mple eneralzed 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 eneralzed decrean aolute rk averon (and no ndvlte), every oluton to the tand-n conumer prolem a pont ma on a nle pont for each. Proof: Let µ e lottere that olve the tand-n conumer prolem. Conder the alternatve deenerate lottery n that put ma one on c ( µ ) ω. Th lottery conume no more reource than µ nce v concave, and, f houehold tell the truth, they yeld exactly the ame utlty. Moreover, the eneralzed decrean rk averon condton mean that any ncentve contrant atfed under µ atfed a well under the new plan. Fnally, f any of the µ are non-deenerate, then, nce the rk premum aumed trctly potve, trctly fewer reource are conumed y the deenerate alternatve, whch contradct Lemma An Example Havn hown that decrean aolute rk averon lead to deenerate lottere, we turn now to an example n whch ncrean aolute 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,, and a nle ood, we denote the endowment a ω > ω. In addton, we aume voluntary pulc endowment. Th mean that houehold can optonally reveal ther endowment. Conequently, the et of feale report are F = {, }, F = {}. Let { x ξ } e a fnte et of pont on whch µ put weht. Stand-n conumer utlty then ven y max π ( ω µ ) µ ( ) ξ v + x S ξ x ξ

26 5 uject to S ξ ξ ξ π x µ ( x ) 0 v( ω + x ) µ x v x x ( ) ( ω + ) µ ( ) ξ ξ 0 ξ ξ ξ ξ ( )= 1 µ ξ xξ µ xξ ( ) 0. Th a lnear proram that can e olved on any rd { x ξ }. Lemma 5.1(c) ay that there wll e a oluton that place weht on at mot 4 dfferent pont for each µ. A the rd refned, the et of approxmate oluton wll approach the et of exact oluton to the prolem; f the ornal rd carefully choen, t e pole to fnd an exact oluton on the rd. Notce that th the eneral ort of lnear prorammn prolem 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 aolute rk averon. A convenent famly wth th property that of quadratc utlty functon. Conder the quadratc utlty functon vc () = 78c c wth endowment ω = 30, ω = 10 and proalte π = π =1/. It can e verfed that a oluton to the lnear prorammn prolem defnn the tand-n conumer utlty a deenerate lottery n the ood tate wth µ ( 1) = 1, and n the ad tate the non-deenerate lottery µ ( 7) = 1/, µ ( 5) = 1/. The mean tranfer n the ood tate 1, wth the mean tranfer n the ad tate of +1. The tranfer n the ad tate nvolve a lare amle etween +7 and 5, however, and the well-endowed houehold prefer to avod th rk. There are multple oluton to th example. For example, µ ( 1) = 1, µ () 1 = 7 / 16, µ ( 9) = 9 / 3, µ ( 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 prolem wll necearly e non-deenerate. If there only one ood, then the only deenerate lottery that atfe the ncentve contrant and doe not lower welfare autarky: µ ( 0) = 1, µ ( 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 µ ( 0) = 1 wth a mall lottery x wth mean x and varance σ x. To e pecfc, let µ ( x ) e a lottery wth mean 0 and varance 1 (for example, µ () 1 = 1/ and µ ( 1) = 1/ ), and let x = x +σ x. We et µ ( x ) = 1, where, to mantan ocal fealty, we requre that x = π π x. x The ncentve contrant holdn exactly can e wrtten π v( ω x) = Ev( ω + x). π To econd order, th contrant can e wrtten approxmately a v( ω ) v ( ω ) π 1 π x+ v ( ω ) ( ) x π π 1 = v( ω ) + v ( ω ) x + v ( ω )( σ 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 ( ω ) π π σ x ( x) = x π v ( ω ) π 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: V( x) = π [ v( ω ) v ( ω ) π 1 π x+ v ( ω )( ) x ] π π 1 + π[ v( ω) + v ( ω) x + v ( ω)( σ x( x) + x )]. Dfferentatn wth repect to x, we fnd π V ( 0 ) = π ( v ( ω) v ( ω)) π v ( ω ) v ( ω ). v ( ω ) 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 e the oluton to the tand-n conumer prolem, and a non-deenerate lottery mut e ued n equlrum. It worth pontn out two feature of th example and our calculaton. Frt, notce that, f utlty exht decrean aolute rk averon, then v ( ω v ) ( ω ) > v ( ω ) v ( ω ) and t mpole for V ( 0 ) to e potve at a mall non-deenerate lottery. Even f there ncrean aolute rk averon, however, there may not e non-deenerate lottere: for th, ncrean aolute rk averon neceary ut not uffcent. Second, notce that, f utlty quadratc, then the formula we otan for V( x) exact, and not jut a ood approxmaton for mall lottere. In fact, t ha een y maxmzn th functon that we have otaned the numercal example. The ntuton for the preence of multple equlra n our numercal example mple: The functon V( x) maxmzed y a lottery n the ad tate wth mean 1 and varance 36. Th pn down expected utlty n oth tate, ut there an nfnte numer of lottere n the ad 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 etween reportn the ood tate and makn a tranfer of 1 and reportn the ad 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 µ ( 1) = 1 and µ ( 7) = 1/, µ ( 5) = 1/ x ( h, σ ) = 1 and x ( h h, ) ( σ + )mod / σ = 7 f f ( σ + h)mod 1< 1/. Th allocaton far from unque, however, nce any way of relatn the unpot varale σ to the ndex of the houehold h that reult n each houehold recevn +7 wth proalty 1/ and 5 wth proalty 1/ work jut a well. Yet another optmal unpot allocaton