The identification of preferences from equilibrium prices under uncertainty 12

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1 The dentfcaton of preferences from equlbrum prces under uncertanty 12 F. Kübler, 3 P. - A. Chappor 4 I. Ekeland 5 H. M. Polemarchaks 6 Dscusson Paper No. 00 (February, 2000) CORE, Unversté Catholque de Louvan 1 Ths text presents results of the Belgan Program on Interunversty Poles of Attracton ntated by the Belgan State, Prme Mnster s Offce, Scence Polcy Programmng. The scentfc responsblty s assumed by ts authors. The Commsson of the European Communtes provded addtonal support through the Human Captal and Moblty grant ERBCHRXCT Donald Brown, Gérard Debreu, Stefano Demchels, John Geanakoplos, Fabrzo Germano, Itzhak Glboa, Roger Guesnere, Werner Hldenbrand, Rosa Matzkn, Chrs Shannon and partcpants n semnars at the Unversty of Bonn, CentER, the Unversty of Chcago, CORE, the Insttuto Veneto durng the 1999 Summer School n Economc Theory, at Purdue Unversty durng the 1999 NBER conference on General Equlbrum, the Unversty of Pars - I, the Unversty of Tel Avv and the Unversty of Toulouse made helpful comments. 3 Department of Economcs, Stanford Unversty, 4 Department of Economcs, Unversty of Chcago 5 CEREMADE and Insttut de Fnance, Unversté de Pars - IX, Dauphne 6 CORE, Unversté Catholque de Louvan

2 Abstract The compettve equlbrum correspondence, whch assocates equlbrum prces of commodtes and assets wth allocatons of endowments, dentfes the preferences and belefs of ndvduals under uncertanty; ths s the case even f the asset market s ncomplete. Key words: equlbrum, uncertanty, dentfcaton. JEL classfcaton numbers: D52,D80

3 1 Introducton Explanaton and predcton requre the behavor of ndvduals, whch s observable, to dentfy, possbly wthn a class, ther characterstcs, whch are not. In a market, t s compettve equlbra that are observable. The theory fals to specfy out of equlbrum behavor and, as a consequence, demand at arbtrary prces and ncomes s not observable. Expermental observatons may be less restrctve. The preferences of ndvduals are unobservable; belefs are unobservable as well, even though they may not be exogenous and mght vary wth equlbrum prces or aggregate behavor. Observatons may nvolve dfferent degrees of aggregaton. At the most dsaggregated level, one can observe the demand of ndvduals as prces and the allocaton of endowments or revenue vares. More approprately, wth less dsaggregaton, observatons are restrcted to equlbrum prces as the allocaton of endowments vares. The endowments of ndvduals may be n part unobservable, though redstrbutons of revenue are mostly observable; producton possbltes may be observable or not. In Chappor et al (1999), under certanty, the compettve equlbrum correspondence, whch assocates equlbrum prces of commodtes and assets wth allocatons of endowments, dentfes the preferences and belefs of ndvduals. Identfcaton obtans n economes wth uncertanty, even f the asset market s ncomplete. Under uncertanty, wth an ncomplete asset market, the dentfcaton of preferences from observed behavor has strong postve as well as normatve mplcatons. A compettve equlbrum allocaton s not optmal; more pertnently, accordng to Geanakoplos and Polemarchaks (1986), redstrbutons of portfolos of assets can result n a Pareto mprovement; alternatvely, followng Herngs and Polemarchaks (1998), the regulaton of prces and the mposton of ratonng n order to attan market clearng can be Pareto mprovng. The queston of the nformatonal requrements for the determnaton of mprovng redstrbutons of portfolos of assets or regulaton of prces mmedately arses. The dentfcaton of preferences from the equlbrum correspondence allows possbly counterntutve dstrbutonal effects of fnancal nnovaton as n Hart (1975) to be predcted. Smlarly t allows for the determnaton of the nvestment decsons of frms as n Drèze (1974) wthout recourse to unobservable characterstcs. Wth an ncomplete asset market, the dentfcaton of preferences even from ndvdual demand behavor, though possble, s not evdent: the frst order condtons for ndvdual optmzaton do not determne, at least mmedately, the margnal rate of substtuton between consumpton at dfferent states of the world. Wth one commodty, restrctons on the structure of payoffs of assets, as n Dybvg and Polemarchaks (1981) and n Polemarchaks and Rose (1984), or on the utlty functon that represents the preferences, as n Green, Lau and 1

4 Polemarchaks (1979) permt dentfcaton. However, wthout such assumpton local dentfcaton s mpossble when only prces and frst- perod ncomes vary. In contrast to ths, wth multple commodtes, the varaton of relatve prce of commodtes at each state of the world permts dentfcaton when preferences are state - separable, as n Geanakoplos and Polemarchaks (1990). The argument of Geanakoplos and Polemarchaks extends to the more restrctve settngs where only aggregate demand as a functon of ndvdual frst perod ncomes and all prces s observable. Ths mples drectly that the equlbrum correspondence dentfes preferences when they are state - separable and when there are several goods traded at each state. In the lmtng case of one commodty dentfcaton from aggregate asset demand s n general not possble. Nevertheless, snce along the equlbrum correspondence consumpton s not restrcted to le n a subspace of lower dmenson than the commodty space, dentfcaton from the equlbrum correspondence s possble. An example shows that the assumpton of separable utlty s needed dentfy preferences when there s only one good. The argument for recoverablty s local. Gven any profle of endowments wth equlbrum prces, one dentfes the assocated consumpton allocaton as well as preferences over consumpton n a neghborhood of ths allocaton. Ths argument extends mmedately to preferences over the whole consumpton set f addtonal assumptons on preferences assure that the assocated allocatons are attaned at some equlbrum. The dentfcaton of unobservable characterstcs here assumes that the behavor of ndvduals s derved from the maxmzaton of utlty subject to budget constrants; whch dstngushes t from the ssue of the ntegrablty of demand functons As n Chappor et al (1999), dentfcaton mples that there are local restrctons on the equlbrum correspondence snce the argument for dentfcaton does not use all restrctons utlty maxmzaton mposes on ndvdual demand. Concernng further research, n a more applcable argument, one needs to take nto account that for tme seres data, prces and ncomes mght be part of one, ntertemporal equlbrum, and not ponts on an equlbrum correspondence. Accordng to K ubler (1999), the assumpton of tme separable expected utlty restores global restrctons n an ntertemporal model. For dentfcaton, the assumpton of separable utlty s not enough, snce suffcently complete asset markets allow ndvduals to smooth ther expendture across dates and states of the world. Also, optmzng ndvduals take all prces and dvdends nto account when choosng ther portfolo and consumpton plans. However, observatons can only consst of one sample path, and t seems unlkely that dentfcaton of preferences s possble wthout any knowledge of equlbrum prces at nodes that that do not le on the sample path. 2

5 2 The economy Indvduals are I= {1,...,I}, a fnte, non - empty set. States of the world, exhaustve and exclusve descrptons of the envronment, are s S= {1,...,S}, a fnte, non - empty set. Commodtes are l L= {1,...,L}, a fnte, non - empty set. At the state of the world s, commodtes are (l, s) L {s}, and a bundle of commodtes s x s =(...,x l,s,...) ; across states of the world, commodtes are (l, s) L S, and a bundle of commodtes s x =(...,x s,...) =(...,x l,s,...). Assets for the transfer of revenue across states of the world are a A= {1,...,A}, a fnte set, and a portfolo of assets s y =(...,y a,...). The payoff of asset a at the state of the world s s r a,s ; across states of the world, the payoffs of an asset are r a =(...,r a,s,...). The payoffs of assets at the state of the world s are R s =(...,r a,s,...); across states of the world, the matrx of payoffs of assets s R =(...,r a,...)=(...,r s,...). The payoff of a portfolo of assets, y, at state of the world s s r s y; across states of the world, the payoffs of a portfolo of assets are Ry =(...,r s y,...). The column span of the matrx of payoffs of assets, [R], s the subspace of attanable reallocatons of revenue across states of the world. Assumpton 1 The asset structure s such that 1. there are at least two assets, A the matrx of payoffs of assets has full column rank; 3. the payoff of asset a =1s postve: r 1 > at every state of the world, the payoffs of assets do not vansh: R s 0. Redundant assets, whose payoffs are lnear combnaton of the payoffs of other assets can be prced by arbtrage. The asset market s ether complete: A = S, or ncomplete: A<S. A portfolo of assets wth postve payoffs serves to elmnate sataton over portfolos; that ths portfolo consst of only one asset, a = 1, smplfes the exposton. The preferences of an ndvdual are descrbed by the utlty functon, w, wth doman the consumpton set. The preferences of the ndvdual admt a representaton that s addtvely separable across states of the world, {u s : s S}: the consumpton set has a product structure; the cardnal utlty functon, at a state of the world, s u s, and 3

6 the utlty functon s w = s S u s. The preferences may, but need not admt a von Neumann - Morgenstern representaton, (u,µ ) : the consumpton sets at dfferent states of the world and the cardnal utlty functons, u, concde; µ =(...,µ s,...), s a subjectve probablty measure on the set of states of the world, and the utlty functon s 1 w =E µ u. Utlty functons, w 1 and w 2, are cardnally equvalent f w 2 s a monotoncally ncreasng, affne transformaton of w 1. The endowment of an ndvdual n commodtes s e, a bundle of commodtes across states of the world; hs endowment of assets s f, a portfolo of assets. The effectve endowment of the ndvdual n commodtes s 2 ẽ =(...,ẽ s,...), where ẽ s = e s + 1 L 1 R s f s the effectve endowment n commodtes at a state of the world. Assumpton 2 For every ndvdual and for every state of the world, 1. the consumpton set s the set of non - negatve bundles of commodtes; 2. the cardnal utlty functon, u s, s contnuous andconcave; n the nteror of the consumpton set, the utlty functon s dfferentably strctly monotoncally ncreasng: Dv s(x) 0, andstrctly concave: y 0 y D 2 u s(x)y <0; for a sequence of strctly postve consumpton bundles, (x s,n 0:n =1,...), andfor x s, a consumpton bundle on the boundary of the consumpton set, (lm n x s,n = x s ) (lm n (( Du (x s,n ) ) 1 Du (x s,n )x s,n )=0), whle lm n Du (x s,n ) = ; 3. ẽ 0:the effectve endowment n commodtes s a consumpton bundle n the nteror of the consumpton set. The dstncton between endowments n commodtes and endowments n assets smplfes the exposton, but s not essental. The profle of utltes functons s and the allocaton of endowments s u I =(...,{u s : s S},...), (e I,f I )=(...,(e,f ),...). Profles of utlty functons, u I 1 and u I 2, are cardnally equvalent f, for every ndvdual, the utlty functons w 1 and w 2 are cardnally equvalent. The profle of utlty functons, u I, and the matrx of payoffs of assets, R, are fxed, whle the allocaton of endowments vares. 1 E µ denotes the expectaton wth respect to the probablty measure µ. 2 1 K k denotes the k th unt vector of dmenson K. 4

7 The aggregate endowment n commodtes s e a = I e, and the aggregate endowment n commodtes s f a = I f ; the aggregate endowment s (e a,f a ). At a state of the world, prces of commodtes are p s =(...,p l,s,...) 0; Across states of the world, prces of commodtes are p =(...,p s,...). Across states of the world, the expendtures assocated wth bundle of commodtes, x, at prces of commodtes p are p x =(...,p s x s,...). Prces of assets are q =(...,q a,...). Prces of assets do not allow for arbtrage f Ry > 0 qy > 0; ths s the case f and only f q = πr, for some π =(...,π s,...) 0. Prces are a par, (p, q), of prces of commodtes and of prces of assets. The optmzaton problem of an ndvdual s max s.t w (x), p x p e + Ry, qy qf. The soluton of the ndvdual optmzaton problem, (x,y )(p, q, e ), exsts and s unque, and the consumpton plan, x (p, q, e ), les n the nteror of the consumpton set; t defnes (x,y ), the demand functon of the ndvdual for consumpton plans and portfolos of assets. The demand functon s contnuously dfferentable; prce effects are ncome effects are D pt x s =(..., x l,s p k,t,...),d q x s =(..., x l,s q a,...), D pl,s y =(..., y a p l,s,...),d q y =(..., y a q b,...); D e 1,t x s =(..., x l,s,...),d e f 1,t 1 x s =(..., x l,s,...),d f1 e 1,s y =(..., y a,...), e 1,s D f 1 y =(..., y a,...). f1 For cardnally equvalent utlty functons, the demand functons concde. Assocated wth the ndvdual optmzaton problem, at each state of the world, there s a condtonal optmzaton problem max u s(z s ), s.t p s z s p s e s + R s y, 5

8 where y > 0 s a fxed portfolo of assets, such that p s e s + R s y > 0. The soluton of the auxlary optmzaton problem, zs(p s,e,y ), exsts, s unque, and les n the nteror of the consumpton set; t defnes zs, the condtonal demand functon of the ndvdual. The condtonal demand functons are contnuously dfferentable; prce effects are D ps zs =(..., z l,s,...); p k,s ncome effects are D e 1,s zs =(..., z l,s e,...), 1,s and D y zs = D e 1,s zsr s. Assumpton 3 For every ndvdual and for every state of the world, 1. the vectors z s =(...,z l,s,...) and D e 1,s z s =(..., z l,s / e 1,s,...) are lnearly ndependent, and 2. (D q y + D f 1 y (y f ) )R s 0, In the condtonal demand functon, the vector of ncome effects and the vector of demands are not co - lnear. Ths excludes homothetc utlty functons. For ths case dentfcaton s stll possble but the argument has to be modfed - the remark at the end of ths secton clarfes ths pont. In the asset demand functon, the sum of the matrx of prce effects and the matrx formed by the product of the column vector of ncome effects and the transpose of the column vector whch s the portfolo of assets does not vansh; ths sum s the matrx of substtuton effects n the demand for assets. To smplfy notaton t s assumed throughout that all dervatves are evaluated at a prce system whch satsfes p 1,s = 1 for all s S. Lemma 1 (Geanakoplos and Polemarchaks (1990)) The demand functon for consumpton plans andportfolos of assets dentfes the utlty functon of the ndvdual up to cardnal equvalence. Proof The argument s developed n two steps. Step 1 The demand functon for consumpton plans and portfolos of assets, (x,y ), determnes the condtonal demand functon, zs, at every state of the world. By the supportng hyperplane theorem, gven prces of commodtes and a portfolo of assets revenue, (p s,y ), there exst commodty prces, p t, for t S\{s}, states of the world other than s, and prces of assets, q, such that y (p, q) =y. It follows that x s(p, q) =z s (p s,e s,y ). 6

9 The necessary and suffcent condtons for a soluton of the condtonal ndvdual optmzaton problem at a state of the world are Du s λ sp s =0, p s z s p s e s R s y =0. Dfferentatng the frst order condtons and settng K s vs = D2 u s p s 0 v s and Ss = λ sks, yelds, by the mplct functon theorem, that b s p s 1, D ps z s = S s v s(z s e s), D e 1,s z s = v s, D y z s = v sr s, D ps λ s = λ sv s + b s(z s e s), D y λ s = b sr s, and, as a consequence dz s =(S s v s(z s e s) )dp + v s(p s de s + R s dy ), dλ s =( λ sv s + b s(z s e s) ))dp s b sv s(p s de s + R s dy ). The matrx, S s, of substtuton effects, s l,k,s =( x l,s / p k,s) v s, s symmetrc and negatve sem - defnte, t has rank (L 1), and satsfes p s S s =0, and the vector, v s, of ncome effects, v l,s = x l,s / t s, satsfes p s v s =1. The margnal utlty of revenue, λ s = v s/ t s, decreases wth revenue: λ s/ t s = b s < 0. The necessary and suffcent frst order condtons for a soluton to the ndvdual portfolo choce problem are λ R = µ q, qy qf =0, where, across states of the world, λ =(...,λ s,...) are the margnal utltes of revenue obtaned from the condtonal optmzaton wth y = y. 7

10 Dfferentatng the frst order condtons and settng K v = s S b sr s R s q v b q 0 and S = λ K, yelds by the mplct functon theorem that D ps y = S R s (λ sv s b sz s ), 1, D q y = S v (y f ) D f 1 y = v, where, for each state of the world, vs and b s are the ncome effects and the dervatve of the margnal utlty of revenue, respectvely, obtaned from the condtonal optmzaton wth y = y. Step 2 The demand functon for assets, y, and ts dervatves wth respect to revenue, D f 1 y, the prces of assets, D q y, determne the vector of ncome effects v, and the S, matrx of substtuton effects. The condtonal demand functon for commodtes at a state of the world, z, and ts dervatve wth respect to revenue, D e 1,s zs, or, alternatvely, D y zs, determne the vector of ncome effects, vs n the latter case, snce R s 0. The dervatves wth respect to the prces of commodtes of the demand for assets, D ps y, determnes the margnal utlty of revenue, λ s and ts frst dervatve b s; ths s the case, snce, by assumpton, the vectors D e 1,s zs = vs and zs and lnearly, whle S R s =(D q y + D f 1 y )R s 0. By the separatng hyperplane theorem, the demand functon for commodtes s surjectve. Snce, at a soluton of the ndvdual optmzaton problem, the gradent of the utlty functon s co - lnear wth the vector λ p, the demand functon for consumpton plans and portfolos of assets dentfes the utlty functon up to a monotoncally ncreasng transformaton. Snce the utlty functon s addtvely separable across states of the world, of whch there are, at least, two, the demand functon for consumpton plans and portfolos of assets dentfes the utlty functon up to a monotoncally ncreasng, affne transformaton. Remark The argument for the dentfcaton of the utlty functon does not requre varatons n the endowments of the ndvdual n commodtes at each state of the world; varatons n the endowment of assets suffce. Wth a sngle commodty endowments of ndvduals at each states have to vary to allow for dentfcaton. 8

11 Across ndvduals, (x a,y a )(p, q, e I )= I(x,y )(p, q, e ), whch defnes (x a,y a ), the aggregate demand functon for consumpton plans and portfolos of assets. For cardnally equvalent profles of utlty functons, the aggregate demand functons concde. At each state of the world, for y I =(...,y,...), a fxed allocaton of portfolos of assets, such that p s e s + R s y > 0, for every ndvdual, zs a (p s,e I s,y I )= zs(p s,e s,y ), I whch defnes z a s, the aggregate, condtonal demand functon. Assumpton 4 For every ndvdual, 1. the ncome effect for every asset, y a/ f 1, s a twce dfferentable functon of revenue, f 1;. Furthermore 2 ya (f1 0, )2 2. there exst assets, d and e, other than the numerare, such that f1 (ln 2 yd ( f1 ) )2 for every state of the world, f 1 (ln 2 ye ( f1 ); )2 3. the ncome effect n the condtonal demand for every commodty, z l,s / e 1,s, s a twce dfferentable functon of revenue, e 1,s; and 2 zm,s (e 0, 1,s )2 4. there exst commodtes, m and n, other than the numerare, such that e 1 (ln 2 z m,s ( e 1,s )2 ) e 1,s (ln 2 zn,s ( e ). 1,s )2 Ths s the analogue of the condton of non - vanshng ncome effects that was employed n the argument under certanty n Chappor et al (1999). However, whle ths assumpton s naturally fulflled as long as preferences are not homothetc, t s not straghtforward to translate the assumpton of non-vanshng ncome effects n the demand for securtes to an assumpton on utlty functons. 9

12 Lemma 2 The aggregate demand functon for consumpton plans and portfolos of assets dentfes the profle of utltes up to cardnal equvalence. Proof It suffces that the aggregate demand functon dentfy, for every ndvdual, the demand functons for portfolos of assets, y, and zs, the condtonal demand functon for commodtes, at every state of the world. The argument s developed n two steps. Step 1 The aggregate demand functon for consumpton plans and portfolos of assets, (x a,y a ), determnes the aggregate, condtonal demand functon, zs a, at every state of the world. For the aggregate, condtonal demand functon, and, as a consequence D ps z a s = s S (S s v s(z s e s) ), D e 1,s z a s = v s, D y z a s = v sr s, dzs a = (Ss vs(z s e s) )dp + vs(p s de s + R s dy ). s S s S Snce D e 1,s zs a = vs, or, alternatvely, D y zs = vsr s, the aggregate, condtonal demand functon dentfes, vs, the ncome effects of every ndvdual n the latter case, snce R s 0. The functons f j,k,s = za j,s za k,s (v p k,s p j,se k,s vk,se j,s), j,k L\{1},j k, j,s I for pars of dstnct commodtes other than the numerare, are dentfed by the aggregate demand functon. By drect substtuton and the symmetry of the matrces of substtuton effects, f j,k,s = k,se I(v j,s vj,se k,s). As n the proof of lemma 2n Chappor et al (1999), the frst and second dervatves of the functons f j,k,s wth respect to revenue, e 1,s, or, alternatvely, y, dentfy zs, the condtonal demand functon of the ndvdual. Step 2 For the aggregate demand functon for portfolos of assets, D q y a = I (S v (y f ) ), D f 1 y a = v, 10

13 and, as a consequence dy a = (S v (y f ) )dp + v df1. I I Snce D f 1 y a = v, the aggregate demand functon for portfolos of assets dentfes v, the ncome effects of every ndvdual. The functons f b,c = ya b q c ya c q b (vbf c vcf b), b,c A\{1},b c, I for pars of dstnct assets other than the numerare, s dentfed by the aggregate demand functon. By drect substtuton and the symmetry of the matrces of substtuton effects, f b,c = I(v cf b v bf c). As n the proof of lemma 2, the frst and second dervatves of the functons f b,c wth respect to revenue, f1, dentfy y, the demand functon of the ndvdual for portfolos of assets. Compettve equlbrum prces are such that (x a (p, e I ),y a (p, e I )) (e a,f a )=0. The compettve equlbrum correspondence assocates compettve equlbrum prces of commodtes and assets to profles of endowments, ω(e I,f I )= {(p, q) :(x a (p, e I,y a (p, e I, ) (e a,f a )=0, and p 1,s =1,s S,q 1 =1}. For cardnally equvalent profles of utlty functons, the compettve equlbrum correspondences concde. Proposton 1 The compettve equlbrum correspondence on an open set of endowments dentfes the assocated subset of the consumpton sets of ndvdual andthe profle of utlty functons, up to ordnal equvalence, on ths set. Proof It suffces that the compettve equlbrum correspondence dentfy the profle of demand functons. The argument s developed n two steps. Step 1 For an allocaton of portfolos of assets, y I =(...,y,...), the aggregate portfolo of assets s y a = I y. 11

14 The graph of the compettve equlbrum correspondence determnes the graph of the condtonal compettve equlbrum correspondence at every state of the world, whch assgns compettve equlbrum prces of commodtes to allocatons of endowments and portfolos of assets, ω s (e I s,y )={p s : z a s (p s,e I s,y I, ) e a s 1 L 1 R s y a =0, and p 1,s =1}. The graph of the condtonal compettve equlbrum correspondence has the structure of a contnuously dfferentable manfold. The tangent space to the condtonal compettve equlbrum manfold s defned by dzs a = de a s + 1 L 1 R s dy a, and, as a consequence, by s I(S vs(z s e s) )dp s + sp I(v I)de s + (vs 1 L 1 )R s dy =0. I The condtonal compettve equlbrum correspondence determnes the compettve equlbrum manfold and, consequently, everywhere, ts tangent space. As n the proof of proposton 1, at every state of the world, the graph of the condtonal compettve equlbrum correspondence dentfes the condtonal demand functon of every ndvdual. Step 2 By substtuton, the tangent space to the compettve equlbrum s S( manfold satsfes SsR s (λ svs b sz ))dp s + I I(S v (y f ) )dq + (v q I)df =0 I As n the proof of proposton 1,the graph of the compettve equlbrum correspondence dentfes the demand functon for assets of every ndvdual and therefore, wth step 1, the entre demand functon. Remark If for every ndvdual, for every commodty and for every state of the world, for any sequence ((p s,n,e s,n,ys):n =1,...), of prces of commodtes, endowments of commodtes and portfolos of assets, lm n e 1,s,n + R s ys = lm n z l,s,n(p s,n,e s,n,ys)=, l L,.e. for every ndvdual, every commodty s normal n a strong sense then the compettve equlbrum correspondence dentfes the utlty functons of ndvduals on ther entre doman of defnton. Remark It s an open queston whether dentfcaton under uncertanty and an ncomplete asset market extends to non-separable preferences when there are several commodtes. It s shown below that ths s mpossble f there s only one commodty. 12

15 2.1 The specal case of a fnance economy The dentfcaton result requres that there are at least 3 commodtes at each state of the world. As s the case under certanty the case of 2commodtes remans an open queston. The case of 1 commodty, vacuous under certanty, s ndeed of nterest under uncertanty. Though recoverablty from ndvdual demand requres addtonal assumptons as ponted out n the earler lterature, recoverablty from the compettve equlbrum correspondence s not problematc. Ths s due to the freedom afforded by the equlbrum correspondence, namely the varaton n the endowments of ndvduals across states of the world. Step 2of the proof of proposton 2mples that as long as parta (1) and (2) of assumpton 6 are satsfed ndvdual asset demand can be dentfed from the equlbrum manfold even f L =1. In ths framework ndvdual asset demand as a functon of ndvdual endowments at all states and of prces does dentfy the utlty: If L = 1,the ndvdual asset demand functon s a soluton to max w (Ry), s.t qy qf. Dfferentatng the frst order condtons and settng K v R sd 2 w R s q = v b q 0 1, one obtans that 2 u s can be recovered by varaton of e s after the dentfcaton of K and v from varaton n prces and endowments n portfolos, snce D 2 w s a dagonal matrx. When preferences are not separable dentfcaton s no longer possble. The followng example clarfes ths. Example There s a sngle ndvdual. Observng the equlbrum correspondence t equvalent of observng the supportng asset prces at all possble consumpton vectors. The matrces H 1 and H 2 are dstnct, negatve defnte symmetrc S S matrces, wth R H 1 = R H 2 ; t s clear that for A much smaller than S these matrces exst. If h =(...,h s,...) 0 s large enough, the utlty functons w 1 (x) =h x x H 1 x and w 2 (x) =h x x H 2 x 13

16 generate dentcal asset prces for all possble allocatons; the equlbrum correspondence under w 1 s ndstngushable from the equlbrum correspondence under w 2 even though these utlty functons represent dfferent preferences. Remark The example shows that t s possble that even though equlbrum allocatons are neffcent a planner who can observe the equlbrum correspondence but not ndvdual preferences mght not be able to ntroduce Paretomprovng assets - a frm mght not be able to choose a constraned effcent producton plan. 14

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