Global identification from the equilibrium manifold under incomplete markets. (Draft for comments)

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1 Global dentfcaton from the equlbrum manfold under ncomplete markets. (Draft for comments) Andrés Carvajal BancodelaRepúblca. Alvaro Rascos BancodelaRepúblca. June 17, 24 Abstract We show that even under ncomplete markets, the equlbrum manfold dentfes ndvdual demands everywhere n ther domans. For ths, we assume condtons of smoothness, nterorty and regularty, but avod mplausble observatonal requrements. It s crucal that there be datezero consumpton. As a by-product, we develop some dualty theory under ncomplete markets. 1 Introducton The transfer paradox, frst ponted out by Leontef (1936), and generalzed by Donsmon and olemarchaks (1994), llustrates the mportance of dentfyng the fundamentals of an economy from observable data. Under the hypothess of general equlbrum, the aggregate demand functon cannot be assumed to be observed: at equlbrum prces aggregate demand s, by defnton, equal to aggregate endowment. Demand, ether ndvdual or aggregate, cannot be observed for out-of-equlbrum prces. One can observe, however, equlbrum prces and ndvdual ncomes. In ths paper we address the problem of dentfyng ndvdual preferences from the equlbrum manfold when asset markets are ncomplete. For the case of complete markets, postve results have been obtaned by Balasko [1999], Chappor et al [2] and Matzkn [23]. Balasko s result has been crtczed for makng very strong observatonal assumptons: that one can observe equlbrum prces n stuatons n whch endowment s zero for all ndvduals but one. Under addtonal assumptons, Chappor et al obtan local dentfcaton of ndvdual demands usng a constructve argument. Matzkn determnes the largest class of fundamentals for whch dentfcaton s possble. Her argument, however, s not constructve. 1

2 The case of ncomplete markets s more cumbersome. Kubler et al [2] and [22] use the mplct functon theorem to dentfy the aggregate demand functon from the equlbrum manfold (hence they obtan a local dentfcaton of the aggregate demand functon). They proceed to dentfy ndvdual demands (locally) from the aggregate demand and fnally, they use Geanakoplos and olemarchaks [199] to dentfy preferences from ndvdual demand functons. Therefore, they are able to obtan local dentfcaton of ndvdual preferences when asset markets are ncomplete. When we have real numerare assets, we dentfy ndvdual demands globally. For general real assets structures, we conjecture that our results hold (genercally on endowments and real asset structures). We extent Balasko s dea on how to recover the aggregate demand functon from the equlbrum manfold to the case of ncomplete asset markets, hence we avod usng the mplct functon theorem. We then use a slghtly dfferent argument than Kubler et al. to dentfy ndvdual demands from the aggregate demand functon and we also avod usng Balasko s strong observatonal assumpton ponted out before. As a by product, we develop some basc dualty theory for ncomplete markets. 2 The Incomplete Markets Model We consder the canoncal, two perod, multgood, ncomplete markets model wth fnancal assets. There are S+1 states of nature, s =,..., S, 1 I ndvduals, =1,..., I, andl > 2 commodtes avalable n each state, l =1,..., L. We denote L(S +1)by n and defne the commodty space as R n +. A fnancal asset s a contract v R S that promses to delver at each state of nature s =1,..., S an amount v s R of the numerare. Let good 1 be the numerare of our economy and let p R n ++ denote the vector of spot prces where p s =(p s,1,.., p s,l ) R++ L and p s,l denotes the (current value of) prce payable n state s for one unt of good l. Snce assets are real, wthout loss of generalty, we normalze prce so that p,1 =1. Defne, S++ n 1 = p R n ++ : p,1 =1 ª. We wrte any p S++ n 1 as p =(p,p 1 ), where p 1 =(p 1,..., p S ). If v 1,..., v J are J 1 fnancal assets, we defne V (p 1 ) as the matrx of ncome transfers: p 1,1 V (p 1 )=... V p S,1 v1 1 v J 1 where V =..... vs 1 vs J the column span of V (p 1 ): hv (p 1 ) = t R S : z R J : t = V (p 1 ) z ª 1 s =s used to denote date zero., and the space of ncome transfers hv (p 1 ), as 2

3 Remark 1 In general, as p 1 changes, hv (p 1 ) changes. If p 1 >>, then the dmenson of hv (p 1 ) remans unchanged. Let q R J be the prce vector at whch each one of these assets can be bought at s =. For (p, q) R++ n R J and w R n +,let B(p, q, w; V )= x R n + : z R J,p (x w ) qz and p 1 (x 1 w 1 )=V (p 1 ) z ª where for every (ρ 1, 1 )=((ρ 1,..., ρ S ), ( 1,..., S )) R LS R LS : ρ 1 1 ρ 1 1 =.. ρ S S Indvdual I = {1,...I} has preferences over consumpton that are represented by utlty functons u : R n + R and endowment denoted by w R n +. Assume the followng: Condton 1 For each ndvdual, u s contnuous, monotone and strongly quas-concave. For each ndvdual, defne the ndvdual demand functon (n fnancal markets) f : S n 1 ++ RJ R n + R n +, as: f (p, q, w) = arg max u (x) :x B(p, q, w; V ) ª Defne also the aggregate demand functon, F : S n 1 ++ RJ R ni + R n +,as: F(p, q, w) = IX f (p, q, w ) Functons f and F are well defned snce for (p, q, w) S++ n 1 R J R n +, B(p, q, w; V ) s nonempty and compact and each u s contnuous and strongly quas-concave. A fnancal markets economy s: E = E {u } I, {w } I,V Defnton 1 A fnancal markets equlbrum for the economy E s (x, z, p, q) R ni + R J S++ n 1 RJ such that: 1. For every, x = f (p, q, w ), p (x w)= qz and p 1 x 1 w1 = V (p 1 )z 2. F (p, q, w) = I w and I z = Remark 2 If V s of full column rank, then prevous defnton. I z = s redundant n the 3

4 Condton 2 Assume V s of full column rank. Defnton 2 The fnancal markets equlbrum manfold M FM s: ( ) IX M FM = (p, q, w) S++ n 1 R J R ni + : F(p, q, w) = w Remark 3 What we observe n the real world s M FM Let R n ++ denote date-zero present value prces (see Magll and Shafer, p. 1534), where =(,..., S ) and for every s, s =( s,1..., s,l ). Let S++ n 1 = R n ++ :,1 =1 ª. Normalzng prces to le n S++ n 1 establshes date-zero frst commodty as the numerare. For S++ n 1 and w R n +,let ( ) SX B(, w; V )= x R n + : z R J, s (x s w s ) and 1 (x 1 w 1 )=V( 1 )z We say that future consumpton x 1 R LS + s fnancally feasble, at prces and endowments ( 1,w 1 ) R LS ++ R LS +, f the second condton n the defnton of B(, w; V ) s satsfed: there s a portfolo of assets, z R J, that delvers the transfers necessary to fnance x 1. Remark 4 If dm hv ( 1 ) = S or equvalently dm hv = S, the second condton that defnes B(, w; V ) s nonbndng. Ths s the case of complete markets. For each ndvdual, defne the ndvdual demand functon f : S n 1 ++ R n + R n +, as: f (, w) = arg max u (x) :x B(, w; V ) ª Defne also the aggregate demand functon, F : S n 1 ++ R ni + R n +,as: F (, w) = IX f (, w ) Remark 5 f and F are well defned snce for (, w) S++ n 1 Rn +, B(, w; V ) s nonempty and compact, and each u s contnuous and strongly quas-concave. Defnton 3 The no-arbtrage equlbrum for the economy E s a par (x, ) R ni + S++ n 1 such that: 1. For every, x = f (, w ) 2. F (, w) = I w 4

5 Remark 6 Snce V has full column rank, n the prevous defnton we do not explctly consder portfolos. Remark 7 It s well known that f (x, ) s a no-arbtrage equlbrum, then there exst portfolos z 1,...,z I,z R J and an asset prce vector q R J such that ((x, z), (p, q)) s a fnancal market equlbrum wth the same asset structure V and vceversa. In fact, f q R J s a no-arbtrage asset prce vector (see Magll and Shafer (1991)) then there exsts a π R++ S+1 such that π =1 and q = π 1 V (p 1 ). It s easy to prove that B(p, q, w; V )=B(, w; V ), where = π p. Defnton 4 The no-arbtrage equlbrum manfold (for short, Equlbrum Manfold) M s: ( ) IX M = (, w) S++ n 1 R ni + : F (, w) = w Henceforth we assume that there s a socety consstng of agents I, wth preferences defned by {u } I that satsfy our assumptons, and a fnancal market V, as descrbed before, and we study whether from the fnancal markets equlbrum manfold M FM ther unobserved fundamentals (.e preferences) can be unquely determned. We do not test the exstence of such socety (under the equlbrum hypothess). The queston posed s one of unqueness of fundamentals (.e preferences) rather than the exstence of some fundamentals (.e preferences). In fact, t s well known that wth stronger assumptons on preferences than the ones we have mposed, and genercally on real assets structures, M FM 6= φ. Therefore, our assumpton s non vacuous. In the followng secton we show that the equlbrum manfold unquely determnes aggregate demand. Then, we show that aggregate demand unquely determnes ndvdual demands. In real lfe we do not observe equlbrum date zero present value prces but rather, we observe (fnancal) equlbrum spot prces for commodtes and assets. We now show how to defne M from M FM consstent wth our prevous defnton of M. roposton 1 Let E = {u } I, {w } I,V be an economy and M FM the assocated fnancal markets equlbrum manfold and M the assocated equlbrum manfold. Defne the set M = {(, {w } I )} S n 1 ++ R ni + :(p, q, {w } I ) M FM,p= and q = Then M = M. roof. Ths follows from remark 7 by settng π =[1,..., 1]. SX V s }. roposton 2 Let E = {u } I, {w } I,V and e E = {eu } I, {w } I,V be two fnancal market economes (wth possble dfferent agents characterstcs but equal endowements and fnancal structure) and let M FM and f M FM be s=1 5

6 the assocated fnancal markets equlbrum manfolds, respectvely. If M FM = fm FM then M = f M (where M and f M are the assocated no-arbtrage equlbrum manfolds of the two economes respectvely). roof. If M FM = f M FM then M = f M, and by proposton 1, M = M and fm = f M therefore M = f M. 3 From the Equlbrum Manfold to the Aggregate Demand Functon Let M be the equlbrum manfold. The next theorem shows that one can unquely recover the aggregate demand functon. Theorem 1 For each (, w) S++ n 1 R ni +,thereexsts bw,...i RnI + such that ³ 1., bw M,...I 2. For all, bw 1 s fnancally feasble at 1,w 1 and bw = w Moreover, where bw,...i condtons. F (, w) = IX bw s any one of the elements of RnI + that satsfy the prevous two roof. There s at least one bw,...i RnI + satsfyng the two condtons: ³ defne bw = f (, w ). Then, bw s a no-arbtrage equlbrum of,...i the economy E(u, bw,...i ; V ) snce, for all, bw 1 s fnancally feasble at 1,w1 and, ³ by strct monotoncty, bw = w. Now, f, bw s any element of M, such that bw,...i 1 s fnancally feasble at 1,w1 and bw = w for all, then, by the defnton of equlbrum, IX IX bw = f (, bw ) and snce B, bw ; V = B, w ; V then, f (, bw )=f (, w ), whch mples that I bw = I f (, w ). 6

7 Remark 8 Ths s Balasko [1999] n ncomplete markets. As n the complete markets case, one makes no use of any topologcal or dfferental property of the manfold M (strctly speakng, set M). Notce that we use that ndvdual demands by defnton unque solutons to an optmzaton problem. One can easly get away of ths by assumng that ndvdual demands are functons unquely defned on prces and ncome, satsfy Walras law and an addtonal condton related to fnancal feasblty at tme t =1(see the workng paper verson of the present artcle). In the case of complete markets ths last restrcton s nonbndng and the frst two are exactly what Balasko assume on ndvdual demands. 4 From the Aggregate Demand to Indvdual Demand If one s wllng to assume that equlbrum prces are observable for stuatons n whch the ncomes of all ndvduals but one are zero, then t s straghtforward that aggregate demand dentfes ndvdual demands: for all, f (, w )=F (, (,,..., w,..., )). That s, when all agents dfferent from, have no ncome, the fact that prces are strctly postve mples no demand for agents dfferent from, and, therefore, that aggregate demand s agent s ndvdual demand. Remark 9 By the defnton of aggregate demand and ndvdual demand, for each set of prces and endowments, w S++ n 1 R n + there s at least one portfolo of assets z such that f (, w ) s fnancally feasble at, w.when the asset structure V has nonredundant assets only, the portfolo of assets s unque (dentfed). We now show that under some addtonal assumptons one can dentfy an ndvdual s demand wthout peggng everybody else s ncome at zero. Condton 3 For each ndvdual, n the nteror of the commodty space R n ++, u s dfferentably strctly monotone and dfferentably strongly quasconcave, and for all x R n ++, x R n + : u (x ) u (x) ª R n ++. Lemma 1 For every (, w) S++ n 1 R n ++, f (, w) R n ++. roof. It suffces to notce that w B (, w; V ) and that x R n + : u (x) u (w) ª R n ++ Lemma 2 f s contnuously dfferentable. roof. Ths follows from Duffe and Shafer (1985, p. 293). As an auxlary result, we frst show that aggregate demand dentfes ndvdual demands up to a functon of prces only. 7

8 Theorem 2 For some ϕ : S++ n 1 R n + R n +, whch s dentfed, and φ : S++ n 1 R n +, f (, w) =ϕ (, w)+φ ( ) for every (, w) S++ n 1 R n +. roof. Let ϕ (p, w) =F (, (1, 1,..., w,..., 1)), wherew occupes the th poston on ts vector. Let φ ( )= f j (, 1). Functon ϕ s IX dentfed. j=1,j6= As n Chappor et al (22) and Kubler et al (22), we mpose the followng: Condton 4 (Regularty) For every ndvdual and every S n 1 ++,there exst w R n +,and(s, l), (s,l ) ({,..., S} {1,..., L}) \{(, 1)}, suchthat: 2 f s,l (w,1) 2 (, w) 3 f s,l (w,1) 3 (, w) 2 f s,l (, w) (w,1) 2 3 f 6= s,l (, w) (w,1) 3 Under regularty, global dentfcaton of ndvdual demands s possble: Theorem 3 Aggregate demand dentfes ndvdual demands. roof. It suffces to prove that the functon φ of theorem 2 s also dentfed. From propostons 6 and 8 n the appendx, gnorng the arguments, t follows that for every w R n + and (s, l), (s,l ) ({,..., S} {1,..., L}) \{(, 1)}: f s,l s,l + fs,l f w s,l s,l w,1 = f s,l + fs,l w fs,l s,l s,l w,1 Substtutng, ϕ s,l s,l + φ s,l + ϕ s,l + φ s,l ϕ w s,l s,l s,l w,1 = ϕ s,l φs,l + + ϕ s,l + φ s,l w ϕ s,l s,l s,l s,l w,1 Takng that (s, l) 6= (, 1) and (s,l ) 6= (, 1) and dervng once and twce wth respect to ncome gves us 2 ϕ s,l w,1 + ϕ s,l + φ s,l 2 ϕ w s,l s,l s,l w,1 2 = 2 ϕ s,l w,1 + ϕ s,l + φ s,l w 2 ϕ s,l s,l s,l w,1 2 8

9 and 3 ϕ s,l w,1 2 + ϕ s,l 2 ϕ s,l s w,l,1 w,1 2 + ϕ s,l + φ s,l 3 ϕ w s,l s,l w,1 3 = 3 ϕ s,l w,1 2 + ϕ s,l 2 ϕ s,l s,l w,1 w,1 2 + ϕ s,l + φ s,l ws,l 3 ϕ s,l w,1 3 where We can rewrte ths system as φ s,l φ = Γ s,l = 2 ϕ s,l (w,1) 2 (, w) 3 ϕ s,l (w,1) 3 (, w) and Γ s a 2 1 matrx wth frst component (, w) (w,1) 2 (, w) (w,1) 3 2 ϕ s,l 3 ϕ s,l 2 ϕ s,l 2 ϕ s,l w,1 s,l w,1 s,l + ϕ s,l ws,l 2 ϕ s,l w,1 2 ϕ s,l 2 ϕ w s,l s,l w,1 2 and second component 3 ϕ s,l 3 ϕ s,l w,1 2 s,l w,1 2 + ϕ s,l 2 ϕ s,l s w,l,1 w,1 ϕ s,l w,1 2 ϕ s,l w,1 2 + ϕ s,l ws,l 3 ϕ s,l w,1 2 3 ϕ s,l w s,l 3 ϕ s,l w,1 Both and Γ are dentfed, as they depend only on ϕ. Moreover, by Regularty, for some w R n +, s, s {1,..., S} and l, l {1,..., L}, matrx s nvertble, whch dentfes φ s,l and φ s,l. For every other φ l,s s (l,s ) ({1,...,L} {,...,S}) \{(, 1)} ³ 2 ϕ l,s w,1 2 ϕ s,l ³ϕ s,l w,1 + l,s s,l + φ s,l ws,l 2 ϕ l,s ϕ (w,1) 2 l,s w l,s 2 ϕ s,l (w,1) ϕ s,l (w,1) 2 where we have assumed, wthout loss of generalty, that φ,1 can be dentfed by Walras law. 2 ϕ s,l (w,1) 2 6=. Fnally, 9

10 Remark 1 Snce V s of full rank by condton 2, the dentfcaton of ndvdual assets demand s straghtforward. So far we have shown that ndvdual fnancal markets demand functons are dentfed but moreover, we can defne them n terms of the the ndvdual demands that we have just dentfed from the no-arbtrage equlbrum manfold. Let (p, q, w) S++ n 1 R J R ni +, where q s a no-arbtrage prce, then f (p, q, w) =f (, w) where = π p and π R S+1 ++ s any vector such that q = π 1 V (p 1 ). Remark 11 Snce V s of full rank by condton 2 and we have dentfed ndvdual fnancal markets demand the dentfcaton of ndvdual assets demand (n fnancal markets) follows. 5 Appendx: dualty n ncomplete markets Fx an ndvdual. Defne U R as the mage of R n ++ under u : U = µ R : x R n ª ++ : u (x) =µ For each (w 1,µ) R LS ++ U, letd (w 1,µ) S++ n 1 be defned as follows: D (w 1,µ)= S++ n 1 : x R n ++ : u (x) =µ and 1 (x 1 w 1 ) hv ( 1 ) ª roposton 3 For each (w 1,µ) R LS ++ U, D (w 1,µ) s dffeomorphc to ((,2,...,,L ), 1 ) R n 1 ++ : x R n ++ : u (x) =µ and 1 (x 1 w 1 ) hv ( 1 ) ª whch s open. roof. Let D denote the latter set. That D (w 1,µ) and D are dffeomorphc s straghtforward. We now show that D s open. Let D. Bydefnton, for some x R n ++, u (x) =µ and 1 (x 1 w 1 ) hv ( 1 ), whereas usng the mplct functon theorem, for some ε>, B ε (x 1 ) R LS ++ and ( ex 1 B ε (x 1 )) ex R L ++ : u (ex, ex 1 )=u (x) Gven that (s, l) {1,..., S} {1,..., L}, there exsts δ s,l > such that δ (w s,l x s,l ) lm δ> s,l = s,1 + δ δ < δ s,l = δ w s,l x s,l s,l + δ s,1 < ε LS 1

11 Defne δ = ª mn δs,l (s,l) {1,...,S} {1,...,L} ³ (,2,...,,L ), and consder the functon φ : R n 1 ++ R n 1 ++, φ( )= 1 1,1,... S The functon h s contnous, therefore there s a δ > such that for all B δ ( ), kh( ) h( )k < δ, npartcular s,l s,l s,1 s,1 < δ. Defne x 1 R LS as follows: (s, l) {1,...,S} {1,..., L}, x s,l = ³ s,l s,1 x s,l + s,l s,1 s,l s,1 s,l s,1 w s,l S,1. Then, x s,l x s,l = and, snce B δ ( ), t follows that s,l s,l s,1 s,l s,1 s,1 ws,l x s,l s,l s,1 ε x s,l x s,l < LS s,l s,1 < δ 6 δs,l, from where and, hence kx 1 x 1k <ε. Thsmplesthatx 1 B ε (x 1 ) and, therefore, that there exsts x R L ++ such that u (x,x 1)=u (x). Fnally, by constructon, ( 1,... 1,1 S ) (x S,1 1 w 1 )=( 1 1,1,... S S,1 ) (x 1 w 1 ) hv, and, hence, D. For each (w 1,µ) R LS ++ U such that D (w 1,µ) 6=, defne the Hcksan demand functon h ( ; w 1,µ):D(w 1,µ) R n ++, as: ( S ) X h ( ; w 1,µ)=argmn s x s : u (x) µ and 1 (x 1 w 1 ) hv ( 1 ) and the expendture functon e ( ; w 1,µ):D (w 1,µ) R as: e ( ; w 1,µ)= h ( ; w 1,µ) Remark 12 By the frst part of condton 3, any soluton les n R n ++ and s unque. Now, defne for each w 1 R LS ++, M (w 1 ) S++ n 1 R ++ as follows: ( M (w 1 )= (, m) S++ n 1 R ++ : x R n S ) X ++ : s x s 6 m and 1 (x 1 w 1 ) hv ( 1 ) 11

12 roposton 4 For each w 1 R LS ++, M (w 1 ) s dffeomorphc to ( (((,2,...,,L ), 1 ),m) R n 1 ++ R ++ : x R n S ) X ++ : s x s 6 m and 1 (x 1 w 1 ) hv ( 1 ) whch s nonempty and open. roof. Ths s straghtforward. For each w 1 R LS ++, defne the condtonal ndvdual demand functon ef (, ; w 1 ):M(w 1 ) R n ++ as ( ) SX ef(, m; w 1 ) = arg max u (x) : s x s m and 1 (x 1 w 1 ) hv ( 1 ) Remark 13 By the frst part of condton 3, any soluton les n R n ++ and s unque. Obvously, SX ef(, s w s ; w 1 )=f (, w) Followng s the standard dualty result, extended to the case of ncomplete markets. It contans three parts: 1. Gven endowments w, Ifx solves the utlty maxmzaton problem at prces and S++ n 1,thenx solves the expendture mnmzaton problem at prces and mnmum utlty u (x ). 2. Gven endowments w 1 and utlty µ, fx solves the expendture mnmzaton problem at prces D (w 1,µ), thenx solves the utlty maxmzaton problem at prces and endowments x. 3. Gven endowments w 1 and utlty µ, fx solves the expendture mnmzaton problem at prces D (w 1,µ), thenx solves the condtonal utlty maxmzaton problem at prces and ncome e (, w, µ).that s roposton 5 and 1. For every w =(w,w 1 ) R n ++ and every S n 1 ++, u (f (, w)) U D w 1,u (f (, w)) h ; w 1,u (f (, w)) = f (, w) 2. Gven (w 1,µ) R LS ++ U, for every D (w 1,µ), f, h ( ; w 1,µ) = h ( ; w 1,µ) 12

13 3. Gven (w 1,µ) R LS ++ U, for every D (w 1,µ),, e (, w, µ) M (w 1 ) and ef, e (, w, µ);w 1 = h ( ; w 1,µ) roof. art (1) s straghtforward gven lemma 1 and condton 3: argue by contradcton and use strct monotoncty of the utlty functon. Gven that u s contnuous, for parts (2) and (3) t suffces to prove that u (h ( ; w 1,µ)) = µ. For ths, suppose not: u (h ( ; w 1,µ)) >µ. Defne x = h ( ; w 1,µ) (ε,,..., ), whereε R ++.Byconstructon,x 1 = h 1( ; w 1,µ), from where 1 (x 1 w 1 ) hv ( 1 ),and S s x s <e( ; w 1,µ), whereas snce h ( ; w 1,µ) R n ++,forε small enough x Rn + and, by contnuty, u (x) > µ, whch s a contradcton. roposton 6 (Shepard s Lemma) For every (w 1,µ) R LS ++ U, thefuncton e ( ; w 1,µ):D(w 1,µ) R ++ s dfferentable and (e ( ; w 1,µ)) = h ( ; w 1,µ) roof. Ths s an mmedate consequence of the Dualty Theorem (see Mas- Colell et al, roposton 3.F.1): let K = {x R n + : u (x) µ and 1 (x 1 w 1 ) hv ( 1 ) Then, K s closed and e ( ; w 1,µ) s the support functon of K. roposton 7 For every w 1 R LS ++, thefuncton e f (, ; w 1 ):M (w 1 ) R n ++ s dfferentable. roof. (1985). Ths can be argued n the same way as fact 5 n Duffe and Shafer roposton 8 (Slutsky Equaton n ncomplete markets).let(, w) S++ n 1 Rn + and µ = u (f (, w)). Then,h ( ; w 1,µ):D(w 1,µ) R n ++ s dfferentable and for all (s, l), (s,l ) ({,..., S} {1,..., L}) \{(, 1)}, wehave: h s,l ( ; w 1,µ) = f s,l (, w) + f s,l (, w) (fs s,l s,l w,l (, w) w s,l ),1 roof. That h ( ; w 1,µ) s dfferentable follows from propostons 5 and 7. Also from proposton 5, we have that h( ; w 1,µ)= e f (, e ( ; w 1,µ);w 1 ). Therefore, h s,l ( ; w 1,µ) s,l = e f s,l (, e ( ; w 1,µ);w 1 ) + f e s,l (, e ( ; w 1,µ);w 1 ) s,l m 13 e( ; w 1,µ) s,l

14 By proposton 6, we have: h s,l ( ; w 1,µ) = f e s,l (, e ( ; w 1,µ);w 1 ) + f e s,l (, e ( ; w 1,µ);w 1 ) h s s,l s,l m,l ( ; w 1,µ) Now, snce f (, w) = f e SX Ã, s w s ; w 1!,then fs,l (, w) = s,l e f s,l Ã,! SX s w s ; w 1 s,l + e f s,l Ã,! SX s w s ; w 1 Under monotoncty (condton 3), at µ = u f (, w), e ( ; w 1,µ)= w s and, hence, m fs,l (, w) = f e s,l (, e (, w, µ);w 1 ) + f e s,l (, e (, w, µ);w 1 ) s,l s,l m Solvng for e f s,l (, e (, w, µ);w 1) w s,l SX s w s,l and replacng gves us s,l h s,l ( ; w 1,µ) = f s,l (, w) + f e s,l (, e (, w, µ);w 1 ) (h s s,l s,l m,l ( ; w 1,µ) w s,l ) By proposton 5, snce µ = u (f (, w)),! e SX f s,l Ã, s w s ; w 1 h s,l ( ; w 1,µ) = f s,l (, w) + (fs s,l s,l m,l (, w) w s,l ) Fnally, notce that! e SX f fs,l s,l Ã, s w s ; w 1 (, w) = w,1 m Substtuton gves us the desred result. References [1] Balasko, Y Dervng ndvdual demand functons from the equlbrum manfold. Manuscrpt. 14

15 [2] Chappor,.A., I. Ekeland, F. Kubler and H. olemarchaks, 22. Testable Implcatons of General Equlbrum Theory: a dfferentable approach. W 2-1. Brown Unversty. [3] Chappor,.A., I. Ekeland, F. Kubler and H. olemarchaks, 2. The dentfcaton of references from Equlbrum rces. CORE Dscusson aper 2/24. [4] Donsmon, M.-. and H. olemarchaks, Redstrbuton and welfare. Journal of Mathematcal Economcs, 23, [5] Duffe, D. and W. Shafer, Equlbrum n Incomplete Markets: I. A Basc Model of Generc Exstence. Journal of Mathematcal Economcs, 14, [6] Geanakoplos, J and H. olemarchaks Observablty and Optmalty. Journal of Mathematcal Economcs, 19, [7] Kubler, F.,.A. Chappor, I. Ekeland and H. olemarchaks, 2. The dentfcaton of references from Equlbrum rces under Uncertanty. CORE Dscusson aper 2/25 [8] Kubler, F.,.A. Chappor, I. Ekeland and H. olemarchaks, 22. The dentfcaton of references from Equlbrum rces under Uncertanty. Journal of Economc Theory, 12, [9] Magll, M. and W. Shafer, Incompelte Markets. In W. Hldenbrand and H. Sonnenschen (Eds.) Handbook of Mathematcal Economcs, Vol. 4, [1] Mas-Colell, A., M. Whnston and J. Green, Mcroeconomc Theory. 15