The identification of preferences from market data under uncertainty

DOCUMENTO CEDE 2005-48

ISSN 1657-7191 (Edición Electrónica)

AGOSTO DE 2005

CEDE

THE IDENTIFICATION OF PREFERENCES FROM MARKET DATA

UNDER UNCERTAINTY

ANDRES CARVAJAL

Royal Holloway, University of London

ALVARO J. RIASCOS

Facultad de Economía, Universidad de los Andes

Abstract

We show that even under incomplete markets, the equilibrium manifold identifies

aggregate demand and individual demands everywhere in their domains.

Moreover, under partial observation of the equilibrium manifold, we we construct

maximal domains of identification. For this, we assume conditions of smoothness,

interiority and regularity, but avoid implausible observational requirements. It is

crucial that there be date-zero consumption. As a by-product, we develop some

duality theory under incomplete markets.

Key words

: Identification, General equilibrium, Incomplete Markets.

JEL classification

: D50, D52, D11

1

Part of this research was done while both authors where at the Banco de la Republica de Colombia and the second was visiting Cowles Foundations for Economic Research at Yale University. Both authors

acknowledge hospitality and financial support. We thank Herakles Polemarchakis, John Geanakoplos and seminar participants at Banco de la Republica, the 2004 North American Summer Meeting of the

Econometric Society (at Brown University), the 2004 Latin American meeting of the Econometric Society (at Santiago de Chile), the Mathematical Economics Workshop at IMPA (Rio de Janeiro) and Cowles

IDENTIFICACIÓN DE PREFERENCIAS CON BASE EN DATOS

OBSERVADOS Y BAJO INCERTIDUMBRE

ANDRES CARVAJAL

Royal Holloway, University of London

ALVARO J. RIASCOS

Facultad de Economía, Universidad de los Andes

Resumen

Mostramos como, aún en mercados incompletes, la variedad de equilibrio

identifica la demanda agregada y las demandas individuales en todas partes de su

dominio. Más aún, bajo observación parcial de la variedad de equilibrio,

construimos dominios máximos de identificación. Utilizamos condiciones de

suavidad y regularidad pero evitamos hacer hipótesis poco plausibles sobre los

observables de la economía. Es crucial la existencia de consumo en el primer

período. Como un subproducto desarrollamos la teoría dual básica de la teoría del

consumidor en mercados incompletos.

Palabras clave

: Identificación, Equilibrio General y Mercados Incompletos.

When there exist uninsurable risks, competitive equilibrium is typically inefficient in a strong sense: a planner could use the existing insurance possibilities to make every individual better off

(Geanakoplos and Polemarchakis [10]). The question immediately arises of how much information

a planner needs to have in order to figure out an improving policy intervention. The question

is not trivial: the transfer paradox, first pointed out by Leontief, and generalized by Donsimoni

and Polemarchakis [8], illustrates how ambiguous the welfare effects of a policy can be when the fundamentals of the economy are unknown.

Under the hypothesis of general equilibrium, the aggregate demand function cannot be assumed

to be observed: at equilibrium prices aggregate demand is, by definition, equal to aggregate

endow-ment. Demand, either individual or aggregate, cannot be observed for out-of-equilibrium prices.

One can observe, however, equilibrium prices and individual incomes. In this paper we address the

problem of identifying individual preferences from the equilibrium manifold of a dynamic economy

withfinancial markets.

For the standard Arrow-Debreu model, positive results have been obtained by Balasko [1],

Chi-appori et al. [6] and [7], Matzkin [17] and Carvajal and Riascos [5]. Balasko’s result has been

criticized for making very strong observational assumptions: that one can observe equilibrium prices

in situations in which endowment is zero for all individuals but one. Under regularity assumptions,

Chiappori et al obtain local identification of individual demands, but their argument has been

crit-icized by Balasko, who has pointed out that it requires extreme smoothness assumptions. Matzkin

determines the largest class of fundamentals for which identification is possible, by excluding

trans-lations of the income expansion paths of individual demands. Carvajal and Riascos obtain local

(maximal) and global identification of individual demands, by combining the methods of Balasko

and Chiappori et al. in a way that avoids their weaknesses: it does not use boundary information,

nor does it require analyticity of preferences.

The case of uncertainty is more cumbersome. Kubler et al. [13] extend the results of Chiappori

et al.[6]: they use the implicit function theorem to identify individual demands (locally) from the

equilibrium correspondence, and then use Geanakoplos and Polemarchakis [11] to identify preferences

from individual demand functions.

This paper extends the results of Carvajal and Riascos [5] -and hence of Balasko and Chiappori et

al.- to the case of uncertainty. We assume an economy with numeraire assets and show that we can

identify individual demands locally (moreover, wefind maximal domains in which local identification

the space of prices and endowments. We extend Balasko’s idea on how to recover the aggregate

demand function from the equilibrium manifold to the case of (possibly incomplete) asset markets,

hence we avoid using the implicit function theorem. We then use a slightly different argument from Kubler et al.’s to identify individual demands from the aggregate demand function and we also avoid

using Balasko’s strong observational assumption. In contrast to Kubler et al., our results do not

assume that preferences are additively separable across states. If that assumption is made, however,

our result suffices to imply, by Geanakoplos and Polemarchakis [11], that the equilibrium manifold locally identifies individual preferences. As a necessary by-product, we develop some basic duality

theory for incomplete markets.

1

No-arbitrage equilibrium

Consider the canonical, two-period, incomplete markets model withfinancial assets. There areS+ 1

states of nature,s = 0, ..., S,1 _{I individuals, i = 1, ..., I, and L>2commodities available in each}

state,l= 1, ..., L. DenoteL(S+ 1)byN and define the commodity space as_RN

+.

Afinancial asset is a contractv_∈RS _{that promises delivery of an amountv}

s∈Rof commodity l= 1at state of nature s= 1, ..., S. Letp= (ps)Ss=0∈RN++ denote the vector of commodity prices,

whereps= (ps,l)L

l=1∈RL++ andps,ldenotes the price in statesof one unit of goodl. Date-1 prices

are denoted byp1= (ps)S_s=1.

Letv1_{, ..., v}J _{beJ ≥1financial assets, defineV(p}

1)as the matrix of income transfers:

V (p1) =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

p1,1 0 · · · 0

0 p2,1 · · · 0

..

. ... . .. ...

0 0 _{· · ·} pS,1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

V = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

V1(p1,1)

V2(p2,1)

.. .

VS(pS,1)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

where

V = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

V1

.. .

VS ⎤ ⎥ ⎥ ⎥ ⎥ ⎦=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

v1

1 · · · vJ1

..

. . .. ...

v_S1 _{· · ·} vJ_S ⎤ ⎥ ⎥ ⎥ ⎥ ⎦.

The space of income transfers is the column span of the matrix of income transfers: _hV(p1)i=

©

t_∈RS¯¯ ¡_∃z∈RJ¢_:V(p

1)z=t

ª

. In general, as p1 changes,hV (p1)i changes. By construction,

however, forp1∈RLS++, the dimension ofhV(p1)iis always equal to the rank ofV.

Assume the following:

Condition 1 V has full column rank.

Take p to denote date-zero present-value prices (see Magill and Shafer, [15, p. 1534]), and let

w= (ws)S_s=0∈RN

++ represent an endowment of commodities.

Given pandw, define the budget2

B(p, w) =

(

x_∈RN₊¯¯ S

X

s=0

ps·(xs−ws)≤0andp1¡(x1−w1)∈hV(p1)i

)

,

wherex1= (xs)Ss=1andw1= (ws)Ss=1. Future consumptionx1isfinancially feasible at future prices

and endowments(p1, w1)if the second condition in the definition ofB(p, w) is satisfied: there is a

portfolio of assets,z_∈RJ_{, that delivers the transfers necessary tofinancex}

1.3

Since there is one degree of nominal indeterminacy, we normalize prices to lie in©p_∈RN

++

¯

¯p0,1= 1

ª

,

a set that we denote by_S++N−1.

Assume that there are I _{∈ N} individuals. Individual i _{∈ I} = _{1, ..., I_} has preferences ui _:

RN

+ −→R, which are assumed to satisfy the following condition:

Condition 2 ui is continuous, monotone and strongly quasi-concave, and satisfies that for allx_∈

RN++,

©

x0_∈RN+

¯

¯ui(x0)_≥ui(x)ª_⊆RN++.4

Define the individual demands,fi_:SN−1

++ ×RN++ →RN++, by fi(p, w) = arg maxx∈B(p,w)ui(x),

and the aggregate demand function F : _S++N−1_×RN

++ →RN++, asF(p,

¡

wi¢I i=1) =

P

i∈Ifi(p, wi).

Functions ¡fi¢

i∈I and F are well defined, since for (p, w) ∈ S N−1

++ ×RN++, B(p, w) is nonempty,

compact and convex, and eachui _{is continuous and strongly quasi-concave. Condition 2 guarantees}

that the range offi _{is contained inR}N

++.

A no-arbitrage equilibriumfor the economy³¡ui_{, w}i¢ i∈I, V

´

is a pair(x, p)_∈RN I

+ × S++N−1

such that:

1. For every i,xi =fi(p, wi);

2. F(p,¡wi¢ i∈I) =

P

i∈Iwi.

2_{For any(ρ, γ)∈R}LS_×RLS_{, we denoteρ¡∆= ρ}

1·∆1 · · · ρS·∆S >.

3_IfdimhV(P

1)i=S, or, equivalently,dimhVi=S, the second condition that defines B(P, w;V)is nonbinding.

This is the case of complete markets.

4_{This condition excludes additively separable preferences of the form u}i_{(x) =} SS

s=0uis(xx), for

ui

s:RL+−→R S

s=0. In this case, our analysis still holds if we introduce the following assumption: for every

Notice that, since V has full column rank, in the previous definition we need not explicitly

consider market clearing of portfolios.

Given¡ui¢

i∈IandV, theno-arbitrage equilibrium manifold(for short,equilibrium man-ifold) is

M=

(

(p, w)_∈S++N−1_×R++N I¯¯F(p,¡wi¢_i

∈I) =

X

i∈I wi

)

.

SinceF is continuous, it is straightforward thatM is closed.

Henceforth, we maintain the assumption that there are an asset structure that satisfies condition 1

and a profile of preferences that satisfies condition 2, and assume that some subset of the equilibrium

manifold is observed.5 We study whether unobserved preferences can be uniquely determined from that subset, but we do not test their existence. Thus, our question is one of identification and not one

of testability or refutability, which has been dealt with elsewhere.6 _{Since, under our assumptions,}

equilibrium is known to exist for every profile of preferences and endowments, our observational

assumption is not vacuous.

2

From the Equilibrium Manifold to Aggregate Demand

Under our assumptions, F is continuous, satisfies Walras’s law (that is, p_·F(p,¡wi¢

i∈I) = p·

¡P

i∈Iwi

¢

),p1¡

³

F1(p,

¡

wi¢ i∈I)−

P

i∈Iwi1

´

∈hV(p1)iand satisfies that

¡

p_·w_bi=p_·wi andp1¡¡wbi1−wi1

¢

∈hV(p1)i¢_i∈I=⇒F(p, w) =F(p,w).b

For anyΦ:_S++N−1_×RN I_++−→RN₊₊ andD_⊆S++N−1_×RN I₊₊, denote byΦ_|Dthe restriction ofΦto

D.

We say that E _⊆ M identifies F over D _{⊆ S++}N−1_×RN I

++ if for every continuous function

Φ:_S++N−1_×RN I

++ −→RN++ that satisfies Walras’s law,p1¡

³

Φ1(p,

¡

wi¢ i∈I)−

P

i∈Iw1i

´

∈hV(p1)i

and is such that

¡

p_·w_bi=p_·wi andp1¡¡wbi1−w1i

¢

∈hV(p1)i¢_i∈I=⇒Φ(p, w) =Φ(p,w)b

5_{So far, we have defined the manifold in terms of present-value prices of commodities only. Section 4 shows how} to identify subsets of the manifold so defined, from observation of the manifold defined in terms of spot commodity prices and asset prices.

and ₍

(p, w)_∈S++N−1_×RN I₊₊¯¯Φ(p, w) =X i∈I

wi

)

⊇E,

it is true thatΦ_|D=F|D.

If in the previous definition we drop the requirement thatΦbe continuous, then we say thatE

strongly identifies F overD. Clearly, strong identification implies identification.

Intuitively, we say that E identifiesF over D if, for any function that cannot be ruled out as

aggregate demand function, we have that, on the restricted domainD, that function is identical to

the true aggregate demand function. A function cannot be ruled out as aggregate demand function

when (i) it satisfies the properties that are necessary for it to be an aggregate demand, and (ii) it is

consistent with the observed data (because all the observed equilibria are equilibrium according to

it).

We say thatthe equilibrium manifold identifies (strongly identifies) aggregate demand

globally if M identifies (strongly identifies) F over _S++N−1_×RN I

++. It is straightforward that the

equilibrium manifold identifies aggregate demand globally if, and only if, F is the only continuous

function Φ: _S++N−1_×RN I

++ −→ RN++, satisfying Walras’s law, p1¡

³

Φ1(p,¡wi¢ i∈I)−

P

i∈Iwi1

´

∈

hV(p1)iandΦis such that

¡

p_·w_bi=p_·wi andp1¡

¡ b

wi₁₋w₁i¢_∈hV(p1)i

¢

i∈I=⇒Φ(p, w) =Φ(p,w)b

and ₍

(p, w)_∈S++N−1_×RN I₊₊¯¯Φ(p, w) =X i∈I

wi

)

=M.

In the case of global identification, every function that satisfies the properties of an aggregate

demand can be ruled out, with the only exception of the true aggregate demand.

Some properties of the concept of identification, that straightforwardly apply here, are shown

by Carvajal and Riascos [5]. For example, (i) identification over a set implies identification over

its subsets; (ii) identification from a subset of the manifold suffices to imply identification from its supersets (over the same, given, set); (iii) identification over a set implies identification over its

For anyE _⊆M, define

DE= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

(p, w)_∈S++N−1_×RN I₊₊¯¯ ¡_∃w_b∈RN I₊₊¢: ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

(p,w)_{b ∈}E

¡

p_·w_bi¢ i∈I =

¡

p_·wi¢ i∈I

¡

p1¡

¡ b

wi

1−w1i

¢

∈hV(p1)i

¢

i∈I ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .

The key result is the following theorem, which generalizes the idea of Balasko to less-than-global

observation under uncertainty.

Theorem 1 E_⊆M identifiesF over any D_⊆DE, strongly over anyD⊆DE.

Proof. It suffices to show thatE strongly identifiesF overDE. LetΦbe such that ifp_·w_bi_=p·wi_andp

1¡¡wbi1−w1i

¢

∈hV(p1)i, for alli, thenΦ(p, w) =Φ(p,w)b ,

and E _⊆©(p, w)_|Φ(p, w) =P_i∈Iwiª_{. Fix (p, w)∈ DE. By definition, we can fix wb ∈ R}N I

++ such

that(p,w)_{b ∈}E, and p_·w_bi =p_·wi and p1¡¡wbi1−wi1

¢

∈hV(p1)ifor alli. Then, by construction,

Φ(p, w) =Φ(p,w)_b and F(p,w) =_b F(p, w). Since E _⊆©(p,_ew)_{e |}Φ(_ep,w) =_e P_i∈Iw_eiª, it follows that

Φ(p,w) =_b P_i∈Iw_bi_{, whereas sinceE ⊆M, F(p,w) =b} P

i∈Iwbi. It follows that Φ(p, w) =F(p, w).

Corollary 1 The equilibrium manifold strongly identifies aggregate demand globally.

Proof. It suffices to show thatDM =S++N−1×RN I++. Letwbi=fi(p, wi)∈RN++. By monotonicity,

p_·w_bi _{= p·w}i_{, whereas, by definition of individual demands, p}

1¡

¡ b

wi

1−w1i

¢

∈ hV(p1)i. That

³

p,¡w_bi¢ i∈I

´

∈M is straightforward, sincefi_(p,wbi_{) =wb}i_.

Lemma 1 If E _⊆M is closed and the closure ofE in _S++N−1_×RN I

+ is contained in S++N−1×RN I++,

thenDE is closed.

Proof. Let(pn, wn)∞_n=1be a sequence defined inDEsuch that(pn, wn)∞_n=1−→(p, w)_∈S++N−1_×

RN I++. By definition, there exists(wn)b ∞n=1such that(pn,wn)b ∞n=1 is a sequence inE such that, for all

n_∈N,

³¡

pn·wbni

¢∞

n=1=

¡

pn·wni

¢∞

n=1 andpn,1¡

¡ b

wi_n,1−w_n,i ₁¢_∈hV(pn,1)i

´

i∈I

For eachi, since¡pnwin

¢∞

n=1is bounded and(pn)

∞

n=1is bounded away from0, it follows that(wbn)∞n=1

is bounded: for every(i, n, s, l)_{∈I ×N× {}0, ..., S_{} × {}1, ..., L_},

0_≤w_bn,s,li _≤sup n∈N

©

pnw_niª/inf

It follows that we can take a convergent subsequence¡pn(k),wbn(k)

¢∞

k=1 such that

¡

pn(k),wbn(k)

¢

−→

(p,w)_b . Since the closure of E in_S++N−1_×RN I

+ is contained inS++N−1×RN I++, then(p,w)b ∈S++N−1×

RN I

++, and, since E is closed (in S++N−1×RN I++), (p,w)b ∈ E. Let

¡

zi n

¢∞

n=1 in R

JI _{be such that}

pn,1¡¡wbin,1−win,1

¢

=V(pn,1)zin for all n∈N. Since

¡

pn(k),wbn(k)

¢

→(p,w)_b and wn _−→w, then

V(pn(k)),1) → V(p1) and, therefore, since V has full column rank, zn(k) → z for some z in RJ I.

In conclusion, (p,w)_{b ∈}E, p_·w_bi _=p·wi _{for alli and p}

1¡¡wb1i −wi1

¢

=V(p1)zi, and, therefore,

(p, w)_∈DE.

As in Carvajal and Riascos [5], if the manifold is known only up to a lower bound on the value

of wealth, then aggregate demand is identified subject to the same restriction. The largest domain

on which, givenE_⊆M, identification is possible, is determined next.7

Theorem 2 1. Suppose E _⊆M is closed and the closure ofE in _S++N−1_×RN I

+ is contained in

S++N−1×RN I++. If E identifies F overD, then D⊆DE.

2. IfE_⊆M strongly identifiesF overD, thenD_⊆DE.

Proof. For the first part, by lemma 1 DE is closed. Now, denote byδ :_S++N−1_×RN I

+ −→R+

the function distance-to-E, which, sinceE is closed, is defined by

δ(p, w) = min

(p,w)∈Ek(p, w)−(p,ew)e k

Closedness also implies thatδ(p, w) = 0_⇐⇒(p, w)_∈E.

Let

B(p, w) =

(

x_∈RN₊¯¯ S

X

s=0

ps·(xs−ws) = 0andp1¡(x1−w1)∈hV(p1)i

)

and define the function∆:_S++N−1_×RN I

++−→[0,1]by

∆(p, w) = min

(

min w∈ i∈IB(p,wi)

δ(p,w)_e ,1

)

,

which is well defined (because δ is continuous and each B¡p, wi¢ _{is compact) and is continuous.}

Moreover, suppose that ∆(p, w) = 0; this implies that min_w∈

i∈IB(p,wi)δ(p,w) = 0e and hence, again by compactness ofQ_i∈IB¡p, wi¢, that for somew_e∈Q_i∈IB¡p, wi¢, δ(p,w) = 0_e ; it follows

that (p,w)_{e ∈} E and hence, since w_{e ∈} Q_i∈IB¡p, wi¢_{, it follows that (p, w) ∈ D}

E; then, it is

immediate that∆(p, w) = 0_⇐⇒(p, w)_∈DE. Also, by construction,

¡¡

p_·w_bi¢=¡p_·wi¢ andp1¡

¡ b

wi₁₋w₁i¢_∈hV(p1)i

¢

i∈I =⇒

Y

i∈I

B¡p, wi¢=Y i∈I

B¡p,w_bi¢

=_⇒ ∆(p,w) =_b ∆(p, w)

Now, define the function Φ:_S++N−1_×RN I

++−→RN++, by

Φ(p, w) =F(p, w) +∆(p, w)F0,2(p, w) 2

∙

p0,2 −1 0 . . . 0

¸>

N×1

.

FunctionΦis well defined: it is continuous, maps into_RN₊₊, satisfies Walras’s law and, by

construc-tion: p1¡

¡

Φ1(p, w)₋P_i∈Iwi1

¢

∈hV(p1)iand

¡¡

p_·w_bi¢=¡p_·wi¢ andp1¡

¡ b

w₁i ₋wi₁¢_∈hV(p1)i

¢

i∈I=⇒Φ(p,w) =b Φ(p, w)

Moreover, for every(p, w)_∈E,Φ(p, w) =F(p, w) =P_i∈Iwi_{, so}©_{(p, w)|Φ(p, w) =}P i∈Iwi

ª

⊇E.

Now, assume thatD _*DE and let(p, w)∈D\DE. By identification over D, Φ(p, w) =F(p, w),

so∆(p, w) = 0 and, hence,(p, w)_∈DE, an obvious contradiction.

For the second part, defineΦb :_S++N−1_×RN I

++−→RN++ simply by

b

Φ(p, w) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(p, w), if (p, w)_∈DE

F(p, w) +F0,2(p,w)

2

∙

p0,2 −1 0 . . . 0

¸>

N×1

, if (p, w)_∈/DE .

Function Φb is well defined, satisfies Walras law and, by constructionp1¡¡Φ1(p, w)−P_i∈Iwi1

¢

∈

hV(p1)iand

¡¡

p_·w_bi¢=¡p_·wi¢ andp1¡¡wb1i −wi1

¢

∈hV(p1)i¢_i∈I=⇒Φb(p,w) =b Φb(p, w).

Also,n(p, w)_|Φb(p, w) =P_i∈Iwio_{⊇E, becauseD}

E∩M ⊇E. Now suppose that there is(p, w)∈ D_\DE. Then, by definition, Φb0,2(p, w) = F0,2(₂p,w), and, since(p, w)∈D andEidentifiesF overD

3

From Aggregate Demand to Individual Demands

If equilibrium prices are observable for situations in which the incomes of all individuals but one

are zero, the argument above still holds and, then, it is straightforward that aggregate demand

identifies individual demands. That result, however, is not surprising: observation of situations in

which the endowments of all consumers but one are pegged at zero amounts, in effect, to assuming that individual information is available.

We now show that, under an additional assumption, one can identify individual demands, without

resorting to boundary analysis. Our proof is somewhat similar to the one presented by Kubler et

al., but simpler: it does not require separability of preferences; it does not require us to claim

uniqueness of the solution to a system of partial differential equations; and it requires a weaker regularity condition than the one used by Kubler et al.

For the sake of simplicity in the presentation, we initially study the global setting introduced in

the previous section. The case in which the aggregate demand is not globally known is presented

afterwards.

3.1

The global case:

In this case, we weaken Kubler et al.’s Regularity assumption as follows:

Condition 3 (Regularity in present value prices) For every individuali,ui_∈C4¡_RN

++

¢

and

is differentiably strictly monotone and differentiably strongly concave, and for everyp_∈S++N−1, there

exist w_∈RN₊₊, and(s, l),(s0, l0)_∈{0, ..., S_{} × {}1, ..., L_{} \ {}(0,1)_}, such that ∂

2

fs,li

∂(wi 0,1)

2 6= 0and

¯ ¯ ¯ ¯ ¯ ¯ ¯

∂2_fi s,l

∂(wi 0,1)

2(p, w)

∂2_fi s0,l0

∂(wi 0,1)

2(p, w)

∂3fs,li

∂(wi 0,1)

3(p, w)

∂3_fi s0l0

∂(wi 0,1)

3(p, w) ¯ ¯ ¯ ¯ ¯ ¯ ¯

6

= 0.

The previous condition is weaker becausewonly needs an existential, and not a universal,

quan-tifier, because it is not assumed state-by-state and because it is not assumed for asset demands.

The condition is indeed restrictive as it requires, for example, that income effects do not vanish. Intuitively, the condition requires that preferences be “complex enough” so as to generate the

inde-pendence of income effects. As Chiappori et al. have pointed out, under complete markets it suffices that individual demands have rank at least two for the condition to be met everywhere. Appendix

Theorem 3 (Slutsky symmetry) For each i, fi _{∈ C}3¡_SN−1 ++ ×RN++

¢

and for every (p, w) _∈

S++N−1×RN++ and(s, l),(s0, l0)∈({0, ..., S} × {1, ..., L})\ {(0,1)},

∂fi s,l ∂ps0_,l0

(p, w) +¡f_si0_,l0(p, w)−ws0,l0¢

∂fi s,l ∂w0,1

(p, w) = ∂f

i s0_,l0

∂ps,l

(p, w) +¡f_s,li (p, w)₋ws,l¢∂f i s0_,l0

∂w0,1

(p, w).

Proof. Thatfi_∈C3_{follows from Duffie and Shafer [9]. Symmetry follows from theorems 4 and}

6 in appendix 2.

We say thataggregate demand identifies individual demands globallyif, givenF,¡fi¢ i∈I

is the only profile of functions ¡ϕi _:SN−1

++ ×RN++→RN++∈C3

¡

S++N−1×RN++

¢¢

i∈I, satisfying

Wal-ras’s law and Slutsky symmetry, and such thatP_i∈Iϕi_=F.

Theorem 4 Aggregate demand identifies individual demands globally.

Proof. Let¡ϕi¢_i

∈I satisfy Walras’s law and Slutsky symmetry and be such thatF=

P

i∈Iϕi.

Fixi_∈Iand defineθi :_S++N−1_×RN

++−→RN++andγi:S++N−1−→RNbyθi(p, w) =F(p,(1,1, ..., w, ...,1)),

wherewoccupies theith _{position, andγ}i_{(p) =−}P

j∈I\{i}ϕj(p,1).

By Slutsky symmetry, for every (s, l),(s0_{, l}0_{) ∈ {0, ..., S} × {1, ..., L} \ {(0,1)}, everywhere in}

S++N−1 ×RN++,

∂ϕi s,l

∂ps0,l0 +

³

ϕi

s0_,l0 −w_si0_,l0

´_∂ϕi s,l

∂wi 0,1 =

∂ϕi s0,l0

∂ps,l + ³

ϕi

s,l−ws,li

´∂ϕi s0,l0

∂wi

0,1 . Since ϕ

i¡_{p, w}i¢ ₌

θi¡p, wi¢_+γi_{(p), substituting,}

∂θi_s,l ∂ps0_,l0

+ ∂γ

i s,l ∂ps0_,l0

+¡θi_s0_,l0 +γ_si0_,l0−wi_s0_,l0

¢ ∂θi_s,l ∂wi

0,1

=∂θ

i s0_,l0

∂ps,l

+∂γ

i s0_,l0

∂ps,l

+¡θi_s,l+γi_s,l−wi_s,l¢∂θ i s0_,l0

∂wi

0,1

.

Fixp_∈S++N−1. Taking that(s, l),(s0_{, l}0_{)6= (0,1) and deriving once and twice with respect tow}i

0,1

gives

∂2_θi s,l ∂wi

0,1∂ps0_,l0

+¡θi_s0_,l0 +γ_si0_,l0−wi_s0_,l0

¢ ∂2_θi s,l ∂¡wi

0,1

¢2 =

∂2_θi s0_,l0

∂wi

0,1∂ps,l

+¡θi_s,l+γi_s,l−w_s,li ¢ ∂

2_θi s0_,l0

∂¡wi

0,1

¢2

and

∂3_θi s,l ∂¡wi

0,1

¢2

∂ps0_,l0

+∂θ

i s0_,l0

∂wi

0,1

∂2_θi s,l ∂¡wi

0,1

¢2 +

¡

θi_s0_,l0 +γi_s0_,l0−wi_s0_,l0

¢ ∂3_θi s,l ∂¡wi

0,1

¢3 =

∂3_θi s0_,l0

∂¡wi

0,1

¢2 ∂ps,l + ∂θ i s,l ∂wi

0,1

∂2_θi s0_,l0

∂¡wi

0,1

¢2 +

¡

θi_s,l+γi_s,l−w_s,li ¢ ∂

3_θi s0_,l0

∂¡wi

0,1

which rewrites as (recall thatpisfixed)

∆(s,l),(s0,l0)¡wi¢

⎡ ⎢ ⎣ γ

i s0_,l0(p)

γi s,l(p)

⎤ ⎥

⎦=Γ(s,l),(s0,l0)¡wi¢,

where

∆(s,l),(s0_,l0₎¡wi¢=

⎡ ⎢ ⎣

∂2θis,l

∂(wi 0,1)

2 ¡

p, wi¢ ₋∂2θis0,l0

∂(wi 0,1)

2 ¡

p, wi¢

∂3θis,l

∂(wi 0,1)

3 ¡

p, wi¢ ₋∂

3

θi_s0,l0

∂(wi 0,1)

3 ¡

p, wi¢ ⎤ ⎥ ⎦

andΓ(s,l),(s0_,l0₎¡wi¢hasfirst component

∂2_θi s0_,l0

∂wi

0,1∂ps,l − ∂2_θi

s,l ∂wi

0,1∂ps0_,l0

+¡θi_s,l−w_s,li ¢ ∂

2_θi s0_,l0

∂¡wi

0,1

¢2 −

¡

θi_s0_,l0−wi_s0_,l0

¢ ∂2_θi s,l ∂¡wi

0,1

¢2

and second component

∂3_θi s0,l0

∂(wi 0,1)

2

∂ps,l −

∂3_θi s,l

∂(wi 0,1)

2

∂ps0,l0 +

∂θi s,l

∂wi 0,1

∂2_θi s0,l0

∂(wi 0,1)

2−

∂θi s0,l0

∂wi 0,1

∂2_θi s,l

∂(wi 0,1)

2

+³θi_s,l−wi s,l

´ ∂3_θi s0,l0

∂(wi 0,1)

3 − ³

θi_s0_,l0 −w_si0_,l0

´ _∂3_θi s,l

∂(wi 0,1)

3

.

Notice that the resulting ∆(s,l),(s0_,l0₎¡wi¢and Γ_(s,l),(s0_,l0₎¡wi¢ do not containγi_s,l or γi_s0_,l0, but

onlyθi, which is determined byF.

By regularity, for some wi _{∈ R}N

++ and (s, l),(s0, l0) ∈ {0, ..., S} × {1, ..., L}, ∆(s,l),(s0_,l0₎¡wi¢ is

invertible, so _⎡

⎢ ⎣ γ

i s0_,l0(p)

γi s,l(p)

⎤ ⎥

⎦=¡∆(s,l),(s0_,l0₎¡wi¢¢−

1

Γ(s,l),(s0_,l0₎¡wi¢, (*)

whereas, for every other (s00_{, l}00_{)∈ {0, ..., S} × {1, ..., L} \ {(0,1)}, by Slutsky symmetry, γ}i s00,l00(p) equals

∂2_θi s00,l00

∂wi

0,1∂ps,l −

∂2θis,l

∂wi

0,1∂ps00,l00 + ³

θi_s,l¡p, wi¢_+γi

s,l(p)−wis,l

´_∂2_θi s00,l00

∂(wi 0,1)

2− ³

θi_s00_,l00

¡

p, wi¢_−wi s00_,l00

´ _∂2

θis,l

∂(wi 0,1)

2

∂2_θi s,l

∂(wi 0,1)

2

,

and, by Walras’s law,γi

0,1(p) =−

³PL

l=2p0,lγi0,l(p) +

PS s=1

PL

l=1ps,lγis,l(p)

´

.

Since ϕi¡_{p, w}i¢_=θi¡_{p, w}i¢_+γi_{(p) and the expression on the right hand side of equation (*)}

depends only onF, it follows thatϕi

3.2

Restricted observation

Notice that the choice of¡wj ₌₁¢

j∈I\{i} in the proof of theorem 4 is arbitrary and that when only F_|D is available, such choice can be modified as needed. If only local information ofF is available,

one must strengthen the second part of the assumption so that, for given prices, the profile of

endowments at which the condition is met lies in the observed domain. For the sake of simplicity,

assume that we strengthen condition 3 by substituting the existential quantifier ofwby the universal

quantifier.

LetD_⊆S++N−1_×RN I

++and, for everyi, letDi⊆S++N−1×RN++. We say thatF|D identifies

¡

fi¢ i∈I over ¡Di¢

i∈I if for every

¡

ϕi_:SN−1

++ ×RN++ →RN++∈C3

¡

S++N−1×RN++

¢¢

i∈I, satisfying Walras’s

law and Slutsky symmetry, and such that:

¡

p_·w_bi=p_·wi andp1¡¡wbi1−w1i

¢

∈hV(p1)i¢_i∈I =⇒¡ϕi(p, w)¢_i∈I=¡ϕi(p,w)b ¢_i∈I

X

i∈I

ϕi¡p, wi¢

|D=F(p, w)|D

it is true that³ϕi |Di

´

i∈I=

³

fi |Di

´

i∈I.

Intuitively, we say thatF_|D identifies

¡

fi¢_i

∈I over

¡

Di¢_i

∈I if, for any profile of functions that

cannot be ruled out as individual demands, we have that, on the restricted domains¡Di¢_i

∈I, those

functions are identical to the true demand functions. A profile of functions cannot be ruled out as

individual demands when (i) it satisfies the properties that are necessary for a profile of individual

demands, and (ii) it is (locally) consistent with aggregate demand.

Theorem 5 Let D_⊆S++N−1_×RN I

++ and, for eachi∈I, denote

Di= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ¡

p, wi¢_∈S++N−1_×RN₊₊¯¯(_∃(p,_bw)_{b ∈}D0) :

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ b

p=p

¡

p_·w_bi¢ i∈I=

¡

p_·wi¢ i∈I

¡

p1¡

¡ b

wi

1−w1i

¢

∈hV(p1)i

¢

i∈I ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ,

whereD0 is the interior ofD. F_|D identifies

¡

fi¢

i∈I over

³

Di´ i∈I.

Proof. Let¡ϕi¢_i

∈I, satisfy Walras’s law, Slutsky symmetry and such that

¡

p_·w_bi=p_·wi andp1¡

¡ b

wi₁₋w₁i¢_∈hV(p1)i

¢

i∈I =⇒

¡

ϕi(p, w)¢ i∈I=

¡

and

X

i∈I

ϕi¡p, wi¢

|D=F(p, w)|D

Fix i _{∈ I} and ¡p, wi¢ _∈ D0i, where Di0 is the projection of D0 into the space of

¡

p, wi¢. By

definition, there exists ¡wj¢

j∈I\{i} such that

³

p,¡wj¢I j=1

´

∈ D0. Since D0 is open, there exists

>0such that

©

(p,_ew)_{e ∈S++}N−1_×RN I₊₊¯¯_k(p,_ew)_{e −}(p, w)_k< ª_⊆D0.

DenoteO=©_ep_∈S++L−1¯¯_kep₋p_k< ªand

Oi=© ¡p,_ew_ei¢_∈S++L−1_×RL₊₊¯¯°°¡ep,w_ei¢₋¡p, wi¢°°< ª,

and define the functionsθi:Oi _−→RL+andγi:O−→RL+, asθi

¡ e

p,w_ei¢=F¡p,_e¡w1, w2, ...,w_ei, ..., wI¢¢,

where w_ei occupies the ith position, and γi(_ep) =₋P_j∈I\{i}ϕj¡p, w_e j¢. By Slutsky symmetry and regularity, as in the proof of theorem 4,ϕi¡_{p, w}i¢_=fi¡_{p, w}i¢_{. That isϕ}i

|Di 0 =f

i |Di

0

Now, let ¡p, wi¢ _{∈ D}i_{. By definition, there exists} ¡_p n, win

¢∞

n=1, in D

i_{, such that} ¡_p n, wni

¢

→

¡

p, wi¢_{. Then, for each n ∈ N, there exists we}

n ∈ RN I++ such that (pn,wen) ∈ D0,

¡

p_·wi n

¢

i∈I =

¡

p_·w_ei n

¢

i∈Iand

¡

p1¡

¡

wi

n,1−wen,i 1

¢

∈hV(p1)i

¢

i∈Itherefore,ϕ i¡_p

n,weni

¢

=ϕi¡_p n, wni

¢

andfi¡_p n,weni

¢

=

fi¡_{pn, w}i n

¢

and, by continuity,ϕi¡_pn,wei n

¢

=ϕi¡_{pn, w}i n

¢

→ϕi¡_{p, w}i¢_andfi¡_pn,wei n

¢

=fi¡_{pn, w}i n

¢

→

fi¡_{p, w}i¢_{. Since}¡_pn,

e

wi n

¢

∈Di

0, it follows from our previous argument thatϕi

¡

pn,w_ei n

¢

=fi¡_pn,wei n

¢

,

and hence thatϕi¡_{p, w}i¢_=fi¡_{p, w}i¢_.

4

Observability: Financial Markets Equilibrium Manifold

Identification results are useful when based on observable data. In real life one does not observe

date-zero present-value equilibrium prices, but, instead, one observes (financial) equilibrium spot

prices for commodities and asset prices.

4.1

From the Equilibrium Manifold to Aggregate Demand

Letq_∈RJ _{be the price vector at which assets can be bought ats= 0.}

For(p, q)_∈RN

++×RJ andw∈RN++, let

B(p, q, w) = ⎧ ⎪ ⎨ ⎪ ⎩x∈R

N

+

¯

¯ ¡_∃z_∈RJ¢: ⎛ ⎜

⎝ p0·(x0−w0)≤ −qz p ¡(x ₋w ) =V (p )z

Since there areS+ 1degrees of nominal indeterminacy, we normalize prices, in each state, to lie

in

S++L−1=

©

ps∈RL++

¯

¯ps,1= 1

ª

.

Asset prices q_{∈ R}J _{are a no-arbitrage price vector if V z >0implies that q·z >0. It is well}

known that no-arbitrage is a necessary condition for optimization, and thatqis a no-arbitrage price

only if for someπ_∈RS

++,q=πV. LetQdenote the set (positive cone) of no-arbitrage price vectors.

Define the individual demand function infinancial markets,fi:¡_S++L−1¢S+1_{× Q ×R}N_++→RN₊₊,

as fi(p, q, w) = arg maxx_∈B(p,q,w)u(x), which is well defined since B(p, q, w) is nonempty, compact

(becauseqis a no-arbitrage price vector) and convex, anduis continuous and strongly quasi-concave.

Define also the aggregate demand function infinancial markets,F:¡_S++L−1¢S+1_{× Q ×R}N I

++→RN++,

asF(p, q, w) =P_i∈Ifi_{(p, q, w}i_).

A financial markets equilibriumfor economy ³¡ui_{, w}i¢ i∈I, V

´

is (x, z, p, q)_∈RN I

+ ×RJ×

¡ S++L−1

¢S+1

× Qsuch that:

1. For every i,xi _=fi_{(p, q, w}i_),p

0·(xi0−wi0) =−qzi andp1¡¡xi1−w1i

¢

=V zi_;

2. F(p, q,¡wi¢I_i=1) =P_i∈Iwi.

SinceV has full column rank,P_i∈Izi _{= 0is unnecessary in the previous definition.}

Given ¡ui¢

i∈I andV, thefinancial markets equilibrium manifoldis

M=

(

(p, q, w)_∈¡_S++L−1¢S+1_{× Q ×R++}N I¯¯_¯F(p, q,¡wi¢I_i=1) =X i∈I

wi

)

.

Under our assumptions,Fis continuous and satisfies:

1. _∃z_∈RJ _:p

0·

³ F0(p, q,

¡

wi¢I i=1)−

P

i∈Iwi0

´

=₋qzandp1¡

³ F1(p, q,

¡

wi¢I i=1)−

P

i∈Iwi1

´

=

V z.

2.

¡

∃zi_∈RJ :p0·¡wb0i−w0i

¢

=₋qzi andp1¡¡wbi1−w1i

¢

=V zi¢_i

∈I

=_⇒ F(p, q, w) =F(p, q,w)_b

For anyΦ :¡_S++L−1¢S+1_{× Q ×R}N I

++ −→RN++ andD⊆

¡ S++L−1

¢S+1

× Q ×RN I

We say that E_⊆M identifies F over D _⊆ ¡_S++L−1¢S+1 _{× Q ×R}N I

++ if for every continuous

functionΦ:¡_S++L−1¢S+1_{× Q ×R}N I

++−→RN++ such that:

1. _∃z_∈RJ _:p

0·

³

Φ0(p, q,¡wi¢

I i=1)−

P

i∈Iw0i

´

=₋qzandp1¡³Φ1(p, q,¡wi¢

I i=1)−

P

i∈Iwi1

´

=

V z

2.

¡

∃zi_∈RJ :p0·¡wb0i−w0i

¢

=₋qzi andp1¡¡wbi1−w1i

¢

=V zi¢_i

∈I

=_⇒ Φ(p, q, w) =Φ(p, q,w)_b

and ₍

(p, q, w)_∈¡_S++L−1¢S+1_{× Q ×R}N I₊₊¯¯_¯Φ(p, q, w) =X i∈I

wi

)

⊇E,

it is true thatΦ_|D=F|D.

If in the previous definition we drop the requirement thatΦ be continuous, then we say thatE

strongly identifiesF over D. Clearly, strong identification implies identification.

Since, what may be observed in the real world is subsets ofM, we now show how to derive subsets

E ofM from subsetsE of M and functions Φon _S++N−1_×RN I

++ from functionΦ on

¡ S++L−1

¢S+1

×

Q ×RN I

++.

For every E_⊆Mdefine

E=

(

(p, w)_∈S++N−1_×RN I₊₊¯¯

õ

1 ps,1

ps

¶S

s=0

, S

X

s=1

Vs(ps,1), w

!

∈E )

,

and for every functionΦ:¡_S++L−1¢S+1_{× Q ×R}N I

++−→RN++ defineΦ:S++N−1×RN I++−→RN++ by:

Φ(p, w) =Φ

õ

1 ps,1

ps

¶S

s=0

, S

X

s=1

Vs(ps,1), w

!

.

Theorem 6 Let E_⊆M.

1. E_⊆M.

2. IfΦ:¡_S++L−1¢S+1_{× Q ×R}N I_{++ −→R}N₊₊ satisfies the properties that characterize identification

(strong identification) in financial markets, then Φ : _S++N−1 _×RN I_{++ −→ R}N₊₊ satisfies the

3. Define

DE=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(p, q, w)_∈¡_S++L−1¢S+1_{× Q ×R}N I++

¯ ¯

¯¡∃(w, z)_{b ∈R}N I++×RJI

¢ : ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

(p, q,w)_{b ∈}E ¡

p0

¡ b

wi

0−wi0

¢

=₋qzi¢ i∈I

¡

p1¡¡wb1i −wi1

¢

=V zi¢ i∈I

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .

Then, E identifiesFover DE, strongly overDE.

Proof. It is well known that ifp_∈RN

++andπ= (π0, π1)∈{1}×RS++, thenB(p, π1V (p1), w) =

B³(πsps)Ss=0, w

´

. By substitution, then, for every p _{∈ S++}N−1 and π = (π0, π1) ∈ {1} ×RS++,

B³¡π−1

s ps

¢S

s=0,1>V(p1), w

´

=B(p, w).

Then, for part 1, let (p, w) _∈ E. By definition, µ³_p1

s,1ps

´S

s=0,

PS

s=1Vs(ps,1), w

¶

∈ E. Let

π= (1,(ps,1)_s=1,...S)then

X

i∈I

wi =X i∈I

fi Ã

¡

π−_s1ps¢S_s=0,

à _S

X

s=1

Vs(ps,1)

!

, wi

!

=X

i∈I

fi(p, wi) =F(p, w).

For part 2, suppose Φ:¡_S++L−1¢S+1_{× Q ×R}N I

++ −→RN++ satisfies properties (1)and (2)in the

definition of identification in financial markets (clearly, if Φ is continuous, then Φ is cotinuous).

We now check thatΦsatisfies all the properties that characterize identification in the no-arbitrage

equilibrium manifold.

Let us check Walras law. By the second part of property(1):

p₁¡

Ã

Φ1(p, q,¡wi¢

I i=1)−

X

i∈I w₁i

!

=V z.

Letp=³_p1

s,1ps

´S

s=0then the previous equation implies which implies

µ

1 ps,1

ps

¶S

s=1

¡

à Φ1(

µ

1 ps,1

ps

¶S

s=0

, q,¡wi¢I_i=1)₋X i∈I

w₁i

!

= V z

⇒

p1¡

à Φ1(

µ 1 ps,1 ps ¶S s=0

, q,¡wi¢I_i=1)₋X i∈I

w₁i

!

and by thefirst part of property(1) :

p0·

à Φ0(

µ

1 ps,1

ps

¶S

s=0

, q,¡wi¢I_i=1)₋X i∈I

w₀i

!

=₋qz

Summing up oversthe last two equations we obtain:

p_·

à Φ(

µ

1 ps,1

ps

¶S

s=0

, q,¡wi¢I_i=1)₋X i∈I

wi

!

= (V (p1)−q)z.

Letq=PS_s=1Vs(ps,1),then we just proved that:

p_· Ã Φ( µ 1 ps,1 ps ¶S s=0 , S X s=1

Vs(ps,1),

¡

wi¢I_i=1)₋X i∈I

wi

!

= 0,

which, by definition, is the same as:

p_·

Ã

Φ(p, w)₋X

i∈I wi

!

= 0

For the next property notice that, by definition,

p1¡

Ã

Φ1(p,¡wi¢_i

∈I)−

X

i∈I w₁i

!

= p1¡

à Φ1

õ

1 ps,1

ps ¶S s=0 , S X s=1

Vs(ps,1),

¡

wi¢_i

∈I

!

−X

i∈I wi₁

!

= (ps,1)S_s=1¡p1¡

à Φ1

³

p, q,¡wi¢ i∈I

´

−X

i∈I w₁i

!

wherep₁=³_p1

s,1ps

´S

s=0andq=

PS

s=1Vs(ps,1).By the second part of property(1), p1¡

³ Φ1

³

p, q,¡wi¢_i

∈I

´

−Pi∈Iw

hV_itherefore,

(ps,1)S_s=1¡p1¡

à Φ1

³

p, q,¡wi¢ i∈I

´

−X

i∈I wi₁

!

∈hV(p1)i

For part 3, it suffices to prove thatE_⊆M strongly identifiesFover DE.LetΦ:¡S++L−1

¢S+1

×

Q ×RN I

++ −→ RN++ such that it satisfies the properties that characterize strong identification and

(p, q, w)_∈DE. By definition, there exists(w, z)b ∈RN I++×RJIsuch that(p, q,w)b ∈E,p0¡wbi0−wi0

¢

=

−qzi andps¡w_bsi₋wi_s¢=V zi, for all i. By no-arbitrage, for someπ_∈{1_{} ×R}S₊₊, q=π1V, which

no-arbitrage equilibrium manifold. Therefore, by theorem 1,Φ³(πsps)S_s=0,w_b´=F³(πsps)S_s=0,w_b´

but Φ(p, q, w) = Φ(p, q,w) =_b Φ³(πsps)Ss=0,wb

´

and F(p, q, w) = F(p, q,w) =_b F³(πsps)Ss=0,wb

´

and henceΦ(p, q, w) =F(p, q, w).

The following corollary stablishes the global result.

Corollary 2 Global knowledge of thefinancial markets manifold identifies aggregate demand

glob-ally.

Proof. It suffices to show thatDE=¡S++L−1

¢S+1

×Q×RN I

++.Let(p, q, w)∈

¡ S++L−1

¢S+1

×Q×RN I

++

and define¡w_bi¢ i∈I=

¡

fi_{(p, q, w}i₎¢

i∈Ithen obviously, there existsz∈R

JI _{such that(w, z)b ∈R}N I

++×

RJI,(p, q,w)_{b ∈}M,¡p0¡wb0i−wi0

¢

=₋qzi andp1¡¡wbi1−w1i

¢

=V zi¢_i

∈I. Therefore(p, q, w)∈DE.

For completness we include the following theorem. Knowledge of thefinancial markets manifold

identifies the no-arbitrage manifold.

Theorem 7 Define

M=

(

(p, w)_∈S++N−1_×RN I₊₊¯¯

õ

1 ps,1

ps

¶S

s=0

, S

X

s=1

Vs(ps,1), w

!

∈M )

.

ThenM=M.

Proof. ThatM _⊆M follows immediately from theorem 4.1. Now, let(p, w)_∈M. By

construc-tion,

X

i∈I

wi = X i∈I

fi(p, wi)

= X

i∈I

fi õ

1 ps,1

ps

¶S

s=0

,

à _S X

s=1

Vs(ps,1)

!

, wi

!

= F

õ

1 ps,1

ps

¶S

s=0

,

à _S

X

s=1

Vs(ps,1)

!

, w

!

,

where the third equality follows from the fact pointed out at the beginning of the previous proof.

This means that, given two profiles of preferences¡ui¢ i∈Iand

¡ e

ui¢

i∈Iand an asset structureV,

ifM(ui₎

i∈I =M(ui)i∈I, thenM(ui)i∈I =M(ui)i∈I, which is to say that the equilibrium manifold is identified.

Lemma 2 SupposeE_⊆Mis closed and the closure ofEin ¡_S++L−1¢S+1_{× Q ×R}N I

+ is contained in

¡ S++L−1

¢S+1

× Q ×RN I

++. Then

1. E_⊆M is closed and the closure ofE in_S++N−1_×RN I₊ is contained in_S++N−1_×RN I₊₊.

2. DEis closed.

Proof. This is straightforward.

We are now ready for the main theorem regarding the identification of the aggregate demand

from thefinancial markets equilibrium manifold. For this, we will need to strenghten condition(2).

Condition 4 ui is continuous,C1¡_RN++

¢

, monotone, strongly quasi-concave, differentiable strictly

monotone (i.e., _∀x_∈RN

++, Dui(x)>> 0) and diferenciable strictly concave (∀x∈RN++, D2ui(x)

is negative definite) and satisfies that for allx_∈RN

++,

©

x0 _∈RN

+

¯

¯ui_(x0_)≥ui_(x)ª_⊆RN

++.

Theorem 8 1. Suppose E_⊆M is closed and the closure of E in ¡_S++L−1¢S+1 _{× Q ×R}N I

+ is

contained in¡_S++L−1¢S+1_{× Q ×R}N I₊₊. IfE identifies FoverD, thenD_⊆DE.

2. IfE_⊆M strongly identifies FoverD, thenD_⊆DE.

Proof. For thefirst part, suppose it’s not true. Let(p, q, w)_∈/DE.Define

¡ b

wi_=fi_{(p, q, w}i₎¢ i∈I,

then(p, q,w)_{b ∈}/E.Now, defineπ:¡_S++L−1¢S+1_{× Q ×R}N

++→RS+++1 by

π(p, q, w) = ⎛ ⎝

∂u1_(fi_(p,q,w))

∂xs,1

∂u1_(f1_(p,q,w))

∂x0,1

⎞ ⎠

s=0,1..S ,

then π is continuous and it is well know that q = π(p, q, w)V. Let π = π(p, q, w1_{) hence q =}

π(p, q, w1_{)V and then}³_(π

sps) S s=0,wb

´

/

∈E_⇒³(πsps) S s=0,wb

´

/

∈DE.By lemma (***) we can define

the following functionΦ(p, q, w)as in the prove of Theorem (***).

Φ(p, w) =F(p, w) +∆(p, w)F0,2(p, w) 2

∙

p0,2 −1 0 . . . 0

¸>

N×1

.

By construction,Φsatisfies all the properties that characterize identification from the no-arbitrage

equilibrium,©(p, w)_∈S++N−1×RN I++

¯

¯Φ(p, w) =P_i∈Iwiª_⊇EandΦ³(πsps) S s=0,wb

´

6

=F³(πsps) S s=0,wb

´

(otherwise,∆³(πsps) S s=0,wb

´

= 0_⇒(πsps) S

s=0,w))b ∈DE.

Now define the following functionΦ:¡_S++L−1¢S+1_{× Q ×R}N I

++ →RN++ from Φ:

Then Φsatisfies all properties that characterize identification in the financial markets manifold

(since there is nothing new in this prove we differ it to an Appendix). Moreover,

(

(p, q, w)_∈¡_S++L−1¢S+1_{× Q ×R}N I₊₊¯¯_¯Φ(p, q, w) =X i∈I

wi

)

⊇E.

To prove this, take(p, q, w)_∈E which is the same as:

õ

1 πs(p, q, w1₎πs

¡

p, q, w1¢ps

¶S

s=0

, π¡p, q, w1¢V, w

!

∈E

and by definition, this implies¡πs¡p, q, w1¢ps¢S_s=0, w)_∈E(recall the normalization,p_∈¡_S++L−1¢S+1)

thereforeΦ³¡πs¡p, q, w1¢ps¢S_s=0, w´=P_i∈Iwi_⇒Φ(p, q, w) =P_i∈Iwi.

Finally, by hyphotesisE⊆M strongly identifiesFoverDtherefore,Φ(p, q, w) =F(p, q, w)but

by construction,Φ(p, q, w) =Φ((p, q,w)_b and F(p, q, w) =F(p, q,w)_b and by definitionΦ(p, q,w) =_b

Φ³(πsps) S s=0,wb

´

andF(p, q,w) =_b F³(πsps) S s=0,wb

´

thereforeΦ³(πsps) S s=0,wb

´

6

=F³(πsps) S s=0,wb

´

a contradiction.

For the second part the argument is similar.

Remark 1 Strictly speaking, part (2) does not require strenghtening condicion(2) on utility

fucn-tions.

4.2

From Aggregate Demand to Individual Demands

In this section we briefly discuss, for simplicity, the case of global identification.

We say thataggregate demand identifies individual demands globallyif, givenF,¡fi¢ i∈I

is the only profile of functions³ϕi_:¡_SL−1 ++

¢S+1

× Q ×RN

++→RN++

´

i∈I, such that

P

i∈Iϕi=F.and

the profile of functions¡ϕi_:SN−1

++ ×RN++→RN++∈C3

¡

S++N−1×RN++

¢¢

i∈Idefined by

µ

ϕi_{(p, w) =ϕ}iµ³ 1

ps,1ps ´S

s=0,

P

Walras’s law and Slutsky symmetry.

Theorem 9 Aggregate demand identifies individual demands globally.

Proof. Suppose it doesn’t. Then there exists a profile of function³ϕi_:¡_SL−1 ++

¢S+1

× Q ×RN

++→RN++

´

i∈I

that satisfy all conditions that characterize individual identification from the aggregate demand

in financial markets, such that for some j _{∈ I}, ϕj ₆= fj.In particular, there exists (p, q, w) _∈

¡ S++L−1

¢S+1

× Q ×RN++ such thatϕj(p, q, w)6=fj(p, q, w).Letπ= (π0, π1)∈{1} ×RS++ be such

that q =π1V then by definition ϕj(

³

ps

πs ´S

s=0, w)6=f

j₍³ps

πs ´S

s=0, w) but f

fjµ³ 1

ps,1ps ´S

s=0,

PS

s=1Vs(ps,1), w

¶

is the same as the individual demands introduced in the first

part of the paper (in terms of present value prices). Therefore, by theorem (***)ϕj₍³ps

πs ´S

s=0, w) =

fj₍³ps

πs ´S

s=0, w)a contradiction.

5

Concluding remarks

We have shown that enough information on how prices respond to income shocks can pin down, in

a unique manner, unobserved individual information.

When there is global information about the equilibria, given unobserved preferences and observed

asset structure, one can identify the aggregate demand function. This is a remarkable property of the

competitive model: the roots of a function (the aggregate excess demand) contain as much

informa-tion as the whole funcinforma-tion itself. Less than global knowledge will, obviously, give less comprehensive

information about the aggregate demand. These results obtain as a consequence of simple

prop-erties of the model, namely (i) Walras’s law, (ii) the fact that, for each individual, if two possible

endowments have the same market value and give the samefinancial constraint, then they give the

same demand, and (iii) no-trade equilibria immediately inform about the aggregate demand.

Then we show that aggregate demand can be used to recover, uniquely, individual demands. This

requires that income effects be different across commodities, which in turn requires that observed domains allow for perturbations of all possible endowments, which is indeed a restrictive assumption

(in particular, as it requires that some income effects do not vanish). Again, local information gives local identification, while the same is true for global information.

The results do not require that preferences be separable across states, but, if one is willing to

assume that they are, then they have the implication that equilibrium prices contain the same

infor-mation as the profile of individual preferences (by Geanakoplos and Polemarchakis [11]). Namely,

when going from preferences to equilibrium prices, wefirst optimize, then aggregate and then solve

for market clearing; the results imply that after all this transformations we still have essentially the

same information as at the beginning, which contrasts with the anything-goes intuition that was

derived from the Sonnenschein-Mantel-Debreu literature.

From a more practical perspective, one would hope that these results implied that readily

avail-able market information suffices for the unambiguous determination of the welfare effects of economic policy, something that is desirable (Geanakoplos and Polemarchakis [10]) but far from obvious

data are necessary. On the one hand, identification of Pareto improving policies has been shown to

fail when price effects across commodities are unknown (Geanakoplos and Polemarchakis [11]) and when only afinite data set is available for a nonstationary economy (work in progress by A. Carvajal

and H.M. Polemarchakis). Moreover, identification, or lack thereof, in the presence of production,

6

Appendix 1: Regularity

It follows from Lewbel [14] and Blanks, Blundell Lewbel [2] that there exists u0 : R3+ −→ Rsuch

thatf :_R3

++×R++−→R3++ is given by

fl(p, m) = m pl

Ã

Al(p) +Bl(p) log

µ

m a(p)

¶

+Cl(p) log

µ

m a(p)

¶2!

, l= 1,2,3,

are the solutions to the problem

max

x u0(x) s.t. p·x=m,

where Al, Bl and Cl are homogeneous of degree zero in p and a is homogeneous of degree 1 in p

(these two conditions guarantee thatfl is homogeneous of degree zero inpandm) and

3

X

l=1

Al(p) +

3

X

l=1

Bl(p) log

µ

m a(p)

¶

+

3

X

l=1

Cl(p) log

µ

m a(p)

¶2

= 1,

for allpandm(which is necessary for Walras’s law).8

Now, suppose that there are three commodities and two states of nature so that, for some

functionsu1 andu2,u(x0, x1, x2) =u0(x0) + min{u1(x1), u2(x2)}.

Suppose that there is only one asset, V = [1 ₋1]> and define

vs(p, m) = max

x us(x) s.t. p·x=m

and

v(p, w) = max

x u(x) st. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

p_·x=p_·w ⎡

⎢

⎣ p1·(x1−w1) p2·(x2−w2)

⎤ ⎥

⎦∈hV(p1)i

.

By construction,

v(p, w) = max

m0,m1,m2

(v0(p0, m0) + min{v1(p1, m1), v2(p2, m2)})

st.

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

m0+m1+m2=p·w

⎡ ⎢

⎣ m1−p1·w1 m2−p2·w2

⎤ ⎥

⎦∈hV (p1)i

8_{For example, take:A}

Claim 1 Let m0(p, w),m1(p, w)andm2(p, w)denote the solution of

max m0,m1,m2

(v0(p0, m0) + min{v1(p1, m1), v2(p2, m2)})

st.

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

m0+m1+m2=p·w

⎡ ⎢

⎣ m1−p1·w1 m2−p2·w2

⎤ ⎥

⎦∈hV(p1)i

and let(p,w)_e be such thatv1(p1, m1(p,w)) =e v2(p2, m2(p,w))e . Then, for every w0>we0,

∂m0

∂w0,1

(p,(w0,we1)) = 1.

Proof. Since w0 > we0, v1(p1, m1(p,(w0,we1))) = v2(p2, m2(p,(w0,we1))). Consider a

per-turbation dw0,1 to w0,1. Notice that by construction of V (p1), dm1 > 0 =⇒ dm2 < 0 =⇒

(dv1>0anddv2<0), whereas dm2 > 0 =⇒ dm1 < 0 =⇒ (dv1<0anddv2>0), which cannot

be optimal, given thatvsis increasing in m.

Now, suppose that for every p, there exists w_e such that v1(p1, m1(p,w)) =e v2(p2, m2(p,w))e ,

and consider onlywwithw0>we0 andw1=we1.

Letfs,l(p, w)denote optimal demands. By construction, for alll,f0,l(p, w) =fl(p0, m0(p, w)),

so

∂f0,l ∂w0,1

(p, w) = ∂fl

∂m(p0, m0(p, w)) ∂m0

∂w0,1

(p, w) = ∂fl

∂m(p0, m0(p, w)),

where the second equality follows from the claim. It then follows (using the claim again) that

∂2f0,l

∂w2

0,1 (p, w) =

∂2fl

∂m2(p0, m0(p, w))and ∂

3

f0,l

∂w3

0,1 (p, w) =

∂3fl

∂m3(p0, m0(p, w)).

Now, the rank of systemf(p0, m)is, by definition, the rank of:

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

A1(p0) B1(p0) C1(p0)

A2(p0) B2(p0) C2(p0)

A3(p0) B3(p0) C3(p0)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦.

IfBlandClare zero then the system is of rank1, the utility function is homothetic and clearly

the regularity condition does not hold. IfB2(p0)C3(p0)−B2(p0)C3(p0)6= 0the system has rank

Setp0,1= 1. Then,

∂fl

∂m(p0, m) = 1 p0,l

Ã

Al(p0) +Bl(p0) log

µ

m a(p0)

¶

+Cl(p0) log

µ

m a(p0)

¶2!

+

1 p0,l

µ

Bl(p0) + 2Cl(p0) log

µ

m a(p0)

¶¶

,

∂2fl

∂m2(p0, m) =

1 p0,lm

µ

Bl(p0) + 2Cl(p0)

µ

log

µ

m a(p0)

¶

+ 1

¶¶

and

∂3_f 1

∂m3 (p0, m) =−

1 p0,l(m)2

µ

Bl(p0) + 2Cl(p0) log

µ

m a(p0)

¶¶

.

It follows that the regularity condition is satisfied if

¯ ¯ ¯ ¯ ¯ ¯ ¯ 1

p0,2m ³

B2(p0) + 2C2(p0)

³

log³ m

a(p0) ´

+ 1´´ 1

p0,3m ³

B3(p0) + 2C3(p0)

³

log³ m

a(p0) ´

+ 1´´

−p0,21m2 ³

B2(p0) + 2C2(p0) log

³

m a(p0)

´´

−p0,31m2 ³

B3(p0) + 2C3(p0) log

³

m a(p0)

´´ ¯ ¯ ¯ ¯ ¯ ¯ ¯6 = 0 ⇔ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ³

B2(p0) + 2C2(p0)

³

log³ m

a(p0) ´

+ 1´´ ³B3(p0) + 2C3(p0)

³

log³ m

a(p0) ´

+ 1´´

³

B2(p0) + 2C2(p0) log

³

m a(p0)

´´ ³

B3(p0) + 2C3(p0) log

³

m a(p0)

´´ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 6 = 0 ⇔ ¯ ¯ ¯ ¯ ¯ ¯ ¯

C2(p0) C3(p0)

³

B2(p0) + 2C2(p0) log

³

m a(p0)

´´ ³

B3(p0) + 2C3(p0) log

³

m a(p0)

´´ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 6 = 0

⇔C2(p0)B3(p0)−C3(p0)B2(p0)6= 0

A rank3system clearly satisfies this condition and, if the condition is satisfied, then the rank is

at least2.9

7

Appendix 2: Duality in Incomplete Markets

Fix an individuali, but ignore its superindex.

DefineU =©µ_∈R:¡_∃x_∈RN

++

¢

:u(x) =µª.

For each(w1, µ)∈RLS++×U, define

D(w1, µ) =©p∈S++N−1

¯

¯ ¡_∃x_∈RN₊₊¢:u(x) =µandp1¡(x1−w1)∈hV(p1)iª.

9_{For the special case considered in previous footnote, the regularity condition is equivalent to:}