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Individual excess demands
Journal of Mathematical Economics 40 (2004) 41–57
Individual excess demands P.A. Chiappori a,∗ , I. Ekeland b a
Department of Economics, University of Chicago, 5801 Ellis Avenue, Chicago, Illinois 60637, USA b University of Paris-Dauphine, France
Abstract We characterize the excess demand function in the one-consumer model. We describe the relation between the direct and indirect utility functions. As an application of our result, we derive a short proof of the celebrated Sonnenschein–Mantel–Debreu theorem on the decomposition of aggregate excess demand. © 2003 Published by Elsevier B.V. Keywords: Optimization; Homogeneous; Quasi-convex
1. Introduction In consumer theory, an individual demand function x(p, y) is defined as the solution to a simple optimization problem, it maximizes some utility function under a linear budget constraint. The properties that stem from this characterization have been known for more than one century. Under standard smoothness assumptions, individual demand functions are fully characterized by three properties: (i) homogeneity, (ii) Walras Law, and (iii) the Slutsky conditions. In addition, one can define the indirect utility and expenditure functions, and there exists a one-to-one relationship between demand, direct utility, indirect utility and expenditure functions, in the sense that the knowledge of any of these function is sufficient to uniquely recover preferences, hence the other three. The results summarized above belong by now to the most standard presentation of consumer theory. Equally standard is the fact that, for many applications (among which general equilibrium theory), it can be helpful to consider excess demand functions instead of Marshallian demands. For any Marshallian demand x(p, y), the excess demand z(p) is defined by z(p) = x(p, p ω) − ω where ω denotes the agent’s initial endowment. A very natural question is whether the standard results on Marshallian demands have a counterpart for excess demands. Specifically, can one find an analogue of the well-known Slutsky conditions, i.e. a system of equations and inequations providing necessary and sufficient condition for ∗
Corresponding author.
0304-4068/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/j.jmateco.2003.11.003
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some arbitrary function z(p) to be an individual excess demand? And does the one-to-one correspondence between the direct utility, indirect utility and demand functions extend to the excess demand framework? Quite surprisingly, these very natural questions have not received an answer so far. One can find in the literature necessary and sufficient conditions expressed in terms of the strong axiom of revealed preferences (see Mas-Colell et al., 1995). One can also find a proof that a certain set of Slutsky-like conditions are necessary (see Geanakoplos and Polemarchakis, 1980). But there is no proof that there are sufficient, and the relationship between direct utility, indirect utility and demand functions has not been investigated for excess demands. Standard consumer theory thus exhibits a gap. The goal of this paper is precisely to fill this gap. We first derive a set of necessary and sufficient conditions that fully characterize excess demand functions. Whenever a smooth function z(p), defined on a compact subset of Rn++ , satisfies these conditions, then it is possible to find an initial endowment and a direct utility function for which z(p) is the excess demand. These conditions, somewhat unsurprisingly, are (i) homogeneity, (ii) Walras Law, and (iii) a variant of the Slutsky conditions; however, the proof of sufficiency (the so-called integrability problem) cannot be transposed from the standard framework, and one has to find a different (and more complex) argument. A second result is that the one-to-one relationship between the direct utility, indirect utility and demand functions that characterizes Marshallian demands is lost in the excess demand context. While it is always possible, starting from some direct utility function (and some initial endowment), to uniquely define the corresponding, indirect utility function, the converse is not true. To any indirect utility function can be associated a continuum of different excess demand functions z(p), each of which satisfies the necessary and sufficient conditions for integrability. Furthermore, for any excess demand function z(p), there exist a continuum of different direct utility functions from which z(p) can be derived. Finally, our results have direct consequences for several problems in consumer theory and aggregation. In the paper, we reconsider the well-known Debreu–Mantel–Sonnenschein theorem on aggregate excess demand. The question, here, is whether it is possible, for any arbitrary, smooth function Z(p) that satisfies homogeneity and the Walras Law, to find n individual excess demand functions z1 (p), . . . , zn (p), such that Z(p) decomposes into the sum of the zi (p). Surprisingly, the existing results do not rely on a formal characterization of individual excess demand functions. Our characterization of individual excess demand yields a three-line proof of the theorem. Not only is this proof short, it also allows a precise characterization of the degrees of freedom one has in constructing the individual utilities. Specifically, we show that there is broad discretion in choosing the indirect utilities V i , and that the choice of the individual excess demands zi corresponding to the chosen indirect utilities is then unique. Moreover, the V i can be chosen independently of the aggregate function at stake, that is, the same functions V i can be used for the decomposition of any function as an aggregate excess demand. Our result thus suggests that the characterization of individual excess demand is indeed a difficult problem, as a matter of fact, it constitutes the core difficulty in the Debreu– Mantel–Sonnenschein excess demand problem. It also provides a better understanding of the fundamental difference between the two versions (market demand versus excess demand) of
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Sonnenschein’s problem, and allows to assess in which sense the market demand problem is considerably more difficult.
2. Individual excess demand 2.1. The necessary conditions Consider a standard consumer model. There are n goods, and the consumer is characterized by a utility function U : Rn → R. The excess demand of the consumer is then defined as the solution z(p) of the individual maximization problem:1 maxz U(z) (IM) p z = pi zi ≤ 0 We shall say that a Ck function U, with k ≥ 2, is standard if: • The gradient Dz U(z) is non-zero everywhere, and the restriction of the second deriva2 U(z) to [D U(z)]⊥ is negative definite for every z; it follows that U is strictly tive Dzz z quasi-concave. • For every p ∈ Rn++ , problem (IM) has a solution z(p) ∈ Rn with p z = 0. Note that the two conditions are independent (neither one implies the other). If U is standard, it follows from the implicit function theorem that the excess demand function z(p) is well-defined and Ck−1 on Rn++ . It is zero-homogeneous and satisfies the Walras Law: z(λp) = z(p) ∀λ > 0, ∀p p z(p) = 0 ∀p and there is a Lagrange multiplier λ(p), which is Ck−1 and homogeneous of degree (−1) on Rn++ , such that Dz U(z(p)) = λ(p)p
(1)
Define the indirect utility V(p), for p ∈ Rn++ , by V(p) = {max U(z)|z ∈ Rn , p z ≤ 0} z
1
In more standard notations, z(p) is defined as x(p, p ω) − ω, where the Marshallian demand x solves ˜ maxx U(x) p x = p ω
Then U is defined from U˜ by ˜ + ω) U(z) = U(z
(IU)
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If U is standard, then V is Ck−1 and satisfies V(p) = U(z(p)). In addition, V(p) is zero-homogenous and quasi-convex. Note that V cannot be strictly quasi-convex because of homogeneity (the level sets {V ≤ a} are cones). However, it can be proved that, if V is 2 V(p) has rank (n − 1) and the restriction of D2 V(p) to [Span{p, D V(p)}]⊥ C2 , then Dpp p pp is positive definite. Also, p Dp V(p) = 0, by the Euler identity, so p and Dp V(p) are orthogonal vectors; if none of them vanishes, their span is truly two-dimensional. The envelope theorem applied to (IU) gives Dp V(p) = −λ(p)z(p)
(2)
Differentiating with respect to p, one gets: 2 V = −λDp z − z(Dp λ) Dpp
(3)
It follows that the restriction of Dp z to [z(p)]⊥ is symmetric, and that its restriction to [Span{p, z(p)}]⊥ = [Span{p, Dp V(p)}]⊥ is negative definite for every p. This condition is the exact equivalent, in the excess demand case, of the fact that the Slutsky matrix for a Marshallian demand is symmetric and negative definite. Note also that if p¯ is, such that z(p) ¯ = 0, then, by Eq. (3), Dp z(p) ¯ is proportional to 2 V(p). ¯ In particular, the matrix Dp z(p) Dpp ¯ is then symmetric and negative semi-definite. Another property of indirect utilities is the following: Lemma 1. If Dp V(p1 ) and Dp V(p2 ) are collinear, then so are p1 and p2 . Proof. If Dp V(p1 ) and Dp V(p2 ) are collinear, then so are z(p1 ) and z(p2 ) by relation (2), so that p 1 z(p2 ) = p 1 z(p1 ) = 0. Set z(p1 ) = λ1 z¯ and z(p2 ) = λ2 z¯ . Since z(p1 ) solves problem (IM) for p = p1 , we have U(z(p1 )) ≥ U(z(p2 )). The converse inequality holds for the same reason, so U(z(p1 )) = U(z(p2 )). Since U is standard, the solution of problem (IM) is unique, so z(p1 ) must be equal to z(p2 ), and p1 , which is collinear to Dp U(z(p1 )) = Dp U(z(p2 )) must be collinear to p2 . 䊐 We summarize these results. They can be found in the existing literature (except, apparently, for the preceding lemma), for instance in Geanakoplos and Polemarchakis (1980). Proposition 2 (Necessary conditions). If the Ck function U(z) is standard, the indirect utility function V : Rn++ → R and the excess demand function z : Rn++ → Rn satisfy the following: 2 V(p) (a) V is Ck−1 , quasi-convex, positively homogenous of degree zero. For every p, Dpp 2 V(p) to [Span{p, D V(p)}]⊥ is positive has rank (n−1); moreover, the restriction of Dpp p definite. If Dp V(p1 ) and Dp V(p2 ) are collinear, so are p1 and p2 . (b) z is Ck−1 and positively homogenous of degree zero. The restriction of Dp z to [z(p)]⊥ is symmetric, and its restriction to [Span{p, z(p)}]⊥ is negative definite for every p.
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2 V(p) and the negativity condition on D z are Note that the positivity condition on Dpp p void if n = 2, but become significant for n ≥ 3.
2.2. Local integrability We now consider the converse. Take some C2 functions V(p) and z(p) satisfying the necessary conditions. Is it possible to find a standard utility function U, such that z(p) solves (IM) for all p, and V(p) is the corresponding, indirect utility? This is the standard ‘integration’ problem in consumer theory, expressed in the case of excess demand. It divides into two subproblems: the local one, where V and z are considered in a suitably small neighborhood of a given point p, ¯ and the global one, where V and z are considered on some arbitrary compact set in the interior of Rn++ . In this section, we address the local problem, and we show that the answer is positive, provided Dp V(p) ¯ = 0 and a nondegeneracy condition is met. The situation near a point p¯ where Dp V(p) ¯ = 0 is mathematically quite interesting, but will not be investigated in this paper. Theorem 3. Let V(p) be a Ck function, k ≥ 3, defined on some neighborhood of p¯ with ¯ = 0. Assume that, on that neighborhood, V(p) is quasi-convex, positively hoDp V(p) 2 V(p) has rank (n − 1) and the restriction of D2 V(p) mogenous of degree zero, that Dpp pp to [Span{p, Dp V(p)}]⊥ is positive definite. Then there is an open convex cone Γ ⊂ Rn++ containing p, ¯ such that V is an indirect utility function on Γ . In fact, take any Ck−1 function λ(p) > 0, homogeneous of degree (−1) on Γ , and define: z(p) = −
1 Dp V(p) λ(p)
(4)
Then there is a quasi-concave function U(z), defined and Ck−1 on a convex neighborhood N of z(p), ¯ with the following properties: 2 U(z) to [D U(z)]⊥ is negative definite. 1. For every z ∈ N, the restriction of Dzz z 2. For every p ∈ Γ , we have
p z(p) = 0
Dz U(z(p)) = λ(p)p,
V(p) = U(z(p)) = {max U(z)|p z ≤ 0, z ∈ N} z
(5) (6)
The proof will be given in the appendix. One can derive a similar result for excess demands: Theorem 4. Let z(p) be a Ck map, k ≥ 2, defined on some neighborhood of p¯ into Rn , such that z(p) ¯ = 0 and Dp z(p) ¯ has rank (n − 1). Assume that, on that neighborhood, z is homogeneous of degree zero, that the restriction of Dp z to [p]⊥ is symmetric, and that its restriction to [Span{p, z(p)}]⊥ is negative definite. Then there is an open convex cone Γ ⊂ Rn++ containing p, ¯ such that z is an excess demand function on Γ . In fact, there is a quasi-concave function U(z), defined and Ck on a convex neighborhood N of z(p), ¯ with the following properties:
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2 U(z) to [D U(z)]⊥ is negative definite 1. For every z ∈ N, the restriction of Dzz z 2. For every p ∈ Γ , we have p z(p) = 0 and:
Dz U(z(p)) = λ(p)p for some λ(p) > 0
(7)
U(z(p)) = {max U(z)|p z ≤ 0, z ∈ N}
(8)
z
The proof will be given in the appendix. Several remarks are in order here. First, unlike the standard case of market demand, one has existence but not uniqueness. This follows from the fact that, while U is a function of n variables, V and z are functions of n − 1 variables only (one dimension being lost because of homogeneity), information is lost in going from U to V and z, and cannot be recovered, in sharp contrast with the Marshallian case. Specifically • For each excess demand function z(p), there exist a continuum of direct utility functions U for which z is the excess demand. The idea is that U is characterized only on the subset M ⊂ Rn generated by z(p) when p varies in Ω ⊂ S n−1 ; in general, M is a submanifold of codimension 1 in Rn . The function U and its derivatives are known only on M, and must be extended outside this set; obviously, this extension can be made in many different ways. • Perhaps more surprisingly, to any homogenous, strictly quasi-convex indirect utility V(p) can be associated a continuum of excess demand functions: just pick up some arbitrary, positive scalar function λ(p), then z(p) = −Dp V(p)/λ(p) is an excess demand. Note that this conclusion can also be viewed as a consequence of a previous result obtained by Debreu, and stating that for any individual excess demand z(p) and any positive scalar function µ(p), the function z˜ (p) = µ(p). z(p) is also an individual excess demand function. It should also be noted that the proof cannot follow the usual path, based on duality. The standard approach, as in Hurwicz and Uzawa (1971), relies on the expenditure function, which has no natural equivalent in the case of excess demand. Similarly, proofs based on indirect utilities (i.e. integration of Roy’s identity) cannot be used here, precisely because the one-to-one correspondence between direct and indirect utilities does not hold in this context. Hence, although the statement of the result is quite similar to the Slutsky characterization of market demand, its proof relies on a completely different (and somewhat more difficult) approach.
3. Global integrability We now address the global problem. Take a C2 function V : Rn++ → R satisfying condition (a) in Proposition 2. We associate with it a function V ∗ : Rn → R defined as follows: V ∗ (z) = inf {V(p)|p ∈ Rn++ , p z ≥ 0} p
(IU*)
The function V ∗ : Rn → R ∪ {−∞} is quasi-concave, positively homogeneous of degree 0, and its domain, domV ∗ = {z|V ∗ (z) > −∞}, is a convex set. However, in contrast with
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the Marshallian demand problem, V ∗ is not a candidate for a direct utility function. Indeed, it is zero homogenous, hence the level sets {V ∗ ≥ a} are cones and the maximization of V ∗ under a linear constraint cannot generate an excess demand function. Still, the function V ∗ will play a key role in the derivation of our result. Introduce the cone: C = {µDp V(p)|µ < 0, p ∈ Rn++ }
(9)
The optimality condition in problem (IU∗ ) is Dp V(p) = µz, with µ < 0. If z¯ = µDp V(p), ¯ for some µ < 0, then p¯ z¯ = 0 by the Euler identity, and the minimum in ¯ In other words, C ⊂ domV ∗ , and if z ∈ C, problem (IU∗ ) with z = z¯ is attained at p = p. so that z = µDp V(p) for some µ < 0 and p ∈ Rn++ , we must have V ∗ (z) = V(p)
(10)
Theorem 5. Let V : Rn++ → R be a a C2 function satisfying condition (a) of Proposition 2. 2 V(p) has Assume that V(p) is quasi-convex, positively homogenous of degree zero, that Dpp 2 ⊥ rank (n − 1) and the restriction of Dpp V(p) to [Span{p, Dp V(p)}] is positive definite. Given any compact subset K in Rn++ where Dp V does not vanish, then V is an indirect utility function on K. In fact, take any C1 function λ(p) > 0, homogeneous of degree (−1) on a neighborhood of K, and define: 1 z(p) = − Dp V(p) (11) λ(p) M = {z(p)|p ∈ K}
(12)
Then there is a quasi-concave function U(z), defined and containing M, with the following properties:
C2
on a convex open set Ω
2 U(z) 1. U is C2 on a neighborhood N of M, and for every z ∈ N, the restriction of Dzz ⊥ to [Dz U(z)] is negative definite 2. For every p ∈ K, the point z(p) is the unique solution of problem (IU), so that:
p z(p) = 0
Dz U(z(p)) = λ(p)p,
V(p) = U(z(p)) = {max U(z)|p z ≤ 0, z ∈ Rn } z
(13) (14)
4. The Sonnenschein–Mantel–Debreu theorem Let Z(p) be a C∞ map from some open cone Ω of Rn+ into Rn , homogeneous of degree zero and satisfying the Walras law p Z(p) = 0. The problem raised by Sonnenschein (1973) is the following: for any k ≥ n, is it possible to find k excess demand functions z1 (p), . . . , zk (p), such that: Z(p) = z1 (p) + · · · + zk (p)
(15)
As it is well known, the answer to this question is positive (see Sonnenschein (1974), Mantel (1974), Debreu (1974), and Shafer and Sonnenschein (1982) for a survey). We
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provide a very short proof. Note that we require that Z be C2 , so that our result is more in the Mas-Colell (1977) line. Following the literature, we normalize prices by imposing that p p = 1, so that prices belong to Rn++ ∩ S n−1 . Since p Z(p) = 0, we can think of Z(p) as a tangent vector to S n−1 at p; the space of all such vectors (the tangent space Tp S n−1 to S n−1 at p) has dimension (n − 1). Also, in what follows K denotes some given compact subset included in the interior of Rn++ ∩ S n−1 . Theorem 6. Pick up k ≥ n arbitrary functions V1 (p), . . . , Vk (p), of class C3 , satisfying condition (a) of Proposition 2, and, such that at every p ∈ K, the set of linear combination of the DVi (p) with non negative coefficients is Tp S n−1 . For any C2 map Z(p) defined on K, homogeneous of degree zero and satisfying the Walras law p Z(p) = 0, and for any k ≥ n, one can find k excess demand functions z1 (p), . . . , zk (p), defined on K, such that the decomposition(15) holds on K and the indirect utility associated to zi is Vi , i = 1, . . . , k. The corresponding, direct utility functions U1 (z), . . . , Uk (z) are quasi-concave, and each Ui is C2 and strictly quasi-concave in a neighborhood of zi (K). Proof. One can find C2 functions µ1 < 0, . . . , µk < 0 on K, such that Z(p) =
k
µi (p)Dp Vi (p)
i=1
Define zi (p) = −µi (p)Dp Vi (p). By Theorem 5, zi is an individual excess demand 䊐 function. The simple intuition of the proof is then that any function Z(p) obtains as a linear combination of the −DVi (p) with non negative coefficients; by Theorem 5, each term in the combination is an individual excess demand. A quite remarkable property is that the same functions Vi can be used for all aggregate demands Z. Note that this proof does not require the full force of our result, but only the fact that if z(p) is an excess demand function, then so is µ(p)z(p) for any µ(p) > 0, a fact that, as we noted above, was already known to Debreu. However, we have been unable to find this proof in the literature. We give two examples of functions V1 (p), . . . , Vn (p), satisfying the conditions of Theorem 6. Note first that the two following conditions are equivalent 1. The convex cone generated by n vectors (e1 , . . . , en ) in Rn−1 is Rn−1 . 2. The convex hull of (e1 , . . . , en ) contains 0. Or first example consists of the functions 2 2 pj pi Vi (p) = −1 + k pk k pk j=i
The Vi satisfy condition (a) of Proposition 2. Consider the simplex Σ = p ∈ Rn++ p1 = 1
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with vertices P1 , . . . , Pn , and denote by ViΣ the restriction of Vi to Σ. Then ViΣ is just the square of the distance from p ∈ Σ to the ith vertex Pi , so that the gradient DVΣ i (p) is directed from p towards Pi . It follows that, for every p in the interior of Σ, the convex Σ hull of the DVΣ i (p) in Σ contains 0 in its interior. Since DVi (p) is the image of DVi (p) by ⊥ the orthogonal projection [p] → Σ, it follows that the convex hull of the DVi (p) in [p]⊥ contains 0 in its interior. So the convex cone spanned by the DVi (p) is Tp S n−1 . Another example, starting from Cobb–Douglas utilities, is given in the appendix. As a corollary of Theorem 6, we then get the celebrated result of Sonnenschein, Mantel and Debreu. It should finally be noted that our approach does not easily allow to characterize the class of ordinal utility functions that generate a given excess demand function. While the topic is left for future research, it is worth noting, in particular, that Mantel’s celebrated (1976) result on the decomposition of aggregate excess demand as sum of individual excess demands stemming from homothetic utility functions is not a direct consequence of Theorem 6. 4.1. Comments An interesting by-product of this proof is that it describes the degrees of freedom available in the choice of individual utility functions. Basically, indirect utilities can be picked almost arbitrarily (the only constraint being that Z(p) belongs to their negative span). However, once they have been chosen, then only one set of λi (p) is acceptable; in other words, among the continuum of individual excess demands that can be associated with each Vi , only one can be used. Then any of the direct utilities corresponding to the zi can be adopted. Note, however, that we have been unable to prove by this method the beautiful result by Mantel (1976), who showed that the individual preferences can be taken to be homothetic. Note also that our method requires that Z(p) be C2 , whereas other proofs work with continuity. We conclude by noting that the previous construct sheds light on the differences between the two problems raised by Sonnenschein, i.e. aggregate excess demand versus aggregate market demand. In the market demand case, a given function X(p) must be decomposed as the sum of n individual Marshallian demands: X(p) = x1 (p) + · · · + xn (p) where xi (p) is the solution of Vi (p) = maxUi (xi ) p xi = 1, xi ≥ 0 In particular, the envelope theorem gives Dp V i (p) = −αi (p)xi (p), where αi is the Lagrange multiplier. It follows that X(p) = −
1 1 Dp Vn (p) Dp V1 (p) − · · · − n α (p) α1 (p)
Again, X(p) must be a convex combination of n gradients. The difference, however, is that while the coefficients of the combination could be freely chosen in the excess demand
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case, in the market demand context they have to satisfy the budget constraint, which implies that αi (p) = −
p D
1 p Vi (p)
∀i
These additional restrictions considerably increase the difficulty of the problem, which now amounts to a system of nonlinear partial differential equations: Dp V1 (p) Dp Vn (p) + ··· + = −X(p) p Dp V1 (p) p Dp Vn (p)
(16a)
This system has been solved in Chiappori and Ekeland (1999), in the case when the right-hand side X(p) is real analytic. The case when X(p) is smooth, C∞ say, but not real analytic, remains open. It is also an open question whether solutions to (16a) can be found globally: the mathematical technique used in Chiappori and Ekeland (1999) only constructs solutions in some neighbourhood of any given point p. ¯
Acknowledgements We thank P. Reny, R. Guesnerie and H. Sonnenschein for useful comments. Financial support from the NSF (grant SBR 9729559) is gratefully aknowledged. Errors are ours.
Appendix A. Proof of Theorem 3 A.1. Step 1: Constructing UA Define z(p) from (4). Then z(p) is Ck−1 , and by the Euler identity, we have: 1 p z = − p Dp V = 0 λ so that z satisfies the Walras law. Differentiating, we get: p Dp z + z = 0
(A.1)
In addition, z(p) is zero-homogeneous of degree zero, so that: (Dp z)p = 0
(A.2)
Without loss of generality, assume that p ¯ = 1. It follows from Eqs. (A.1) and (A.2) that p ∈ KerDp z(p) and z(p) ∈ Range[Dp z(p)] , and it follows from relation (3) that Dp z(p) has rank (n − 1). Consider the map φ defined on S n−1 × R by φ(p, t) = tz(p). We have: Dp,t φ(p, 1) = (Dp z(p), z(p))
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We claim that z(p) ∈ / Range [Dp z(p)], so that Dp,t φ(p, 1) has rank n. Indeed, assume otherwise, so that we have: z(p) = Dp z(p)ξ
(A.3)
for some ξ. Substituting (A.3) into (A.1) yields p z = z ξ = 0 by the Walras law. So ξ ∈ [z(p)]⊥ . Taking another vector η ∈ [z(p)]⊥ , we have: η z(p) = η (Dp z(p)ξ) = 0 But we already know that η (Dp z(p)p) = 0 for all η ∈ [z(p)]⊥ , with p ∈ [z(p)]⊥ . Since the restriction of Dp z(p) to [z(p)]⊥ is symmetric and has rank (n − 1), it follows that ξ = αp for some scalar α, and Dp z(p)ξ = 0 by (A.2), which is the desired contradiction. So Dp,t φ(p, 1) has rank n, and it follows from the inverse function theorem that φ is a Ck−1 diffeomorphism from some neighborhood N×[1 − ε, 1 + ε] of (p, ¯ 1) in S n−1 × R n onto a neighborhood of z(p) ¯ in R . In particular, the image M = {z(p)|p ∈ N} = {φ(p, 1)|p ∈ N} is diffeomorphic to N. In particular, if we pose ˜ U(p, t) = U[φ(p, t)] then defining U on M is equivalent to defining U˜ on N, and ˜ Dp,t U(p, 1) = Dz U ◦ (Dp z, z) We want a function U(z), such that U(z(p)) = V(p) and Dz U(z(p)) = λ(p)p. In other words, we are prescribing the values of U and its first derivatives on M. There is a compatibility condition to be satisfied, namely that we get the same value for Dp U(z(p)). This yields Dp V = λ(Dp z) p Substituting relations (A.1) and (4), we find an identity, so the compatibility condition ˜ holds. Now, let us look at the problem in the (p, t)-coordinates. We know U(p, 1) and the ˜ ˜ ˜ gradient Dp,t U(p, 1) (actually, the knowledge of U(p, 1) fully defines Dp U(p, 1) under the compatibility conditions checked above, so the knowledge of the gradient only pins ˜ ˜ down the partial Dt U(p, 1)). What we want is to define U(p, t) in a neighborhood of t = 1. This can be done in many ways, for instance by setting U˜ A (p, t) = V(p) + (t − 1)A(p, t)
(A.4) ˜
k−1 function, such that A(p, 1) = ∂U (p, 1). In particular, where A : Rn−1 ++ ×R → R is any C ∂t if we evaluate the situation at p, ¯ we find that the matrix of second derivatives takes the form a1,n 2 U ˜A ... Dpp D2 U˜ A (p, (A.5) ¯ 1) = an−1,n an,1 . . . an,n−1 ann
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where the n terms an,i = ai,n , 1 ≤ i ≤ n, can be chosen freely. We now show that if A, and hence the ai,n , are appropriately chosen, the extension UA = U˜ A ◦ φ−1 is quasi-concave. A.2. Step 2: UA is strictly quasi-concave 2 U (z) on the tangent space T M A.2.1. Substep 2a: investigating Dzz A z We have U = UA on M. Differentiating the relation Dz UA (z(p)) = λ(p)p, we get: 2 Dzz UA (z)Dp z = pDp λ + λI
and hence, for any ξ ∈ Tp S n−1 : 2 UA (z)(Dp zξ) = (p Dp zξ)(Dp λξ) + λ(ξ Dp zξ) (Dp zξ) Dzz
Now, by relation (A.1), p Dp zξ = −z ξ. So the first term on the right vanishes if Dp zξ ∈ [p]⊥ , or, equivalently, by relation (A.1), if ξ ∈ [z(p)]⊥ . We are then left with the second term, which we compute: 1 2 1 ξ Dp zξ = ξ − Dpp V + 2 Dp V(Dp λ) ξ λ λ 1 2 1 = − ξ Dpp Vξ + 2 (ξ , Dp V)(ξ, Dp λ) λ λ 1 2 1 = − ξ Dpp Vξ − (ξ , z)(ξ, Dp λ) λ λ and the last term on the right vanishes again if z ξ = 0. So we are left with the following: 1 2 2 ξ ∈ Tp S n−1 , z ξ = 0 ⇒ (Dp zξ) Dzz UA (z)(Dp zξ) = − ξ Dpp Vξ λ and the quadratic form on the right-hand side is negative definite on [Span {p, z(p)}]⊥ , by condition (a). Since ξ ∈ Tp S n−1 , we have (ξ, p) = 0, and so: 0 = ξ ∈ Tp S n−1 ,
2 z ξ = 0 ⇒ (Dp zξ) Dzz UA (z)(Dp zξ) < 0
Set η = Dp zξ. Note that Dz UA (z)η = Dz UA Dp zξ = Dp Vξ = −λz ξ, so that z ξ = 0 if and only if Dz UA (z)η = 0. Rewrite the preceding result in terms of η: 0 = η ∈ Tp M,
2 Dz UA (z)η = 0 ⇒ η Dzz UA (z)η < 0
2 U (z) on the whole space A.2.2. Substep 2b: investigating Dzz A >From U˜ A = UA ◦ ϕ we get, at z = ϕ(p, t): 2 2 (Dz ϕ) Dzz UA (z)Dz ϕ = D2 U˜ A (p, t) − Dz UA (z)Dzz ϕ
Note that the last term, which involves only first derivatives of UA , does not depend on the choice of A, provided it is evaluated at a point z ∈ M; this is because of the relations
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UA (z(p)) = V(p) and Dz U(z(p)) = λ(p)p. Evaluating everything at z¯ = φ(p, ¯ 1), the above equation takes the form: 2 UA (¯z)Dz ϕ = D2 U˜ A (p, ¯ 1) + M (Dz ϕ) Dzz
(A.6)
where M is a fixed operator and D2 U˜ A (p, ¯ 1) is given by formula (A.5). We have shown, in 2 U (z) to T M∩[p]⊥ is negative definite. the preceding substep, that the restriction of Dzz A z 2 ¯ 1) + M to E ∩ F is negative definite, where This means that the restriction of D U˜ A (p, ¯ 1)]−1 (Tz M) and F = [Dz ϕ(p, ¯ 1)]−1 ([p]⊥ ). From the definition of ϕ, it E = [Dz ϕ(p, follows that E is simply the hyperplane t = 0. Going back to formula (A.5), we find that: a1,n + m1,n ... Q ¯ 1) + M = D2 U˜ A (p, an−1,n + mn−1,n a1,n + m1,n . . . an−1,n + mn−1,n an,n + mn,n where the restriction of Q to F ∩ E is negative definite. It is then easy to pick the ai,n so that the restriction of D2 U˜ A (p, ¯ 1)+M to F is negative definite. Going back to Eq. (A.6), we find 2 U (¯ ¯ ⊥ is negative definite. Since [p] ¯ ⊥ is just [Dp UA (¯z)]⊥ , that the restriction of Dzz A z) to [p] this means that UA is strictly quasi-concave. Appendix B. Proof of Theorem 4 Proof. Introduce the differential one-form ω= zi (p)dpi Since the restriction of Dp z to [z(p)]⊥ is symmetric, we must have ω ∧ dω = 0. By the Darboux theorem, there are Ck+1 functions µ(p) and V(p), defined in some neighborhood of p, ¯ such that ω = −µdV. By the convex Darboux theorem (see Chiappori and Ekeland (1997), Ekeland and Nirenberg (2002)), and the references therein; the proof has to be adapted to the homogeneous case, which is straightforward), we can take the functions µ 2 V positive definite on [p]⊥ . and V to be homogeneous, with µ > 0 and Dpp We can therefore apply Theorem 3, taking λ(p) = 1/µ(p). We find a Ck utility function U satisfying (a) and (b), the corresponding excess demand function being precisely z(p).䊐
Appendix C. Proof of Theorem 5 Pick up an arbitrary positive function λ. By Theorem 3, for every point z ∈ M there is a neighborhood Nz ⊂ C and a C2 function Uz : Nz → R satisfying conditions (1) and (IU) in that neighborhood. Since M is compact, it can be covered by finitely many such neighborhoods, say N1 , . . . , Nm , associated with the functions U1 , . . . , Um . Now, take a smooth partition of unity associated with this covering, that is, m functions ϕ1 , . . . ., ϕm of class C2 , such that ϕi ≥ 0, ϕi (z) = 0 if z ∈ / Ni and ϕi = 1. Set
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U = ϕi Ui ; the function U then is defined over ∪Ni , which is a neighborhood of M. Let us show that it satisfies conditions (1) and (IU) in that neighborhood. Looking at formula (9), which defines the cone C, we find that M ⊂ C. If z ∈ M, then z = z(p) for some p, and ϕi V(p) = V(p) = V ∗ (z) U(z) = ϕi Ui = In addition, for any z(p) ∈ M we have that DUi = λ(p)p, so that: Dz ϕi V(p) = λp ϕi Dz Ui + U i D z ϕi = ϕi λp + Dz U = We summarize: z = z(p) ∈ M ⇒ [U(z) = V ∗ (z)
and
Dz U(z) = λ(p)p]
(C.1)
We now claim that: / M, z ∈ ∪Ni , z ∈
p z = 0 ⇒ U(z) < V ∗ (z)
(C.2)
/ M. Since z ∈ ∪Ni , the value U(z) is To see this, pick a point z ∈ ∪Ni with z ∈ well-defined. If p is, such that p z = 0, we must have U(z) < U(z(p)) = V(p), otherwise z would be the maximizer of problem (IM), and hence z = z(p) ∈ M, a contradiction. On the other hand, since z ∈ ∪N ⊂ C, we must have z = µDp V(p) for some µ < 0 and p, and V ∗ (z) = V(p), as noted in relation (10). So U(z) < V ∗ (z), as announced. In other words, although V ∗ is not a direct utility, it provides an upper envelope, a fact that will be crucial to prove global concavity. Lemma 7. There is a neighborhood N of M, with N ⊂ ∪Ni , such that the restriction of U to N is quasi-concave: z = αz1 + (1 − α)z2 , 0 < α < 1, z1 ∈ N,
z2 ∈ N, ⇒ U(z) ≥ min{U(z1 ), U(z2 )} (C.3)
Proof. Argue by contradiction. If condition (B.3) does not hold, there must be three sequences zk1 , zk , zk2 , k → ∞, with the following property: zk1 → z1 ∈ M⇔zk → z ∈ M, zk = (1 − αk )zk1 + αk zk2 , U(zk ) < min{U(zk1 ),
zk2 → z2 ∈ M
0 < αk < 1
U(zk2 )}
(C.4) (C.5) (C.6)
There are three possible cases: 1. The three points z1 , z, z2 are distinct. Then we get U(z) ≤ min{U(z1 ), U(z2 )} in the limit. Since U = V ∗ on M, and V ∗ is quasi-concave, the reverse inequality holds as well, and we get V ∗ (z) = min{V ∗ (z1 ), V ∗ (z2 )}. Without loss of generality, assume V ∗ (z1 ) ≤ V ∗ (z2 ), so that V ∗ (z) = V ∗ (z1 ). From the quasi-concavity of V ∗ it follows that V ∗ (z1 + t(z−z1 )) = V ∗ (z1 ) for all t ∈ [0, 1]. Differentiating once, we get Dz U(z1 )(z2 −z1 ) = 0, 2 U(z )(z −z ) = 0. so (z2 −z1 ) ∈ [Dz U(z1 )]⊥ . Differentiating twice, we get (z2 −z1 ) Dzz 1 2 1 2 ⊥ But Dzz U(z1 ) is positive definite on [Dz U(z1 )] , and we have a contradiction.
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2. Two of the points z1 , z, z2 coincide. The middle one, z, must be one of them; without loss of generality, say the other one is z1 . So z = z1 = z2 . Condition (B.6) then yields U(z1 ) ≤ U(z2 ),
(C.7)
and condition (C.5) yields αk → 0. Substituting into condition (C.6), we get Dz U(z1 )(z2 − z1 ) ≤ 0,
(C.8)
If the strict inequality holds in (C.8), then there is a point z3 between z1 and z2 where U(z3 ) < U(z1 ) ≤ U(z2 ), contradicting the quasi-concavity of U. If Dz U(z1 )(z2 − 2 U(z ) is negative definite on [D U(z )]⊥ , again there will be z1 ) = 0, then, since Dzz 1 z 1 a point z3 between z1 and z2 where U(z3 ) < U(z1 ) ≤ U(z2 ), and again we derive a contradiction. 3. The three points coincide: z1 = z = z2 . But then, for k large enough, the points zk1 , zk , zk2 enter a convex neighborhood of z where U is strictly quasi-concave, contradicting relation (C.6). 䊐
¯ defined on a convex set containing N, We now extend U to a quasi-concave function U, and coinciding with U on N. This is an easy matter: for any number a, consider the set Ω(a) = [z ∈ N|U(z) > a] ⊂ Rn and let co[Ω(a)] be its convex hull. It is an open and convex subset of Rn++ . If a > b, then co[Ω(a)] ⊂ co[Ω(b)]. Set Ω = ∪co[Ω(a)], which is a convex open set, and define a function U¯ : Ω → R by: ¯ U(z) = sup{a|z ∈ co[Ω(a)]} It is quasi-concave by construction. The Lemma implies that if z ∈ N and z ∈ co[Ω(a)], ¯ then z ∈ Ω(a), so U(z) = U(z). This concludes the proof. Appendix D. Direct utilities: an explicit example Let K be a compact subset included in the interior of Rn++ ∩ S n−1 . We give an example of n arbitrary functions V1 (p), . . . , Vn (p), of class C3 , on Rn++ , satisfying condition (a) of Proposition 2, and, such that, at every p ∈ K, the set of linear combination of the DVi (p) with non negative coefficients is Tp S n−1 . Our construction, starting from Cobb–Douglas utility functions, is based on Balasko (1988) (see also Mas-Colell et al., 1995, p. 55). Define, for 1 ≤ j ≤ k and 1 ≤ i ≤ n: Uj (x) = αj,i log xi with i αj,i = 1. The corresponding excess demand is zj (p), with: zij (p) = αj,i
p ω j − ωji pi
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and the indirect utility Vj (p) =
αj,i log
i
αj,i
p ωj pi
= log (p ωj ) −
αj,i log pi +
i
αj,i log αj,i
i
(D.1) j
Let us first take α¯ j,i = 1 if i = j, and 0 otherwise. Assume moreover that ω¯ ji = ω¯ i for all i, j. Then: p ω¯ j j − ω¯ j if i = j i pj zj (p) = −ω¯ ji otherwise Since the zj (p), 1 ≤ j ≤ n, all belong to the (n − 1)-dimensional space Tp S n−1 = [p]⊥ , j the rank of the system is at most (n − 1). By an appropriate choice of the ω¯ ji = ω¯ i , we may assume that this rank is exactly (n − 1). Proceeding with the calculations, we have: j
pj zij (p) = pi
p ω¯ i j − pi ω¯ ii − pj ω¯ ji = p ω¯ i − pj ω¯ ji = p ω¯ i − pj ω¯ i = 0 pi j=i
j
j
In other words, with these particular choices of αj,i = α¯ j,i and ωji = ω¯ ji , we have ¯ the family α¯ j,i , 1 ≤ j pj zj (p) = 0 for all p. To shorten notations, let us denote by α i, j ≤ n, and by ω¯ the family ω¯ ji , 1 ≤ i, j ≤ n. By the implicit function theorem, for every ε > 0, there is a neighborhood N of (α, ¯ ω), ¯ such that, for all (α, ω) ∈ N, the corresponding demand functions zj are, such that the family zj (p), 1 ≤ j ≤ n, has rank (n − 1), and there is a vanishing linear combination qj (p)zj (p) = 0 j
with |qj (p) − pj | < ε. In particular, ε can be chosen so small that qj (p) > 0 for all p ∈ K. It follows that the positive cone spanned by the zj (p) coincides with the linear subspace they span, namely Tp S n−1 . In other words, for all (α, ω) ∈ N, the functions Vj (p) given by (D.1) answer the question.
References Balasko, Y., 1988. Foundations of the Theory of General Equlibrium, Academic Press. Chiappori, P.A., Ekeland, I., 1999. Aggregation and market demand: an exterior differential calculus viewpoint. Econometrica 67 (6), 1435–1458. Chiappori, P.A., Ekeland, I., 1997. A Convex Darboux Theorem. A nnali della Scuola Normale Superiore di Pisa 4.25, 287–297. Debreu, G., 1974. Excess demand functions. Journal of Mathematical Economics 1, 15–23. Ekeland, I., Nirenberg, L., 2002. The Convex Darboux Theorem. Methods and Applications of Analysis 9 (3), 329–344.
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Geanakoplos and Polemarchakis, 1980. On the disaggregation of excess demand functions, Econometrica 48 (2), 315–331. Hurwicz and Uzawa, 1971. On the integrability of demand functions. In: Chipman, J., Hurwicz, L., Sonnenschein, H. (Eds.), Utility and Demand, Harcourt Brace Jovanovich, New York (Chapter 6). Mantel, R., 1974. On the characterization of aggregate excess demand. Journal of Economic Theory 7, 348–353. Mantel, R., 1976. Homothetic preferences and community excess demand Functions. Journal of Economic Theory 12, 197–201. Mas-Colell, A., Whinston, M., Green, J., 1995. Microeconomic Theory, Oxford University Press, Oxford. Mas-Colell, A., 1977. On the equilibrium price set of an exchange economy, Journal of Mathematical Economics, 117–126. Shafer, W., Sonnenschein, H., 1982. Market Demand and Excess Demand Functions. In: Kenneth Arrow, Michael Intriligator (Eds.), Handbook of Mathematical Economics, vol. 2, North Holland, Amsterdam, Chapter 14, pp. 670–693. Sonnenschein, H., 1973. Do Walras’ identity and continuity characterize the class of community excess demand functions, Journal of Economic Theory, 345–354. Sonnenschein, H., 1974. Market excess demand functions. Econometrica 40, 549–563.
Individual excess demands P.A. Chiappori a,∗ , I. Ekeland b a
Department of Economics, University of Chicago, 5801 Ellis Avenue, Chicago, Illinois 60637, USA b University of Paris-Dauphine, France
Abstract We characterize the excess demand function in the one-consumer model. We describe the relation between the direct and indirect utility functions. As an application of our result, we derive a short proof of the celebrated Sonnenschein–Mantel–Debreu theorem on the decomposition of aggregate excess demand. © 2003 Published by Elsevier B.V. Keywords: Optimization; Homogeneous; Quasi-convex
1. Introduction In consumer theory, an individual demand function x(p, y) is defined as the solution to a simple optimization problem, it maximizes some utility function under a linear budget constraint. The properties that stem from this characterization have been known for more than one century. Under standard smoothness assumptions, individual demand functions are fully characterized by three properties: (i) homogeneity, (ii) Walras Law, and (iii) the Slutsky conditions. In addition, one can define the indirect utility and expenditure functions, and there exists a one-to-one relationship between demand, direct utility, indirect utility and expenditure functions, in the sense that the knowledge of any of these function is sufficient to uniquely recover preferences, hence the other three. The results summarized above belong by now to the most standard presentation of consumer theory. Equally standard is the fact that, for many applications (among which general equilibrium theory), it can be helpful to consider excess demand functions instead of Marshallian demands. For any Marshallian demand x(p, y), the excess demand z(p) is defined by z(p) = x(p, p ω) − ω where ω denotes the agent’s initial endowment. A very natural question is whether the standard results on Marshallian demands have a counterpart for excess demands. Specifically, can one find an analogue of the well-known Slutsky conditions, i.e. a system of equations and inequations providing necessary and sufficient condition for ∗
Corresponding author.
0304-4068/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/j.jmateco.2003.11.003
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some arbitrary function z(p) to be an individual excess demand? And does the one-to-one correspondence between the direct utility, indirect utility and demand functions extend to the excess demand framework? Quite surprisingly, these very natural questions have not received an answer so far. One can find in the literature necessary and sufficient conditions expressed in terms of the strong axiom of revealed preferences (see Mas-Colell et al., 1995). One can also find a proof that a certain set of Slutsky-like conditions are necessary (see Geanakoplos and Polemarchakis, 1980). But there is no proof that there are sufficient, and the relationship between direct utility, indirect utility and demand functions has not been investigated for excess demands. Standard consumer theory thus exhibits a gap. The goal of this paper is precisely to fill this gap. We first derive a set of necessary and sufficient conditions that fully characterize excess demand functions. Whenever a smooth function z(p), defined on a compact subset of Rn++ , satisfies these conditions, then it is possible to find an initial endowment and a direct utility function for which z(p) is the excess demand. These conditions, somewhat unsurprisingly, are (i) homogeneity, (ii) Walras Law, and (iii) a variant of the Slutsky conditions; however, the proof of sufficiency (the so-called integrability problem) cannot be transposed from the standard framework, and one has to find a different (and more complex) argument. A second result is that the one-to-one relationship between the direct utility, indirect utility and demand functions that characterizes Marshallian demands is lost in the excess demand context. While it is always possible, starting from some direct utility function (and some initial endowment), to uniquely define the corresponding, indirect utility function, the converse is not true. To any indirect utility function can be associated a continuum of different excess demand functions z(p), each of which satisfies the necessary and sufficient conditions for integrability. Furthermore, for any excess demand function z(p), there exist a continuum of different direct utility functions from which z(p) can be derived. Finally, our results have direct consequences for several problems in consumer theory and aggregation. In the paper, we reconsider the well-known Debreu–Mantel–Sonnenschein theorem on aggregate excess demand. The question, here, is whether it is possible, for any arbitrary, smooth function Z(p) that satisfies homogeneity and the Walras Law, to find n individual excess demand functions z1 (p), . . . , zn (p), such that Z(p) decomposes into the sum of the zi (p). Surprisingly, the existing results do not rely on a formal characterization of individual excess demand functions. Our characterization of individual excess demand yields a three-line proof of the theorem. Not only is this proof short, it also allows a precise characterization of the degrees of freedom one has in constructing the individual utilities. Specifically, we show that there is broad discretion in choosing the indirect utilities V i , and that the choice of the individual excess demands zi corresponding to the chosen indirect utilities is then unique. Moreover, the V i can be chosen independently of the aggregate function at stake, that is, the same functions V i can be used for the decomposition of any function as an aggregate excess demand. Our result thus suggests that the characterization of individual excess demand is indeed a difficult problem, as a matter of fact, it constitutes the core difficulty in the Debreu– Mantel–Sonnenschein excess demand problem. It also provides a better understanding of the fundamental difference between the two versions (market demand versus excess demand) of
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Sonnenschein’s problem, and allows to assess in which sense the market demand problem is considerably more difficult.
2. Individual excess demand 2.1. The necessary conditions Consider a standard consumer model. There are n goods, and the consumer is characterized by a utility function U : Rn → R. The excess demand of the consumer is then defined as the solution z(p) of the individual maximization problem:1 maxz U(z) (IM) p z = pi zi ≤ 0 We shall say that a Ck function U, with k ≥ 2, is standard if: • The gradient Dz U(z) is non-zero everywhere, and the restriction of the second deriva2 U(z) to [D U(z)]⊥ is negative definite for every z; it follows that U is strictly tive Dzz z quasi-concave. • For every p ∈ Rn++ , problem (IM) has a solution z(p) ∈ Rn with p z = 0. Note that the two conditions are independent (neither one implies the other). If U is standard, it follows from the implicit function theorem that the excess demand function z(p) is well-defined and Ck−1 on Rn++ . It is zero-homogeneous and satisfies the Walras Law: z(λp) = z(p) ∀λ > 0, ∀p p z(p) = 0 ∀p and there is a Lagrange multiplier λ(p), which is Ck−1 and homogeneous of degree (−1) on Rn++ , such that Dz U(z(p)) = λ(p)p
(1)
Define the indirect utility V(p), for p ∈ Rn++ , by V(p) = {max U(z)|z ∈ Rn , p z ≤ 0} z
1
In more standard notations, z(p) is defined as x(p, p ω) − ω, where the Marshallian demand x solves ˜ maxx U(x) p x = p ω
Then U is defined from U˜ by ˜ + ω) U(z) = U(z
(IU)
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If U is standard, then V is Ck−1 and satisfies V(p) = U(z(p)). In addition, V(p) is zero-homogenous and quasi-convex. Note that V cannot be strictly quasi-convex because of homogeneity (the level sets {V ≤ a} are cones). However, it can be proved that, if V is 2 V(p) has rank (n − 1) and the restriction of D2 V(p) to [Span{p, D V(p)}]⊥ C2 , then Dpp p pp is positive definite. Also, p Dp V(p) = 0, by the Euler identity, so p and Dp V(p) are orthogonal vectors; if none of them vanishes, their span is truly two-dimensional. The envelope theorem applied to (IU) gives Dp V(p) = −λ(p)z(p)
(2)
Differentiating with respect to p, one gets: 2 V = −λDp z − z(Dp λ) Dpp
(3)
It follows that the restriction of Dp z to [z(p)]⊥ is symmetric, and that its restriction to [Span{p, z(p)}]⊥ = [Span{p, Dp V(p)}]⊥ is negative definite for every p. This condition is the exact equivalent, in the excess demand case, of the fact that the Slutsky matrix for a Marshallian demand is symmetric and negative definite. Note also that if p¯ is, such that z(p) ¯ = 0, then, by Eq. (3), Dp z(p) ¯ is proportional to 2 V(p). ¯ In particular, the matrix Dp z(p) Dpp ¯ is then symmetric and negative semi-definite. Another property of indirect utilities is the following: Lemma 1. If Dp V(p1 ) and Dp V(p2 ) are collinear, then so are p1 and p2 . Proof. If Dp V(p1 ) and Dp V(p2 ) are collinear, then so are z(p1 ) and z(p2 ) by relation (2), so that p 1 z(p2 ) = p 1 z(p1 ) = 0. Set z(p1 ) = λ1 z¯ and z(p2 ) = λ2 z¯ . Since z(p1 ) solves problem (IM) for p = p1 , we have U(z(p1 )) ≥ U(z(p2 )). The converse inequality holds for the same reason, so U(z(p1 )) = U(z(p2 )). Since U is standard, the solution of problem (IM) is unique, so z(p1 ) must be equal to z(p2 ), and p1 , which is collinear to Dp U(z(p1 )) = Dp U(z(p2 )) must be collinear to p2 . 䊐 We summarize these results. They can be found in the existing literature (except, apparently, for the preceding lemma), for instance in Geanakoplos and Polemarchakis (1980). Proposition 2 (Necessary conditions). If the Ck function U(z) is standard, the indirect utility function V : Rn++ → R and the excess demand function z : Rn++ → Rn satisfy the following: 2 V(p) (a) V is Ck−1 , quasi-convex, positively homogenous of degree zero. For every p, Dpp 2 V(p) to [Span{p, D V(p)}]⊥ is positive has rank (n−1); moreover, the restriction of Dpp p definite. If Dp V(p1 ) and Dp V(p2 ) are collinear, so are p1 and p2 . (b) z is Ck−1 and positively homogenous of degree zero. The restriction of Dp z to [z(p)]⊥ is symmetric, and its restriction to [Span{p, z(p)}]⊥ is negative definite for every p.
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2 V(p) and the negativity condition on D z are Note that the positivity condition on Dpp p void if n = 2, but become significant for n ≥ 3.
2.2. Local integrability We now consider the converse. Take some C2 functions V(p) and z(p) satisfying the necessary conditions. Is it possible to find a standard utility function U, such that z(p) solves (IM) for all p, and V(p) is the corresponding, indirect utility? This is the standard ‘integration’ problem in consumer theory, expressed in the case of excess demand. It divides into two subproblems: the local one, where V and z are considered in a suitably small neighborhood of a given point p, ¯ and the global one, where V and z are considered on some arbitrary compact set in the interior of Rn++ . In this section, we address the local problem, and we show that the answer is positive, provided Dp V(p) ¯ = 0 and a nondegeneracy condition is met. The situation near a point p¯ where Dp V(p) ¯ = 0 is mathematically quite interesting, but will not be investigated in this paper. Theorem 3. Let V(p) be a Ck function, k ≥ 3, defined on some neighborhood of p¯ with ¯ = 0. Assume that, on that neighborhood, V(p) is quasi-convex, positively hoDp V(p) 2 V(p) has rank (n − 1) and the restriction of D2 V(p) mogenous of degree zero, that Dpp pp to [Span{p, Dp V(p)}]⊥ is positive definite. Then there is an open convex cone Γ ⊂ Rn++ containing p, ¯ such that V is an indirect utility function on Γ . In fact, take any Ck−1 function λ(p) > 0, homogeneous of degree (−1) on Γ , and define: z(p) = −
1 Dp V(p) λ(p)
(4)
Then there is a quasi-concave function U(z), defined and Ck−1 on a convex neighborhood N of z(p), ¯ with the following properties: 2 U(z) to [D U(z)]⊥ is negative definite. 1. For every z ∈ N, the restriction of Dzz z 2. For every p ∈ Γ , we have
p z(p) = 0
Dz U(z(p)) = λ(p)p,
V(p) = U(z(p)) = {max U(z)|p z ≤ 0, z ∈ N} z
(5) (6)
The proof will be given in the appendix. One can derive a similar result for excess demands: Theorem 4. Let z(p) be a Ck map, k ≥ 2, defined on some neighborhood of p¯ into Rn , such that z(p) ¯ = 0 and Dp z(p) ¯ has rank (n − 1). Assume that, on that neighborhood, z is homogeneous of degree zero, that the restriction of Dp z to [p]⊥ is symmetric, and that its restriction to [Span{p, z(p)}]⊥ is negative definite. Then there is an open convex cone Γ ⊂ Rn++ containing p, ¯ such that z is an excess demand function on Γ . In fact, there is a quasi-concave function U(z), defined and Ck on a convex neighborhood N of z(p), ¯ with the following properties:
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2 U(z) to [D U(z)]⊥ is negative definite 1. For every z ∈ N, the restriction of Dzz z 2. For every p ∈ Γ , we have p z(p) = 0 and:
Dz U(z(p)) = λ(p)p for some λ(p) > 0
(7)
U(z(p)) = {max U(z)|p z ≤ 0, z ∈ N}
(8)
z
The proof will be given in the appendix. Several remarks are in order here. First, unlike the standard case of market demand, one has existence but not uniqueness. This follows from the fact that, while U is a function of n variables, V and z are functions of n − 1 variables only (one dimension being lost because of homogeneity), information is lost in going from U to V and z, and cannot be recovered, in sharp contrast with the Marshallian case. Specifically • For each excess demand function z(p), there exist a continuum of direct utility functions U for which z is the excess demand. The idea is that U is characterized only on the subset M ⊂ Rn generated by z(p) when p varies in Ω ⊂ S n−1 ; in general, M is a submanifold of codimension 1 in Rn . The function U and its derivatives are known only on M, and must be extended outside this set; obviously, this extension can be made in many different ways. • Perhaps more surprisingly, to any homogenous, strictly quasi-convex indirect utility V(p) can be associated a continuum of excess demand functions: just pick up some arbitrary, positive scalar function λ(p), then z(p) = −Dp V(p)/λ(p) is an excess demand. Note that this conclusion can also be viewed as a consequence of a previous result obtained by Debreu, and stating that for any individual excess demand z(p) and any positive scalar function µ(p), the function z˜ (p) = µ(p). z(p) is also an individual excess demand function. It should also be noted that the proof cannot follow the usual path, based on duality. The standard approach, as in Hurwicz and Uzawa (1971), relies on the expenditure function, which has no natural equivalent in the case of excess demand. Similarly, proofs based on indirect utilities (i.e. integration of Roy’s identity) cannot be used here, precisely because the one-to-one correspondence between direct and indirect utilities does not hold in this context. Hence, although the statement of the result is quite similar to the Slutsky characterization of market demand, its proof relies on a completely different (and somewhat more difficult) approach.
3. Global integrability We now address the global problem. Take a C2 function V : Rn++ → R satisfying condition (a) in Proposition 2. We associate with it a function V ∗ : Rn → R defined as follows: V ∗ (z) = inf {V(p)|p ∈ Rn++ , p z ≥ 0} p
(IU*)
The function V ∗ : Rn → R ∪ {−∞} is quasi-concave, positively homogeneous of degree 0, and its domain, domV ∗ = {z|V ∗ (z) > −∞}, is a convex set. However, in contrast with
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the Marshallian demand problem, V ∗ is not a candidate for a direct utility function. Indeed, it is zero homogenous, hence the level sets {V ∗ ≥ a} are cones and the maximization of V ∗ under a linear constraint cannot generate an excess demand function. Still, the function V ∗ will play a key role in the derivation of our result. Introduce the cone: C = {µDp V(p)|µ < 0, p ∈ Rn++ }
(9)
The optimality condition in problem (IU∗ ) is Dp V(p) = µz, with µ < 0. If z¯ = µDp V(p), ¯ for some µ < 0, then p¯ z¯ = 0 by the Euler identity, and the minimum in ¯ In other words, C ⊂ domV ∗ , and if z ∈ C, problem (IU∗ ) with z = z¯ is attained at p = p. so that z = µDp V(p) for some µ < 0 and p ∈ Rn++ , we must have V ∗ (z) = V(p)
(10)
Theorem 5. Let V : Rn++ → R be a a C2 function satisfying condition (a) of Proposition 2. 2 V(p) has Assume that V(p) is quasi-convex, positively homogenous of degree zero, that Dpp 2 ⊥ rank (n − 1) and the restriction of Dpp V(p) to [Span{p, Dp V(p)}] is positive definite. Given any compact subset K in Rn++ where Dp V does not vanish, then V is an indirect utility function on K. In fact, take any C1 function λ(p) > 0, homogeneous of degree (−1) on a neighborhood of K, and define: 1 z(p) = − Dp V(p) (11) λ(p) M = {z(p)|p ∈ K}
(12)
Then there is a quasi-concave function U(z), defined and containing M, with the following properties:
C2
on a convex open set Ω
2 U(z) 1. U is C2 on a neighborhood N of M, and for every z ∈ N, the restriction of Dzz ⊥ to [Dz U(z)] is negative definite 2. For every p ∈ K, the point z(p) is the unique solution of problem (IU), so that:
p z(p) = 0
Dz U(z(p)) = λ(p)p,
V(p) = U(z(p)) = {max U(z)|p z ≤ 0, z ∈ Rn } z
(13) (14)
4. The Sonnenschein–Mantel–Debreu theorem Let Z(p) be a C∞ map from some open cone Ω of Rn+ into Rn , homogeneous of degree zero and satisfying the Walras law p Z(p) = 0. The problem raised by Sonnenschein (1973) is the following: for any k ≥ n, is it possible to find k excess demand functions z1 (p), . . . , zk (p), such that: Z(p) = z1 (p) + · · · + zk (p)
(15)
As it is well known, the answer to this question is positive (see Sonnenschein (1974), Mantel (1974), Debreu (1974), and Shafer and Sonnenschein (1982) for a survey). We
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provide a very short proof. Note that we require that Z be C2 , so that our result is more in the Mas-Colell (1977) line. Following the literature, we normalize prices by imposing that p p = 1, so that prices belong to Rn++ ∩ S n−1 . Since p Z(p) = 0, we can think of Z(p) as a tangent vector to S n−1 at p; the space of all such vectors (the tangent space Tp S n−1 to S n−1 at p) has dimension (n − 1). Also, in what follows K denotes some given compact subset included in the interior of Rn++ ∩ S n−1 . Theorem 6. Pick up k ≥ n arbitrary functions V1 (p), . . . , Vk (p), of class C3 , satisfying condition (a) of Proposition 2, and, such that at every p ∈ K, the set of linear combination of the DVi (p) with non negative coefficients is Tp S n−1 . For any C2 map Z(p) defined on K, homogeneous of degree zero and satisfying the Walras law p Z(p) = 0, and for any k ≥ n, one can find k excess demand functions z1 (p), . . . , zk (p), defined on K, such that the decomposition(15) holds on K and the indirect utility associated to zi is Vi , i = 1, . . . , k. The corresponding, direct utility functions U1 (z), . . . , Uk (z) are quasi-concave, and each Ui is C2 and strictly quasi-concave in a neighborhood of zi (K). Proof. One can find C2 functions µ1 < 0, . . . , µk < 0 on K, such that Z(p) =
k
µi (p)Dp Vi (p)
i=1
Define zi (p) = −µi (p)Dp Vi (p). By Theorem 5, zi is an individual excess demand 䊐 function. The simple intuition of the proof is then that any function Z(p) obtains as a linear combination of the −DVi (p) with non negative coefficients; by Theorem 5, each term in the combination is an individual excess demand. A quite remarkable property is that the same functions Vi can be used for all aggregate demands Z. Note that this proof does not require the full force of our result, but only the fact that if z(p) is an excess demand function, then so is µ(p)z(p) for any µ(p) > 0, a fact that, as we noted above, was already known to Debreu. However, we have been unable to find this proof in the literature. We give two examples of functions V1 (p), . . . , Vn (p), satisfying the conditions of Theorem 6. Note first that the two following conditions are equivalent 1. The convex cone generated by n vectors (e1 , . . . , en ) in Rn−1 is Rn−1 . 2. The convex hull of (e1 , . . . , en ) contains 0. Or first example consists of the functions 2 2 pj pi Vi (p) = −1 + k pk k pk j=i
The Vi satisfy condition (a) of Proposition 2. Consider the simplex Σ = p ∈ Rn++ p1 = 1
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with vertices P1 , . . . , Pn , and denote by ViΣ the restriction of Vi to Σ. Then ViΣ is just the square of the distance from p ∈ Σ to the ith vertex Pi , so that the gradient DVΣ i (p) is directed from p towards Pi . It follows that, for every p in the interior of Σ, the convex Σ hull of the DVΣ i (p) in Σ contains 0 in its interior. Since DVi (p) is the image of DVi (p) by ⊥ the orthogonal projection [p] → Σ, it follows that the convex hull of the DVi (p) in [p]⊥ contains 0 in its interior. So the convex cone spanned by the DVi (p) is Tp S n−1 . Another example, starting from Cobb–Douglas utilities, is given in the appendix. As a corollary of Theorem 6, we then get the celebrated result of Sonnenschein, Mantel and Debreu. It should finally be noted that our approach does not easily allow to characterize the class of ordinal utility functions that generate a given excess demand function. While the topic is left for future research, it is worth noting, in particular, that Mantel’s celebrated (1976) result on the decomposition of aggregate excess demand as sum of individual excess demands stemming from homothetic utility functions is not a direct consequence of Theorem 6. 4.1. Comments An interesting by-product of this proof is that it describes the degrees of freedom available in the choice of individual utility functions. Basically, indirect utilities can be picked almost arbitrarily (the only constraint being that Z(p) belongs to their negative span). However, once they have been chosen, then only one set of λi (p) is acceptable; in other words, among the continuum of individual excess demands that can be associated with each Vi , only one can be used. Then any of the direct utilities corresponding to the zi can be adopted. Note, however, that we have been unable to prove by this method the beautiful result by Mantel (1976), who showed that the individual preferences can be taken to be homothetic. Note also that our method requires that Z(p) be C2 , whereas other proofs work with continuity. We conclude by noting that the previous construct sheds light on the differences between the two problems raised by Sonnenschein, i.e. aggregate excess demand versus aggregate market demand. In the market demand case, a given function X(p) must be decomposed as the sum of n individual Marshallian demands: X(p) = x1 (p) + · · · + xn (p) where xi (p) is the solution of Vi (p) = maxUi (xi ) p xi = 1, xi ≥ 0 In particular, the envelope theorem gives Dp V i (p) = −αi (p)xi (p), where αi is the Lagrange multiplier. It follows that X(p) = −
1 1 Dp Vn (p) Dp V1 (p) − · · · − n α (p) α1 (p)
Again, X(p) must be a convex combination of n gradients. The difference, however, is that while the coefficients of the combination could be freely chosen in the excess demand
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case, in the market demand context they have to satisfy the budget constraint, which implies that αi (p) = −
p D
1 p Vi (p)
∀i
These additional restrictions considerably increase the difficulty of the problem, which now amounts to a system of nonlinear partial differential equations: Dp V1 (p) Dp Vn (p) + ··· + = −X(p) p Dp V1 (p) p Dp Vn (p)
(16a)
This system has been solved in Chiappori and Ekeland (1999), in the case when the right-hand side X(p) is real analytic. The case when X(p) is smooth, C∞ say, but not real analytic, remains open. It is also an open question whether solutions to (16a) can be found globally: the mathematical technique used in Chiappori and Ekeland (1999) only constructs solutions in some neighbourhood of any given point p. ¯
Acknowledgements We thank P. Reny, R. Guesnerie and H. Sonnenschein for useful comments. Financial support from the NSF (grant SBR 9729559) is gratefully aknowledged. Errors are ours.
Appendix A. Proof of Theorem 3 A.1. Step 1: Constructing UA Define z(p) from (4). Then z(p) is Ck−1 , and by the Euler identity, we have: 1 p z = − p Dp V = 0 λ so that z satisfies the Walras law. Differentiating, we get: p Dp z + z = 0
(A.1)
In addition, z(p) is zero-homogeneous of degree zero, so that: (Dp z)p = 0
(A.2)
Without loss of generality, assume that p ¯ = 1. It follows from Eqs. (A.1) and (A.2) that p ∈ KerDp z(p) and z(p) ∈ Range[Dp z(p)] , and it follows from relation (3) that Dp z(p) has rank (n − 1). Consider the map φ defined on S n−1 × R by φ(p, t) = tz(p). We have: Dp,t φ(p, 1) = (Dp z(p), z(p))
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We claim that z(p) ∈ / Range [Dp z(p)], so that Dp,t φ(p, 1) has rank n. Indeed, assume otherwise, so that we have: z(p) = Dp z(p)ξ
(A.3)
for some ξ. Substituting (A.3) into (A.1) yields p z = z ξ = 0 by the Walras law. So ξ ∈ [z(p)]⊥ . Taking another vector η ∈ [z(p)]⊥ , we have: η z(p) = η (Dp z(p)ξ) = 0 But we already know that η (Dp z(p)p) = 0 for all η ∈ [z(p)]⊥ , with p ∈ [z(p)]⊥ . Since the restriction of Dp z(p) to [z(p)]⊥ is symmetric and has rank (n − 1), it follows that ξ = αp for some scalar α, and Dp z(p)ξ = 0 by (A.2), which is the desired contradiction. So Dp,t φ(p, 1) has rank n, and it follows from the inverse function theorem that φ is a Ck−1 diffeomorphism from some neighborhood N×[1 − ε, 1 + ε] of (p, ¯ 1) in S n−1 × R n onto a neighborhood of z(p) ¯ in R . In particular, the image M = {z(p)|p ∈ N} = {φ(p, 1)|p ∈ N} is diffeomorphic to N. In particular, if we pose ˜ U(p, t) = U[φ(p, t)] then defining U on M is equivalent to defining U˜ on N, and ˜ Dp,t U(p, 1) = Dz U ◦ (Dp z, z) We want a function U(z), such that U(z(p)) = V(p) and Dz U(z(p)) = λ(p)p. In other words, we are prescribing the values of U and its first derivatives on M. There is a compatibility condition to be satisfied, namely that we get the same value for Dp U(z(p)). This yields Dp V = λ(Dp z) p Substituting relations (A.1) and (4), we find an identity, so the compatibility condition ˜ holds. Now, let us look at the problem in the (p, t)-coordinates. We know U(p, 1) and the ˜ ˜ ˜ gradient Dp,t U(p, 1) (actually, the knowledge of U(p, 1) fully defines Dp U(p, 1) under the compatibility conditions checked above, so the knowledge of the gradient only pins ˜ ˜ down the partial Dt U(p, 1)). What we want is to define U(p, t) in a neighborhood of t = 1. This can be done in many ways, for instance by setting U˜ A (p, t) = V(p) + (t − 1)A(p, t)
(A.4) ˜
k−1 function, such that A(p, 1) = ∂U (p, 1). In particular, where A : Rn−1 ++ ×R → R is any C ∂t if we evaluate the situation at p, ¯ we find that the matrix of second derivatives takes the form a1,n 2 U ˜A ... Dpp D2 U˜ A (p, (A.5) ¯ 1) = an−1,n an,1 . . . an,n−1 ann
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where the n terms an,i = ai,n , 1 ≤ i ≤ n, can be chosen freely. We now show that if A, and hence the ai,n , are appropriately chosen, the extension UA = U˜ A ◦ φ−1 is quasi-concave. A.2. Step 2: UA is strictly quasi-concave 2 U (z) on the tangent space T M A.2.1. Substep 2a: investigating Dzz A z We have U = UA on M. Differentiating the relation Dz UA (z(p)) = λ(p)p, we get: 2 Dzz UA (z)Dp z = pDp λ + λI
and hence, for any ξ ∈ Tp S n−1 : 2 UA (z)(Dp zξ) = (p Dp zξ)(Dp λξ) + λ(ξ Dp zξ) (Dp zξ) Dzz
Now, by relation (A.1), p Dp zξ = −z ξ. So the first term on the right vanishes if Dp zξ ∈ [p]⊥ , or, equivalently, by relation (A.1), if ξ ∈ [z(p)]⊥ . We are then left with the second term, which we compute: 1 2 1 ξ Dp zξ = ξ − Dpp V + 2 Dp V(Dp λ) ξ λ λ 1 2 1 = − ξ Dpp Vξ + 2 (ξ , Dp V)(ξ, Dp λ) λ λ 1 2 1 = − ξ Dpp Vξ − (ξ , z)(ξ, Dp λ) λ λ and the last term on the right vanishes again if z ξ = 0. So we are left with the following: 1 2 2 ξ ∈ Tp S n−1 , z ξ = 0 ⇒ (Dp zξ) Dzz UA (z)(Dp zξ) = − ξ Dpp Vξ λ and the quadratic form on the right-hand side is negative definite on [Span {p, z(p)}]⊥ , by condition (a). Since ξ ∈ Tp S n−1 , we have (ξ, p) = 0, and so: 0 = ξ ∈ Tp S n−1 ,
2 z ξ = 0 ⇒ (Dp zξ) Dzz UA (z)(Dp zξ) < 0
Set η = Dp zξ. Note that Dz UA (z)η = Dz UA Dp zξ = Dp Vξ = −λz ξ, so that z ξ = 0 if and only if Dz UA (z)η = 0. Rewrite the preceding result in terms of η: 0 = η ∈ Tp M,
2 Dz UA (z)η = 0 ⇒ η Dzz UA (z)η < 0
2 U (z) on the whole space A.2.2. Substep 2b: investigating Dzz A >From U˜ A = UA ◦ ϕ we get, at z = ϕ(p, t): 2 2 (Dz ϕ) Dzz UA (z)Dz ϕ = D2 U˜ A (p, t) − Dz UA (z)Dzz ϕ
Note that the last term, which involves only first derivatives of UA , does not depend on the choice of A, provided it is evaluated at a point z ∈ M; this is because of the relations
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UA (z(p)) = V(p) and Dz U(z(p)) = λ(p)p. Evaluating everything at z¯ = φ(p, ¯ 1), the above equation takes the form: 2 UA (¯z)Dz ϕ = D2 U˜ A (p, ¯ 1) + M (Dz ϕ) Dzz
(A.6)
where M is a fixed operator and D2 U˜ A (p, ¯ 1) is given by formula (A.5). We have shown, in 2 U (z) to T M∩[p]⊥ is negative definite. the preceding substep, that the restriction of Dzz A z 2 ¯ 1) + M to E ∩ F is negative definite, where This means that the restriction of D U˜ A (p, ¯ 1)]−1 (Tz M) and F = [Dz ϕ(p, ¯ 1)]−1 ([p]⊥ ). From the definition of ϕ, it E = [Dz ϕ(p, follows that E is simply the hyperplane t = 0. Going back to formula (A.5), we find that: a1,n + m1,n ... Q ¯ 1) + M = D2 U˜ A (p, an−1,n + mn−1,n a1,n + m1,n . . . an−1,n + mn−1,n an,n + mn,n where the restriction of Q to F ∩ E is negative definite. It is then easy to pick the ai,n so that the restriction of D2 U˜ A (p, ¯ 1)+M to F is negative definite. Going back to Eq. (A.6), we find 2 U (¯ ¯ ⊥ is negative definite. Since [p] ¯ ⊥ is just [Dp UA (¯z)]⊥ , that the restriction of Dzz A z) to [p] this means that UA is strictly quasi-concave. Appendix B. Proof of Theorem 4 Proof. Introduce the differential one-form ω= zi (p)dpi Since the restriction of Dp z to [z(p)]⊥ is symmetric, we must have ω ∧ dω = 0. By the Darboux theorem, there are Ck+1 functions µ(p) and V(p), defined in some neighborhood of p, ¯ such that ω = −µdV. By the convex Darboux theorem (see Chiappori and Ekeland (1997), Ekeland and Nirenberg (2002)), and the references therein; the proof has to be adapted to the homogeneous case, which is straightforward), we can take the functions µ 2 V positive definite on [p]⊥ . and V to be homogeneous, with µ > 0 and Dpp We can therefore apply Theorem 3, taking λ(p) = 1/µ(p). We find a Ck utility function U satisfying (a) and (b), the corresponding excess demand function being precisely z(p).䊐
Appendix C. Proof of Theorem 5 Pick up an arbitrary positive function λ. By Theorem 3, for every point z ∈ M there is a neighborhood Nz ⊂ C and a C2 function Uz : Nz → R satisfying conditions (1) and (IU) in that neighborhood. Since M is compact, it can be covered by finitely many such neighborhoods, say N1 , . . . , Nm , associated with the functions U1 , . . . , Um . Now, take a smooth partition of unity associated with this covering, that is, m functions ϕ1 , . . . ., ϕm of class C2 , such that ϕi ≥ 0, ϕi (z) = 0 if z ∈ / Ni and ϕi = 1. Set
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U = ϕi Ui ; the function U then is defined over ∪Ni , which is a neighborhood of M. Let us show that it satisfies conditions (1) and (IU) in that neighborhood. Looking at formula (9), which defines the cone C, we find that M ⊂ C. If z ∈ M, then z = z(p) for some p, and ϕi V(p) = V(p) = V ∗ (z) U(z) = ϕi Ui = In addition, for any z(p) ∈ M we have that DUi = λ(p)p, so that: Dz ϕi V(p) = λp ϕi Dz Ui + U i D z ϕi = ϕi λp + Dz U = We summarize: z = z(p) ∈ M ⇒ [U(z) = V ∗ (z)
and
Dz U(z) = λ(p)p]
(C.1)
We now claim that: / M, z ∈ ∪Ni , z ∈
p z = 0 ⇒ U(z) < V ∗ (z)
(C.2)
/ M. Since z ∈ ∪Ni , the value U(z) is To see this, pick a point z ∈ ∪Ni with z ∈ well-defined. If p is, such that p z = 0, we must have U(z) < U(z(p)) = V(p), otherwise z would be the maximizer of problem (IM), and hence z = z(p) ∈ M, a contradiction. On the other hand, since z ∈ ∪N ⊂ C, we must have z = µDp V(p) for some µ < 0 and p, and V ∗ (z) = V(p), as noted in relation (10). So U(z) < V ∗ (z), as announced. In other words, although V ∗ is not a direct utility, it provides an upper envelope, a fact that will be crucial to prove global concavity. Lemma 7. There is a neighborhood N of M, with N ⊂ ∪Ni , such that the restriction of U to N is quasi-concave: z = αz1 + (1 − α)z2 , 0 < α < 1, z1 ∈ N,
z2 ∈ N, ⇒ U(z) ≥ min{U(z1 ), U(z2 )} (C.3)
Proof. Argue by contradiction. If condition (B.3) does not hold, there must be three sequences zk1 , zk , zk2 , k → ∞, with the following property: zk1 → z1 ∈ M⇔zk → z ∈ M, zk = (1 − αk )zk1 + αk zk2 , U(zk ) < min{U(zk1 ),
zk2 → z2 ∈ M
0 < αk < 1
U(zk2 )}
(C.4) (C.5) (C.6)
There are three possible cases: 1. The three points z1 , z, z2 are distinct. Then we get U(z) ≤ min{U(z1 ), U(z2 )} in the limit. Since U = V ∗ on M, and V ∗ is quasi-concave, the reverse inequality holds as well, and we get V ∗ (z) = min{V ∗ (z1 ), V ∗ (z2 )}. Without loss of generality, assume V ∗ (z1 ) ≤ V ∗ (z2 ), so that V ∗ (z) = V ∗ (z1 ). From the quasi-concavity of V ∗ it follows that V ∗ (z1 + t(z−z1 )) = V ∗ (z1 ) for all t ∈ [0, 1]. Differentiating once, we get Dz U(z1 )(z2 −z1 ) = 0, 2 U(z )(z −z ) = 0. so (z2 −z1 ) ∈ [Dz U(z1 )]⊥ . Differentiating twice, we get (z2 −z1 ) Dzz 1 2 1 2 ⊥ But Dzz U(z1 ) is positive definite on [Dz U(z1 )] , and we have a contradiction.
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2. Two of the points z1 , z, z2 coincide. The middle one, z, must be one of them; without loss of generality, say the other one is z1 . So z = z1 = z2 . Condition (B.6) then yields U(z1 ) ≤ U(z2 ),
(C.7)
and condition (C.5) yields αk → 0. Substituting into condition (C.6), we get Dz U(z1 )(z2 − z1 ) ≤ 0,
(C.8)
If the strict inequality holds in (C.8), then there is a point z3 between z1 and z2 where U(z3 ) < U(z1 ) ≤ U(z2 ), contradicting the quasi-concavity of U. If Dz U(z1 )(z2 − 2 U(z ) is negative definite on [D U(z )]⊥ , again there will be z1 ) = 0, then, since Dzz 1 z 1 a point z3 between z1 and z2 where U(z3 ) < U(z1 ) ≤ U(z2 ), and again we derive a contradiction. 3. The three points coincide: z1 = z = z2 . But then, for k large enough, the points zk1 , zk , zk2 enter a convex neighborhood of z where U is strictly quasi-concave, contradicting relation (C.6). 䊐
¯ defined on a convex set containing N, We now extend U to a quasi-concave function U, and coinciding with U on N. This is an easy matter: for any number a, consider the set Ω(a) = [z ∈ N|U(z) > a] ⊂ Rn and let co[Ω(a)] be its convex hull. It is an open and convex subset of Rn++ . If a > b, then co[Ω(a)] ⊂ co[Ω(b)]. Set Ω = ∪co[Ω(a)], which is a convex open set, and define a function U¯ : Ω → R by: ¯ U(z) = sup{a|z ∈ co[Ω(a)]} It is quasi-concave by construction. The Lemma implies that if z ∈ N and z ∈ co[Ω(a)], ¯ then z ∈ Ω(a), so U(z) = U(z). This concludes the proof. Appendix D. Direct utilities: an explicit example Let K be a compact subset included in the interior of Rn++ ∩ S n−1 . We give an example of n arbitrary functions V1 (p), . . . , Vn (p), of class C3 , on Rn++ , satisfying condition (a) of Proposition 2, and, such that, at every p ∈ K, the set of linear combination of the DVi (p) with non negative coefficients is Tp S n−1 . Our construction, starting from Cobb–Douglas utility functions, is based on Balasko (1988) (see also Mas-Colell et al., 1995, p. 55). Define, for 1 ≤ j ≤ k and 1 ≤ i ≤ n: Uj (x) = αj,i log xi with i αj,i = 1. The corresponding excess demand is zj (p), with: zij (p) = αj,i
p ω j − ωji pi
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and the indirect utility Vj (p) =
αj,i log
i
αj,i
p ωj pi
= log (p ωj ) −
αj,i log pi +
i
αj,i log αj,i
i
(D.1) j
Let us first take α¯ j,i = 1 if i = j, and 0 otherwise. Assume moreover that ω¯ ji = ω¯ i for all i, j. Then: p ω¯ j j − ω¯ j if i = j i pj zj (p) = −ω¯ ji otherwise Since the zj (p), 1 ≤ j ≤ n, all belong to the (n − 1)-dimensional space Tp S n−1 = [p]⊥ , j the rank of the system is at most (n − 1). By an appropriate choice of the ω¯ ji = ω¯ i , we may assume that this rank is exactly (n − 1). Proceeding with the calculations, we have: j
pj zij (p) = pi
p ω¯ i j − pi ω¯ ii − pj ω¯ ji = p ω¯ i − pj ω¯ ji = p ω¯ i − pj ω¯ i = 0 pi j=i
j
j
In other words, with these particular choices of αj,i = α¯ j,i and ωji = ω¯ ji , we have ¯ the family α¯ j,i , 1 ≤ j pj zj (p) = 0 for all p. To shorten notations, let us denote by α i, j ≤ n, and by ω¯ the family ω¯ ji , 1 ≤ i, j ≤ n. By the implicit function theorem, for every ε > 0, there is a neighborhood N of (α, ¯ ω), ¯ such that, for all (α, ω) ∈ N, the corresponding demand functions zj are, such that the family zj (p), 1 ≤ j ≤ n, has rank (n − 1), and there is a vanishing linear combination qj (p)zj (p) = 0 j
with |qj (p) − pj | < ε. In particular, ε can be chosen so small that qj (p) > 0 for all p ∈ K. It follows that the positive cone spanned by the zj (p) coincides with the linear subspace they span, namely Tp S n−1 . In other words, for all (α, ω) ∈ N, the functions Vj (p) given by (D.1) answer the question.
References Balasko, Y., 1988. Foundations of the Theory of General Equlibrium, Academic Press. Chiappori, P.A., Ekeland, I., 1999. Aggregation and market demand: an exterior differential calculus viewpoint. Econometrica 67 (6), 1435–1458. Chiappori, P.A., Ekeland, I., 1997. A Convex Darboux Theorem. A nnali della Scuola Normale Superiore di Pisa 4.25, 287–297. Debreu, G., 1974. Excess demand functions. Journal of Mathematical Economics 1, 15–23. Ekeland, I., Nirenberg, L., 2002. The Convex Darboux Theorem. Methods and Applications of Analysis 9 (3), 329–344.
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