Pierre-André Chiappori∗ February 2003 Ivar Ekeland†
Abstract The paper provides an introduction to exterior diﬀerential calculus, and an application to the standard problem of the characterization of aggregate demand in an economy in which the number of agents is smaller than the number of goods.
∗ †Department of Economics, University of Chicago; firstname.lastname@example.org CEREMADE and Institut de Finance, Université de Paris - IX, Dauphine;
1. Introduction : aggregation and gradient structures
In many situations, economists are interested in the behavior of aggregate variables that stem from the addition of several elementary demand or supply functions. In turn, each of theseelementary components results from some maximizing decision process at the ’individual’ level.1 From a mathematical standpoint, the two ideas of maximization and aggregation have a natural translation in terms of combination of gradients. Speciﬁcally, they require that some given function X(p), mapping Rn to Rn and representing aggregate behavior, can be decomposed as a + linear combination of gradientsDp V k (p), where the V k , k = 1, ..., K are functions deﬁned on Rn . Formally: + X(p) = λ1 (p) Dp V 1 (p) + ... + λK (p) Dp V K (p) (1.1)
To give an example, consider an economy with K consumers, each of them characterized by some nominal income that can, without loss of generality, be normalized to 1. For a given price vector p, consumer k solves the program max U k (xk ) p.xk = 1 (1.2)The value of this program, denoted V k (p), is k’s indirect utility; under adequate assumptions, it is decreasing, quasi-convex, and diﬀerentiable. The envelope theorem implies that Dp V k (p) = −αk (p).xk (p), where αk (p) is the Lagrange multiP plier associated with the budget constraint. Substituting into the sum X = xk , we ﬁnd the economy’s aggregate demand at prices p: X(p) = −
1 1 DpV 1 (p) − ... − Dp V K (p) α1 (p) αK (p)
A standard illustration is the characterization of aggregate market or excess demand in an exchange economy, a problem initially raised by Sonnenschein (1973a,b) and to which a number of author contributed, including Debreu (1974), McFadden et al. (1974), Mantel (1974, 1976, 1977), Diewert (1977), Geanakoplos and Polemarchakis (1980) andChiappori and Ekeland (2000). A diﬀerent but related example is provided by Browning and Chiappori (1994) and Chiappori and Ekeland (2002) who consider the demand function of a two-person household, where each member is characterized by a speciﬁc utility function and decisions are only assumed to be Pareto-eﬃcient.
which is a linear combination of K gradients and satisﬁes p0 X(p) = K. Inaddition : • the V k are (quasi) convex and decreasing • the αk are negative • furthermore, the budget constraint implies : p.Dp V k (p) = αk (p) ∀k (1.4)
A natural question, initially raised by Sonnenschein (1972), is the following : what does relation (1.1) imply upon the form of the function X ? In particular, are there testable necessary restrictions on the aggregate function X(p) that reﬂect itsdecomposability into individual maximizing behavior? And is it possible to ﬁnd suﬃcient conditions on X(p) that guarantee the existence of a decomposition of the type (1.1)? In practice, it is useful to decompose this problem into two subproblems. One, called the mathematical integration problem, can be stated as follows: given some function X, when is it possible to ﬁnd functions V k , k = 1,..., K, such that the decomposition (1.1) holds? The second problem, the economic integration problem, requires in addition that the V k arise from maximizing some concave utilities, i.e. satisfy the conditions (quasi-convexity, (1.4), etc.) listed above. In a previous contribution, Chiappori and Ekeland (1999) have argued that a particular subﬁeld of diﬀerential topology, developped in the ﬁrst...