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On the Disaggregation of Excess Demand Functions
J. D. Geanakoplos
H. M. Polemarchakis
Abstract
We solve the problem of the restrictions imposed on the Jacobian A at prices $\bar{p}$ of the aggregate excess demand function $x(p)$ of m agents in an exchange economy with $l$ commodities, under the assumption of individual rationality. Given an arbitrary differentiable function $x(p)$ satisfying homogeneity and Walras' law, we attribute rational individual excess demand functions $x^{1} (p), \ldots, x^{m} (p)$ to the $m$ agents such that at any arbitrarily specified vector $\bar{p}$ aggregate excess demand is equal to $x(\bar{p})$ and the following condition is satisfied: There exists a subspace $M$ of dimension $m$ such that the Jacobian at $\bar{p}$ of $x(p)$ and the Jacobian at $\bar{p}$ of the aggregate excess demand function define the same linear function on $M$. If $x(\bar{p}) \not={0}, M$ can be taken to have dimension ($m$+1). As an immediate consequence of our proof for $m$=1 we show that even if $\bar{p}, x(\bar{p}),$ and $Dx(\bar{p})$ are known for the excess demand function of a single agent, the substitution effect and the income effect cannot be unambiguously determined without knowledge of the utility function. We extend the results proved at a point to large open neighborhoods. We show that if $x(p)$ is an arbitrary function which bounded from below and satisfies homogeneity and Walras' law, and if $x(\bar{p}) /not={0}$, then we can find an open neighborhood $G$ of $\bar{p}$ and ($l$1) individually rational excess demand functions $x^{1}(p), \ldots, x^{ll} (p)$, such that $\Sigma_{k=1}^{l1} x^{k} (p) = x(p)$ everywhere on $G$.
