[We solve the problem of the restrictions imposed on the Jacobian A at prices p̄ 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), ..., x^m (p) to the m agents such that at any arbitrarily specified vector p̄ aggregate excess demand is equal to x(p̄) and the following condition is satisfied: There exists a subspace M of dimension m such that the Jacobian at p̄ of x(p) and the Jacobian at p̄ of the aggregate excess demand function define the same linear function on M. If x(p̄) ≠ 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 p̄, x(p̄), and Dx(p̄) 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(p̄) ≠ 0, then we can find an open neighborhood G of p̄ and (l-1) individually rational excess demand functions x^1(p), ..., x^l-l (p), such that Σ_k=1^l-1 x^k (p) = x(p) everywhere on G.]

We establish conditions which (in various settings) guarantee the existence of equilib-ria described by ergodic Markov processes with a Borel state space S. Let 9(S) denote the probability measures on S, and let s- G(s) c 4?(S) be a (possibly empty-valued) correspondence with closed graph characterizing intertemporal consistency, as prescribed by some particular model. A nonempty measurable set J c S is self-justified if G(s) n 9?(J) is not empty for all s E J. A time-homogeneous Markov equilibrium (THME) for G is a self-justified set J and a measurable selection TI: J-9 _(J) from the restriction of G to J. The paper gives sufficient conditions for existence of compact self-justified sets, and applies the theorem: If G is convex-valued and has a compact self-justified set, then G has an THME with an ergodic measure. The applications are (i) stochastic overlapping generations equilibria, (ii) an extension of the Lucas (1978) asset market equilibrium mnodel to the case of heterogeneous agents, and (iii) equilibria for discounted stochastic games with uncountable state spaces.