Bernard Cornet

Valuation equilibrium and pareto optimum in non-convex economies

In this paper, we report an extension of the second welfare theorem when both convexity and differentiability assumptions are violated. Our model allows various formalizations of the marginal rule and considers the general setting of a topological vector space of commodities.

Existence of marginal cost pricing equilibria : the nonsmooth case

In this article, the authors report a general existence theorem of a marginal (cost) pricing equilibrium for an economy that may exhibit increasing returns to scale or more general types of nonconvexities in the production sector. Their model considers an arbitrary number of firms and no smoothness assumption is made on the production sets or on the aggregate production set. Copyright 1990 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Existence of financial equilibria in a multi-period stochastic economy

We consider the model of a stochastic financial exchange economy with finitely many periods. Time and uncertainty are represented by a finite event-tree \( \mathbb{D} \) and consumers may have constraints on their portfolios. We provide a general existence result of financial equilibria, which allows to cover several important cases of financial structures in the literature with or without constraints on portfolios.

Arbitrage and price revelation with asymmetric information and incomplete markets

This paper deals with the issue of arbitrage with differential information and incomplete financial markets, with a focus on information that no-arbitrage asset prices can reveal. Time and uncertainty are represented by two periods and a finite set S of states of nature, one of which will prevail at the second period. Agents may operate limited financial transfers across periods and states via finitely many nominal assets. Each agent i has a private information about which state will prevail at the second period; this information is represented by a subset Si of S. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩iSi, hence assumed to be non-empty. Our analysis is two-fold. We first extend the classical symmetric information analysis to the asymmetric setting, via a concept of no-arbitrage price. Second, we study how such no-arbitrage prices convey information to agents in a decentralized way. The main difference between the symmetric and the asymmetric settings stems from the fact that a classical no-arbitrage asset price (common to every agent) always exists in the first case, but no longer in the asymmetric one, thus allowing arbitrage opportunities. This is the main reason why agents may need to refine their information up to an information structure which precludes arbitrage.

Lipschitzian solutions of perturbed nonlinear programming problems

We prove that if a second order sufficient condition and a constraint regularity assumption hold, then for sufficiently small perturbations of the constraints and the objective function, the set of local minimizers reduces to a singleton. Moreover, the minimizes and the associated multipliers are Lipschitzian functions of the parameter.

The second welfare theorem in nonconvex economies

Publisher Summary This chapter presents an extension of the second welfare theorem when both convexity and differentiability assumptions are violated. This theorem has asserted that the identity of the marginal rates of substitution and the marginal rates of transformation holds at a pareto-optimal allocation. However, this assertion has been formulated and proved by means of convex analysis, that is, a separation theorem. In the absence of convexity and differentiability assumptions, Guesnerie proved a theorem asserting that at a Pareto-optimal allocation, it is associated a nonzero price vector such that each consumer satisfies the first order necessary conditions for expenditure minimization and each firm satisfies the first order necessary conditions for profit maximization. The chapter also discusses different versions of the second welfare theorem that are in the spirit of the assertion of Guesnerie but differ from it in the way the necessary conditions and the marginal rule are mathematically formalized.

Lipschitz properties of solutions in mathematical programming

This paper deals with the dependence of the solutions and the associated multipliers of a nonlinear programming problem when the data of the problem are subjected to small perturbations. Sufficient conditions are given which imply that the solutions and the multipliers of a perturbed nonlinear programming problem are Lipschitzian with respect to the perturbations.

Existence of Marginal Cost Pricing Equilibria in Economies With Several Nonconvex Firms

This paper considers a general equilibrium model of an economy where some firms may exhibit increasing returns to scale or more general types of nonconvexities. The firms are instructed to follow the standard marginal cost pricing rule or to fulfill the first-order necessary conditions for profit maximization. A general existence theorem of equilibria is proved in the case of an arbitrary number of firms. No assumption is made to imply the aggregate productive efficiency of equilibria, a condition that must be excluded in the nonconvex case. Copyright 1990 by The Econometric Society.

Existence of generalized equilibria

We consider the problem of the existence of equilibria, generalized equilibria, and 8xed-points of a correspondence F , de8ned on a compact subset M of Rn, with values in Rn, when the set M is neither assumed to be convex, nor smooth. 1 The general framework of the article is the following. Let F be a correspondence from M to Rn, that is, a map from M to the set of all the subsets of Rn; the correspondence F is said to be upper semicontinuous (u.s.c.), if the set {x∈M |F(x)⊂V} is open in M for every open set V ⊂Rn. Let N be a “normal cone” correspondence, i.e., at this stage, N is a correspondence from M to Rn such that, for every x∈M; N (x)

Economic Equilibrium: Optimality and Price Decentralization

Mathematical economics has a long history and covers many interdisciplinary areas between mathematics and economics. At its center lies the theory of market equilibrium. The purpose of this expository article is to introduce mathematicians to price decentralization in general equilibrium theory. In particular, it concentrates on the role of positivity in the theory of convex economic analysis and the role of normal cones in the theory of non-convex economies.