Back to Fundamentals: Convex Geometry and Economic Equilibrium

Journal of Mathematical Analysis and Applications, 2008

B-convexity was introduced in [W.

arXiv (Cornell University), 2023

This paper examines the relationship between resource reallocation, uniqueness of equilibrium and efficiency in economics. We explore the implications of reallocation policies for stability, conflict, and decision-making by analysing the existence of geodesic coordinate functions in the equilibrium manifold. Our main result shows that in an economy with M = 2 consumers and L goods, if L coordinate functions, representing policies, are geodesics on the equilibrium manifold (a property that we call the finite geodesic property), then the equilibrium is globally unique. The presence of geodesic variables indicates optimization and efficiency in the economy, while non-geodesic variables add complexity. Finally, we establish a link between the existing results on curvature, minimal entropy, geodesics and uniqueness in smooth exchange economies. This study contributes to the understanding of the geometric and economic properties of equilibria and offers potential applications in policy considerations.

Journal of Mathematical Economics, 2011

We prove an equilibrium existence theorem for economies with externalities, general types of nonconvexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L ∞ (M, M, µ), the space of all µ-essentially bounded M-measurable functions on M. As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough. Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.

2010

This paper provides an innovative axiomatic analysis of the notion of exploitation as the unequal exchange of labour, focusing on the relation between exploitation and profits. General convex economies with heterogeneous agents endowed with unequal amounts of physical and human capital are considered. An axiomatic characterisation of the class of definitions that preserve the Fundamental Marxian Theorem (FMT) in this general context is derived. It is shown that none of the main received definitions preserves the FMT. Instead, a definition related to the `New Interpretation' (Dumenil, 1980; Foley, 1982) is presented which preserves the FMT and allows one to generalise a number of key insights of exploitation theory to complex advanced economies.

Positivity, 2002

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.

Positivity, 2002

Metroeconomica, 1968

2013

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of incomplete financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence of an equilibrium is established with techniques that include bounds derived from the duals to problems of utility maximization. Composite variational inequalities furnish the modeling platform. Models with and without shortselling are handled, moreover in the absence of any requirement that agents must initially have a positive amount of every asset, as is typical in equilibrium work in economics.

RePEc: Research Papers in Economics, 1992

An economy has a robustly efficient marginal cost pricing equilibrium (mcpe) if it has an mcpe that is Pareto efficient and if this property is preserved under small variations in preferences endowments and technologies. We consider economies in which there is a finite number of equilibria, each of which varies continuously with preferences and endowments. We prove that there exist no robustly efficient marginal cost pricing equilibria.

Journal of Mathematical Economics, 1979

This article represents an attempt to deal with singular economies and related subjects. We use a geometric approach based on intersecting suitable manifolds and afline spaces embedded in some Euclidean space. This purely geometric theory is in fact in duality with Walrasian equilibrium theory. This approach provides a deeper insight in several subjects such as the structure of the set of critical equilibria, the characterization of economies with a unique equilibrium, the global properties of economies with several equilibria.