Linear Programming and General Equilibrium Theory

The general equilibrium model of market pricing and the duality theory of linear programming (LP) are paradigmatic expressions of the duality relation between quantities and prices. While LP is canonical, the market duality found in general equilibrium theory is regarded as the more comprehensive in the sense that LP models of markets are special cases of the general equilibrium model. This paper shows that by adopting an infinite-dimensional perspective, the general equilibrium model of market pricing can be formulated to fit inside LP theory. When utility is quasi-linear (transferable), LP solutions to the primal and dual coincide with market equilibria. When utility is ordinal, we show that LP theory can nevertheless be used to characterize market equilibrium through the money metric utility representation of preferences. A key feature of the LP formulation is that not only commodities, but also individuals, are ``right-hand side constraints'' of the LP problem. An implication of this feature, not explicitly found in the market equilibrium model, is the emergence of the individual as a margin of analysis and, dually, the prices of individuals. The individual as a margin of analysis is important in highlighting the properties of perfectly competitive equilibrium not emphasized in the general equilibrium model. It is also central to the relation between market equilibrium and game-theoretic concepts such as the core and the Shapley value in models with a continuum of individuals. Using the LP perspective, we demonstrate the connections among market equilibrium, the core and the Shapley value for models with both transferable and ordinal utility.