Competitive Equilibria in Semi-Algebraic Economies

This paper examines the equilibrium correspondence in exchange economies with semi-algebraic preferences. We develop the foundation for a systematic analysis of multiplicity and for robust calibration in applied general equilibrium. Given a class of semi-algebraic exchange economies parametrized by individual endowments and possibly other exogenous variables such as preference parameters, asset-payoffs or tax-rates there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters. This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational.