Competitive equilibria in semi-algebraic economies

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, 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. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium. ∗We thank seminar participants at various universities and conferences and in particular Alex Citanna, Egbert Dierker, George Mailath, Andreu Mas-Colell, Paolo Siconolfi and Harald Uhlig for helpful comments. We thank Gerhard Pfister for help with SINGULAR and are grateful to Gerhard Pfister and Bernd Sturmfels for patiently answering our questions on computational algebraic geometry.