A Note on Exchange Market Equilibria with Leontief’s Utility: Freedom of Pricing Leads to Rationality

We extend the analysis of [26] to handling more general utility functions: linear substitution functions, which include the Leontief utility. We show that the problem reduces to the general analytic center model discussed in [26]. Thus, the same complexity bound applies to approximating the Fisher equilibrium problem with linear substitution utilities. More importantly, we apply the model to show that the solution to a (pairing) class of ArrowDebreu problems with Leontief’s utility, a more difficult exchange market problem, can be decomposed to solutions of two systems of linear equalities and inequalities, and the price vector is the Perron-Frobenius eigen-vector of a scaled Leontief utility matrix. Consequently, if all input data are rational, then there always exists a rational Arrow-Debreu equilibrium, that is, the entries of the equilibrium vector are rational numbers. Furthermore, the size (bit-length) of the equilibrium solution is bounded by the size of the input data. The result is interesting since rationality does not hold for Leontief’s utility in the general model, and it implies, for the first time, that this class of Leontief’s exchange market problems can be solved as a linear complementarity problem. ∗Department of Management Science and Engineering, Stanford University, Stanford, CA 94305. E-mail: [email protected]. Research supported by NSF grant DMS-0306611. I would like to thank Kamal Jain and Kasturi Varadarajan for insightful discussions on this subject.