Cybernetics and Systems Analysis, Vol. 33. No. 6, 1997 SYSTEMS ANALYSIS CONVERGENCE OF A METHOD FOR COMPUTING ECONOMIC EQUILIBRIA V. I. Norkin, Yu. M. Ermol'ev, and G. Fischer UDC 519.6 1. INTRODUCTION The method Of sequential joint maximization (of consumer utility functions) [1] has been proposed as a heuristic procedure for finding equilibria in applied economic models. The method exploits the equivalence of the model of equilibrium with fixed consumer incomes to some optimization problem (see Eisenberg and Gale [2], Gale [3], Eisenberg [4], and Polterovich [5]). It involves iterative nonlinear aggregation of consumer utility functions and solution of a sequence of optimization problems with an aggregated utility function as the objective. The method is effective for solving fairly complex temporal equilibrium models intended for integrated assessment of international environmental protection strategies (see Manne [6], Manne and Rutherford [7]). In the present article, we investigate the convergence conditions of the method. We show that this method fits the general scheme of Dafermos [8] for solving variational inequalities (the equilibrium problem is reducible to variational inequalities). Convergence conditions are formulated in terms of a parametric aggregated excess demand function. In the consumer income space, the proposed method is a simple iterative method for Finding the fixed points of some multivalued map. For Cobb-Douglas utility functions, the properties of the method can be studied analytically. We show, for instance, that the convergence of the method is linked with the convergence of some homogeneous and nonhomogeneous Markov chains. We also prove convergence of the method without assuming strict gross substitutability. 2. ECONOMIC EQUILIBRRYM MODEL We introduce some notation (see [9-11]). Consider an economy consisting of m consumers and l producers. Each consumer k is characterized by the utility function Uk(xk), the consumption vector xk E Qk C Rn, the initial commodity stock m w k E R+n, and the shares Ski in the profit of producer i, ~ Ski "- 1. Producer i is characterized by the production activity k=l vector Yi E Yi C R n and the vector production function gi(Yi) = = (gil(,Yi) ..... gin(Yi))" Let p E R+ n be the vector of commodity prices in the economy, x = (.If 1 ..... Xm), Y = (Yl ..... Yt), Q = Q t x ... x Qm, Y = Y1 x ... x YI. The demand for commodities in the economy is generated according to the maximum utility principles: consumer k chooses the consumption vector xk by maximizing the utility function (1) subject to the budget constraint (2) and other constraints (3): Uk,(xO --,.max, (1) xk Xk ~ Qk C R n - ' (3) Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 127-142, November-December, 1997. Original article submitted December 24, 1996. 854 1060-0396/97/3306-0854518.00 9 Plenum Publishing Corporation