Mathematical economics

One more group of results concerns methods for searching economic equilibrium. We develop two approaches, the reduction of the equilibrium problem to a sequence of optimization problem (Norkin, Ermoliev and Fischer (1997)) and decomposition of a general equilibrium problem by means of stochastic quasi-gradient method (Ermoliev, Norkin and Keyzer (2000)).

It is shown that under certain conditions (fixed consumer budgets and etc., see Norkin (1999)) general equilibrium model is reduced to convex (joint/aggregate utility) optimization problem and thus can be effectively solved. General case is reduced to iterative application of utility aggregation and optimization. It was discovered that convergence of this method for searching economic equilibrium is closely related to the ergodicity of certain Markov chains.

The second approach consists in randomization of a classical Walrasian tâtonement process such that price adaptation is made not in the direction aggregate excess demand but in a random direction reflecting excess demand of a small number randomly sampled market agents. Some sufficient convergence conditions and rate of convergence to equilibrium are established.

Paper by Ermoliev , Norkin and Keyzer (2001) develops a practical modeling framework for land use planning and presents the associated stochastic algorithms for numerical implementation. We focus on the case in which transfers among social group adjust to support social welfare optimization. It appears that the problem becomes more tractable if it is treated as the minimization of a dual welfare function, that solely depends on prices but is evaluated after integration over space. Next, we allow for (nonrival) demand that simultaneoulsy benefits several agents, in order to represent general informational infrastructure as well as investments with uncertain outcomes. This leads to a minimax problem, with a dual welfare function to be minimized with respect to prices and maximized with respect to nonrival demand.

Selected references

Norkin V.I., Ermoliev Y.M. and Fischer G. (1997). On convergence of one method for searching economic equilibrium, Kibernetika i sistemnyi analiz, 1997, N 6 (In Russian, English translation in Cybernetics and Systems Analysis, see also IIASA Working Paper WP-96-118).

Norkin V.I. (1999). On a possibility to reduce a general equilibrium model to optimization problems, Kibernetika i sistemnyi analiz (in Russian, English translation in Cybernetics and Systems Analysis), 1999, N 5, 75-86.

Ermoliev Yu., Norkin and Keyzer M.A. (2000). Global convergence of the stochastic tatonement process, J.of Mathematical Economics, 2000, V.34, P.173-190.

Ermoliev Yu., Norkin and Keyzer M.A. (2001). General equilibrium and welfare modeling in spatial continuum: a practical framework for land use planning (Abstract), Interim Report IR-01-033, Int. Inst. for Appl. Syst. Anal., August 2001, 28p.