Vladimir A. Bulavsky

Complementarity, equilibrium, efficiency and economics

1. Introduction. 2. Optimization Models. 3. General Economic Equilibrium. 4. Models of Oligopoly. 5. Oligopoly with Leaders. 6. Complementarity Problems with Respect to General Cones. 7. Pseudomonotone and Implicit Complementarity Problems. 8. Complementarity Pivot Methods. 9. Scarf Type Algorithms. 10. Newton-Like Methods. 11. Parameterization and Reduction To Nonlinear Equations. 12. Efficiency. 13. Approximative Efficiency. Index.

Application of Topological Degree Theory to Complementarity Problems

The topological degree theory is applied to study the problem of existence of solutions to complementarity problems of various kinds. A notion of an exceptional family of elements is introduced, and assertions of a non-strict alternative type are obtained. Namely, for a continuous mapping, there exists at least one of the following two objects: either a solution to the complementarity problem, or an exceptional family of elements. Hence, if there is no exceptional families, then at least one solution exists.

AN ALTERNATIVE MODEL OF SPATIAL COMPETITION

In this paper, an alternative network model of oligopolistic markets of homogeneous product is developed. The agents sell their product at several independent markets taking into account the prices of the product unit at different markets, production expenditures, and transportation costs. The unit price at a market depends upon the total supply, whereas the production expenditures may grow along with the total volume of output by all producers. The latter ones choose production volumes and distribution of the output fractions sold at the markets. In order to do that, each agent uses conjectures about the total market supply variations depending upon those of his own supply. Under general enough assumptions concerning the market inverse demand functions and the producers’ cost and transportation functions, the equilibrium existence and uniqueness theorems are formulated and proven. Thus, the paper could be considered as a contribution to the analysis of the structuring effect of transportation network on markets and society as a whole.

Structure of demand and consistent conjectural variations equilibrium (CCVE) in a mixed oligopoly model

We study conjectured variations equilibrium (CVE) in a model of mixed oligopoly with not necessarily continuous demand functions. The agents’ conjectures concern the price variations depending upon their production output increase or decrease. We establish the existence and uniqueness results for the CVE (called exterior equilibrium) for any set of feasible conjectures. To introduce the notion of interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as CVE with consistent conjectures, or CCVE). In addition, we also examine the behavior of the consistent conjectures as functions of a parameter representing the demand’s derivative with respect to the market price. The latter results allow one to predict the behavior of groups of consumers with different consumption abilities. The proposed techniques permit one to develop a qualitative description of the dependence of the market price on the active demand component, too. It should be noticed that due to the non-smoothness of the demand function, there is possibly a path dependency and indeterminacy of equilibria in certain cases. This is, on the one hand, a theoretically inconvenient result (multiple equilibria), but on the other hand, it may happen to be extremely useful for applications. Indeed, the latter multiplicity of equilibria might serve as a rationale for the regulatory intervention to induce a change of equilibrium whenever the total welfare could be improved (cf., for example, the Keynesian stimulus).

Consistent conjectures are optimal Cournot-Nash strategies in the meta-game

In this paper, we investigate the properties of consistent conjectural variations equilibrium developed for a single-commodity oligopoly. Although, in general, the consistent conjectures are distinct from those of Cournot-Nash, we establish the following remarkable fact. Define a meta-game as such where the players are the same agents as in the original oligopoly but now using the conjectures as their strategies. Then the consistent conjectures of the original oligopoly game provide for the Cournot-Nash optimal strategies in the meta-game.

Existence of the Nash-Optimal Strategies in the Meta-Game

In this paper, we investigate the properties of consistent conjectural variations equilibrium (CCVE) developed for a single-commodity oligopoly. Although, in general, the consistent conjectures are distinct from those of Cournot-Nash, we establish the following remarkable fact. Define a meta-game as such where the players are the same agents as in the original oligopoly but now using the conjectures as their strategies. Then the consistent conjectures of the original oligopoly game provide for the Cournot-Nash optimal strategies for the meta-game.

Parametrization and Reduction to Nonlinear Equations

This chapter is dedicated to two different types of methods solving nonlinear complementarity problems. The first one is a method of approximate solution of nonlinear complementarity problem with parameters: Given a continuous mapping f : R n × R m → R n , and a fixed vector of parameters u = (u 1, ..., u m ) T , find a x ∈ R n such that

Newton-Like Methods

x \ge 0,\quad f(x,u) \ge 0,\quad and\quad {x^T}f(x,u) = 0