A mathematical programming-based analysis of a two stage model of interacting producers

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Virginia Polytechnic Institute and State University

This dissertation is concerned with the characterization, existence and computation of equilibrium solutions in a two-stage model of interacting producers. The model represents an industry involved with two major stages of production. On the production side there exist some (upstream) firms which perform the first stage of production and manufacture a semi-finished product, and there exist some other (downstream) firms which perform the second stage of production and convert this semi-finished product to a final commodity. There also exist some (vertically integrated) firms which handle the entire production process themselves.

In this research, the final commodity market is an oligopoly which may exhibit one of two possible behavioral patterns: follower-follower or multiple leader-follower. In both cases, the downstream firms are assumed to be price takers in purchasing the intermediate product. For the upstream stage, we consider two situations: a Cournot oligopoly or a perfectly competitive market.

An equilibrium analysis of the model is conducted with output quantities as decision variables. The defined equilibrium solutions employ an inverse derived demand function for the semi-finished product. This function is derived and characterized through the use of mathematical programming problems which represent the equilibrating process in the final commodity market. Based on this analysis, we provide sufficient conditions for the existence (and uniqueness) of an equilibrium solution, under various market assumptions. These conditions are formulated in terms of properties of the cost functions and the final product demand function.

Next, we propose some computational techniques for determining an equilibrium solution. The algorithms presented herein are based on structural properties of the inverse derived demand function and its local approximation. Both convex as well as nonconvex cases are considered.

We also investigate in detail the effects of various integrations among the producers on firms' profits, and on industry outputs and prices at equilibrium. This sensitivity analysis provides rich information and insights for industrial analysts and policy makers into how the foregoing quantities are effected by mergers and collusions and the entry or exit of various types of firms, as well as by differences in market behavior.