Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

Abstract

In this paper, we consider a model of mixed duopoly with Conjectured Variations Equilibrium (CVE). The agents’ conjectures concern the price variations depending on the increase or decrease in their production outputs. We establish existence and uniqueness results for the conjectured variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an 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 a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function derivative with respect to the market price.

Partially Mixed Duopoly and Oligopoly: Consistent Conjectural Variations Equilibrium (CCVE). Part 1

“…The following theorem is an extension of Theorem 4.2 from [20] to the case of a partially mixed duopoly. In our future research, we are going to extend the obtained results to the case of non-differentiable demand functions.…”

“…In our recent papers [20] and [18], we extended the results of [19] to the mixed duopoly and oligopoly cases, respectively. In both papers, we defined the concept of an exterior equilibrium, i.e., a conjectural variations equilibrium (CVE) state with the influence coefficients fixed in an exogenous mode.…”

“…Existence and uniqueness theorems for this kind of CVE were established, serving as a base for the concept of an interior equilibrium, which is defined as an exterior equilibrium state with consistent conjectures (influence coefficients). The consistency criteria, a consistency verification procedure, and the existence theorems for the interior equilibrium were also formulated and proved in [20] and [18].…”

“…In order to present the verification procedure we need another notion of equilibrium called exterior (cf., [18] and [20]) with parameters v i given exogenously. The set (p, q 0 , q 1 ) is called an exterior equilibrium state for given influence coefficients (v 0 , v 1 ), if the market is balanced, i.e., condition (1) is satisfied, and the maximum conditions (7) and (9) are valid.…”

“…To do that, we first describe the procedure of verification of the influence coefficients v i as it was given in [18] and [20]. Assume that we have an exterior equilibrium state (p, q 0 , q 1 ) that has occurred for some v 0 , v 1 , and D. One of the producers, say k, temporarily changes its behavior by abstaining from maximization of the conjectured profit (or domestic social surplus, as it is in case k = 0) and making small fluctuations around its output volume q k .…”