Michael Kopel

Simple and complex adjustment dynamics in Cournot duopoly models

Abstract We are investigating microeconomic foundations of Cournot duopoly games such that the reaction functions are unimodal. We demonstrate that cost functions incorporating an interfirm externality lead to a system of coupled logistic equations. In the situation where agents take turns, we observe periodic and complex behavior. A closer analysis reveals some well-known local bifurcations. In a more general situation, where agents move simultaneously, we observe global bifurcations which typically occur in two-parameter families of two-dimensional endomorphisms.

Equilibrium selection in a nonlinear duopoly game with adaptive expectations

Abstract We analyze a nonlinear discrete time Cournot duopoly game, where players have adaptive expectations. The evolution of expected outputs over time is generated by the iteration of a noninvertible two-dimensional map. The long-run behavior is characterized by multistability, that is, the presence of coexisting stable consistent beliefs, which correspond to Nash equilibria in the quantity space. Hence, a problem of equilibrium selection arises and the long run outcome strongly depends on the choice of the players’ initial beliefs. We analyze the basins of attraction and their qualitative changes as the model parameters vary. We illustrate that the basins might be nonconnected sets and reveal the mechanism which is responsible for this often-neglected kind of complexity. The analysis of the global bifurcations which cause qualitative changes in the topological structure of the basins is carried out by the method of critical curves.

Multistability in a dynamic Cournot game with three oligopolists

The time evolution of a dynamic oligopoly game with three competing firms is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional non-invertible map. For the symmetric case of identical players a complete analytical study of the stability conditions for the fixed points, which are Nash equilibria of the game, is given. For the situation of several coexisting stable Nash equilibria a numerical study of their basins of attraction is provided. This gives, evidence of the occurrence of global bifurcations at which the basins are transformed from simply connected sets into non-connected sets, a basin structure which is peculiar of non-invertible maps. The presence of several coexisting attractors (or multistability) is observed even when complex attractors exist. Two different routes to complexity are presented: one related to the creation of more and more complex attractors; the other related to the creation of more and more complex structures of the basins. Starting from the benchmark case of identical players, the effects of heterogeneous behavior of the players, causing the loss of the symmetry properties of the dynamical system, are investigated through numerical explorations.

Socially Responsible Firms and Endogenous Choice of Strategic Incentives

In this paper we are analyzing a mixed quantity-setting duopoly consisting of a socially concerned firm and a profit maximizing firm. The socially concerned firm considers one group of stakeholders in its objective function and maximizes its profit plus a share of consumer surplus. Both firms have the option to hire a manager who determines the production quantity on behalf of the firm’s owner. We find that in the subgame-perfect equilibrium of this game both firms hire a manager and delegate the production choice. If the unit production costs of the firms are similar, then the socially concerned firm has a higher market share and even higher profit. Interestingly, we observe that as the share of consumer surplus taken into account by the socially concerned firm increases, also its profit might increase. The conclusion is that it pays off to take stakeholder interests into account, but not too much.

Analysis of global bifurcations in a market share attraction model

In this paper we demonstrate how the global dynamics of an economic model can be analyzed. In particular, as an application, we consider a market share attraction model widely used in the analysis of interbrand competition in marketing theory. We analyze the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in discrete time. The main result of this paper is given by the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps. ( 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: E32; M30

Evolutionary Competition in a Mixed Market with Socially Concerned Firms

In this paper we study an oligopoly market where profit-maximizing firms and socially concerned firms compete in quantities. Confronting remarks by Milton Friedman and Gary Becker, we are using an evolutionary setting to investigate the endogenous choice of the proper objective of business firms and the influence of product differentiation on the long run survival of firms which pursue non-profit motives. We find that firms which consider a combination of profit and consumer welfare can indeed have larger market shares and profits than their profit-maximizing rivals. One insight is that it might pay off for shareholders to consider stakeholder welfare, but that the level of social concern should not be too high. Based on a strategy׳s profitability, we consider asynchronous evolutionary updating with firms selecting Nash quantities or choosing best replies to the expected market quantity. Here we observe that the consumers׳ willingness to pay a price premium for products is crucial for the long run survival of socially concerned firms. Depending on the degrees of product differentiation and social concern, long run outcomes consist either of both types of firms or only one type of firm. If the firms׳ propensity to switch between a social or a profit-maximizing strategy is sufficiently large, steady states are unstable and even complicated dynamics can occur.

Spillover effects and the evolution of firm clusters

Abstract In this paper we consider the impact of spillovers occurring within each of two groups of firms on the long run agglomeration patterns in a market. In each period every single firm can either produce for this market or choose some outside option (e.g. a risky asset). Firms switch between the two options based on information about the relative profitability of the market and the outside option. In the market, due to spillovers, the production costs are influenced by the number of firms from the same population which are in the market. The resulting model describes the evolution of the size of the two firm clusters and their market shares over time. We provide a global analysis of the existence and basins of attraction of equilibria to address the question what impact different constellations of spillover effects have on the growth of dominant respectively incoming clusters. We demonstrate that the basins of attraction of coexisting long run equilibria do not depend continuously on the size of the spillover effects. Furthermore, an increase in the initial cluster size is not necessarily beneficial if the switching behavior of firms is fast.

Expectation-Stock Dynamics in Multi-Agent Fisheries

In this paper we consider a game-theoretic dynamic model describing the exploitation of a renewable resource. Our model is based on a Cournot oligopoly game where n profit-maximizing players harvest fish and sell their catch on m markets. We assume that the players do not know the law governing the reproduction of the resource. Instead they use an adaptive updating scheme to forecast the future fish stock. We analyze the resulting dynamical system which describes how the fish population and the forecasts (expectations) of the players evolve over time. We provide results on the existence and local stability of steady states. We consider the set of initial conditions which give non-negative trajectories converging to an equilibrium and illustrate how this set can be characterized. We show how such sets may change as some structural parameters of our model are varied and how these changes can be explained. This paper extends existing results in the literature by showing that they also hold in our two-dimensional framework. Moreover, by using analytical and numerical methods, we provide some new results on global dynamics which show that such sets of initial conditions can have complicated topological structures, a situation which may be particularly troublesome for policymakers.

Gaining the competitive edge using internal and external spillovers: a dynamic analysis

This paper studies the evolution of two clusters of firms competing on a common market. Firms exit and enter a cluster based on the perceived chances for profits inside and outside the cluster. Information about profits are diffused by direct communication between firms. Internal and external spillover effects reduce the overall costs of firms in the clusters depending on the number of firms in the own and the competing cluster. A discrete time deterministic dynamical system describing the evolution of cluster sizes is derived. An analysis of the long run attractors of the system and their basins of attraction is used to compare the effects of advantages of a cluster with respect to the size of internal and external spillover effects, respectively. Furthermore, the implications of slow and fast exit and entry behavior of firms for the long run survival and the size of the clusters are studied.

THE LONG RUN OUTCOMES AND GLOBAL DYNAMICS OF A DUOPOLY GAME WITH MISSPECIFIED DEMAND FUNCTIONS

In this paper we study a model of a quantity-setting duopoly market where firms lack knowledge of the market demand. Using a misspecified demand function firms determine their profit-maximizing choices of their corresponding perceived market game. For illustrative purposes we assume that the (true) demand function is linear and that the reaction functions of the players are quadratic. We then investigate the global dynamics of this game and characterize the number of steady states and their welfare properties. We study the basins of attraction of these steady states and present situations in which global bifurcations of their basins occur when model parameters are varied. The economic significance of our result is to show that in situations where players choose their actions based on a misspecified model of the environment, additional self-confirming steady states may emerge, despite the fact that the Nash-equilibrium of the game under perfect knowledge is unique. As a consequence the long run outcome of the game and overall welfare is highly dependent upon initial conditions.