Discrete Dynamics in Nature and Society

Volume 2015, Article ID 394810, 13 pages

http://dx.doi.org/10.1155/2015/394810

## Market Share Delegation in a Bertrand Duopoly: Synchronisation and Multistability

^{1}Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi 10, 56124 Pisa, Italy^{2}Department of Law, University of Genoa, Via Balbi 30/19, 16126 Genoa, Italy^{3}Department of Economics and Law, University of Macerata, Via Crescimbeni 20, 62100 Macerata, Italy

Received 21 October 2014; Revised 2 March 2015; Accepted 9 March 2015

Academic Editor: Jinde Cao

Copyright © 2015 Luciano Fanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

#### 1. Introduction

Strategic delegation is a relevant topic in both oligopoly theory and industrial organisation, and several papers have contributed to clarify questions related to the differences between the behaviour of profit-maximising firms and managerial firms (e.g., [1–4]). In the former kind of firms, ownership and management coincide, and consequently the main aim they pursue is profit maximisation. In the latter, ownership and management are separate and managers may be driven by incentive schemes that only partially take into account profit and the other objectives of the firms, such as output, revenues, relative performance evaluation, and market share [3, 5–10]. In addition to the above-mentioned theoretical papers, there also exist some empirical works that stress the importance of market share delegation contracts in actual economies [11, 12].

The present paper studies a nonlinear duopoly game with price competition and market share delegation and extends the study carried out by Fanti et al. [13] to the case of complementary or independent products. To this end, by following an established literature led by Bischi et al. [14], we assume that players have limited information and analyse how a managerial incentive scheme based on market share affects the local and global dynamics of a two-dimensional discrete time system. The paper stresses the differences with the analysis carried out by Fanti et al. [13] on the substitutability between products in the case with managerial firms and market share contracts and compares the results achieved.

The rest of the paper is organised as follows. Section 2 describes the model. Section 3 shows some preliminary global properties of the two-dimensional dynamic system (feasible set). Section 4 studies the fixed points of the system, the invariant sets, and local stability. Section 5 is concerned with multistability and shows that synchronisation may arise when managers receive the same bonus. It also stresses the differences with Fanti et al. [13] and takes into account the asymmetric case in which bonuses are not equally weighted in the managers’ objective function. Section 6 outlines the conclusions.

#### 2. The Model

Consider a duopoly game with price competition, horizontal differentiation, and market share delegation contracts (see [13] for details). Market demands of goods and are, respectively, given bywhere is the degree of differentiation of (complementary) products, while and are quantity and price per unit of good of firm ().

Both the firms have the same marginal cost and hire a manager, who receives a bonus based on market share (, ), where is total supply. The objective function of manager iswhere are profits and is the (constant) delegation variable of player . Hence, by using (1), (2) becomesfrom which we get the following marginal bonus:

We now assume a discrete time () dynamic setting, where each player has limited information, as in Bischi et al. [14], and uses the following behavioural rule to set the price for the subsequent period:where . We want to describe the qualitative and quantitative long-term price dynamics when products are complementary or independent, that is, , and underline the similarities and differences with the case of substitutability investigated in Fanti et al. [13].

Assume , , , and . By using (4) and (5), the two-dimensional discrete time dynamic system is as follows:

#### 3. The Feasible Set

It is of importance to observe that system (6) is economically meaningful only whether, at any time , the two state variables and are not negative; that is, they belong to , where is the convex polygon with vertices , , , and .

Let , , denote the th iterate of system for a given initial condition . Then, the sequence is called* trajectory*. A trajectory is said to be* feasible* for if for all ; otherwise, it is* unfeasible*. The set whose points generate feasible trajectories is called* feasible set*. A point belonging to the feasible set is called* feasible point*.

The feasible set of system is depicted in white in Figures 1(a) and 1(b) for two different parameter constellations, while the unfeasible points belonging to are depicted in grey. The following evidence can be immediately observed: similarly to the substitutability case, (i) set is nonempty such that and (ii) set may have a simple structure (as in Figure 1(a)) or a complex structure (as in Figure 1(b)). The first evidence can be easily demonstrated by considering that the origin is a feasible point and that there exists a small enough such that is not a feasible point.