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Decisions in Economics and Finance

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27-11-2018

Heterogeneous players in a Cournot model with differentiated products

Authors: Andrea Caravaggio, Mauro Sodini

Published in: Decisions in Economics and Finance | Issue 2/2018

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Abstract

In this article, we analyse a duopolistic Cournotian game with firms producing differentiated goods, marginal costs are constant and demand functions are microfounded. We consider firms adopting different decisional mechanisms which are based on a reduced degree of rationality. In particular, we assume that a firm adopts the local monopolistic approximation approach, while the rival adjusts its output level according to the gradient rule. We provide conditions for the stability of the Nash equilibrium and investigate some bifurcation scenarios as parameters vary. The main finding of the article is that both a high level and a low level in goods differentiation may have a destabilising role in the system.

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In Cavalli and Naimzada (2014), the authors consider the speed of adjustment as exogenous, i.e. independent on the level of production, while in Cavalli et al. (2015) it is taken as endogenous.

The Bertrand paradox describes the situation in which a price war is waged between firms, leading the system on a state of perfect competition where the extra-profits of both firms are zero.

For \(\alpha =0\), we notice that from the consumer problem, any pair on the budget constraint is a solution of the optimisation problem. This causes problems in defining demand functions. Ultimately, in a static context, this phenomenon generates a not interesting problem from an economic point of view, in the sense that the definition of supplies is irrelevant in the utility of the agents.

We notice that, for \(\alpha =0\), Eq. 10 becomes \(q_{1,t+1}=q_{1,t}(1-c_{1}q_{1,t})\) that defines dynamics converging to zero. This paradoxical result, typical of isoelastic demands, can be overcome by considering bounded demand functions (see Agliari et al. 2002).

Trajectories of (11) may become negative. However, the analysis focuses only on initial values and parameters for which \(q_{2,t}\) assumes a positive value.

We note that, because of the homogeneity of degree \(-2\) of functions \(F_{l}\), the relations \(F_{l}(\alpha ,c_{1},c_{2})=\frac{f_{l}(\alpha ,x)}{c_{2}^{2}}\) with \(l=1,2\) hold.

The configurations in Panels (c) and (d) of Fig. 1 can be obtained only when \(c1>c2\).

In dynamic exercises, we have considered values of \(\alpha \) values such as to avoid negativity problems.

In this context, the marginal profit that drives the decision-making mechanism is such as to induce strong fluctuations in the decisions of firm 2. Indeed, in order to maintain the nonnegativity of the produced quantities, Eq. (11) (and therefore also the first equation in the map M) should be rewritten as \(q_{2,t+1}=max\bigg (0,q_{2,t}+k\frac{\partial \pi _{2}(q_{1,t},q_{2,t})}{\partial {q_{2,t}}}\bigg )\).

We have verified such result by assuming that our two firms as LMA.

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Title: Heterogeneous players in a Cournot model with differentiated products
Authors: Andrea Caravaggio
Mauro Sodini
Publication date: 27-11-2018
Publisher: Springer International Publishing
Published in: Decisions in Economics and Finance / Issue 2/2018
Print ISSN: 1593-8883
Electronic ISSN: 1129-6569
DOI: https://doi.org/10.1007/s10203-018-0228-x

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