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Edited by: Laura Gardini, University of Urbino, Italy

Reviewed by: Fabio G. Lamantia, University of Calabria, Italy; Luca Gori, University of Genoa, Italy

*Correspondence: Ugo Merlone

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Single-product oligopolies without product differentiation are examined with linear production, production adjustment, flexible workforce and investment costs. The price function is assumed to be hyperbolic which makes the non-linearity of the model much stronger than in the case of linear price function examined earlier in the literature. The best responses of the firms are determined which are not monotonic in contrast to the linear case. The set of all steady states is then characterized and in the case of a duopoly it is illustrated. The asymptotical behavior of the steady states is examined by using simulation. We analyze the effects of the different types of costs on the industry dynamics and compare them to the prediction by the well known model with hyperbolic price function and no product adjustment and investments costs.

Oligopoly theory and its applications became one of the central issues in the literature of mathematical economics since the pioneering work of Cournot [

A complete equilibrium analysis was offered in Zhao and Szidarovszky [

The flexibility of workforce is well known to be an important aspect in terms of manufacturers competitivity [

This paper develops as follows. The mathematical model is introduced and the best responses of the firms are determined in Section 2. The set of all steady states is characterized in Section 3, and the asymptotic behavior of the steady states is examined in Section 4 by using simulation. The last Section 5 concludes the paper with future research directions.

An _{k} denote the output of firm _{k}(_{k}) = _{k} + _{k}_{k} with _{k}, _{k} > 0. In addition to these production costs we consider the following cost types. Hiring new workers requires their training and possibly higher wages. Layoff of workers costs the company the unemployment insurance and usually severance pays. The decrease in production levels requires layoffs and any increase is possible only by increasing the workforce. It is assumed that the additional cost of production level changes linearly depend on the levels of decrease or increase in production. This can be modeled as
_{k} is the production level of the firm in time period _{k} is the maximum possible capacity limit that cannot be increased further. There is a difference between _{k}(_{k}. While _{k} is the maximum possible production level that firm _{k}(_{k} and δ_{k}), the cost of adding new machinery (parameters α_{k}) and, finally some structural limits which bound the firms capacity. While we assume that workforce flexibility and new machinery costs are piece-wise linear, as it concerns the structural limits they would need more time to be overcome. For the sake of simplicity let φ_{1}, φ_{2}, and φ_{3} denote these functions.

Clearly
_{k}, _{k} is a strictly concave, piece-wise differentiable, continuous function.

In order to find out the shape of the profit function Π_{k} and determine the best responses _{k} we have to consider the following cases.

_{k} > δ_{k}. For the sake of simplified notation let _{k} = 0, then in the first terms of Π_{k}, _{k} cancels out and _{k} = 0 is the best choice. However, at _{k} = _{k} = 0, Π_{k} is undefined, so this is only a fictitious best response. So this case occurs when _{k} = 0. At

_{k}:
_{k}, in which case define _{k} is the stationary point in interval (0, _{k} (

_{k} and so we can select _{k} (

_{k} (_{k} (

_{k} (

_{k} (_{k}:

And finally, the case of _{k} = _{k}.

The different segments of Π_{k} are summarized in Figure _{k} are illustrated in

_{k} when

For easier understanding of the different cases notice that

The computation of _{k} can be done in the following algorithm:

_{k} is determined from Figure

_{k} is determined from Figure

_{k} is determined from Figure

_{k} is determined from Figure

_{k} is determined from Figure

_{k} is determined from Figure

_{k} when

_{k} when

_{k} when

_{k} when

_{k} when

By denoting the best response function of firm _{k}(_{k}, _{k}, _{k}), the dynamic model with positive adjustment toward best responses can be written as

Notice that in case (b), _{3} is eliminated. The condition of case (b) can be rewritten as
_{k}, which is the case when (29) is violated.

Figure

The model introduced and analyzed in the previous sections is a clear generalization of the duopoly model of Puu [_{1} = α_{2} = γ_{1} = γ_{2} = δ_{1} = δ_{2} = 0 and sufficiently large values of _{1} and _{2}. In this particular case, the only fixed point, except the origin, is, of course, the Cournot equilibrium point:

In Puu [_{1} = α_{2} = γ_{1} = γ_{2} = δ_{1} = δ_{2} = 0 and sufficiently large _{1} and _{2}, we obtain the bifurcation diagram reported in Figure _{1} = _{2} = 0, _{1} = 1 and initial condition _{1}(0) = _{2}(0) = 0.1 when _{2} varies in the interval [5.75, 6.25].

The model we present here is much more complex. To understand the effects of the different costs and production constraints we introduced, we analyze each of them separately and compare the dynamics to the one originally presented in Puu [_{2}/_{1}, α = α_{1} = α_{2}) in which the regions of different periodicity are represented by different colors. Bifurcation diagrams are also shown in which amplitudes are plotted vs. marginal cost ratio as in the original figure in Puu [

_{2}/_{1}, α = α_{1} = α_{2}); the regions of different periodicity are represented by different colors. The horizontal yellow lines represent the line on which, in _{1} = α_{2} = 0.20 and in _{1} = α_{2} = 0.48.

Let us start our analysis by considering the effects of the investment costs α_{k}. As the investment costs are small (Figure _{1} = α_{2} = 0.48) the trajectory becomes less complex than in the case of smaller costs (α = α_{1} = α_{2} = 0.2). This can be explained because larger investment costs dampen firms' reply.

The effects of the hiring costs γ_{k} are much more pronounced. In fact even for small values of hiring costs the dynamics is much simpler as it can be seen in Figures

_{2}/_{1}, γ = γ_{1} = γ_{2}); the regions of different periodicity are represented by different colors. The horizontal yellow line represents the line on which, in _{1} = γ_{2} = 0.05, and in _{1} = γ_{2} = 0.4.

When considering layoff costs δ_{k}, the effects are similar to those of investment costs. For small values of layoff costs the dynamics is similar but not identical to the one with no costs. In fact, although Figures

_{2}/_{1}, δ = δ_{1} = δ_{2}); the regions of different periodicity are represented by different colors. The horizontal yellow lines represent the line on which, in _{1} = δ_{2} = 0.1 and in _{1} = δ_{2} = 0.2.

Finally, when considering capacity limits _{k} we can see that, unless they are influencing firms' response, they have no effect on the dynamics. As a matter of fact, Figures

_{2}/_{1}, _{1} = _{2}); the regions of different periodicity are represented by different colors. The horizontal yellow line represents the line on which, in _{1} = _{2} = 0.15, and in _{1} = _{2} = 0.2.

Non-linear single product oligopolies without product differentiation were introduced and examined where the production cost was linear, and the piecewise linear production adjustment and investment costs made the model non-linear. The non-linearity of the model became even stronger by assuming hyperbolic price function. These models are equivalent with rent-seeking and market-share attraction games as well. The profit functions of the firms are continuous, piece-wise differentiable and strictly concave implying the uniqueness of the best responses. The best response functions of the firms were then determined which are not monotonic in contrast with the case of linear price functions assumed earlier in the literature. The set of all steady states were characterized and illustrated in the case of a duopoly. The asymptotical properties of the steady states were investigated by using simulation. Comparing the dynamics to the one of Puu's original model, showed the different roles these costs have on the dynamics. Although some of them seem to have little effects on the dynamics, others –such as the hiring costs– have a deep impact. In particular, as recruitment and selection cost can be staggering, see for instance Gusdorf [

It will be interesting to consider the case of a generic isoelastic function. Also, further non-linearities can be introduced into the models by assuming non-linear cost functions and different types of the price functions. We will elaborate on these ideas in our next research project.

UM: Original Idea and Simulations. FS: Mathematical Derivation. AM: General Comments.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AM has received research grants from the MEXT-Supported Program for the Strategic Research Foundation at Private Universities 2013–2017 and Chuo University (Joint Research Grant). UM has developed this work in the framework of the research project on “Dynamic Models for behavioural economics” financed by DESP-University of Urbino.