Cournot’s Oligopoly Equilibrium under Different Expectations and Differentiated Production
1. Introduction
Microeconomic theory recognizes two extreme cases of market structures, namely the market structure of monopoly and the market structure of perfect competition. A duopolistic and then oligopolistic market can be considered as the initial stage of transition from a monopoly to a market with perfect competition. The analytical expression of oligopoly is more complex than the market extremes mentioned above. This complexity arises from the fact that a firm operating in an oligopolistic market must take into account not only the behavior of consumers, as reflected in the market demand curve, but also the behavior of its competitors, including their response to any significant actions that affect their market position.
The subject of this study is an oligopolistic market in which three firms operate in an environment of quantitative competition known as the Cournot oligopoly model. The firms and their production are differentiated, i.e., each of the firms uses its own production processes and technologies (resulting in different cost functions), and production is not homogeneous. This assumption brings the theoretical model closer to real market conditions, since there are rarely firms whose output is completely homogeneous, i.e., whose products are perfect substitutes. Simultaneously, it was assumed that the products are not perfectly complementary, but also not perfectly independent.
The assumptions of the classical Cournot model can be considered unrealistic under today’s market conditions. It assumes that players set their quantity independently. Since there are only a limited number of players, they would usually react very strongly to the strategies of their competitors. Under real market conditions, competition is not only based on quantity and price, but also on differentiation. The models presented in this article eliminate these shortcomings of the original Cournot model to a considerable extent.
The main objective is to extend the well-known model of duopoly—the Cournot duopoly—by another firm leading to an oligopolistic market structure under the assumption of a partially differentiated production and coalition strategy between two firms, which could bring the theoretical model even closer to real market conditions and show new perspectives for oligopolistic market strategies.
As firms adjust their decisions over time based on different expectations of future performance, the goal is to find and verify the stability of the oligopolists’ net equilibrium. The Slovak mobile network operators (MNO) market was chosen to demonstrate the application of the developed models. The presented models consider three types of expectations, i.e., naïve, adaptive, and real expectations.
This article has the following structure. The introduction contains the identification and description of the research subject, the underlying theory and general assumptions, the description of the Cournot oligopoly model and the possibilities of oligopolists to adapt over time. The subsection on the underlying theory also contains the derivation of the inverse demand functions and general assumptions regarding the model variables that were further applied. The second subsection of the introduction presents the Cournot oligopoly model as a basis for further modifications. The last subsection of the introduction describes the firm’s adaptation to new market situations, including naïve, adaptive, and real expectations of firms. The second section is entitled Proposed model modifications and presents proposed extensions and modifications of the basic oligopoly model, aimed at incorporating differentiation of output and firms, cartelization of the oligopoly by two players and entry of a new firm. It involves the derivation of static quantities and prices of the oligopolists and the adjustment of firm decisions by defining and applying naïve, adaptive, and real expectations about the future market situation. We show the dynamic map for each type of expectation and also eight fixed points of the dynamic map for real expectations. In the Results section, we present the numerical solution of the proposed model for the case of the Slovak mobile operators’ market. This section starts with the naïve expectations, where the results are the same as for the static model, then continues with the results for the adaptive expectations, where a stable solution was found, and finally with the results for the real expectations, where only one of the fixed points gives a non-negative quantity for all three firms. The discussion section discusses the stability of the equilibria found under all three types of expectations when the exogenous parameters change. The conclusion summarizes the results presented and points the way for future research.
1.1. Underlying Theory and General Assumptions
where
—quantity of purchased goods,
,
—price of the i-th product,
—coefficient measuring the quality of the i-th consumed product, and
—coefficient measuring the degree of product differentiation.
where
—the production price of the k-th company,
—the production volume of the k-th company, and
—coefficient measuring the quality of production of the k-th company.
the products of oligopolists are neither perfect substitutes nor complements,
the production of oligopolists is interdependent, and
Note: If or , products of oligopolists are perfect substitutes, complements.
We can rewrite these rules in a more common notation as
. From this notation, we can see that for:
the right side of this inequality is negative, so that the condition holds for all values of for three firms,
the right sight is positive and increasing in , and
the right-hand side is equal , what means for we have .
From the above, it is clear that if the products are identical in terms of product quality (case ), the left-hand side of the inequality
is equal to , so the condition holds for all values of . However, if there are differences in products quality (there are differences in the values of ’s), the left-hand side is less than . In this case, the condition will not be satisfied for sufficiently large values of .
For satisfying Assumption 1:
- –
-
The three products are not too different from each other in terms of quality and/or
- –
-
The products are not extremely substitutable for each other.
-
Assumption 2.Firms have linear cost functions with a zero constant, i.e., their marginal costs (, ) are constant for any quantity of output. To exclude negative output, ruleandwas applied. The second inequality represents the net quality of the firm’s output [7].
1.2. Cournot Oligopoly Model
The Nash equilibrium is such a solution where if one of the players does not adhere to his optimal strategy while his opponent does, his profit is reduced (in the best case it remains the same), i.e., whoever deviates from the optimal strategy cannot achieve a higher profit.
1.3. Adaptation Possibilities of Oligopolists over Time
Currently, there is increasing research on how oligopolists adapt to new circumstances. In this article, we will address adaptation models in the case of the Cournot oligopoly model, but nevertheless provide at least a brief overview of the authors who have studied the oligopoly adaptation model.
2. Proposed Model Modifications
The necessary condition of the second order for the existence of the extreme of the function is satisfied for all the companies . Function (5) form the oligopolists’ response functions, which can also be written as
,
,
where index represents the basic Cournot model. The price at which the oligopolists offer their output on the market is then
Subtracting Equation (7) from Equation (6), we obtain , after adjustment , and substituting the profit function of the oligopolists, we obtain their maximum profit at the optimal output level
2.1. Cartel Oligopoly Model
In this article, we consider an oligopoly model with three firms, as described in the previous section. We assume a situation, where two firms form a coalition () against the last firm. For this model, we later assume naïve, adaptive, and real expectations of the oligopolists.
2.2. Naïve, Adaptive, and Real Expectations
2.2.1. Naïve Expectations
2.2.2. Adaptive Expectations
where is a matching coefficient that applies an optimal production, .
2.2.3. Real Expectation
where is a positive parameter describing the adaption rate of a rational player.
If a non-zero firm production is assumed, only the last fixed point at which all firms offer a non-zero quantity needs to be examined.
3. Results
The services offered by mobile operators were assumed to be substitute products. The mobile call or mobile data transmission is the same for all three operators but is not considered a perfect substitute, as there may be differences in signal coverage, data transmission speed, additional services for customers, or customer loyalty.
The quality of the firm’s output was measured by the ARPU (Average Revenue Per User). Other exogenous parameters of the model are the firm’s marginal costs (). We can determine the constant marginal cost as the ratio between the firm’s operating cost and the number of active SIM (Subscriber Identity Module) cards. The operating costs of each firm were obtained from their annual reports and financial data published in the Register of Financial Statements. The coefficient was set due to the fact that the services offered by each mobile operator can be considered substitutes rather than independent. The core attributes of the services are similar, but the marketing approach applied by the competitors is different, resulting in different market positions and consumer perceptions. However, it should be noted that the applied value of is close to the largest value of for which
for the chosen values of (for these values, this condition is violated for ). Thus, the above condition is satisfied.
In the model presented, the 1st and 2nd firm form a coalition against a new mobile operator—the 3rd firm. The following table shows the exogenous variables of the model.
Moreover, the stability of the presented model under naïve, adaptive, and real expectations is analyzed.
3.1. Naïve Expectations
.
3.2. Adaptive Expectations
3.3. Real Expectations
.
4. Discussion
.
If we focus only on quantity, without regard to the profit function or stability, the 3rd firm, new to the market, would achieve a higher market share than the 2nd firm
, which is part of the coalition, under adaptive and real expectations, resulting in a potential advantage for the 3rd firm under real market conditions.
5. Conclusions
The subject of this study was an oligopolistic market in which three firms operate in an environment of quantitative competition, the so-called Cournot oligopoly model. The firms and their production were partially differentiated, which brought the theoretical model closer to real market conditions. The main objective was to extend the Cournot duopoly and add another firm, resulting in an oligopolistic market structure assuming partially differentiated production and a coalition strategy between two firms.
This paper presents an oligopolistic model specifically designed for three different types of expectations. It was applied to find and verify the stability of the net equilibrium of oligopolists in the market of telecommunication operators in Slovakia. An indefinite time interval and three types of future expectations were considered: a simple dynamic model with naïve expectations, a model with adaptive expectations and a model with real expectations.
The dynamic system of three companies under naïve expectations proved to be stable, as the eigenvalues of the matrix were less than one and the variables converged to a stable point after approximately 30 time periods. The presented model also proved stable in the case of adaptive expectations, as the quantities converged to a stable point very quickly. In contrast, equilibrium quantities under real expectations were not stable and did not converge to a stable point.
Further analysis proved that the presented model under naïve expectations was stable for any value of and for the exogenous parameters , while the model under adaptive expectations was stable under all exogenous variables except . The model under real expectations was not only unstable, but there was further evidence of the chaotic behavior of the system.