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This paper introduces a new Cournot duopoly game and gives an applied study for price discrimination in a market by dynamic methods. One of two oligopolies has two different prices for a homogeneous product, while the other charges one kind of price. It is found that there is only one stable equilibrium for the discrete dynamic system, and a corresponding stable condition is given. Using a discriminative price is not always beneficial to a firm in equilibrium. If both oligopolies carry out price discrimination, the market’s average price is lower than when only one oligopoly does it. The results are verified by numerical simulations.

Cournot duopoly assumes that there are two oligopolists who compete in a market by offering a homogeneous commodity. This was introduced by Cournot [

Since Cournot dynamic competition is related to the behavior of consumers, expectations of manufacturer demands, the number of oligopolists, and so on, there are extensive works from different points of view to study Cournot competition. For instance, see the complexity of solutions for Cournot duopoly [

In a static situation, Hazledine extended the homogeneous product Cournot–Nash oligopoly model to allow for price discrimination [

In addition, in [

Noting that the current dynamic games have no consideration of price discrimination, in the present paper, we examine the equilibrium in a discrete dynamic Cournot game with price discrimination, where players adjust their strategies through local estimates of marginal profits. All equilibrium points of the dynamic system are unstable except for one Nash equilibrium. The sufficient condition to guarantee the local stability of the equilibrium is given. It is shown that, compared with a firm with only one kind of price, a firm does not necessarily benefit more from having two different prices. Numerical examples are presented.

The rest of this article is organized as follows. Section

This section aims to establish a dynamic Cournot model with price discrimination. Suppose that there are two firms in a market and they sell a homogeneous product with a constant marginal cost

The decision of each firm is made in discrete time. For any discrete adjustment process, the firm

To find the stationary points of system (

Clearly, the solution of (

We have

Then, the market share of firm 1 is three times that of firm 2 at the equilibrium. Moreover, firm 1 benefits much more than firm 2 from selling the good with two different prices.

The solution

To analyze the stability of these stationary points, the following two lemmas are needed.

The spectral radius

Given a polynomial equation

From Section

It is known that if the modulus of all the eigenvalues of

The stationary point

(i) At the stationary point

(ii) For the equilibrium point

(iii)

(iv)

From the above (i), (ii), (iii), and (iv), the equilibrium solutions

There exists

The Jacobian matrix

Next, we check the eigenvalues of

The next steps, (i) and (ii), show that the eigenvalues are not

(i) Suppose that

(ii) Assuming that

Let

Noting that

The stationary point

From Theorem

Figure

Blue + signs denote cases for

Numerical solutions for system (

Subgraphs (a), (b), and (c) show the solutions of (

As shown in Figures

We assume that one firm sells the product with a certain price while another firm sells the same good for two different prices. This paper contributes to applied research on price discrimination by introducing a new dynamic Cournot competition model. For the dynamic system, each firm makes a decision with bounded rationality in each discrete period to maximize its profit. The stability analysis shows that only one static Nash equilibrium is stable for the dynamic in a large parameter range. The seven other stationary points of the dynamic system are all unstable. At the stable Nash equilibrium, the firm with two prices benefits greatly due to having a major market share; the firm with only a low price has one-third of the market share of the other firm.

If firm 2 only charges a high price

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

This project is supported by the National Natural Science Foundation of China (11661030, 61763008) and Guangxi Natural Science Foundation (2016GXNSFAA380059).