The Impact of Cost Uncertainty on Cournot Duopoly Game with Concave Demand Function

Abstract

It is reported in the literature that the most fundamental idea to address uncertainty is to begin by condensing random variables. In this paper, we propose Cournot duopoly game where quantity-setting firms use nonlinear demand function that has no inflection points. A random cost function is introduced in this model. Each firm in the model wants to maximize its expected profit and also wants to minimize its uncertainty by minimizing the cost. To handle this multiobjective optimization problem, the expectation and worst-case approaches are used. A model of two rational firms that are in competition and produce homogenous commodities is introduced using an unknown demand function. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation, and chaos are investigated. Complete stability and bifurcation analysis are provided. The obtained theoretical results are verified by numerical simulation.

1. Introduction

In economic markets, competed firms often face considerable uncertainty upon their productions. This uncertainty may be concerned with information on the exact demand function, rivals’ costs, or even components of firms own production cost. The effects of uncertainty in the optimizing behavior of economic models have been previously examined by many researchers. Sandmo [1] has introduced a systematic study of the theory of firms under a price uncertainty and risk aversion. In [2], some properties of class of models involving optimization under uncertainty have been analyzed. Dardononi [3] has presented the impact of uncertainty on an agent that has a two-argument utility function under plausible normality conditions. A simple noncooperative game with common uncertainty in which the firms have two-argument utility functions has been discussed by Gradstein et al. [4]. In [4], the authors have shown that the effect of increased uncertainty on the equilibrium strategies is similar to its effect on the optimal strategy of a single agent.

In the literature, several studies concerning duopolistic models under uncertainty have investigated that in such models each firm is in risk neutral and may share or exchange its information on market uncertainty with its rival [5–11]. In these studies, it has been investigated how market uncertainty with either unknown market demand or unknown constant marginal cost can affect firms’ behavior.

Everyone agrees that risk is associated with having to make a decision without fully knowing its consequences, due to future uncertainty, but also knowing that some of those consequences might be bad or at least undesirable relative to others [12]. To many people, the amount of risk in a random variable representing a cost of some kind is the degree of uncertainty in it, that is, how much it deviates from being constant. To other people, risk must be quantified in terms of a surrogate for the overall cost, such as its mean value, median value, or worst possible value [12]. An alternative approach for risk measurement has been introduced in detail by Ahmed et al. [13]. In [13], Ahmed et al. have modeled risk in Cournot game based on multiobjective method. Ahmed et al. [13] have used the standard deviation for modeling risk in Cournot game.

The goal of this paper is to set up a discrete time dynamic Cournot model in which duopoly firms use an unknown nonlinear demand function with cost uncertainty. We assume that each firm does not know the other firm’s cost function at the time when the output is produced. It is assumed that the cost function used in this model is generated based on a random variable with zero mean and standard deviation equal to one. Therefore, with the given situation of uncertainty, the existence and uniqueness of the equilibrium point and its asymptotic behavior are proved and investigated.

The paper is devoted as follows. In Section 1, an introduction and related works are presented. Next, some basic concepts which can be found elsewhere [14–17] are provided in Section 2 to make the current paper self-contained. The model and the main results are illustrated and discussed in detail in Section 3. Finally, some concluding results are presented.

2. Basic Concepts

This section provides some basic concepts that will be used in the sequel.

Definition 1. A payoff is a function 𝜋∶𝔸→ℝ that associates a numerical value with every action 𝑎∈𝔸.

Definition 2. An action 𝑎^∗∈𝔸 is an optimal action if 𝜋(𝑎^∗)≥𝜋(𝑎) for all 𝑎∈𝔸.

Definition 3. A solution is called a Pareto-optimal solution if no player’s payoff can be increased without decreasing the payoff to the other player. Such solutions are also termed socially efficient or just efficient.

Definition 4. A Nash equilibrium (for the two player games) is a pair of strategies (𝜎^∗₁,𝜎^∗₂) such that 𝜋₁𝜎^∗₁,𝜎^∗₂≥𝜋₁𝜎₁,𝜎^∗₂,∀𝜎₁∈Σ₁,𝜋₂𝜎^∗₁,𝜎^∗₂≥𝜋₂𝜎^∗₁,𝜎₂,∀𝜎₂∈Σ₂, (1) where Σ_𝑖 is the set of all possible strategies for player 𝑖.

3. Model and Main Results

Here, we propose a new demand function that has not been used in the literature before. The purpose of using this function is that the function has no inflection points. In addition, it has been investigated elsewhere that when the monopolistic firms use such function in the market it seems that Nash equilibrium goes from stability through double bifurcation to chaos [18]. To start our work in the present paper, we consider that the market includes two firms producing homogeneous commodities to sale in the market. Let 𝑞^𝑡_𝑖 be the output of the firm 𝑖, 𝑖=1,2, at discrete time 𝑡=0,1,2,…. This simple situation is called a Cournot duopoly game. Apparently, one should be aware that those two firms are in conflict (more precisely, in competition). Now, let us assume that both firms adopt the following inverse demand function: 𝑃𝑄_𝑡=𝑎−𝑏𝑄⁴_𝑡, (2) where 𝑄_𝑡=∑²_𝑖=1𝑞^𝑡_𝑖, 𝑞_𝑖, 𝑖=1,2, is the quantity of commodity produced by the firm 𝑖. In addition, 𝑎, 𝑏 are positive constants.

Few approaches of uncertainty in multiobjective optimization problems have been reported in the literature [19]. The most fundamental idea to address uncertainty in such problems is to begin by condensing random variables. Rockafellar [19] has reported shortcomings on those approaches. In this section, we apply the expectation and worst-case approaches described by Rockafellar [19]. Let us now suppose that the two firms use the following random cost function: 𝐶𝑞^𝑡_𝑖=𝑐+𝛾𝜀_𝑖𝑞^𝑡_𝑖,𝑖=1,2, (3) where 𝑐≥0 is the fixed marginal cost, 𝜀_𝑖≥0, and 𝛾 is a random variable with zero mean and variance equal to 1. The main aim of each firm is to maximize its following profit: 𝜋₁𝑞^𝑡₁,𝑞^𝑡₂=𝑎−𝑐−𝛾𝜀₁−𝑏𝑄⁴_𝑡𝑞^𝑡₁,𝜋₂𝑞₁,𝑞₂=𝑎−𝑐−𝛾𝜀₂−𝑏𝑄⁴_𝑡𝑞^𝑡₂. (4)

Now, a repeated Cournot duopoly game is considered. It consists of two bounded rational players (firms) in which they update their production strategies at discrete time periods. They use an adjustment mechanism based on a local estimation of the marginal profit. This dynamical adjustment mechanism is described by the following dynamical system: 𝑞_𝑖^𝑡+1=𝑞^𝑡_𝑖+𝛼𝑞^𝑡_𝑖𝜕𝜋_𝑖𝜕𝑞^𝑡_𝑖,𝑖=1,2. (5) It is known elsewhere [13, 18] the significance of speed of adjustment 𝛼(𝑞^𝑡_𝑖). It is a positive function which gives the extent of production variation of 𝑖th firm following a given profit signal. Moreover, it captures the fact that relative effort variations are proportional to the marginal profit.

Here, two scenarios are considered as follows.

Scenario 1 (worst-case approach). This case relies on determining the worst that might happen. For example, if the uncertainty in 𝛾 has to do with market fluctuation due to future circumstances, there may be no available choice of the fixed cost 𝑐 that guarantees absolute compliance. Therefore, if the target of a firm is to maximize its profit, it may have to live with a balance between the practicalities of the fixed cost 𝑐 and the chance that fluctuation of the market could be controlled by some extreme efforts.

Substituting (4) in (5), one gets 𝑞₁^𝑡+1=𝑞^𝑡₁+𝑘𝑞^𝑡₁𝑎−𝑐−𝛾𝜀₁−4𝑏𝑞^𝑡₁𝑄³_𝑡−𝑏𝑄⁴_𝑡,𝑞₂^𝑡+1=𝑞^𝑡₂+𝑘𝑞^𝑡₂𝑎−𝑐−𝛾𝜀₂−4𝑏𝑞^𝑡₂𝑄³_𝑡−𝑏𝑄⁴_𝑡, (6) where 𝜀_𝑖>0, 𝑖=1,2, is the cost control parameters. In the literature many authors have assumed a constant speed of adjustment inducing that a positive variation of profits will make the monopolist to change the quantity in the same direction from that in the previous one. Therefore, no changes will occur if profits are constant. Bischi et al. [20, 21] have chosen a speed of adjustment that is linear function of the quantities produced. In this paper, we assume a linear speed of adjustment, where 𝛼(𝑞^𝑡_𝑖)=𝑘𝑞^𝑡_𝑖, 𝑖=1,2, 𝑘 is constant. The above system has the following equilibrium points: 𝐸_𝑜=(0,0),𝐸₁=𝑎−𝑐−𝛾𝜀₁5𝑏^1/4,𝐸,0₂=0,𝑎−𝑐−𝛾𝜀₂5𝑏^1/4,𝐸₃=4(𝑎−𝑐)−𝛾5𝜀₁−𝜀₂246𝑏2𝑎−2𝑐−𝛾𝜀₁+𝜀₂^31/4,4(𝑎−𝑐)+𝛾𝜀₁−5𝜀₂246𝑏2𝑎−2𝑐−𝛾𝜀₁+𝜀₂^31/4. (7) The fixed points 𝐸₁ and 𝐸₂ can be denoted as monopoly fixed points such that𝑎>𝑐+𝛾𝜀_𝑖, 𝑖=1,2. Assuming 𝑎>𝑐+𝛾𝜀_𝑖, 𝑖=1,2, guarantees positive equilibrium productions. To get more insights about these two monopolistic points, some analytical and numerical discussions are carried out.

Proposition 5. Letting𝑎>𝑐+𝛾𝜀_𝑖, 𝑖=1,2, then(i)the fixed point 𝐸_𝑜 is a repelling node,(ii)the monopoly point 𝐸₁ is stable if 𝑘<2/(𝛾𝜀₂−(1/5)(4𝑎−4𝑐+𝛾𝜀₁)),(iii)the monopoly point 𝐸₂ is stable if 𝑘<2/(𝛾𝜀₁−(1/5)(4𝑎−4𝑐+𝛾𝜀₂)).

Proof. The proof of the above proposition is based on the standard analysis of the eigenvalues as follows. It is easy to check that the point 𝐸_𝑜 is an unstable point. The Jacobian matrix at the point 𝐸₁ becomes 1+4𝑘𝛾𝜀₁−4𝑘(𝑎−𝑐)−16𝑘5𝑎−𝑐−𝛾𝜀₁01−𝑘𝛾𝜀₂+𝑘54𝑎−4𝑐+𝛾𝜀₁, (8) whose eigenvalues are 𝜆₁=1+4𝑘𝛾𝜀₁−4𝑘(𝑎−𝑐) and 𝜆₂=1−𝑘𝛾𝜀₂+(𝑘/5)(4𝑎−4𝑐+𝛾𝜀₁). It is easy to see that these eigenvalues are real given𝑎>𝑐+𝛾𝜀_𝑖, 𝑖=1,2. Hence, for the point 𝐸₁ to be stable, the eigenvalues should satisfy |𝜆_𝑖|<1, 𝑖=1,2, that is, 𝑘<1/(2(𝑎−𝑐−𝛾𝜀₁))<2/(𝛾𝜀₂−(1/5)(4𝑎−4𝑐+𝛾𝜀₁)), which can be reduced to 𝑘<2/(𝛾𝜀₂−(1/5)(4𝑎−4𝑐+𝛾𝜀₁)). For the stability of the point 𝐸₂, the same considerations and arguments hold and this completes the proof. Regarding 𝐸₃, numerical simulation is employed.
To illustrate the stability of the point 𝐸₃, some numerical cases are assumed to make the discussion economically interpreted. Let us assume that the system’s parameters are given as follows: 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀₁=0.3, 𝜀₂=0.7, and a randomly selected value 𝛾=0.7254. We graphically show how the dynamical behavior of the map (6) changes for different values of the parameter 𝑘. Figure 1(a) presents different values of the quantities for different value of the parameter 𝑘. It is easily illustrated that both quantities move from stability through a sequence of period doubling bifurcation to chaos. It is also noted that the marginal cost at the market should be less than the fixed price (𝑎>𝑐+𝛾𝜀_𝑖, 𝑖=1,2). Interestingly, it is numerically observed that when 𝑎≤1, 𝑏<0.5, 𝑐<0.3, and the other parameters are left without change the period doubling bifurcation of the map (6) is going to be reduced. Figures 1(b) and 2 simulate the dynamic behavior of the map for different values of the parameters. Figure 3 shows the chaotic behavior of the relationship between quantities produced by both firms at different value of the parameter 𝛾.

(a)

(b)

Figure 1 

Bifurcation and chaos of the map (6) at the parameters: (a) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=0.3, 𝜀2=0.7, and 𝛾=0.7254; (b) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=0.3, 𝜀2=0.7, and 𝛾=2.7694.

(a)

(b)

Figure 2 

Bifurcation and chaos of the map (6) at the parameters: (a) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=2, 𝜀2=6, and 𝛾=0.3192; (b) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=2, 𝜀2=6, and 𝛾=.3714.

(a)

(b)

Figure 3 

The relationship between 𝑞𝑡1 and 𝑞𝑡2 at the parameters: (a) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=2, 𝜀2=6, and 𝛾=2.7694; (b) 𝑎=3, 𝑏=1.5, 𝑐=0.3, 𝜀1=2, 𝜀2=6, and 𝛾=0.7254.

Scenario 2 (expectation approach). Expectation is primarily suitable for situations where the interest lies in long-range operation and where stochastic ups and downs can safely average out. Since the expectation of 𝛾 is zero, (3) becomes 𝐶(𝑞^𝑡_𝑖)=𝑐𝑞^𝑡_𝑖, 𝑖=1,2. With this information, the system (6) is reduced to the following form: 𝑞₁^𝑡+1=𝑞^𝑡₁+𝑘𝑞^𝑡₁𝑎−𝑐−4𝑏𝑞^𝑡₁𝑄³_𝑡−𝑏𝑄⁴_𝑡,𝑞₂^𝑡+1=𝑞^𝑡₂+𝑘𝑞^𝑡₂𝑎−𝑐−4𝑏𝑞^𝑡₂𝑄³_𝑡−𝑏𝑄⁴_𝑡. (9) That admits the following fixed points: 𝐸_𝑜=(0,0),𝐸₁=𝑎−𝑐5𝑏^1/4,𝐸,0₂=0,𝑎−𝑐5𝑏^1/4,𝐸₃=𝑎−𝑐48𝑏^1/4,𝑎−𝑐48𝑏^1/4. (10) The proof of stability and instability conditions of the above fixed points are similar to those in Proposition 5. Here we briefly give only numerical simulation for the dynamical behavior of the system (9) at certain value parameters. Figure 4 illustrates the dynamic behavior of the system (9) at those parameters.

(a)

(b)

Figure 4 

Bifurcation and chaos of the map (9) at the parameters: (a) 𝑎=3, 𝑏=1.5, 𝑐=0.3; (b) 𝑎=0.5, 𝑏=0.3, 𝑐=0.4.

4. Conclusions

In this paper, we have proposed a Cournot duopoly model based on an uncertain cost and a demand function without inflection point. For this model, the existence and uniqueness of the equilibrium have been proved. The asymptotic behavior of the equilibrium point has been investigated. Complete stability and bifurcation analysis have been illustrated. The obtained theoretical results have been verified by numerical simulation.

Acknowledgment

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.