# Nonlinear Phenomena in Cournot Duopoly Model

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## Abstract

**:**

## 1. Introduction

- there is more than one company, and all the companies produce a homogenous product (there is no product differentiation);
- the companies do not cooperate (there is no collusion);
- the companies have a market power (each company’s output decision affects the goods’ price);
- the number of companies is fixed;
- the companies compete in quantities and choose the quantities simultaneously;
- the companies are economically rational and act strategically, usually seeking to maximize the profit given by their competitors’ decision.

- there are at least two companies producing a homogeneous product, and they cannot cooperate in any way;
- the companies compete by setting the prices simultaneously, and consumers want to buy everything from the company that offers a lower price;
- both companies have the same constant unit cost of the production, then marginal and average costs are the same, and they are equal to the competitive prices (as long as the price is above the unit cost, the company is willing to supply any amount that is demanded; if the price is equal to the unit cost, then it is indifferent to how much the company sells; the company will never want to set a price below the unit cost).

- the leader must know ex ante that the follower observes his/her action;
- the follower must have no means of reacting to the next leader’s action, and the leader must know that (if the follower could react to the leader’s action and the leader knew this, the leader’s best response would be to play the follower’s action).

- In the Cournot model, the quantity produced by every duopoly will be $a/3b,$ and the price on the market will be $a/3$; therefore, the profit of every duopolist will be ${a}^{2}/9b;$
- In the Bertrand model, the quantity produced by every duopolist will be $a/2b,$ and in a very absurd way, the market price will be zero (therefore, the profit of every duopolist will also be zero);
- In the Stackelberg model, the leader will sell a quantity $a/2b$ at the price $a/4$ (and the profit will be ${a}^{2}/8b$), when the follower will sell a quantity $a/4b$ at the same price as the leader (and the profit will be ${a}^{2}/16b$).

## 2. Materials and Methods

- Firstly, a stationary solution to (1) is found. For the given ${\mu}_{0}\in M$, the stationary solution ${x}^{\circ}\in X$ can be found as a solution to the equation ${x}^{\circ}=F({x}^{\circ},{\mu}_{0}).$
- Next, the stationary solution having been found ${x}^{\circ}$, the behavior of solutions that start near this solution can be examined. Therefore, we consider the following difference $y\left(t\right)=x\left(t\right)-{x}^{\circ}$ that measures the deviation from ${x}^{\circ}$ at time $t.$ The corresponding difference equation with stationary solution of 0 is $y(t+1)=F({x}^{\circ}+y\left(t\right),{\mu}_{0})-{x}^{\circ}.$ The simplification of the latter equation is its linear approximation by the Taylor theorem $y(t+1)={D}_{x}F({x}^{\circ},{\mu}_{0})y\left(t\right)+o\left(y\left(t\right)\right).$ Neglecting the higher order terms of the following linear system of difference, we can consider the equations:$$y(t+1)=J\left({x}^{\circ}\right)\xb7y\left(t\right),$$
- Finally, the stability of the stationary solution ${x}^{\circ}$ can be assessed. When using Jacobian matrix $J\left({x}^{\circ}\right)$, its eigenvalues can be found. The values of these eigenvalues determine the character of the stationary solution ${x}^{\circ}.$ For details, see [15].

#### 2.1. Bifurcation Diagram

- A limit set X of a point $x\in W,$ where W is an open set in state space, is the set of all points $a\in W,$ for which a sequence ${t}_{i}$ of natural numbers characterized by ${t}_{i}\to \infty $ exist, and ${lim}_{{t}_{i}\to \infty}{F}^{{t}_{i}}(x,\mu )=a.$ By the notation ${F}^{{t}_{i}}$, it is meant that the map (1) is composed of itself ${t}_{i}-1$ times.
- A compact set $A\subseteq W$ is called the attractor if there is such a neighborhood U of A in which A is the limit set of all initial values $x\left(0\right)\in U.$

- (i)
- (ii)
- Randomly choose the initial value $x\left(0\right)$ of the map (1).
- (iii)
- Calculate several first iterations of (1) and ignore them.
- (iv)
- Calculate several next iterations of (1) and plot them.
- (v)
- Increment the value of the parameter $\mu $ of the map (1), and repeat all the above given steps until you reach the end of the parameter sets $M.$

#### 2.2. Lyapunov Exponent

- If ${L}_{k}>1$, then ${h}_{k}>0,$, which means that two initially close trajectories can move away from each other, and the system (1) is sensitively dependent on the initial conditions.
- On other side, if $0<{L}_{k}<1$, then ${h}_{k}<0,$, which means that two initially close trajectories can stay close to each other, and the system (1) is not sensitively dependent on the initial conditions.

- (i)
- Start with an initial orthonormal basis $\{{w}_{1}^{0},{w}_{2}^{0}\}$ of the space ${\mathbb{R}}^{2}$ that sufficiently characterizes the initial circle $\mathbf{S}.$
- (ii)
- Compute the vectors ${z}_{1}={D}_{x}F({x}_{0},{\mu}_{0}){w}_{1}^{0}$ and ${z}_{2}={D}_{x}F({x}_{0},{\mu}_{0}){w}_{2}^{0}.$
- (iii)
- Use the vectors $\{{z}_{1},{z}_{2}\}$ and the Gram–Schmidt orthogonalization method to find a new orthogonal basis $\{{y}_{1}^{1},{y}_{2}^{1}\}$.
- (iv)
- Set ${w}_{1}^{1}={y}_{1}^{1}$, ${w}_{2}^{1}={y}_{2}^{1}$ and consider the basis $\{{w}_{1}^{1},{w}_{2}^{1}\}$ for further computation.
- (v)
- Repeat (ii), (iii) and (iv) for a sufficiently large number of steps $m.$ At each step, use (1), compute the following state $x(t+1)$ and use it instead of ${x}_{0}$ in the Jacobian matrix in Step (ii).
- (vi)
- The good approximation for the total expansion ${r}_{k}^{m}$ in the direction $k,$ where $k\in \{1,2\},$ is vector ${w}_{k}^{m}.$ Thus, a good approximation of the Lyapunov number is $\left|\right|{w}_{k}^{m}{\left|\right|}^{1/m},$ where $\left|\right|\xb7\left|\right|$ is the Euclidean norm.

## 3. Model

#### 3.1. Fundamental Principles of the Dynamics Cournot Oligopoly Model

#### 3.2. Dynamics of Cournot Model

#### 3.3. Linear Model

#### 3.4. Nonlinear Model

- the quantity demanded is reciprocal to the price,
- the companies operate under constant unit costs.

## 4. Discussion

#### 4.1. The Equilibrium of the Linear Model and Its Properties

#### 4.2. Equilibrium of the Nonlinear Model and Its Properties

#### 4.3. Bifurcation Diagram of the Duopoly Model

#### 4.4. Lyapunov Exponent of the Duopoly Model

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. MATLAB Implementation of the Bifurcation Diagram for the Duopoly Map

## Appendix B. MATLAB Implementation of the Lyapunov Exponent for the Duopoly Map

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**Figure 1.**The bifurcation diagram of the map (23) with the inverse demand function (15), where $a=0$ and $b=1$. The left diagram describes the dependence of the limit points of the variable ${q}_{1}$ on the parameter $\mu ={a}_{1}/{a}_{2}$, and similarly, the right diagram describes the limit points of the variable ${q}_{2}.$ Source: computation in MATLAB; cf. Appendix A.

**Figure 2.**Diagram of the largest Lyapunov exponent of the map (23) with the inverse demand function (15), where $a=0$ and $b=1$. Bifurcation parameter $\mu \in [5.8,6.25].$ Source: computation in MATLAB; essential steps of the algorithm are given in Appendix B.

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Pražák, P.; Kovárník, J.
Nonlinear Phenomena in Cournot Duopoly Model. *Systems* **2018**, *6*, 30.
https://doi.org/10.3390/systems6030030

**AMA Style**

Pražák P, Kovárník J.
Nonlinear Phenomena in Cournot Duopoly Model. *Systems*. 2018; 6(3):30.
https://doi.org/10.3390/systems6030030

**Chicago/Turabian Style**

Pražák, Pavel, and Jaroslav Kovárník.
2018. "Nonlinear Phenomena in Cournot Duopoly Model" *Systems* 6, no. 3: 30.
https://doi.org/10.3390/systems6030030