Answered: Consider a Cournot duopoly with the…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.5P

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Question

Consider a Cournot duopoly with the inverse demand P = 200 − 2Q.

Firm 1 and 2 compete by simultaneously choosing their quantities. Both firms have constant marginal and average cost MC = AC = 20.

Find each firm’s best response function. Plot the best response functions (label the x-axes as ?1 and y-axes as ?2 ).

Expert Solution

Step 1

Cournot duopoly model is a duopoly in which two firms produce homogeneous products and decide their outputs at the same time. Each firm considers that the competitor firm will behave rationally and treats their competitor’s output as fixed.

Step 2

In order to determine the best response function, we will have to estimate the marginal revenue of each duopolist firm by equating it with the marginal cost so that the profits can be maximized. Let Q1 and Q2 be the quantities sold by firm 1 and firm 2 respectively where Q represents the sum of each firm’s output. Here, the total revenue function of firm 1 will compute as-

$\begin{array}{rcl} T R_{1} & = & P . Q_{1} \\ = & (200 - 2 Q) . Q_{1} \\ = & 200 Q_{1} - 2 (Q_{1} + Q_{2}) Q_{1} \\ = & 200 Q_{1} - 2 {Q_{1}}^{2} - 2 Q_{1} Q_{2} \end{array}$

Now the marginal revenue of firm 1 will be-

$\begin{array}{rcl} M R_{1} & = & \frac{∆ T R_{1}}{∆ Q_{1}} \\ = & \frac{∆ (200 Q_{1} - 2 {Q_{1}}^{2} - 2 Q_{1} Q_{2})}{∆ Q_{1}} \\ = & 200 - 4 Q_{1} - 2 Q_{2} \end{array}$

By using the profit-maximizing condition, we have-

$\begin{array}{rcl} M R_{1} & = & M C \\ 200 - 4 Q_{1} - 2 Q_{2} & = & 20 \\ 180 & = & 4 Q_{1} + 2 Q_{2} \\ 90 & = & 2 Q_{1} + Q_{2} \\ \Rightarrow & Q_{1} = 45 - \frac{1}{2} Q_{2} \end{array}$

It means that the best response function of firm 1 will be Q₁ = 45 − ½ Q₂. The best response function of firm 2 can be found in a similar way such that it will be Q₂ = 45 − ½ Q₁.

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