Answer to Question #140703 in Microeconomics for Gany

Question

In lecture we saw the Cournot competition model for two firms with the same cost function. Now, we are going to consider asymmetric cost functions. Assume that demand for a good is given by

Answer 1

Reaction function of firm1:

$2Q{_1}+Q{_2}=\frac{a-C{_2}}{b}$       

Reaction function of firm2:

 $Q{_1} - 2Q{_2} = \frac{(a - C2)}{b}$

Explanation:

Demand function:

P = a - bQ

Q = Q₁ + Q₂

Thus,

P = a - b(Q₁ + Q₂)

MC₁ = C₁

TC = C₁Q₁

MC₂ = C₂

TC₂ = C₂Q₂

Profit of the firm 1:

Profit₁ = TR₁ - TC₁

= PQ₁ - C₁Q₁

= [a - b(Q₁ + Q₂)]Q₁ - C₁Q₁

=aQ₁ - b(Q₁)² - bQ₁Q₂ - C₁Q₁

dProfit1/dQ₁ = a - 2bQ₁ - bQ₂ - C₁

dProfit₁/dQ₁ = 0

a - 2bQ₁ - bQ₂ - C₁ = 0

2Q₁ + Q₂ = (a - C₁)/b              ... (Reaction function of firm 1)

This reaction function shows that optimal quantity of the firm 1 depends on the optimal quantity of the firm two.

Putting the given information in the reaction function of firm 1 as follows:

$2Q{_1} + Q{_2} = \frac{(a - C1)}{b }$   

          

$2Q{_1} + 100 = \frac{(4 - 2)}{0.01}$

2Q₁ = 200

Q₁ = 100

Thus, if firm 2 produces the 100 units then the firm 1 will produce 100 units.

Profit of the firm 2:

Profit₂ = TR₂ - TC₂

= PQ₂ - C₂Q₂

= [a - b(Q₁ + Q₂)]Q₂ - C₂Q₂

=aQ₂ - bQ₁Q₂ - b(Q₂)² - C₂Q₂

dProfit₂/dQ₂ = a - bQ₁ - 2bQ₂ - C₂

dProfit₂/dQ₂ = 0

a - bQ₁ - 2bQ₂ - C₂ = 0

Q₁ - 2Q₂ = (a - C₂)/b           ... (Reaction function of firm 2)