Question

In the initial Cournot duopoly​ equilibrium, both firms have constant marginal​ costs, m, and no fixed​ costs, and there is a barrier to entry. Show what happens to the​best-response function of firms if both firms now face a fixed cost of F.
Let market demand be

p=a-−​bQ,

where a and b are positive parameters with 2 firms.  

Let q1 and q2 be the amount produced by firm 1 and firm​ 2, respectively. Assuming it is optimal for the firm one to​ produce, its best-response function is

q1=??

Answer 1

Answer:

$q_1=\frac{a-m}{2b}-\frac{q_2}{2}$

Explanation:

Note that Total Costs are given by fixed costs (F) and marginal costs (m) that depend on the production level of the firm

$CT_i=F+mq_i$

for i=1,2. The market demand is given by

$p=a-bQ$

where $Q=q_1+q_2$ is the total production, so it's the sum of each firms production

Firm 1 will maximize it's own profits

$max\,\Pi_1=p=(a-b(q_1+q_2))q_1-F-mq_1$

The first order conditions (take derivative of the profit with respect to $q_1$ are given by

$a-2 b q_1-b q_2-m=0$

Then the best-response function for Firm 1 will be

$q_1=\frac{a-m}{2b}-\frac{q_2}{2}$

and the solution for Firm 2 would be symmetric.

Note that only marginal costs are relevant for getting the best-response function, so adding fixed costs (F) don't change the results