Cournot Duopoly Game: Best Response and Profits Under Cartel Allocation

In the cartel allocation, the two firms act as a monopoly and jointly maximize profits. The total quantity produced is given by q*=2.5. Each firm produces q_i*=1.25.

To calculate the static best-response quantity of a firm when the other firm produces according to the cartel allocation, we can use the first-order condition of the Cournot model:

q_i = (1/2) * (5 - q_j - q_i - c)

where q_j is the quantity produced by the other firm.

Substituting q_j = q* - q_i, we get:

q_i = (1/2) * (5 - q* + q_i - c)

Solving for q_i, we get:

q_i = (3/4) * (5 - q* - c)

Substituting q* = 2.5 and c = 1, we get:

q_i = 0.875

Therefore, each firm produces 0.875 units of output when the other firm produces according to the cartel allocation.

To calculate the profits of the firms under this strategy profile, we can use the profit function:

π_i = (5 - q_i - q_j) * q_i - cq_i

Substituting q_i = q_j = 0.875, we get:

π_i = (5 - 1.75) * 0.875 - 0.875 = 2.109

Therefore, each firm earns a profit of 2.109 under this strategy profile.