Cournot Duopoly: Static Nash Equilibrium and Infinitely Repeated Game

In a Cournot duopoly, each firm chooses its quantity of output given the quantity of output produced by the other firm. The profit function for each firm is given by:

π_i(q_i,q_j) = (5 - q_i - q_j)q_i - cq_i

where i = 1,2 and j = 2,1 respectively. Taking the derivative of this profit function with respect to q_i and setting it equal to zero, we get:

∂π_i(q_i,q_j)/∂q_i = 5 - 2q_i - q_j - c = 0

Solving for q_i, we get:

q_i = (5 - q_j - c)/2

Substituting this expression for q_i into the profit function, we get the profit function for firm i:

π_i(q_i,q_j) = (5 - q_i - q_j)q_i - cq_i = [(5 - q_j - c)/2 - q_j/2]^2 - c[(5 - q_j - c)/2 - q_j/2]

Simplifying this expression, we get:

π_i(q_i,q_j) = (25 - 7q_j - 3c + c^2)/4

To find the static Nash equilibrium quantities and profits, we need to solve for the values of q_1 and q_2 that maximize the profits of each firm, given the quantity produced by the other firm. Since the firms are identical, we can assume that q_1 = q_2 = q. Substituting this into the profit function for firm i, we get:

π_i(q,q) = (25 - 7q - 3c + c^2)/4

Taking the derivative of this profit function with respect to q and setting it equal to zero, we get:

∂π_i(q,q)/∂q = -7/2 + q = 0

Solving for q, we get:

q = 7/2

Substituting this value of q into the profit function, we get:

π_i(q,q) = (25 - 7q - 3c + c^2)/4 = (25 - 49/2 - 3 + 1)/4 = 15/4

Therefore, the static Nash equilibrium quantities are q_1 = q_2 = 7/2 and the profits of each firm are π_1 = π_2 = 15/4.

Note that this is also the Cournot-Nash equilibrium of the infinitely repeated game, since there is no incentive for either firm to deviate from this strategy in any period, given that the other firm will continue to produce the same quantity in all future periods.