Dominant Firm and Fringe Firms in an Oligopoly: Price, Output, and Profit Maximization

(a) To find the supply curve for an individual fringe firm, we need to determine the quantity of output q that the firm will produce at a given price P.

The total cost function for a fringe firm is given as: TC = 20q + 2q^2.

To find the quantity q that a fringe firm will produce at a given price P, we can set the firm's marginal cost (MC) equal to the price P.

MC = dTC/dq = 20 + 4q.

Setting MC = P, we have:

20 + 4q = P.

Solving for q, we get:

4q = P - 20.

q = (P - 20)/4.

Therefore, the supply curve for an individual fringe firm is: q = (P - 20)/4.

(b) To find the residual demand curve for the dominant firm, we need to subtract the total quantity supplied by the fringe firms from the total quantity demanded.

Total quantity supplied by the fringe firms is given by: Q_fringe = 4q.

Total quantity demanded is given by: Q_demand = Q.

Therefore, the residual demand for the dominant firm is: Q_residual = Q - 4q.

(c) The dominant firm maximizes its profit by equating its marginal cost (MC) to the marginal revenue (MR).

MC = dTC/dq = 30 + 2q.

MR is given by the derivative of the demand curve: MR = dP/dQ.

P = 100 - Q.

dP/dQ = -1.

Setting MC = MR, we have:

30 + 2q = -1.

2q = -31.

q = -31/2.

Since output cannot be negative, the dominant firm's profit-maximizing choice of output is 𝑞𝐷∗ = 0.

(d) The price at which the firms sell their outputs is given by the demand curve: P = 100 - Q.

Substituting q = 0 (from part c), we have:

P = 100 - Q.

P = 100 - q - Q_fringe.

P = 100 - 0 - 4q.

P = 100 - 4q.

Therefore, the firms sell their outputs at a price of P = 100 - 4q.

(e) Each fringe firm produces the quantity of output q given by the supply curve: q = (P - 20)/4.

Substituting P = 100 - 4q, we have:

q = (100 - 4q - 20)/4.

4q = 80 - 4q.

8q = 80.

q = 10.

Therefore, each fringe firm produces an output of q = 10 units.