Dominant Firm and Fringe Firms: Output, Pricing, and Profit Maximization

(a) To find the supply curve for an individual fringe firm, we need to determine the quantity of output that each fringe firm is willing to supply at different prices.

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

To find the quantity that a fringe firm is willing to supply at a given price P, we can equate the marginal cost (MC) to the price (P):

MC = P

Differentiating the total cost function with respect to q, we get the marginal cost function:

MC = dTC/dq = 20 + 4q

Setting MC equal to P, we have:

20 + 4q = P

Rearranging the equation, we can express q in terms of P:

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 in the market.

The total quantity supplied by the fringe firms is given by:

Q_fringe = 4q

The total quantity demanded in the market is given by:

Q_demand = Q

Therefore, the residual demand for the dominant firm is:

Q_res = Q - Q_fringe = Q - 4q

(c) To find the dominant firm's profit-maximizing choice of output, we need to equate the marginal cost (MC) to the marginal revenue (MR) and solve for qD.

MR is given by the derivative of the revenue function with respect to qD:

MR = d(P*qD)/dqD = P - 2qD

Setting MC equal to MR, we have:

20 + 4qD = P - 2qD

Rearranging the equation, we can express qD in terms of P:

6qD = P - 20

qD = (P - 20)/6

Therefore, the dominant firm's profit-maximizing choice of output is:

qD* = (P - 20)/6

(d) The price at which the firms sell their outputs is given by:

P = 100 - Q

Substituting Q = Q_res = Q - 4q into the equation, we have:

P = 100 - (Q - 4q)

Simplifying the equation, we get:

P = 100 - Q + 4q

(e) To find the output q produced by each fringe firm, we can substitute the price P into the supply curve equation for an individual fringe firm:

q = (P - 20)/4

Using the equation from part (d), we have:

q = (100 - Q + 4q - 20)/4

Simplifying the equation, we get:

4q = 80 - Q + 4q

Q = 80

Therefore, each fringe firm produces q = (100 - 20)/4 = 20 units of output.