Question

A monopolist with total cost function ?(?) = ??? + ?² faces a market demand function of ?_D(?) = ??? − ?/ ? ?.

a)Is the monopolist in the short run or the long run? Explain.

b)Find the profit maximizing price that the monopolist should set and find the level of profits they will earn at that price.

Please show, step by step solution.

Answer 1

?(?) = ??? + ?²

The marginal cost is first order derivative of cost function, so MC = dC/dQ = 2Q, also Average Cost (AC) = C/q = 100/Q+Q

?_D(?) = ??? − ?/ ? ? or P = 600 - 2Q (the inverse demand function)

Now we know that total revenue TR=Price*Quantity = (P*Q)

TR = 600Q - 2Q^2, MR=dTR/dQ = 600-4Q (The marginal revenue is first order derivative of revenue function)

Equilibrium output is determined at the intersection of MR and MC curves or a point where MR=MC

so, 600-4Q = 2Q

6Q = 600 or Q = 100 units (Profit maximizing output)

P = 600-2*100 = $400 (Profit maximizing price)

AC = 100/Q+Q = 100/100+100 = 101

Profits = (P-ATC)*Q = (400-101)*100 = $29900

The firm is earning positive economic profits means that the firm is operating in the short run. in the long run the firm earns only normal profits and ZERO economic profits.