Consider a market with demand given by Q=400P2Q=400P2. To enter the market a firm must first pay an entry cost of κκ, thereafter it can produce at a constant marginal cost of 2 with no other fixed costs. (a) If a government grants a firm an exclusive monopoly in this market, how low must κκ be for the declared monopolist to enter the market? (b) If instead there are two potential entrants who will engage in symmetric Cournot competition if they both enter, how low must κκ be for them both to enter the market?

(a) The fee cannot exceed $50.

Once the monopoly pays the fee, it will produce until marginal revenue equals marginal cost to maximize profit. Total revenue for the monopoly is PQ=√400QQ=20√Q,PQ=400QQ=20Q, so marginal revenue is 10Q−0.510Q−0.5. Setting marginal revenue equal to marginal cost we have 10Q−0.5=210Q−0.5=2, which yields Q = 25, and P = 4. Total profit for the monopoly is 25*(4 - 2) = 50. So the fee cannot exceed $50.

(b) Then the fee cannot exceed 18.84.

Conditional on entering, let Q1 and Q2 denote the quantity produced by the two firms. The total revenue of firm 1 is PQ=√400Q1+Q2Q1=20Q1√Q1+Q2,PQ=400Q1+Q2Q1=20Q1Q1+Q2, and marginal revenue is 20√Q1+Q2−10Q1(Q1+Q2)−3/220Q1+Q2−10Q1(Q1+Q2)−3/2. In a symmetric Cournot equilibrium, two firms will produce the same quantity, i.e., Q1 = Q2 = Q. Substitute this into the profit maximizing condition, we have 20√2Q−10Q(2Q)−3/2=2202Q−10Q(2Q)−3/2=2, which yields Q= 28.125, and price P = 2.67. Each firm will earn profit =28.125*(2.67 - 2) = 18.84. Then the fee cannot exceed 18.84.