## (solution) Question 1 A monopolist sells a product in two different markets.

Question 1 A monopolist sells a product in two different markets. Demand in the firstmarket is Q = 120?2P. Demand in the second market is Q = 120?8P. The firm?s cost function is given by C(Q) = 10Q.

(a) Suppose the firm can charge different prices in the two markets. Determine theprices and the firm?s profit.

(b) Now suppose that the firm must charge the same price in both markets. Whatis the firm?s profit now.

(c) Determine the the total change in surplus (consumer + firm) between (a) and(b).

Question 2 Suppose there are 100 households whose demand for electricity is given byQ = 40 ? P. The local power company has a constant marginal cost of 2, and afixed cost of 10,000.

(a) Suppose the power company charges a price per unit of electricity that maximizesprofits. Determine the price, the demand of each household, and thecompany?s profit (Recall that you must use aggregate demand of all 100households. Q = 40 ? P is the demand of a single household).

(b) Now suppose that the power company uses two part pricing. It selects a priceP per unit and a fixed fee F that maximizes profit. Determine P, F, and thefirm?s profit (Now you must use individual demand Q = 40 ? P to determineP and F. Of course, in order to determine profits you must use the fact thatthere are a total of 100 such households.)

(c) Determine the efficiency gain from using two-part pricing instead of a priceper unit. This efficiency gain is measured by the sum of the following: (a)the change of firm profits; (b) the change of consumer utility (see your classnotes).

(d) Now suppose that the government wants to regulate the power company. Inparticular, the government wants (a) production to be efficient and (b) firmprofits to be zero. This objective can be achieved by charging a fixed fee F inaddition to a price per unit P. Determine F and P.

Question 3 Suppose there are three firms producing the same product. The firm?s costfunctions are C1(Q) = 10Q, C2(Q) = 18Q and C3(Q) = 12Q. Demand for theproduct is given by Q(P) = 1, 000 ? 10P. Determine the equilibrium price, eachfirm?s market share, and each firm?s profit. Note: The market shares will not be thesame since marginal costs differ.

Question 4 Suppose there are two firms. The demand for firm 1?s product is given byQ1(P1, P2) = 10 ? P1 + 0.5P2, and the demand for firm 2?s product is Q2(P1, P2) =10 ? P2 + 0.5P1. Both firms have cost functions C(Q) = 2Q.

(a) Suppose that each firm charges a pricer per unit that maximizes profits takenthe price of the other firm as given. Determine P1, P2, Q1, Q2 and each firm?sprofit.Hint: Determine the price elasticity of demand for each firm, and then use theformula that MC = Pi(1 + 1/?iP), where ?iPis the price elasticity of demandfor firm i?s product and Piis the price firm i charges. You get two equationsin two unknowns, P1, and P2 which you can solve.

(b) Now suppose that both firms use two-part-pricing, i.e., each firm i charges a fixed fee Fi and a price per unit Pi that is equal to marginal costs (charging a price equal to marginal cost is still optimal in this setting). Determine Fi, Pi,and the firm?s profit.

Question 5 There are 200 firms in an industry. Half of the firms use newer technologyresulting in a cost function c(Q) = 200 + Q2, while the the remaining firms? costfunctions are c(Q) = 200 + 2Q2. Market demand is QD(P) = 3, 450 ? 40P. Theindustry is competitive (i..e., P = MC.)

(a) Determine the equilibrium price and quantity and the profits of both types offirms, assuming perfect competition.

(b) Now suppose that the firms with the inferior technology exit the market (theirprofit in (a) should be negative). Determine the new equilibrium price, quantityand firm profits (again, assuming perfect competition).

(c) Determine the loss to the consumers when the high-cost firms exit the market(recall that the area underneath the inverse demand curve P(Q) betweentwo values Q1 and Q2 measures the benefit to all consumers from increasingconsumption from Q1 to Q2).

(d) Taking the effect of firm profits into account determine by how much welfare increases or decreases when the high-cost firms exit the market.

(e) Should the government subsidize the industry such that all producers will remainin the market? Your answer should be based on the argument in (d).Does your answer change if the high-cost firms are domestic firms and thelow-cost firms are foreign, and all the demand for the product is from domesticconsumers only?

Question 6 Suppose a market demand function is given by QD(P) = 1, 000 ? 10P. Theproduct can be produced with a cost function C(Q) = 10, 000 + 20Q.

(a) Determine Q and P and the firm?s profit if there is a single firm.

(b) Determine total output Q, the equilibrium price, and the profit of each firm ifthere are two firms (i.e., a Cournot oligopoly).2(c) Determine Q, the equilibrium price, and the profit of each firm if there are 10firms.(d) Determine the welfare gain or loss between situations (a) and (b).

Question 7 A firm has a cost function c(Q) = 10Q. The demand function is given byQD(P) = 40 ? 2P. Determine the following graphically

1. The firm?s marginal revenue curve (assuming the firm is a monopolist).

2. The monopolist?s optimal P and Q.

3. P and Q if the market were competitive, i.e., there are many firms with costfunction c(Q) = 10Q.

4. The total net-benefit of consumers from the competitive market.

5. The total net loss in industry profits from the competitive market.

6. The total surplus from the competitive market.

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