Problem set 5 solutions PDF

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Problem set 5 solutions

Problem 1. A small town is served by many competing supermarkets, which have the same constant marginal cost. 1. Using a diagram of the market for groceries, show the consumer surplus, producer surplus, and total surplus. 2. Now suppose that the independent supermarkets combine into one chain. Using a new diagram, show the new consumer surplus, producer surplus, and total surplus. Relative to the competitive market, what is the transfer from consumers to producers? What is the deadweight loss? Solution. 1. The figure below illustrates the market for groceries when there are many competing supermarkets with constant marginal cost. Output is Q C , price is PC , consumer surplus is area A, producer surplus is zero, and total surplus is area A.

2. If the supermarkets merge, the figure below illustrates the new situation. Quantity declines from Q C to Q M and price rises to PM . Area A in the previous figure is equal to area B + C + D + E + F in this figure. Consumer surplus is now area B + C, 1

producer surplus is area D + E, and total surplus is area B + C + D + E. Consumers transfer the amount of area D + E to producers and the deadweight loss is area F. Problem 2. Larry runs the only pub in town. The following equations describe Larry’s demand, marginal revenue, and marginal cost: Demand: P = 10 − Q Marginal Revenue: MR = 10 − 2Q Total Cost: TC = 3 + Q + 0.5Q2 Marginal Cost: MC = 1 + Q Where Q is the quantity of drinks and P is the price. 1. How many drinks does Larry sell? At what price are they sold? What is the Larry’s profit? 2. What is the price elasticity of demand at Larry’s monopoly price and quantity? Is the demand curve elastic at the monopoly quantity? (Hint: price elasticity of demand = (∆Q/Q ) / (∆P/P) = (∆Q/∆P) × ( P/Q ) = (1/slope of demand curve) ×

( P/Q )). Solution. 1. Set MR = MC and solve for Q. That is, 10 − 2Q = 1 + Q, or 3Q = 9, or Q = 3. Larry sells 3 drinks. Substitute Q = 3 into the demand equation to find the price. That is, P = 10 − Q = 10 − 3 = $7. Total revenue is thus P × Q = 7 × 3 = 21, while total cost is TC = 3 + Q + 0.5 Q2 = 3 + 3 + 0.5 × 32 = 10.5. Larry makes a profit of $10.5. 2. The slope of the demand curve is 1. Thus, the price elasticity of demand is 1/1 × P/Q = 7/3 = 2.33. The demand curve is elastic at the monopoly quantity. Problem 3. A company is considering building a bridge across a river. The bridge would cost $2 million to build and nothing to maintain. The following table shows the company’s anticipated demand over the lifetime of the bridge:

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Price per crossing Number of crossings, in thousands 8 7 6 5 4 3 2 1 0

0 100 200 300 400 500 600 700 800

1. If the company were to build the bridge, what would be its profit-maximizing price? Would that be the efficient level of output? Why or why not? 2. If the company is interested in maximizing profit, should it build the bridge? What would be its profit or loss? 3. If the government were to build the bridge, what price should it charge? 4. Should the government build the bridge? Explain. Solution. The following table shows total revenue and marginal revenue for the bridge: Price per crossing Number of crossings, Total revenue, Marginal revenue in thousands in thousands 8 7 6 5 4 3 2 1 0

0 100 200 300 400 500 600 700 800

0 700 1200 1500 1600 1500 1200 700 0

– 7 5 3 1 -1 -3 -5 -7

1. The profit-maximizing price would be where total revenue is maximized, which will occur where marginal revenue equals zero, since marginal cost equals zero. This occurs at a price of about $4. The efficient level of output is 800, since that’s where price equals marginal cost equals zero. The profit-maximizing quantity is lower than the efficient quantity because the firm is a monopolist. 3

2. The company shouldn’t build the bridge because its profits are negative. The most revenue it can earn is $1.6 million and the cost is $2 million, so it would lose $400,000. 3. If the government were to build the bridge, it should set price equal to marginal cost to be efficient. But marginal cost is zero, so the government shouldn’t charge people to use the bridge.

4. The government should build the bridge, because it would increase society’s total surplus. As shown in the figure above, total surplus has area 1/2 × 8 × 800, 000 = $3.2 million, which exceeds the cost of building the bridge.

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