Problem Set 9



1. Nicholson 20.2.

2. Suppose that a firm produces in two markets, a protected domestic market where P1 = 80 - q1, and a competitive world market, where P2 = 30. Suppose that the monopolist's total costs are given by TC = 1200+.25(q1+q2)2, where qj is the quantity supplied to market j.

(a) What is the firm's total profit function (including both markets)?

(b) What are the optimal quantities supplied to each market? The corresponding prices?

(c) At the optimal quantities, what is the marginal revenue in each market? What is the marginal cost of production?

(d) What are the firm's average costs?

(e) Many international trade agreements explicitly prohibit "dumping," where a firm sells in a foreign market at a price below its average costs in order to drive foreign firms out of the market. Based on your above analysis, should we presume that pricing below average cost is necessarily motivated by a desire to drive out competitors?

3. Suppose that demand for tickets for the MIT-Cal Tech battle-of-the-robots competition is given by PP=20 qP/6000 for the general public, and by PS=8 qS/2500 for students. MIT can distinguish between students and the general public.

(a) If the arena has an unlimited number of seats, then what prices maximize total revenue for MIT? How many seats are sold? What is the deadweight loss?

(b) Suppose that one end of the stadium is closed for repairs, so that there are only 53,000 seats available. Then, what are the revenue-maximizing prices and quantities? [Hint: set up a Lagrangian].

(c) What is the maximum amount any student is willing to pay? At what price can the university sell out the whole stadium to the general public?

(d) Evaluate the following statement: altruism and a desire to generate future donations are the only possible explanations for universities (or other institutions, for that matter) to sell tickets at low prices to students.

4. Suppose that an internet service provider knows that there are two types of potential subscribers, heavy users and light users, but the company can't identify which users are which. The marginal cost of access time is constant at 0. There are 1000 identical heavy users and 500 identical light users in the population. The demand curve for hours online is given by P = 10 - .5 q for heavy users and by P = 8 - .8 q for light users.

(a) What is the total surplus-maximizing (not profit-maximizing) number of hours used by each type of user (call these qH* and qL*)? What is the total surplus generated by each type of user at those quantities (call these TSH(qH*) and TSL(qL*)?

(b) The service provider considers announcing the following price schedule: anyone can buy qH* hours of time for a price of TSH(qH*), or else they can buy qL* hours of time for TSL(qL*). No other options are available. What will the light users choose? The heavy users?

(c) Suppose that the service provider is committed to targeting qL* to the light users and qH* to the heavy users. (i) Does lowering the price of qL* hours make the heavy users more or less likely to purchase qH*? (ii) What is the highest price for qL* which induces the light users to buy it? (iii) What is the highest price for qH* which induces the heavy users to buy qH* instead of qL*? (iv) Illustrate graphically the surplus received by each type of user (following the diagrams from class).

(d) Now suppose that the provider can choose any two price-quantity combinations it wants. (i) What are the optimal choices? (ii) What are firm profits? (iii) How much surplus does each type of user receive? (iv) Compare your answer to part (c). (v) Show graphically the surplus each user receives, and argue (referring to your graph) that each user finds it optimal to choose the bundle which the firm targets to that user's type. Also show on your graph the condition which makes the firm indifferent between lowering and raising qL (as in class). [Hint: Proceed by picking any qL and qH, and figuring out what the optimal price to charge the low user will be given any qL, and then figuring out what the price to charge the high user must be to make the high user exactly indifferent between the low and high bundles. Then, choose quantities to maximize total revenue. Remember that there are different numbers of each type of user.]

5. Suppose that Mathematica can be used to solve problem sets by undergraduates and to solve very hard problems by physics professors (or easier problems by economics professors who have a high outside value of their time, so that the demand for speed is identical!). Suppose that 1/4 of the potental Mathematica customers at MIT are undergraduates, and 3/4 are professors, and that there are no relevant substitutes. However, Wolfram research cannot distinguish between students and professors at the MCC (or, each professor has a UROP who can be sent over to buy the student version if the prices are different). Mathematica is considering eliminating the calls to the math coprocessor in a student version and packaging the student version separately, at a marginal cost c for special handling of student versions. The marginal cost of producing the standard version is 0. Professors value the goods (H for standard and L for student) at vP(L) = 1/4 and vP(H) = 1/2. Undergraduates value the goods at vS(L) = 1/5 and vS(H) = 1/3.

(a) In the case where the monopolist produces only the standard version, does she serve the whole market or just professors? What is the price charged and the profits?

(b) Suppose that the monopolist decides to produce the damaged good. At what price will students be just indifferent between buying the student version and not buying? At what price for the standard version are professors just indifferent between the student version and the standard version? What are the monopolist's profits at those prices (as a function of c)?

(c) At what cost c is the monopolist just indifferent between offering both versions, and offering only the standard version (assuming that the monopolist maximizes profits in each case)?

(d) In this problem, should a government regulator be concerned about the prospect of Wolfram Research offering a damaged good?

6. Suppose that the time it takes for a person to drive from Boston to Cape Cod on Hwy. 1 is given by 3 + .002 X1, where X1 is the total number of people on the highway. On Hwy. 2, the time is 4 + .001 X2. There are 2000 people who want to go to the Cape for the weekend.

(a) If a social planner wanted to minimize the total time spent getting all 2000 people to the beach, how many people would the social planner ask to drive on each highway?

(b) If people decide to stay home if it will take more than 6 hours to get to the beach, but each individual makes his own personal decision about which road to use taking as given the decisions of others, what is the equilibrium number of people who will take each road to the Cape?

(c) Compare your answers to (a) and (b), and briefly comment on similarities and/or differences.

(d) Suppose that the government could put in a toll on Hwy. 1. What amount of toll would induce the socially efficient number of people to take each highway, but still have all 2000 people make it to the Cape?


(Minimum Wage, Comparable Worth, and Monopsony)

1. In our study of minimum wage and our study of comparable worth, we argued that perfect competition was critical for our conclusions that regulated wage increases would necessarily reduce employment.

Illustrate using mathematical formulas and a graph of labor supply, labor demand (by the monopsonist) , and "marginal expense" (the analog of marginal revenue) how a monopsonist decides how much labor to purchase. Prove that a small regulated wage increase can raise employment and total welfare in a monopsonistic labor market, but that a large mandatory increase might decrease employment.

(Krugman AER and Monopoly)

2. We have argued that one reason for a "natural monopoly" is increasing returns to scale. Suppose that there is free trade in an increasing returns industry. Suppose further that two countries have identical resources and available technologies, and that transportation costs are zero.

(a) Do we necessarily expect to see both countries producing in this industry?

(b) Do we expect neoclassical results about the efficiency of perfect competition to hold?

(c) What does Krugman think about the relevance of these kinds of considerations for trade policy?