Introductory Microeconomics: Problem Set 7

Market failure; return to price discrimination. Problems 2-4 of this set are from the Perloff text, with minor modifications.
  1. The main economic activity in the villages around lake Isis is fishing. Anyone can send a boat out to fish but doing so incurs a cost equal to thirty percent of the price of a ton of fish. The total number of tons of fish caught depends on the number of boats b fishing in the Lake: f (b) = ln(b + 1). Total catches are distributed equally across boats.

    1. Write the profit function for each boat as a function of the price of a ton of fish pf and the number of boats fishing b.
    2. A person thinking of sending a boat out to fish will do so until returns are driven to zero. Assuming pf = 1, how many boats will there be fishing in the lake?
    3. If one profit-maximising, perfectly competitive firm owned the lake, how many boats would the firm hire at a cost of c = 0.3 per boat to fish in the lake? In what sense is this the optimal number of boats?
    4. A social planner wants to impose a tax per boat in order to restore efficiency in the de-centralised case (b). Find the optimal t that the planner will charge each boat.

  2. Consider a monopolist that wishes to price discriminate by means of a "block pricing" scheme: buyers must pay p1 for purchases up to amount q1, then benefit from a lower price p2 for any further purchases. (Let the consumer's total, freely chosen purchases be denoted "q2"). The firm's customers are identical and have inverse demand functions P = 90 - Q. The firm has costs MC = AC = 20.

    1. If charging a single price for all purchases (by a single customer), what price should the firm charge to maximise profits? (Would your answer be different for a larger market of many identical such customers?)
    2. Illustrate in a diagram how the firm's new block pricing scheme can be used to increase profits earned.
    3. What are the optimal choices of p1, q1, and p2 to maximise profits earned from an individual consumer?

  3. Anna and Bess are assigned write a joint essay in a 24 hour period about the Pareto optimal provision of public goods. Let tA be the number of hours Anna contributes and similarly for Bess. The essay score is a function of the total hours invested in the project by both students: S = 23 ln(tA + tB). Anna derives utility from her economics essay score (S) and hours of leisure (R) according to UA = S + ln R, and similarly for Bess.

    1. If they choose simultaneously and independently how many hours to contribute, what is the Nash equilibrium investment of time in the project?
    2. What is the number of hours each should contribute in order to maximise the average of their utilities?

  4. Suppose the inverse demand curve for paper is P = 200 - Q, the private marginal cost is MCP = 80 + Q, and the marginal harm from pollutants emitted by paper manufacture MCG = Q. (Following Perloff, G stands for "gunk".)

    1. What is the unregulated competitive equilibrium?
    2. What is the social optimum? What specific (i.e. per-unit) tax results in the social optimum?
    3. What is the unregulated monopoly equilibrium?
    4. How could the monopoly be optimally regulated? What is the resulting equilibrium?

  5. Be prepared to discuss public goods and the Coase theorem for your tutorial, starting with a clear definition and examples of both before considering any more technical issues.

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