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.
- 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.
- 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.
- 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?
- 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?
- 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
- 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.
- 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?)
- Illustrate in a diagram how the firm's new block pricing scheme can be used to
increase profits earned.
- What are the optimal choices of p1, q1, and
p2 to maximise profits earned from an individual consumer?
- 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.
- If they choose simultaneously and independently how many hours to contribute, what
is the Nash equilibrium investment of time in the project?
- What is the number of hours each should contribute in order to maximise the average
of their utilities?
- 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".)
- What is the unregulated competitive equilibrium?
- What is the social optimum? What specific (i.e. per-unit) tax results in the social optimum?
- What is the unregulated monopoly equilibrium?
- How could the monopoly be optimally regulated? What is the resulting equilibrium?
- 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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