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
*p*_{f}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
*p*_{f}= 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 each boat. - Consider a monopolist that wishes to price discriminate by means of a "block pricing"
scheme: buyers must pay
*p*_{1}for purchases up to amount*q*_{1}, then benefit from a lower price*p*_{2}for any further purchases. (Let the consumer's total, freely chosen purchases be denoted "*q*_{2}"). 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
*p*_{1},*q*_{1}, and*p*_{2}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
*t*_{A}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(*t*_{A}+*t*_{A}). Anna derives utility from her economics essay score (*S*) and hours of leisure (*R*) according to*U*_{A}=*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*MC*^{P}= 80 +*Q*, and the marginal harm from pollutants emitted by paper manufacture*MC*^{G}=*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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