Introductory Microeconomics: Problem Set 7
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. 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?

3. You observe a consumer who has a budget of 4 and is choosing from two products (Good 1 and Good 2) which have prices p1 = 2 and p2 = 1. You see that she is buying one unit of Good 1 and two units of Good 2 (x1 = 1, x2 = 2).

1. Draw and fully label the consumer's budget constraint illustrating her demands and sketch in an indifference curve which would rationalise her observed choice (Good 1 on the horizontal axis).
2. You think that Good 1 is a product for which income effects are likely to be negligible and therefore you hypothesise that this consumer has preferences of the form u = log(x1) + βx2, where β > 0. Derive the Marshallian demand curves consistent with these preferences.
3. Show that, conditional on your hypothesis, the consumer must have a parameter value of β = 1/2.
4. Calculate the income elasticities of demand for each good.
5. Good 2 is going to be withdrawn from sale. If this happens what will the consumer's demand be? Draw an appropriate diagram illustrating her demands and her indifference curves before and after the withdrawal of this product.
6. Calculate a money-metric measure of the loss in economic welfare resulting from this reduction in choice. Illustrate your calculations using an appropriate diagram.
7. Suppose that this consumer is a member of a population of consumers, all of whom have identical tastes but who differ in their incomes. Explain the likely distributional effects of the withdrawal of this product.

Question 3 appeared on the 2016 prelims examination.

4. Consider the Dictator Game. Player 1 has a stake of size 1 which she can divide between herself and Player 2. Player 2 is passive recipient. Let x1 denote Player 1's retained portion and let x2 (which equals 1 - x1) denote her offer to Player 2. Suppose that Player 1 has a utility function given by:

u1 = x1 - (α/2) · {(x1 - x)2 + (x2 - x)2},

where α > 0 and x is the average pay-off for the two players.

1. What general properties does this utility function have? How is inequality-aversion displayed? What restrictions on this model would give you the homo economicus, egoistic, special case?
2. Show that this utility function can be written as: u1 = x1 - (α/4) · (x1 - x2)2.
Hint: you can think geometrically rather than algebraically, if you wish, and use the fact that in this case x = 1/2 and x1 and x2 must be located symmetrically around it.
3. Derive the offer (x2) as a function of the inequality aversion parameter α.
4. Show that the offer to Player 2 is increasing in α.
5. Show that this utility function can rationalise any offer less than or equal to 1/2, but could not explain why anyone would wish to give away more than half the stake.
6. Suppose that you observe
1. someone offering 1/4. What is their α?
2. someone offering nothing. What is their α?