# Project #59489 - Advanced Economics

Problem 1. Consider an island with Tom Hanks and Wilson, and one good - coconuts. There is NO endowment of coconuts, and to have something to eat Tom Hanks and Wilson have to work climb the palm trees and gather coconuts. In one hour, Tom Hanks can gather wTH and Wilson wW coconuts. Both Tom Hanks and Wilson have T hours at disposal. Suppose that the utility functions for person i with i ∈{TH,W} is

Ui(ci,`i) = logci + Bi log`i

where ci is consumption of coconuts, `i is the time spent lying on beach and surfing, and BTH and BW are positive constants.

(a)    Write down the problem of person i ∈{TH,W}, if Tom Hanks and Wilson are on their own and no trade is possible.

(b)    Use results from the worked example in the math revision to write the expressions for optimal choices ciand `i. How do they depend on wi and Bi? Explain the intuition behind this.

(c)    Suppose that Tom Hanks and Wilson could trade if they wanted. Would they trade? Hint: Think about what the person who would like to buy a coconut could offer in return for the coconut bought.

(d)    Given your answer what can we say about the competitive equilibrium allocation? How is it different from the allocation cTH,`TH,cW,`Wfrom part (c)?

(e)    Suppose now that Tom Hanks and Wilson decide to pool the coconuts they gather. Write down the aggregate resource constraint for coconuts on this island - an equation that shows how total consumption of coconuts by Tom Hanks and Wilson cTH + cW is limited by the amount of coconuts available that depends on `TH,`W,wTH,wW and T.

(f)     Suppose that the social welfare function of the social planner is

W(UW,UTH) = φWUW + φTHUTH

where UW = U(cW,`W) and UTH = U(cTH,`TH). Write down the social planner’s problem, as a problem of maximizing this social welfare function subject to the resource constraint.

(g)    Solve the social planner’s problem using the Kuhn-Tucker theorem.

(h)    Suppose that φTH = φW = 1 and that Tom Hanks and Wilson have same preferences and time endowment and so BTH = 3,BW = 2, T = 20. Suppose further that it is much easier for Tom Hanks to climb the palm trees: wTH = 7, wW = 2. Calculate cTH,`TH,cW,`Wand

the social planner allocations cTHSPP,`THSPP,cWSPP,`WSPPusing the above results. Compare these allocations. Who benefits from redistribution in this example?

1

Note : Problems 2 and 3 are adaptations of questions found in the Gruber textbook.

Problem 2: The private marginal benefit associated with a product’s consumption is PMB = 300 − 5Q and the private marginal cost associated with its production is PMC = 3Q. Furthermore, the marginal external damage associated with the good’s production is 2Q. What tax should the government impose to achieve the efficient allocation? Illustrate your answer with a diagram.

Problem 3: Firms A and B each currently produce 100 units of pollution. The federal government wants to reduce pollution levels. The marginal costs associated with pollution reduction are MCA = 100 + 2QA and MCB = 40 + 6QB for firms A and B, respectively, where QA and QB denote the amount of reduction. Society’s marginal benefit of reduction is MB = 640−4QT where QT is the total level of pollution reduction. Answer the following:

1.    What is the socially optimal level of each firm’s production?

2.    How much total pollution is there in the social optimum?

3.    Explain why it is inefficient to give each firm an equal number of pollution permits.

4.    Explain the the social optimum can be achieved if firms are given equal permits but are allowed to trade them.

5.    Can the social optimum be achieved using a tax on pollution?                                                                                                                                                                                THERE ARE MINOR MISTAKES IN THE QUESTION

 Subject Mathematics Due By (Pacific Time) 02/25/2015 11:00 am
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