UNIVERSITY OF CALGARY DEPARTMENT OF ECONOMICS ECONOMICS 357 (01) PROBLEM SET 4. Dr. J. Church Fall 2011

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1 UNIVERSITY OF CALGARY DEPARTMENT OF ECONOMICS ECONOMICS 357 (01) PROBLEM SET 4 Dr. J. Church Fall 2011 DUE: December 9, 2011 at 4:30 p.m. in SS 454 Please note that you are encouraged to work with your classmates. However, I take a very dim view of assignments which are identical: you are referred to the section on Student Misconduct in the University of Calgary Calendar for the possible consequences of not handing in your own work. You should assume that the person grading this assignment has no previous knowledge of economics. Hence your answers need to be intelligible, coherent, and comprehensive. All graphs should be graphed to scale (use graph paper), readily decipherable, and clearly labelled. You might find it helpful to read through the assignment, then go and read your class notes and the parts of the text which seem relevant and only then try and do the questions. Attempting this assignment without consulting your notes and the text is hazardous to your health and your grade. Please read the preceding sentence again! Part 1: Short Answer 1. The Government of Alberta claims that high voltage electric transmission lines are a public good. Explain how you would evaluate this claim. 2. Do you agree that traffic lights are efficient because they allocate property rights to an intersection? Explain. 3. What is the famous Arrow Impossibility Theorem? Why is it important? How is it related to the Second Fundamental Theorem of Welfare Economics? What would you say about a proposal to determine the social state by majority voting? 1

2 F11 Econ 357(01) Problem Set 4 2 Part 2: Problems 1. Consider the Robinson Crusoe Economy in the Post-Walras Arrival Age. Robinson s utility function over yams (y) and leisure (l) is U (y, l) = 2y+ l. Robinson s endowment of time is L. The production function for yams is y = αh where h is the amount of labour used. Let yams be the numeraire. Useful information: (i) the marginal product of labour is dy = α ; the dh marginal utility of yams is du = 1 dy 2y ; the marginal utility of leisure is 1; and the marginal disutility of labour is 1. (a) Find the supply curve of yams for Robinson the Manager. (b) Find the demand curve of yams for Robinson the Consumer. Illustrate with a diagram. (c) Explain the conditions required for general equilibrium and illustrate with a diagram. (d) What are the general equilibrium prices? [Hint: This is very simple if you think about your answer in (a) and understand that competition and the special nature of the production function means that wage rate is determined only by costs of production. Why?] (e) Suppose that Friday introduces a flat tax on labour income, i.e., for every hour that Robinson works he pays the government t w and receives (1 t) w. The entire proceeds of the tax are rebated back to Robinson as a lump sum (LS). Using a diagram illustrate the new general equilibrium. [HINT: Robinson the consumer does not realize that the lump sum rebate depends on how much he works.] (f) Find the general equilibrium allocation for a tax equal to t: yams, labour, profits, and Robinson s income. (g) Explain why the equilibrium without a tax is Pareto Optimal but the equilibrium with a tax is not. 2. Consider an individual with the following utility function: U(r, y) = ln(r + 1) + y. The marginal utility of r is 1/(1 + r); the marginal utility of y is 1. (a) Find the marginal rate of substitution of y for r. Is there anything unusual about the MRS yr? What does this imply about the indifference map?

3 F11 Econ 357(01) Problem Set 4 3 (b) What is the definition of compensating variation (CV)? (c) Find the CV for any level of r where the change is that r is banned. (d) Find the inverse demand curve for r. (e) On a graph show the change in consumer surplus for a ban on r assuming that the price of r is zero and initial consumption is r = 4. (f) For any r what is the change in consumer surplus from a ban, assuming that the price of r is zero? (g) Show that the change in consumer surplus in these circumstances equals the CV! (h) Find the CV for an individual with the following preferences if q is banned and the price of q is zero: U(q, y) = 2q 1/2 + y. 3. Tale of Two Houses. Two houses are adjacent. In one lives a group of students who love to listen to Rock N Roll at volumes which are rather loud. In the other lives Professor O. Fogy, who considers any tune written after the middle of the seventeenth century noise. Her current research project requires all of her limited mental powers: peak performance requires silence. The utility function of the students is U(r, y s ) = ln(r + 1) + y s, where r is hours spent listening to Rock N Roll and y s is expenditure on all other goods. The Professors utility function is U (r, y p ) = 2q 1/2 + y p, where q is hours of quiet and y p is expenditure on all other goods. The students marginal rate of substitution of other goods for hours of Rock N Roll is 1/(r + 1). The Professor s marginal rate of substitution of other goods for hours of quiet is 1/ q. The number of total hours that the Professor and the students are both at home is 5. (a) If the students were presently listening to Rock N Roll for r hours per day, What is the amount they would have to be compensated if they were prohibited from listening to any Rock N Roll? What is this called? [HINT: See preceding question.] (b) If the Professor was currently enjoying 5 hours of quiet and was forced to listen to r hours of Rock N Roll, How much would she have to be compensated?[hint: See preceding question.]

4 F11 Econ 357(01) Problem Set 4 4 (c) What are the marginal social benefit and marginal social cost functions for Rock N Roll? On a graph show the marginal social benefit of another hour of r and the marginal social cost of another hour of r. How are these functions related to the functions you found in parts a) and b)? What is the socially optimal amount of Rock N Roll? Why? Indicate it on your graph. (d) Indicate on your graph the endowment or starting point if the students have the property rights to noise pollution. Calculate and indicate on your graph the following magnitudes: i. the maximum the professor will pay the students; ii. the minimum the students will accept; iii. the amount the students and the professor are jointly better off by if the professor and the students exchange cash for the rights to noise pollution. (e) Indicate on your graph the endowment or starting point if the professor has the property rights to noise pollution. Calculate and indicate on your graph the following magnitudes: i. the maximum the students will pay; ii. the minimum the professor will accept; iii. the amount the students and the professor are jointly better off by; if the professor and the students exchange cash for the rights to noise pollution. (f) Suppose that the students have the property rights to noise pollution but the professor is unable to make a deal with the students due to transaction costs. The professor lobbies City Hall to impose a tax on hours of Rock N Roll. What is the optimal tax? What is the optimal amount of Rock N Roll? Who benefits and who losses from the tax? By how much? Is this a Pareto improvement? Why? Indicate the benefits, costs, the optimal tax and the optimal r on an appropriate diagram. 4. Mick and Keith only consume crackers and music Chuck Berry records to be exact. Crackers are a pure private good and music is a pure public good. Their utility functions are:

5 F11 Econ 357(01) Problem Set 4 5 and U M (c M, M) = c M M (1) U K (c K, M) = c K M, (2) where c M is Mick s cracker consumption, c K is Keith s cracker consumption, and M is the amount of music consumed by the two of them. The marginal utility of i with respect to good j is MU i j = x q where i = Mick or Keith; j= crackers or music; q is the opposite of j; and x q is the quantity of q. Music is measured in hours and the production function for music is constant returns to scale: 10 crackers produces one hour of music. Mick s endowment is 50 crackers; Keith s 50. (a) What is the optimal amount of Music? (b) How much will be provided if they each voluntarily decide how much Music to provide? [HINT: To solve this question you must follow these steps: First find how much each is willing to provide as a function of how much the other provides, i.e., find the best-response function of each Stone. Then find a Nash equilibrium in amounts provided. You may appeal to symmetry.] (c) Explain why the amount of music provided in parts (a) and (b) are different.