API 111 – Econ 2020a – HBS 4010 Fall 2007 Problem Set #3

Due: Wednesday, October 17, 2007 Due to the Columbus Day Holiday, you have two weeks to complete this problem set. Depending on how much we cover, we may add one or two additional problems, also to be due on October 17th. Please turn in your problem set by noon on the date the assignment is due. Assignments not handed in in class should be put in the assignment drop box on the second floor of KSG, around the corner from room L239 (near the elevator on the second floor). Although you may work in groups on you assignment, each person should write up and hand in their own assignment. Also, note the names of your group members on your problem set. Problem sets submitted late will not be given credit unless prior arrangements are made. 1. Expenditure Minimization: Cobb-Douglas Utility Consider the utility function:

u (x1, x2) = x1a1 x2

a2

Let p1 and p2 be the (strictly positive) prices of x1 and x2, respectively. a) Derive the consumer’s Hicksian demand functions. b) Derive the consumer’s expenditure function. c) Let w denote the consumer’s wealth. Using the solution to last week’s problem set, verify that h1(p,v(p,w)) = x1(p,w) and e(p,v(p,w)) = w. 2. Utility Maximization: The Labor-Leisure Tradeoff A consumer has utility function:

U(L,F) = LAF1-A Where L is the number of hours of leisure the consumer has during the day, F is the amount of food he consumes, and 0 < A < 1. The consumer has a total of 24 hours per day which he must divide between leisure and labor. For each hour of labor, the consumer earns wage s, and each unit of food costs p dollars. The consumer has initial wealth w. 3a) Write down an expression for the number of hours the consumer spends on labor in terms of the notation above. Write down the consumer’s budget constraint. 3b) Write down the consumer’s utility maximization problem. Hint: do not forget the appropriate resource constraints on the availability of labor and leisure time. Derive the Kuhn-Tucker conditions. 3c) Derive the consumer’s Walrasian demand functions for leisure, L(p,s,w), and food, F(p,s,w). You may assume an interior solution (i.e. that L(p,s,w) and F(p,s,w) are strictly positive for all levels of p, s, and w.) 3d) In words, what is the meaning of the optimal value of the Lagrange multiplier λ(p,s,w)? 3e) How will the consumer’s consumption of food be affected by changes in s, p, or w?

3f) How will the consumer’s consumption of leisure be affected by changes in s, p, or w? 3) (Based on Varian, Microeconomic Analysis, ex. 10.2) Consider a consumer with utility function

( ) 1/ 2 1/ 21 2 1 2,u x x x x= and income of $500 per week. Suppose that prices are p1 = 1 and p2 = 1 and

that the consumer spends his entire paycheck each week. The consumer’s boss asks him to move to a new city that is identical to his current city except that p1 = 1 and p2 = 2. The boss offers no raise in pay. The consumer tells his boss: “Asking me to move is just like if I stayed here and you cut my pay by $A. I would be willing to move to the new city, but you would have to pay me $B more for me to be willing to do it.” Find values for A and B, and relate them to concepts we discussed in class. If A = B, explain why. If not, explain why not.

4) Based on empirical data, you believe the following about a “typical” consumer’s consumption of

widgets (widgets are an imaginary good economists use – you do not need to know what a widget is to correctly answer this question). Currently, the price of widgets is p0 = $8, and a typical consumer consumes x0 = 11 widgets. You have estimated the following parameters of the consumer’s utility function.

dx

dp= −0.5 and

dx

dw= 0.01.

Suppose the government is considering imposing a tax that will raise the price of widgets $2 to p1 = $10. The remainder of this question asks you to use this information to estimate the welfare impact of this price change. In estimating the various quantities, you may assume that Walrasian and Hicksian demand curves are linear.

5a. Estimate how many widgets this consumer consumes after the price increase. 5b. Estimate the equivalent variation of this price change. 5c. Estimate the compensating variation of this price change. 5d. Estimate the change in consumer surplus associated with this price change. 5e. Why is CV the most appropriate measure of the welfare change? 5f. Is the change in consumer surplus a good estimate of CV in this case? 5g. Compute the tax revenue. Also compute the deadweight losses associated with each of the three

welfare measures. Is the deadweight loss computed using the change in consumer surplus a good approximation of the deadweight loss computed using CV?

5. MWG question: Please answer MWG 3.I.6. Include with your answer to MWG 3.I.6. a response to the following: 3.I.6. continued: Economists frequently evaluate policy changes based on the sum of compensating (or equivalent) variations. In particular, they consider a policy good for society as a whole if and only if the sum of CV over all consumers is positive. Explain why this is a good indicator of whether a policy is good for society. Also, provide one or two reasons why this measure may not be a good indicator.