Theory

1. Suppose that a consumer's indirect utility function is given as follows:

V(P_x,P_y,I) = - (P_x + sqrt(P_x P_y)) / I - (P_y + sqrt(P_x P_y)) / I

(a) What are the uncompensated demands d_x(P_x,P_y,I) and d_y(P_x,P_y,I)?

(b) What is the expenditure function E(P_x,P_y,U₀)?

(c) What are the compensated demands h_x(P_x,P_y,U₀) and h_y(P_x,P_y,U₀)?

2. On the last problem set, you analyzed the consumer's problem with U(X,Y)=X^(1/4)Y^(3/4). Now continue this exercise. Let I=16, P_x=1, and P_y=3.

(a) What are the (Marshallian) demands for X and Y (you may use your answers from the previous problem set)?

(b) Suppose that the government gives the consumer an income subsidy of 16. What are the demands now?

(c) Suppose that the government instead decides to give an in-kind transfer of 16 units of X, which cannot be sold. What are the new consumption levels?

(d) What utility does the consumer attain at the consumption levels of part c?

(e) What is the minimum expenditure required to attain the utility level calculated in part d (if the consumer was buying the goods on the market at P_x=1, P_y=3)?

(f) What is the cash equivalent of the in-kind transfer, that is, how much do we need to subsidize the consumer's income (beyond the 16 she started with) in order to make her as well off as with an in-kind transfer of 16 units of X?

(g) In class, I gave some economic arguments against in-kind transfers. Suppose you are a benevolent dictator who can pass any law you like. Name three reasons you might choose to use in-kind transfers anyway (you may introduce additional considerations or argue why the assumptions made in class don't hold).

3. (continued from above; same utility function) We wish to compute the "ideal" change in consumer surplus from a change in the price of good X. Fix P_y=3 and I=16. Let the initial price be equal to P_x⁰ = 1, and consider an increase to P_x¹ = 16.

(a) Calculate the change in consumer's surplus from the change in prices setting U₀ equal to the indirect utility at the low price, and then repeat the calculation setting U₀ equal to the indirect utility at the high price.

(b) Use the integral "under" the Marshallian demand curve to provide an intermediate estimate of the change in consumer surplus. [Hint: follow Nicholson examples on pp. 162-167.]

4. Now suppose that the consumer's utility is U(X,Y)=sqrt(XY).

(a) Fix prices and income. Suppose that the government taxes good X with a tax t, but compensates the consumers with a rebate so that the consumer can still remain on the originial indifference curve U(X,Y)=V(P_x,P_y,I). This is a "compensated" tax.

(b) Given the prices and the tax, what "rebate" is required to keep the consumer on the indifference curve U(X,Y)=V(P_x,P_y,I)? That is, what is the difference in expenditures required to attain the old utility with the old prices, and with the tax-distorted prices?

(c) How much tax revenue is raised with the compensated tax, as a function of prices, income, and t? This can be calculated directly or as the difference between consumer expenditures at the post-tax consumption bundle and the income received by the stores at that bundle.

(d) What is the deadweight loss from taxation? Explain why it must always have the same sign. Can it ever be zero? If so, under what conditions?

5. Continuation from #4: U(X,Y)=sqrt(XY). Suppose that initially, prices are P_x⁰ = 1, P_y⁰ = 2, and then they increase P_x¹ = 2, P_y¹ = 3. We use different price indices to estimate the change in the cost of living. For each, give the formula for arbitrary prices and income, and then plug in.

(a) What is the ideal cost of living index?

(b) What are the Laspeyres and Paasche indices?

(c) Draw two graphs, each of which illustrates the bias of Laspeyres and Paasche. Do you use the same "reference utility" in each exercise? Explain.

Applications: CPI articles

1. The industry commission for Boston puts out a brochure. It announces: "Move your corporation to Boston and make your workers happy. The cost of living is lower than many other cities. We looked at the bundles of goods our citizens purchased, and then computed the cost of buying that bundle in San Francisco. It was more expensive in San Francisco. Furthermore, there are some things you just can't get in San Francisco: you can walk the historical Freedom Trail whenever you like." Does this claim follow sound economic logic? Be very precise about why or why not.

2. How does the presence of different retail outlets lead to bias in the CPI, according to the Boskin report? What is the assumption which is embodied in the CPI calculation that leads to this problem? What evidence is there that the assumption is wrong?

3. Discuss three economic implications of an upward bias in the CPI. How do each of these implications affect which political factions support the CPI?

4. The editorial in the St. Louis Post-Dispatch argues that "if the average family can afford only chicken instead of beef, does this mean inflation is being overstated? Hardly." Does this argument make sense? Explain your answer precisely.

5. The next argument in the editorial says: "if lower quality alternatives no longer exist, the ability of consumers to obtain a given product at all may require more income. Again, that's hardly an argument that inflation is being overstated." Do you find this compelling? Name some examples of quality improvements where this argument fails to apply. Can you think of any where it does apply?

6. Is it possible that the introduction of new products might coincide with a understatement of the change in the cost of living? Explain.

7. What three new products or quality changes improve your life over what it might have been 30 years ago at this same age?