Taxing Durable Goods

One place where compensating variation is often used is in the analysis of tax policies. To see how it works, consider the following example.

Suppose people buy two things: durable goods such as cars, D, and all other goods, O. In addition, suppose that consumers' preferences between D and O can be represented by a Cobb-Douglas utility function of the form:

U(D,O) = D^a * O^(1-a)

The demand equations consistent with this utility function can be shown to be:

D = a*M/Pd

O = (1-a)*M/Po

Please answer the following questions:

1. Show that the consumers' expenditure function is given by:

   M(U,Pd,Po) = U * (Pd/a)^a * (Po/(1-a))^(1-a)

2. In 1995, total consumer spending, M, was about $5 trillion. About $600 billion of that was durable goods and $4400 billion was spent on everything else. Suppose the prices of durables, Pd, and everything else, Po, are both initially $1. Use this information to determine the value of parameter a. Use the value you obtain in the remainder of the problem.
3. Suppose the government decides to raise revenue by imposing a 20% tax on durables (so that Pd rises to $1.20). How much does durable demand decrease? What happens to the demand for other goods? How much revenue will the tax raise?
4. Calculate the compensating variation for the 20% tax. Discuss how this compares to the revenue raised.
5. Decompose the change in D caused by the tax into components due to the income and substitution effects. (That is, calculate the numbers.) Show all your work. Which effect is larger? Why does that make sense? How does this relate to your answer to the previous question?