Solution

1. The steps are as follows:

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

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

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

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

2. Inserting the numbers given into the demand equation for durables allows a to be calculated:

   D = a*M/Pe

   600 = a*5000/1

   a = 600/5000 = 0.12

   Check by using the demand for O:

   O = (1-a)*M/Po

   O = 0.88*5000/1

   O = 4400

   The value of a passes the check: the resulting value of O matches the data given in the problem.

3. D = 0.12 * 5000 / 1.2 = 500, a 17% drop from the original value of 600. The demand for other goods is unaffected. Revenue will equal $0.2 times 500 billion units = $100 billion.
4. The first step is to calculate the original utility, U1. After that, the expenditure function can be used to calculate the amount of money needed to achieve the original utility at the new prices, M3:

   U1 = 600^0.12 * 4400^0.88 = 3464.31

   M3 = 3464.31 * (1.2/0.12)^0.12 * (1/0.88)^0.88

   M3 = $5111 billion

   The CV is $5111-$5000 = $111 billion. That's bigger than the tax revenue by $11 billion: the $11 billion is the DWL of the tax.

5. Under full compensation, the demand for D would be:

   D3 = 0.12 * M3 / Pd

   D3 = 0.12 * 5111 / 1.2

   D3 = 511

   The substitution effect is 511-600 = -89. The income effect is 500-511 = -11. The total effect is -89-11 = -100. The substitution effect is much larger than the income effect because people regard D and O as fairly good substitutes (D and O are far from being perfect complements). That's why the CV for the policy is relatively small: $111 billion is large in absolute terms but it is only about 2% of the overall $5000 expenditure.