Problem: Suppose a consumer has utility function u(x1, x2) = x1x2, and therefore demand functions
x1 = 3/2pb1
x2 = m/2pb2
Recall that pb indicate a buyer's price. Let ps1 = 2 and ps2 and m = 72. Also let r1 = 1be a per-unit tax on the consumption of good 1.
a. Draw the consumer's budget constraint and optimal choice when three is no tax. Calculate the utility the consumer gets.
b. On the same diagram as (a), draw the consumer's budget constraint and optimal choice when the tax is imposed. Calculate (i) the tax collected, T, and the new utility the consumer gets, ut.
c. Supposed we directly removed T from part (b) as lump-sum tax. Calculate what amounts of each good the consumer would choose, and show that their utility would behiger than part (b).
d. How muchlump-sum tax could be removed from the consumer, such that their utility would be the same as in part (b) [i.e., with the per-unit tax on good 1 only]? Hint: use the defination of the utility function and demands. Use your answer to calculate the excess burden of the per-unit tax.
e. What is the excess burden per dollar of tax collected?
f. Suppose we imposed the per-unit tax on good 1 and collected $T. We then immediately returned $T to the consumer as a lump sum, and allowed othem to spend this T on good 1 for $2 and on good 1 for $1. Show that the consumer's utility would still not rise to the pre-tax level. Why is this? Explain in words.
g. Show using a diagram that we could have raised $T using as equal tax rate t on both goods, and that there would have been no excess burden in this case. Calculate the necessary tax rate, t.
