Ad valorem taxes Throughout this chapter our analysis of taxes has assumed that they are imposed on a per-unit basis. Many taxes (such as sales taxes) are proportional, based on the price of the item.

In this problem you are asked to show that, assuming the tax rate is reasonably small, the market consequences of such a tax are quite similar to those already analyzed. To do so, we now assume that the price received by suppliers is given by P and the price paid by demanders is P 11 1 t2, where t is the ad valorem tax rate (i.e., with a tax rate of 5 percent, t 5 0.05, the price paid by demanders is 1.05P). In this problem then the supply function is given by Q 5 S 1P2 and the demand function by Q 5 D 3 11 1 t2P4.

a. Show that for such a tax d ln P dt 5 eD,P eS,P 2 eD,P .
(Hint: Remember that d ln P/dt 5 1 P # dP dt and that here we are assuming t < 0.)

b. Show that the excess burden of such a small ad valorem tax is given by:
DW 5 −0.5 eD,P eS,P eS,P 2 eD,P t 2 P∗Q∗.

c. Compare these results to those derived in this chapter for a per-unit tax. Can you make any statements about which tax would be superior in various circumstances?