AS/ECON 4070 3.0AF     Answers to Midterm Exam     October 2000

1. If there is an excise tax at the rate t on the ( net-of-tax ) price of the good, then

PD = ps(1+t)

where p_s is the price received by sellers and P^D is the price paid by buyers. The equilibrium condition, that quantity demanded equal quantity supplied, then becomes

6ps = 144 - 10ps(1+t)

or

[16 + 10 t]ps = 144

When there is no tax, t = 0, so that the equilibrium price solves

16 ps = 144

or

ps = 9 = PD

With a twenty percent excise tax, then t = 0.2, so that

[16+10(0.2)]ps = 144

or

18 ps = 144

So that

ps = 8

when the tax is imposed. The tax therefore makes the price received by sellers fall from 9 to 8, and the price paid by buyers increase from 9 to 8(1.2) = 9.6. Since the buyers' price rises by 0.6 and the sellers' price falls by 1, then sellers are bearing a share 1/1.6 = 5/8 » 0.62 of the tax.

Another way of approximating this result is to use elasticities. Without any taxes, p_s = P^D = 9. From the definitions of the demand and supply functions

eD = - ¶QD ¶PD PD QD = 10 9 54 = 1.6667

es = ¶Qs ¶ps ps Qs = 6 9 54 = 1

Therefore, sellers should bear a share of approximately

eD es + eD = 10 10+6 = 5 8

of the cost of the tax.

2. This was a question about the output and factor substitution effects in the Harberger model, as outlined, for example, on pages 443 and 444 of the text. The factor substitution effect of any specific factor tax on the use of capital in one sector of the economy is a reduction in capital's net return, relative to labour's. The direction of the output effect depends on the factor intensity in the taxed sector, relative to the untaxed sector. If the taxed sector is more capital-intensive, then the output effect is a reduction in capital's net return relative to labour's, reinforcing the factor substitution effect. On the other hand, if the taxed sector is more labour intensive, then the output effect is an increase in capital's net return relative to labour's, an effect working in the opposite direction to the factor substitution effect.

It seems most plausible to assume that the service sector is more labour intensive than the rest of the economy. If that were true, then the output and factor substitution effects of a tax on the use of capital in the service sector would work in opposite directions from each other. It is not necessarily true that capital would bear more of the tax than labour. [ That is, the answer would be ``uncertain''.] Assuming the service sector is more labour intensive, capital would bear more of the tax only if the factor substitution effect were bigger than the output effect.

3. An individual consumer's budget constraint is

PX X + PY Y = m

where P_X and P_Y are the prices paid by the consumers for goods X and Y, and m is her disposable income. A comprehensive tax on all her consumption at the rate t is equivalent to a reduction in her disposable income, to a fraction 1/(1+t) of what it was. ( That is, raising all prices by 10 percent is the same as lowering your income to a fraction 1/1.10 of what it was. ) Thus a comprehensive sales tax is equivalent to a proportional tax on a person's income.

If a person's income did not depend on the choices she herself made, then a tax on her income would be a lump-sum tax, and would have no excess burden. So the circumstances under which the sales tax would have no excess burden would be the exogeneity of all of a person's income. On the other hand, if a person's income is affected by her own decisions ( how much to work, how much to invest, what training to acquire ), then both the sales tax and the income tax would have a positive excess burden.