Through the income tax code, governments typically tax most interest income—but, through a variety of retirement programs, they often subsidize at least some types of interest income.
A. Suppose all capital is supplied by individuals that earn income now but don’t expect to earn income in some future period — and therefore save some of their current income. Suppose further that these individuals do not change their current consumption (and thus the amount they put into savings) as interest rates change.
(a) What is the economic incidence of a government subsidy of interest income? What is the economic incidence of a tax on interest income?
(b) In the text, we illustrated the deadweight loss from a subsidy on interest income when savings behavior is unaffected by changes in the interest rate. Now consider a tax on interest income. In a consumer diagram with current consumption c1 on the horizontal and future consumption c2 on the vertical axis, illustrate the deadweight loss from such a tax for a saver whose (uncompensated) savings supply is perfectly inelastic.
(c) What does the size of the deadweight loss depend on? Under what special tastes does it disappear?
(d) On a separate graph, illustrate the inelastic savings (or capital) supply curve. Then illustrate the compensated savings supply curve that allows you to measure the deadweight loss from the tax on interest income. Explain where in the graph this deadweight loss lies.
(e) What happens to the compensated savings supply curve as consumption becomes more complementary across time—and what happens to the deadweight loss as a result?
(f) Is the special case when there is no deadweight loss from taxing interest income compatible with a perfectly inelastic uncompensated savings supply curve?
B. Suppose everyone’s tastes and economic circumstances are the same as those described in part B of exercise 19.4—with α = 0.5 and I = 100,000.1
(a) Suppose further that there are 10,000,000 consumers like this—and they are the only source of capital in the economy. How much capital is supplied regardless of the interest rate?
(b) Suppose next that demand for capital is given by Kd = 25,000,000,000/r . What is the equilibrium real interest rate r ∗ in the absence of any price distortions?
(c) Suppose that, for any dollar of interest earned, the government provides the person who earned the interest a 50 cent subsidy. What will be the new (subsidy-inclusive) interest rate earned by savers, and what will be the interest rate paid by borrowers? What if the government instead taxed 50% of interest income?
(d) Consider the subsidy introduced in (c). How much utility V will each saver attain under this subsidy?
(e) How much current income would each saver have to have in order to obtain the same utility V at the pre-subsidy interest rate r∗ ? In terms of future dollars, how much would it therefore cost the government to make each saver as well off in a lump sum way as it does using the interest rate subsidy?
(f) How much interest will the government have to pay to each saver (in the future) under the subsidy? Use this and your previous answer to conclude the amount of deadweight loss per saver in terms of future dollars. Given the number of savers in the economy, what is the overall deadweight loss?
(g) Derive the compensated savings function (as a function of r ) given the post-subsidy utility level V .
(h) Use your answer to (g) to derive the aggregate compensated capital supply function — and then find the area that corresponds to the deadweight loss. Compare this to your answer in part (f).
(i) Repeat parts (d) through (h) for the case of the tax on interest income described in part (c).
(j) You have calculated deadweight losses for interest rates that are reasonable for 1-year time horizons. If we consider distortions in people’s decisions over longer time horizons (such as when they plan for retirement), a more reasonable time frame might be 25 years. With annual market interest rates of 0.05 in the absence of distortions, can you use your compensated savings function (given in the footnote to the problem) to estimate again what the deadweight losses from a subsidy that raises the effective rate of return by 50% and from a tax that lowers it by 50% would be?