Question

Suppose that the government introduces a tax on interest earnings. That is, borrowers face a real interest rate of r before and after the tax is introduced, but lenders receive an interest rate of (1-x)r on their savings, where x is the tax rate. Therefore, we are looking at the effects of having x increase from zero to some value greater than zero, with r assumed to remain constant.

a) Show the effects of the increase in the tax rate on a consumer’s lifetime budget constraint.

b) How does the increase in the tax rate affect the optimal choice of consumption (in the current and future periods) and saving for the consumer? Show how income and substitution effects matter for your answer, and show how it matters whether the consumer is initially a borrower or a lender.

Answer 1

Solution

(i) Define savings in the first period as s1 = y1 ? c1. Then

c2 = ( (1 + r)(y1 ? c1) + y2 if s1 ? 0,

?(1 + (1 ? x)r)(y1 ? c1) + y2 if s1 > 0

What is the effect on the budget constraint? As long as the consumer is not lending in the first period, there is no effect, so the points where c1 ? y1 are not affected (blue line). If the consumer is lending (c1 < y1), then the savings are taxed, so effectively, effectively she is facing a lower real interest rate. This pivots teh budget constraint to the left along the point (y1, y2) (if (c1, c2) = (y1, y2) then the consumer is neither lender nor saver and hence this corresponds to the ‘highest’ savings where she is not affected by the taxation).

(ii) There are 2 possible cases; the consumer can be either borrower or lender before the change in the taxation. • Assume that the consumer was a borrower before the change in the taxes. Then the set feasible consumptions is smaller (lending is less profitable), however, the allocation which she chose before the change in the taxes is still feasible, hence it still must be the best allocation in the smaller feasible set. Hence the consumer chooses the same allocation and she remains a borrower in the first period. Finally, as she chooses the same allocation, her utility is not affected. • On the other hand, if the consumer was originally a lender, then the original allocation is no longer feasible, hence she must be worse off. The substitution effect will induce the consumer to consume more today, ase saving has became less profitable. However, the income effect is negative and hence both c1 and c2 are reduced. The resulting c2 will be lower after the change in the taxes (both income and substitution effect are negative). The change in c1 is ambiguous as it depends on the relative magnitude of the income and substitution effect, which in turn depend on the particular utility fucntion

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