1. A consumer’s income in the current period is y = 100, and income in the future period is y’= 120. He or she pays lump-sum taxes t = 20 in the current period and t’= 10 in the future period. The real interest rate is 0.1, or 10%, per period.
(a) Determine the consumer’s lifetime wealth.
(b) Suppose that current and future consumptions are perfect complements for the consumer and that he or she always wants to have equal consumption in the current and future periods. Draw the consumer’s indifference curves.
c) Determine what the consumer’s optimal current-period and future-period consumptions are, and what optimal saving is, and show this in a diagram with the consumer’s budget constraint and indifference curves. Is the consumer a lender or a borrower?
(d) Now suppose that instead of y = 100, the consumer has y = 140. Again, determine optimal consumption in the current and future periods and optimal saving, and show this in a diagram. Is the consumer a lender or a borrower?
(e) Explain the differences in your results between parts (c) and (d).
4. 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.