Say a consumer has the following endowment of income in each of two time periods, (m1, m2) = (10, 5). Further, she can borrow or lend at interest rate, r, and she has an inter-temporal utility function of u(c1, c2) = ln(c1) + ln(c2).
(a) What is the equation of her budget constraint?
(b) How much does the consumer decide to consume in each of the two time periods?
(c) If the interest rate is 10%, is she a borrower or a lender?
(d) Let's complicate things a little. Say that the consumer will need to pay taxes on any interest income that she accrues. Let the tax rate be t. Hence, if she saves X dollars she will earn rX in interest income but will have to pay trX in taxes. How does this change the consumer's consumption in the two periods (i.e. calculate the new consumption levels where the interest rate is r)?
(e) Now let's see if the consumer is has changed her behavior from (c). Let the interest rate again be 10%, but now let the tax rate be 65%. Is she a borrower or a lender?