Suppose we return to the example from the text in which you earn $5,000 this summer and expect to earn $5,500 next summer.
A: In the real world, banks usually charge higher interest rates for borrowing than they will give on savings. So, instead of assuming that you can borrow and lend at the same interest rate, suppose the bank pays you an interest rate of 5% on anything you save but will lend you money only at an interest rate of 10%. (In this exercise, it helps to not draw everything to scale much as we did not draw inter temporal budgets to scale in the chapter.)
(a) Illustrate your budget constraint with consumption this summer on the horizontal and consumption next summer on the vertical axis.
(b) How would your answer change if the interest rates for borrowing and lending were reversed?
(c) A set is defined as “convex” if the line connecting any two points in the set also lies in the set. Is the choice set in part (a) a convex set? What about the choice set in part (b)?
(d)Which of the two scenarios would you prefer? Give both an intuitive answer that does not refer to your graphs and demonstrate how the graphs give the same answer.
B: Suppose more generally that you earn e1 this year and e2 next year and that the interest rate for borrowing is rB and the interest rate for saving is rS. Let c1 and c2 denote consumption this year and next year.
(a) Derive the general expression for your inter temporal choice set under these conditions.
(b) Check that your general expression is correct by substituting the values from A(a) and (b) and checking that you get a choice set similar to those you derived intuitively.