In end-of-chapter exercise 6.10. we analyzed cases where the interest rates for borrowing and saving are different. Part of the reason they might be different is because of government policy.
A. Suppose banks are currently willing to lend and borrow at the same interest rate. Consider an individual who has income e1 now and e2 in a future period, with the interest rate over that period equal to r. After considering the tradeoffs, the individual chooses to borrow on his future income rather than save. Suppose in this exercise that the individual’s tastes are homothetic.
(a) Illustrate the budget constraint for this individual —and indicate his optimal choice.
(b) Now suppose the government would like to encourage this individual to save for the future. One proposal might be to subsidize savings (through something like a 401K plan) — i.e. a policy that increases the interest rate for saving without changing the interest rate for borrowing. Illustrate how this changes the budget constraint. Will this policy work to accomplish the government’s goal?
(c) Another alternative would be to penalize borrowing by taxing the interest the banks collect from loans—thus raising the effective interest rate for borrowing. Illustrate how this changes the budget. Will this policy cause the individual to borrow less? Can it cause him to start saving?
(d) In reality, the government often does the opposite of these two policies: Savings (outside qualified retirement plans) are taxed while some forms of borrowing (in particular borrowing to buy a home) are subsidized. Suppose again that initially the interest rate for borrowing and saving is the same—and then suppose that the combination of taxes on savings (which lowers the effective interest rate on savings) and subsidies for borrowing (which lowers the effective interest rate for borrowing) reduce the interest rate to r ′ < r equally for both saving and borrowing. How will this individual respond to this combination of policies?
(e) Suppose that, instead of taxing or subsidizing interest rates, the government simply “saves for” the individual by taking some of the individual’s current income e1 and putting it into the bank to collect interest for the future period. How will this change the individual’s behavior?
(f) Now suppose that, instead of taking some of the person’s current income and saving it for him, the government simply raises the social security benefits (in the future period) without taking anything away from the person now. What will the individual do?
B. Suppose your tastes can be captured by the utility function u (c1,c2) = c1α c2(1−α).
(a) Assuming you face a constant interest rate r for borrowing and saving, how much will you consume now and in the future (as a function of e1, e2 and r.)
(b) For what values of α will you choose to borrow rather than save?
(c) Suppose that α = 0.5, e1 = 100,000, e2 = 125,000 and r = 0.10. How much do you save or borrow?
(d) If the government could come up with a “financial literacy” course that changes how you view the tradeoff between now and the future by impacting α, how much would this program have to change your α in order to get you to stop borrowing?
(e) Suppose the “financial literacy” program had no impact on α. How much would the government have to raise the interest rate for saving (as described in A(b)) in order for you to become a saver?
(f) Verify your conclusion about the impact of the policy proposal outlined in A(c).
(g) Verify you conclusion to A(d).
(h) Verify your conclusion to A(e); i.e. suppose the government takes x of your current income e1 and saves it—thus increasing e2 by x(1+r ).
(i) Finally, suppose the increase in social security benefits outlined in A(f ) is implemented; i.e. suppose e2 is increased by x. How and by how much does your borrowing change?