EXERCISE 11: Intertemporal Choice and Present Value
J. Wahl -- Micro Principles
1. Why do we typically model the interest rate as positive?
2. Suppose the interest rate is i, consumption in period 1 is c1, consumption in period 2 is c2, income
in period 1 is Y1, and income in period 2 is Y2. Please draw the budget constraint. Denote the range of the
budget constraint pertaining to borrowers and the range pertaining to lenders. What happens to utility if the
interest rate rises?
3. Suppose on Feb. 1, 2009, the real interest rate r is 3% and everyone expects prices to rise at 6%
per year. Your econ teacher asks you for a loan of $100, which she will repay you in 2 years (Jan. 31,
2011). You are certain that she will not default on the loan.
a. What interest rate will you charge her and why? (Assume simple interest paid at the end of each
fiscal year (Feb-Jan). I’ve already told you that the principal will be repaid at the end of the
second fiscal year.)
b. What amounts of money will you receive from her and when?
c. What is the present value (as of Feb. 1, 2009) of the stream of payments you described in part b?
d. Suppose you discover new information on Feb. 1, 2010, that prices will rise by 8% during the
second year of your loan. You knew at the time the loan was made that your prediction about
prices might be wrong, and you wrote into your loan agreement that you could change the interest
rate after one year. What interest rate will you ask your teacher to pay you for the second year of
the loan?
e. Suppose your teacher agrees with you about the new prediction on inflation. What amounts will
she end up paying you over the life of the loan?
f. You decide to leave Carleton on Feb. 1, 2010, and you sell the rights to your loan to your
roommate. What is the present value (as of Feb. 1, 2010) of the remaining payments of the loan?
What will your roommate be willing to pay you for the rights to the remaining payments?
g. Go back to the statements in part d. Suppose your teacher doesn't believe your information, sticks
to her belief that inflation during the second year will be 6%, and refuses to pay you a different
interest rate the second year. What is the present value (as of Feb. 1, 2010) of the remaining
payments of the loan IN YOUR OPINION?
h. In a moment of weakness, you confided in your roommate about your expectations on inflation.
Your roomie believes you, even though your prof doesn't. If you try to unload the loan on your
roommate on Feb 1, 2010, what is the most that this savvy economist will be willing to pay you?
4. Suppose banks are paying 8% interest, compounded monthly. What simple interest rate would
give you the same amount of interest at the end of a year?
5. You are contemplating financing a car that costs $10,000. GMAC will allow you to finance the
car in three equal payments of $3500, payable 1 year from today, 2 years from today, and 3 years
from today. Wells Fargo offers you a loan where you will pay only interest for the first 2 years,
then interest plus principal the third year.
a. If the interest rate Wells Fargo charges is 5% (and represents your best estimate of the
real interest rate plus expected inflation), is the GMAC loan a good deal for you? (Hint:
find the present value of the payments to GMAC.)
b. What if Wells Fargo charges 2%?
6. You are thinking about renting a house in Minneapolis for 4 years. Your landlord gives you the
option of buying his used snowblower for $175 (which you both think will last for the next 4 years
and then will be useless), or of contracting with his son, who will shovel you out for $50 a year,
payable starting 1 year from now, 2 years from now, and so forth. If the interest rate is now 8%
and you expect it to stay the same, which will you choose?
