CHAPTER 10: Intertemporal Choice
TRUE/FALSE
1. An increase in the interest rate cannot make a lender who satisfies WARP become a borrower.
2. If the real interest rate is positive, then a unit of future consumption can be had for the sacrifice of less
than 1 unit of current consumption.
3. The real interest rate is the interest rate that one receives net of brokerage costs or fees imposed by
financial intermediaries.
4. An increase in the interest rate will necessarily result in a decrease in the present value of a given
stream of positive incomes.
5. In a graph that has current consumption on the horizontal axis and future consumption on the vertical
axis, the horizontal intercept of the budget line is the present value of all one’s income in the two
periods.
6. If a consumer can borrow and lend at the same interest rate, then he can exactly afford a consumption
plan if the present value of his consumption equals the present value of his income.
7. It would be a mistake to choose the investment that maximizes the present value of your income
stream unless you planned to spend your entire wealth in the current time period.
8. If the interest rate at which you can borrow is higher than the interest rate at which you can lend, your
budget for current and future consumption is a convex set.
9. If apples today are perfect substitutes for bananas today, then apples today must also be perfect
substitutes for bananas tomorrow.
10. Isaiah is a net borrower when the interest rate is 5% and a net saver when the interest rate is 25%. An
increase in the interest rate from 5 to 25% may make Isaiah worse off.
11. If the interest rate is less than the inflation rate, a rational person will never save money.
12. An increase in the interest rate can make a utility-maximizing lender become a borrower.
13. The intertemporal budget constraint for a consumer can be expressed by setting the present value of
her lifetime consumption equal to the future value of her endowment.
14. The nominal interest rate is 5% and the inflation rate is 6%. A rational consumer will not choose to
save.
15. If the inflation rate doubles and the nominal interest rate remains constant, the real interest rate must
be halved.
16. If the nominal interest rate is 3% and if prices fall by 2% per year, then the real rate of interest is
approximately 5%.
17. If the nominal interest rate is 5% and if prices fall by 3% per year, then the real rate of interest is
approximately 8%.
18. A utility-maximizing consumer would not choose the investment that maximizes the present value of
her income stream unless she planned to spend her entire wealth in the first period.
19. Susan is a net borrower when the interest rate is 10% and a net saver when the interest rate is 20%. A
decrease in the interest rate from 20% to 10% may make Susan worse off.
20. A newspaper article claims that more students are choosing 1-year M.B.A. programs instead of 2-year
programs because the 2-year programs no longer guarantee a well-paid job. If the length of your
M.B.A. program doesn’t matter to employers, and you take a job right after completing your M.B.A.,
the present value of your lifetime earnings is the same whether you take a 1-year or 2-year program.
21. If the interest rate is 5% and will be 5% forever, the present value of an income stream consisting of
$10 a year paid to you on February 11 of every year, starting right now, is $210.
MULTIPLE CHOICE
1. If current and future consumption are both normal goods, an increase in the interest rate will
necessarily
a.
cause savers to save more.
b.
cause borrowers to borrow less.
c.
reduce everyone’s current consumption.
d.
make everyone worse off.
e.
None of the above.
2. Harvey Habit has a utility function U(c1, c2) = minc1, c2, where c1 and c2 are his consumption in
periods 1 and 2 respectively. Harvey earns $189 in period 1 and he will earn $63 in period 2. Harvey
can borrow or lend at an interest rate of 10%. There is no inflation.
a.
Harvey will save $60.
b.
Harvey will borrow $60.
c.
Harvey will neither borrow nor lend.
d.
Harvey will save $124.
e.
None of the above.
3. Harvey Habit has a utility function U(c1, c2) = minc1, c2, where c1 and c2 are his consumption in
periods 1 and 2 respectively. Harvey earns $147 in period 1 and he will earn $63 in period 2. Harvey
can borrow or lend at an interest rate of 10%. There is no inflation.
a.
Harvey will save $102.
b.
Harvey will borrow $40.
c.
Harvey will neither borrow nor lend.
d.
Harvey will save $40.
e.
None of the above.
4. Mr. O. B. Kandle will live for only two periods. In the first period he will earn $100,000. In the second
period he will retire and live on his savings. Mr. Kandle has a Cobb-Douglas utility function U(c1, c2)
= c21c2, where c1 is his period 1 consumption and c2 is his period 2 consumption. The real interest rate
is r.
a.
If the interest rate rises, Mr. Kandle will save more.
b.
If the interest rises, Mr. Kandle will save less.
c.
He will consume the same amount in each period.
d.
A change in the interest rate won’t affect his saving.
e.
None of the above.
5. Suppose that a person can borrow and lend at an interest rate of 10%. But there is a 5% rate of
inflation and the person has to pay an income tax of 30% on all interest income. If the person borrows
money, he can deduct interest as an expense. Where current consumption is on the horizontal axis and
future consumption is on the vertical axis, the budget line will
a.
have a kink at the point of no saving or lending.
b.
be a straight line with a slope of about −1.02.
c.
be a straight line with a slope of about −1.05.
d.
be a straight line with a slope of about −1.35.
e.
None of the above.
6. For every two boxes of strawberries that she consumes, Millicent insists on having one pitcher of
cream. She does not, however, insist on consuming the same amount every week. Her utility functions
is U = mins1, 2c1 mins2, 2c2, where s1 and s2 are the number of boxes of strawberries she
consumes this week and next week and c1 and c2 are the number of pitchers of cream she consumes
this week and next. Strawberries cost $2 a box and cream costs $1 a pitcher. She has a present value of
$100 to spend on these goods in the next two weeks. The weekly interest rate is 1%. How many boxes
of strawberries will she consume this week?
a.
10
b.
20
c.
22
d.
14.1
e.
6.06
7. Roger’s utility function is U = mina1, a2 minb1, b2, where a1 and a2 are the number of piano
lessons he consumes this year and next and b1 and b2 are the number of ice skating lessons he
consumes this year and next. The price of piano lessons is $10 each and the price of ice skating lessons
is $4 each. The prices won’t change, but the interest rate is 7%. If Roger consumes 20 piano lessons
this year, how many ice-skating lessons will he consume next year?
a.
50
b.
20
c.
40
d.
30
e.
There is not enough information to be able to tell.
8. If a consumer views a unit of consumption in period 1 as a perfect substitute (one for one) for a unit of
consumption in period 2 and if the real interest rate is positive, the consumer will
a.
consume only in period 1.
b.
consume only in period 2.
c.
consume equal amounts in each period.
d.
consume more in period 1 than in period 2 if income elasticity exceeds 1, otherwise
consume more in period 2 than in period 1.
e.
equalize expenditures but not consumption in the two periods.
9. If the price level increases by 80% in one year, then for the real rate of interest to be 10%, the nominal
rate of interest would have to be
a.
98%.
b.
70%.
c.
18%.
d.
88%.
e.
72%.
10. Kenny Kink’s utility function is u(c1, c2) = minc1, c2, where c1 is his consumption in period 1 and c2
is his consumption in period 2. He earns $200 in period 1 and $220 in period 2. Kenny can borrow and
lend at an interest rate of 10%, and there is no inflation. The number of dollars that Kenny spends on
consumption in the first period must be
a.
more than 200 but less than 220.
b.
exactly 200.
c.
more than 220.
d.
exactly 180.
e.
more than 180 but less than 200.
11. The nominal interest rate is 5% and the inflation rate is 6%. A rational consumer
a.
will not save since the real interest rate is negative.
b.
will save less than 1% of her income.
c.
will save the same amount regardless of the inflation rate; only the nominal interest rate
matters.
d.
might save despite the negative real interest rate.
e.
will necessarily save less if the inflation rate rises and the nominal interest rate does not
change.
12. If the real rate of interest is 8% and the nominal rate of interest is 28%, then the rate of inflation must
be about
a.
36%.
b.
24.26%.
c.
3.50%.
d.
18.52%.
e.
23%.
13. If the real rate of interest is 12% and the nominal rate of interest is 31%, then the rate of inflation must
be about
a.
27.26%.
b.
43%.
c.
2.58%.
d.
16.96%.
e.
22%.
14. In an isolated mountain village, the only crop is corn. Villagers plan for two time periods. In the first
time period each villager will harvest 100 bushels. In the second time period, no corn will be
harvested. There is no trade with the rest of the world and no stocks of corn remain from before the
first period. Corn can be stored from one time period to the next, but rats eat 25% of what is stored.
The villagers all have Cobb-Douglas utility functions U(C1, C2) = C1C2 and can allocate their own corn
between consumption and storage as they wish. If the introduction of cats to the village reduces the
rats’ predations to 10% of what is stored,
a.
consumption in the first time period will not change.
b.
villagers will consume 5% more corn in each time period.
c.
consumption in the first time period will increase but by less than 5%.
d.
consumption in the second time period will not change.
e.
consumption in the first time period will decrease.
15. Minnie has an income of $300 in period 1 and will have an income of $625 in period 2. Her utility
function is U(c1, c2) = c0.801c0.202, where c1 is her consumption in period 1 and c2 is her consumption in
period 2. The interest rate is 0.25. If she unexpectedly won a lottery which pays its prize in period 2 so
that her income in period 2 would be $1,250 and her income in period 1 would remain $300, then her
consumption in period 1 would
a.
double.
b.
increase by $400.
c.
increase by $150.
d.
stay constant.
e.
increase by $120.
16. Mandy has an income of $800 in period 1 and will have an income of $500 in period 2. Her utility
function is U(c1, c2) = c0.801c0.202, where c1 is her consumption in period 1 and c2 is her consumption in
period 2. The interest rate is .25. If she unexpectedly won a lottery which pays its prize in period 2 so
that her income in period 2 would be $1,000 and her income in period 1 would remain $800, then her
consumption in period 1 would
a.
stay constant.
b.
double.
c.
increase by $320.
d.
increase by $400.
e.
increase by $320.
17. Holly’s utility function is U(c1, c2) = c1/21 + 0.87c1/22, where c1 is her consumption in period 1 and c2 is
her consumption in period 2. In period 2, her income is twice as large as her income in period 1. At
what interest rate will she choose to consume the same amount in period 2 as in period 1? (Choose the
closest answer.)
a.
0.30
b.
0.08
c.
0.23
d.
0
e.
0.15
18. Vanessa’s utility function is U(c1, c2) = c1/21 + 0.83c1/22, where c1 is her consumption in period 1 and c2
is her consumption in period 2. In period 2, her income is 4 times as large as her income in period 1.
At what interest rate will she choose to consume the same amount in period 2 as in period 1? (Choose
the closest answer.)
a.
0.10
b.
0
c.
0.30
d.
0.80
e.
0.20
19. Will Wisp will live for exactly two periods. His utility function is U(c1, c2) = c1c2, where c1 is
consumption in period 1 and c2 is consumption in period 2. He will have no income in period 2. His
income in period 1 is $80,000. If the interest rate rises from 10 to 12%:
a.
his savings will increase by 2% and his consumption in period 2 will increase.
b.
his savings will not change but his consumption in period 2 will increase by $800.
c.
his consumption in both periods will increase.
d.
his consumption in both periods will decrease.
e.
his consumption in period 1 will decrease by 12% and his consumption in period 2 will
increase.
20. Will Wisp will live for exactly two periods. His utility function is U(c1, c2) = c1c2, where c1 is
consumption in period 1 and c2 is consumption in period 2. He will have no income in period 2. His
income in period 1 is $40,000. If the interest rate rises from 10 to 14%:
a.
his savings will increase by 4% and his consumption in period 2 will increase.
b.
his consumption in both periods will increase.
c.
his consumption in both periods will decrease.
d.
his savings will not change but his consumption in period 2 will increase by $800.
e.
his consumption in period 1 will decrease by 14% and his consumption in period 2 will
increase.
21. Peregrine consumes ($700, $880) and earns ($600, $990). If the interest rate is 0.10, the present value
of his endowment is
a.
$1,590.
b.
$1,500.
c.
$1,580.
d.
$3,150.
e.
$3,750.
22. Peregrine consumes ($1,300, $1,320) and earns ($1,000, $1,680). If the interest rate is 0.20, the present
value of his endowment is
a.
$2,400.
b.
$2,680.
c.
$2,620.
d.
$5,280.
e.
$6,280.
23. Molly has income $400 in period 1 and income $600 in period 2. Her utility function is ca1c1−a2, where
a = 0.40 and the interest rate is 0.20. If her income in period 1 doubled and her income in period 2
stayed the same, her consumption in period 1 would
a.
double.
b.
increase by $160.
c.
increase by $80.
d.
stay constant.
e.
increase by $400.
24. Molly has income $200 in period 1 and income $920 in period 2. Her utility function is ca1c1−a2, where
a = 0.80 and the interest rate is 0.15. If her income in period 1 doubled and her income in period 2
stayed the same, her consumption in period 1 would
a.
increase by $160.
b.
double.
c.
increase by $80.
d.
stay constant.
e.
increase by $200.
25. Mr. O. B. Kandle has a utility function c1c2, where c1 is his consumption in period 1 and c2 is his
consumption in period 2. He will have no income in period 2. If he had an income of $80,000 in period
1 and the interest rate increased from 10 to 19%,
a.
his savings would increase by 9% and his consumption in period 2 would also increase.
b.
his savings would not change but his consumption in period 2 would increase by $3,600.
c.
his consumption in both periods would increase.
d.
his consumption in both periods would decrease.
e.
his consumption in period 1 would decrease by 19% and his consumption in period 2
would also decrease.
26. Mr. O. B. Kandle has a utility function c1c2, where c1 is his consumption in period 1 and c2 is his
consumption in period 2. He will have no income in period 2. If he had an income of $70,000 in period
1 and the interest rate increased from 10 to 17%,
a.
his savings would not change but his consumption in period 2 would increase by $2,450.
b.
his consumption in both periods would decrease.
c.
his consumption in both periods would increase.
d.
his savings would increase by 7% and his consumption in period 2 would also increase.
e.
his consumption in period 1 would decrease by 17% and his consumption in period 2
would also decrease.
27. Harvey Habit has a utility function U(c1, c2) = minc1, c2. If he had an income of $880 in period 1
and $1,320 in period 2 and if the interest rate were 0.20, how much would Harvey choose to spend on
bread in period 1?
a.
$1,620
b.
$360
c.
$540
d.
$2,160
e.
$1,080
28. Harvey Habit has a utility function U(c1, c2) = minc1, c2. If he had an income of $1,230 in period 1
and $615 in period 2 and if the interest rate were 0.05, how much would Harvey choose to spend on
bread in period 1?
a.
$1,860
b.
$465
c.
$1,395
d.
$310
e.
$930
29. In an isolated mountain village, the harvest this year is 3,000 bushels of grain and the harvest next year
will be 1,100 bushels. The villagers all have utility functions U(c1, c2) = c1c2, where c1 is consumption
this year and c2 is consumption next year. Rats eat 10% of any grain that is stored for a year. How
much grain could the villagers consume next year if they consume 1,000 bushels of grain this year?
a.
2,900 bushels
b.
1,800 bushels
c.
4,100 bushels
d.
4,350 bushels
e.
1,200 bushels
30. In an isolated mountain village, the harvest this year is 6,000 bushels of grain and the harvest next year
will be 900 bushels. The villagers all have utility functions U(c1, c2) = c1c2, where c1 is consumption
this year and c2 is consumption next year. Rats eat 40% of any grain that is stored for a year. How
much grain could the villagers consume next year if they consume 1,000 bushels of grain this year?
a.
5,850 bushels
b.
3,000 bushels
c.
3,900 bushels
d.
6,900 bushels
e.
1,000 bushels
31. Patience has a utility function U(c1, c2) = c1/21 + 0.83c1/22, where c1 is her consumption in period 1 and
c2 is her consumption in period 2. Her income in period 1 is twice as large as her income in period 2.
At what interest rate will she choose to consume the same amount in period 1 as in period 2?
a.
0.40
b.
0.10
c.
0.20
d.
0
e.
0.30
32. Patience has a utility function U(c1, c2) = c1/21 + 0.80c1/22, where c1 is her consumption in period 1 and
c2 is her consumption in period 2. Her income in period 1 is twice as large as her income in period 2.
At what interest rate will she choose to consume the same amount in period 1 as in period 2?
a.
0
b.
0.25
c.
0.13
d.
0.50
e.
0.38
33. Samantha Smoothie’s utility function is U(c1, c2) = c1c2, where c1 is her consumption in period 1 and c2
is her consumption in period 2. She earns $200 in period 1 and $220 in period 2. Samantha can borrow
and lend at an interest rate of 10% and there is no inflation. The number of dollars that Samantha
spends in the second period must be
a.
more than 200 but less than 220.
b.
exactly 220.
c.
more than 220.
d.
exactly 200.
e.
less than 200.
34. I am always indifferent between a unit of consumption today and tomorrow and the interest rate is 5%.
a.
My intertemporal indifference curve is horizontal.
b.
My intertemporal indifference curve is a straight line with slope 21.
c.
I will spend all of my current and future income on consumption today.
d.
I will spend 5% more on consumption today than on consumption tomorrow.
e.
I will spend 5% more on consumption tomorrow than on consumption today.
PROBLEM
1. Ophelia says, “If I could lend money at the rates I must pay to borrow, I would. And if I could borrow
money at the rates I receive when I lend, I would again. But forsooth, although I spend, I neither
borrow nor lend.” Contrary to common belief, Ophelia is entirely rational. Draw a diagram to show
how Ophelia’s remarks can be consistent with rational behavior and smooth convex preferences if she
pays a different interest rate when she borrows than she gets when she lends. Explain what happens in
words.
2. Patience has the utility function U(c1, c2) = c1/21 + 2c1/22 where c1 is her consumption in period 1 and c1
is her consumption in period 2. She will earn 100 units of the consumption good in period 1 and 100
units of the consumption good in period 2. She can borrow or lend at an interest rate of 10%.
a. Write an equation that describes Patience’s budget.
b. If Patience neither borrows nor lends, what will be her marginal rate of substitution between
current and future consumption?
c. If Patience does the optimal amount of borrowing or saving, what will be the ratio of her period 2
consumption to her period 1 consumption?
3. Buzz is a chicken farmer. His earnings will be $100 this year and $100 next year. He can lend money
at an interest rate of 20%. Because of a subsidized loan program for chicken farmers, he can borrow
money at an interest rate of 10%. No matter what he borrows or lends, his earnings will still be $100
each year.
a. If he is not allowed to either borrow or lend, draw a graph showing his budget between
consumption this year and consumption next year. Put numerical labels on the vertical and
horizontal intercepts of the budget set.
b. Suppose that Buzz is allowed to borrow up to the present value of next year’s earnings at 10% and
is also allowed to make loans. Draw Buzz’s budget constraint in this case.
4. Ymir Larson farms near Niffleheim, Minnesota. He works 80 hours a week. He can either grow
rutabagas or raise pigs. Every hour that he spends growing rutabagas gives him $2 of income this year.
Every hour that he spends raising pigs this year will add $4 to his income next year. In fact, next year’s
weekly income will be 100 + 4H dollars where H is the number of hours he spends raising pigs this
year. Ymir’s utility function is U(c1, c2) = minc1, c2, where c1 and c2 are his consumption
expenditures this year and next year. Ymir doesn’t believe in banks and will neither lend money nor
borrow money.
a. Draw Ymir’s budget line for current and future consumption, labeling key points on it.
b. How many hours a week will he choose to spend raising pigs?
c. How much money will he spend per week on consumption in each year?
5. Luella has to pay an interest rate of 50% to borrow. She only gets an interest rate of 5% if she lends.
She is currently endowed with $1,000 in period 1 and $1,050 in period 2. She considers two alternative
investment projects. She can only choose one of them. For project A she would have to pay $500 in
period 1 and would be paid back $630 in period 2. For project B, she would be paid $500 in period 1
and would have to pay back $525 in period 2.
a. Diagram her budget set if she chooses project A. Also show her budget if she chooses project B.
b. If she neither borrows nor lends, which project has the higher present value at the interest rate
50%? Which has the higher present value at an interest rate of 5%?
c. Draw indifference curves such that she should choose A.
d. With different preferences might she choose B?
6. In an isolated peasant village, the only crop is corn. Good harvests alternate with bad harvests. This
year the harvest will be 1,000 bushels. Next year it will be 150 bushels. There is no trade with the
outside world. Corn can be stored, but rats will eat 25% of what is stored in a year. The villagers have
the Cobb-Douglas utility function U(c1, c2) = c1c2, where c1 is consumption this year and c2 is
consumption next year.
a. Draw a budget line for the village with this year’s consumption on the horizontal axis and next
year’s consumption on the vertical axis. On your graph show the quantities at which the budget
line intercepts the vertical and horizontal axes.
b. How much will the villagers consume this year?
c. How much will the rats eat?
d. How much will the villagers consume next year?