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Chapter 10 NAME
Intertemporal Choice
Introduction. The theory of consumer saving uses techniques that you
have already learned. In order to focus attention on consumption over
time, we will usually consider examples where there is only one consumer
good, but this good can be consumed in either of two time periods. We
will be using two “tricks.” One trick is to treat consumption in period 1
and consumption in period 2 as two distinct commodities. If you make
period-1 consumption the numeraire, then the “price” of period-2 con-
sumption is the amount of period-1 consumption that you have to give
up to get an extra unit of period-2 consumption. This price turns out to
be 1/(1 + r), where ris the interest rate.
The second trick is in the way you treat income in the two different
periods. Suppose that a consumer has an income of m1in period 1 and
m2in period 2 and that there is no inflation. The total amount of period-
1 consumption that this consumer could buy, if he borrowed as much
money as he could possibly repay in period 2, is m1+m2
1+r.Asyou
work the exercises and study the text, it should become clear that the
consumer’s budget equation for choosing consumption in the two periods
is always
c1+c2
1+r=m1+m2
1+r.
This budget constraint looks just like the standard budget constraint that
you studied in previous chapters, where the price of “good 1” is 1, the
price of “good 2” is 1/(1 + r), and “income” is m1+m2
(1+r). Therefore
if you are given a consumer’s utility function, the interest rate, and the
consumer’s income in each period, you can find his demand for consump-
tion in periods 1 and 2 using the methods you already know. Having
solved for consumption in each period, you can also find saving, since the
consumer’s saving is just the difference between his period-1 income and
his period-1 consumption.
Example: A consumer has the utility function U(c1,c
2)=c1c2.Thereis
no inflation, the interest rate is 10%, and the consumer has income 100
in period 1 and 121 in period 2. Then the consumer’s budget constraint
c1+c2/1.1 = 100 + 121/1.1 = 210.The ratio of the price of good 1 to the
132 INTERTEMPORAL CHOICE (Ch. 10)
You will also be asked to determine the effects of inflation on con-
sumer behavior. The key to understanding the effects of inflation is to
see what happens to the budget constraint.
Example: Suppose that in the previous example, there happened to be
an inflation rate of 6%, and suppose that the price of period-1 goods is
1. Then if you save $1 in period 1 and get it back with 10% interest,
you will get back $1.10 in period 2. But because of the inflation, goods
in period 2 cost 1.06 dollars per unit. Therefore the amount of period-1
consumption that you have to give up to get a unit of period-2 consump-
10.1 (0) Peregrine Pickle consumes (c1,c
2) and earns (m1,m
2)inperiods
1 and 2 respectively. Suppose the interest rate is r.
(a) Write down Peregrine’s intertemporal budget constraint in present
(1+r)
(b) If Peregrine does not consume anything in period 1, what is the most
(c) If Peregrine does not consume anything in period 2, what is the most
he can consume in period 1? m1+m2
10.2 (0) Molly has a Cobb-Douglas utility function U(c1,c
2)=ca
1c1−a
2,
where 0 <a<1andwherec1and c2are her consumptions in periods 1
and 2 respectively. We saw earlier that if utility has the form u(x1,x
2)=
xa
1x1−a
2and the budget constraint is of the “standard” form p1x1+p2x2=
m, then the demand functions for the goods are x1=am/p1and x2=
(1 −a)m/p2.
NAME 133
(a) Suppose that Molly’s income is m1in period 1 and m2in period 2.
(b) We want to compare this budget constraint to one of the standard
(c) If a=.2, solve for Molly’s demand functions for consumption in
each period as a function of m1,m2,andr. Her demand function for
10.3 (0) Nickleby has an income of $2,000 this year, and he expects an
income of $1,100 next year. He can borrow and lend money at an interest
rate of 10%. Consumption goods cost $1 per unit this year and there is
no inflation.
134 INTERTEMPORAL CHOICE (Ch. 10)
0123
4
1
2
3
Consumption this year in 1,000s
Consumption next year in 1,000s
4
e
a
Squiggly
line
Red line
Blue
line
ink, show the combinations of consumption this year and consumption
next year that he can afford. Label Nickelby’s endowment with the letter
E.
(b) Suppose that Nickleby has the utility function U(C1,C
2)=C1C2.
Write an expression for Nickleby’s marginal rate of substitution between
consumption this year and consumption next year. (Your answer will be
diagram.
NAME 135
(f) On your graph use red ink to show what Nickleby’s budget line would
be if the interest rate rose to 20%. Knowing that Nickleby chose the
point Aat a 10% interest rate, even without knowing his utility function,
you can determine that his new choice cannot be on certain parts of his
new budget line. Draw a squiggly mark over the part of his new budget
line where that choice can not be. (Hint: Close your eyes and think of
WARP.)
(g) Solve for Nickleby’s optimal choice when the interest rate is 20%.
units in period 2.
10.4 (0) Decide whether each of the following statements is true or
false. Then explain why your answer is correct, based on the Slutsky
decomposition into income and substitution effects.
(a) “If both current and future consumption are normal goods, an increase
(b) “If both current and future consumption are normal goods, an in-
crease in the interest rate will necessarily make a saver choose more
10.5 (1) Laertes has an endowment of $20 each period. He can borrow
money at an interest rate of 200%, and he can lend money at a rate of
0%. (Note: If the interest rate is 0%, for every dollar that you save, you
get back $1 in the next period. If the interest rate is 200%, then for every
dollar you borrow, you have to pay back $3 in the next period.)
136 INTERTEMPORAL CHOICE (Ch. 10)
(a) Use blue ink to illustrate his budget set in the graph below. (Hint:
The boundary of the budget set is not a single straight line.)
0102030
40
10
20
30
C1
C2
40
Red line
Blue line
Black line
(b) Laertes could invest in a project that would leave him with m1=30
and m2= 15. Besides investing in the project, he can still borrow at 200%
interest or lend at 0% interest. Use red ink to draw the new budget set
in the graph above. Would Laertes be better off or worse off by investing
in this project given his possibilities for borrowing or lending? Or can’t
one tell without knowing something about his preferences? Explain.
(c) Consider an alternative project that would leave Laertes with the
endowment m1= 15, m2= 30. Again suppose he can borrow and lend
as above. But if he chooses this project, he can’t do the first project.
Use pencil or black ink to draw the budget set available to Laertes if he
chooses this project. Is Laertes better off or worse off by choosing this
project than if he didn’t choose either project? Or can’t one tell without
10.6 (0) The table below reports the inflation rate and the annual rate
of return on treasury bills in several countries for the years 1984 and 1985.
NAME 137
Inflation Rate and Interest Rate for Selected Countries
% Inflation % Inflation % Interest % Interest
Country Rate, 1984 Rate, 1985 Rate, 1984 Rate, 1985
United States 3.6 1.9 9.6 7.5
Israel 304.6 48.1 217.3 210.1
Switzerland 3.1 0.8 3.6 4.1
W. Germany 2.2 −0.25.3 4.2
Italy 9.2 5.8 15.3 13.9
Argentina 90.0 672.2 NA NA
Japan 0.6 2.0 NA NA
(a) In the table below, use the formula that your textbook gives for the
exact real rate of interest to compute the exact real rates of interest.
(b) What would the nominal rate of return on a bond in Argentina have
to be to give a real rate of return of 5% in 1985? 710.8%.What
would the nominal rate of return on a bond in Japan have to be to give
(c) Subtracting the inflation rate from the nominal rate of return gives
a good approximation to the real rate for countries with a low rate of
inflation. For the United States in 1984, the approximation gives you
Argentina in 1985, the approximation would tell us that a bond yielding
138 INTERTEMPORAL CHOICE (Ch. 10)
Real Rates of Interest in 1984 and 1985
Country 1984 1985
10.7 (0) We return to the planet Mungo. On Mungo, macroeconomists
and bankers are jolly, clever creatures, and there are two kinds of money,
red money and blue money. Recall that to buy something in Mungo you
have to pay for it twice, once with blue money and once with red money.
Everything has a blue-money price and a red-money price, and nobody
is ever allowed to trade one kind of money for the other. There is a blue-
money bank where you can borrow and lend blue money at a 50% annual
interest rate. There is a red-money bank where you can borrow and lend
red money at a 25% annual interest rate.
A Mungoan named Jane consumes only one commodity, ambrosia,
but it must decide how to allocate its consumption between this year and
next year. Jane’s income this year is 100 blue currency units and no red
currency units. Next year, its income will be 100 red currency units and
no blue currency units. The blue currency price of ambrosia is one b.c.u.
per flagon this year and will be two b.c.u.’s per flagon next year. The red
currency price of ambrosia is one r.c.u. per flagon this year and will be
the same next year.
(a) If Jane spent all of its blue income in the first period, it would be
