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 r is the interest rate.The second trick is in the way you treat income in the two di�erent periods. Suppose that a consumer has an income of m1 in period 1 and
m2 in period 2 and that there is no in�ation. 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 + 1+rm2. As you
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 + (1+r)m2 . Therefore
if you are given a consumer’s utility function, the interest rate, and the consumer’s income in each period, you can �nd 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 �nd saving, since the consumer’s saving is just the di�erence between his period-1 income and his period-1 consumption.
Example:
A consumer has the utility function U(c1, c2) = c1c2. There isno in�ation, 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
price of good 2 is 1 + r = 1.1. The consumer will choose a consumption bundle so that MU1/M U2 = 1.1. But MU1 = c2 and MU2 = c1, so the
consumer must choose a bundle such that c2/c1= 1.1. Take this equation
together with the budget equation to solve for c1 and c2. The solution is
c1 = 105 and c2 = 115.50. Since the consumer’s period-1 income is only
100, he must borrow 5 in order to consume 105 in period 1. To pay back principal and interest in period 2, he must pay 5.50 out of his period-2 income of 121. This leaves him with 115.50 to consume.
sumer behavior. The key to understanding the e�ects of in�ation is to see what happens to the budget constraint.
Example:
Suppose that in the previous example, there happened to be an in�ation 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 in�ation, 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- tion is 1.06/1.10 = .964 units of period-2 consumption. If the consumer’s money income in each period is unchanged, then his budget equation is c1 + .964c2 = 210. This budget constraint is the same as the budgetconstraint would be if there were no in�ation and the interest rate were r, where .964 = 1/(1 + r). The value of r that solves this equation is known as the real rate of interest. In this case the real rate of interest is about .038. When the interest rate and in�ation rate are both small, the real rate of interest is closely approximated by the di�erence between the nominal interest rate, (10% in this case) and the in�ation rate (6% in this case), that is, .038 ∼ .10 − .06. As you will see, this is not such a good approximation if in�ation rates and interest rates are large.
10.1 (0) Peregrine Pickle consumes (c1, c2) and earns (m1, m2) in periods
1 and 2 respectively. Suppose the interest rate is r.
(a) Write down Peregrine’s intertemporal budget constraint in present value terms.
c
1+
c2(1+r)
= m
1+
(1+r)m2.
(b) If Peregrine does not consume anything in period 1, what is the most he can consume in period 2?
m
1(1 + r) + m
2.
This is the (future value, present value) of his endowment.Future value.
(c) If Peregrine does not consume anything in period 2, what is the most he can consume in period 1?
m
1+
(1+r)m2.
This is the (future value, present value) of his endowment.Present value.
What is the slope of Peregrine’s budget line?−(1 + r).
10.2 (0) Molly has a Cobb-Douglas utility function U(c1, c2) = ca1c12−a,
where 0 < a < 1 and where c1 and c2 are her consumptions in periods 1
and 2 respectively. We saw earlier that if utility has the form u(x1, x2) =
xa1x12−a and the budget constraint is of the “standard” form p1x1+p2x2 =
m, then the demand functions for the goods are x1 = am/p1 and x2 =
(a) Suppose that Molly’s income is m1 in period 1 and m2 in period 2.
Write down her budget constraint in terms of present values.
c
1+
c
2/(1 + r) = m
1+ m
2/(1 + r).
(b) We want to compare this budget constraint to one of the standard form. In terms of Molly’s budget constraint, what is p1?
1.
Whatis p2?
1/(1 + r).
What is m?m
1+ m
2/(1 + r).
(c) If a = .2, solve for Molly’s demand functions for consumption in each period as a function of m1, m2, and r. Her demand function for
consumption in period 1 is c1 =
.2m
1+ .2m
2/(1 + r).
Herdemand function for consumption in period 2 is c2 =
.8(1+ r)m
1+
.8m
2.
(d) An increase in the interest rate will
decrease
her period-1 consumption. It willincrease
her period-2 consumption andincrease
her savings in period 1.10.3 (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 in�ation.
0 1 2 3 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
(a) What is the present value of Nickleby’s endowment?
$3,000.
What is the future value of his endowment?$3,300.
With blue ink, show the combinations of consumption this year and consumption next year that he can a�ord. Label Nickelby’s endowment with the letter E.(b) Suppose that Nickleby has the utility function U(C1, C2) = C1C2.
Write an expression for Nickleby’s marginal rate of substitution between consumption this year and consumption next year. (Your answer will be a function of the variables C1, C2.)
MRS
= −C
2/C
1.
(c) What is the slope of Nickleby’s budget line?
−1.1.
Write an equation that states that the slope of Nickleby’s indi�erence curve is equal to the slope of his budget line when the interest rate is 10%.1.1 =
C
2/C
1.
Also write down Nickleby’s budget equation.C
1+
C
2/1.1 = 3, 000.
(d) Solve these two equations. Nickleby will consume
1,500
units in period 1 and1,650
units in period 2. Label this point A on your diagram.(e) Will he borrow or save in the �rst period?
Save.
How much?500.
(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 A at 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%. Nickleby will consume
1,458.3
units in period 1 and1,750
units in period 2.(h) Will he borrow or save in the �rst period?
Save.
How much?541.7.
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 e�ects.
(a) “If both current and future consumption are normal goods, an increase in the interest rate will necessarily make a saver save more.”
False.
Substitution effect makes him consume less
in period 1 and save more. For a saver,
income effect works in opposite direction.
Either effect could dominate.
(b) “If both current and future consumption are normal goods, an in- crease in the interest rate will necessarily make a saver choose more consumption in the second period.”
True. The income
and substitution effects both lead to more
consumption in the second period.
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.)
(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.)
0 10 20 30 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 o� or worse o� by investing in this project given his possibilities for borrowing or lending? Or can’t one tell without knowing something about his preferences? Explain.
Better off. If he invests in the project,
he can borrow or lend to get any bundle he
could afford without investing.
(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 �rst project. Use pencil or black ink to draw the budget set available to Laertes if he chooses this project. Is Laertes better o� or worse o� by choosing this project than if he didn’t choose either project? Or can’t one tell without knowing more about his preferences? Explain.
Can’t tell. He
can afford some things he couldn’t afford
originally. But some things he could afford
before, he can’t afford if he invests in
this project.
10.6 (0) The table below reports the in�ation rate and the annual rate of return on treasury bills in several countries for the years 1984 and 1985.
In�ation Rate and Interest Rate for Selected Countries
% In�ation % In�ation % 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 −0.2 5.3 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 a real rate of return of 5% in 1985?7.1%.
(c) Subtracting the in�ation rate from the nominal rate of return gives a good approximation to the real rate for countries with a low rate of in�ation. For the United States in 1984, the approximation gives you
6%
while the more exact method suggested by the text gives you5.79%.
But for countries with very high in�ation this is a poor approximation. The approximation gives you−87.3%
for Israel in 1984, while the more exact formula gives you−21.57%.
For Argentina in 1985, the approximation would tell us that a bond yielding a nominal rate of677.7%
would yield a real interest rate of 5%. This contrasts with the answer710.8%
that you found above.Real Rates of Interest in 1984 and 1985 Country 1984 1985 United States
5.7
5.5
Israel−21.57
109.4
Switzerland0.5
3.33
W. Germany3.0
4.4
Italy5.6
7.6
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 �agon this year and will be two b.c.u.’s per �agon next year. The red currency price of ambrosia is one r.c.u. per �agon this year and will be the same next year.
(a) If Jane spent all of its blue income in the �rst period, it would be enough to pay the blue price for
100
�agons of ambrosia. If Jane saved all of this year’s blue income at the blue-money bank, it would have150
b.c.u.’s next year. This would give it enough blue currency to pay the blue price for75
�agons of ambrosia. On the graph below, draw Jane’s blue budget line, depicting all of those combinations of current and next period’s consumption that it has enough blue income to buy.0 25 75 100 25
50 75
Ambrosia this period Ambrosia next period
100
50
(b) If Jane planned to spend no red income in the next period and to borrow as much red currency as it can pay back with interest with next period’s red income, how much red currency could it borrow?
80.
(c) The (exact) real rate of interest on blue money is
−25%.
The real rate of interest on red money is25%.
(d) On the axes below, draw Jane’s blue budget line and its red budget line. Shade in all of those combinations of current and future ambrosia consumption that Jane can a�ord given that it has to pay with both currencies.
0 25 75 100 25
50 75
Ambrosia this period Ambrosia next period
100 50 Blue line Red line c
(e) It turns out that Jane �nds it optimal to operate on its blue budget line and beneath its red budget line. Find such a point on your graph and mark it with a C.
(f) On the following graph, show what happens to Jane’s original budget set if the blue interest rate rises and the red interest rate does not change. On your graph, shade in the part of the new budget line where Jane’s new demand could possibly be. (Hint: Apply the principle of revealed preference. Think about what bundles were available but rejected when Jane chose to consume at C before the change in blue interest rates.)
0 25 75 100 25
50 75
Ambrosia this period Ambrosia next period
100 50 Blue line Shaded region c New blue line Red line
10.8 (0) Mr. O. B. Kandle will only live for two periods. In the �rst period he will earn $50,000. In the second period he will retire and live on his savings. His utility function is U(c1, c2) = c1c2, where c1 is con-
sumption in period 1 and c2 is consumption in period 2. He can borrow
and lend at the interest rate r = .10.
(a) If the interest rate rises, will his period-1 consumption increase, de-
crease, or stay the same?
Stay the same. His demand
for c
1is .5(m
1+ m
2/(1 + r)) and m
2= 0.
(b) Would an increase in the interest rate make him consume more or
less in the second period?