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To illustrate this idea, suppose that the government cuts Prudence's current taxes by

300.

This tax reduction increases Prudence's current income by

300,

which (all else being equal) would cause her to consume more. Because the government's revenue has been reduced by

300

and its expenditures have not changed, however, the government must increase its current borrowing from the public by

300

(per taxpayer). Furthermore, the government must pay interest on its borrowings. For example, if the real interest rate that the government must pay on its debt is

10°/o,

in the future period the govern­ ment's outstanding debt will be

330

greater than it would have been without the tax cut.

1 60 Part 2 Long-Run Economic Performance

As a taxpayer, Prudence is ultimately responsible for the government's debts. Suppose that the government decides to repay its borrowings and accumulated interest in the future period (Chapter 15 discusses what happens if the govern­ ment's debt is left for Prudence's descendants to repay). To repay its debt plus interest, the government must raise taxes in the future period by

330,

so Prudence's expected future income falls by

330.

Overall, then, the government's tax program has raised Prudence's current income by

300

but reduced her future income by

330.

At a real interest rate of

10°/o,

the present value of the future income change is

-300,

which cancels out the increase in current income of

300.

Thus Prudence's

PVLR

is unchanged by the tax cut, and (as the Ricardian equivalence proposition implies) she should not change her current consumption.

Excess Sensitivity and Borrowing Constraints

A variety of studies have confirmed that consumption is affected by current income, expected future income, and wealth, and that permanent income changes have larger effects on consumption than do temporary income changes all of which are outcomes implied by the model. Nevertheless, some studies show that the response of consumption to a change in current income is greater than would be expected on the basis of the effect of the current income change on

PVLR.

This tendency of consumption to respond to current income more strongly than the model predicts is called the

excess sensitivity

of consumption to current income.

One explanation for excess sensitivity is that people are more short-sighted than assumed in our model and thus consume a larger portion of an increase in current income than predicted by it. Another explanation, which is more in the

spirit of the model, is that the amount that people can borrow is limited. A restric­ tion imposed by lenders on the amount that someone can borrow against future income is called a

borrowing constraint.

The effect of a borrowing constraint on the consumption-saving decision depends on whether the consumer would want to borrow in the absence of a borrowing con­ straint. If the consumer wouldn't want to borrow even if borrowing were possible, the borrowing constraint is said to be

nonbinding.

When a consumer wants to borrow but is prevented from doing so, the borrowing constraint is said to be

binding.

A consumer who faces a binding borrowing constraint will spend all available current income and wealth on current consumption so as to come as close as possible to the consumption combination desired in the absence of borrowing constraints. Such a consumer would consume the entire amount of an increase in current income. Thus the effect of an increase in current income on current consumption is greater for a consumer who faces

a binding borrowing constraint than is predicted by our simple model without bor­ rowing constraints. In macroeconomic terms, this result implies that if a significant number of consumers face binding borrowing constraints the response of aggregate consumption to an increase in aggregate income will be greater than implied by the basic theory in the absence of borrowing constraints.

In

other words, if borrowing con­

straints exist, consumption may be excessively sensitive to current income.8

8 Although we have no direct way of counting how many consumers are constrained from borrowing, several studies estimate that, to account for the observed relationship between consumption and cur­ rent income, during any year some 20°/o to 50°/o of U.S. consumers face binding borrowing constraints. See, for example, John Y. Campbell and N. Gregory Mankiw, "Consumption, Income, and Interest

Rates: Reinterpreting the Time Series Evidence," in 0. Blanchard and S. Fischer, eds., NBER Macroeco­ nomics Annual, Cambridge, Mass.: MIT Press, 1989; and Robert E. Hall and Frederic S. Mishkin, "The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households,"

Figure 4.A.6

The effect of an

increase in the real interest rate on the budget line

The figure shows the effect on Prudence's

budget line of an increase in the real interest rate, r, from 1 0°/o to 7 6 °/o. Because the slope of a budget line is -(1 + r) and the initial

real interest rate is 10°/o, the slope of Prudence's

initial budget line, BL _1, is

-1.10. The initial budget line, BL _1, also passes

through the no-borrow­ ing, no-lending point, E, which represents the con­

sumption combination

that Prudence obtains by spending all her current income and wealth on current consumption.

Because E can still be obtained when the real interest rate rises, it also lies on the new budget line, BL 2. However, the

slope of BL 2 is -1.76,

reflecting the rise in the real interest rate to 76°/o. Thus the higher real inter­ est rate causes the budget line to pivot clockwise

around the no-borrowing, no-lending point.

§

138,600 ·� � � 120,000 Q) 105,000 = 99,000 �

&

90,000 75,000 60,000 45,000 BL 2 (slope = -1.76) BL1 (slope = -1.10)

Chapter 4 Consumption, Savi ng, and I nvestment 1 61

Real interest

rate increases

from 10°/o to 76°/o

33 000 · · · . . · · . . . . . . No-borrowing, no-lending point

15,000

0 15,000 30,000 45,000 60,000 / 90,000 105,000 120,000 135,000

78,750

Current consumption, c

The Rea l I nterest Rate and the Consumption-Savin g Decision

To explore the effects of a change in the real interest rate on consumption and saving, let's return to the two-period model and Prudence's situation. Recall that Prudence initially has current real income, _y, of

42,000,

future income, _y

f

, of

33,000,

initial wealth,

a,

of

18,000,

and that she faces a real interest rate,

r,

of

10°/o.

Her budget line, which is the same as in Fig.

4.A.1,

is shown in Fig.

4.A.6

as

BL

1 . Now let's see what

happens when for some reason the real interest rate jumps from

10°/o

to

76°/o.9

The Real Interest Rate and the Budget Line

To see how Prudence's budget line is affected when the real interest rate rises, let's first consider point E on the budget line

BL

1 . Point E is special in that it is the only

point on the budget line at which current consumption equals current income plus initial wealth

(c =

_y +

a = 60,000)

and future consumption equals future income

(cf

=

yf

=

33,000).

If Prudence chooses this consumption combination, she

doesn't need to borrow (her current income and initial wealth are just sufficient to pay for her current consumption), nor does she have any current resources left to deposit in (lend to) the bank. Thus E is the

no-borrowing, no-lending point.

Because E

involves neither borrowing nor lending, the consumption combination it represents is available to Prudence regardless of the real interest rate. Thus the no-borrowing, no-lending point remains on the budget line when the real interest rate changes.

Next, recall that the budget line's slope is

-(1

+

r).

When the real interest rate,

r,

jumps from

10°/o

to

76°/o,

the slope of the budget line changes from

-1.10

to

-1.76;

that is, the new budget line becomes steeper. Because the budget line becomes steeper but still passes through the no-borrowing, no-lending point, E, it pivots clockwise around point E.

1 62 Part 2 Long-Run Economic Performance

F i g u re 4.A.7

The substitution effect of an increase in the

real interest rate

We assume that Pru­

dence's preferences are such that when the real

interest rate is 10°/o she

chooses the consumption combination at the no­

borrowing, no-lending point, E, on the initial

budget line BL 1. Point E

lies on the indifference

curve IC1. An increase in the real interest rate to

76°/o causes the budget line to pivot clockwise from BL 1 to BL 2, as in

Fig. 4.A.6. By substitut­ ing future consumption for current consumption along the new budget

line, BL _2, Prudence can

reach points that lie

above and to the right of IC1; these points repre­

sent consumption combi­ nations that yield higher utility than the consump­ tion combination at E.

Her highest utility is

achieved by moving to point V, where the new budget line, BL 2, is tan­

gent to indifference

curve IC2. The drop in

current consumption (by 9000) and the resulting

equal rise in saving that occur in moving from _E to V reflect the substitu­ tion effect of the increase in the real interest rate.

§

138,600 . """" ..., = 120,000 (1) 105,000 = 99,000 ...,

�

90,000 75,000 60,000 Real interest rate increases

from _10°/o to _76°/o

48,840 . . .

/New

consumption point

v • 33 000 . .. . ... . . ' . . . ... .. . . .. . . .. . : ' . . . I :

y�

15,000 •

Old consumption point :

• • • • • • • • • .. • • • 0 15,000 30,000 I 60 000 I 90 000 105,000 120,000 135,000 51,000 I 78,750 1 Current consumption, c

The Substitution Effect

As we discussed in Chapter 4, the price of current consumption in terms of future consumption is

1

+ r, because if Prudence increases her consumption by one unit today, thereby reducing her saving by one unit, she will have to reduce her future con­ sumption by

1

+ r units. When the real interest rate increases, current consumption becomes more expensive relative to future consumption.

In

response to this increase in

the relative price of current consumption, Prudence substitutes away from current con­ sumption toward future consumption by increasing her saving. This increase in saving reflects

the substitution effect of the real interest rate on saving,

introduced in Chapter 4.

The substitution effect is illustrated graphically in Fig. 4.A.7. Initially, the real interest rate is

10°/o

and the budget line is BL 1. Suppose for now that Prudence's preferences are such that BL 1 is tangent to an indifference curve, IC1, at the no- bor­ rowing, no-lending point, E.l0 At a real interest rate of

10°/o,

Prudence chooses the consumption combination at E.

When the real interest rate rises from

10°/o

to

76°/o,

the budget line pivots clock­ wise to BL2. Because Prudence's original consumption point the no-borrowing, no-lending point, E also lies on the new budget line, BL2, she has the option of remaining at E and enjoying the same combination of current and future con­

sumption after the real interest rate rises. Points along BL 2 immediately above and to the left of E lie above and to the right of IC1, however. These points represent consumption combinations that are available to Prudence and yield a higher level of utility than the consumption combination at E. Prudence can attain the highest level of utility along BL 2 at point V, where indifference curve IC2 is tangent to BL 2. In response to the increase in the relative price of current consumption, Prudence reduces her current consumption, from

60,000

to

51,000,

and moves from E to V on BL 2. Her reduction of

9000

in current consumption between E and V is equivalent

Chapter 4 Consumption, Savi ng, and I nvestment 1 63

to an increase of 9000 in saving. The increase in saving between E and

V

reflects the substitution effect on saving of a higher real interest rate.

The Income Effect

If Prudence's current consumption initially equals her current resources (current income plus initial wealth) so that she is neither a lender nor a borrower, a change in the real interest rate has only a substitution effect on her saving, as shown in Fig. 4.A.7. If her current consumption initially is not equal to her current resources, however, then an increase in the real interest rate also has an income effect. As we discussed in Chapter 4, if Prudence is initially a saver (equivalently, a lender), with current con­

sumption less than her current resources (current income plus initial wealth), an increase in the real interest rate makes her financially better off by increasing the future interest payments that she will receive. In response to this increase in future interest income, she increases her current consumption and reduces her current saving. On the other hand, if Prudence is initially a borrower, with current consump­ tion exceeding her current resources, an increase in the real interest rate increases the

interest she will have to pay in the future. Having to make higher interest payments in the future makes Prudence financially worse off overall, leading her to reduce her current consumption. Thus, for a borrower, the income effect of an increase in the real interest rate leads to reduced current consumption and increased saving.

The Substitution Effect and the Income Effect Together

Figure 4.A.8 illustrates the full impact of an increase in the real interest rate on Prudence's saving, including the substitution and income effects assuming that Pru­

dence initially is a lender. As before, Prudence's original budget line is

BL 1

when the real interest rate is 10°/o. We now assume that Prudence's preferences are such that

BL 1

is tangent to an indifference curve,

IC1,

at point D. Thus, at a 10°/o real interest rate, Pru­

dence plans current consumption of 45,000 and future consumption of 49,500. Her cur­ rent resources equal 60,000 (current income of 42,000 plus initial assets of 18,000), so if she enjoys current consumption of 45,000 she will have resources of 15,000 to lend. Her chosen point, D, is located to the left of the no-borrowing, no-lending point, E (current consumption is lower at D than at E), showing that Prudence is a lender.

The increase in the real interest rate from 10°/o to 76°/o causes Prudence's budget line to pivot clockwise through the no-borrowing, no-lending point, E, ending at

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