The formal model developed in this appendix provides a helpful insight: The effect on consumption of a change in current income, expected future income, or wealth depends only on how that change affects the consumer’s present value of lifetime resources, or PVLR.
an Increase in Current Income
Suppose that Prudence receives a bonus at work of 12,000, which raises her current real income from 42,000 to 54,000. Her initial assets (18,000), future income (33,000), and the real interest rate (10%) remain unchanged; hence the increase of 12,000 in current income implies an equal increase in Prudence’s present value of lifetime resources, or PVLR. If she hasn’t yet committed herself to her original consumption–
saving plan, how might Prudence revise that plan in light of her increased current income?
We use the graph in Fig. 4.A.4 to answer this question. In Fig. 4.A.4, BL1 is Prudence’s original budget line, and point D, where c = 45,000 and cf = 49,500, represents Prudence’s original, pre-bonus consumption plan. Prudence’s bonus will allow her to consume more, both now and in the future, so the increase in her income causes her budget line to shift. To see exactly how it shifts, note that the increase of 12,000 in Prudence’s current income implies that her PVLR also increases by 12,000. Because the horizontal intercept of the budget line occurs at c = PVLR, the bonus shifts the horizontal intercept to the right by 12,000. The slope of the budget line, -(1 + r) = -1.10, remains unchanged because the real interest rate r is unchanged. Thus the increase in current income of 12,000 causes a parallel shift of the budget line to the right by 12,000, from BL1 to BL2.
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That shift demonstrates graphically that, after receiving her bonus, Prudence can enjoy greater current and future consumption. One strategy for Prudence, represented by point K on the new budget line BL2, is to use the entire bonus to increase her current consumption by 12,000 while leaving her future consumption unchanged. Another strategy, represented by point H on BL2, is to save all of her bonus while keeping her current consumption unchanged, and then use both the bonus and the interest of 1,200 earned on the bonus to increase her future con-sumption by 13,200.
If Prudence operates under a consumption-smoothing motive, she will use her bonus to increase both her current consumption and (by saving part of her bonus) her future consumption, thereby choosing a point on BL2 between point K (consume the entire bonus) and point H (save the entire bonus). If her indifference curves are as shown in Fig. 4.A.4, she will move to J, where her new budget line, BL2, is tangent to the indifference curve IC**. At J, current consumption, c, is 51,000, future con-sumption, cf, is 56,100, and saving, s, is 54,000 - 51,000 = 3000. Both current and future consumption are higher at J than at D (where c = 45,000 and cf = 49,500).
Prudence’s current saving of 3000 at J is higher than her saving was at D (where she dissaved by 3000) because the increase in her current consumption of 6000 is less than the increase in her current income of 12,000. This example illustrates that an increase in current income raises both current consumption and current saving.
an Increase in Future Income
Suppose that Prudence doesn’t receive her bonus of 12,000 in the current period, so that her current income, y, remains at its initial level of 42,000. Instead, because of an improved company pension plan, she learns that her future income will increase by 13,200, so yf rises from 33,000 to 46,200. How will this good news affect Prudence’s current consumption and saving?
FIgure 4.a.4
an increase in income or wealth
An increase in current income, future income, and/or initial wealth that raises Prudence’s PVLR by 12,000 causes the budget line to make a parallel shift to the right by 12,000, from BL1 to BL2. If Prudence’s original consumption plan was to consume at point D, she could move to point H by spending all the increase on future consumption and none on current consumption;
or she could move to point K by spending all the increase on current consumption and none which has both higher current consumption and higher future consump-tion than D. Point J is optimal because it lies where the new budget line BL2 is tangent to an indifference curve, IC**.
Future consumption, cf
Current consumption, c 45,000 51,000 57,000 90,000 102,000
0
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At a real interest rate of 10%, the improvement in the pension plan increases the present value of Prudence’s future income by 13,200/1.10, or 12,000. So, as in the case of the current-period bonus just discussed, the improved pension plan raises Prudence’s PVLR by 12,000 and causes a parallel shift of the budget line to the right by that amount. The effects on current and future consumption are there-fore exactly the same as they were for the increase of 12,000 in current income (and Fig. 4.A.4 applies equally well here).
Although increases in current income and expected future income that are equal in present value will have the same effects on current and planned fu-ture consumption, the effects of these changes on current saving are different.
Previously, we showed that an increase in current income raises current saving.
In contrast, because the increase in future income raises current consumption (by 6000 in this example) but doesn’t affect current income, it causes saving to fall (by 6000, from –3000 to –9000). Prudence knows that she will be receiving more income in the future, so she has less need to save today.
an Increase in Wealth
Changes in wealth also affect consumption and saving. As in the cases of current and future income, the effect of a change in wealth on consumption depends only on how much the PVLR changes. For example, if Prudence finds a passbook sav-ings account in her attic worth 12,000, her PVLR increases by 12,000. To illustrate this situation, we use Fig. 4.A.4 again. Prudence’s increase in wealth raises her PVLR by 12,000 and thus shifts the budget line to the right by 12,000, from BL1 to BL2. As before, her optimal consumption choice goes from point D (before she finds the passbook) to point J (after her increase in wealth). Because the increase in wealth raises current consumption (from 45,000 at D to 51,000 at J) but leaves current income (42,000) unchanged, it results in a decline in current saving (from –3000 at D to –9000 at J). Being wealthier, Prudence does not have to save as much of her cur-rent income (actually, she is increasing her dissaving) to provide for the future.
The preceding analyses show that changes in current income, future income, and initial wealth all lead to parallel shifts of the budget line by the amount that they change the PVLR. Economists use the term income effect to describe the impact of any change that causes a parallel shift of the budget line.
the Permanent Income theory
In terms of our model, a temporary increase in income represents a rise in current income, y, with future income, yf, held constant. A permanent increase in income raises both current income, y, and future income, yf. Therefore a permanent one-unit increase in income leads to a larger increase in PVLR than does a temporary one- unit increase in income. Because income changes affect consumption only to the extent that they lead to changes in PVLR, our theory predicts that a permanent one-unit increase in income will raise current and future consumption more than a temporary one-unit increase in income will.
This distinction between the effects of permanent and temporary income changes is emphasized in the permanent income theory of consumption and saving, developed in the 1950s by Nobel laureate Milton Friedman. He pointed out that income should affect consumption only through the PVLR in a many-period ver-sion of the model we present here. Thus permanent changes in income, because they last for many periods, may have much larger effects on consumption than
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temporary changes in income. As a result, temporary income increases would be mostly saved, and permanent income increases would be mostly consumed.7 Consumption and Saving over many Periods: the Life-Cycle model
The two-period model suggests that a significant part of saving is intended to pay for retirement. However, it doesn’t reflect other important aspects of a consumer’s lifetime income and consumption patterns. For example, income typically rises over most of a person’s working life, and people save for reasons other than re-tirement. The life-cycle model of consumption and saving, originated in the 1950s by Nobel laureate Franco Modigliani and his associates, extends the model from two periods to many periods and focuses on the patterns of income, consump-tion, and saving throughout an individual’s life.
The essence of the life-cycle model is shown in Figure 4.A.5. In Fig. 4.A.5(a), the typical consumer’s patterns of income and consumption are plotted against the con-sumer’s age, from age twenty (the approximate age of economic independence) to age ninety (the approximate age of death). Two aspects of Fig. 4.A.5(a) are significant.
First, the average worker experiences steadily rising real income, with peak earnings typically occurring between the ages of fifty and sixty. After retirement, income (excluding interest earned from previous saving) drops sharply.
Second, the lifetime pattern of consumption is much smoother than the pat-tern of income over time, which is consistent with the consumption-smoothing motive discussed earlier. Although shown as perfectly flat in Fig. 4.A.5(a), con-sumption, in reality, varies somewhat by age; for example, it will be higher during years of high child-rearing expenses. An advantage of using the life-cycle model to study consumption and saving is that it may be easily modified to allow for various patterns of lifetime income and consumption.
The lifetime pattern of saving, shown in Fig. 4.A.5(b), is the difference between the income and consumption curves in Fig. 4.A.5(a). This overall hump-shaped pattern has been confirmed empirically. Saving is minimal or even negative during the early working years, when income is low. Maximum saving occurs when the worker is between ages fifty and sixty, when income is highest. Finally, dissaving occurs during retirement as the consumer draws down accumulated wealth to meet living expenses.
An important implication of the hump-shaped pattern of saving is that national saving rates depend on the age distribution of a country’s population. Countries with unusually young or unusually old populations have low saving rates, and countries with relatively more people in their middle years have higher saving rates.
bequests and Saving
We have assumed that the consumer plans to spend all of his or her wealth and income during his or her lifetime, leaving nothing to heirs. In reality, many people leave bequests, or inheritances, to children, charities, and others. To the extent that consumers desire to leave bequests, they will consume less and save more than when they simply consume all their resources during their lifetimes.
7Friedman also provided some of the first empirical evidence for this theory. For example, he found that the consumption of farm families, on average, responded less to changes in income than the consumption of nonfarm families. Friedman’s explanation was that, because farm incomes depend heavily on weather and crop prices, both of which are volatile, changes in farm incomes are much more likely to be temporary than are changes in nonfarm incomes. Current changes in farm incomes have a smaller effect on the PVLR and therefore have a smaller effect on current consumption.
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ricardian equivalence
One of the most significant results of analyzing our model is that changes in income or wealth affect desired consumption only to the extent that they affect the consumer’s PVLR. The point made by advocates of Ricardian equivalence, discussed in Chapter 4, is that, holding current and future government purchases constant, a change in current taxes does not affect the consumer’s PVLR and thus should not affect desired consumption, Cd, or desired national saving, Y - Cd - G.
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%, in the future period the government’s outstanding debt will be 330 greater than it would have been without the tax cut.
FIgure 4.a.5
Life-cycle consumption, income, and saving (a) Income and consump-tion are plotted against age. Income typically rises gradually throughout most of a person’s work-ing life and peaks shortly before retirement. The desire for a smooth pat-tern of consumption means that consumption varies less than income over the life cycle.
Consumption here is constant. (b) Saving is the difference between income and consump-tion; the saving pattern is hump-shaped. Early in a person’s working life consumption is larger than income, so saving is negative. In the middle years saving is positive;
the excess of income over consumption is used to repay debts incurred earlier in life and to provide for retirement.
During retirement people dissave.
20
Age Income and consumption (dollars per year)
Consumption
0 Saving (dollars per year)
40 60 90
20
Age
40 60 65 90
Dissaving Dissaving
Saving
Income
Saving 65
(b) (a)
ChaPter 4 | Consumption, Saving, and Investment 163
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 in-terest, 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%, 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 in-come, expected future inin-come, and wealth, and that permanent income changes have larger effects on consumption than 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 bor-rowing constraint. If the consumer wouldn’t want to borrow even if borbor-rowing 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 con-sumption is greater for a consumer who faces a binding borrowing constraint than is predicted by our simple model without borrowing 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 constraints exist, consumption may be excessively sensitive to current income.8
8Although we have no direct way of counting how many consumers are constrained from borrow-ing, several studies estimate that, to account for the observed relationship between consumption and current income, during any year some 20% to 50% 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 O. Blanchard and S. Fischer, eds., NBER Macroeconomics 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,” Econometrica, March 1982, pp. 461–481.
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