A THEORY OF TIME PREFERENCE.

TROSTEL, PHILIP A. ; TAYLOR, GRANT A.

GRANT A. TAYLOR [*]

This article proposes that people generally prefer present consumption to future consumption because their expected utility from consumption (eventually) falls as their mental and physical abilities (eventually) decline with age. Moreover; contrary to the ubiquitous intertemporal formulation with a constant rate of time preference and contrary to three recent theories of time preference that predict decreasing discounting as people age, this article asserts that discounting increases over the life cycle. This hypothesis is supported by data from the Panel Study of Income Dynamics as well as evidence from numerous previous studies. (JEL D91)

1. INTRODUCTION

Despite the almost universal assumption of time preference in models of intertemporal choice, an adequate explanation of why people discount the future has not been provided. [1] Olsen and Bailey (1981) make a convincing case for positive time preference by contradiction (i.e., without time preference models of intertemporal choice do not yield predictions consistent with observed behavior), but they do not explain why individuals would rationally discount the future. [2] Three recent studies have attempted to provide the explanation. Rogers (1994) argues that societies that discount the future (at a rate of about 2%) will be the long-run survivors in an evolutionary fitness situation. [3] Posner (1995) argues that discounting occurs because individuals have "multiple selves," that is, "people are weighting their present consumption far more heavily than their future consumption... [because] the present self and the future self are, in some meaningful sense, separate persons" (92). Becker and Mulligan (1997) argue that discounting occurs because of a rational "defective recognition of future utilities" (730). But, as will be shown, these three theories are inconsistent with empirical evidence on consumption over the life cycle in at least one important aspect.

This article constructs and empirically supports a new theory of time preference. The hypothesis is that people do not have an intrinsic preference for present consumption relative to future consumption. Instead, the instantaneous utility function (also known as the felicity function) is expected to vary with age, just as people's physical and mental abilities vary with age. In other words, the ability to enjoy consumption eventually deteriorates over the life cycle along with other abilities. This life-cycle view of the expected utility function causes people to rationally devalue consumption in periods of low expected felicity, that is, discount future consumption. Perhaps this is what is meant by the cliche "don't wait until you are too old to enjoy it." Moreover, this causes individuals' implicit discount rates to vary systematically with age. [4] In fact, this provides the testable implication of the theory. The standard specification of intertemporal preferences holds the discount rate constant over th e life cycle. [5]

To be more precise, in the early stage of the life cycle the capacity to enjoy consumption may be expected to rise. Hence a young individual's discount rate may be negative. Figure 1 illustrates this possibility with a hypothetical adult life-cycle marginal felicity function for a given level of consumption, [U.sub.C](C, a) (C denotes consumption and a denotes age), and its corresponding discount rate function, [rho](a). The vast majority of wealth is held by older people, that is, those in the declining stage of their abilities. Thus the preference for present consumption over future consumption appears to hold in the aggregate.

This view of intertemporal preferences may at first seem inconsistent with practically all dynamic analyses, which are based on the notion of stationary utility. On further consideration, however, the hypothesis is quite consistent with modern consumer theory. Since Becker (1965) it has been widely accepted that households derive utility from commodities that are produced with market goods combined with household time and skills (i.e., home human capital). Moreover, since Becker (1964) (and many others) it has also been widely accepted that people's skills in the work place (i.e., market human capital) depreciate. Thus, it does not seem inconsistent to assume that people's home skills depreciate as well. In fact, it seems inconsistent not to assume so. Moreover, during adulthood people experience substantial losses in hearing, eyesight, sense of smell, sense of touch, sense of taste, strength, dexterity, stamina, memory, intellectual powers, health, and practically every other conceivable ability as they gro w old. [6] It seems reasonable that these abilities would influence people's expected capacity for enjoyment. [7]

Furthermore, although this is the first study to explicitly formulate this theory, this idea is not completely novel. The notion that the utility function could vary with age is mentioned in passing in numerous studies, including Rae (1905), Thurow (1969), Ghez and Becker (1975), Olsen and Bailey (1981), Blinder et al. (1983), Ehrlich and Chuma (1990), and Banks et al. (1998) [8] Borsch-Supan and Stahl (1991) investigate the similar notion that the utility function could be constrained by health deterioration during old age. Johannesson and Johansson (1997b) find evidence that the discounting of life-saving programs rises significantly with the age of the person saved. Similarly, Cropper et al. (1994) provide evidence that the discounting of life-saving programs falls and then rises dramatically with age. Yaari (1965) suggests the analogous concept that the discounting of bequests is likely to fall and then rise with age because bequests are particularly important during middle age. Attanasio et al. (1995) present evidence that discounting decreases initially and then increases substantially during the life cycle. Rogers (1994), Posner (1995), and Becker and Mulligan (1997) also contend that the discount rate should vary with age (but in the opposite direction) . This idea is also implicit in most of the (micro) empirical work on the intertemporal allocation of consumption that includes age as an ad hoc explanatory variable. [9]

Thus, the hypothesis that the expected utility function varies systematically with age can not only explain the existence of rational discounting, but it can also provide an explicit theoretical justification for the common practice of including age as an explanatory variable in econometric work on consumption. The theory is now expressed more explicitly.

II. HYPOTHESIS

Most of the typical simplifications in analyzing intertemporal choice are followed. [10] Lifetime expected utility, V, is a time-separable and concave function of consumption. [11] The departure from the usual intertemporal framework is that there is no intrinsic rate of pure time preference; instead, the felicity function is age-dependent. Thus the utility function is given by

(1) [V.sub.t] = [E.sub.t] [[[[integral].sup.T].sub.t] U([C.sub.a], a)da],

where [E.sub.t] is the mathematical expectation conditional on information available at age t, T is the terminal age, and [C.sub.a] is consumption at age a. [12] The expected life-cycle felicity function obeys the following assumptions:

(a1) [E.sub.t][d[U.sub.C]/da] is continuous in age,

(a2) [E.sub.t][[lim.sub.a[right arrow]T] d[U.sub.c]/[da\.sub.dC=0]] [less than] 0,

(a3) [E.sub.t][[d.sup.2][U.sub.C]/[[da.sup.2]\.sub.dC=0]] [less than] 0,

where [U.sub.C] is the marginal ulitity of consumption. Implicit in the above specification is the assumption that abilities and goods are complements. [13] Obviously some goods can substitute for some abilities (e.g., eye glasses, medical care, various personal services, etc.). Although there is clearly some room for such substitution, its scope seems limited. For example, someone cannot be hired to enjoy a show, a meal, or a trip for another. Thus, abilities and composite consumption are assumed to be complementary. The potential complementarity and substitutability of various types of consumption goods and abilities is clearly an interesting issue, but is beyond the scope of this article.

The budget constraint is

(2) dA/da = [r.sub.a][A.sub.a] + [Y.sub.a] - [C.sub.a],

(3) C = - ([U.sub.C]/[CU.sub.CC])(r + [U.sub.C]),

and the terminal condition is [A.sub.T] [greater than or equal to]0, where A is nonhuman wealth, r is the after-tax real interest rate, and Y is (exogenous) after-tax real labor income. The first-order condition for intertemporal utility maximization is

where [sim] represents the instantaneous rate of change. This Euler equation is very similar to the typical one. In fact, the typical Euler equation is a special case of equation (3). The first term on the right side of the equation is the negative of the intertemporal elasticity of consumption. The last term in the equation, the expected rate of change in the marginal utility of consumption, can be interpreted as the (nonconstant) expected depreciation rate of felicity. Or, more traditionally, [U.sub.C] can be interpreted as the negative of the discount rate, -[rho](a). In fact, if U(C, a) is replaced with the usual [e.sup.-[rho]a] u(C), then [U.sub.C] is exactly -[rho]. Assumption (a2) (marginal utility is expected to decline in old age) requires that [rho](a) eventually becomes negative, but this does not preclude (or require) it being positive (i.e., an appreciation rate) early in adult life.

It is instructive to compare the expected felicity function described by assumptions (a1)-(a3) and illustrated in Figure 1 to the discounted felicity function in the usual framework, [e.sup.[rho]a] [u.sub.c](C). A corresponding illustration is given in Figure 2 for the typical framework. The marginal value of future consumption declines in both cases, thus the typical specification first-order approximates the life-cycle felicity specification. The standard framework, however, differs from the second-order behavior of the expected lifecycle felicity function. The expected marginal value of future consumption declines exponentially in the life-cycle framework, but the typical specification imposes a constant rate of decline (hence, discounted marginal utility is convex over time, i.e., [d.sup.2] [U.sub.C]/[da.sup.2] [greater than] 0). If expected felicity declines at a constant rate, then the idea that future consumption is discounted because of falling expected felicity is consistent with and observationally equ ivalent to the idea of intrinsic preference for current consumption.

It seems unlikely, however, that felicity declines at a constant rate. [14] An increasing rate of decline is consistent with observed age variations in human abilities. In other words, extensive research on aging shows that abilities generally change in the concave manner shown in Figure 1. [15] Although regression analysis is rarely used in this field, and many of the data sets are too small to determine the precise nonlinear relationship between age and abilities, our search was unable to uncover any clear evidence of a constant (or decreasing) rate of decline of an ability (nor were we able to find evidence of any ability that does not decline noticeably after middle age). As far as we can ascertain, an increasing rate of decline of human abilities (on average) is a natural law. Moreover, the more precise data on aging indicate a concave relationship between age and abilities. Some of this evidence suggests that the relationship might become convex in very old age, but this appears to be the result of sel ection bias, that is, the more functionally able are more likely to be test subjects in old age because they are more likely to be alive and healthy (most of these studies exclude unhealthy subjects). [16]

It is also instructive to compare the consumption path predicted from the life-cycle-utility model to that from the standard constant-time-preference model. In the simple standard framework, the consumption path depends only on the difference between the rates of interest and time preference, both of which are assumed constant over the life cycle. [17] In the life-cycle-utility framework, however, the consumption trajectory changes over the life cycle. In other words, life-cycle factors drive the consumption path, as opposed to an arbitrary relation between r and [rho]. In particular, an individual will consume relatively less late in the life cycle, which we believe explains why age is consistently a significant explanatory variable in econometric analyses of consumption.

An important implication of our hypothesis is that there does not have to be any intrinsic preference for current consumption versus future consumption if there is a life cycle of expected utility. A declining expected felicity function undermines Olsen and Bailey's (1981) argument to the contrary. Olsen and Bailey's argument uses the intertemporal first-order condition evaluated over a long time horizon, say, [tau] years:

(4) [U.sub.C]([C.sub.[tau]])/[U.sub.C]([C.sub.0]) = [(1 + r).sup.-[tau]].

If r [greater than] 0, then the right-hand side of (4) approaches zero as [tau] increases. If there is no intrinsic time preference, then [C.sub.0] must approach zero and [C.sub.[tau]] must approach satiation consumption. This type of behavior is not observed; thus, time preference is revealed. The argument does not hold, however, if the felicity function declines with age. In this case, the left-hand side of (4) also approaches zero as [tau] increases, even if [C.sub.0] = [C.sub.[tau]]. Intrinsic time preference does not have to be invoked to yield a plausible consumption profile.

III. ECONOMETRIC MODEL

The theory is tested by following Lawrance (1991) and Dynan (1993) in estimating the implicit discount rate using consumption data from the Panel Study of Income Dynamics (PSID). [18] Their approach is modified by explicitly modeling the rate of time preference as a function of age. Their approach is further modified by including survival uncertainty (in this respect Skinner [1985] is followed). As shown by Yaari (1965), survival risk can affect the discount rate. Moreover, this risk varies with age, hence failure to account for it could bias the estimate of the age coefficient.

To make the model econometrically tractable, utility function (1) takes the following form:

(5) [V.sub.i,t] = [E.sub.t] [[[[sigma].sup.D].sub.a=t][gamma][[pi].sub.[tau]]([a.sub.i])exp(-[eta ]([a.sub.i])[a.sub.i]) X [([C.sub.i,a][[F.sup.-0].sub.i,a]).sup.([gamma]-1)/[gamma]]/([gamma] - 1)],

(7) [[pi].sub.a+1]([a.sub.i])(1 + [r.sub.i,a+1])[([C.sub.i,a+1]/[C.sub.i,a]).sup.-1/[gamma]] X [([F.sub.i,a+1]/[F.sub.i,a]).sup.0/[gamma]]exp(-[alpha] - 1/2 [beta] - [beta][a.sub.i] - [[sigma].sub.k][[delta].sub.k][X.sub.k,i]) = 1+ [[epsilon].sub.i,a+1],

where i indexes an individual family, D is the maximum possible length of life, [[pi].sub.[tau]](a) is the probability of living to age [tau] conditional on surviving to age a, [eta](a) is the implicit discount factor (instantaneous, rather than annual, discounting is specified to simplify the estimation), [gamma] is the intertemporal elasticity of consumption, C is household consumption, F is family size, and 1 [greater than or equal to] [theta] [greater than or equal to] 0 allows for economies of scale in family consumption. To test the theory in a simple way, the linear approximation [alpha] + [beta]a is specified for [rho](a). [19] The discount rate is also assumed to depend on permanent income, Y, education, E, and race, R. Thus the discount factor is

(6) [eta]([a.sub.i]) = [alpha] + 1/2[beta][a.sub.i] + [[sigma].sub.k][[delta].sub.k][X.sub.k,i], [X.sub.k] = Y, E, R. [20]

The Euler equation resulting from (5) with (6) is

where [[epsilon].sub.i,a+1] is the random forecast error. Taking the logarithm of (7) and rearranging yields

(8) ln([C.sub.i,a+1]/[C.sub.i,a])

=[theta]ln([F.sub.i,a+1]/[F.sub.i,a])

+[gamma][ln([[pi].sub.a+1]([a.sub.i])) + ln(1+[r.sub.i,a+1])-[alpha]

+1/2([[[sigma].sup.2].sub.[epsilon]] - [beta]) - [beta][a.sub.i] - [[sigma].sub.k] [[delta].sub.k] [X.sub.k,i]

-[[epsilon].sub.i,a+1] + 1/2([[[epsilon].sup.2].sub.i,a+1]) - [[[sigma].sup.2].sub.[epsilon]])]. [21]

The model to this point has ignored several factors that can remove the effect of survival uncertainty on consumption. [22] Rather than explicitly model all these factors, a separate parameter, 1 [greater than or equal to] [lambda] [greater than or equal to] 0, is placed on ln([pi]) (in this respect the model is more general than Skinner's [1985]). This parameter summarizes the extent that these factors do not eliminate the effect of (the log of) survival risk. Another complication arises because age and survival probability for couples are not well defined unless an assumption is made about choices within households. We believe the most reasonable assumption is that the wife and husband share equally in making family decisions. [23] This assumption coupled with the linear approximation of [rho](a) implies that the appropriate age for couples is their average age. This assumption also implies that [[pi].sub.a+1](a) for couples is their average probability of surviving for another year. [24] The parameter on ln([pi]) for couples, [[lambda].sub.M], however, will differ from that of singles, [[lambda].sub.S](i.e., [[lambda].sub.M] [less than or equal to] [[lambda].sub.S]). This difference is captured using an interactive dummy variable, m (1 for married, 0 for single), and allowing [lambda] to differ between groups. Incorporating these factors into (8) yields the equation to be estimated:

(9) ln([C.sub.i,a+1]/[C.sub.i,a])

= [[beta].sub.0] + [[beta].sub.1] ln(1 + [r.sub.i,a+1]) + [[beta].sub.2][a.sub.i]

+ [[beta].sub.3S](1 - [m.sub.i]) ln([[pi].sub.a+1]([a.sub.i]))

+ [[beta].sub.3M][m.sub.i] ln([[pi].sub.a+1]([a.sub.i]))

+ [[beta].sub.4] ln([F.sub.i,a+1]/[F.sub.i,a])

+ [[sigma].sub.k] [[beta].sub.5k][X.sub.k,i] + [e.sub.i,a+1],

where [[beta].sub.0] = [gamma](1/2 [[[sigma].sup.2].sub.[epsilon]] - [alpha] - 1/2 [beta]), [[beta].sub.1] = [gamma], [[beta].sub.2] = -[gamma][beta], [[beta].sub.3S] = [gamma][[lambda].sub.S], [[beta].sub.3M] = [gamma][[lambda].sub.M], [[beta].sub.4] = [theta], [[beta].sub.5k] = -[gamma][[delta].sub.k], [e.sub.i,a+1] = [gamma](1/2 [[epsilon].sub.i,a+1] - 1/2 [[[sigma].sup.2].sub.[epsilon]] - [[epsilon].sub.i,a+1]).

The estimation of equation (9) is complicated slightly by two additional issues. First, there are likely to be aggregate expectational errors caused by macroeconomic shocks. This can potentially bias the estimates because the time series is somewhat short. Following Chamberlain (1984), this problem is addressed by including year dummy variables in equation (9). Second, the PSID consumption data appears to be subject to substantial measurement error. Under the reasonable assumption that ln(C) is subject to random measurement error, there will be an additional MA(1) error in (9). This can potentially bias the standard errors of the least-squares parameter estimates. To correct for this problem, White's (1984) method is used to produce a variance-covariance matrix that is consistent under unknown heteroscedasticity and MA(1) errors. [25]

IV. DATA

The only source of household consumption data over time is the PSID. It contains 14 years of consistently measured food consumption data (1974 to 1987 surveys). This measures the previous year's annual spending (including food stamps) on food consumed at home and at restaurants. [26] These components are deflated by the appropriate Consume Price Index (CPI).

The PSID also contains income and demographic data. To avoid the possible problem of reverse causality, permanent income is measured as average after-tax real labor income prior to the sample period (practically identical results are obtained using total income). The PSID contains income data back to 1967. Thus, household after-tax labor income for the years 1967-1972 is deflated by the CPI and averaged. Similarly, education is defined as education prior to the sample period. Thus, E is a dummy variable for holding a college degree in 1973 (there are more detailed education variables in the PSID, but college degree is the only one that is consistently reported). This dummy variable is allowed to take three possible values for couples: one if both possess a degree, one-half if one possesses a degree, and zero if neither have a degree. Analogously, R is a dummy variable for nonwhite (black and hispanic) households which can take values of one, one-half, and zero for couples.

The variable used for family size is what the PSID calls the "annual food standard" (AFS), which measures the real minimum cost of food, adjusted for family size and ages of family members. In other words, AFS is meant to measure family consumption needs and presumably is a slightly better adjustment for household consumption than simply family size. The AFS allows for some economies of scale in family consumption by making the following adjustments to families of size: one, +20%; two, +10%; three, +5%; four, 0; five, -5%; six or more, -10%. It adjusts for age using the values shown in Table 1. This age adjustment for food needs is somewhat similar to the hypothesized expected life-cycle utility function. Hence, the data to some extent already incorporate the hypothesis of this paper. Thus, the parameter estimate for age is likely to be conservatively biased.

The real after-tax interest rate is calculated from the average yield on one-year Treasury bills (similar results are obtained using the interest rate on passbook saving accounts). This rate is adjusted by the marginal tax rate calculated by the PSID and the CPI for overall food consumption. [27] Survival probabilities by age, gender, and race for 1974 through 1986 are from U.S. life tables.

Several adjustments were made to the PSID data. To obtain a representative sample, the extra poverty subsample was removed. Only households with no change in marital or household-head status were included to remove those with possible major changes in family preferences. Cases where a "major assignment" was made to food consumption (when the PSID editors felt that an estimate was made with a probable error of at least 10%) were also eliminated. These adjustments leave a sample of 11,895 observations of consumption growth. The mean age in the sample is 53, the minimum and maximum ages are 23 and 93, and almost 95% of the observations lie in the range of 30 to 78.

V. EVIDENCE

The Euler equation estimates are reported in Table 2. The primary case examined is model (1), the model outlined in the previous two sections. Three other cases are reported in Table 2 to show the robustness of the results over different estimation strategies. Model (2) drops the year dummy variables from the regression. This case is reported because, although the year dummies are jointly significant, they remove all the intertemporal variation in the interest rate (there is only cross-section variation due to differences in marginal tax rates). Thus, removing the year dummies substantially reduces the standard error of the estimated intertemporal elasticity of substitution. Model (3) uses consumption dated concurrently with the survey year. In other words, this model follows the interpretation of the data employed by several previous authors, rather than the PSID's interpretation of the data. This case shows that the results are somewhat (but not greatly) affected by the different interpretations of the con sumption data. Model (4) uses the husband's age, survival probability, education, and race for couples, rather than their joint variables. This is the approach taken in similar studies. The results, however, are practically identical to those in model (1). [28]

The results are consistent with those found in other studies, particularly Lawrance (1991) and Dynan (1993) because of the similarity of the data and econometric procedures employed. [29] The estimated intertemporal elasticity of substitution is 0.88 and is marginally statistically significant. The parameter estimate on family size is significantly less than one, which indicates substantial economies of scale in family consumption (it suggests that a doubling in family size causes family consumption to increase by only 35%). The variables that are typically believed to be associated with lower rates of time preference (higher permanent income, higher education, and nonminority status) and hence higher consumption growth have the expected signs, but are not statistically significant. The important coefficient estimate for this study, however, is that for age. It is negative and statistically significant at the 99% confidence level, which supports the thesis of this article.

There is perhaps one unexpected result. The coefficients on survival probabilities are not statistically significant and have the wrong a priori signs. This outcome may not be that surprising, however, given that planned bequests, annuities, and imperfect annuities can eliminate the effect of survival uncertainty. This outcome also suggests that the results of earlier studies, which do not include survival uncertainty, are not significantly biased. More important, though, this indicates that it is age, rather than mortality uncertainty, that causes the discount rate to increase with age. In other words, these results suggest that it is aging which causes discounting, not the probability of dying.

There are, however, four very plausible hypotheses besides aging that could explain the significant negative correlation between consumption growth and age. One, as argued by Heckman (1974) and Ghez and Becker (1975), if consumption and leisure are substitutes, then the life-cycle path of wages may be driving the path of consumption. Two, as argued by Borsch-Supan and Stahl (1991), bad health, which occurs more frequently in old age, may constrain consumption. Three, as argued by Thurow (1969), binding borrowing constraints may force the consumption path to follow the life-cycle profile of income. Four, as emphasized by Nagatani (1972), Deaton (1991), and Carroll (1997), precautionary saving motives may cause the consumption path to mimic the income path over the life cycle. To see whether these four factors are driving the age-consumption relationship, model (1) is modified slightly. The results of these modifications are reported in Table 3.

Model (5) includes the growth of the real net wage rate in the Euler equation. This case shows that including changes in the opportunity cost of leisure has no perceptible effect on the results. [30] It is quite plausible, however, that leisure does not respond quickly to changes in the wage rate. Thus a more appropriate way to observe the effect of substitutions between consumption and leisure may be to include hours of work directly (after instrumenting because it is an endogenous variable). Indeed, this is the approach taken by Blundell et al. (1994), Attanasio and Browning (1995), Attanasio and Weber (1995), and Attanasio et al. (1995). Thus, in model (6) the (instrumented) growth of hours worked is included in the regression. [31] This case confirms that changes in hours of work are significant in explaining changes in consumption. But it does not remove the effect of age on consumption.

The effect of ill health on consumption is investigated in model (7). The only consistently reported health-related variable in the PSID is weeks of work missed by the household head due to illness (there are better health measures in the PSID, but unfortunately, not in many of the years of our sample). The growth rate of this variable is included in the regression in model (7). As in models (5) and (6), the additional parameter estimate has the correct a priori sign (and, in this case, is not quite statistically significant), but the effect of age on consumption growth is unaffected.

The influence of liquidity constraints on consumption is examined in model (8). The PSID does not contain information on financial assets, but it does have data on financial income. Thus, in model (8) the observations that correspond to periods of zero asset income (i.e., the observations that are the most likely to be affected by binding liquidity constraints) are removed from the sample.

Because each observation of consumption growth could be influenced by borrowing constraints in two periods, this restriction reduces the sample by over 28%. Nonetheless, the results are little affected. [32] This case shows that removing the observations where liquidity constraints are the most likely to be binding reduces the estimated discount rate (because it raises the intercept and nonwhite coefficient, and it reduces the coefficients on permanent income and college degree). It does not, however, remove the effect of age on discounting (indeed, it is slightly stronger in this case).

Carroll (1997) shows that precautionary saving motives and the resulting "buffer-stock" behavior will be particularly important for younger households. He argues that most households follow buffer-stock behavior until about age 45. [33] Thus, all observations from households under the age of 45 are removed from the sample in model (9). This also reduces the sample by over 28%. But more important, it also substantially truncates the variation in ages. Not surprisingly, the precision of the age coefficient estimate is reduced considerably. [34] The age coefficient is noticeably lower as well. In addition, the intercept is much lower and is less precisely estimated (this is noted because it affects the subsequent estimates shown in Tables 4 and 5). This case suggests that precautionary saving motives may indeed explain some (but apparently not all) of the age variation in the implicit discount rate.

Table 4 reports the estimates of the underlying model parameters that are derived from the coefficient estimates in Tables 2 and 3. [35] In the interest of brevity, the parameter estimates from the models with results very similar to model (1) (models [4]-[7]) are not shown. [36] The parameter estimates generally seem reasonable. The estimates of the survival parameters, [[lambda].sub.S] and [[lambda].sub.M], are usually below their a priori range between zero and one, but are never close to being significantly different from zero. The estimate of [[delta].sub.Y] in model (1) indicates that the discount rate falls by about 0.45 percentage points per $10,000 of permanent income. [37] The estimate of [[delta].sub.E] implies that someone with a college degree has a discount rate that is 1.8 percentage points lower than someone without a degree. The estimate of [[delta].sub.R] implies that nonwhites have a discount rate that is 1.9 percentage points higher than that of whites. These parameter estimates are not s tatistically significant, however.

The important parameter estimates for this article are [alpha] and [beta]. The standard intertemporal formulation implies that [alpha] (the coefficient reflecting pure time preference) is positive and [beta] (the coefficient reflecting the effect of aging on discounting) is zero. The theories of Rogers (1994), Posner (1995), and Becker and Mulligan (1997) predict (for different reasons) that [alpha] is positive and [beta] is negative. The hypothesis of this article, however, predicts that [beta] is positive and [alpha] is theoretically indeterminate. [38] The results in Table 4 clearly support the hypothesis of this study and not the alternatives. In all models [alpha] is not significantly different from zero, and [beta] is statistically greater than zero in every case except model (9). The estimated effect of intrinsic discounting (i.e., a positive [alpha]) is clearly distinguished from the estimated effect of aging (i.e., a positive [beta]) in all cases but model (9), and this case is not particularly surpri sing given that it substantially truncates the distribution of ages. The estimate of [beta] in model (1) indicates that discounting increases by 0.19 percentage points per year of age.

The evolution of the estimated implicit discount rate over the adult life cycle is shown in Table 5. This evolution is similar in all models, even in model (9) when the effect of aging cannot be clearly disentangled from the effect of intrinsic discounting. These results show that, contrary to the popular notion that the old are more patient than the young, aging causes increasing discounting of future consumption.

VI. CORROBORATING EVIDENCE

In some sense these results are hardly surprising. The estimated increase in the discount rate with age is driven by the humpshaped pattern of consumption over the life cycle. Figure 3 illustrates this relationship. It shows the mean log (food) consumption over the life cycle in our data. This sort of empirical pattern has been well known since being documented by Thurow (1969), Ghez and Becker (1975), and Blinder et al. (1983). Thus, although the results come from a data set which is far from ideal, [39] there is little reason to suspect that the results are unique to the PSID data. Figure 4 demonstrates this. Using data from the Consumer Expenditure Survey (CES), this figure plots mean log total consumption against age (for comparison, it also shows food consumption). The life-cycle pattern of food consumption is the same as that of total consumption. This empirical pattern holds across data sets, countries, and categories of consumption. [40]

Moreover, there is a great deal of previous evidence to corroborate the results shown here. The evidence in total may not be overwhelming at this point, but it is certainly very considerable.

For instance, although they do not explore or argue this point, Blinder et al. (1983) conclude that "the data [from the Longitudinal Retirement History Survey] are consistent with the life cycle theory only if it is assumed that people's utility functions shift systematically by age in such a way as to produce an optimal consumption stream with quite low consumption levels late in life" (92). Effects of age on consumption growth similar to that found here are reported in other studies using the PSID, namely, Zeldes (1989), Lawrance (1991), and Dynan (1993). Using the CES, Attanasio et al. (1995) estimate the discount rate over the life cycle and find that it declines until about age 35 and then increases substantially. [41] Using the UK Family Expenditure Survey, Banks et al. (1998) find a positive effect of aging on discounting that is about three times as large and more precisely estimated than the effect found in this study. But like the studies using the PSID, Banks et al. (1998) do not attempt to explai n this discounting behavior.

Very different and compelling pieces of corroborating evidence come from Cropper et al. (1994) and Johannesson and Johansson (1997a, 1997b). Cropper et al. (1994) find that age significantly increases the discount rates derived from survey data. They also find that people discount older people's lives much more than can be explained by accounting for their fewer life years saved. They speculate that "this might reflect the view that quality of life diminishes as one ages" (258). Moreover, they find that "the utility attached to saving an anonymous life is a hump-shaped function of the age of the person saved" (245) (with the maximum at age 28). [42]

Johannesson and Johansson (1997b) confirm the finding that people significantly discount older lives more than can be explained by accounting for the fewer life years saved and discounting. Moreover, Johannesson and Johansson (1997a) provide convincing evidence to support Cropper et al.'s (1994) (and our) conjecture that the heavy discounting of older lives is due to expected diminishing quality of life in old age. In particular, Johannesson and Johansson find that the willingness to pay for a life-extending program in old age is very strongly correlated with the expected quality of life in old age. They show that the expected monetary value of an additional year of life at age 75 decreases dramatically with the expected quality of life at 75.

VII. CONCLUSION

This article has put forth an explanation of why future consumption is rationally discounted. The hypothesis is that a lower marginal value is placed on consumption in the future because the ability to enjoy consumption is expected to be lower in the future. In other words, discounting occurs because the expected marginal utility of consumption is (eventually) declining, just as all other human abilities are (eventually) declining. No intrinsic time preference is necessary to explain observed intertemporal behavior. Moreover, if the ability to enjoy consumption changes in the manner that other human abilities change, then discounting should increase as people age because the expected rate of decline of marginal utility should increase with age. This prediction stands in contrast to the decrease in discounting with age predicted in three recent attempts to explain time preference.

This testable prediction of theory was supported by consumption data from the PSID. The empirical work in this study extended earlier work on consumption over the life cycle by explicitly examining the effect of age on the implicit discount rate. The rate of discount was estimated to increase by between 1.1 percentage points and 2.8 percentage points every ten years of adult age. The empirical work also showed that it is not increasing mortality risk over the life cycle that is causing this increase in discounting.

In addition, these results were shown to be robust after attempting to control for other factors that could explain the observed hump-shaped pattern of consumption over the life cycle. The results hold after controlling for changes in family size, the opportunity cost of time, labor supply, and health. Similarly, the results were not affected after dropping the observations that are the most likely to be influenced by binding borrowing constraints. A noticeable effect on the results was found only when eliminating the observations that were the most likely to be affected by strong precautionary-saving motives, and this only removed some of the estimated effect of aging on discounting.

Obviously, however, these results are far from definitive. Clearly there is scope for much further investigation using alternative data sources and methods. The evidence presented here, as well as the corroborating evidence from several other studies, however, do at least suggest some validity to the hypothesis that aging is an important cause of discounting behavior.

Moreover, although the discounting behavior hypothesized in this study is approximated by the standard framework, which assumes a constant rate of pure time preference, there are numerous intertemporal issues that may be better explained within the hypothesized framework. In other words, the approximation implicit in the typical framework may be misleading in some applications of intertemporal choice theory. We conclude with some brief speculation on three of the more obvious examples of this.

The life-cycle theory of discounting may improve the understanding of bequest behavior. In particular, if parents' expected utility of consumption is falling, then the relative value of leaving bequests may increase with age. [43] This could help explain the anomaly that bequests do not appear to decrease with age as much as predicted by the standard life-cycle model. [44] Similarly, the theory could help explain the timing and composition of bequests. For instance, it could help explain why bequests occur late in life (usually death). Furthermore, if bequest motives in the hypothesized framework do indeed differ from those in the standard framework, then this could influence the understanding of the effects of intergenerational redistributions associated with fiscal policy changes (i.e., the Ricardian equivalence issue).

The theory may also provide an explanation of why few people buy annuities. The traditional model suggests that annuities should be an attractive way to insure for longevity. Yet very few people buy annuities. [45] The life-cycle felicity hypothesis provides a simple explanation. Consumption in old age is expected to yield low felicity. Thus, actuarially fair annuities are far from being fair in terms of felicity.

Perhaps the most important ramification of the theory is that it lends support to Pigou's (1932) well-known contention that the social rate of time preference should be zero (or at least less than the private discount rate) because the preferences of future generations are not taken into account in the market place. [46] According to this view, the social discount rate should only reflect the expected rising levels of consumption by future generations. If individuals have no intrinsic time preference and discount the future because of expected declining felicity, then there is no reason to believe that there is intrinsic social time preference. A society is generally not going to expect declining felicity.

Trostel: Associate Professor, Department of Economics, and Research Associate, Margaret Chase Smith Center for Public Policy, University of Maine, 5715 Coburn Hall, Orono, ME 04469-5715. Phone 1-207-581-1646, Fax 1-207-581-1266, E-mail [email protected]

Taylor: Lecturer, Division of Applied Economics, Nanyang Technological University, 50 Nanyang Ave., Singapore. Phone +65-790-5691, Fax +65-792-4217, E-mail [email protected]

(*.) For helpful comments we are grateful to Paul Chen, Mike Ellis, Emily Lawrance, Andrew Oswald, the referees, and seminar participants at Australian National University, University College London, University of East Anglia, Federal Reserve Bank of Dallas, Hong Kong University of Science and Technology, Keele University, Kent State University, University of New South Wales, University of Texas at Arlington, University of Texas at Austin, and University of Warwick.

(1.) Friedman (1969), for example, argues that none of the "reasons for discounting the future relative to the present... appeals to me strongly as a satisfactory explanation. Yet I must confess that I have found no other" (21-23). This led Stigler and Becker (1977) to conclude "that the assumption of time preference impedes the explanation of life cycle variations in the allocation of resources, the secular growth in real incomes, and other phenomena" (89).

(2.) Indeed, many of the initial writers on the subject (e.g., Rae [1905], Jevons [1965], Bohm-Bawerk [1959], Marshall [1920], Pigou [1932], Ramsey [1928], Fisher [1930], and Harrod [1948]) argued that discounting is to some extent irrational.

(3.) On the other hand, Hansson and Stuart (1990) contend that societies that do not have an intrinsic preference for current consumption will be the evolutionary winners.

(4.) This specification of variable discounting does not lead to the inconsistency of plans (i.e., reoptimizing) as shown by Strotz (1956), because the relative value of consumption at a particular date does not change over time.

(5.) Not all models of intertemporal choice hold time preference constant. A number of studies assume that the rate of time preference depends on the level of consumption (e.g., Fisher [1930], Koopmans [1960], Uzawa [1968], Obstfeld [1981, 1990], Epstein and Hynes [1983], and Lucas and Stokey [1984]). This recursive specification of intertemporal preferences, however, neither rules out nor is ruled out by our theory. Similarly, our theory does not contribute to the literature on evidence of hyperbolic discounting (this literature is surveyed in Loewenstein and Elster [1992], Harvey [1994], and Sozou [1998]).

(6.) See, for example, Corso (1981), Verrillo and Verrillo (1985), Aiken (1989), Kenney (1989), Salthouse (1991), and the references in Posner (1995).

(7.) Gfellner's (1989) survey of elderly adults (from 80 to 96 years of age) supports this argument. Her respondents expected substantial deterioration in their health, functional abilities, and life satisfaction over their next five years. The average expected deterioration over the their next five years was about 24%, 20%, and 16%, respectively. Johannesson and Johansson (1997a) report similar findings. The nonelderly adults (from 18 to 69 years old) in their survey expected an average quality of life at age 75 that is 47% less than the average current quality of life from a comparable survey of adults.

(8.) Rae (1905) expressed the idea particularly eloquently: "The approaches of old age are at least certain, and are dulling, day by day, the relish of every pleasure. A mere reasonable regard to their own interest, would therefore, place the present very far above the future, in the estimation of most men" (p. 54).

(9.) See the recent survey by Browning and Lusardi (1996).

(10.) The hypothesis could be demonstrated more elegantly using the home production framework developed by Becker (1965). The hypothesis can be formalized more succinctly, however, using the simpler standard framework.

(11.) There are several natural extensions of the subsequent analysis. The assumptions of time separability and no bequests could be relaxed to provide a more complete analysis of discounting and saving behavior. The life-cycle utility of leisure and its interactions with lifecycle labor productivity and life-cycle utility of consumption could also be an interesting extension.

(12.) The expected path of the life-cycle felicity function is assumed to be exogenous. Another interesting extension of this study would be to allow people's choices (such as exercise, diet, lifestyle, etc.) to alter their expected health and lifespan.

(13.) This is similar to the assumption in Ehrlich and Chuma (1990) that health and goods are complements.

(14.) Grossman (1972) and Ehrlich and Chuma (1990) make similar arguments about the depreciation of health capital.

(15.) See, for example, Corso (1981), Verrillo and Verrillo (1985), Stones and Kozma (1985), Aiken (1989), Kenney (1989), Schaie (1989), Salthouse (1991), Strauss et al. (1993), and Fair (1994).

(16.) This type of selection bias is also likely to be present in the consumption data examined later in this article.

(17.) Other factors may also matter in more complicated models with borrowing constraints, various sources of uncertainty, etc.

(18.) Zeldes (1989) and Runkle (1991) are followed quite closely as well.

(19.) We considered including [a.sup.2] as well. But this would clutter the analysis, and more important, our data is not precise enough to distinguish the separate effects of a and [a.sup.2], i.e., neither coefficient is statistically significant (they are jointly significant, however).

(20.) When discounting is a function of age the discount factor in equation (5), [eta](a), is not the same as the discount rate, [rho](a). The discount rate is -[V\.sub.dC=0]. Hence, [eta](a) = [alpha] + 1/2 [beta]a yields [rho](a) = a + [beta]a.

(21.) This equation is derived using the second-order Taylor approximation ln(1 + [epsilon]) [similar/equal] [epsilon] - 1/2 [[[sigma].sup.2].sub.[epsilon]]. The term 1/2 [[[sigma].sup.2].sub.[epsilon]] is added and subtracted in (8) to preserve a zero mean error.

(22.) Yaari (1965) demonstrated that the effect of survival uncertainty can be eliminated by annuities and/or planned bequests. In addition, private pensions, Social Security, and families can act as imperfect annuities (see Davies [1981], Abel [1985], and Kotlikoff and Spivak [1981]).

(23.) In this respect we differ from Zeldes (1989), Lawrance (1991), Runkle (1991), and Dynan (1993), who implicitly assume that the husband's characteristics determine household decisions.

(24.) This is a linear approximation around [[pi].sub.a+1] (a) = 1. See Skinner (1985).

(25.) Specifically, the estimated variance-covariance matrix is where [(X'X).sup.-1] X'[omega]X[(X'X).sup.-1], where X is the matrix of regressors and [omega] is a NT x NT block-diagonal symmetric matrix. The ith block of [omega] (i = 1 to N) has the following form: It is a (bandwidth = 3) Toeplitz matrix; the diagonal elements are [[e.sup.2].sub.i,a] (a = [a.sub.1] to [a.sub.T]), which are the squared residuals from the Euler equation estimation for family i at age a; and the secondary diagonal elements are [e.sub.i,a][e.sub.i,a-1].

(26.) It is unclear if the respondents correctly interpreted the questions as referring to the prior year's consumption, and some authors (e.g., Zeldes [1989], Runkle [1991], and Dynan (1993]) interpret the data as the current year's consumption.

(27.) Marginal tax rates were not calculated by the PSID until 1976. The PSID did, however, report taxes paid in prior years, which allows the marginal tax rates to be inferred from tax tables.

(28.) We also considered including various demographic dummy variables (in particular, for the presence of children at various ages, for not working, and for retirement). Blundell et al. (1994), Attanasio and Browning (1995), and Banks et al. (1998) find significant effects from these demographics. In our data, however, these demographic dummy variables did not noticeably affect the results in any of the models. Thus, in the interest of brevity, only the more streamlined results are reported.

(29.) The substantive differences are that Lawrance (1991) and Dynan (1993) do not include the survival probability variable, and they use the extra poverty subsample.

(30.) In the cases where both the wife and husband work, their average real net wage rate is used. The results are practically the same when using the husband's wage rate and when including a dummy variable for the wife's working status. Similarly, the inclusion of a retirement dummy did not perceptibly affect the results.

(31.) Following Blundell et al. (1994), Attanasio and Browning (1995), Attanasio and Weber (1995), and Attanasio et al. (1995), the instruments for growth in hours of work are the independent variables in equation (8), their lags, and the second lag of consumption growth. Practically identical results are obtained using: hours of leisure instead of hours of work, the husband's hours of work rather than the couple's average, and dummy variables for the wife's working status and for retirement.

(32.) Similar, although less precise results were also found when excluding observations associated with real asset income below various thresholds.

(33.) It should be emphasized that this result depends on the assumption of significant intrinsic time preference.

(34.) A large distribution of ages appears to be essential to identify the effect of age on consumption growth. After experimenting with various subsamples, we found that any sizable truncation of ages (from either the young or old) removes the statistical significance of the age coefficient. Moreover, the coefficient estimates vary considerably within subsamples (although not in a systematic way). Perhaps this difficulty is to be expected given that the regression equation has to identify the separate effects of aging, survival uncertainty, intertemporal substitution, and trend.

(35.) The parameter estimates and variances are computed using the delta method. The estimate of [alpha] also requires an estimate of the forecast error variance, which is [[[sigma].sup.2].sub.[epsilon]] = [{1 + 2[[gamma].sup.-2](var[[e.sub.i,a]]+2 cov[[e.sub.i,a], [e.sub.i,a-1]])}.sup.1/2] - 1 (see Lawrance [1991] or Dynan [1993] for the derivation).

(36.) The estimates of [alpha] and [beta] (and their standard errors) in these models are: (4) 0.0137 (0.0431) and [0.0020.sup.*] (0.0011), (5) 0.0125 (0.0387) and [0.0019.sup.**] (0.0010), (6) 0.0116 (0.0417) and [0.0019.sup.*] (0.0010), (7) 0.0124 (0.0385) and [0.0020.sup.**] (0.0010).

(37.) Some care should be taken in interpreting [[delta].sub.Y] as well as [[delta].sub.E] and [[delta].sub.R]. Dynan (1993) argued that their coefficient estimates may be biased due to group-specific shocks.

(38.) Even if there is no intrinsic time preference, there is little reason to believe that the implicit discount rate estimated from a linear approximation on adult data is zero at birth.

(39.) See, for example, Runkle (1991) on the high degree of measurement error, and Attanasio and Weber (1995) on the inadequacy of data on food consumption only.

(40.) See, for example, Carroll and Summers (1991), Borsch-Supan and Stahl (1991), and Banks et al. (1998).

(41.) Attanasio et al. (1995) attribute this phenomenon to changing family size and labor supply over the life cycle. But they do not control for the separate influence of age. Thus the underlying cause of the changes in the discount rate over the life cycle is unclear.

(42.) Some possibly conflicting evidence should also be noted. Cropper et al. (1994) also found that lower discount rates are applied to the distant future, that is, hyperbolic discounting (a result found in other surveys as well, see Loewenstein and Elster [1992], Harvey [1994], and Sozou [1998]), while the theory presented here suggests that the opposite should occur, ceteris paribus.

(43.) A similar argument could be made regarding charitable donations.

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TABLE 1
Individual Food Standard +
Age (years) Male Female
Under 4 3.90 3.90
4-6 4.60 4.60
7-9 5.50 5.50
10-12 6.40 6.30
13-15 7.40 6.90
16-20 8.70 7.20
21-35 7.50 6.50
36-55 6.90 6.30
56 and older 6.30 5.40
(+) 1967 USDA Low-Cost Plan estimates of weekly food costs.
TABLE 2
Euler Equation Estimates
Model (1) (2)
Intercept 0.0122 0.0662 **
 (0.0304) (0.0250)
Interest rate 0.8814 * 0.5898 *
 (0.5307) (0.1003)
Age -0.0017 ** -0.0017 **
 (0.0006) (0.0005)
Survival probability -0.1434 -0.0643
 (single) (0.4827) (0.4815)
Survival probability -0.1106 -0.0414
 (married) (0.2660) (0.2648)
Growth in AFS 0.3521 ** 0.3505 **
 (0.0222) (0.0223)
Permanent income 0.0040 0.0027
 (in tens of thousands of 1983$) (0.0037) (0.0029)
College degree 0.0159 0.0135
 (0.0108) (0.0105)
Nonwhite -0.0169 -0.0161
 (0.0163) (0.0163)
Model (3) (4)
Intercept 0.0092 0.0136
 (0.0426) (0.0308)
Interest rate 1.3625 * 0.8175
 (0.7429) (0.5274)
Age -0.0024 ** -0.0016 **
 (0.0009) (0.0006)
Survival probability -0.1811 -0.1313
 (single) (0.6757) (0.3704)
Survival probability -0.1505 -0.0491
 (married) (0.3724) (0.2716)
Growth in AFS 0.5801 ** 0.3515 **
 (0.0366) (0.0223)
Permanent income 0.0100 0.0040
 (in tens of thousands of 1983$) (0.0103) (0.0037)
College degree 0.0262 0.0159
 (0.0181) (0.0129)
Nonwhite -0.0425 -0.0177
 (0.0411) (0.0161)
Notes: Standard errors are in parentheses.
(*) and (**) denote significance at the 90% and 95% confidence
levels, respectively. Model (1) uses the variables discussed in
sections III and IV. Model (2) does not use year dummies. Model
(3) uses consumption dated concurrently with the survey year. Model
(4) uses the husband's age, survival probability, degree, and race.
TABLE 3
More Euler Equation Estimates
Model (5) (6)
Intercept 0.0120 0.0119
 (0.0305) (0.0304)
Interest rate 0.8806 * 0.8217
 (0.5308) (0.5316)
Age -0.0017 ** -0.0016 **
 (0.0006) (0.0006)
Survival probability -0.1417 -0.0943
 (single) (0.4828) (0.4834)
Survival probability -0.1094 -0.0949
 (married) (0.2662) (0.2661)
Growth in AFS 0.3520 ** 0.3524 *
 (0.0222) (0.0222)
Permanent income 0.0040 0.0037
 (in tens of thousands of 1983$) (0.0037) (0.0037)
College degree 0.0159 0.0154
 (0.0108) (0.0108)
Nonwhite -0.0168 -0.0165
 (0.0163) (0.0163)
Growth in wage rate 0.0003
 (0.0020)
Growth in hours worked 0.0245 *
 (0.0133)
Growth in weeks sick
Model (7) (8)
Intercept 0.0121 0.0395
 (0.0304) (0.0330)
Interest rate 0.8847 * 0.9094 *
 (0.5309) (0.5483)
Age -0.0017 ** -0.0019 **
 (0.0006) (0.0007)
Survival probability -0.1468 -0.3546
 (single) (0.4828) (0.5187)
Survival probability -0.1126 -0.0690
 (married) (0.2662) (0.2808)
Growth in AFS 0.3526 ** 0.2889 **
 (0.0223) (0.0277)
Permanent income 0.0040 0.0028
 (in tens of thousands of 1983$) (0.0037) (0.0037)
College degree 0.0155 0.0087
 (0.0108) (0.0108)
Nonwhite -0.0173 -0.0086
 (0.0163) (0.0283)
Growth in wage rate
Growth in hours worked
Growth in weeks sick -0.0060
 (0.0043)
Model (9)
Intercept -0.0202
 (0.0594)
Interest rate 0.9040
 (0.6285)
Age -0.0010
 (0.0013)
Survival probability 0.1989
 (single) (0.6786)
Survival probability 0.0755
 (married) (0.3478)
Growth in AFS 0.3599 **
 (0.0267)
Permanent income 0.0048
 (in tens of thousands of 1983$) (0.0040)
College degree 0.0093
 (0.0135)
Nonwhite -0.0290
 (0.0207)
Growth in wage rate
Growth in hours worked
Growth in weeks sick
Notes: Standard errors are in parentheses. (*) and (**) denote
significance at the 90% and 95% confidence levels, respectively. Model
(5) includes the percentage change in the real net wage rate. Model
(6) includes the percentage change in (instrumented) hours of work.
Model (7) includes the percentage change in weeks of work missed due
to illness. Model (8) does not use any observations associated with
zero asset income. Model (9) does not use any observations with age
less than 45.
TABLE 4 Parameter Estimates
Model (1) (2) (3) (8)
[alpha] 0.0122 -0.0541 0.0146 -0.0197
 (0.0387) (0.0428) (0.0331) (0.0492)
[beta] 0.0019 ** 0.0028 ** 0.0017 ** 0.0021 **
 (0.0009) (0.0009) (0.0008) (0.0010)
[[lambda].sub.S] -0.1627 -0.1089 -0.1330 -0.3899
 (0.5470) (0.8159) (0.4943) (0.5812)
[[lambda].sub.M] -0.1255 -0.0702 -0.1105 -0.0759
 (0.3189) (0.4497) (0.2861) (0.3147)
[[delta].sub.Y] -0.0045 -0.0046 -0.0073 -0.0031
 (0.0032) (0.0049) (0.0059) (0.0034)
[[delta].sub.E] -0.0180 -0.0229 -0.0192 -0.0096
 (0.0145) (0.0182) (0.0150) (0.0122)
[[delta].sub.R] 0.0191 0.0273 0.0312 0.0095
 (0.0208) (0.0279) (0.0332) (0.0316)
Model (9)
[alpha] 0.0476
 (0.0696)
[beta] 0.0011
 (0.0012)
[[lambda].sub.S] 0.2200
 (0.8115)
[[lambda].sub.M] 0.0836
 (0.3972)
[[delta].sub.Y] -0.0054
 (0.0039)
[[delta].sub.E] -0.0102
 (0.0148)
[[delta].sub.R] 0.0321
 (0.0292)
Notes: Standard errors are in parentheses.
(**) denotes significance at the 95% confidence level. Model (1) uses
the variables discussed in sections III and IV. Model (2) does not
use year dummies. Model (3) uses consumption dated concurrently
with the survey year. Model (8) does not use any observations
associated with zero asset income. Model (9) does not use
any observations with age less than 45.
TABLE 5 Estimates of [rho] +
Model (1) (2) (3) (8) (9)
Age
 20 0.0359 -0.0126 0.0271 0.0111 0.0548
 (0.0232) ** (0.0239) (0.0219) (0.0297) (0.0448)
 40 0.0743 0.0442 ** 0.0617 ** 0.0533 ** 0.0772 **
 (0.0182) (0.0096) (0.0194) (0.0177) (0.0282)
 60 0.1127 ** 0.1009 ** 0.0963 ** 0.0954 ** 0.0997 **
 (0.0295) (0.0165) (0.0280) (0.0241) (0.0260)
 80 0.1511 ** 0.1577 ** 0.1309 ** 0.1375 ** 0.1221 **
 (0.0464) (0.0333) (0.0412) (0.0410) (0.0406)
Sample 0.0996 ** 0.0815 ** 0.0845 ** 0.0829 ** 0.0989 **
Mean (0.0245) (0.0117) (0.0243) (0.0203) (0.0258)
Notes: Standard errors are in parentheses.
(**) denotes significance at the 95% confidence level, Model (1) uses
the variables discussed in sections III and IV. Model (2) does not
use year dummies. Model (3) uses consumption dated concurrently
with the survey year. Model (8) does not use any observations
associated with zero asset income. Model (9) does not use any
observations with age less than 45.
(+) Evaluated at the sample means of Y, E, and R.

ABBREVIATIONS

AFS: Annual Food Standard

CES: Consumer Expenditure Survey

CPI: Consumer Price Index

PSID: Panel Study of Income Dynamics

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[Graph Omitted]