Can mortality risk explain the consumption hump?

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Journal of Macroeconomics 30 (2008) 844–872 www.elsevier.com/locate/jmacro

Can mortality risk explain the consumption hump? James Feigenbaum

*

Department of Economics, University of Pittsburgh, 230 South Bouquet Street, Pittsburgh, PA 15260, USA Received 6 July 2006; accepted 8 August 2007 Available online 28 August 2007

Abstract A lifecycle consumption profile with a hump of roughly the same relative size and peak location as empirical consumption profiles can be obtained in a general equilibrium model where mortality risk is the only active mechanism that can account for the hump. Moreover, the key preference parameter, the elasticity of intertemporal substitution, is close to that estimated in a buffer-stock saving model by Gourinchas and Parker [Gourinchas, Pierre-Olivier, Parker, Jonathan A., 2002. Consumption over the life cycle. Econometrica 70, 47–89], where borrowing constraints primarily account for the consumption hump. Since borrowing is virtually eliminated in the model with mortality risk, mortality supplants the borrowing constraint as the explanation for the hump with these parameters. If a pay-as-you-go Social Security system is also incorporated in the model, mortality risk can no longer account for the observed properties of the hump. However, the set of intertemporal elasticities for which mortality risk disables the borrowing constraint in the neighborhood of peak consumption extends to any value greater than 1/3. Ó 2007 Elsevier Inc. All rights reserved. JEL classification: E21 Keywords: Consumption hump; Borrowing constraints; Mortality risk; Elasticity of intertemporal substitution; Social security

*

Tel.: +1 412 383 8157; fax: +1 412 648 1793. E-mail address: [email protected] URL: http://www.pitt.edu/~jfeigen

0164-0704/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.08.010

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0. Introduction Macroeconomics essentially begins with an understanding of the tradeoff between consumption and investment, so it is rather troubling that the simplest and most widely used models of consumption are at odds with salient facts about lifecycle consumption. In the familiar additively separable model of consumption preferences with a constant discount rate, consumption should rise or fall monotonically over the lifecycle, depending on whether the rate of return on saving is larger or smaller than the discount rate. However, empirical consumption profiles are not monotonic. They are hump-shaped with a peak around age 50.1 Fortunately, there is no lack of explanations as to why consumption profiles should be hump-shaped.2 The puzzle is not how the hump could possibly occur but rather which mechanism – or combination of mechanisms – can best account for the hump while also being consistent with other macroeconomic data. Of the candidate explanations, borrowing constraints and precautionary saving, usually studied in tandem, have received the most attention in the literature (Carroll, 1997; Carroll and Summers, 1991; Gourinchas and Parker, 2002; Hubbard et al., 1994; Nagatani, 1972; Thurow, 1969). Several other explanations have also been studied. Variations in household consumption might simply reflect variations in household size (Attanasio et al., 1999; Browning and Ejrnæs, 2002), although several researchers (Ferna´ndez-Villaverde and Krueger, 2002; Gourinchas and Parker, 2002) argue that the hump persists even after correcting for household size. If we add another good to the model, such as leisure, agents will not smooth consumption by itself but rather they will smooth out the utility derived from bundles of the two goods. If leisure and consumption are substitutes, agents will substitute away from consumption when productivity and the marginal cost of working are low and substitute towards it when the marginal cost is high (Becker and Ghez, 1975; Bullard and Feigenbaum, forthcoming; Heckman, 1974), so a hump-shaped wage profile can lead to a hump-shaped consumption profile. As a variation on the theme that market frictions can account for the hump, Ferna´ndez-Villaverde and Krueger (2001) have shown that, if consumer durables are used as collateral to secure loans, this can lead to a hump-shaped pattern of consumption in both durables and nondurables. The monotonicity result is also strongly dependent on the assumption of rationality, so time-inconsistent preferences can easily account for a hump in consumption (Caliendo and Aadland, 2004; Laibson, 1997). Another very simple explanation for the consumption hump is a time-varying discount rate. If the discount rate increases over the lifecycle, the rate of consumption growth will decrease, and conceivably it could transition from positive to negative as is seen in the data. Rising mortality risk is a natural explanation for why the discount rate should increase with age. If we account for the uncertainty in an agent’s lifespan, the effective discount rate becomes the sum of the intrinsic discount rate coming from preferences, which we may assume is constant, and the hazard rate of dying, which does indeed increase with age. While several researchers have considered the effects of mortality risk in the context of explaining why the elderly save more than would be expected in a basic lifecycle model (Davies, 1981; Hurd, 1989) or to study questions pertaining to Social Security

1 2

This was first observed by Thurow (1969). Browning and Crossley (2001) provides a good review of this literature.

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_ ˘ lu et al., 1995), only a handful of papers in the consumption-hump literature (Imrohorog incorporate mortality risk.3 The reason for this is quite simple. Prior to age sixty, the hazard rate of dying remains quite small. Since the peak of the consumption hump occurs before then, one would expect, a priori, that mortality risk should not be able to contribute much to the consumption hump. _ ˘ lu (2006) have recently argued that Contrary to this intuition, Hansen and Imrohorog perhaps mortality risk can plausibly account for the consumption hump.4 The objective of the present paper is to assess the potential for mortality risk to account for the consumption hump and to assess the effects of the interaction between a borrowing constraint and mortality risk. Working in a simple, continuous-time environment, I find in principle it is theoretically possible that mortality risk could explain the consumption hump. However, in general equilibrium, mortality risk only produces a consumption hump that matches the relative size and peak location of the empirical consumption hump for a knife-edge set of parameters, unlike other mechanisms that produce far more robust consumption humps. Thus mortality risk is an unlikely answer to the question of the consumption hump. Nevertheless, mortality risk can still have significant effects by interfering with the ability of a borrowing constraint to explain the consumption hump. _ ˘ lu (2006) considered a model with both mortality risk and leiHansen and Imrohorog sure-consumption substitution. Bullard and Feigenbaum (forthcoming) have previously shown that when such a model is optimally calibrated to fit Gourinchas and Parker’s (2002) data on lifecycle consumption that mortality risk has a negligible effect compared to leisure-consumption substitution. Consistent with this finding, mortality risk also has _ ˘ lu’s calibrations only a small effect on the consumption profile for Hansen and Imrohorog that come closest to matching their consumption data. In the family of preferences studied in these two papers, leisure-consumption substitution is absent at just one point in the parameter space, where the elasticity of intertemporal substitution is exactly 1, whereas matching consumption data requires a much lower elasticity. _ ˘ lu’s (2006) work in three respects. First, I This paper expands on Hansen and Imrohorog consider a different family of preferences without leisure-consumption substitution that allows us to isolate the effect of mortality risk while keeping the elasticity of intertemporal substitution a free parameter. Second, I carry out a more detailed sensitivity analysis. Third, I also include in the model a competing and more widely studied explanation for the consumption hump, a borrowing constraint, so we can weigh the relative importance of these two mechanisms. Previously, Hubbard et al. (1994) ran counterfactual experiments corroborating the prevailing wisdom that mortality risk could not account for the hump. Comparing what happens in a model with no uncertainty to a model with just lifespan uncertainty, they found that the rate of consumption growth does decrease with age in the latter model, but this decrease is not apparent until long after the peak of the empirical consumption hump. However, they only considered a narrow region of the parameter space, and they also limited their attention to a partial equilibrium analysis.

3

Of the aforementioned papers on the consumption hump, only Ferna´ndez-Villaverde and Krueger (2001) and Hubbard et al. (1994) account for mortality risk. 4 Bu¨tler (2001) also considers mortality risk along with several of the other explanations mentioned above, though in partial equilibrium.

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This last point is important because the interest rate is crucial to determining whether or not mortality risk can produce a hump. The consumption profile will have a local maximum at any age when the rate of consumption growth vanishes, which will occur if the effective discount rate equals the interest rate. In a model where agents begin their economic life in their twenties, the effective discount rate will increase without bound from its initial starting value. Therefore, mortality risk can produce a hump if and only if the interest rate is higher than the initial value of the effective discount rate. Meanwhile, since the rate of consumption growth (or decay) is proportional to the elasticity of intertemporal substitution, the size of any consumption hump will be a strictly increasing function of this elasticity. By fine tuning the intertemporal elasticity and the interest rate, one can trivially obtain a partial equilibrium where mortality risk produces a consumption hump of the same relative size and peak location as the empirical consumption hump. Thus, there is nothing puzzling about the consumption hump if we are content to treat the interest rate as a free parameter. The nontrivial question is whether we can replicate the properties of the consumption hump in a general equilibrium model while also maintaining reasonable values of other macroeconomic variables. Surprisingly, the answer is yes. Moreover, one can do this with parameters quite similar to those of Gourinchas and Parker’s (2002) baseline consumptionhump model. Although Gourinchas and Parker incorporate both a borrowing constraint and idiosyncratic risk (but not mortality risk) into their model, Feigenbaum (2007) demonstrates that risk aversion has a negligible effect on this consumption hump, leaving the borrowing constraint as its primary cause. Using Gourinchas and Parker’s value for the elasticity of intertemporal substitution, mortality risk – or more precisely, the relative immortality of the young – virtually shuts down all demand for borrowing by young agents after the first couple years of life, yet the hump still persists. Thus, accounting for mortality risk would make a borrowing constraint ineffectual under Gourinchas and Parker’s parameters and supplant the constraint as the explanation for the consumption hump. Adding Social Security to the model reinforces this finding as mortality risk then interferes with the borrowing constraint over a much larger subset of the parameter space. Where mortality risk falters as an explanation for the consumption hump is its lack of robustness. Other explanations for the hump, such as buffer-stock saving or leisure-consumption substitutability, yield properties that are relatively insensitive to the choice of _ ˘ lu (2006) found that introparameters. In their sensitivity analysis, Hansen and Imrohorog ducing Social Security into their model moves their consumption peak later in time. Here we show that Social Security is not special. Any factor that influences the equilibrium interest rate can drastically alter the size and location of the consumption hump. Mortality risk gives properties close to those seen in the data for only a specific choice of parameters. With those parameters, the path of consumption away from the peak does not match the data well. Because the model is so simple, it has few degrees of freedom, so the fit to data away from the peak cannot be improved upon while preserving the properties at the peak. Consequently, the argument that mortality risk alone accounts for the consumption hump is not compelling. On the other hand, the findings here demonstrate why it is important to consider mortality risk when modeling lifecycle consumption. Even though mortality risk is presumably not the primary cause of the consumption hump, the lesson here is that it can still have significant interaction with other mechanisms that might contribute towards the existence of a consumption hump. The argument that borrowing constraints are the primary explanation for the consumption hump is substantially weakened after we account for mortality

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risk, and the same may hold for other mechanisms. A plausible theory of lifecycle consumption should, therefore, include mortality risk. The paper is organized as follows. The model is described in Section 1. In Section 2, I summarize the known results on the time dependence of consumption and derive the properties of a consumption hump generated by mortality risk alone. The calibration of the model is discussed in Section 3. Section 4 describes the properties of the consumption profile in the calibrated model without Social Security. Then in Section 5, I consider the sensitivity of these properties to the model parameters. In Section 6, I show how introducing Social Security can amplify the effects of mortality risk. Finally, in Section 7, I offer some concluding remarks. 1. The model Consider a continuous-time model in which agents have preferences over consumption and live to a maximum age of T. We will denote a quantity A relating to an agent at time t who was born at s by A(t; s). Such an agent will also be said to belong to cohort s. The probability of surviving until age s = t  s or beyond, i.e. the survivor function, is Q(s), which is cohort-independent. Agents of cohort s maximize expected utility Z sþT Qðt  sÞ expðqðt  sÞÞuðcðt; sÞÞ dt; s

where c(t; s) is consumption, q is the discount rate, and  1 1r c r 6¼ 1; uðcÞ ¼ 1r ln c r ¼ 1: A worker at t has an endowment of e(t  s) labor efficiency units that are supplied inelastically. The wage per labor efficiency unit at time t, w(t), is determined in a labor market, and the worker earns income w(t)e(t  s). Retirement is exogenously imposed at age Tret, so, for s P Tret, e(s) = 0. The government implements a pay-as-you-go Social Security system financed by a payroll tax at the rate g. Beginning at age Tret, agents receive benefits S(t). There is no aggregate uncertainty in the model, and we will assume a stationary equilibrium with a constant rate of return. Agents can invest in capital, which pays the riskless net return r. An agent holding capital k(t; s) from age t to age t + dt will earn interest rk(t; s)dt during that interval. As is typical in the literature, because few consumers purchase annuities, we assume there is no market for claims contingent on an agent’s lifespan, so an agent cannot, for example, purchase an annuity that only pays out if he continues to live.5 Finally, we assume that the assets of agents who die are spread uniformly over the surviving population. At time t, a surviving agent will receive the bequest B(t).6 5

Note this is not an innocuous assumption as is discussed in Section 6. Uniform allocation of bequests is a common assumption because it is straightforward to implement, though it is obviously not realistic. The present value of bequests is negligible relative to labor income, so for most macroeconomic issues their impact should be negligible. Nevertheless, the specification of the bequest function will affect the equilibrium interest rate, so this is another of the many factors that the properties of the consumption hump will be sensitive to. A more realistic bequest function would presumably depend on both age and time with a peak in middle age (when the agent’s parents are most likely to die). This would result in a redistribution of wealth to the elderly and should affect the consumption hump much as Social Security does in Section 6. 6

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The consumer’s problem can then be stated as follows: Z sþT max Qðt  sÞ expðqðt  sÞÞuðcðt; sÞÞ dt

849

ð1Þ

s

subject to cðt; sÞ þ

dkðt; sÞ ¼ rkðt; sÞ þ ð1  gÞwðtÞeðt  sÞ þ BðtÞ þ SðtÞHðt  T ret Þ dt

ð2Þ

cðt; sÞ P 0; kðt; sÞ P 0; kðs; sÞ ¼ kðT þ s; sÞ ¼ 0;

ð3Þ

where H is the step function  1 s P 0; HðsÞ ¼ 0 s < 0: We impose a hard no-borrowing constraint to facilitate comparisons with buffer-stock saving models that also make this assumption. This is also a simple way to avoid the issue of what happens to a deceased agent’s debts. In the United States and most of the developed world, a person’s heirs inherit his net assets but do not have to accept net debt. Thus, if a borrower dies without enough assets to pay off his creditors, his creditors will have no recourse to collect any outstanding debt, which would make lending a risky enterprise.7 To avoid this complication, we assume there is no borrowing.8 To close the model in general equilibrium, we assume a constant population and use a standard Cobb-Douglas technology with exogenous growth g in labor productivity and depreciation rate d. Gross output is a

Y ðtÞ ¼ F t ðKðtÞ; N ðtÞÞ ¼ KðtÞ ðegt N ðtÞÞ

1a

;

where K(t) is the aggregate capital stock and N(t) is the aggregate labor supply, measured in efficiency units. In equilibrium, the aggregate demand for capital of all agents alive at t (who will be members of cohorts s 2 [t  T, t]) must equal the total capital stock at t, so Z t KðtÞ ¼ Qðt  sÞkðt; sÞ ds: ð4Þ tT

Likewise, the aggregate labor supply is the aggregate of the efficiency units supplied to the market Z t N ðtÞ ¼ Qðt  sÞeðt  sÞ ds: ð5Þ tT

7

This might seem like an unimportant issue, but for many parameterizations of the model, if we do not impose the borrowing constraint, agents in their 90s will start to borrow, knowing they can probably escape the obligation of paying back the debt. Since this behavior is not observed in reality, the borrowing constraint is needed to shut it down. 8 Although this is the simplest solution to the problem of the debt of the dead, it is a tighter constraint than is necessary. Alternatively, we could impose a soft borrowing constraint where the interest rate paid by borrowers includes a mortality risk premium so the expected interest rate to be paid by borrowers equals the certain interest rate received by lenders.

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Factor markets are competitive, so rðtÞ ¼

oF t ðKðtÞ; N ðtÞÞ  d; oK

ð6Þ

wðtÞ ¼

oF t ðKðtÞ; N ðtÞÞ : oN

ð7Þ

and

The hazard rate of dying is hðsÞ ¼ 

d ln QðsÞ; ds

ð8Þ

meaning that h(s)ds is the probability of dying between age s and age s + ds for a small interval ds. The size of the bequest B(t) at t is then determined by the balance equation Z t Z t Qðt  sÞBðtÞ ds ¼ Qðt  sÞhðt  sÞkðt; sÞ ds; ð9Þ tT

tT

where the left-hand side is the total of bequests received by surviving agents at t and the right-hand side is the total amount of capital held by agents who die at t. Likewise the size of the Social Security benefit is determined so the government has a balanced budget: Z tT ret Z t Qðt  sÞSðtÞ ds ¼ gwðtÞ Qðt  sÞeðt  sÞ ds: ð10Þ tT

tT ret

An equilibrium consists of demand functions for consumption c(t; s) and capital k(t; s), a bequest function B(t), a Social Security benefit S(t), an interest rate r, and a wage per efficiency unit w(t) such that (i) the demand functions solve the consumer’s optimization problem (1) given B(t), S(t), r, and w(t), (ii) the marginal product conditions (6) and (7) are satisfied, given the aggregate demands (4) and (5), (iii) the bequest B(t) satisfies the balance equation (9) given k(t; s), and (iv) the Social Security benefit S(t) satisfies (10). We will consider only steady-state equilibria in which the capital stock and the total amount of bequests grows at the rate g and the labor supply is constant, so KðtÞ ¼ expðgtÞK 0 ; N ðtÞ ¼ N ; SðtÞ ¼ expðgtÞS 0 ; and BðtÞ ¼ expðgtÞB0 : Then the rate of return will be  a1 K0  d; r¼a N which is constant, and the wage  a K0 wðtÞ ¼ expðgtÞð1  aÞ N grows at the rate g.

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2. Time evolution of consumption Let us now consider the theoretical properties of the lifecycle consumption profile. In this section, we assume for simplicity that the no-borrowing constraint never binds.9 This condition will be relaxed in the computational sections that follow. Bu¨tler (2001) showed that the optimal consumption allocation will satisfy the differential equation d r  q 1 d ln Qðt  sÞ ln cðt; sÞ ¼ þ : dt r r dt

ð11Þ

Employing the definition of the hazard rate (8), we can write this as d r  qeff ðt  sÞ ln cðt; sÞ ¼ ; dt r

ð12Þ

where qeff ðt  sÞ ¼ q þ hðt  sÞ

ð13Þ

is the effective discount rate. Note that (12) describes the time evolution of consumption over the lifecycle of an agent from cohort s. A cross-sectional lifecycle profile of all agents alive at t will behave slightly differently. Since the economy grows over time, the consumption of an agent of a fixed age s will grow at the rate g with respect to time t: ccs ðs; tÞ  cðt; t  sÞ ¼ egðtsÞ cðs; 0Þ: Thus, a cross-sectional lifecycle profile for consumption will vary according to d d r  qeff ðsÞ  gr ln ccs ðs; tÞ ¼ ln cðs; 0Þ  g ¼ : ds ds r

ð14Þ

In the absence of mortality risk, (12) simply reduces to the familiar result d rq ln cðt; sÞ ¼ : dt r Thus, the only effect that mortality risk has on the rate of consumption growth is that the intrinsic discount factor q is replaced by the effective discount factor qeff(t  s). For most of the lifecycle, this replacement would appear to be a negligible change since the hazard rate in the United States remains under 1% per annum until age 60. As we will see though, if r is low enough, for certain values of the interest rate r mortality risk actually can have a large effect on the lifecycle consumption profile even before age 60. Suppose that r > qeff ð0Þ þ rg:

ð15Þ

In that case, if we assume mortality risk increases monotonically with age (which is reasonable for an adult population) and that lim hðsÞ ¼ 1; s!T

9

The borrowing constraint can affect the level of consumption but not its growth rate at times when it does not bind, so the differential equations obtained in this section will still be valid where the constraint does not bind.

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there must be some critical age sc at which r ¼ qeff ðsc Þ þ rg:

ð16Þ

Then since we have assumed q0eff ðsc Þ ¼ h0 ðsc Þ > 0, Eq. (14) shows that the consumption profile is strictly concave with a global maximum at sc. The size of the peak relative to the initial consumption value will then be given by   Z sc Z sc ccs ðsc ; tÞ d ln ccs ðs; tÞ r  gr  q  hðsÞ ds ¼ ds ln ¼ ccs ð0; tÞ ds r 0 0 Z hðsc Þ 1 sc ¼ sc  hðsÞ ds; r r 0 where we use (16) in the last step. Eq. (8) implies       Z sc ccs ðsc ; tÞ 1 d 1 Qðsc Þ hðsc Þsc þ ln QðsÞ ds ¼ hðsc Þsc þ ln ln : ¼ ccs ð0; tÞ r ds r Qð0Þ 0

ð17Þ

Thus, once the location of the peak is determined, the relative size of the peak depends only on r and actuarial data. Moreover, the relative size is proportional to r1, the elasticity of intertemporal substitution. Note also that     d ccs ðsc ; tÞ 1 0 d h0 ðsc Þsc h ðsc Þsc þ hðsc Þ þ > 0; ð18Þ ln ln Qðsc Þ ¼ ¼ dsc ccs ð0; tÞ r dsc r so a consumption hump caused by mortality risk alone will get larger if the peak is pushed to later in life. While the introduction of a small degree of mortality risk has only a negligible effect on the rate of consumption growth at a given point of time, the cumulative effect integrating over several years can still be significant. This is not entirely surprising because if r was much larger than the intrinsic discount rate q then consumption should grow at a rapid rate until the hazard rate of death shoots up near the end of life. In that case, consumption would be hump-shaped, but it would only taper off at the very end of life. This could not account for the consumption hump seen in the data because the empirical hump peaks in middle age where the hazard rate is still relatively small. Holding r constant, an earlier peak would imply a hump of smaller relative size since consumption would have less time to grow. But with a high elasticity of intertemporal substitution, the rate of consumption growth can still be large enough to produce a substantial hump as we will see in Section 4. On the other hand, regardless of r, a consumption hump that peaks in middle age cannot be robust. From Eq. (16), osc 1 ¼ 0 > 0: h ðsc Þ or

ð19Þ

The equilibrium interest rate r depends on all the exogenous parameters of the model. Since h 0 (sc) will be small early in life, a small change in the equilibrium interest rate, caused by a change in any exogenous parameter, can have a huge impact on the location of the consumption peak. A change in the interest rate of even a few tenths of a percent can shift the peak by a decade. For this reason, mortality risk would have to work almost perfectly at a plausible value of r to be seriously considered as the primary explanation for the consumption hump. With other explanations for the hump adding refinements to a model to

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improve its fit to the data will not significantly alter the properties of the hump, but any refinement of the present model will cause disproportionate changes to the hump. Thus, we would have to be confident our model captures all the effects that might influence r in order to take a firm stand in favor of mortality risk as the answer. 3. Calibration Now that we have seen it is at least theoretically possible that mortality risk alone can produce a consumption hump with a peak in middle age, let us now ask whether this can happen in a model with plausible parameters. The model described in Section 1 has the following parameters: a discount rate q, the elasticity of intertemporal substitution r1, the share of capital in the production function a, the depreciation rate of capital d, the growth rate of output g, the payroll tax rate g, the age-productivity profile e(s), and the survivor function Q(s). The average lifecycle consumption profile from ages 25–65 reported by Gourinchas and Parker (GP) (2002) is plotted in Fig. 1 along with a corresponding polynomial fit:10 6 4 2 3 cGP cs ðsÞ ¼ 1:062588 þ 0:015871s  0:00184s þ 0:000109s þ 4:13  10 s

 5:6  107 s5 þ 1:63  108 s6  1:475  1010 s7 ;

ð20Þ

where the units are such that the initial income is unity. Gourinchas and Parker corrected for family size, so (20) represents a household of a constant size. Since most households with multiple members will not remain constant in size for 40 years, we view this as the average consumption of a household of one individual. Consistent with this interpretation, we use mortality data for individuals, taken from life tables for the United States in 2001 (Arias, 2004). Following Gourinchas and Parker (2002), we model agents from age 25 onward. Defining the model age s = 0 to correspond to actual age 25, I fit the probability of surviving to actual age s + 25 to the sextic polynomial ln QðsÞ ¼ :01943039  :00030548s þ ð5:998  106 Þs2  ð3:279  106 Þs3  ð3:055  108 Þs4 þ ð3:188  109 Þs5  ð5:199  1011 Þs6 :

ð21Þ

These data were top cut at age 100, so the model age s ranges from 0 to T = 75. The hazard rate implied by (21), defined by (8), is plotted in Fig. 2 along with the measured probability of surviving from one year to the next obtained from the life tables. The inferred hazard rate does not deviate far from the discrete probability of dying except at low ages between 25 and 35, and the graph exaggerates the difference between the two rates there since the scale is logarithmic. Notice that the hazard rate stays below 1% per annum until age 60. Labor is supplied inelastically in the model, so income and productivity are proportional. Since I am comparing to Gourinchas and Parker’s (2002) baseline model, it is necessary to use their income profile. Therefore, to obtain the age-productivity profile, I fit their estimate of the average after-tax income profile to the quartic polynomial ð1  gÞweðsÞ ¼ 1 þ 0:018095s þ 0:000817s2  5:1  105 s3 þ 5:36  107 s4 ; 10

ð22Þ

A septic polynomial was used to eliminate secondary peaks from the polynomial that are not apparent in the data.

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Consumption 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 25

30

35

40

45

50

55

60

65

Age Fig. 1. Average lifecycle consumption profile from Gourinchas and Parker (2002) along with the polynomial fit cgp(s). Consumption is measured in units such that initial income we(0) = 1.

where after-tax labor income at s = 0 is normalized to 1. Average income data and (1  g)we(s) are both plotted in Fig. 3. Following convention, in the baseline model retirement will occur at age 65, so the model retirement age Tret = 40. I initially set g = 0.11 We will consider the model with Social Security in Section 6. Of the remaining scalar parameters, the average growth rate of output g (due to technological growth as opposed to population growth) is directly observed, and I will use the value of g = 1.56% per annum from Bullard and Feigenbaum (forthcoming). The four remaining parameters are more difficult to measure, so they will be chosen to minimize a loss function of the distances between endogenous macroeconomic variables and corresponding data. Three convenient and commonly used targets are the capital–output ratio K/Y, the consumption rate C/Y, and the real interest rate r. Following Rios-Rull (1996), I target the consumption rate C/Y to be 0.748. For the interest rate, I use McGrattan and Prescott’s (2000) estimate of r = 3.5% as the target, which is also close to Gourinchas and Parker’s (2002) estimate of 3.44%. Estimates of the capital–output ratio tend to range between 2.9 and 3.3,12 so I will use a target value of 3 for K/Y. Since we have four unobserved parameters, we then need one more target to pin down those parameters. The only other predictions that the model makes pertain to consumption, so for the last target we will try to match properties of the consumption data. The 11 Although in principle Gourinchas and Parker (2002) wrote a model general enough to include Social Security payments after retirement, their baseline model is calibrated with a replacement ratio of a tenth of a percent. 12 Cooley and Prescott (1995) report a value of 3.32 for the United States during the postwar era. In an experiment involving a lifecycle model similar to the present model, Rios-Rull (1996) used a value of 2.94, correcting for the absence of a government.

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1

0.1

hazard rate

0.01

discrete probability of death

0.001

0.0001 25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

Age Fig. 2. The hazard rate assumed in the model as a function of age along with the measured probability of surviving from one discrete period of a year to the next.

smoothed consumption series (20) peaks at age 45, so sc = 20, and the ratio of peak consumption to initial consumption is ccs(sc)/ccs(0) = 1.148. Note that if we consider a partial equilibrium (PE) model where r is a free parameter then we can fix r at the target value of 3.5% and solve Eqs. (16) and (17) for the remaining free parameters q = 0.027 and r = 0.294 (since a and d do not play a role in determining the consumer’s behavior) needed to get sc = 20 and ccs(sc)/ccs(0) = 1.148. The resulting consumption profile is shown in Fig. 4, and it almost exactly matches the profile in the data. Thus, mortality risk can perfectly account for the consumption hump in a partial equilibrium model, in which case there is no puzzle to be resolved.13 However, with these parameters, the model is out of equilibrium at r = 3.5%.14 Using the baseline values of a and d obtained below, the equilibrium interest rate is r = 3.78%. While this difference might seem slight, the hypersensitivity of the peak location sc to r is such that the

13

This is not specific to mortality risk. Most other explanations for the hump can also account for the hump almost perfectly if r is a free parameter. 14 It also requires the inheritance of debts from the elderly since the borrowing constraint is not imposed in Eqs. (16) and (17).

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Age Fig. 3. Average after-tax income data from Gourinchas and Parker (2002) and fitted polynomial (1  g)we(s) with (1  g)we(0) normalized to 1.

corresponding general equilibrium model has a much more pronounced hump with a peak several years later as is also shown in Fig. 4. We will consider two parameterizations for the general equilibrium model.15 For the MSE parameterization, we minimize the mean squared error of the lifecycle consumption 16 profile ccs(s) with respect to the smoothed data cGP For the CP parameterization, we cs ðsÞ. match the relative size of the consumption peak, i.e. the ratio of peak consumption to initial consumption. The two parameterizations are listed in Table 1. Aside from the value of r, which is the parameter we are most interested in varying, they are essentially the same, so we will use the CP parameterization as our baseline model. The model predictions for the target observables under both parameterizations are listed in Table 2. 4. A mortality-induced consumption hump Now let us consider the lifecycle consumption profiles. First we look at the MSE parameterization, in which r = 0.580, as is shown in Fig. 5. The consumption hump has a relative size ccs(sc)/ccs(0) = 1.09 with a peak at age 41. Note that the model has two mechanisms that can account for the hump: mortality risk and a borrowing constraint. To distinguish the effects of these two mechanisms, the cross-sectional bond demand 15 To deal with the borrowing constraint, I actually solve the model in discrete time using the method described in Feigenbaum (2007). Since the Gourinchas and Parker (2002) data is annual, time is divided into 75 year long periods. 16 More precisely, I minimize the sum of the mean squared error and a quartic loss function of the distance between the three other macroobservables and their targets.

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Age Fig. 4. Lifecycle consumption profile for partial equilibrium model with q = 0.027, r = 0.294, and r = 3.5% along with the corresponding general equilibrium model with r = 3.78% and the GP data. Consumption is measured in units such that initial income we(0) = 1.

profile bcs(s; t) = b(t; t  s) is plotted in Fig. 6. The consumer is borrowing constrained wherever the bond demand is zero. For the MSE parameterization, consumers are borrowing constrained from ages 25 to 30, so the consumption profile is very steep in that regime. Using Eq. (17), which was derived assuming no borrowing constraint, we estimate the relative size of the peak in the absence of a borrowing constraint to be only 1.042, so more than half of the rise in consumption from ages 25 to 41 can be attributed to the action of the borrowing constraint in those first five years.17 The borrowing constraint can only determine the location of the peak if the bond demand becomes positive at the peak, so the main contribution of mortality risk is to determine the location of the peak via Eq. (16). Note that this is a significantly smaller peak than what is observed in the data. The MSE consumption profile matches the empirical profile fairly well at the edges of the data set, but not so well in the middle, where the peak of the hump occurs. As we saw in the previous section, in partial equilibrium we can exactly match both the location of the peak and its relative size. In general equilibrium, we cannot match both simultaneously. However, if we decrease r, we can bring the parameters closer to the

17 Note that Eq. (17) only provides a rough gauge of how big the consumption hump would be in the absence of the borrowing constraint. For an exact computation, we would have to compute a general equilibrium for a model without the borrowing constraint. Since, as previously discussed, the borrowing constraint was included to eliminate counterfactual behaviors, some care would have to be taken to construct a model without borrowing constraints that also avoids these behaviors.

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Table 1 Two parameterizations of the model: One that minimizes the mean squared error of the consumption profile (MSE) and one that matches the relative size of the consumption peak (CP) Parameter

MSE

CP

FL

PE

SS

a d q r

0.308 0.0705 0.0241 0.580

0.312 0.0707 0.0241 0.425

0.312 0.071 0.040 0.514

N/A N/A 0.027 0.294

0.312 0.0710 0.0071 2.5

The parameters of a fixed lifetime (FL) model without mortality risk, a partial equilibrium (PE) model with mortality risk, and a mortality risk model with Social Security (SS) are also given.

Table 2 Model predictions and corresponding targets for the MSE, CP, FL, PE, and SS parameterizations Observable

Target

MSE

CP

FL

PE

SS

r (%) C/Y K/Y ccs(sc)/ccs(0) sc + 25 ccs RMSE

3.5 0.748 3.0 1.15 45 0

3.562 0.7526 2.952 1.09 41 0.0388

3.5 0.748 3.0 1.15 48 0.0555

3.5 0.748 3.0 1.31 40 0.1478

3.5 N/A N/A 1.15 45 0.0307

3.5 0.747 2.996 1.065 44 0.0650

PE parameters, and then we can match the relative size of the empirical profile. This is demonstrated by the CP parameterization, which is also depicted in Fig. 5. In this case, young consumers exploit the high return on savings to the point where they do not wish to borrow at all, as is shown in Fig. 6. Consumption grows at a rapid rate, reaching a maximum 15% larger than the initial value. Where this profile deviates from the empirical profile is that consumption starts out at a lower point. The CP consumption profile roughly parallels the GP empirical profile up to age 40, and they both ultimately grow by the same proportion. However, the CP profile grows at a slightly slower rate, so it peaks at age 48, three years later than the peak of the empirical profile. To compare how well the mortality risk model does to another leading model of the consumption hump, let us consider Gourinchas and Parker’s (2002) baseline model. This is a partial equilibrium buffer-stock saving model with a borrowing constraint and income uncertainty but no mortality risk.18 Using the Method of Simulated Moments, Gourinchas and Parker estimate r = 0.514, and this is the value they use to parameterize their baseline model. Their consumption profile is shown in Fig. 7. It reaches its maximum at age 39 with a peak to initial consumption ratio of 1.24. Fig. 5 shows what happens in the present model if r = 0.514 (with all other parameters kept at the CP values) as compared to the MSE and CP profiles. Like the empirical profile, this profile has a peak at age 45, but the relative size is only 1.09. Since r = 0.514 falls in between the MSE and CP parameterizations of r, this profile is a compromise between the CP and MSE profiles. 18 The interest rate in Gourinchas and Parker’s (2002) baseline model is r = 3.44% and the discount factor b = 0.96. All agents live with certainty to age 90. The cross-sectional mean of income as a function of age follows the profile of Fig. 3. In addition, agents experience idiosyncratic income shocks with a permanent component that has a mean of 1 and a variance of 0.0212 and a temporary component that has a mean of 1 and a variance of 0.440. (There is also a 0.3% probability of a zero-income realization.)

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Age Fig. 5. Lifecycle consumption profiles for the MSE and CP parameterizations and the CP parameterization with r = 0.514.

What would happen if we turned on mortality risk in Gourinchas and Parker’s baseline model? Feigenbaum (2007) has shown that, between the borrowing constraint and precautionary saving, it is the borrowing constraint that is responsible for the hump in Gourinchas and Parker’s model – i.e. the consumption hump persists (and actually gets more pronounced) if one turns off income uncertainty, but it disappears if one turns off the borrowing constraint. This means that Gourinchas and Parker’s baseline model essentially differs from the model presented here only by the absence of mortality risk and by the presence of income heterogeneity. Indeed, the consumption profile for this model can be reasonably approximated by the corresponding model without intracohort heterogeneity, although the consumption hump is, again, more pronounced in this simpler model. Consider a fixed lifespan (FL) model where all agents live to age 90 as in the GP baseline model.19 If we set the parameters to those given in the FL column of Table 1, we can match the three macroeconomic calibration targets used in Section 3. The resulting consumption profile is also depicted in Fig. 7. The FL profile has a relative size of 1.3 and 19 Note that there are two ways that one could turn off mortality risk in this model. The approach taken here is to set Q(t) = 1 for all t, which is the approach taken in virtually all papers that ignore mortality risk. This approach is necessary to compare to Gourinchas and Parker’s (2002) baseline model, as I do here. The other _ ˘ lu (2006), is to continue to match Q(t) to the empirical survivor approach, exemplified by Hansen and Imrohorog function but then to allow agents to perfectly annuitize wealth. The upside of this approach is that it still produces a demographically realistic model, though with the counterfactual prediction that all consumers save via annuities.

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has a peak at age 40. The FL profile differs from Gourinchas and Parker’s profile mainly in that consumption decays at a much faster rate after the peak, so the FL profile has a much higher root mean squared error of 0.15.20 However, the FL and baseline GP profiles are quite similar up to the peak. So we will view the FL profile as a proxy for the GP baseline model in the absence of mortality risk. Then the mortality risk model with CP parameters except r = 0.514 is the corresponding model with mortality risk switched on. Fig. 8 shows the lifecycle bond demand profiles for the FL model and the mortality risk model with the CP parameterization and r = 0.514. Within the FL model, consumers are borrowing constrained until age 40, and, indeed, in Fig. 7 we see that consumption in the FL model equals labor income up until age 40. In this case, the consumption hump is clearly caused by the borrowing constraint. In contrast, agents in the corresponding model with mortality risk are only borrowing constrained until age 27. Using Eq. (17), we find that the peak to initial consumption ratio would still be 1.076 (as opposed to 1.095) in the absence of the borrowing constraint, so in this model the consumption hump is primarily caused by mortality risk. Thus, if Gourinchas and Parker incorporated mortality risk into their model, there would still be a consumption hump, but the economics would completely change.21

20

Note that this is not an entirely fair comparison because the interest rate and the decay rate of consumption for the FL model are determined by general equilibrium conditions. In Gourinchas and Parker’s partial equilibrium model, they were not. 21 With intracohort heterogeneity, more young agents may hit the borrowing constraint. However, the majority of agents would still be unconstrained, so the effect of the borrowing constraint on the consumption profile should still be small.

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Age Fig. 7. Lifecycle consumption profiles for models with r fixed at 0.514 both with and without mortality risk. The latter include the FL parameterization with a fixed lifespan of 90 years and Gourinchas and Parker’s (2002) baseline profile for a buffer-stock saving model with the same lifespan. The mean labor income profile is also shown.

5. Sensitivity analysis In the previous section, we showed that it is possible to choose parameters for this model such that the lifecycle consumption profile roughly matches the empirical consumption profile, and in general equilibrium. However, as we will see now, slight changes in some of the parameters away from this choice can produce a substantial change in the properties of the consumption hump. Thus the consumption hump studied above is not robust. Recall that in the CP parameterization, r = 0.425, and the consumption hump achieves its maximum at age 48 with a peak to initial consumption ratio of 1.15. If we increase r to Gourinchas and Parker’s value of r = 0.514, the consumption peak moves back three years to age 45 and the relative size falls to 1.09 while the other macroeconomic variables remain essentially unchanged. Figs. 9 and 10 respectively show more generally how the relative size and age of the consumption peak vary as a function of r, holding all other parameters at their CP values. For r < 1, both graphs are quite steeply sloped, indicating these variables are quite sensitive to r. Meanwhile, Figs. 11–13 confirm that other endogenous macroeconomic variables are relatively insensitive to r for r between 0 and 2. A small change in r will cause only a tiny change in the interest rate r. Yet, according to Eq. (19), this tiny change in r will cause a large change in the age of the consumption peak, sc, since the hazard rate varies quite slowly prior to age 60. Meanwhile, Eq. (17) implies that the increase in consumption from the initial level to its peak is proportional to r1, so the size is even more responsive to changes in r.

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fixed lifespan mortality risk

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Age Fig. 8. Lifecycle bond demand profiles for the CP parameterization with r = 0.514 and for the FL parameterization with a fixed lifespan of 90 years.

3 2.8 2.6 2.4 2.2

ccs ( sc ) ccs (0)

2 1.8 1.6 1.4 1.2 1 0

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σ

6

7

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Fig. 9. The relative size of the hump, ccs(sc)/ccs(0), as a function of the inverse elasticity of intertemporal substitution r with all other parameters held fixed at their CP values. The shaded area is the region of the state space where the size of the hump falls between 1.1 and 1.5 as in most empirical studies.

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Fig. 10. Age of the consumption peak sc (+25) as a function of the inverse elasticity of intertemporal substitution r. All parameters besides r are at their CP values. The shaded area is the region of the state space where the age of the peak falls between ages 45 and 55 as in most studies.

Although there is much variation between studies regarding the relative size of the consumption peak, most researchers find a relative size between 1.1 and 1.5. The region of the parameter space where the relative size satisfies these loose bounds is shaded in Fig. 9. Likewise, most researchers find that the peak occurs between ages 45 and 55, and the corresponding region of the parameter space is also shaded in Fig. 10. Only a narrow band with r between 0.25 and 0.5 can satisfy both sets of bounds. Note that there are two mechanisms in this model that can produce a consumption hump. Which mechanism dominates also varies with r. For r less than about 0.475, there is no demand for borrowing by young agents, so any consumption hump must purely be the result of mortality risk. For r roughly between 0.475 and 0.7, the location of the consumption peak is determined by mortality risk but the borrowing constraint does cause a steeper rise in consumption during the first few years. Then for r greater than 0.7, the consumption hump is determined almost entirely by the borrowing constraint as the peak occurs where the optimal unconstrained consumption profile intersects total income. Notice that for r > 1, both Figs. 9 and 10 exhibit broad plateaus where the size and location of the consumption peak are relatively insensitive to r. Borrowing constraints provide a more robust explanation for the consumption hump than mortality risk.22

22 Likewise, Bullard and Feigenbaum (forthcoming) find the properties of the consumption hump to be quite robust in a model that also includes a labor-leisure decision. Since they have more endogenous variables in their model, they do not have to use the properties of the consumption profile to determine the exogenous parameters. Without any tinkering, their model yields a consumption hump with a relative size and peak location comparable to the hump in Gourinchas and Parker’s (2002) empirical consumption profile.

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Fig. 11. Equilibrium rate of return r as a function of the inverse elasticity of intertemporal substitution r. All parameters besides r are at their CP values. 5 4.75 4.5 4.25 4

K/Y

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Fig. 12. Capital to output ratio K/Y as a function of the inverse elasticity of intertemporal substitution r. All parameters besides r are at their CP values.

Of the four scalar parameters, a, d, q, and r, r determines the shape of the consumption profile while having little effect on the macrovariables K/Y, C/Y, and r while the reverse is true for the other three parameters. This was evident in the determination of the MSE and

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10

Fig. 13. Consumption to output ratio C/Y as a function of the inverse elasticity of intertemporal substitution r. All parameters besides r are at their CP values.

CP parameterizations, which produce macrovariables with nearly the same values and also have nearly the same values of a, d, and q. The main distinction between these parameterizations lies in which aspect of the consumption profile we used to calibrate the model and this led us to different values of r. As such, the sensitivity of a mortality-induced consumption hump to r should not be surprising. If the profile were not sensitive to r, we could not have used the properties of the profile to calibrate r. The sensitivity of the hump extends to the other parameters also. Table 3 summarizes the results for a series of experiments. We have already discussed what happens if we turn off mortality risk or vary r. The parameter that has the least effect on the consumption hump is the discount rate q. For the most part, an increase in q simply causes r to increase by roughly the same amount, which effectively cancels out any large changes in consumption behavior and macroeconomic variables. Perturbations of the technology parameters a and d have a more pronounced effect on the consumption profile.23 One parameter that the consumption profile is especially sensitive to is the retirement age Tret. A small increase in the retirement age will cause a small increase in the interest rate since consumers will have fewer post-retirement years to save for. As we have seen, a small change in r can have a big impact on the consumption peak if the peak falls in the region where the hazard rate increases slowly with age, and this is what happens. The dependence of the relative size and location of the peak on the retirement age are portrayed in Figs. 14 and 15 respectively. Increasing the retirement age from 65 to 70 causes the growth in 23 The specification is also sensitive to the specification of the survivor function. For the ‘‘double household’’ experiment, we consider the survivor function Qd(s) = 1  (1  Q(s))2 that would arise if the household consisted of two individuals instead of one. (We ignore the effect that sharing consumption between two individuals would have on the utility function.) In this case, mortality risk plays a negligible effect in shaping the consumption profile, with a hump caused by the borrowing constraint that peaks at 32.

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Table 3 Sensitivity of the endogenous variables to alternative parameter values Model

r (%)

C/Y

K/Y

sc

ccs ðsc Þ ccs ð0Þ

ccs RMSE

MSE CP CP except r = 0.514 CP except r = 1.0 ‘‘ ’’ and no BC CP except a = 0.29 CP except a = 0.33 CP except d = 0.06 CP except d = 0.08 CP except q = 0.015 CP except q = 0.035 CP except Tret = 60 CP except Tret = 67 CP except Tret = 70 CP double household FL GP baseline CP with SS SS FL except r = 2.5

3.56 3.50 3.54 3.56 3.74 3.42 3.57 3.58 3.44 2.60 4.60 3.12 3.63 3.80 2.81 3.50 – 3.97 3.50 3.50

0.752 0.748 0.752 0.750 0.754 0.764 0.735 0.756 0.742 0.723 0.773 0.738 0.751 0.755 0.728 0.748 – 0.759 0.747 0.747

2.952 3.000 2.958 2.981 2.930 2.812 3.156 3.292 2.786 3.294 2.702 3.118 2.962 2.912 3.235 3.000 – 2.865 2.996 2.990

41 48 44 33 25 45 50 51 46 48 48 30 52 56 32 40 39 59 44 35

1.091 1.150 1.095 1.148 1.000 1.112 1.194 1.203 1.115 1.154 1.155 1.063 1.241 1.400 1.140 1.310 1.240 1.595 1.065 1.219

0.0388 0.0555 0.0409 0.0590 0.0718 0.0393 0.0724 0.0760 0.0424 0.0706 0.0659 0.1210 0.0953 0.1618 0.1025 0.1478 0.0381 0.2071 0.0650 0.0607

Note that the ages are measured in the table (as opposed to the rest of the paper) with agents being born at 25.

1.8

1.7

1.6

1.5

ccs ( sc ) ccs (0)

1.4

1.3

1.2

1.1

1 55

57

59

61

63

65

67

69

71

73

75

Tret ( + 25) Fig. 14. The relative size of the hump, ccs(sc)/ccs(0), as a function of the retirement age, Tret, with all other parameters held fixed at their CP values.

consumption from age 25 to the peak to quadruple in size and pushes the peak back from 48 to 56. This is troubling because the retirement age is a parameter to which the literature has

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Tret ( + 25) Fig. 15. Age of the consumption peak, sc (+25), as a function of the retirement age, Tret, with all other parameters held fixed at their CP values.

given very little attention. Since 65 used to be the age when American workers typically became eligible to receive Social Security benefits, it has become standard practice with an inelastic labor supply to set retirement at 65. However, since 2003, the age of eligibility has been gradually ramping upward to 67. Moreover, empirically one does not find a sudden, discontinuous drop in average hours worked at some age around 65, but instead one finds a gradual reduction between ages 55 and 80. This suggests that it would be better to model the retirement decision as in Bullard and Feigenbaum (forthcoming). Yet while this would remove the dependence on an exogenous retirement age, a mortality-induced consumption hump would still be sensitive to the introduction of any element that can modify the equilibrium interest rate. As such, it would be extremely unlikely that all the forces that shape the economy would align so the parameters match up to what is necessary to get mortality risk to work as an explanation for the consumption hump. While the examples presented here show that mortality risk can have a substantial impact on the shape of the consumption profile, it is doubtful that mortality risk is the predominant force that shapes the consumption hump. 6. The consumption profile with social security The effects of time-varying mortality risk on the consumption profile will be eliminated if workers can fully annuitize their wealth.24 Suppose agents at age s invest not in risk-free bonds that pay a fixed return but in an asset that pays a net return ra ðsÞ ¼ r þ hðsÞ _ ˘ lu (2006) measure the effect of mortality risk by introducing annuities rather than Thus Hansen and Imrohorog by eliminating uncertainty about the consumer’s lifespan. 24

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to an agent who does not die whereas the asset is forfeit if the agent dies. In that case, the hazard rate will cancel out of the Euler equation (11). Meanwhile the expected net return, inclusive of mortality risk, will still be r. An annuity here then has two essential properties: (i) the rate of return for a survivor increases with the hazard rate of dying, and (ii) it leaves no wealth to be passed on to the agent’s survivors after he dies. While annuities are rarely used by investors in the United States today, Social Security is often viewed as a way of partially annuitizing wealth because it provides a stream of income at the end of life that shares the second property of an annuity, though not the first. _ ˘ lu (2006) have demIn a model without a borrowing constraint, Hansen and Imrohorog onstrated, if we hold all other exogenous parameters constant, the introduction of Social Security moves the consumption peak later and also makes it larger, effectively nullifying the possibility that mortality risk can account for the consumption hump. They argue this happens because Social Security is a partial annuity. However, as we saw in the previous section, the properties of the consumption profile are hypersensitive to the interest rate, and Social Security can certainly affect the interest rate. Indeed, absent a borrowing constraint, Social Security can only affect the consumption hump through its effect on the interest rate. The rate of consumption growth over the entire lifecycle is specified by Eq. (14), and everything that appears on the right-hand side is an exogenous variable independent of Social Security, except the interest rate r. Social Security can also affect the initial level of consumption ccs(0), but the shape of the normalized consumption profile ccs(s)/ ccs(0) is determined entirely by integrating Eq. (14). Thus we can understand the impact of Social Security on the shape as follows. A transfer of income from the young to the old reduces the demand for saving so, barring income effects, the equilibrium interest rate will have to go up. From Eq. (19), we know that increasing r also increases the peak age sc since the effective discount rate will have to be higher at the point where it matches the interest rate and the rate of consumption growth vanishes. From Eq. (18), this in turn implies a higher peak to initial consumption ratio since consumption will be increasing over a longer time. Let us now consider the interactions of mortality risk and Social Security in the model presented here, which does include a borrowing constraint. Following Findley and Caliendo (2006), we set the Social Security labor income tax rate to g = 0.106. The macro-observables for this model are listed in Table 3. For the baseline CP parameters, the interest rate increases from 3.50% to 3.97%, which might not seem like much. However, because of the hypersensitivity of the consumption profile to r, the peak age increases from 48 to 59, and the peak to initial consumption ratio increases from 1.15 to 1.60. These _ ˘ lu (2006) report. findings are similar to what Hansen and Imrohorog The CP value of r is required to obtain a peak to initial consumption ratio close to what is observed in the data. For the CP calibration without Social Security, the equilibrium interest rate is not exactly what is needed to match the location of the peak, but it is close enough so the resulting consumption hump resembles the hump in the data. However, when Social Security is added, the equilibrium interest rate at this value of r increases enough to move the peak much later than what is in the data. Thus adding Social Security completely disrupts the ability of the mortality model to fit the data. Nevertheless, even if mortality risk cannot explain the consumption hump, it can still have important interactions with Social Security that might affect other explanations for the hump. Recall that unconstrained consumption will be nonmonotonic iff the condition (15) r > qeff(0) + rg is satisfied. Without Social Security, setting other parameters to their

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Age Fig. 16. Lifecycle consumption profiles for the baseline mortality model with Social Security (SS) and the corresponding fixed lifespan model.

CP values, we found that mortality risk only determined the location of the consumption peak for the narrow range r 6 0.7 (which albeit did include Gourinchas and Parker’s (2002) baseline model). By increasing the equilibrium interest rate, introducing Social Security expands the set of parameters that satisfy (15). If we fix r to a given value and set a, d, and q to match our target values of K/Y, C/Y, and r, mortality risk determines the location of the consumption peak for r 6 2.9, which covers most of the range favored by macroeconomists today.25 As an example, consider the SS calibration listed in Table 1, where we set r = 2.5 so as to match the location of the consumption peak to the data. The consumption profile for this calibration is shown in Fig. 16 along with the consumption profile that would be obtained in a fixed lifespan model with the same parameters (except q = 0.0239 is set so the fixed lifespan model still matches the three macrotargets). Macroobservables for both models are listed at the bottom of Table 3. Notice that the two consumption profiles are quite different. Whereas in the fixed lifespan model the consumption profile has a huge peak, in the mortality model the consumption profile is nearly flat. There is an initial steep rise in the mortality profile for the first three years where the borrowing constraint binds, but the peak to initial consumption ratio is still only 1.06, as compared to the ratio of 1.22 that would arise in the absence of mortality risk. Thus mortality risk, enhanced by Social

25

From estimates of labor supply elasticity, Chetty (2006) concludes that r < 2 while Ljungqvist and Sargent (2004) state that the ‘‘prejudice’’ of most macroeconomists is that r ‘‘should not be much higher than 2 or 3.’’

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Security, can effectively wipe out the pronounced consumption peak that a borrowing constraint would otherwise produce in a fixed lifespan model. 7. Conclusion Although empirical lifecycle consumption profiles are hump-shaped with a peak around age 50, the most basic models of consumption and saving cannot account for this behavior. If consumers have a constant discount factor then, barring any frictions, consumption should rise or fall at a constant rate, depending on whether the interest rate is larger or smaller than the discount rate. Allowing for mortality risk can change this story because then it is the effective discount rate, adjusted for the probability of dying, that gets compared to the interest rate. If the probability of dying increases with age, consumption can rise early in life when the interest rate is greater than the effective discount rate. Later on, consumption will fall after the effective discount rate surpasses the interest rate. Most research on the consumption hump has disregarded mortality risk, turning instead to other explanations such as borrowing constraints, precautionary saving, and leisure-consumption substitution. Prior to age 60, the probability of dying is quite small, so it has generally been assumed that variation in mortality risk during middle age is not large enough to produce a significant hump in consumption. Here, we have shown in a general equilibrium model that mortality risk alone can produce a consumption hump with roughly the same relative size and peak location as empirical consumption profiles if the elasticity of intertemporal substitution is high enough. However, since the profile of mortality risk over the lifecycle has been fairly well estimated, the model makes a rigid prediction regarding the shape of the consumption profile given the other parameters, so there is little room to improve the fit. Moreover, mortality risk only works as an explanation for the hump for a very specific choice of parameters, and the introduction of any new element into the model, particularly elements that affect the interest rate, can completely throw off the properties of this hump. Thus, in a general equilibrium setting, we cannot make any definite predictions about a consumption hump whose properties are primarily determined by mortality risk (or any other source of time-varying discount factors) without accounting for all forces that determine the interest rate. In contrast, other explanations for the hump are far more robust. So mortality risk alone is presumably not the explanation for the hump, but, acting in concert with other frictions, mortality risk may still play an important role in shaping the consumption hump. Accounting for mortality risk may also weaken the case for alternate explanations for the hump that would otherwise seem plausible. Feigenbaum (2007) has demonstrated that the hump in Gourinchas and Parker’s (2002) baseline model, with an intertemporal elasticity  2, is principally caused by its borrowing constraint. Here we have shown that if we account for mortality in this model there would still be a consumption hump, but the story of what produces it would change since young consumers would essentially stop borrowing. More generally, if we include Social Security in the model, mortality risk can shut down borrowing even for intertemporal elasticities as small as 0.4. This opens up the question of whether a lifecycle model with time-consistent preferences is capable of explaining the borrowing behavior of most consumers.26 Using 26

Borrowing is also absent from Bullard and Feigenbaum (forthcoming).

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