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Journal of Monetary Economics 51 (2004) 609–
Consumption and risksharing over the
life cycle
Kjetil Storesletten
a
, Christopher I. Telmer
b,
*, Amir Yaron
c a (^) Department of Economics, University of Oslo, N-0317 Oslo, Norway b (^) Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213, USA c (^) Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA
Received 7 May 2001; received in revised form 2 December 2002; accepted 30 June 2003
Abstract
A striking feature of U.S. data on income and consumption is that inequality increases with age. This paper asks if individual-specific earnings risk can provide a coherent explanation. We find that it can. We construct an overlapping generations general equilibrium model in which households face uninsurable earnings shocks over the course of their lifetimes. Earnings inequality is exogenous and is calibrated to match data from the U.S. Panel Study on Income Dynamics. Consumption inequality is endogenous and matches well data from the U.S. Consumer Expenditure Survey. The total riskhouseholds face is decomposed into that realized before entering the labor market and that realized throughout the working years. In welfare terms, the latter is found to be more important than the former. r 2003 Elsevier B.V. All rights reserved.
JEL classification: E21; D
Keywords: Risksharing; Buffer-stocksavings; Consumption inequality
$We thankDave Backus, RickGreen, Burton Hollifield, Dean Hyslop, MarkHuggett, Bob Miller,
Christina Paxson, Ed Prescott, B. Ravikumar, V!ıctor R!ıos-Rull, Tom Sargent, NickSouleles, Kenneth Wolpin, Stan Zin, and an anonymous referee for helpful comments and suggestions. We have benefited from the support of NSF Grant SES-9987602 and the Rodney White Center at Wharton. *Corresponding author. Department d’Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fragas 25-27, Barcelona 08005, Spain. Tel.: 412-268-8838. E-mail address: chris.telmer@cmu.edu (C.I. Telmer).
0304-3932/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2003.06.
- Introduction
Understanding the determinants of economic inequality is important for many questions in economics. It bears directly on issues as wide ranging as education policy, economic growth and the equity premium puzzle. Most existing workon inequality has focused on income, wealth and a variety of individual-specific characteristics such as educational attainment and labor market status. Relatively little attention has been paid to inequality in what these items ultimately lead to: consumption. This is unfortunate. The reason, presumably, that income and wealth inequality are of such interest is that they have an important impact on consumption inequality and, as a result, on inequality in economic welfare. We focus on how inequality in consumption and labor earnings change with age. The data reveal three salient facts: (a) age-dependent inequality in earnings and consumption increases substantially between ages 23 and 60, (b) the increase in consumption is less than the increase in earnings, and (c) the increase in both is approximately linear. 1 We askif these facts can be explained by the existence of noninsurable idiosyncratic shocks to labor earnings. We use a general equilibrium overlapping generations model in which households face earnings shocks over the course of their lifetimes. Direct insurance of these shocks is ruled out. Agents can invest in a single financial asset with a fixed rate-of-return. There is a pay-as-you-go social security system. The model is calibrated so that the age-profile of earnings inequality matches that of the data. Our main finding is that the model’s profile of consumption inequality matches the data, both qualitatively and quantitatively. Given this, we decompose the riskwhich households face into that realized before entering the labor market and that realized throughout the working years. In welfare terms, we find that the latter is more important than the former. We start by characterizing a process for the idiosyncratic component of labor earnings risk. This process is calibrated using individual data from the PSID, allowing for fixed-effects, persistent shocks, and transitory (i.i.d.) shocks. Based on age-dependent cross-sectional variances, we estimate the autocorrelation coefficient of the persistent shocks to be close to unity. This is driven by the linear shape of the empirical age profile. Less than unit-root shocks would yield a concave-shaped age profile. This persistence is a robust feature of the data, even when considering education groups separately or adding autocovariance moments of individual earnings. Given the income process, we solve for equilibrium allocations and examine the implications for consumption inequality. General equilibrium considerations are important as they pin down the aggregate amount of wealth in the economy. The level of wealth governs the amount of risksharing that is feasible, since trading in capital is the means by which agents ‘self-insure.’ This in turn determines the model’s pattern of consumption inequality. We find that, absent a social security system, consumption inequality is roughly 20% too high relative to the data. Incorporating
(^1) Consumption data are from Deaton and Paxson (1994), who use the Consumer Expenditure Survey (CEX). Labor earnings data are from the Panel Study on Income Dynamics (PSID).
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Mace, 1991). Our paper, corroborates these empirical findings in that agents in our model are able to self-insure against only a limited amount of the shocks they face, shocks which are calibrated to PSID data. In contrast, Altug and Miller (1990) use PSID data and are unable to reject the restrictions implied by complete markets. Finally, a large literature—including Aiyagari (1994), Chatterjee (1994), Huggett (1996), Quadrini (2000), Krusell and Smith (1998) and Castan˜ eda et al. (2003)— focuses on wealth inequality using a class of models similar to ours. Huggett (1996) paper is of particular interest in that he shows that a model similar to ours can account for how wealth inequality varies over the life cycle. The remainder of the paper is organized as follows. Section 2 presents stylized facts on labor earnings and consumption, and estimates a model for idiosyncratic earnings shocks. Section 3 outlines and parameterizes our life-cycle model, Section 4 reports quantitative results, and Section 6 concludes.
- Evidence
We begin with empirical evidence on how inequality in labor income and consumption vary with age. Labor income data (and other types of non-wage income data) are from the PSID, 1969–1992. Consumption data are from Angus Deaton and Christina Paxson (used in Deaton and Paxson, 1994), whose source is the CEX, 1980–1990. We use both the PSID and the CEX for reasons of data quality. The PSID is arguably the best source for income-related data, but is very narrow in terms of consumption, being limited to food. The CEX, in contrast, covers a broad set of consumption categories but is inferior to the PSID in terms of income. Merging the two datasets, therefore, is an attempt to use the best-available data. In Appendix A we elaborate further, discuss potential inconsistencies, and ultimately argue that, at least for our purposes, the PSID and the CEX are compatible. Our measure of consumption is defined as nonmedical and nondurable expenditures on goods and services by urban U.S. households. See Deaton and Paxson (1994) for further details. Our measure of labor earnings is defined as total household wage income before taxes, plus unemployment insurance, workers compensation, transfers from nonhousehold family members, and several additional categories listed in Appendix A (we define earnings inclusive of these ‘transfers’ because the implicit risk-sharing mechanisms they represent are absent from our theory). Unlike many previous studies, our selection criterion includes both male- and female-headed households, and also allows for within-sample changes in family structure such as marriage, divorce and death. This relatively broad sampling criteria—motivated in part to be consistent with CEX consumption data— incorporates many idiosyncratic shocks that might otherwise be omitted (e.g., divorce), while still allowing for the identification of time-series parameters. The cost, of course, is increased sensitivity to measurement error, something which is mitigated by our focus on cross-sectional properties of the data. Fig. 1 reports the cross-sectional variance of the logarithm of earnings and consumption by age. These moments will be the focal point of our theory. In
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computing them we follow Deaton and Paxson (1994) in removing ‘cohort effects’ via dummy-variable regressions (details are provided in Appendix A). That is, if we define a ‘cohort’ to be all households with a head of a given age born in the same year, then our measures of cross-sectional dispersion are net of dispersion which is unique to a given cohort. These cohort effects turn out to be quantitatively important, something which Deaton and Paxson (1994) also document for CEX data. By not removing them, for instance, our estimate of the cross-sectional variance for the young (old) increases (decreases) by roughly 50% (20%), thereby making for a substantially flatter age profile. The important features of Fig. 1 are as follows. First, both earnings and consumption inequality increase over the working part of life cycle, but only in the case of earnings does it decline at retirement. Second, inequality among the young is roughly the same for earnings and consumption, but the former increases faster with age. For example, over the working years the cross-sectional standard deviation of
25 30 35 40 45 50 55 60 65 70 75
1
Age
Variance of Log
Consumption
Earnings
Fig. 1. The graphs represent the cross-sectional variance of the logarithm of earnings and consumption. The basic data unit is the household. Consumption data are from the CEX and are taken directly from Deaton and Paxson (1994). Earnings data are taken from the PSID. The variances are net of ‘cohort effects:’ dispersion which is unique to a group of households with heads born in the same year. This is accomplished, as in Deaton and Paxson (1994), via a cohort and age dummy-variable regression. The graphs are the coefficients on the age dummies, scaled so as to mimic the overall level of dispersion in the data. Further details are in Appendix A.
K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–633 613
0.2 25 30 35 40 45 50 55 60 65 70
1
No High School
High School
College
Age
Variance of Log
Earnings
0.2 20 25 30 35 40 45 50 55 60 65
1
Variance of Log
Age
College
High School
No High School
Consumption
Fig. 2. Each line represents the cross-sectional variance of the logarithm of earnings (consumption) for the specified educational cohort. The basic data unit is the household. Consumption data are from the CEX and are taken directly from Deaton and Paxson (1994). Earnings data are taken from the PSID. All graphs represent variances which are net of ‘cohort effects:’ dispersion which is unique to a group of households with heads born in the same year. This is accomplished, as in Deaton and Paxson (1994), via a cohort and age dummy-variable regression. The graphs are the coefficients on the age dummies, scaled so as to mimic the overall level of dispersion in the data. Further details are in Appendix A.
K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–633 615
2.1. A Parametric model for earnings
Our fundamental data unit is y (^) ih; the logarithm of annual real earnings for household i of age h: From y (^) ih; we use a dummy-variable regression—analogous to that used in Deaton and Paxson (1994) and in the construction of Figs. 1 and 2—to control for cohort effects and extract u (^) ih; the idiosyncratic component of earnings. Details are provided in Appendix A. We then specify a time-series process for u (^) ih:
u (^) ih ¼ ai þ Eih þ z (^) ih; ð 1 Þ
z (^) ih ¼ rz (^) i;h 1 þ Zih; ð 2 Þ
where aiBNð 0 ; s^2 aÞ; EihBNð 0 ; s^2 E Þ; ZihBNð 0 ; s^2 ZÞ; z (^) i 0 ¼ 0 and, therefore, Eðu (^) ihÞ ¼ 0 in the cross-section for all h (the latter makes precise the language ‘idiosyncratic shock’). The random variable ai—commonly called a ‘fixed effect’—is realized at birth and then retained throughout life. The variables z (^) ih and Eih are realized at each period over the life cycle and are what we refer to as persistent and transitory ‘life- cycle shocks,’ respectively. We estimate the parameters of process (1) directly from the cross-sectional variances in Fig. 1. The population moments are
Varðu (^) ihÞ ¼ s^2 a þ s^2 E þ s^2 Z
Xh^1
j¼ 0
r^2 j^ : ð 3 Þ
For jrjo 1 ; the summation term converges to the familiar s^2 Z=ð 1 r^2 Þ; the unconditional variance of an AR(1). What distinguishes our approach is that we do not take this limit, but instead condition on age h and make strong assumptions on initial conditions (i.e., the distribution of ai and z (^) i 0 ¼ 0). Increasing cross- sectional variances with h; therefore, map directly to a relatively large value for r: Inspection of Eq. (3), alongside the earnings profile in Fig. 1, indicates that the sum s^2 a þ s^2 E can be identified by the profile’s intercept, the conditional variance s^2 Z by its slope, and the autocorrelation r by its curvature. A graphical depiction of this (admittedly impressionistic) algorithm is provided in Fig. 3. The result is s^2 a þ s^2 E ¼ 0 : 2735 ; s^2 Z ¼ 0 : 0166 and r ¼ 0 : 9989 : In Storesletten et al. (2000) we use one autocovariance to exactly identify s^2 a from s^2 E : We find s^2 a ¼ 0 :2105 and s^2 E ¼ 0 : 0630 : This graphical approach is obviously informal. In Storesletten et al. (2000) we develop a formal GMM-based framework. We show that the above, exactly identified estimates are very precise. We go on to incorporate a host of overidentifying restrictions, using additional age-dependent cross-sectional variances as well as additional autocovariances. We find very similar estimates of the variances and slightly lower estimates of the autocorrelation, but none below r ¼ 0 : 977 :^4
(^4) Our approach is unorthodox in the sense that we estimate a time-series parameter, r; with relatively little reliance on time-series moments. In Storesletten et al. (2000), however, we argue that our approach nests more conventional approaches in that an overidentified GMM system, one which includes (more conventional) autocovariances, yields similar estimates as long as the age-dependent cross-sectional variances are included.
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- The model
The economy is populated by H overlapping generations, each generation consisting of a continuum of agents. Lifetimes are uncertain. We use fh to denote the unconditional probability of surviving up to age h; with f 1 ¼ 1 ; and use xh ¼ fh=fh 1 ; h ¼ 2 ; 3 ; y; H; to denote the probability of surviving up to age h; conditional on being alive at age h 1 : The fraction of the total population attributable to each age cohort is fixed over time and the population grows at constant rate. Preferences are identical across agents and are represented by
E
XH
h¼ 1
bhfh UðcÞ; ð 4 Þ
where U is isoelastic; UðcÞ ¼ c^1 g=ð 1 gÞ: Agents begin working at age 22 and, conditional on surviving, retire at age 65. Prior to retirement an agent of age h receives an annual endowment, n (^) h; of labor hours (or, equivalently, productive efficiency units) which they supply inelastically. Individual labor earnings are then determined as the product of hours worked and the wage rate. Aside from age, heterogeneity is driven by idiosyncratic labor market risk. We adopt the following process for the logarithm of hours worked,
log n (^) h ¼ kh þ a þ z (^) h þ Eh; ð 5 Þ
where kh govern the average age-profile of earnings, aBNð 0 ; s^2 aÞ is a fixed effect, determined at birth, EhBNð 0 ; s^2 E Þ is a transitory shockreceived each period, and z (^) h is a persistent shock, also received each period, which follows a first-order autoregression:
z (^) h ¼ rz (^) h 1 þ Zh; ZhBNð 0 ; s^2 ZÞ; z 0 ¼ 0 : ð 6 Þ
This process is a direct analog of what we estimated in the previous section, the only difference being that, here, we specify a process for hours worked and not labor earnings. The difference, however, will be an additive time trend (the wage rate in our stationary equilibrium will be growing at a constant rate), thereby making the distinction innocuous. As a normalization, the average labor endowment, across all workers, is EðnÞ ¼ 1 : After retirement, agents receive a pension, given by a fraction Bð n%Þ > 0 of average labor earnings in a particular year, where n% is the average labor endowment during an agent’s working life. Thus, B is a replacement rate and n% corresponds to indexed average annual earnings. The pension outlays are financed by a flat tax on labor income, t: There is a single asset–capital—which pays a return R plus a survivor’s premium, representing an actuarially fair annuity. Capital is used, along with labor, as inputs to a Cobb–Douglas production function for a representative firm,
Y ¼ ZKyN^1 y; ð 7 Þ
where K and N denote aggregate capital and labor, respectively. The level of technology, Z; is growing so that the economy exhibits a steady-state growth rate of
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g: The firm rents capital and labor at rental rates W and R; respectively. Given a rate of depreciation d; the law of motion for K is K^0 ¼ Y C þ ð 1 dÞK; where C is aggregate consumption. Let Vh denote the value function of an h year old agent, given a constant interest rate R and a wage rate W growing at rate g: After properly normalizing for growth, (which involves redefining the discount factor to b# bð 1 þ gÞ^1 g), and defining VHþ 1 0 ; the choice problem of the agents can be recursively represented as
V (^) hða; z (^) h; Eh; a (^) h; n% (^) hÞ ¼ max a^0 hþ 1
fUðc (^) hÞ þ bx# hþ 1 E½V h^0 þ 1 ða; z^0 hþ 1 ; E^0 hþ 1 ; a^0 hþ 1 ; n%^0 hþ 1 Þg ð 8 Þ
subject to the following constraints. Before retirement,
c (^) h þ ð 1 þ gÞa^0 hþ 1 pa (^) h R=xh þ n (^) hð 1 tÞW ð 9 Þ
a^0 hþ 1 X %
aða; z; hÞ; ð 10 Þ
n %^0 hþ 1 ¼ n% (^) h þ n (^) h=I; ð 11 Þ
where a (^) h denotes beginning of period asset holdings, a^0 hþ 1 denotes end of period asset holdings, %
aða; z; hÞ denotes a state-dependent borrowing constraint, and I is the number of years before retirement. After retirement, constraints are given by a^0 hþ 1 X 0 and
c (^) h þ a^0 hþ 1 pa (^) h R=xh þ Bð n% (^) hÞW ð 12 Þ
n %^0 hþ 1 ¼ n% (^) h: ð 13 Þ
Our timing convention is that savings decisions are made at the end of the current period, and returns are paid the following period at the realized capital rental rate. Fair annuity markets are captured by the survivor’s premium, 1=xh; on the rate of return on savings. A stationary equilibrium is defined as prices, R and W ; a set of cohort-specific functions, fVh; a^0 hþ 1 gHh¼ 1 ; aggregate capital stock K and labor supply N; and a cross- sectional distribution m of agents across ages, idiosyncratic shocks, asset holdings, and past earnings, such that (a) prices W and R are given by the firm’s marginal productivity of labor and capital (i.e., market clearing for capital and labor), (b) individual optimization problems are satisfied (so that fVh; a^0 hþ 1 gHh¼ 1 satisfy Eqs. (8)), (c) the pension tax t satisfies the pay-as-you-go budget constraint W
R
S Bð^ n%Þ^ dm^ ¼ WNð 1 tÞ; (d) the distribution m is stationary, given individual decisions, and (e) aggregate quantities result from individual decisions: K ¼
R
S a^ h^ dm^ and^ N^ ¼^
R
S n^ h^ dm: Because our economy does not feature aggregate shocks and preferences are of the CRRA class, there exists a unique stationary equilibrium, and any initial distribution eventually converges to m (Huggett, 1993). We solve the individuals’ optimization problems based on piecewise linear approximation of the decision rules (with 80 points on the wealth-grid and up to 20 points on the grid for accumulated earnings), and follow Huggett (1993) and Aiyagari (1994) in solving for the stationary equilibrium.
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The borrowing constraint is set so that agents cannot borrow in excess of expected earnings next period, i.e., %
aða; z; hÞ ¼ Wekhþ^1 þaþz: In Section 4.4 we show that the choice of borrowing constraint is of negligible importance for consumption inequality. The pension replacement rate is based on the Old Age Insurance of the U.S. social security system and given by
Bð n% (^) hÞ ¼
0 : 9 n% for n%p 0 : 3 ; 0 : 27 þ 0 : 32 n% for n%Að 0 : 3 ; 2 ; 0 : 81 þ 0 : 15 n% for n%Að 2 ; 4 : 1 ; 1 : 1 for n% > 4 : 1 :
ð 14 Þ
- Results
We begin with an economy without a social-security system, B ¼ 0 ; which we call the ‘benchmarkeconomy.’ The model’s implications for average consumption over the life cycle are broadly in line with U.S. data. Fern!andez-Villaverde and Krueger (2002) use data from the CEX to estimate the age-profile of per-adult-equivalent consumption, correcting for age, seasonal, and cohort effects. They find that the profile is hump-shaped, peaking at roughly age 50, with the peak30–40% higher than at age 23. In our benchmark economy, the peakoccurs at age 55, roughly 40% higher than at age 23. The hump would be even more pronounced with higher riskaversion, and the timing of the peakwould be earlier if the retirement age were before 65. Fig. 4 displays the main result of the paper: the age-profile of consumption dispersion implied by the benchmarkmodel. The model is successful at capturing two important qualitative features of the data. First, consumption inequality is everywhere less than earnings inequality. Second, the slope of the consumption profile is everywhere less than that of the income profile. Quantitatively, theoretical inequality coincides with the data at age 27 but then grows slightly faster with age, ending up 20% higher at retirement (0.65 versus 0.54). 8 The rise in consumption inequality is of particular importance for risksharing. Given our specification for preferences, the discrepancy between theory and data can be interpreted as a measure of the insurance arrangements available to actual agents, but not present in our model. 9 In order to quantify this, we askhow much we must
(^8) The model does not fit the data particularly well before age 27, when empirical consumption dispersion
is falling. However, the initial fall is not a robust feature of the data. For instance, there is no fall when considering inequality in per-capita household consumption (Fig. 8 in Deaton and Paxson, 1994). (^9) An alternative—and equally valid—interpretation is that agents have more knowledge than the econometrician regarding future income changes. Thus, the econometrician would attribute some income changes to risk, even though agents’ consumption should not respond to such changes. According to this interpretation, the excess rise in consumption inequality implied by the model measures the degree to which income riskis over-estimated.
K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–633 621
reduce the conditional variance of the permanent and transitory earnings shocks in order to match the observed increase in consumption inequality between age 27 and retirement. The answer is 20%. That is, we reduce s (^) Z to 0.0129 and se to 0.05. We conclude that—relative to ‘self-insurance’ via financial markets—actual agents have additional insurance arrangements available to them which insure against 20% of the variability in the life-cycle shocks that they face.
4.1. The role of wealth
What we mean by ‘risksharing’ is an activity which aligns marginal rates of substitution relative to what they would be under autarky. In our model, ‘self- insurance’ or ‘buffer-stocksavings’ plays this role. In aggregate, the amount of self- insurance which is feasible is limited by the amount of aggregate wealth which working-age agents hold as a buffer-stock. The amount of aggregate wealth is what pins down where the theoretical consumption-inequality profile in Fig. 4 lies, relative to the earnings profile.
20 30 40 50 60 70 80
1
Age
Cross
−Sectional Variance
U.S. Consumption
Theoretical Consumption
U.S. Earnings
Theoretical Earnings
Fig. 4. This graph compares the population moments from our benchmarkmodel with those of the data. The solid lines (without dots) represent the theoretical and empirical cross-sectional variance of log earnings. The dashed line represents the empirical cross-sectional variance of consumption and the solid- dotted line represents the theoretical cross-sectional variance of consumption from the benchmark economy.
622 K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–
relatively unimportant and have a relatively small impact on individual consumption and cross-sectional inequality. The inequality profile for the low-wealth economy in Fig. 5 reflects these effects. In contrast to the benchmarkeconomy, the average young household in the low-wealth economy is borrowing against future expected increases in earnings (because their discount factor is lower). That is, financial wealth is negative and the ratio of human to total wealth exceeds unity. Accordingly, earnings shocks have an exceptionally large impact and consumption inequality increases at a faster rate than earnings inequality. Later in life, once the average household begins to save for retirement, human wealth becomes less important and, while consumption inequality still increases, it does so at a slower rate than earnings inequality. A natural question to ask, then, is what changes can be made to the model in order to generate a more realistic, linear consumption profile. In the previous version of this paper (Storesletten et al., 2000) we considered an earnings process with ro1 and heteroskedastic innovations with s (^) Z increasing with age. This approach, with 0 : 85 oro 0 : 90 ; was successful in accounting for linearity in both earnings and consumption, as well as the respective rise and level.
20 30 40 50 60 70 80
1
Age
Cross
−Sectional Variance of Consumption U.S. Consumption
Benchmark Consumption
Theoretical Earnings
Low−wealth Economy
Social Security Economy
Fig. 5. The dashed line represents the empirical cross-sectional variance of consumption. The solid-dotted line represents the theoretical variance for the benchmarkeconomy, whereas the dotted and triangle- marked lines represent the low wealth and social security economies, respectively, discussed in Section 4 of the text. All the economies have an identical pattern of earnings inequality, given by the solid line (which closely matches its empirical counterpart).
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4.4. What matters for consumption inequality?
We now analyze a number of additional economic factors and parameterizations of our model which are (potentially) important for consumption inequality. In order to preserve transparency and minimize the complexity of the experiments, we abstract from social security hereafter (i.e., B ¼ 0).
4.4.1. Persistence Our analysis to this point has assumed unit-root shocks. However, in related work—Storesletten et al. (2000, 2004)—we find evidence that r is less than unity but greater than 0.92. Fig. 6 explores the implications for consumption inequality. It plots the increase in inequality between age 27 and retirement for each value of r between zero and unity. In order to maintain a sensible comparison, we set, for each r; the conditional variance, s^2 Z; so that the variance of the persistent shock, averaged over age groups, is the same as in the benchmarkeconomy. The variances s^2 a and s^2 e
(^0) .25 .5 .75 .91 1
Persistence ρ
Rise in Consumption Inequality
Rise in U.S. Consumption inequality
Fig. 6. This figure shows the impact of persistence of shocks on the rise in consumption inequality. For each degree of persistence, r; the solid graph indicates the rise in consumption dispersion between age 27 and retirement implied by the model. The dashed line illustrates the empirical counterpart (from Fig. 1). In each of the experiments, the (homoskedastic) conditional variance of persistent shocks, s^2 Z; is set so that overall inequality of persistent shocks corresponds to that in the benchmark unit-root economy, while the variance of fixed effects and transitory shocks are held constant. The borrowing constraint is set to
% a ¼ W ; average earnings per-worker.
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young have more wealth (relative to the benchmark), which tends to reduce inequality by a small amount.
4.4.3. Risk aversion Riskaversion in the benchmarkeconomy is set to g ¼ 2 : An increase in g; ceteris paribus, will increase the level of precautionary savings and, therefore, decrease consumption inequality. Aggregate wealth, however, will be unrealistically high, thus defeating the entire purpose of our exercise (i.e., asking how much risk sharing a given amount of aggregate wealth will support). A more interesting question asks what happens when riskaversion is increased, holding aggregate wealth constant. We achieve this—setting g as high as 7—by simultaneously reducing the value of the discount factor b: The implications for the increase in consumption inequality are inconsequential. The shape of the profile, however, becomes slightly more linear. The reason is that, as the fraction of total wealth attributable to precautionary savings increases, agents hold more wealth when young and less when old. Thus, consumption inequality grows more slowly (relative to the benchmarkeconomy) over the younger ages and faster during the close-to-retirement ages.
4.4.4. Initial wealth In the benchmarkeconomy all agents are born with zero financial wealth. We find that economies with a more realistic dispersion in initial financial wealth (as reported in D!ıaz-Gim!enez et al., 1997), but with average wealth for newborn kept equal to zero, generates consumption inequality which is qualitatively similar to our benchmarkeconomy. The variance is slightly higher for agents aged 23–29, but slightly lower for the remaining age cohorts. The magnitude of these differences is not large and the average variance is quite similar to our benchmarkmodel.
4.4.5. Annuity markets Our model incorporates perfect annuity markets via the term in the budget constraint (9) which reflects the survivor’s premium. We find that the elimination of annuities (and taxing all assets at death), again holding fixed the level of wealth, makes the consumption profile slightly less concave than our benchmark economy. The size of this effect, however, is quite small and bridges very little of the gap between the concavity in our theory and the linearity in data.
- Life-cycle shocks versus fixed effects
The model with social security provides an accurate account of consumption inequality over the life cycle. We now use it to asktwo normative questions:
- (^) What is the welfare cost of not being able to insure against life-cycle shocks?
- (^) How important are life-cycle shocks relative to fixed-effect shocks?
For the first question, consider an economy where agents receive a deterministic life- cycle profile of earnings, but still have heterogeneous fixed-effects. Let cl denote the
K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–633 627
percentage increase in per-period consumption (in the social security economy) that would make an individual indifferent between living in the social security economy and this alternative economy. 11 We find that cl ¼ 27 : 4 %: For the second question we consider two approaches. First, we compare cl to the corresponding welfare gain of removing the fixed effects. We find that ca ¼ 20 : 2 %; where ca is defined as the gain, under the veil of ignorance, of receiving a ¼ 0 with certainty. Thus, fixed-effects are somewhat less important than life-cycle shocks. Second, we follow the approach of Keane and Wolpin (1997). They estimate a dynamic model of occupational choice and decompose the variability in lifetime utility into a component which is realized early in life and a component which is realized along the life cycle. They conclude that
According to our estimates, unobserved endowment heterogeneity, as measured at age 16, accounts for 90 percent of the variance in lifetime utility. Alternatively, time-varying exogenous shocks to skills account for only 10 percent of the variation. Keane and Wolpin (1997, p. 515).
With this in mind, denote realized utility along some random path as wða; Z; EÞ:
wða; Z; EÞ ¼
XH
h¼ 1
bhfh Uðc (^) hÞ: ð 15 Þ
If w is evaluated at the optimum, then the conditional mean Eðw j aÞ is the value function V conditional on some realized fixed effect, a: We denote Va ¼ Eðw j aÞ: The total variance in realized utility can therefore be decomposed into variance attributable to fixed effects (i.e., variance in these conditional value functions) and the average variance, conditional on fixed effects:
VarðwÞ ¼ VarðE½w j aÞ þ EðVar½w j aÞ ð 16 Þ
¼ VarðVaÞ þ EðVar½w j aÞ: ð 17 Þ
Our interpretation of Keane and Wolpin (1997) is that VarðVaÞ=VarðwÞ is roughly 0.90. For our economy, after converting w into monetary equivalents (to be consistent with Keane and Wolpin, 1997) we find this ratio to be 0.47. Fixed effects, therefore, account for slightly less of the variation in lifetime utility than do the variation in the present value of lifetime earnings. This corroborates the previous result that life-cycle shocks are at least as important as fixed-effects. We attribute the difference relative to Keane and Wolpin (1997) to two factors. First, they assume, for computational reasons, that
(^11) We compute this equivalent variation welfare gain numerically as
cl ¼ 1 EV^1 ða;^ z;^ E;^0 Þ E V# 1 ða; 0 j no life-cycle riskÞ
1 =ð 1 gÞ ;
where V 1 ða; z; E; 0 Þ is the value function for a newborn agent (from Eq. (8)) and V# 1 is the value function of a newborn agent who does not face life-cycle shocks (i.e., sZ ¼ sE ¼ 0), given the prices of the social security economy.
628 K. Storesletten et al. / Journal of Monetary Economics 51 (2004) 609–
