NBER WORKING PAPER SERIESSOCIAL SECURITY AND INEQUALITY OVER THE LIFE CYCLEAngus DeatonPierre-Olivier GourinchasChristina PaxsonWorking Paper 7570 BUREAU OF ECONOMIC RESEARCH1050 Massachusetts AvenueCambridge, MA 02138February 2000Prepared for NBER Conference on the Distributional Effects of Social Security Reform, , October 22–23, 1999. Deaton and Paxson acknowledge support from the National Institute ofAging through a grant to NBER, and from the John D. and Catherine T MacArthur Foundation within theirnetwork on inequality and poverty in broader perspectives. We are grateful to Martin Feldstein, LaurenceKotlikoff, Jeffrey Liebman, and James Poterba for helpful comments and discussions. The views expressedherein are those of the authors and are not necessarily those of the National Bureau of Economic Research.© 2000 by Angus Deaton, Pierre-Olivier Gourinchas, and Christina Paxson. All rights reserved. Shortsections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that fullcredit, including © notice, is given to the source.
Social Security and Inequality over the Life CycleAngus Deaton, Pierre-Olivier Gourinchas, and Christina PaxsonNBER Working Paper No. 7570February 2000JEL No. H2, H5, E2, E6ABSTRACTThis paper examines the consequences of social security reform for the inequality ofconsumption across individuals. The idea is that inequality is at least in part the result of individualrisk in earnings or asset returns, the effects of which accumulate over time to increase inequalitywithin groups of people as they age. Institutions such as social security, that share risk acrossindividuals, will moderate the transmission of individual risk into inequality. We examine howdifferent social security systems, with different degrees of risk sharing, affect consumptioninequality. We do so within the framework of the permanent income hypothesis, and also usingricher models of consumption that incorporate precautionary saving motives and borrowingrestrictions. Our results indicate that systems in which there is less sharing of earnings risk–such assystems of individual accounts– produce higher consumption inequality both before and afterretirement. However, differences across individuals in the rate of return on assets (including socialsecurity assets held in individual accounts) produce only modest additional effects on DeatonPierre-Olivier Gourinchas221 Bendheim Hall311 Fisher HallPrinceton UniversityPrinceton UniversityPrinceton, NJ 08544Princeton, NJ 08544and NBERand NBERdeaton@@ Paxson219 Bendheim HallPrinceton UniversityPrinceton, NJ 08544and NBERcpaxson@
0. IntroductionThis paper explores the consequences of social security reform for the inequality of consumptionacross individuals. The basic idea is that (at least part of) inequality is the consequence ofindividual risk in earnings or asset returns. In each period, each person gets a different draw, ofearnings or of asset returns, so that whenever differences cumulate over time, the members ofany group will draw further apart from one another, and inequality will grow. Inequality at amoment of time is the fossilized record of the history of personal differences in risky institution that shares risk across individuals, the social security system being the case inpoint, will moderate the transmission of individual risk into inequality, and it is this process thatwe study in the paper. Note that we are not concerned here with what has been one of the centralissues in social security reform, the distribution between different generations over the , we are concerned with the equilibrium effects of different social security arrangementson inequality among members of any given concrete and readily analyzed example is when the economy is composed of autarkicpermanent income consumers, each of whom has an uncertain flow of earnings. Each agent sconsumption follows a martingale (. consumption today equals expected consumptiontomorrow), and is therefore the cumulated sum of martingale differences, so that if shocks toearnings are independent over agents, consumption inequality grows with time for any groupwith fixed membership. The same is true of asset and income inequality, though not necessarilyof earnings inequality, see Deaton and Paxson (1994), who also document the actual growth ofincome, earnings, and consumption inequality over the life-cycle in the United States andelsewhere. An insurance arrangement that taxes earnings and redistributes the proceeds equally,1
either in the present or the future, reduces the rate at which consumption inequality complete insurance, marginal utilities of different agents move in lockstep, andconsumption inequality remains constant. Social security pools risks, and thus limits the growthof life-cycle inequality. Reducing the share of income that is pooled through the social securitysystem, as envisaged by some reform proposals such as the establishment of individualaccounts with different portfolios or different management costs but not by others such assetting up a provident fund with a common portfolio and common management costs increasesthe rate at which consumption and income inequality evolve over life in a world of permanentincome consumers. Even if inequality is not inherited from one generation to the next, and eachgeneration starts afresh, partial privatization of social security will increase average much of the discussion about limiting portfolio choice in new social security arrangementshas (rightly) focused on limiting risk, such restrictions will also have effects on the reform is structured so that the poor are made no worse off, it can be argued thatthe increase in inequality is of no concern, see for example Feldstein (1998), so that our analysiswould be of purely academic interest. Nevertheless, the fact remains that many people perhapsmistaking inequality for poverty find inequality objectionable, so that it is as well to be awareof the fact if it is the case that an increase in inequality is likely to be an outcome of socialsecurity reform. There are also instrumental reasons for being concerned about inequality; thereare both theoretical and empirical studies implicating inequality in other socially undesirableoutcomes, such as low investment in public goods, lower economic growth, and even poorhealth, Wilkinson (1996). The paper is organized as follows. Section 1 works entirely within the framework of the2
permanent income hypothesis (PIH). We derive the formulas that govern the spread ofconsumption inequality, and show how inequality is modified by the introduction of a stylizedsocial security scheme. The baseline analysis and preliminary results come from Deaton andPaxson (1994), which should be consulted for more details, refinements, and reservations, aswell as for documentation that consumption inequality grows over the life-cycle, not only in theUS, but at much the same rate in Britain and Taiwan. The PIH is convenient because it permitsclosed-form solutions which show explicitly how social security is related to , it is not a very realistic model of actual consumption in the US, and it embodiesassumptions that are far from obviously appropriate for social security analysis, for example thatconsumers have unlimited access to credit, and that intertemporal transfers that leave the presentvalue of lifetime resources unchanged have no effect on consumption. In consequence, inSection 2, we consider richer models of consumption and saving that incorporate bothprecautionary motives for saving and borrowing restrictions. These models help replicate whatwe see in the data, which is consumers endogenously switching from buffer-stock behavior earlyin life to life-cycle saving behavior in middle age. The presence of the precautionary motive andthe borrowing constraints breaks the link between consumption and the present value of lifetimeresources, which both complicates and enriches the analysis of social security. Legal restrictionsprevent the use of social security as a collateral for loans, and for at least some people, suchrestrictions are likely to be binding. Solutions to models with precautionary motives and borrowing constraints are used todocument the how social security systems with differing degrees of risk-sharing affectinequality. We first consider the case in which all consumers receive the same rate of return on3
their assets. Our results indicate that systems in which there is less sharing of earningsrisk such as systems of individual accounts produce higher consumption inequality bothbefore and after retirement. An important related issue is whether differences across consumersin rates of return will contribute to even greater inequality. Somewhat surprisingly, we find thatallowing for fairly substantial differences in rates of return across consumers has only modestadditional effects on inequality. The bulk of saving, in the form of both social security and non-social security assets, is done late enough in life so that differences in rates of return do notcontribute much to consumption inequality. 1. Social security and inequality under the permanent income hypothesis Section introduces the notation and basic algebra of the permanent income hypothesis, whileSection reproduces from Deaton and Paxson (1994) the basic result on the spread ofconsumption and income inequality over the life-cycle. Both subsections are preliminary to themain analysis. Section introduces a simplified social security system in an infinite horizonmodel with PIH consumers and shows how a social security tax at rate reduces the rate of2spread of consumption inequality by the factor (1&). Sections discusses what happenswhen there is a maximum to the social security tax, and Section extends the model to dealwith finite lives and retirement and shows that the basic result is Preliminaries: notation and the permanent income hypothesisIt is useful to start with the algebra of the PIH; the notation is taken from Deaton (1992). Realearnings at time t are denoted y. Individual consumption is c and assets A;when it isttt4
necessary to do so we shall introduce an i suffix to denote individuals. There is a constant realrate of interest r. These magnitudes are linked by the accumulation identityA'(1%r)(A%y&c)(1)tt&1t&1t&1Under certainty equivalence, with rate of time preference equal to r, and an infinite horizon,consumption satisfies the PIH rule, and is equal to the return on the discounted present value ofearnings and assets:4rr1c'A%'E(y)(2)tttt%kk1%r1%rk'0(1%r)for expectation operator E conditional on information available at time t. It is convenient to starttwith the infinite horizon case; the finite horizon case is dealt with in Section consumption follows a martingale follows from manipulation of (2);4r1c'/'(E&E)y(3)tttt&1t%kk1%rk'0(1%r)d Disposable income y is defined as earnings plus income from capitaltdry'A%y(4)ttt1%rSaving is the difference between disposable income and consumptionds'y&c(5)tttwhich, enables us to rewrite the PIH rule (2) in the equivalent form, see Campbell (1987),41s'&'Ey(6)ttt%kkk'1(1%r)Assets are linked to saving through the identity (implied by (1) and (5))5
A'(1%r)s(7)tt&1Finally, it is convenient to specify a stochastic process for earnings y. It is convenient to do thistby assuming that(L)(y& )'(L)(8)ttfor lag operator L and polynomials (L) and (L) and white noise . As written, and under thetusual conditions on the roots, earnings is stationary (around ) and invertible. In fact, we canallow a unit root in (L) with essentially no modification. (In the more realistic models inSection 2, we will work with a process with a unit root but specified in logarithms.)Given (8), we can derive explicit forms for the innovation to consumption, Flavin (1981):1r1%rc''(9)ttt1%r11%rso that consumption is a random walk and the innovation variance of consumption is tied to theinnovation variance of earnings by the autocorrelation properties of the Spreading inequalityStart from the simplest illustrative case where earnings are white noise, and add an i suffix for anindividualy' %' %w%z(10)itiitititwhere is the individual-specific mean of earnings, w is a common (macro) component, and zititis an idiosyncratic component. The macro component w is also . over time. Given (10),tequation (3) implies6
rc'c%(w%z)(11)itit&1tit1%rAs a result, if the idiosyncratic components are orthogonal to lagged consumption in the cross-section (which need not be true in each year but is true on the average by the martingaleproperty), the cross-sectional variance of consumption satisfies2222rtrzzvar(c)'var(c)%'var(c)%(12)tt&1022(1%r)(1%r)so that consumption inequality is increasing over that although (12) is derived for the variance of consumption, the increase inconsumption variance is general, not specific to a particular measure of inequality. According to(11), the household distribution of consumption at t is the distribution of consumption at t&1plus uncorrelated white noise. Given that the mean is not changing, the addition of noise impliesthat the distribution of consumption at t is second-order stochastically dominated by thedistribution of consumption at t&1, so that any transfer-respecting measure of inequality, suchas the gini coefficient, the Theil inequality measure, or the coefficient of variation (but notnecessarily the variance in logarithms), will show an increase of inequality over the . case, saving is given by, see (6), its'(13)it1%rwhile assets satisfyA'A%.(14)itit&1it&1Because disposable income is the sum of consumption and saving, the change in disposable7
income satisfiesrdititit&1it&1y'%&'&(15)itit1%r1%r1%r1%rwhich implies, after some manipulation, that2222rtrzzdddvar(y)'var(y)%'var(y)%(16)tt&1022(1%r)(1%r)Because the consumption variance is spreading, and because saving is stationary by (13),disposable income variance must spread at the same rate as the consumption variance. Note thatearnings variance is constant given the stationarity assumption (10), so that22vary'%'constant(17) ztFrom (14), the variance of assets satisfies2var(A)'var(A)%t(18)zt0The rate of spread of the variance of assets is the variance of the idiosyncratic component of theinnovation of earnings. At a real interest rate of 5 percent, this is 400 times faster than the rate ofspread of the variance of consumption and of disposable income. From any given starting point,asset inequality among a group of individuals grows much faster than does consumption ordisposable income the US, the data on consumption, earnings, and income are consistent with the predictionsof the theory. Deaton and Paxson (1994) use repeated cross-sections from the ConsumerExpenditure Survey to trace birth cohorts through the successive surveys, and find that cross-sectional consumption inequality for any given birth cohort increases with the age of the example, the Gini coefficient for family consumption (family income) increases (on average8
over all cohorts) from () at age 25 to about () at age 55. We shall return tothese findings in Section 2 Social security and the spread of inequalitySuppose that the government enacts a simple social security system. A proportionate tax onearnings is levied at rate , and the revenues are divided equally and given to everyone. We thinkabout the (partial) reversal of this process as a stylized version of reform proposals that payssome part of each individual s social security tax into personal saving accounts; the precisemechanisms will be presented in Section . We recognize that the establishment of personalaccounts has other effects, some of which are not captured under our simple assumptions. Butour concern here is with the reduction in the pooling or risk sharing that is implied by removingpart of social security tax proceeds from the common pool and placing it in individual accounts provide smoothing benefits for autarkic agents who would not or cannot save ontheir own accounts, but they reduce the risk-sharing elements of the current of the infinite horizon and certainty equivalence assumptions, dividing up therevenues and returning them immediately is the same as giving them back later. The modelbassumes no deadweight loss. Denote before tax earnings as y and retain the notation y forititbafter tax income, (1&)y. In the . case we haveity'(1&)( %)% fl(19)itiitwhere the last term is the average revenue of the tax, which is given back to everyone. Equation(19) can also be writteny' &( & fl)%(1&)(20)itiiit9
Compared with the original earnings process (10), there is a shift toward the grand mean theredistributional effect of the social security system together with a scaling of the innovation by1&, which is the risk-sharing component of the social security system. The redistribution willchange consumption levels for everyone not at the mean, but will not affect the innovation ofconsumption equations (11), nor the saving equation (13), the asset equation (14), and thedisposable income equation (15), except that the original innovation must be rescaled by 1&.In consequence, the variances of consumption, disposable income, and assets all evolve as2before, but at rate that is (1&) times the original rate. If the social security tax is percent,inequality (measured by the variance) will spread at percent of the rate that it would in theabsence of the system. Imagine an economy in equilibrium, with no inheritance of inequality,and no growth in lifetime resources, so that the cross-section profile of consumption by age isidentical to the lifetime profile of consumption for each cohort, and all consumption inequality iswithin-cohort inequality. With a working life of 40 years, the imposition of a social security taxat percent will reduce the cross-sectional standard deviation of consumption by a factor (the square root of the 40 year average of to the power of t from 0 to 40.)In (19) and (20), we have not explicitly distinguished the macro common component of theinnovation w from the idiosyncratic component . If we substitute to make the decompositiontitexplicit, (20) becomesy' &( & fl)%(1&)z%w(21)itiiittwhich shows that the common component is not insured. The change in consumption warrantedby (21) isrc'w%(1&)z(22)ittit1%r10
but only the second term in the bracket contributes to the spread in consumption variance, andthe results are as stated Social security with a maximumThe permanent income hypothesis is not well-suited to modeling a social security system wheretaxes are paid only up to the social security maximum. The nonlinearity complicates theforecasting equations for earnings and eliminates the analytical tractability that is the mainattraction of the formulation. However, in the spirit of a system with a maximum it is worthnoting what happens when there are two classes of people, one whose earnings never gets abovethe social security maximum, and one whose earning never gets below the social securitymaximum. Equation (20) still gives after tax income for the poor group, and inequality amongthem spreads as in the previous section. For the rich group, after tax income isy'(1&)( %)%( %&m)%( fl%m)/2(23)itiitiit1where m is the social security maximum, fl is mean earnings of the poorer group, and we have1assumed that there are equal numbers in the two groups. (The first term is what is left if tax waspaid on everything, the second term is the rebate of tax above the maximum, and the last term isthe shared benefit.) Equation (23) can be rewritteny' &(m& fl)/2%(24)iti1itwhich makes the straightforward point that those above the maximum no longer participate inthe risk-sharing, only in the redistribution. As a result, the social security system with the twogroups will limit the rate of spread of inequality among the poorer group, but not among thericher group, though it will bring the two groups closer together than they would have been in11
the absence of the Finite lives with finitely lived consumers we can have a more realistic social security system, in which thetaxes are repaid in retirement rather than instantaneously. One point to note about retirement isthat it induces a fall in earnings at the time of retirement, a fall that enters into the determinationof saving, see (6). When there is a unit root in earnings, earnings immediately prior to retirementhas a unit root, and so does the drop in earnings at retirement. In consequence saving, which hasto cover this drop in earnings, is no longer stationary but integrated of order one, so that assets,which are cumulated saving, are integrated of order two. The spread of inequality in assets istherefore an order of integration faster than the spread of inequality in consumption anddisposable income. But this seems more a matter of degree than an essential work until age R and die at T. The consumption innovation formula is only slightlydifferentR&tr1c'''(E&E)y(25)ttttt&1t%kk1%rk'0(1%r)where the annuity factor is given byt1/1&'(1%r)&rtt&1(26)(T&t%1)(1%r)From (25), we can writet&1c'c%'(27)st0ts'012
Hence in the . case previously considered,t2r2&2var(c)'var(c)%'(28)zst02s'0(1%r)With the social security scheme, after tax earnings while working isy'(1&)( %)'(1&) %(1&)(w%z).(29)itiitititWith a uniform distribution of ages, the benefits while retired in year R%s areR( fl%w)R%s(30)T&RWith certainty equivalence, only the expected present value of this matters, which is a constantgiven the . assumption so that, once again, although the levels of consumption are altered,there is no change to the innovation of consumption, nor to the rate at which the variousinequalities results would clearly be different if either (a) the autocorrelation structure of the macrocomponent of earnings were such that current innovations had information about what willhappen in retirement; this seems like an issue that is hardly worth worrying about, or (b) withprecautionary motives or borrowing restrictions, where transactions that leave net present valueunaffected can have real effects on the level and profile of consumption. Without quadraticpreferences, and without the ability to borrow, we cannot even guarantee the basic result thatuncertainty in earnings causes consumption and income inequality to increase with age. Inconsequence, we have little choice but to specify a model and to simulate the effects ofalternative social security policies, and this is the topic of Section 2. Of course, it mightreasonably be argued that the purpose of social security is not well-captured within any of these13
models, and that present-value neutral forced savings has real effects, not because ofprecautionary motives or borrowing restrictions, but because people are myopic or otherwiseunable to make sensible retirement plans on their own. We are sympathetic to the generalargument, but have nothing to say about such a case; without a more explicit model of behavior,it is not possible to conclude anything about the effects of social security reform on . Social security with precautionary saving or borrowing Describing the social security systemWhen consumers cannot borrow, or when they have precautionary motives for saving, the timingof income affects their behavior. In consequence, we need to be more precise about thespecification of the social security system and its financing. We assume that there is a constantrate of social security tax on earnings during the working life, levied at rate , and that duringretirement, the system pays a two part benefit. The first part, G, is a guaranteed floor that is paidto everyone, irrespective of their earnings or contribution record. The second part, V, isiindividual-specific and depends on the present value of earnings (or contributions) over theworking life. We write S for the annual payment to individual i after retirement, so thatiR&1R&1bR&jR&jS'G%V'G% 'y(1%r)'G%'y(1%r)(31)ijiiijj'1j'1where ' /(1&). The size of the parameter determines the extent of the link betweenearnings in work and social security payments in retirement. When we consider the effects ofdifferent social security systems on inequality, we shall consider variations in and G whileholding the tax rate constant. As we shall see below, this is equivalent to devoting a larger orsmaller share of social security tax revenues to individual accounts. When is high relative to G14
(personal saving accounts,) the system is relatively autarkic and there is relatively little sharingof risk. Conversely, when G is large and small (the current system,) risk sharing is moreimportant and we expect inequality to be government finances the social security system in such a way as to balance the budget inpresent value terms within each cohort. If we use the date of retirement as the base fordiscounting, the present value of government revenues from the social security taxes levied onthe cohort about to retire is given byR&1NR&1bR&jR&j''y(1%r)''Y(1%r)(32)ijtj'1i'1j'1where N is the number of people and Y is aggregate before-tax earnings for the cohort in year must equal the present value at R of social security payments, which isNTR&1bR&jR&j(33)''(1%r)G% 'y(1%r).iji'1j'Rj'1The budget constraint that revenues equal outlays, that (32) equal (33), gives a relationshipbetween the three parameters of the social security system, , G, and , namely(yfl(G% yfl'.T(34)R&j'(1%r)j'R(where yfl is the average over all consumers of the present value of lifetime earnings, R&11(R&jyfl''Y(1%r).(35)jNj'1Equation (34) tells us that we can choose any two of the three parameters, G, (or ), and , andwhat is implied for the third. It also makes clear that, after appropriate scaling, and holding the15
guarantee fixed, increases in the earnings-related or autarkic part of the system areequivalent to increases in the rate of the social security tax, given that the government ismaintaining within-cohort budget link between earnings-related social security payments and individual accounts can beseen more clearly if we reparametrize the system. Suppose that V, the earnings relatedicomponent of the social security payment, is funded out of a fraction of social security taxes setaside for the purpose, or equivalently, that a fraction of the tax is used to build a personalaccount, the value of which is used to buy an annuity at retirement. Equating the present value ofthe each annuity V to the present value of contributions gives the relationship between and ,iT R&j''(1%r).(36)j'RHence, any increase in the earnings related component of social security through an increase in (or ) can be thought of as an increase in the fraction of social security taxes that is sequesteredinto personal accounts. Equation (34), which constrains the parameters of the social securitysystem, can be rewritten in terms of as(yfl(1&)G'.T(37)j&R'(1%r)j'RNote also that the individual social security payment (31) can be rewrittenR&1b(R&jS'(1&)yfl%'y(1%r)ijiT(38)j'1R&j'(1%r)j'Rso that each person s social security benefits are related to a weighted average of their own16
lifetime earnings and the average lifetime earnings of their entire the above scheme were implemented for permanent income consumers who are allowed toborrow and lend at will, the component of social security taxes that does goes into personalaccounts would have no effect on individual consumption nor therefore on its distribution acrossindividuals. Although the scheme forces people to save, it is fair in present value terms, and sohas no effect on the present value of each individuals lifetime resources. And although taxes arepaid now and benefits received later, such a transfer can be undone by appropriate borrowingand lending. If the social security tax rate is , and a fraction is invested in a personal account,it is as if the tax rate were reduced to (1&), and the rate of increase in the consumption andincome variance will be higher. Of course, none of these results hold if consumers are notallowed to borrow, or if preferences are other than Modeling consumption behaviorAlthough we shall also present results from the permanent income hypothesis, our preferredmodel is one with precautionary motives based on that in Gourinchas and Parker (1999) andLudvigson and Paxson (1998), with the addition of retirement and a simple social securitysystem. The specification and parameters are chosen to provide a reasonable approximation toactual behavior so that, even though it is not possible to derive closed-form solutions for theresults, we can use simulations to give us some idea of the effects of the have intertemporally additive isoelastic utility functions and, as before, theywork through years 1 through R&1, retiring in period R and dying in period T. The real interestrate is fixed, but the rate of time preference is (in general) different from r, so that consumers17
satisfy the familiar Euler equation&&c'(1%r)E(c)(39)tt%1t&1where is the inverse of the intertemporal elasticity of substitution and '(1%). After taxearnings, where taxes include social security taxes, evolves according to the (also fairlystandard) non-stationary processlny'lny%%&(40)tt&1tt&1which derives from a specification in which log earnings is the sum of a random walk with drift and white noise transitory earnings. The quantity is the parameter of the moving averageprocess for the change in earnings, and is an increasing function of the ratio of the variances ofthe transitory and random walk components respectively. Consumers are assumed not to be ableto borrow, which requires a modification of (39), see below. One reason is to mimic the US,where it is illegal to borrow against prospective social security income. A second reason is torule out the possibility that people borrow very large sums early in life to finance a decliningconsumption path over the life-cycle. This prohibition could be enforced in other ways, such asthe voluntary borrowing constraints in Carroll (1998) that result from isoelastic utility coupledwith a finite probability of zero earnings. We do not find Carroll s income process empiricallyplausible, and it seems simpler to rule out borrowing explicitly rather than to choose the form ofthe earnings process to do so. Our calculations for the permanent income case are done with andwithout borrowing constraints, which will give some idea of the effects of the borrowingconstraints in the other procedure is as follows. Given values for the real interest rate, the rate of timepreference, the intertemporal elasticity of substitution, the moving average parameter in income18
growth, and two out of three parameters of the social security system, we calculate a set ofpolicy functions for each year of a 40 year working life. After retirement, there is no furtheruncertainty and consumption can be solved analytically for each of the 20 years remaining. Weassume that the social security system presented in Section has been in place for a long time,that its parameters are fixed, and that people understand how it works, including thegovernment s intertemporal budget constraint. In particular, they understand the implications ofinnovations to their earnings for the value of their annuities in retirement. We do not requireconsumers to take into account the effects of successive macroeconomic shocks on the size ofthe social security guarantee G. Instead, we assume that the government sets G to the value thatsatisfies the budget constraint in expectation for each cohort, and that deficits and surpluses fromcumulated macro shocks are passed on to future generations. There are, however, no macroshocks in the simulations reported each period of the working life, the ratio of consumption to earnings can be written asfunction of three state variables. These are defined as follows. Define cash on hand x'A%ytttwhich, by (1), evolves during the working life t<R according tox'(1%r)(x&c)%y.(41)tt&1t&1tDuring retirement, for t$R,x'(1%r)(x&c)%S.(42)tt&1t&1If w is the ratio of cash on hand to earnings, and the ratio of consumption to earnings, thentt(34) becomes, for t<R,(1%r)(w&)t&1t&1w'%1(43)tgt19
where g is the ratio of current to lagged income, y/y. To derive corresponding equations forttt&1the dynamics of social security, define S as the current present value of the annual socialtsecurity payment to which the consumer would be entitled if he or she earned no more incomebetween year t and retirement. Hence, for t<R,t&(R&t)t&jS'G(1%r)%'y(1%r)(44)tjj'1while for t$R, S is constant and given by (31). Noting that earnings in year R is zero, (44)tsatisfies, for t#R,S'(1%r)S%Y(45)tt&1tand is constant thereafter. If we define , the ratio of S to current earnings and thus the socialttsecurity replacement rate so that, the corresponding evolution equation isStt&1''(1%r)%.(46)tygttWith borrowing constraints, which imply that consumption cannot be greater than cash onhand, or that the consumption ratio be no larger than the cash on hand ratio, the Euler equation(39) is modified to&&&&'max(1%r)E(g),w.(47)tt%1t%1ttWe write the consumption ratio as a function of the cash on hand ratio w, the social securityttreplacement rate , and the current innovation to earnings (which is required because, withttpositive , high earnings growth in one period predicts low earnings growth in the next) and thenuse (47) to solve backwards for the policy function in each period, starting from the closed-formsolution for consumption in the first year of
Armed with the policy functions, we simulate lifetime stochastic earnings profiles for each of1,000 people. The logarithm of initial earnings is drawn from a normal distribution with meanln(20,000) and a standard deviation of , the latter chosen to give an initial gini coefficientthat roughly corresponds to what we see in the data from the CPS. The drift (expected rate ofgrowth) of earnings is set at 2 percent a year. For any given value of the replacement parameter and the social security tax rate , the corresponding value of the social security guarantee G is setfrom (34) using actual realized earnings, which as we have already noted, is potentiallyproblematic if macro shocks are important. The value of G also gives the initial value of at thetbeginning of life. The calculated policy functions are then used to simulate life-cycleconsumption for each of the 1,000 people, and these trajectories are used to assess lifetimeinequality as a function of the design of the social security system. Different simulations use thesame 1,000 sets of earnings realizations, so that comparisons across social security regimesreflect the regime parameters and not the specific Social security design and inequality: results with constant interest ratesThe model is solved under the following assumptions. The interest rate r is set at 3 percent, andthe rate of time preference at either 3 or 5 percent. The drift of the earnings process is set at 2percent a year, the moving average parameter to , and the standard deviation of theinnovation (in logs) to be . The coefficient of relative risk aversion is set to 3, so that theintertemporal elasticity of substitution is one-third. We also include a certainty equivalent case,with and without borrowing restrictions, in which the rate of interest is set equal to the rate oftime preference at 3 percent. There are four cases carried through the analysis: (1) isoelastic21
preferences, no borrowing, r', '; (2) isoelastic preferences, no borrowing,r', '; (3) quadratic preferences, no borrowing, r', '; (4), quadraticpreferences, borrowing allowed, r', '. The social security tax rate is set at itscurrent value of percent of before-tax earnings and there are no other taxes or benefits. Thesocial security systems we consider are indexed on the level of the social security guarantee G,which takes the values (0, $5,000, $10,000, $15,000, $20,000); given the tax rate, these valuestranslate into corresponding values for , or perhaps more revealingly, into values for , theshare of the tax devoted to personal accounts (1, , , , ). These different setsof parameters have quite different implications for the dispersion in social security paymentsamong retirees. For example, our simulation results indicate that with a guarantee of 0, thethperson at the 10 percentile (ranked by the present value of lifetime earnings) receives an annualthsocial security payment of $6,405, in contrast to a payment of $52,639 for the person at the 90percentile. When the guarantee is increased to $20,000, this spread declines to $21,569 for theth10 percentile, and $32,896 for the 1 shows the averages over the 1,000 consumers of the simulated trajectories ofincome (earnings prior to retirement and receipts from social security after retirement),consumption, and cash on hand (earnings plus assets excluding social security assets) for thefour models all with G set at $5,000. These graphs are shown to demonstrate that the variousmodels do indeed generate standard life-cycle profiles. Earnings are the same in each of the threegraphs. Consumption is flat over the life-cycle in the certainty equivalent case when borrowingis allowed, but rises in the models with precautionary motives and borrowing constraints, and inthe quadratic case with borrowing constraints. Indeed the quadratic case with no borrowing (on22
the bottom left) and the isoelastic impatient case with no borrowing (top left) generate similarprofiles. With more patient (lower ) consumers in the top right panel, there is moreaccumulation during the working life, and assets prior to retirement are higher. The certaintyequivalent consumers in the bottom right panel have expectations of earnings growth and soengage in substantial borrowing early in life but even so, have some net assets prior 2 shows the average consumption profiles for the four different models (in the fourpanels, as before) and for the five different social security schemes (in each panel). To a firstapproximation, and with the tax rate held fixed, the choice of system has no effect on the lifetimeprofile of consumption. Figure 2 also shows more clearly than Figure 1 the lifetime shape ofconsumption in the four models; precautionary motives or borrowing restrictions drive theincrease in consumption over the working period; in the top left panel, where impatience isgreater than the interest rate, consumption declines after retirement once all uncertainty isresolved. For the cases with precautionary motives and/or borrowing restrictions, averageconsumption during retirement is somewhat higher in the regimes with the higher minimumguarantee. This appears to be a consequence of the borrowing constraints. Those consumers whohave poor earnings draws throughout their lives, and who would like to borrow against theirsocial security but cannot, have higher consumption in retirement when the guarantee becomesavailable. In effect, such consumers are being forced to save for higher consumption inretirement than they would choose if left to themselves. Such effects are absent in the purecertainty equivalent case where borrowing is 3 plots the gini coefficients of consumption by age and shows how consumption23
inequality evolves in the various models and for the different social security systems. The ginicoefficients, together with interquartile ranges of the logarithm of consumption, are given innumerical form in Table 1. In all of the models, consumption inequality is higher at all ages thelower the social security guarantee (the higher the fraction of taxes invested in personalaccounts) and the more autarkic the system. A higher guarantee with its associated lower limit tolifetime earnings causes consumption inequality to be lower from the start of the life-cycle,though the early effects are strongest in the pure certainty equivalence case, and manifestthemselves only later in life in the models with borrowing constraints. With a higher guarantee,and less in individual accounts, the system has more sharing, so that individual earningsinnovations have less effect on consumption because the good (or ill) fortune will be shared withothers. Although this sharing is implemented only after retirement, because consumption issmoothed over the life-cycle, the effect on inequality works at all ages to an extent determinedby the assumptions about preferences, growth, and borrowing constraints. When borrowingconstraints are imposed in an environment with earnings growth, consumption smoothing isinhibited, and the effects of risk sharing on inequality are more apparent in the later than in theearlier phases of the life-cycle. These results are not sensitive to the choice of inequalitymeasure. The interquartile ranges, although somewhat jumpier, display patterns that are similarto the gini 3 also shows a sharp drop in consumption inequality after retirement, particularlywhen the guarantee in the social security system is relatively large. Once again, this comes fromthe borrowing constraints and the inability of life-time unlucky consumers to borrow against thesocial security system. These people have very low consumption immediately prior to24
retirement, which exaggerates inequality. The effect vanishes as social security becomesavailable and their consumption rises. In the cases where the guarantee is large, there is alsosome decline in inequality prior to retirement. While there is no theoretical reason that prohibitsthis, we have not so far developed a convincing explanation of why it should occur. Panel 1 ofFigure 3 also shows a small decline in consumption inequality during retirement. This is due tothe combination of borrowing constraints and impatience (r<). Unconstrained consumerschoose declining consumption paths during retirement (at a constant and common rate of percent per year), while those who are constrained simply consume their constant socialsecurity income. The result is a compression of the distribution of , the results in Figure 3 and Table 1 show that as we move from one extreme to theother, from putting everything into individual accounts and giving no guarantee (a social securitysystem than confines itself to compulsory saving) to a guaranteed floor of $20,000 with only aquarter of social security taxes going to personal accounts, the gini coefficient of consumptionincrease by between 5 and 6 percentage points on average over the life-cycle, less among theyoung, and more among the old. This is a large increase, exceeding the increase in consumptioninequality in the US during the inequality boom from the early to the mid-1980s. For example,the gini coefficient of total consumption for urban households from the . ConsumerExpenditure rose from in 1981 to in 2 shows poverty rates by age for the different models and social security individual is defined to be in poverty if annual consumption is less than $10,000. Thispoverty threshold was arbitrarily chosen, but it delivers total poverty rates are not too differentfrom those in the United States. For example, with G equal to $5,000, the total poverty rate is25
% for the first model. We are more concerned with how poverty varies with age than with itslevel. The age profiles of poverty are similar for the first three models, in which there areborrowing constraints. Poverty rates decline up to retirement age: constrained consumers aremore likely to be poor when they are young, and earnings are low. Poverty in retirement dependson the value of the social security guarantee. When the guarantee is greater than or equal to thepoverty threshold, poverty in retirement must equal zero. For smaller values of the guarantee, thepoverty rate in retirement is generally less than during working years. However, in onecase that of isoelastic preferences and r< the poverty rate grows during retirement. In thiscase, impatient consumers reduce consumption over time, and increasingly fall below thethreshold. The fourth model, with quadratic preferences and no borrowing constraints, yields verydifferent results. Poverty rates increase with age up to retirement. Average consumption isconstant over the life-cycle, and the increasing dispersion in consumption with age implies thatconsumers will increasingly fall below the threshold. Increases in the poverty rate cease atretirement. However, social security guarantees in excess of the poverty threshold do noteliminate poverty, since (in this model) individuals are free to borrow against the guaranteeduring working years. Higher social security guarantees do, in fact, reduce poverty, but they doso at all ages, by making life-time wealth more equal across 4 compares our simulated patterns of inequality over the life-cycle with thosecalculated from the data in the Consumer Expenditure Survey and reported in Deaton andPaxson (1994). By construction, the life-cycle profile of simulated earnings inequality is similarto the actual profile. Simulated consumption inequality (from the impatient isoelastic case) is26
too high relative to the actuals; perhaps the borrowing restrictions are preventing consumptionfrom being sufficiently smoothed. Nevertheless, the upward drift of consumption inequality isvery much the same in the data as in the simulations, which also show the effects on inequalityof the different social security Figures 5 and 6 we turn to the life-cycle pattern of inequality in assets, in Figure 5 forassets excluding social security wealth, and in Figure 6 including social security wealth. Thepermanent income model is excluded from these comparisons since average wealth is negativefor much of the life-cycle. Total wealth at any given age is defined as the sum of non social-security assets A and the present discounted value at t of receiving G from retirement R to deathtT, plus the accumulated balance in the personal saving account, if any. Making the socialsecurity system more autarkic by holding the social security tax constant and devoting more ofthe revenue to personal accounts and less to a universal guarantee has the opposite effect oninequality of non-social security wealth than it does on the inequality of consumption. This isbecause of the substitutability between saving for retirement inside and outside the socialsecurity system. If we examine the profiles of asset accumulation by age (not shown here)average non-social security accumulations are larger the smaller is the fraction of the socialsecurity tax invested in individual accounts. There is a similar substitutability in asset inequality;when there is a large social security floor for everyone, the resulting equality is partially offsetby inequality in private different patterns of asset inequality for the quadratic case in the bottom left panel, asopposed to the isoelastic cases in the top panels, are associated with the fact that a substantialfraction of the quadratic consumers are credit constrained up until around age 40, so that27
inequality is high at early ages, because so many consumers have exactly nothing. The offsettingof private wealth against social security wealth only shows up once the majority of consumersare accumulating private assets, at which point they are no longer credit constrained. In the caseswith isoelastic preferences, the borrowing constraints are binding for only a small fraction ofyoung consumers; the variability of earnings and the convexity of marginal utility is enough toovercome impatience and the expected growth of we come Figure 6, which shows the inequality of all assets, we see the standard pattern restored; the more autarkic the system and the larger the fraction of social security taxesdevoted to private accounts, the larger is the inequality of assets. Note that the gini coefficientsfor all assets are much lower than those for private assets; even with personal accounts, theaddition of social security to private wealth makes the distribution of wealth much more with consumption, asset inequality rises with age, but does so most rapidly in the cases whereinsurance is greatest, so that the differences in asset inequalities across the various schemesdiminishes with age. Even so, the most autarkic systems are the most unequal at all Social security design and inequality: results with variable interest ratesThe results on asset inequality, and to a lesser extent those on consumption inequality, are likelyto be seriously affected by our assumption that everyone earns the same rate of return on theirassets. Under some of the early proposals for reform, for example those from the largest group inthe Gramlich report, one of the great virtues of personal accounts was seen as the freedom givento individual consumers to choose their own portfolios. More recent proposals have tended tofavor severe restrictions on portfolio choice, perhaps restricting consumers to a limited menu ofapproved funds which themselves must adhere to strict portfolio rules. Clearly, allowing28
different people to obtain different returns adds a new source of inequality, in both assets and inconsumption. If, for example, the funds for the minimum guarantee G were invested in acommon fund at rate r, as above, but the personal accounts obtained different rates of return fordifferent individuals, either because of their individual portfolio choice or because of differentialmanagement fees, then a move to personal accounts can be expected to increase inequality bymore than in the analysis so is not obvious how to construct a model with differential asset returns that is both realisticand computationally tractable. We have so far considered only one simple case. Personalaccounts are invested in one of eleven mutual funds, and consumers must choose between themat the outset of their working life. The eleven mutual funds have rates of return from to a year. One can think of the funds as having identical (S&P 500) portfolios, butmanagement fees range from zero to 1 percentage point; the equilibrium is maintained bydifferential advertising and reporting services. We allocate our 1,000 consumers randomly to theeleven mutual funds, with equal probability of receiving any one interest rate; this is aconservative procedure and inequality would presumably be higher if those with higher earningswere more financially sophisticated and systematically chose the no load funds. We assume thatconsumers are forced to convert their retirement accounts into annuities at retirement (using theinterest rate to which they have been assigned), and also that the social security system giveseach consumer a guaranteed amount of $5,000 per year after retirement in addition to results indicate that there is virtually no increase in consumption inequality beforeretirement, and very little after retirement, associated with assigning different consumers to29
different fixed interest rates. The top panel of Figure 7 shows the gini coefficient forconsumption for the cases described above, with dispersion in interest rates, and the case inwhich all consumers receive the same interest rate of percent. This result may not be notsurprising, considering that most saving (whether private or through the social security system)is done late in life, when income is high, so that those that receive lower interest rates do nothave wealth at retirement that is much lower than those with higher interest rates. Consider, forexample, a group of 1,000 consumers whose incomes follow the process described above, eachof whom pays percent in taxes, 81 percent of which is allocated to private social securityaccounts (thereby generating enough government revenue to fund a $5,000 guaranteed paymentto each during retirement.) If each member of the group receives an interest rate of percent,the average private social security account balance upon retirement will equal $278,597. Thisnumber will be percent higher, or $341,014, with an interest rate of percent. Thepercentage difference in total retirement wealth, including the equalizing guarantee of $5,000 peryear, is even smaller, and the difference in average consumption in the first year of retirement forthe two interest rates is less than $5,000. This difference is for a spread of a full 1 percentagepoint; in the exercise conducted above, most consumers have interest rates between theextremes, so there is even less of an effect on overall with a much wider spread of returns, there is only a modest effect on inequality. Thebottom panel of Figure 7 shows the case where consumers are distributed over (fixed) rates ofreturn from 1 percent to 7 percent, compared with the case when all get 3 percent. This can bethought of as the case where consumers make a choice between equities and bonds at thebeginning of their working careers, and may never change thereafter. Because the spread is30
wider, there is more inequality than before, but the effects are modest compared with the otherissues examined in this is important to note that assigning consumers to different but fixed rates of interest will notnecessarily have the same affects as allowing the interest rate to vary randomly over time forindividual consumers. In future work, we plan to examine how interest rate risk, as opposed tointerest rate dispersion, affects . List of works cited: Campbell, John Y., 1987, Does saving anticipate declining labor income? An alternative test ofthe permanent income hypothesis Econometrica, 55, 1249 , Christopher, 1997, Buffer-stock saving and the life-cycle permanent incomehypothesis, Quarterly Journal of Economics, 112, 1 55Deaton, Angus, 1992, Understanding consumption, Oxford. Clarendon , Angus and Christina Paxson, 1994, Intertemporal choice and inequality, Journal ofPolitical Economy, 102, 437 , Martin S., 1998, Is income inequality really a problem? in Income inequality: issuesand policy options: a symposium sponsored by the Federal Reserve Bank of Kansas City,357 , Marjorie, 1981, The adjustment of consumption to changing expectations about futureincome, Journal of Political Economy, 89, 974 , Pierre-Olivier, and Jonathan A. Parker, 1999, Consumption over the life-cycle, NBER Working Paper , Cambridge, MA. (Jul.)31
Ludvigson, Sydney and Christina H. Paxson, 1998, Approximation bias in linearized Eulerequations, NBER Technical Paper 236, Cambridge, MA. (Mar.)Wilkinson, Richard, 1996, Unhealthy societies: the afflictions of inequality, London,
Table 1a Gini coefficients for consumption and interquartile ranges for logarithm ofconsumption, with different social security plans. Isoelastic preferencesG=0G=$5000G=$10,000G=$15,000$G=20,000 Age giniiqrginiiqrginiiqrginiiqrginiiqrIsoelastic preferences, r=.03, =.05, borrowing preferences, r=.03, =.03, borrowing : Gini refers to the gini coefficient for consumption. Iqr is the interquartile range of the logarithm
Table 1b Gini coefficients for consumption, with different social security plansQuadratic preferencesG=0G=$5,000G=$10,000G=$15,000G=$20,000 Age giniiqrginiiqrginiiqrginiiqrginiiqrQuadratic preferences, r=.03, =.03, borrowing : Quadratic Preferences, r=.03, =.03, no borrowing
Table 2 Poverty rates (fraction of age group with consumption less than $10,000) withdifferent social security plans. G:$0$5,000$10,000$15,000$20,000$0$5,000$10,000$15,000$20,000ageIsoelastic preferences, r=.03,=.05Isoelastic preferences, r=.03, =. preferences, r=.03, =.03Quadratic preferences, r=.03, =.03, no borrowing
(1) isoelastic, δ=, r=(2) isoelastic, δ=, r=-200(3)quadratic, δ=, r=(4)quadratic, δ=, r= allowed2000-2002040608020406080ageFigure 1: Age profiles of consumption, earnings (inclusive of transfers), and assets for different specifications, G=$5,00036consumption, earnings, and assets in $,000
(1) isoelastic, δ=, r=(2) isoelastic, δ=, r=(4) quadratic, δ=, r=(3)quadratic, δ=, r= allowed50040030020002040600204060ageFigure 2: Consumption profiles under different specifications and alternative social security rules37consumption, $,000
(2) isoelastic, δ=, r=(1) isoelastic, δ=, r=(3)quadratic, δ=, r=(4) quadratic, δ=, r= 3: Consumption inequality for different specifications and social security systemsgini coefficient of consumption
SimulationsUS with alternativeconsumptionsocial security 4: Actual and simulated inequality of earnings and consumption39gini coefficient
(1) isoelastic, δ=, r=(2) isoelastic, δ=, r==$20,=$5,000G=$(3)quadratic, δ=, r==$15,000G=$10, 5: Gini coefficients for assets excluding social security assets40gini coefficient
(1) isoelastic, δ=, r=(2) isoelastic, δ=, r==$=$5,=$20,0000010203040(3)quadratic, δ=, r==$10,=$15,0000010203040age2030405060Figure 6: Gini coefficients for total assets, including social security assets41gini coefficient
with r from to with r=3 with r from 1 to 7 with r=4 7: The effects on consumption inequality of a distribution of interest rates42gini coefficient for consumption
