how does a pay-as-you-go system affect asset returns and the equity premium?

Review of Economic Dynamics 17 (2014) 131–149

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Review of Economic Dynamics

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How does a pay-as-you-go system affect asset returns and theequity premium?

Conny Olovsson 1

Institute for International Economic Studies, Stockholm University, SE-106 91 Stockholm, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 May 2011Received in revised form 18 February 2013Available online 5 March 2013

JEL classification:G12H55

Keywords:Social securityAsset pricesThe equity premium puzzle

When applying a differences-in-differences approach, equity returns and the equitypremium are both estimated to be more than four percentage points higher after theintroduction of a pay-as-you-go (PAYGO) system. In a realistically calibrated model, thePAYGO system is also found to increase the returns and the premium, although the effectsare smaller than in the data. Intuitively, the system lowers asset prices, which in turnincreases the importance of dividend risk. Since only equity is subject to dividend risk,equity returns become more volatile relative to bond returns.

© 2013 Elsevier Inc. All rights reserved.

Nearly three decades ago, Mehra and Prescott (1985) showed that the equity premium, i.e., the risk premium on stocksversus government bonds, generated in a representative consumer framework is at most 0.35 percent. This modest premiumis in stark contrast to the historical value of 6 percent for the U.S. economy. Since then, their result has given rise to a largeliterature that still is trying to solve the equity premium puzzle. So far, however, the vast majority of these studies havebeen based on U.S. data and the U.S. experience.2 One question that has not yet received much attention is whether anyspecific institutional factors might be important determinants of the equity premium. This paper addresses this issue byanalyzing the effects of a pay-as-you-go (PAYGO) social security system on asset returns and the equity premium.

The focus on social security is motivated by theoretical and empirical evidence that PAYGO systems tend to crowd outprivate savings.3 In addition, as shown by Constantinides et al. (2002), bond and equity demands vary with age. A redistri-butional system that transfers income and aggregate risk between generations will then influence the age-specific demandfor bonds and equity.4 These demand effects can be expected to have an impact on both absolute and relative prices andreturns.

The paper begins with an empirical analysis of the effects of a PAYGO system on asset returns and the equity premiumbased on a panel that includes asset returns for 16 countries over the period 1900–2000. Due to a lack of sufficiently long

E-mail address: [email protected] I am grateful for comments from two anonymous referees, Kjetil Storesletten, Per Krusell and Masayuki Kudamatsu. All remaining errors are, of course,

my own. Financial support from Mistra-SWECIA is greatly acknowledged.2 Some exceptions are Campbell (1998) and Dimson et al. (2000, 2002).3 In the life-cycle model, private savings will depend on the amount of available resources (such as pensions) after retirement. See, for example, Feldstein

(1974, 1996), Kotlikoff (1979), Attanasio and Rohwedder (2001) and Attanasio and Brugiavini (2003).4 For example, if an economy is subject to aggregate risk and social security benefits are completely safe, taxpayers are exposed to a higher level of

income risk under a PAYGO system, since taxes must then be contracyclical. Retirees, on the other hand, will instead collect a relatively high risk-freeincome under the system, so that it effectively transfers risk from the old to the young. See also Bohn (2009) and Olovsson (2010).

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132 C. Olovsson / Review of Economic Dynamics 17 (2014) 131–149

and detailed data series on the size of the social security systems, the PAYGO system is treated as a binomial variable,taking either the value of zero (no system) or one (a system). By applying a differences-in-differences approach with fixedtime and country effects, I find both the equity premium and the average return to equity to be more than four percentagepoints higher in economies with a PAYGO system. The point estimate for average bond returns is just above zero, but thisresult is insignificant.

The correlations found in the empirical analysis are then examined by setting up an overlapping-generations model toquantify the effects of a PAYGO system on asset returns and the equity premium. Agents are assumed to live for threeperiods as young, middle-aged and old. They can invest in stocks and a consol bond (both assets are infinitely lived). Severaldifferent social security arrangements, that differ in the way they allocate risk between generations, are compared to thecase with no system. The PAYGO system is found to increase asset returns and the equity premium. Specifically, the averagereturn to equity in the model is about three percentage points higher and the equity premium is about one percentagepoint higher under this system. Since equity is more volatile in the data than in the model, a smaller effect on the equitypremium is expected. Hence, the model goes part of the way in accounting for the data. I also find that the exact design ofthe social security system is of minor quantitative importance for both the size of the premium and asset returns. Marketincompleteness is important for all of the results, however. When markets are close to complete, agents can differentiateaway age-specific risk and unless the redistributional system makes markets less complete, the system has only a marginaleffect on the premium.

The higher equity premium is due to the fact that agents have a lower marginal willingness to pay for private assetswhen they are forced to save in an unfunded system. For given coupon payments and dividends, asset prices thus fall. Thisreduction in prices has asymmetric effects on the volatilities of bond and equity returns. In fact, even an equal reduction inboth prices has asymmetric effects on the volatilities of the returns. This is because the only risk associated with holding abond between two periods is the price risk (since coupon payments are constant), whereas equity is also subject to dividendrisk, in addition to the price risk. A lower bond price therefore implies that a larger share of the total payoff of the bond isconstant, but a lower equity price instead implies that a larger share of the total payoff of equity is due to risky dividends.Lower average asset prices thus result in not only higher average returns, but also that equity returns become more volatilerelative to bond returns. Hence, equity becomes more risky relative to a bond and agents require a higher risk premium forholding it.

Another reason for the higher premium is that retirement benefits provide relatively safe income. The middle-agedrespond to a less uncertain future by holding a relatively more equity-based portfolio, thereby implying that their old-ageconsumption is less correlated with bond returns. As a result, the risk premium that the middle-aged require in order tohold bonds is reduced, which also increases the equity premium.

To the best of my knowledge, this is the first paper to estimate empirically the effects of the PAYGO system on theequity premium, as well as to quantify the full effect on returns and the premium of a social security system.5 Thus, it canhopefully provide a better understanding of saving behavior, asset prices and the equity premium, all of which are issuesof first-order importance in economics. The paper is organized as follows. The empirical analysis is outlined in Section 1.In Section 2, the life-cycle model is set up to study the effects of the PAYGO system on the equity premium. The modelresults are presented in Section 3. Section 4 provides intuition for why the equity premium is higher with a PAYGO system.Section 5 contains a discussion of potentially important issues that are excluded from the analysis. Section 6 concludes.

1. An empirical analysis

A panel data set on asset prices for 16 countries over the period 1900–2000 is used to estimate the effects of a PAYGOsystem on asset prices and the equity premium. The data is taken from Dimson et al. (2000, 2002). Asset returns and theequity premium are denoted as averages over the decades (1900–1910, 1910–1920, etc.).6

There are several potential concerns regarding the estimation procedure. First, for most countries, the PAYGO system hasbeen introduced and expanded only gradually. However, owing to a lack of sufficiently long and detailed data series on thesize of the social security systems, here the PAYGO system is instead treated as a binomial variable. For each country, thismethod requires a judgment on the year in which the system can be considered as having been introduced. Admittedly, thisdecision is somewhat arbitrary. However, in many countries there has in fact been one major reform that generalized socialsecurity for the entire active population and/or significantly increased the benefits. A country is defined as having a PAYGOsystem if it covers a majority of the population and benefits are not subject to a means test.

Second, the regressions rely on the assumption that the introduction of a PAYGO system is not anticipated in advance.The assumption might not be too unreasonable as long as the system was not anticipated “too long” in advance. Since the

5 Krueger and Kubler (2006) analyze the welfare effects of a marginal increase in the social security tax rate in an economy where this tax rate isinitially zero and report the effects of this reform on asset returns of this reform. Storesletten et al. (2007) and Constantinides et al. (2002) analyze theequity premium in a life-cycle setting, but abstract from social security. My paper is also related to Abel (2003) who analyzes the price of capital in thepresence of social security.

6 Ideally, the anticipated equity premium should be used. However, since I do not have such data, the realized equity premium is used instead.

C. Olovsson / Review of Economic Dynamics 17 (2014) 131–149 133

Table 1Panel regression results.

Prm bonds Prm bills Equity returns Bond returns Bills returns

PAYGO 4.70∗∗ 5.7∗∗∗ 4.9∗∗ 0.07 −0.8(2.3) (1.9) (2.2) (2.2) (0.8)

p-value 0.060 0.048 0.100 0.984 0.254

Note: Robust standard errors from the standard regressions in parentheses; ∗ significant at 10%; ∗∗ significant at 5%; ∗∗∗ significant at 1%. The stars denotethe significance levels in the standard regressions whereas the p-values are from the bootstrap estimations. Fixed country and time effects are included inthe regressions but not displayed in the table.

equity premium is averaged over decades, anticipation up to a year in advance is unlikely to affect the decade average. Ifthe system is anticipated several years in advance, the results will be biased, but the direction of the bias is unclear.7

Third, the analysis abstracts from transitional dynamics. In particular, it might take some time before the introductionof a PAYGO system fully affects the equity premium. Since the system can be expected to crowd out private savings, anunanticipated system will imply that individuals unexpectedly have more savings than they desire. Agents can then consumetheir excessive savings in the current period (which would have no effect on future asset returns and premia). Alternatively,they may spread out excessive savings over their remaining life cycle. Part of the surplus is then saved, which wouldincrease aggregate savings. Therefore, if transitional dynamics are regarded as important, the estimates for average returnsin Section 1.1 are biased upwards.8 Since the amount of excessive savings that agents would hold decreases with age,abstracting from transitional dynamics might not be too harmful.

Fourth, since the estimation procedure treats the countries as isolated from each other, a potential concern is cross-sectional dependence in terms of return data and/or in the timing of a introduction of the PAYGO system. Such dependencemay imply inefficient estimators. For this reason, I test for cross-sectional dependence in Section 1.3 and find that thehypothesis of cross-sectional independence cannot be rejected at any reasonable significance level in any of the regressions.Hence, cross-sectional dependence should not constitute a direct problem in the estimations. Appendix A.1 contains a shortdescription of social security expansion in each of the 16 countries (Australia, Belgium, Canada, Denmark, France, Germany,Ireland, Italy, Japan, the Netherlands, South Africa, Spain, Sweden, Switzerland, the U.K. and the U.S.).

1.1. The econometric model

The econometric model is

yit = λ + λi + λt + β ∗ PAYGOit + uit,

where PAYGOit is the independent variable, λ is the intercept, λi denotes country-fixed effects and λt denotes time (decade)fixed effects. The dependent variables of interest, yit , are the returns to equity Re , bonds Rb and the equity premium,Re − Rb . Since the data include information on bills, they are also included in the regressions. Equity returns incorporatereinvested income. The definition of a bond differs slightly between countries, but in many cases, it refers to a 3–4 percentcoupon bond.9 If social security was introduced in the middle of a decade, PAYGOit takes the value equal to the share of thenumber of years with a PAYGO system within that interval.

Since PAYGOit is highly positively autocorrelated—it takes the value of zero for all periods before the treatment and 1for all periods after—the standard errors are clustered at the country level (rather than on country/year). The default OLSstandard errors that ignore such clustering can significantly underestimate the true OLS standard errors.10 The method ofclustering standard errors generally works well when the number of clusters is large (roughly above 30). Here, the numberof clusters is only 16, thereby implying that the cluster-robust standard errors are likely to be biased downwards. Onesolution to this problem is bootstrapping, as suggested by Cameron et al. (2008). Basically, bootstrap methods generate anumber of pseudo samples from the original sample. The statistic of interest is calculated for each pseudo sample and thedistribution of this statistic is used to infer the distribution of the original statistic. Specifically, Cameron, Gelbach and Millershow that the wild cluster bootstrap-t method performs excellently even with as few as six clusters. In addition, there is nonoticeable loss of power after accounting for size. Hence, this method is applied here.11

The results are displayed in Table 1. The equity premium is more than four percentage points higher in economies withPAYGO systems than in economies without. This result is robust at the ten-percent level in the bootstrap estimation.12

Similarly, the premium of equity over bills is even higher and this coefficient is significant at the five-percent level. The

7 One the one hand, individuals value savings less because they are entitled to retirement benefits later, which lowers asset prices. On the other hand,anticipation of high future returns may drive up current asset prices.

8 The direction of the bias for the premium is unclear, however.9 For details about the data, see Dimson et al. (2000, 2002).

10 See Moulton (1990) and Bertrand et al. (2004).11 The wild cluster bootstrap-t method is described in detail in Cameron et al. (2008) who also provide an algorithm for its implementation.12 Note that the significance levels (given by the p-values) are lower in the bootstrap estimations than in the standard regressions (where the significance

levels are given by the stars).


Table 2Regression coefficients and p-values with different controls.

I II III IV V VIPrm bonds Prm bonds Prm bonds Prm bonds Prm bonds Prm bonds

PAYGO 0.12 0.41 1.94 4.70 4.81 5.07p-value 0.909 0.735 0.344 0.06 0.06 0.02

Note: Specifications: I—no fixed effects, II—only fixed country effects, III—only fixed time effects, IV—both fixed time and country effects, V—same as IV plusthat the average growth rate of GDP per capita (per decade) is included as a control, and VI—same as IV plus that the level of GDP per capita is includedas a control. The data on GDP per capita are from Maddison. Data are missing for Ireland 1900–1920 and for South Africa 1900–1949.

Fig. 1. The difference in the equity premium before and after the introduction of a PAYGO system for individual countries. The solid line shows the meandifference for all countries. The order of the countries is JP, IT, DE, CA, U.K., FR, BE, U.S., DK, CH, SE, ES, ZA, NL and IR.

point estimate for equity returns is largely positive and it is significant at the ten-percent level in the bootstrap estimation.The point estimate for bonds is positive, but insignificant in both the standard and bootstrap regressions. The size of theeffect of a PAYGO system on the premia and equity return is surprisingly large.

1.2. The effects of the controls

Here, I analyze the importance of controlling for fixed time and country effects. Table 2 shows how the point estimatesand the significance levels change when fixed time and country effects are added to the econometric specification. Withoutany controls, the effect of the treatment, i.e., the introduction of the PAYGO system, is close to zero and the estimate ishighly insignificant. This is also shown in the upper left-hand graph in Fig. 1, which displays differences in premia beforeand after treatment for individual countries.

Since the equity premium may vary across countries for reasons not related to the PAYGO system, failure to account forfixed country effects may bias the results. In particular, when the fixed country effect (λi) is correlated with the independentvariable (PAYGOit ) this leads to an omitted variable bias. The correlation in the data is –0.34 without any controls, thusimplying that the estimate is downward biased because the regression (that should only estimate within effects) also picksup cross-sectional variation.13 Controlling for only fixed country effects increases the point estimate (as shown in Table 2)—but only slightly.

Similarly, the equity premium may vary over time for reasons not related to the PAYGO system; thus failure to accountfor fixed time effects may also bias the estimate. In fact, the median country introduces a PAYGO system in 1960 and theaverage premium is relatively high before 1960 and relatively low afterwards.14 Unless controlled for, these time trendsimply a downward bias because it can artificially seem like the equity premium is reduced by the treatment. However, thelower left-hand graph in Fig. 1 reveals that after controlling for fixed time trends, almost all countries experience a higher

13 For instance, Australia has the highest average premium in the sample and did not have a PAYGO system throughout the period under study.14 The average is 4.9 before 1960 and 3.95 afterward. Specifically, the premia for the ten decades are {3.7,5.5,2.7,−1.93,5.8,13.6,5.7,0.1,8.5,1.6}.


Table 3Panel regression results.

std(Re) std(Rbond) std(Rbills)

PAYGO 1.97 −1.77∗∗∗ −2.45∗∗∗(16.76) (0.93) (0.6)

p-value 0.38 0.10 0.02

Note: Robust standard errors from the standard regressions in parentheses; ∗ significant at 10%;∗∗ significant at 5%; ∗∗∗ significant at 1%. The stars denote the significance levels in the standardregressions whereas the p-values are from the bootstrap estimations.

Table 4Testing for cross-sectional dependence.

Regression p-value

PAYGO on equity returns 0.9885PAYGO on bond returns 0.8695PAYGO on bills returns 0.4485

Note: The p-values give the significance levels where the nullhypothesis of no cross-sectional dependence can be rejected.

premium after the treatment.15 When time trends alone are removed, the point estimate raises from 0.12 to almost twopercentage points and has an extensive influence on the significance level.

The estimate with fixed time effects is still downward biased, however, unless it is also possible to control for fixedcountry effects. Controlling for both time and individual effects removes the cross-sectional variation, which has a substan-tial effect on both the point estimate and the significance level. Note also from the lower right-hand graph in Fig. 1 that thelarge point estimate is not driven by any specific country. In fact, 11 of the 15 countries that introduced a PAYGO systemat some point have an equity premium that is two percentage points or higher after the introduction. Moreover, half of thecountries experienced an increase larger than five percentage points.16

In Appendix B.1, I show how the results are affected when lags, leads and interaction terms are included in the re-gression. The point estimate does not change to any considerable extent, but turns insignificant when lags and leads areincluded.17

Finally, in order to evaluate how social security affects the volatility of returns, I regress the variable (Reit − Re

i,PAYGO)2 on

PAYGO, where Rei,PAYGO is the country-specific average return with and without the treatment, respectively. Table 3 shows re-

sults for the standard deviation instead of the variance. As can be seen, equity returns are more volatile after the treatment,but this effect is highly insignificant. However, both bond and bills returns are, less volatile with the treatment.

1.3. Testing for cross-sectional dependence

As mentioned above, the estimation procedure treats the countries as isolated from each other. This assumption is prob-lematic if there is cross-sectional dependence in the return data and/or in the timing of the introduction of a PAYGO system.Such dependence may imply inefficient estimators. For this reason, I test for cross-sectional dependence by performing aFriedman test in each of the regressions. Specifically, the test is carried out by testing the following hypothesis

H0: There is no cross-sectional dependence in the data.

The results are displayed in Table 4.The high p-values imply that the null of cross-sectional independence cannot be rejected at any reasonable significance

level in any of the regressions. In other words, dependence should not constitute a direct problem. This result might beunderstood against the strong home bias observed in portfolio decisions over most of the data period covered.18

2. An economic model

Let us now consider a simple three-period overlapping-generations model where each generation is modeled as a rep-resentative consumer. Production is exogenous and the index i = 0,1 and 2 is used to denote the young, the middle-aged

15 In addition, the difference in the premium before and after is also significantly higher for all countries (except Switzerland).16 Individual countries and decades were also sequentially excluded from the regression to check the robustness of the results. The estimate for the

premium never falls below 3.52.17 This is not surprising since the leads and lags (as well as the interaction terms) are all highly correlated with PAYGO. In fact, the correlation is around

0.90 and such collinearity inflates the standard errors.18 The home-bias puzzle refers to the fact that individuals and institutions in most countries only hold very modest amounts of foreign equity. The

suggested reasons are additional difficulties associated with investing in foreign equities, such as legal restrictions and transaction costs.


and the old, respectively. Agents inelastically supply labor for two periods and then retire in their third period when old.Specifically, an agent born in period t receives a deterministic wage income wt,0 > 0 when young, a stochastic wage incomewt+1,1 > 0 when middle-aged and a social security benefit ϕt+2,2 � 0 when old.

There are two types of securities in the economy: bonds and equity. Both are in unit supply and infinitely lived. Thebond—which may be considered a proxy for long-term government debt—is default free and pays a fixed coupon b > 0 inevery period in perpetuity. The ex coupon bond price in period t is denoted by qb

t . Equity is a claim to a net dividend stream{dt}, the sum total of all private capital income. The ex dividend price of equity in period t is denoted by qe

t .Wages, consumption, dividends and coupons as well as bond and equity prices are denominated in units of the non-

storable consumption good. The consumer born in period t has zero endowment of assets. This consumer makes a portfoliodecision zt,0 = (zb

t,0, zet,0) when young; adjusts this decision to zt+1,1 = (zb

t+1,1, zet+1,1) when middle-aged, and sells the

portfolio in period t + 2 when old. As usual, a negative position in bonds or stocks denotes a short position in that asset.The payroll tax rate and the social security benefit in period t are denoted by τt and ϕt , respectively. The budget

constraints in period t are then given by

ct,0 + zbt,0qb

t + zet,0qe

t +� wt,0(1 − τt), (1)

ct,1 + zbt,1qb

t + zet,1qe

t � wt,1(1 − τt) + zbt−1,0

(qb

t + b) + ze

t−1,0

(qe

t + dt), (2)

ct,2 � ϕt + zbt−1,1

(qb

t + b) + ze

t−1,1

(qe

t + dt), (3)

where (1), (2) and (3) are the budget constraints faced by the period t young, middle-aged and old, respectively.Period utility takes the following form

u(c) = c1−γ − 1

1 − γ, (4)

where γ > 0 is the constant coefficient of relative risk aversion. In order to predict future prices, agents need to know thedistribution of assets. Since only two age groups trade at any point in time, the distribution is perfectly characterized by theasset holdings of the middle-aged. The Bellman equation for the consumer’s problem for an agent of age i ∈ {0,1} is thengiven by

V i(st , zb

t−1,0, zet−1,0

) = maxct ,zb

t,i ,zet,i

u(ct,i) + βEt V i+1(st+1, zb

t,0, zet,0

), (5)

subject to the relevant budget constraints given by (1)–(3). Parameter β is the subjective discount factor and s is an aggre-gate state that contains information about current wages, dividends, taxes and benefits.

It is also assumed that

Vt,3 ≡ 0 ∀t

which implies that the old do not buy any assets (and that altruistic bequests are ruled out).Aggregate income is defined as

yt = wt,0 + wt,1 + dt . (6)

2.1. The government sector

Four different types of PAYGO systems are considered. Specifically, regime A features no social security system so bothsocial-security taxes and benefits are zero. Under system B , agents receive a safe benefit that does not depend on incomehistories or aggregate wages. In regime C , benefits are based on previous income, but they are not indexed to aggregatewages. Under D , agents collect benefits that are both wage indexed and based on previous income. Regime D is supposedto reflect the U.S. social-security system. As a reference, a case with constant lump-sum taxes is also considered. Thesearrangements differ according to the way in which they allocate risk between generations. For instance, safe benefits andhistory-dependent benefits both increase the risk for workers while reducing it for the old (because taxes have to be higherin low-wage states and vice versa).

There are three main factors that determine the social security benefits received by an agent in the U.S. system: his/heraverage income, the replacement ratio and the average wages.19 The first factor determines the level of benefits. Eachworker’s so-called Average Indexed Monthly Earnings is calculated. This amounts to the average of the worker’s earningsover the best 35 years of her career and is hereafter referred to as reference earnings. The second determinant of socialsecurity benefits is the replacement ratio, i.e., the rate at which social security replaces past earnings. Third, the rate of

19 See, for instance, the Social Security Handbook (2004).


Table 5Social security systems.

System E[ ϕw0+w1

] ψt,2 Wt

A No social security 0 0 0B Safe benefits 0.40 w0+E[w1]

2 1

C Benefits based on previous income 0.40 w0+wt−1,12 1

D Wage-indexed benefits based on previous income 0.40 w0+wt−1,12

WtE[W ]

return on social security is related to aggregate labor income since an individual’s earnings are indexed to the average wagelevel at the time of retirement.20

In the model, the reference earnings of an old agent in period t , ψt,2, are given by ψt,2 ≡ w0+wt−1,12 .21 Denoting the

replacement ratio by η, the social security benefit of an agent who retires in period t is

ϕt = ηψt,2Wt, (7)

where Wt is the wage index. Formally, Wt is given by

Wt = Wt

E[W ] . (8)

Hence, if the aggregate wage rate in period t is above its unconditional mean, benefits are increased proportionately andvice versa. The government is assumed to balance its budget in each period. Budget balance, in combination with definedbenefits, implies that the tax rate is a stochastic variable that takes on whatever value is needed to keep the governmentbudget in balance. Since benefits are not taxed, the government budget constraint is given by

τt(w0 + wt,1) � ϕt + b. (9)

It follows from (7) and (9) that the tax rate in period t is given by

τt = ηψt,2Wt + b

w0 + wt,1. (10)

Table 5 lists all of the social security systems considered and their implied parameter values.

2.2. Short-selling constraints

Short-selling constraints are potentially important since they can have a major impact on portfolio choices. Cases bothwith and without short-selling constraints are therefore considered. The short-selling constraints for all t are given by

zbt,i � 0 and ze

t,i � 0, i = 0,1. (11)

2.3. Equilibrium

I now focus on searching for a stationary equilibrium where decisions made in a given period are determined by theaggregate state s j = (y j, w j), s j ∈ S , the current wealth of the middle-aged z−1 ≡ (zb−1, ze−1) and the reference earn-ings of the currently old ψ2(s−1).22 Since the wage income of the young is constant, the reference earnings dependonly on the shock in the preceding period. In this economy, a stationary equilibrium is given by time-invariant pric-ing functions qb(s, z−1,ψ2(s−1)), qe(s, z−1,ψ2(s−1)), and the decision rules for agents’ asset holdings, zb

0(s, z−1,ψ2(s−1)),zb

1(s, z−1,ψ2(s−1)), ze0(s, z−1,ψ2(s−1)) and ze

1(s, z−1,ψ2(s−1)), such that the following conditions hold:

1. the decision rules solve the agents’ maximization problem, which is to maximize (5) subject to (1)–(3) for i = {0,1};2. zb

0 + zb1 = 1 and ze

0 + ze1 = 1; and

3. the government budget constraint (9) is satisfied.

20 Actually, an individual’s earnings are indexed to the average wage level two years prior to the year of eligibility, i.e., when the agent reaches the age of62.21 The computation of reference income in the model includes 40 years (rather than 35 years as in the U.S.), since one period is assumed to be 20 years

and each worker supplies labor for two periods.22 The young are born with zero assets. The specification follows Constantinides et al. (2002), who also prove that an equilibrium exists in a similar model

without social security.


The first condition ensures that each consumer’s consumption and investment policy maximizes her expected utilityfrom the set of admissible policies (while taking the price process as given) and the second condition ensures that the bondand equity markets clear in all periods. Note that by Walras’ law, two and three combined imply that the goods marketclears. In the presence of short-selling constraints, the equations in (11) must also hold.

2.4. Calibration

It is somewhat problematic to calibrate a model where a period is 20 years. Even a century-long time series providesonly five non-overlapping observations, thereby resulting in large standard errors in the point estimates. Given the datalimitations, the model is calibrated to the U.S., for which at least some relevant earlier studies are available. For instance,some calibration targets are taken from Constantinides et al. (2002), who also consider a model with 20-year periods. Animportant difference, however, is that I calibrate the life-cycle pattern of labor income to match PSID data. This implies thatthe young always hold a fraction of the equity.23

The joint process for detrended aggregate income and wage income of the middle-aged (yt , wt,1) is modeled as a time-stationary Markov process. Specifically, the state space is given by S = [s1, s2, s3, s4] and the associated transition probabilitymatrix is denoted by π = (πi j). The four states are given by

S =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(y − ν), (w1(1 − ζ )),

(y + ν2 ), (w1(1 − ζ )),

(y − ν2 ), (w1(1 + ζ )),

(y + ν), (w1(1 + ζ )),

where y is average aggregate income and w1 is the average wage income of a middle-aged agent. The stochastic process isassumed to be i.i.d. over time. Even though aggregate productivity shocks are highly autocorrelated at annual and quarterlyfrequencies, there does not seem to be any evidence that suggests such positive serial correlation at generational frequencies(i.e., 20–30 year periods).

With the above representation, state s1 is characterized by a low aggregate income and a low wage income of themiddle-aged, whereas s4 has a high aggregate income and a high wage income of the middle-aged. In aggregate states s2and s3, aggregate income and the wage w1 move in opposite directions. In addition, parameter ν is introduced to allow fora small and a large shock to aggregate income because it is difficult for the model to match all of the eight desired targetsbelow with ν = 1. It is also assumed that π1 = π4 and π2 = π3.

The wages of the young are constant and the wages of the middle-aged can take on two different values, which impliesthat the variable ψ2 (reference earnings) can also take on two different values. With four possible income states and twopossible income histories in each period, the total number of discrete exogenous states is eight.24

Parameter η is set at 0.328, which implies a replacement ratio of 0.40, in the sense of benefits replacing 40 percentof the average life-time wage.25 The expected tax rate needed to finance the replacement ratio is 0.16 in the model (and0.153 in the U.S., including medicare).26 The coefficient of relative risk aversion is set at γ = 6, which leaves eight additionalparameters to be determined: y, v , w0, w1, ζ , π1, β , and b.27,28 These parameters are chosen to satisfy the eight targetmoments described below.

1. The average share of income which goes to labor is set at 0.7, which is consistent with the U.S. historical experience.2. The average share of wage income which goes to the young E[ w0

w1] is set at 0.75. This number is based on income

profiles estimated by Storesletten et al. (2004).3. The average share of income which goes to interest on government debt b

E(y)is set at 0.025, the mean in the U.S. for

the period 1960–2009.29

4. The coefficient of variation of twenty-year aggregate income σ(y)E(y)

is a first challenge to calibrate. Bohn (1999) estimatesthe long-run variances of output, earnings and returns using data from 1871–1996. He finds the generational varianceof output to be 0.124, which implies a coefficient of variation above 0.35.30 I follow Constantinides et al. (2002) and setσ(y)E(y)

= 0.25.

23 In contrast, in Constantinides et al. (2002), the young are calibrated to be so income poor that they never buy any equity unless they can borrow.24 In addition, the state space includes two continuous endogenous dimensions, i.e., the bond and equity holdings of the middle-aged.25 The U.S. replacement rate has fluctuated between 38.7 percent and 51.7 percent over the last 30 years. Today however, benefits replace roughly

42 percent of average wage income.26 The tax rate is slightly higher because the ratio of retired individuals to those who work is somewhat higher in the model than in the U.S.27 As a sensitivity check, the results were computed with γ = 4. The effects of the PAYGO system are qualitatively similar but somewhat smaller.28 Since π1 + π2 + π3 + π4 = 1, the transition matrix is uniquely defined by π1.29 Computed from Tables B-1 and B-84 in the Economic Report of the President (2010).30 Four different specifications were used and the estimates were found to be similar for all specifications.


5. The coefficient of variation in the twenty-year wage income of the middle-aged σ(w1)E(w1)

represents yet another calibrationchallenge. In a study of women, Cox (1982) estimates the coefficient of variation in wages to be 0.25 in the cross section.Constantinides et al. (2002) set the coefficient of variation in the twenty-year wage income of the middle-aged at 0.25.I choose a more moderate value and set σ(w1)/E(w1) = 0.20.

6. The correlation between equity returns and labor income of the middle-aged is calibrated to be in the interval(−0.12,0.20) as in Davis and Willen (2000).31 In the model, the correlation has an exogenous and an endogenouscomponent. Dividends are exogenous, whereas prices are endogenous and the return includes both of these terms. Thecalibrations feature correlations that lie within those intervals.

7. The correlation between the labor income of the middle-aged and aggregate income is also difficult to calibrate. Due toa lack of sufficient time-series data to estimate this correlation, I simply consider two different correlations: one lowand the other one high. The high and the low correlations are set at the highest and lowest possible values which canensure that the correlation between equity returns and labor income of the middle-aged remains within the intervalsprovided by Davis and Willen. In the benchmark calibrations, the correlations considered are corr(yt , wt,1) = 0.05 andcorr(yt , wt,1) = 0.30. The calibration with a correlation of 0.30 features a correlation between equity returns and laborincome of the middle-aged that is at the upper bound of these intervals, whereas the calibration with a correlation of0.05 features a correlation that is close to the lower bound.

8. The average real return to equity E[Re] is set at between six and seven percent, which was found by Constantinideset al. (2002) when estimating the mean of the real annualized, twenty-year holding-period return from the S&P 500total return series. I set β so that this number is matched on average.

The parameters are chosen to match the eight targets above. Six parameters are constant between the calibrationsand they are set at y = 10, w0 = 2.8514, w1 = 3.8014, ζ = 0.20, β = 0.70 and b = 0.20. The rest of the parametersare in the economies with corr(yt , wt,1) = 0.05 and corr(yt, wt,1) = 0.30, respectively, set at {v = 3.45,π1 = 0.177} and{v = 3.25,π1 = 0.242}.

2.5. Numerical computation of the equilibrium

Since there is no analytical solution to the problem outlined in Section 2, the equilibrium has to be solved for numer-ically. Specifically, the state variables are the productivity shock, the bond and stock holdings of the middle-aged and theincome history of the old. The grid consists of two continuous endogenous dimensions, i.e., bond and stock holdings ofthe middle-aged, and it has eight discrete states.32 Portfolio policy functions and pricing functions are approximated byB-splines of order 4 (piecewise cubic polynomials). A spline-collocation algorithm is used to approximate the equilibriumnumerically. This implies starting out with a guess for the unknown functions. This guess is then assumed to govern behav-ior in future periods. Given the guess, it is straightforward to solve for agents’ optimal behavior in the current period (onall collocation points). The algorithm then checks whether the resulting functions are sufficiently close to the guess. If thedifference is smaller than a chosen tolerance level, the solution has been found; otherwise, the guess is updated with thenew functions, followed by iteration until there is convergence.

3. Simulation results

All economies were simulated for 20 000 periods and the results are reported below.33 The mean return of an assetis defined as 100 × [{mean of the 20-year holding-period return}1/20 − 1]. The standard deviation of the equity or bondreturn is defined as 100 × [std{20-year holding-period return}1/20]. The mean premium of equity return over the bondreturn is defined as the difference between the mean return on equity and the mean return on the bond. The stan-dard deviation of the premium of equity return over the bond return is defined as 100 × [std{20-year equity return} −std{20-year bond return}1/20].

Table 6 (which gives the results for corr(yt , wt,1) = 0.05) shows that the PAYGO system significantly increases both assetreturns and the equity premium.34 The average return to equity is somewhat more than three percentage points higher, thebond returns are around two percentage points higher and the equity premium is about one percentage point higher undera PAYGO system. The exact design of the social security system is of marginal quantitative importance for the size of thepremium. In fact, the difference in equity premia between the different PAYGO regimes is at most 0.2 percentage points (seealso Table 8 in Appendix B). Short-selling constraints affect the level of the equity premium but not the difference betweenpremia in economies with a PAYGO system and those without.

31 See Figs. 1 and 2 in Davis and Willen (2000).32 There are four aggregate shocks and two possible income histories of the old.33 All economies were simulated for 21 000 periods, but the first 1000 periods were discarded.34 Table 8 in Appendix B shows results for the alternative correlation between aggregate income and the wages of the middle-aged corr(yt , wt,1) = 0.30.

The results are similar to those in Table 6. Table 11 in Appendix E also shows that the effect of social security is to increase the premium also when agentshold T-bills instead of bonds.


Table 6Simulation statistics.

ConstraintsPAYGO system

Unconstrained Constrained

A B C D A B C D

Mean equity return 2.90 6.24 6.17 5.92 4.81 8.08 8.02 7.81Mean bond return 2.04 4.13 4.09 3.99 1.84 4.02 3.98 3.77

Mean prm/bond 0.86 2.10 2.08 1.93 2.96 4.06 4.04 4.03

Std of equity return 8.85 8.71 8.74 8.88 4.43 10.93 10.45 9.68Std of bond return 8.35 7.83 7.74 7.99 2.56 4.08 3.72 3.44E(ze

0) 2.85 1.19 1.21 1.35 0.22 0.17 0.17 0.17

E(zb0) −9.77 −4.35 −4.42 −4.88 0 0 0 0

R f1 −2.09 2.96 2.92 2.66 0.79 3.35 3.46 3.22

Note: corr(yt , wt,1) = 0.05. A—no social security, B—safe benefit, C—benefits are based on previous income, no wage indexation, D—benefits are based onprevious income and are wage indexed. E(zi

0) denotes the share of asset i that is held by the young, i = e, b. The risk-free rate is computed by using theshadow price of a risk-free bond, determined by the marginal rate of substitution of the middle-aged consumer.

Thus, the model is to some extent able to account for the data. However, since bonds are much more volatile than inthe data, a smaller effect on the equity premium is to be expected. It is perhaps somewhat surprising that the differencesin equity premia between social security systems are small. One explanation is simply that the differences between PAYGOsystems are also relatively small. For example, the coefficient of variation in the real wage is only 10 percent higher in theeconomy with history-based benefits than in the economy with safe benefits. In addition, the share of total old-age incomereceived from social security is only around 10 percent in the model. The exact way in which these 10 percent vary doesnot matter to any considerable extent for the overall variability of old-age consumption.

4. Why is the equity premium higher under a PAYGO system?

Since the PAYGO system simultaneously affects levels and volatilities of prices, returns and asset holdings, it is crucialto determine the factors which are quantatively important for the higher equity premium. A natural starting point here isthe Consumption Capital Asset Pricing Model (CCAPM), which specifies that the return to a specific asset is given by therisk-free rate plus a risk adjustment that depends on the covariance between the return of the asset and the marginal utilityof consumption (hereafter marginal utility). According to the model, an investor should be willing to pay a higher price foran asset with a high covariance between the payoff of the asset and the future marginal utility of the investor than for anasset with a low such covariance. The reason is that the asset with the high covariance provides a better insurance (since itis more likely to pay off in bad states of the world). The risk-adjustment term is given by

−cov(c−γ , R)

E[c−γ ] . (12)

So as to identify the mechanisms behind the covariances, we make use of the fact that (12) can be written as

covc−γ ,R ≡ −cov(c−γ , R)

E[c−γ ] = −σc ∗ σR ∗ ρc,R j , (13)

where σc is the standard deviation of marginal utility over expected marginal utility, σR is the standard deviation of thereturn of the asset and ρc,Re is the correlation between them. If markets are incomplete (as they always are in OLG models),the covariances in (13) will differ between agents.35 Since the young only buy equity whereas the middle-aged buy both as-sets, there are three relevant covariances to consider: covc

−γ1 ,Re , covc

−γ2 ,Re and covc

−γ2 ,Rb . Table 7 shows how the covariances

are affected by a PAYGO system. Specifically, it reports ratios of covariances and components in the decomposition with aPAYGO system that uses constant lump-sum taxes, relative to no system. As expected, the risk-adjustment term is higher forequity and lower for bonds when agents are entitled to social security.36 The table also reveals that the covariances withequity returns are higher mainly because equity returns are significantly more volatile relative to bond returns (even thoughbond returns are also more volatile), and that the covariance with bond returns is lower mainly because bond returns areless correlated with old-age marginal utility.

35 If markets are complete, individual consumption is perfectly correlated with aggregate consumption for all agents, thus implying that the covariancein (12) is the same for all agents. This is not true for incomplete markets. For instance, with short-selling constraints, the bond is exclusively priced bythe middle-aged agents, since they are the only ones who buy the asset. The young want to short-sell bonds but are prevented from doing so. Hence, the

risk adjustment for bonds depends only on the risk-adjustment term−cov[c−γ

2 ,Rb ]c−γ2

. The quantitative importance of market incompleteness is analyzed in

Appendix D.36 Both covariances with equity returns are higher, thereby implying that the risk premium for equity increases unambiguously.


Table 7Decomposition of the effects of PAYGO on covariance terms.

Ratios

covc−γ1 ,Re σc1

σRe ρc1,Re

1.81 1.05 1.93 0.90covc

−γ2 ,Re σc2

σRe ρc2,Re

1.58 0.91 1.93 0.94covc

−γ2 ,Rb σc2

σRb ρc2,Rb

0.90 0.91 1.57 0.61

Note: All numbers are ratios of outcomes with a PAYGO system that uses lump-sum taxes rela-tive to outcomes without a social security system. For example, corr(c−γ

2 , Re) denotes the ratiocorr(c

−γ2 ,Re )PAYGO

corr(c−γ2 ,Re )No PAYGO

. Since all correlations and covariances are negative, a higher ratio implies a higher

risk adjustment. The returns are 20-year returns, i.e., not annual returns.

Finally, the table shows that the equilibrium effect of the system is to transfer consumption risk between generations.Even though the results in Table 7 are for a specific PAYGO system (that uses constant lump-sum taxes), they are similar forall social security systems considered in the paper. The intuition for the findings in Table 7 are discussed in detail below.

4.1. Asset prices and return volatilities

An important reason for the higher equity premium is that equity returns are relatively more volatile under a PAYGOsystem. To understand this result, note that agents’ marginal willingness to pay for private assets will generally be lowerwhen they are forced to save in an unfunded system. As a result, average asset prices will be lower and average returnswill be higher. This is true in the model, as can be seen from the last row in Table 6, which shows the risk-free rate R f ,with and without social security.37 For given coupon payments and dividends, asset prices thus fall and this reductionin prices has asymmetric effects on the volatilities of bond and equity returns. In fact, even an equal reduction in bothprices has asymmetric effects on the volatilities of the returns. This is because the only risk associated with holding a bondbetween two periods is the price risk (since coupon payments are constant), whereas equity is also subject to dividend risk,in addition to the price risk. A lower bond price therefore implies that a larger share of the total payoff of the bond isconstant, but a lower equity price instead implies that a larger share of the total payoff of equity is due to risky dividends.Thus, the reduction in asset prices results not only in higher returns, but also that equity returns become more volatilerelative to bond returns.

To see this explicitly, consider for simplicity a reduction in all asset prices by a factor x, with 0 < x < 1. Since b isconstant, the variance of the payoff of the bond is given by x2var(qb) (which goes to zero as x goes to zero). The variance ofthe payoff of equity is instead given by var(xqe + d), which can be written as x2 var(qe)+ var(d)+ 2x std(qe) std(d) corr(qe,d)

(which goes to var(d) as x goes to zero). A general reduction in asset prices thus also lowers the variance of bond payoffsby a factor of x2. However, since dividends are unaffected by the level of the equity price, the variance of equity payoffs isreduced by less than x2.

The PAYGO system that uses constant lump-sum taxes (with the results displayed in Table 10 in Appendix C) does, in fact,have an almost neutral effect on relative prices. Both asset prices are reduced by the system and by the same magnitude,but the volatility of equity returns increases by 25 percent more than for bond returns in response to the reduction inprices.38 In addition, without social security, the equity price is roughly 2.5 times more volatile than the dividends. Butwhen agents are entitled to social security, both the levels and the standard deviations of asset prices are lower, averagereturns are higher and the main risk of holding equity is instead due to dividend risk.

When benefits are history based and/or wage indexed, relative prices may also change after the introduction of a socialsecurity system, but in all cases, lower asset prices heighten the importance of dividend risk. I have verified numerically thatthe reduction in asset prices increases the importance of dividend risk more when the volatility of dividends is initially largerelative to the volatility of the equity price. Specifically, the effect on returns in the benchmark calibration is compared tothe effects in an economy where the coefficient of variation of dividends is 40 percent lower. In the benchmark calibration,the increase in the standard deviation of equity returns is roughly 23 percent higher than the increase in the standarddeviation of bond returns (see Table 7). In the economy with less volatile dividends, the corresponding number is fivepercent, i.e., significantly smaller. As a result, the effect of social security on the equity premium is also smaller: about halfa percentage point (compared to one percentage point in the benchmark case).

37 In fact, there is no risk-free asset in the model. The risk-free rate is instead computed by using the shadow price of a risk-free bond, determined by themarginal rate of substitution of the middle-aged consumer.38 The influence of the PAYGO system on relative prices can be established by comparing qe(s, ·) ≡ qe

No PAYGO(s,·)qe

PAYGO(s,·) to qb(s, ·) ≡ qbNo PAYGO(s,·)qb

PAYGO(s,·) , where · refers to

the asset holdings of the middle-aged. If the system does not distort the relative prices between bonds and equity, the ratio qe(s, ·)/qb(s, ·) should equalone. The average ratio of qe /qb is 0.993, thus implying that relative prices are basically unaffected by a system with lump-sum taxes.


It may also be noted from Table 7 that bond and equity returns are both more volatile under social security. Thisis because asset prices respond differently to income shocks when agents receive social security. In fact, the higher thetax rate, the more asset prices respond to shocks to untaxed income (such as savings, which are untaxed in the model).Similarly, the higher the tax rate, the less asset prices respond to shocks to taxed income (such as labor). These results areproved in Appendix C (in a simple two-period model) and they contribute to understanding why both returns are morevolatile with a PAYGO system.

4.2. Portfolio holdings and risk-sharing

Consider now first the intuition behind the life-cycle portfolios. In short, the young find equity to be an attractive assetbecause the correlation between equity returns and their future wage is low, i.e., equity is a hedge against future wage fluc-tuations.39 When unconstrained, the young short-sell bonds (to increase current consumption) and buy equity (to smoothfuture consumption). When prevented from borrowing, they instead accumulate a small buffer of equity for precautionaryreasons. From the perspective of the middle-aged, future income is either zero or a relatively small share of past income.Hence, a strong position in equity implies a high correlation between equity income and future consumption. Therefore, eq-uity is not a good hedge against fluctuations in consumption for the middle-aged. Instead, they hold a diversified portfolioconsisting of both bonds and equity. As shown in Tables 6 and 8, a PAYGO system induces the middle-aged to reduce theshare of bonds and increase the share of equity in their portfolios.40

According to Table 7, the risk-adjustment term is lower for bonds under a PAYGO system primarily because old-agemarginal utility and bond returns are less correlated. In fact, old-age consumption is less correlated with both returns whenagents receive retirement benefits, because the middle-aged effectively hold one more asset in their portfolio. Their futureconsumption then depends less on individual assets. The correlation with bond returns is particularly low however, becausethe middle-aged choose to hold a relatively more equity-based portfolio when they are entitled to relatively safe retirementbenefits.

Note also from Table 7 that the equilibrium effect of the system is to transfer the consumption risk between generationsby making old-age consumption less volatile and middle-aged (and young) consumption more volatile. This also contributesto the higher premium, since the young then require a higher return for holding equity, whereas the middle-aged require alower return for holding bonds.

5. Discussion

As has been shown above, in both the data and the model, an unfunded system increases the equity premium. Thisraises the question as to whether the reason found in the model is the same as in the data? In other words, what aspects ofthe data does the model actually fit? There is a good fit for the average levels of returns for equity and T-bills. In the data,these returns are, respectively, about 3 and −0.65 percent before the treatment, and about 8 and 1.55 after. The results inTables 6 and 11 (for T-bills) show that the model produces numbers close to these observed values. The return to bonds,however, is roughly two percentage points too high in the model and the increase is also too high, so the model does notmatch bond returns equally well.

One reason for the higher premium in the model is that the volatility of equity returns increases more than for bondreturns. As shown in Table 3, the empirical estimate of the volatility of equity returns is insignificant and both bond andbill returns are less volatile. Therefore, the model does not match the data in this dimension. Note however, that in thedata as well, equity returns are more volatile relative to bond and T-bill returns after the treatment. The importance of thiseffect should be regarded with caution, since the empirical estimates are only marginally significant. Another dimensionwhere it would be interesting to compare the model to the data is to examine price-to-dividend ratios before and after thetreatment. Unfortunately, I do not have access to such data, so this question has to be left for future research.

The theoretical results all rely on the assumption that markets are incomplete. If markets were complete, age-specificrisk would be efficiently shared among agents. Undesired policies (that do not decrease the level of market completeness)would then not have any effects because agents could simply “undo” them whenever they wished. This is not true if mar-kets are more incomplete because individual consumption is then only imperfectly correlated with aggregate consumption,thus implying that the different covariances between age-specific consumption and the return to different assets combineddetermine the equity premium. In Appendix D, the effect of the PAYGO system on the equity premium is shown to be closeto zero when markets are almost complete.

The paper abstracts from several potentially important issues. For example, both the empirical and the theoretical sec-tions abstract from the transition associated with implementation of a PAYGO system. The potential bias of the empiricalresults for abstracting from transitions is discussed in Section 1. As for the omission of transitions in the model, this isclearly a limitation in the analysis. Nevertheless, comparison of steady states is a natural starting point that can help us

39 The correlation is calibrated to be in the interval (−0.12,0.20).40 These tables show asset demand by the young, but because both assets are in unit supply, the demand of the middle-aged is simply one minus the

demand of the young for both assets.


identify the main mechanisms and the long-run effects. The process of taking transitions into account is an important issuefor future research.

Many other changes have taken place during the 100 years under consideration. The retirement age has been reduced,individuals live longer and longer and the elderly do not live with their extended families to the same extent as they did atthe beginning of the period. None of these changes is taken into account in the model. The paper also abstracts from incomeheterogeneity within age groups. For instance, Parker and Vissing-Jorgensen (2009) find that high-income households bear alarge share of the aggregate fluctuations after 1980. This should not qualitatively change the results, but since most socialsecurity systems have a cap on retirement benefits, social security may be of less importance for the rich. If those who arerich have a sizable impact on the equity premium, the importance of social security is likely to be smaller. These are allimportant issues for future research.

6. Concluding remarks

This paper begins by estimating the effects of a PAYGO system on asset returns and the equity premium with adifferences-in-differences approach, using panel data on asset returns in 16 countries over the period 1900–2000. The sys-tem is found to be associated with substantially higher equity returns and a higher equity premium relative to no system.In fact, both the premium and average equity returns are more than four percentage points higher under a PAYGO system.The effect on average bond returns is just above zero, but this result is insignificant.

In the next part of the paper, a three-period overlapping-generations model is calibrated and it is found that a PAYGOsystem significantly increases asset returns and the equity premium. Specifically, the average return to equity is aroundthree percentage points higher and the equity premium is about one percentage point higher. The model thus goes part ofthe way in accounting for the data, although bond returns in particular increase too much in the model. The exact design ofthe social security system is found to be of minor quantitative importance, but market incompleteness is important for all ofthe results. When markets are close to complete, agents can differentiate away age-specific risk. Unless the redistributionalsystem makes markets less complete, the system has only marginal effects on the premium.

The higher equity premium is due to the fact that agents have a lower marginal willingness to pay for private assetswhen they are forced to save in an unfunded system. The resulting reduction in prices has asymmetric effects on thevolatilities of bond and equity returns. Specifically, the only risk associated with holding a bond between two periods isthe price risk (since coupon payments are constant), whereas equity is also subject to dividend risk (in addition to theprice risk). A lower bond price then implies that a larger share of the total payoff of the bond is constant, whereas a lowerequity price instead implies that a larger share of the total payoff of equity is due to risky dividends. When the variance ofdividends remains unaffected, lower prices result not only in higher average returns, but also in more volatile equity returnsrelative to bond returns.

Appendix A

A.1. The introduction of social security41

AustraliaUnlike pension payments in many other countries, workers do not contribute to pensions or insurance in Australia.

Moreover, pension are subject to means testing, to ensure that only those who require assistance actually receive it. Hence,a typical PAYGO system—as in the theoretical model—has not yet been introduced in Australia.42

Belgium (1956)In 1944, a “social pact” was signed to regulate social security, which was controlled with equal representation of workers

and employees. Pension insurance did not became mandatory until 1956.43

Canada (1952)The Canadian pension system dates back to 1927, when the Parliament enacted legislation according to which the federal

government would pay part of the costs of provincial means-tested benefits for the elderly.44 After various extensions ofthe system, the federal Old Age Security Act took effect on January 1, 1952, with provision for all individuals in Canada whomet the age requirements.

Denmark (1964)Denmark was the second country in the world (after Germany) to introduce an old-age security system in 1891. This

system targeted the poor on a highly discretionary basis. After the so-called people’s pension reform of 1956, eligibilitybecame universal but entitlements remained means-tested until 1964, when an almost flat-rate pension scheme for allcitizens was adopted.45

41 The year of publication in parentheses.42 Herscovitch and Stanton (2008).43 Everything you have always wanted to know about social security (2009).44 Tamagno (2005).45 Andersen (2008).


France (1972)A social security system was created in 1945. Supplementary schemes were introduced in order to enhance basic cov-

erage. Both the basic and the supplementary schemes operate a PAYGO system. In 1972, membership in a supplementarypension scheme became compulsory.46

Germany (1957)The German pension system was originally introduced by Bismarck in the late 1800s. The current PAYGO system came

into effect after an extensive pension reform in 1957.47

Ireland (1960)The so-called Old Age Contributory Pension system in Ireland was legislated by the Social Welfare Act in 1960.48

Italy (1969)The pension system in Italy evolved over a long period of time. It was not until 1969 that financial distress finally led to

a move from a funded system toward a PAYGO system for the major funds.49

JapanJapan’s public pension system has a long history, although universal coverage was not implemented until 1961.50

Netherlands (1957)Since the introduction of the Dutch General Old Age Pension Act (AOW) in 1957, the intention has been to entitle all

people aged 65 and over to full AOW pension rights.51

South Africa (1960)A pension scheme was introduced in 1944. Sagner (2000) argues that the Nationalist Party expressed open hostility

towards the South African pension program, but that the feasibility of discriminating through the pension system becameexhausted between 1948 and 1960. The benefits increased by almost 60 percent in real terms between 1951 and 1955.

Spain (1963)The origins of the current Spanish pension system can be traced to the legislation in 1963, which initiated the unification

of different types of benefit and welfare mechanisms.52

Sweden (1960)A so-called people’s pension was introduced in 1913 but the benefits were extremely low. In a referendum in 1957, a

majority voted for a general pension system with higher benefits which was implemented in 1960.Switzerland (1948)Like most OECD countries, Switzerland has a multi-pillar pension system. The first (national federal public) pillar for the

elderly and survivors was introduced in 1948.53

U.K. (1959)In 1946, the National Insurance Act introduced a basic state pension (BSP) which took effect as of 1948.54 However, the

BSP was not strictly a social insurance scheme. Even though the BSP provided a basic retirement income, the level of thebenefits did not keep pace with the growth in average earnings. As a result, private employers started to offer workers’pensions. In 1959, a graduated retirement benefit was introduced in another National Insurance Act.

U.S. (1954)The idea of a U.S. social security system arose in the 1930s.55 The first payments were made as lump-sum refund

payments between 1937 and 1940. In January 1940, benefits became monthly. Between 1940 and 1950 social securitybenefits were low and only about 50 percent of America’s workers were covered. Amendments in 1950, 1952 and 1954substantially expanded the scope of the Old-Age and Survivors Insurance (OASI) program by extending coverage to around10 million additional workers and by greatly increasing the benefit levels.

Appendix B. Model results

Table 8 shows results for the alternative correlation between aggregate income and the wages of the middle-agedcorr(yt, wt,1) = 0.30.

B.1. Regressions with lags, leads and time interactions

In terms of the time dimension, one way of analyzing whether the effect of a PAYGO system is uniform over calendartime is to include an interaction term between time and PAYGO in the regression. For instance, the variable Time50 takes

46 Ribémont (2003).47 Potrafe (2007).48 Salter et al. (2009).49 Brugiavini (1997).50 Horioka and Yuji (1999).51 The Old Age Pension system in the Netherlands (2008).52 Patxot et al. (2009).53 Queisser and Vittas (2000).54 See Bozio et al. (2010).55 See “Social security: a brief history” at www.socialsecurity.gov/history (2005).

www.socialsecurity.gov/history



Social security system Unconstrained Constrained

A B C D A B C D

Mean equity return 3.09 6.42 6.39 6.10 4.79 8.20 8.18 7.94Mean bond return 2.34 4.62 4.58 4.41 2.19 4.65 4.62 4.39

Mean prm/bond 0.75 1.81 1.81 1.69 2.60 3.55 3.56 3.54

E(ze0) 2.92 1.23 1.25 1.37 0.21 0.17 0.17 0.17

E(zb0) −11.50 −4.99 −5.05 −5.56 0.00 0.00 0.00 0.00

Note: corr(yt , wt,1) = 0.30. A—no social security, B—safe benefit, C—benefits based on previous income, no wage indexation, D—benefits based on previousincome and wage-indexed.

Table 9Regressions including time interactions, lags and leads.

Prm bonds

PAYGO 7.65∗ 3.52 4.60∗ 4.24 3.11 5.33 4.82(4.16) (3.21) (2.54) (2.84) (3.27) (3.47) (2.92)

Time50 ∗ PAYGO −5.91(6.95)

Time60 ∗ PAYGO 3.61(4.55)

Time70 ∗ PAYGO 1.65(2.54)

Prm bonds−2 0.31(1.87)

Prm bonds−1 3.60(2.30)

Prm bonds+1 −0.71(2.87)

Prm bonds+2 −4.81(3.42)

N 159 159 159 128 144 143 127

the value of zero before 1950 and one afterwards. If the effect is then uniform over time, the coefficient on the variableTime50 ∗ PAYGO should be zero. Another interesting issue is whether the effect of PAYGO is uniform over time since itsimplementation. This can be tested by including leads and lags, where a uniform effect requires the coefficients on theleads and lags to be zero.

The results are shown in Table 9. The interaction terms affect the estimate of PAYGO, but all the interaction termsconsidered are highly insignificant. Similarly, all the leads and lags included are insignificant. Thus, it cannot be ruled outthat the effect of a PAYGO system is uniform over calendar time and over time since its implementation. Including leads andlags does not change the point estimate of PAYGO to any considerable extent. However, the estimate is no longer significant,mainly because all controls in Table 9 are highly correlated with PAYGO. In fact, the correlation is around 0.90 and suchcollinearity inflates the standard errors. As an additional check, I excluded the main variable (PAYGO) from the regressionsand except for the one-period lag, none of the lags and leads has a significant effect on the equity premium. Hence, PAYGOis significant without leads and lags, but not the contrary.

Appendix C. An analytical example to illustrate the effects of the PAYGO system on asset prices

Here, a simple two-period economy with two agents, young and middle-aged, is set up to illustrate how taxes affect theresponse of asset prices to a change in income. In the second period, the agents are middle-aged and old. Subscripts 0 and1 are used to denote the age of the agents. There is a risk-free asset z, which is in unit supply, has the price q and pays offone unit of consumption in the subsequent period. Allowing for uncertainty does not qualitatively change the results, butsince it complicates the expressions, the simple model is set up with a risk-free asset. Utility is assumed to be logarithmicin consumption. Two simple PAYGO systems are considered. In the first case, a constant lump-sum tax is imposed on theagents and the proceeds are redistributed to the old. In the second case, the taxes are instead proportional. This simplemodel captures the important mechanisms that are also present in the more complicated three-period OLG model, but ithas the advantage of delivering closed-form solutions.

Denoting taxed and untaxed income by w and m, respectively (recalling that labor income is proportionally taxedwhereas non-labor income is untaxed in the three-period model), the model in cases 1 and 2 below shows that the elas-ticity of the asset price with respect to untaxed income, ∂q

∂mmq , is increasing in the tax rate τ , whereas the elasticity with

respect to taxed income, ∂q w , is decreasing in τ . The intuition for these results is as follows:
∂ w q


1. The asset price is increasing in the ratio of aggregate income in the first period relative to income in the second period.This is merely consumption smoothing, i.e., when current income is high relative to future income, the desire to smoothconsumption between periods is also strong. The marginal willingness to pay for savings is then higher.

2. In a given period, high taxes reduce current income. It then follows from point 1 that the asset price is decreasingin the current tax rate. Moreover, shocks to untaxed income lead to larger increases in the asset price than shocks totaxed income, simply because agents get to keep more of the additional income when it is not taxable. With log utility,∂q/∂m is positive and independent of the tax rate, whereas ∂q/∂ w is weakly positive and decreasing in the tax rate.

3. The elasticity of the asset price with respect to untaxed income ∂q∂m

mq is increasing in the tax rate τ . From point 2, it

follows that ∂q/∂m > 0 and that the level of the asset price q is decreasing in the tax rate. Dividing ∂q/∂m by a lowerasset price q then implies a relatively larger elasticity. Intuitively, the income shock is large relative to the asset pricewhen it is untaxed, so the effect on the price is also large.

4. The elasticity of the asset price with respect to taxed income ∂q∂ w

wq is decreasing in τ . According to point 2, ∂q/∂ w is

positive and decreasing in τ . The asset price q is also decreasing in the tax rate. Thus, two opposing effects influencethis elasticity, but the net effect is negative.

Two PAYGO systems are considered in the full three-period model. The first system features constant lump-sum taxes.In the second system labor income is proportionally taxed whereas non-labor income is untaxed. According to the analysis,asset prices respond relatively more to shocks to both sources of income in an economy with lump-sum taxes, as comparedto no system. Similarly, asset prices respond relatively more to shocks to non-labor income and relatively less to wageshocks in an economy with a social security system of the second type, as compared to no system. Since returns are morevolatile under both systems, the higher elasticity with respect to untaxed income dominates the lower elasticity with respectto taxed income in the second system. More volatile prices then imply more volatile returns because the spreads betweenbuying and selling prices are then larger.

C.1. Case 1: Lump-sum taxes

The budget constraints for the young and the middle-aged in the two periods are given by

c0,t = w0,t − τt − z0,tqt and c1,t+1 = w1,t+1 − τt+1 + z0

and

c1,t = w1,t + mt − τt − z1,tqt and c2,t+1 = 2τt+1 + z1,

where mt is non-labor income in period t .The maximization problems respectively, are given by

maxz0,t

log(w0,t − τt − z0,tqt) + β log(w1,t+1 − τt+1 + z0)

and

maxz1,t

log(w1,t − τt + mt − z1,tqt) + β log(2τt+1 + z1).

The resulting first-order conditions yield

z0,t = β(w0,t − τt) − qt(w1,t+1 − τt+1)

qt(1 + β)and z1,t = β(w1,t − τt + mt) − qt2τt+1

qt(1 + β).

Market clearing requires that z0,t + z1,t = 1. Hence, we get

qt = β(w0,t + w1,t + mt − 2τt)

(1 + β) + τt+1 + w1,t+1, (14)

implying that the asset price is increasing in current income and decreasing in future income. It is straightforward to verifythat

∂qt

∂ w j,t

w j,t

qt= w j,t

w0,t + w1,t + mt − 2τt, j = 0,1

and

∂qt

∂mt

mt

qt= mt

w0,t + w1,t + mt − 2τt.

Hence, q responds relatively more to changes in wages and non-labor income when the tax rate (τt) is higher.


Table 10The relevance of complete markets.

System 2 aggregate states 4 aggregate states

No PAYGO PAYGO No PAYGO PAYGO

Mean prm/bond 0.53 0.43 0.75 1.79corr(c0, C) 0.9975 1.000 0.95 0.90corr(c1, C) 0.9979 1.000 0.96 0.94corr(c2, C) 0.9926 1.000 0.95 0.95

Note: The PAYGO system uses constant lump-sum taxes. With two states, dividends and wages are perfectly negatively correlated, whereas with four statesthe correlation is close to zero.

C.2. Case 2: Proportional taxes

Here, the four budget constraints are instead given by

c0,t = w0,t(1 − τt) − z0,tqt; c1,t+1 = w1,t+1(1 − τt+1) + z0,t;c1,t = w1,t(1 − τt) + mt − z1,tqt and c2,t+1 = τt+1(w0,t+1 + w1,t+1) + z1,t .

Maximizing utilities and imposing market clearing gives

qt = β[(w0,t + w1,t)(1 − τt) + mt](1 + β) + τt+1 w0,t+1 + w1,t+1

.

It then follows that

∂qt

∂ w j,t

w j,t

qt= w j,t(1 − τt)

(1 − τt)(w0,t + w1,t) + mt, j = 0,1 (15)

and

∂qt

∂mt

mt

qt= mt

(1 − τt)(w0,t + w1,t) + mt. (16)

Hence, with proportional lump-sum taxes, q responds relatively less to changes in wages and relatively more to changesin non-labor income, the higher the tax rate. If mt is also taxed proportionally at the same tax rate, Eqs. (15) and (16)instead become ∂q

∂ w j

w jq = w j

w0+w1+mt, and ∂q

∂mmq = m

w0+w1+mt. Then the tax rate does not affect the elasticities of the asset

price with respect to income.

Appendix D. Complete markets

Table 10 shows the effects of a PAYGO system on the equity premium in economies that differ with respect to marketcompleteness. Specifically, the table compares results in economies with only two aggregate states to results in economieswith four aggregate states. When there are two aggregate states, markets are close to complete, but not fully. There willalways be a degree of incompleteness in OLG models due to the fact that only agents who are alive at a certain point intime can trade. With two states, a PAYGO system then has only has a marginal effect on the equity premium (in fact, theeffect is slightly negative).

Intuitively, when markets are (almost) complete, agents basically consume a share of aggregate consumption and sinceaggregate income is exogenous in the model, the PAYGO system cannot increase the volatility of aggregate consumption.Hence, social security is of little importance in such a case. As in Table 7, the returns are more volatile with social securityalso when markets are close to complete, but then the correlations between age-specific consumption and asset returns arelow. In fact, markets are actually somewhat more complete with a PAYGO system, because it provides the agents with onemore asset (see Krueger and Kubler, 2006).56

When markets are less complete, i.e., with four aggregate states, agents can no longer share age-specific risk efficiently,thus implying that individual consumption is only imperfectly correlated with aggregate consumption. Instead, agents haveto rely on assets to reduce the volatility of their consumption. This implies a higher correlation between individual con-sumption and the return to individual assets, and that more volatile returns translate into more volatile consumption. Inaddition, policies that transfer risk and resources between agents cannot be fully “undone”. The PAYGO system may stillhave a positive effect and make markets somewhat more complete but, quantitatively, this effect is small (i.e., σRe and σRb

increase by more than σciand ρc,R decrease for one or more agents after the introduction of a social security system).

56 Markets are more complete in the sense that the correlations between individual and aggregate consumption are higher.



Constraints Unconstrained Constrained

PAYGO system A D A D

Mean T-bill return -1.23 1.88 0.03 0.10Mean equity return 3.09 6.62 4.80 8.10Mean prm/T-bills 4.32 4.74 4.77 7.99

Note: A—no social security, B—safe benefit.

Appendix E. T-bills instead of bonds

Since the equity premium literature has focused on stocks and bills, I also outline model results for bills and equityinstead of bonds and equity. Specifically, the holder of a T-bill is entitled to a payoff b in the subsequent period. Thegovernment issues one T-bill in each period and, as with bonds, the payment b is financed through taxes. The revenuesfrom the T-bill are redistributed to the old, so as to make the arrangement resemble government debt. The results arepresented in Table 11.

As can be seen, the effects of the PAYGO system increase the returns and the equity premium when agents hold equityand T-bills.

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