IDIOSYNCRATIC RISK, AGGREGATE RISK, AND THE WELFARE EFFECTS OF SOCIAL SECURITY

• Author: Alexander Ludwig,Daniel Harenberg
• Published date: 01 May 2019
• Date: 01 May 2019
• DOI: http://doi.org/10.1111/iere.12365

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 2, May 2019 DOI: 10.1111/iere.12365

IDIOSYNCRATIC RISK, AGGREGATE RISK, AND THE WELFARE EFFECTS

OF SOCIAL SECURITY∗

BYDANIEL HARENBERG AND ALEXANDER LUDWIG1

Center for Economic Research, ETH Z ¨

urich, Switzerland, Oxford Economics Ltd., Germany;

SAFE, Goethe University Frankfurt, Germany

We ask whether a pay-as-you-go financed social security system is welfare improving in an economy with

idiosyncratic and aggregate risk. We show that the whole welfare benefit from insurance against both risks

is greater than the sum of benefits from insurance against the isolated risks. One reason is the convexity

of the welfare gain. The other reason is a direct risk interaction amplifying the utility losses from risk. Our

quantitative evaluation shows that introducing a minimum pension leads to sizeable welfare gains, despite

substantial crowding out. About 60% of these gains would be missing from summing up the isolated benefits.

1. INTRODUCTION

Many countries operate large social security systems. One reason is that social security can

provide insurance against risks for which there are no private markets. However, these systems

also impose costs by distorting prices and decisions. The question arises whether the benefits of

social security outweigh the costs.

We address this question in a model economy featuring two types of risk, aggregate business

cycle risk in the form of aggregate wage and asset return risk as well as idiosyncratic productivity

risk. We follow the literature and assume that insurance markets for both types of risk are

incomplete. In such a setting, social security can increase economic efficiency by providing

partial insurance. However, it also distorts decisions leading to welfare losses from crowding

out (CO) of capital formation. Our analysis differs from the previous literature in that prior

studies characterized social security’s welfare effects in models with only one type of risk. One

strand of the literature examined social security when only aggregate risk (AR) is present, for

example, Krueger and Kubler (2006). In that setting, social security—by pooling aggregate wage

and asset return risks across generations—can improve intergenerational risk sharing. The other

strand only considered idiosyncratic risk (IR); see, for example, ˙

Imrohoro˘

glu et al. (1995, 1998)

and Conesa and Krueger (1999). There, social security provides intragenerational insurance

by redistributing ex post from high- to low-productivity households. Broadly speaking, both

strands of this literature conclude that the costs of introducing social security outweigh the

benefits.

∗Manuscript received June 2016; revised June 2018.

1An earlier version of this paper circulated under the title “Social Security and the Interactions between Aggregate

and Idiosyncratic Risk.” We thank the editor Harold Cole and three anonymous referees as well as Klaus Adam, Alan

Auerbach, Martin Barbie, Tino Berger, Antoine Bommier, Johannes Brumm, Georg D¨

urnecker, Ays¸e Imrohoro ˘

glu,

David Jaeger, Philip Jung, Tom Krebs, Dirk Krueger, Per Krusell, Felix Kubler, Marten Hillebrand, Mich´

ele Tertilt,

Fabrizio Zilibotti, and various seminar participants at several places for helpful discussions. Daniel Harenberg gratefully

acknowledges financial support by the German National Research Foundation (SFB 884), by Swiss Re Foundation,

and by ETH Z ¨

urich Foundation. Alex Ludwig gratefully acknowledges financial support by the German National

Research Foundation (SPP 1578) and by the Research Center SAFE, funded by the State of Hessen initiative for

research LOEWE. Please address correspondence to: Alexander Ludwig, SAFE House of Finance, Goethe University

Frankfurt, Theodor-W-Adorno-Platz 3, Frankfurt am Main, Hessia 60629, Germany (DE). E-mail: ludwig@safe.uni-

frankfurt.de.

661

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

662 HARENBERG AND LUDWIG

Such a segregated view is incomplete because households face both types of risk over the life

cycle and because social security, when appropriately designed, can (partially) insure both types

of risk. We also argue that simply combining the findings from previous studies leads to severe

biases in the overall welfare assessment. Our theoretical contribution is to show analytically

why the whole insurance benefit exceeds the sum of the benefits from insurance against isolated

risk components. Our quantitative contribution is to establish that joint insurance against both

types of risk leads to large net welfare gains, thereby turning previous findings in the literature

upside down: social security is welfare improving from the ex ante perspective.

We emphasize that two important biases emanate from simply combining previous findings.

The first arises even when the two types of risk are statistically independent. This bias is a

consequence of the convexity of the welfare gain (CWG) in total risk. The welfare gain is convex

in the amount of total risk because the marginal utility of insurance increases disproportionately

as risk increases. Joint presence of idiosyncratic productivity and aggregate business cycle risk

strongly fans out the earnings and consumption distributions. If social security is designed as

a Beveridgean system with flat pension benefits, it provides partial insurance against this total

life-cycle risk exposure. Because of CWG, the whole benefit from insurance is therefore greater

than the sum of benefits from insurance against the single risk components. We show that this

difference in welfare assessments, the “CWG bias,” increases in the total amount of risk. Since

total life-cycle risk is large, we can expect this bias to be large.

The second bias stems from a direct interaction of risks in form of a countercyclical cross-

sectional variance (CCV ) of idiosyncratic productivity risk: The variance of persistent idiosyn-

cratic shocks is higher in a downturn than in a boom. The CCV has been documented in the data

(Storesletten et al., 2004b) and analyzed with respect to its asset pricing implications (Mankiw,

1986; Constantinides and Duffie, 1996; Storesletten et al., 2007).2It leads to a high variance of

the idiosyncratic income component when the aggregate income component is low. Due to con-

cavity of the utility function, this amplifies the welfare gains from insurance against both risks.

To expose these biases, we start our analysis by employing an analytically tractable two-period

life-cycle model in which a household faces idiosyncratic and aggregate wage risk in the first

period of life. In the absence of social security, retirement consumption is financed by private

savings that bear aggregate return risk. We study the welfare consequence of introducing a

pay-as-you-go (PAYG) financed social security system with flat, unconditional pension pay-

ments. By pooling idiosyncratic wage risks within and ARs across generations, this Beveridgean

system jointly provides partial insurance against IR and AR. We measure welfare gains by a

consumption equivalent variation (CEV). Abstracting from CCV , we derive a term capturing

the welfare difference between the whole insurance benefit and the sum of the benefits from

insurance against the isolated risk components. This difference reflects the CWG bias. We sub-

sequently modify the two-period model to account for the CCV mechanism and show how an

additional welfare difference emerges.

Our arguments so far ignore behavioral reactions, that is, the reduction of savings caused

by social security. In general equilibrium (GE), this savings reaction leads to crowding out of

aggregate capital, which entails corresponding changes in relative prices. Therefore, both the

sign and the size of the welfare effects of introducing social security in a model with both risks

have to be determined in a quantitative GE analysis.

To conduct such a quantitative analysis, we build a large-scale overlapping generations (OLG)

model in the tradition of Auerbach and Kotlikoff (1987), extended by idiosyncratic productivity

risk and aggregate wage and asset return risk. Households can save privately by investing in

a risk-free bond and a risky stock. Including this portfolio choice is important. It allows us

to appropriately calibrate the risk–return structure of the private savings technologies, which

directly affects the value of social security. The possibility to save in two assets also implies

2Based on Guvenen et al. (2014), a recent paper by Busch and Ludwig (2017) finds that in addition to the variance,

the skewness of persistent idiosyncratic shocks is countercyclical. Adding such a countercyclical left-skewness would

strengthen our results, because more households would find themselves in situations with high marginal utility.

WELFARE EFFECTS OF SOCIAL SECURITY 663

that households have additional means of self-insurance. In our computational experiment,

we consider a stylized social security reform by introducing a pure Beveridgean PAYG social

security system—like in our analytic two-period model—with a contribution rate of 2%. This is

the size of the U.S. system when first introduced in 1935. We hence study the welfare implications

of introducing a flat rate minimum pension.3

By calibrating the model to the U.S. economy, we find that such a marginal introduction

of social security leads to a strong welfare gain of 2.6% in terms of a CEV. This welfare

improvement is obtained because strong partial equilibrium (PE) insurance gains of 5.2%

outweigh the substantial welfare losses from CO of capital of 2.6% in GE. Our key finding of

net welfare gains stands in stark contrast to the previous literature. When instead replicating

the earlier literature by considering economies with only one type of risk, we indeed observe

net welfare losses. We therefore conclude that it is of crucial quantitative importance to jointly

consider both risks.

To uncover the sources of the PE welfare gain of 5.2%, we decompose it into the components

that are attributable to insurance against the isolated risks as well as the two bias terms, CCV

and CWG. We find that the combined effect of the two bias terms scales up the PE welfare

gains by 60%. This strong effect reemphasizes our key finding on the quantitative importance

of jointly considering both risks. Finally, we investigate how much of the GE welfare effects

stem from changes in mean consumption and from changes in the intra- and intergenerational

distribution of consumption.

The notion that social security can insure against ARs dates back to Diamond (1977) and

Merton (1983). They demonstrate how it can partially complete financial markets, thereby

increasing economic efficiency. Building on these insights, Shiller (1999) and Bohn (2001, 2009)

show that social security can reduce consumption risk of all generations by pooling labor income

and capital income risks across generations. Gordon and Varian (1988), Matsen and Thogersen

(2004), Krueger and Kubler (2006), and Ball and Mankiw (2007) use a two-period PE model

in which households only consume in the second period of life, that is, during retirement. For

our analytical results, we extend this model by adding IR. Among the few quantitative papers

with AR and social security, Krueger and Kubler (2006) is the most similar to our work. They

conclude that the introduction of a small PAYG system does generally not constitute a Pareto

improvement.4The concept of a Pareto improvement requires that they take an ex interim

welfare perspective, whereas we calculate welfare from an ex ante perspective. Our analysis

also differs substantially because we include IR and analyze interactions between the risks.5

Many quantitative papers consider IR and social security, for example, Conesa and Krueger

(1999), ˙

Imrohoro˘

glu et al. (1995, 1998), Huggett and Ventura (1999), and Storesletten et al.

(1999). One general conclusion from this literature is that welfare in a stationary economy

without social security is higher than in one with a PAYG system. More recently, Nishiyama

and Smetters (2007), Fehr and Habermann (2008), and Golosov et al. (2013) focus on modeling

the institutional features of existing social security systems in detail, which we abstract from.

Our results demonstrate the benefits of a flat minimum pension. Like all these papers, we

conduct a limited policy design experiment by restricting attention to insurance through the

social security system, taking insurance through taxes and transfers during the working period

as given.6

3Nearly, all OECD countries feature either a minimum or a basic pension that is independent of previous income

(OECD, 2015).

4The recent work by Hasanhodzic and Kotlikoff (2015) mirrors these findings.

5Other related papers are Ludwig and Reiter (2010), who assess how pension systems should optimally adjust

to demographic shocks, Olovsson (2010), who contends that pension payments should be highly risky because this

increases precautionary savings and thereby capital formation, Peterman and Sommer (2015, 2016), who discuss the

insurance benefits of social security in the Great Depression and the Great Recession, respectively, modeling each

event as a one-time macroeconomic shock, and, finally, Gomes et al. (2012), who use a model similar to ours to study

how changes in fiscal policy and government debt affect asset prices and the wealth distribution.

6Huggett and Parra (2010) point out the importance of analyzing the optimal design of social security and the income

tax system jointly.