Optimal Social Insurance for Heterogeneous Agents With Private Insurance

• DOI: http://doi.org/10.1111/jori.12050
• Author: Brandon Lehr
• Date: 01 June 2016
• Published date: 01 June 2016

©2014 The Journal of Risk and Insurance. Vol.83, No. 2, 301–333 (2016).

DOI: 10.1111/jori.12050

Optimal Social Insurance for Heterogeneous

Agents With Private Insurance

Brandon Lehr

Abstract

This article analytically characterizes optimal social insurance in an economy

with both ex ante heterogeneity and ex post risk, decomposing the benefits

of social insurance into a redistributive and insurance benefit. Agents exert

effort to increase the likelihood of high outcome events and are additionally

supplied actuarially fair private insurance for this earnings risk. This article is

novel in its joint consideration of two sources of heterogeneity, two potential

sourcesof insurance, and an endogenous ex post distribution of outcomes. The

introduction of optimal private insurance eliminates the insurance benefit of

social insurance, but leaves the redistributive benefit intact. An income effect

induced by the crowding out of private insurance generates an additional

benefit to social insurance when it takes the form of a linear income tax.

Finally,numerical simulations illustrate how the relative contributions of ex

ante and ex post risk differentially impact the welfare loss associated with

setting optimal social insurance without recognizing the presence of private

insurance.

Introduction

Social insurance provides the government with the ability to insure individuals

against risk and also to redistribute across agents in the economy. To account for

each of these effects in an economic model, we must allow for an ex ante heteroge-

nous population of individuals, each of whom face idiosyncratic risk. Moreover, it

is reasonable to assume that in a market economy the social planner cannot directly

control or exclude the provision of actuarially fair private insurance, which is of value

to each individual. Thus, it is important for the government to internalize the en-

dogenous response of private insurance markets and individual behavior to social

policy.1Failure to recognize the presence of privately provided insurance can lead to

incorrect prescriptions for optimal policy that may result in substantial welfarelosses.

This article therefore takes up the task of characterizing, analytically and numerically,

Brandon Lehr is at the Department of Economics, Occidental College. Lehr can be contacted

via e-mail: lehr@oxy.edu.Thanks for helpful comments and suggestions from Emmanuel Saez,

Peter Diamond, Amy Finkelstein, Ivan Werning, the participants at the MIT Public Finance

Seminar, WEAI Conference 2012, and two anonymous referees.

1Chetty and Saez (2010) make this observation in a setting with only one source of heterogeneity.

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the impact of introducing both ex ante heterogeneity across agents and ex post risk on

optimal social insurance formulas in the presence of endogenously supplied private

insurance.

Specifically,this article builds a model of the economy in which a heterogeneous pop-

ulation of workers who differ with respect to productivity each face an ex post shock

to earnings. Workers exert effort to increase the likelihood of a high shock. This ef-

fort is unobservable to insurers, both private and public, generating moral hazard.

The important assumption, however, is that private insurers have better information

than the government and in particular can observe the productivity of each agent.

Thus, private insurance contracts are type contingent and vary across the population,

while a single social insurance contract applies to everyone. I consider two cases of

social insurance, depending on whether or not the social insurance contract can be

state dependent for an individual. For example, social insurance for unemployment

is state dependent since unemployment is verifiable, whereas in the case of produc-

tivity shocks, social insurance in the form of an income tax is state independent. In

addition, I conclude with numerical simulations for the income tax as social insurance

case.

I establish four main results in this article: two analytically and two via numerical

simulations. First, without private insurance, expressions are derived decomposing

the social insurance benefit into separable insurance and redistributive components.

In the state-dependent social insurance setting, the redistributive benefit, which de-

pends on the covariance between risk and income, is new relative to standard models

of social insurance (e.g., Baily,1978; Chetty, 2006) that assume a representative agent.

With ex ante heterogeneity, this redistributive benefit shows that the case for social

insurance is greater when risk and income are negatively correlated across the pop-

ulation. Intuitively, when higher ability types exhibit lower risk of negative income

shocks, the variance in the ex post income distribution is increased and a utilitarian so-

cial planner can increase welfare with greater redistribution. In the state-independent

social insurance setting (social insurance as linear income tax), however, it is the in-

surance benefit that is new relative to standard models of taxation (e.g. Saez, 2001).

By introducing ex post risk for each productivity type in the population, the income

tax achieves not only redistribution but provides a consumption-smoothing benefit

across states.

Second, I show that with the introduction of endogenously supplied optimal private

insurance, only the redistributive benefit of social insurance remains. This is due

to the fact that private insurance can redistribute consumption across states for each

individual, but only social insurance can redistribute across agents. This redistributive

benefit implies that even with optimal private insurance, it is still optimal for the

government to provide social insurance, as opposed to the case with only one source

of heterogeneity (as in Chetty and Saez, 2010). I also identify an income effect benefit to

the income tax as social insurance case in the presence of optimal private insurance.

Intuitively, a high government tax rate crowds out private insurance, reducing the

income of agents and subsequently increasing effort, thereby reducing the behavioral

distortion of the tax.

Optimal Social Insurance for Heterogeneous Agents 303

In addition to the analytical results, numerical simulations of optimal government tax

rates in the presence of endogenous taxable earnings help to illustrate the impact of ex

ante and ex post heterogeneity with and without private insurance. My third finding is

that when ex ante heterogeneity is large, optimal social insurance can be larger in the

presence of private insurance. This is due to the impact of insurance on effort, which

in turn determines the ex post distribution of earnings. Private insurance reduces

effort, increasing the likelihood of low states of nature, and increasing the benefits of

social insurance. Finally,I show how the difference between optimal social insurance

with and without private insurance depends on the ex ante variance of productivity

types. When ex ante heterogeneity is large, private firms provide little in the way of

insurance across the economy, so the optimal social insurance is largely unaffected by

the presence or absence of private insurance. However, when ex ante heterogeneity

is small, private insurance for a single individual, who is largely representative of

the average worker in the economy, obviates the need for social insurance across

individuals.

The analysis here extends a previous literature on optimal insurance. There exists a

class of models in which there is only a single source of heterogeneity. For example,

agents are ex ante identical but experience exogenous “luck," which introduces uncer-

tainty, and taxation therefore serves as social insurance (Diamond et al., 1980; Eaton

and Rosen, 1980; Varian,1980). Subsequent extensions in this setting include an exam-

ination of the role of commitment to consumption or labor prior to the realization of

uncertainty (Cremer and Gahvari, 1999) and the introductionof endogenous outcomes

determined via effort provision (Low and Maldoom, 2004). Kaplow (1991) allows for

both private and social insurance, where unobservable actions impact the probability

of a high outcome event, but all agents are ex ante identical. Chetty and Saez (2010)

build a similar model with a single source of earnings heterogeneity, but they provide

expressions for the welfare gains from government intervention in the presence of

optimally and suboptimally provided private insurance. Another literature considers

optimal social insurance in the presence of two sources of heterogeneity due to dif-

ferences in earnings ability and exogenous ex post shocks to income. Mirrlees (1990)

is able to characterize the optimal linear tax in such a setting, while Rochet (1991)

and Cremer and Pestieau (1996) consider jointly optimal income taxation and social

insurance without moral hazard.

The article most closely related to the analysis of this article is Boadway et al. (2003)

in which the authors also consider two sources of heterogeneity and both social and

private insurance in the presenceof moral hazard. Their application is health insurance

and they allow for the government to offer both a linear income tax system and an

insurance program, with the assumption that insurance benefits and taxes are not

included in taxable income. This may be justifiable in the healthcare context, but is

certainly not a reasonable assumption when considering unemployment insurance

(UI), for which benefits are taxable, and more general insurance in the form of wage

compression for agents facing uncertain productivity outcomes. I restrict attention to

a single government social insurance program for two reasons. The first is that in the

second half of the article, the social insurance program is in fact a linear income tax,

making an additional income tax redundant. Second, although an income tax could be