Tax Sharing in Insurance Markets: A Useful Parameterization

• DOI: http://doi.org/10.1111/j.1539-6975.2013.01528.x
• Author: Christelle Viauroux
• Date: 01 December 2014
• Published date: 01 December 2014

©

DOI: 10.1111/j.1539-6975.2013.01528.x

907

Tax Sharing in Insurance Markets:

A Useful Parameterization

Christelle Viauroux

Abstract

We use a principal–agent framework to evaluate the economic impacts of

imposing a tax on insurance payment in presence of moral hazard using

a Gamma conditional distribution of losses. Our results show that any tax

paid by the insured would lower his effort to prevent loss, hence increasing

insurance payments and decreasing profits. This result is reinforced as the

insured becomes more risk averse unless the distribution of losses is uni-

form. We find that any decrease in the insurer’s tax share would generate

an overall decrease in welfare unless the insured characteristics prevent him

from reacting to the policy.

Introduction

During the past few years, much attention has been placed on possible revenue

sources to pay for health care. The recent political interest has been to tax insurance

companies: the Baucus plan levies a nondeductible excise tax of 40 percent on

insurance companies. In many insurance markets, the insurance provider determines

the premium based on a coverage amount chosen by the insured. As a result, a tax on

the premium amount corresponds to a proportional tax on the coverage amount. For

example, in the long-term care insurance market, the future insured is prompted to

choose a monthly benefit during the contract negotiation. In this article, we analyze

the economic impacts of imposing a tax on payments out to the insured. The question

then is: should the insurance provider pay the full amount of the tax as is proposed

or should the insured bear a portion of it? And in case the insured bears a portion

of it, should this portion be attributed uniformly across contract types? To answer

these questions, we generalize the analysis to a case where the tax could be shared

between the insurer and the insured.

Unfortunately,the presence of asymmetric information in insurance markets compli-

cates the analysis (Chiappori and Salanie, 1997, 2000; Dionne, Doherty,and Fombaron,

2000,Forthcoming;Villeneuve, 2000; Abbring, Chiappori,andPinquet, 2003, to name a

Christelle Viaurouxis in the Department of Economics at the University of Maryland, Baltimore

County. She can be contacted at ckviauro@umbc.edu. The author is thankful to G. Dionne,

Editor,and an anonymous referee for their very useful comments. She is also grateful to Vilmos

Komornik, Richard Pollard, and WendyTakacs for very helpful discussions and feedback.

1

The Journal of Risk and Insurance, 2013, Vol. 81, No. 4, 907–941

908 THE JOURNAL OF RISK AND INSURANCE

few). The lack of care enforcementon the part of the insurer, that is, moral hazard, com-

plicates the welfare outcome of any policy aiming at redistributing health care cove-

rage (see, e.g., Dionne et al., 1997; Doherty and Smetters, 2005, for empirical evidence

of moral hazard in insurance markets; see Ketsche, 2004, for an empirical analysis of

the impact of a subsidy on welfare). Indeed, the more health insurance an individual

acquires, the lower his risk. This improves the social welfare of those with greater

coverage. But as his coverage increases, the insured subsequently invests less in self-

protection and consequently increases his use of health care services. This increased

demand causes increases use of services, which results in higher prices, thus having

the opposing effect on social welfare. Because risk-averse consumers would not pur-

chase this additional care if they had to pay the full cost, the value of the extra service to

consumers falls short of the social cost of producing that care. Therefore,although risk

sharing increases social well-being, the change in moral hazard induces welfare loss.

Early theoretical studies on moral hazard in insurance markets (Arrow, 1963; Pauly,

1968; Shavell, 1979) proposed solutions to the moral hazard issue. The solutions inclu-

ded (1) an incomplete coverage against loss, which gives the individual an incentive

to prevent loss by exposing him to some financial risk (see Wang,Chung, and Tzeng,

2008; Chiappori et al., 1997; Koc, 2011, for empirical evidence of this solution in the

market for automobile insurance and for physician services insurance, respectively)

and (2) the observation of care, to link it to the premium or coverage in the event of a

claim. The impossibility to fully observe care, however, has led to an increasing lite-

rature on the design of optimal contracts (Lewis and Sappington, 1995; Winter, 1992,

2000; Gollier, 2000; Doherty and Smetters, 2005).

In this article, we use a framework allowing for a distribution of losses with care

reducing both the probability of a loss and the size of a loss (MasColell, Whinsten, and

Green, 1995).1In our model, an agent insures with a principal. Both have property

rights to an uncertain income stream that represents a possible loss from current

wealth. The random income stream depends on care or effort on the part of the agent,

to be taken in the future. The principal establishes a sharing rule on how to share

the random income stream. The principal–agent relationship involves moral hazard,

because the agent’s effort to avoid any loss is unobservable by the principal, whereas

it ultimately affects the expected profit. Therefore, the principal wants to use the

contract to induce the agent to exert optimal effort to invest in self-protection and/or

loss reduction.

1Economic models of moral hazard in insurance markets differ on their assumptions about the

impact of greater care on the distribution of losses faced by the insured (Winter, 2000). When

the loss is assumed to be random, the literature distinguishes the effects of moral hazard on

two types of expenditures on the part of the insured (Ehrlich and Becker,1972). The first type

of expenditures, called self-protection, reduces the probability of an accident. It refers to an

increase in the probability of a zero loss, with no change in the conditional distribution. The

second type, called loss reduction, refers to a first-order stochastic reductionwith no change in

the probability of a loss. In these two cases, the optimal insurance contract is characterized by a

premium and a coverage amount that may depend on the loss amount. Weuse a third approach

allowing for an arbitrary distribution of losses with care reducing both the probability of a loss

and the size of a loss.

TAX SHARING IN INSURANCE MARKETS 909

It is well known, however, that the principal–agent problem (Laffont and Martimort,

2002) is difficult to solve when effort is a continuous variable. Its tractability depends

on the ability to simplify an infinite number of global incentive constraints corres-

ponding to an infinite number of possible effort levels and replace them by a local

incentive constraint to induce the maximum effort from the agent. This last condi-

tion states that the agent is indifferent between choosing a given level of effort and

increasing it by a slight amount when he receives the risk premium. This so-called

“first-order approach” has been one of the most debated issues in contract theory.

Mirrlees (1975) shows that the problem may sometimes have no optimal solution

in the class of unbounded sharing rules. Rogerson (1985) shows that the “original”

first-order approach (with an infinite number of global incentive constraints) gives

the same solutions as the first-order approach when the following two properties are

satisfied. The first, the maximum likelihood ratio property (MLRP), ensures that the

agent is rewarded in the state of nature that is most informative in that he has exer-

ted positive effort. The second property is the convexity of the distribution function

condition (CDFC), which ensures that higher profit (of the principal) is a signal of

higher effort on the part of the agent. The problem is that the CDFC property is very

restrictive and tremendously limits the list of possible distributions that can be used.

Simple distributions, such as the exponential distribution function, do not verify the

latter property. Jewitt (1988), however, shows that the CDFC can be relaxed, provided

that the agent’s utility function satisfies certain properties.

We show that the Gamma conditional distribution of loss verifies the first-order vali-

dation conditions of Jewitt (1988). We characterize the optimal insurance contract in

presence of taxation using this distribution as well as a representation of the agent’s

preferences also satisfying Jewitt’s conditions. We then analyze how the tax affects

equilibrium outcomes. More specifically, we analyze how increasing the agent’s share

of the tax impacts his investment in self-protection/loss reduction, the principal’s ave-

rage payouts, the principal’s profit, and the overall social welfare. We simulate the

above effects for different measures of risk aversion of the agent and for different

characteristics of his preferences and his likelihood to contract with alternative pro-

viders. We present the characteristics of the optimal contract and the impact of an

increase in the insured tax share in case of a simple utility function. The closed-form

solutions to the problem emphasize the good properties of the Gamma distribution.

Still, we check the robustness of our results by using alternative loss distributions:

one proposed by Rogerson (1985) and one proposed by LiCalzi and Spaeter (2003).

To the best of our knowledge, this is the most general, yet most detailed, analysis of

the impact on welfare of imposing a tax on insurance output.

Our results show that maximum effort of the insured is achieved when the insurer

pays the full amount of the tax. An exception to the result occurs when losses happen

to be uniformly distributed. In this case, the insured may increase his effort because he

is unable to anticipate the size of his future losses and of his insurance tax amount. If

the tax share paid by the insuredwere to increase, we find average insurance payments

out to the insured would be higher, unless the provider recognizes the insured has

made only marginal effort to self-protect and the insured is uncertain about keeping

insurance with them. This suggests that the insurance provider would compensate