Equilibrium in Insurance Markets With Adverse Selection When Insurers Pay Policy Dividends

• DOI: http://doi.org/10.1111/jori.12243
• Author: Pierre Picard
• Date: 01 December 2019
• Published date: 01 December 2019

©2018 The Journal of Risk and Insurance (2018).

DOI: 10.1111/jori.12243

Equilibrium in Insurance Markets With Adverse

Selection When Insurers Pay Policy Dividends

Pierre Picard

Abstract

We show that an equilibrium always exists in the Rothschild–Stiglitz in-

surance market model with adverse selection and an arbitrary number of

risk types, when insurance contracts include policy dividend rules. The

Miyazaki–Wilson–Spence state-contingent allocation is an equilibrium allo-

cation (defined as a set of type-dependent lotteries sustained at a symmetric

equilibrium of a market game), and it is the only one when out-of-equilibrium

beliefs satisfy a robustness criterion. It is shown that stock insurersand mutu-

als may coexist, with stock insurers offering insurance coverage at actuarial

price and mutuals cross-subsidizing risks.

Introduction

The fact that no equilibrium may exist in the Rothschild–Stiglitz (1976) model of

insurance markets under adverse selection has been at the origin of an abundant lit-

erature in economic theory. In one way or another, most articles in this area have

moved away from the basic premise of the Rothschild–Stiglitz approach. This ap-

proach consisted of modeling the strategic interactions between insurers who simul-

taneously offer contracts under hidden information about the risk types of insurance

seekers.

An important avenue of research that follows the seminal contribution of Rothschild

and Stiglitz (1976) has its origin in the article by Wilson (1977). It focuses attention on

competitive mechanisms when insurers interact in a dynamic way. This includes the

“anticipatory equilibrium” of Miyazaki (1977), Wilson (1977), and Spence (1978); the

“reactive equilibrium” of Riley (1979); and the variations on the equilibrium concept

introduced by Hellwig (1987) and Engers and Fernandez (1987); and in more recent

Pierre Picard is at the CREST research center of Ecole Polytechnique (France). He can be con-

tacted via e-mail: pierre.picard@poytechnique.edu. This article has benefited from comments

and suggestions at various stages of its preparation. Special thanks are due to a referee whose

constructive insights were particularly helpful. The author is also grateful to Richard Arnott,

Renaud Bourl`

es, Keith Crocker,Georges Dionne, Rida Laraki, WandaMimra, Patrick Rey, Casey

Rothschild, Bernard Salani´

e, Franc¸ois Salani´

e, Arthur Snow,and Achim Wambach, for valuable

exchanges of views.

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. Vol. 86, No. 4, 887–914 (2019).

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articles surveyed by Mimra and Wambach(2014), in particular, Mimra and Wambach

(2011) and Netzer and Scheuer (2014). Another line of research, illustrated by the

works of Dubey and Geanakoplos (2002) and Bisin and Gottardi (2006) among others,

departs from the strategic dimension and considers atomistic insurance markets under

adverse selection in line with the approach by Prescott and Townsend (1984). Unlike

these two strands of research,1our purpose is to reexamine the equilibrium issue in a

perspective that remains framed within the initial Rothschild–Stiglitz approach. This

requires a few preliminary explanations.

Rothschild and Stiglitz (1976) consider a simple setting in which each insurer is con-

strained to offering a single contract, with a free entry equilibrium concept, but they

emphasize that such an equilibrium could be viewed as a Nash equilibrium of a game

in which insurers interact by offering contracts simultaneously. They also note that a

next step is to test a less restrictive definition of insurers’ strategies. In particular,they

observe that allowing insurers to offer menus of contracts would make the condition

under which an equilibrium exists even more restrictive. When commenting on the

approach by Wilson (1977), they note that “the peculiar provision of many insurance

contracts, that the effective premium is not determined until the end of the period

(when the individual obtains what is called a dividend), is perhaps a reflection of

the uncertainty associated with who will purchase the policy, which in turn is asso-

ciated with the uncertainty about what contracts other insurance firms will offer.”

In other words, many insurance contracts, mostly those offered by mutuals, have a

participating dimension that should not be ignored when we seek to understand how

competition works in the real word.2

Our objective in the present article is to move forward in that direction.

Picard (2014) studies how allowing insurers to offer either participating or

nonparticipating contracts, or in other words to act as mutuals or as stock

1The fact that there may be no equilibrium in the Rothschild–Stiglitz model is related to the

discontinuity of insurers’ payoff functions, as small changes in their contract offers may lead

all individuals of a given type to switch to other insurers, with a possible jump in the in-

surers’ expected profits. Dasgupta and Maskin (1986a, 1986b) establish existence theorems

for mixed-strategy equilibria in a class of games where payoff functions have discontinu-

ity points, and as shown by Rosenthal and Weiss (1984) in the case of the Spence model of

education choices, such a mixed strategy equilibrium exists in the Rothschild–Stiglitz insur-

ance market model. Recently,Luz (2017) has fully characterizes these mixed-strategy equilib-

ria in this model. However, assuming that firms play mixed strategies at the contract offer

stage has not been considered as reasonable in most of the literature on markets with ad-

verse selection. In addition, as shown by Rosenthal and Weiss (1984), at a mixed-strategy

equilibrium, a potential entrant could make positive profit. This reinforces the fundamental

conclusion of Rothschild and Stiglitz, that is, that an entry-deterring equilibrium may not

exist.

2Mutuals differ according to the role of the premium chargedat the start of each policy period.

“Advance premium mutuals” set premium rates at a level that is expected to be sufficient

to pay the expected losses and expenses while providing a margin for contingencies, and

policyholders usually receive dividends. In contrast, “assessment mutuals” collect an initial

premium that is sufficient only to pay typical losses and expenses and levy supplementary

premiums whenever unusual losses occur.

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