Endogenous Insolvency in the Rothschild–Stiglitz Model

• Published date: 01 March 2019
• DOI: http://doi.org/10.1111/jori.12206
• Author: Achim Wambach,Wanda Mimra
• Date: 01 March 2019

©2017 The Journal of Risk and Insurance. Vol.86, No. 1, 165–181 (2019).

DOI: 10.1111/jori.12206

Endogenous Insolvency in the

Rothschild–Stiglitz Model

Wanda Mimra

Achim Wambach

Abstract

Even 30 years after Rothschild and Stiglitz’s (1976) seminal work on competi-

tive insurance markets with adverse selection, existence and characterization

of the equilibrium outcome are still an open issue. We model a basic exten-

sion to the Rothschild and Stiglitz model: we endogenize up-front capital

of insurers. Under limited liability, low up-front capital gives rise to an ag-

gregate endogenous insolvency risk, which introduces an externality among

customers of an insurer (Faynzilberg, 2006). It is shown that an equilibrium

with the second-best efficient Miyazaki–Wilson–Spence allocation always

exists.

Introduction

The Rothschild and Stiglitz (1976) (RS) model on competitive insurance markets with

adverse selection is considered as one of the major contributions to the literature on

markets with asymmetric information to date. Yet, the RS model entails a puzzle: if

the share of high-risk types in the population is low,an equilibrium in pure strategies

fails to exist. The reason is that if there are relatively few high-risk types, a pooling

contract would be preferred by all types over the candidate separating, contract-wise

zero profit-making RS contracts. However, a pooling contract cannot be tendered in

equilibrium as insurers would try to cream skim low risks.1This equilibrium inex-

istence result has spurred extensive research, and although several modifications to

the RS model have been brought forward, the debate is still ongoing and has gained

in interest in recent years.2

Wanda Mimra is at the ETH Zurich, CER-ETH, Zuerichbergstr, Zurich, Switzerland. Mimra

can be contacted via e-mail: wmimra@ethz.ch. Achim Wambach is at the Centre for

European Economic Research (ZEW), Mannheim, Germany. Wambach can be contacted via

e-mail: wambach@zew.de. We would like to thank Vitali Gretschko, Christian Hellwig, Wolf-

gang Leininger, Pierre Picard, Ray Rees, and two anonymous referees and the editor for very

helpful comments and discussions.

1The original RS model assumes that insurance firms only offersa single contract each. However,

if firms are allowed to offer contract menus, the equilibrium nonexistence problem is even

aggravated as now cross-subsidizing contracts can pose a threat to the RS equilibrium.

2A review of the literature is given in Mimra and Wambach(2014).

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166 The Journal of Risk and Insurance

In the literature, mainly two candidates for the equilibrium allocation have emerged

in competitive insurance markets with adverse selection. One is the original RS al-

location, even if it is not an equilibrium in the RS model (e.g., Riley, 1979; Engers

and Fernandez, 1987; Inderst and Wambach, 2001; Ania, Tr¨

oger, and Wambach, 2002;

Mimra and Wambach, 2016). The second is the Miyazaki–Wilson–Spence (MWS) al-

location (e.g., Asheim and Nilssen, 1996; Mimra and Wambach, 2011; Netzer and

Scheuer, 2010; Netzer and Scheuer, 2014; Picard, 2014). Now the RS allocation is only

efficient if it is an equilibrium in the RS model with contract menus, whereas the MWS

allocation is generally second-best efficient; thus, it matters which allocation prevails.

Besides equilibrium existence, we therefore also contribute to the discussion on the

necessity of efficiency-based regulation of insurance markets by providing another

foundation for the MWS allocation.

We model a basic extension to the RS model: instead of being exogenously endowed

with sufficiently high assets as in RS, insurers choose the level of up-front capital

before entering the market stage. Now under limited liability, low up-front capital

gives rise to an endogenous insolvency risk as, depending on contract offers and the

distribution of risk types over the contracts of an insurer, there might not be suffi-

cient assets to fulfill all claims.3Endogenous insolvency risk introduces an externality

among customers of a firm as an individual’s expected utility now does not only de-

pend on contract parameters but also on the risk profile of the other customers of

the insurer. This externality guarantees equilibrium existence—we show that with

capital choice under limited liability, an equilibrium in pure strategies that yields the

second-best efficient MWS allocation exists. The MWS contracts are separating and

cross-subsidizing; high-risk types are fully insured at a more than fair premium and

low risks are partially insured at an unfair premium offsettinglosses from high risks.4

The equilibrium is sustained as cream-skimming offers aimed at attracting low risk

types lead to a deterioration of high risks’ MWS contract due to insolvency when

up-front capital is low.High risks then prefer to choose the deviating contract as well,

rendering the deviation unprofitable. Now putting in more capital is not profitable

for an insurer,as this only increases incentives of competitors to cream skim low risks

away from this insurer. Hence, low up-front capital and the MWS allocation are an

equilibrium outcome. One implication of our analysis is that minimum capital re-

quirements might have unintended consequences: if imposed capital requirements

are too strict, a second-best efficient equilibrium fails to exist.

Webuild upon the work by Faynzilberg (2006). Faynzilberg discusses the set-up with

exogenous capital and argues that an equilibrium with the MWS allocation exists.

3Insolvency has been analyzed in insurance markets without adverse selection. Doherty and

Schlesinger (1990) model insurance demand under an exogenous insolvency risk and show

that less than full insurance will be purchased at the actuarially fair premium if default is total;

however, if default is partial, overinsurance might occur and there is generally no monotonic

relationship between default payout rate and insurance coverage.

4We concentrate on the case with a low share of high risks where the MWS contracts do not

coincide with the RS contracts and exhibit the above features. The analysis is trivial for the

case where MWS and RS contracts coincide.