Moral Hazard, Risk Sharing, and the Optimal Pool Size

• Author: Frauke Bieberstein,Jörg Schiller,Florian Kerzenmacher,Eberhard Feess,José F. Fernando
• Date: 01 June 2019
• DOI: http://doi.org/10.1111/jori.12211
• Published date: 01 June 2019

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©2017 The Journal of Risk and Insurance (2017).

DOI: 10.1111/jori.12211

Moral Hazard, Risk Sharing, and the

Optimal Pool Size

Frauke von Bieberstein

Eberhard Feess

José F. Fernando

Florian Kerzenmacher

Jörg Schiller

Abstract

We examine the optimal size of risk pools with moral hazard. In risk pools,

the effective share of the own loss borne is the sum of the direct share (the

retention rate) and the indirect share borne as residual claimant. In a model

with identical individuals with mixed risk-averse utility functions, we show

that the effective share required to implement a specific effort increases in

the pool size. This is a downside of larger pools as it, ceteris paribus, reduces

risk sharing. However, we find that the benefit from diversifying the risk in

larger pools always outweighs the downside of a higher effective share. We

conclude that, absent transaction costs, the optimal pool size converges to

infinity. In our basic model, we restrict attention to binary effort levels, but

we show that our results extend to a model with continuous effort choice.

Introduction

Formal risk pools such as mutual insurance arrangements, partnerships of lawyers,

farmers, and physicians benefit from risk sharing, but are encumbered by free riding

(moral hazard). In contrast to traditional insurance arrangements where risks are

Frauke von Bieberstein is at the University of Bern, IOP, Engehaldenstr. 4, Bern 3012,

Switzerland. Bieberstein can be contacted via e-mail: vonbieberstein@iop.unibe.ch. Eberhard

Feess is at the Frankfurt School of Finance & Management, Sonnemannstr.9-11, Frankfurt 60314,

Germany.Feess can be contacted via e-mail: e.feess@fs.de. José F.Fernando is at the Universidad

Complutense de Madrid, Facultad de Matemáticas, Departamento de Álgebra, Plaza de Cien-

cias, 3, Madrid 28040, Spain. Fernando can be contacted via e-mail: josefer@mat.ucm.es. Florian

Kerzenmacher is at the Frankfurt School of Finance & Management, Sonnemannstr.9-11, Frank-

furt 60314, Germany. Kerzenmacher can be contacted via e-mail: f.kerzenmacher@fs.de. Jörg

Schiller is at the Universitaet Hohenheim, Chair in Insurance and Social Systems, Fruwirthstr.

48, Stuttgart 70593, Germany.Schiller can be contacted via e-mail: j.schiller@uni-hohenheim.de.

We are grateful to Harris Dellas, Marc Möller, Martin Nell, Richard Peter, Harris Schlesinger,

and two anonymous referees for valuable comments.

Vol. 86, No. 2, 297–313 (2019).

2The Journal of Risk and Insurance

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transferred to an insurance company and its stockholders, the members of risk pools

are the residual claimants of the transferred risks. Our article addresses the following

question: In a world with independent risks, risk-averse participants, moral hazard,

and perfect enforceability of contracts, does the optimal size of the risk pool converge

to infinity if the risk-sharing arrangement is properly designed?

As a first intuition, the answer would be a straightforward “yes” due to better risk

sharing in larger pools. However, the issue is more involved in case of moral hazard.

In risk pools, part of the own loss is borne directly as a retention rate, and an additional

share indirectly as residual claimant. We will refer to the sum of these two parts as

the effective share. As the part of the own loss borne as residual claimant decreases in

the pool size, the retention rate needs to increase in order to keep effort incentives

constant. For the special case with linear marginal utility such as quadratic utility

functions, we show that it suffices to increase the retention rate to an extent that keeps

the effective share constant. It is then, indeed, straightforward to show that the utility

increases in the pool size due to better risk sharing when effort incentives are kept

constant.

For individuals with mixed risk-averse utility functions where higher-order deriva-

tives weakly alternate in sign (see, e.g., Caballé and Pomansky, 1996),1however, im-

plementing the high effort requiresthat the effective share increases in the pool size. In

other words, a larger part of the own loss needs to be borne by each individual. To see

the reason, consider the case with a binary effort choice. A binding incentive compati-

bility constraint (ICC) for choosing the high effort requiresthat the expected utility dif-

ferencewithoutown loss and with own loss isconstant in the pool size, and equalto the

cost difference of high and low efforts. Due to the diversification effect of larger pools

(mean-preserving contraction), extremelylow income levels become less likely even in

case with own loss. For individuals with mixed risk-averse utility functions, this ceteris

paribus decreases the incentive to avoid the own loss and thereby also decreasesthe in-

centive to choose the high effort. This incentive-reducing impact of larger pools needs

to be balanced by a higher effective share. This yields a countervailing effectto the ben-

efits of risk sharing in larger pools, so that it is ex ante unclear whether larger pools are

superior.

Our main result is that the benefit from improved risk sharing in larger pools always

dominates. The intuition is that the higher effective share is only needed because the

expected utility in case with an own loss increases faster than in the case without

own loss. Thus, the higher effective share just redistributes a part of the utility gain

from larger pools to the case without own loss, so that the expected utility difference

between the two cases remains constant. This ensures that the ICC is binding and that

expected utility increases in both states of the world. In our main model, we consider

only two effort levels, but we show that the superiority of larger pools carries over to

the case of continuous effort. The intuition follows from the fact that our finding does

1Mixed risk-averse utility functions include most of the commonly used von Neumann–

Morgenstern utility functions (see Eeckhoudt and Schlesinger, 2006).