quality, self-regulation, and competition: the case of insurance

Insurance: Mathematics and Economics 32 (2003) 267–280

Quality, self-regulation, and competition: the case of insurance

Fredrik Anderssona,∗, Göran Skoghba Department of Economics, Lund University, P.O. Box 7082, S-220 07 Lund, Sweden

b Department of Management and Economics, Linköping University, Linköping, Sweden

Received 1 February 2000; received in revised form 1 March 2002; accepted 16 January 2003

Abstract

In this paper, insurers’ credibility problems explain contracting, co-operation, and regulation in the insurance industry.First, it is noted that cheating by policyholders may be eliminated if the insurer withholds compensation on the basis ofdetecting careless behaviour with high enough probability. Then, assuming that care taken is imperfectly observable andnon-contractible, the problem that insurers may deceive policyholders is addressed. In a repeated game, insurers’ buildinga reputation for beinggenerouscan sustain an efficient outcome. Finally, intra-industry co-operation is considered; it isshown that the industry’s monitoring care and “fair” compensation, while sustaining monopolistic pricing, mitigates negativecredibility externalities.© 2003 Elsevier Science B.V. All rights reserved.

JEL classification:D43; K20; L14; L15

Keywords:Generosity; Self-regulation; Credibility externalities; Competition

1. Introduction

An insurer is usually reserved the right to withhold or reduce compensation if the policyholder has been fraudulentor careless. The purpose of this rule is to limit cheating. It may, however, also result in the insurer deceiving thepolicyholder by withholding compensation in cases where correct information has been disclosed and due carehas been taken. Insurance is an extreme experience good in that the quality of the good cannot be ascertaineduntil claims are made; moreover, low-probability events result in few claims. The policyholders’ experience will,therefore, remain limited and one might expect cheating by insurers. The policyholder may go to court but thatis unusual; in practice, policyholders usually accept claims adjustments made by the insurer. One reason for notgoing to the court is, of course, the costs, but another is that the courts may be restricted to using information that isfully verifiable and judging according to the wording of the contract. Agenerousinsurer may compensate althoughit cannot be fully verified that the policyholder has taken due care. Indeed, there is evidence that insurers honourcontracts to alarger extent than is enforceable in court. Hence, there seems to exist some form of self-regulationby individual firms and/or industry.

∗ Corresponding author. Tel.:+46-46-222-8676; fax:+46-46-222-4613.E-mail address:[email protected] (F. Andersson).

0167-6687/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0167-6687(03)00111-2

268 F. Andersson, G. Skogh / Insurance: Mathematics and Economics 32 (2003) 267–280

Fraudulent interpretations by insurers may reduce demand throughout the entire industry, and the firms thus havea common interest in controlling “unfair” competition. The existence of externalities in maintaining credibility iswell known but has not been emphasised in an insurance context; we will approach it in a simple but arguablygermane fashion by a set of assumptions that lead to acost of re-establishing credibilitythat must be borne by allinsurers after any default within the industry.

To mitigate the problem with fraud—and policyholders’ fear of fraud—the industry may set up a body withexperts from the industry to settle disputes and control business ethics. We will model such a body, and it willbe called the Claim Settlement Board (CSB) below. The CSB is thought of as a private court which settles allclaims. It is not restricted to verifiable facts; its verdicts are based on observations, common experience, and whatmaximises total profit for the industry. The insurers submit to the CSB in order to gain credibility, or in other words,to maintain a collective reputation.1 However, the presence of self-regulatory organisations raises the question ofanti-competitive practices. We study how the control of “unfair” competition and cartelisation interact. We alsoconsider entry, showing that the externality between insurers’ credibility gives rise to a social cost of entry and thusprovides the industry with an argument (beyond maintaining rents) for regulation.

The notion of self-regulation and the importance of externalities in maintaining credibility have received someattention, in particular in the context of financial contracting.Mayer and Neven (1991)andFletcher (1993)considermodels with an intrinsic “firm quality” which is positively correlated with a firm’s financial reserves; there is also amoral-hazard problem. Both contributions consider the relative benefits of external regulation and self-regulation;regulation consists of a capital requirement and a fine for a firm cheating consumers. They consider reputationmechanisms that are different from ours, however.2 None of the mentioned contributions consider the specificinstitutional environment of insurance.3

Some attention has been paid to related moral-hazard problems in insurance.Picard (1996)demonstrates theexistence of a commitment problem in insurers’ auditing of potentially fraudulent claims. An insurer would likeex ante to commit to auditing more frequently than is optimal ex post. Interestingly, Picard finds that a commonagency, i.e. a self-regulatory body similar to our CSB below, improves the outcome; the reason is quite differentfrom ours however.Bond and Crocker (1997)provide an insurance model in the presence of rich possibilities forpolicyholders to manipulate the information about the cost of an insurable loss. Our model is quite distinct fromthese models since we consider the moral-hazard problem faced byinsurersdue to the possibility that they default.

In this paper we analyse quality and self-regulation in an industry with experience goods where sellers have anability to cheat the buyers, and the cheating cannot be verified in court. We limit our study to property and liabilityinsurance.Quality is defined as the degree of compensation to non-cheating policyholders. The policyholder maybe uncompensated for two reasons: (i) insolvency; (ii) biased or fraudulent interpretation of the contract at claimssettlements. While insolvency is thoroughly analysed in insurance theory, claims adjustments are not. We focus,therefore, on the claims adjustments.

1.1. Background—claims adjustment in practice

An insurer may, according to our theory, pay fair compensation although the care taken by the policyholder is notenforceable in court. Indeed,generosity, meaning that insurers pay more than a court would stipulate, is an observedphenomenon in the industry, although scientific investigations are rare.4 The only article to our knowledge is thestudy byRoos (1981)on “attractive goods” in Swedish tenant and home insurance. Attractive goods, such as gold,

1 We thus advance a notion of a collective reputation quite distinct from that ofTirole (1996); this will be discussed inSection 6.2 Scarpa (1999)provides a good survey. Other relevant contributions areGehrig and Jost (1995)who consider self-regulation in a somewhat

different environment where competition issues do not arise, andNuñez (2000)who consider a general repeated-game model of self-regulation.Zweifel and Eichenberger (1992)focus on the cartelisation potential of medical associations.

3 There is also work on “double moral hazard” in the context of warranties; see, e.g.Cooper and Ross (1985)andEmons (1988)for earlycontributions.

4 Coulanceis the French word for the insurer’s willingness to pay more than the wording of the contract requires.

F. Andersson, G. Skogh / Insurance: Mathematics and Economics 32 (2003) 267–280 269

silver, cameras, furs, weapons, etc. are not protected against theft when stored in cubicles in cellars and attics, orwhen taken along on journeys. The study concerns claims settlements, where loss due to theft of attractive goodswas compensated, although the loss was excluded in the policy. Roos examines all such claims handled during theyears 1977 and 1978 by 10 out of the 12 claims adjusters in the local office of a large national insurance company.He discovers 205 cases where it was beyond question that the insurer was not legally obliged to pay. Nevertheless,in 50 of these cases the policyholders were compensated.

The practice may be illustrated by the following example from Roos’ study. The holder of a home insurancepolicy claims that three furs in her attic have been stolen. The locker is broken. The insurer compensates for theloss according to the estimate of a fur expert. The estimate is based on the policyholder’s description of the furs.Yet, according to a clause in the contract stating that “attractive” property should be stored safely and with care,the insurer could refuse to pay. Moreover, the existence of the furs is unproved and, thus fraud on the part of thepolicyholder cannot be ruled out. The policyholder may bring the case to court if the insurer does not pay; however,this is costly, and the court will rule according to the wording of the contract. Still, the insurer accepted the claimof the insured.

A generous compensation may, as in the example, be a policy of an individual firms, or as we model it, by theindustry. The modelled profit-maximising, central CSB is an extreme. It is of interest, however, as a special caseof self-regulation. In reality, the institutions of conflict-solving vary greatly between branches of insurance andbetween nations. Arbitration and CSBs set up by industry and/or public authorities often exist jointly. Bookersacting repeatedly in the market with knowledge of the performance of different insurers may also play a role inconflict-solving and, thus in the establishment of credibility.5

1.2. Outline

In Section 2, a simple static insurance model is introduced, and inSection 3we introduceimperfect observability.In Section 4the model is extended to a repeated game, where the insurer may establish a reputation for beinggenerous, i.e. for honouring legitimate claims. InSection 5we examine competition among insurers, and inSection6 a CSB that controls settlements and quality is modelled.Section 7discusses practice and regulation in the industryandSection 8concludes.6

2. A static model

A sudden loss may strike a potential policyholder, him, in a way that can be described completely as a reductionof his wealth. The likelihood of a loss depends on theprecautionary effortexerted by the policyholder, which iscostly for him in terms of utility. There are many policyholders, a continuum of unit measure with independent risks,i.e. insurers end up facing no variance. Furthermore, there is one (several fromSection 5on) risk-neutral insurer, tobe referred to asshe, who is willing to assume the risks at a price.

5 For instance, in Sweden complaints about consumer insurance can be brought free-of-charge to a consumer representative at the municipallevel. In 1991, these local services handled 3164 insurance cases. An insurance can also be taken up in the National Board of Consumer Complaint(Allmänna Reklamationsnämnden). The Board is a governmental body, but three of the six members in the section of insurance are electedfrom the insurance industry and three represent the consumers. Out of the 400 cases handled by the Board in 1991/1992, approximately 100had been dealt earlier by local agencies. There are also a number of voluntary bodies within the industry. In 1991, 10 such officially listedbodies handled 7404 cases submitted by policyholders. In addition, the industry may bring up cases in the Property Insurance Condition Board(Skadeförsäkringen villkorsnämnd), which has been set up by the industry on the national level. The last resort is to bring the case to the court.Although most of the litigation costs of the policyholder are covered, conflicts rarely reach this stage (seeBjuggren et al., 1994).

6 Our approach is similar in spirit to that ofShapiro (1982, 1983), who considers market equilibrium in models where quality is non-contractibleand important. His approach differs from ours in that there is no moral hazard concerning consumer behaviour, and in that he considers perfectcompetition and monopoly, respectively. A survey of much of the relevant literature is part III ofStiglitz (1989). Although we limit our studyto insurance markets, the structure is more general; it encompasses goods with guarantee arrangements where consumer care is important butsubject to moral hazard.


All policyholders are identical, with a von Neumann–Morgenstern utility function that is additively separableover utility of wealth and disutility of effort. They enjoy wealth according to the strictly increasing, strictly concave,and twice differentiable functionv, and they suffer disutilitye from exerting effort. They have initial wealthw, andthey risk suffering a monetary loss,L. The likelihood of a loss depends on the precautionary effort exerted by thepolicyholder according toq(e) ∈ [0, 1], satisfying

q ′(e) < 0, q ′′(e) > 0, e ∈ [0, ∞].

An insurance contract with the premiump and which paysL − D, D being adeductible, in case of an accidenthence gives the policyholder utility:

u(p, D, e) = q(e)v(w − p − D) + (1 − q(e))v(w − p) − e.

The insurer is a risk-neutral profit maximiser with reservation profit zero. Profits areπ = p − q(e)(L − D).We first consider the case wheree is observable and contractible, and where the insurer designs the contract

subject to a reservation utility,u ≥ u¯, whereu

¯is defined by the absence of insurance. The reservation utility is

known by the insurer.7 The insurer’s problem is to maximise profits subject to the individual attaining her reservationutility, i.e. q(e)v(w − p − D) + (1 − q(e))v(w − p) − e ≥ u. The solution implements a first-best level of effort,e∗, with a fully insuring contract, i.e. a contract withD = 0.8

3. Imperfect observability

The origin of the problem studied here is that the behaviour by the policyholder cannot easily be observed andverified by external parties. Therefore, a contract on due care (e∗) cannot be costlessly enforced. In the case effort,e, is completely unobservable by others than the policyholder, a deductible trading off risk-sharing and incentives,and producing a second-best level of effort results (Shavell, 1979). We, on the other hand, assumeeto beimperfectlyobservablein the sense that if the insurer chooses to inspect a claim, she learnse with probabilityη, while withprobability 1− η she learns nothing. There is no cost of observation, but this is only shorthand for the cost beingsmall enough for observation to be worthwhile.9 The detection probability is independent of care taken; this is asimplification, but arguably a mild one—relaxing it would not change the results.

In this section, the observation ofe is assumed to be contractible. The policy may thus include a clause statingthat the insurer has the right to reduce or withhold compensation if she observes less effort than contracted. Theinsurer cannot punish the policyholder beyond not paying anything if he has been found to have exerted too littleeffort, i.e. the insurer cannot set the penalty aboveL.10

If a fully efficient solution is to be attained, i.e. if a policyholder who takes due care is to be fully insured,compensation must be paid whene is not identified at inspection. This may be possible.

Claim 1. There isη ∈ (0, 1) such that forη ≥ η the first-best solution is implementable, i.e. e∗ is chosen by thepolicyholder, andD = 0, unless the policyholder is found cheating.

7 The formulation may seem to allocate all bargaining power to the insurer, but it is without loss of generality since efficient solutions aredistinguished qualitatively only by the distribution of the surplus, parameterised byu.

8 This fact is simple and well known (see, e.g.Shavell, 1979).9 It is a general result (given the timing) that random inspection is never strictly desirable (seeBaiman and Demski, 1980a). For other instances

of imperfect observability of a hidden action seeHolmström (1979)(where thesufficient-statistic conditionfor when information is valuable isproved),Baiman and Demski (1980b), Dye (1986), andJewitt (1988). For applications in insurance contexts, seeBond and Crocker (1997)andPicard (1996).10 It is well known and easy to see that there will be no reason to punish less and that with unlimited punishments, the first best would be

implementable. Unlimited punishments are unrealistic, and they can be dismissed with by assuming the policyholder to have limited wealth andruling out non-monetary punishments; for simplicity the upper bound is equal toL.


Proof. SeeAppendix A. �

We will confine the analysis to the case whereη ≥ η, and hence the first-best implementable. Furthermore, weassume that the maximum profits resulting are strictly positive. We wish to stress, however, that our restriction onη is for convenience. Ifη is allowed to fall moderately short ofη, the solution will differ only moderately with asmall deductible andeclose toe∗.11

4. A repeated game

Above, we assumed that the insurer is given the right to enforce the contract, neglecting the problem that theinsurer may cheat the policyholder. Now, we drop the assumption that the effort identified by the insurer can beobserved and verified by a court, i.e. we assume that the contract is too costly to enforce publicly.12 Instead, thecontract is enforced by the parties themselves. Either the insurer or the policyholder may then be given discretionin assessinge. If, in a one-period setting, the right is given to the policyholder, it is clear that he will claim that duecare was taken and, anticipating this he will have no incentive to take care; giving discretion to the insurer producesan analogous result. In a multi-period framework, however, reputation may come to rescue.

Although policyholders may demand insurance for several periods, their time horizon is likely to be shorterthan that of insurers; for low-probability losses there are, in addition, few claims where the policyholder coulddemonstrate honesty. It seems easier for an insurer, who adjusts a large number of claims, to establish a rep-utation. These circumstances indicate why the insurer in practice usually is given the right to interpret thecontract.13

We consider a multi-period model with an infinitely long-lived insurer, and a sequence of policyholders whodemand insurance for one period only. When a long-lived player of a repeated game in equilibrium refrains fromplaying one-shot best responses to his opponents’ strategies in order to receive a better payoff in the long run, wewill say that he bears areputation.14

We are going to investigate whether there is a reasonable equilibrium resulting in an efficient outcome. Thelong-run player, the insurer, is the only player who can bear a reputation, and the obvious candidate equilibrium isone that mimics the first-best outcome of the static game. Such an equilibrium entails the insurer behavinggenerouslyin the following sense.

Definition 1. An insurer is generous if she uses the evidence to deny reimbursement only if she identifies withcertainty effort less than contracted on.15

Each period, the insurer faces a new set of policyholders with the preferences specified above, and a reservationutility u ≥ u that the insurer cannot influence. Each policyholder knows the full history of the market in the sense thathe observes the proportion of honoured claims (this should be interpreted as due to word-of-mouth communication,media coverage and the like). He also knows the incentives for other policyholders to cheat, and hence he inferscorrectly whether the insurer has behaved generously in the past, i.e. if she has honoured legitimate claims, includingthose whereewas not fully identified.

11 For a proof, seeAndersson (1996). It may be considered unrealistic that there is no cheating by thepolicyholderin equilibrium; however, atrivial extension allowing heterogeneity among policyholders could produce that feature without affecting anything else.12 The costs of going to court are two-fold; first, there are administrative costs, secondly, there are costs of mis-interpretation of the (implicit)

contract—if the court follows the wording of the contract, it may be too restrictive, and if the “deep pocket” rule is generally applied, the verdictmay be too generous.13 This point is made inRoos et al. (1980).14 There are two notions of reputation: the one used here, where reputation concerns merely the choice of strategies, and the one employing

incomplete information about the players’ preferences (see, e.g.Fudenberg and Tirole, 1991, Chapters 5 and 9).15 We have assumed care to unverifiable, but the same logic would apply if care was verifiable when observable.


The insurer maximises present values of streams of profits,{πt }∞t=0, discounted by a factorδ; Π = ∑∞t=0δ

tπt . Theefficient solution to the static problem is denoted by(p∗, D∗, e∗); D∗ = 0. The implied profit isπ∗ = p∗ −q(e∗)L.Consider now an arbitrary contract,(p, D, e), giving profit,π = p − q(e)(L − D), each period. If the followinginequality isviolated, the insurer would prefer receiving premiums,p, and then defecting by not reimbursing evenif she thereby lost all prospects of future sales:

∞∑t=0

δtπ = π

1 − δ≥ p ⇔ p ≥ q(e)

δ(L − D).

This possibility is allowed because of the insurer’s discretion—she can always claim that the policyholder exertedtoo little effort—and it imposes the above inequality as acredibility constrainton her contract. The condition canbe restated:

−q(e)(L − D) + δ

1 − δπ ≥ 0,

i.e. as a positive-profit condition at the point where premiums have been collected. We will call the contractthat maximises profits subject to the policyholders’ reservation utility and the credibility constraint themonopolycontract; note that the deductible below results not from moral hazard, but from its mitigating the insurer’s credibilityproblem.

Claim 2. If the credibility constraint does not bind, the monopoly contract coincides with the static optimum. Ifcredibility binds, it has a deductible if the resulting profit is positive; otherwise, the null contract is optimal.


There are many equilibria of the game, but we focus on the following, which is efficient.

Proposition 1. The following is a subgame perfect Nash equilibrium. The insurer offers the monopoly contract foreach period and honours all legitimate claims. Provided the policyholders receive at least their reservation utility,all policyholders accept in the first period and, then, if and only if the insurer has honoured all legitimate claims inthe past.


This equilibrium, which is preferred by the long-lived party in our game, is simple in that it involves only purestrategies. The optimality of this follows from the insurer being risk neutral and policyholders’ being risk averse.

5. Competition among insurers

We now consider a situation with several insurers who may compete for customers or collude. Even though ourfocus is on the arrangements for co-operation in the next section, competition in their absence is the natural referencecase; we will also show how the game begins.16

In order to capture the credibility problem, we introduce the possibility that an insurer “goes crazy”. We takethis to mean that a random event makes her completely myopic, immediately implying that she will default on anyobligations; in reality, this may correspond to her being bought by a myopic or fraudulent investor. The probability

16 For interesting treatments of the building of reputation, seeShapiro (1982, 1983). See alsoHägg (1994)for an analysis of the first-moverproblem.


of an insurer going crazy in any period isε, and it is independent across insurers and periods. We assume that goingcrazy involves someone else capturing an insurer (e.g. buying her at the price defined by expected future profits),and hence that the payoff of a crazy insurer does not enter the expected profits directly.

When an insurer defaults, the rest of the industry is hurt—the defaulting insurer imposes some kind ofcoston the other insurers. We capture this by assuming that when an insurer defaults, this will be known—through,e.g. the media—but that policyholders who do not have a loss that period will not immediately knowwhich in-surer it was, i.e. we assume that policyholders do not keep track of each insurer. The underlying tenet is thatpolicyholders are lexicographically lazy in preferring to keep track of fewer rather than more insurers. To defendthemselves, non-defaulting insurers may provide policyholders with hard evidence of their innocence; such evi-dence is available (e.g. through a reputable rating firm) and may be disclosed at costa. We call the disclosure ofsuch information acampaign. For simplicity, we will also assume that the defaulting insurer is re-organised by anew owner and that she can prove this fact at the same cost (this is purely for convenience: to keepn constant).In the absence of insurers’ evidence the defaulter would continue trying to sell insurance; thus a campaign isnecessary.17

A campaign must take place prior to the collection of premiums, and hence does not affect the credibilityconstraint except by reducing future profit expectations. A campaign is necessary ifat least oneinsurer defaults,and the expected future cost of campaigns at the beginning of a period is

∞∑t=1

δt (1 − (1 − ε)n)a = δ

1 − δ(1 − (1 − ε)n)a,

it is, obviously but importantly, increasing inn (and ε). Note that the credibility constraintis affected throughfuture expected profits. In order to focus on interesting cases, we assume thata is small enough for a cam-paign not to induce exit upon a default, and thatε is small enough that the credibility constraint is satisfied forsomen > 1.

5.1. Competitive environment

We will assume that the insurers do not face capacity constraints and that there is no product differentiation; thus,the credible one with the lowest price captures the entire market. When several insurers offer equally good terms,the market is split equally among them. The timing within a period is:

• if necessary, insurers organise campaigns to prove previous record;• insurers contract with policyholders;• other insurers observe contracts; accidents occur and settlements are made.

Thus, an insurer who undercuts an anticipated equilibrium offer will not meet responses until the next period.

5.2. Competition

The timing makes clear that the insurers, if competing without formal arrangements, compete in contracts inthe fashion of Bertrand. We first consider a fixed set of insurers at period zero (and thus abstract from profits’depending onn through campaign costs). Policyholders purchase if they benefit from doing so according to theirrational beliefsabout whether the insurer will honour legitimate claims or not. We will show that long-run profitsare bounded above and below, and we will characterise equilibria that produce these profit bounds.

The first equilibrium we describe corresponds to the static equilibrium of standard Bertrand models. Sincecredibility towards policyholders is necessarily violated at zero profits, the most competitive outcome is the

17 See the end ofSection 6for a discussion of the specification.


solution that maximises consumer surplus subject to profits being non-negativeand to credibility. The prop-erties of this solution follow fromProposition 1, and we denote it by(p, D, e); it entails a deductible andgives total profit (inclusive of expected campaign costs)π , 0 < π < π∗; this is the lower bound on long-runprofits.

The other extreme is successful tacit collusion among insurers. If there arenfirms trying to sustain and split equallythe monopoly profit,π∗, by threatening with infinite repetition of the competitive outcome upon any deviation, thecondition for sustainability of monopoly profits is

(1 − 1

n

)≤ δ

(1 − π

nπ∗

),

which is stricter than the “standard” one for Bertrand competition with homogeneous products whereπ = 0 (seeTirole, 1988, p. 248).

It follows from the folk theorem for repeated games, e.g. Proposition 5.4 inFudenberg and Tirole (1991), that allconfigurations of long-run per-firm profits in the (product) interval×n

i=1[π/n, π∗/n] can be supported in equilibriumif the parties are patient enough.18

However, the literature onrenegotiationin supergames has cast serious doubt on the plausibility of collusiveequilibria.Farrell (2000)has shown that sustainability of monopoly profits in repeated Bertrand games applies onlyfor small numbers of firms when renegotiation proofness is imposed.19 This is one reason why we believe that theexplicit arrangement to be studied in the next section may be an important force in sustaining collusion.

5.3. Entry

Since we have chosen not to work with incomplete information about insurers’ preferences, the credibility problemis no larger for an entrant than for an incumbent; the competition described may start in period zero.

If there are many potential entrants and if entry were costless, it would go on until profits were minimal (equalto π ) and an unlimited number of firms would enter. If, on the other hand, there is a fixed cost,c, which must beincurred by any entrant, things are more complicated. We do not provide a general characterisation of equilibriumwith entry; MacLeod (1987)shows that the equilibrium set of a homogeneous-good Bertrand model with entrygenerally is very large, but that high enough entry costs quite unambiguously lead to the monopoly outcome. In ourframework, a simple argument shows that ifc is large enough, there is an equilibrium where firms earn monopolyprofits and the number of firms is the largestn for whichπ∗/n ≥ c.

6. A CSB

We now introduce a CSB which acts as a private court. Members (insurers), as well as policyholders, may bringa case to the CSB. The CSB is assumed to be able to monitor each insurer’s claims adjustment, including theobservation of care taken; it thus has an informational advantage over, e.g. the government.

The main purpose of the CSB is to control the externality firms exert on each other through the risk of their goingcrazy. We assume that each member effectively commits to following the decisions of the CSB. In order to makethis commitment, the members of the CSB must deposit a large enough collateral or premium reserve coveringdefaults. Further, we assume that the CSB is governed by majority rule. Hence, a claim can be dismissed only if themajority so wishes, and the CSB defaults only if more than half of the members go crazy in a given period. This is

18 It also follows from the folk theorem that equilibria may be very complicated; in our case, non-stationary equilibria may be even morecomplicated because of the credibility constraint, but this does not affect the bounds.19 Farrell and Maskin (1989)consider repeated Bertrand duopoly, and find that renegotiation proofness shrinks the set of equilibria (in that

sufficiently unequal distributions of the surplus are ruled out).


the key difference between this case and the Bertrand case (where one default triggered campaigns). The expectedcost of campaigns is

δ

1 − δh(ε|n)a <

δ

1 − δ(1 − (1 − ε)n)a,

whereh(ε|n) is the probability that more than half the members go crazy in a given period (it is a straightforwardsum of binomial probabilities). Note, again, that since insurers’ preferences are known, the CSB may start fromperiod zero. An insurer may opt out of the CSB, and she can then recap her premium reserve at a cost which makesup (part of) her entry cost when re-entering the market.

The CSB not only evaluates the effort exerted by a policyholder, it may also decide what deductibles are admissible.The CSB thus protects the industry and the policyholders from “unfair” competition. We assume that there are nomeans of playing supergames within the CSB, i.e. that no punishments affecting anyone other than the deviator areenforceable. This is in the spirit of renegotiation arguments, but not formalised. We stretch the power of collusionwithin the CSB as modelled very far, whereas we abstract from other forms of punishment and harassment; however,the power of the CSB even without these forces rather strengthens the results.

6.1. Collusion

Suppose first that there aren insurers, all committed members of the CSB; the possibility that some insurersleave the CSB is equivalent to entry and discussed below. Ifπ∗ is the profit (inclusive of expected campaign costs)from the monopoly contract,(p∗, 0, e∗), then there is anotherp < p∗ such that the contract(p, 0, e∗) satisfiesthe credibility constraint and gives positive profits. An insurer that lowers her price will thus increase her profit byobtaining all policyholders. If disputes are settled by the CSB, it can retaliate byalwaysforcing the insurer whohas cut her price to pay the claims she faces (given the power of the CSB to set the standard of compensation, itwill obviously be optimal only to support contracts with the prescribed deductible, zero). In equilibrium, this wouldinduce policyholders, anticipating this outcome, to exert no effort if buying from her. This may or may not deterprice cuts, but since demand is perfectly elastic and since the least upper bound on profits when undercutting andfacing policyholders responding withe = 0 isπ(p∗|e = 0), this shows the following proposition.

Proposition 2. If π(p∗|e = 0) ≤ π∗/n, then it is not possible to do better by undercutting p∗ infinitesimally, evenif thereby obtaining the whole market.

It is clear that if the condition of this proposition is satisfied, it is a Nash equilibrium for each member of theCSB to offer the monopoly contract. If, on the other hand, the condition is violated, the monopoly contract cannotbe an equilibrium. Note that if we let policyholders be heterogeneous so that there is cheating in equilibrium, themechanism will be punitive also in the absence of equilibrium reactions by policyholders, i.e. even if no policyholderadjustse. The power of the mechanism depends on the importance of due care. If the outcome depends stronglyone—which it does ifq′ andL are large in magnitude—the inequality is likely to be satisfied (the left-hand side islikely to be negative). If, on the other hand, the outcome is independent ofe, the mechanism has no power.

If the condition ofProposition 2is violated so that undercutting is profitable, it can be satisfied by making theinsurance contract more favourable to the policyholders (since both sides are decreasing withu, and the left-handside reaches zero before the right-hand side). In that case, the equilibrium that maximises profits is characterisedby the condition of the proposition being an equality.

Claim 3. The optimal contract with the condition ofProposition 2binding and with the credibility constraint notbinding satisfiesD < 0.

Proof. See the proof of Claim 4. �


The logic is as follows: if due care is to be as important as possible to theinsurer, which empowers themechanism, her payoffs in the two states should be as different as possible. Even though literally negative de-ductibles are rare, the claim points out a force that makes deductibles smaller than they otherwise would havebeen.

The condition ofProposition 2may render the credibility constraint binding (by construction, credibility doesnot bind at the monopoly contract).

Claim 4. If both the condition ofProposition 2and the credibility constraint bind, a solution may not exist. If itdoes, the sign of the deductible is ambiguous, and the insurers will have an incentive to distort care upwards.


The possibility of non-existence may seem troubling, but each insurer will then either stay out of business orbuild a reputation of her own and play the Bertrand game.

If the CSB supports all contracts, i.e. the CSB does not restrict deductibles, it will be optimal for a deviator to offerthe contract where care,e, is determined only by the deductible (the same holds for an entrant). The above conclusionsremain for that case, except for the fact that negative deductibles do not mitigate the constraint ofProposition 2in that case. Note that although it may seem emphatic to assume that the CSB can constrain deductibles, there arestrong incentives to do so since it empowers the delegation of punishments.

6.2. Entry

Let us now consider entry in a situation withn incumbents organised in the CSB. The externality is present inthis case as well, and with one entrant, expected campaign costs increase to(h(ε) + ε(1 − h(ε)))aδ/(1 − δ).

The CSB has very strong incentives to prevent entry, not only—and probably not primarily—because of thisnegative externality but also because the dilution of rents. The negative externality, however, provides incum-bents with an argument for entry being impeded and for CSB membership being compulsory by regulation,this because the costs of re-establishing confidence (as well as the risk of default itself) entail a loss ofwelfare.

Since there is a cost of withdrawing from the CSB, an insurer who withdraws will be in the same position asan entrant, but may face a different entry cost. Whether entry is possible or not depends on entry costs, regula-tion, and entry deterrence. If entry is not regulated, the CSB can try to deter it. If the threat of entry is repeatedthe CSB may build a reputation for deterring entry by playing the competitive Bertrand strategy against any en-trant. Analogously to the situation with entry into the Bertrand game there will be entry in the absence of entrycosts and no entry with high enough entry costs, while the outcome is highly indeterminate for intermediate entrycosts.

6.3. Collective reputation

Our means of modelling credibility externalities differ from those ofTirole (1996); he considers an environmentwhere (as in our case) outsiders only observe the behaviour of the group, and (contrary to our case) where the onlymeans of re-establishing reputation for the group is exclusion of members. The implication of his specification isthat once a collective reputation is lost, it can be re-established only over a longer period of time. While relevantin many contexts—his key applications are corruption and “firm quality”—we believe that in insurance markets(as well as in financial markets in general) it is often a prerequisite for an active market that reputation can beregained quickly by some kind of re-organisation such as our “campaigns”; this model is an attempt to capturethis.


7. Results, practice, and regulation

7.1. Moral hazard

The insurer’s distrust of the policyholder is treated in the literature on moral hazard—deductibles and bonussystems are used to restrict careless behaviour and cheating. We show that cheating by the policyholder may beeliminated if the insurer detects care taken with a high enough probability and withholds compensation. The intuitionis that a risk-averse individual is severely punished by being uninsured. It is, therefore, rational to ensure coverage bytaking due care at a detection rate smaller than 1. The analysis shows how the policyholders may be controlled, andit throws light on why most insurance contracts (as well as insurance laws) include clauses on the policyholders’duty to disclose information and take due care, as well as the insurer’s right to inspect, and reduce or withholdcompensation if the policyholder has not behaved according to the contract.

7.2. Insolvency

The policyholders’ distrust of the insurer is due to possible insolvency, and potential cheating at claim adjustments.We show that the policyholders’ distrust imposes a credibility constraint on insurers, and that the constraint requiresinsurers to earn a rent equal to the foregone interest on one period’s expected compensation payments. This resulthas a straightforward interpretation in relation to insolvency: credibility can be established only if the premiumincome from the first period, in effect, is used as a collateral for claims, i.e. as apremium reserve. A financialreserve solves the credibility problem in our model and, apparently, also the insolvency problem in practice.20 Inmost countries, premium reserves are required and enforced in solvency regulation.21

7.3. Industry-wide dispute settlement and competition

A premium reserve may eliminate distrust due to insolvency, but not distrust due to cheating at claims settlements.The insurer’s intent is not, as a specified fund, observable. Moreover, the policyholder is unlikely to know theinsurer’s records accurately, and the care taken by the policyholder may be unobservable by a court. The industrymay, therefore, act jointly in organising dispute settlement to reduce the policyholders’ distrust.

We model a profit-maximising CSB that offers and enforces efficient insurance policies. We also show that theCSB’s control of claims paid may provide a glue for price control. Altogether, the CSB offers the first-best contract,and charges a premium that transfers all rents to the insurer. The main regulatory matter is thus distributive.Redistribution may be achieved by regulation of the premium. Indeed, price control aiming at fair prices is commonin consumer insurance.22 The success may be questioned however. The industry often claims that prices must besufficiently high to promote large enough reserves to avoid insolvency.23 The objectives of solvency and fair (zeroprofit) prices are, thus, in conflict.

The efficient monopoly contract depends on the assumption of identical policyholders. Monopoly pricing gener-ally leads to inefficiency, unless the seller can price-discriminate perfectly. Moreover, the “quiet life” of a cartelisedindustry without newcomers causes well-known dynamic inefficiency. Free entry may both vitalise the industryand reduce prices in favour of the consumers. New firms may offer low-price second-best contracts attractive tocustomers, including due-care requirements only for efforts enforceable in public courts, combined with deductibles

20 We disregard the variance in outcomes that makes larger reserves necessary. We also neglect the problem that fraud may be a reality also inrelation to funds.21 For a recent survey of regulation in Anglo-American countries, seeAdams and Tower (1994).22 Insurance regulation is characterised by multiple goals. Solvency regulation is usually seen as the prime motive, while fair pricing and

conditions are the second (Meier, 1991).23 The industry has apparently been successful in counteracting price control and promotion of large reserves—insolvencies harming consumers

are very rare (seePauly et al., 1986; Finsinger and Pauly, 1986, Chapter 1).


for the incentive to take non-enforceable care. Outsiders belonging to other CSBs may be competitive. Independentnon-profit mutuals may also be a trustworthy alternative.

Entry may be restricted for two reasons in our model: first to maintain rents, and second to limit defaults. Limitson competition to sustain monopoly profits is thoroughly analysed in the literature. The default risk of outsidersharming the industry is less analysed but often stressed by industry.

Since control of the claim settlement process may be sufficient for monopoly power, it may be argued that aCSB should be regulated or outlawed. Yet, there are several arguments against this. First, if care taken by thepolicyholder is important but incompletely observable, and public courts are unable to judge in an efficient way,a CSB or similar institution may be a necessity in order for trade to occur. Second, an industry-wide CSB hasan interest in efficient (generous or fair) claims adjustments. Third, there is usually competition among claimssettlement institutions.

Hence, there is no obvious advantage in intervening in the industry’s claim settlement and provision practice.Indeed, industry-wide co-operation on contractual provisions and claims settlement practice prevails in many coun-tries, as well as in the European Union.24 The arguments for non-intervention appear to be understood in manycountries where legislation is limited to codification of industry practice. The EU anti-trust exception for insurance(EEG, no. 1534/91), which allows insurers in the union to co-operate, can be viewed in this perspective.

8. Conclusions

In this article insurers’ credibility problems explain contracts, structures, and regulations that are frequent in theinsurance industry. In a repeated-game model, we show how an efficient solution is attainable through reputationalmechanisms, also when care is less than perfectly observable and not enforceable in court.

We show that a CSB organised by insurers can mitigate the problems with negative externalities of defaults, aswell as provide a glue facilitating collusion—it can maintain a cartel by forcing a firm that undercuts the monopolyprice to compensate careless policyholders. The control by the CSB is likely to work when precautionary efforton the part of the policyholder is important, incompletely observable, and not verifiable in court. Inefficient publicjustice increases the industry’s benefit of the CSB.

In the model, efficient monitoring of care and “fair” compensation is due to the monopolistic industry’s interestin efficient contracts sold at a monopoly price. Regulation is thus first of all a matter of redistribution. Redistributionin favour of the consumers, as well as dynamic efficiency, may be achieved by a public policy that supports freeentry into the market.

Appendix A

Proof of Claim 1. We will introduce some simplifying, but somewhat abusive, notation letting

u(e) = q(e)v(w − p − D) + (1 − q(e))v(w − p), e ≥ e,

u(e) = q(e)[ηv(w − p − L) + (1 − η)v(w − p − D)] + (1 − q(e))v(w − p) − e, e < e,

whereu(e) is the utility obtained by a buyer exerting at least as much effort as specified in the contract andu(e) thecorresponding utility if he does not. The insurer’s optimisation problem is now

maxp,D,e p − q(e)(L − D),

s.t. u(e) ≥ u, u(e) ≥ u(e), q ′(e)(ηv(w − p − L) + (1 − η)v(w − p − D) − v(w − p)) − 1 = 0,

24 For a cross national study, seeFinsinger and Pauly (1986).


where the first constraint is the participation constraint as before, the second is the constraint that the buyer mustprefer exerting the contracted level of effort to exerting less (which, obviously, is without loss of generality), andthe third simply states thatif the buyer defects, he exerts the level of effort that is unconstrained optimal from hisown point of view.25 Once the above problem is stated, the proof is simple. Forη close enough to one, the secondconstraint is clearly not binding and the problem is identical to the full information one. Forη close enough to zero,on the other hand, the third constraint implies that (forD = 0, i.e. at the optimum)e is close to zero as well, and it isclear that the second constraint binds and hence that the solution is affected. For a more complete characterisationof what happens when the second constraint becomes binding, seeAndersson (1996). �

Proof of Claim 2. The first part is obvious. If the constraint binds, the optimal static strategy is given by

maxp,D,e π = p − q(e)(L − D),

s.t. q(e)v(w − p − D) + (1 − q(e))v(w − p) − e ≥ u (λ), p ≥ q(e)(L − D)

δ(µ).

The first-order conditions with respect toD andp are

∂L

∂D=q − λqv′(w − p − D) + µ

q(e)

δ=0,

∂L

∂p= 1 − λ(qv′(w − p − D) + (1 − q)v′(w − p)) + µ = 0

and they imply that

q

[1 + µ

δ− 1 + µ

qv′(w − p − D) + (1 − q)v′(w − p)v′(w − p − D)

]= 0

from which we immediately see thatD > 0 if the credibility constraint binds andδ < 1. �

Proof of Proposition 1. It is clear by definition that the profit from the monopoly contract,π∗, is an upper boundon profits if the insurer always honours claims unless she has evidence that insufficient care was taken. The insurermight in principle be able to do better by randomising between honouring and not honouring, but that is strictlyfalse in our model since the agent is strictly risk averse. Hence,π∗ is a strict upper bound on payoffs in each period,and it is clear that it is attainable as a Nash equilibrium if the insurer cannot gain by defecting, i.e. precisely if thecredibility constraint is satisfied

(∑δtπ = π/(1 − δ) ≥ p

). To verify that the equilibrium is subgame perfect, we

need to only check that there are no profitable one-time deviations upon any history (Fudenberg and Tirole, 1991,Theorem 4.2). But this is obvious for the buyers, and once this is clear it is equally obvious for the insurer since sheis credible by construction. �

Proof of Claim 4. We perform the analysis forε = 0 to simplify notation; the result does not depend on it. Theoptimisation problem is then

maxp,D,e π = p − q(e)(L − D),

s.t. q(e)v(w − p − D) + (1 − q(e))v(w − p) − e ≥ u (λ), p ≥ q(e)(L − D)

δ(µ),[

q(0) − q(e)

n

](L − D) −

(1 − 1

n

)p ≥ 0 (ξ),

25 The formulation of the last constraint exploits the fact that thefirst order approachis valid because of our assumptions aboutq and thedisutility of effort (i.e. the buyer’s objective function being strictly concave). For discussions of this issue, which is quite intricate in its generality,seeKreps (1990)or Jewitt (1988).


where the last constraint is that ofProposition 2. Manipulating the first-order conditions with respect top andD asin Claim 2 gives us

q(e)v′(w − p − D) + (1 − q(e))v′(w − p)

v′(w − p − D)= 1 + µ − (1 − 1/n)ξ

1 + µ/δ − (q(0)/q(e) − 1/n)ξ

from which we immediately see thatD < 0 if only ξ > 0, while it is ambiguous ifµ is positive too. By comparingthe two last constraints, we see that once the credibility constraint binds too, the only way to relax both constraintsis to increasee, and we also see that this may not be possible. �

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