Full Insurance, Bayesian Updated Premiums, and Adverse Selection

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P1: VTL/EHE/NJH P2: KCU

The Geneva Papers on Risk and Insurance Theory KL517-06 October 20, 1997 4:13

The Geneva Papers on Risk and Insurance Theory, 22: 135–150 (1997) c

°1997 The Geneva Association

Full Insurance, Bayesian Updated Premiums,

and Adverse Selection

RICHARD WATT [email protected]

Departamento de Análisis Económico, Universidad Autónoma de Madrid, 28049-Madrid, Spain

FRANCISCO J. VAZQUEZ [email protected]

Department of Economic Analysis, University Aut´onoma of Madrid, 28049-Madrid, Spain, and

C.U. Francisco de Vitoria, Ctra. Pozuelo-Majadahonda Km. 1,800, 28223-Pozuelo de Alarc´on, Madrid, Spain

Abstract

In the classic Rothschild-Stiglitz model of adverse selection in a competitive environment, we analyse a “no-claims bonus” type contract (bonus-malus). We show that, under full insurance coverage, if the insurance company applies Bayes’s rule to learn about client probability types over time and uses this information in premium calculations for contract renewals, then there exist conditions under which all client types strictly prefer the Bayesian updating contract to the classic Rothschild-Stiglitz separating equilibrium.

Key words: insurance, adverse-selection, Bayesian learning

1. Introduction

One of the most interesting conclusions to come out of the adverse-selection literature is that where separating contracts are possible, they are more efficient than any possible pooling equilibria1_{. In general, by ensuring that the incentive compatibility constraints are}

satisfied, the principal can be sure that each agent type will be interested in accepting only that contract designed for precisely his type. That this type of equilibrium, if available, is optimal depends critically on the one-shot aspect of much of principal-agent theory. When recontracting is possible, there may exist other contracts, involving possible initial pooling of agent types, that Pareto dominate the separating contract.

Casual observation of real-world insurance contracts reveals that the optimal adverse-selection contract menu is not always offered or at least it is not the only contract choice offered. Instead, it is rather common to see insurance companies that recontract with their clients on terms that depend on the client’s past history of accidents. This seems to be par-ticularly common in automobile insurance. The no-claims bonus (or bonus-malus contract) is the most common of this type of contract, in which the client is offered a reduction in premium if the monetary amount of his claims does not surpass some prespecified quantity. There are many possible ways in which a no-claims bonus can be modeled. The most obvious is to consider that the company is searching for efficiency, it is often the case that JEL Classification: D8

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136 RICHARD WATT AND FRANCISCO J. VAZQUEZ

small losses are more easily born by the client, and so the potential revenue loss from offering the premium discount should offset the additional costs of the small claims that would otherwise have been presented. In this way, a no-claims bonus is very similar to a voluntary deductible. The difference is that instead of the client paying the first part of all claims, the claims are divided into two groups, large and small, and the insurance company pays for the large claims (entirely) while the client pays for the small ones2_{. Second, a}

no-claims bonus can be interpreted as the company offering a solution to moral hazard; the client is given the chance to alter his type (thus suffering accidents less frequently and presenting fewer claims) in exchange for a saving in his premium. Third, a no-claims bonus can be thought of as the company updating its beliefs as to the client’s type according to his accident history. This article concentrates exclusively on the third interpretation.

The idea can be introduced using the typical undergraduate insurance setting, in which there exist two types of client, differentiated only by their probability of suffering the insurable accident. When this probability is unobservable by the insurer, and unalterable by the client, an adverse-selection problem occurs. As was first noted in the famous article by Rothschild and Stiglitz [1976], a separating equilibrium may exist in which the low-risk individuals signal their type accepting a partial insurance contract. In an economy in which the insurer is risk neutral and acts in a competitive environment (thus earning zero expected profits), it is easy to show that, if a separating equilibrium exists, then at this equilibrium the participation constraints of both high- and low-risk individuals will not be saturated (both types of individual strictly prefer to participate), while the only incentive compatibility constraint that saturates is that of the high-risk type (the low-risk type strictly prefers the contract designed for him, but the high-risk type will be indifferent between the two contracts on the market.)

This equilibrium will exist only if the expected utility that the low-risk individual would obtain under a pooling equilibrium with full insurance and a premium that is calculated using the market average (the expected) accident probability is strictly less than the expected utility that he obtains under the separating equilibrium3.

In a multiperiod setting, it is not obvious that simple repetition of the separating equilib-rium is optimal. In this article, we take the benchmark contract that would be offered as an alternative to any Bayesian information gathering type of contract as being periodic repe-titions of the Rothschild-Stiglitz single-period equilibrium contract menu. In a two-period model, Hosios and Peters [1989] show that this is not necessarily optimal for a monopolist, mainly since the high-risk client may play strategically in the first period by not reporting an accident, in an attempt to influence the second-period probability that the principal attaches to him being low risk. In our setting, such strategic play will be ruled out by assumption (all accidents are public information), but anyway, since (as Hosios and Peters show) in the last period the principal will definitely offer the Rothschild-Stiglitz single-period equilibrium, and since under perfect competition this does not depend on the probability of being low risk (except in its very existence), no strategic first-period play in a perfect-competition model is optimal anyway. Second, under the assumption that commitment is only for one period, the greatest second-period penalty that the insurer can impose on the client is zero coverage (and hence reservation utility) in a monopoly, but when other companies are in existence, this is not so (any company can just offer the Rothschild-Stiglitz equilibrium in any period),

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and therefore the only feasible equilibrium that is not directly related to Bayesian infor-mation gathering is periodic repetition of the single-period Rothschild-Stiglitz separating equilibrium.

The most relevant articles on multiperiod insurance contracts and adverse selection gen-erally consider models with precommitment to the contract over time and initial revelation of type, and so this type of repeated insurance can be thought of as a single transaction in which the incentive compatibility constraints are modified in a dynamic sense, in order to incite truthful type revelation (separation of types) at the outset of the contract (see Dionne [1983]). These models tend to concentrate on situations in which the insurer is a monopolist and thus is interested in profit maximization. In this article we are concerned with Pareto dominance in a perfectly competitive insurance market in which commitment is not an issue, mainly since commitment to a contract over time does not seem to be a feature often included in real-life insurance contracts4.

It may be possible that contract formats other than the Rothschild-Stiglitz equilibrium menu exist, some involving a learning process but no commitment and still offering zero expected profits to the insurer but greater expected utility to both client types.

Consider the following multiperiod contract format:

• All insurance contracts have full coverage of the insurable loss.

• All insurance contracts begin with a premium corresponding to the market average loss probability.

• All insurance contract renewals reflect the client’s accident history in a rational manner.

• Any client is free to not renew his contract, but his accident history is assumed to be public knowledge and will be used in any new contracts drawn up between him and any other insurers.

Graphically, this multiperiod contract with learning corresponds to all contracts starting out at the intersection of the market fair-odds line and the certainty line (the 45 degree line). Then the slope of the fair odds line is recalculated, for each individual client, after each contract period using the client’s current accident history, and the contract renewal is where the updated fair-odds line intersects the certainty line. Given an adequate learning procedure, all clients who accept this contract format can expect that their contract will, over time, converge to their full information contract.

Of course, we do not argue that this is the only possible multiperiod updating contract that could be considered or in fact that it is in any sense optimal. In fact, there are two important variables in an insurance contract—the premium and the coverage, both of which could be subjected to updating. Specifically, an insurer could offer the contract that is optimal for the low-risk client at the market average premium (which would imply partial coverage) and then go updating both the coverage and the premium according to observed accidents. Doing so would, of course, introduce a much greater degree of complication into any theoretical analysis. Besides, the analysis contemplated in the current article would appear to be more closely related to most empirically observed bonus-malus contracts, which usually imply full insurance always but with a premium that may vary over time. If we show that our proposed updating contract Pareto dominates the nonupdating alternative, then surely the optimal updating contract will also.

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Note the following characteristics of this contract format. First, the insurer must be earning zero expected profits in all periods, since each client is charged a fair premium corresponding to his expected type, given his accident history. Second, any high-risk clients must receive greater expected utility under this contract format than under the separating equilibrium, since under both contract formats they get full coverage, but the learning format offers a gradual convergence (from above) to the full information contract, which is the same contract that they would receive under the separating contract (we prove this formally in Theorem 2). The high-risk individuals will suffer excess claims penalties during the life of their updating contract. Hence, our assumption that any client’s accident history is public knowledge (in the same way as an individual’s credit history is public knowledge when applying for a bank loan or a credit card) is vital, or else the high-risk individuals could simply move from one company to another once their current insurer begins to learn the true type. It is interesting to note that this type of information sharing among insurance companies is a normal feature of contracts in France. This assumption also means that claim underreporting is not an issue5. Third, this contract is equally valid under conditions of total ignorance—that is, when even the clients are unaware of their type. Hence, while we are considering an adverse-selection environment, this is by no means the only environment that this contract could be considered under, and that seems to be realistically reasonable.

It may be argued that the first-period pooling situation is vulnerable to competition from rebel contracts that are below the corresponding indifference curve of high-risk clients and above that of the low-risk clients. However, the proposed contract is certainly defendable under the “reactive equilibrium” concept of Riley [1979]. If the company that offers the Bayesian contract infers that some (or all) of the low-risk clients are going to defect to some rebel contract, then that company will just revise the initial probability that it attaches to the market average loss probability accordingly, to take into account the average loss probability expected from the pool of clients accepting the Bayesian contract. In the limit, the Bayesian contract will collapse onto the full-coverage, high-premium contract of the Rothschild-Stiglitz menu, and hence the only credible rebel contract must collapse onto the partial-coverage, low-premium contract of the Rothschild-Stiglitz menu.

The only individuals that we must look closely at are the low-risk clients. Assuming existence of a separating equilibrium contract, the updating contract will necessarily start out offering them less expected utility than the separating equilibrium, but on the other hand it can be expected to improve over time as the full-information contract is approached. In particular, since the full-information contract must necessarily offer greater expected utility than the separating-equilibrium contract, the updating contract can be thought of as an initial investment in signals that should pay off in cheaper contract renewals as time goes on (no-claims bonuses).

This article proposes to explore the comparisons between the two possible solution formats; the optimal-contract menu versus an information-updating model based on full coverage and premiums that vary according to accident histories. The second model will, logically, be Bayesian in nature, in that the insurer uses Bayes’s rule to rationally update beliefs on the probability that any particular client is of one type or another. In particular, our intention is to consider the necessary and sufficient conditions for the Pareto dominance

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The existing literature on Bayesian learning and adverse selection in insurance contracts seems to be rather scarce. The most relevant reference manual as far as insurance and Bayesian statistics is concerned would appear to be Klugman [1992], but he does not treat the adverse-selection aspect of insurance at all. Cyert and DeGroot [1987] analyze many aspects of economic decisions under uncertainty and learning, but they do not treat insurance at all. In two separate articles, Holt and Shore [1980] analyze an abundance of related topics, using techniques that are similar to those to be applied here, but they do not touch on insurance problems either.

The repeated contract self-selection models do not always specifically use Bayesian learn-ing as the basis for premium updates (see, for example, Dionne [1983], Dionne and Lasserre [1985], Cooper and Hayes [1987], and Dionne and Lasserre [1987]). More specifically to this article, Gal and Landsberger [1988] use binomial statistics to show that the expected profits of the insurer increase if repeated contracts are based on longer experience histories (which is related to our Theorem 3), but Bayesian updating is not explicit in their model. Lastly, Hosios and Peters [1989] do use Bayesian learning and binomial statistics to con-sider contract renegotiation and underreporting of claims (underreporting is also the theme of Hey [1985] but is of no relevance here as we are considering pure adverse selection, when all accidents are public information).

In what follows, the classic Rothschild-Stiglitz separating equilibrium will be referred to as RS contract menu, while the suggested Bayesian updating contract will be referred to as BUC.

2. A general two-client type model

We shall maintain the basic assumptions of the Rothschild and Stiglitz setting; there will only be two agent types (a and b), differentiated only by their probability of accident (state 2), pa>pb. Wealth in state 1, x1is strictly greater than wealth in state 2, x2. Hence

the insured loss is just L=x1−x2. The insurance company is assumed to be risk neutral and

to operate in a competitive market (and hence is restricted to zero expected profits), and the agents have the same utility function for money, U(x), which is assumed to be continuous, differentiable at least twice, increasing, and concave. Assume that each contract period implies one new observation is added to the client’s accident history and that a new contract is signed after each contract period, taking into account the entire accident history to date. Letλdenote the insurance company’s initial estimation of the probability that any par-ticular client is of type b, and so the initial prior as to the probability of an accident for any particular individual is

p≡(1−λ)pa+λpb.

We assume that 0< λ <1 since an extreme value (0 or 1) for this probability indicates that there is no adverse-selection problem. The initial pooling contract will be just that contract found at the intersection of the certainty line and the iso-expected value contour passing through the initial wealth distribution, x, with slope−(1−p)/p. This contract

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zero expected profits to the insurance company given the initial beliefs. Note also that since the high-risk individual will be fully insured under both the contract menu and the initial Bayesian contract, but that the second is necessarily cheaper, he must enjoy greater expected utility at the initial pooling contract than under the contract menu solution. Since we are assuming that a separating equilibrium exists, the low-risk client must receive less expected utility from the initial pooling contract than from the separating contract menu6_.

After the first contract period, the prior,λ, can be updated according to whether or not an accident occurred. The posterior forλafter the first contract period will be

λ(1,1)= λpb

(1−λ)pa+λpb

if an accident occurred and

λ(0,1)= λ(1−pb)

(1−λ)(1−pa)+λ(1−pb)

if not. It is easy to see thatλ(1,1) < λ < λ(0,1)if and only if pa > pb, as has been

assumed. Hence, if there has been an accident, the client’s contract becomes a little bit more expensive, while if there has not been an accident, the contract becomes a little bit cheaper (there is a no claims bonus).

After n contract periods have gone by with, say s accidents, the posterior forλwill be

λ(s,n)= λp s b(1−pb) n₋s (1−λ)ps a(1−pa)n−s+λpsb(1−pb)n−s . (1)

To see this, let S stand for the event; from n attempts, s are observed to be accidents. From binomial statistics, the probability of event S given an agent of type i =a,b,p(S|i)is

p(S|a)=Cn_sp_as(1−pa)n−s p(S |b)=Csnp s b(1−pb)n−s, where C_sn = µ n s ¶ = n! s!(n−s)!.

Given that the prior probability for the agent being type b isλ, Bayes’s rule tells us that

λ(s,n)≡ p(b|S)= λp(S|b)

(1−λ)p(S|a)+λp(S|b).

Substituting for p(S | i)and cancelling common terms leaves us with the equation given forλ(s,n).

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Theorem 1: Let E [λ(s,n),pi] be the expected value ofλ(s,n)for a type i client:

E[λ(s,n),pi]≡E [λ,n,pi]= n X s=0 n! s!(n−s)!p s i(1−pi) n−s_λ(_s_,_n_); _i ₌_a_,_b then E[λ(s,n),pa]→0 and E[λ(s,n),pb]→1 as n→ ∞.

Proof. See the Appendix. 2

Theorem 1 is an entirely expected, but nevertheless reassuring result: if the insurer uses

Bayes’s rule to update over client types,then expected type converges to true type over time.

Given Eq. (1) we can calculate the probability of an accident given history(s,n):

p(s,n)=(1−λ(s,n))pa+λ(s,n)pb. (2)

Note that from Theorem 1, Eq. (2) implies that the expected value of the probability of an accident approaches its true value as n increases. Using Eq. (1), it is easy to show that

p(s,n)can be written as p(s,n)=(1−λ)p s₊1 a (1−pa)n−s+λpbs+1(1−pb) n₋s (1−λ)psa(₁−_pa)n−s+λ_p_b(s ₁₋_p_b)n−s .

Note that p(0,0)= p, so that the assumed probability of accident without a history is just

the initial prior.

The expected utility of the BUC, up to z contract renewals for a type i agent, is

EU_iBUC(z,W,L, γ ) = z X n=0 γn n X s=0 n! s!(n−s)!p s i(1−pi) n₋s_{U [W}₋_p₍_s_,_n₎_L]_; _i ₌_a_,_b_, ₍₃₎

where z is the number of permitted contract renewals,γis the psychological discount factor per period(0< γ <1),W is the initial periodic wealth, and L is the amount of the possible

loss.

This can then be compared with the expected utility, over z periods, of the optimal RS contract menu for a type i agent:

EU_iRS(z,W,L, γ ) = z X n=0 γn_{_p

iU [W−L(1−αi(1−pi))]+(1−pi)U [W −αipiL]}; i =a,b,

(4) whereαiis the proportion of the loss that is insured—that is,αa=1, αb<1.7

For given values of initial wealth (W ) and possible loss amount (L), define

Hi(z, γ )=EUiBUC(z, γ )−EU

RS

i (z, γ ); i =a,b. (5)

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Theorem 2: For any values of the parameters,a type a individual will always prefer the BUC—that is,Ha(z, γ ) >0∀z∈IN;0< γ <1.

Given Theorem 2, in order to consider the Pareto dominance of the BUC, it is sufficient to analyze only the low-risk agent’s preferences.

Theorem 3: Given U [·] strictly concave,ifγ =1,then there exists a value z0∈IN such

that Hb(z, γ =1) >0∀z∈IN,z≥z0.

Theorem 3 considers a special case—that in which the low-risk agent is perfectly patient. In words, the result is, if the agent is perfectly patient, then he will always prefer the BUC so long as the insurer offers sufficient contract renewals. The intuition behind the result is clear; since (from Theorem 1) the low-risk agent’s expected premium approaches the full-information level over time, so his expected utility increases over time under the BUC. While his expected utility starts out lower than that corresponding to the contract menu, it must converge to something greater, and so long as the sum of the positive expected utility gains outweighs the sum of the negative part, the agent is better off. This can be guaranteed under perfect patience, so long as there are sufficient contract renewals.

Corollary 1: a. For each z∈IN;z≥z0,there exists aγ0(z) <1 such that Hb(z, γ ) >0 ∀γ ∈IR, γ0(z) < γ <1.

b. γ0(z)is a strictly decreasing function for z≥z0.

This result can be observed in figure 1, in which the graph of a typical Hb(z,·)function is shown8. Note that as z is reduced, the critical valueγ0(z)gets closer to 1—that is, if

fewer contract renewals are offered, then only the more patient consumers benefit from the BUC; hence, there is a kind of tradeoff between the number of renewals and the discount factor.

As an example of how the above calculations could work, consider the following pa-rameter values; pa=1/2,pb=1/6, γ=0.98,U [x]=

√

x, λ=0.9,W=20,000, and L= 10,000. The optimal RS adverse-selection contract menu corresponds toαb=0.1010205, and it gives the low-risk agent expected utility of 134.708048 in each contract period. Hence, the (ex ante) expected utility of z repetitions (note that, from Eqs. (3) and (4), the initial period is called z=0; hence, the number of repetitions is the number of periods, after the initial one, that the contract is renewed) of the optimal adverse-selection contract for this agent (Eq. (3)) is just

134.708048 µ 1−γz+1 1−γ ¶ =6735.4024(1−0.98z+1).

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Figure 1. The graph of Hb(z,·)for different values of z.

For selected values of z, this expected utility(EURS₎_{is shown in Table 1, together with the}

corresponding values of the expected utility of the BUC (EUBUC_{) calculated from Eq. (4).}

As can be seen, the RS contract menu, as expected, is better for shorter-length contracts, but the BUC does better for longer-term contracts (more renewals), since it gives the client enough time for his premium to settle down close to the optimal full-information level. The BUC overtakes the RS contract menu after nineteen renewals are possible. At eigh-teen renewals, the RS contract menu gives 2147.026548 as against 2146.846511 from the BUC, but at nineteen renewals (not in the table) the BUC gives 2238.900073 as against

Table 1. Expected utilities of the RS and BUC contracts according to contract length.

EURS _EUBUC _EURS _EUBUC z=0 134.708048 134.1640787 z=9 1232.088256 1230.06272 z=1 266.721935 265.7181625 z=10 1342.154538 1340.234478 z=2 396.0955443 394.7210192 z=12 1555.727154 1554.113634 z=3 522.8816815 521.21987 z=14 1760.842293 1759.637104 z=4 647.1320958 645.2597343 z=16 1957.834874 1957.11429 z=5 768.8975019 766.8838065 z=18 2147.026548 2146.846511 z=6 888.2275999 886.1344213 z=20 2328.726231 2329.126461 z=7 1005.171096 1003.05358 z=22 2503.230608 2504.237897 z=8 1119.775722 1117.682667 z=24 2670.824611 2672.455524

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2238.794065 from the RS contract menu. After the BUC overtakes the RS contract menu, it never falls behind again (see the Appendix, proof of corollary).

There is an important point to note about the dynamics of the BUC. As we have pointed out, conditional on the only contracts on the market being the RS contract menu and the BUC, all high-risk individuals will purchase the BUC, and there exist conditions under which the low-risk ones will also purchase the BUC. Recalling that our initial assumption was that all individuals are symmetric in all but the probability of accident, thus requiring that they all have the same discount factor, the Pareto dominance theorem is true for a population of individuals with a given discount factor. In reality, there may well exist a range of discount factors in the population, which can be modeled by simply decreasing the discount factor for some clients9_{. If this were so, then we would have a model in which,}

for a given number of contract renewals, all high-risk types (independent of discount rate) would purchase the BUC, and all low-risk types with a discount rate greater than or equal to some critical number (determined by the number of contract renewals) would also purchase the BUC, while the other low-risk types, for whom the BUC does not payoff, will purchase the RS partial coverage contract.

This implies the following dynamic situation10, since not all low-risk individuals are going to purchase the BUC, the initial market odds line is below that corresponding to the entire population of individuals. Hence, the number of contract renewals required increases, and so the most impatient of current BUC buyers will drop out, preferring the RS partial coverage contract, which leads to a further lowering of the average loss probability, and so a further increase in the contract renewal horizon, and so a further defection of impatient BUC buyers to the RS contract. The question that this obviously raises is if the process converges to a stable equilibrium in which the market is divided among impatient low-risk clients at the RS partial coverage contract, and the rest of the clients (all high-risk ones plus patient low-risk clients) at the BUC.

The answer is almost always. In fact, the insurance company would be simply reinter-preting the initial prior to be not the proportion of high-risk types in the entire population but rather the proportion of the final mix of clients desired at the BUC contract. In effect, the above model does not depend on the initial prior,λ, being in fact the population proportion, it could just as well be any other number. Hence, the insurance company could just decide, ex ante, on the client mix that it wanted to purchase the BUC (that is, it does not necessarily want that the entire population participate in the BUC, as we have done above), calculate the implied initial prior, and then offer the corresponding number of contract renewals that will attract exactly the anticipated client mix, given the critical cutoff discount rate. Note that this does not require the insurance company to be able to observe discount factors, but it does require that the distribution of discount factors, at least over low-risk types, be known. Vazquez and Watt [1997] have formally analyzed the nonhomogeneous discount factor case. The above argument makes the following prediction: if the BUC bonus-malus contract is offered along with the RS contract menu, then the full-coverage, high-premium contract in the RS menu will not sell. Hence, presumably, it will not be offered. Furthermore, we should see only impatient clients purchasing the RS partial coverage contract—older people, people with a high propensity to emigrate to another country, and so on. Both of these predictions would be easily testable empirically.

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3. Conclusions

In this article we considered the comparison between two different contract settings for in-surance when the insurer acts in a competitive environment. On the one hand, we considered the standard separating equilibrium contract menu, and on the other, a format under which all contracts have full coverage, and the premium is endogenously determined according to the client’s accident history. It is claimed that this second format corresponds to usual real-world insurance practice, and it was noted that in order for such a contract format to be viable, all accident histories must be made public information among insurers.

We have shown that, all high-risk types will prefer the BUC format to the RS contract menu, and, so long as there are sufficient contract renewals, and insurance consumers are sufficiently patient, so will all low-risk types. Since both contract formats offer expected profits of 0 to the insurer, we have shown that there exist conditions under which the BUC with endogenous premiums Pareto dominates repetition of the RS contract menu.

Furthermore, if we assume that discount rates are not homogeneous over consumers, then the above model would appear to predict that insurance companies can design their bonu-malus contracts such that there is a separation of types over the RS partial insurance contract, and the BUC. Specifically, all high-risk types, independent of their discount factors, and the most patient low-risk types can be attracted to a BUC, while impatient low-risk types will opt for the RS partial-coverage contract.

The analysis presented here suggests plenty of further studies. Since the critical values of either z0orγ0(z)depend on all the other parameters, it would be interesting to consider

the comparative statics exercise of changes in initial wealth (W ), loss amount (L), the probabilities of loss ( paand pb), the (ex ante) probability of any particular client being type

b(λ), and perhaps most important, the shape of the utility function U [·]—that is to say, the characteristics of risk aversion11_.

Lastly, the analysis here is valid for any increasing concave utility function. Given this, the results achieved could certainly be recast in a setting based solely on second-order stochastic dominance. Doing so may well be rewarding empirically, since it would not require estimations of utility functions in order to calculate the exact terms of the BUC.

Appendix

Proof of Theorem 1

We make use of the well-known mathematical fact

n

X

s₌0

C_snxsyn−s =(x+y)n. (A.1) Part 1: limn→∞E[λ,n,pa]=0.

Note that we can write

psa(1−pa)n−sλ(s,n)= λp s a(1−pa)n−spsb(1−pb)n−s (1−λ)ps_a(₁₋_p_a)n₋s₊_λ_ps b(1−pb)n−s =A×B,

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where A= √ λ√ps a p (1−pa)n−s p ps_bp(1−pb)n−s 2√1−λ B =2 √ λ√1−λ√ps a p (1−pa)n−sp_ps b p (1−pb)n−s (1−λ)ps a(1−pa)n−s+λpsb(1−pb)n−s ≤1. Hence E [λ,n,pa]≤ n X s=0 C_sn √ λ 2√1−λ ¡√ pa √ pb ¢s¡p 1−pa p 1−pb ¢n−s . (A.2)

Together, (A.1) and (A.2) imply

E[λ,n,pa]≤ √ λ 2√1−λ ¡√ pa√pb+ p 1−pa p 1−pb ¢n , (A.3)

which approaches 0 as n→ ∞, since

√ pa √ pb+ p 1−pa p 1−pb<1.

Hence, from (A.3) it is obtained that limn→∞E[λ,n,pa]=0.

Part 2: limn→∞E[λ,n,pb]=1.

Note that it is always true that

(1−λ)E [λ,n,pa]+λE [λ,n,pb]= n

X

s=0

λC_snp_bs(1−pb)n−s=λ.

Hence it is also always true that

(1−λ) ½ lim n→∞E[λ,n,pa] ¾ +λ ½ lim n→∞E[λ,n,pb] ¾ =λ.

Since limn→∞E[λ,n,pa]=0, it must be true that limn→∞E[λ,n,pb]=1. 2

Proof of Theorem 2

Since, for the high-risk individual, in the contract menuαa =1, we can write

EU_aRS(z,W,L, γ )= n X s₌0 γn_{U [W}₋_p aL]. (A.4)

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Second, note that, using the equation for p(s,n)in the text

p(s,n)=(1−λ)p

s+1

a (1−pa)n−s+λpbs+1(1−pb) n−s

(1−λ)psa(₁−_pa)n−s+λ_p_b(s ₁₋_p_b)n−s < pa. (A.5)

Hence, it is satisfied that U [W − p(s,n)L] > U [W − paL] for any values of s and

n(0≤s≤n), since U [·] is a strictly increasing function.

Thus, (A.1), (A.4), and (A.5) imply that, for any values of z ∈ IN,W,L ∈ IR, and 0< γ <1, EU_aBUC(z,W,L, γ ) > z X n=0 γn n X s=0 C_snp_as(1−pa)n−sU [W −paL]=EUaRS(z,W,L, γ ). 2 Proof of Theorem 3

First note that we can write

Hb(z, γ )= z X n=0 γn_h₍_n_), where h(n)= n X s=0 C_snpsb(1−pb)n−sU [W −p(s,n)L]−K,

with K≡pbU [W −L(1−αb(1− pb))]+(1− pb)U [W −αbpbL], which is just the

single-period utility that corresponds to the RS. Given U [·] strictly concave, it is easy to see that K<U [W −pbL].

Since p(s,n) > pb, it is true that

h(n) <

n

X

s=0

C_snpsb(1−pb)n−sU [W −pbL]−K =U [W−pbL]−K. (A.6)

On the other hand,

U [W−p(s,n)L]=U [(1−λ(s,n))(W−paL)+λ(s,n)(W −pbL)],

and since U [·] is strictly concave and 0≤λ(s,n)≤1, then

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Hence, we obtain

h(n) > (1−E(λ,n,pb))U [W−paL]+E(λ,n,pb)U [W −pbL]−K. (A.7)

Thus, (A.6) and (A.7) imply that

(1−E(λ,n,pb))U [W −paL]+E(λ,n,pb)U [W−pbL]−K <h(n) <U [W −pbL]−K

and Theorem 1 yields

r≡ lim

n→∞h(n)=U [W−pbL]−K >0.

Moreover, h(n)is a bounded monotone increasing sequence for n sufficiently large. Therefore, it is satisfied that

1. Ifγ =1,P∞_n₌₀γn_h₍_n₎₌P∞

n=0h(n)= +∞since r >0.

2. If 0< γ <1, sinceP∞_n₌₀γn_<_+∞_{and h}₍_n₎_{is a bounded monotone increasing sequence}

for n sufficiently large, the series convergence criterion of Abel yieldsP∞_n₌₀γn_h₍_n_{) <} +∞.

Hence, Hb(z = ∞, γ =1)= +∞and Hb(z = ∞, γ < 1) < +∞. In consequence,

there exists a value z0∈IN such that Hb(z, γ =1) >0∀z∈IN,z≥z0. 2

Proof of Corollary 1

a. Given Theorem 3, standard continuity arguments are sufficient for this part of the corollary to be true.

b. First note that

h(n+1)−(1−pb)h(n)= n+1 X s=0 C_sn+1psb(1−pb)n−s+1U [W−p(s,n+1)L] − n X s=0 C_snpsb(1−pb)n−s+1U [W−p(s,n)L]+pbK > n X s₌0 C_snpsb(1−pb)n−s+1{U [W −p(s,n+1)L] −U [W−p(s,n)L]},

since C_sn+1 ≥C_sn. Hence, using that p(s,n)is strictly decreasing in n, it is true that

h(n+1)−(1−pb)h(n) >0,

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FULL INSURANCE 149

Secondly, note that if Hb(z, γ )≥0 then h(z) >0 and

Hb(z+1, γ )−Hb(z, γ )=γz+1h(z+1) >0,

which implies that Hb(·, γ )is increasing for z≥z0andγ ≥γ0(z0). In particular, for any

given discount factor, once we have found a z such that Hb(z, γ )≥0, then further increases in z can never cause Hb(·, γ )to go negative again.

As a direct consequence it is obtained thatγ0(z)is a strictly decreasing function for

z≥z0. 2

Acknowledgments

We wish to thank the participants of the Twenty-third Seminar of the European Group of Risk and Insurance Economists, and the seminar participants of Carlos III University in Madrid. In particular, we thank Harris Schlesinger for valuable comments. Any remaining errors are the responsibility of the authors.

Notes

1. Unless, off course, the optimal separating contract happens to coincide with a pooling contract. In the standard 2 agent-type model, this would require that the principal attaches zero prior probability to the any particular agent being of the most risky type, which is inconsistent with the existence of an adverse-selection problem. 2. When this type of coverage is explicitly contracted, it is usually known as a disappearing deductible. 3. Strictly speaking, this is only the necessary condition. The sufficient condition is that no part of the indifference

curve of the low-risk individual at the separating equilibrium lie geometrically below the market fair odds line (see Rothschild and Stiglitz [1976]).

4. In Spain, the law regulating damage insurance contracts (as opposed to life insurance), allows a maximum contract duration of ten years. However, there is a commitment problem since the law stipulates that the contract will become null and void twenty days after any agreed premium payment date should the premium not be paid. Second, with respect to repetition of the Rothschild-Stiglitz contract menu, according to the Spanish Actuarial Association, it is normal practice for both life and nonlife insurance (for example, homeowner’s insurance) to be of annual (but renewable) duration, and after correcting for changes in risk and inflation, the premium is, in many cases constant over periods.

5. Claim underreporting is a moral-hazard problem, and we are only interested in adverse selection. For a good treatment of claim underreporting, see Hey [1985].

6. Note that if the Rothschild-Stiglitz equilibrium did not exist because of there being too many low-risk types, then the Pareto dominance of the suggested BUC over RS becomes even more probable.

7. It is a well-known result (see, for the original exposition, Rothschild and Stiglitz [1976]) thatαb <1 and

αa =1—that is to say, low-risk clients must signal their type by accepting partial coverage, and high-risk individuals purchase full insurance (the insurer has no informational difficulties in offering a fair premium to these individuals in the separating contract.)

8. Although for the purposes of the exposition, the actual shape of the function is irrelevant, it can be mathemati-cally shown (by analyzing the first- and second-order derivatives) that it corresponds to the representation presented as Figure 1.

9. For example, some people may be more likely to emigrate. It may also be useful to model older people as being more impatient (having lower discount factors), but on this point we must be careful as we have modeled the discount factor as a constant for the z periods of the BUC. It would undoubtedly be interesting (all-be-it complicated) to allow the discount factor to diminish as the client ages. However, so long as the contract

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length is rather insignificant in relation to the entire life of a client, our assumption of constant discount factor seems acceptable. We thank an anonymous referee for pointing this out.

10. We thank Art Snow for pointing this out.

11. This is bound to be a complicated but most intriguing question. Since updating yields, ex ante, greater variance of possible outcomes, one may be led to believe that more risk-averse individuals might prefer the more stable contract menu, but then on the other hand, in the longer term the Bayesian contract settles down (converges) to the full-information contract and thus offers less variance together with greater periodic utility.

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