The Legal Constraint of Misrepresentation in Insurance Market

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2 The Legal Constraint of Misrepresentation in Insurance Market Shinichi Kamiya Nanyang Business School Nanyang Technological University Joan T. Schmit Wisconsin School of Business University of Wisconsin-Madison First draft: April 2011 This draft: August 2013 Corresponding author, Tel.: , Fax :

3 The Legal Constraint of Misrepresentation in Insurance Market Abstract We consider competitive insurance markets in which insurers induce information revelation of applicant s risk type with self-selection mechanisms and investigate the effect of legal penalties imposed if individuals choice of a contract is considered as misrepresentation. We first provide a brief overview of contract law and discuss the effect of misrepresentation on an individual s self-selection, and then demonstrate that a market equilibrium, if it exists, is unique and characterized by a set of separating full-coverage contracts. We also show that the market equilibrium approximates the first-best under feasible conditions, which tend to be satisfied by the intrinsic nature of insurance contracts. Use of sufficient claim management against misrepresentation is found to be not always Pareto improving as ex ante risk classification. Keywords: misrepresentation, self-selection, adverse selection, insurance JEL Codes: D81, D82, G22 1 Introduction In the advance of contract theory, the main theoretical contributions to competition under adverse selection and a large fraction of empirical literature deal with insurance contracts. Despite the efforts of making models more realistic since the seminal work by Rothschild and Stiglitz (1976), competitive insurance market models ignore the existence of post-loss testing, assuming no potential penalty against policyholder s misrepresentation on its choice of a contract. Assuming no post-loss testing, however, may not be appropriate because it virtually ignores the presence of contract law. What if I lied about my smoking habit on my life insurance application? And what happens if the insurance company finds out after a claim is filed? Such questions may come across an applicant s mind when he/she answers questions on an insurance application. Since life insurance rates for smokers are significantly higher than that for non-smokers, smokers could be motivated to lie on their policy applications. Meanwhile, an applicant would expect that if a claim event happens within some given period as a result of a smoking-related illness, such as lung cancer or 1

4 heart disease, the insurer simply would deny the claim. This economic tradeoff must be considered in insurance applicant s choice of a policy in many types of insurance. Under conditions of information asymmetry in which individuals have private information regarding their risk type (e.g., high risk or low risk), a well-known theoretical prediction is derived; high-risk individuals demand more coverage than low-risk individuals, although the existence of the separating equilibrium depends on the proportion of high-risk individuals (Rothschild and Stiglitz, 1976). While this result is generally accepted and empirically tested in many markets, some underlying assumptions of insurance market models have been shown to be overly strong and may not be realistic. For instance, empirical studies in insurance markets support the argument that appropriate underwriting does not require individuals to self-select a policy, which causes a demand differential between risk types (e.g., Dionne, Gourieroux, and Vanasse, 2001). Furthermore, Browne and Kamiya (2012) theoretically show that an equilibrium may exist where insurers offer full coverage even to low-risk type through use of an underwriting test instead of simply relying on self-selection. Thus, an insurer s effort at gathering information related to private information, and hence successful testing, may make the first-best outcome possible. As noted above, one aspect of the market not yet examined, is the existence of post-loss testing for misrepresentation. The purpose of this paper is to investigate the effect of post-loss testing on market equilibrium. Such testing is intended to distinguish high-risks from low-risks, both for equitable outcomes and as signals to place positive value on proper classification. 1 Prior literature on the general value of testing after entering a contract, beginning with Mirrlees (1974), offers a set of conditions for the existence of the first-best outcome. Much of that literature reveals that severe punishment is a key component to attaining the first-best outcome. For example, Nalebuff and Scharfstein (1987) show that the first-best is attained where agent s private information is revealed with some positive probability and severe punishment occurs (see also Guasch and Weiss, 1980, 1981, 1982). Riordan and Sappington (1988) show that even uninformative signals are sufficient to ensure the first-best outcome under some conditions. The intuition behind the first-best outcome is that an agent s fear of being caught and receiving severe punishment prevents that agent from being untruthful. Given that an arbitrarily large penalty is unrealistic in many settings, recent literature such as 1 Insurance intermediaries such as brokers and agents may play an important role in the choice of a contract, but we investigate the effect of post-loss test in the simple relationship between insurers and individuals for consistency with the literature. 2

5 Kessler, Lülfesmann, and Schmitz (2005) for endogenous punishments and Kofman and Lawarrée (1993) for exogenous punishments has considered the impact of limited liability on the impact of ex-post testing. Here we extend the literature by identifying a particular setting in which feasible conditions exist to achieve a Nash equilibrium with ex-post testing in insurance. 2 Specifically, the availability of contract law rules surrounding misrepresentations generates the necessary conditions for our results. The effect of ex-post testing is particularly important in insurance markets because misrepresentation/concealment of material information may be identified just when coverage is needed, that is, when the policyholder suffers loss that is to be covered by the policy. In other words, a penalty of misrepresentation could cause financial disaster. Thus, the effect of investigating misrepresentations may be amplified by a potential large penalty. 1.1 How does Post-Loss Testing Affect an Individual s Incentive? A failure to answer application questions truthfully in such a way that an insured is misclassified for rating or coverage purposes may result in misrepresentation by the policyholder. Misrepresentation is a contract law concept involving a false statement (or omission, known as concealment ) of fact made by one party to another to induce the other party into a contract under terms that would not be accepted had the true fact been known. In insurance contracts, this legal principle presumably binds applicants to provide relevant private information regarding their risk characteristics. The legal principle binds especially when applicants are aware of their risk type because a finding of intention may be more likely to allow insurers to void a contract, a particularly harsh penalty; hence, rational applicants should expect claims not to be paid when misrepresentation is identified by an insurer. If misrepresentation is known to be monitored by insurers, applicants would have an incentive to respond accurately, thereby being classified appropriately with respect to their true risk type, and hence the first-best allocation is attained. Such a complete outcome may be unrealistic, however, because contract law is not perfectly enforced. In many types of insurance, misrepresentation may be investigated only when post-loss verifiable 2 We investigate the existence of Nash equilibria for robustness. Our argument can be extended easily to other equilibrium concepts where insurers non-myopic behavior is assumed. 3

6 information is available. 3 Specifically, an applicant s misrepresentation would be revealed when claims are filed and additional policyholder characteristics are revealed. 4 For instance, suppose that homeowner s insurance applicants know that storing explosives on the premises affects the price as well as the terms and conditions of an insurance policy. Because premises inspections generally are not done at the time of application, however, the insurer typically will not know whether the applicant stores explosives on the premises unless the applicant truthfully discloses the fact to the insurer at the time of application. The applicant s misrepresentation may be revealed only when a covered loss event occurs. 5 Thus, it is reasonable to assume that contract law enforcement could provide individuals an incentive to act honestly. Here we focus on the insurer s post-loss test, and evaluate the effect of potential claim denial. The question of interest to us is how imperfect enforcement of contract law affects individuals self-selection and thus market equilibrium in a competitive insurance market settings. We consider an insurer s investigation of misrepresentation as a conditional post-loss test to uncover an applicant s choice of policy offered at a premium lower than its actuarially fair premium. 6 The effect of introducing the legal constraint is straightforward. A post-loss test discourages high-risk individuals from deviating from fair premiums because claims may be denied if the deviation is observed by an insurer in the process of adjusting the claim. The effectiveness of the test as a self-selection device lies in the performance of the test, the size of loss, a penalty that yields infinite negative utility, and a test fee charged to each policy. The primary results are summarized as follows. First, adverse selection is completely eliminated by utilizing a post-loss test, but the existence of the equilibrium depends on the accuracy and the cost of the test. Second, if a total loss of wealth causes an infinite utility penalty, the market outcome approximates the first-best equilibrium with an arbitrarily small expenditure on testing. A notable difference from the result of testing in a labor market model demonstrated by Nalebuff and Scharfstein (1987) is that the outcome results from the intrinsic nature of insurance risk, which 3 Even in life insurance, policies with small indemnity amount tend to be offered without medical examination in contrast to policies with large indemnity amount which usually requires taking medical examination as a part of underwriting process. 4 It is also possible that individuals are not fully aware of their risk type. 5 This paper ignores the potential moral hazard problem to focus on investigating the effect of enforcement of identifying misrepresentation on the self-selection mechanism. 6 We focus not on insurance claim fraud but on misrepresentation in contracting. See, for instance, Katja Müller and Wagner (2013) for insurer s auditing strategies against fraudulent claims. 4

7 could cause a total loss of wealth (e.g., a huge liability loss in an auto accident). As an implication to adverse selection literature, these results predict that a standard argument of positive correlation between risk and coverage in self-selection models (Chiappori, Jullien, Salanié, and Salanié, 2006) is mitigated if insurers are able to identify an individual s misrepresentation at a reasonable cost or if the individual s fear (expected utility penalty) is large enough in the state where the loss is not paid by a policy. Thus, the theoretical analysis provides another answer as to why empirical research on the existence of adverse selection in insurance markets is mixed, given that insurers spend more on claim management in some insurance markets and because some insurance policies cover larger losses than others. 7 The remainder of the article is organized as follows. A brief overview of contract law and the effect of misrepresentation on an individual s self-selection are presented in the next section. In Section 3 we consider equilibrium when misrepresentation is identified by an imperfect and costly post-loss test. A summary of our findings in our work is contained in Section 4. 2 Effect of the Legal Constraint of Misrepresentation Insurance underwriting is the process of classifying potential insureds according to future loss potential, thereby determining both whether or not to accept an applicant as an insured and if so the appropriate pricing category for that policyholder. In estimating future potential losses, insurers rely on information supplied by the applicants. Inaccuracies with this information will lead to costly decisions about who to accept as insureds and at what price. The temptation to offer inaccurate information, however, can be substantial. Some potential policyholders would be denied coverage altogether; others would be accepted but only at a far higher price or restricted coverage terms. To discourage such inaccuracies, legal systems around the globe have implemented punitive mechanisms when those inaccuracies are discovered. In general, the punishment is to make the policy void; that is, as if it did not exist. The circumstances under which the policy becomes void, however, differ depending on the legal jurisdiction. (See Ingram, 2005, for a discussion) A common approach to punishment of misrepresentation is to void an insurance contact for any material misrepresentation. A misrepresentation has been defined as a statement of 7 See Cohen and Siegelman (2010) for an extensive survey of this literature. 5

8 something as a fact which is untrue and affects the risk undertaken by the insurer (Methodist Medical Center of Illinois v. American Medical Security, Inc., 38 F. 316, 1994). Often incomplete answers or a failure to disclose material information on an application for insurance may constitute a misrepresentation when the omission prevents the insurer from adequately assessing the risk involved (Ibid.). The inaccuracy is considered material if it would have altered the insurer s decision to accept the applicant and/or to do so at a different price. In this severe approach, intent or knowledge is irrelevant to the voidance of the policy. Furthermore, in most jurisdictions the inaccuracy need not be related to the loss situation itself. Courts, however, may deny a claim of misrepresentation if knowledge of the fact is imputed to the insurer. For instance, an application may ask if prior policies have been cancelled. Even if the insured answered no when a prior policy has been cancelled, if it is with the same insurer or agent, the courts may choose not to consider this a misrepresentation because the insurer knew or should have known of the false statement at the time the policy was issued. (see Graphic Arts Mutual Insurance Company v. Pritchett, 469 S.E. 2d 199 (GA. Ct. App. 1995)) In some jurisdictions, the insurer must prove that the applicant intentionally provided inaccurate (and usually material) information. This, of course, is a higher standard for the insurer to meet in order to void the contract. Proof of intent often is difficult and costly. In many jurisdictions, intent is required only in the case of concealment, which has been defined as the fraudulent failure to reveal information which someone knows and is aware that in good faith he/she should communicate to another (Legal Dictionary, 2005). 8 The key element associated with concealment is that the applicant to insurance does not answer questions falsely; rather s/he omits various information that would be relevant to the underwriter s decision. As with misrepresentation, in most instances, the actual loss need not relate to the facts that have been concealed. In general, the effect of a misrepresentation or concealment is, as mentioned above, to void the policy. Voidance indicates that in effect the contract was never in force. Any claim, from any cause, therefore, is denied. Exceptions exist in most life insurance policies (and some health insurance policies) where an incontestability clause applies. The incontestability clause typically states that after some given period (usually up to two years), applicant misrepresentations or concealments will be ignored in 8 Legal Dictionary (2005) 6

9 disputing (i.e., contesting ) a claim. These provisions exist because of the reliance on benefits by someone other than the individual who made the representations in most instances, as well as the very long time period of coverage. A misstatement of age clause is found in most life insurance policies as well, and overrides the misrepresentation rules for an inaccurate age. Instead, if the insured life is older than stated, the payout will equal what would have been purchased at the accurate age. The effect of policy cancellation on insured s financial status can be illustrated as follows. In 2002, US medial family net worth is $92,500, which is calculated by linear interpolation of Federal Reserve Bank, Survey of Consumer Finances in 2001 and And the Insurance Research Council 2002 auto bodily injury liability claim payment data, which lists 34,255 claim payments, shows that 1.3% of the claim payments exceed the medial family net worth. Furthermore, from the fact that 5% of the claim payments reach policy limits and the claim payments include only bodily injury liability payments, the actual probability of losing family net worth conditional on an auto accident is higher than 1.3% if a policy is voided. This illustrates that a severe punishment against misrepresentation is an intrinsic nature of an insurance policy. 3 Model Setting 3.1 Competitive Insurance Market Consider the paradigm of risk-neutral insurers operating in a competitive market where individuals who have a standard risk-averse utility function (i.e., u > 0, u < 0) live in one period with initial endowment W 0 > 0. Their terminal wealth is uncertain with two states: W L withafixedlossd>0 ( W 0 )andw NL with no loss. Individuals differ in their likelihood of loss: the probability that loss state W L occurs for a high-risk is π H and that for a low-risk is π L,where0<π L <π H < 1. Individuals know their risk type but insurers do not. Both parties know the probability of loss and the proportion of high-risk individuals and that of low-risk individuals in the market denoted by λ and 1 λ, respectively. Let π M = λπ H +(1 λ)π L be the average probability of loss in this economy. We assume that insurers can identify misrepresentation/concealment when policyholders file claims. Note that tasks in insurers claim management include investigating the loss and deter- 7

10 mining whether the policy covers the loss. In the process of adjusting claims, insurers may find verifiable information of policyholder s misrepresentation/concealment on its choice of a contract. We introduce a post-loss test defined by the probability of identifying an individual s misrepresentation and as a function of an insurer s expenditure on the test, denoted by p(e) wherethe expenditure, e, is insurers choice. It is assumed that insurers have access to an identical testing technology that the probability is zero at e = 0 (i.e., no post-loss test) and increases (at a decreasing rate) as the expenditure increases (i.e., p(0) = 0, p (e) > 0andp (e) < 0, p(e) < 1). Thisisareasonable setting because testing is, of course, costly, and while the first dollar spent on testing adds significant value, each new dollar adds a diminishing value to policyholder information accuracy. Furthermore, no test will be perfectly accurate in identifying policyholder misrepresentation. In our two risk type setting, the misrepresentation of a high-risk policyholder who chooses coverage at a rate lower than would be fair given its risk type, will be identified by the insurer with the probability, p, upon the filing of a claim. It may be also possible that a low-risk policyholder who purchased a policy at its fair premium would be misclassified as a high-risk and the policy is voided. However, the probability should not be large enough to affect low-risk individuals choice of a contract because an insurer needs to prove that low-risk individual s misrepresentation in the absence of misrepresentation by collecting verifiable information. Therefore, we assume away the possibility. The structure of the market is as follows: In stage 1, risk-neutral insurers offer a single contract including their decision on post-loss test. In stage 2, each individual chooses one contract. And in stage 3, insureds who file a claim are tested for misrepresentation if the contract imposes post-loss test. If the test reveals misrepresentation, the policy is voided. Otherwise, the claim is paid. Both the insurer and the applicant are bound to follow the terms of the contract. A policy offered by insurers specifies: a testing fee denoted by t, charged to applicants who apply for a policy with a post-loss test; coverage denoted by I, which is less than or equals to a loss D; and premium rate, π. Note that a per policy testing fee, t, is different from an insurer s per application expenditure on a test, e, because the test is carried out only for those whofile a claim. For instance, when only low-risk individuals purchase a policy which investigates misrepresentation after a claim is filed, the resource constraint, t = eπ L, holds. Therefore, insurer s per-claim expenditure for a post-loss test is defined by e = t/π L. It is assumed that all of these variables are publicly observable 8

11 and verifiable. Thus, a contract offered by an insurer is defined by C (t, I, π), where the premium for coverage I is πi + t. The high-risk individual s expected utility by accepting C L is defined by: U H (C L )=π H (1 p(e))u(w L1 )+π H p(e)u(w L2 )+(1 π H )u(w NL ). (1) where we use shorthand notations for terminal wealth in the states: W L1 = W 0 +(1 π L )I t D W L2 = W 0 D W NL = W 0 π L I t W L1 represents the state where insurance indemnity is paid according to the policy, given loss occurs; W L2 represents the state where insurance indemnity is not paid and the premium paid is refunded due to policy cancellation given loss occurs; and W NL represents the state where no loss occurs. 9 Note that the penalty state W L2 does not depend on insurance indemnity. Thus, with probability π H (1 p), the high-risk individual has a loss and receives indemnity from the policy because with probability (1 p) his misrepresentation is not identified by the insurer. With probability pπ H, the high-risk individual has a loss and does not receive indemnity from the policy because with probability p, her misrepresentation is identified by the insurer and the policy is voided. In contrast to the standard self-selection constraint, which does not penalize a high-risk individual s choice of low-risk policy, with probability pπ H, the high-risk individual has a loss and does not receive indemnity from the contract. This self-selection constraint provides two major implications. First, as test accuracy increases, the weight of expected penalties for high-risk individuals (the second term) increases and it becomes easier to offer a policy that will attract only low-risk individuals. Second, as the amount of loss increases, the expected penalty for a high-risk individual increases as well. It is of special interest that the equilibrium is affected when D W 0 and u(0) =. These cases will be discussed later. 9 For simplicity, it is assumed that premiums paid by a policyholder are fully refunded in the case of policy cancellation. Relaxing the assumption by allowing insurers to retain a portion of premium slightly increases the possible penalty of misrepresentation but does not significantly affect the results of the analysis. 9

12 Consider a sequential game in which a new firm decides to enter the existing insurance market by offering a contract. When individuals maximize their expected utility, Cournot-Nash equilibrium identified in this game can be characterized by the non-existence of a loss-producing contract in equilibrium and the non-existence of a non-negative profit contract outside of equilibrium, if offered. Without loss of generality, we assume that: 1) low-risk (high-risk) individuals choose a contract with (without) post-loss test when they are indifferent between a contract with a post-loss test and one without a test, 2) they choose a contract with larger coverage when they are indifferent between two contracts, and 3) all individuals of the same risk type choose the same policy when they are indifferent between contracts. 3.2 Competitive Equilibrium There are two possible types of insurance policies in equilibrium. A separating policy is a contract that only one type of individual accepts. In contrast, a pooling policy is demanded by individuals of differing risks, negating the need for varied contracts. Rothschild and Stiglitz (1976) show that a separating equilibrium identified as a set of policies (C H,C L ) may be attained (see Figure 1). If the equilibrium exists, high-risk individuals acquire full coverage, while low-risk individuals obtain only partial coverage. This is the case where the population of high-risk individuals is sufficiently large relative to that of low-risk individuals. [Insert Figure 1 Here] In terms of individuals freedom of choice of a contract, separating policies and pooling policies may be treated differently. Consider that a self-selection model describes an insurance market where uninformed insurers offer a menu of policies and informed individuals characterized by either high-risk or low-risk choose a policy that maximizes their utility. Separating policies are offered for the purpose of sorting individuals, with low-risk individuals given opportunities for lower-cost and/or broader coverage insurance. In this scenario, high-risk types may cause an externality on low-risk types by misrepresenting themselves as low-risk types. Since insurers do not intend to attract high-risk individuals to a policy offered at a low-risk premium, insurers may examine individuals choice of a policy. Since contract law itself is not binding without 10

13 the insurer s effort to identify misrepresentation, it is natural that they allocate some resources to enforce the law to reduce the effects of misrepresentation. In contrast, a pooling policy is supposed to attract both risk types. In this instance, high-risk individuals should not to be exposed to any penalty for choosing a pooling policy. Although there may be a policy with a post-loss test that attract both risk types, it is straightforward to see that such a policy does not hold in equilibrium as follows. Lemma 1. No pooling equilibrium exists. Proof. Let a pooling policy C P = (t, I P < D,π P ), which solves max t,i U L (C P )whereπ P = λ(1 p)π H +(1 λ(1 p))π L. Suppose C P is offered and attracts both low-risks and highrisks. Since the policy attracts both risk types, the coverage of the policy is determined at partial coverage. Note that in the plane of terminal wealth (W NL,W L1 ), the indifference curve has a slope of dw L1 dw NL = 1 π L π L u (W NL ) u (W L1 ) for a low-risk and dw L1 dw NL = 1 π H u (W NL ) π H (1 p) u (W L1 ) for a high-risk at any given contract. 10 This means that the single-crossing property holds but the marginal benefit of coverage for any given contract is not necessarily larger for a high-risk insured; any differences in marginal benefit of coverage between the two risk types depends on the post-loss test accuracy, p, which appears in the slope for a high-risk. Thus, high-risk s preference in coverage is determined by the accuracy of misrepresentation test. To see if there is profitable deviation from the pooling policy offered with a post-loss test at p, we need to consider three cases: (1) 1 π L π L > 1 π H π H (1 p),(2) 1 π L π L < 1 π H π H (1 p),and(3) 1 π L π L = 1 π H π H (1 p). The first case where the accuracy of the post-loss test satisfies p < π H π L π H (1 π L ), creates creamskimming opportunities at smaller coverage than CP because high-risk individuals have a larger marginal benefit of coverage (e.g., Rothschild and Stiglitz, 1976). Similarly, the second case where the accuracy of the post-loss test satisfies p> π H π L π H (1 π L ), allows insurers to offer a new contract that attracts only low-risks at greater levels of coverage than CP, which is a partial coverage. This happens because the highly accurate post-loss test reduces high-risk policyholders marginal benefit of coverage. Thus, the first two cases allow insurers profitable deviation by offering a new policy without post-loss test. In the third case defined by the equality, p = 10 The state W L2 is not affected by a choice of coverage. π H π L π H (1 π L ), both risk types have the same marginal 11

14 benefit of coverage. Therefore, there is no profitable deviation by offering a policy without post-loss test. However, given C P is offered, insurers can make a profit by offering a policy with a post-loss test at p π H π L π H (1 π L ). Consider low-risk s indifference curve passing through C P. Offering a policy on the indifference curve with slightly smaller levels of coverage than CP and with a post-loss test at p< π H π L π H (1 π L ) attracts only low-risks because high-risks prefer C P to the new contract. Similarly, a policy with slightly greater coverage on the indifference curve can attract only low-risks by imposing a post-loss test at p> π H π L π H (1 π L ). Thus, given that a policy with a post-loss test is offered and attracts both high-risk and low-risk individuals, there always exists a policy with or without a post-loss test that attracts only low-risk individuals and makes positive profits. Of course, such a policy also cannot survive in the long run because C P will be withdrawn and then the separating policy immediately makes a loss. Thus, any pooling policy cannot be sustained in equilibrium. Lemma 1 implies that if equilibrium exists, it must separate different risk types of individuals. Therefore, attention here is confined to a separating equilibrium. Furthermore, the lemma also implies that an equilibrium contract is full coverage with post-loss test defined at p = π H π L π H (1 π L ), because insurers offer full coverage and minimize the cost of testing in a competitive market. To formally state the existence of the equilibrium, we first consider a separating policy for high-risk individuals followed by that for low-risk individuals. Lemma 2. In a competitive separating equilibrium, insurers offer a policy C H =(0,D,π H ) to be accepted only by high-risk individuals. Proof. Suppose that a separating contract with post-loss test attracts only high-risk individuals and holds in equilibrium. With costly testing, the policy is C H =(t > 0,D,π H ). Given that the policy is offered, an insurer could enter and offer a contract without test at a slightly lower premium such as C H =(t ε, D, π H ) with arbitrarily small ε>0. This contract attracts all high-risk individuals and earns positive profits, contradicting the assumption of equilibrium. The equilibrium policy that attracts low-risk individuals is determined where their expected utility is maximized under the self-selection constraints: only low-risk individuals choose the policy offered at low-risk individuals actuarially fair premium. Let the equilibrium contract with post- 12

15 loss test be CL =(t,i,π L ), which solves max t>0,i U L (C L ) subject to u H (C H ) U H (C L )where U L (C L ) is the low-risk individual s expected utility by accepting C L under competitive market condition and is defined by U L (C L ) π L u(w 0 +(1 π L )I t D)+(1 π L )u(w 0 π L I t). The self-selection constraint guarantees that high-risk individuals prefer C H to C L.Onecharacteristic of this program is that the objective function is affected only by the post-loss test fee. The premium rate is not affected by the accuracy of the test because cost is zero in an application pool if the post-loss test successfully discourages high-risk applicants from applying for the policy. Therefore, the test fee simply reduces low-risk individual s terminal wealth in both states. The accuracy of the test, however, affects the high-risk type s self-selection constraint. Lemma 3. A competitive equilibrium, if it exists, is unique and insurers offer a set of full-coverage policies: C H =(0,D,π H ) for high-risks and C L =(t,d,π L ) for low-risks where t = e π L is defined by the accuracy of test: p(e )= π H π L π H (1 π L ). (2) Proof. See Appendix. To illustrate the unique post-loss test, we identify the test accuracy where high-risk individuals are worse off as coverage increases and low-risk individuals are better off: p(e) 1 π L(1 π H ) u (W NL ) π H (1 π L ) u (W L1 ) (3) Since a high-risk individual s expected utility monotonically falls as coverage increases, there is a unique solution to solve the constraint. Note that the expenditure on a post-loss test can be reduced by increasing the coverage. In a competitive market, insurers minimize the expenditure on conditional tests by offering full coverage (i.e., u (W NL ) u (W L1 ) Equation (2) (See Figure 1). = 1) where the test accuracy is defined as The unique accuracy of the post-loss test is simply a function of the loss probabilities of both risk types. When both loss probabilities are close, the required accuracy is close to zero. Further, given a fixed probability of loss for the low-risk type, the p π H the accuracy of the post-loss test in terms of π H given π L =0.1 > 0and 2 p π 2 H < 0. Figure 2 illustrates [Insert Figure 2 Here] 13

16 Although the lemma above shows a post-loss test is carried out in the separating policy, whether an individual is willing to buy the policy depends on the level of the test fee, e π L. If the fee for testing is high enough, an individual prefers a policy without a post-loss test. It is clear that without a post-loss test, the optimization problem is reduced to the standard self-selection model demonstrated by Rothschild and Stiglitz (1976). A policy for low-risk individuals C L =(0,I L,π L ) is characterized by partial coverage I L <D. To make the following definition tractable, following Crocker and Snow (1985), we introduced the term λ RS to represent the proportion of high-risk individuals such that the separating equilibrium holds if and only if λ λ RS in the absence of a conditional test. And if λ<λ RS, there is no Nash equilibrium. Definition: Consider low-risk individual s certainty equivalent determined by a policy offered by insurers in the absence of a conditional test. The value of risk premium, μ, is defined by the certainty equivalent: U L (C L )=u(w 0 π L D μ) ifλ λ RS and U L (C M )=u(w 0 π L D μ) if λ<λ RS where the pooling contract C M =(0,I M <D,π M ) is defined by C M =argmax I U L (C M ). Hence, the risk premium depends on the proportion of high-risk individuals. For λ λ RS, C L is simply a low-risk individuals separating policy (see Figure 1). In contrast, for λ<λ RS,low-risk individuals prefer a pooling policy to C L. In these cases, the value of μ becomes strictly smaller. Proposition 1. A competitive Nash equilibrium exists if e π L μ, and offers two full-coverage separating contracts: C H =(0,D,π H ) and C L =(e π L,D,π L ). It is obvious from lemmas that low-risk individuals accept CL if the test fee is small enough and highrisk individuals accept C H. Thus, insurers may choose stringent claim management by spending e per claim which discourages high-risk individuals to purchase a policy from the insurer offering CL. Proposition 1 implies that, compared with the market in the absence of post-loss test, the separating equilibrium is more likely to exist when an efficient test is available. In other words, an accurate and low cost misrepresentation test helps a full coverage separating equilibrium to exist in equilibrium. 14

17 3.3 Infinite Utility Penalty The effectiveness of both unbounded utility and unbounded penalties is demonstrated in the labor markets literature (Guasch and Weiss, 1982; Nalebuff and Scharfstein, 1987). It is of particular interest to investigate how the equilibrium configuration is altered if an infinite utility penalty is allowed in the insurance context. The following condition placed on the potential loss amount and utility function guarantees that a separating equilibrium exists and that it approaches the first-best competitive allocation. Condition 1: D = W 0 and u(0) = This potential loss and unbounded utility ensure that the expected utility penalty in case of a total loss of wealth is sufficiently large. While the validity of unbounded utility is an empirical question, the potential total loss of wealth with positive probability can be observed in claim payment data as discussed in Section 2. Proposition 2. Under Condition 1, a competitive Nash equilibrium always exists and approximates the first-best outcome. Two separating policies are offered: one offers C H =(0,D,π H ) without test; the other offers C L =(ε, D, π L) with arbitrarily small test fee ε. Proof. A high-risk individual s expected utility for taking C L approaches because a utility penalty goes to negative infinity in the state W L2 where the individual s misrepresentation is identified. To see this, suppose that the loss amount is D = W 0, which could cause a total loss of wealth. Then, for any test that has a positive accuracy, the expected utility is negative infinity. Therefore, the self-selection constraint always holds, and high-risk individuals always choose C H without test. A low-risk individual s unconstrained optimization problem is solved at I = D. Thus, the first-best outcome is attained. As one may expect, the fixed loss condition, D = W 0, can be relaxed by replacing the fixed loss with a continuous loss distribution in the support of 0 D L where L W 0. Consider that the loss amount is unknown for both insurers and individuals ex ante but both know the loss distribution. The loss amount is revealed at the end of a period. The key assumption is applicants knowledge that at least one possible outcome is the total loss of their initial wealth with positive probability. With the support of the loss distribution, it is straightforward to show that 15

18 the high-risk individual s incentive constraint always holds. A low-risk individual s unconstrained optimization problem is solved at full insurance, I = D. To approximate the first-best, the loss distribution does not need to be particular for risk types (see Doherty and Jung, 1993, for the cases that heterogeneous loss distributions separate risk types). This proposition implies that adverse selection can be eliminated solely by an individual s expected utility penalty. In contrast to the finite utility penalty case demonstrated in Proposition 1, here we observe that an arbitrarily small expenditure on testing can yield an outcome that approximates a full information separating equilibrium. Notably, this consequence results from the intrinsic nature of some insurance contracts, which provide protection against total loss of wealth and does not depend on an insurer s investment in claim management to verify the accuracy of applications. Another remarkable property is that this equilibrium always exists regardless of the underlying market and the accuracy of the conditional test, while the existence of the proposition 1 equilibrium depends on factors including the proportion of high-risk individuals and post-loss test efficiency. This equilibrium simply relies on high-risk individual s fear of being caught resulting in high punishments, and the existence of the first-best equilibrium overrules the non-existence problems raised by Rothschild and Stiglitz (1976). The welfare implication for Proposition 1 and Proposition 2 is consistent with that for ex ante risk classification demonstrated by Hoy (1982). Consider the case where the proportion of highrisk individuals is large enough for a separating equilibrium to hold in the absence of a test. A post-loss test makes low-risk types strictly better off because they can purchase a full-coverage separating policy, while high-risk types are not affected because they accept a separating contract C H. Therefore, introducing a conditional test is Pareto improving. In contrast, consider the case where the proportion of high-risk individuals is small enough for both risk types to prefer a pooling policy in the absence of a test. Since introducing a test leads to a separating equilibrium, high-risk types are worse off. Hence, a test is not Pareto improving in welfare. 16

19 4 Conclusion Insurance market models often assume that uninformed insurers offer a menu of contracts such that high-risk individuals choose the expensive, high-coverage contract, while low-risk types choose the cheaper, low-coverage contract. This self-selection mechanism relies on individuals freedom of choice regarding a contract in that there is no penalty of being untruthful regarding private information. We investigate the effect of the self-selection constraint of misrepresentation on the self-selection mechanism, under which insurers may test individuals contract choice and impose a penalty if their choice is considered as misrepresentation. The effect of the legal constraint is straightforward in that the constraint discourages deviation from a contract offered at an actuarially fair premium corresponding to their risk type due to a potential utility penalty. The change in the equilibrium configuration is striking in two respects. First, regardless of the underlying market, market equilibrium, if it exists, is unique and consists of full-coverage contracts. Considering the fact that insurers allocate significant resources to claim management, we may conclude that applicants in some markets are encouraged to behave truthfully by the fear of policy cancellation in case of loss. As a result, full-coverage contracts can be obtained. Second, the market outcome approximates the first-best outcome when: (1) there is an arbitrarily small probability that misrepresentation is caught when a claim is made; (2) a contract cancellation causes a total loss of wealth (i.e., sufficiently large loss); and (3) the total loss of wealth leads to a negative infinite utility. The first condition tends to be satisfied by insurance markets because insurers review a contract when claims are processed. And the second condition tends to be met in some types of loss such as personal liability where the loss distribution extends to extremely large values. For instance, it can be expected that an auto accident involving bodily and property damages could cause personal bankruptcy of a negligent party in the absence of sufficient liability coverage. Thus, the intrinsic nature of an insurance contract may satisfy the first two conditions, though the third condition remains an empirical question. Overall, our results suggest that a positive correlation between risk and coverage may not be empirically observed in some types of insurance markets. 17

20 Acknowledgements We thank Mike Hoy, Katja Müeller, Harris Schlesinger, Hato Schmeiser, Daniel Schwarcz, and Justin Sydnor for helpful comments. All errors are our own. 18

21 Appendix ProofofLemma3 Consider the Lagrangian to the optimization problem: l = π L u(w 0 +(1 π L )I t D)+(1 π L )u(w 0 π L I t) +γu(w 0 π H D) γ [π H (1 p)u(w 0 +(1 π L )I t D)+π H pu(w 0 D)+(1 π H )u(w 0 π L I t)] where γ is Lagrangian multiplier for the self-selection constraint. The first-order condition implies that the multipliers must be positive as follows: l I =(1 π L)π L u (W L1 ) π L (1 π L )u (W NL ) γ [π H (1 p)(1 π L )u (W L1 ) π L (1 π H )u (W NL )] = 0. To see if the self-selection constraint is binding, set γ =0. l I = π L(1 π L ) [ u (W L1 ) u (W NL ) ] > 0. Hence, γ must be strictly positive, which implies that the constraint must be binding at an optimum. To show that for a given t there is a unique coverage I that solves the constraint. From the self-selection constraint, we have subject to max U L (c L ) I u(w 0 π H D)=π H (1 p)u(w L1 )+π H pu(w L2 )+(1 π H )u(w NL ) The constraint can be rewritten as: u(w L1 )= u(w 0 π H D) π H pu(w L2 ) (1 π H )u(w NL ) π H (1 p) 19

22 Substituting the constraint into the objective function, we have [ ] U L u(w0 π H D) π H pu(w L2 ) (1 π H )u(w NL ) (C L )=π L +(1 π L )u(w NL ) π H (1 p) Taking a derivative in terms of coverage and setting UL (C L ) I =0, U L (C L ) I [ πl (1 π H )u ] (W NL ) = π L π L (1 π L )u (W NL )=0 π H (1 p) This first order condition is solely determined by the accuracy p and is indifferent to coverage. Thus, the separating contract that satisfies a self-selection constraint is determined where the accuracy of the test is: p(e )=1 π L(1 π H ) π H (1 π L ) = π H π L π H (1 π L ) C L : Alternatively, we consider the high-risk individual s expected utility obtained by taking contract U H (C L )=π H (1 p)u(w L1 )+π H pu(w L2 )+(1 π H )u(w NL ). Taking a derivative of high-risk individual s expected utility in terms of coverage, U H (C L ) I = π H (1 π L )(1 p)u (W L1 ) π L (1 π H )u (W NL ) It is straightforward to show that 2 U H (C L ) I < 0 holds. To see the relationship between the expected 2 utility and coverage, UH (C L ) I 0 holds where: π H (1 π L )(1 p)u (W L1 ) π L (1 π H )u (W NL ) Solving this in terms of p, wehave p(e) 1 π L(1 π H ) u (W NL ) π H (1 π L ) u (W L1 ) Thus, when the test is accurate enough to satisfy the condition, a high-risk individual s expected utility monotonically falls as coverage increases. Note that p is minimized at u (W NL ) u (W L1 ) =1where 20

23 I = D. Hence, there is a unique solution to solve the constraint. 21

24 References Browne, M. J., and S. Kamiya, 2012, A theory of demand for underwriting, Journal of Risk and Insurance 79 (2), Chiappori, P.-A., B. Jullien, B. Salanié, and F. Salanié, 2006, Asymmetric information in insurance: General testable implications, The RAND Journal of Economics 37 (4), Cohen, A., and P. Siegelman, 2010, Testing for adverse selection in insurance markets, Journal of Risk and Insurance 77 (1), Crocker, K. J., and A. Snow, 1985, The efficiency of competitive equilibria in insurance markets with asymmetric information, Journal of Public Economics 26 (2), Dionne, G., C. Gourieroux, and C. Vanasse, 2001, Testing for evidence of adverse selection in the automobile insurance market: A comment, Journal of Political Economy 109 (2), Doherty, N. A., and H. J. Jung, 1993, Adverse selection when loss severities differ: First-best and costly equilibria., Geneva Papers on Risk and Insurance Theory 18 (2), Guasch, J. L., and A. Weiss, 1980, Wages as sorting mechanisms in competitive markets with asymmetric information: A theory of testing, The Review of Economic Studies 47 (4), Guasch, J. L., and A. Weiss, 1981, Self-selection in the labor market, The American Economic Review 71 (3), Guasch, J. L., and A. Weiss, 1982, An equilibrium analysis of wage productivity gaps, The Review of Economic Studies 49 (4), Hoy, M., 1982, Categorizing risks in the insurance industry, The Quarterly Journal of Economics 97 (2), Katja Müller, H. S., and J. Wagner, 2013, The impact of auditing strategies on insurers profitability, Working Paper, University of St. Gallen. Kessler, A. S., C. Lülfesmann, and P. W. Schmitz, 2005, Endogenous punishments in agency with verifiable ex post information, International Economic Review 46 (4), pp

25 Kofman, F., and J. Lawarrée, 1993, Collusion in hierarchical agency, Econometrica 61, Mirrlees, J., Notes on Welfare Economics, Information, and Uncertainty. In: Balch, M., McFadden, D. and Wu, S. (eds.) Essays on Economic Behavior under Uncertainty. Amsterdam: North Holland. Nalebuff, B., and D. Scharfstein, 1987, Testing in models of asymmetric information, Review of Economics Studies 54 (2), Riordan, M. H., and D. E. Sappington, 1988, Optimal contracts with public ex post information, Journal of Economic Theory 45 (1), Rothschild, M., and J. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 90 (4),

26 I H I C u H (C H H ) C L * L I+t * L I t * C L u L (C L* ) 0 D I Figure 1: Full-Coverage Separating Equilibrium Without Post-Loss Test Test Accuracy: p H Figure 2: The Accuracy of Post-Loss Test in Equilibrium (π L =0.1) 24

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