(PDF) Advantageous Effects of Regulatory Adverse Selection in the Life Insurance Market*

This paper finds evidence for the presence of adverse selection in the life insurance market, a conclusion contrasting with the existing literature. In particular, we find a significant and positive correlation between the decision to purchase life insurance and subsequent mortality, conditional on risk classification. Individuals who died within a 12year time window after a base year were 19 percent more likely to have taken up life insurance in that base year than were those who survived the time window. Moreover, we find that individuals are most likely to obtain life insurance four to six years before death. Methodologically, we address sample-selection and omitted-variables issues overlooked in the previous literature.

A standard problem of applied contracts theory is to empirically distinguish between adverse selection and moral hazard. We show that dynamic insurance data allow to distinguish moral hazard from dynamic selection on unobservables. In the presence of moral hazard, experience rating implies negative occurrence dependence: individual claim intensities decrease with the number of past claims. We discuss econometric tests for the various types of data that are typically available. Finally, we argue that dynamic data also allow to test for adverse selection, even if it is based on asymmetric learning. (JEL: D82, G22, C41, C14)

The Economic Journal, 116 (January), 327–354. Ó Royal Economic Society 2006. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. ADVANTAGEOUS EFFECTS OF REGULATORY ADVERSE SELECTION IN THE LIFE INSURANCE MARKET* Mattias K. Polborn, Michael Hoy and Asha Sadanand We analyse the effects of regulations prohibiting the use of information to risk-rate premiums in a life insurance market. New information derived from genetic tests is likely to become increasingly relevant in the future. Many governments prohibit the use of this information, thereby generating regulatory adverse selection. In our model, individuals early in their lives know neither their desired level of life insurance later in life nor their mortality risk, but learn both over time. We obtain both positive and normative results that differ qualitatively from those in standard, static models. Legislation prohibiting the use of genetic tests for ratemaking may increase welfare. The effort to sequence the human genome completely is now effectively completed. Although a comprehensive understanding of the effects of most genes on health and mortality risks will still take substantial further effort, it is likely that in the medium-term future more and more information relevant to ratemaking in the life, health, and long-term care insurance markets will become available. In particular, we will learn the effects of many genes on the likelihood of various illnesses and consequently individuals life expectancy.1 Moreover, new genetic tests for these genes will likely become much cheaper.2 Allowing this information to be used by insurers raises the prospect of substantial changes in future pricing of insurance which, from an ex ante perspective, imposes a premium or classification risk on potential insurance buyers who do not buy their insurance before their risk type is realised. One might counter that individuals who are currently young could make their life insurance purchases early in life and avoid this premium risk. However, it is not reasonable to suppose that individuals early in life necessarily know their future life insurance needs, which we refer to as their demand type. Our main objective in this article is to consider how new information about mortality risk and demand type together influences insurance holding over time and to analyse the effect of different forms of regulation related to whether information about risk type can be used for tarification. At present, there are many regulatory responses to the prospects of genetic information for insurance pricing. These range from no action to voluntary * We thank two anonymous referees and David DeMeza (the editor) for very helpful comments, as well as seminar participants at the 2001 Risk Theory Society Seminar in Montreal, Australian National University, University of Melbourne and Iowa State University. We are also grateful for research funding of this project by SSHRC, CIHR, and Genome Canada. 1 For a brief discussion of the scientific aspect of these possibilities, see for example Rowen et al. (1997). 2 News reports (BBC News service http://news.bbc.co.uk - Sept. 23, 2002, Your Genetic Code on a Disc) suggest that currently the cost of determining an individual’s entire genome is about $(US)700,000, while the company Solexa expects to do this at the cost of about $(US)1,000 in the near future. Drell and Adamson (2002) speculate that by the year 2020 it will be commonplace for individuals complete genomes to be recorded in their health records. [ 327 ] 328 THE ECONOMIC JOURNAL [JANUARY moratoria by the insurance industry to government guidelines or legislation. Several countries in Europe have legislation which regulates the use of genetic information by the insurance industry. Many of these, such as Austria, Belgium, Estonia, Luxembourg, Norway, and Denmark, prohibit insurers from requesting or using the test results of applicants that are available in medical records. Legislation in Belgium takes an even stricter stance in that insurance applicants are prohibited from voluntarily submitting favourable results of genetic tests to insurers. An interesting feature of some regulations is the use of a cap or ceiling on the cumulative amount of life insurance purchases that an individual can hold without insurers being allowed to use genetic test results in setting premiums. The moratorium in the UK, for example, includes a cap set at £500,000 (approximately $780,000) while in the Netherlands a cap was set at D.Fl. 300,000 in 1998 (approximately $150,000), to be adjusted every three years according to the costof-living index.3 In this article we address the efficacy of these sorts of regulations. We show that prohibiting insurers from using future information about risk type may increase welfare despite the fact that such regulations create adverse selection costs. We develop a dynamic three period model of a life insurance market. At the beginning of period 3, individuals face the risk of death. They can buy life insurance either early in life (period 1), before they know their demand type and before information about their risk type becomes available (either to themselves or insurers), and/or they may buy insurance later in life (period 2), after they have received information about both their mortality risk and their demand type. Consider the cases with only one type of uncertainty. First, we show that if an individual knows her risk type at the outset (period 1) but not her demand type, then it is optimal for her to make all of her insurance purchases only after demand type uncertainty is resolved (i.e., in period 2). This follows because there is no premium risk to be concerned about and so there is no advantage to buying insurance early in life. Moreover, if there is any transaction cost, no matter how small, to purchasing insurance, then one should wait until demand uncertainty is resolved in order to end up holding the correct amount of life insurance at the lowest possible cost. Second, suppose an individual knows his demand type early in life but that his risk-type information will not be resolved until later in life. In this scenario, buying insurance early in life (period 1) is optimal. The reason is that, if information about mortality risk is symmetric in the second period of life, then waiting until then to make insurance purchases exposes the individual (ex ante) to classification risk (i.e., the premium is lower for those with good news about their death probability, but higher for those with bad news). On the other hand, if information about risk type in the second period is asymmetric, there will be adverse selection in the second period market and hence the premium will be higher for all individuals than in the first period. 3 According to the UK agreement there is a restricted list of types of genetic tests that insurers are allowed to use. In order that caps be effective, the amounts must apply to total insurance purchased by any individual, not simply to a single policy. For a discussion of these and various other regulations see Lemmens (2003) and Knoppers et al. (2004). Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 329 In the presence of both demand and risk type uncertainty, we show that there will be a trade off between buying insurance later rather than earlier in life. This has not yet been studied in the literature. Probably the most interesting set of results in the general setting of our model with demand and risk type uncertainty pertains to the effects of regulation preventing insurers from using genetic information that the individual has obtained. This regulation creates adverse selection in the life insurance market at some point in the future but also ameliorates premium risk. In the short run, this form of regulation is often justified by equity considerations: an individual who has inherited unfavourable genes, it is argued, should not also be punished in the life insurance market by having to pay a much higher premium. In the scenario that we consider, the case for this form of regulation appears at first glance to be weaker. After all, people can buy life insurance before new genetic information becomes available and so the equity aspect of regulation becomes less relevant in comparison to the likely efficiency losses due to adverse selection. However, this argument is valid only if people know initially how much life insurance they desire to hold later in life for a given probability of death and given premium rates. If demand uncertainty exists, then buying life insurance early in life in order to eliminate classification risk is problematic in that the individual may buy insurance that he later finds he does not need. Even if unwanted insurance can be sold ex post, the purchase of too much life insurance nevertheless generates an undesirable risk, as the future value of life insurance bought early in life is a random variable. In such a case, creating an environment of asymmetric information by restricting the insurers use of information about individuals risk types can provide an alternative type of insurance against classification risk. As long as there is some participation by low risk types in the market for insurance after risk-type information is revealed and equilibrium contracts involve pooling, the premium for insurance purchases will be lower for those who are discovered to be high risk types than it would be under symmetric information. On the other hand, there is an inefficiency created by adverse selection. We compare these two effects using individuals ex ante expected utility (i.e., from their perspective early in life before information about demand type and risk type is known). In general, either effect may dominate. However, in a specific setting we are able to show that if there are sufficiently few individuals who receive bad news about their genetic type, then a ban on using genetic information in rate-making in combination with a cap which limits adverse selection is welfare improving. Although there has been some recent attention in addressing the impact of restricting categorical information in general insurance markets, the models usually adopted are property-liability insurance models which do not apply to the life insurance setting.4 These models apply to a setting where the loss is of a fixed 4 Papers that investigate the properties of equilibria under asymmetric information for the standard model include Rothschild and Stiglitz (1976), Miyazaki (1977), Spence (1978) and Wilson (1977). Several papers, including Crocker and Snow (1985, 1986), Doherty and Posey (1998), Doherty and Thistle (1996), Hoy (1984, 1989), Hoy et al. (2003), Ligon and Thistle (1996), Tabarrock (1994), analyse the value of alternative information sets in insurance markets. Ó Royal Economic Society 2006 330 THE ECONOMIC JOURNAL [JANUARY and identifiable size (e.g., the value of an automobile) and so exclusivity of provision, and hence prices which are nonlinear (i.e., convex) in the coverage level are plausible. On the other hand, in a life insurance context no obvious value of a lost life exists, or at least it is likely to be highly subjective. Hence, the quantity of life insurance demanded is not a good indicator of risk type. In fact, Pauly et al. (2003, p. 4) note that term life insurance providers do not seek information about total life insurance holdings of their clients and so do not try to induce different risk types to self-select contracts based on a price-quantity constrained menu of contracts. Cawley and Philipson (1999) even find modest quantity discounts rather than convex price schedules in the life insurance market, presumably because of some administrative fixed costs of providing a consumer with a life insurance contract, combined with the absence of a significant adverse selection problem. Moreover, insureds often increase their coverage level as time passes as a result of changing family circumstances etc, so exclusivity of coverage would be an unattractive feature of an insurance contract for many people. It is also interesting to note that in his survey of empirical research on the existence of asymmetric information in the automobile insurance market, Chiappori (1999, p. 10) finds little evidence of adverse selection in unregulated markets but does point out that government regulations, similar to those that we are discussing in this article, could create significant adverse selection. In this case a government regulation that creates adverse selection may trigger insurers into exclusive contracting and nonlinear pricing. Therefore, although we adopt a model with linear pricing, we note where applicable the possible complications that could arise in the presence of regulatory adverse selection.5 Our model is more similar to that of Brugiavini (1993)6 who develops a dynamic model of the annuity market where there is no preconceived notion of the size of the loss but in which demand is determined by the individual’s state dependent utility. Brugiavini shows that if there is only heterogeneity about risk type and information about this becomes available only in the second period, then the full amount of annuities desired for period two is purchased in the first period, a result which is parallel to our Proposition 1. However, that paper does not analyse the interaction between risk type and demand type information7 nor compare welfare levels under different informational regimes, which are the key contributions of this article. We present our model in the next Section. Sections 2, 3 and 4 contain the main results. This is followed by discussion and conclusions in the final Section. 5 However, we will argue that, for parameter values that are realistic for the application of genetic information, linear pricing with a cap is consistent with exclusivity of provision if one adopts the so-called Wilson foresight equilibrium concept (Wilson, 1977). 6 See also Abel (1986), Villeneuve (1999) and Hoy and Polborn (2000). In the last of these papers the welfare effects of additional information are characterised for a life insurance model but in a static setting. For early models of insurance purchasing decisions with state dependent utilities, see Cook and Graham (1977) and Dehez and Dreze (1982). 7 See Smart (2000), Villeneuve (2003) and Wambach (2000) for implications of multi-dimensional adverse selection in the Rothschild-Stiglitz model. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 331 1. The Model We use three time periods in our model, t ¼ 1, 2, 3. The only mortality risk we consider is whether the individual dies at the beginning of period t ¼ 3, or lives for this period.8 At t ¼ 1, each individual has a fixed amount y to be eventually transferred to period 3, either through (term) life insurance (denoted L) which pays only in the death state, or unconditionally to both states, which we call saving (S).9 Expected utility of an individual with risk type pi 2 fpL, pHg and demand type hj 2 fh1, h2g is given by EUij ¼ pi vðC 3D ; hj Þ þ ð1 pi ÞwðC 3L Þ: ð1Þ 3D We denote consumption in the third period by C ¼ S þ L in the death state and C3L ¼ S in the life state, where S and L represent cumulative savings and life insurance purchases made in periods 1 and 2. Subscripts i and j refer to the individual’s risk and demand type, respectively. v(Æ, Æ) is the utility function in the death state; it depends not only on how much money is left for the beneficiaries but also on the individual’s demand type.10 For the high demand type (h ¼ h2), the marginal utility of income in the death state is greater than for the low demand type (h ¼ h1); i.e., ov(C, h2)/oC > ov(C, h1)/oC. The utility function in the life state is w(Æ), which is assumed to be the same for all individuals. We further assume that at least the h2 consumers have a strictly positive demand for life insurance if it is offered for a risk type specific fair premium. If this assumption were not satisfied, there would be no demand for life insurance in equilibrium. Note, however, that we impose no restriction on the low demand types demand for life insurance; this may be positive or zero. The timing of the model is as follows. People can buy life insurance for the first time in period 1, at which time they, as well as insurers, are uninformed about their demand and risk type. In period 2, individuals learn both their mortality risk and their demand type. Concerning the risk type, a proportion qL of the population learn that they are low risk types with mortality risk pL and the remainder (qH ¼ 1 qL) learn that they are high risk types with mortality risk pH. Concerning demand type, a proportion r1 of the population learn that they are low demand types (h1), while the rest (r2 ¼ 1 r1) learn that they are high demand types h2. Demand type and risk type are statistically independent, so the probability of being a low risk and low demand type is qLr1 and so on. In period t ¼ 2, people can recontract on the basis of the new information. Of course, the price of insurance will generally be different in these two time periods. We always assume that the demand type is private information, so that insurance contracts cannot be written contingent on the demand type. We analyse two 8 In order to simplify the model as much as possible, we assume no risk of death in period 1 or 2. This possibility could be added without difficulty or change in the qualitative nature of the results. 9 In a previous version of this article (Polborn et al., 2001), we have explicitly modelled consumptionsavings decisions in the first two periods as well, and results are qualitatively equivalent. 10 Our use of state dependent utilities allows implicitly for loss of income due to death; that is, if some income amount y3 were earned in the life state in period t ¼ 3, loss of this in the death state could be included simply by writing vðcÞ ¼ v~ðc y3 Þ. Ó Royal Economic Society 2006 332 THE ECONOMIC JOURNAL [JANUARY different scenarios relative to risk type. Under symmetric information, the price of additional units of life insurance bought in the second period corresponds to the individual’s risk type. Under asymmetric information, all individuals can buy additional insurance for the same price that reflects the demand weighted average risk type of life insurance buyers. Note that the case of asymmetric information can be interpreted as an outcome of regulation in the insurance market in that information could in principle be available for insurers but laws prohibit its use in insurance pricing decisions.11 (As mentioned in the introduction, there are several countries which have a ban or a moratorium on the use of genetic information for insurance purposes.) We also allow individuals to resell insurance that they bought in period 1 if they find that they do not want to keep it. We will discuss this assumption further below. Given these assumptions, letting S1 and L1 represent first period savings and life insurance purchased, the first period budget constraint is S 1 þ p1 L 1 y; ð2Þ where p1 ¼ qLpL þ qHpH is the insurance premium in the first period, equal to the population weighted average risk type. In the second period, the budget constraint is S þ p2 ðL L 1 Þ S 1 ; ð3Þ where p2 is the relevant second period insurance premium which may depend on the individual’s risk type (under symmetric information) and on whether the individual wants to buy or sell life insurance (under asymmetric information).12 We restrict cumulative purchases to be non-negative (i.e., L 0).13 We assume that insurers are risk neutral, have no transaction costs and are in perfect competition. We also assume insurers do not employ exclusivity of contracts, which is a standard assumption for models of life insurance markets. Thus, linear pricing applies. We will go into more detail on second period prices in the Sections that deal with the different informational scenarios. Summing up, the first period is the uninformed or ex ante stage, the second is the informed stage, and the third is the outcome stage. The reason we include the informed stage (t ¼ 2) is that it allows us to model the effects of information that is revealed before the final state of the world in period 3 is known, but after an initial period in which uninformed insurance purchases can be made. 11 In the absence of laws that restrict insurance companies from properly classifying risks, McCarthy and Mitchell (2002) find no evidence of adverse selection in markets for life insurance or annuities while Cawley and Phillipson (1999) draw a similar conclusion, even demonstrating that insurers appear to know the mortality risks of individuals better than they do themselves. 12 Note that with Lt, St representing period t decisions and L, S representing cumulative values, we have S ¼ S1 þ S2 and L ¼ L1 þ L2. 13 Although one could allow negative life insurance purchases (i.e., annuities), we want to focus on the life insurance market. The market for annuities for people in the age group relevant to our analysis (say 30 to 50) is very small relative to the life insurance market. Various reasons proposed for the lack of participation in the annuity market include the fact that, given the small death probabilities for this age group, savings is a very close substitute for annuities and also considerably more flexible, and pensions tend to crowd out annuities. See Villeneuve (2003b) for a comprehensive analysis of the annuity market. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 333 We reserve until later (Section 3) a discussion on how consumers may implement various strategies allowed in our model; in particular, buying insurance early in life (in period 1) for coverage later in life (in period 3) and also the ability to resell insurance coverage if it is determined to be unwanted later in life (in period 2). Although explicit market mechanisms for such transactions do not exist in a literal way, consumers can nonetheless use various real world contractual options to approximate the transactions we have modelled.14 It is interesting to note that, in our model, buying life insurance in period 1 is similar to what Tabarrok (1994) calls genetic insurance; that is, an insurance policy bought before obtaining information on the risk type that covers future insurance payments resulting from being a high risk type. In our model, life insurance bought in the first period is equivalent to an insurance that pays out a fixed amount depending on whether the genetic test turns out to be positive or negative. 2. Polar Cases It is useful to start with an analysis of the two polar cases in which there is only uncertainty about the risk type p or only uncertainty about the demand type h. In Section 3 we analyse the most interesting and difficult case where there is uncertainty about both risk type and demand type. Our general procedure is as follows. First, we derive each type’s optimal decisions concerning saving and life insurance in the second period conditional on a general, unspecified level of first period life insurance purchases (and savings). When we substitute the optimal second period decisions into the second period objective functions, we get the type specific value functions (there are different solutions and different objective functions for each type). These depend on the first period’s saving and life insurance purchases. We then use these value functions in order to compute the optimal decisions in the first period. 2.1. Uncertainty Only About Risk Type In this subsection, we assume that individuals know their demand type (h) and only receive information concerning their risk type in the second period. For any h, an individual of risk type i faces the following optimisation problem in the second period (recall that S ¼ S1 þ S2 and L ¼ L1 þ L2): max pi vðS þ L; hÞ þ ð1 pi ÞwðSÞ ð4Þ s.t. S þ p2i ðL L 1 Þ y p1 L 1 : ð5Þ S;L The right-hand side of (5) is the money (S1) which is still left in the second period; that is, initial income y minus first period insurance purchases. Since there is no risk-type specific information available in the first period, p1 ¼ p ¼ 14 In any case we analyse what happens if resale of insurance contracts is not possible. Ó Royal Economic Society 2006 334 THE ECONOMIC JOURNAL [JANUARY qLpL þ qHpH (i.e., the pooled fair premium which results from our assumption of zero expected profits for firms). On the left-hand side of (5), the available money in period 2 can be spent to supplement first period savings and insurance purchases. Thus, L L1 is the additional insurance coverage purchased in period t ¼ 2 and S S1 is the additional savings. Here, we do not constrain L to be greater or equal to L1 and so L < L1 is to be interpreted as the individual reselling insurance which was purchased in period 1. It turns out for this case that in the optimum the insured does not wish to resell any insurance purchased in the first period and so this assumption is not of any consequence here. The individual’s decisions in period 2 will result in a consumption of S in the life state in period 3 and of S þ L in the death state. As noted earlier, if death involves loss of income in period 3, then this is implicitly included in the definition of the death state utility function v(Æ, Æ). Thus, we allow for the possibility that consumption may well be less in the death state than in the life state. We now distinguish two cases concerning the nature of information about risk type in the second period; that is, whether it is symmetric or asymmetric. Symmetric information: pL ¼ pL, pH ¼ pH. In the case of symmetric information, the premium in the second period will reflect the buyer’s risk type. The first order conditions for the second period optimisation problem are (where in a slight abuse of notation v 0 denotes the derivative of v with respect to the first argument, consumption) pi v 0 ðS þ L; hÞ þ ð1 pi Þw 0 ðSÞ ki ¼ 0; ð6Þ pi v 0 ðS þ L; hÞ ki pi ¼ 0: ð7Þ From this we can conclude that, in the optimum, the marginal utilities for both the life state and the death state of period 3 are equal: v 0 (S þ L, h) ¼ w 0 (S). This equality, together with the budget constraint, determines the optimal allocation of the available money to second period saving and life insurance purchases. Do high risk individuals or low risk individuals buy more insurance? This relation is not a priori clear. For high risks, the death state is more probable (pH > pL), making insurance purchases more attractive; but the premium is higher (pH > pL), which has the opposite effect. Let Zi ½y þ ðp2i p1 ÞL 1 denote the value function of the optimisation problem given in (4) and (5). From the envelope theorem, we have Zi0 ¼ ki . In the first period, the individual maximises max qL ZL ½y þ ðp2L p1 ÞL 1 þ qH ZH ½y þ ðp2H p1 ÞL 1 : L1 ð8Þ The first order condition for this problem is: qL ðp2L p1 ÞZL0 þ qH ðp2H p1 ÞZH0 ¼ 0: ð9Þ Using p1 ¼ qLpL þ qHpH, it follows from (9) that ZL0 ¼ ZH0 and, hence, kL ¼ kH. Returning to the second period problem, it follows that consumptions in the life state and in the death state are the same for both risk types (i.e., the vectors Ó Royal Economic Society 2006 2006 ] 3L EFFECTS OF REGULATORY ADVERSE SELECTION 335 3D (C , C ) are independent of risk type i). The only way that this can happen is for LL ¼ LH ¼ L1; that is, the entire amount of insurance coverage is purchased in period t ¼ 1, and neither risk type wants to buy additional coverage (or resell any insurance) in period t ¼ 2. The intuition for this result is that since demand type is known at the outset, insurance buyers can avoid any premium risk by purchasing all of their insurance needs in the first period. There is no off-setting benefit to waiting until period 2 when risk type becomes known to themselves and, more importantly, to insurers. Asymmetric information: p2L ¼ p2H ¼ p2 . First consider the scenario in which individuals cannot resell any first period purchases of insurance in the second period. If information about the individual’s risk type cannot be used by insurers for the purpose of rate-making, say because of regulatory constraints, then the premium an individual faces in the second period will not depend on her risk type. In the second period market equilibrium, the premium will be equal to the average clientele risk, which is a weighted average of the death probabilities of insurance buyers, where the weights take into account the relative amounts of insurance purchased by the two risk types. The average clientele risk (p2) will be higher than the population weighted average risk (p1) because, when faced with the same price, high risks buy more insurance under asymmetric information than do low risks.15 Thus, if insurers do not purchase all of their insurance needs in the first period they will have to pay a higher price in the second period (i.e., p2 > p1). Since individuals know their insurance needs in period 1, the optimal decision is to purchase all of their insurance early in life and avoid the higher second period price. If no reselling is possible, then naturally insurance buyers do not want to purchase more than the optimal amount of insurance in period one when they already know their insurance needs. Suppose reselling is possible and insureds buy more than their insurance needs in period one. If they can sell some of their insurance in the second period, then one might think that at least there is no cost or disadvantage to holding too much insurance from period one. However, at any given price for reselling insurance to insurers, the low risk types will wish to sell more than the high risk types and so the reselling price will be lower than the actuarially fair price of p1. This is just the usual type of adverse selection in reverse. Since buying unwanted insurance coverage at a higher price than one then resells it simply reduces one’s consumption in both possible states of the world in period 3, it is not optimal to do so. Thus, whether or not reselling is possible in the second period market for insurance, individuals should buy exactly all of their desired insurance purchases in period 1.16 More details concerning this outcome are provided in the relevant sections covering the general case as this is simply a special case of that analysis. We gather our findings of this Section in the following Proposition: Proposition 1. If all individuals are identical in period t ¼ 1 and it is known that in period t ¼ 2 new information about the individuals risk type will arrive, then all 15 16 See Abel (1986), Hoy and Polborn (2000). This is a special case of the no re-contracting result in Milgrom and Stokey (1982). Ó Royal Economic Society 2006 336 THE ECONOMIC JOURNAL [JANUARY individuals buy the complete amount of life insurance in period 1 (there is no recontracting in period 2); in particular, high risks and low risks end up with the same amount of life insurance coverage. These results are valid independent of whether insurance companies in period 2 can use the individual’s risk type for ratemaking purposes (symmetric information) or not (asymmetric information). 2.2. Uncertainty only about Demand Type In this subsection it is assumed that all individuals are of the same risk type (so pL ¼ pH ¼ p) but that individuals do not know ex ante (i.e., in period 1) what will be their demand type. Information about h becomes known to them in the second period. After receiving this information, an individual of demand type j faces the following optimisation problem: max pvðS þ L; hj Þ þ ð1 pÞwðSÞ ð10Þ s.t. S þ pðL L 1 Þ y pL 1 : ð11Þ S;L Since firms are competitive and the individuals risk types are known and unchanging, the price of insurance is the same in periods one and two and is just equal to the actuarially fair price (i.e., p2 ¼ p1 ¼ p). Thus, there is no role for first period insurance purchases to play in terms of reducing premium risk. Formally, L1 cancels from the constraint. Moreover, there is no disadvantage in waiting until period 2 to make insurance purchases because there is no inefficiency created by adverse selection costs. Therefore, the utility maximising choice for consumers does not involve any interesting timing issues concerning when to purchase insurance. This is described formally in the following Proposition.17 Proposition 2. (i) The final life insurance demand of type h2 consumers is higher than that of type h1 consumers. (ii) For both types h1 and h2, the optimal final insurance demand is independent of the first period insurance purchases L1. Proof. Since we assumed that @vðC 3D ; h2 Þ=@C 3D > @vðC 3D ; h1 Þ=@C 3D , the optimal C 3D for high demand types must be higher than for low demand types. The only way that this can be satisfied is that high demand types have a larger L and a smaller S. The proof of the second claim is immediate from the fact that (11) is independent of L1. ( 17 Note also that if individuals did differ by risk type but that this was symmetric information ex ante (in period 1), then Proposition 2 would be applicable separately for each case of p ¼ pL and p ¼ pH. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 337 Thus, in the case of demand type risk only, there are (infinitely) many ways to implement the consumer’s optimal decision, involving different combinations of first and second period insurance purchases. However, this is a knife edge case and we argue that the most reasonable interpretation is that optimal first period insurance purchases in this scenario are equal to zero because if there were some fixed transaction cost incurred for buying or selling life insurance, no matter how small, then it would clearly be better not to buy any insurance in the first period. 3. The General Case We will now consider the case in which each individual knows neither her demand type nor her risk type until the second period; hence there will be a trade-off between the two effects of buying insurance early or late in life, as was studied in the previous Section. We will first analyse the case of symmetric information between insurer and insured in the second period and consider the asymmetric information case afterwards. 3.1. Symmetric Information In this case individuals face risk-type specific actuarially fair prices in the second period insurance market (i.e., p2i ¼ pi , i ¼ H, L). Thus, the second period problem for type ij is max pi vðSij þ Lij ; hj Þ þ ð1 pi ÞwðSij Þ ð12Þ Sij þ pi ðLij L 1 Þ S 1 ð13Þ Lij ;Sij subject to where the substitution pi ¼ p2i has been made in the constraint (13). Differentiating with respect to Sij and Lij yields, respectively pi v 0 þ ð1 pi Þw 0 kij ¼ 0 ð14Þ pi v 0 kij pi ¼ 0: ð15Þ and Combining (14) and (15) yields v 0 (Sij þ Lij, hj) ¼ w 0 (Sij), so in the second period, all types under symmetric information choose to equalise marginal utilities in the death state and the life state. Two other important observations for later reference are summarised in the following lemma. Note that the substitution p ¼ p1 has been made in following value functions Zij. Lemma 1. (i) The value function of problem (12), Zij(S1 þ piL1) ¼ Zij[y þ (pi p)L1], is strictly concave. (ii) If LiH > L1, then kiH > kiL. (iii) If LiL < L1, then kiL > kiH. Ó Royal Economic Society 2006 338 THE ECONOMIC JOURNAL [JANUARY Proof. (i) This follows immediately from v and w being strictly concave. (ii) In this case, we must have SiL > SiH. (Suppose that, to the contrary, SiL SiH; then v 0 ¼ w 0 implies SiL þ LiL SiH þ LiH; but this cannot be the optimal savings and life insurance demand of an iL type, since she could afford to choose (SiH þ e, LiH þ e), for some e > 0, and this would increase her utility). Since kiH ¼ w 0 (SiH) and kiL ¼ w 0 (SiL), and w(Æ) is strictly concave, SiL > SiH implies the claim. (iii) The proof of this claim is analogous to the last one and so is omitted. ( We now go back to the first period in order to answer the question how much life insurance is bought in the first period. The optimisation problem is to choose L1 to max r1 qL Z1L ½y þ ðpL pÞL 1 þ r1 qH Z1H ½y þ ðpH pÞL 1 þ r2 qL Z2L ½y þ ðpL pÞL 1 þ r2 qH Z2H ½y þ ðpH pÞL 1 : ð16Þ Taking the derivative with respect to L1 and noting that Zij0 ¼ kij , where kij is the Lagrange multiplier in the optimisation problem (12), gives r1 qL k1L ðpL pÞ þ r1 qH k1H ðpH pÞ þ r2 qL k2L ðpL pÞ þ r2 qH k2H ðpH pÞ: ð17Þ Using qL(p pL) ¼ qH(pH p), (17) becomes qL ðp pL Þ½ðr1 k1H þ r2 k2H Þ ðr1 k1L þ r2 k2L Þ: ð18Þ We have Proposition 3. The optimal solution of problem (12), (13) and the optimal first period insurance purchases can be characterised as follows: (i) The optimal L1 is chosen to equalise the expected marginal utility of income for high risk and low risk individuals in the second period: r1kL1 þ r2kL2 ¼ r1kH1 þ r2kH2. (ii) First period insurance purchases, L1, are chosen such that, in the second period, high demand types buy additional life insurance, and low demand types sell some or possibly all of their life insurance holdings. (iii) Consumption in both states (S, S þ L) is a normal good: oS/oy > 0 and o(S þ L)/oy > 0. Proof. (i) Assuming an interior solution L1 > 0 for the moment (which we will confirm further below), (18) must be equal to 0, which implies the claim. (ii) Observe that Lemma 1 implies that the second period additional life insurance demand of both risk types of the same demand type are either both positive or both negative: either LiH > L1 and LiL > L1, or LiH < L1 and LiL < L1, for i 2 f1, 2g. Furthermore, it cannot be true that all types buy additional insurance in period 2: If this were the case, then k1H > k1L and k2H > k2L, and hence the expression in (18) would be positive, indicating that it would be optimal to Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 1 339 1 increase L (in particular, this shows that L ¼ 0 cannot be optimal, as claimed earlier). An analogous argument shows that not all types can sell life insurance in the second period. (iii) An increase of y must increase (at least) either S or S þ L, otherwise the individual would not use all her resources. Then, the claim follows immediately from v 0 ¼ w 0 in the solution of the period 2 problem. ( The first result in this Proposition is a generalisation of the result from the case presented in Section 2 where only risk type is uncertain. The marginal utility of income for low and high risk individuals must be equal in the optimum. Note that no insurance is possible against the risk concerning demand type. Therefore, the optimal strategy in period 1 is to buy a level of insurance, L1, such that the expected marginal utilities of high risks and of low risks in the second period are equal, where the expectation is taken with respect to the demand type risk. High demand types have a higher marginal utility of income in the death state; they will therefore buy more life insurance and have less savings than the corresponding low demand types of the same risk type. Consequently, high demand types also have a higher overall marginal utility of income. The third result implies that, among high demand types, low risks consume more in both the death and the life states because they are effectively richer than high demand-high risk individuals (both high demand types buy additional insurance, and this insurance is cheaper for low risk types). However, it is not possible to decide in general whether, among high demand types, the low risks necessarily buy more life insurance than the high risks; the advantage that low risks have from their lower premium costs could, in principle, go to 100% (or even more) into higher savings.18 A graphical illustration of the results of Proposition 3 for an interior optimum is given in Figure 1. The individual’s endowment without any life insurance (i.e., savings only) is given by point E, where the horizontal axis represents consumption in the death state and the vertical axis represents consumption in the life state. The price of insurance for purchases made in the first period, p1 ¼ p, determines the slope of the budget line p/(1 p) or trading opportunities for the individual, as denoted by the line EP. The full insurance curves for the two demand types, h1 and h2, represent the loci of points where such individuals would purchase insurance if it were offered at an actuarially fair price.19 If the individuals knew their demand types in period 1, then the low demand types (h1) would purchase 18 It is in fact possible to construct examples where this happens. Suppose that the v function has a kink so that both risk types buy exactly so much insurance that they have S þ L as determined where the kink occurs. Then, since low risks need to pay less for the additional insurance bought in the second period, they can afford higher savings and consequently need less life insurance L than high risks. It is equally easy to construct examples in which, among high demand types, low risks end up buying more insurance than high risk types. 19 With state dependent utilities, full insurance means buying that amount of insurance in order to equalise marginal utilities of income across states. From the first-order conditions of the consumer optimisation problem, this means pv 0 /(1 p)w 0 ¼ p/(1 p), which implies v 0 ¼ w 0 . Note that this condition is independent of the specific value p (i.e., whether the person is in the uninformed scenario ex ante or the informed scenario ex post - having received either good or bad news about her risk type). Ó Royal Economic Society 2006 340 [JANUARY THE ECONOMIC JOURNAL Life State 45o Line Total premium paid by type H1 E PB B AL1 AH1 * B AL2 B L AH2 P LH1 PS H Death State Fig. 1. Equilibria Under Alternative Information Sets the amount of insurance as indicated by point B while the high demand types (h2) would purchase the greater amount of insurance as indicated by point B. Since in period 1 individuals do not know demand type, the optimal choice is to buy some intermediate amount of insurance as indicated by the point B . In period 2 each individual’s risk type is revealed and the second period price of insurance will reflect the risk-type specific probabilities of death. Thus, in period 2 the low risk type’s budget line becomes flatter while high risk types face a steeper budget line. Since B is the allocation corresponding to the optimal first period insurance purchase, trading possibilities in period 2 are given by the H line (with slope pH/(1 pH)) for high risk individuals and by the L line (with slope pL/ (1 pL)) for low risk individuals. Low demand types resell some of their first period insurance purchases while high demand types purchase additional insurance coverage, as determined by the intersection of their respective budget lines and risk-type specific full insurance curves. Hence, low risk–low demand individuals will realise allocation AL1, and so on. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 341 Suppose for institutional reasons it is not possible for individuals to resell any of their first period insurance purchases in period 2. The qualitative nature of the results of Proposition 3 would still hold in that the optimal first period purchase would be an intermediate amount of insurance (i.e., between B and B). However, only additional purchases could occur in period 2 and so low demand types would end up holding too much insurance. Nevertheless, from an ex ante (period 1) perspective, it is optimal for an individual to balance the risk of holding too much insurance, should she be determined to be a low demand type, against premium risk from wanting to purchase additional insurance, should she be determined to be a high demand type.20 3.2. Asymmetric Information Under asymmetric information, we denote the price at which individuals can purchase insurance in the second period by p2S and, if a market for reselling insurance exists, then the price at which individuals can resell some or all of their first period insurance purchases is p2B. Since information is asymmetric, these prices are not differentiated by risk type. However, insurers can observe whether an individual is buying or reselling insurance, and so it will not generally be the case that p2S ¼ p2B. If insurance can be resold in the second period, then the second period optimisation problem for an individual of type ij is given by (12) and (13) but noting that the second period price of insurance is not dependent on risk-type (i.e., replacing p2i with p2S).21 In equilibrium, the price at which additional insurance is sold to consumers is determined by the zero expected profit condition, which implies that: p2S ¼ ½r1 qL ðL~L1 L 1 Þ þ r2 qL ðL~L2 L 1 ÞpL þ ½r1 qH ðL~H 1 L 1 Þ þ r2 qH ðL~H 2 L 1 ÞpH ; ½r1 qL ðL~L1 L 1 Þ þ r2 qL ðL~L2 L 1 Þ þ ½r1 qH ðL~H 1 L 1 Þ þ r2 qH ðL~H 2 L 1 Þ ð19Þ where L~ij ¼ maxðL 1 ; Lij Þ; so L~ij L 1 is the effective demand of an individual in the second period market, given that additional insurance is sold at price p2S. The right-hand side of (19) is known in the literature as average clientele risk and is a weighted average of the death probabilities pL and pH, where the weights are the quantities of insurance bought by each type. The price at which insurers buy back insurance from consumers is given by the solution to p2B ¼ ½r1 qL ðL 1 L^L1 Þ þ r2 qL ðL 1 L^L2 ÞpL þ ½r1 qH ðL 1 L^H 1 Þ þ r2 qH ðL 1 L^H 2 ÞpH ; ½r1 qL ðL 1 L^L1 Þ þ r2 qL ðL 1 L^L2 Þ þ ½r1 qH ðL 1 L^H 1 Þ þ r2 qH ðL 1 L^H 2 Þ ð20Þ 20 This is shown formally in Polborn et al. (2001). The S superscript is added here to distinguish between the price at which consumers buy and sell insurance in period 2, which is not relevant when information is symmetric. 21 Ó Royal Economic Society 2006 342 THE ECONOMIC JOURNAL [JANUARY where L^ij ¼ minðL 1 ; Lij Þ; so L 1 L^ij is the effective amount of life insurance an individual of type ij would like to sell on the second period market, given that he gets a price for his policy of p2B. Recall that p1 ¼ p; that is, the first period price of insurance is equal to the population weighted average of the probability of death. It is shown in Villeneuve (1999) in a similar, but static model,22 that the price at which insurance is sold to consumers exceeds the average probability of death while the price at which consumers resell insurance to insurers is less (i.e., p2B < p < p2S). The intuitive reason for this result is that, among buyers of additional insurance, at any given price high risk types will want to buy more than will low risk types and so p2S > p1 ¼ p. Analogously, among individuals who wish to resell insurance, low-risk types will view the reselling price as more favourable than will high risk types, and so low risk types resell a larger quantity, with the result that p2B < p1 ¼ p. In the context of a continuum of possible types, Villeneuve (1999) shows that there will, in general, be some types who are inactive (neither buying a positive amount of insurance nor selling a positive amount of insurance). The reason for this is that the existence of a wedge between the buying and the (re)selling price means for some individuals the price at which they could buy additional insurance may be too high to make this worthwhile but, at the same time, the price for reselling insurance may be considerably lower and thus it may also not be attractive for them to resell any of the life insurance they purchased in period 1. If there is no opportunity in the second period for individuals to resell insurance purchased in the first period, then we must add the following constraint to the optimisation problem: L L1: ð21Þ It turns out that the qualitative nature of the general demand properties, as described in the following Proposition, are the same whether or not reselling of insurance is possible. Proposition 4. If there is asymmetric information in the second period insurance market, the equilibrium has the following properties: (i) LL1 min(LH1, LL2) max(LH1, LL2) LH2. That is, type L1 has the smallest final life insurance demand, type H2 has the greatest life insurance demand and the other two types cannot be ranked in general. (ii) The optimal first period insurance purchases, L1, are chosen such that type L1 does not buy additional insurance in the second period and type H2 does not resell insurance. The intuition and proofs for both situations of reselling possible and not possible are almost identical and so only the proof for the case in which reselling is possible is included here. 22 Villeneuve’s model is a one period model of the life insurance and annuity market, where individuals just differ in their risk type (i.e., there is no variation in demand type). See also Hoy and Polborn (2000) for a similar model and results. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 343 Proof. (i) As in Proposition 3, high demand demand consumers must have an at least weakly higher final life insurance demand than low demand consumers of the same risk type. Since the premium is the same for both risk types, short inspection of the optimisation problem (12), (13) shows that a higher p also increases piv 0 (S2 þ L, hj) and hence has the same effects as an increase in h: high risk types have a higher final insurance demand than low risk types of the same demand type. An example showing that LH1 and LL2 cannot be ranked in general is given in the text. (ii) Suppose first that in the optimum L1 < LL1. A feasible plan (for an individual) is to increase L1 to LL1, and to leave the final life insurance demands for all 4 types unchanged (this will in general not be optimal, but is clearly feasible). As more expensive second period life insurance purchases (p2 > p) have been replaced by cheap first period purchases (p1 ¼ p), each type can save more than in the supposed optimum, and hence has a higher consumption in both states. This contradiction proves L1 LL1. The proof that L1 LH2 is analogous. ( The intuition explaining claim 1 of Proposition 4, for the case in which reselling is not possible, is very straightforward. As in Proposition 3, high demand individuals have a higher life insurance demand than do low demand individuals (of the same risk type). Under asymmetric information, however, all consumers pay the same premium and so, since the death state is more probable for high risk consumers, it follows that high risk individuals have a higher insurance demand than do low risk individuals. The following example shows that the final life insurance demand of type H1 and of type L2 cannot in general be ranked. Suppose v(S2 þ L, h) can be written as hv(S2 þ L); then type H1 has a higher expected marginal utility from a unit of money transferred to the death state than type L2 if and only if h1pH > h2pL. Since the budget constraint is the same for both types, type H1s final life insurance demand is greater than type L2s final life insurance demand if and only if this inequality holds. Since h2 > h1 and pL < pH, the inequality can, but need not, hold. For claim 2, suppose that all types were buying insurance in period 2. In Proposition 3 we argued that if this were so in the symmetric information case, then increasing life insurance purchases in period 1 and reducing life insurance purchases in period 2 would reduce the individual’s premium risk which results from possibly receiving bad news about health status before period 2 purchases are made. In the case of asymmetric information, this risk is already reduced by the asymmetric information in period 2; that is, provided some insurance purchases are made by low risk types in period 2, high risk types will face a price of insurance less than pH for their additional life insurance purchases in period 2. However, since high risk individuals have a higher life insurance demand than do low risk individuals of the same demand type, the second period life insurance market is subject to adverse selection. Therefore, the premium which has to be paid in period 2 is higher than the premium which has to be paid in period 1 (i.e., p2S > p1) and so waiting to purchase additional life insurance in period 2 is in one sense not as effective in dealing with premium risk as simply buying more insurance in period 1. Ó Royal Economic Society 2006 344 THE ECONOMIC JOURNAL [JANUARY On the other hand, waiting until one knows one’s demand type before deciding on how much insurance to purchase is also advantageous because one does not want to hold too much insurance. Thus, one has to balance these two conflicting goals. However, if all individuals are certain that they will want to buy additional life insurance in period 2 regardless of which demand type or risk type is revealed, then buying it in period 1 dominates buying additional insurance in period 2. So it is not possible that all types buy additional insurance in period 2. Similarly, it cannot be true that all individuals want to sell life insurance in period 2, since then it would be strictly better to buy less life insurance in the first period. The fact that individuals cannot sell life insurance in period 2 simply reinforces this point. Now we consider briefly the case in which reselling is possible. For the same reasons as the above case (i.e., p2S > p1), individuals do not want to purchase an amount of insurance in period 1 that is too little for all possible demand and risk types that could be realised in period 2. On the other hand, it cannot be true that all individuals choose to buy so much insurance in period 1 that they end up all wanting to resell life insurance in period 2 because in that scenario the reselling price is less than the price at which they purchased insurance (i.e., p2B < p1). Allowing for reselling of insurance may ameliorate the cost of this bad decision, in that people who overinsure can at least sell off some of the excess. However, it is still not a sensible decision ex ante to buy so much insurance in period 1 that she knows she will be selling some of it at a lower price than she paid for it. Thus, the qualitative nature of the demand characteristics under asymmetric information do not depend on whether reselling of insurance is possible. The specific quantities of insurance purchased are not, however, the same in the two cases. A graphical illustration of the results of Proposition 4 for the case in which reselling is allowed is included in Figure 1. As for the case of symmetric information, if the individual knew her demand type in period 1, then she would purchase all of her insurance needs at that time, as represented by the point B for a low demand type and the point B for a high demand type. However, since she does not have this information at the outset, she purchases an intermediate amount in period 1, as indicated by point B .23 If in period 2 she turns out to be a high demand type, then she purchases additional insurance along the line PS (i.e., at price p2S). This price line will be steeper than the P-line (i.e., p2S > p1) due to adverse selection. For those who turn out to be low demand types, B represents too much insurance and these individuals will resell insurance along the line PB (i.e., at price p2B). This price line will be flatter than the P-line (i.e., p2B < p1). The optimal decision is to balance these two adverse selection costs in the purchasing and reselling markets for insurance in period 2. The critical aspect of the demand properties derived in this Section and illustrated in Figure 1 is that, as a result of not knowing one’s demand type in period 1, an individual’s optimal first period decision is to purchase an amount of insurance 23 Of course, the actual location of the point B would not generally be the same in the cases of asymmetric and symmetric information. We use a single point here for both scenarios in order to simplify the diagram. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 345 which is between what is ideal for a low versus a high demand type. If reselling is possible and symmetric information prevails, then the consumer suffers from premium risk in period 2 whether one is a buyer (high demand type) or seller (low demand type). If reselling is possible and asymmetric information prevails, say through regulation, then the consumer faces some adverse selection costs in period 2 whether one is a buyer (high demand type) or a seller (low demand type).24 In the following Section we develop circumstances under which the individual would prefer to face adverse selection costs rather than premium risk. In such circumstances a regulation prohibiting the use of risk-type information by insureds will increase welfare despite creating regulatory adverse selection. 4. Welfare Comparison As noted earlier in this article, several European countries have regulations which prohibit the use of genetic test results by insurance companies for rate-making purposes. The UK and the Netherlands also have caps on how much life insurance an individual may purchase, above which genetic test results can be used for pricing. In this Section we develop a theorem that demonstrates that, under certain conditions, such a regulation will enhance welfare. Moreover, in the context of how much genetic information is likely to be available over the next, perhaps, 10–15 years in the future, it seems reasonable to assume that information could, in the absence of regulation, be symmetric and that the parameters under which the theorem hold are plausible. We consider a regulation which prohibits insurance companies from using this genetic information. Given that genetic tests for specific diseases are likely to become much cheaper, pervasive genetic information at least on a subset of diseases is likely to be available. Given such a scenario, in order for regulation to be effective, genetic information may not even be used for tarification if it is supplied voluntarily by the individual – otherwise, every individual with favourable test results will reveal them and people who do not supply test results are rationally expected by the insurers to be high risk types. What are the welfare effects of such regulatory adverse selection? Note firstly that it is not possible to apply the first theorem of welfare economics here (Hart, 1975) to argue that the laissez-faire allocation under symmetric information must be a Pareto optimum. The reason is that the insurance market between the first and the second period is incomplete as no insurance policy can condition on both demand type and risk type (i.e., the former is intrinsically private information and hence non-contractible). In the following Proposition 5, we consider a scenario that is very relevant for serious genetic diseases, namely that there are very few carriers in the population.25 If we assume in addition that low demand types derive no utility at all 24 If reselling is not possible, then the disutility cost of holding too much insurance should one turn out to be a low demand type must simply be optimally balanced against the premium risk or adverse selection costs of buying more insurance if one turns out to be a high demand type. 25 Strachan and Reid (1999) note that most serious genetic diseases caused by a single gene affect fewer than one in a thousand individuals. Ó Royal Economic Society 2006 346 THE ECONOMIC JOURNAL [JANUARY 26 from consumption in the death state, then regulatory adverse selection in combination with an appropriately chosen cap on insurance purchases increases expected utility in comparison to a scenario with symmetric information. Proposition 5. Assume v(y, h1) 0 for all y. For any given parameter vector pL, pH, r1, r2 and any utility functions w(Æ) and v(Æ, h2) satisfying conditions stated above, there exists qH > 0 and a cap on insurance purchases under asymmetric information such that, for all qH < qH , expected utility is higher under asymmetric than under symmetric information. Proof. Consider first the case that qH ¼ 0. In this case, ex ante expected welfare is obviously the same under symmetric and asymmetric information in the second period, and an optimal level of first period insurance purchases is L1 ¼ 0. Our objective for the rest of the proof is to show that the derivative of expected utility with respect to qH is larger (less negative) for asymmetric than for symmetric information. Given qH > 0, expected utility at the start of the first period under symmetric information is r1 ½qL ð1 pL ÞwðyÞ þ qH ð1 pH ÞwðyÞ þ r2 fqL ½ð1 pL ÞwðS2L Þ þ pL vðS2L þ L2L ; h2 Þ þ qH ½ð1 pH ÞwðS2H Þ þ pH vðS2H þ L2H ; h2 Þg: ð22Þ Taking the derivative with respect to qH (and remembering that qL ¼ 1 qH) yields r1 ðpL pH ÞwðyÞ þ r2 f½ð1 pH ÞwðS2H Þ þ pH vðS2H þ L2H ; h2 Þ ½ð1 pL ÞwðS2L Þ þ pL vðS2L þ L2L ; h2 Þg: ð23Þ For comparison purposes later, it is helpful to write this as r1 ðpL pH ÞwðyÞ þ r2 f½ð1 pH ÞwðS2L Þ þ pH vðS2L þ L2L ; h2 Þ ½ð1 pL ÞwðS2L Þ þpL vðS2L þ L2L ; h2 Þg þ r2 f½ð1 pH ÞwðS2H Þ þ pH vðS2H þ L2H ; h2 Þ ½ð1 pH ÞwðS2L Þ þ pH vðS2L þ L2L ; h2 Þg: ð24Þ The third line in (24) is the difference between the expected utility of a high demand, high risk type when she consumes her optimal (affordable) bundle and when she consumes the bundle that high demand low risk types consume. Note that this difference is negative (because low risk types are effectively richer, given L1 ¼ 0, they can afford a higher consumption in both states). Since the Z2H(Æ) function is concave, the third line of (24) is smaller than r2w 0 (S2L)(pH pL)L2L: here, (pH pL)L2L is the additional cost that a 2H type would have to spend to afford the 2L life insurance demand and w 0 (S2L) is the marginal utility of low types (which must be lower than the marginal utility of high risk types). 26 For a household to hold no life insurance is not uncommon. From the 1998 US Survey of Consumer Finances, 31% of families did not own any life insurance (American Council of Life Insurers, http://www.acli.org). Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 347 Let us now turn to expected utility in the case of asymmetric information. We also impose a cap on life insurance purchases here, equal to the amount that low risks are willing to buy in the second period (note that, as qH ! 0, this amount approaches the life insurance demand for high demand types in the case that qH ¼ 0, which we simply denote by L2L). Expected utility is r1 fqL ð1 pL ÞwðyÞ þ qH ð1 pH ÞwðyÞg þ r2 fqL ½ð1 pL ÞwðSÞ þ pL vðS þ L; h2 Þ þ qH ½ð1 pH ÞwðSÞ þ pH vðS þ L; h2 Þg: ð25Þ Differentiating with respect to qH and evaluating at qH ¼ 0 gives r1 ðpL pH ÞwðyÞ þ r2 f½ð1 pH ÞwðS2L Þ þ pH vðS2L þ L2L ; h2 Þ ½ð1 pL ÞwðS2L Þ þ pL vðS2L þ L2L ; h2 Þg r2 ðpH pL ÞL2L k2L : ð26Þ The third line of (26) is due to the fact that high risks who buy L2L cost society (pH pL)L2L more than low risk types and this additional resource cost must be multiplied with the marginal utility of income in order to get the effect on ex ante expected utility. Comparing (24) and (26) completes the proof. ( It is instructive to note where the conditions in the Proposition are required in the proof. Firstly, we assume that low demand types do not have any utility in the death state, so that their life insurance demand is zero. This is important because otherwise adverse selection in the second period market from low demand high risk types buying more insurance than their low risk counterparts would arise; moreover, these types would (at least initially) not be constrained by the cap on life insurance purchases, which is set at the optimal level for high demand low risk types. Secondly, we also use the fact that qH is small, when we argue that the life insurance demand of high demand low risk types under asymmetric information is close to their optimal symmetric information demand. Then, (essentially) the only cost due to the presence of high risk types is the resource cost (their contracts are more expensive), but there is only a small distortion as to how much life insurance low risk types buy (and the welfare loss from that distortion is a second order effect). On the other hand, when qH is considerably bigger than 0, the additional welfare loss from low risk high demand types buying too little insurance (i.e., having a higher marginal utility in the death state than in the life state) would have to be considered and favour symmetric information. Of course, in this case a higher cap would alleviate this source of inefficiency. Although the number of genetic diseases for which cheap and reliable tests become available are likely to increase substantially in the foreseeable future, it is unlikely that the number of these will cumulate so quickly that the proportion of people who know they have high mortality risk (qH) will become large. Moreover, as the list of such diseases lengthens, it is presumed that more and more will become known about those diseases so that some will be effectively treated further Ó Royal Economic Society 2006 348 THE ECONOMIC JOURNAL [JANUARY into the future (e.g., through developments in gene therapy techniques). Not only does this limit the amount of actuarially significant information about mortality risk for a population but also it means uncertainty about future risk type will persist after genetic causes of diseases are identified. The uncertainty about risk type will arise because of uncertainty about future developments for treatment of genetic diseases. Thus, it is possible that our model will remain applicable and, in particular, Proposition 5 will be relevant for a significant length of time. An assumption used for the proof of Proposition 5 is that a cap on life insurance purchases can be enforced. This raises three important sets of questions. First, how can caps be enforced in practice, and how does this assumption square with our earlier observation that linear pricing obtains in life insurance contracts sold today? If there is a substantial interest to do so, it should be reasonably straight forward to implement a cap on life insurance coverage bought by individuals (i.e., which is priced independently of any genetic test result). All that is needed is some information pooling which is checked at the time of death before payments are made, for example through a national clearing house to which insurance companies report all payments and insureds’ stated medical information (in particular genetic test results).27 In principle, the information collected would also enable other forms of nonlinear pricing that are feasible only if there is exclusive contracting between an insured and only one insurer. The fact that life insurers today use only linear pricing and generally neither limit the amount of coverage bought nor prohibit customers from buying additional life insurance coverage from other providers does not imply that the necessary information pooling for implementing a cap is impossible. Rather, it seems likely that linear pricing today is a consequence of the absence of significant asymmetric information problems in the life insurance market existing today.28 In any case, operation of such a database, while ensuring pricing independent of genetic test results (up to the cap amount) and continued linear pricing, would presumably require government oversight. Second, in Proposition 5, we assume that the high demand types of both low and high risk types buy the same quantity of coverage in period 2. If the no genetic discrimination law only specifies that an insurance company cannot discriminate based on genetic tests but does not forbid nonlinear exclusive contracts, insurance companies could design separating contracts for high risks and low risks. How would our results of Proposition 5 be affected in such a setting? In the setting of Proposition 5 where there are many low risk types and few high risk types, a Rothschild and Stiglitz (1976) separating equilibrium is unlikely to exist. Intuitively, since a separating contract must give an incentive for low risk types to selfselect the contract designed for them, it needs to limit the low risks amount of coverage severely, otherwise high risk types would take the same contract. How27 An example of a similar type of organisation is the Medical Information Bureau (MIB, see MIB.com). North American Insurers may become members of the MIB which, among other services, shares medical information on insurance applicants. Although the MIB does not maintain records on total insurance purchases of individuals, it does maintain an Insurance Activity Index which records the number of times an MIB check has been made on a particular applicant. 28 We have referred to empirical evidence justifying this claim earlier in the article, although we have also argued that there are also reasons besides the lack of significant adverse selection that could induce firms not to use exclusive contracting. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 349 ever, if there are many low risks and few high risks, the low risk premium is only slightly smaller than the pooling premium and so even low risk types prefer the pooling contract to an incentive compatible separating pair of contracts. This pooling contract does not satisfy the equilibrium definition of Rothschild and Stiglitz (1976) either, since there is a contract that, if offered in addition to the pooling contract, attracts only low risk types and makes a profit (and also leads to losses for the insurance companies that offer the pooling contract). However, as argued by Wilson (1977), the pooling contract that maximises the low risks expected utility (which is the allocation that we use in Proposition 5) can be considered a reasonable equilibrium under adverse selection with few high risk types, if insurance companies are forward-looking.29 Third, suppose that parameters are such that the regulatory adverse selection solution provides a higher ex ante expected utility than the solution that obtains under symmetric information in the second period and linear pricing in both periods. If insurance companies can use exclusive contracting already in period 1, is there a way how a private contract can implement the regulatory adverse selection allocation? In other words, is regulation really necessary in this market, or would enforcing exclusivity of contracts do? In principle, an insurer could offer the following contract: in period 1, customers commit to an insurance company. In the second period, they can then choose whether they want to pay the (prespecified) premium equal to the average clientele risk to get the amount of coverage high demand types get in the regulatory adverse selection solution of Proposition 5, or whether they want to forgo their option and stay without life insurance coverage (from any insurer). Such a contract can implement exactly the same allocation as the regulatory adverse selection solution and will therefore be preferred by first period customers to linear contracts. However, there are several disadvantages of this private solution, compared with the regulatory adverse selection solution of Proposition 5. First, one of the advantages of the regulatory adverse selection solution in the real world that is not formally modelled is that no contract needs to be written in the first period and thus the transactions cost can be saved. This would not be the case with the private solution described above. Second, the first period contract requires a lot of information that may not be easily available in practice. In particular, the first period contract needs to specify the second period premium in nominal terms, which does not only require an accurate estimate of the mortality table in period 2 (i.e., 10 to 15 years from now) but also of the inflation rate over the same time period (so that the optimal coverage level can be specified in nominal terms). Similarly, it may be necessary to know the interest rate prevailing in period 2 in order to determine the appropriate premium for such a long term contract. All this information will be difficult to specify ex ante. Third, in practice people often change the amount of life insurance they hold over time, for example, because 29 Some readers may feel uneasy with Wilson’s (1977) anticipatory equilibrium concept. However, it should be stressed that, if equilibrium non-existence really generated problems in our setting, in the sense that the Pareto better regulatory adverse selection solution cannot be implemented any more if insurance companies use nonlinear contracts, one could easily write into the law that only linear pricing is allowed for insurance contracts. Ó Royal Economic Society 2006 350 THE ECONOMIC JOURNAL [JANUARY their income or other demand-influencing parameters change. If consumers are restricted by an exclusive contract to buying additional insurance only from one company (or to forgo the upfront loading if they change to another company), insurers get some monopoly power over existing clients and so anti-trust concerns arise. In our view, all these problems make it much easier to implement the asymmetric information solution through regulation rather than through a first period exclusive contract. 5. Discussion and Conclusion In this Section, we revisit some of the assumptions made in the article and analyse the applicability of the model for other insurance markets. Resellability Our model is based on the presumption that people can buy term insurance covering the risk of death in some later period both early in life and in that period. In period 2 they can resell any coverage they bought in period 1 if they find they do not need it, although we also develop results where this is not feasible. The reader may wonder whether people in actual insurance markets have these options. Individuals in today’s markets can buy life insurance with a guaranteed renewability option, meaning that the customer can choose to renew the contract at a predetermined premium after the first period has elapsed even if her health state has deteriorated. In effect, buying guaranteed renewable insurance is a way to buy life insurance which is valid at some later time. The difference with our model is that with renewable insurance the coverage valid for tomorrow has to be bought bundled together with life insurance valid for today. For simplicity, in our model, no risk of death in period 2 exists, so life insurance against this event is not relevant. However, it would be easy to introduce such a risk in our model and nothing would change qualitatively. By mixing renewable and non-renewable policies, individuals can adjust their life insurance coverage valid for period 2. People who discover later in life that they do not need life insurance (or at least not all of their renewable insurance) can let the option lapse. This corresponds to selling insurance in the asymmetric information scenario. By the same arguments as in the model, the average risk of death among consumers who let their renewable contract lapse should be lower than among those who continue, and so an insurer has to charge p1 p2B more in period 1 for a policy which can be renewed later at a guaranteed premium of p1 than for a non-renewable one in order to cover its adverse selection costs for the contracts that are renewed.30 30 Hendel and Lizzeri (2003) analyse this empirically in life insurance markets. They find that, the higher the front loading of a long term life insurance contract (i.e., the more premium is paid in the early periods), the better the risk types that choose to prolong the contract. The reason why, in their model, not everyone buys a life insurance contract with lots of front loading is that some individuals in their model are cash constrained and suffer if the first period cost of their life insurance policy is too large. While they do not analyse the question of optimal regulation in their model, it appears that there would be some scope in their model, too, for regulatory adverse selection to be welfare improving for some parameter settings. Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 351 The case of reselling insurance under symmetric information is more complicated. A high risk low demand individual cannot effect the equivalent transaction of reselling insurance coverage as we model it by simply letting her renewability option lapse because that action will not yield her risk type equivalent price. However, there are so called viatical companies who buy life insurance from people who can demonstrate that they have a high mortality risk. Currently, viatical companies mainly buy policies from people who are terminally ill and have a remaining live expectancy of less than 2 years. Since rapid medical progress in the area of genetics will make possible considerably more accurate prognoses for persons whose life expectancy is not so short, we may see this market expand substantially. One important condition for viaticals to be non-threatening to insurance market opportunities altogether, however, is that the ability to sell such coverage before death must apply only to insurance purchased before genetic information becomes available to the individual (i.e., period 1 purchases in our model). Otherwise, an obvious arbitrage opportunity exists in which a high risk insurance buyer can buy as much insurance as she wishes and then immediately resell it to a viatical company for a higher price. Such opportunities must be restricted to avoid excessive adverse selection in the market. The Prudential Insurance Company, for example, is reported to be planning to include such opportunities within their future policies, thus internalising this resale possibility.31 Low risk individuals in the symmetric information scenario can, as in the asymmetric information case, effect the reselling of insurance again by letting their renewability options lapse. Since insurers will not want to make a loss due to low risk individuals renewing their policies at a lower average rate of intensity than do high risks, it is necessary that the guaranteed premium for which people can renew be equal to pL in order to attract renewals by low risks. The difference with the average population risk, p pL, must be charged in the first period (i.e., renewable policies must be that much more expensive than non-renewable ones). Summing up, while the instruments available in the model do not completely coincide with those available in reality, we can nevertheless conclude that the insurance opportunities available today correspond quite closely to those available in the model. Transaction costs When individuals choose to buy and possibly resell insurance in our model, they can do so without incurring any kind of transaction cost. As far as the welfare ranking is concerned, this assumption works in favour of a symmetric information organisation because consumers can insure against the second period classification risk by buying insurance in the first period, without being afraid that they end up with unwanted insurance that they cannot get rid of at a fair price. The fact that, even in a setting with no such transactions costs, regulatory adverse selection in the second period might be welfare improving is therefore an even more surprising result. 31 For a brief discussion about viatical markets, see Black and Skipper (2000, p. 233). Given the novelty and rapid growth in the resale market for life insurance, the most informative source for details would appear to be the internet; for example, http://www.insure.com/life/viatical. Ó Royal Economic Society 2006 352 THE ECONOMIC JOURNAL [JANUARY If each time an individual buys or sells insurance, transaction costs arise, there is an additional reason why it would not be socially desirable that everyone buys life insurance before knowing his demand type. In such a setting, regulatory adverse selection can again provide a social insurance against the classification risk, and this is socially beneficial as long as the welfare loss associated with adverse selection in the second period is not too large. In particular, this will be the case if there are only few people who get significantly bad news, while most individuals receive favourable results that, however, do not change their perceived risk type very much. Interpretation as a short-run model The main application of our model is the medium-term future, a time frame of about 15–20 years in which much more genetic information than is available today is likely to arrive. Our model gives those individuals living today (who are almost all uninformed about their genetic type, let alone the consequences of certain genes for their life expectancy) the chance of buying insurance before they receive further information. However, it is also interesting to think what the model implies for a society in which, up to now, individuals did not buy life insurance until they learned their demand type, but now agents have been surprised by the arrival of new information (some have received favourable, and others unfavourable information). It is clear that in this scenario, symmetric and asymmetric information are generally not Pareto comparable. Those individuals with favourable news certainly prefer that symmetric information be allowed in the insurance market, while high risk individuals prefer that regulatory adverse selection be imposed. The only case in which high risk types are indifferent with respect to regulation is if adverse selection in the insurance market would be so severe that all low risk types choose not to buy and, therefore, the premium in the insurance market is pH. In this case, high risk types do not benefit from regulatory adverse selection and it would be Pareto better to let insurance companies use the new information. However, even though symmetric and asymmetric information allocations cannot be Pareto compared, one can still compare them using a utilitarian approach. As in our Proposition 5, as long as the percentage of people receiving bad news is small (and consequently, adverse selection costs are small), the welfare increase from regulatory adverse selection among high risk types is likely to be larger than the welfare cost to low risk types. Other insurance markets There are also other important insurance markets (in particular, the health insurance market) that could be affected considerably by new genetic information in the future. Although our model is specifically designed to reflect the workings of the life insurance market, our analysis provides some insight also for long-term care and health insurance markets. Most analyses of such markets have been guided by the classic Rothschild-Stiglitz model, and associated models noted in footnote 4, which assume that there is a fixed financial loss associated with an illness.32 However, it is also plausible that in health insurance markets, individuals have heterogeneous tastes concerning service quality and that these tastes may be changing over time, so that individuals possibly do not yet know their future demand type. 32 An exception to this is Strohmenger and Wambach (2000). Ó Royal Economic Society 2006 2006 ] EFFECTS OF REGULATORY ADVERSE SELECTION 353 If we apply the Wilson-Miyazaki-Spence equilibrium as the equilibrium concept in such a market, then arguments similar to the ones in this paper should apply; that is, regulatory adverse selection may be beneficial for high risk types and, therefore, it may provide insurance against classification risk that may be ex ante welfare improving. 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