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The Theory of Risk ClassificationbyKeith J. CrockerSmeal College of BusinessThe Pennsylvania State UniversityUniversity Park, PA 16802-3603andArthur SnowDepartment of EconomicsUniversity of GeorgiaAthens, GA 30602-6254JEL: D82, G22Keywords: risk categorization, classification, informational asymmetry, information, insurance.
8.1INTRODUCTIONThe efficiency and equity effects of risk classification in insurance markets have been a source ofsubstantial debate, both amongst economists and in the public policy arena.1 The primaryconcerns have been the adverse equity consequences for individuals who are categorizedunfavorably, and the extent to which risk classification enhances efficiency in insurancecontracting.While equity effects are endemic to any classification scheme that results inheterogeneous consumers being charged actuarially fair premiums, whether such classificationenhances market efficiency depends on specific characteristics of the informational environment.In this contribution we set out the theory of risk classification in insurance markets andexplore its implications for efficiency and equity in insurance contracting. Our primary concernis with economic efficiency and the role of risk classification in mitigating the adverse selectionthat arises when insurance applicants are better informed about their riskiness than insurers. Weare also interested in the role of classification risk, that is, uncertainty about the outcome of aclassification procedure. This uncertainty imposes a cost on risk averse consumers and is thus apotential cause of divergence between the private and social value of information gathering. Inaddition, the adverse equity consequences of risk classification bear directly on economicefficiency as they contribute to the social cost of classification risk.8.2RISK CLASSIFICATION IN THE ABSENCE OF HIDDEN KNOWLEDGEWe begin by considering as a benchmark the case in which both insurers and insuranceapplicants are symmetrically uninformed about the applicants’ propensities for suffering aninsurable loss.1
8.2.1Homogeneous AgentsFormally, the insurance environment consists of a continuum of risk averse consumers, each ofwhom possesses an initial wealth W and may suffer a (publicly-observed) loss D with knownprobability p . Each consumer’s preferences are represented by the von Neumann-Morgensternutility function U(W), which is assumed to be strictly increasing and strictly concave, reflectingrisk aversion.A consumer may purchase insurance against the loss by entering into a contractC ! (m, I ) , which specifies the premium m paid to the insurer and the indemnification I receivedby the insured when the loss occurs. A consumer’s expected utility under the insurance contractC is given byV ( p, C ) " pU (WD ) (1 ! p)U (WN ) ,(1)where WD " W ! m ! D I and WN " W ! m denote the consumer’s state-contingent wealthlevels. The expected profit of providing the insurance contract C is given by# ( p, C ) " m ! pI .(2)In order to be feasible, a contract must satisfy the resource constraint" ( p, C ) ! 0 ,(3)which requires that the premium be sufficient to cover the expected insurance indemnity.In this setting, an optimal insurance contract is a solution to the problem of maximizing(1) subject to the feasibility constraint (3), which results in full coverage for losses (I D) at theactuarially fair premium (m pD). This contract, which is depicted as F in Figure 1, is also the2
competitive equilibrium for an insurance market with free entry and exit when all consumershave the same (publicly observed) probability p of suffering loss.8.2.2Classification with Heterogeneous AgentsWe now turn to the case in which both insurers and insurance applicants have access to a costlessand public signal that dichotomizes applicants into two groups. After the signal has beenobserved, a proportion ! of the agents are known to be high risk with probability pH of sufferingthe loss, while 1-! are low risk with loss propensity pL, wherep H p L andp !p H (1 " ! ) p L . When each individual’s type (pH or pL) is publicly observable, insurers ina competitive market equilibrium offer full coverage (I D) to all consumers, and charge theactuarially fair premium m" p"D appropriate for the p" – types. These contracts are depicted asH* (L*) for pH – types (pL – types) in Figure 1.Notice that competitive pressures force firms to implement risk classification based uponthe insureds’ publicly observed characteristic, p". Any insurer attempting to offer a contract thatwould pool both high and low risks (such as F) recognizes that a competitor could offer aprofitable contractual alternative that would attract only the low risks. The exodus of low riskscaused by such cream-skimming would render the pooling contract unprofitable.The introduction of symmetric information about risk type accompanied bycategorization based on this information increases the utility of some of the insured agents (lowrisks, who receive L*), but reduces the utility of others (high risks, who receive H*) relative tothe pre-classification alternative (when both types receive F). From an efficiency perspective,however, the relevant question is whether the insureds expect to be better off when moving froma status-quo without information and risk-based categorization to a regime with information and3
risk classification. If an individual who is classified as a p" – type receives the contract C", thenthe expected utility of the insured in the classification regime isE{V} # !VH (1-!)VL(4)where V i " V ( p i , C i ) for i ! {H , L}. The corresponding resource constraint is! (pH, CH) (1-!) (pL, CL) ! 0,(5)requiring that premiums collected cover expected indemnity payments per capita.An efficient classification contract is a solution to the problem of maximizing (4) subjectto (5), which turns out to be the pooling contract, depicted as F in Figure 1, and which providesfull coverage at the pooled actuarially fair premium pD . The intuition behind this result isstraightforward. From an ex ante perspective, there are four possible payoff states: The two lossstates and the two risk types.Since individuals are risk averse, ex ante expected utilitymaximization (3) subject to the resource constraint (4) requires equal consumption in all states,and F is the only zero-profit contract with this property.The technical rationale for this result can be illustrated with reference to Figure 2, whichillustrates the utilities possibilities frontier for the classification regime as locus XFY. Theconcavity of XFY is dictated by the risk aversion of consumers, and movement along the frontierfrom X towards Y makes L-type (H-types) better (worse) off. From equation (4), we infer thatthe slope of an indifference curve for the expected utility of an insured confronting classificationrisk, dVH/dVL, is –(1-!)/!. By the concavity of U and Jensen’s inequality, the pool F is theunique optimum for the consumer anticipating risk classification.We conclude that the efficient contract in the classification regime ignores the publiclyobserved signal, and treats all insureds the same independently of their types. Put differently,when information is symmetric between insurers and insureds, uniformed insureds prefer to4
remain uninformed if they anticipate that the information revealed will be used to classify therisks. The reason is that the pooling contract F provides full coverage against two types of risk,the financial risk associated with the occurrence of the loss state, and the classification risk facedby insurance applicants, who may find out that they are high risk. The competitive equilibriumcontracts H* and L* satisfy the resource constraint (5) and, therefore, are candidate solutions foroptimal classification contracts.However, while they provide complete protection fromfinancial risk, they leave consumers wholly exposed to classification risk. Thus, insurers woulduse public information to classify insurance applicants, even though risk classification based onnew information actually reduces efficiency in this setting, and is therefore DDENKNOWLEDGEWe now turn to an environment in which the individuals to be insured all initially possess privateinformation about their propensities for suffering loss, as in the model introduced by Rothschildand Stiglitz (1976). Each consumer has prior hidden knowledge of risk type, pH or pL, butinsurers know only that they face a population of consumers in which a proportion ! (1-!) havethe loss probability pH (pL). Given the nature of the informational asymmetry, in order to beattainable a pair of insurance contracts (CH, CL) must satisfy the incentive compatibility (selfselection) constraintsV(p", C" ) % V(p", C"' )for every ", "' " {H, L}(6)as a consequence of the Revelation Principle exposited by Myerson (1979) and Harris andTownsend (1981).5
In this informationally constrained setting, an efficient insurance contract can becharacterized as a solution to the problem of maximizing the expected utility of low-riskconsumers V(pL, CL) subject to the resource constraint (5), the incentive constraint (6), and autility constraint on the welfare of high-risk typesV(pH, CH) ! V H .(7)As discussed by Crocker and Snow (1985a), a solution to this problem yields full (partial)coverage for H-types (L-types); both the resource constraint (5) and the utility constraint (7) holdwith equality; and the incentive constraint (6) binds (is slack) for high (low) risks.One element of the class of efficient contracts is depicted in Figure 3 as {Cˆ H , Cˆ L } . Byconstruction, the locus FA depicts the set of contracts awarded to low risks that, when coupled! the resource constraintwith a full-insurance contract to which high risks are indifferent, satisfieswith equality.2 The full class of efficient contracts is obtained by varying the utility level VHinHconstraint (7). Setting V V ( p H , F) yields the first-best pooling allocation F as a solution to!Hthe efficiency problem. Setting lower values for V results in a redistribution away from H!types toward L-types and a solution in which the types receive distinct contracts, as described!above, which entail a deductible for L-typesand so are strictly second-best. The particularsolution depicted in Figure 3, {Cˆ H , Cˆ L } , is obtained when the constraint level of utility for the HHtypes, V , is set equal to V ( p H , Cˆ H ) and results in the efficient contract most preferred by the!L-type individuals. The allocation {Cˆ H , Cˆ L } will be referred to in the discussion below as the M-!W-S allocation.3!!Also depicted in Figure3 is the Rothschild-Stiglitz separating allocation (H*, A), whichis the Pareto dominant member of the family of contracts that satisfy the incentive constraints (6)6
and the requirement that each type of contract break even individually. The Rothchild-Stiglitzallocation is not an element of the (second-best) efficient set when the proportion of H-types (!)is sufficiently small. Such a situation is depicted in Figure 3, since both types of customers canbe made strictly better off at {Cˆ H , Cˆ L } than they would be at {H*, A} . In this particular case, allof the efficient contracts involve a cross-subsidy from L-types to H-types. Only when " is! that {H*, A} is contained in !the class of efficient allocation, is there ansufficiently large, so!efficient contract that does not entail a cross-subsidy. The utility possibilities frontier associatedwith the solutions! to the efficiency problem is depicted in Figure 4. At one end is the utilitiesdistribution associated with the first-best pooling contract F which involves a large cross-subsidybut no inefficiency since the L-types are not subject to a deductible. As one moves along theefficiency frontier toward the point associated with the M-W-S allocation, the degree of crosssubsidy is reduced and the amount of inefficiency increases as the L-types are choosing contractswith higher deductibles.4At this juncture, it is useful to elaborate on the differences between the efficiencyapproach that we have adopted in this chapter, and the equilibrium analyses that havecharacterized much of the insurance literature. The potential for the non-existence of a Nashequilibrium in pure strategies that was first observed by Rothschild and Stiglitz is an artifact ofthe incentives faced by uninformed insurers who compete in the offering of screening contractsto attract customers. This result has spawned a substantial body of work attempting to resolvethe nonexistence issue, either through the application of non-Nash equilibrium concepts (Wilson(1977); Riley (1979); Miyazaki (1977)) or by considering alternative extensive form models ofthe insurance process with Nash refinements (Hellwig (1987); Cho and Kreps (1987)).7
Unfortunately, the insurance contracts supported as equilibrium allocations generally differ, anddepend on the particular concept or extensive form being considered.In contrast, the characterization of second-best efficient allocations that respect theinformational asymmetries of the market participants is straightforward. The model is that of asocial planner guided by the Pareto criterion, and who has the power to assign insuranceallocations to the market participants.5 While the planner is omnipotent, in the sense of havingthe ability to assign any allocation that does not violate the economy’s resource constraints, it isnot omniscient, and so is constrained to have no better information than the market participants.6Hence, the issue of how firms compete in the offering of insurance contracts does not arise, sincethe social planner assigns allocations by dictatorial fiat subject to the (immutable) informationaland resource constraints of the economy. This exercise permits an identification of the bestoutcomes that could, in principle, be attained in an economy. Whether any particular set ofequilibrium mechanics can do as well is, of course, a different issue, and one that we consider inmore detail in Section 8.5 below.Finally, as we close this section, notice that risk classification, accomplished throughself-selection based on hidden knowledge of riskiness, is required for efficient contracting in thisenvironment. Specifically, with the exception of the first-best pooling allocation F, all efficientallocations are second best, as they entail costly signaling by low-risk types. These consumersretain some risk by choosing a contract that incorporates a positive deductible, but in so doingthey are best able to exploit opportunities for risk sharing given the potential adverse selection oflow-risk contracts by high-risk consumers.8.3.1Categorization Based on Immutable Characteristics8
We suppose for the purposes of this section that consumers differ by an observable trait that isimmutable, costless to observe, and correlated with (and, hence informative about) theunobservable risk of loss.Examples of such categorizing tools are provided by, but notrestricted to, an insured’s gender, age or race, which may be imperfectly correlated with theindividual’s underlying probability of suffering a loss. The interesting question is whether theinformation available through categorical discrimination, which can be used by insurers to tailorthe contracts that are assigned to insureds based upon their observable characteristics, enhancesthe possibilities for efficiency.In the first attempt to examine the implications of permitting insurers to classify risks inthis environment, Hoy (1982) considered the effects of categorization on market equilibria.Since there was, and still is, little consensus on the identity of the allocations supported byequilibrium behavior, Hoy considered the pure strategy Nash equilibrium of Rothschild andStiglitz, the “anticipatory” equilibrium of Wilson (1977), and the equilibrium suggested byMiyazaki (1977) which assumes anticipatory behavior but permits cross-subsidization within aninsurer’s portfolio of contractual offerings. Hoy found that the efficiency consequences ofpermitting risk classification were ambiguous, depending on the particular equilibriumconfiguration posited. The primary reason for this ambiguity is that, with the exception of theMiyazaki equilibrium, none of the allocations supported by the equilibrium behaviors consideredis guaranteed to be on the efficiency frontier.7 Thus, a comparison of the equilibrium allocationspre- and post-categorization provides no insights regarding whether permitting categorizationenhances the efficiency possibilities for insurance contracting.A more fruitful approach is explored by Crocker and Snow (1986), who compare theutilities possibilities frontier for the regime where categorization is permitted to the one in which9
it is not. Throughout the remainder of this section, we assume that each insurance applicantbelongs either to group A or to group B, and that the proportion of low-risk applicants is higherin group A than in group B. Letting !k denote the proportion of H-types in group k, we have0 !A !B 1, so that group membership is (imperfectly) informative. Assuming that aproportion & of the population belongs to group A, it follows that &!A (1 - &)!B !.()Let C k ! C kH , C kL denote the insurance contracts offered to the members of group k.Since insurers can observe group membership but not risk type, the contractual offerings mustsatisfy separate incentive constraints for each group, that is,V ( p! , Ck! ) " V ( p! , Ck! ' ) for all " ," ' !{H , L}(8)for each group k "{A, B}. In addition, contracts must satisfy the resource constraint%[ A# ( p H , C AH ) (1 " A )# ( p L , C AL )] (1 " % )[ B # ( p H , C BH ) (1 " B )# ( p L , C BL )] ! 0 ,(9)which requires that the contracts make zero profit on average over the two groups combined.To demonstrate that risk categorization may permit Pareto improvements8 over the nocategorization regime, it proves useful to consider the efficiency problem of maximizingV ( p L , CBL ) subject to the incentive constraints (8), the resource constraint (9), and the utilityconstraintsV ( p! , C !A ) " V ( p! , Cˆ ! ) for # " {H, L}; and(10)V ( p H , C BH ) ! V ( p H , Cˆ H ) ,(11)where Ĉ ! (Ĉ H , Ĉ L ) is an efficient allocation in the no-categorization regime. By construction,we know that this problem has at least one feasible alternative, namely the no-categorizationcontract Ĉ which treats the insureds the same independently of the group (A or B) to which they10
belong. If Ĉ is the solution, then the utilities possibilities frontier for the categorization and theno-categorization regimes coincide at Ĉ . However, if Ĉ does not solve the problem, thencategorization admits contractual opportunities Pareto superior to Ĉ and the utilities possibilitiesfrontier for the categorization regime lies outside the frontier associated with the nocategorization regime.Let ' denote the Lagrange multiplier associated with the utility constraint (7) for theefficiency problem in the no-categorization regime, and let µH be the multiplier associated withthe incentive constraint (6) for " H. The following result is from Crocker and Snow (1986, p.329).Result: Categorization permits a Pareto improvement to be realized over efficient contractswithout categorization if and only if! # !A". µ H ! A (1 # ! )(12)For the inequality to hold, it is sufficient that ' 0, which necessarily obtains whenever theutility constraint, VH, in (7) is set sufficiently low. When ' 0, the location of the utilitiespossibilities frontiers depends on the informativeness of the categorization. When categorizationis more informative, !A is smaller and the right hand side of (12) is larger. If categorization wereuninformative (! !A), then (12) could never hold, and if categorization were perfectlyinformative (!A 0), then (12) would always be satisfied. Finally the inequality can never holdwhen µH 0, which occurs when the incentive constraint (6) for the efficiency problem in theno-categorization regime is slack.Contract F is the only efficient contract for which theincentive constraint is slack, so that the utilities possibilities frontiers always coincide at F11
regardless of the degree of informativeness of the categorization. The relative positions of theutilities possibilities frontiers for the categorization and the no-categorization regimes for thosein group A are depicted in Figure 5, while a similar diagram applies to those in group B.To evaluate the efficiency of categorization, we employ the Samuelson (1950) criterionfor potential Pareto improvement. Risk classification through a priori categorization by insurersis defined to be efficient (inefficient) if there exists (does not exist) a utility distribution in thefrontier for the no-categorization regime Pareto dominated by a distribution in the frontier for thecategorization regime, and there does not exist (exists) a distribution in the categorizationfrontier Pareto dominated by one in the no-categorization frontier. Since costless categorizationshifts outward the utilities possibilities frontier over some regions and never causes the frontierto shift inward, we conclude that categorization is efficient.Crocker and Snow (1985b) show that omniscience is not required to implement thehypothetical lump-sum transfers needed to effect movement along a utilities possibilities frontier.Although the appropriate lump-sum transfers cannot be assigned directly to individualconsumers, since their risk types are hidden knowledge, these transfers can be built into thepremium-indemnity schedule so that insurance applicants self-select the taxes or transfersintended for their individual risk types. In this manner, a government constrained by the sameinformational asymmetry confronting insurers can levy taxes and subsidies on insurancecontracts to implement redistribution, while obeying incentive compatibility constraints andmaintaining a balanced public budget. Our application of the Samuelson criterion is thusconsistent with the informational environment.The situation is somewhat different when consumers differ by an observable, immutabletrait that is correlated with the unobservable risk of loss, but is costly to observe. Crocker and12
Snow (1986) show that the utilities possibilities frontiers cross in this case, so long as the cost isnot too high. Intuitively, the cost of categorization amounts to a state-independent tax on eachconsumer’s wealth. As a result, when the adverse selection externality is not very costly andlow-risk types are nearly fully insured, categorization costs dominate the small efficiency gainsrealized by the winners leaving no possibility of compensating the losers. Conversely, if theadverse selection externality imposes sufficient cost on the low-risk consumers, then gains fromcategorization realized by the winners are sufficient for potential Pareto improvement providedcategorization is not too costly. This situation is depicted in Figure 6.If categorization were required, then insurers would sometimes categorize insuranceapplicants even when the result is not a potential Pareto improvement over not categorizing. Inthis scenario the efficiency effects of costly categorization are ambiguous. As Rothschild (2011)points out, however, the second-best efficient allocations when categorizing is costly do notrequire the use of categorization. Consider a social planner with the power to assign insurancecontracts to applicants subject to the economy’s resource and informational constraints, and whohas access to the same costly categorizing technology as insurers. Because the social planner canchoose not actually to employ the categorizing technology, the second-best Pareto frontier for theplanner is the outer envelope of utility possibilities. The Samuelson criterion therefore leads tothe conclusion that allowing costly categorization is more efficient than either banning orrequiring categorization.Rothschild further shows that this application of the Samuelson criterion is againconsistent with the informational environment. Specifically, for any allocation in the nocategorization regime, a government constrained by the same informational asymmetryconfronting insurers can simultaneously provide a properly calibrated social insurance policy and13
also legalize categorization so that, in response, insurers choose to employ categorizationprecisely when doing so yields a Pareto improvement over not categorizing. In this sense, theno-categorization regime is inefficient.8.3.2An Empirical Estimate: The Case of AnnuitiesFinkelstein, Poterba and Rothschild (2009) adapt the basic framework of Hoy (1982) and Crocker andSnow (1986) to facilitate empirical estimates of the efficiency and distributional consequences ofprohibiting categorical discrimination in real-world insurance markets. Their approach is to use anempirically calibrated model to estimate the welfare consequences restricting gender-based pricing inthe compulsory annuities market of the United Kingdom. In this market, which is described in greaterdetail in Finkelstein and Poterba (2002, 2004), retirees are required to annuitize a substantial portion oftheir accumulated tax-preferred retirement savings, but there is scope for annuity providers to screendifferent risk types by offering annuity contracts with different lifetime payout structures.Their adaptation requires two significant modifications of the standard insurance model. First,the model is extended to allow many “indemnity” states that correspond to annuity payments in futureyears, where the uncertainty arises because the annuity is paid only if the annuitant survives. From theinsurer’s perspective, low-risk (high-risk) individuals are those that have a lower longevity (higherlongevity), and this is assumed to be private information known only to the annuitant. Second, themodel allows for the possibility that individuals could, in a fashion that is not observable to the insurer,save a portion of their annuity income to supplement the consumption provided by the annuity at laterages, in effect, permitting individuals to engage in a form of “self-insurance”.14
Using mortality data from a major insurer, maximum likelihood estimation is used to calibratea model with two unobservable types (high-risk and low-risk) and two observable categories (male andfemale). The categories are observable to the insurer and each category contains both high- and lowrisk types, although the female category contains a higher proportion of high-risk (longer-lived)individuals. When insurers are permitted to categorize their insurance offerings on observable gender,the market segments into male and female sub-markets in which insurers screen each category forunobservable risk type through their contractual offerings. The result is screening of types in bothgender categories in the manner of Figure 3, but with different contracts offered to male and femaleapplicants. In contrast, when such categorical discrimination is prohibited, insurers still screen types asin Figure 3, but now are constrained to offer the same screening contracts to both genders. As a result,when calculating the efficiency costs of prohibiting gender-based pricing, there are in principle threeefficiency frontiers that must be considered: those associated with each of the two genders whencategorical discrimination is permitted, and the one associated with the regime in which suchdiscrimination is prohibited.The goal in Finkelstein et al. is to calculate bounds on the welfare costs associated with a banon gender-based pricing. Their approach is to assume that, when gender discrimination is allowed, thesegmented markets provides a second-best efficient allocation to each category, and that there is nocross-subsidy between the two observable categories. In contrast, when gender-based pricing isbanned, the market is assumed to attain an allocation on a no-categorization efficiency frontier of thetype described by Figure 4. As noted by Crocker and Snow (1986, p. 329), starting from an efficientcontract on the no-categorization frontier, it is possible to make the category composed of fewer highrisks better off, and at a lower resource cost, if risk categorization were to be introduced. This saving inresources represents the efficiency cost of the categorization ban. Thus, the potential efficiency cost of15
a ban on gender-based pricing ranges from zero if the post-ban market achieves the first best poolingallocation F (which results in maximal across-gender redistribution) to its maximum value when thepost-ban result is the M-W-S allocation (which result in the minimal across-gender redistribution).Figure 7 (which is Figure 4 from Finkelstein et al.) depicts the efficient annuity contractsassociated with the W-M-S allocation in the presence of a ban on gender-based pricing.High-risk(long-lived) types receive a full insurance annuity that provides constant real payments for the durationof their retirements. Low-risk types, by contrast, receive a front-loaded annuity. This front loadingallows them to receive substantially higher annuity payments for most of their expected lifetimes whilestill effectively discouraging the high-risks from selecting the annuity targeted to the low-risk types.Moreover, the efficient annuities involve a cross-subsidy from low- to high-risk types since the latterobtain a better than actuarially fair annuity payment. Since the high-risks are the recipients of thesubsidy, and the female category contains a disproportionate share of the high-risk annuitants, the effectis to generate a cross subsidy from males to females. Column (9) of the Table (which is Table 3 fromFinkelstein et al.) quantifies the cross-subsidy associated with the post-ban W-M-S allocation, which ison the order of a two to four per-cent transfe
5 remain uninformed if they anticipate that the information revealed will be used to classify the risks. The reason is that the pooling contract F provides full coverage against two types of risk, the financial risk associated with the occurrence of the loss state, and the classification risk faced by insurance applicants, w
