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Evidence on Adverse Selection: Equilibrium Signaling and Cross-Subsidization in the Insurance Market
by Robert Puelz, Arthur Snow
Evidence on Adverse Selection: Equilibrium Signaling and Cross-Subsidization in the Insurance Market
Robert Puelz, Arthur Snow
The Journal of Political Economy
Updated: October 25th, 2012
Evidence on Adverse Selection: Equilibrium Signaling and Cross-Subsidization in the Insurance Market
Southern Methodist University
University of Georgia
The configuration of equilibrium in the market for automobile colli- sion insurance is examined empirically by representing the pre- mium-deductible menu and the demand function as a standard he- donic system. Using contractual data from a representative insurer, we estimate a reduced-form hedonic premium equation and the inverse of the marginal bid equation for insurance coverage. The data reveal an equilibrium with adverse selection and market signal- ing but lead us to reject the hypothesis that high risks receive con- tracts subsidized by low risks.
The "lemons" principle, as elucidated by Akerlof (1970), predicts that "bad" products drive "good" products from the marketplace when knowledge is asymmetrically distributed between buyers and sellers. Various theories of market signaling, based on Spence's (1973, 1978) original insights, predict that sellers of good products in such an environment undertake some costly activities designed specifically to
We gratefully acknowledge helpful comments received on previous versions of the manuscript from Keith Crocker, Bev Dahlby, Georges Dionne, George Mailath, and anonymous referees.
[Journal of Polittcal Economy, 1994, vol. 102, no. 21 O 1994 by The University of Chicago. All rights reserved. 0022-380819410202-0005$01.50
signal buyers that they offer products of high quality. While adverse selection and market signaling are now widely accepted as conse- quences of hidden knowledge, few empirical studies to date have tested for their presence in any market and found outcomes consis- tent with these phenomena.'
In this paper we present evidence on adverse selection from the automobile collision insurance market. We provide an empirical char- acterization of the market equilibrium that allows us to determine whether an adverse selection externality is present and to discrimi- nate among some alternative theories of an adverse selection equilib- rium. Using individual data from a representative insurer domiciled in the state of Georgia, we find strong evidence of adverse selection and market signaling, with no cross-subsidization between the con- tracts of different risk classes.
Asymmetric information about product quality arises in insurance markets when firms have difficulty judging the riskiness of those who demand insurance coverage. Firms respond to the adverse selection externality through screening, categorizing, and sorting. In the mar- ket for automobile insurance, firms engage in screening activities by assigning each insurance applicant to a particular risk category on the basis of observable traits. An insurer offers the premium-deductible menu appropriate to the applicant's risk category, with profitable premiums forecast by the insurer given the observable characteristics defining the category. Faced with a single menu, each applicant chooses a premium-deductible pair on the basis of traits that cannot be observed by the insurer. In this fashion, customers within a given risk category may further sort through a self-selection mechanism.
In the presence of hidden knowledge and unobservable heteroge- neity, low-risk insurance applicants have an incentive to signal their quality by selecting high deductibles. Market signaling theories pre- dict that insurers, recognizing this incentive, charge a higher price per unit for insurance as the amount of coverage increases. Thus, in these theories, equilibrium with adverse selection satisfies a mono- tonic signaling property; namely, within each risk category, individu- als of a lower risk type choose a contract with a higher deductible, and contracts with higher deductibles are associated with lower aver- age prices for coverage. We develop and estimate a system of hedonic
' Using individual data from the market for used pickup trucks, Bond (1982) re- jected the hypothesis of adverse selection. The only empirical work relating to automo- bile insurance markets is that by Dahlby (1983, 1992), who reported evidence of ad- verse selection based on aggregate data in the Canadian collision insurance market. Additionally, Beliveau (1984) estimated a price equation for the life insurance market and found evidence of signaling.
238 JOURNAL OF POLITICAL ECONOMY premium and demand equations representing the insurance market that allows us to test both of these predictions. Given the categorizing practices of insurers and the loading of in- surance premiums to cover operating costs, a number of market equi- librium configurations are possible. Our empirical model discrimi- nates among these alternative possibilities, and our data indicate the presence of adverse selection and market signaling. We also imple- ment a test to determine whether the equilibrium entails the cross- subsidization of high risks by low risks within a given risk category, as predicted by Miyazaki's (1977) theory of adverse selection. In this type of signaling equilibrium, high risks pay a premium with negative loading subsidized by positive loading in the premium of low risks. Our data reject the hypothesis of cross-subsidization. We conclude that the market for collision insurance represented by our data attains a separating, adverse selection equilibrium in which each type of contract breaks even individually and the average price of coverage, net of fixed charges, is inversely related to a con- tract's deductible. These findings are consistent with the market sig- naling theories developed by Rothschild and Stiglitz (1976), Riley (1979, 1985), and Cho and Kreps (1987), which predict that equilibria in markets with adverse selection in which signaling of hidden knowl- edge is possible, as through the choice of deductible, entail separa- tion, nonlinear pricing, and no cross-subsidization. Our findings are inconsistent with the linear-pricing equilibrium suggested by Arrow (1970) and developed by Pauly (1974) and Schmalensee (1984) in which insurers engage in pure price competition, customers separate by riskiness, and low risks subsidize high risks, but all risk types none- theless pay the same average price for insurance. The evidence we present is also inconsistent with predictions of a pooling equilibrium, in which customers of different innate riskiness choose the same deductible level, and with predictions of a separating equilibrium that incorporates cross-subsidization. In this sense, our empirical findings do not provide support for the theories of Wilson (1977), Grossman (1979), and Hellwig (1987) that predict pooling, or for Miyazaki's (1977) theory of signaling with cross-subsidization, although the results are not inconsistent with these theories since each of them also predicts separation with nonlinear pricing and no cross-subsidization for at least some population distributions of risk. Section I1 of the paper sets out the economic environment and presents the competing predictions of alternative theories. In Section I11 we introduce the empirical model and the data. The evidence on equilibrium signaling and on cross-subsidization is presented in Sections IV and V, respectively. Section VI provides a summary and concluding remarks.
11. Alternative Theories of Adverse Selection
Consider the following simple model of an insurance market. For a customer who incurs a loss X with probability T,, an insurance con- tract with a deductible D, yields an expected profit of zero for the insurer if the premium for the policy P, is given by
where ko is a cost proportional to the net premium, necessitated by commission payments to agents and ceded reinsurance charges; k, is the fixed cost of bookkeeping; k2 is the cost of processing a claim; and k,, is a charge for cross-subsidization among the contracts chosen by different risk types2 For zero expected profit, the charges for cross-subsidization must net out to zero across risk types, so that
where A, is the proportion of customers of risk type 7.
With an applicant's risk type unobservable, insurers categorize indi- viduals on the basis of traits that are observable and correlated with risk type, such as age. Potential customers within a given risk category are offered the opportunity to choose from a menu of premium- deductible choices
where z is a vector of observable characteristics indicative of the loss to be insured. A customer of risk type T chooses the deductible level that maximizes expected utility,
(1 -T,) u(w -g(D, 2)) + T,U(W -g(D, z) -D), given the endowed wealth W and the von Neumann-Morgenstern utility function U(.), which is assumed to be strictly concave reflecting risk aversion. The chosen deductible D = D,satisfies the first-order condition
where -gD is the marginal price of coverage. In an equilibrium, equa- tions (1)-(4) must be satisfied.
According to the relation in (4), a customer's choice of deductible
As suggested by Riley (1983), the loss X can be interpreted as the expected loss X = zp(s)X(s), where X(s) is the loss in state s that occurs with a probability a,p(s) for an individual of risk type 7.
depends on the following factors: risk type, through the odds ratio (1 -T,)/T,; the degree of risk aversion, through the marginal utili- ties of income; and the marginal price of coverage, through the rela- tive price ratio (1 + gD)/ -gD. Thus we can write the demand for insurance implied by (4), expressed as the demand for a deductible, as
where p indicates the degree of risk aver~ion.~
If organizing an insurance pool were costless and risk type were observable, then insurance would be fairly priced, in which case -gD(D, z) = n, and each customer chooses full coverage, D = 0. Alternatively, if providing insurance were a costly activity with risk type observable, then the marginal price of coverage would be con- stant although unfair, so that -gD(D, z) = (1 + k,)~,, but the average price of coverage, net of the fixed charge k,, would equal the mar- ginal price. In either case in which a customer's risk type is observable, the premium-deductible schedule is linear, with slope -(1 + k,)n,, and there is no cross-subsidization of contracts, so that k,, = 0.
In a market in which risk type is not observable, the equilibrium configuration of contracts depends on the precision of insurers' cate- gorization schemes and the manner in which insurers and customers engage in competition. If categorization is perfect, then the equilib- rium outcome is the same as that attained when risk type can be observed. However, if categorization is imperfect, then the adverse selection described by Rothschild and Stiglitz (1976) precludes the perfect-categorization outcome. Figure 1 illustrates the adverse selec- tion problem in the case of two risk types with no loading charges. The high-risk and low-risk fair-odds lines are represented by the loci EH and EL, respectively. When a customer's probability of loss is hidden knowledge, the full-information equilibrium (H,L), in which both risk types are optimally insured, is unattainable because of the adverse selection of L by high risks.
In the model suggested by Arrow (1970) and developed by Pauly (1974) and Schmalensee (1984), categorization is imperfect and insur- ers engage in pure price competition, so that in equilibrium the mar- ginal price of insurance is constant. In this linear-pricing equilibrium, customers of different risk types separate by choice of deductible, with the higher risk type choosing a lower deductible. In addition, while there is cross-subsidization of high risks by low risks (k,, # O),
We assume that differences in risk aversion are attributable to differences in wealth, age, and gender, and we do not account for any effects of wealth on demand other than through risk aversion.
the average price of insurance, net of the fixed charge k,,is the same for all risk types. Figure 2 illustrates the linear-pricing equilibrium with two risk types and positive proportional loading (k, > O),but no common fixed contracting charge (k, = 0)and no claims fee (k2 = 0).Note that the separation that occurs in the linear-pricing equilib- rium does not constitute signaling since all customers pay the same average price for coverage. Thus, while low-risk types select the higher deductible, they are not compensated for doing so by paying a lower average price for their insurance ~overage.~
Rothschild and Stiglitz (1976) suggested that, in markets in which risk type is unobservable and categorization is imperfect, insurers would engage in price-quantity competition and would compete by offering a nonlinear menu of policies that incorporate no charges for cross-subsidization. In this equilibrium, customers separate by risk type through their deductible choices as in the linear-pricing equilib- rium, but the average price of insurance, net of the fixed charge k,, differs according to risk, being lower at the higher deductible levels
Since, as indicated by Schmalensee (1984), the linear-pricing equilibrium explicitly rules out signaling, which entails differentiated pricing as well as separation, this equi- librium may not be relevant to insurance markets in which the choice of deductible presents a natural signaling instrument. However, linear pricing could arise as a device for economizing on the transaction costs of discovering, communicating, and pro- cessing a nonlinear premium schedule.
chosen by the lower-risk types. Hence, the equilibrium entails separa- tion with nonlinear pri~ing.~
The theories of competition with ad- verse selection developed subsequently by Riley (1979) and Cho and Kreps (1987) provide models that always yield this signaling equilib- rium as the market o~tcome.~
Specifically, these theories predict that a competitive market with unobservable risk types and imperfect cate- gorization attains as an equilibrium the Pareto-dominant member of the set of separating allocations among those in which each type of contract breaks even individually. This signaling configuration is illus- trated as the pair (H,A) in figure 1.
In an equilibrium with separation, customers of different risk types receive con- tracts with different deductible levels, and the equilibrium contractual configuration satisfies the incentive compatibility constraints that require that each risk type does not prefer the contract received by any other risk type.
As reported by Rothschild and Stiglitz (1976), a Nash equilibrium in which each insurer announces a premium schedule from which customers choose does not exist in the case of two risks types if the proportion of high risks is below a critical level. The theories developed by Riley (1979) and by Miyazaki (1977) and Wilson (1977) propose non-Nash equilibria that exist regardless of the population distribution of risks. In the theories developed by Cho and Kreps (1987) and by Grossman (1979) and Hellwig (1987), the insurance market is modeled as a game with three stages wherein Nash equilibria always exist. Cho and Kreps assume that the informed parties (insurance customers) move first and show that the strategically stable equilibrium entails separation, whereas Grossman and Hellwig assume that the uninformed parties (insurers) move first, in which case the strategically stable equilibrium entails pooling if pooling is Pareto superior to the separating equilibrium.
Miyazaki (1977) presents a theory that predicts signaling with cross- subsidization if this results in an allocation of risk bearing that is Pareto superior to the Pareto-dominant, break-even separating allo- cation. In Miyazaki's equilibrium, customers separate by riskiness through their deductible choices, resulting in positions H" and AM in figure 1, and insurer profits earned on the low-risk contracts just offset losses that accrue on high-risk contracts, so that low risks subsi- dize high risks. In addition, the average price of insurance, net of any fixed charge k,, is lower at higher deductible levels, so the pre- mium-deductible schedule is nonlinear.'
In contrast, the theories presented by Wilson (1977), Grossman (1979), and Hellwig (1987) predict a pooling equilibrium if pooling is Pareto superior to the Pareto-dominant, break-even separating allo- cation. Cross-subsidization occurs as all risk types receive the break- even pooling contract most preferred by the lowest risks, resulting in position G on the pooled fair-odds line EF in figure 1.'
On the face of it, the observation that insurers offer contracts at more than one deductible level provides evidence supporting the the- ories that predict separation, with or without nonlinear pricing, and runs counter to the theories of Wilson, Grossman, and Hellwig that predict pooling, although these latter theories do predict separation for at least some population distributions of risk.g However, the alter- native deductible levels represented in an insurer's contract portfolio could reflect pooling if customers with different potential losses are offered different contractual opportunities resulting in the pooling of risk types at different deductible levels. Alternatively, a multiplicity of deductible levels could reflect perfect categorization by the insurer
'We note that Miyazaki's theory may not apply to insurance markets. Miyazaki examined the assignment of workers to jobs within a firm as dictated by the production technology. In the labor market context, cross-subsidization may survive the competi- tive pressure to drop losing contracts because of a technological constraint not present in insurance markets.
While Miyazaki's equilibrium is always Pareto efficient over the attainable set of incentive-compatible contracts, the pooling equilibrium is never efficient. Crocker and Snow (1985~) provide an efficiency analysis of alternative equilibria based on price- quantity competition. The linear-pricing equilibrium, based on pure price competition, is not Pareto efficient among the attainable incentive-compatible contracts and may be Pareto inferior to alternative linear-pricing equilibria attainable either through com- pulsory insurance (governmentally enforced pooling) as suggested by Pauly (1974) or through commodity taxation as suggested by Greenwald and Stiglitz (1986).
In the case of two risk types and costless insuring, the theories of Miyazaki (1977), Wilson (1977), Grossman (1979), Riley (1979), Cho and Kreps (1987), and Hellwig (1987) all predict the equilibrium outcome (H, A) if the proportion of high risks in the customer population is sufficiently large. As this proportion declines, first the Miyazaki equilibrium and then the equilibrium of Wilson, Grossman, and Hellwig differs from (H, A).
and the self-sorting of customers by risk aversion, in which event there is neither pooling nor separation of risk types.''
To discriminate among the alternative possibilities, we investigate the market equilibrium empirically by estimating the premium- deductible schedule (3) and the demand equation (5) using contrac- tual data from a representative insurer operating in the automobile collision insurance market. The estimation of (5)requires proxies for the unobservable variables 7,indicating risk type, and p, indicating risk aversion. We use wealth variables described below, age, and gen- der as proxies for p, and for T we use the ex post observation of whether the customer filed a claim. This proxy for risk type is sup- ported by the casual observation that insurers use experience rating to categorize renewal customers, and more formally by the work of Boyer and Dionne (1989),who show that past accidents are a good predictor of an individual's loss probability.
With the influence of risk aversion on deductible choice accounted for through various proxies and the influence of observable charac- teristics of the loss insured accounted for through the marginal price of coverage, an empirical finding that riskiness has a statistically sig- nificant influence on deductible choice (f,< 0) would be indicative of an equilibrium with separation. If, in addition, the estimated pre- mium-deductible schedule is nonlinear (gDD# 0), then the empirical evidence would indicate an equilibrium with market signaling. Alter- natively, if the estimated premium-deductible schedule is linear (gDD
= 0) and there is no separation by risk type (f,= 0), then the evi- dence would be consistent with either a pooling equilibrium or an equilibrium with perfect categorization. Finally, evidence of a linear premium-deductible schedule with customers of different risk type separating by choice of deductible (g, = O and f,< 0) would be consistent with the linear-pricing equilibrium associated with pure price competition.
111. The Empirical Model and Data
In this section we introduce the empirical model and the data used to test for separation, nonlinear pricing, and cross-subsidization. Our data set contains a finite number of deductible choices, with a suffi- cient number of sample observations at three deductible levels to carry out our tests. Although we are limited to three deductibles, they
'O Note that since risk aversion does not affect contract profitability, separation by risk aversion does not entail an adverse selection inefficiency.
are adjacent in the premium schedule, consisting of the lowest ($100) and the next two higher levels ($200 and $250).11
We interpret the premium-deductible schedule (3) and the demand function (5) as a standard hedonic system of the type discussed by Rosen (1974), and we estimate the empirical counterparts to (3) and
(5) using a procedure described by Bartik (1987) and Epple (1987). In the first step, the premium-deductible schedule is estimated as a reduced-form hedonic premium equation that is assumed to be determined by the market interaction of insurers and insurance de- manders. This schedule is taken as exogenous by each demander, who, in choosing a deductible level, selects both an amount of cover- age and the marginal price of coverage. Each individual's marginal bid for a deductible equals the marginal price paid.
The inverse of the marginal bid function, the functional (5), is assumed to depend on the marginal price paid, risk type, and other demand shifters. We avoid the consistency problem identified by Ep- ple that arises in estimating the demand equation in a hedonic equilib- rium by choosing a plausible set of instruments for marginal pre- mium, representing an individual's wealth and automobile, that exogenously shift the individual's budget constraint. We assume that the unobserved taste parameter for risk type is uncorrelated with the choice of automobile type.
A. The Emfirical Model
Equations (3) and (5) are implemented by specifying the following two-equation system to represent the market for automobile insur- ance:
+ p, MALE + P8.PERAGE + €1,
l1 The remaining deductible levels in the data are $300, $500, and $1,000. Although we have developed the analysis as though the premium-deductible schedule is continu- ous, insurers typically offer only a discrete set of deductible options. Presumably, this marketing practice economizes on the transaction costs of discovering, communicating, and processing a continuous nonlinear schedule. In the bunching that occurs as a result, risk type T is the average risk type in the 7th most risky bunch.
+ a,.W, + a6 MALE + a7 PERAGE + E,,
where Ais the age of the automobile; MR = 1 for a multirisk contract and 0 otherwise; SYM is the symbol of the automobile; T is the terri- tory; b = 0 for D = $100, b = 1 for D = $200, and b = 2 for D = $250; W1, W,, W3 = wealth dummy variables; MALE = 1 for a male and 0 for a female; and PERAGE is the age of the individual.
In the empirical hedonic premium equation (6), the price of insur- ance P is the gross premium for insurance coverage. On the right- hand side of the premium equation, the dummy variables Dl and D, represent deductible categories. The variable Dl equals one for the $200 deductible and equals zero otherwise; the variable D2 equals one for the $250 deductible and zero otherwise; Dl and D, are both set equal to zero in the benchmark case, which is the $100 deductible.
Given this benchmark and the fact that deductible variables interact with other regressors, we are able to perform a test for the nonlinear- ity of the premium-deductible menu. When information from the estimated equation (6) is used, the marginal effects on the premium of moving from the $100 deductible to the $200 and $250 deductibles are, respectively,
Over the range of deductibles in our sample, nonlinearity is present when the marginal premium between the $100 and $200 deductibles differs from the marginal premium between the $200 and $250 deductible~. Thus we calculate
and compare the result with B 1. A value for B 3 different from B 1 is evidence of a nonlinear premium-deductible schedule, whereas B3 = B 1 indicates linear pricing. Figure 3 illustrates a concave schedule, where B 3 is less than B 1 .12
l2 The statistical relationship between prices and deductibles is verified by using an incrementalR' test to evaluate, under the null hypothesis, that the relationship between premiums and deductibles is linear. Using an F-test, we evaluate the differences in
slope = B 1
slope = B2
Our test for separation is carried out by estimating the inverse marginal bid equation (7) in two stages. In the first stage, we utilize Bartik's instrumental variables procedure to evaluate the marginal price of coverage, g,. The instruments used are the wealth proxies, the individual's age, and the age and type of automobile the individ- ual drives.
Consistent estimates of the coefficients on risk type and other de- mand shifters are recovered when, in the second stage, the inverse marginal bid equation is estimated via ordered logit. The ordered, discrete nature of our data on deductible choice dictates such model- ing. As noted by Greene (1990), however, care must be taken in interpreting the signs of the coefficients of ordered models. Thus, for our variable of primary concern, RT, we are interested not only in whether the coefficient is significantly different from zero but also in whether its effect on the predicted probabilities of choosing certain deductible amounts indicates that lower-risk types are more likely to choose the higher deductibles. For the estimation procedure, RT is set equal to one for those who incurred a loss during the year and set equal to zero otherwise.13 We carry out our test for separation by
explanatory power between eq. (6)and a model that accounts for a simple linear effect between deductibles and prices. We define high risks (RT = 1)as those who incurred a loss during the year greater than $250 to account for the potential bias in the estimation attributable to individuals
evaluating the estimated equation (7) at the mean sample values of the other demand shifters. An increase in the probability of choosing a $250 deductible for low-risk types is evidence supporting an equilib- rium in which risk types separate through their deductible choices.
Taken together, support for the nonlinearity hypothesis tested by the estimation of equation (6)and support for the separation hypoth- esis tested by the two-stage estimation of equation (7) would provide evidence consistent with the monotonic signaling predictions of sepa- ration with nonlinear pricing. In this event, we may conclude that lower-risk types signal by purchasing the higher-deductible contracts and are compensated for doing so as insurers charge lower premiums for insurance at lower levels of coverage.
The control variables in equation (6) reflect the prospective loss insured. We expect the premium to be negatively related to the ages of the automobile and the individual, and positively related to male drivers (MALE = 1).The variable MR indicates that the individual is a holder of a multirisk contract with more than one automobile insured under a single insurance policy, and we expect the premium to be lower as a result of reduced administrative expenses associated with multirisk contracts. The variables SYM and T are two sets of controls used by the insurer to rate the contract representing, respec- tively, the symbol of the automobile and the territory in which the automobile is principally garaged. Finally, we have included a set of interaction terms of deductible choice with age, symbol, and territory that reflect our suspicions that the marginal premium of coverage varies with the automobile's characteristics and location.
The variables Wi,PERAGE, and MALE in equation (7) control for variation in risk preferences. We expect the deductible choice to be related to wealth in a manner consistent with decreasing absolute risk aversion for all insured individuals. This is equivalent to asserting that wealthier individuals retain the risk of small losses. Wealth is proxied by the amount of bodily injury liability coverage per accident (coverage A in the Georgia Personal Auto Policy) purchased by an individual, since neither income nor net worth data across insureds are available. One expects that wealthier insureds choose to hold more automobile liability insurance because of the retained limit re- quirement in the excess personal umbrella contract.14 Hence, the
not filing claims less than their deductible amounts. No individual in our sample in- curred more than one loss.
l4 The retained limit requirement is a policy condition in the personal umbrella contract whereby insureds are required to maintain certain liability limits in their automobile insurance contract. If the condition is satisfied and a loss occurs, then the automobile contract is primary coverage and the umbrella contract is secondary coverage.
choice of personal liability insurance as a proxy for wealth is based on the assumption that wealthier individuals purchase more liability insurance to fully protect such wealth, ceteris paribus.
Since this proxy for wealth is discrete, four wealth categories are considered. The minimum required amount of bodily injury liability insurance per accident in Georgia is $30,000. Thus the dummy vari- able W, equals one when an individual's wealth proxy is less than $3 1,000 and equals zero otherwise. Similarly, the dummy variable W, equals one for an individual whose wealth proxy is at least $31,000 but less than $51,000 and equals zero otherwise, and W3 equals one for an individual whose wealth proxy is at least $51,000 but less than $10 1,000 and equals zero otherwise. The variables W, , W,, and W3 are all set equal to zero in the benchmark case of an individual whose wealth proxy is at least $101,000.
B. Descrzftion of the Data
The model is estimated with 1986 data from a representative insurer domiciled in Georgia. The data pertain to the individual insured and consist of information about the driver's age, gender, marital status, and past driving record and the automobile's age, use, performance type, symbol, and territory of principal location. Any losses incurred during the year and their severity are also reported by individual insured. The data set numbers 3,280 insureds, each of whom pur- chased either a $100, $200, or $250 deductible in collision insurance coverage.
From the perspective of the underwriting insurer, who controls for an individual's age and gender, multirisk contracting, automobile age and type, and territory of principal location, each insured in the data set is homogeneous with respect to his or her loss-producing charac- teristics relative to any other insured in the category.15 Specifically, each individual in the data set we examine satisfied the informational criteria of the insurer necessary to be placed in risk category A.16
Table 1 summarizes descriptive statistics of the variables used in the regression analysis. Mean values are reported with standard devi- ations in parentheses. Although there is no explicit recognition in the
'' In preliminary testing we included marital status as a regressor in our price equa- tion. We found, however, that this variable was unrelated to the price of insurance for the category of individuals in our data.
l6 Risk category A includes those individuals insuring a standard performance auto- mobile for pleasure use, for which the principal operator is between the ages of 25 and 65 with zero driving record points. The definition of the risk category in our data comes directly from the primary rating factor classification table, which is a component of the 161-class rating plan mandated in the State of Georgia during the time period of our study.
descriptive data of principal territory and automobile symbol, the mean statistics for the variables across deductible levels do provide initial evidence of the hypothesized relationships in our model of the insurance market. For example, the loss frequency, calculated as the number of losses divided by the number of exposures, increases as the deductible declines. Moreover, the mean collision premium is higher for those choosing the $100 deductible, and the proxy variable for wealth is higher, but the mean automobile age is lower, for those choosing the $250 deductible. The majority of policyholders at both deductible levels are insured under a multirisk contract, which is not unexpected.
IV. Empirical Evidence on Signaling
The hypotheses to be tested are that equilibrium in the insurance market entails lower-risk types selecting contracts with higher deduct- ibles and insurance firms offering nonlinear pricing of insurance cov- erage. Each observation in our data set corresponds to an individual who is the primary driver of an automobile in one of five symbol classifications and one of five territories. The estimated coefficients for equations (6) and (7) and their t-statistics are reported in table 2.
Consider first the estimated effect of risk type on deductible choice from equation (7). The significant value for this coefficient reveals a strong relationship between risk type and deductible choice. To ex- plore this relationship more fully, we compare the predicted probabil- ities for each deductible choice when RT = 1 and when RT = 0, while holding the other demand shifters at their sample mean values.
DEPENDENTVARIABLE:P DEPENDENT D
VARIABLE:ESTIMATEDEQUATION(6) ESTIMATED (7)
Variable Coefficient t-Ratio Variable Coefficient t-Ratio
A A.Dl w;A.D2 w3 MR MALE SYM, PERAGE SYM8 P SYMg Log L SYMln Restricted log L*
SYM~.D; SYM9 ' Dl SYMlo. Dl SYM, .D2 SYM8 . D2 SYMg. D2 SYMIo. D2
TI1 .Dl Tl2 .Dl T13 .Dl 7-14 .Dl T11 .D2 Tl2 .D2
T13 ' D2 ' D2
* Slopes = 0.
The information in table 3 indicates that those who filed a claim are more likely to have chosen the lower-deductible contract, which is expected if there is imperfect categorization and separation, since those who have accidents are more likely to be high risks and have a greater incentive than low risks to select contracts with low deduct- ible~."
"The direction of change is the crucial element in assessing the marginal effect of RT on deductible choice. The insignificant differences between probabilities reflect the low predictive power of the overall model.
RT= 1 ,56592 ,20535 ,22873 RT = 0 .41938 .23197 .34865 A in probability +,14654 -,02662 -.I1992
(.63618) (-,11556) (-,52062)
NOTE.-1-statistics are in parentheses.
Although evidence that lower risks tend to choose higher deduct- ibles is consistent with theories that predict separation by risk type in an adverse selection equilibrium, this finding is not alone sufficient to demonstrate that lower risks effectively signal and implicitly pro- vide the insurer with more information than is observable at con- tracting, as would be indicated by a nonlinear premium-deductible schedule. Thus in the premium equation (6) we are interested in testing for nonlinearity with respect to the deductible choice. The estimated values of B 1 and B2 at sample means are -26.0795 and
-36.2065, respectively, indicating that the slope of the premium- deductible schedule declines when one moves from the $100 to the $200 deductible and declines at a faster rate when one moves from the $100 to the $250 deductible. The change in the marginal pre- mium when one moves from the $200 to the $250 deductible is mea- sured by B3, for which we obtain -72.4129.These estimated values indicate that the insurer charges an average premium for insurance coverage, net of fixed charges, that varies with the deductible chosen."
Evidence of a concave premium-deductible schedule, taken together with the finding that lower-risk types are more likely to choose higher deductibles, supports the hypothesis that equilibrium in the automobile collision insurance market entails imperfect categoriza- tion, adverse selection, and market signaling. Thus our data reveal that lower-risk types signal by selecting higher deductibles and are being compensated for doing so by paying a lower average premium for insurance coverage.
V. Empirical Evidence on Cross-Subsidization
Empirical support for the predictions of the market signaling theories developed by Rothschild and Stiglitz, Riley, Cho and Kreps, and Mi-
We performed the incremental R~ test and rejected the null hypothesis that the premium-deductible menu is linear at the .O1 level.
yazaki indicates that insurance customers sort through a self-selection mechanism, but does not provide evidence on the cross-subsidization hypothesis that high risks are subsidized by low risks. To test for cross-subsidization, we accept the estimated equation (6) as a valid representation of the equilibrium premium-deductible schedule and calculate the loading charges k, by using the zero-profit conditions
(1) and (2) in the following manner. The documentation presented in the preceding section supporting the separation prediction justifies our using the empirical relative frequency of loss for those choosing each deductible D, as an estimate of the loss probability T,for the associated risk type. The values obtained for T,are given in table 4. We note that, although T, exceeds T,, which in turn exceeds T,in accord with the separation hypothesis, only the difference between T, and T,is statistically significant.
The fixed loading charge per policy k, is obtained from informa- tion provided by the insurer, and each premium P, is calculated from the estimated equation (6). Thus, once we account for the propor- tional loading factor ko, equations (1) for the three risk types and equation (2) constitute a system of four equations that can be solved for the four unknowns, the loss to be insured, X + k,, and the three cross-subsidization charges, k,,.
For the loading factor ko we use a commission rate typical for the automobile insurance market of 15 percent, but our conclusions are not sensitive to the magnitude of k0.l9 Estimates of the gross premi- ums are obtained from equation (6) using the sample means of the independent variables and considering each automobile symbol and territory combination separately. The resulting values for a represen- tative subset of the cross-subsidy loading charges, k;, for symbol i in territory j for each gender, are displayed in table 5.20
Across all symbollterritory combinations, the cross-subsidy charge for each risk type is less than $18 in absolute value and is insignifi- cantly different from zero. These findings indicate that insurers do not offer loss-making contracts to high risks, and lead us to reject predictions of an equilibrium configuration with signaling and cross- subsidization. Combining these results with our evidence on market signaling, we conclude that the collision insurance market under study attains an adverse selection equilibrium with signaling but with- out the cross-subsidization of high risks by low risks.
l9 When we let commission rates vary over the interval (0, l), the predicted values of X + k2 change; however, the cross-subsidy values, k!T, are not much affected.
20 TO carry out the calculations using the estimated eq. (6),we set MR = 1, A = 3, and PERAGE = 44.48. The average age of automobiles for the full sample is 3.58 years.
High Medium Low
NOTE.-t-statistics are in parentheses
-3.73 (-,193) -2.89 (-,142) -4.37 (-.185) -4.17 (-,169)
NOTE.-1-statistics are in parentheses
In this paper we present empirical verification that equilibrium in the market for automobile collision insurance entails adverse selection and market signaling without cross-subsidization. The data reveal in- dividuals of different risk type self-selecting through the purchase of incentive-compatible, unsubsidized collision insurance coverage of- fered by the insurer, with lower risks being compensated for selecting higher deductibles by paying a lower average premium for their cov- erage.
Our investigation provides empirical support for the adverse selec- tion theories based on price-quantity competition developed by Roth- schild and Stiglitz (1976), Riley (1979), and Cho and Kreps (1987), while providing evidence inconsistent with the linear-pricing equilib- rium considered by Arrow (1970), Pauly (1974), and Schmalensee (1984), which is based on pure price competition. The documented equilibrium configuration of contracts could be the Nash equilibrium described by Rothschild and Stiglitz (1976), if the fixed fee per policy is high enough relative to self-insurance alternatives, as suggested by Riley (1983, 1985), or if the screening practiced by insurers yields a sufficiently high proportion of high-risk types in each risk category. Finally, since we do not find evidence of pooling or of separation with cross-subsidization, our results are consistent with, but do not confirm, the theories of price-quantity competition developed by Mi- yazaki (1977), Wilson (1977), Grossman (1979), and Hellwig (1987).
Evidence of adverse selection with market signaling is consistent with the conclusions reached by Crocker and Snow (1985a, 1985b, 1986) and Bond and Crocker (1991), which indicate that ex ante screening of insurance applicants into risk categories and interim self-sorting by applicants according to risk type are characteristic of efficient contracting in insurance markets with adverse selection. Cross-subsidization of high-risk by low-risk types is necessary for ef- ficiency only if the lower risks are sufficiently prevalent in the popula- tion. Thus the empirical results presented in this paper are consistent with efficient contracting, provided that insurers use categories in which higher risks predominate.
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