choice of purchasing arrangements in insurance markets

Journal of Risk and Uncertainty, 2:405-414 (1989) © 1989 Kluwer Academic Publishers

Choice of Purchasing Arrangements in Insurance Markets

KEVIN D. COTI'ER Department of Economics, Wayne State University, Detroit, MI 48202

GAIL A. JENSEN* Institute of Gerontology and Department of Economics, Wayne State University, Detroit, MI 48202

Key words: insurance purchasing, risk pooling, competitive equilibrium, loss probabilities

Abstract

A simple dynamic model helps explain why risk-pooling purchasing arrangements evolved for health, disability, and term life insurance but not for property, automobile, or homeowners' insurance, and why whole-life policies typify life insurance purchased on an individual basis. We show that risk- pooling purchases facilitate insurance against unpredictable changes in one's risk type, but such contracts prevail in competitive equilibrium only when the loss probabilities increase with age, as they do for health, disability, and life insurance. In contrast, when the loss probability declines with age (as it does for automobile insurance), then competitive equilibrium entails separating insurance contracts.

Most health insurance, as well as a substantial portion of life and disability insurance, is purchased on a risk-pooling basis. Of the 83% of nonelderly persons with private health insurance in the United States, 89% have employment-related group coverage (Employee Benefit Research Institute, 1987). Approximately 43% of all life insurance and 70% of all disability income insurance is provided through group contracts as well (American Council of Life Insurance, 1986; Health In- surance Association of America, 1985). Virtually all group contracts pool risks, in which all members of the group have identical premiums and benefits regardless of their loss experience or risk factors. Individually purchased life insurance is usually either whole-life, renewable term, or extended-period term, all of which guarantee coverage over an extended period with no change in premium other than for age (American Council of Life Insurance, 1986). This contrasts with the purchase of automobile, personal property, and personal liability insurance. Con-

*We thank Roger Feldman, James Peck, Richard Rogerson, and William White for useful com- ments. They are not responsible for any errors that remain. Partial research support was provided by an Alfred P. Sloan grant from the Economics Department, University of Minnesota.

406 KEVlN CO'ITER AND GAIL JENSEN

tracts in these lines are typically short-term (one year or less), renewal is rarely guaranteed, and premiums are based on loss experience.

The pattern that emerges is that long-term pooling (group and individual long- term) contracts are offered when the average probability of a loss increases with the age of the insured. This is the case for health, life, and disability income insurance. Automobile insurance, however, entails a loss probability that generally decreases over an individual's lifetime, 1 and other personal insurance does not involve risks that are correlated with age. One possible reason for these differences in purchasing arrangements is that the classification risk of future uncertain changes in one's risk type provides an incentive for long-term risk-pooling contracts. The ability of risk-pooling group contracts to insure against classification risk was recognized by Arrow (1963) and later by Pauly (1970), who noted (p. 412) that "because the group premium is uniform for all members, each individual pays a higher [than experience-rated] premium when their risk type is low in ex- change for the opportunity to pay a lower premium when their risk type is high." Such contracts are feasible, however, only when the loss probabilities increase with age. When loss probabilities decrease with age, it is generally not possible to write a long-term pooling contract that discourages older individuals with a favorable loss history from leaving the contract in lieu of coverage on more favorable terms. Consequently, sequential short-term contracts arise as a second-best arrangement when neither group nor individual long-term pooling contracts are feasible. Property and liability insurance do not involve loss probabilities related to age, but a lack of classification risk in these lines of insurance decreases the incentive for long-term arrangements.

In this article, we present a simple dynamic insurance model that formalizes this explanation. For an environment with symmetric but incomplete information on the future risk type of each consumer, we derive a correspondence between the life cycle properties of the loss probabilities and the equilibrium premiums and indemnities over the life cycle. The development of different purchasing mechanisms for different lines of insurance, both in the United States and other coun- tries, can be explained by appealing to our model. The reasons why group purchases were linked historically to employment, first through workers themselves and later through employer fringe-benefit plans, are also explored, though the model does not include any mechanisms for group formations, and there is no employment.

Other studies of multiperiod contracting in the presence of classification risk in- elude Harris and Holmstrom (1982), Holmstrom (1983), and Palfrey and Spatt (1985). As in Harris and Holmstrom (1982) and Palfrey and Spatt (1985), our model is of overlapping generations in which all parties have common but imperfect information about consumer risk types, but learn over time by conditioning current beliefs on past events. Adverse selection is not present, and credit markets are ruled out. Following Harris and Holmstrom (1982) and Holmstrom (1983), we assume that risk-neutral insurers (employers in their models) must honor long- term contracts, but consumers (workers in their models) need not. A consumer can walk away from a contract at any time if a better insurance arrangement presents

CHOICE OF PURCHASING ARRANGEMENTS IN INSURANCE MARKETS 407

itself. However, our model addresses two aspects of equilibrium not considered previously. First, in our model, long-term risk-pooling contracts compete directly with separating short-term contracts. Other studies have not explored pooling as a distinct phenomenon. Second, we explore the role of various life cycle patterns of the loss distribution in determining the structure of contracts. Others have assumed either that an individual's risk (productivity) class is the same in each period (Holmstrom, 1983; Palfrey and Spatt, 1985), or that assignment to a risk class is random (Harris and Holmstrom, 1982).

The reasons usually given for employer-based group purchases of health, term life, and disability insurance are the favorable tax treatment of premiums paid by employers, economies of scale in group purchases, and the avoidance of adverse selection problems that could occur with individual purchases. Employer-paid premiums for these three coverages are exempt from employees' federal and state personal income tax, social security tax for both employers and employees, cor- porate profit tax, and other payroll taxes for employers. The premiums for other types of insurance, even if paid by the employer, are not tax-exempt. The average annual tax savings on health insurance alone, for those employees with health insurance, was estimated at $499 in 1983 (Taylor and Wilensky, 1983). Administra- tive cost savings from group insurance are also considerable. Underwriting costs in health insurance, for example, compose about 40% of premiums for individual policies compared to 5%-10% for large group policies. Group arrangements, in which persons of different risk types buy the same contract, also facilitate risk pooling in the insurance market, which could never occur in an individual purchase market (Rothschild and Stiglitz, 1976). The employer can act as a sen- tinel, preventing other insurers from offering any contract that would attract lower risks away from the group plan. The presence of significant transactions costs to job switching and finn-specific human capital help prevent risk types from segregating themselves across finns to create a separating equilibrium at the finn level.

None of these arguments, however, explains why insurance for health, life, and disability income historically evolved along group-purchasing lines while personal property and casualty insurance did not. Group economies of scale and the avoidance of adverse selection problems would accrue with any coverage purchased on a group basis. Moreover, the Internal Revenue Service (IRS) ruling on the treatment of employer-paid life insurance occurred in 1920 and the ruling on health and disability insurance occurred in 1943. Both occurred after group purchasing arrangements began to emerge in the lines of insurance to which the rulings pertained. While tax treatment may explain why many firms currently offer health, disability and life insurance coverage as fringe benefits, it cannot explain why group and individual long-term policies evolved for these lines historically. During the nineteenth century in the United States and Europe, it was not uncom- mon for wage earners to belong to a fraternal or friendly society providing group insurance that covered lost income due to illness or death. 2 By 1940, most health insurance in the United States was provided through employee groups, 3 though the proportion of the population insured was smaller than it is now. In Europe, the

408 KEVIN cOTrER AND GAlL JENSEN

friendly societies, which were widespread at the time, were eventually either dis- placed by or merged into the national social insurance programs.

• I. The model

Consider an overlapping generations models with a single consumption good, identical consumers, and an insurance market. In each period t a new generation of N consumers is born. Each consumer lives for two periods. Consumer preferen- ces are given by a single von Neumann-Morgenstern utility function U for the consumption good, which is strictly increasing, strictly concave, and twice dif- ferentiable. In each period the consumer receives a constant endowment of C units of the consumption good and is exposed to a possible loss of X units. The losses of different individuals are independent. A loss when young occurs with probability Py. Each consumer is also exposed to classification risk through the realization of one of J events E', E2, . . . , E ~ when young that provides information about the probability of a loss when old. These events include whether a loss occurred when young as well as any other information that affects the probability of a loss when old, such as occupation and personal lifestyle. Since the same information is observable by all, there is no asymmetric information about whether a loss occurred and there is no adverse selection. Let qJ be the prior probability of event EL An old consumer for whom E j was observed is said to be of risk typej. Let the loss pro- ability of a type j consumer be PJo .4 Without loss of generality, assume P~ </~o < . . .

Assumptions regarding the insurance market are standard. Insurers are risk- neutral and face no underwriting or administrative costs. Consequently, they offer any policy to an observable risk class of consumers (i.e., young, and each old risk type) that those consumers desire, and that at least breaks even on average given the actions of other insurers. Insurers can sell contracts directly to groups as well, though in this model there are no incentives to organize a group for the purchase of purchasing insurance.

Each young person chooses, with perfect foresight about premiums and indemnities, a contingency plan for insurance purchases to maximize ex ante expected utility. This plan consists of a choice of insurance when young as well as choices when old contingent on the events E 1 . . . . , E J. Borrowing and saving are not per- mitted. An equilibrium is defined to be a set of insurance contracts, each of which breaks even over the set of consumers who purchase it, such that no consumer may improve his or her expected utility by choosing a different contract available at that time, and such that there exist no opportunities for insurers to earn a profit on average given the contracts offered by other insurers. In this article, we restrict at- tention to stationary equilibria in which the contracts do not depend on t. s

The consumers lifetime choice is a vector L = (LyN~Ly~,LoN~LoI' ' . . . . #LoN~LoI)~: ~ where L w is the premium paid when young, Lvi is the indemnity net of premium paid when young if a loss occurs, L~N is the premium paid when old if the consumer turns out to be of type j, and LJoi is the indemnity net of premium in that

C H O I C E O F P U R C H A S I N G A R R A N G E M E N T S IN INSURANCE MARKETS 4 0 9

case. The arrangement L may be made either through a sequence of short-term contracts, a single individual lifetime contract, or a group contract. In a sequential short-term contract, the individual purchases a single-period contract in each period, with the contract purchased when old depending on his or her risk type. In a lifetime contract, the consumer purchases a single contract L that provides the specified premiums and indemnities that may depend on both age and risk type. Clearly, any sequence of short-term contracts can be replicated by a single lifetime contract with the same contingencies. 6 Unlike a sequence of short-term contracts, however, a lifetime contract can also insure against classification risk by setting premiums and indemnities to be independent of the consumer's risk type when old. Insurers are required to honor long-term contracts, but an old consumer may choose to leave the plan if a better short-term contract is available to his or her risk type.

In a group contract, a single contract is sold to a group of consumers. An old consumer may participate only if he or she joined the plan when young, but as with an individual long-term contract, the consumer may choose to leave the plan if a better short-term contract is available. Since all consumers are identical when young, all choices of L in equilibrium must yield the same lifetime ex ante utility. Since utility is strictly concave, this implies that all young consumers make the same lifetime choice of L. In particular, any group contract includes all consumers who are alive at the time. Since there is no population growth and the individual contract break-even constraint uses an interest rate of zero, any group contract can be duplicated by a single individual long-term contract that is offered to each young consumer in the group.

Therefore both sequential short-term and group contracts are special cases of individual lifetime contracts in this model, so the choice of purchasing arrangement reduces to the choice of features for the individual lifetime contract. Since all consumers are identical and purchase the same contract when young, the equilibrium arrangement is one that maximizes ex ante lifetime utility over all contracts that break even and that do not induce old consumers to leave and join another plan. That is, ~ w ~ o N ~ o i , t r ~ r l 'rl .. • J~gN~Lgi) must be chosen to solve the following problem:

Maximize (1 - P y ) U ( C - LyN) + PyU(C - X + L,a)

J

+ ~ q J[(1 - PJo)U(C - LioN) + PJoU(C - X + L J o i ) (1) j = l

subject to

J J

(1 -- Py)LyN + Y. qY(1 - PYo)LJoN = PyLyI + ~. qJPYoL~I, j =1 j =1

(2)

410 KEVlN COTYER AND GAIL JENSEN

(1 - P:o)U(C - L:oN) + P:oU(C - X + LYoI)

> U(C - P~>k r) j = 1 , . . . ,J. (3)

Equation (1) is the expected lifetime utility of a consumer from a given contract L. Equation (2) requires that, on average, L breaks even over the consumer's l i fe t ime, 7 or equivalently, that the insurance company breaks even in each period. Equation (3) is a set of self-selection constraints. In equilibrium, the terms of the contract must be such that an old consumer of typej will not desire to leave the contract in favor of a short-term contract that is actuarially fair for his or her risk class. The best such short-term contract would yield a utility of U(C - PJoX), which is in equation (3).

In the absence of the self-selection constraint (3), the maximization problem also describes the first-best Pareto efficient outcome. Let P = [Py + ~]=l qJPJo]/2 be the average probability of a loss for the population. The solution to equations (1) and (2) is L * = L~* = PX, L~ = L~* = (1 - P)X) As in Rotschild and Stiglitz (1976) and Palfrey and Spatt (1985), optimal coverage would entail full coverage at fair odds, which in this article includes coverage against classification risk.

The first-order conditions for an interior solution to equations (1)-(3) are

U ' ( C - L w ) = U'(C - X + Lw) = ~o, (4)

V ' ( C -- LioN) = U ' ( C -- X + L / o i ) = Xo

1 + ~//qJ j = 1 . . . . . J, (5)

where k0 and kJ are Lagrange multipliers associated with equations (2) and (3). Equation (4) implies L ~ + L,a = X, and equation (5) implies Lion + LJoi = X, so all consumers receive full coverage in both periods. Therefore equation (2) reduces to

1~I _jL j I L w + j=l ~ or~ = PYX + Zj=l qJPJoX, and equation (3) reduces to LYoN < PJoX, so L ~ > PvX. If ~/ > 0, then equation (3) is binding, so L~N = PJoX, and using equations (4) and (5), L~N < Lvs. I f ~ / = 0, thenL~N = Lys from equation'(5). Com- bining these statements yields LJos = min{PJoX#LyN}.

The equilibrium arrangement takes a simple form. All consumers receive full coverage in both periods. Young people pay a uniform premium L ~ , which is also paid by old people except for those whose risk class is sufficiently favorable that they would otherwise defect to a short-term contract at fair odds. The contract of- fers those consumers an actuarially fair premium. Thus experience rating is one- sided; it can only cause an individual's premium to fall. 9

Let H C { 1 . . . . . J} be the set of pooled old risk types, i.e., thosej for which Lion = Lw. The different types of arrangements can be characterized by H. In a pooled equilibrium, H = {1, J}, and LyN = (PY + Y-,J qJPJo)X/2 = L,~, so first-best • " ' , j = l

Pareto efficiency is achieved. In a separating equilibrium, H = ~ with LyN = PyX and LJos = P ~ . Any other H is a partial pooling arrangement. The main result of this article gives conditions for which each type of contract occurs. For notational convenience, we use the convention/~o = 0, a n d j and k refer to risk types.


Theorem. In an equilibrium, exactly one of the following cases occurs:

PJo < Py if and only if complete separation occurs, (s)

J P/o-t< ev + ~. qk(e~ _ P/o) < Pb

k =j

if and only if risk t,)pes k with k

> j are pooled. (G j)

Proof." Suppose H = ~l. Then PJoX = L ~ for each j, so L ~ = PvX. Since Lion < Lw, we must have P{yg < PvN for all j, proving (S). If (S) holds, then Lion < P~7 < PyX < LyN for all j, so L2oN = PYot,X, and separation occurs.

Suppose H 4= 9[. Then [1 + Y.kenqk]LyN + Y~g~nqkP~oX = PyX + ~Jk=tqkl#oX, SO

LVN = [ ( P y + k~__t~qkPk)/(l+ k~_.~qk)]x. (6)

For k C H, L~N < PkoX, while for k' ~ H, Lkot~ = P~f. Therefore,

(7)

so k' < k. Therefore, i f j C H, all h igher j are also in H, so H = {jj+ 1 , . . . , J} for somej. Then

p j o I < ( e l dr" ~ q~:p~l/(1 + ~. qk)<PJo, k C H / \ k ~ l ' l

(8)

and rearranging terms yields condition (G J), completing the proof.

Complete risk pooling is equivalent to condition (Gt), which can be written P < P~o. In this case, all old people are riskier than average. As noted earlier, health and life insurance fall into this category, and the theorem correctly predicts that these lines will be characterized by long-term or group contracts, since all old consumers are of sufficiently high risk that they could not obtain better insurance terms on their own. The condition (S) for separating contracts requires that all old people be of lower risk than young people. This is essentially the case for automobile insurance, where short-term risk-rated contracts indeed prevail. The conditions (G y) for contracts with partial pooling cover all intermediate cases. In summary, a set of risk classes H = {/,j + I , . . . , J} will pool with all young consumers if and only if each old member of the pool has a loss probability greater than the average of all consumers (including young) in the pool. In this case, the old consumers separate into those of sufficiently high risk that they are better off staying in the pool, which consists of H, and those of suffiicently low risk that they are better off leaving the pool.

412 KEVlN COTI 'ER AND GAIL JENSEN

Two extensions are worth mentioning. The presence of a growing population would sharpen the effects of life cycle loss probabilities. If loss probabilities increase with age, then there would be more young to subsidize the old, so pooling contracts become easier to achieve. If the reverse holds, then pooling becomes har- der to achieve, since there is a greater proportion of high-risk young. In addition, the ability to save in a separate credit market would not affect the results reported in this article, since any saving could be duplicated through a lifetime insurance contract. However, borrowing would allow young consumers to shift extra consumption into subsidizing the old in a long-term arrangement to prevent low risks from defecting. Therefore, the assumption that borrowing is prohibited is crucial in this article. In practice, however, consumers face more obstacles to borrowing than saving, so the prohibition against borrowing is not unreasonable.

2. Conclusions

The above model only describes the types of purchasing arrangements that occur in equilibrium, and says nothing about the institutions that would implement a particular equilibrium. In particular, it does not explain why group contracts for health, life, and disability income insurance were traditionally provided through employment-based groups, even before there was a tax subsidy for employer spon- sorship. One possible explanation is that relatively healthy persons could easily exclude uninsurables, such as the disabled and the chronically ill, from becoming members of the poolJ ° By tying coverage to employment, better risks could segregate themselves from those who would have required a subsidy. Tax advan- tages also explain why employers rather than other infinite-lived employment- related groups, such as unions and professional associations, are now the main brokers for coverage. An extension of the model, in which workers receive income through inelastically supplied labor, would allow for employment and the provi- sion of insurance fringe benefits by firms. If workers are mobile among firms when young but immobile when old due to firm-specific human capital, then employer- sponsored purchasing arrangements can implement group equilibrium. The im- mobility of old workers reduces self-selection and facilitates pooling.

Notes

1. The rate of traffic accidents decreases with respect to the driver's age up to age 75, then increases for drivers over 75 (Insurance Information Institute, 1987, p. 83).

2. For a detailed discussion of the history of health, fife, and disability insurance, see Isle (1952), Mehr and Cammack (1972), and Stair (1982).

3. Of coverage provided by insurers other than Blue Cross Blue Shield (BCBS) in 1940, 80% was provided through groups (Health Insurance Association of America, 1985). Statistics on the extent of group coverage within BCBS plans are unavailable for the years prior to 1961. In 1961, however, 73% of BCBS subscribers had group coverage (Blue Cross Blue Shield Association, 1986). Data for the earlier years on group-based coverage were not recorded because during that period the BCBS plans practiced community rating whereby the premium was based on the claims experience of the entire community


within the plan's catchment area. However, since the marketing of the plans took place mainly in the personnel offices of firms (Goldman, 1948), it is likely that employment-based coverage was dominant even then.

4. Our assumptions regarding the loss distribution describe health, disability income, and property and casualty coverages better than they describe life insurance. To apply the model to life insurance requires the assumption that individuals have a bequest motive.

5. This replaces the need for a transversality condition in the maximization problem that follows. Otherwise, an insurer could duplicate an existing contract except that in the beginning period in which only young people are insured, their premium is reduced by $1. Then the contract makes a profit in the first period and breaks even in each period thereafter, but only as long as the insurer remains in busi- ness forever. In reality, an employer and insurance regulators would not permit this practice. Our thanks to James Peck for this observation.

6. If moral hazard is present, then short-term purchasing arrangements are no longer a special case of the general lifetime contract. Group and long-term contracts become less attractive, since guaranteed renewal can increase moral hazard. This may explain why property insurance is typically offered on a short-term basis.

7. For simplicity, we assume an interest rate of zero. The generalization to a positive interest rate is straightforward and does not significantly affect the qualitative results.

8. If consumers discount future consumption, then young and old no longer are pooled even in the first-best Pareto efficient outcome. The old pay a higher premium, since young consumers are less will- ing to subsidize the old. The other qualitative features of the model remain unchanged.

9. This type of one-sided experience rating is typical of long-term contracts in which moral hazard is not present. In particular, the long-term wage contract of Harris and Holmstrom (1982) follows a com- parable pattern, in which wages may rise but cannot fall over an indixddual~s lifetime in a long- term contract.

10. The use of inpatient hospital services is known to vary according to employment status. Persons not in the labor force have the highest rates of admission, followed by unemployed persons. Length of hospital stay per admission follow the same pattern. Men in poor physical health are significantly less likely to be employed than men in good health, controlling for other personal characteristics (Benham and Benham, 1982). Also, Berk, Cafferata, and Hagan (1984) report that persons who are limited in the amount or kind of activity they can perform are almost three times more likely than other persons to be admitted to a hosptial and on average stay about four days longer once admitted.

References

American Council of Life Insurance. (1986). LifeInsuranceFactBook, 1986. Washington, DC: American Council of Life Insurance.

Arrow, IC (1963). "Uncertainty and the Welfare Economics of Medical Care," American Economic Review 53, 941-973.

Benham, L. and A. Benham. (1982). "Employment, Earnings, and Psychiatric Diagnosis." In V. Fuchs (ed.), Economic Aspects of Health. Chicago: University of Chicago Press, pp. 203-220.

Berk, M., G. Cafferata, and M. Hagan. (1982). "Persons With Limitations of Activities: Health In- surance, Expenditures and Use of Services." National Medical Care Expenditures Study, Data Preview 19. Washington, DC: U.S. Department of Health and Human Services.

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Goldmann, F. (1948). Voluntary Medical Care Insurance in the United States. New York: Columbia Uni- versity Press.

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