Stiglitz 1976

Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information Michael Rothschild; Joseph Stiglitz The Quarterly Journal of Economics, Vol. 90, No. 4. (Nov., 1976), pp. 629-649. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28197611%2990%3A4%3C629%3AEICIMA%3E2.0.CO%3B2-N The Quarterly Journal of Economics is currently published by The MIT Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/mitpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Sun Dec 2 15:37:31 2007 EQUILIBRIUM IN COMPETITIVE INSURANCE MARKETS: AN ESSAY ON THE ECONOMICS OF IMPERFECT INFORMATION* MICHAELROTHSCHILDAND JOSEPHSTIGLITZ Introduction, 629.-I. The basic model, 630.-11. Robustness, 638-111. Conclusion, 648. Economic theorists traditionally banish discussions of infor- mation to footnotes. Serious consideration of costs of communication, imperfect knowledge, and the like would, it is believed, complicate without informing. This paper, which analyzes competitive markets in which the characteristics of the commodities exchanged are not fully known to a t least one of the parties to the transaction, suggests that this comforting myth is false. Some of the most important con- clusions of economic theory are not robust to considerations of im- perfect information. We are able to show that not only may a competitive equilibrium not exist, but when equilibria do exist, they may have strange prop- erties. In the insurance market, upon which we focus much of our discussion, sales offers, a t least those that survive the competitive process, do not specify a price a t which customers can buy all the in- surance. they want, but instead consist of both a price and a quan- tity-a particular amount of insurance that the individual can buy at that price. Furthermore, if individuals were willing or able to reveal their information, everybody could be made better off. By their very being, high-risk individuals cause an externality: the low-risk indi- viduals are worse off than they would be in the absence of the high-risk individuals. However, the high-risk individuals are no better off than they would be in the absence of the low-risk individuals. These points are made in the next section by analysis of a simple model of a competitive insurance market. We believe that the lessons gleaned from our highly stylized model are of general interest, and attempt to establish this by showing in Section I1 that our model is robust and by hinting (space constraints prevent more) in the con- clusion that our analysis applies to many other situations. * This work was supported by National Science Foundation Grants SOC 74-22182 a t the Institute for Mathematical Studies in the Social Sciences, Stanford University and SOC 73-05510 a t Princeton University. The authors are indebted to Steve Salop, Frank Hahn, and Charles Wilson for helpful comments, and to the participants in the seminars a t the several universities a t which these ideas were presented. 1976 by the President and Fellows of Harvard College. Published by John Wiley & Sons. Inc. 630 QUARTERLY JOURNAL OF ECONOMICS Most of our argument can be made by analysis of a very simple example. Consider an individual who will have an income of size W if he is lucky enough to avoid accident. In the event an accident occurs, his income will be only W - d. The individual can insure himself against this accident by paying to an insurance company a premium al, in return for which he will be paid &:! if an accident occurs. Without insurance his income in the two states, "accident," "no accident," was (W, W - d); with insurance it is now (W - al, W - d + a2), where a:! = &2 - al. The vector a = (al, a2)completely describes the insurance c0ntract.l 1.1 Demand for Insurance Contracts On an insurance market, insurance contracts (the a's) are traded. To describe how the market works, it is necessary to describe the supply and demand functions of the participants in the market. There are only two kinds of participants, individuals who buy insurance and companies that sell it. Determining individual demand for insurance contracts is straightforward. An individual purchases an insurance contract so as to alter his pattern of income across states of nature. Let W1 denote his income if there is no accident and W2 his income if an accident occurs; the expected utility theorem states that under relatively mild assumptions his preferences for income in these two states of nature are described by a function of the form, where U( ) represents the utility of money income2 and p the probability of an accident. Individual demands may be derived from (I). A contract a is worth V(p, a)= P(p, W - al, W - d + a:!).From 1.Actual insurance contracts are more complicated because a single contract will offer coverage against many potential losses. A formal generalization of the scheme above to cover this case is straightforward. Suppose that an individual will, in the ab- sence of insurance, have an income of W, if state i occurs. An insurance contract is simply an n-tuple (ul,. . . , a,) whose i-th coordinate describes the net payment of the individual to the insurance company if state i occurs. We confine our discussion to the simple case mentioned in the text, although it could be trivially extended to this more complicated case. Many insurance contracts are not as complicated as the n-tuples described above-Blue Cross schedules listing maximum payments for specific illnesses and operations are an isolated example-but are instead resolvable into a fixed premium and a payment schedule that is in general a simple function of the size of the loss such as F(L )= Max [O, c(L-D)], where c X 100% is the co-insurance rate and D is the de- ductible. With such a contract when a loss occurs, determining its size is often a serious problem. In other words, finding out exactly what state of the world has occurred is not always easy. We ignore these problems. A large literature analyzes optimal insurance contracts. See, for example, Arrow (1971) and Borch (1968). 2. We assume that preferences are not state-dependent. THE ECONOMICS OF IMPERFECT INFORMATION 631 all the contracts the individual is offered, he chooses the one that maximiies V(p, a) . Since he always has the option of buying no in- surance, an individual will purchase a contract a only if V(p, a ) 2 V(p, 0 ) = Q(p, W,W - d). We assume that persons are identical in all respects save their probability of having an accident and that they are risk-averse (U" < 0);thus V(p, a ) is quasi-concave. 1.2Supply of Insurance Contracts I t is less straightforward to describe how insurance companies decide which contracts they should offer for sale and to which people. The return from an insurance contract is a random variable. We as- sume that companies are risk-neutral, that they are concerned only with expected profits, so that contract a when sold to an individual who has a probability of incurring an accident of p , is worth Even if firms are not expected profit maximizers, on a well-organized competitive market they are likely to behave as if they maximized Insurance companies have financial resources such that they are willing and able to sell any number of contracts that they think will make an expected p r ~ f i t . ~ The market is competitive in that there is free entry. Together these assumptions guarantee that any contract that is demanded and that is expected to be profitable will be sup- plied. 3. Since the theory of the firm behavior under uncertainty is one of the more unsettled areas of economic theory, we cannot look to it for the sort of support of any assumption we might make, which the large body of literature devoted to the expected utility theorem provides for equation (1)above. Nonetheless, two arguments (and the absence of a remotely as attractive distinguishable alternative) justify (2): the first is the rather vaguely supported but widely held proposition that companies owned by stockholders who themselves hold diversified portfolios ought to maximize their ex- pected profits; management that does not follow this policy will be displaced. The second supposes that insurance companies are held by a large number of small share- holders each of whom receives a small share of the firm's profits. If the risks insured against are independent or otherwise diversifiable, then the law of large numbers guarantees that each shareholder's return will be approximately constant and any in- dividual insurance contract contributes to his profits only through its expected value. In this case stockholders' interests will be well served if, and only if, management maximizes expected profits. A variant of the second argument is obtained by considering the case in which shareholders and policyholders are the same people, or in more familiar terms, when the insurance company is a mutual company. In this case the insurance company is just a mechanism for risk pooling. Under conditions where diversification is possible, each contract's contribution to the company's dividend (or loss) is proportional to its expected value. 4. The same kinds of arguments used to justify @-in particular the appeal to the law of large numbers-can be used to justify this assumption. Weaker conditions than independence will suffice. See Revesz (1960), p. 190, for a theorem that states roughly that, if insurance contracts can be arranged in space so that even though con- 632 QUARTERLY JOURNAL OF ECONOMICS 1.3 Information about Accident Probabilities We have not so far discussed how customers and companies come to know or estimate the parameter p, which plays such a crucial role in the valuation formulae (1)and (2). We make the bald assumption that individuals know their accident probabilities, while companies do not. Since insurance purchasers are identical in all respects save their propensity to have accidents, the force of this assumption is that companies cannot discriminate among their potential customers on the basis of their characteristics. This assumption is defended and modified in subsection 11.1. A firm may use its customers' market behavior to make infer- ences about their accident probabilities. Other things equal, those with high accident probabilities will demand more insurance than those who are less accident-prone. Although possibly accurate, this is not a profitable way of finding out about customer characteristics. In- surance companies want to know their customers' characteristics in order to decide on what terms they should offer to let them buy in- surance. Information that accrues after purchase may be used only to lock the barn after the horse has been stolen. I t is often possible to force customers to make market choices in such a way that they both reveal their characteristics and make the choices the firm would have wanted them to make had their charac- teristics been publicly known. In their contribution to this symposium, Salop and Salop call a market device with these characteristics a self-selection mechanism. Analysis of the functioning of self-selection mechanisms on competitive markets is a major focus of this paper. 1.4 Definition of Equilibrium We assume that customers can buy only one insurance contract. This is an objectionable assumption. I t implies, in effect, that the seller of insurance specifies both the prices and quantities of insurance purchased. In most competitive markets, sellers determine only price and have no control over the amount their customers buy. Nonethe- less, we believe that what we call price and quantity competition is more appropriate for our model of the insurance market than tradi- tracts that are close to one another are not independent, those that are far apart are approximately independent, then the average return from all contracts is equal to its expected value with probability one. Thus, an insurance company that holds a large number of health policies should be risk-neutral, even though the fact that propinquity carries illness implies that not all insured risks are independent. Some risks that cannot be diversified; i.e., the risk of nuclear war (or of a flood or a plague) cannot be spread by appeal to the law of large numbers. Our model applies to diversifiable risks. This class of risks is considerably larger than the independent ones. THE ECONOMICS OF IMPERFECT INFORMATION 633 tional price competition. We defend this proposition a t length in subsectibn 11.2 below. Equilibrium in a competitive insurance market is a set of con- tracts such that, when customers choose contracts to maximize ex- pected utility, (i) no contract in the equilibrium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a nonnegative profit. This notion of equilibrium is of the Cournot-Nash type; each firm assumes that the contracts its competitors offer are independent of its own actions. 1.5Equilibrium with Identical Customers Only when customers have different accident probabilities, will insurance companies have imperfect information. We examine this case below. To illustrate our, mainly graphical, procedure, we first analyze the equilibrium of a competitive insurance market with identical customers." In Figure I the horizontal and vertical axes represent income in 5. The analysis is identical if individuals have different p's, but companies know the accident probabilities of their customers. The market splits into several sub- markets-one for each different p represented. Each submarket has the equilibrium described here. 634 QUARTERLY JOURNAL OF ECONOMICS the states: no accident, accident, respectively. The point E with coordinates (wl, w2)is the typical customer's uninsured state. In- difference curves are level sets of the function of equation (1). Pur- chasing the insurance policy a = (al,a2)moves the individual from E to the point ( ~ 1 al, w 2 + a2).- Free entry and perfect competition will ensure that policies bought in competitive equilibrium make zero expected profits, so that if a is purchased, The set of all policies that break even is given analytically by (3) and diagrammatically by the line E F in Figure I, which is sometimes re- ferred to as the fair-odds line. The equilibrium policy oi*maximizes the individual's (expected) utility and just breaks even. Purchasing a* locates the customer at the tangency of the indifference curve with the fair-odds line. a* satisfies the two conditions of equilibrium: (i) it breaks even; (ii) selling any contract preferred to it will bring in- surance companies expected losses. Since customers are risk-averse, the point a* is located at the intersection of the 45"-line (representing equal income in both states of nature) and the fair-odds line. In equilibrium each customer buys complete insurance at actuarial odds. To see this, observe that the slope of the fair-odds line is equal to the ratio of the probability of not having an accident to the probability of having an accident ((1- p)lp), while the slope of the indifference curve (the marginal rate of substitution between income in the state no accident to income in the state accident) is [U'(W1) (1-p)]l[U'(W2)p], which, when income in the two states is equal, is (1 - p)lp, independent of U. 1.6Imperfect Information: Equilibrium with Two Classes of Customers Suppose that the market consists of two kinds of customers: low-risk individuals with accident probability pL, and high-risk in- dividuals with accident probability pH >pL. The fraction of high-risk customers is A, so the average accident probability is p = ApH + (1 - X)pL. This market can have only two kinds of equilibria: pooling equilibria in which both groups buy the same contract, and separating equilibria in which different types purchase different contracts. A simple argument establishes that there cannot be a pooling equilibrium. The point E in Figure I1 is again the initial endowment of all customers. Suppose that a is a pooling equilibrium and consider T@,a).If T@, a)< 0, then firms offering a lose money, contradicting THE ECONOMICS OF IMPERFECT INFORMATION 635 the definition of equilibrium. If ~ ( p , a ) > 0, then there is a contract that offers slightly more consumption in each state of nature, which still will make a profit when all individuals buy it. All will prefer this contract to a , so a cannot be an equilibrium. Thus, ~ ( p , a ) = 0, and a lies on the market odds line EF (with slope (1 -p)lp) . I t follows from (1) that a t a the slope of the high-risk indifference curve through a , UH, is (p L/l-p L, (1 -p H/pH, times the slope of UL, the low-risk indifference curve through a. In this figure uHis a broken line, and UL a solid line. The curves intersect a t a ; thus there is a contract, /3 in Figure 11, near a , which low-risk types prefer to a . The high risk prefer a to P. Since P is near a , it makes a profit when the less risky buy it, ( r (pL, p) N r(pL, a ) > r (p , a ) = 0). The exis- tence of p contradicts the second part of the definition of equilibrium; a cannot be an equilibrium. If there is an equilibrium, each type must purchase a separate contract. Arguments, which are, we hope, by now familiar, demon- strate that each contract in the equilibrium set makes zero profits. In Figure I11 the low-risk contract lies on line EL (with slope (1 - pL)lpL), and the high-risk contract on line EH (with slope (1 - pH)lpH). AS was shown in the previous subsection, the contract on EH most preferred by high-risk customers gives complete insurance. QUARTERLY JOURNAL OF ECONOMICS This is aHin Figure 111; it must be part of any equilibrium. Low-risk customers would, of all contracts on EL, most prefer contract P which, like an, provides complete insurance. However, 0 offers more con- sumption in each state than aH,and high-risk types will prefer it to aH.If and aH are marketed, both high- and low-risk types will purchase p. The nature of imperfect information in this model is that insurance companies are unable to distinguish among their customers. All who demand P must be sold P. Profits will be negative; (aH,P ) is not an equilibrium set of contracts. An equilibrium contract for low-risk types must not be more attractive to high-risk types than aH;it must lie on the southeast side of UH,the high-risk indifference curve through aH.We leave it to the reader to demonstrate that of al