A Model of an Oligopoly in an Insurance Market

The Geneva Papers on Risk and Insurance Theory, 23: 41 48 (1998) c 1998 The Geneva Association A Model of an Oligopoly in an Insurance Market MATTIAS K. POLBORN polborn@lrz.uni-muenchen.de. University of Munich, Seminar für Versicherungswissenschaft, Ludwigstr. 28, 80539 München, Germany Abstract This article analyzes the behavior of an oligopoly of risk-averse insurers that insure many consumers facing identical independent risks; however, the probability of a loss is ex ante not known with certainty. It is shown that there is a continuum of equilibria in the Bertrand game. The most plausible equilibrium can be obtained by requiring that all insurers are content with the number of policies they sell given the equilibrium premium. Key words: insurance, oligopoly, imperfect competition 1. Introduction The theoretical insurance literature is full of models assuming either a competitive or a monopolistic market for insurance. It is therefore quite surprising that there are to my knowledge virtually no theoretical models of oligopolistic interaction in an insurance market. 1 The model of this article is concerned with an oligopoly of risk-averse insurers who compete in premium and do not know the average accident probability in the next period with certainty. The assumption that the accident probability is ex ante uncertain seems quite reasonable. Many accident probabilities change from time to time and in a manner that seems to be not predictable with certainty. The main idea of this article is that an uncertain accident probability introduces a systematic per capita risk that is not reduced by selling more policies. If insurers are risk averse, this effect relaxes price competition. It is shown that in the game where insurers compete in premiums there exists a continuum of symmetric equilibria. The most plausible one has the additional property that (given the premium) each firm is content with the number of policies it sells. An analogy to a marginal cost curve is drawn. Consider an insurer who already has a portfolio of n 1 policies and writes an additional policy. Denote as the marginal cost of the nth policy the amount of money that has to be given to the insurer as premium so that the insurer s expected utility does not change in comparison to the case in which it sells only n 1 policies. We show that marginal costs are constant for an insurer with constant absolute risk aversion if the accident probability is certain 2 and increasing if it is uncertain. In the most plausible equilibrium, the last policy sold has marginal costs equal to the premium received. Since intramarginal consumers pay the same premium but cause only lower costs, firms in this market have an expected utility that is higher than their outside option. The effects of an uncertain accident probability on the premium determination of insurance companies have been analyzed by Hogarth and Kunreuther [1989, 1992]. Their work is

42 MATTIAS K. POLBORN mainly empirical; they show that actuaries in laboratory experiments tend to increase the recommended premium for an insurance policy considerably if the situation described has an ambiguous (rather than a certain) accident probability. 3 It has been shown by Sandmo [1971] that risk aversion and price uncertainty are sufficient to explain the presence of expected profits even in a competitive setting, since expected profits are a necessary and sufficient condition for a positive output. While Sandmo s article looks for the optimal behavior of a firm under exogenous price uncertainty, this article is mainly concerned with a determination of the equilibrium price in an oligopolistic market. Wambach [1996] analyzes the equilibrium in an oligopolistic market under cost uncertainty and Bertrand competition. His model is similar to this one but focuses on another, and as is argued in the following, less plausible equilibrium. In the next section the model is presented. The third section contains an analysis of the model and the main results. The last section concludes. 2. The model There are N consumers and I insurance companies in the economy, where N is large relative to I. Each consumer has the same strictly concave expected utility function 4 and faces the risk to lose l in an accident. The accident probability p is equal for each consumer in the economy, and conditional on p accidents of different consumers are statistically independent. However, p is not known ex ante with certainty but is a random variable distributed on [p, p ] with cumulative density function F(p). All insurers have the same constant absolute risk-aversion utility function v(y) = exp ( ry). Each insurer could have an arbitrary initial certain income, but given constant absolute risk aversion one can show that everything in this article is independent of this initial income, and hence we normalize it to zero; we assume, however, that an insurer is never bankrupt, which will be approximately satisfied if the insurer s wealth is large. Insurance policies (C, )consist of a cover payment to the insured in case of an accident and a premium that has to be paid to the insurance company in any state of the world. We assume that C is given exogenously and concentrate on the determination of the equilibrium premium; this is no restriction since all propositions of this article will be valid for all values of C, so the exact determination of C plays no role. In particular, we can let C be the Pareto optimal level of cover. 5 Insurers compete by quoting a premium for which they are willing to sell policies, so we have Bertrand competition. The firm that quotes the lowest premium sells to all consumers; it is assumed that this insurer cannot refuse to sell to a consumer who wants to buy for the quoted premium. If several insurers quote the same lowest premium, they share the market equally. 3. Analysis of the equilibrium For the moment, we disregard the consumers participation constraint, assume that it is fulfilled, and concentrate on the supply side of this market.

A MODEL OF AN OLIGOPOLY IN AN INSURANCE MARKET 43 As first shown by Freund [1956] for a utility function v(y) = exp( ry) and y normally N(μ, σ 2 ) distributed, expected utility is exp ( rμ + σ 2 r 2 ). (1) 2 Foragivenp,μ = x( pc) and σ 2 = xp(1 p)c 2, and if the number of insureds is large, losses are approximately normally distributed. Ex ante (before we know p), the expected utility is therefore V (x, ) p p exp( rx( pc rp(1 p)c 2 /2)) df(p), (2) where pc rp(1 p)c 2 /2 is premium minus costs, here costs include expected cover payments pc and the subjective costs of taking risks rp(1 p)c 2 /2. Observe that V (x, )is increasing in and concave in x. We define L and H by the following identities: V (N/I, L ) = V (0, L ) (3) V (N/I, H ) = V (N, H ), (4) where L is the premium that makes an insurer indifferent between selling policies to N/I consumers and selling no policies at all; note that, since V (0, )= 1 for all, the second argument ( L ) of V on the right-hand side of (3) does not matter. At the premium H the insurer is indifferent between serving N/I consumers and serving the whole market that is, N consumers. The following two lemmas provide useful information about L, H, and V (x, ). In a symmetric equilibrium, each insurer will sell to N/I consumers. Lemma 1 (besides uniqueness) states that the premium that makes an insurer indifferent between selling to 1/I of the market and leaving the market is lower than the premium at which he is indifferent between selling to 1/I of the market and selling to the whole market. Lemma 2 states that an insurer prefers selling to 1/I of the market to selling to the whole market for all premiums lower than H. The proofs of the lemmas are in the appendix. Lemma 1: L and H are uniquely defined by (3) and (4), respectively. L < H. Lemma 2: V (N/I, ) V (N, )for all H. We can now turn to the first proposition, which deals with the equilibria in this market. Proposition 1: For each, where L H there is a symmetric Nash equilibrium in which all insurers demand a premium and sell to N/I consumers each.

44 MATTIAS K. POLBORN Proof. Lemma 1 shows that the interval [ L ; H ] exists and is nontrivial. We have to show that, starting from the equilibrium described by, no insurer has an incentive to deviate. Suppose insurer i increases his premium i over while all other insurers stick to ; then insurer i loses all customers and receives a utility of V (0, i ) = V (N/I, L ) V (N/I, ), where the equality follows from (3) and the inequality follows from the fact that V ( ) is increasing in and L. Hence increasing the premium over does not give insurer i a higher utility than sticking to the stipulated equilibrium behavior. Suppose insurer i decreases his premium i while all other insurers stick to ; then insurer i has to take over the whole market and receives a utility of V (N, i )<V(N, ) V (N/I, ). Here, the first inequality follows from i < and the second inequality follows from Lemma 2. Hence, decreasing the premium lowers the insurer s utility. Therefore, all insurers quoting is an equilibrium, for all [ L, H ], as claimed. Proposition 1 proves that there is a continuum of equilibria with premiums between L and H. In the L -equilibrium, each insurer receives his outside option utility level; in all other equilibria (with higher premiums) insurers enjoy a rent. Deviating by increasing the premium is not profitable since the insurer then loses all customers and hence its rent. If he decreased his premium, the insurer would take over the whole market, and although each policy is profitable in expectation, more policies also increase the risk. The costs of increased risk outweigh the benefit of increased expected profit for the following intuitive reason: If the accident probability is known, expected profit and risk increase proportionally as the number of policies sold increases, and given the reduced form utility function (1) it is clear that when selling N/I policies for a premium increases expected utility, selling N policies for the same premium is profitable, too. If, however, the accident probability is uncertain, there is a common systematic risk (the risk that the accident probability is high for all individuals as opposed to the individuals idiosyncratic risks of having an accident); this systematic risk is for [ L ; H ] so large as to make it unattractive for each single insurer to take over the whole market. When there are infinitely many equilibria, this raises the question of which one will be played; hence we would like to single out a unique equilibrium as the most plausible one. A first candidate seems the highest-price equilibrium since it is the best equilibrium for insurers. 6 The problem with this equilibrium is that all firms sell less than they want to given the price (although they still refrain to take over the whole market). If insurers have in reality some control over the number of policies they sell apart from premium, it seems reasonable to require that firms are content with the number of policies they sell in equilibrium. It is, however, important that insurers do not have too much control, either. For example, if insurers were free to choose how many policies to sell after the premiums have been quoted, there is no pure strategy equilibrium; see Tirole [1988]. The next proposition proves that there is exactly one equilibrium in the set of equilibria characterized in Proposition 1 in which insurers are content with the number of policies they sell in equilibrium.

A MODEL OF AN OLIGOPOLY IN AN INSURANCE MARKET 45 Proposition 2: There is a unique equilibrium characterized by such that each insurer is content with the number of policies he sells in equilibrium: N I = arg max V (x, ). (5) x Proof. Consider the problem max x V (x, ). The first-order condition is V x = p p r( pc rp(1 p)c 2 /2) exp( rx( pc rp(1 p)c 2 /2)) df = 0. (6) An easy calculation shows that the second-order condition is satisfied. This defines x( ), the optimal x as a function of. Applying the implicit function theorem to (6) yields p dx d = p r exp( rx( pc rp(1 p)c2 /2)) df(p) rxv x. (7) V xx Using the optimality condition V x = 0 and observing that V xx < 0 and that the integrand in the numerator is always positive show that this expression is positive. Hence, is unique. To show that characterizes an equilibrium, we still have to show that L H.At L, the insurer is indifferent between x = 0 and x = N/I, and hence, 0 < x( L )<N/I. By a similar argument, N/I < x( H )<N. Therefore, L < < H. It is useful to interpret the inverse function of x( ) as a marginal cost function: when insurers are content with the number of policies they sell, the premium for the last policy sold exactly compensates the insurer for his obligation to pay cover in case of an accident of this last customer. The slope of this marginal cost curve is positive. Since the last policy is sold for marginal costs and inframarginal policies cause lower costs, a profit is made on inframarginal policies. This argument corresponds to the usual textbook analysis concerning profits in competitive markets if marginal costs are increasing. Note, however, the difference: while the textbook models assume exogenously that marginal costs are increasing, increasing marginal costs are a result in this model. Suppose that there are fixed costs for each insurer; then we get a U-shaped average cost curve. In a market with free entry, the profits derived in this model have to be equal to the fixed costs since otherwise there would be an incentive for entry or exit. When new insurers enter the market, N/I sinks, and hence, the equilibrium premium decreases. It is also clear that the profit each single insurer can make decreases in this case. If the accident probability p is certain, (6) can be satisfied by a finite x only for = pc + rp(1 p)c 2 /2, so we have a horizontal marginal cost curve. In this case, the equilibrium premium is independent of the number of customers per insurer. Insurers receive only their reservation utility in this case. The formal analysis of the model largely ignored the demand side of the market. In Polborn [1997] it is shown that if C is the Pareto optimal cover, the consumers participation

46 MATTIAS K. POLBORN constraint is satisfied in the equilibrium. Though we assumed all consumers to be equal, it is easy to see that admitting heterogeneous consumers would not change much except that the Pareto optimal cover is no longer equal for all consumers; if consumers differ not too much in risk aversion, still all will buy the equilibrium policy. 4. Conclusion The model of this article showed that in insurance oligopolies supernormal profits could be earned and that the transition from monopoly to perfect competition in terms of premiums, profits, and so on may be a smooth one and not the abrupt one suggested by simple Bertrand models, in which two firms yield a fair premium and zero profit for insurers at once. A problem of the approach taken in this article was that we had to assume that insurers have constant absolute risk aversion in order to keep the analysis mathematically tractable. Non-CARA utility functions in general do not yield a closed-form solution for the slope of the marginal cost curve. Moreover, for utility functions with decreasing absolute risk aversion one can construct examples 7 for a certain accident probability that yield increasing marginal cost and other examples where marginal cost decrease. The same is true if the accident probability is uncertain but this systematic risk is small; I conjecture however that, if the systematic risk is sufficiently large, the marginal cost curve should be increasing for all concave utility functions. Although it is possible to interpret the model as analyzing systematic cost uncertainty in an arbitrary industry, the insurance interpretation is the most plausible one. A major difference between insurance and other markets is that in insurance markets sales take place before costs are realized while in most other industries it is the other way round. Therefore, the assumption that the probability of accident is unknown when policies are written is quite satisfactory for insurance markets, but for most other markets cost uncertainty will be resolved before sales take place. This fact makes this model insurance specific. What will happen in the model if insurers have access to risk-neutral reinsurers or a CAPM-world capital market and the risk of p is independent of the systematic risk in the capital market? Then, of course, marginal costs of policies are equal to the fair premium. However, Greenwald and Stiglitz [1990] have shown how asymmetric information between managers and owners can yield a risk-averse behavior of the firm even if the risk of the firm is purely idiosyncratic. Also, the results of Hogarth and Kunreuther [1989, 1992] indicate that actuaries behave in a risk-averse way and are especially averse to situations with an uncertain accident probability; this would not be the case if they could sell all risks that they do not like to risk-neutral organizations for a fair price. A reason may also be that insurers can sell only a part of their risks to reinsurers under standard reinsurance treaties. In this case, we can interpret C in the model as that part of cover that the primary insurer must pay; if C > 0, marginal costs are increasing. A possible reinterpretation of the model in the presence of reinsurance is that the insurers of the model are those parties that ultimately must keep the risk for example, reinsurers or insurers who keep some part of the risk of policies under standard reinsurance treaties. There is also another method to diminish the systematic risk for insurers. Sometimes, insurance contracts determine some profit-sharing arrangement; the insurer quotes a rather

A MODEL OF AN OLIGOPOLY IN AN INSURANCE MARKET 47 large premium and pays out a part of his profit to the policyholders. Effectively, consumers thus share a large part of the insurer s systematic risk. Appendix Lemma 1: L and H are uniquely defined by (3) and (4), respectively. L < H. Proof. V (x, )increasing in implies uniqueness of L satisfying (3). H is uniquely determined since V (N, H ) > V (N/I, H ). Observe that V (N/I, H )>V(0, H ) = V (0, L ) = V (N/I, L ), where the inequality follows from V (x, H ) strictly concave and hence increasing in x at x = N/I, since N/I < arg max x V (x, H ); the first equality follows from the definition of V (x, ) and the second from (3). Now, since V is increasing in, L < H. Lemma 2: g( ) V (N/I, ) V (N, ) 0 for < H. Proof. By definition of H, the lemma holds for = H. Assume to the contrary that the lemma does not hold for some < H. Then, by continuity of g( ), there exists a value such that g( ) = 0 and g( ) < 0 for. But the derivative of g evaluated at is g ( ) = r N I V (N/I, ) + rnv(n, ) < rn[v (N, ) V (N/I, )] = 0, (8) where the inequality sign follows from V ( ) <0and I > 1. Hence, for slightly smaller than, g( ) > 0, a contradiction that proves Lemma 2. Acknowledgments This article is based on a chapter of my Ph.D. thesis (Polborn [1997]). I would like to thank Ekkehard Kessner, Mathias Kifmann, Matthias Messner, Ray Rees, Harris Schlesinger, Achim Wambach, seminar participants at the 1996 Seminar of the European Group of Risk and Insurance Economists held at Hanover, and two anonymous referees for helpful comments on this article. The usual disclaimer applies. Notes 1. An exception is Schlesinger and Schulenburg [1991], who analyze a model with search and switching costs in a horizontally differentiated insurance oligopoly; consumers here have imperfect information about prices, and they view policies from different insurers not as perfect substitutes.

48 MATTIAS K. POLBORN 2. A similar result has been obtained for this case by Schneeweiss [1967]. 3. In most scenarios, risk aversion and expected utility theory in general cannot account for different premiums in the ambiguous and the nonambiguous case; however, Hogarth and Kunreuther [1992] note that selling several insurance policies under ambiguity results in a higher variance of the insurer s wealth and that a risk-averse insurer with a mean-variance utility function requires a higher premium in this case. 4. The assumption that all consumers have the same utility function and the same initial income is by no means essential for the results but simplifies the exposition because the number of policies sold is independent of the equilibrium premium (as long as consumers are willing to buy). 5. For the Pareto optimal cover to be equal for all consumers, they all must have the same utility function. Details are derived in Polborn [1997]. 6. The properties of this equilibrium are analyzed in Wambach [1996]. 7. See Polborn [1997]; the examples are available from the author on request. References FREUND, R. [1956]: The Introduction of Risk into a Programming Model, Econometrica, 24, 253 264. GREENWALD, B., and STIGLITZ, J. [1990]: Asymmetric Information and the New Theory of the Firm: Financial Constraints and Risk Behavior, American Economic Review, Papers and Proceedings, 80, 160 165. HOGARTH, R., and KUNREUTHER, H. [1989]: Risk, Ambiguity, and Insurance, Journal of Risk and Uncertainty, 2, 5 35. HOGARTH, R., and KUNREUTHER, H. [1992]: Pricing Insurance and Warranties: Ambiguity and Correlated Risks, Geneva Papers on Risk and Insurance Theory, 17, 35 60. POLBORN, M. [1997]: Three Essays in Insurance Economics, Ph.D. thesis, University of Munich. SANDMO, A. [1971]: On the Theory of the Competitive Firm Under Price Uncertainty, American Economic Review, 61, 65 73. SCHLESINGER, H., and SCHULENBURG, M.v.d. [1991]: Search Costs, Switching Costs and Product Heterogeneity in an Insurance Market, Journal of Risk and Insurance, 58, 109 119. SCHNEEWEISS, H. [1967]: Entscheidungskriterien bei Risiko, Springer, Berlin. TIROLE, J. [1988]: The Theory of Industrial Organization, MIT Press, Cambridge, MA. WAMBACH, A. [1996]: Oligopoly and Uncertainty, Münchener Wirtschaftswissenschaftliche Beiträge 96-04.