Oligopoly in Insurance Markets

Transcription

1 Oigopoy in Insurance Markets June 3, 2008 Abstract We consider an oigopoistic insurance market with individuas who differ in their degrees of accident probabiities. Insurers compete in coverage and premium. Different to the price setting behavior of firms in Bertrand competition, the market demand for insurance varies aong an undercutting process in premiums because the range of individuas who areinterestedinbuyinginsurancechangeswitheveryreductioninpremium. As a resut, insurers choose to offer a premium according to a we-specified distribution function. With respect to wefare, we compare the case of competition with a monopoistic insurer as we as with a reguated insurance market. We find that under competition industry profits as we as the range of insured individuas decrease compared to the monopoy case. Moreover, a monopoistic insurer offers fu coverage for a ower premium than the insurers in a competitive market woud do. Comparing the monopoy outcome with a wefare-maximizing insurance we show that the premium for coverage is higher under monopoy than in a reguated market and that the range of contracted insurance poicies is smaer under monopoistic insurance than under a wefare-maximizing contract.

2 Introduction For a broad range of insurance sectors competition is characterized by an oigopoistic market structure. In the US market, for exampe, Nissan and Caveny (200) find that some ines of property and iabiity insurance are significanty more concentrated than a number of other industries. In the UK market, the Association of British Insurers documents that the argest ten property insurers have a market share of 85%. And in the Austraian market, the empirica study by Murat et a. (2002) for the genera insurance industry suggests that competition is ess than perfect and that insurers have some extent of market power. Interestingy, in the theoretica iterature on competition in insurance markets amost a artices consider either a monopoy insurer or perfect competition with two or more insurers. Oigopoistic modes, however, in which ony reativey few firms compete and make positive profits are very rare. 2 This is even more surprising because strategic behavior in competition can ony be anayzed in oigopoistic modes. The mode of this artice is concerned with the strategic impications of competition between two insurance companies. They compete by quoting a premium for which they are wiing to pay a coverage to an insured individua if an accident takes pace. Individuas are uniformy distributed over a continuum of types and differ with respect to their risk-attitudes. We appy a oss-averse utiity function for individuas. Insurers face neither adverse seection nor mora hazard probems. We first show that in equiibrium coverage is aways compete, hence insurers compete in the premium per unit of coverage. We then argue that there cannot exist a Nash-equiibrium in pure strategies in which it is optima for an insurer to offer a fixed premium for coverage. The intuition behind this resut is an immediate consequence of the foowing undercutting process in premium offers: Suppose, for exampe, that one of the two insurers woud offer the monopoistic premium. But then the other insurer woud have an incentive to undercut this premium sighty, thus recapturing individuas who want to buy insurance. But then the first insurer woud once again have an incentive to set a premium sighty beow the opponent s premium and so on. However, different to the price setting behavior of firms in Bertrand competition, the market demand for insurance varies aong this undercutting process because the range of individuas which are interested in buying insurance changes with every reduction in premium. In fact, we show that this undercutting See for the Association of British Insurers. 2 See Sonnenhozner and Wambach (2004) for an overview on oigopoistic modes in insurance markets. Studies in which two or more insurers enjoy positive profits are Schesinger and von der Schuenburg (99) who consider insurance product differentiationinthepresence of search and switching costs, Rothschid-Stigitz (976), Vieneuve (997) Smart (2000), Wambach (2000) and Gerber (2003) who examine asymmetric information between the insurer and consumers, Zeckhauser (970) and Bennardo and Chiappori (2002) who examine competition under mora hazard or Poborn (998) who assumes risk-averse insurers. See aso Wison (976), Miyazaki (977) or Spence (978) for the use of aternative equiibrium concepts in adverse seection modes of insurance. 2

3 process stops, if a certain critica premium is reached. It is then optima for an insurer to raise his premium to a certain higher eve. In sum, both insurers best responses to a premium offer of the competitor are aways between two critica premiums. We then show that there exists a (unique) Nash-equiibrium in mixed strategies in which each insurer chooses to offer a premium according to a we-specified distribution function. With respect to wefare, we compare the case of duopoy with a monopoistic insurer as we as with a reguated insurance market. Of course, under competition industry profits as we as the range of insured individuas decrease compared to the monopoy case. Moreover, a monopoistic insurer offers fu coverage for a ower premium than the insurers in a competitive market woud do. Comparing the monopoy outcome with a wefare-maximizing insurance, we find that the premium for coverage is higher under monopoy than in a reguated market and that the range of contracted insurance poicies is smaer under monopoistic insurance than under a wefare-maximizing contract. The paper is organized as foows. In Section 2 we present the mode. Section 3 contains the equiibrium anaysis of insurance poicies for three cases. In Section 3. we first investigate the optima insurance contract a socia panner woud dictate, Section 3.2 then considers the case of a monopoistic insurer and finay, in Section 3.3, we consider competitive insurance. Section 4 concudes. 2 The Basic Mode of the Insurance Competition To formaize our arguments we start with a simpe mode of an insurance industry with profit-maximizing insurance companies and a continuum of individuas. Each individua faces an individua risk of accident. To protect himsef against the oss of an accident, an individua can insure himsef against this accident by buying an insurance contract. 2. Demand for Insurance Individuas are assumed to be uniformy distributed on the ine segment [0, ]. The ocation π [0, ] of a particuar individua on the unit interva describes his individua probabiity of an accident. This probabiity is out of the individua s contro so that no mora hazard probem arises. Moreover, individuas probabiities of an accident are common knowedge, hence no adverse seection probem arises. Except for their probabiities of an accident, individuas are assumed to be identica. Each individua has an initia weath w > 0. In the event an accident occurs, an individua has a financia oss >0. We assume that his weath exceeds the potentia oss, that is, w. The individua can purchase insurance. An insurance poicy is characterized by a premium p>0 and an indemnity c>0 the insurer pays to the individua if an accident takes pace. We assume that overinsurance is not possibe, that is, the eve of coverage is 3

4 smaer than the financia oss, c. Moreover, an individua can buy ony one insurance contract. We appy a oss-averse utiity function for individuas. 3 An individua indexed by π, π [0, ], has a utiity defined by u π (p, c) = ½ w π if he buys no insurance (p, c) =(0, 0) w p π ( c) if he buys an insurance poicy (p, c) () where. For =, an individua evauates his oss by an accident to be exacty. Therefore, we interpret this case as oss neutraity. For >, however, an individua evauates the risk of accident and osing as generating a disutiity measured by ( ), in addition to his financia oss. This additiona disutiity might, for instance, be a representation of the idiosyncratic pain an accident inficts on the individua. 4 In this case, the individua exhibits oss aversion. Hence, we interpret as the degree of oss aversion. Note that the coverage c paid by the insurer not ony reduces his financia oss but aso his additiona disutiity ( ). Athough in practice the reduction of additiona disutiity is not compete, this assumption can be justified by the observation that coverage incudes caim management, which in turn incudes services for the costumer to reduce the stress connected with his accident. 5 Individuas are homogeneousy oss-averse and face an identica degree of oss aversion. When deciding whether or not to buy insurance, and if yes, which insurance poicy is best, each individua maximizes expected utiity based on (). 2.2 Suppy of Insurance We consider one, respectivey, two insurance companies operating in the market. Insurers compete for individuas by offering insurance poicies. One company can offer ony one or two contracts. Let (p i,c i ) be an insurance contract offered by insurer i {, 2} which specifies the premium p i to be paid by an insured individua and the eve of coverage c i paid to the insured if an accident takes pace. We assume that insurance companies are risk-neutra, that they are concernedonywithexpectedprofits. If a poicy (p i,c i ) is sod to an individua 3 See Shy and Stenbacka (2004) who introduce oss-averse individuas in the context of banking where each individua can make a deposit but faces the risk that the bank gets bankrupt. Other behavioria formuations of individuas risk preferences in the context of insurance demand are given in Jeeva and Vieneuve (2004) and Chassagon and Vieneuve (2005) who assume subjective risk perception or De Feo and Hindriks (2005) who examine risk aversion with different weights for good and bad outcomes. 4 Aternativey, the prospect theory by Kahneman and Tversky (992) suggests that a potentia oss woud have a higher negative vaue for an individua according to the concave shape of the vaue function in the oss region. The payment of a premium p before an accident takes pace then estabishes the reference point. 5 Note that in prospect theory this assumption is aways satisfied since a reduction in the financia oss by coverage c naturay reduces the additiona oss an accident inficts upon the individua by some amount vaued greater than ċ. 4

5 who has a probabiity on incurring an accident of π [0, ], this contract is worth ( π) p i + π (p i c i )=p i πc i. (2) Since each insurer can identify the accident probabiity π of an individua that demands its contract, insurance is given ony if the contract is expected to be profitabe. For a given insurance poicy (p i,c i ) et D i [0, ] be the insured individuas of insurer i. Then, the overa profits of insurer i are given by Z Π i (p i,c i )= (p i πc i ) dπ. D i 3 Equiibrium Insurance Poicies We begin the anaysis by investigating the optima insurance contract a socia panner woud dictate. We then turn a situation in which a singe profitmaximizing insurance company offers insurance. Finay, we consider an industry with two insurers and anayze the competitive forces that determine equiibrium insurance poicies. 3. Wefare-Maximizing Insurance First consider the case in which a socia panner can decide on a wefaremaximizing insurance poicy. For simpicity, she vaues the wefare of each individua equay. She woud choose an insurance contract (p S,c S ) to maximize individuas expected utiities, whie ensuring nonnegative profits for the insurer. To compare this wefare-maximizing insurance poicy with the performance of profit-maximizing insurance contracts we assume that the socia panner aows both parties, the individuas and the insurer, to decide autonomousy whether to effect insurance or not. 6 That is, given the contract (p S,c S ), an individua can freey decide whether or not to buy coverage, and the insurance company can decide whether or not to accept an appication for insurance. To sove for the wefare-maximizing insurance contract, we use backward induction. Suppose the socia panner dictates a contract (p S,c S ). Given his accident probabiity π, an individua then wants to buy insurance, u π (p S,c S ) u π (0, 0), if w p S π( c S ) w π, hence, if π p S := π (p S,c S ). c S That is, an individua wi buy coverage ony if his probabiity of an accident is higher than some critica vaue π. This critica vaue is determined by the premium per unit of coverage and the degree of oss aversion: Let p 0 S = ps c S be 6 But see Proposition 2 for the case in which the socia panner can aso dictate the range of insured individuas. 5

6 the premium per unit of coverage. Then the ower p 0 S, respectivey, the higher the additiona disutiity of an accident, the higher an individua s incentives to buy insurance. However, not a individuas interested in buying insurance are actuay profitabe for the insurer. In fact, using (2) the insurance company rejects an appication for insurance, if the accident probabiity of the individua is too high, that is, if π p S := π (p S,c S ). c S Hence, if the premium per unit of coverage is smaer than one, p 0 S <, the accident probabiity π of an individua such that the insurer is indifferent between seing the poicy to him or not is determined by the fair premium rate p S = πc S for which the company just breaks even. Athough higher risk individuas prefer to buy coverage they are refused by the insurer. Anticipating this behavior the socia panner chooses an insurance contract (p S,c S ) to maximize the sum of expected utiities of a individuas not being insured pus the expected utiities of a insured individuas subject to the constraint that the insurer makes nonnegative profits. Let W R denote individuas wefare under (p S,c S ). Then formay, the panner s probem is max (p S,c S ) W R = Z π(ps,c S) π(p S,c S ) Z π(ps,c S) 0 such that [w p S π( c S )] dπ + (w π) dπ + Z π(ps,c S ) π(p S,c S) Z π(p S,c S) [p S πc S ] dπ 0 (w π) dπ The first term represents the expected utiity of a individuas with an accident probabiity π [π (p S,c S ), π (p S,c S )]. They want to buy coverage and are insured. The second term is the expected utiity of a individuas π [0,π(p S,c S )] who have no incentive to buy insurance, whereas the third term refects the expected utiity of a individuas π [π (p S,c S ), ] which want to buy coverage but are rejected by the insurer. The insurer s expected profits are given by the right hand side of the constraint. Proposition In case of a reguated insurance market, the wefare-maximizing insurance poicy (p S,c S ) offers fu coverage c S = for a premium p S =. Individuas with an accident probabiity π higher than wi buy insurance and are accepted for insurance by the insurer. Individuas wefare then is WS (2 ) = w 2 and equiibrium profits of the insurer are Π ( )2 S =

7 Proof. See Appendix. The wefare-maximizing insurance contract is designed such that those individuas with the highest risks are insured. In particuar, the sociay optima insurance poicy offers a fair premium for those individuas who wi have an accident with certainty. The reason for this resut foows immediatey from the fact that it is sociay desirabe to give insurance to those individuas which bear the highest oss. Since individuas are identica except for their probabiity of an accident, the expected disutiity of an individua from an accident is ower the higher his risk of an accident. Hence, individuas with the highest risks shoud be insured getting fu coverage. Since the individua which is just indifferent between buying coverage or not is characterized by the degree of oss aversion, the range of contracted insurance poicies is increasing in the individuas oss aversion. Three remarks as worth noting: First, the insurance company aways makes positive profits. These profits come from those individuas who are wiing to buy insurance athough the premium is not fair given their accident probabiity. As a consequence, the insurer s profits are increasing in the degree of oss aversion. Second, suppose the socia panner has the possibiity to dictate two insurance contracts (p S,c S ) and (p S2,c S2 ). Straightforward argumentation then shows that µ (p S,c S )=(, ) and (p S2,c S2 )=, are the optima wefare-maximizing insurance contracts. As before, it is sociay desirabe to ensure that individuas with the highest risks can buy insurance. The second contract therefore foows the wefare-maximizing singe contract (p S,c S )=(, ) and ensures that a individuas with the remaining highest risks can buy coverage. The second insurance contract therefore offers a fair premium for those individuas who wi have an accident with probabiity. The range of individuas contracting such an insurance poicy is determined by an accident probabiity π,. 2 Third, suppose that the socia paner not ony reguates the insurance contract but aso determines the range of individuas the insurer has to insure under this poicy. Let [π, π] be this range, that is, a individuas with an accident probabiity π [π, π] are forced to buy insurance and must be accepted by the insurer. The panner s probem then is to max (p S,c S,π,π) Z π π Z π 0 [w p S π( c S )] dπ + (w π) dπ + such that Z π π Z π (w π) dπ [p S πc S ] dπ 0. As before, the optima eve of coverage is identica to individuas oss, c S =. Moreover, since socia wefare now is decreasing in the insurance premium, the 7

8 nonnegative profit constraint must be binding and the insurer s expected profits are zero in equiibrium. This, however, impies that the profits the insurer makes with ow risk individuas in [π, π] are just baanced with the osses he makes with high risk individuas. Since individuas are uniformy distributed, the premium per unit of coverage then is p S = (π + π). 2 Using this observation, socia wefare reads as µw 2 (π + π) (π π)+ µ wπ 2 π2 + µw ( π) 2 π 2 = w 2 ( ) (π π)(π + π) 2 Hence, it is sociay optima to choose (π π) as high as possibe, that is, the socia panner introduces a mandatory insurance, π =0, π =. The resuting socia optima premium per unit of coverage then just equas 2, p S = 2. In particuar, it is sociay desirabe that ow risk individuas subsidize the coverage of high risk individuas. Proposition 2 In case of a reguated insurance market in which the socia panner can determine the range of insured individuas, the wefare-maximizing insurance poicy (p S,c S ) offers fu coverage c S = for a premium p S = 2. Insurance is mandatory and a individuas independent of their accident probabiity have to buy insurance and are accepted for insurance by the insurer. The insurer makes zero profits, Π S = Monopoistic Insurance Assume now that there is one singe profit-maximizing insurance company in an unreguated insurance market. Let (p M,c M ) be the insurance contract the monopoist offers to the individuas. Simiar to the argumentation above an individua with accident probabiity π then wants to buy insurance, if π p M := π (p M,c M ). c M And, the insurance company accepts an appication for insurance, if the accident probabiity π of the individua is not too high π p M c M := π (p M,c M ). 8

9 Hence, if the premium per unit of coverage is smaer than one, p 0 M <, ony an individua with accident probabiity π buys coverage at a fair premium rate p M = πc M. If, on the other hand, the premium per unit of coverage is higher than one, p 0 M, a individuas seeking coverage wi be insured. Atogether, the monopoist s overa profits when offering an insurance poicy (p M,c M ) are then given by Π M (p M,c M )= Z min{π(p M,c M ),} (p M πc M ) dπ. π(p M,c M ) Proposition 3 In case of a monopoistic insurance market, the optima insurance poicy (p M,c M ) offers fu coverage c M = for a premium 2 p M = 2. Individuas with an accident probabiity π higher than 2 wi buy insurance and are accepted for insurance by the insurer. Equiibrium monopoy profits are Π ( )2 M = 2(2 ). To prove this resut consider for a given insurance poicy (p M,c M ) the insurer s monopoy profits " Π M = c M "p 0 M min {p 0 M, } p0 M µ ## (min {p 0 2 M, }) 2 p 0 2 M which are increasing in the eve of coverage c M. Hence, the monopoistic insurer maximizes profits by offering fu coverage c M =. Moreover, as ong as the premium per unit of coverage is smaer than one, p 0 M, expected profits Π M = c M (p 0 ( ) 2 M )2 2 2 are increasing in the premium per unit of coverage p 0 M. This foows from three observations: First, the saes voume, that is, the range of individuas who demand insurance and wi be suppied, p 0 M p0 M = p0 M ( ). increases with the premium per unit of coverage for >, hence, gross profits on insurance saes c M (p 0 M) 2 ( ) 9

10 as we. Second, the h average i probabiity of an accident within the range of p 0 insured individuas M,p0 M Zp 0 M p 0 M πdπ =(p 0 M ) 2 2 aso increases with the premium per unit of coverage and, hence, the expected costs of the insurer for coverage c M (p 0 M) Third, the trade-off for the insurer between additiona profits on insurance saes and expected costs for coverage is aways positive. This simpy foows from the fact that additiona profits come with certainty whereas additiona costs not aways reaize. Since margina profits are positive at a premium per unit of coverage equa to one, p 0 M =, the monopoistic insurer maximizes profits by offering fu coverage c M = for an unfair premium p M >.Maximizing profits for p0 M, Π M = c M "p 0 M p0 M " µ p 0 M 2 ## then eads to the optima premium per unit of coverage p M as proposed.7 The equiibrium shows that as in the case of a socia panner a monopoist ony insures individuas with high risks and, thereby, offers fu coverage. However, the premium for coverage is higher under monopoy than in a reguated market, that is, the premium for those individuas who wi have an accident with certainty is unfair. The range of contracted insurance poicies then is smaer than under a wefare-maximizing contract and the profits of the insurance company are higher than in the socia optimum. Of course, individuas wefare in a monopoistic equiibrium is ower than wefare in case of a wefare-maximizing insurance contract. To see this, consider first wefare under the monopoistic contract (p M,) whichcanbewrittenas Z Z WM 2 µ 2 = (w π) dπ + w dπ 2 0 = w 2 (3 2) 2(2 ) This foows immediatey from the first-order condition p 0 Π M = c M 2 M 2 p 0 M (2 ) =0. Note that p M = 2 (, ). 2 0

11 Then WS >W M if and ony if 2 (2 ) 2 (3 2) > (2 ) which is aways satisfied. Two extensions of our resut are worth mentioning: First, suppose that the monopoistic insurer offers two insurance poicies (p M,c M ) and (p M2,c M2 ). Aong the argumentation above it is aways optima to offer fu coverage, c M = c M2 =. Moreover, if the second contract is the more expensive one, that is, p M2 >p M, the monopoist optimay sets p M = pm2 : Under contract (p M2,c M2 ) a individuas with an accident probabiity higher than p M2 wi buy coverage. Because expected profits are increasing in the premium per coverage the monopoist offers the ess expensive contract to those individuas with the remaining highest accident probabiities θ< pm2. Atogether, the monopoist then chooses p M2 to maximize his overa expected profits 8 Π M2 = p M2 Z p M2 2 ³ pm2 π dπ + Z p M2 (p M2 π) dπ. Proposition 4 In case of a monopoistic insurance market in which the monopoist offers two insurance contracts, the optima poicies (p M,c M ), (p M2,c M2 ) give fu coverage c M = c M2 = for a premium 3 p M = 2 2 ( ) + 2 and 4 p M2 = 2 2 ( ) + 2 Individuas with an accident probabiity π between p M and p M buy insurance contract (p M,), individuas with an accident probabiity higher than p M buy contract (p M2,). Overa equiibrium monopoy profits are Π ( ) M2 = ( ) Writing expected profits as Π M2 = p2 M p M2 4 2 the first order condition reads as Π M2 = p 2M p 2M 4 =0 which eads to the optima p M2.

12 Compared to the case in which the monopoists offers ony one insurance contract, more individuas wi be insured. That is because p M < p M. 9 However, the premium per unit of coverage for the individuas with the highest risks increases, 0 p M2 >p M. Hence, additiona insurance for individuas previousy uninsured is bought by a higher premium for those individuas aready insured. To anayze the trade-off between these two effects, consider the overa individuas wefare W M2 = Z p M 0 Z p M (w π) dπ + p M (w p M) dπ + = w Z p M (w p M) dπ Then a monopoy insurer offering two contracts eads to higher wefare than a monopoist with ony one contract, WM2 >W M if 2 (3 2) 2(2 ) 2 > which is aways satisfied since ( ) > 0 for >. A second extensions concerns the insurer s costs for providing insurance. So far we assumed that he has no costs for acquisition or handing of caims. Let α denote his acquisition costs and β his costs for handing of caims, α, β > 0. Then the insurer wi reject an individua with an accident probabiity π appying for insurance, if p M π ( + β) α<0, thatis,π< p M α + β. (3) Proposition 5 If the monopoistic insurer has acquisition costs α and costs β for handing of caims of, no insurance wi be given if α + β ( ). 9 Using p M = 2 2, p M < is equivaent to 2 (2 ) < 2 2 ( ) + 2 which is aways satisfied since 0 < 2 ( ) +. 0 p M2 >p M is equivaent to ( +)( ) (2 ) > 2 2 ( ) which is aways satisfied since > 0. 2

13 If overa costs α + β are ower than ( ), the optima insurance poicy (p M,c M ) offers fu coverage c M = for a premium p ( + α) M (α, β) = (2 ) β. +α (2 ) β Individuas with an accident probabiity π higher than wi buy insurance and are accepted for insurance by the insurer. Equiibrium monopoy profits are Π ( ( ) α β)2 M (α, β) = 2( (2 ) β). The introduction of costs then impies that ess individuas wi actuay be insured in equiibrium. In fact, because the premium is increasing in α as we as β, the critica accident probabiity is increasing in α and β as we. 3.3 Competitive Insurance + α (2 ) β We now anayze an industry structure with two insurance companies, caed insurer and insurer 2. Suppose that insurer i {, 2} offers an insurance poicy (p i,c i ). What is the optima insurance contract? How does competition affect the insurance market? To answer these questions, suppose that the two insurance poicies are equiibrium contracts (p D,c D ) and (p D2,c D2 ).Thenwefirst show that both equiibrium insurance contracts aways impy fu coverage for an insured individua, that is, c D = c D2 =. To see this, we argue to a contradiction. Assume, for exampe, that insurer does not offer fu coverage, c D <.Then we caim that insurer 2 can cover the whoe segment of insured individuas by offering a contract (p D2,c D2 ) that gives an individua more coverage by an identica premium per unit of coverage. To see this, et c D2 = c D + ε, withε>0 but ε c D, and p D2 = c p D D2 c. Then both insurers offer the same premium per D unit coverage p D c = p D2 D c D2 hence, a individuas who want to buy poicy (p D,c D ) are aso potentia buyers of poicy (p D2,c D2 ), and vice versa, that is π (p D,c D) =π (p D2,c D2) and π (p D,c D) =π (p D2,c D2). How do individuas decide from which insurer to buy? We caim that a individuas who want to buy insurance prefer the contract (p D2,c D2 ) which offers the 3

14 higher coverage. To see this consider the expected utiity of such an individua π in either case. Then u π (p D2,c D2 ) >u π (p D,c D ), if w p D2 π ( c D2) >w p D π ( c D) that is, if µ (π (p Di,c Di) π) c D2 c > 0. D Since π π (p Di,c Di ) - otherwise this individua won t buy any coverage at a - this inequaity is satisfied. As a consequence, a individuas seeking coverage choose insurer 2, and, insurer s equiibrium profits woud be zero. Of course, insurer coud make positive profits by offering a contract that gives these individua more coverage than insurer 2, thereby eaving the premium per unit of coverage constant. This, however, contradicts our assumption that (p D,c D ) is an equiibrium contract. To sum up, as ong as coverage is not compete, insurers compete in the eve of coverage. Hence, c D = c D2 =. Given that equiibrium coverage is compete, insurers compete in the premium per unit of coverage. In the foowing we wi argue that there cannot exist a Nash-equiibrium in pure strategies in which it is optima for an insurer to offer a fixed premium for coverage. Instead, we show that there exists ony a (unique) Nash-equiibrium in mixed strategies in which each insurer chooses to offer a premium according to a we-specified distribution function. To understand the reasoning behind this finding et us consider the competitive forces in the insurance market in more detai, see Figure : Suppose that insurer 2 thinks that insurer wi offer a contract (p D,) with a premium p D ower than p M. Then insurer 2 woud maximize his profits by offering the monopoistic contract (p M,). This guarantees monopoy profits for insurer 2. However, if insurer 2 demanded the monopoistic premium p M insurer s best response woud be to undercut this premium sighty. Since a individuas who intended to buy insurance from insurer 2 now switches to insurer, the atter s profits are amost identica to monopoy profits. Hence, insurer woud never offer a premium ower than p M. Of course, if insurer offered a premium beow or equa to the monopoistic premium, insurer 2 woud have an incentive to undercut insurer s premium sighty, thus recapturing individuas who want to buy insurance. But, if insurer 2 did this, insurer woud once again have an incentive to set a premium sighty beow insurer 2 s new, ower, premium. This reasoning seems simiar to the price setting behavior of firms in Bertrand competition but it is not: Different to Bertrand competition, the market demand for insurance varies because the range of individuas who are interested in buying insurance changes with every reduction in premium. In fact, suppose that the undercutting process of premiums comes to a premium p with the property Z p p (p π) dπ = Z p (p π) dπ. 4

15 The premium p is characterized by the property thath the expected i profits from offering an insurance poicy (p, ) with a demand p, p is identica to the h i expected profits from offeringapoicy(p, ) with a demand p,. Cacuation shows that this critica premium p is higher than p M. Since an insurer s profits are increasing, respectivey decreasing, in the premium as ong as the premium is beow, respectivey above, the monopoistic premium p M, the critica premium p has an additiona property: If an insurer reduces p sighty by some sma number ε>0, the resuting expected profits wi be ess than the ones in case he reduces the premium p bythesamenumberε>0. As a consequence, if the undercutting process of premiums comes to p, saybyinsurer,itwibe optima for insurer 2 to raise the premium to p ε. By owering his premium once again, insurer can then recapture those individuas who want to buy insurance from insurer 2. This undercutting process continues ti one insurer comes to the premium p and they are back in a situation they aready had. Figure : Undercutting process in premium offers In sum, both insurers best responses to a premium offer of the competitor are aways between p and p. Moreover, there exists no combination of premiums such that these premiums are mutuay best responses. Hence, there cannot be an equiibrium in which an insurer offers a poicy with a fixed premium. The foowing resut characterizes the unique equiibrium in mixed strategies Proposition 6 In case of a duopoistic insurance market, the optima insurance poicies offer fu coverage c D = c D2 =. Moreover, insurers offer premiums within the range [p, p] according to the cumuative distribution functions G ( ) that describe the equiibrium distribution of premium choices 0 for p<p ³ ³ c p m 2 +2 G c2 p m2 2 ( )2 2 for p [p, ] (p) = ( )2 mc+m2c2 2 2(p p) for p [, p] p 2 ( ) 2 (p ) 2 for p>p with we-defined constants m,m 2,c,c 2 and p. Expected equiibrium duopoy profits are identica for both insurers and equa q Π D = Π D2 = 2 ( ) 2 + ( ) 3 2 ( +) The foowing figure iustrates the equiibrium cumuative distribution of premium choices for different specifications of. See the proof of this proposition for the definitions of these parameters. 5

16 Figure 2: Equiibrium cumuative distribution of premium choices for different specifications of, red =., back =2, bue =5 Using such simuations the shape of the equiibrium distribution of premium choices exhibits the foowing features: If the degree of risk aversion is sufficienty high, an insurer chooses his premium in equiibrium according to a convex function on the range [p, p] where the probabiity for choosing p is higher than for p. If, on the other hand, the degree of risk aversion is ower than some critica vaue, the equiibrium distribution of premiums is convex on [p, ] as we as on [,p] such that the probabiity for choosing p is ower than for p. With respect to the previous proposition severa remarks are worth noting. First, industry profits decrease under competition. This foows immediatey from the observation that a monopoistic insurer who offers two insurance poicies can aways set insurance premiums identica to the competitive premiums but is not restricted by competitive forces when doing so. Second, competition aways eads to ess insured individuas than monopoy. To see this consider the maxima range of individuas that wi be insured under both regimes: Using Proposition 4 a monopoistic insurer offering two poicies insures a individuas with an accident probabiity π p M 2 = 2 2 ( ) + 2, whereas under competition the previous proposition reveas that at most those individuas wi be insured whose accident probabiity is q π p = 2 + ( ) 3 ( +) Simpe cacuation then shows that p<p M. Third, a monopoistic insurer offers fu coverage for a ower premium than the insurers in a competitive market woud do. This finding is a direct impication of the second remark since the range of individuas who wi be insured directy depends on the owest premium for coverage offered in the market. Since the ast two effects are both beneficia for the overa individuas utiities an important question then is how individuas wefare is effected by competition compared to the monopoy case. Athough it is not possibe to cacuate individuas wefare in a genera expression, numerica estimations suggest that overa individuas utiities are greater under competition. In fact, athough more individuas are insured by a monopoistic insurer this increase in wefare is in genera bought by a higher premium for individuas with a higher accident probabiity. A fina remark concerns the costs of providing insurance. Suppose that insurers are not identica but that insurer is ess productive than insurer 2. For simpicity, assume that insurer faces positive acquisition costs α > 0 and positive costs of handing of caims of β > 0, whereas the costs of insurer 2 are zero, α 2 = β 2 =0. 6

17 Of course, the presence of costs for insurer changes his behavior in the undercutting process of premiums. 2 In fact, the critica premium p (α,β ) such that it is not optima for insurer to ower the premium again but to raise the premium to p (α,β ) is now characterized by the property p(α,β ) α +β Z p(α,β ) {p (α,β ) α π ( + β )} dπ = Z {p (α,β ) α π ( + β )} dπ. p(α,β ) The LHS refects insurer s expected profits from offering an insurance poicy (p (α,β ),). This poicy is demanded by individuas with an accident probabiity higher than p(α,β ) but the insurer rejects appication for insurance if his expected profits are negative, that is, if an individua s accident probabiity is higher than p(α,β ) α +β. The RHS of the equation above characterizes insurer s expected profits from offering an insurance poicy (p (α,β ),). In this case, insurance wi be sod to a individuas with an accident probabiity higher than p(α,β ). Note that for insurer 2 his critica premium p (0, 0) is identica to p. In the presence of costs, the undercutting process between the two insurer then changes as foows: As soon as one of the insurer reaches his critica premium p (α,β ), respectivey p (0, 0), it is optima for this insurer to raise the premium to p (α,β ), respectivey p (0, 0). Hence, in equiibrium insurers offer premiums within the range max {p (α,β ), p (0, 0)} and min {p (α,β ), p (0, 0)}. Which of the two critica premiums then is higher or ower than the other depends cruciay on the eve of these costs for insurer. In fact, suppose that insurer is confronted ony with positive administrative costs α and β =0. Then the critica premium can be cacuated as µ 2α ( ) q( ) α 2 2α p (α, 0) = Cacuation shows that p (α, 0) is increasing in α as ong as these costs are sufficienty sma. If, on the other hand, costs are sufficienty high, p (α, 0) decreases in α. 3 To understand this shape consider the effect of administrative costs on 2 As before, it is aways optima to offer fu coverage in equiibrium. 3 The first derivative is ( ) p (α, 0) = α 2 +2α (2 ( 2 ) 2α 2 2α )

18 profits. Since profitspercustomerdecreasebyα with every additiona contract signed, the overa costs depend on the range of insured individuas. Now consider the reative cost effects if the insurer offers poicies (p, ) and (p, ). For α =0the range of costumers for poicy (p (0, 0),) is greater than for poicy (p (0, 0),), hence the insurer adjusts his critica vaue p (α, 0) toahigherpremium eve. This, in turn, increases the range of insured individuas under the premium p (α, 0) and owers the range of individuas insured under the premium p (α, 0). Hence, if α gets sufficienty high, the range of customers for a premium p (α, 0) is ower than the one for p (α, 0). But then the insurer adjusts his critica vaue p (α, 0) to a ower premium eve. This, in turn, increases the range of customers under the premium p (α, 0) and owers the range insured under the premium p (α, 0). An additiona increase in administrative costs then makes insurance so expensive for the insurer that he reduces the range of customers by decreasing the critica premium even more. Cacuation then shows that if administrative costs are equa to p α = 2( +2)(2 ) the critica premium p (α, 0) wi be identica to the monopoy premium p M. Hence, if α > α³ insurer, 2 earns monopoy profits and insurer offers an p insurance poicy M. 4 This resut changes if we consider the costs of insurer for handing of caims. Suppose that these costs β are positive but that administrative costs are zero, α =0. The critica premium can now be cacuated as µ q (β + ) ( ( ) β ) p (0,β )= β ( ) ( +)(2 ( ) β ) Cacuation then shows that the critica premium p (0,β ) is increasing in β for a cost eves. 5 This is different to the infuence of administrative costs on the critica premium and refects the fact that settement costs have profit impication. In fact, the overa settement costs for a given poicy now not ony depend on the range of insured individuas but aso on their accident probabiities. 4 In fact, insurer sti makes positive profits since ( ) > α, see Proposition 5. 5 The first derivative ( ) ( +)( + β) (β ( ) + (2 )) ( + β ( +)) p (0,β β )= β 3 2β 2 2β +2β β β 2 2 is aways positive since ( ) ( +)( + β) > 2 (β ( ) + (2 )) ( + β ( +)) which is equivaent to β 3 2β 2 2β +2β β β 2 2 > 0. 8

19 Consider the reative cost impications if the insurers offer poicies (p, ) and (p, ). For β =0not ony the range of costumers for poicy (p (0, 0),) is greater than for poicy (p (0, 0),) but aso their corresponding risks. Hence, the insurer adjusts his critica vaue p (0,β ) to a higher premium eve. As with administrative costs, this increases the range of insured individuas under the premium p (0,β ) and owers the range insured under the premium p (0,β ). Different to the infuence of administrative costs, however, the settement costs under p (0,β ) are sti higher than under p (0,β ). Hence, an increase in settement costs β eads to an even higher critica premium. This process stops if p (0,β ) is identica to the coverage, which impies that insurer makes zero profits. 6 Note that, in equiibrium the two insurers now offer premiums within the range [p (0,β ),p (0, 0)]. To sum up, the next proposition summarizes our discussion on cost impications: Proposition 7 In case of a duopoistic insurance market suppose that ony one insurer has positive acquisition costs α or positive costs for handing of caims of β. Then equiibrium has the foowing properties:. There exist two critica eves of costs α 0 < α <( ) such that for α < α 0 equiibrium profits are Π D = µ 2 2 p (α, 0) 2 p (α, 0) 2 µ 2 α 2 2 Π D2 = µ 2 2 p (α, 0) 2 where p (α, 0) is a critica premium. For α > α 0 but α < α equiibrium of both insurers profits are constant and identica to Π D = µ 2 2 p (0, p (0, 0) 2 µ 2 0)2 α 2 2 Π D2 = µ 2 2 p (0, 0)2. ³ For α, > α but α <( ) insurer offers in equiibrium the poicy p M and insurer 2 the poicy (p M,) eading to equiibrium profits Π ( ) 2 D = 2(2 ) 2 α 2 2(2 ) 2 Π D2 = 2 ( ) 2 2(2 ) 2. If α ( ), insureroffers no insurance and insurer 2 offers the monopoy poicy (p M,) making monopoy profits. 6 In fact, p (0,β )= is identica to β = ( ) which impies zero profits according to Proposition 5. 9

20 2. For β <( ) there exists a critica premium p (0,β ) such that the equiibrium distribution of premium choices is within the range [p (0,β ),p]. Equiibrium profits are Π D = µ 2 µ 2 p (0,β ) 2 β p (0,β ) Π D2 = 2 p (0,β ) 2 µ and decreasing, respectivey increasing, in β for insurer, respectivey insurer 2. If β ( ), insurer offersnoinsuranceandinsurer2 offers the monopoy poicy (p M,) making monopoy profits. 4 Concusion How do competitive insurance markets function when insurers compete in coverage and premium for individuas? The standard argument, see Sonnenhozner and Wambach (2004), is that "Bertrand competition is the most pausibe mechanism... As a resut, there is no space eft between the two extremes of a monopoy (one insurer) and perfect competition (two or more insurers)." In a simpe mode of an insurance industry with two profit-maximizing insurance companies and a continuum of individuas who differ in their individua risks of accident we showed that athough the undercutting process in premiums seems simiar to the price setting behavior of firms in Bertrand competition there exists a crucia difference: Since market demand for insurance varies with every reduction in premium, the undercutting process of premiums ony works within a we-defined range of premiums. Anayzing these competitive forces eads to a unique Nash-equiibrium in mixed strategies in which each insurer chooses to offer a premium according to a we-specified distribution function. In equiibrium both insurers earn positive profits. Severa extensions of the previous mode are possibe. First, one coud extent the anaysis to an oigopoistic situation with more than two insurers. Second, we assumed that insurers do not face adverse seection and mora hazard probems. The introduction of either of these aspects woud incorporate other reevant characteristics of insurance in our mode. Third, individuas in an insurance market often do not ony differ with respect to their risk attitudes but aso in their vauations of coverage. One woud expect that under these circumstances thedegreeofproductdifferentiation rises compared to our mode. 5 Literature Bennardo, A., and P.-A. Chiappori (2003), "Bertrand and Waras Equiibria under Mora Hazard," Journa of Poitica Economy, (4),

21 Chassagnon, A. and B. Vieneuve, (2005), "Optima Risk-Sharing under Adverse Seection and Imperfect Risk Perception", Canadian Journa of Economics, 39, De Feo, G. and J. Hindriks (2005): "Efficiency of Competition in Insurance Markets with Adverse Seection", CORE-DP Université Cathoique de Louvain. Gerber, A. (2003), "Cream Skimming and the Vaue of Information", DP Universität Zürich. Jeeva, M. and B. Vieneuve, (2004), "Insurance Contracts with Imprecise Probabiities and Adverse Seection", Economic Theory, 23, Kahneman, D. and A. Tversky (992), "Advances in Prospect Theory: Cumuative Representation of Uncertainty", Journa of Risk and Uncertainty 5, Miyazaki, H., (977), "The Rat Race and Interna Labor Markets", Be Journa of Economics, 8, Murat, G., R. Tonkin, and D. Jüttner (2002), "Competition in the Genera Insurance Industry", Zeitschrift für die gesamte Versicherungswissenschaft, 3, Nissan, E., and R. Caveny (200), "A Comparison of Large Firm Dominance in Property and Liabiity Insurance with Major Industries," Journa of Insurance Issues, 24, Poborn, M. (998), "A Mode of Oigopoy in an Insurance Market" Geneva Papers on Risk and Insurance Theory 23, Ross, S. L. (984): Differentia Equations, Wiey, New York. Rothschid, M., and J. E. Stigitz (976), "Equiibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," Quartery Journa of Economics, 90, Schesinger, H., and J.-M. G. von der Schuenburg (99), "Search Costs, Switching Costs and Product Heterogeneity in an Insurance Market," Journa of Risk and Insurance, 8, Shy, O. and R. Stenbacka (2004), "Market Structure and Risk Taking in the Banking Industry", Journa of Economics 82, Smart, M. (2000), "Competitive Insurance Markets with Two Unobservabes," Internationa Economic Review, 4, Sonnenhozner, M. and A. Wambach(2004), "Oigopoy in insurance markets", in: J. Teuges and B. Sundt (eds.), Encycopedia of Actuaria Science, New Jersey: John Wiey & Son,

22 Spence, M. (978), "Product Differentiation and Performance in Insurance Markets", Journa of Pubic Economics 0, Wambach, A. (2000), "Introducing Heterogeneity in the Rothschid-Stigitz Mode", Journa of Risk and Insurance 67, Wison, C., (977), "A Mode of Insurance Markets with Incompete Information", Journa of Economic Theory, 2, Zeckhauser, R. (970), "Medica Insurance: A case study of the tradeoff between risk spreading and appropriate incentives", Journa of Economic Theory 2, Appendix ProofofProposition. Note first that the insurer s profits µ p S p 0 S µ 2 c S (p 0 S) = c S 2 (p0 S) 2 ( ) 2 2 are aways positive, independent of the panner s contract (p S,c S ), with p 0 S = p S cs. Hence the constraint in the panner s probem is not binding. Using the definitions for π (p S,c S ) and π (p S,c S ), the panner s probem can then be written as max (p S,c S) µ (w c S p 0 S) p 0 S µ Ã +w p 0 S + p0 S 2 µ 2 ( c S)(p 0 S) 2 2! 2 (p 0 S) 2 + µ p 0 S Since the derivative with respect to c S, µ p 0 Sp 0 S + µ 2 (p0 S) 2 2 =(p 0 S) 2 ( ) is stricty positive, it is optima to set c S =. This reduces the panner s probem to µ max (w p 0 S) p 0 p 0 S w S ( p0 S ( )) ³ 2 (p 0 2 S) 2 2. Taking the derivative with respect to p 0 S, p 0 S ( ) 2,. 22

23 which is aways positive we find that p 0 S shoud be maxima. Since an unfair premium p S >reduces wefare, p S =. Individuas wefare then amounts to W S = Z 0 = w (w π) dπ + (2 ). 2 Z (w ) dπ Q.E.D. Proof of Proposition 6:. Given that the two insurers offer contracts (p,) and (p 2,) expected profits, e.g. for insurer canbecacuatedas p for p <p 2,p p for p = p 2 EΠ (p,p 2 )= 2 (p a) 2 for p >p 2,p 2 ( b)(2p, + b) for p <p 2,p 2 ( c)(2p + c) for p > p 2 0 for p >p 2 with a =max ª p 2, p,b=min p, ª and c =min, max p ªª,p 2. Define the critica premium p> as EΠ (p, ) = Z p p (p π) dπ = EΠ (p, ) Z Simpe cacuation shows that these profits are identica for µ q 2 + ( ) 3 ( +) p = p (p π) dπ = EΠ (p, ). Then p<and p> p M. 7 Since expected profits EΠ (p, ) of an insurer are increasing as ong as p<p M and decreasing for p>p M, it foows that EΠ (p, ) < EΠ (p, ) for p<p and EΠ (p, ) > EΠ (p, ) for p>p. Now consider, say for insurer, the best response function p ( ) if insurer 2 offers an insurance contract (p 2,): 7 Note that p <,thatis 2 ( ) 3 ( +) < since 0 < ( ) 2. Moreover p > p M = 2 is equivaent to (2 ) ( ) 3 ( +) > ( ) 2 which is aways satisfied for >. 23

24 If p 2 p M, insurer s best response is to set p (p 2)=p M which guarantees monopoy profits. If p 2 > p M but p 2 < p, it is optima for insurer to set p (p 2 )=p 2. This is optima since undercutting insurer 2 s premium eads to ess profits and contract (p 2,) attracts a individuas with π [p 2, ] with the highest possibe premium. If p 2 p but p 2 p M, insurer s best response is to set p (p 2)=p 2 ε for ε>0 arbitrary sma. Undercutting insurer 2 s premium now is optima since it eads to higher profits than contract (p 2,). If p 2 >p M, insurer s best response again is to set p (p 2 )=p M which guarantees monopoy profits. The reaction function p ( ) of insurer iustrates the foowing figure: Figure 3: Reaction function p ( ) of insurer Since both insurers are identica no equiibrium in pure strategies exists. To determine the equiibrium in mixed strategies, et G ( ) be the insurers cumuative distribution functions that describe the equiibrium distribution of premium choices and g ( ) the corresponding density function. The argumentation above impies that G (p) =0for p<p and G (p) =for p p. Moreover, if G ( ) determines a mixed equiibrium insurer is indifferent between a premiums p [p, p] if insurer 2 foows G ( ).That is, if EΠ denotes insurer s expected equiibrium profits then EΠ (p,) = EΠ for a p [p, ] and EΠ (p,) = EΠ for a p [, p]. Consider first insurer s equiibrium profits for p [p, ]: EΠ (p,) = Z p p + 2 (p p 2 ) 2 g (p 2 ) dp 2 p Z 2 p ³ p p 2 g (p 2 ) dp 2 The first term represents expected profits if insurer 2 s choice of a premium is ower than p - in this case insurer wi insure ony individuas with an accident probabiity θ p 2, p resuting in profits 2 (p p 2 ) 2. The second term represents insurer s expected profits if insurer 2 s choice of a premium is higher than p - in this case insurer wi insure a individuas with an accident probabiity θ p, p resuting in profits 2 p p 2. 24