Characterization and uniqueness of equilibrium in competitive insurance

Transcription

1 Theoretica Economics ), / Characterization and uniqueness of equiibrium in competitive insurance Vitor Farinha Luz Vancouver Schoo of Economics, University of British Coumbia This paper provides a compete characterization of equiibria in a game-theoretic version of Rothschid and Stigitz s 1976) mode of competitive insurance. I aow for stochastic contract offers by insurance firms and show that a unique symmetric equiibrium aways exists. Exact conditions under which the equiibrium invoves mixed strategies are provided. The mixed equiibrium features i) crosssubsidization across risk eves, ii) dependence of offers on the risk distribution, and iii) price dispersion generated by firm randomization over offers. Keywords. Asymmetric and private information, mechanism design, oigopoy, economics of contracts, insurance. JEL cassification. C72, D43, D82, D86, G Introduction This paper provides a compete characterization of equiibria in a game-theoretic version of Rothschid and Stigitz s 1976) henceforth, RS) mode of competitive insurance with private information. I aow for stochastic contract offers by insurance firms and show that a unique symmetric equiibrium aways exists, extending the cassica resut of RS to mixed strategies. The unique equiibrium is expicity presented and its comparative static resuts are discussed. The equiibrium simutaneousy features i) crosssubsidization across risk eves, ii) dependence of offers on the risk distribution, and iii) price dispersion. The iterature on competitive insurance mosty restricts attention to equiibria with deterministic contract offers. 1 This restriction is probematic, as it rues out important economic phenomena present in insurance markets. Vitor Farinha Luz: vitor.farinhauz@ubc.ca I woud ike to thank Dirk Bergemann, Johannes Horner, and Larry Samueson for their encouraging support throughout this project. I aso thank Marceo Sant Anna, Jean Tiroe, Juuso Toikka, Piero Gottardi, Eduardo Azevedo, Andrea Attar, the co-editor, Nicoa Persico, three anonymous referees, and seminar participants at Yae, the European University Institute, Humbodt, Bonn, and Aato for hepfu comments. Financia support from the Anderson Prize Feowship and the Max Weber programme is gratefuy acknowedged. 1 See Rothschid and Stigitz 1976), Dubey and Geanakopos 2002), Dubey et a. 2005), Bisin and Gottardi 2006), Guerrieri et a. 2010). Copyright 2017 The Author. Theoretica Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercia License 4.0. Avaiabe at DOI: /TE2166

2 1350 Vitor Farinha Luz Theoretica Economics ) First, the focus on deterministic contract offers impies that a static equiibrium cannot feature cross-subsidization. 2 Cross-subsidization means that firms may make profits from ow-risk agents so as to subsidize high-risk agents. In a deterministic equiibrium, any such set of contracts is vunerabe to cream-skimming deviations by one of the competing firms, which ony attract ow-risk agents and eave the high risks to its competitors. However, the construction of such cream-skimming deviations hinges on firms knowing exacty which offer they are competing against, which is not true when firms use mixed strategies. This observation is reevant for poicy anaysis. The fact that cross-subsidization might be wefare improving has been used as a justification for government intervention see Bisin and Gottardi 2006). In the mode considered in this paper, cross-subsidization may arise in equiibrium without governmenta intervention. Second, the absence of cross-subsidization means that the contract consumed by each risk type is priced at an actuariy fair rate. Hence equiibrium contracts are independent of the reative share of each risk in the market. However, the dependence of market outcomes on risk distribution is a centra theme in the poicy arena. 3 The equiibrium characterized in this paper is continuous with respect to the risk distribution. In particuar, when amost a agents share the same risk eve, the contracts chosen by consumers on-path are very simiar to the fu information outcomes with high probabiity. Under fu information, the competitive mode proposed in this paper eads to efficient outcomes: the consumer obtains fu insurance at actuariy fair prices. In other words, our characterization is in ine with the statement that markets with sma frictions generate approximatey efficient outcomes. This property is not present in severa modes that resove the existence issue presented in RS such as Dubey and Geanakopos 2002 and Guerrieri et a. 2010). We consider a competitive market where firms offer contract menus to an agent who is privatey informed about his own risk eve. I foow Dasgupta and Maskin 1986) in modeing competition as a simutaneous offers game with a finite number of firms. The consumer or agent) has private information about having high or ow risk of an accident. Dasgupta and Maskin 1986, Theorem5) proved the existence of equiibria for this game, but provided ony a partia characterization and present no resuts regarding mutipicity of equiibria. The main contributions of this paper are i) to estabish uniqueness of symmetric equiibria, ii) to sove expicity for this equiibrium, and iii) to derive properties and comparative statics of the equiibrium. In Section 4, I expicity describe an equiibrium for a prior distributions. Equiibrium offers ie on a critica set of separating offers that generate zero expected profits 2 This caim refers excusivey to static modes of competitive insurance. In semina papers, Wison 1977), Miyazaki 1977), and Riey 1979) obtain equiibria with cross-subsidization whie considering equiibrium notions incorporating anticipatory behavior behavior akin to dynamic modes. 3 During the impementation of the heath care exchanges foowing the approva of the Affordabe Care Act in the United States, the presence of young aduts with ower risk eve was considered a necessary condition for the successfu roout and stabiity of the program for exampe, see Levitt et a. 2013). In reguated markets such as the exchanges, observabe conditions such as age and previous diagnostics are treated as private information since they can affect the coverage choice of consumers whie not being used or having imited use) expicity in pricing contracts.

3 Theoretica Economics ) Equiibrium in competitive insurance 1351 in the market as a whoe, referred to as cross-subsidizing offers. The equiibrium offers coincide with the zero cross-subsidization offers described in RS whenever a pure strategy equiibrium exists. An equiibrium in pure strategies exists whenever crosssubsidization cannot ead to Pareto improvements. This occurs whenever the probabiity of high risks is sufficienty high 4 Coroary 1). Equiibria necessariy invove mixed strategies whenever the RS menu of contracts cannot be sustained as an equiibrium. If the RS separating contracts fai to be an equiibrium outcome, the equiibrium invoves each firm offering cross-subsidizing offers, with a random amount of cross-subsidization between zero and a Pareto efficient positive) eve. Offers in the support of equiibrium strategies have the foowing properties: i) high-risk agents aways receive a fu insurance contract; ii) ow-risk agents aways receive partia insurance, which eaves the high-risk agent indifferent between this contract and his own; iii) a the menus of contracts in the support of the equiibrium strategy are ordered by attractiveness. The firm that deivers the most attractive menu of contracts attracts the customer, no matter what his type is. Moreover, firms aways earn zero expected profits. 5 The equiibrium distribution over the possibe eves of cross-subsidization comes from a oca condition that guarantees that, for any menu offer in the support of the equiibrium strategy, there is no oca profitabe deviation. I show that this condition impies there is no goba profitabe deviation by a firm. In Section 5, I show that the equiibrium described is the unique symmetric equiibrium. Equiibrium offers can be described by the utiity vector they generate to both possibe risk types. Describing offers in terms of utiity profies means that the offer space is essentiay two dimensiona. The main chaenge in the anaysis ies in showing that equiibrium offers necessariy ie in a one-dimensiona subset of the feasibe utiity space. The crucia step uses properties of the equiibrium utiity distribution to show that expected profits are supermoduar in the utiity vector offered to the consumer, i.e., there is a compementarity in making more attractive offers to both risk types. This property is used to show that equiibrium offers are necessariy ordered in terms of attractiveness, i.e., a more attractive offer provides higher utiity to both risk types. Since firms make zero profits, this ordering of offers impies they generate zero profits even if they are accepted by both risk types with probabiity 1. Hence, offers can be indexed by the amount of subsidization that occurs across different risk types. The use of supermoduarity and the zero profits condition to reduce the dimensionaity of the equiibrium support is nonstandard in the iterature. 4 The competitive equiibrium concept considered in RS is different from the game-theoretic) equiibrium concept considered here. Nevertheess, the RS pair of contracts is an equiibrium outcome of my game if and ony if it is an equiibrium of their mode, provided that entering firms are aowed to propose a pair of contracts. In their main definition of competitive equiibrium Section I.4), outside firms are ony aowed to offer a singe contract, whie it is acknowedged that a new pair of contracts might be more profitabe than a deviating pooing contract Section II.3). As a consequence, the exact condition for existence of a pure strategy equiibrium is reated to separating, and not pooing, offers. 5 In fact, each firm earns zero expected profits for any reaization of its opponents randomization but in expectation with respect to the agent s type).

4 1352 Vitor Farinha Luz Theoretica Economics ) Our uniqueness resut aows us to discuss comparative static exercises in a meaningfu way. I anayze two reevant comparative statics exercises: with respect to the prior distribution and with respect to the number of firms. With respect to the prior distribution over types, equiibrium offers have monotone comparative statics. If the probabiity of ow-risk agents increases, firms make more attractive offers in the sense of first-order stochastic dominance and both agent types are better off. The equiibrium features mixing whenever cross-subsidization benefits a consumers in the market but the extent of cross-subsidies is discipined by the absence of profitabe cream-skimming deviations. On one hand, cream-skimming deviations are ess attractive to high-risk types and hence attract them with smaer probabiity risk seection). On the other hand, cream-skimming deviations introduce more risk in the consumption profie of ow-risk agents risk inefficiency) and hence deiver any expected utiity eve with ower profits for the offering firm. An increase in the share of ow-risk agents in the popuation reduces the gain from risk seection and increases the osses from risk inefficiency. As a consequence, more cross-subsidization arises in equiibrium, which benefits consumers. As mentioned before, the equiibrium outcome is continuous with respect to the risk distribution, even around the fu information imits. When the probabiity of owrisk agents converges to 1, the distribution of offers converges to a mass point at the actuariay fair fu insurance aocation of the ow-risk agent. For such distributions, the sma share of high-risk agents in the popuation impies that the potentia gains from cream-skimming deviations are sma as we. Hence a very sma amount of contract uncertainty is enough to deter such deviations. Aternativey, when the probabiity of ow-risk agent is sufficienty sma, the RS pair of contracts is an equiibrium. Hence a consumers obtain actuariy fair contracts given their risk eve, with high-risk consumers obtaining fu coverage and ow-risk agents obtaining partia coverage. Obviousy, as the probabiity of high-risks converges to 1, the consumed contract on-path converges in probabiity to the fu insurance contract consumed by ow-risk consumers. Equiibrium strategies aso feature monotone comparative statics with respect to the number of firms, N 2. The support of the equiibrium strategies does not change with the number of firms, but the distribution does. Surprisingy, the wefare of both types decreases with the number of firms. Each firm s offers converge to the worst pair of offers in the support: the pair of RS separating contracts. The distribution of the best offer in the market converges, as N, to the equiibrium offer of a singe firm in a duopoy. This resut carifies the impossibiity of construction of mixed equiibrium when there are infinitey many firms and sheds ight on the nonexistence resuts for the competitive equiibrium concept considered in RS. A comparative statics resuts hod with strict inequaities whenever the equiibrium invoves mixed strategies. Our mode provides a rationae for the existence of cross-subsidization and has reevant empirica predictions for markets with adverse seection. First, insurance firms can be ranked in terms of attractiveness, with more attractive firms offering better contract choices for a risk types. Second, our anaysis shows how risk distribution can affect market offers, increasing the presence of cross-subsidization in a wefare increasing

5 Theoretica Economics ) Equiibrium in competitive insurance 1353 way. Finay, we show how an increase in the eve of competition, which can be interpreted as a arger number of firms, can have adverse effects in the presence of adverse seection. The paper is organized as foows. The next section discusses the reated iterature. Section 3 describes themode. Section 4 constructs a specific symmetric strategy profie and shows that it is an equiibrium. Section 5 shows that the constructed equiibrium is the unique symmetric one. Section 6 presents the comparative static resuts. Finay, Section 7 concudes. 2. Reated iterature Severa papers have considered aternative modes or equiibrium concepts that dea with the nonexistence probem in the RS mode. Maskin and Tiroe 1992) consider two aternative modes: the mode of an informed principa and a competitive mode in which many uninformed firms offer mechanisms to the agent. In the informed principa mode, the agent proposes a mechanism to the uninformed) firm. The equiibrium set consists of a incentive compatibe aocations that Pareto dominate the RS aocation. The equiibrium outcome in my mode is contained in the equiibrium set of the informed principa mode. Maskin and Tiroe 1992) aso consider a competitive screening mode in which firms simutaneousy offer mechanisms to a privatey informed agent. A mechanism is a game form in which both the chosen firm and the agent choose actions. The equiibrium set of this mode is aways arge: it contains any aocations that are incentive compatibe and satisfy individua rationaity for the agent and firms. Hence the equiibrium set aso contains the unique equiibrium outcome of my mode. 6 More recenty, severa modes of adverse seection with price taking firms have been studied. Bisin and Gottardi 2006), Dubey and Geanakopos 2002), and Dubey et a. 2005) consider genera equiibrium modes with adverse seection that aways have a unique equiibrium, which has the same outcome as RS. Guerrieri et a. 2010) consider a competitive search mode in which the chance of an agent getting a given insurance contract depends on the ratio of insurance firms offering and agents demanding it. They show that equiibrium aways exists, can be tractaby characterized, and reduces to the Rothschid and Stigitz contracts in this framework. Hence their mode does not present the key empirica predictions discussed in this paper, which hinge on the presence of cross-subsidization. Since their equiibrium prediction is prior-independent, the equiibrium correspondence has a discontinuity at the perfect information case in which a agents have ow risk and efficient provision of insurance occurs. In the mixed strategy equiibria described here, the probabiity of attracting any type of consumer varies continuousy with the menu offered by a firm. This is in sharp contrast to competitive search modes: if the contract consumed by ow-risk 6 The distinguishing feature of their mode is the richness of the strategy set. A firm can react to moves by its opponents by offering a mechanism that contains a subsequent move by it. In equiibrium, a firm can respond to a cream-skimming attempt by an opponent by choosing to offer no insurance if the mechanism aows for such a move by the firm.

6 1354 Vitor Farinha Luz Theoretica Economics ) agents is perturbed to feature sighty more coverage, it is expected to be consumed excusivey by high-risk agents a beief restriction is imposed on off-path contracts). In the eary contributions of Wison 1977), Miyazaki 1977), and Riey 1979), it was shown that cross-subsidization can arise without randomization if firms are aowed to change their contract offers as a response to a deviation by a competitor. The equiibrium notions proposed in these papers aways exist and aso ead to equiibrium outcomes that are prior-dependent. These equiibrium notions have been found to be equivaent to equiibria of specific extensive forms with mutipe stages by Engers and Fernandez 1987), Hewig 1987), Mimra and Wambach 2011, 2016), and Netzer and Scheuer 2014). Whie the introduction of dynamics eads to interesting equiibrium outcomes with cross-subsidies, the objective of this paper is to highight that the introduction of stochastic offers by itsef eads to the same economic insights in a static framework. Rosentha and Weiss 1984) present an anaysis of a competitive version of the Spence mode that shares severa common feature with ours. They characterize a mixed equiibrium of the mode whenever a pure equiibrium does not exist. They have no resuts regarding uniqueness and dependence on the prior distribution. The effect of the number of firms on the constructed equiibrium is discussed, and is very simiar the one presented here. Chari et a. 2014) characterize a mixed strategy equiibrium in a inear competitive screening mode where firms are privatey informed about their asset quaities Mode A singe agent faces uncertainty regarding his future income. There are two possibe states {0 1} and his income in state 0 1) isy 0 = 0 y 1 = 1). The agent has private information regarding his risk type, which determines the probabiity of each state. For an agent of type t {h }, the probabiity of state 0 is p t. Assume that 0 <p <p h < 1. This means that the -type ow-risk) agent has higher expected income than h-type highrisk) agents. The prior probabiity of type t is denoted μ t.definep μ p + μ h p h.there are N identica firms i = 1 N that compete in offering menus of contracts. I assume that the reaization of the state is contractibe. A contract is a vector c = c 0 c 1 ) R 2 + that specifies, respectivey, the consumption eve for the agent in case of ow or high reaized income. Contracts are excusive. A menu of contracts is a compact subset of R 2 + that is denoted Mi. The set of a compact subsets of R 2 + is defined as M 2 R2 +. A specia case of a menu of contracts is a pair of contracts. I show ater that one can focus without oss on pairs of contracts, with each one of them targeted for one specific type. Timing is as foows. A firms simutaneousy offer menus of contracts M i M. Nature draws the agent s type according to probabiities μ and μ h. After observing his own type t and the compete set of contracts M 1 M N, the agent announces a choice 7 They obtain a partia uniqueness resut under the assumption that contract offers are ordered in terms of attractiveness. In my paper, this property is a centra part of my anaysis as it aows one to move from a two-dimensiona strategy space to a one-dimensiona subset.

7 Theoretica Economics ) Equiibrium in competitive insurance 1355 a i Mi {i}) { }. A choice a = c i) indicates that contract c M i is chosen from firm i, whie choice a = means that the agent chooses to get no contract and maintains his own income). A fina outcome of the game is M 1 M N t a) everything is evauated before the income reaization is reveaed). Given outcome M 1 M N t c i)),thereaized profit by firm j is zero if j i and otherwise is c t) 1 p t )1 c 1 ) p t c 0 The agents have instantaneous utiity function u ), which is stricty concave, increasing, and continuousy differentiabe. Finay, the utiity achieved by the agent is Uc t) 1 p t )uc 1 ) + p t uc 0 ) Given outcome M 1 M N t ), the reaized profit by a firms is zero and the utiity achieved by the agent is Uy t). A pure) strategy profie is a menu of contracts for each firm M i ) i and a choice strategy for the agent, which is a measurabe function s :{h } i M) R 2 + {1 N}) such that st M i ) i ) i Mi {i}) { }. A mixed acceptance rue is a Markov kerne 8 s :{h } i M) [ R 2 + {1 N}) ] with the restriction s i Mi {i}) { } t M i ) i ) = 1 with abuse of notation). Whenever the acceptance rue has a singeton support, we aso use st M i ) i ) to refer to the chosen contract. A mixed strategy profie is a probabiity measure over menus of contracts i for each firm i and a mixed acceptance rue s. 9 A mixed strategy profie defines a probabiity distribution over outcomes in the natura way; expected profits are defined by integrating reaized profits across outcomes according to this probabiity distribution. The equiibrium concept is subgame perfect equiibrium. 10 This means that i) each firm i maximizes expected profits, given the strategies used by its opponents and the acceptance rue used by the agent, and ii) the agent ony chooses contracts that maximize 8 The Markov kerne definition incudes the requirement that, for any measurabe set A R 2 + {1 N}, the function t M 1 M N ) sa t M 1 M N ) is measurabe. 9 IendowM with the Bore sigma agebra induced by the open bas in the Hausdorff metric. I use ony two properties from this sigma agebra: i) it contains any singe contract and ii) the function that eads to the best avaiabe utiity to any fixed risk type must be measurabe. 10 In this game, perfect Bayesian equiibrium PBE) is outcome equivaent to subgame perfection. Considering a game tree in which the firms act sequentiay, each subgame perfect equiibrium has a corresponding PBE with the same strategy profies and firms beiefs about the earier firms pay) given directy by equiibrium strategies. Notice that the agent, who moves ast, has perfect information because he knows a the offers and his type as we. In this game, the concept of Nash equiibrium aows the agent to behave irrationay to menu offers off the equiibrium path. This enabes many additiona cousive equiibria. In fact, I can sustain any individuay rationa aocation as a Nash equiibrium outcome.

8 1356 Vitor Farinha Luz Theoretica Economics ) his interim) utiity, i.e., c i) supp s t M i) i )) c arg max c Uc t) i Mi {y} and supp s t M i) i )) Uy t) max c i Mi Uc t) The optimization probem faced by the agent aways has a soution because the set of avaiabe contracts, i Mi {y}, iscompact. 4. Equiibrium construction In this section, I construct an equiibrium of the described mode. The existence issue raised in Rothschid and Stigitz 1976) is overcome by the use of mixed strategies by insurance firms. This is the first characterization of a mixed strategies equiibrium in an insurance setting. The nove feature of this equiibrium is the potentia presence of cross-subsidization, which generates a dependence of the equiibrium aocation on the risk distribution in this market. In Section 5, this equiibrium is shown to be unique. In the first part of this section, I assume that N = 2. I show, in the end of this section, how to adjust the equiibrium to the case N> Equiibrium offers In equiibrium, firms compete away profit opportunities. However, zero expected profits are consistent with cross-subsidization from ow-risk to high-risk agents: the presence of osses generated from high-risk individuas, which in turn get subsidized by profits from ow-risk individuas. In what foows, I construct a famiy of offers, indexed by the amount of cross-subsidization across types, and show that an equiibrium using these offers aways exists. Low-risk agents have higher expected income and, as a consequence, receive more attractive offers from firms. The ony way to respect incentive constraints is by offering partia insurance contracts i.e., with c 1 >c 0 ) to ow-risk agents. High-risk agents, aternativey, receive ess attractive contracts that do not confict with incentive constraints. As a consequence, they receive fu insurance contracts i.e., c 1 = c 0 ). This impies that the set of contracts that arise in equiibrium ies in a restricted ocus, which is described in the foowing paragraph. For a eve k [0 p h p] of subsidies received by high-risk individuas, the fu insurance contract received by high-risk agents has consumption c = 1 p h + k, which is above their actuariy fair consumption eve by k. Asodefineγk) = γ 1 k) γ 0 k)) to be the partia insurance contracts that can be offered to the ow-risk agent together with subsidy k to the high-risk agent. These contracts eave the high-risk agent indifferent between partia and fu insurance, which provides incentives efficienty, and generate zero expected profits.

9 Theoretica Economics ) Equiibrium in competitive insurance 1357 Formay, I define the set-vaued) function γ :[0 p h p] 2 R2 + by γk) { c R 2 ++ Uc h) = u1 p h + k); μ c ) + μ h k) = 0; c 1 c 0 } Lemma 1. For any k [0 p h p], γk) is a singeton, i.e., there exists a unique c R 2 + such that c γk). Proof. Letus defineζ = sup{c 1 c 0 such that Uc h) = u1 p h + k)}. The strict concavity of u impies that ζ>1 ζ = is possibe). Consider the path ι : I =[0 w] R 2 that starts at 1 p h + k)1 1) and moves aong the indifference curve of U h) by increasing c 1, i.e., ι 1 t) = 1 p h + k + t and define w ζ k 1 + p h ). Let tota profit generated by point t in the path, when the high-risk agent consumes 1 p h + k)1 1) and the ow-risk agent consumes ιt), be denoted as πt). We know that π0) 0 because k p h p. Ifζ<, continuity impies that πw) μ 1 p )1 ζ) < 0 If ζ =, it foows that im t πt) =. Therefore, in both cases continuity of πt) impies that there is t 0 such that πt 0 ) = 0. It aso foows from concavity of u ) that π t) < 0 for a t>0, which means that πt 0 ) = 0 for at most one point t 0. From now on, I refer to γ ) as a singe-vaued function. Figure 1 iustrates the ocus of {γk) k [p p]}. From now on, I refer to offers M k { 1 p h + k 1 p h + k) γk) } for k [0 p h p] as cross-subsidizing offers. cross-subsidization eve k as Aso define the utiities obtained from U k) U γk) ) U h k) u1 p h + k) The pair of contracts with zero cross-subsidization coincides with the unique equiibrium aocation in RS. Given their importance on the anaysis of this mode, we introduce notation to refer to these contracts. Definition 1. The Rothschid Stigitz RS) contracts are the pair { c RS ch RS } M 0 We aso define u RS t U t 0) for t = h and u RS u RS u RS h ). The Pareto efficiency of cross-subsidization pays a crucia roe in equiibrium anaysis. Cross-subsidization aways benefits high-risk agents, since their compete coverage comes at ower prices. What is more surprising is that ow-risk agents can aso benefit from cross-subsidization when the prior probabiity of high-risk is sufficienty ow. The reason for that it is that subsidizing high-risk is cheap when the probabiity of such a

10 1358 Vitor Farinha Luz Theoretica Economics ) Figure 1. The contract space and the image of the γ ) function. Notice that the RS partia insurance contract, c RS,isequatoγ0) and coincides with the owest eve of cross-subsidization. The other extreme point in the image of γ ) is γp h p), which features fu insurance at the correct price for the average popuation risk. state is sma. In the foowing emma, we show that the gains from cross-subsidization are negative for arge subsidization eves and are potentiay positive for ow subsidization eves. Lemma 2. There exists k [0 p h p) such that U ) is stricty increasing for k<k and stricty decreasing for k>k. More specificay, k is the unique peak of U ) in [0 p h p]. Proof. The impicit function theorem impies that γ is continuousy differentiabe and satisfies u p h 1 p h + k) + u γ 0 k) ) μ h γ 1 = p μ [ 1 p ) p h u γ 0 k) ) 1 p h) u γ 1 k) ) ] p p h u 1 p h + k) + μ h 1 p h ) μ 1 p ) u γ 1 k) ) γ 0 = 1 p ) p p h [ 1 p ) p u γ 0 k) ) 1 p h) p h u γ 1 k) ) ]

11 Theoretica Economics ) Equiibrium in competitive insurance 1359 which impies that γ 1 k) < 0 and γ 0 k) > 0. Simpe differentiation gives us U k) = { [u γ 1 k) ) 1 u γ 0 k) ) 1] 1 p )u 1 p h + k) [ 1 p ) 1 p ]) h) p p h μ h 1 μ p h / [ 1 p ) 1 p h p u γ 1 k) ) 1 p h) 1 p h u γ 0 k) ) ])} The numerator in the ast expression is stricty positive. The denominator is stricty decreasing for k [0 p h p] and negative for k = p h p in which case, γ 1 p h p) = γ 1 p h p) = 1 p). Whenever cross-subsidization eads to interim efficiency gains, which happens when k>0, quasi-concavity of U ) impies that the ow-risk utiity increases with the eve of subsidization at any eve k [0 k]. We define this restricted set of subsidization eves, which have the property that ow-risk agents benefit from it, as R [0 k] A comment on the iterature is in order. The Pareto efficient cross-subsidization eve k defines the separating aocation that maximizes the utiity offered to the owrisk agent subject to zero profits. It coincides with the aocation described by Miyazaki 1977), who obtains this aocation as an equiibrium outcome when using a reactive equiibrium notion. As described in Section 4.2, the equiibrium described here generates a cross-subsidization eves between zero and the efficient one. If cross-subsidization is not optima k = 0), the no cross-subsidization contracts presented in RS are indeed an equiibrium. This foows from the fact that there is no way that a firm can attract both types of agents and make expected positive profits. However, when cross-subsidization eads to interim gains k>0), then it has to arise in equiibrium. When facing contracts M 0 with no cross-subsidization, a firm can offer a menu with optima subsidization M k that wi generate zero expected profits whie eading to a stricty positive utiity gain for both risk types. This means that a sighty ess attractive offer can make positive profits. The presence of cross-subsidizing contract offers in a pure strategy equiibrium is rued out because they are vunerabe to cream-skimming deviations. If firms make positive profits from ow-risk agents and take osses on high-risk agents, one firm can offer a contract with sighty ess coverage that attracts ony ow-risk agents whie eaving the osses from high-risk agents to its competitors. The construction of such deviations, however, ony appies to pure strategies, as a firm facing a nondegenerate distribution of competing contracts is not abe to design a oca deviation that attracts the ow-risk agents with probabiity 1 whie attracting high-risk agents with probabiity 0. Infact,we show here that firms randomize continuousy between a Pareto efficient eve of crosssubsidization k and the RS contracts, which feature zero cross-subsidization.

12 1360 Vitor Farinha Luz Theoretica Economics ) Formay, every firm mixes over menu offers: M R { M k k R } The set M R has the property that if a firms offer menus within this set, then a offers in this set guarantee zero profits to a firm. This occurs because the firm that makes the offer M k with the highest eve of cross-subsidization attracts the agent, regardess of his risk type. However, the defining property of cross-subsidizing offers is that they make zero expected profits if both types consume the same menu. Firms that offer a crosssubsidizing eve beow their opponents make zero profits, as they never serve the agent. It is worth noting that the contract chosen by a high-risk agent in a cross-subsidizing offer aways has compete coverage, ony varying in the premium charged. The contract chosen by a ow-risk agent, however, has degree of coverage increasing with the amount of cross-subsidization. 4.2 Equiibrium distribution The goa of this section is to describe the equiibrium distribution over cross-subsidization eves k R, which is denoted as F, and to show that firms have no profitabe deviations outside of M R. The strategy set of firms contains a possibe contract menus and, hence, is very arge. The first step in our anaysis is to describe menu offers by the expected utiity it generates to each risk type. This description is usefu for the foowing reason. For each type, the utiity generated by a menu determines the probabiity with which it is chosen. This is the probabiity that the best aternative offer is ess attractive than the menu considered, which is determined in equiibrium. Aso, within the set of menus that deiver a specific utiity profie, firms wi ony offer the one that minimizes expected profits. This aows us to focus on a subset of menus that are indexed by utiity profies. The profit, or oss, that is made from each type in case he joins a firm is determined by a cost minimization probem that considers the utiity vectors as constraints. Define ϒ as the set of incentive feasibe utiity profies. 11 For each risk type t { h} and utiity profie u = u u h ) ϒ, define the ex post profit function P t u) as the soution to the probem subject to generating utiity u t to type t, max c t) c R 2 + Uc t) = u t and respecting incentive constraints regarding type t t, U c t ) u t 11 The foowing notation is important for the proof. Let ϒ { u u h ) R 2 Uc h ) Uc ) = u ; Uc h) Uc h h) = u h for some c c h R 2 +}

13 Theoretica Economics ) Equiibrium in competitive insurance 1361 Aso, define as χu) = χ u) χ h u)) the unique soution to probems P u) and P h u), respectivey. Contracts that maximize ex post profits are the most profitabe ones that deiver a specific utiity profie. In a mixed strategy equiibrium, the utiity offered by a contract determines the probabiity it is chosen. Hence, there are other menus that attract both types with the same probabiity and generate more profits. The foowing properties of the optima ex post profit function are key to our resuts. Lemma 3 Ex post profit characterization). The ex post profit function has the foowing properties: i) It is continuousy differentiabe. ii) Utiity is costy: P tu) u t < 0. iii) Separation is costy: and P t u) u t P t u) u t > 0 if u t >u t = 0 if u t u t P iv) Supermoduarity: t u t is continuousy differentiabe and 2 P t u) u t u 0, and the inequaity is strict if u t >u t t. TheproofisgivenintheAppendix. From the point of view of a singe firm, the offers made by other firms can be treated as a stochastic type-contingent outside option to the agent. The distribution of outside options for a given firm is determined by the equiibrium contract distribution in the foowing way. On the support of equiibrium offers, higher subsidization benefits both types, so the distribution is a monotone transformation of the distribution over crosssubsidization eve k R: and G U k) ) Fk) G h Uh k) ) Fk) Aso et G t u) = 0 for u<u t 1 p h ) and G t u) = 1 for u>u t k). The distributions G and G h constructed in this section are absoutey continuous. In Section 5, it is shown that this is necessariy the case in equiibrium. Expected profits are determined by the probabiity of attracting each type, which is given by distributions G t ) t= h, and the ex post profits made from each type, which is determined by the function P t ) t= h. Now we define this function formay: for any u = u u h ) ϒ, πu) μ G u )P u) + μ h G h u h )P h u)

14 1362 Vitor Farinha Luz Theoretica Economics ) When contempating a more attractive offer to a specific type, firms have to consider the foowing trade-off. When increasing the utiity promised to such agent, he wi be attracted with higher probabiity, which entais a gain if the firm makes profits out of such agent. However, to make a more attractive offer the firm has to make ess profits out of this agent, if indeed he ends up accepting the offer. In equiibrium, firms can ony make positive profits from ow-risk agents. For any utiity pair u = u u h ) with u u h, we define the margina profit from attracting the ow-risk agent as Mu) πu) u = μ [ g u )P u) + G u ) P ] u) 1) u where g t u t ) G t u t) is the density of the equiibrium utiity distribution. For cross-subsidizing offers M k for k R to arise in equiibrium, it must be optima for a firm to offer utiity profie Uk)= U k) U h k) ) for any k R. This means that G has to satisfy the equaity M Uk) ) = 0 for a k R 2) Hence, oca deviations around any offer in the support shoud not be optima. This is a necessary condition to sustain an equiibrium with this support. Using the equaity fk)= g U k))u k), wedefinef as the soution to the differentia equation impied by 2). The necessary condition for an equiibrium with support M R is for F to satisfy with fina condition Fk) = 1. fk) Fk) = P ) Uk) u U k) P Uk) ) 3) Lemma 4. The differentia equation 3) has a unique soution. Moreover, F is given by [ Fk)= exp k k ] φz)dz where P ) Uz) U u z) φz) = ) P Uz) Moreover, F puts no mass at zero if k>0, i.e., F0) = 0

15 Theoretica Economics ) Equiibrium in competitive insurance 1363 Proof. Integration of 3)impiesthat [ k ] 1 = Fk) = Fk)exp φz)dz k Finay notice that P Uk)) = γk) ) = μ h μ k. This means that φz) is on the order of 1 k around 1 p h. 12 Then it foows that im k 1 p h ) + k k φz)dz = This impies that F0) = im k 0) + Fk)= 0. As mentioned, condition 3) impies that any offer M k is ocay optima for k R. However, in equiibrium firms aso consider nonoca deviations. To rue out such deviations, we show that the expected profits are supermoduar in the utiity pair offered to the agent. This means that increasing the utiity offered to the high-risk agent makes it more profitabe, at the margin, to make a higher utiity offer to the ow-risk agent. Lemma 5 Supermoduarity of profits). For any u = u u h ) such that u u h, we have that Mu u h ) is nondecreasing in u h and it is stricty increasing if u >u h and G u )>0. The proof foows directy from the definition of M ) in 1), and properties iii) and iv) of Lemma 3. The reevance of the supermoduarity conditions is as foows. In equiibrium, the firm must find it equay optima to offer eves of subsidization k k R with k >k, which generate utiities Uk)= U k) U h k) ) U k ) U h k )) = U k ) This mean that, in our candidate equiibrium, firms are indifferent between offering Uk) or stricty increasing the utiity offered to both types to Uk ). But then supermoduarity impies that increasing ony the utiity offered to the ow-risk agent eads to a oss: π U k ) U h k) ) π U k) U h k) ) = < U k ) U k) U k ) U k) M U s) U h k) ) U s) ds M U s) U h s) ) ds = 0 Offers that deiver to a type t { h} utiity u t outside of support [U t 0) U t k)] can easiy be rued out: an offer that generates utiity u <U 0) never attracts ow-risk types, 12 Notice that U 0)>0 if k>0, P u0)) u > 0, and both U ) and P u )) u are continuous.

16 1364 Vitor Farinha Luz Theoretica Economics ) whie making profits from high-risk types is impossibe. A potentia offer that generates utiity u >U k) is dominated by another offer that generates exacty utiity U k) to ow-risk types: it sti attracts ow risks with probabiity 1 and makes higher profits in the case of acceptance. In the Appendix we provide the compete proof that offering cross-subsidizing contracts according to distribution F is indeed an equiibrium. The proof uses the supermoduarity property of the profit function in a simiar way to show that a possibe deviations are unprofitabe. Proposition 1. There exists a symmetric equiibrium such that i) every firm randomizes over offers in {M k k R},wherek [0 k] is distributed according to F ), and ii) after observing menu offers M k 1 M k 2) with k i >k j, the agent chooses according to s h M k 1 M k 2 )) = 1 p h + k i 1 p h + k i ) s M k 1 M k 2 )) = γk i ) After observing offers that are not of the form M k 1 M k 2), the agent chooses any arbitrary seection from his best response set Pure strategy equiibrium The equiibrium describedin Proposition 1 coincides with the pure strategy equiibrium characterized in Rothschid and Stigitz 1976), whenever it exists. The equiibrium invoves no mixing if and ony if k = 0 in which case the set of cross-subsidization eves offered in equiibrium is R ={0} and firms offer the zero cross-subsidization menu M 0, which coincides with the contract pair {c RS ch RS}. Lemma 2 shows that the benefits from cross-subsidization, from the ow-risk agent s point of view, are quasi-concave. As a consequence zero cross-subsidization is Pareto optima if and ony if U 0) 0 4) The uniqueness resut discussed in Section 5) impies that 4) is a necessary and sufficient condition for the existence of a pure strategy equiibrium. The exact condition in terms of the prior distribution is presented in the foowing coroary With the restriction that s is sti a mixed strategy, as defined in Section The conditions for the existence of pure strategy equiibria woud be different if each firm is restricted to offering a singe contract. In the menu game discussed here, firms can design more profitabe deviations invoving cross-subsidies; hence, the conditions under which a pure equiibrium exists are more demanding. It is worth noting that the characterization resut for mixed equiibria used in this mode does not extend to the singe-contract game mentioned. The crucia feature of the anaysis presented in this paper is the supermoduarity of the mapping between the utiity profie offered to different types and expected profits obtained by each firm. This property does not extend to the mode where each firm offers a singe contract. I thank one referee for pointing this out.

17 Theoretica Economics ) Equiibrium in competitive insurance 1365 Coroary 1. The equiibrium described invoves pure strategies if and ony if u ch RS ) [ ] 1 1 u c 1 RS ) u c 0 RS ) μ [ h ph 1 p ] h μ p 1 p If this is the case, a firms offer the pair of contracts {c RS ch RS } in equiibrium. A pure strategy equiibrium fais to exist whenever the share of ow-risk agents is sufficienty high. In this case, the cost of cross-subsidization is very ow since there are few high-risk agents to be subsidized. 4.4 The case N>2 In the anaysis of the duopoy case, I have shown that one can find a distribution over the set of menu offers M R such that each firm finds it optima to make any offer in this set. This support M R has the foowing property: the utiity obtained by both types, U ) and U h ), is stricty increasing in the cross-subsidization eve k for k R. Thismeans that if firm i = 1 faced two firms, 2 and 3, that were choosing offers M k according to continuous distributions F 2 and F 3, the reevant random variabe for firm 1 woud be k 23 = max{k 2 k 3 }, which determines the ony reevant threat from the offers of firms 2 and 3. The distribution of this variabe is given by Fk) = F 1 k)f 2 k). This aows us to adapt the arguments above by equaizing the distribution of the best among N 1 firms with the singe firm distribution in the duopoy. Proposition 2. InthegamewithN firms, the foowing strategy profie represents an equiibrium: every firm randomizes over offer set M R with distribution over crosssubsidization eve k R given by F i ),where F i k) = Fk) 1 N 1 The equiibrium described has the foowing properties. First, whenever there is randomization, ties occur with zero probabiity: there is aways a firm that offers M k i such that k i > max j i k j. The agent gets a contract from this firm, independent of which type is reaized. If the type is h, theagentendsupwithcontract1 p h + k i 1 p h + k i ).If the agent is of type, he chooses contract γk i ). Second, whenever the pure equiibrium with the RS contracts exists, R ={1 p h } and the support of strategies reduces to the RS contracts. In the next section, I show that this is the unique symmetric equiibrium. Section 6 presents monotone comparative statics resuts regarding the prior distribution and the number of firms. 5. Uniqueness In this section, we show that the equiibrium constructed in Section 4 is the unique symmetric equiibrium. We start by showing that equiibrium offers can be fuy described

18 1366 Vitor Farinha Luz Theoretica Economics ) by the utiity they generate to both possibe risk types. This means that describing the equiibrium strategies used by firms reduces to describing the equiibrium distribution of utiity eves generated by equiibrium offers. Describing offers in terms of utiity profies means that the offer space is essentiay two dimensiona. The main chaenge in the anaysis ies in showing that equiibrium offers necessariy ie in a one-dimensiona subset of the feasibe utiity space. The crucia step uses properties of the equiibrium utiity distribution and the ex post profit functions to show that expected profits are supermoduar in the utiity vector offered to the consumer. There is a compementarity in making more attractive offers to both risk types. Supermoduarity is used to show that equiibrium offers are necessariy ordered in terms of attractiveness, i.e., a more attractive offer provides higher utiity for both risk types. Firms inthismarketmake zeroexpectedprofits asshown in Proposition 3). The ordering of offers is used to show that equiibrium offers necessariy generate zero profits even if they are accepted by both risk types with probabiity Hence, offers can be indexed by the amount of subsidization that occurs across different risk types. The remaining anaysis foows standard steps in the iterature on games with one-dimensiona strategy spaces see Lizzeri and Persico 2000 and Maskin and Riey 2003). For an arbitrary mixed strategy φ for insurance firms, we denote as G the distribution of the highest utiity for each type t { h} induced by N 1 offers generated according to distribution φ. Formay, define the utiity obtained from offer M M by an agent of type t { h} as u M t max { Uc t) c M } as for any u R 2, Gu u h ) [ φ { M M u M t u t for t { h} }] N 1 In equiibrium, G is reevant because it determines the distribution of outside options that any given firm faces when trying to attract a consumer. Aso, we define as G t the margina of G over u t for t { h}. The equiibrium outcome distribution is denoted as P. Proposition 3 Zero profits). In any symmetric equiibrium, the foowing statements hod: i) Firms make zero expected profits. ii) Fix t { h}. If a firm i makes an offer M that generates utiity profie u, then χ t u) M and this is the ony possibe offer accepted by type t from firm i: for any c M \{χ t u)}, 15 Or if they are accepted in the market as a whoe. P [ t s t M M i ) )) = t c i) ] = 0

19 Theoretica Economics ) Equiibrium in competitive insurance 1367 iii) Firms make nonnegative profits from ow-risk agents and nonpositive profits from high-risk agents: any equiibrium utiity offer u satisfies u intϒ), u u h,and P u) 0 P h u) 0 The proof is provided in the Appendix. The previous statement contains three main resuts. First, it shows that firms never make positive expected profits in equiibrium. This foows from the assumption of Bertrand competition without differentiation. The main chaenge in the proof is to dea with the mutidimensionaity of the offer space. Second, it shows that equiibrium offers can be described in terms of the utiity profie they generate. If an offer M generates utiity profie u, then the ony contracts from this set that can be consumed in equiibrium are {χ u) χ h u)} M. This comes from the fact that a firm s maximization probem can be spit into choosing the attractiveness of the contract given by the utiity eves) and the minimization of costs for a fixed eve of utiity. 16 Finay, a consequence of zero profits is that firms necessariy take potentiay zero) osses on high-risk agents and make profits on ow-risk agents. The reason is that if an offer generates stricty positive profits from high-risk agents, a firm can aways guarantee ex ante expected profits since ow-risk agents are aways at east as profitabe as high-risk agents. Given Proposition 3, our uniqueness proof consists of showing that there exists ony one possibe equiibrium utiity distribution generated by each firm. From part ii) of the proposition, describing the distribution over utiity profies provides a description of equiibrium offers, namey χu). 17 The foowing intermediary emma provides a characterization of the equiibrium utiity distribution G. It shows that this distribution is absoutey continuous except at one point: the utiity eve generated by Rothschid Stigitz offers. For each t = h, define u t inf{u G t u) > 0} as the owest equiibrium utiity and, simiary, define u t sup{u G t u) < 1} as the highest equiibrium utiity. Lemma 6 Utiity distribution). In any symmetric equiibrium, the utiity distribution G satisfies the foowing conditions: i) Lower bound on utiity: u t u RS t. ii) Mass points for ow risk: G has support [u u ] and is absoutey continuous on [u u ]\{u RS }. iii) Mass points for high risk: the ony possibe mass point of G h is u RS h. 16 One has to consider the possibiity of the agent being indifferent between two contracts. However, this is soved in the proof of Proposition 3 by showing that a firm can aways break such indifferences at infinitesima cost. 17 Obviousy any equiibrium offer that generates utiity u can incude other contracts beyond {χ u) χ h u)}, as ong as they are unattractive. This means they satisfy Uc t) Uχ t u) t). Proposition 3 shows that if such contracts are present, they are irreevant, meaning they are never chosen on the equiibrium path.