An Efficient Competitive Insurance Market with Adverse Selection

Transcription

1 An Efficient Competitive Insurance Market with Adverse Selection Anastasios Dosis Preliminary and Incomplete September 21, 2014 Abstract We propose a simple market structure for general competitive insurance environments with adverse selection. An insurance company o ers a menu of insurance policies and decides whether to inflict the rest of the active rival companies in the market to include these policies in the menus they o er. Consumers with private information rationally select which company to join and what policy to buy. Unlike what is usually believed, we show that in our market equilibria generically exist and are e cient. 1. Introduction We study general competitive insurance environments with adverse selection. Rothschild and Stiglitz [22] (henceforth RS) have shown that in this type of environments the market fails to generically sustain a competitive equilibrium. We show that their argument significantly relies on the special market structure they examine. In particular, in RS, any new insurance company is free to enter the market anytime and o er any policy without taking into account those policies that are already o ered by incumbent companies. This, under certain conditions, can be problematic since for any menu of policies o ered by incumbent companies, an entrant can attract away of incumbent companies a su cient number of consumers and realise positive profits. This open-ended process is the main cause for the lack of existence of equilibrium in RS. Some of the ideas in this paper initiated in previous work (Dosis [7]), which was part of my PhD dissertation. Nonetheless, the mechanism examined in this paper fundamentally di ers with the one there which better fits into the informed principal literature. Address: Department of Economics, ESSEC Business School, Av. Bernard Hirsch, B.P , Cergy, 95021, France, dosis@essec.com, Tel:+33 (0)

2 We show that once one allows incumbent companies to influence the set of insurance policies that entrants can o er equilibria generically exist and are e cient. The environment we study consists of a continuum of risk-averse consumers with private information about the probabilities of su ering wealth-damaging accidents. The type and accident spaces can be arbitrary finite sets. 1 Unlike RS and follow-up studies in competitive insurance markets with adverse selection, 2 we do not impose any conditions in the space of utility functions such as monotonicity or single-crossing. Insurance companies, acting as pools of risk, o er insurance policies and consumers select which company to join and which policy to buy. A demand system for a company is defined as a set of demand functions one for each possible type. Notably, in our model, when an insurance company decides which policies to o er, it also decides if the rest of the companies in the market also have to make these policies available to consumers, a strategy that we call infliction. In other words, company A can inflict company B to o er policies that it (company A) o ers if company B decides to enter the market. We define a competitive equilibrium as a set of insurance policies o ered by all companies, infliction strategies and system of demands such that: (i) No company has an incentive to unilaterally deviate taking as given the demand system and the set of actions of rivals, and, (ii) Demand systems satisfy a weak subgame-perfection criterion. The latter means that demands of each of the insurance companies is in accordance with the utility maximisation problem of each type even for policies that are not actually o ered in equilibrium. In that sense, a company forms beliefs about the measure of types that a policy will attract if supplied and these beliefs must be consistent in equilibrium. The main result we establish is that competitive equilibrium always exists and is e cient. The key point that allows us to always sustain equilibria, even when the RS market does not, is that in our market any company A can always protect itself from rivals (entrants) through infliction. Consider for example the case where company A o ers an e cient menu of policies and inflicts the rest of the companies. In this case, all rival companies (including new entrants) are obliged to o er, along with the policies they o er, those policies o ered by company A. Hence, one can construct a rational demand system in such a way that any introduction of new policies in the market that try to attract good types will turn out to be unprofitable. On the contrary, company A can always select to o er policies without inflicting its rivals, which allows it to steal all customers from all companies if these have not maximised consumers surplus. In other words, our market, in accordance with e cient competitive markets, does not allow companies to earn strictly positive profits in equilibrium. 1 Our analysis and all results go through even if there is a continuum of possible types and a continuum of possible states. This stands in stark contrast with RS who argue that with if the type space is a continuum, equilibrium never exists. See also Riley [20]. 2 See also Related Literature. 2

3 Albeit the generic existence, our market is characterised by a generic indeterminacy of equilibria. Specifically, the set of equilibrium payo s is equivalent to the set of payo s from all individually rational and constrained Pareto optimal allocations. That is, even for these parameter values for which an equilibrium in the RS market exists and is unique, our model sustains a continuum of equilibria. In terms of the literature, apart from RS, Wilson [24] andriley[20] examinealternative notions of competitive equilibrium, known as reactive equilibrium, to guarantee that equilibrium always exists. Miyazaki [15] extends the reactive equilibrium of Wilson [24] bylettingcompaniestoo ermenusandshowsthatequilibriumexistsandis constrained e cient. 3 A serious drawback in Wilson [24], Riley [20] andmiyazaki[15] is that they do not examine explicit games but rather define properties of competitive equilibrium. 4 Hellwig [11] and Engers and Fernadez [9] specifiedgamesforthemodels of Wilson [24] andriley[20] respectively.theyexaminedi erentmarketmechanisms in which equilibrium exists but the equilibrium set includes allocations that are not zero-profit or constrained e cient. Only few papers tackle both existence and e ciency of equilibria simultaneously. For instance, Asheim and Nilssen [1], allow insurance companies to renegotiate the contracts they have signed with their customers, but prohibiting them to discriminate among the di erent types in the renegotiation stage. When e ciency requires cross-subsidisation, in equilibrium companies o er a menu of contracts. Renegotiation creates credible o -the-equilibrium path threats that make any deviation that tries to attract low-risk types unprofitable. The unique equilibrium coincides with MW. Picard [17] examines a model in which insurance companies can o er participating contracts such that any consumer who signs a contract needs to participate in the profits of the firm who o ered it. When e ciency requires cross-subsidisation, then in equilibrium companies o er menus of participating contracts. Deviations that are supposed to attract only low-risk types, and they are responsible for the lack of existence of equilibrium in RS, would attract also high-risk types since the latter have to pay the losses made by their companies and therefore are better-o switching to the new contract. With two types, the equilibrium coincides with MW. With more than two types, the equilibrium is characterised based on the method provided by Spence [23]. Netzer and Scheuer [16] andmimraandwambach[14] alsoproposealternative mechanisms for insurance markets with adverse selection. In Netzer and Scheuer [16], insurance companies play the following three-stage game: In the first stage, they o er 3 The specific allocation that results as a unique equilibrium in Miyazaki [15] is also known as Miyazaki-Wilson allocation (MW). 4 As a matter of fact, RS itself does not specify an explicit game among insurance companies. However, this is possible as it is shown in MasColell et al. [13] and Jehle and Reny [12]. In this game that resembles Bertrand competition, a pure strategy equilibrium does not exist. Nonetheless, mixed strategy equilibrium exists as it is shown in Rosenthal and Weiss [21], and Dasgupta and Maskin [4, 5]. 3

4 menus of policies. In the second stage, each company decides whether to stay in the market, or become inactive. In the latter case, it pays an exogenously-given withdrawal cost. In the last stage, consumers select from the set of policies o ered by all active firms. Existence of equilibrium crucially depends on the ad hoc specification of the withdrawal cost. If the cost is high, an equilibrium fails to exist for the same reason it fails to exist in RS. For certain (small) values of the withdrawal cost equilibrium exists and coincides with the MW. Mimra and Wambach [14] allow insurance companies, instead of becoming inactive, to eliminate individual contracts from those they have o ered in an endogenously ending number of rounds. Consumers choose from those policies that survive elimination. Without further restrictions, the equilibrium set of this game contains every incentive compatible and positive profit allocation. Mimra and Wambach [14] thenshowthat the only equilibrium that is robust to entry is the MW. Entry in this model is rather ad hoc, since entrants cannot make initial o ers of policies along with the incumbent companies but only in the elimination stage. In our model all companies are active players from the beginning and our results are robust to any form of entry. 5 There are several advantages of our mechanism compared to those discussed in the previous paragraphs. First, all the aforementioned papers follow the rather restricted environment of RS, according to which types can be clearly ranked according to the following monotonicity condition: The profit (cost) of every insurance policy is increasing in the same order of the set of types. Moreover, all papers impose the rather strict condition that for every policy, indi erence curves cross only once (single-crossing condition). Based on those two conditions, the usual procedure is to characterise equilibrium by showing that this coincides with the MW allocation. This leaves open the question of what happens when these conditions fail. For instance, what is the Miyazaki-Wilson allocation in more general environments when the monotonicity and single-crossing conditions are not satisfied? Do equilibria exist in these more general environments? If yes, what is their structure? All these remain unclear in all these papers. In contrast, all our results hold in much more general environments with general type spaces and utility functions. Second, our market is way simpler than all those discussed above. In fact, our market resembles a simple competitive market for insurance policies with a slight twist and the solution concept is a simple modification of subgame perfect Nash equilibrium. Our paper is also related to the literature in general equilibrium with adverse selection. Prescott and Townsend [19, 18], Gale [10], Dubey and Geanakoplos [8], Bisin and Gottardi [2] and Citanna and Siconolfi [3] whoanalysewalrasianmarketswith adverse selection. For instance Bisin and Gottardi [2] allowconsumerstotradeproperty rights that permit them to buy insurance from insurance companies. They show 5 Another paper which extends the game of Hellwig [11] is Diasakos and Koufopoulos [6] whoshow that the MW allocation is the unique equilibrium of the game. However, neither the action/strategy space nor the contract space are well-defined in this paper. 4

5 that with the right ex ante allocation of property rights, e ciency can be achieved in acompetitiveequilibriumwithexternalities. Inourmarket,thereiscompetitionbut it is imperfect in the sense that companies are not price-takers. The paper is organised as follows: In Section 2 we present the insurance environment, insurance policies, payo s and allocations. We also characterise the market structure and the set of actions and payo s. A competitive equilibrium is defined. In Section 3, we analyse the equilibria of the market. First, we construct equilibrium demand systems and actions. Last, we show that any equilibrium is necessarily e cient. 2. The Insurance Market Consumers and Insurance Companies. There is a continuum of risk-averse consumers of di erent types. The set of all types is finite and denoted by T with a representative element t 2 T. 6 The measure of type t consumers is t with P t2t t = 1. The type of each consumer is private information. There is a finite number of individual states with representative element! 2. Each consumer starts with initial wealth W and can su er state-dependent losses l :! R +. Type t s objective probability distribution function over the states is denoted by t :! [0, 1], with P!2 t (!) =1. The individual state is perfectly observable and verifiable by a court of law. The von Neuman-Morgenstern utility index of type t is u : R! R; astrictlyincreasingand strictly concave function. As opposed to most of the literature in insurance markets with adverse selection, we do not make any further assumptions on the space of utility functions and therefore our results are fairly general. There is a finite set I of profit-maximising, risk-neutral, private insurance companies. A representative company is denoted as i with i 2 I. Insurance Policies and Allocations. Consumers and insurance companies exchange insurance policies. An insurance policy is denoted by =(p, b) andspecifiesapremium and state-dependent benefits b :! R +. The set of feasible insurance policies is denoted by. The expected utility of type t from insurance policy is given by U t ( ) =!2 (!)u t [W l(!) p + b(!),!] We denote the null contract as o and the (status quo) utility of type t as: U t =!2 t (!)u[w l(!),!] 6 A bit of explanation on notation might be useful. We denote sets with capital letters. We denote elements of sets with small letters. We denote as c(k) the power of set K. We denote a set that consists of sets with a bold capital letter. A set whose elements have been indexed using some index k 2 K is denoted as = {A k } k2k or = {a k } k2k. 5

6 The net expected profit (cost) of insurance policy when taken up by type t is denoted as t ( ) andgivenby t ( ) =p t (!)b(!)!2 When c(t )=c( ) = 2 and u t [,!]=u[ ] foralltand!, themodelisidentical to RS. 7 In this model, there are only two individual states, no accident and accident respectively, and two possible types, high-risk and low-risk type respectively, and all types have the same type and state independent utility index. An allocation = { t } t2t is a set of policies one for each possible type. 8 An allocation is incentive compatible (IC) if for all t, t 0 2 T, U t ( t ) U t ( t0 ). The set of all incentive compatible allocations in denoted by IC.Thenetexpectedprofit (cost) of an IC allocation is denoted by Z( ) = P t2t t t ( t ). An IC allocation is individually rational (IR) if for all t 2 T, U t ( t ) U t. An allocation is a strong Pareto optimum if (i) it is incentive compatible, (ii) positive-profit or Z( ) 0, and, (iii) there exists no other incentive compatible and positive-profit allocation such that for each t 2 T, U t ( t ) U t ( t )withtheinequalitybeingstrictforatleastone t 2 T. An allocation is a weak Pareto optimum if we take all inequalities to be strict. SPO The set of individually rational, strong and weak Pareto optima is denoted by and WPO respectively. In Appendix A, we provide properties of incentive compatible allocations, which will be used in order to prove the results of the paper. Market Structure, Strategies and Payo s. An action is denoted by =(Q, c) and consists of a set of a finite number of policies denoted by Q and a decision c 2{0, 1}, where c =1isinterpretedas infliction. An action for company i is denoted by i. The RS market always imposes c i =0foreveryi 2 I. The set of all possible actions is given by A = {Q {0, 1}} [ {;}, whereq is the set of all menus of policies. If i = ;, company i decides not to enter the market. A profile of actions is denoted by = { i } i2i. 9 Aprofileofactionsforallcompaniesotherthani is denoted by i = { i } i2i/{i}. When all companies play, thesetofpoliciessuppliedbycompany i becomes S i = Q i [{ 2 Q j :(j 2 I/{i})(c j =1)}. For each company, a demand system is a set of functions one for each possible type of the form Dt i :! [0, t ]. When company i is contemplating to supply a new policy, it forms beliefs about the measure of types that this policy will attract. These beliefs must be consistent in equilibrium as we will see in our definition of equilibrium. The expected profit of company i from action i when all other companies play 7 A more general model for c(t ) > 2 and c( ) = 2 was analysed at the same time by Wilson [24]. 8 A menu of policies is an unordered set of policies indexed by the set of types. 9 We apologise for slightly abusing notation and denoting with a small letter aset. 6

7 i is denoted as i ( i, i )andgivenby: 8 >< 0 if i = ; (?) i ( i, i )= >: P 2S Pt2T i Di t( ) t ( ) if i 6= ; If company i decides to o er the empty set (and therefore not to enter the market) its payo is zero. However, if company i decides to enter the market, its profit for its supplied policies S i depends on the profit of each of the policies and the demand from each of the types. Abestresponsecorrespondenceforcompanyi 2 I is denoted as B i ( i )andis given by: B i ( i )=argmax i 2A i ( i, i ) We are now ready to define a competitive equilibrium in our market. Definition-Equilibrium: Acompetitiveequilibrium(ˆ, ˆD)isaprofileofactions ˆ = {ˆ i } i2i,withˆ i =(ˆQ i, ĉ i ), and demand systems ˆD = { ˆD i t} t2t i2i such that: (I) For every i 2 I, ˆ i 2 B i ( ˆ i ). (II) For every i 2 I and i =(Q i,c i ) 2 A, (a) If /2 i t = S i \{ : 2 (b) If i t = ; then ˆD i t( ) =0forevery 2 S i. (III) For every i 2 I and i =(Q i,c i ) 2 A, ˆD t( i )+ ˆD j t ( ) = t, arg max U t ( )} 6= ;, then ˆD t( i ) =0 2Q i [([ ˆQj j2i/{i} )[{ o} 2 i t j2i/{i} 2 j t If j t = ; for all j 2 I, then ˆD t ( o )= t,where ˆD t ( o )representsdemandfor the null policy.. It is useful to denote as Û t the equilibrium utility level of type t consumers from equilibrium ( ˆ, ˆD). Condition (I) is the usual no unilateral deviation condition for each of the insurance companies. Companies entertain Nash (myopic) expectations about the behaviour of the rival companies in the market when deciding their strategies. Conditions (II) and (III) stem from subgame perfection. According to Condition (II), in equilibrium, given 7

8 the actions of the rival companies, company s i demand system for any policy that is not in the set of payo -maximising policies for type t must be necessarily zero. This is also true for policies that are not supplied in equilibrium. Similarly, given the equilibrium actions of the rival companies, the sum of the demands of all companies for policies that are payo -maximising for type t must be t ;theexantemeasureoftypet in the economy. Notice that the definition requires that this is also true for policies that are not supplied in equilibrium. If no policy supplied in the market is payo maximising for type t, thentypet selects his status quo policy and therefore D t ( o )= t. Our definition of competitive equilibrium, even though game theoretic in flavour, has a twist of the theory of competitive markets. Companies supply policies and form beliefs about the measure of types these policies will attract. The market is in equilibrium if given those beliefs and the actions of the rival companies, which can be considered as the state of the economy, no company has an incentive to unilaterally vary its strategy. 3. Equilibria Main Result. As it is known by RS, in insurance markets with adverse selection equilibrium does not generically exists. In our market, as we show in the following theorem, this is never the case. Theorem 3.1: Equilibrium generically exists. Proof: Consider the following profile of actions ˆ = {ˆ i } i2i such that for every i 2 I, ˆ i =(ˆ, 1), where ˆ 2 SPO,anddemandsystems ˆD = { ˆD t} i t2t i2i such that for every i 2 I: 8 t, c(i) if Qi = ˆ and 2 arg max 2 ˆ U t ( ) >< ˆD t( i ) = t, if Q i 6= ˆ and 2 arg max 2Q i [ ˆ U t ( ) >: 0, if Q i 6= ˆ and /2 arg max 2Q i [ ˆ U t ( ) It is straightforward to verify that ( ˆ, ˆD) satisfyconditions(i),(iii)and(iii)of our definition of equilibrium. Q.E.D. When an insurance company is contemplating to o er new policies, it needs to form expectations about the demands for these policies. The reason for the universal 8

9 existence of equilibrium in our market is that for every company, a demand system can be constructed that makes any deviation from a SPO menu of policies unprofitable. In other words, if a company introduces a di erent menu of policies than the one offered by rivals, it may expect to attract a su ciently number of all types of consumers such that its aggregate profit is strictly negative. Recall that all types of consumers lose nothing switching from one company to the other since in the characterisation of equilibrium we have assumed that all companies inflict. A possible intuitive explanation why a company entertains such beliefs is that when consumers observe a deviation from the equilibrium menu of policies by some of the companies, they expect that the rest of the companies may go bankrupt.. Anticipating such a behaviour, no company has an incentive to supply new policies that would attract any of the types. Theorem 3.2: If ( ˆ, ˆD) isanequilibrium,thenû t = U t ( t )forsome 2 SPO. The intuition behind Theorem 3.2 could be simply explained in the following adjustment process. If Û t = U t ( t )forsome 2 SPO then some of the companies has an incentive to supply a new menu of policies that is strictly preferred by all types and makes strictly higher profits and select that the rest of the companies should not include these policies in their already supplying menu. Therefore, regardless of the actions of the rivals at least one company will always have an incentive to undercut prices and increase profits. This process terminates when at least one company has supplied a menu of policies that is SPO. We now provide a formal proof of Theorem 3.2. Proof of Theorem 3.2: Assume to the contrary that there exists an equilibrium ( ˆ, ˆD) withû t <U t ( t )forall 2 SPO.Therearetwocasestobeconsidered: Case 1. Assume that P i2i i (ˆ i, ˆ i ) > 0. There exists at least one j 2 I such that j (ˆ j, ˆ j ) < P i2i i (ˆ i, ˆ i ). Consider allocation ={ t } t2t such that: u[w d(!) p t + b t (!),!]= 1 ˆD i t t( )u[w d(!) p t + b t (!),!] i2i where ˆ t i is the set of equilibrium payo -maximising policies of type t supplied by company i. Straightforwardly, U t ( t )=Ût. Some of the properties of allocation are summarised in the two auxiliary lemmas. Lemma 3.3: isincentivecompatible. Proof of Lemma 3.3: By definition, for any t 2 T, Û t U t ( ) forany 2 [ i2i S i. Multiplying and summing up over all payo -maximising contracts of any of 9 2 ˆ i t

10 the types t 0 2 T/{t}, U t ( t )=Û t 1 t 0 for all t 0 2 T/{t}, or U t ( t ) U t ( t 0 ) for all t 0 2 T/{t}. Q.E.D. Lemma 3.4: Z( ) i2i 2 ˆ i t 0 ˆDi t 0( )U t ( ) P i2i i (ˆ i, ˆ i ). Proof of Lemma 3.4: By definition of allocation u[w d(!) p t + b t (!),!]= 1 t Since for every! 2, u[,!]isstrictlyconcavewehavethat p t bt (!) 1 t i2i 2 ˆ i t i2i 2 ˆ i t ˆD i t( )(p t b t (!)) ˆD i t( )u[w d(!) p t + b t (!),!] By linearity of t ( ) ( p t bt (!))!2 1 t i2i 2 ˆ i t ˆD i t( )!2 (p t b t (!)) or t t ( t ) ˆD i t( ) t ( ) i2i 2 ˆ i t Summing up over t, t t ( t ) t2t t2t i2i which can be rewritften as t t ( t ) t2t i2i t2t 2 ˆ i t 2 ˆ i t ˆD i t( ) t ( ) ˆD i t( ) t ( ) 10

11 From (?) thisisequalto Z( ) (ˆ i, ˆ i ) Q.E.D. i2i We can now complete the proof. Consider action j =(, 0) for company j, such that issuchthatu t ( t ) >U t ( t )foreveryt 2 T and Z( ) = Z( ) for some >0 arbitrarily small. From Lemmas A.1 and A.2 of Appendix A, we know that such an allocation exists. From Condition (III) of our definition of equilibrium, given that U t ( t ) > Û t,itmustbethatdt( i t )= t for every t 2 T.From(?) we have that j ( j, ˆ j )=Z( ) > i2i (ˆ i, ˆ i ) > j (ˆ j, ˆ j ) which contradicts that ˆ j 2 B i ( ˆ j ). Case 2. Assume that P i2i i (ˆ i, ˆ i )=0whichnecessarilymeansthatforevery i 2 I, i (ˆ i, ˆ i ) = 0. With arguments similar to Case 1 and Lemma A.4 of Appendix A, we can show that there exists some =(, 0) such that U t ( t ) > Û t for every t 2 T with Z( ) > 0whichcontradictsthethesisthatˆ j 2 B i ( ˆ j )foralli 2 I. Q.E.D. Corollary 3.5: The set of equilibrium payo s {Û t } t2t is such that Û t = U t ( ˆt ) for any ˆ 2 SPO. 4. Conclusion In this paper, we proposed a market structure for general competitive environments with adverse selection. Unlike RS, we showed that in our competitive market equilibria generically exist and are e cient. One of the advantages of the market structure we proposed lie on its simplicity. The market is a straightforward variant of a simple market in which companies compete in insurance policies. Even though our contribution is mainly theoretical, it is not di cult to imagine situations in which the mechanism we propose is applicable. For instance, in many private health insurance markets like Switzerland, The Netherlands, and recently the US with the A ordable Care Act (ACA), insurance provision is highly regulated. For instance in the US, the rather controversial Healthcare Exchange Marketplace (HEM), was an initiation from the Federal government in order to promote competition in the market for health insurance and to give access to health insurance to more individuals. HEMs are regulated, online marketplaces in which insurance companies and consumers 11

12 meet and trade insurance policies. HEMs objective is twofold: First, it accommodates more fierce competition among insurance companies, and, second its use facilitates the provision of federal subsidies. HEM provides a first class opportunity towards the e cient regulation of the health insurance market in the US and an ideal way for the implementation mechanism. This is because of the transparency it provides as well as its organisation. The regulator can easily monitor the set of menus of policies o ered by companies and can force companies to include in their menus the set of policies of other companies who have decided to make their policies common. Lastly, the structure can be applied to many other competitive environments with adverse selection such as credit or labour environments in which one side of the market hold private information about the commodity traded. Appendix A: Properties of Incentive Compatible Allocations Lemma A.1: For every, 2 there exist 0 < < 1and 2 Z( ) > Z( ) + (1 )Z( ). IC, such that strictly Pareto dominates, IC that also strictly Pareto dominates with Proof: Take, 2 IC,suchthat strictlyparetodominates.considerthe following random allocation: Every type t is o ered a contract that after the realisation of the state of nature!, thereisalotterywhichwithprobability pays p t +b t (!) and with probability 1, p t + b t (!). The expected utility of type t from this random contract can be written as: U t = t (!) u[w!2 l(!) p t + b t (!),!]+(1 )u[w l(!) p t + b t (!),!] For every 0 < <1, every t 2 T and every! 2 wecanfind equivalent) such that: p t + b t (!) (thecertainty u[w l(!) p t +b t (!),!]+(1 )u[w l(!) p t + b t (!),!]=u t [W l(!) p t + b t (!),!] Because of the strict concavity of the utility function and by Jensen s inequality, W l(!) p t + b t (!) < [W l(!) p t + b t (!)] + (1 )[W l(!) p t + b t (!)] or p t bt (!) > [p t b t (!)] + (1 )[ p t bt (!)] Multiplying by t (!) andsummingupover!, p t t (!) b t (!) > [p t t (!)b t (!)] + (1 )( p t t (!) b t (!)]!2!2!2 12

13 or t ( t ) > t ( t )+(1 ) t ( t ) Summing up over t: t t ( t ) > t2t t t ( t )+(1 t t ( t ) t2t ) t2t Given that, 2 IC,forany, (, 1 ) 2 IC. Since u( ) istype independent, 2 IC. Moreover, for any 0 < <1, U t ( t ) <U t ( t ) <U t ( t ), therefore strictlyparetodominates.q.e.d. Lemma A.2: For every 2 IC and >0, there exists 2 Pareto dominates with Z( ) >Z( ). IC that strictly Proof: Take allocation 2 IC.Considerthefollowingcomplete-risk-poolingallocation, where t =( p, (b(!))!2 )foreacht 2 T and p < P P t t2t!2 t (!)l(!). From Lemma A.1, there exists 0 < <1and thatstrictlyparetodominates such that Z( ) > Z( )+(1 )Z( ). For =(1 )[Z( ) Z( )], and p appropriately chosen, we obtain the result. Q.E.D. Corollary A.3: Every 2 WPO is such that Z( ) =0. Lemma A.4: For every 2 IC,withZ( ) = 0 and /2 2 IC that strictly Pareto dominates with Z( ) > 0. SPO,thereexists Proof: Case 1. Assume first that /2 WPO,withZ( ) = 0. By definition, there exists 2 WPO that strictly Pareto dominates. From Corollary A.1, Z( ) = 0. From Lemma A.1, there exists 2 IC that strictly Pareto dominates with Z( ) > Z( ) + (1 )Z( ) = 0. Case 2. Assume that 2 WPO but /2 SPO. There exists 2 SPO that weakly Pareto dominates. Let the set of types whose utility remains the same in both allocations be T 1 and those whose utility is strictly higher under bet 2. By following the same logic as in the proof of Lemma A.1, we can find 2 IC with U t ( t ) > U t ( t )forallt 2 T 2 and U t ( t ) = U t ( t ), for all t 2 T 1. Moreover, Z( ) > 0. From Lemma A.2, for any >0, there exists ˆ 2 IC that strictly Pareto dominates andz( ) >Z( ) 0, for small enough. Q.E.D. Corollary A.5: The sets of strong and weak Pareto optima coincide, or: WPO. SPO = 13

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