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ENDOGENOUS INSOLVENCY IN THE ROTHSCHILD-STIGLITZ MODEL.

INTRODUCTION

The Rothschild and Stiglitz (1976) (RS) model on competitive insurance markets with adverse selection is considered as one of the major contributions to the literature on markets with asymmetric information to date. Yet, the RS model entails a puzzle: if the share of high-risk types in the population is low, an equilibrium in pure strategies fails to exist. The reason is that if there are relatively few high-risk types, a pooling contract would be preferred by all types over the candidate separating, contract-wise zero profit-making RS contracts. However, a pooling contract cannot be tendered in equilibrium as insurers would try to cream skim low risks. (1) This equilibrium inex-istence result has spurred extensive research, and although several modifications to the RS model have been brought forward, the debate is still ongoing and has gained in interest in recent years. (2)

In the literature, mainly two candidates for the equilibrium allocation have emerged in competitive insurance markets with adverse selection. One is the original RS allocation, even if it is not an equilibrium in the RS model (e.g., Riley, 1979; Engers and Fernandez, 1987; Inderst and Wambach, 2001; Ania, Troger, and Wambach, 2002; Mimra and Wambach, 2016). The second is the Miyazaki-Wilson-Spence (MWS) allocation (e.g., Asheim and Nilssen, 1996; Mimra and Wambach, 2011; Netzer and Scheuer, 2010; Netzer and Scheuer, 2014; Picard, 2014). Now the RS allocation is only efficient if it is an equilibrium in the RS model with contract menus, whereas the MWS allocation is generally second-best efficient; thus, it matters which allocation prevails. Besides equilibrium existence, we therefore also contribute to the discussion on the necessity of efficiency-based regulation of insurance markets by providing another foundation for the MWS allocation.

We model a basic extension to the RS model: instead of being exogenously endowed with sufficiently high assets as in RS, insurers choose the level of up-front capital before entering the market stage. Now under limited liability, low up-front capital gives rise to an endogenous insolvency risk as, depending on contract offers and the distribution of risk types over the contracts of an insurer, there might not be sufficient assets to fulfill all claims. (3) Endogenous insolvency risk introduces an externality among customers of a firm as an individual's expected utility now does not only depend on contract parameters but also on the risk profile of the other customers of the insurer. This externality guarantees equilibrium existence--we show that with capital choice under limited liability, an equilibrium in pure strategies that yields the second-best efficient MWS allocation exists. The MWS contracts are separating and cross-subsidizing; high-risk types are fully insured at a more than fair premium and low risks are partially insured at an unfair premium offsetting losses from high risks. (4) The equilibrium is sustained as cream-skimming offers aimed at attracting low risk types lead to a deterioration of high risks' MWS contract due to insolvency when up-front capital is low. High risks then prefer to choose the deviating contract as well, rendering the deviation unprofitable. Now putting in more capital is not profitable for an insurer, as this only increases incentives of competitors to cream skim low risks away from this insurer. Hence, low up-front capital and the MWS allocation are an equilibrium outcome. One implication of our analysis is that minimum capital requirements might have unintended consequences: if imposed capital requirements are too strict, a second-best efficient equilibrium fails to exist.

We build upon the work by Faynzilberg (2006). Faynzilberg discusses the set-up with exogenous capital and argues that an equilibrium with the MWS allocation exists. We recast the model, show equilibrium existence, and derive conditions for uniqueness when firms do not hold any up-front capital and then proceed to show that an equilibrium with firm capital choice in which consumers obtain their respective MWS contract exists.

Externalities that guarantee equilibrium existence in competitive markets with adverse selection are modeled in Picard (2014) and von Siemens and Kosfeld (2014). Picard modifies the structure of insurance contracts and shows that with participating contracts as, for example, implemented in mutuals, the MWS contracts constitute an equilibrium allocation. Interestingly, in Picard, as in our work, externalities are not superimposed but do arise endogenously. In Kosfeld, direct externalities between workers are modeled, high-productivity workers do not want to be pooled in teams with low-productivity workers. This sustains the RS allocation as equilibrium allocation as potentially profitable deviations involve pooling at a firm, which makes it unattractive for high-productivity workers to join this firm.

THE MODEL

There is a continuum of individuals I of mass 1 in the market, representing a large population of consumers. Each individual faces two possible states of nature: in state 1, no loss occurs and the endowment is [w.sub.0]; in state 2, a loss D > 0 occurs and the endowment is [w.sub.0] - D with [w.sub.0] > [w.sub.0] - D > 0. There are two types of individuals: an individual may be a high-risk type (H) with loss probability [p.sub.H], or a low-risk type (L) with loss probability [p.sub.L], with 0 < [p.sub.L] < [p.sub.H] < 1. Individuals have a twice continuously differentiable strictly concave von Neumann-Morgenstern utility function v(w). Insurance is provided by firms in the set F := {1, ..., f, ...m}. An insurance contract [omega] = (P, I) specifies a premium P and an indemnity I. The set of feasible contracts is [OMEGA] := {(P, I) [member of] [R.sup.2]|I [less than or equal to] D}. Firms are risk neutral and do not know, ex ante, any individual's type.

The timing of the market interaction follows RS; that is, insurers offer contracts first, then insurees choose. The difference from RS is that as RS assume insurers to be exogenously endowed with sufficiently high capital such that insolvency cannot occur, insurers in this model choose the level of up-front capital before entering the market stage. Firms will be subject to limited liability; that is, if a loss occurs and the insurer does not have enough assets, full indemnity payment might not be possible. Formally, the timing of the game is as follows:

Stage 0: The risk type of each individual is chosen by nature. Each individual has a chance of [gamma], 0 < [gamma] < 1 to be a H-type, and of (1 - [gamma]) to be a L-type.

Stage 1: Each firm f [member of] F decides on the level of its up-front capital [K.sup.f]. There are non-negative opportunity costs r to holding capital. (5)

At the end of stage 1, each individual and each firm observes each firm's up-front capital.

Stage 2: Each firm f [member of] F offers a finite set of contracts [[OMEGA].sup.f] = {[[OMEGA].sup.f.sub.1], [[OMEGA].sup.f.sub.2]}, ..., [[OMEGA].sup.f.sub.k].

Stage 3: Individuals choose their insurance contract. (6)

Stage 4: Losses are realized. Insurance firms pay out indemnities. If total claims [C.sup.f] at firm f are less than total assets [A.sup.f], the insurance firm fully settles claims. If total claims [C.sup.f] exceed total assets [A.sup.f], then the insurance firm pays out the assets and defaults on the remaining claims. The ex ante known insolvency rules specify proportional payout, that is, a customer at firm f who has bought contract [[OMEGA].sup.f.sub.g] and has realized a loss receives a fraction [[beta].sup.f]= [A.sup.f] / [C.sup.f] of her indemnity claim [I.sup.f.sub.g]. (7)

For expositional convenience, we will henceforth call the subgame starting with firms' contract offers in stage 2 the "market game."

Firm assets are composed of two components: up-front capital and premium income. When up-front capital is low, depending on the offered contracts and distribution of agents across the contracts, insolvency might occur. However, the risk of insolvency also impacts a customer's expected utility from choosing a contract at a specific insurer. Assume that contract [omega.sup.f.sub.i] [member of] [OMEGA.sup.f] is taken out by the mass of individuals [[lambda].sup.f.sub.i] and let [[sigma].sup.f.sub.l] denote the share of high risks in [[lambda].sup.f.sub.i] Then the expected utility of a J-type individual J [member of] {H, L} from choosing contract [[omega].sup.f.sub.i] [member of] [[OMEGA].sup.f], [[lambda].sup.f.sub.l] > 0 for some I, is

[mathematical expression not reproducible] (1)

where

[mathematical expression not reproducible] (2)

Firms profits from the insurance business are given by

[mathematical expression not reproducible] (5)

Consider the market game with exogenous capital profile [kappa] = ([K.sup.1], [K.sup.2], ..., [K.sup.n]) and denote the set of all capital profiles by K. Furthermore, denote a market contract offer by [delta] = ([[OMEGA].sup.1], [[OMEGA].sup.2], ...[[OMEGA].sup.n]). The set of all market offers is denoted by A. A (restricted) strategy in the market game for insurer f is a map f where [s.sup.f] maps capital profiles in sets of contracts, that is,

[mathematical expression not reproducible]

In the full game with capital choice, a strategy for insurer f is a capital choice in stage 1 and in stage 2 a map [s.sup.f] where [s.sup.f] maps capital profiles in sets of contracts as above. A pure strategy of consumer l [member of] I is a map from a capital profile and a market offer to a contract at an insurer, that is,

[mathematical expression not reproducible]

EQUILIBRIUM ANALYSIS

Candidates for Equilibrium Allocations

In the RS model, firms are assumed to be exogenously endowed with sufficient capital such that insolvency does not occur. The expected utility derived from a contract then depends solely on the contract parameters. We denote expected utility of a J-type individual from contract [w.sub.l] in the RS setting with large capital holdings by:

[mathematical expression not reproducible] (6)

The equilibrium RS contracts, if an equilibrium in pure strategies exists, are then such that the high-risk-type contract specifies full coverage as the expected zero-profit condition for insurers on this contract holds, and the low-risk-type contract maximizes low risk utility subject to the expected zero-profit condition for the insurer on this contract and the H-type incentive compatibility constraint, which both become binding. We denote the H-type and L-type RS contracts by [[omega].sup.H.sub.RS] and [[omega].sup.L.sub.RS], respectively.

The MWS contracts, which we denote by [[omega].sup.H.sub.MWS] and [[omega].sup.L.sub.MWS] obtained as the unique solution (9) to the following maximization problem:

[mathematical expression not reproducible]

Hence, the MWS contracts are obtained by maximizing L-type utility subject to an overall zero profit condition allowing for cross-subsidization. Note that this implies that the MWS contracts are second-best efficient. (10) When the second constraint on H-type utility is binding, the MWS contracts correspond to the RS contracts. When it is not binding, MWS contracts are such that the fully insured H-types are subsidized by the partially insured L-types. We will focus on this more interesting case for the remainder of this article. (11)

The MWS and RS contracts are shown in Figure 1. The straight lines correspond to the H-type and L-type fair insurance contracts with the dashed line indicating the fair insurance contracts for an average risk. The dotted curve gives all L-types contracts that combined with a full insurance contract that lies on the same H-type indifference curve as the L-type contract jointly yield zero profits if taken out by the whole population. The MWS contracts are then the contracts that maximize L-type utility.

Equilibrium in the Market Game With No Up-Front Capital

In this section, we will consider equilibria of the market game when firms do not hold any up-front capital, that is, for [K.sup.f] = 0 for all f [member of] F. Faynzilberg (2006) discusses the setup with exogenous capital and argues that an equilibrium with the MWS allocation exists and is unique for low capital. However, the game is not fully specified. We provide this specification and solve for the equilibrium. As it turns out, many equilibria are possible, and to obtain uniqueness, further restrictions are required. Assume that [K.sup.f] = 0 for all f [member of] F. Note that the expected utility of a customer from any contract at a firm when being the only customer at this firm is always lower than the expected utility from remaining uninsured, as neither risk diversification across insurance policyholders nor insurance capital is available to compensate the risk. With this observation, it is easy to show that many allocations can be sustained in equilibrium as single customers would not deviate to a contract offer of a firm at which they are the only customer.

Lemma 1: Any allocation that is not loss making, incentive compatible, and yields an expected utility at least as high as expected utility when remaining uninsured for all customers can be sustained as equilibrium allocation of a perfect Bayesian equilibrium.

Proof: Consider the following strategies: all firms offer the proposed equilibrium contracts as contract menu and believe that if they offer a different menu, no customer purchases any of their contracts. Customers choose their respective contract at some firm offering as contract menu the proposed equilibrium contracts and believe that no customer will purchase a contract at a firm that offers a different menu from the proposed equilibrium contracts. We need to show that these strategies form a perfect Bayesian equilibrium. Consider a deviation by a firm offering a menu different from the proposed equilibrium contract, for example, including an undercutting contract (menu). First, the proposed equilibrium contracts remain on offer from other firms. Now, as noted above, the expected utility of any customer i [member of] I from choosing a contract at a firm that is offering some contract menu different from the proposed equilibrium allocation is less than the expected utility of remaining uninsured if this customer is the only customer at this firm. Furthermore, the expected utility of any customer of purchasing a contract at a firm offering the proposed equilibrium menu is higher than the expected utility of remaining uninsured. Thus, a customer is better off sticking to buying his respective contract at a firm offering the proposed equilibrium allocation. Then, a firm offering a contract menu different from the proposed equilibrium allocation cannot make a positive profit with this deviation. Beliefs are consistent as no customer chooses a contract at a firm offering a menu different from the proposed equilibrium menu.

An insurance company with no or little capital is unattractive for a single consumer. However, it might become attractive if more consumers go to this firm, as there will be risk diversification across policyholders. Thus, to rule out equilibria that are based on the failure of coordination of a positive mass of customers to take out a contract with better contract terms, a further requirement is needed.

Assumption 1: No coordination failure

For each risk type i there is a positive mass [member of] > 0, which decides jointly which contract to choose.

Before looking at the contract offers of firms, we first show existence of equilibrium in the subgame among individuals in stage 3, following any market contract offer [delta] when Assumption 1 holds.

Lemma 2: For all capital choices and contract choices of the firms, in the subgame where individuals choose contracts, an equilibrium that satisfies Assumption 1 exists.

Proof: The proof proceeds in two steps. First we simplify the subgame by assuming that there are only two players, a low-risk type (with mass 1 - [gamma]) and a high-risk type (with mass [gamma]). Both players choose among the contracts on offer. The payoff for each player from choosing a contract is given by Equation (1). Thus, the game boils down to a static game of complete information with a finite number of players and a finite number of actions. Therefore, Nash's theorem (Nash, 1950) applies and an equilibrium in mixed strategies exists. Second, we return to the general subgame with a continuum of players. Given an equilibrium strategy of the reduced subgame, we argue that it is an equilibrium strategy of the general subgame where every individual chooses the contract (or that mixture of contracts) that the according type chooses in the reduced subgame. No individual has an incentive to deviate from this strategy for the following reasons: either he deviates by choosing a contract from a firm with a positive mass of other customers or he chooses a contract from a firm without any other customer. For a deviation to a contract from a firm with a positive mass of other customers, note that the indemnity payout factor at any firm f with a positive mass of other customers (which is either 1 or [[beta].sup.f]) does not depend on the individual's contract choice and is the same as the one for the same type in the reduced game at firm f, as all other individuals chooses the contract (or that mixture of contracts) that the according types choose in the reduced subgame. Thus, it cannot be the case that such a contract makes this customer strictly better off. If that were the case, in equilibrium of the reduced game the same type would have bought this contract at least with some small probability. The other possible deviation is that the individual chooses a contract from a firm without any other customer. But then his payoff is the same or smaller (see footnote 8) than the payoff of the same type in the reduced game, so also this deviation cannot make him better off. Note that by construction of step 1, this equilibrium satisfies Assumption 1.

Now, we can analyze the role of an aggregate insolvency risk. Insolvency will play a role when contracts and the distribution of risks among contracts are such that in the aggregate losses are made. To see how endogenous insolvency risk plays a key role in the analysis, suppose that firm f has capital [K.sup.f] = 0, offers only the H-type MWS contract [[Omega].sup.H.sub.MWS], and attracts all H-types. Then, this firm would go insolvent with final assets of [mathematical expression not reproducible] and final claims of [mathematical expression not reproducible]. Thus, every insured with a loss obtains [mathematical expression not reproducible], so the expected utility of a customer of firm f is as follows:

[mathematical expression not reproducible] (7)

Endogenous insolvency in this case lowers the indemnity such that the effective contract shifts downward to the point where it crosses the H-type fair insurance line. To see this, note that, due to insolvency, the insurer makes exactly zero profits as he pays out the assets and defaults on the remaining claims. Thus, the effective contract lies on the H-type zero profit line. On the other hand, the customer still has to pay the premium in case of no loss; thus, the effective contract is a vertical shift downward from the initial contract. The resulting expected utility is lower than that of the H-type RS contract as there is no full insurance. This deterioration of the contract is illustrated in Figure 2.

Proposition 1: Let [K.sup.f] = 0 [for all]f [member of] F. Then there exists an equilibrium in the market game where every individual of type J obtains contract [[omega].sup.J.sub.MWS] in stage 3 and no insurer goes insolvent.

Proof: [K.sup.f] = 0 [for all]f [member of] F. Let F := {f [member of] F|[[OMEGA].sub.f] = {[[omega].sup.H.sub.MWS], [[omega].sup.L.sub.MWS]}}. We claim that an equilibrium strategy for the firms is [s.sup.f] = {[[omega].sup.H.sub.MWS], [[omega].sup.H.sub.MWS] for all f [member of] F.

If all firms offer MWS contracts, and all other customers choose their respective MWS contract, no customer has an incentive to deviate, as no insolvency occurs and all are served with the best possible contract on offer. It remains to show that there is no profitable firm deviation. Consider the case that all firms follow the above strategy apart from firm f, which offers [[OMEGA].sup.f] = {[[OMEGA].sup.f.sub1], [[OMEGA].sup.f.sub.2], [[OMEGA].sup.f.subk]}. We will proceed in two steps. We will first show that there is no profitable deviation if f does not offer a cream-skimming contract. We will then show that f offering a cream-skimming contract cannot be a profitable deviation either.

Assume to the contrary that in the subgame following the market offer f is making a strictly positive profit. Note that this implies that f does not go insolvent.

For the first part, assume that [[OMEGA].sup.f] [intersection] [[OMEGA].sub.CS] = [empty set] where [[OMEGA].sub.CS] is defined as follows:

[mathematical expression not reproducible] (8)

that is, [[OMEGA].sub.CS] is the cream-skimming region with respect to the MWS contracts as displayed in Figure 3. (12) There are two possible cases:

Case 1: There exists a contract [omega] [member of] [[OMEGA].sub.f] such that [u.sup.L]([omega]) > [u.sup.L]([[omega].sup.L.sub.CS]). As [[OMEGA].sup.f] [intersection] [[OMEGA]sub.CS] = [empty set], this implies that [u.sup.H] (omega) [u.sup.H]([[OMEGA].sup.H.sub.MWS]) as well. Then, both L- and H-types would choose the deviating contract offer. (13) However, by construction of the MWS contracts, there is no contract (set) preferred by both types to the MWS contracts that is profitable.

Case 2: There does not exist a contract [omega] [member of] [[OMEGA].sup.f] such that [u.sup.L](omega) > [u.sup.L]([[omega].sup.L.sub.MWS]). In this case, we can assume that in equilibrium no L-type chooses a contract offer of the deviator, as in this case it is more attractive to remain with the nondeviating insurers. As no L-type deviates, no MWS insurer goes insolvent. However, then either [u.sup.H]([omega]) < [u.sup.H]([[omega].sup.H.sub.MWS]), that is, there is no customer at f, or [u.sup.H]([omega]) [greater than or equal to] [u.sup.H]([[omega].sup.H.sub.MWS]) and some H-types choose the deviating contract. However, in this latter case, as no L-type deviates, [mathematical expression not reproducible] would not be making a strictly positive profit.

Now, the more interesting part, assume that [mathematical expression not reproducible]. We will consider three cases.

Case 1: MWS insurers do not sell any contract.

As [mathematical expression not reproducible], each type prefers taking a contract at [mathematical expression not reproducible] to remaining uninsured. However, as [mathematical expression not reproducible] construction of the MWS contracts [mathematical expression not reproducible] cannot be making a strictly positive profit when attracting the whole population.

Case 2: MWS insurers sell contracts and there is no insolvency.

As [mathematical expression not reproducible] there exists a contract [mathematical expression not reproducible] such that [mathematical expression not reproducible] and [mathematical expression not reproducible]. Then some L-types deviate from [[omega].sup.L.sub.MWS]. It follows that either MWS insurers make losses, which is ruled out by assumption, or some H-types deviate as well. The proportion of H-types to choose the contract from [mathematical expression not reproducible] has to be at least [gamma]/(1 - [Gamma]) times the mass of L-types choosing [mathematical expression not reproducible], as otherwise at least one of the MWS insurers would go insolvent. For some H-types to deviate as well, a contract [mathematical expression not reproducible] such that [mathematical expression not reproducible] has to exist. However, by construction of the MWS contracts, there is no profitable contract set preferred by both types to the MWS contracts given that the mass of H-types choosing its contract is not smaller than [gamma]/(1-[gamma]) the mass of L-types choosing its respective contract. Hence, [mathematical expression not reproducible] does not make strictly positive profit.

Case 3: Some MWS insurers go insolvent with positive probability.

As [mathematical expression not reproducible] there exists a contract [mathematical expression not reproducible] such that [mathematical expression not reproducible] and [mathematical expression not reproducible]. Now, as [mathematical expression not reproducible] for any [beta] [less than or equal to] 1, it is optimal for an L-type to choose the deviating offer if [mathematical expression not reproducible] and it is optimal to either choose the deviating offer or the L-type MWS contract if [mathematical expression not reproducible]. We define the strategy of an L-type such that he chooses the deviating contract offer. Hence, all L-types deviate. As all L-types deviate, the H-type MWS contract at MWS insurers deteriorates such that [mathematical expression not reproducible]

[mathematical expression not reproducible] (9)

As [mathematical expression not reproducible], it follows that [mathematical expression not reproducible] by construction of the MWS contracts: the MWS contracts maximize L-type utility. Hence, when they do not coincide with the RS contracts, [u.sup.L]([[omega].sup.L.sub.MWS]) > [u.sup.L]([[omega].sup.L.sub.RS]) AS [u.sup.H] ([[omega].sup.H.sub.RS]) = [u.sup.H]([[omega].sup.L.sub.RS]) and using that H-type indifference curves are less steep than L-type indifference curves in the two-states wealth space, this implies that [mathematical expression not reproducible] and thus [mathematical expression not reproducible]. Then all H-types would choose the deviating offer. (14) However, as all types deviate, this contradicts the assumption that some MWS insurers go insolvent.

The intuition behind Proposition 1 works as follows: as high risks are cross-subsidized and the MWS contracts maximize low-risk utility subject to overall nonnegative profits, a deviation has to aim at cream skimming low risks. That is an insurer offers a contract in the set [[OMEGA].sub.CS] as shown in Figure 3. In the RS model, this deviation is profitable as high risk's expected utility is not affected by cream skimming. Here however, due to endogenous insolvency risk, a deviation aimed at cream skimming low risks leads to a contract deterioration for high risks. When they correctly anticipate this contract deterioration, high risks prefer the deviating contract as well, at least as long as the deviator makes a profit and does not go insolvent. However, this implies that the deviator will go insolvent, which renders a deviation unprofitable.

Remark 1: The result of Proposition 1 does not depend on the particular insolvency rule. Instead of the proportional insolvency rule where each customer receives the same share of her indemnity claim, consider, for example, an equal payout rule. With an equal payout rule, all insureds receive the same payment independent of their claim, as long as the specified indemnity is larger than this payment. The expected utility of an H-typefrom his respective MWS contract when there is insolvency is never higher with an equal payout rule than with proportional rationing: either there are only high-risk claimants, then there is no difference, or there are also low-risk claimants, but then high risks are worse of as under proportional rationing they have a higher indemnity claim and thus a higher payout. As contract deterioration for high risks is the crucial part that guarantees equilibrium existence, equal payout rationing would therefore not affect our results.

Equilibrium existence is guaranteed via the externality that the expected utility of a customer may depend on the distribution of customers across contracts of an insurer. In Picard (2014), there is also an externality between customers choosing the same contract at an insurer. However, the origin of the externality differs: whereas it is a property of the insurance company, namely, that the company has too little capital to cope with losses in our model and Faynzilberg (2006), the externality in Picard (2014) is created through the contractual form.

Next it is shown that the equilibrium allocation in the market game is unique if [K.sup.f] = 0 [for all]f [member of] F.

Proposition 2: Let [K.sup.f] = 0 [for all] f [member of] F. Any equilibrium of the market game that satisfies Assumption 1 yields the MWS allocation.

Proof: Suppose that an equilibrium exists that does not yield the MWS allocation. From the set of contracts traded in equilibrium, select the menu of the H-type and L-type contracts with the highest indemnity respectively denoted by [mathematical expression not reproducible]. (15) These are precisely the contracts that yield insurers the highest per contract profit. We distinguish two cases:

Case 1: All insurers make nonnegative profits if the allocation is A.

Now, as insurers make nonnegative profits, [mathematical expression not reproducible] and [mathematical expression not reproducible]. As [mathematical expression not reproducible] and [mathematical expression not reproducible] there exist contracts [mathematical expression not reproducible] such that [mathematical expression not reproducible], and for some firm [mathematical expression not reproducible] offering [mathematical expression not reproducible] and attracting the whole population, [mathematical expression not reproducible] for all j [member of] F under A. As [mathematical expression not reproducible], a deviating insurer offering [mathematical expression not reproducible] would not go insolvent and, with Assumption 1, all types would choose their respective deviating offer. Hence, there is a profitable deviation and A cannot be an equilibrium allocation.

Case 2: Some insurers go insolvent if the allocation is A.

As some insurers make negative profits, there is insolvency. Now, as A are equilibrium contracts and customers take insolvency into account, that is, they maximize expected utility with insolvency, A can be converted to a contract set [mathematical expression not reproducible] without insolvency providing individuals the same expected utility and yielding zero expected profit for insurers with [mathematical expression not reproducible]. Now we can apply the reasoning from case 1 above to show that there is a profitable deviation.

Endogenous Capital

So far in the analysis, we assumed that capital endowment is exogenous and in particular that firms do not own any assets. Now, we are ready to analyze the complete game with up-front capital choice.

Proposition 3: In the complete game with endogenous capital, an equilibrium always exists in which [K.sup.f] = 0 [for all] f [member of] F, every individual of type J obtains contract [[omega].sup.J.sub.MWS] in stage 3 and no insurer goes insolvent.

Proof: Consider the following firm strategy: firm f sets K = 0 in stage 0. In stage 1, if [K.sup.j] = 0 [for all]j [member of] F, firm f sets both MWS contracts. If [K.sup.f] > 0 and [K.sup.j] = 0 [for all] j [not equal to] f, then firm f sets the RS contracts. If [K.sup.l] > 0, l [not equal to] f and [k.sup.j] = 0 [for all] j [not equal to] l, firm f sets both MWS contracts. (16) For the case that all firms f [member of] F offer {[[omega].sup.H.sub.MWS], [[omega].sup.L.sub.MWS]} In stage 2 the strategy of a J-type individual specifies to choose [[omega].sup.J.sub.MWS] at firm f [member of] F with probability 1/k.

On the equilibrium path no individual has an incentive to deviate, as no insolvency occurs and all are served with the best possible contract on offer. Firms make zero expected profits. It remains to show that there is no profitable firm deviation. Consider the case that all firms follow the above strategy apart from firm [mathematical expression not reproducible] which sets [mathematical expression not reproducible] and offers [mathematical expression not reproducible].

Now, as all firms f [member of] F\ [mathematical expression not reproducible] set both MWS contracts each, a profitable deviation has to involve cream skimming, as shown in the proof of Proposition 1. However, as also [K.sup.f] = 0 [all for]f [member of] F \[mathematical expression not reproducible], cream skimming is not profitable for [mathematical expression not reproducible] for any [mathematical expression not reproducible] following the reasoning laid out in the proof of Proposition 1. Hence, there is no profitable deviation.

From Propositions 1 and 2, we know that if firms do not hold any capital, an equilibrium with the MWS allocation exists and is unique under Assumption 1. If now all firms set zero up-front capital in stage 1, then the MWS allocation will result. A single firm has no incentive to deviate to more capital, as the competitors will still offer the MWS contracts in stage 1, thus preventing any profitable contract offer even with more capital.

Remark 2: We consider up-front capital choice but do not model the possibility to recapitalize ex post if claims exceed assets. However, note that for the same reasoning that makes it unattractive for an insurer to put in up-front capital, an insurer does not have any incentive to commit to a recapitalization policy.

If capital costs are zero, then there also exists an equilibrium of the game where all insurers choose large up-front capital (such that they never go bankrupt). They then offer the contracts according to the mixed-strategy equilibrium in Farinha Luz (forthcoming). Customers choose the best contract available, as long as that insurer has sufficient enough capital such that bankruptcy does not occur. Insurers make zero profit in expectation (see Farinha Luz, forthcoming), and play best responses both in their contract offer as well as in their capital choice. But note that this equilibrium does not exist if capital is costly. We conjecture that under positive opportunity costs of holding capital, the only candidate equilibrium allocation in the market game that satisfies Assumption 1 is the MWS allocation.

DISCUSSION

Minimum Capital Requirements

From the analysis above, it is clear that too stringent capital requirements impede the existence of a second-best efficient equilibrium. Let K be implicitly defined by

[mathematical expression not reproducible] (10)

where [[omega].sub.LCS] is the unique contract that satisfies

[mathematical expression not reproducible]

Corollary 1: Under a minimum capital requirement [mathematical expression not reproducible], an equilibrium with the MWS allocation fails to exist.

Proof: Let required up-front capital be [mathematical expression not reproducible] for any firm operating in the market and assume an equilibrium exists that yields the MWS allocation. Then, there is at least one insurer who offers both MWS contracts, hereafter called a MWS insurer, sells contracts to both risk types and does not go insolvent.

Now, consider contract [mathematical expression not reproducible] with [mathematical expression not reproducible] and

[mathematical expression not reproducible]

that is, contract [mathematical expression not reproducible] is preferred by L-types over their respective MWS contract even without MWS insurer insolvency but not preferred by H-types over their respective MWS contract even if there is MWS insurer insolvency. As [mathematical expression not reproducible], such a contract exists. Now consider a firm offering [mathematical expression not reproducible]. As [mathematical expression not reproducible], all L-types prefer [mathematical expression not reproducible] to their respective MWS contract at the MWS insurer. As [mathematical expression not reproducible], all H-types at the MWS insurer prefer to be insured with the MWS insurer and would not choose [mathematical expression not reproducible]. A firm offering [mathematical expression not reproducible] would thus attract L-types and make a positive expected profit. However, then the MWS insurer does not sell contracts to both risk types, which is a contradiction.

If there is sufficiently much capital in the industry, then cream skimming becomes attractive, as high risks will remain with their insurer, even if this insurer makes a loss, as long as capital is sufficient to guarantee the indemnity payments.

Note that from this result of nonexistence of an equilibrium with the MWS allocation--and analogously any second-best efficient allocation as these all require cross-subsidization--it however cannot be concluded that capital requirements are necessarily bad. For this, it would have to be shown that equilibrium allocations with capital requirements lead to lower welfare than those without, which is beyond the scope of the current article. (17)

Note, furthermore, that the above result is derived under the assumption of no aggregate uncertainty such that there is no generic role of capital and solvency regulation to protect market participants against insolvency. With some aggregate uncertainty, for example, due to correlated losses, the logic of the model should continue to hold if losses from aggregate uncertainty are small relative to losses if a large part of the low-risk segment moves to another insurer. In this case, reserves prevent insolvency from residual uncertainty without eliminating the externalities between the customer risk type segments.

Rationality of Consumers

The analysis relies on rational consumers that (1) observe capital of all insurers and the set of offered contracts and (2) anticipate that their expected utility from choosing a contract might depend on that due to the risk of insolvency (18)

It is an empirical question whether in real insurance markets customers are aware of the possibility of insolvency of insurers, and how this depends on the contribution of customers at this insurer. Currently, it seems that in many countries due to heavy regulation insolvencies are relatively unlikely such that customers might not always take possible bankruptcies into account when choosing their insurer. However, as a consequence of the financial crisis, with banks going bankrupt and the increased reporting on risks in the financial system, for example, due to low interest rates that are a severe problem for life and health insurers, awareness among customers will presumably increase.

CONCLUSION

We build upon Faynzilberg (2006) and modify the RS model by allowing insurers to decide on the level of up-front capital and possibly go insolvent. This introduces an externality among the customers of a firm that guarantees equilibrium existence: an equilibrium with the second-best efficient MWS allocation always exists. In such a market, cream skimming becomes unattractive as an insurer trying to attract low risks has to fear attracting high risks as well as the high risks' contract at another insurer deteriorates if low risks do not buy from that particular insurer. When insurers choose the level of their up-front capital, this externality is present because any insurer will opt for a low amount of capital simply because putting in more capital only increases the incentive of competitors to engage in cream skimming. Interestingly, solvency regulation aiming at minimizing insolvency risk might entail unintended consequences: if imposed minimum capital requirements are too strong, the externality from contracting disappears and there is no equilibrium with a second-best efficient allocation.

REFERENCES

Ales, L., and P. Maziero, 2009, Adverse Selection and Non-Exclusive Contracts, Working paper.

Ania, A. B., T. Troger, and A. Wambach, 2002, An Evolutionary Analysis of Insurance Markets With Adverse Selection, Games and Economic Behavior, 40(2): 153-184.

Asheim, G. B., and T. Nilssen, 1996, Non-Discriminating Renegotiation in a Competitive Insurance Market, European Economic Review, 40(9): 1717-1736.

Attar, A., T. Mariotti, and F. Salanie, 2014, Non-Exclusive Competition Under Adverse Selection, Theoretical Economics, 9:1-40.

Crocker, K. J., and A. Snow, 1985, The Efficiency of Competitive Equilibria in Insurance Markets With Asymmetric Information, Journal of Public Economics, 26(2): 207-219.

Doherty N. A., and H. Schlesinger, 1990, Rational Insurance Purchasing: Consideration of Contract Nonperformance, Quarterly Journal of Economics, 105(1): 243-253.

Engers, M., and L. Fernandez, 1987, Market Equilibrium With Hidden Knowledge and Self-Selection, Econometrica, 55(2): 425-439.

Farinha Luz, V., forthcoming, Characterization and Uniqueness of Equilibrium in Competitive Insurance, Theoretical Economics.

Faynzilberg, P. S., 2006, Credible Forward Commitments and Risk-Sharing Equilibria, Working Paper.

Inderst, R., and A. Wambach, 2001, Competitive Insurance Markets Under Adverse Selection and Capacity Constraints, European Economic Review, 45(10): 1981-1992.

Mimra, W., and A. Wambach, 2011, A Game-Theoretic Foundation for the Wilson Equilibrium in Competitive Insurance Markets With Adverse Selection, CESifo Working Paper No. 3412.

Mimra, W, and A. Wambach, 2014, New Developments in the Theory of Adverse Selection in Competitive Insurance, Geneva Risk and Insurance Review, 39(2): 136-152.

Mimra, W., and A. Wambach, 2016, A Note on Uniqueness in Game-Theoretic Foundations of the Reactive Equilibrium, Economics Letters, 141: 39-43.

Miyazaki, H., 1977, The Rat Race and Internal Labor Markets, Bell Journal of Economics, 8(2): 394-418.

Netzer, N, and F Scheuer, 2010, Competitive Markets Without Commitment, Journal of Political Economy, 118(6): 1079-1109.

Netzer, N, and F Scheuer, 2014, A Game Theoretic Foundation of Competitive Equilibria With Adverse Selection, International Economic Review, 55(2): 399-422.

Picard, P., 2014, Participating Insurance Contracts and the Rothschild-Stiglitz Equilibrium Puzzle, Geneva Risk and Insurance Review, 39(2).

Riley, J. G., 1979, Informational Equilibrium, Econometrica, 47(2): 331-359.

Rothschild, C, 2015, Nonexclusivity, Linear Pricing, and Annuity Market Screening, Journal of Risk and Insurance, 82(1): 1-32.

Rothschild, M., and J. Stiglitz, 1976, Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information, Quarterly Journal of Economics, 90(4): 629-649.

von Siemens, F, and M. Kosfeld, 2014, Team Production in Competitive Labor Markets With Adverse Selection, European Economic Review, 68:181-198.

Wanda Mimra is at the ETH Zurich, CER-ETH, Zuerichbergstr, Zurich, Switzerland. Mimra can be contacted via e-mail: wmimra@ethz.ch. Achim Wambach is at the Centre for European Economic Research (ZEW), Mannheim, Germany. Wambach can be contacted via e-mail: wambach@zew.de. We would like to thank Vitali Gretschko, Christian Hellwig, Wolfgang Leininger, Pierre Picard, Ray Rees, and two anonymous referees and the editor for very helpful comments and discussions.

DOI: 10.1111/jori.12206

(1) The original RS model assumes that insurance firms only offers a single contract each. However, if firms are allowed to offer contract menus, the equilibrium nonexistence problem is even aggravated as now cross-subsidizing contracts can pose a threat to the RS equilibrium.

(2) A review of the literature is given in Mimra and Wambach (2014).

(3) Insolvency has been analyzed in insurance markets without adverse selection. Doherty and Schlesinger (1990) model insurance demand under an exogenous insolvency risk and show that less than full insurance will be purchased at the actuarially fair premium if default is total; however, if default is partial, overinsurance might occur and there is generally no monotonic relationship between default payout rate and insurance coverage.

(4) We concentrate on the case with a low share of high risks where the MWS contracts do not coincide with the RS contracts and exhibit the above features. The analysis is trivial for the case where MWS and RS contracts coincide.

(5) Equilibrium existence does not depend on whether opportunity costs of capital are zero or positive.

(6) An individual can only sign one contract with one firm; that is, we consider exclusive contracting. See Ales and Maziero (2009) and Attar, Mariotti, and Salanie (2014) for nonexclusive contracting in RS environments. Rothschild (2015) provides an analysis of constrained efficient allocations in nonexclusive linearly priced compulsory insurance markets.

(7) We will discuss the effects of alternative insolvency rules in the "Equilibrium Analysis" section.

(8) We evoke the law of large numbers to identify the average indemnity payment of a contract with the expected indemnity of a customer randomly drawn at the contract. Note that we assume that claimants have priority over creditors. This representation of expected utility is valid as long as it is not the case that [[lambda].sup.f.sub.i] = 0 [for all]i. If [[lambda].sup.f.sub.i] = 0 [for all]i, the expected utility of a customer choosing contract [[omega].sup.f.sub.i] [member of] [[OMEGA].sub.f] at firm f is

[mathematical expression not reproducible] (3)

for K > 0, and for K = 0, it is

[mathematical expression not reproducible] (4)

that is, if [[lambda].sup.f.sub.i] = 0 [for all]i and [K.sup.f] = 0, the expected utility of a customer choosing a contract at firm f is less or equal than her expected utility from remaining uninsured.

(9) See, for example, Miyazaki (1977).

(10) This is shown by Crocker and Snow (1985).

(11) This is precisely when equilibrium in pure strategies fails to exist in the RS screening game when firms are allowed to offer contract menus. Our results hold trivially for the case that the MWS contracts correspond to the RS contracts.

(12) Note that [[OMEGA].sub.CS] includes [[omega].sup.L.sub.MWS] and contracts that, under solvency, give the L-types the same expected utility as[[omega].sup.L.sub.MWS].

(13) Note that as [u.sup.H]([[omega].sup.H.sub.MWS]) [greater than or equal to] [u.sup.H]([[omega].sup.L.sub.MWS], [beta]) and [u.sup.L]([[omega].sup.L.sub.MWS]) [greater than or equal to] [u.sup.L]([[omega].sup.L.sub.MWS], [beta]) this is true irrespective of MWS insurer insolvency, that is, for any possible [beta] of MWS insurers.

(14) Note that expected utility for H-types at an MWS insurer would even be lower if some or all take out the L-type MWS contract.

(15) It might well be the case that [mathematical expression not reproducible] and [mathematical expression not reproducible] coincide.

(16) A complete specification of the strategy includes the contract sets by firm f if two or more firms choose [K.sup.j] > 0 at stage 1. As this is not required for the existence proof, we do not specify the strategy further.

(17) With (high) capital requirements, candidate equilibria are in mixed strategies in the market game.

(18) Similar requirements apply to the analysis of participating contracts in Picard (2014). However, in terms of practical informational requirements, in our model a customer does not have to worry about an externality on her expected utility as long as she observes that her insurer's capital is sufficiently high, whereas in Picard she needs to understand profits and losses generated at her chosen contract.
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Author:Mimra, Wanda; Wambach, Achim
Publication:Journal of Risk and Insurance
Date:Mar 1, 2019
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