Asymmetric Information, Transaction Cost, and Externalities in Cometitive Insurance Markets * Jerry W. iu Deartment of Finance, University of Notre Dame, Notre Dame, IN 46556-5646 wliu@nd.edu Mark J. Browne School of Business, University of Wisconsin, Madison, WI 53706 m b r o w n e @ b u s. w i s c. e d u Version: March, 00 Preliminary Draft - Comments are very much welcome. * We thank Georges Dionne, Edward Frees, Donald ausch, Robert Puelz, Cheng-zhong Qin, arry Samuelson, and Virginia Young for helful comments on an early version of this aer. All errors are strictly ours.
Abstract The Rothschild and Stiglitz adverse selection model is considered a classic among aers on the theory of markets with asymmetric information. owever, this model relies on the unrealistic assumtion that there are no rofits or transaction costs. This aer modifies the model by incororating various forms of transaction costs, which are either a fixed er olicy fee or variable costs roortional to the remium and exected indemnity amount. Proortional costs bring distinct consequences to insurance buyers behavior and artial coverage becomes the otimal choice for consumers. Dearture from full coverage significantly changes the dynamics of Rothschild and Stiglitz s arguments. We can see a new searating equilibrium if customer grous have heterogeneous risk aversion or heterogeneous endowment wealth. In these two scenarios, neither high-risk individuals nor low-risk individuals cause any externality to the other grou in the market. It is demonstrated that, under more realistic hyotheses, the standard result in the analysis of the insurance market under adverse selection can be reversed. Journal of Economic iterature Classification Numbers: G, G4, D8 Key Words: information asymmetry, insurance market, transaction costs, risk aversion, wealth effect.
I. Introduction In their seminal aer, Rothschild and Stiglitz (976) rovide a framework for analyzing the roblem of adverse selection in an insurance market. The model has become a standard in related literature. Joseh Stiglitz, together with George Akerlof and Michael Sence, was awarded the Nobel Prize in Economics in 00 for their analyses of markets with asymmetric information. Assuming there is zero cost (or exected rofit) and agents buy full coverage, Rothschild and Stiglitz make the conclusion that information asymmetry may lead to market failure: The highrisk (low ability, etc.) individuals exerted a dissiative externality on the low- risk (high ability) individuals Insurance olicies, however, seldom rovide comlete rotection against otential losses. On the other hand, emirical evidence to date has not always been consistent with the major imlications of the Rothschild and Stiglitz model (976). In a recent study, Chiaori and Salanié (000) test whether contracts with more comrehensive coverage are chosen by agents with higher accident robabilities. They find no statistical evidence to suort this rediction. Several factors constitute exlanations for artial insurance coverage. First of all, insurance comanies bear significant roortion costs that could be as high as 30% of their income. Many aers, such as Arrow (97) and Mossin (968), roose that artial coverage is otimal when the remium contains a fixed-ercentage loading. Secondly, as the required return for their caital investments, insurers do take rofit out of their oerations. Moreover, the risk-neutral hyothesis on the insurer art may not always hold. When risks are correlated with market risk, firms
will demand a risk remium, since they are not diversifiable, for shareholders to take the risk. Finally, moral hazard, which refers to the imact of insurance on incentives to reduce risk, also leads to less than full insurance. To summarize, artial insurance coverage is the reality of the world and adverse selection should be looked at in such context. In addition to the zero-rofit hyothesis, other assumtions in the Rothschild and Stiglitz (976) model are simlified. Peole may differ not only by their risk, but also by their references, risk aversion, wealth, loss severity, etc. de Meza and Webb (00) assume that risk-averse eole tend to buy more coverage while the reckless eole often ut less value on insurance rotection. In their setting, ooling equilibrium becomes ossible. Wambach (000) adds exogenous, unobservable wealth differences in addition to the unobservable accident robabilities. When differences in assets are large, artial-ooling equilibria could exist and might involve ositive rofits. The case of severity risk is studied in Doherty and Schlesinger (995). They conclude that adding this severity term will decrease the likelihood of the market achieving equilibrium as suggested in the Rothschild and Stiglitz model. This study integrates transaction costs into Rothschild and Stiglitz s framework by systemically introducing linear cost functions. These costs include both constant costs and roortional costs as a ercentage of remium or indemnity amount. Insurance rofit can also be incororated into this framework if we describe rofit as a transaction cost. Our model shows a clear icture of the transaction cost effects on insurance buyers behavior. To insurance consumers, the aftermath of aying
transaction costs can be considered to be (a) Fixed cost that reduces the endowment wealth of the insurance buyers, (b) Proortional costs that change the sloe of the fair-market line, (c) Customers still buy comlete insurance for a reduced fee when they must ay a constant cost, (d) Peole tend to choose artial coverage when they must ay a roortional fee. Rothschild and Stiglitz (976) conclude that high-risk individuals always cause an externality on low-risk individuals in a market with asymmetric information. In their setting, both tyes of individuals buy comlete insurance and high-risks always refer the low-risk contract, which is located on the certainty line and rovides more wealth in both states of nature. Incororating transaction costs forces both tyes to move away from the the certainty line and brings significant change to the dynamics of Rothschild and Stiglitz s arguments. With the resence of roortional costs, we introduce heterogeneous risk aversions into the model. When high-risks have a much higher level of risk aversion than lowrisks, the two customer grous otimal oints become farther aart, which could lead to a new searating equilibrium. In what we call NE equilibrium, no grou causes any negative externality to the other grou in the market. Both tyes of customers will automatically urchase the first-best olicy designed for them and the information roblem disaears. In the scenario that high-risks are less risks averse than low-risks, the negative externality is aggravated as it is the high-risks who move closer to the endowment oint. owever, when difference in risk aversion is substantial and loss robabilities are close, the information roblem disaears again. 3
Not only eole can have different levels of risk aversion, they can have distinct amounts of endowment wealth. With the resence of roortional costs, we introduce heterogeneous wealth into the standard adverse selection model. There are two ossible combinations high-risks are wealthier than low-risks or high-risks are simly less affluent. In the first case, NE equilibrium exists when the difference in wealth is large enough while in the second case the information roblem becomes worse under most circumstances. owever, even in the second case, we can still find NE equilibrium, when the difference in wealth is large and the difference in loss robabilities is small. Our study rovides new insights into adverse selection models. We give concrete examles in which the Rothschild and Stiglitz externality result can be reversed. Our aer shows that transaction costs can be one exlanation for emirical findings. With the resence of transaction cost, high-risk tyes do not have to be the ones that choose more comrehensive coverage. The remainder of this aer is organized as follows: Section II gives an overview of Rothschild and Stiglitz model assumtions. Section III analyzes the effect of having both constant cost and roortional costs. The case of heterogeneous risk aversion and the new searating equilibrium are resented in Section IV. Section V studies wealth effect. Finally, conclusions are resented in Section VI. 4
II. The Model The following model is based on that of Rothschild and Stiglitz (976) and only the skeleton necessary to understand the question is resented here. The reader could consult the original work for a more detailed account. The market consists of two tyes of customers: low-risks and high-risks with different accident robabilities >. Customers know their accident robabilities, but the insurance comanies do not. Insurance comanies cannot discriminate among their otential customers on the basis of their characteristics. Both low-risks and high-risks have same endowment wealth W. This assumtion is relaxed in section V in which we allow one risk grou to have more endowment wealth than the other grou in the market. If an accident occurs, the loss will be d. To avoid roblems of bankrutcy, the value of d is assumed to be lower than endowment wealth W. An insurance comany writes a contract = (, ), where is the remium amount and is the net indemnity amount (remium deducted). If an accident occurs, the individual will receive a total indemnity of. Neither the insurance remium nor indemnity could become be a negative number and we call an insurance contract (, ) admissible if and only if 0 and 0. The insurance olicy is an exclusive contract, which means that consumers cannot buy insurance from multile insurers. The nature result of this 5
assumtion is that an insurer can observe the total amount of coverage urchased by any customer. Insurers are risk neutral. One suorting argument is that comany stocks are owned by those eole who themselves are well diversified in stock holdings. Individual s wealth in two states of nature (accident or no accident) is indicated as a vector ( W, W ). If no accident haens, individual s wealth is the original endowment wealth minus the remium aid to insurance comany. That is, W = (.) If an accident haens, eole will suffer a loss of d and receive a reimbursement of amount from the insurance comany. W = d (.) The equilibrium concet to be alied is taken from Rothschild and Stiglitz (976), which is defined as a set of contracts such that, when customers choose contracts to maximize exected utility, (i) no contract in the equilibrium set makes negative exected rofits and (ii) there is no contract outside the equilibrium set that, if offered, will make a nonnegative rofit. This definition is equivalent to Nash equilibrium of a static game where insurers are the only layers. Other equilibrium concets, such as Wilson (977) and Miyazaki (977), have been develoed in the literature. Our result can be extended under these equilibrium concets. owever, to focus on our main oint, we restrict the equilibrium to be the original Rothschild and Stiglitz definition. Both tyes of customers have the same state-indeendent utility function U. The customers refer more wealth than less ( U > 0), and they are risk-averse 6
( U < 0 ). This assumtion imlies that all the customers in the market have the same risk reference and level of risk aversion, which is not necessary true in the real world. ater, in section IV, we modify this assumtion and introduce heterogeneous risk aversion into the model. Absolute risk aversion is reresented by U ( x) r( x) = (.3) U ( x) If the absolute risk aversion r( x) is a decreasing function of wealth level x, then the utility is showing decreasing absolute risk aversion (DARA). Constant absolute risk aversion (CARA) corresonds to r( x) being a constant. Similarly, we can define increasing absolute risk aversion (IARA). The realistic situation is that absolute risk aversion is decreasing with wealth. owever, in section IV, we use CARA in order to eliminate the wealth effect. The exected utility of a tye t individual ( t =, ) who urchases contract is written as V t ( ) : V t t t = ( ) U ( ) U ( d ) (.4) By (.4), the marginal rate of substitution for the two tyes is MRS t t dw ( ) U ( W ) = = (.5) t dw U ( W ) Obviously, >, the low-risk indifference curve is everywhere steeer than the high-risk indifference curve. 7
III. Transaction Costs In the Rothschild and Stiglitz model, insurance firms make zero exected rofits. Consequently, for each insurance olicy sold, insurers charge a remium of the amount, which is equal to the exected claim, ). The insurer s rofit can be exressed as ( π = ) 0 (3.) ( = This hyothesis is clearly unrealistic. Most insurance olicies have a rice tag that is more than the cost of claims. According to the Insurance Information Institute: in 998, net claims accounted for merely $59 of every $00 earned in rivate assenger auto insurance remiums in the United States. Commissions, however, accounted for $8, and on average, $4 were aid for other costs of settling claims. The insurer usually ket a rofit of $ and the remaining amount is attributable to taxes, olicy issuance costs, and comany oeration exenses. To consider the effort of exense loading, a term reresenting cost function should be added to the right side of the budget equation (3.). In the rest of this section, we discuss ossible forms of transaction costs and how they will change an individual s behavior. A. Constant Cost Insurers normally charge their customers for their daily exenses, which include salaries and exenses associated with secretarial hel, comuters, and other asects 8
of comany oerations. These costs should be averaged into each insurance contract. For each olicy, this is a fixed amount. cost = a (3.) When there are no transaction costs, as indicated in Rothschild and Stiglitz (976), eole buy full insurance at actuarial odds. Do eole ay for full insurance coverage under a fixed cost? With fixed transaction costs, the individual s otimization roblem becomes (thereafter Problem I): Max : E U ) = ( ) U ( ) U( d ) ( Subject to : ( = a ) ) U ( ) U( d ) ( ) U ( W ) U( d) ( 0, 0 In Problem I, the first condition is the budget constraint and the second condition states that eole need to see an imrovement in their welfare before they ay for any amount of insurance. The final admissible condition guarantees that insurance remium will not become negative. * * PROPOSITION. Otimal solution for Problem I is W = W = d a Proof. See Aendix. when a d ( ) ( d ) ( d ) ; Otherwise r r * * W = W and W = d. Proosition states that, when the fixed charge is not too large, eole still buy full insurance, which means that they will have the same amount of wealth in both states 9
of nature. owever, when the burden of fixed charge outweighs the benefit of having insurance rotection, eole choose not to ay for any insurance. ********************** Insert FIG.. about here ********************** As we can see in figure, the increase in the remium relocates the individual s starting oint from E to F as if the individual s endowment wealth is reduced by the amount a. Without any cost, the otimal oint is at d in both states of nature, and it moves down to the oint with the wealth level β ( d a, d a ) when a fixed charge is imosed. owever, the sloe of the fair-odds line ( ) does not change. The recirocal of this sloe, q ( ) =, is defined in Rothschild and Stiglitz (976) as cost er unit coverage or rice of insurance. The new market odds line E. In figure, otimal oint F β is arallel to the original fair-odds line β is located at the intersection of the o 45 line and the new market rice line. Each customer will buy comlete insurance and will have equal income (reduced by the amount a comared to the no cost case) in both states of nature. This result confirms many earlier works in insurance literature. The effect of a constant cost can be viewed as a reduction in eole s wealth. A rational customer will not buy a small amount of insurance since he would become worse off should he urchase insurance near oint F. From another ersective, a fixed transaction cost lowers an individual s highest attainable exected utility level. 0
As we can see, the higher the fixed cost, the lower is the highest attainable exected utility level. The trend goes on until the fixed cost exceeds a i m i t in the following equation, in which case customers will buy no insurance. a imit = d ( ) ( d ) ( d ) r r Obviously, a imit decreases when d ( ) becomes smaller, which means that the incentive to urchase coverage, under a constant cost function, is the lowest when eole have a small risk (in both loss amount and loss robability). B. Proortional Cost Exense loading in insurance rates also includes commissions and other marketing exenses, remium taxes and fees, loss adjustment and litigation exenses, and insurance rofits. Sales commissions for agents and brokers deend on the total remium collected while remium taxes vary from two to four ercent of the remium in different states. When an accident does haen and claims are filed, claim related exenses, such as loss adjustment and litigation exenses, could become substantial. Exenses associated with the claim settlement are closely correlated with the total claim amount,. Moreover, normal insurance rofits are also art of the bill, and they are often exressed as a fraction of the remium or coverage amount. In this subsection we consider the case where Cost = ba c( a ) (3.3) When transaction costs are roortional to the insurance remium and claim amount, the sloe of the fair-odds line (absolute value) becomes
b ( c) = ( c) Proortional costs turn the fair-odds line counter-clockwise downward. In figure, we can see that the budget line is turned downward around the endowment oint from EA to EB. For consumers, this downward turn of the budget line means they are facing a hike in the rice of insurance. ********************** Insert FIG.. about here ********************** Under roortional transaction costs, the individual seeks to maximize exected utility (Problem II): M a x : E( U ) = ( ) U( ) U( d ) Subject to : ( ) = b c( ) 0, 0 * * PROPOSITION. In any admissible otimal solution to Problem II, W >. Proof. The first order condition for Problem II is W U ( ) U ( d b ( c) = ) ( c)( ) (3.4) Reorganize the budget constraint in Problem II, [ b ( c)] = ( c)
To make any non-zero otimal contract admissible, it is necessary to have b > ( c). Under this condition, with b and c not equal to zero at the b ( c) same time, it is trivial to show that <. Then we have ( c)( ) U ( ) < U ( d ) Since U < 0, we can conclude that, for any admissible otimal contract * * W > d Proosition shows that individuals will have more wealth in the non-loss state than in the loss state and they will not buy full insurance. With a roortional transaction cost, otential losses are not fully covered since market rice for insurance is above the actuarial fair value. In figure, otimal coverage moves from the certainty line, to the oint β., which is on C. Other Cost Functions Besides comany oerating exenses and marketing exenses, exense loading in insurance rates include insurance rofits, loss adjustment and litigation exenses. The real exense loading might be more comlicated than the two simle cases that we have discussed. For examle, exense loading in most insurance olicies are most likely not in a linear form. We feel that our oints are best illustrated in the stylized linear situation and therefore we will not let readers be distracted by the more comlicated nonlinear functions. owever, it is indicated in insurance literature, such as Raviv (979), that full coverage is not otimal under nonlinear costs. 3
As a matter of fact, insurance olicies urchased in rivate markets seldom rovide olicyholders with comlete coverage against losses. For instance, a common olicy contract often contains uer limits on the coverage, with loss reimbursement only u to that limit, and a deductible, with which the insurer is only resonsible for the excess of loss over a certain, redetermined level. The incomlete coverage henomena have been extensively discussed in the literature, for examle, Mossin (968), Arrow (97), Raviv (979), uberman, Mayers, and Smith (983), and Young and Browne (997). In the real world we hardly observe an insurance olicy that rovides full coverage. Besides transaction costs, there are several other considerations that lead to less-than full coverage. First of all, if full rotection is rovided, the insured erson has no incentive to avoid the misfortune and may act to bring it on under certain circumstances. Insurers incororate deductible, co-ayment, and uer-limit clauses into the olicy to reduce the moral hazard roblem. Secondly, insurers are not riskneutral as stated in the model, as they often imose safety loading into the remium. Finally, as an investor, insurers do exect returns to their caital and, as we have seen in the automobile statistics, an insurer s rofit share could become substantial. As one of the major toics in insurance economics, the roblem of adverse selection has long been studied under the zero-rofit assumtion of the Rothschild and Stiglitz (976) model. Because of this unrealistic hyothesis, full insurance coverage becomes the choice of the universe. A second look at the Rothschild and Stiglitz (976) model, under non-zero rofit assumtion, is necessary since we can hardly 4
find an insurance olicy that rovides full coverage. Moreover, moving away from the full coverage can significantly change the dynamic of the Rothschild and Stiglitz arguments. As we can see in the rest of this aer, the well-regarded externality result could be reversed under many scenarios when heterogeneity is introduced into the model. IV. Risk Aversion Rothschild and Stiglitz (976) assume eole ossess identical utility function, which imlies that all individuals have the same degree of risk aversion. Nevertheless, insurance customers can differ in both loss robability and attitude toward risk. Multidimensional adverse selection roblem of such kind has drawn the attention of many economists since the birth of Rothschild and Stiglitz (976) model. In the second art of their aer, Rothschild and Stiglitz conclude that ooling equilibrium cannot exist when high-risk eole are more risk-averse than low-risk individuals. An oosite assumtion is made by de Meza and Webb (00). They argue that more risk-averse eole tend to buy more coverage while the reckless eole often ut less value on insurance rotection. In their setting, ooling equilibrium becomes ossible. Many other articles, such as Smart (000) and Villeneuve (00), examine the same multidimensional roblem under different assumtions. owever, none of these aers questions the externality conclusion of the canonical model. In this section, we extend the model under the following suositions: 5
) The market consists of two tyes of customers high-risks and low-risks with different loss robabilities >. ) Two tyes of customers have the same utility function but different levels of risk aversion. igh-risk utility function is U ( x) and the low-risk utility function is U ( x ), with r (x) and r (x) as the resective absolute risk aversion. 3) One grou is, on average, more risk averse than the other grou and there are two ossibilities. In Scenario A, high-risk individuals are more risk averse than low-risks, r(x) > r ( x) ; while in Scenario B the relation is reversed. 4) Insurance firms face roortional costs in the form of Cost = ba c( a ) (4.) 5) Obviously, the absolute risk aversion deends on the wealth level. At this stage, the wealth effect, which is analyzed in the next section, brings only background noise. We delete the wealth factor by assuming eole have constant absolute risk aversions (CARA). According to Pratt (964), constant risk aversion leads to the negative exonential utility function, t t -r x U (x) = -e,, t = (4.) This section roceeds as follows: As the first ste of our study, we examine basic roerties of high-risk and low-risk indifference curves. Then we insect the multidimensional adverse selection roblem under no transaction cost. Finally, Although our discussion is based on CARA, our result can be easily generalized to other utilities like DARA. All we get from the CARA assumtion is the simlicity, which is necessary for our illustration. 6
roortional cost is added into the question and the concet of non-externality searating equilibrium is introduced. At the end, we give an illustrative examle. A. Indifference Curves Our rocedure is mainly grahical. We need a clear icture of the indifference curves of two risk tyes. The sloe of the indifference curve is reresented by the marginal rate of substitution, defined as, MRS t t t dw ( ) U ( W ) = =, t =, (4.3) dw t t U ( W ) Under CARA, MRS is t r ( W W ) t t [( ) / ] e and the ratio of high-risk MRS vs. low-risk MRS becomes, MRS MRS ( r r ) ( W W ) = ( / ) e (4.4) Obviously, / is less than and the icture will become clear, once we know about the second art of (4.4). As we know, eole are not allowed to have more wealth in the loss state than in the no-loss state, which means W W. Now we will discuss the relative sloe of two tyes indifference curves under the two scenarios roosed in assumtion (3). Scenario A - igh-risks are more risk averse than low-risk eole ( r > r ). In this case the second factor, e ( r r ) ( W W ), is less than one and MRS < MRS. The high-risk indifference curve is always flatter than low-risk indifference curves. Scenario B - igh-risks are less risk averse than low-risks ( r < r ). 7
Our major interest is whether high-risk indifference curve is still flatter than low-risk indifference curves. The answer, nevertheless, can be either a yes or a no. In Scenario B, the second factor of the equation (4.4), e ( r r ) ( W W ), is greater than one. But the first factor, /, is less than. It becomes difficult to decide which indifference curves are flatter. If we hold factor increases as the differences between r and, constant, the second r, W and W become larger. At MRS some oint, the second factor overowers the first factor,, and the high- MRS risk indifference curve becomes steeer. Otherwise, when the differences are small, the high-risk curve is flatter. Our second interest lies in whether the two indifference curves cross only once. If the indifference curve of one class is always flatter than the other class s indifference curve, the two indifference curves will cross only once. Otherwise, the indifference curves cross twice. In Scenario B, double-cross becomes a ossibility. B. eterogeneous risk aversion with no cost In this subsection, we introduce different risk aversions into the Rothschild and Stiglitz model, assuming transaction costs. A simle argument, similar to that of Rothschild and Stiglitz (976), establishes that there cannot be a ooling equilibrium. Under most circumstances, at any ossible ooling oint, two indifference curves have distinct sloes. Even in the least likely case that two indifference curves have the same sloe at the otential ooling oint, 8
the two indifference curves will diverge in the area next to the ooling oint. Another contract, which lies in between these two indifference curves, will attract one tye away from the ooling oint. It could be either the low-risks or the highrisks who are attracted away. The argument for searating equilibrium remains almost the same. Although two tyes have different degrees of risk aversion, both of them regard full insurance, and contract in figure 3, as the first-best choice. The high-risks refer the low-risk, because it rovides more consumtion on both states of nature. Since insurance firms cannot tell who is the bad tye, they can only offer the low-risks another artial coverage contractγ, at the intersection of high-risk indifference curve and low-risk budget line. In this way, high-risk individuals inflict negative externality to low-risks. The argument for searating ooling equilibrium has a slight change in another unique circumstance. In Scenario B, low-risk indifference curve could become flatter than that of high-risks at γ. In this case, low-risks will refer another oint η, the tangent oint of two difference curves. The contract η is rofitable to the insurer since it is located below the low-risk budget line. C. Otimal choice with transaction costs Now we introduce the roortional cost into the model. We assume the cost function is ba c( a ). In Rothschild and Stiglitz (976), eole choose full coverage 9
under zero cost and the degree of risk averse cannot affect individual behavior. Will risk aversion affect the otimal choice under roortional cost? PROPOSITION 3. With cost function ba c( a ), increases in the risk aversion level will lead to increases in otimal insurance coverage amount u n d e r C A R A. P r o of. When U(x) = -e rx, the otimal solution for Problem II is ( c) ( )( c) = [ d log ] b r b c (4.5) b c ( )( c) = [ d log ] ( c) r b c (4.6) d Obviously, > 0 dr d and > 0. Both dr and are increasing with the risk aversion r. Proosition 3 states that, under roortional costs, eole who are more risk averse tend to urchase more insurance. ess risk-averse individuals will buy a small amount of insurance that rovides incomlete rotection. Although we rove this roerty under CARA, this result alies to other utility functions. This conclusion has been comfirmed in the insurance literature in aers such as Schlesinger (000). D. No-Externality Searating Equilibrium We now move on to the multidimensional adverse selection roblem with both roortional costs and heterogeneous risk aversion. The argument for ooling 0
equilibrium is the same as the no cost case and we will concentrate on the mechanism of searating equilibrium. First, we consider searating equilibrium under Scenario A. In Scenario A, highrisks are more risk averse than low-risks and high-risk indifference curve is flatter everywhere than the low-risk indifference curve. The searating equilibrium, however, is different from the standard Rothschild and Stiglitz searating equilibrium in the following asects: First of all, both tyes of individuals, under roortional costs, will not view full insurance as the first-best choice. Both tyes move away from the 45-degree certainty line. Secondly, the less risk averse, low-risk eole tend to move closer to the endowment oint. The highrisk stay close to the certainty line since they are more risk averse. Will high-risks still refer low-risks otimal contract to their own contract? To answer this question, we need to set u a formal framework. For a tye t individual ( t =, ), following roblem (Problem III): t (, ) is defined as the solution to the t t t t Max : E( U ) = ( ) U ( ) U ( d ) t t Subject to : ( ) = b c( ) 0, and 0 A tye s individual s ( s =, ) exected utility at tye t otimal oint is written s t as V ( ). For examle, a high-risk individual s exected utility at a low-risk otimal oint is: V [ ] = ( ) U[ ( )] U[ d ( * )]
DEFINITION. A searating equilibrium in which neither grou causes any negative externality to the other grou in the market, thereafter NE equilibrium, satisfies the following conditions: V * [ ] V [ ] (4.7) * * V [ ] V [ ] (4.8) If V * [ ] is greater than V [ ], high-risks will not refer low-risks otimal consumtion bundle. Meanwhile, we need low-risks to stay with. If both conditions (4.7) and (4.8) are satisfied, the insurance market will reach a state of equilibrium by an automatic searation of two tyes of consumers. In that case, neither high-risk individuals nor low-risk individuals will cause any externality to the other grou. * * EMMA. V [ ] V [ ] always holds under linear cost function. Proof. See Aendix. emma states that low-risks will always refer their own otimal consumtion bundle to that of the high-risks. This result should hold as long as the rice of insurance q( ) is lower for the low-risk tye. There is no need to worry about the ossibility that low-risks could bring negative effects on the high-risks. It only remains to see whether high-risks affect low-risks in the market.
Existence of NE Equilibrium PROPOSITION 4. Under cost function C o s t = b c ), when high-risks ( are more risk averse than low-risk and the difference in risk aversion is large enough, NE equilibrium exists. Instead of offering a roof, we show how to construct the NE Equilibrium: Ste - Find out the high-risks otimal coverage contract 4. The high-risk indifference curve that asses market odds line at some oint, and call it γ. Ste - According to roosition 3, low-risk otimal choice, shown in figure must intersect the low-risk decreases as the low-risks reduce their risk aversion. et the low-risk risk aversion to be small enough and make fall between γ and the endowment oint E. At this time, high-risks will refer its own otimal consumtion bundle to, the low-risk s otimal consumtion bundle. By lemma, we have V * * [ ] V [ ], which means low-risks will not refer high-risk contract. Therefore, this is a NE equilibrium. ********************** Insert FIG. 4. about here ********************** Proosition 4 states that, in Scenario A, when the difference in risk aversions is large enough, the negative externality disaears and the low-risks can enjoy their first best choice as if there were no high-risks in the market. 3
In other cases, where the high-risk indifference curve asses below the low-risk otimal oint, high-risks nonetheless ose negative externality to the low-risk. owever, the degree of externality is alleviated since endowment oint. is already very close to the aving examined Scenario A, we move on to Scenario B. In Scenario B, high-risk individuals are less risk averse than low-risks. igh-risk indifference curve could be either steeer or flatter than the low-risk indifference curve deending the arameters. Since high-risks are more risk averse, they tend to urchase less insurance and their otimal oint is closer to the endowment oint, as shown in figure 5. Under most circumstances, the negative externality is aggravated since low-risks have to take a much lower insurance coverage contract. In this case, the mechanism for searating equilibrium remains same as that of the standard model. A secial case that needs attention is when the low-risk indifference curve is flatter than high-risk at γ. Under such circumstance, low-risks will refer another oint, η, on the high-risk indifference curve. Contract since it is located below the low-risk budget line. η is rofitable to the insurer owever, interesting enough, it is still ossible to find a NE equilibrium when highrisk individuals are more risk averse. PROPOSITION 5. In Scenario B, when high-risks are less risk averse than lowrisks, the following conditions lead to NE equilibrium: ) Difference in risk aversion is large. ) Difference in loss robabilities is small. 4
The NE Equilibrium can be constructed by the following stes: Ste - Find the high-risks otimal coverage contract Ste - The high-risk indifference curve that asses degrees line at some oint. Find this oint and call it A., shown in figure 5. must intersect 45 Ste 3 Turn the low-risk budget line counter clock wise. When the low-risk loss robability is close enough to the high-risk loss robability, the low-risk budget line intersect the 45 degree line below A. Name the intersection of highrisk indifference curve and low-risk budget line as β. Ste 4 Move low-risks otimal oint toward the 45-degree certainty line. We can always do this because, according to Proosition 3, eole buy more insurance when they become more risk averse. When risk aversion is high enough, the low-risk otimal consumtion oint low-risk budget line. can fall above β on the At this time, high-risks will refer its own otimal consumtion bundle,, to the low-risk s otimal consumtion bundle,. By lemma, we have V * * [ ] V [ ], which means low-risks will not refer high-risk contract. Therefore, this is a NE equilibrium for this air of individuals. ********************** Insert FIG. 5. about here ********************** 5
Illustrative Examle for NE Equilibrium ere, we rovide an illustrative examle with which we show NE eqilibrium exists under both Scenario A and Scenario B. et every customer in the insurance market have an initial wealth of W =. When an accident occurs, the individual suffers a loss of d = 0.9. Both the low- and high-risks have utility function of the form U rx ( x) = e. First consider Scenario A - high-risks are more risk averse. Suose that low-risks have loss robability of = 0. and that high-risks have = 0.. igh-risks are less tolerant toward risk and their absolute risk aversion is set as 6 r = 4. On the other side the low-risks have a lower absolute risk averse, which is assumed to be r = 0.5. When insurance comanies rovide actuarially fair olicies, i.e. C o s t = 0, low- and high-risks want full coverage and they are willing to ay 0.09 and 0.8 for the remium, resectively. In case a loss of 0.9 haens, they will get full reimbursement and receive a ayment of 0.9 from the insurance comany. owever, the high-risk individuals will have a gain of 0.0 in the exect utility if they buy the low-risk olicy and ay a lower remium. Insurance comanies are forced to offer artial coverage to low-risk individuals due to the existence of highrisks in the market. igh-risk individuals exert a negative externality on the lowrisk individuals making them worse off. Now, let us consider the more realistic case that insurance comanies include a roortional transaction cost, of the form b c ), in the remium. ( Suose both b and c are 0%. Thus, both the high- and low-risks will not buy full
insurance since the remum is not actuarially fair. While high-risks have to ay a remium of 0. and in case a loss of 0.9 haens, they will receive a reimbursement of 0.84 from the insurance comany. ow-risks will scale down their remium ayment ( ) to 0.04 and can only receive a check of 0.45 if they suffer a loss of 0.9. Comared to high-risks, the best coverage that low-risks can receive offers much lower rotection. Thus, high-risk eole do not like the low-risk olicy any more. They will have a higher exected utility at their own olicy ( V * [ ] = 0.044) than switching to the low-risk olicy ( V [ ] = 0.046). The low-risks, however, always refer their own olicy ( V [ ] = 0.63) to the highrisk olicy ( V [ ] = 0.66). In this case, no grou interferes with the other grou in the market and the informational roblem disaears. The market achieves a NE equilibrium. Similary, under Scenario B when low-risks are more risk averse, the market can reach a state of NE equilibrium. Suose that high-risks have risk aversion of r = 0.4 and the low-risk absolute risk averse is r = 4. ow-risk loss robability is assumed to be = 0. and high-risk loss robabilty is close to the high-risk one with = 0.05. Again, NE equilibrium exists. In this case, high-risks will send 0.04 on remium, while low-risks are willing to aying a remium of 0.03. In case a loss of d = 0.9 haens, high-risks will only receive a ayment of 0.33 and the low-risks will get a During the searching rocess for the NE equilibrium, we have to set the high-risk loss robabilty very close to that of low-risks. Otherwise such NE equilibrium is not available. 7
ayment of 0.84. Again, low-risk customers refer their own olicy ( V [ ]- V [ ] = 0.0), and high-risk customers will automatically choose a high-risk olicy ( V * [ ] - V [ ] = 0.000). Again we find NE equilibrium exists when difference in risk aversion is large. V. Wealth In Rothschild and Stiglitz (976), all agents have same amount of endowment wealth. But, in reality, eole may differ in their financial status. In this section, we extend the model under the following set of assumtions: ) The market consists of two tyes of customers high-risks and low-risks with different loss robabilities >. ) One tye of customers is, on average, wealthier than the other grou and there are two ossibilities. In Scenario I, low-risk individuals are richer, W > W ; while in Scenario II this relation is reversed. 3) Insurance firms face roortional costs in the form of Cost = ba c( a ) (5.) 4) Two tyes of customers have same utility function and the same level of risk aversion. 5) Peole have decreasing absolute risk aversion (DARA). This is a common hyothesis used in economic studies. 8
In the rest of this section, we will first exam the sloe of indifference curves for the two tyes. Then, under the zero-rofit assumtion, we analyze the multidimensional adverse selection roblem with heterogeneous wealth. In the end, we show that NE equilibrium exists. A. Indifference Curves Our grahical rocedure requires that we have a clear icture of the indifference curves of the two tyes. Instead of looking at the indifference curves in the ( W, W ), we move into the (, ) sace, in which our grah is greatly simlified. In the (, ) sace, the marginal rate of substitution for the two tyes is reresented by, MRS t d = d( t ( ) U ( ) = t ) U ( d ), t =, (5.) The ratio of high-risk MRS vs. low-risk MRS turns out to be, MRS U ( ) U ( ) = ( / ) [ / ] (5.3) MRS U ( d ) U ( d ) Obviously, the first art of (5.3) is less than. et U ( ) Z ( W ) = U ( d, and we ) will find that dz( W ) U ( W ) = [ r( W ) r( W )] (5.4) dw U ( W ) dz( W ) Under roortional costs, W > W, and, with DARA, > 0. The relative dw sloes of two indifference curves can be analyzed under two scenarios. 9
Scenario I - igh-risks are oorer than low-risks. W < W In this case, the second factor in equation (5.3) is less than one. Therefore, MRS < MRS. The indifference curve of the high-risks is flatter everywhere than that of low-risks. Their indifference curves can across once. Scenario II - igh-risks are richer than low-risks. W > W In this case, the second factor in equation (5.3) is greater than one. Therefore, it is difficult to judge which indifference curve is flatter. owever, we can hold and constant, the second factor increases when the difference between W and W become larger. As a result, the second factor could overower the first factor and the high-risk indifference curve becomes steeer. On the other side, when the difference between W and W is small, the high-risk indifference curve is flatter. Based on what we have seen so far, wealth affect eole s behavior through risk aversion. Under DARA, more wealth means less risk averse, and less wealth more higher degree of risk aversion. B. eterogeneous wealth with no cost The situation here is almost exactly same as the heterogeneous risk aversion case in Section IV. Pooling equilibrium is not ossible. For searating equilibrium, both tyes regard full insurance as the first-best choice and the low-risks have to take a second-best choice due to the resence of high-risks. A secial case is that, in Scenario II, low-risk indifference curve is flatter than that of high-risks. The lowrisk contract is rofitable to the insurer. 30
C. Otimal choice with transaction costs We now examine, under roortional costs, how wealth difference will affect eole s choice. PROPOSITION 6 Under DARA, rich eole buy less insurance, when there is a roortional cost of the form ba c( a ). Proof. Rewrite the first order condition of Problem II as U ( ( W )) b ( c) = U ( d ( W )) ( c)( ) (5.5) Notice that [ b ( c)] ( W ) = ( W ) ( c) Take derivative with resect to W on both side of (5.5), we can get d dw = r( W ) r( W ) b c r( W ) r( W ( c) ) (5.6) d Since W > W, we have r ( W ) < r( W ) under DARA. Therefore, < 0. dw Proosition 6 states that, under roortional costs, the richer eole are, the less they urchase insurance. On the other side, oor individuals, everything else equal, will buy more insurance. 3
D. No-Externality Searating Equilibrium We now roceed to analyze the Rothschild and Stiglitz (976) arguments for equilibrium under roortional costs with heterogeneous wealth levels. Again, the argument for ooling equilibrium is the same as the no cost case and we will concentrate on the mechanism of searating equilibrium. First, we discuss the searating equilibrium under Scenario I, in which high-risk individuals are oorer and low-risks are richer. In this case, high-risk indifference curve is flatter everywhere than the low-risk indifference curve. The searating equilibrium, however, is totally different from RS arguments. The most dramatic change is that both tyes, under roortional costs, will not regard full insurance as their first-best choice. Both tyes move away from the 45-degree certainty line. The low-risks, who are wealthier and less risk averse, tend to move close to the endowment oint. The high-risks, on the other hand, move down by a smaller distance. Now the question is: Will high-risks refer low-risks otimal contract to their own contract? PROPOSITION 7. Under cost function C o s t = b c ), when high-risks ( are oorer than low - risks and the difference in wealth level is l a r g e e n o u g h, NE equilibrium exists. Proosition 7 is, in nature, the same as Proosition 4. All we did is substituting risk aversion with wealth level. Being oorer is equivalent to being more risk averse while being wealthier means that the erson is more tolerant toward risk. When difference in wealth levels is large enough, the externality roblem disaears and we will reach NE equilibrium. 3
********************** Insert FIG. 6. about here ********************** This case is illustrated in figure 6. The high-risks, who are also less wealthy eole, refer to get more rotection and buy the contract wealthier, refer to buy a small amount of coverage, reresented by. The low-risks, who are in figure 6. Since is too close to the endowment oint, high-risk will not urchase although they can do so under the asymmetric information condition. Therefore, low-risks can enjoy their first best choice as if there were no high-risks in the market. In other cases, where the high-risk indifference curve still asses below the low-risk otimal oint, high-risks ose negative externality to the low-risk. owever, the degree of externality is weakened since oint. is already very close to the endowment In Scenario II, where high-risks are affluent, they are less risk averse than low-risks. Generally seaking, they urchase less insurance and their otimal consumtion oint is very close to the endowment oint. Under most circumstances, the negative externality is aggravated. owever, it is still ossible to find a NE equilibrium in Scenario II, but the following two conditions have to hold, (i) The difference in loss robability is small. ( i i ) The difference in endowment wealth is large. 33
Illustrative Examle for NE Equilibrium ere, we rovide an illustrative examle with which we show NE eqilibrium exists. Suose both the low- and high-risks have log utility, U ( x) = log( x). Assume that low-risks have loss robability of = 0. and they are relatively wealthy, with a endowment wealth of W =. igh-risks, on the other hand, have a higher loss robability, = 0., and they are less wealthy, W = 0. 75. In case of an accident, both tyes will suffer a loss of d = 0.5. When insurance comanies rovide actuarially fair olicies, i.e. Cost = 0, low- and high-risks are willing to ay u to 0.05 and 0. for the remium, resectively, for full coverage. In case a loss of 0.5 haens, they will get full reimbursement and receive a ayment of 0.5 from the insurance comany. owever, the high-risk individuals will have a gain of. in the exect utility if they buy the low-risk olicy and ay a lower remium. Insurance comanies are forced to offer artial coverage to low-risk individuals due to the existence of high-risks in the market. igh-risk individuals exert a negative externality on the low-risk individuals making them worse off. Now, let us consider the more realistic case that insurance comanies include a roortional transaction cost, of the form b c ), in the remium. ( Suose both b and c are 0%. Thus, both the high- and low-risks will not buy full insurance since the remum is not actuarially fair. While high-risks have to ay a remium of 0.085 and in case a loss of 0.5 haens, they will receive a reimbursement of 0.349 from the insurance comany. ow-risks will scale down 34
their remium ayment ( ) to 0.0 and can only receive a check of 0.098 if they suffer a loss of 0.5. Comared to high-risks, the best coverage that low-risks can receive offers much lower rotection. Thus, high-risk eole do not like the low-risk olicy any more. They will have a higher exected utility at their own olicy ( V * [ ] = 0.46) than switching to the low-risk olicy ( V [ ] = 0.46). The low-risks, however, always refer their own olicy ( V [ ] = 0.66) to the high-risk olicy ( V [ ] = 0.64). In this case, no grou interferes with the other grou in the market and the informational roblem disaears. The market achieves a N E equilibrium. Similary, under Scenario II, when low-risks are less wealthy, the market can reach a state of NE equilibrium. We assume that high-risks have an endowment wealth of W =.4 and the low-risks financial status is reresented by W =. ow-risks have loss robability is assumed to be = 0.. igh-risks loss robabilty has to be very close to, = 0.05. We find the NE equilibrium does exist: high-risks will send = 0.006 on remium ayment, while low-risks is aying a remium equal to 0.037. In case a loss of d = 0.5 haens, high-risks will only receive a net ayment of 0.07 and the low-risks will get a ayment of 0.306. Still, low-risk customers will refer their own olicy ( V 3 [ ]- V [ ] = 7.9 0 ), and highrisk customers will automatically choose a high-risk olicy ( V * [ ] - V 4 [ ] =.7 0 ). 35
VI. Conclusion This article demonstrates that the externality conclusion of Rothschild and Stiglitz (976) can be reversed when transaction costs are resent. The long-held zero-rofit hyothesis is in direct contradiction to reality. As we know that insurance olicies seldom rovide comlete coverage. Market fictions are simly too imortant to be ignored. In our view, analysis of markets with informational roblems should be conducted in a more realistic setting. Incororating transaction costs brings significant change to consumers behavior. They move away from their full demand amount and urchase less than what they wanted to urchase in fictionless world. Diversity of customers means that care has to be taken on whether one grou will cause negative effect to other grous in the market. It is illustrated that the secial non-externality searating equilibrium, in which no grou causes any negative externality to the other grou in the market, does exist. The world is more comlicated than the two simle cases that we have analyzed. For examle, high risk grou can be both less wealthy and more risk averse. Rothschild and Stiglitz (976) model is a construct of simlicity -- the market is cometitive, firms are risk neutral and make zero rofit. Furthermore, the model is a construct of homogeneity -- eole have same endowment wealth, utility function (therefore risk aversion), and loss amount. These hyostheses lead to the externality result. Introducing comlexity and heterogeneity, as we have showed with heterogenous risk aversion and endowment wealth, leads to deviations from their standard conclusion. 36