INTRODUCING HETEROGENEITY IN THE ROTHSCHILD-STIGLITZ MODEL

Transcription

1 Te Journal of Risk and nsurance, 2000, Vol. 67, No. 4, NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL Acim Wambac ABSTRACT n teir seminal work, Rotscild and Stiglitz (1976) ave sown tat in competitive insurance markets, under asymmetric information, pooling contracts cannot exist in equilibrium, firms make zero profit, and, under some circumstances, equilibrium does not exist. n te present work, te model is extended by introducing unobservable wealt in addition to te differing risks. Te study sows tat if te differences in wealt are sma, different wealt types are pooled wile different risks are separated. For large wealt differences, partial risk pooling contracts, in wic one type cooses different contracts in equilibrium, are feasible. Furtermore, equilibria wit profitmaking contracts can exist. Complete risk pooling contracts can occur only under very restrictive assumptions. Te effect of te extra dimension of asymmetric information on te nonexistence problem is ambiguous. NTRODUCTON Since te seminal work by Rotscild and Stiglitz (1976), it as become clear tat competitive insurance markets under asymmetric information display features not known from te conventional competitive analysis. Contracts specify bot quantity and price, and it may appen tat equilibrium does not exist. Te nonexistence problem as led to a lengty debate in te literature. Solutions proposed are oter equilibrium concepts (Wilson, 1977; iyazaki, 1977; Spence, 1978), extensions to multi-stage games (Grossman, 1979; Hewig, 1987; Aseim and Nilssen, 1996), or te introduction of mixed strategies (Dasgupta and askin, 1986). Altoug te equilibrium nonexistence problem as been te main focus of researc, some oter results by Rotscild and Stiglitz deserve attention. First, tey sowed tat in a competitive environment, pooling contracts cannot exist in equilibrium, and second (maybe less surprisingly), tey demonstrated tat firms make zero profits. n oter equilibrium concepts, te existence of pooling contracts is possible; owever, te no-profit result always olds. Acim Wambac is a member of te Department of Economics, University of unic, Germany. He tanks Klaus Scmidt, Ray Rees, Ricard acinn, one anonymous referee, and participants of te 1997 Seminar of te European Group of Risk and nsurance Economists for elpful comments and suggestions. 579

2 580 THE JOURNAL OF RSK AND NSURANCE As was already argued by Spence (1978, p. 427), not only may individuals differ in te expected cost tey impose upon te insurer, tey may also differ in teir preferences wit respect to insurance coverage. Foowing up on tis remark, te autor considers individuals wo differ in risk aversion in addition to teir risks. Tere are two dimensions of asymmetric information te insurer observes neiter te risk nor te degree of risk aversion of te individual. Te autor models te different degrees of risk aversion by assuming tat individuals ave decreasing absolute risk aversion and different wealt levels. Oter studies on multidimensional adverse selection problems in te context of insurance markets include te ones by Fluet and annequin (1997) and Landsberger and eilijson (1996). Te first autors model two types of individuals wit multiple risks. Teir main result is tat bundling insurance contracts can be efficiency enancing. Landsberger and eilijson analyze te case in wic two types of individuals differ wit respect to teir distribution of losses. However, tey are mainly concerned wit te derivation of assumptions under wic a first-best result can be acieved or approximated. n contrast to tese studies, tis article assumes tat tere are four different types of individuals: tose wit ig or low risks wit eiter ig or low wealt. To te autor s knowledge, tis is te first study tat analyzes te insurance market wit four unobservable types. t derives te foowing new results. First, in te standard case, were te single crossing property olds in te relevant region of te contract space, different wealt types are pooled, wile different risk types are separated. Tis pooling is te same as in te first-best situation, in wic different risks obtain fu insurance at teir fair premium, wic is independent of te wealt. Second, te effect of te additional dimension on te equilibrium nonexistence problem is ambiguous: t may appen tat if wealt is observable, no equilibrium would exist, but it does exist for unobservable wealt, and vice versa. Tird, if te single crossing property does not old, ten even under perfect competition wit free entry an equilibrium may exist in wic firms make positive profits. 1 Fourt, if te single crossing property does not old, partial risk pooling, were one type mixes between two contracts and anoter risk type cooses one of te two contracts, can exist in equilibrium. Fift, complete risk pooling, were different risk types coose te same contract wit a probability of one, exists only under very restrictive assumptions. Generalizing from te results, te autor is led to conclude tat an equilibrium in a competitive insurance market under multidimensional adverse selection is eiter wit a pooling of types tat are also pooled in te first best and at te same time wit complete separation of risks, or wit a partial risk pooling, were some types wit te same risk are separated. Te study is structured as foows. Te autor first presents te model and discusses te slope of te indifference curves as a function of risk and wealt. Te study ten deals wit te standard case, were te single crossing property olds in te relevant region of te contract space, before te violation of te single crossing property is considered. Finay, te results are summarized. 1 Tis result as been independently sown by Vieneuve (1997) and Smart (2000). Vieneuve uses a model wit only two types and very general utility functions. Smart, wo also works wit two types only, employs a different equilibrium concept.

3 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 581 THE ODEL Consider an individual wo as te probability Q of losing L. Her initial wealt is X, and er utility function displays decreasing absolute risk aversion. Te insurer offers a contract tat specifies a premium and an indemnity tat wi be paid if a loss occurs. Tus te expected utility of te individual is: V ÉQX,, L,, É1Q U ÉX QU ÉX L (1) were U is a concave increasing function tat satisfies decreasing absolute risk aversion, i.e., déubb Éx/ Ub Éx/ dx 0. Te literature so far as been concerned wit an adverse selection problem on te probability of losses ÉQ. n tis study, te discussion wi be extended for unobservable X. 2 Te slope of te indifference curve in (, ) space is given by: 1 Q b( X ) 1 QU b( X1) µ 1 < b> Q b( X2 ) d U L d E U U, (2) were X1 X and X2 X L. t is easy to sow tat te second derivative is negative; tus te indifference curves are increasing and concave. f te loss probability increases, te term É 1 Q / Q decreases, and wit tat te slope of te indifference curve increases. Tus at every point in an (, ) diagram, ig-risk individuals ave a steeper indifference curve, given tat te wealt and te loss are te same. Wit a similar argument, one can sow tat individuals wit a iger loss level ave ceteris paribus a steeper indifference curve. Finay, consider a cange in wealt d U ( X ) U ( X ) dx U X U X b µ 1 b 1 < A( X1) A( X2) > b( 2) b( 2), (3) were A(X ) is te coefficient of absolute risk aversion. t is assumed in te foowing tat a individuals ave decreasing absolute risk aversion, so individuals wit lower wealt (and tus wit larger risk aversion) ave a steeper indifference curve as long as X1 X2, i.e., as long as tere is under-insurance. For over-insurance, te result is reversed. t is important to keep in mind tat if = L, te slope of te indifference curve is Q. (A risk-averse individual wi buy fu insurance if te premium is fair.) Te equilibrium concept used in tis study is te one Rotscild and Stiglitz (1976) employed: Equilibrium in a competitive market is a set of contracts suc tat, wen 2 Similar results can be obtained if it is observable as to weter an accident occured; owever, te exact size of L is unobservable.

4 582 THE JOURNAL OF RSK AND NSURANCE consumers coose contracts to maximize expected utility, (i) no contract in te equilibrium set makes negative expected profits; and (ii) tere is no contract outside te equilibrium set tat, if offered, wi make a non negative profit. t is we known tat te Rotscild-Stiglitz definition of equilibrium in insurance markets is equivalent to Nas s equilibrium of te foowing game: At stage one, risk-neutral firms offer one contract eac. At stage two, customers coose between contracts. Te expected profit of a firm from a contract (, ) tat is taken by someone wo is wit probability H a ig risk is given by ( H Q (1 H) Ql). Note tat ere, firms are required to offer one contract at most, wile in a more general model, one would like to ave firms offering a menu of contracts instead. Some of te foowing results depend on tis restriction. Te autor wi return to tis point in te final section. f te standard equilibrium concepts are used, namely te Rotscild-Stiglitz equilibrium, te Wilson equilibrium (Wilson, 1977), te Wilson- iyazaki-spence (WS) equilibrium (Wilson, 1977; iyazaki, 1977; and Spence, 1978), or te Riley equilibrium (Riley, 1979), te autor s results old in a cases except te WS concept, in wic cross-subsidization becomes possible. Differing Risks and Differing Wealt Levels: Te Standard Case Te autor now discusses diagrammaticay te possible scenarios if tere is asymmetric information about risks and wealt and were in te relevant region of te contract space te single crossing property olds. Tis tecnical requirement is explained below in more detail. Consider four types of individuals wit different risks and wealt É Qi, Xj i, j Ž \ l, ^, were Q l Q and Xl X. Ca tese te (, l, l, ) types. Te proportion of eac type in society is ij, Éi, j Ž{(,), (,l), (l,), (l,l)}. Define as te Rotscild-Stiglitz (RS) pair of contracts tese contracts tat constitute te RS equilibrium if only te type and te l type are present. Ca tese contracts A and B. Tey are sown in Figure 1. On te two axes are te premium and indemnity. Te straigt lines are te zero-profit lines for te ig- and te low-risk types. FGURE 1 Equilibrium Contracts in te Standard Case A Q l l Q l B L

5 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 583 Te type of te individuals denotes te indifference curves. Hig risks receive fu insurance at teir fair premium [contract A ÉQ L, L ], wile low risks receive partial insurance [contract B = (*, *)] at teir fair premium, suc tat te type is indifferent between A and B. Consider te situation were te slope of te indifference curve of te type at B is lower tan tat of te type. Foowing Equation (2), tis implies tat 1 Q Ub( X *) 1 Ql Ub( Xl *) c Q Ub( X * L *) Q Ub( X * L *). (4) l l f tis condition olds, it wi be caed te standard case. n te next section, te autor sows tat if te wealt levels are not too far apart, tis condition is satisfied. 3 roposition 1: f Condition (4) olds, and wen equilibrium exists, ten types wit different wealt but te same risk are pooled, wile different risks are separated. Te equilibrium contracts are te RS contracts A and B. roof: Te derivation of te equilibrium is a straigtforward extension of te discussion in Rotscild and Stiglitz. Tis is displayed in Figure 1. Hig risks obtain teir fair fu insurance contract, wile low risks obtain te best fair contract available suc tat no ig risk as an incentive to buy tis contract. As te indifference curve of te type is less steep tan tat of te l type, it is te former tat determines te position of te low-risk contract. Q.E.D. n tis equilibrium, only te type is indifferent between two contracts. Tis is for two reasons. First, as is usual in tese models, it is te ig-risk type tat considers buying te ceaper contracts tat are offered to te low risks. Second, te type is less risk averse tan te l type. Terefore, te type is less distracted by a partial insurance contract tat is offered to te low-risk types tan te l type. n te standard Rotscild-Stiglitz (RS) model, te equilibrium in pure strategies does not exist if te number of ig risks is sufficiently low, so tat te indifference curve of te low-risk type cuts te zero-profit line for a pooling contract. For two types, tis line is defined by ÉQ Q l l, were is te proportion of te ig-risk type in society, given tat tere is only one wealt level. Accordingly, l 1 is te proportion of te low-risk type. Aowing for additional eterogeneity introduces two furter effects, wic ave different consequences for te existence of te equilibrium. First, te introduction of a different type wit larger wealt sifts te contract for te low risks to lower indemnity levels. n tose cases, te low risks migt find a pooling fu insurance contract more attractive, wic makes te RS equilibrium less likely to exist. However, if tere are sufficiently many ig risks wit larger wealt, ten tis pooling fu insurance contract as a large premium, wic makes it less attractive for te low risks. Tis ten makes te RS equilibrium more likely to exist. 3 One can sow tat if Equation (4) is satisfied, te inequality wi old for a *, * wit * *. Terefore, in te standard case te indifference curve of te type wi be less steep tan te indifference curve of te type in te relevant region of te contract space.

6 584 THE JOURNAL OF RSK AND NSURANCE roposition 2: (a) t is possible to find parameter values for ij,, Éi, j Ž\ É,, É, l, Él,, Él, l^ X j, j Ž \ l, ^ and Q i, i Ž \ l, ^ suc tat an equilibrium exists if one could condition te contracts on wealt, but not if wealt is unobservable. (b) For different parameter values it could be te case tat no equilibrium exists if contracts can condition on wealt, but it does so for unobservable wealt. FGURE 2 Observable and Unobservable Wealt A Q l C l Q l B B' L roof: (a) Consider a situation in wic for bot ric and poor types equilibrium exists, if te contracts can condition on wealt. Ten te ig risks would bot obtain contract A, as drawn in Figure 2, wile te l () type obtains contract BBb É. f wealt is unobservable, te candidate equilibrium contracts are te same contracts for a types apart from te type. Her contract sifts to te left (from Bb to B), as oterwise te type would coose te contract. Te indifference curve of te type migt ten cut te zero-profit line of te and types 4 (te dased line in te diagram). f tis occurs to te left of te indifference curve of te l type, wic goes troug A e.g., at point C ten an equilibrium does not exist. Tis wi occur if te number of types is very sma compared to te number of types, wic sifts te pooling line of te and te types downwards, or if te wealt of te type is large, wic sifts contract B to te left and, terefore, te indifference curve of te type upwards. 5 (b) Consider now a situation in wic l is sufficiently low compared to, suc tat an equilibrium for te l and types does not exist, if wealt is observable. f wealt is unobservable, and if tere are sufficiently many of te type, te zeroprofit line of te pooling contract wi sift upwards. f tis effect is strong enoug, equilibrium wi exist in te unobservable case. Q.E.D. 4 A zero-profit line for types ij and mn is given by ij mn µ Q Q i m ij mn ij mn. 5 For different parameter values, a profit-making pooling contract for te,, and l types migt be feasible. Also, in tat case equilibrium does not exist.

7 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 585 Relaxing te Single Crossing roperty So far te autor assumed tat at contract B te slope of te indifference curve of type is steeper tan te slope of te type. Tis is not necessarily te case. As B is a point of under-insurance, it is a priori not clear wo wi pay more for one more unit of coverage, te person wit a ig risk but low risk aversion or te one wit a low risk but ig risk aversion. f te type is very risk averse, i.e., very poor, e or se migt even prefer contract A (fu insurance wit a ig premium) to contract B (under-insurance wit a low premium). Formay, te single crossing property, i.e., indifference curves cross only once, wic is usuay assumed in adverse selection models, does not old in tis case. n te foowing, te autor derives one necessary and one sufficient condition for tat to be te case. Foowing Equation (4), te violation of te single crossing property at contract B implies tat 1 Q Ub( X *) 1 Ql Ub( Xl *) Q Ub( X * L * Q Ub( X * L *), (5) were * Q * and * is given by l l l (1 Q ) U( X *) Q U( X * L *) U( X Q L). (6) f along te Q l line te left side of nequality (5) is smaer tan te rigt side, it is easy to verify tat it wi stay smaer for a larger values of. So, a necessary condition for nequality (5) to be true is tat it olds at = = 0. Tis implies: 1 Q Ub( X) 1 Ql Ub( Xl) Q Ub( X L) Q Ub( X L). (7) l l By using te result from Equation (3), a sufficient condition can be derived: 6 1 Q Ub( X) 1 Ql Ub( Xl). Q Ub( X L) Ql Ub( Xl L *) (8) As * is independent of Xl, * L, and given te assumption on te slope of U, one can always find a sufficiently sma X l suc tat te inequality in Equation (8) is satisfied. Tis just implies tat if te low-risk type is risk averse enoug, te result olds. However, if te difference in wealt (risk) is sma (large), or if te size of te loss is sma, nequality (7) sows tat te slope of te indifference curve of te type wi not be steeper tan te slope of te type. However, it is not correct to conclude tat 1 Q Ub( X *) 1 Q Ub( X) 6 Tis olds because Q Ub( X * L *) Q Ub( X L) 1 Q Ub( X *) 1 Q Ub( X ) l l l l. Q Ub( X * L *) Q Ub( X L *) l l l l and

8 586 THE JOURNAL OF RSK AND NSURANCE for a sufficiently sma difference in risks or for sufficiently large losses nequality (8) wi be satisfied, as * depends on Q, Q, and L. l Anoter parameter influences te differences in slopes, namely te cange in te degree of absolute risk aversion induced by different wealt levels. To see tis, take for example a utility function wit constant relative risk aversion S. Ten nequality (7) transforms into S 1 Q X 1 Ql Xl Q X L Q X L S µ µ l l. (9) So if S is large te inequality is satisfied. Te inequality in Equation (8) olds only for large S if LÉX Xl * X. Having establised tat te violation of te single crossing property may old for large differences in wealt, te autor now turns to te analysis of tis case. roposition 3: f te single crossing property does not old, conditions for equilibrium existence depend on te relative proportion of agents. Wen equilibrium exists, only one equilibrium fuy separates low and ig risks. n tis particular equilibrium, firms make positive profit. roof: Suc an equilibrium is sown in Figure 3. For = L, te slope of te indifference curve for te type is Q l, wic is smaer tan te slope of te type at = L. Tus it foows tat tere must be one point between A and B at wic te indifference curves of te and te types are tangential to eac oter (point C). 7 Now consider te case in wic contracts A, B, and C are offered and were te l type cooses contract B; te and l types, contract A; and te type, contract C. FGURE 3 Equilibrium Contracts if te Single Crossing roperty Does Not Hold A Q l C l Q l B L 7 Tis olds even if te type prefers contract A to contract B.

9 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 587 f te proportion of te type is sufficiently large, i.e., C lies below te line, µ Q Q l, any deviating contract, designed to attract te type, wi also attract te type and make a loss. 8 Terefore, tese contracts can be sustained as an equilibrium. Q.E.D. n tis equilibrium, a firms offering contract C make positive expected profit. 9 Deviating from tis point is not profitable, as it is not possible to offer anoter contract tat wi be taken up only by te type. Even in a perfectly competitive environment wit free entry, no oter firm wi opt to under-bid tis contract, as not only te desired low-risk individuals wi coose tis new contract, but also te undesired igrisk individuals, wic wi lead to negative expected profits. 10 For te foowing proposition, te autor defines partial risk pooling: f a contract is cosen in equilibrium by more tan one risk type, were one type mixes between contracts, we ca tis situation partial risk pooling. roposition 4: f te single crossing property does not old, partial risk pooling migt occur. roof: As firms make profits wit contract C, and ig-risk, ig-wealt types are indifferent between contracts A and C, some but not a of te individuals migt coose contract C as we in equilibrium. Tis olds only if some of te ig risks sti coose contract A, because oterwise C would, for not making a loss, ave to lie above te pooling line of te and types. But, foowing te proof of roposition 3, tis would violate te equilibrium-existence condition. Q.E.D. One sould remark tat altoug tis partial pooling is a possible equilibrium of te game, it wi be areto-dominated by te complete separating scenario, as discussed in roposition n contrast to partial risk pooling, complete risk pooling is defined by te autor as a situation in wic more tan one risk type buys a particular contract and only tat contract. roposition 5: Complete risk pooling occurs only for very special parameter values. roof: Tis is sown in Figure 4 and proved below. Te autor wi sow tat risk pooling can occur only at a point at wic te indifference curves are tangential; insurers make zero profit; and te indifference curve of te l type, wic goes troug A, intersects. Tis combination occurs only under very restrictive assumption on te parameters. Te autor proceeds in four steps: 8 We only need to consider pooling between te and te type, as te oter types strictly prefer teir contracts to contract C. 9 nterestingly, te profit is made wit a poor type. 10 Note tat for tis argument to go troug one only requires two types of individuals, i.e., te and te types. However, for te oter results of tis study, more tan two types are needed, so te autor sticks to te four types representation. 11 Te autor tanks Arnold Cassagnon for pointing tis out.

10 588 THE JOURNAL OF RSK AND NSURANCE 1. As was argued by Rotscild and Stiglitz (1976), pooling contracts cannot exist if indifference curves of different risk types cut eac oter at tis point. n tat case, one can always find a new contract close to te old one, wic wi be taken up only by te low-risk types. Tis is sown in Figure 4(a), were te pooling contract (b) is dominated by te contract a. So one needs te indifference curves of te and te types to be tangential at a possible point of pooling. 2. A contract in wic te indifference curves are tangential but tat makes a profit also cannot exist in equilibrium, as one could easily under-bid suc a contract. Tis is displayed in Figure 4(b), were te pooling contract (b) lies above te (dased) zero-profit line for te two types. Te zero-profit line as te equation µ Q Q l. A contract like a wi be taken up by bot types and sti make a profit for te firm offering it. So a possible equilibrium contract as to lie on te zero-profit line. 3. Ten it must be te case tat te indifference curves cut te zero-profit line, as te slope of te ig-risk type is always larger tan Q for under-insurance contracts and is larger tan te slope of te zero-profit line, wic is given by Q Ql. Tus one could find a profitable contract to te rigt of te old contract (b) tat would be preferred by bot types and would make strictly positive expected profit (contract a), as is displayed in Figure 4(c). 4. Te only possible stable pooling contract is were te indifference curves of te and te types are tangential at a point tat lies on te zero-profit line and at te same time on te indifference curve of te l type tat goes troug A [see Figure 4(d)]. 12 A contract like a would also attract te l types, wic makes it unprofitable. However, tis is genericay not possible, as can be seen from te foowing argument. Let % Q Q Ql be te pooling premium. Te slope of te indifference curve of te l type is larger tan Q in te region of under-insurance. As Q is larger tan Q%, te indifference curve of te l type tat goes troug point A cuts te zero-profit pooling line at exactly one point. Now consider any point on te pooling line were te slope of te indifference curves of te type and te type are tangential, i.e., for some b we ave 1 Q Ub( X Q% b) 1 Ql Ub( Xl Q% b) 0. Q Ub( X L (1 Q% ) b) Q Ub( X L (1 Q% (10) ) b) l l Define tis expression for a general as S(). Taking te derivative of S() wit respect to, using tat S É b 0 and reformulating gives 12 Note tat te indifference curve of te type is not required to go troug A.

11 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 589 FGURE 4 Nonexistence of Complete Risk ooling Contracts b b a a (a) (b) A l b a b a (c) (d) L ds() 1 Q Ub( X Q% b) ( ( ) ( )) ( (1 ) ) [ Q% AX Q% b A Xl Q% b d Q U b X L Q% (11) b b (1 Q% )( A( X L (1 Q% ) b) A( X L (1 Q% ) b)) ] 0, were A(X ) is te degree of absolute risk aversion. Because tis expression is negative, tere is at most one point on te zero-profit pooling line at wic te indifference curves are tangential to eac oter. t olds only for very special values of (,, Q, Ql, X, X l) tat tis point coincides wit te intersection of te indifference curve of te l type. Q.E.D. l SUARY AND CONCLUSONS Te autor as extended te work by Rotscild and Stiglitz to aow for different wealt in addition to different risks of te insured. f te difference in wealt is sma, different wealt types are pooled wile different risks are separated. f te difference in wealt is large, equilibria may exist at wic firms make positive profits wit separating contracts. Separating different wealt types wit te same risk may occur

12 590 THE JOURNAL OF RSK AND NSURANCE in equilibrium. artial risk pooling contracts can exist, wile complete pooling contracts of two types wit different risks exist only under very special circumstances. Te introduction of a furter unobservable parameter did not aow te autor to solve te equilibrium nonexistence problem. Te study as sown tat in some cases, te problem may be strengtened and in oters, it may be weakened. t is left to te reader to verify tat similar results can be obtained if te individuals differ in te size of teir losses instead of wealt in addition to te different risks. n tis model, te autor considered firms tat offer one contract eac. f one aows for te possibility of a menu of contracts and tus for cross-subsidization, propositions 1, 2, and 5 remain to old, but 3 and 4 do not. To see tis, note tat if firms make a profit wit te poor, low-risk type and zero profit wit te ig risks, ten anoter firm migt offer two contracts tat are sligtly better for bot types, but sti suc tat te ig risks do not prefer to buy te contract designed for te low risks. Ten te contract for te ig risks wi make a loss, but tis loss is muc smaer tan te profit tat wi be made wit te low-risk type. So altogeter suc a deviation is profitable. However, recently nderst and Wambac (2001) ave argued tat if insurers face capacity constraints, ten te Rotscild-Stiglitz contracts are equilibrium outcomes even if a areto-improving pair of cross-subsidizing contracts exists. ntuitively, if a firm can serve only a limited number of customers, ten by offering two new contracts, it cannot be sure tat te low risks wi turn up. ndeed, as tere wi be rationing at te firm, one can sow tat only ig risks wi queue, as tey gain more from coosing te new contracts. Tis in turn makes any deviating offer unattractive. Applying tis logic to te present model implies tat also propositions 3 and 4 remain to old, even if firms offer more tan one contract. One limitation of te model is te restriction to four types only. However, from figures 1 and 3, one can get an idea of a possible solution wit many different degrees of risk aversion. Te set of contracts wi lie along te indifference curve of te ig-risk type wit te igest wealt, wic goes troug te point of fair fu insurance for tis type. ndividuals wit ig risks wi coose te fu insurance contract. Lowrisk types wit different wealt can coose any contract along tis contract line. f teir contract lies above teir zero-profit line (i.e., = Q L l ), teir indifference curve must be tangential at tis point. Tis argument wi break down if te proportion of types is suc tat deviations from tese points are profitable. An extension to a finite number of types wit respect to risks is also possible. As was argued by Rotscild and Stiglitz, in a continuum of risk types, equilibrium does not exist. t migt be argued tat in equilibrium, te type as an incentive to demonstrate is wealt, as te insurer could in tat case offer im a contract better tan B. However, it is not clear tat one can credibly reveal tat one as low wealt, and in any case, te autor as used wealt to demonstrate te effects of different risk aversion. Tat risk aversion by itself is not verifiable is surely not contentious. n tis study, te autor as limited te insurer to offering only deterministic price quantity contracts. t can easily be seen tat random contracts wi not cange te results, because if an accident as appened, it is te type wo is more risk averse

13 NTRODUCNG HETEROGENETY N THE ROTHSCHLD-STGLTZ ODEL 591 tan te type. 13 However, in some cases, te insurer migt ave more instruments available to deal wit multidimensional asymmetric information. For example, life insurance contracts are often sold togeter wit saving contracts tat migt be used to screen between ric and poor customers. Tis wi be an interesting topic for future researc. 14 REFERENCES Aseim, G. B., and T. Nilssen, 1996, Non-Discriminating Renegotiation in a Competitive nsurance arket, European Economic Review, 40: Dasgupta,., and E. askin, 1986, Te Existence of Equilibrium in Discontinuous Economic Games, : Applications, Review of Economic Studies, 53(1): Fluet, C., and F. annequin, 1997, Complete vs. ncomplete nsurance Contracts Under Adverse Selection wit ultiple Risks, Geneva apers on Risk and nsurance Teory, 22: Grossman, H.., 1979, Adverse Selection, Dissembling and Competitive Equilibrium, Be Journal of Economics, 10: Hewig,., 1987, Some Recent Developments in te Teory of Competition in arkets wit Adverse Selection, European Economic Review, 31: nderst, R., and A. Wambac, 2001, Competitive nsurance arkets Under Adverse Selection and Capacity Constraints, European Economic Review, fortcoming. Laffont, J. J., and J. C. Rocet, 1988, Stock arket ortfolios and te Segmentation of te nsurance arket, Scandinavian Journal of Economics, 90(3): Landsberger,., and. eilijson, 1996, Extraction of Surplus Under Adverse Selection: Te Case of nsurance arkets, Journal of Economic Teory, 69: iyazaki, H., 1977, Te Rat Race and nternal Labor arkets, Be Journal of Economics, 8: Riley, J. G., 1979, nformational Equilibrium, Econometrica, 47: Rotscild,., and J. E. Stiglitz, 1976, Equilibrium in Competitive nsurance arkets: An Essay on te Economics of mperfect nformation, Quarterly Journal of Economics, 90: Smart,., 2000, Competitive nsurance arkets wit Two Unobservables, nternational Economic Review, 41: Spence,., 1978, roduct Differentiation and erformance in nsurance arkets, Journal of ublic Economics, 10: Vieneuve, B., 1997, Random Contracts and ositive rofits in an nsurance arket Under Adverse Selection, mimeograp. Wilson, C., 1977, A odel of nsurance arkets wit ncomplete nformation, Journal of Economic Teory, 16: Te ig risks obtain a nondistorting contract. f te low risks are more risk averse, by offering tem a random contract, te ig risks are not furter deterred from coosing tis contract, wile te low risks are made worse off. A more general discussion on random contracts is provided in Vieneuve (1997). 14 Laffont and Rocet (1988) ave taken a first step in tis direction by discussing a monopolist insurer, wo conditions er contracts in a linear pricing sceme on te risky assets eld by te insured.

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