Adverse selection in the annuity market when payoffs vary over the time of retirement


 Roderick Randall
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1 Adverse selecton n the annuty market when payoffs vary over the tme of retrement by JOANN K. BRUNNER AND SUSANNE PEC * July 004 Revsed Verson of Workng Paper 0030, Department of Economcs, Unversty of nz. Abstract Ths study deals wth a specfc mplcaton of adverse selecton on annuty prcng. Varyng the tmepath of the payoffs over the retrement perods affects annuty demand and welfare of ndvduals wth low and hgh lfe expectancy n dfferent ways. Therefore they can be separated by nsurance frms through approprate contract offers. We show that n ths framework a NashCournot equlbrum may not exst; f one exsts, t wll be a separatng equlbrum. On the other hand, even f a separatng equlbrum does not exst, a Wlson poolng equlbrum exsts. (JE: D8, D9, G) * Address: Department of Economcs, Unversty of nz, Altenberger Straße 69, A4040 nz, Austra. Phone: , 8593, FAX: 98. Emal:
2 Introducton Prvate lfeannuty markets are frequently recognzed as beng weak. That s, less lfeannutes are demanded than one could expect, gven the need to nsure aganst uncertanty about the duraton of lfe, n order to smooth consumpton approprately over one's lfetme. Emprcal evdence for ths fact, whch s sometmes called the "annuty puzzle", has been establshed n varous studes for the US (see, e.g., MOORE AND MITCE [000], FRIEDMAN AND WARSAWSKY [990]), but also for the U.K., Canada and other countres (for an overvew see BROWN [00]). To the extent that the low demand s explaned by a bequest motve or by the exstence of a publc penson system, the weakness s not attrbuted to an ntrnsc problem of ths market. owever, there s a further reason put forward n the lterature, namely asymmetrc nformaton whch leads to adverse selecton: The fact that ndvduals have more nformaton about ther lfe expectancy than annuty companes leads to an overrepresentaton of persons wth a hgh survval probablty among the buyers of annuty contracts, whch n turn drves down the rate of return on annutes below the rate correspondng to the average probablty of survval. As a consequence of ths phenomenon, a loss of welfare arses for persons who cannot buy an approprate annuty contract. Ths shortcomng of the annuty market s supposed to become ncreasngly mportant, because n many countres the exstng publc penson system, organzed Emprcal evdence suggests that none of these three reasons alone, but only the nteracton of adverse selecton, publc penson system and bequest motves can explan the weakness of the market. See, e.g., FRIEDMAN AND WARSAWSKY [988, 990], WAISER [000], MITCE ET A. [999].
3 accordng to the payasyougo method, s expected to allow only a reduced replacementrato n the future, hence ncreased prvate nsurance wll be requred. In the present paper we focus on the fact that annuty contracts provde perodc payouts for the duraton of the annutants' lfe (or at least for a fxed number of years). We pont out a further consequence of the asymmetrc nformaton problem, n addton to the adverseselecton problem descrbed so far: The tme structure of the payoffs matters. Indvduals wth low lfe expectancy wll put less weght on the payment they may not receve n the last perod of lfe than ndvduals wth hgh lfe expectancy do. Ths fact can be used by frms to offer annuty contracts whch are favourable for lowrsk ndvduals but not for hghrsk ndvduals. Indeed, n two recent emprcal papers, FINKESTEIN AND POTERBA [00, 004] have found evdence for such selecton effects n the U.K. annutes market. They analyzed three types of annuty contracts, whch dffer n the tmepath of payoffs: constant nomnal payoffs, annually escalatng nomnal payoffs and nflatonndexed payoffs. They showed that for the latter two contracts the expected present value of the payoffs, based on the average populaton mortalty, s sgnfcantly lower than that for fxed nomnal annutes. Ths result suggests that those two contracts, whch provde the hgher payoffs n later years, are selected by ndvduals wth a hgh lfeexpectancy: Only these ndvduals have an ncentve to buy such contracts, because for them the expected present value of the payoffs, based on ther low mortalty rates, s hgher and may exceed that of annutes wth decreasng real (.e. fxed nomnal) payoffs; the latter
4 are favourable for ndvduals wth lower lfeexpectancy. In fact, estmatng a hazard model regardng the annutants' lfespans, FINKESTEIN AND POTERBA [004] found clear evdence for such an annutant selfselecton wth respect to the tme profle of payoffs. Moreover, the selecton effects turned out to be qute large. In the present contrbuton we provde a theoretcal analyss of the functonng of annuty markets, when selecton through the tmng of payoffs takes place. In partcular, we nvestgate the reacton of nsurance demand and the consequences for the exstence of equlbra, f nsurers offer contracts whch vary wth respect to the tmepath of the payoffs. In the model usually employed for the analyss of annuty markets (see PAUY [974], ABE [986] and WAISER [000]), there s one perod of retrement, and there are two groups of ndvduals wth dfferng lfe expectancy. Competton takes place va prces (.e. va the rate of return, that s the penson payment per unt of annuty), whch are fxed by the frms. Indvduals can buy as many annutes as they want. As s wellknown, n ths framework only a poolng equlbrum s possble, where all ndvduals receve the same rate of return. We extend ths model by ntroducng two perods of retrement, to whch the ndvduals may or may not survve, and by assumng that the payoffs need not be the same n both The lower expected present dscounted value of the real annuty, based on average mortalty, may partly also arse, because a premum for the nsurance aganst nflaton has to be pad. The market for real annutes s analyzed n BROWN, MITCE AND POTERBA [00], who study the role of governmentssued nflaton ndexed bonds and other securtes as nstruments, whch nsurance companes use to hedge prce level rsks (prmarly n the UK and US). owever, the authors do not consder selecton effects. 3
5 perods. Ths mples that contracts are characterzed by two prces, set by the frms. The mportant aspect n ths extended model s that  n accordance wth the observaton mentoned above  annuty demand as well as welfare of the ndvduals are senstve wth respect to the tme structure of the payoffs, and the possblty arses for frms to separate buyers accordng to ther survval probabltes. Ths addtonal separaton effect, whch was up to now neglected n the theoretcal lterature, may represent a further explanaton for the fact that annuty markets are not well developed. Indeed, t turns out that n such a market no NashCournot equlbrum may exst. If one exsts, t wll be a separatng equlbrum. The NashCournot equlbrum n nsurance markets was studed by ROTSCID AND STIGITZ [976]. In ther framework frms offer a number of dfferent contracts whch specfy both a prce and a quantty. Indvduals who prefer a hgher quantty are wllng to pay a hgher prce for t. A prerequste for the exstence of prce and quantty competton s that ndvduals can buy at most one contract, whch may be a reasonable assumpton for some nsurance markets, e.g. nsurance aganst accdents, but seems dffcult to apply to the annuty market. 3 Consequently, n our model ndvduals are free to buy as many annutes as they want. Separaton becomes possble because frms can fx two prces nstead of a prce and a quantty. As a potental answer to the queston of what happens n an nsurance market, f no NashCournotequlbrum exsts, WISON [977] ntroduced a dfferent equlbrum 3 ECKSTEIN, EICENBAUM AND PEED [985] make ndeed the assumpton of a prce and quantty competton for the annuty market wth one perod of retrement only. In ths framework they derve the same results as ROTSCID AND STIGITZ [976]. 4
6 concept, whch s based on specfc belefs of frms concernng the reacton of other frms to new contract offers. We show that a Wlson equlbrum always exsts n our model. Other studes whch qut the assumpton of a sngle perod of retrement are by TOWNEY AND BOADWAY [988] and FEDSTEIN [990]. Feldsten consders a publc penson system, organzed accordng to the payasyougo method, and dscusses the tme structure of the benefts. e assumes two perods of retrement, but only survval to the second s uncertan. In ths framework current populaton prefers to receve benefts ether n the frst or n the second perod of retrement, dependng on whether the return on socal securty s lower or hgher than the expected return on prvate savng. owever, steadystate welfare s maxmzed by payng benefts only n the frst retrement perod, snce ths ncreases savngs and therefore unntended bequests. The paper by TOWNEY AND BOADWAY [988] deals wth the market for prvate annutes and s, thus, more related to the present contrbuton. The authors model the lfespan from retrement to death n contnuous tme and consder termnsured annuty contracts,.e. contracts whch guarantee a stream of payoffs for a lmted tme, ether untl the nsured ndvdual des or untl the term of the annuty expres. In ther analyss of equlbra, Townley and Boadway take the stream of payoffs as constant over the whole duraton; hence the contracts are characterzed by two parameters: the term (duraton) and the payoff (per unt of money nvested). In contrast to our model, where frms can separate costumers through a varaton of the payment over tme, Townley and Boadway study separaton effects wth respect to the term of the annuty: 5
7 Indvduals wth longer expected lfespan estmate a contract wth a longer duraton hgher than ndvduals wth shorter expected lfespan. In the framework of ther model, wth asymmetrc nformaton concernng lfeexpectancy, no equlbrum may exst, f t exsts, t s ether a poolng equlbrum or a separatng equlbrum. In TOWNEY AND BOADWAY [988] ndvduals can make provson for the tme after expry of the annuty through prvate savngs only. In a related study (BRUNNER AND PEC [00]), we have consdered a model whch also takes the exstence of tmelmted annuty contracts nto account, but allows ndvduals to provde for the tme after expry of the annuty by purchasng another annuty. That s, ndvduals need not make ther decson concernng oldage provson for the whole tme of retrement at once, but can do so sequentally. In ths framework t turns out that only a stuaton, where all ndvduals decde sequentally, represents an equlbrum, whch s to the dsadvantage of the shortlvng ndvduals. The rest of our paper proceeds as follows: In Secton we ntroduce the basc model of consumpton behavour under asymmetrc nformaton wth two perods of retrement, where ndvduals provde for oldage by buyng annutes. We analyze the effect of a varaton n the tme structure of the payoffs on annuty demand and on welfare of an ndvdual under uncertan lfetme. In Secton 3 we turn to the nvestgaton of equlbra. Frst, we derve all results concernng the exstence and characterzaton of the equlbra n the basc model. Then we extend the model and allow ndvduals to save n rskless bonds n addton to annutes. Secton 4 contans concludng remarks. 6
8 Annuty demand n a model wth two perods of retrement. The basc model wth asymmetrc nformaton Consder an economy wth N ndvduals who lve for a maxmum of three perods t = 0,,. In the workng perod t = 0 ndvdual earns a fxed labour ncome w, spends an amount A on annutes and consumes an amount for perod 0: c o. Ths gves the budget equaton () c0 = w A. The ndvduals retre at the end of perod 0. Through the purchase of annutes they make provson for future consumpton n the two perods of retrement t =,. An annuty contract s characterzed by the payoffs (q,q ): An annuty A = pays q t unts of money to the ndvdual n the retrement perods t =,, f she survves. ence, for ndvdual the budget equatons for the two retrement perods are () c = qa, (3) c = qa. The budget equatons () (3) are bult on the assumpton that the ndvduals do not save and buy other assets, n addton to annutes. At ths stage of the analyss we exclude holdng other assets, n order to concentrate on the desgn of the annuty 7
9 contracts. owever, the possblty of buyng bonds n the workng perod and n the frst perod of retrement s explctly consdered n Secton 3.4, and t wll be shown that ths does not change the man results derved n the basc model. Further, for the sake of smplcty, the assumpton s made that no publc penson system exsts. 4 Survval to perod t = s uncertan and occurs wth probablty π, 0< π <. In the same way, gven that an ndvdual s alve n perod, survval to perod occurs wth probablty π, 0< π <. Each ndvdual decdes on her consumpton plan over the uncertan duraton of her retrement by maxmzng expected utlty from a tmeseparable utlty functon U, (4) U ( π ( ) ( ) u( c 0) π ( π ) u( c 0) αu( c ) π π u( c 0) αu( c ) α u( c ) ) = , subject to condtons (), () and (3). In (4) uc ( t ) descrbes utlty of consumpton per perod, where we assume that u ( c t ) > 0, u ( c t ) < 0 and lm u ( c ) = c 0. α denotes the oneperod dscount factor of utlty, wth 0< α. Notce that the specfcaton n (4) means that the ndvduals dscount future consumpton for two reasons, rsk averson and tme preference. (4) can be reduced to (4 ) U = u( c ) + παu( c ) + ππα u( c ). 0 4 In Secton 4 we dscuss the consequences for our results, f a publc penson system s ntroduced n the model. 8
10 Insertng (), () and (3) nto (4 ) and dfferentatng wth respect to A yelds the frst order condton of ths maxmzaton problem as (5) u'( c ) + παqu'( c ) + ππα q u'( c ) = 0. 0 From () and (3) we know that c >_ < annuty demand determned by (5), for gven (q,q ). c corresponds to q >_ < q. et A (q,q ) be the From now on we assume that the otherwse dentcal ndvduals are dvded nto two groups =,, characterzed by dfferent rsks of a long lfe,.e. by dfferent probabltes of survval π t > π for t =,. et γ and ( γ) denote the share of the t hghrsk and lowrsk ndvduals, resp., wth 0 < γ <. The probabltes π t and the share γ are publc nformaton, known by the annuty companes. But t s the prvate nformaton for each ndvdual to know her type,.e. her probablty of survval. As a consequence, there s an adverseselecton problem n the annuty market. Ths s llustrated by the followng lemma, whch shows that hghrsk ndvduals buy more annutes than lowrsk ndvduals, gven any contract (q,q ). emma : For any contract (q,q ) an ndvdual wth hgh survval probabltes wll demand a larger quantty of annutes than an ndvdual wth low survval probabltes,.e. A (q,q ) > A (q,q ). 5 5 The proofs of all emmas and Propostons are relegated to the Appendx A.. 9
11 Ths result mples that f there s only a sngle contract offered n the annuty market, wth some gven payoffs (q,q ) per perod, then the share of annuty purchases of hghrsk ndvduals n total annuty demand s larger than γ, whch s the share of hghrsk ndvduals n the economy.. Separatng and poolng contracts An annuty contract ( q ) s sad to be ndvdually far for an ndvdual of type =,, f expected payoffs equal the prce,.e. f t fulflls q q = 0, (6) π ππ gven the assumpton of a zero nterest rate, whch s chosen for the sake of smplcty; a postve nterest rate would not affect the qualtatve results. Obvously, (6) mples that the annuty companes make zero expected profts, gven that solely ndvduals of type buy ther ndvdually far contracts. owever, as there exst many contracts ( q ) whch fulfll (6), t s nterestng to nvestgate whch of the ndvdually far annuty contracts s the most preferred one by an ndvdual of type. The next emma provdes a characterzaton. emma : Among all ndvdually far contracts ( q ) for an ndvdual of type, the most preferred s characterzed by 0
12 (7) u'( c) = αu'( c), whch mples that q > q, f α < and q = q, f α =. That s, n case of a zero rate of tme preference (α = ), an equal dstrbuton of the payoffs ( q/ q = ) over the two perods of retrement s optmal, for both types =,, gven ther respectve ndvdually far contract. For α <, however, the optmum rato q / q ( > ), determned by (7), wll n general be dfferent for the two types, because n ths case the optmum rato depends on the respectve annuty demand A and A, whch wll be dfferent. (But one checks easly that the most preferred rato q / q s ndependent of A and thus dentcal for both types, n case of a perperod utlty functon u whch exhbts a constant relatve rsk averson, rrespectve of the rate of tme preference.) Note, moreover, that wth the most preferred ndvdually far contract the relaton (7), whch characterzes the optmum dvson of consumpton between the two perods of retrement, also apples for the allocaton decson between consumpton n the workng perod and the frst perod of retrement, namely (8) u'( c0) = α u'( c). Ths can be seen when elmnatng u'( c ) n (5) by use of (7), whch yelds u'( c ) = αu'( c )( π q + π π q ). Substtutng (6) nto ths condton, t reduces to (8). It 0
13 follows that an ndvdual, who does not dscount future consumpton due to tme preference (α = ), consumes the same amount n all three perods of lfe,.e. c0 = c = c. Otherwse (α < ), she chooses 0 > > c c c. The assumpton π t < π, t =,, mples that ndvdual farness (condton (6)) for t each group can be fulflled only wth two separate contracts. If each s bought by the respectve rsk group, both produce zero profts. On the other hand, a contract (q,q ) whch s bought by both groups, s called a poolng contract. In order that a poolng contract produces zero profts, t must fulfll the condton (for shortness we use A nstead of A (q,q )) (9) γ A qπ qπ π γa qπ qπ π ( ) ( ) + ( ) = 0. Zeroproft contracts (whether separate or poolng) are of specal nterest, because under the assumpton of perfect competton n the annuty market, only such contracts can persst. (9) can also be wrtten as (9') ρ q q q( π ρ q q π ) q( π π ρ q q π π ) + (, ) + (, ) + (, ) = 0, where ρ s defned by (, ) ( ρ q q γa ( q) ) (( γ) A ( q) ), that s the rato of annuty demand of both groups. Note that ρ depends on (q,q ), but for shortness, we usually do not ndcate ths dependency. Of course, our assumptons on the survval probabltes mply that for the lowrsk ndvduals expected returns from a zeroproft
14 poolng contract are lower than requred for ndvdual farness ( qπ q ππ > 0), whle for the hghrsk ndvduals they are hgher ( qπ q π π < 0)..3 Varyng the payoffrato of a poolng contract In the emmas 3 and 4 below, we consder a zeroproft poolng contract and nvestgate the effect of a margnal change n the payoffs on ndrect utlty and on annuty demand of an ndvdual of type =,. Clearly, f q (or q ) s ncreased alone, then both groups beneft and buy more annutes. owever, such an ncrease would produce a loss for the annuty companes. ence, the nterestng case s when q s ncreased at the expense of q (or vce versa), such that the zeroproft condton (9) remans fulflled. We characterze the frstround effect on ndrect utlty and on annuty demand of a margnal ncrease of q, when the assocated change of q, such that (9') remans fulflled, s calculated under the assumpton of a constant rato ρ of annuty demand of the two groups. Moreover we dscuss the condtons necessary for the secondround effects (.e. the nfluence of the payoffs on ρ) not to outwegh the frstround effects. In emma 3() we consder a contract wth a rato of the payoffs, whch s optmal for an ndvdual of type accordng to condton (7) for an ndvdually far contract. We show that ths rato s no longer optmal n case of a zeroproft poolng contract: The lowrsk ndvdual benefts, f q s ncreased. An analogous result s found for a hghrsk ndvdual (emma 3()): She benefts f q s reduced. 3
15 emma 3: Consder two poolng contracts ( q ), ( q ) where each, together wth annuty demand of the two groups, fulflls the zeroproft condton (9'). () If the payoff rato q q satsfes the condton (7) for an optmal ndvdually far contract for type, a margnal ncrease of q (and thus a margnal decrease of q ) where (9') for fxed ρ remans fulflled, makes an ndvdual of type better off. () If the payoff rato q q satsfes the condton (7) for an optmal ndvdually far contract for type, a margnal ncrease of q (and thus a margnal decrease of q ) where (9') for fxed ρ remans fulflled, makes an ndvdual of type worse off. Ths frstround effect descrbed n emma 3 s of partcular nterest, because t reveals the mechansm whch s responsble for the negatve result concernng the exstence of a poolng contract n equlbrum (see Secton 3.). For an llustraton, consder the case α =, whch means as we know from emma  that both ndvduals prefer an equal dstrbuton of the payoffs over the two perods of retrement, gven ther respectve ndvdually far contract. owever, n case of a zeroproft poolng contract, such an equal dstrbuton of the payoffs s no longer optmal: Indvduals wth low lfeexpectancy are better off, f q, the payoff n the frst perod of retrement, s ncreased at the expense of q, whle the opposte holds for ndvduals wth hgh lfeexpectancy. Thus, the annuty companes have an ncentve to desgn separate contracts for the two groups. The ntutve reason why a lowrsk ndvdual fnds a shft of consumpton from perod to perod attractve can easly be explaned for α = (and thus startng from q = q ) 4
16 as follows: If q s ncreased by one s decreased by dq / dq, whch s determned by the requrement that the zeroproft condton (9') be preserved. Snce wth a poolng contract the assocated decrease of q goes more to the expense of the hghrsk ndvduals, t turns out from (9') that dq / dq < / π (for constant ρ). As a result, for type ndvduals the expected loss n perod, π dq / dq, s lower than one and they beneft from a shft towards ncreasng q. (Note that due to q = q, margnal utlty s equal n both perods.) By the same reasonng and observng that, on the other hand, dq / dq > / π holds, type ndvduals, who expect to lve longer, are better off by a shft towards reducng q. Smlar consderatons apply for the case of α <. Obvously also the secondround effect, that s the effect ρ q of the change of the t payoffs on (the rato of) annuty demand of the two groups, matters, as can be seen from (A6) n the Appendx. The above consderaton certanly mantans, f both ρ q, t t =,, are suffcently small, otherwse an approprate relaton between them must hold. (For nstance, a suffcent, but not necessary condton s π ρ q ρ q π ρ q, whch ensures that the secondround effect goes nto the same drecton as the frstround effect.) Remark: Inspecton of the proof of the foregong emma shows that an ncrease of q at the expense of q mproves welfare of lowrsk ndvduals also f ntally the rato q / q s lower than that determned by the optmalty condton (7) for ndvdually far contracts. It follows that ther most preferred poolng contract exhbts a hgher rato (that s, n case of α = : q > q ). By smlar reasonng one fnds for the hghrsk 5
17 ndvduals that ther most preferred poolng contract exhbts a lower payoffrato than that determned by (7) (whch means q < q n case of α = ). A characterzaton of the effect of a margnal change of q (and q ) on annuty demand s gven n the followng emma, agan startng from a zeroproft poolng contract wth a payoffrato whch satsfes (7) for the respectve optmal ndvdually far contract. We restrct attenton to the case where the dscount factor α equals one or the perperod utlty functon exhbts a constant coeffcent of relatve rsk averson n the sense of ArrowPratt, defned as R cu ( c ) u ( c ). Then, as mentoned above, the optmal t t t payoffrato, gven an ndvdually far contract, s the same for both rsktypes (and equal to, f α = ). emma 4: Assume that α = or that R s constant. Consder a poolng contract ( q ) whch, together wth annuty demand of the two groups, fulflls the zeroproft condton (9') and whose payoff rato q q s determned by the condton (7) for an optmal ndvdually far contract for both types =,. Then the effect of a margnal ncrease of q on the annuty demand of each ndvdual =,, where (9') for fxed ρ remans fulflled, depends on the relatve rsk averson n the followng way: Iff R < > _, then da dq < _ da > 0 and dq <> _ 0. Ths result follows from the fact that, per defnton, the effect of an ncrease of q on q u'(q A ),.e. on the margnal utlty of A n perod, can be wrtten as ( R), and the 6
18 same apples to perod. ence, whether an ncrease of q at the expense of q ncreases or decreases expected margnal utlty of A (n both retrement perods together) depends on ( + π dq / dq) ( R), where, as argued above, π dq / dq descrbes the expected loss n perod, f q s ncreased and the zeroproft condton s preserved. (Note that, by assumpton, ether q = q whch means that R s equal n both perods, or R s constant at all.) We know from above that + π dq / dq s postve for = and negatve for =, gven a fxed rato ρ of annuty demand of both groups. Thus we fnd that, n case of R <, for type ndvduals the expected margnal utlty of A n the two perods of retrement ncreases, f q s ncreased at the expense of q. On the other hand, the decson on annuty demand s made by balancng the (negatve) margnal utlty of A n the workng perod aganst the expected (postve) margnal utlty n retrement. It s ntutvely clear that demand ncreases, f the latter ncreases (the former s unaffected by a change of q and q ). Moreover, n case of R >, the effect obvously goes towards a decrease of A, and smlar consderaton hold for type ndvduals. 3 Equlbra Introducng two nstead of one retrement perod n the model allows annuty companes to offer contracts whch dffer n the dvson of the payoffs over tme. In ths secton t s shown that ths mples the possblty of a separatng equlbrum, whch means that annuty companes separate ndvduals accordng to ther survval probabltes. To obtan ths result we make use of the wellknown concept of a NashCournot equlbrum, whch was studed by ROTSCID AND STIGITZ [976] n the context of nsurance markets. Our result s n contrast to studes consderng one perod of 7
19 retrement only, whch fnd that under prce competton there wll be a poolng equlbrum. In Subsecton 3.3 we extend the analyss by ntroducng the concept of the WISON [977] equlbrum, where t s assumed that frms antcpate reactons of the other frms to new contract offers, vz. that they wll wthdraw unproftable exstng contracts. Frst we derve all results concernng the exstence and characterzaton of equlbra n the model consdered n Secton, where ndvduals provde for retrement by buyng annutes only, then we ntroduce, n Secton 3.4, the possblty of savng n rskless bonds. 3. The nonexstence of a poolng equlbrum We call a contract (q,q ) a poolng equlbrum, f together wth A (q,q ), =,, the zeroproft condton (9) s fulflled and f no other contract exsts, whch s preferred to (q,q ) by at least one group {,} and whch allows a nonnegatve proft. Our man result s that n general no poolng equlbrum exsts. As a preparaton we show: emma 5: et (q,q ) be a poolng contract whch together wth A (q,q ), =,, fulflls the zeroproft condton (9). Any contract ( q + δ q + δ q ), whch s close enough to (q,q ) and whch s chosen only by group (.e. A = 0) allows a nonnegatve proft. Ths result follows from the observaton n Secton. that a zeroproft poolng contract offers less expected returns to lowrsk ndvduals than requred for ndvdual 8
20 farness. Ths n turn mples postve profts, f only the lowrsk ndvduals buy ths contract or one close to t. We now ntroduce a further assumpton on U, n addton to strct concavty of the nstantaneous utlty functon u. et ndrect utlty U (q,q ) for any contract (q,q ) be defned n the usual way as utlty attaned wth annuty demand A (q,q ). We assume that ndfference curves n the (q,q )space satsfy the snglecrossng condton U q U q (0) < U q U q for all (q,q ). Ths condton, whch s famlar from other models wth asymmetrc nformaton, requres that the slope of an ndfference curve of a lowrsk ndvdual s always steeper than that of a hghrsk ndvdual. ence, ndfference curves of the two groups can cross only once. Usng the Envelope Theorem, (0) reduces to u'(q A )/(απ u'(q A )) > u'(q A )/(απ u'(q A )), and one observes that, as π < π, the condton s certanly fulflled for any utlty functon whch exhbts a constant coeffcent of relatve rsk averson, hence, n partcular, for logarthmc utlty. Snglecrossng s needed for a concse formulaton of the followng Proposton only; n the remark afterwards t wll be argued that n general the Proposton holds wthout ths assumpton. Proposton : No poolng equlbrum exsts, gven the snglecrossng condton (0). 9
21 Ths result can be llustrated n a dagram where the payoffs q and q are drawn on the axs (see Fgure ). The dashed lne ZP denotes the zeroproft condton (9) for a poolng contract, wth slope dq dq, as determned by (A6) n the Appendx. Consder any contract (q,q ) fulfllng (9),.e. any pont on ZP. Due to the snglecrossng condton the slope of the ndfference curve U correspondng to the lowrsk group s steeper than that of U, the ndfference curve of the hghrsk group. Therefore one can fnd a contract (q + δ q, q + δ q), close to (q,q ), whch s preferred by the lowrsk ndvduals only  and s, therefore, proftable for the annuty companes, as emma 5 tells us. ence (q,q ) does not represent a poolng equlbrum. Fgure The nonexstence of a poolng equlbrum q ( q ) U U (q + q + q ) ZP q Remark: By means of Fgure the sgnfcance of the snglecrossng condton can be dscussed. One observes mmedately that the result of Proposton certanly holds as 0
22 long as the slopes of U and U dffer n (q,q )space, ndependently of whch one s steeper. Even f U and U have the same slope, the result holds, gven that the slope of ZP s dfferent. In ths case one can fnd another poolng contract ( q + δ q + δ q ) close to (q,q ) whch s preferred by both groups and produces nonnegatve profts. Only f there exsts a pont on ZP n whch the slopes of ZP, U and U are dentcal, ths represents a poolng equlbrum. Clearly, ths case can occur for very specfc parameter constellatons only, a small perturbaton of γ or of π t would destroy the equlbrum. From these consderatons we can conclude that n general Proposton holds wthout assumng the snglecrossng condton. 3. The possblty of a separatng equlbrum We call a set of two contracts ( q ), ( q ) a separatng equlbrum, f each fulflls the respectve zeroproft condton (6), f group does not prefer ( q ) to ( q, q ) and vce versa,.e. f U ( q ) U ( q ), () () U q q U q q (, ) (, ), and f no other contract exsts, whch s preferred to ( q ) by at least one group {,} and whch allows a nonnegatve proft.
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