Adverse selection in the annuity market with sequential and simultaneous insurance demand. Johann K. Brunner and Susanne Pech *) January 2005

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1 Adverse selecton n the annuty market wth sequental and smultaneous nsurance demand Johann K. Brunner and Susanne Pech *) January 005 Revsed Verson of Workng Paper 004, Department of Economcs, Unversty of nz. Abstract Ths paper nvestgates the effect of adverse selecton on the prvate annuty market n a model wth two perods of retrement and two types of ndvduals, who dffer n ther lfe expectancy. In order to ntroduce the exstence of lmted-tme penson nsurance, we consder a model where for each perod of retrement separate contracts can be purchased. Demand for the two perods can be decded ether sequentally or smultaneously. We show that only a stuaton where all rsk types choose sequental contracts can be an equlbrum and that ths outcome s favourable for the long-lvng, but s unfavourable for the short-lvng ndvduals. Keywords: annuty market, adverse selecton, uncertan lfetme, equlbrum. JE codes: D8, D9, G. *) Address: Department of Economcs, Unversty of nz, Altenberger Straße 69, A-4040 nz, Austra. Phone: , -8593, FAX: -98. E-mal: johann.brunner@jku.at, susanne.pech@jku.at.

2 . Introducton Socal securty systems, whch n many ndustralsed countres are organsed accordng to the pay-as-you-go method, are threatened by the ageng of the populaton due to a decrease n fertlty and an ncrease n lfe expectancy. Ths problem s recognsed by academcs as well as by poltcans, and several possble measures to mantan the fnancal stablty of the system are suggested. One of these measures s a reducton of the penson payments, and t seems n fact unavodable that t wll be mplemented to some degree. If ths s the case, then a natural strategy for the ndvduals s to rase prvate provson for retrement, n partcular by an ncreased purchase of lfe annutes. As governments want to prevent old-age poverty, they tend to encourage prvate penson nsurance through tax ncentves. owever, there are concerns that the market for annutes does not offer a sutable supplement to the publc penson system. One obvous argument s that t cannot ncorporate redstrbuton, as the publc system does for several reasons. Another argument concentrates on the phenomenon of adverse selecton, whch s a common problem that affects the effcent workng of nsurance markets. The present paper studes ths problem n the context of specfcally desgned contracts for old-age nsurance. Generally, adverse selecton occurs wth asymmetrc nformaton, that s, when the nsurer has less nformaton than the ndvdual as to the probablty that the nsured event occurs. In case of annutes, ths means that companes have less nformaton on lfe expectancy of an annutant than the ndvdual herself. As a consequence, returns from annutes cannot reflect ndvdual lfe expectancy but only overall lfe expectancy, whch n turn wll nduce hgh-rsk ndvduals (that s, the long-lvng) to buy more annutes than low-rsk ndvduals. Ths s the standard observaton, dscussed n varous contrbutons to the lterature (see, e.g., Pauly 974, Ecksten et al. 985, Abel 986, Mtchell et al. 999, Wallser 000). owever, there s a further consequence of the adverse-selecton problem, namely that the tme structure of the benefts matters. Indvduals wth low lfe expectancy put less weght on penson payouts n later perods than ndvduals wth hgh lfe expectancy. Ths aspect s Tax ncentves are granted n many ndustralsed countres, e. g. n Great Brtan, U.S.A, Canada and Sweden. Moreover, the recent reform of socal securty n Germany ams at cuttng publc pensons and nducng ndvduals, by grantng a tax release, to contrbute four percent of ncome to prvate old-age nsurance. Smlarly, n Austra contrbutons to prvate old-age nsurance are subsdsed by a premum snce 000.

3 neglected n the usual overlappng-generatons model wth one workng perod and one perod of retrement. But n realty the tme of retrement must not be seen as a sngle, homogeneous perod, for whch provson can be made through a once-and-for-all contract only, wth a fxed and constant (n nomnal or real terms) payout. Plannng ndvduals, beng aware of some estmate of ther lfe expectancy, wll attempt to make provson n accordance wth ths estmate, whch means that they want to use more dfferentated nstruments. In practce, they can buy an nsurance contract wth payouts ncreasng or decreasng over tme, or they can buy a lmted-tme contract for the earler phase of retrement and then use another nstrument to provde for the rest of ther lfetme. In order to analyse the consequences on the functonng of the annuty market of the fact that the tme structure of the payouts matters, one has to extend the standard model by assumng that retrement conssts of more perods and that provson can be made separately for each of them. Brunner and Pech (005) ntroduce a model wth one workng perod and two perods of retrement, where two groups of ndvduals wth dfferng lfe expectancy buy an annuty contract whch runs for the whole tme of retrement, but wth payouts possbly varyng over tme. 3 It s shown that n ths framework an equlbrum n the sense of Nash- Cournot may but need not exst. 4 In the present contrbuton we consder a smlar model, but wth dfferent types of contracts. We agan assume that ndvduals lve for one workng perod and for at most two perods of retrement, but now contracts run for one perod only; for the second perod, a new contract has to be bought. By ths formulaton we take account of the fact that n realty term-nsured penson contracts exst, whch provde payouts only for a lmted tme, gven that the ndvdual s alve. For the rest some other form of provson must be made Poterba (997) emphaszes the mportance of the wde range of dfferent annuty products for the growth of the U.S. annuty market. e provdes a typology of ndvdual annutes wth respect to the terms under whch accumulated captal s dspersed durng the lqudaton phase. In partcular, he dstngushes between two broad classes of ndvdual annutes, that are deferred and mmedate annutes, dependng on whether there s a watng perod between the premum payment and the begnnng of the annuty payouts or not. The role of annuty contracts wth escalatng payouts n the U.K. annuty market s studed by Fnkelsten and Poterba (00). Yag and Nshgak (993) also employed a model wth one workng perod and two perods of retrement n order to dscuss optmal nsurance demand of a representatve ndvdual. They showed that constant annuty payouts over tme are neffcent, gven that the ndvdual rate of tme preference dffers from the nterest rate. owever, they dd not consder the adverse-secton problem and ts mpact on the exstence of equlbra. Wth these lfe annutes, frms can separate ndvduals accordng to ther lfe expectancy by a varaton of the payouts over tme. In fact, only a separatng equlbrum (compare Rothschld and Stgltz 976) can occur. Townley and Boadway (987) studed the functonng of the annuty market when ndvduals save out of ther payouts from the lmted-tme penson contract. In contrast, we consder the case that they can buy a second annuty to provde for the remanng tme.

4 The mportant ssue whch we address s that ndvduals can choose between two strateges to provde for the second perod of retrement: smultaneously, that s, ndvduals buy an addtonal contract already n the workng perod, or sequentally, that s, only those ndvduals who have survved to the frst perod of retrement, purchase an addtonal contract on the spot market. We show that n our model ndvduals n general chose only one of these alternatves, dependng on the prces. owever, n a frst-best equlbrum, prces that then correspond to the ndvdual lfe expectances assume such values whch make ndvduals ndfferent between the two alternatves, because each provdes the same consumpton path over lfetme. Ths s no longer true f asymmetrc nformaton, where prces are dstorted by adverse selecton, s ntroduced n our model. Then, under the assumpton of prce competton between annuty companes, the prce for any contract s the same for both rsk-groups, and only a stuaton where both groups buy the same type of second-perod contract s feasble. Further, t turns out that the type of contract chosen to provde for the second perod of retrement also affects the prce of the frst-perod contract. In partcular, we fnd that the two strateges have dfferng consequences for the welfare of the ndvduals, because they allow dfferent consumpton paths over the tme of retrement: long-lvng ndvduals, who put more weght on consumpton n the second perod, prefer the regme when all ndvduals make sequental provson, whle short-lvng ndvduals prefer the regme wth smultaneous provson. Assumng that nsurance companes can credbly commt n the workng perod to offer contracts at a pre-specfed prce, we fnd that only the former regme, favourable to the long-lvng ndvduals, represents a Nash-Cournot equlbrum. Ths result s puzzlng because emprcal evdence shows that n fact term-lmted contracts represent a small share of annuty contracts (see, e.g., Mtchell et al., 999). We dscuss possble explanatons for ths puzzle n the concludng remarks. In the followng Secton we ntroduce the basc model and show that ether smultaneous or sequental annuty contracts are chosen. We characterse demand n both cases. In Secton 3 we analyse the consequences of adverse selecton for annuty prces and for the exstence of an equlbrum. Moreover, we study consumpton and welfare of the ndvduals. Secton 4 provdes a dscusson of the results. 3

5 . Sequental and smultaneous demand for annutes Consder an economy wth ndvduals who lve for a maxmum of three perods t = 0,,. In the workng perod t = 0, each ndvdual earns a fxed labour ncome w 0. At the end of perod 0 she retres and lves for at most two further perods. Survval to the retrement perod t = 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 <. Provson for old age can be made through three types of annuty contracts, whch are offered by nsurance companes: - A denotes the quantty of a contract, whch s bought at a prce Q n workng perod 0 and offers an mmedate payout A n retrement perod. - A denotes the quantty of a contract, whch s bought at a prce Q n retrement perod and offers an mmedate payout A n retrement perod. - D denotes the quantty of a contract, whch s bought at a prce R n workng perod 0 and offers a deferred payout D n retrement perod. That s, each type of contract offers payouts for one perod of retrement, but they dffer n the date of purchase and the watng perod for the payout to begn: provson for retrement perod s made through A, whle provson for retrement perod can be made through A (bought by those only, who survve to retrement perod ) and/or through D (bought already n the workng perod). Q, Q and R are the correspondng prces (premums, resp.) per unt of annuty payout; the recprocal of each prce represents the rate of return on each contract type. We assume that the ndvduals have no bequest motve, whch means that savng s not an attractve strategy for them to provde for old-age. Ths follows from the fact that the rate of return of annutes s hgher than the nterest rate, as annutes allow to avod (and redstrbute) unntended bequests (see Yaar 965). Further, n order to concentrate on the desgn of the annuty contracts and to smplfy the analyss, the assumpton s made that no publc penson system exsts. The budget equaton of an ndvdual for the workng perod 0 s 0 0 c = w Q A R D. (.) 4

6 Moreover, gven that ndvdual s alve n the retrement perod, she can spend an amount QA from her ncome A n order to make addtonal provson for consumpton n the retrement perod, and consumes an amount c. Ths gves us the budget equatons for the two retrement perods t =,: =, (.) c A Q A c = A + D. (.3) Preferences over lfetme consumpton of an ndvdual are tme-separable and are represented by expected utlty wth a per-perod utlty functon u dependng on consumpton. An ndvdual s confronted wth the followng two-stage decson problem: n the workng perod 0, she decdes on the quanttes A and D of annutes, thus on her consumpton level n perod 0 and on her ncome w t n each of the two retrement perods t =,. For ths decson she takes nto account her optmal annuty demand A and her optmal consumpton levels n perods and, about whch she wll decde n perod, gven that then she s alve. Formally, ths two-stage problem can be wrtten as: t = 0: o max u(c ) +πϕ (A,Q,D ), (.4) s. t. (.), π t = : max u(c ) + u(c ), (.5) s. t. (.) and (.3), where c,c,a ϕ (A,Q,D ) max {u(c ) +π u(c ) c = A Q A, c = D + A }. Concernng the A -contract, we n fact assume that the ndvduals are nformed about ts prce Q already n the workng perod 0, n other words, that the nsurance companes can credbly commt to offer those contracts at a prce Q one perod later. Otherwse ϕ would not t > be well-defned. 6 Further, we assume u (c ) 0, u (c ) 0 and lm u (c) =. Notce that the 0 specfcaton of the decson problem means that the ndvduals do not dscount future consumpton for any reason other than rsk averson. 7 t < c 6 7 We leave t to the concludng secton to dscuss the consequences of relaxng ths assumpton, e.g. that the ndvduals are uncertan n the workng perod about the future prce level Q, due to mssng nstruments of credble commtment by the frms. To smplfy notaton, we do not nclude a tme preference parameter explctly n the utlty functon. To do so, would mean that a per-perod dscount factor enters (.4) and (.5) just n the same way as the survval probabltes. Nothng would change wth the results. 5

7 By nsertng (.) nto (.4) and dfferentatng wth respect to (.) and (.3) nto (.5) and dfferentatng wth respect to condtons of ths maxmzaton problem: A and D as well as nsertng A, we obtan the Kuhn-Tucker ϕ 0 A ϕ 0 D (A,Q,D ) Qu'(w QA R D ) +π = 0, (.6) D > 0 and D = 0 and (A,Q,D ) Ru'(w QA RD) +π = 0or ϕ 0 D (.7a) (A,Q,D ) Ru'(w QA RD) +π 0, (.7b) A > 0 and A = 0 and Qu'(A QA ) +π u'(d + A ) = 0or (.8a) Qu'(A QA ) +π u'(d + A ) 0, (.8b) where by applcaton of the Envelope Theorem A ϕ (A,Q,D ) = u'(c ), (.9) D ϕ (A,Q,D ) =π u'(c ). (.0) Obvously, an ndvdual always has a postve annuty demand A for the frst-perod contract, snce ths s the only possblty to provde for frst-perod consumpton. But she can decde ether to buy the mmedate annuty contract (.e. A > 0, D = 0 ) or the deferred contract (.e. A = 0, D > 0 ) or both knd of contracts (.e. A > 0, D > 0 ) n order to make provson for consumpton n the second retrement perod. The followng emma shows that the latter case s n general excluded. emma : In general, t s not optmal for an ndvdual to choose both D > 0 and A > 0. The nequalty Q Q > R (Q Q < R ) mples D > 0 and A = 0 ( A > 0 and D = 0, resp.). Proof: See the Appendx. In order to receve one unt of payout n the second retrement perod, the ndvdual has to nvest Q unts nto the A -contract n the frst retrement perod and hence Q Q unts of 6

8 ncome nto the A -contract n the workng perod. On the other hand, R unts of ncome nvested (n the workng perod) nto the D -contract transforms nto one unt of payout n the second retrement perod. Therefore, the decsve relaton s Q Q < > R, whether provson for the second perod of retrement s made through an A - or a D -contract. Only n case of Q Q = R, the ndvduals would be ndfferent between both types of contracts. For the remander of ths secton we rule out ths specfc parameter constellaton, but we dstngush between the two dfferent stuatons whether an ndvdual expresses annuty demand sequentally or smultaneously. In the frst case of sequental annuty demand, the purchase of A t, t =,, arses from a two-stage decson process, whose optmal soluton s charactersed by (.6) and (.8a). In the second case of smultaneous annuty demand, the purchased amounts A, D are determned by the frst-order condtons (.6) and (.7a). The followng two emmas characterse how prces and the survval probabltes nfluence annuty demand n both cases. emma : () In case of sequental annuty demand,.e. A > 0, A > 0, D = 0, we have A(Q,Q ) < 0, Q A (Q,Q ) < 0, Q A (Q,Q ) Q < _ > 0, f ε < > _, A (Q,Q ) < 0, Q where ε denotes the prce elastcty of annuty demand A for constant () In case of smultaneous annuty demand,.e. A > 0, A = 0, D > 0 A., we have: A(Q,R ) < 0, Q D(Q,R) Q < _ > 0 f η >_ <, A (Q,R ) R < _ > 0 f η >_ D(Q,R) <, < 0, R where η denotes the prce elastcty of annuty demand prce elastcty of annuty demand D for constant A. A for constant D and η the Proof: See Appendx. 7

9 We fnd the usual result that demand for an annuty contract decreases, f ts prce rses. owever, demand may but need not react n the same way, f the prce of the other contract rses. The reasonng for the cross-prce effects n case of sequental demand s the followng: An ncrease n Q reduces demand for the A -contract and hence ncome w = A n retrement perod, out of whch the A -contract has to be fnanced. The ndvdual adapts to the decrease n w (whch would mean a reducton of c n case of unchanged A ) by decreasng demand for the A -contract (and thus consumpton c as well). The cross-prce effect Q A follows from related arguments: In ths case, t s essental how a change n Q affects consumpton c holdng obvously, the expendtures A for the moment. Note that, although QA for the A -contract and hence c for fxed A Q < 0, A may de- or ncrease, dependng on the prce elastcty ε of demand A for fxed w = A. If Q ncreases and ε s larger than -, then QA ncreases, whch means a reducton of c. Then t s optmal for the ndvdual to shft part of the reducton to the workng perod 0 by decreasng c 0 and ncreasng demand for the A -contract. By the same argument, t s optmal for an ndvdual to leave demand for the A -contract unchanged, f ε =, and to decrease demand A, f ε < A. Analogous consderatons apply for the cross-prce effects n the case of smultaneous demand: The relevant ssue s how an ncrease of the prce Q and R, resp., drectly affects expendtures QA and RD, resp. (wth demand for the other contract held constant). If t expendtures ncrease ( η > ), then part of the mpled reducton of consumpton c 0 n the workng perod s shfted to the respectve other perod through a reducton of the correspondng annuty demand; and analogous for elastctes η t. emma 3: () In case of sequental annuty demand,.e. A > 0, A > 0, D = 0, we have π A(Q,Q ) > 0, π A(Q,Q ) > 0, A (Q,Q ) > 0, π A (Q,Q ) > 0. π () In case of smultaneous annuty demand,.e. A > 0, A = 0, D > 0, we have π A(Q,R ) > 0, D(Q,R) > 0, π 8

10 π A(Q,R ) < 0, D(Q,R) > 0. π Proof: See Appendx. We fnd that generally annuty demand reacts postvely, f any probablty of survval ncreases. owever, there s an essental dfference between the two cases, whch concerns the cross effect of π on the frst-perod contract. Wth sequental decsons, an ncrease of the probablty of survval to the second perod of retrement ncreases demand A, because ths allows to buy more nsurance for perod. On the other hand, wth smultaneous decsons an ncrease of π means that nsurance for the frst perod of retrement s substtuted by nsurance for the second perod of retrement. Note further that an ncrease n π clearly ncreases the probablty π π of survval to the second perod as well, hence demand for the second-perod contracts rses n both cases. 3. Adverse selecton n the annuty market avng descrbed the two possble strateges to provde for old age, namely through sequental and smultaneous annuty demand, we now study the mplcatons of asymmetrc nformaton on the functonng of the annuty market. et from now the otherwse dentcal ndvduals be dvded nto two groups =,, charactersed by dfferent rsks of a long lfe,.e. by dfferent probabltes of survval t t π > π for t =,. et γ 0 and γ 0, resp., denote the shares of the hgh-rsk and low-rsk ndvduals n perod 0, wth 0 < γ 0 <. Frst, as a pont of reference we consder the case that there s perfect nformaton about the survval probabltes. Then, obvously, perfect competton among the nsurance frms, ensures that each type of ndvduals receve ther ndvdually far contracts. An annuty s sad to be ndvdually far, f expected payouts equal ts prce. Ths requres for the A -, A - and D -contracts that the respectve condtons t t Q π = 0, t =,, (3.) R ππ = 0. (3.) 9

11 hold. Clearly, ths mples that the annuty companes make zero expected profts, gven that dentcal ndvduals buy these contracts. emma 4: Gven ndvdually far contracts, any ndvdual s ndfferent between choosng an A - or D -contract for the second perod of retrement. She chooses the same level of consumpton n every perod t = 0,,. Proof: The zero-proft condtons (3.), (3.) mply Q Q = R, whch s the condton for ndfference, as mentoned after emma. Consderng (.6) (.0), one observes that the 0 = zero-proft condtons also mply u'(c ) = u'(c ) u'(c ), rrespectve of the chosen contracts. Q.E.D. In a frst-best world, where every ndvdual can buy an annuty contract whose prce s precsely adjusted to her lfe expectancy, t does not matter, whch type of contract s chosen for provson for the second perod of retrement. Each offers an optmal smoothng of consumpton. owever, n realty, lack of nformaton prevents the supply of frst-best contracts. We ntroduce asymmetrc nformaton nto the model n the usual way: The probabltes and γ 0 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 adverse-selecton problem n the annuty market. Moreover, we assume that there s perfect competton among the annuty companes and that they cannot montor whether consumers buy annutes from other nsurance companes, whch seems to be a reasonable assumpton frequently made for the annuty market (see e.g. Pauly 974, Abel 986, Brugavn 993, Wallser 000, Brunner and Pech 005). 8 π t Ths assumpton means that frms fx the prce of a contract and ndvduals can buy as many annutes of each contract as they want. It follows that n equlbrum for each contract only one prce, pad by both types of ndvduals, can exst n each perod t =,. As a consequence, the frst-best 8 Prce and quantty competton, where frms offer a number of dfferent contracts whch specfy both a prce and a quantty, needs as a prerequste that ndvduals can buy at most one contract. Ths s regarded to be approprate for some nsurance markets, e.g. nsurance aganst accdents, but not for the annuty market. owever, prce and quantty competton generates the possblty of a separatng equlbrum (see Rothschld and Stgltz, 976; Wlson, 977). 0

12 soluton s unsustanable, because the ndvdually far prces of the A -, A - and D -contract as defned by (3.) and (3.) are lower for type- ndvduals than for type- ndvduals. 9 Both types of ndvduals buy the same contract, whch s called a poolng stuaton. Moreover, the same argument mples that only a stuaton, where both groups use the same type of contract n order to provde for the second perod of retrement, can preval. That s, ether both groups use the A -contract for the second perod of retrement or both groups use the D -contract. Ths follows from the fact that only one prce Q for the frst-perod contract A can exst, and that each group chooses ether the A - or the D -contract, dependng on whether Q Q < > R (see emma ). Thus we dstngush between two dfferent regmes, where all ndvduals demand ether sequental or smultaneous poolng contracts: sequental regme: A > 0, A > 0, D = 0 for =,, smultaneous regme: A > 0, A = 0, D > 0 for =,. As a next step we dscuss, to whch extent the adverse-selecton problem matters n the two regmes, that s, whether ndvduals wth a long lfe expectancy buy a larger amount of the dfferent types of contracts. emma 5: () In the sequental regme, for any prces Q,Q, an ndvdual wth hgh survval probabltes demands larger quanttes of annutes than an ndvduals wth low survval probabltes,. e. t t A (Q,Q ) > A (Q,Q ), t =,. () In the smultaneous regme, for any prces Q,R, an ndvdual wth hgh survval probabltes demands a larger quantty D than an ndvdual wth low survval probabltes,.e. D (Q,R ) > D (Q,R ). The rato of demand for the A -contract s undetermned. Proof: Follows mmedately from emma 3 and t t π < π, t =,. Q.E.D. 9 That both types of ndvduals buy the same contract could be called a poolng stuaton. owever, n a strct sense, ths term refers to a framework where n prncple the groups could be separated through approprate nstruments, e.g. through prce and quantty competton.

13 If the problem of adverse selecton s defned by the crteron that the rato of aggregate group- demand to aggregate group- demand exceeds the rato of group shares γ 0 /( γ 0 ), we fnd that ths problem certanly occurs for both contracts n the sequental regme; n the smultaneous regme t occurs for the second-perod contract, whle for the frst-perod contract t s mtgated by the fact that an ncrease of π decreases demand for the A -contract. 3. Prces n both regmes The consequence of the over-representaton of hgh-rsk ndvduals among aggregate annuty demand s that n equlbrum nsurance companes charge a prce whch s hgher than the actuarally far prce correspondng to the average probablty of survval of the populaton. The respectve prces are determned by the condton that, due to the assumpton of perfect competton n the annuty market, the expected profts of a contract, bought by both groups and, must be equal to zero. As π t s the expected payout for group, the zero-proft condton for the A -contract n ether regme reads 0 0 ( γ )A (Q π ) +γ A (Q π ) = 0. (3.3) Snce type- ndvduals have a hgher probablty to survve to retrement perod,.e. π > π, ther share n perod wll rse to γ0π 0 ( 0) γ γ π + γ π, (3.4) whle the share of type- ndvduals reduces to ( γ ). Thus relatvely more type- ndvduals wll buy an A -contract for the retrement perod, and ts zero-proft condton reads ( γ )A (Q π ) +γ A (Q π ) = 0. (3.5) In the smultaneous regme, where the A -contract s supplemented by the D -contract, the expected payout from the latter s ππ, and the zero-proft condton reads 0 0 ( γ )D (R π π ) +γ D (R π π ) = 0. (3.6)

14 Note that the prces cannot be computed explctly from (3.3), (3.5) and (3.6), because n each equaton annuty demand depends on the respectve prces. Nevertheless, f one takes the rato of aggregate demand of group to that of group as exogenous for the moment, one observes that the respectve prce s hgher, the larger ths rato. For a more detaled study of the functonng of the annuty market, when there s asymmetrc nformaton and annuty companes can offer both knds of second-perod contracts, namely mmedate as well as deferred annutes, we assume from now on that nstantaneous utlty s logarthmc,.e. u(c t ) = ln(c ) for t = 0,,. (3.7) t ogarthmc utlty has the convenent property that most cross-prce effects are zero (as t ε, η =, see emma ). Ths property keeps explct computaton of the zero-proft prces n ether regme smple. 0 In the followng, a tlde refers to the sequental regme, whle a bar refers to the smultaneous regme. Sequental regme: The condtons (.6) and (.8a) together wth (.9) determne annuty demand A ~ and A ~ for each sngle-perod mmedate contract. For logarthmc utlty one computes +π = Q ( )X A, πx = QQ A. (3.8) where 0 X π w ( +π +ππ ). Solvng (3.3) and (3.5), together wth (3.8), gves Q π +π ρ +ρ, Q π +π ρ +ρ, (3.9) where ρ t, t =,, denotes the the rato of annuty demand of both groups: γ0a ( γ0)a ρ, ρ γa ( γ)a, (3.0) 0 In the concludng secton we wll dscuss the consequences for our results of assumng a general utlty functon. 3

15 Smultaneous regme: By use of (A0), (A0) n the Appendx and (3.7) we obtan annuty demand A for the frst-perod contract and annuty demand D for the second-perod contract as A X =, Q D πx R =. (3.) Solvng (3.3) and (3.6), together wth (3.) yelds Q π +π ρ +ρ, R π π +π π ρ +ρ, (3.) where ρ t, t =,, s defned by γ0a ( γ0)a ρ, γ0d ( γ0)d ρ. (3.3) Wth these formulas, we are able to compare the composton of aggregate demand and the prces n the two regmes. emma 6: The followng relatons hold () between the ratos of aggregate annuty demand of group to that of group for the dfferent types of contracts: ~ ρ > ρ, ~ ρ > ρ, ~ ρ < ρ. () between the prces for the dfferent types of contracts: Q > Q, Q > R, QQ < R. Proof: See Appendx. The nequaltes ~ ρ > ρ and Q > Q ndcate that the adverse-selecton problem for the frst-perod contract s more severe n the sequental regme than n the smultaneous regme (compare the dscusson after emma 5). The ntutve reason for ths result s the followng: n the smultaneous regme, annuty demand A for the frst-perod contract satsfes only the need for future consumpton n perod. In contrast, n the sequental regme, annuty demand A ~ for the frst-perod contract has to satsfy the need for future consumpton n both retrement perods and, snce part of the payouts A ~ s used for the demand A ~. ghrsk ndvduals choose a hgher demand A ~ than low rsk-ndvduals (see emma 5), whch n turn ntensfes adverse selecton for the frst-perod contract. 4

16 The essental reason, why ~ ρ > ρ and Q > R hold, s that from perod 0 (when the D - contract s bought) to perod (when the A -contract s bought) the share of the hgh-rsk < π ndvduals n the populaton rses,.e. γ 0 < γ, because of π. As the rato of ndvdual demand s the same for both contracts n case of logarthmc utlty [ A ~ A ~ = D D, see (3.8) and (3.)], these shares ndeed are responsble for the hgher prce of the A -contract compared to that of the D -contract. A further mportant result of emma 6 s the nequalty QQ < R. Remember that QQ s the prce n the sequental regme whch must be pad n the workng perod n order to receve one unt of payout n the second retrement perod. It s smaller than R, the correspondng prce n the smultaneous regme. Ths can be explaned by the fact that n the former regme provson for perod s made va the frst-perod contract A, whch s bought by the low-rsk ndvduals to a larger extent than the D -contract (note that ~ ρ < ρ ). In other words, n the sequental regme the hgh-rsk ndvduals, when nsurng for the second perod, beneft from beng for the frst perod n a pool wth the low-rsk ndvduals, who put partcular weght on nsurance for ths perod, due to ther short lfe expectancy. In a sense, ths result represents the counterpart to the above argument explanng why Q > Q. 3. Equlbrum Now we turn to an analyss of whether ether or both of the two regmes consttute an equlbrum. We call a set of contracts an equlbrum n the sense of Nash-Cournot, f together wth annuty demand of both groups =, the respectve zero-proft condton for each contract s fulflled and f no other contract exsts, whch s preferred by at least one group {,} and whch allows a nonnegatve proft. Proposton : The sequental contracts wth prces Q, Q represent an equlbrum. Proof: If the A -contract were offered at a prce Q < Q, both groups would buy that and the nsurance company would make a loss. (Note that ρ s ndependent of Q (see (3.0)), hence Q s the unque payout whch fulfls the zero-proft condton (3.5)). By the same argument, an nsurance company offerng an A -contract wth prce Q < Q would make a loss. 5

17 Fnally, f an alternatve D -contract wth a prce R < QQ was offered, agan both groups would buy that (see emma ) and the nsurance company would make a loss. Ths follows from the fact that R does not depend on Q (see (3.)) and the zero-proft condtons mply R > QQ (see emma 6 ()). Q.E.D. Proposton : The smultaneous contracts wth prces Q, R do not consttute an equlbrum. Proof: Gven the smultaneous contracts, an nsurance company can addtonally offer a sequental A -contract wth prce Q. Indeed, as ρ and consequently Q do not depend on Q [see (3.9) and (3.0)], frms make a nonnegatve proft by ths offer. We know that Q < Q and QQ < R from emma 6. ence QQ < R, whch means that any ndvdual wll accept the offer (see emma ). Q.E.D. Remember that t s n the workng perod 0 when an ndvdual opts ether for the D - or the A -contract. ence, the assumpton made n Secton. that n perod 0 nsurance companes can credbly commt to offer the sequental contract wth prce Q one perod later, s essental for these results. Ths ssue wll be dscussed further n the concludng secton 4. Intutvely there are two reasons why the sequental regme wth prces Q, Q consttutes an equlbrum: ) From the above results we know that the sequental regme allows provson for the second retrement perod at a lower prce. Thus, t s plausble that no better D -contract can be offered wthout makng a loss. ) No lower prce than Q can be granted for the A - contract, n vew of the fact that ndvduals use part of the returns from ths A -contract to provde for the second perod va the A -contract. Conversely, the smultaneous regme wth prces Q, R s not an equlbrum, because frms can addtonally offer an A -contract at a prce Q, whch combned wth the exstng A - contract wth prce Q allows provson for both retrement perods at a lower prce. (Obvously however, the exstng A -contract wth return Q would make a loss n ths case, because Q s the lowest prce compatble wth sequental contracts.) 6

18 3.3 Welfare analyss In a fnal step of our analyss, we study welfare of both types of ndvduals =, n the two regmes, n order to fnd out whether the equlbrum outcome - the sequental contracts - s a favourable soluton for one or both rsk groups. We start the analyss by comparng optmal consumpton levels n each regme. emma 7: () In workng perod 0, the consumpton level c 0 of any ndvdual,, s the same rrespectve whether she chooses sequental or smultaneous annuty contracts. () In retrement perod, consumpton of any ndvdual =,, s lower n the sequental regme than n the smultaneous regme,.e. c(q,q ) < c(q,r ) for =,. () In retrement perod, consumpton of any ndvdual =,, s hgher n the sequental regme than n the smultaneous regme,.e. c (Q,Q ) > c (Q,R ) for =,. Proof: See Appendx. Note that part () of emma 7 holds for any contracts wth prces Q,Q and Q,R, snce for logarthmc utlty the expendtures for annutes n workng perod do not depend on prces,.e QA = same amount QA + w RD for any Q, Q, R. ence t s optmal for an ndvdual to nvest the 0 c0 nto old-age provson n ether regme. owever, the prces nfluence the level of consumpton n both retrement perods. Partcularly, we fnd that the relatons Q > Q, QQ < R, as shown n emma 6 (), are n fact decsve for the dstrbuton of consumpton over the two perods of retrement n the two regmes. Wth the sequental regme, more consumpton s postponed to the second perod of retrement, whle the smultaneous regme nduces ndvduals to consume relatvely more n the frst perod of retrement. Altogether, t follows that t s unclear from the results of emma 7, n whch regme an ndvdual of type s better off. In order to answer ths queston, we frst determne the consumpton possblty curves for perod and. The consumpton possblty curves, abbrevated by CPC SI for the smultaneous regme, are obtaned by elmnatng and combnng budget equatons (.) (.3), where CPC SE for the sequental regme and by D = 0 and A, A and A, D, resp., A = 0, resp.: CPC : w c = Q c + Q c. (3.4) SE 0 0 7

19 { { CPC SI : 0 0 w c = Q c + R c (3.5) We use the convenent property that n ether regme an ndvdual consumes the same amount 0 c c n the workng perod to draw the consumpton possbltes curves n the ( c, )- SE 0 0 space (see Fgure and ). CPC descrbes the feasble consumpton bundles ( c, c ) for an ndvdual who nvests the fxed amount w0 c0 = QA nto the frst-perod contract. She can consume all payouts c = (w c ) Q n perod or transform part of t nto secondperod consumpton, by buyng the sequental second-perod contract at prce Q. If she transforms everythng, then c = (w c ) (Q Q ) results. On the hand, the 0 0 CPC SI represents all feasble consumpton bundles for an ndvdual who nvests the same amount w 0 c0 (n the workng perod 0!) nto the A - and D -contract. ence, n ths regme, the trade-off between consumpton n perod and n perod s Q R, whch s prce rato for one unt payout n the second and n the frst retrement perod. In ether regme, an ndvdual c π chooses ( c, ) by maxmzng u(c ) + u(c ) subject to respectve consumpton possblty set. c w 0 - c0 ~ ~ Q Q CPC SE Fgure Fgure c ( c~, c~ ) w _ - c R 0 0 CPC SI ~ A { ~ ~ Q A - ~ Q w 0 - c0 ~ Q c _ D { _ A ( - - c, c ) - Q R w - c _ Q 0 0 c A comparson of the consumpton possblty curves n both regmes demonstrates that CPC SI s flatter than CPC SE, because QQ < R due to emma 6. Ths nequalty s responsble for the fact that relatve consumpton c c s lower n the smultaneous regme than n the sequental regme, as shown n emma 7. Further, the curve CPC SI crosses the c-axs at a hgher level than the curve CPC SE, snce Q < Q. The opposte holds for ts 8

20 crossng wth the c -axs, snce QQ < R. It follows that the CPC's ntersect and that the sequental regme allows hgher consumpton to the left of the pont of ntersecton, but lower consumpton to the rght. Ths property s essental for followng result on welfare: Proposton 3: An ndvdual of type s better off n the smultaneous regme wth prces Q, R, whle an ndvdual of type s better off n the sequental regme wth prces Q,Q. Proof: See Appendx. We gve a graphcal llustraton of Proposton 3 n Fgure 3, where the consumpton possblty curves (3.4) and (3.5) of both types of ndvduals =, are drawn, denoted by CPC SE and CPC SI for a low-rsk type and by CPC SE and CPC SI for a hgh-rsk type. Note that due to adverse selecton the long-lvng ndvduals make more provson for retrement, therefore ther consumpton possblty curves are above those of the short-lvng. c CPC SE Fgure 3 CPC SI CPC CPC SI c Essental for the result of Proposton 3 s that at any combnaton ( c, c ) the slope c /( π c ) of the ndfference curve s steeper for a type- ndvdual than for a type- ndvdual, as π < π. As one can show, ths property mples that, rrespectve of the regme, the optmal combnaton for a type- ndvdual s to the rght of the pont of ntersecton of her consumpton possblty curves CPC SE and CPC SI, whle the optmal 9

21 consumpton bundle for a type- ndvdual les to the left of the ntersecton of CPC SE and CPC SI. Consequently, snce the smultaneous regme allows hgher consumpton possbltes to the rght of the pont of ntersecton, t s preferred by a type- ndvdual. The opposte holds for a type- ndvdual. Ths result conforms wth the ntuton that the short-lvng ndvduals, who put more weght on consumpton n perod one, are ndeed better off wth that regme whch provdes more consumpton n ths perod (emma 7). Conversely, the long-lvng ndvduals are better off wth the sequental regme, whch provdes more consumpton n perod two. 4. Concludng Remarks Provson for old age can be made through a varety of annuty products, whch dffer n the terms concernng asset accumulaton and the payout path. In the present paper we have concentrated on annutes whch run over a lmted tme only and have to be supplemented by a second contract. Ths addtonal contract can ether be bought smultaneously wth the frst or later, when an ndvdual knows that she has survved some years of retrement. We have charactersed demand, gven these two possbltes, and we have studed the consequences of the adverse-selecton phenomenon n ths market. The results show that only a stuaton, where all ndvduals demand sequental contracts represents an equlbrum. Ths s favourable for the hgh-rsk group, whle the low-rsk group would be better off wth the smultaneous regme. Ths result, though derved n a specfc framework, shows some smlarty to conclusons from other models wth asymmetrc nformaton, where typcally the low-rsk groups do not receve ther frst-best contract. The man concluson from our contrbuton s that adverse selecton has more severe consequences on the annuty market than recognsed n studes usng the standard overlappng-generatons model. These manly concentrate on the nfluence of adverse selecton on a sngle rate of return for a unform perod of retrement. By extendng ths model and makng the realstc assumpton that provson for retrement need not be made through a once-and-for-all annuty contract, but can be made through dfferent contracts for earler and later phases of retrement, one fnds that adverse selecton also affects the choce of contracts as well as the exstence and propertes of equlbra. 0

22 Two mportant assumptons were used n ths study n order to derve the results: logarthmc utlty and the possblty of credble commtment. The former one s essentally a techncal assumpton, allowng us to construct a framework, whch s suffcently smple to derve the defnte results n Secton 3. Consderng the ntuton provded below emma 6, t appears qute lkely that the relatons stated n ths emma hold for a broader class of utlty functons. Ultmately, the mportant relaton that drves the fnal result s QQ < R, that s, for perodtwo contracts the problem of adverse selecton s releved n the sequental regme, compared to the smultaneous regme. The reason s that n the sequental regme nsurance goes va perod where the hgh-rsk ndvduals are n a common pool wth the low-rsk ndvduals, who put more weght on the frst perod of retrement than on the second. Indeed, numercal smulatons usng an soleastc per-perod utlty functon have shown that the relaton QQ < R holds generally for any soleastc per-perod utlty functon. On the other hand, the assumpton that nsurance companes can credbly commt to offer an nsurance contract at a fxed prce one perod later s essental to establsh the sequental regme as an equlbrum. Wthout commtment, one would have to ntroduce some way of how belefs concernng the expected prce of a future contract are formulated. At fst glance, n such a formulaton the occurrence of uncertanty would make future contracts less attractve, and, as a consequence, the sequental regme less lkely to represent an equlbrum. It s well-known that the observed data from the prvate annuty market reveal puzzlng peculartes. The most frequently mentoned s that people buy consderably less annuty contracts than one would expect, gven ther hgher return compared to that on other forms of wealth (see, e.g., Fredman and Warshawsky, 990). Another one s the wde-spread use of so-called "years certan" contracts, whch s also dffcult to explan wthn a standard model of household decson. Gven the result of our analyss that usng tme-lmted contracts n the sequental regme represents an equlbrum, the observed fact, mentoned n the ntroducton, that tme-lmted contracts make up only a small share of all actually purchased annutes adds another puzzle: n fact the majorty of ndvduals buy lfe annutes for the whole perod of retrement altogether, whch can be nterpreted as the smultaneous regme n our model. A possble explanaton could be seen n the lack of commtment as dscussed These contracts offer guaranteed for a fxed number perods (possbly to descendants), and then regular payouts untl death. Obvously, ther prce s hgher than that of annutes wthout a guaranteed payout phase.

23 above. A further crucal ssue of our analyss s the range of annuty contracts or ther combnatons avalable to the ndvduals. Further research s needed n order to clarfy the functonng of the market, f addtonal types of contracts, for nstance packages of frst- and second-perod nsurance, are ntroduced. Appendx Proof of emma : By use of the equatons n (.8a), (.9) and (.0), equaton (.6) can be wrtten as 0 QQ u(w QA R D ) +π ϕ(a,q,d ) D = 0, where the frst term on the S n general wll not be equal to the term 0 Ru(w QA RD) of the equaton n (.7a). As we know that (.6) must always be fulflled, ths means that the equatons n (.7a) and (.8a) cannot hold smultaneously. By the same reasonng, one observes that f (.8a) holds, (.7b) can be fulflled only f R Q Q, and analogously for (.7a) and (.8b). Q.E.D. Proof of emma : Part (): We denote by where We fnd that W the S of (.6) and by D = 0. Implct dfferentaton gves V the S of the equaton n (.8a), A A W W W W Q Q A A Q Q =. (A) A A V V V V Q Q A Q A Q V A V A ϕ = 0 +π = 0 +π = Q u"(c ) > 0, (A) = Qu"(c) +π u"(c ) < 0, (A3) W (A,Q ) A (A,Q ) Qu"(c ) Qu"(c ) ( Q ). (A4) A w A due to (.9), where A A (A,Q ) denotes annuty demand A for fxed A A (.8a). ence, A (A,Q ) s derved from mplct dfferentaton of (.8a) as =, determned by

24 = A Qu"(c). (A5) A V A By use of (A3) - (A5) we obtan Further, we have: 0 W u"(c )u"(a ) = Qu"(c ) +π π < 0. (A6) A V A W A V Q V Q W Q W Q = 0, (A7) = 0, (A8) 0 0 ϕ =π = π + A Q Q due to (.9). By use of A (A,Q ) for fxed we obtan = u(c ) + Q A u (c ) < 0, (A9) = u(c ) + Q Au(c ) < 0, (A0) (A,Q ) A (A,Q ) u(c )(A Q ), (A) ε A (A,Q ) Q (A,Q ) Q A, denotng the prce elastcty of demand w, together wth (dfferentate (.8a) mplctly and use (A) and (A3)) Q V A A (A,Q ) V Q = < 0, (A) W Q ( ) = π A u(c ) +ε (A,Q ) >_ < 0 ε (A,Q ) >_ <, ε (A,Q ) < 0. (A3) Now let W V W V N > 0. (A4) A A A A < 0 < 0 = 0 Invertng the frst matrx on the RS of (A) and multplyng gves (note the nequaltes from above): A(Q,Q ) V W W V = ( ) < 0, (A5) Q N A Q Q A < 0 < 0 = 0 3

25 A (Q,Q ) V W W V = ( + ) < 0, (A6) Q N A Q Q A > 0 < 0 = 0 A(Q,Q ) V W W V = ( ) Q N A Q Q A < > _ 0 ε (A,Q ) >_ <, (A7) < 0 = 0 A (Q,Q ) V W W V = ( + ), (A8) Q N A Q Q A > 0 < 0 whch can rearranged to [use (A), (A6), (A9), (A) and (A)] < 0 = + Q N Q A A A (Q,Q ) W W W [Q u (c )( A ) u (c )] πqu(c)u(c) 0 W = [Qu(c)( + QAu(c )) u(c)] < 0. N V A A (A9) Part (): et now denote W and V the S's of (.6) and (.7a), resp. Wth A = 0 these equatons reduce to (see (.9), (.0)) 0 Qu'(w QA R D ) +π u'(a ) = 0, (A0) 0 Ru'(w QA RD) +π π u'(d) = 0. (A) The formula for mplct dfferentaton of these equatons s the same as (A), when replaced by D and Q by R. A s We fnd that W = Qu"(c 0 ) +π u"(c) < 0, A V A W Q V Q 0 = QR u"(c ) < 0, 0 0 = u(c ) + Q A u"(c ) < 0, 0 = RAu"(c) < 0, W = QR u"(c 0 ) < 0, (A) D V D 0 = Ru"(c ) +ππ u"(c ) < 0, (A3) W = QD u"(c 0 ) < 0, (A4) R V R 0 0 = u'(c ) + R D u"(c ) < 0. (A5) Now let W V W V M > 0, (A6) A D D A 4

26 where the postve sgn follows mmedately from easy calculaton by use of (A) and (A3). One derves mmedately, usng (A5) (A9), together wth (A) (A6) and by some straghtforward computatons: Q M D Q D Q A(Q,R ) V W W V = ( ) < 0, (A7) R M A R A R D(Q,R) V W W V = ( + ) < 0. (A8) Further, we fnd that = + = 0 + Q M A Q A Q M Q A D(Q,R) V W W V W W ( ) R u (c )( Q A ). (A9) Substtutng the prce elastcty of demand A (D,Q ) for fxed D, defned as A (D,Q ) Q A(D,Q) W Q η, together wth = < Q 0 (dfferentate (A0) mplctly) A Q W A nto (A9) yelds 0 Q M A D(Q,R) W = Ru(c )A ( η + ) >_ < 0 η < > _, η < 0. Analogously, we use the prce elastcty A, together wth sgn of D(A,R)R η R D R V D R A(Q,R ) : of demand D (A,R ) for fxed D(A,R) V R = < 0 (dfferentate (A) mplctly) to determne the = ( ) R M D R D R = 0 M D V = 0 +η M D A(Q,R ) V W W V V V Qu(c )(D R ) R Qu(c )D ( ) >_ < 0 η < > _, η < 0. (A30) Q.E.D. Proof of emma 3: Part (): We defne W and V as n the proof of emma, part (). The formula for mplct dfferentaton of (.6) and the equaton n (.8a) s the same as n (A), where Q,Q are π replaced by π,, resp. 5

27 We fnd that W π ϕ = > 0, w ϕ(w,q ) A (w,q ) πqu (c )u(c ) π (A3) W =π = π Qu(c) = > 0, (A3) π w π Qu(c ) + π u(c ) due to (.9), where A (w,q )/ π (denotng the change of annuty demand for fxed π ncreases), s determned by mplct dfferentaton of the equaton n (.8a). w, f Furthermore: V π = 0, V π = u'(c ) > 0. (A33) Usng these computatons (A3) (A33), together wth (A), (A3), (A6), (A7) and (A4) n (A5) (A8) gves π N A π A(Q,Q ) V W = > 0, π N A π A(Q,Q ) V W = > 0, π N A π A (Q,Q ) V W = > 0, π N A π A π A (Q,Q ) V W W V = ( + ) > 0. Part (): We defne W and V as n part () of emma and use the same formula (A) for mplct dfferentaton of (A0), (A), where A, Q, Q are replaced by D, π, π resp. We fnd that W = u'(c ) > 0, π V π =π u'(c ) > 0, W π V π = 0, (A34) = π u'(c ) > 0. (A35) Usng these computatons (A34) and (A35), together wth (A), (A4) and (A6), n (A7) (A30), t follows [note that R u'(c ) =π Q u'(c ) due to (A0) and (A)]: π M D π D π A(Q,R ) V W W V = ( ) = ππ u (c 0)u(c ) > 0, M π M A π A π D(q,r) V W W V = ( + ) = πu (c ) π u(c ) > 0, M 6

28 π M D π A(Q,R ) W V = < 0, π M A π D(Q,R ) W V = > 0. Q.E.D. Proof of emma 6: () By use of the frst equaton n (3.8), (3.0), (3.) and (3.3), ~ ρ can be wrtten as ( +π ) ρ ( +π ) ρ =, (A36) whch s larger than ρ, as π > π. By use of (3.4) and the second equaton n (3.8), (3.0), (3.) and (3.3), ~ ρ can be wrtten as π π ρ = ρ, (A37) whch s larger than ρ, as π > π. By use of the frst equaton n (3.8) and (3.0) and the second equaton n (3.) and (3.3) ~ ρ can be wrtten as ρ π +ππ ( π +ππ) ( ) ρ =, (A38) whch s smaller than ρ, as π > π. () Frst, we calculate the dfference ( Q Q ) from the frst equatons n (3.9) and (3.) as Q ( π π )( ρ ρ ) Q =, (A39) ( +ρ )( +ρ ) whch s postve due to ~ ρ > ρ and π > π. Next, we show that Q > R : Usng the second equatons n (3.9), (3.) and (A37), the prce rato R Q can be wrtten as R Q π +πρ = +ρ, (A40) < whch s smaller than due to π, for =,. 7

29 Fnally, we show that QQ < R, whch s equvalent to Q R Q < 0. Substtutng the frst equaton of (3.9) and (A40) nto the dfference Q R Q yelds (after some easy steps of calculatons) Q R ( π π)( ρ ρ) =, (A4) Q ( +ρ )( +ρ ) whch s negatve, snce π > π and ρ > ~ ρ. Q.E.D. Proof of emma 7: () Ths result follows from the fact that QA = QA + RD : substtutng the frst equaton n (3.8) and D = 0 nto (.) gves the same consumpton level c 0 as substtutng the both equatons n (3.3) nto (.). () Substtutng both equatons n (3.8) nto (.) as well as the frst equaton n (3.) and A = 0 nto (.) gves c(q,q ) = A(Q,Q ) Q A (Q,Q ) = X Q ; (A4) c(q,r ) = A(Q,R ) = X Q. (A43) Due to emma 6 (), Q > Q, and t follows that c(q,q ) < c(q,r ). () Substtutng the second equaton n (3.7) and D = 0 nto (.3) as well as (3.4) and A = 0 nto (.3) yelds ( ) c (Q,Q ) = A (Q,Q ) = π X QQ ; (A44) c (Q,R ) = D (Q,R ) = π X R. (A45) Due to emma 6 (), QQ < R, and t follows that c (Q,Q ) > c (Q,R ). Q.E.D. Proof of Proposton 3: We show that for an ndvdual of type, the optmal consumpton bundles ( c, c ) n both regmes le to the rght of the pont of ntersecton of the consumpton possblty curves (3.4) and (3.5), whle the opposte holds true for an ndvdual of type. From ths we conclude that an ndvdual of type prefers the smultaneous regme wth prces Q,R, whle an ndvdual of type prefers the sequental regme wth prces Q,Q. 8

30 We calculate the pont of ntersecton of the possblty curves of an ndvdual =,, under both regmes by solvng (3.4) and (3.5) for c and substtutng (3.5), whch gves R QQ c(s) ( +π )X. (A46) Q(R QQ ) As a preparaton, we show n step () that c (Q,Q ) c (S) > 0 and n step () that c (Q,R ) c (S) < 0. () By use of (A4) and (A46) for =, we calculate the dfference c (Q,Q ) c (S), whch can be wrtten as X R c (Q,Q ) c (S) = [(Q Q ) +π(q )] Q(R QQ ) Q (A47) Frst note from emma 6 () that R Q Q > 0 and Q < Q, hence R QQ > 0. It follows that c (Q,Q ) c (S) has the same sgn as the term n the squared bracket on the RS of (A47), whch we denote by Ω (Q Q ) +π (Q R Q ). By use of (A39) and (A4), together wth (A36) and ρ ρ = π π [mmedate by use of (3.) and (3.3)], Ω can be rewrtten after some easy transformatons as where +π π +π π Ω=λ[( )( + ρ ) +π ( )( +ρ)] +π π +π π (A48) ( π π ) ρ λ, (A49) ( +ρ )( +ρ )( +ρ ) whch s postve, as π > π. Further computatons of (A48) yelds +π +π +π π +π Ω =λ[( +π π ) +ρ ( +π π )], (A50) +π +π +π π +π whch reduces to (note that the frst term n the parenthess on the RS of (A40) s equal to zero) ρ Ω=λ ( π π ) ( +π ) π. (A5) From (A5), together wth (A49), t s mmedate that Ω > 0, whch proves that c (Q,Q ) c (S) > 0. 9