Optimum Taxation of Life Annuities. Johann K. BRUNNER and Susanne PECH

Transcription

1 DEPARTMENT OF ECONOMICS JOANNES KEPER UNIVERSITY OF INZ Optmum Taxaton of fe Annutes by Johann K. BRUNNER and Susanne PEC Workng Paper No November 2005 Johannes Kepler Unversty of nz Department of Economcs Altenberger Strasse 69 A-4040 nz - Auhof, Austra

2 Optmum Taxaton of fe Annutes Johann K. Brunner and Susanne Pech * November 2005 Abstract The market for prvate lfe annutes s charactersed by adverse selecton, that s, contracts offer lower than far payoffs to ndvduals wth low lfe expectancy. Moreover, lfe expectancy and ncome have been found to be postvely correlated. The paper shows that a lnear tax on annuty payoffs, whch rases more revenues from long-lvng ndvduals than from short-lvng, represents an approprate nstrument for redstrbuton, n addton to an optmally desgned labour ncome tax. Further, we fnd that a nonlnear tax on annuty payoffs can be drectly employed to correct the dstorton of the rate of return caused by asymmetrc nformaton. These results are contrasted wth theoretcal fndngs concernng the role of a tax on captal ncome. JE Classfcaton: 2, G2. Keywords: Optmum taxaton, lfe annutes, adverse selecton. * Address: Department of Economcs, Unversty of nz, Altenberger Straße 69, A-4040 nz, Austra. Phone: , FAX: E-mal: johann.brunner@jku.at, susanne.pech@jku.at.

3 I. Introducton As n many ndustralsed countres socal securty systems are under pressure because of an agng populaton, governments are tryng to establsh a so-called thrd pllar of old-age provson. The dea s that ndvduals should compensate a possble declne of the publc pensons by ncreased prvate savng, that s, by shftng part of ther ncome n the workng perod to the perod of retrement. Among the varous ways of how savngs can be nvested and how wealth, avalable for the fnancng of retrement consumpton, can be accumulated, the purchase of prvate lfe annutes has the advantage of provdng approprate nsurance aganst a long lfe. It allows the ndvdual to avod runnng out of assets before death as well as leavng unntended bequests. By transferrng wealth from those, who de early, to those, who lve long, annutes provde a hgher rate of return than nvestments n the captal market (Yaar 965). There s a tendency among governments to grant some way of preferental taxaton to ndvduals who purchase prvate lfe annutes. The arguments for such a preferental treatment are rarely formulated explctly; n the publc dscusson there seems to preval a mert-good vew, sayng that ndvduals are myopc and therefore save too lttle n ther actve perod of lfe. 2 In partcular, they mght not have a suffcent percepton of the lkely reducton of the replacement rate offered by the publc penson. 3 In accordance wth ths vew, several studes have nvestgated the ncentve effect of taxes on the demand for prvate lfe annutes and on savngs n general (for an overvew see Bernhem 2002). Usually, economsts have reservatons aganst arguments based on rratonalty of the ndvduals. If at all, these arguments mght serve as a justfcaton for publc For nstance, n Austra the state subsdses the premum and, n addton, guarantees tax exempton of the payoffs. A smlar regulaton was ntroduced n Germany, the so-called Rester-Rente. In OECD countres the prevalng system seems to be that contrbutons to a penson fund are tax exempt, up to some lmt, and penson payments are taxed, see, e.g., Whtehouse 999 or OECD For the UK, Dsney et al. 200a,b fnd some evdence, especally for low-wage earners, that the savngs rate s too low n order to transfer suffcent ncome to the perod of retrement. 3 A smlar argument rests on the suspcon that ndvduals mght delberately save too lttle, because they expect to receve some socal assstance anyway. 2

4 programs of lmted sze only, whch means n the case of prvate pensons that preferental taxaton for mert-goods reasons should be restrcted to a mnmum provson to cover basc needs. In prncple, taxaton of prvate pensons should follow the same rules that are gudng the desgn of the tax system as a whole. Ths leads one to the queston of what s the approprate tax treatment of annutes wth regard to effcency and equty. In partcular, the dstrbutve effect of the way of how annutes are taxed deserves a more thorough analyss than has been provded by the theoretcal lterature so far. 4 To consder that together wth the effect on economc effcency n a coherent model s the subject of the present contrbuton. Obvously, besdes provdng nsurance, the purchase of annutes represents an alternatve to an nvestment on the captal market. It s therefore nterestng to refer to results concernng the queston of how such an nvestment should be taxed. On ths, t seems far to say that most theoretcal studes have lead to the concluson that captal ncome should be left tax-free, at least f some other nstrument s appled for (redstrbutve) taxaton n an optmal way and preferences are separable between lesure and consumpton n dfferent perods. Such a result can be derved n a statc representatve-consumer model of the Ramsey-type (see, e.g., Atknson and Stgltz 980), but also rather generally n dynamc models wth many households, whch dffer n ther ablty to earn ncome or n ther captal endowment. 5 The mportant queston s then whether ths result changes, f the nsurance aspect of prvate pensons s taken nto account. To answer ths, one has to consder the functonng of the annuty market, n partcular the problem of asymmetrc nformaton: nsurance companes cannot dstngush between ndvduals wth low and hgh lfe expectancy. Ths fact, together wth the assumpton of prce competton among nsurance companes, 6 mples that frms are forced to offer the same rate of return to 4 Some studes concentrate on smulaton results concernng to dstrbutonal effects of dfferent taxtreatments of annutes, see, e.g., Brown et al. (999) and Burman et al. (2004). 5 See Chamley (986) and Judd (985) for nfnte-horzon models, and Ordover and Phelps (979) for OG-models. On the other hand, n both frameworks there are a few studes whch establsh the desrablty of a tax on captal ncome, e.g. f credt constrants (Chamley 200, Ayagar 995) or human captal accumulaton (Jacobs and Bovenberg 2005) are ncorporated. 6 Prce competton follows from the assumpton that nsurance frms cannot montor whether costumers hold annutes also from other frms. In contrast, prce and quantty competton, whch 3

5 all customers, rrespectve of ther expected duraton of lfe. Ths n turn nduces an adverse-selecton effect, whch was estmated to make prvate penson more costly, to the extent of 7-5 percent. 7 What we focus on n the present study are the dstrbutve consequences of ths phenomenon and whether t provdes a specfc ratonale for taxaton. As a frst step of the analyss, we compare, by means of a smple example, the effects of two forms of tax exempton, namely ether a wage tax, whch leaves annuty payoffs tax-free, or a consumpton tax, whch leaves savng untaxed but taxes dssavng (.e., annuty payoffs) fully. In partcular, we show that, though both forms are equvalent as to consumpton and welfare of the ndvduals, the latter extracts less revenue from the short-lvng ndvdual than the former. Next, we turn to a model wth fxed labour ncome and endogenous annuty demand and demonstrate that the ntroducton of a proportonal tax on annuty payoffs ndeed redstrbutes ncome, whch s due to the larger annuty demand of the longer-lved ndvduals. ence, gven that socety wants to treat ndvduals wth hgh mortalty better, t can do so and use ths nstrument to compensate them for ther dsadvantage. We offer some deas on the normatve ssue, whch relatve weghts should be gven to short- and long-lvng ndvduals n the socal objectve. For a more accurate analyss we have to allow for ncentve effects on labour supply, where we take nto account the emprcal fndng that ncome and lfe expectancy are postvely correlated (see, among others, Attanaso and oynes 2000, llard and Pans 998). We ntroduce ths nto the smplest possble model of optmum ncome taxaton, consstng of two types of ndvduals, who lve for two perods and dffer n ther wage rate and n ther probablty of survval to the second perod. In ths framework we consder two cases, that of a lnear and of a nonlnear tax on annuty was studed frst by Rothschld and Stgltz (976) and Wlson (977), requres that costumers can buy only one contract. Snce ths s regarded to be napplcable for the annuty market, prce competton s usually adopted for the analyss of the annuty market, see e.g. Pauly (974), Abel (986), Brugavn (993), Wallser (2000), Brunner and Pech (2005). 7 Compared to a stuaton wthout adverse selecton,.e., where mortalty of costumers s dentcal to that of the average populaton. See, e.g. Mtchell et al. (999), Wallser (2000), Fnkelsten and Poterba (2002). 4

6 ncome, resp. For a lnear tax on annuty payoffs (n addton to the optmum nonlnear labour-ncome tax), we can prove that t ncreases socal welfare, gven that the weght of the low-wage (and low lfe expectancy) ndvdual s suffcently large n the socal objectve and that demand for annutes does not decrease too much wth lesure. ence, also n case of weak separablty between lesure and consumpton n both perods, taxaton of annuty payoffs s optmal. Ths fndng s n contrast to the results on captal ncome taxaton mentoned above. Concernng the case of an optmum tax system, whch s nonlnear not only wth respect to wage ncome, but also wth respect to annuty payouts, a remarkable, new feature turns out mportant: Such a fully nonlnear tax system has the advantage over the one wth lnear taxaton of annuty payouts that t not only allows redstrbuton, but also a correcton of the market falure arsng from the adverse-selecton problem. In partcular, we fnd that the annuty payout of the long-lvng (and hgh-ncome) ndvdual s reduced by the margnal tax rate to her ndvdually far payout. For the short-lvng ndvdual the analogous effect ncreases her payout, however, a dstorton may occur famlar from optmum labour ncome taxaton - and mpede the (full) realsaton of the frst-best payout accordng to her low survval probablty. Ths correctve role s characterstc for nonlnear annuty taxaton, t dffers markedly from the correspondng results for the optmum nonlnear tax on captal ncome. These typcally show a zero margnal rate for the hgh-ncome ndvdual and the same, but possbly dstorted, for the low-ncome ndvdual (see, e.g., Ordover and Phelps 979, Brett 998). The paper proceeds as follows: Secton II provdes an ntutve example, llustratng that taxaton of annuty payoffs ndeed extracts more tax revenue from the long-lvng ndvduals, compared to an equvalent wage tax. Secton III contans the detaled analyss, whch frst consders the effect of a tax on annuty payoffs vs-à-vs a proportonal tax on fxed labour ncome. In the second part of ths secton a Mrrleestype model s formulated n order to study the optmum propertes of a lnear as well as of a nonlnear tax on annuty payoffs. Secton IV provdes concludng remarks. 5

7 II. Equvalent taxaton As s well-known from straghtforward textbook analyss (see, e.g., Stgltz 2000), a proportonal tax on wages (leavng captal ncome untaxed) s equvalent to a proportonal consumpton tax (leavng savng untaxed but burdenng dssavng). We study some smple examples n order to derve an ntuton of how ths equvalence extends to the case of annutes. Consder a group of N dentcal ndvdual who lve for two perods. Each ndvdual survves to the second perod wth probablty π = ½. et wage ncome n perod be w = 300. Suppose that after payng a wage tax wth rate t = 0.2 each ndvdual spends one thrd of net ncome w n = 240 on annutes, that s, a = 80. We assume, for smplcty and n order to concentrate on the nsurance aspect of annutes, that the nterest rate s zero. Gven the far payoff rate q = /π = 2 per unt of annuty, the ndvdual receves 60 n perod 2. Altogether we have w = 300, w n = 240, a = 80, c 0 = 60, c = 60, () and total tax revenue s 60N. On the other hand, f expenses for annutes are deductble, but payoffs are fully taxed wth rate t = 0.2, the ndvdual s exactly n the same stuaton as before, f she chooses a hgher annuty demand of a = 00; then we have: w = 300, w n = 260, a = 00, c 0 = 60, c = 60. (2) Note that the government receves 40 as a wage tax from every ndvdual and another 40 from the tax on annuty payouts, but only from the survvng ndvduals. Therefore, total tax revenue s 40N N = 60N, as before. Next assume that there are two groups, of ndvduals, where each group s of equal sze N and charactersed by a dfferng survval probablty: π =, π 3 = 2, 3 otherwse they are dentcal. Insurance frms cannot dstngush between the groups, hence there exst only - so-called - poolng contracts wth the same payoff rate for 6

8 each ndvdual. We assume for the moment that both groups buy the same amount of annutes, then the payoff rate, whch allows zero profts, s q = /((π + π )/2) = 2. In case of a wage tax wth rate t = 0.2 the stuaton for each group s as descrbed n () above, and (2) contnues to reflect the effect of a consumpton tax. owever, there s one nterestng aspect to observe: tax revenues T from group are 60N wth the ncome tax, but only 40N + 40N/3 < 60N wth the consumpton tax. Stll, consumpton s the same wth both knds of taxes. Ths leads us to the queston: Why s payng less taxes not to the advantage of group? The answer s that wth the consumpton tax ths group has to nvest more nto annutes, compared to the stuaton wth the ncome tax (namely 00 nstead of 80), whch provde a lower than far rate to them. If the tax payment of 60N were unchanged, ths ncreased annuty demand a = 20, gven q < /π, would reduce the short-lvng group s lfetme ncome by the amount of a( - qπ )N = 20( - 2 )N. From these consderatons, t s obvous 3 that the smaller tax payment T = 20N/3 just compensates the short-lvng group for the dsadvantage arsng from the ncreased demand, gven the lower than far rate of return. Moreover, one could say that, va ths ncreased annuty purchase, group mplctly fnances the addtonal tax amount, whch n fact the group wth the hgher lfe expectancy (for whch the stuaton s vce versa, as q > /π ) has to pay n case of consumpton taxaton. Altogether, both groups are as well off n ether tax regme. We can generalse ths example by assumng that the share α of group =,, α + α =, and the frst-perod ncome w need not be the same for each group. In an annuty market, whch s charactersed by asymmetrc nformaton, a sngle rate of return, offered to both types of ndvduals, prevals n equlbrum. Under the assumpton of perfect competton, ths poolng rate of return s mplctly defned by the zero-proft condton that aggregate expected payoffs must equal the aggregate spendng (the nterest rate s stll assumed to be zero),.e. q( πα a +πα a ) = ( α a +α a ). (3) Moreover, we can determne the lfetme budget constrant of an ndvdual, n case of an ncome tax, by combnng c 0 = w ( t) a and c = qa and, n case of a 7

9 consumpton tax, by combnng c 0 = ( t)(w a ) and c = q( t)a. In ether tax regme one gets (by elmnaton of a ) 0 c c + = w ( t). q From ths t s obvous that an ndvdual chooses the same consumpton path over her lfetme n ether tax regme (for any arbtrary utlty functon rate of return s the same. In that case, and co ndcate the respectve tax regme. n co 0 π u(c,c ; ) ), f the a = a ( t), where the superscrpts n As a result, total tax revenues from group are co co are T ( tw t(q )a ) T n =α w t wth an ncome tax and =α + π wth a consumpton tax. One fnds co T T n dependng on π /q. Thus, as above, group pays less wth a consumpton tax than wth an ncome tax n case of a poolng payout rate, whle the opposte s true for group. owever, as wth the former tax group has to buy more nsurance, whch offers unfavourable condtons, t s equally well-off, though t pays less taxes, and vce versa for group. 8 Fnally, we observe that total revenue tax equals total tax revenue T co co + T of both groups from the consumpton n n T + T = t( α w +α w ) from the ncome tax, f q s the poolng payoff rate determned by (3). Obvously, f q s below that rate, then a consumpton tax rases less revenue than an ncome tax: the taxable base s reduced because of admnstratve costs and profts of the nsurance companes, whch are not accounted for n the present model. III. Taxaton of annuty payoffs From the example n Secton II we learn that taxng payoffs from annuty contracts not only changes the tme path of tax payments, but also the tax burden fallng on a 8 co n As above we fnd that the change n tax revenues T T of each group equals exactly the co n change of lfetme ncome α (a a )(qπ ) for the respectve group, whch would occur due to co n co n co ther ncreased annuty demand a a = ta (use the above result that a = a ( t) ) for unchanged T. n 8

10 partcular group, gven that survval probabltes dffer across groups and that there s a poolng rate of return. owever, n ths example, the two tax systems stll were equvalent, leavng welfare of both groups unaltered, due to a compensatng effect from ncreased annuty purchases. In the present secton we ask whether the fact that taxaton of annuty payoffs reduces the revenue rased from the low lfe-expectancy group, compared to an ncome tax, can be used for redstrbutve purposes. The ntuton s that n a tax system where a tax on annuty payoffs s employed, n addton to a tax on labour ncome (whose rate can then be reduced), some burden mght be shfted from the short-lvng group to the long-lvng group through a related mechansm as descrbed above. We consder two dfferent knds of models: the frst assumes a fxed frstperod ncome, as n Secton II, and s desgned to answer the queston of whether a dstortng tax on annuty payoffs s desrable n addton to a proportonal tax on labour ncome. The other model allows for ncentve effects on labour supply n order to deal wth the role of a tax on annutes, gven the optmum nonlnear ncome tax n the tradton of Mrrlees. III. A tax on annuty payoffs n addton to a proportonal ncome tax As n Secton II we consder an economy that conssts of two groups of ndvduals, wth shares α, α (α + α = ), who lve for at most two perods and dffer n the probablty to survve to the second perod, wth π < π. In order to concentrate on the mplcatons of dfferent longevty rsks, the fxed frst-perod ncome w s assumed 0 to be dentcal for both types of ndvduals. Preferences over consumpton c,c, = 0,, n both perods are descrbed by a utlty functon u(c,c ; π ), whch s strctly 0 concave wth respect to c,c, and depends postvely on the survval probablty,.e. u π > 0. Note that n case of expected utlty, whch s typcally assumed for the study of old-age provson under longevty rsk (see e.g., Abel 986, Brugavn 993, Wallser 2000), 0 u(c,c ; π ) reads u(c,c ; ) u(c ) u(c ), (4) 0 0 π = +π 9

11 where u s (strctly concave) per-perod utlty derved from consumpton. et furthermore t denote a proportonal tax on labour ncome, whch s lump-sum (as ncome s fxed) and τ be the tax rate on annuty payouts. Indrect utlty v( π,t, τ ) of an ndvdual s gven as the soluton of the optmsaton problem { } max u(c,c ; π ) c + c (q( τ)) w( t),c,c 0. The budget constrant results from the two separate condtons: 0 c + a w( t), c q( τ) a. As n Secton II, we assume for the sake of smplcty that the nterest rate s zero and that ndvduals have no other savngs nstrument. et S(t,τ) be the weghted utltaran socal welfare functon (wth weghts ρ, ρ ), (5) =, S(t, τ) ρ α v ( π,t, τ) dependng on the tax rates t,τ 0 that generate the requred tax revenue G, G= tw +τ( παqa () +παqa ()), (6) where a ( ) denotes the demand functon for annutes for =,. Insertng a = c /(q( τ)), the constrant (6) can be wrtten as G= tw +σ( π α c () +π α c ()), (6') wth σ τ /( τ). et throughout Secton III q be the poolng rate of return n equlbrum, mplctly determned by (3), for some gven τ. Note that annuty demands a, a depend on q (and on τ), therefore q cannot be computed explctly from (3). We start wth τ = 0 and ask whether the ntroducton of a postve tax rate τ on annuty payoffs and a correspondng decrease n the ncome tax rate t, such that (6 ) remans fulflled, ncreases socal welfare. In the followng we only consder the frst-round effect, that s, we take q as constant. The modfcaton, f the effect of an ncrease of τ on the rate of return (va changes n annuty demand) s taken nto account, wll be dscussed later on. Dfferentaton of (5) wth respect to τ gves 0

12 S t = ρ α + τ τ τ ( ) (7) =, t where t τ s obtaned by mplct dfferentaton of (6 ) as σ c c ( π α c +π α c ) +σ( π α +π α ) t = τ τ τ τ w (8) Usng the abbrevaton Q /(q( τ)) and Roy's emma, (7) can be transformed to n ( w denotes net ncome and λ n v/ w) S Q t = ρ α λ λ τ τ τ ( c w ). (7 ) =, Insertng (8) nto (7'), we derve for σ / τ =, Q / τ = / q ) S / τ at τ = 0 (whch also means σ = 0 and τ= 0 =, c S = ρ α λ ( +π α c +π α c ). (9) τ q Substtutng the poolng rate of return, determned by (3), together wth nto (9) yelds τ= 0 = a c q S τ τ= 0 ( )( a a ) =α α ρ λ ρ λ, (0) whch allows us to formulate condtons under whch the ntroducton of a dstortng tax on annuty payoffs s desrable. Proposton : Suppose that annuty demand ncreases wth the survval probablty. Then the ntroducton of a tax τ on annuty payoffs and a correspondng decrease n the ncome tax rate t, such that (6) remans fulflled, mproves socal welfare, f the

13 socal margnal valuaton of net ncome for ndvdual s larger than that for ndvdual,.e. f ρλ >ρλ. Proof: Immedate from (0) and the assumpton that a π > 0. QED. Frst note that a larger annuty demand of ndvduals wth hgh lfe expectancy s ndeed a reasonable assumpton. It holds, for nstance, f preferences are of the expected-utlty type (4). 9 Moreover, from many emprcal studes t s well known that adverse selecton n the prvate annuty market n fact occurs,.e. longer-lved ndvduals do purchase a larger amount of annutes. 0 An ntuton for Proposton can be found by observng from (8) that τq( πα a +πα a ) w represents the reducton of t n case of a margnal ncrease τ (at τ = 0) to keep tax revenues constant. On the other hand, from the total dfferental of the ndrect utlty functon v one fnds that utlty of an ndvdual remans constant or ncreases, f t s reduced at least by τ a w. It follows, usng equaton (3) - whch defnes the poolng rate of return - that the short-lvng ndvduals are made better off through such a change, as clearly ( α a +α a ) < a, whle the long-lvng ndvduals are made worse off. The relatve weghts n the socal objectve determne whether ths s desrable. owever, f q were suffcently small (below the poolng rate of return), then not even the short-lvng ndvduals would beneft, because q( πα a +πα a ) may eventually be larger than a. Wth any dstortng tax the deadweght loss grows at second order wth the tax rate, hence there s a upper bound to the tax rate on annuty payouts, above whch an addtonal ncrease makes both types of ndvduals worse off. Takng nto account the effect of τ on the poolng rate of return means that an addtonal term occurs n Q/ τ at τ = 0, vz. 2 Q/ τ= /q q'/q (where q' q/ τ 9 Follows from mplct dfferentaton of the frst-order condton n u(w a) + π qu(qa) = 0. 0 Mortalty tables for voluntary annutants n the well-developed U.S. and U.K. markets suggest that lfe expectancy for a typcal 65-year-old male annutant s about 20 percent longer than for a typcal 65-year-old male (see, e.g., Fnkelsten and Poterba 2002, Mtchell et. al. 999). 0 Usng the envelope theorem we get Dv = tw u c τqa u c = 0. From ths and the frstorder condton for annuty demand u c + q u c = 0, t can be 0 computed. 2

14 can n prncple be determned by mplct dfferentaton of (3)). As a consequence, the term ( ραλ a +ραλ a )q'/q has to be added n formula (0). In general, the sgn of q' s undetermned. 2 Obvously, Proposton remans true as long as q' s not too negatve,. e., as long as the composton of annuty demand does not change too much n favour of the hgh-rsk group, f τ s ntroduced. Further note that the fndng of Proposton s not restrcted to an annuty market whch s charactersed by asymmetrc nformaton, such that (3) determnes the poolng payout rate n equlbrum. Even n a frst-best world wth perfect nformaton, n whch the ndvdually far rates of return q = π, =,, can be realsed as an equlbrum, the ntroducton of a tax on annuty payouts mples redstrbuton from the hgh-demand ndvduals to the low-demand ndvduals, due to the same arguments as above. Ths s straghtforward to show by usng q = π nstead of q, and τ= 0 a = c q n (9), whch yelds S τ τ= 0 ( )( a (q ) a (q )) =α α ρ λ ρ λ. () Moreover, we fnd that under the assumpton of expected utlty (4) a (q ) a (q ) > 0 3, whch mples that, as above, t s the group wth low lfe expectancy that benefts from the ntroducton of τ at the expense of the group wth the hgh lfe expectancy. In ether case, t s the relatve socal evaluaton ρ λ ( ρ λ ) of the rsk-groups, whch s decsve, whether a postve annuty tax wth the correspondng ncome-tax decrease should be mplemented. Obvously, establshng the magntude of ρλ /( ρλ ) brngs us to a dffcult normatve queston; to tackle the ssue, note frst that wth expected utlty (4) the margnal utlty of net ncome of the long-lvng ndvduals s hgher than that of the short-lvng ndvduals,.e. λ >λ : Gven equal 2 If the per-perod utlty functon u ~ n expected utlty (4) exhbts constant relatve rsk averson, Pech (2004) has shown that a tax on payoffs aggravates adverse selecton for certan parameter values, otherwse the effect s undetermned. 3 To proof ths result, substtute the ndvdually far rate nto the frst-order condton n footnote 9 to obtan u(w n a ) + u(a π ) = 0 and dfferentate ths term mplctly. 3

15 net ncome, ndvduals wth a hgher lfe expectancy value an addtonal unt of money more. Consequently, assumng an unweghted utltaran socal welfare functon ( ρ =ρ ), the ntroducton of an annuty tax (wth a correspondng decrease n the ncome tax) s not desrable. Instead a subsdy on the annuty payoffs (wth a correspondng ncrease n the ncome tax) could serve as a possble nstrument to ncrease socal welfare. On the other hand, t s a common crtcsm on (unweghted) utltaransm, that t mples dstrbuton to be drven by ndvdual margnal utltes, whch may have nothng to do wth an ethcal objectve. Such a poston s taken, e. g. by Sen (973, p. 6f). e llustrates ths crtcsm by consderng a handcapped ndvdual, who derves less utlty from addtonal ncome, compared to a healthy person, whle for ethcal reasons we mght prefer to favour the handcapped person. Transferrng Sen s vew to the present case, socety may want to treat those ndvduals wth poor health and hgh mortalty 4 better, whch means that ths group s gven a hgher weght ρ >ρ n the utltaran welfare functon (5). If ρ s suffcently large, t may mply redstrbuton, va the mplementaton of a proportonal tax on annuty payouts. Fnally, consder the even stronger egaltaran concept of Rawls, whch ams at maxmsng the utlty of the worst-off group. It can be shown that n case of a poolng rate of return ndeed the group wth low expectancy s worse off than the one wth hgh lfe expectancy. 5 In the present model, we would then have λ = 0 and t follows that accordng to the maxmn crteron a postve tax rate on annuty payouts ncreases socal welfare. owever, note that wth ndvdually far payouts the comparson of both groups utlty levels s not clear-cut: On the one hand, the ndvdual wth the low lfe expectancy s worse off wth any gven consumpton bundle 4 Obvously, what one has n mnd here s that hgher mortalty occurs largely for bologcal or genetc reasons respectvely and not rather out of personal choce (such as smokng or other rsky behavour). 5 Immedate from the followng consderatons: Wth an equal rate of return, the budget set s the same for both types of ndvduals. Due to ther hgher lfe expectancy, ndvdual attan a hgher 0 utlty level at type s optmal consumpton bundle (c,c ) than ndvdual (remember that u π > 0). Consequently, ndvdual s utlty s obvously hgher wth her optmal choce 0 (c,c ). 4

16 than the long-lvng ndvdual; on the other hand, as q > q, the budget set of the short-lvng ndvdual s larger. Altogether, we found n the present model that the socal desrablty of a tax on annuty payoffs leads one to the dffcult normatve ssue, whether lfe expectancy tself can be an argument for a dfferentated treatment of ndvduals. owever, one should recognse that n any dscusson of socal equty and justce, llness and mortalty cannot be gnored. Moreover, t should be clear that value judgments of ths sort are nvolved n any poltcal decsons, partcular n the desgn of the tax system. III.2 A tax on annuty payoffs n addton to an optmum nonlnear ncome tax A more realstc model of the problem of tax desgn has to take nto account the ncentve effects on labour supply. Therefore, n the present subsecton, the assumpton of a fxed wage ncome s dsmssed. Instead we consder an economy wth two types of ndvduals, who dffer n ther wage rate b, =,, wth b < b, and, as before, n ther probablty of survval to the second perod. We agan assume π < π and, thus, a postve correlaton between the wage rate and lfe expectancy, whch s plausble from emprcal studes, as mentoned n the Introducton. 0 π Preferences are now descrbed by the utlty functon u(l,c,c ; ), where l denotes labour supply. As s usual n optmum-taxaton theory n the tradton of Mrrlees, we assume that the authorty does not know ndvdual abltes, but only gross ncome. Thus, the tax system conssts of a (nonlnear) tax on gross ncome and, n addton, of a tax on annuty payoffs. Concernng the latter, we consder two dfferent models: In the frst one, we ask whether the ntroducton of a lnear tax mproves welfare, whle n the second we analyse an optmum tax system whch s fully nonlnear wth respect to ncome from both labour and annutes. III.2. near taxaton of annutes Ths case could be seen as a dual ncome tax system, where annuty payoffs are taxed separately from labour ncome, wth a unform rate τ. et z b l denote gross 5

17 ncome and x net ncome. We defne the ndrect utlty functon, dependng on z and x, v (x,z,τ) max{ u(z b,c,c ; ) c c (q( )) x,c,c 0} π + τ >, for any tax rate τ on annuty payoffs. As s usual n models of optmum nonlnear ncome taxaton, we have to assume that preferences fulfll the sngle-crossng condton AM: / z )/( / x ) > ( / z )/( / x ), ( for any x, z, and τ. 6 Otherwse, redstrbuton va the tax on labour ncome could go from the less to the more able ndvdual, because the latter would choose to work so lttle that her gross ncome s below that of the former. As already mentoned, the basc element of ths Mrrlees-type model s asymmetrc nformaton concernng ndvdual abltes,.e., wage rates. In the model, ths s accounted for through the so-called "self-selecton constrants": the government assgns two gross and net ncome postons to the ndvduals n such a way that each does not prefer the poston assgned to the other. Assumng agan a weghted utltaran socal objectve, the problem of the optmum nonlnear ncome tax reads, for any gven tax τ on annuty payoffs: max v(x,z,) v(x,z,), s. t. ρ τ +ρ τ (2) x + x z + z +σ( π c () +π c ()) G, (3) v (x,z, τ) v (x,z, τ ), (4) x,z 0,,. (5) = (4) represents the self-selecton constrant for the long-lvng and hgh-wage ndvdual. In prncple, an analogous restrcton has to be formulated for the shortlvng ndvdual. owever, one can show that wth approprate weghts on the latter n the objectve functon (compare the dscusson n III.), the self-selecton constrant for the short-lvng ndvduals s not bndng n the optmum, whle (4) s. Equaton 6 The condton was called "agent monotoncty" by Seade 982. It s a "sngle crossng condton", because t mples that ndfference curves of the two ndvduals n (z,x)-space cross only once. See also Brunner

18 (3) s the resource constrant. For smplcty, we have neglected - possbly dfferng - group shares α, α n (2) - (3). Introducng them would not change the results. The frst-order condtons for an optmum soluton of (2) - (5) wth respect to x,z, =, are gven n Appendx A. As n Subsecton III., we denote by S(τ) the optmum value of the objectve functon (2), for gven τ, and derve, by some manpulatons (see Appendx A) S τ =υ τ= 0 x (a [] a ), (6) where υ > 0 s the agrange multpler assocated wth (4), and a [] denotes annuty demand, whch the hgh-wage ndvdual would choose n case of mmckng,. e. f optng for gross and net ncome assgned to the low-wage ndvdual. (6) follows from (A.9) n Appendx A, f the zero-proft condton (3) s used to show that the term n square brackets vanshes. Wth (6) we can formulate our man result, whch refers to the frst-round effect (holdng q fxed), as before: Proposton 2: Assume that demand for annutes ncreases wth the survval probablty and does not ncrease wth labour tme. Then a lnear tax on annuty payoffs, n addton to the optmum nonlnear ncome tax, mproves socal welfare. Proof: a [] > a follows from the assumptons that () demand for annutes ncreases wth the survval probablty, whch s hgher for ndvdual and that () demand s not postvely assocated wth labour tme, because, n case of mmckng, ndvdual works less than ndvdual. υ > 0 and v / x 0 complete the proof. > QED. avng n mnd Proposton, t may appear surprsng that the weghts of the ndvduals do not play a role n the above proposton. The explanaton for ths fact s that, when formulatng the model (2) - (5) we have already assumed that wth the optmum soluton the self-selecton constrant for ndvdual s bndng. That s, the government would lke to redstrbute more ncome, but s restrcted because of 7

19 asymmetrc nformaton. Ths assumpton, whch corresponds to the usual way the problem s formulated, n turn hnges on a suffcent mportance of the dsadvantaged ndvdual n the objectve functon. Proposton 2 shows that n such a stuaton, gven a postve correlaton between the wage rate and lfe expectancy, a tax on annuty payoffs allows addtonal redstrbuton, as t makes mmckng less attractve for the more able ndvdual. Further, remember from Secton III. that for expected utlty the frst assumpton requred for Proposton 2, namely that annuty demand ncreases wth lfe expectancy s ndeed fulflled. Moreover, the second assumpton - a non-postve assocaton of annuty demand wth labour tme - s clearly guaranteed, f preferences are weakly separable between labour (lesure) and consumpton n both perods,.e. f annuty demand s ndependent of labour tme, for gven net ncome. It s nterestng to compare the above result wth a correspondng one concernng the role of a tax on captal ncome. As s well-known, gven that preferences are weakly separable between labour (lesure) and consumpton n dfferent perods, no tax on captal ncome n addton to the optmum nonlnear ncome tax s desrable (see, e.g. Atknson and Stgltz 980, Ordover and Phelps 979). A tax on captal ncome s only desrable, f savng s postvely assocated wth lesure (compare also Corlett and ague 953). In the present model, however, the desrablty of the tax on annuty payoffs n fact results, as long as savng (that s, annuty demand) does not decrease too much wth lesure, so that ths effect s outweghed by the ncrease n annuty demand of ndvdual due to her hgher lfe expectancy. Note also that exactly the same formula (6) would arse n a frst-best world, wth ndvdual far rates of return q, q used nstead of q n (A.8) and (A.9). We have argued n Secton III. that n the expected-utlty case a (q ) > a (q ) holds, thus a result smlar to Proposton 2 apples. Fnally t should be mentoned that takng nto account the (second-round) effect of τ on the (poolng) equlbrum rate of return q means that an addtonal term dependng on q/ τ occurs n (6), analogous to the stuaton n Secton III.. 8

20 III.2.2 Nonlnear taxaton of ncome from labour and annutes A fully nonlnear system s based on the dea that n fact, by choosng the approprate bundle of gross ncome and net ncome, an ndvdual reveals her type (see, e. g., Brett 998, Prttla and Tuomala 200 n the context of captal ncome taxaton). Ths means that n the second perod the tax on ncome from annutes can be mposed separately on each type. Consequently, we consder a tax system, whch conssts of a (nonlnear) tax T(z), mposed on gross ncome, and two (nonlnear) taxes T (qa ), T (qa ), dependng on annuty payoffs. Techncally, ths means that the self-selecton constrant now has the form v (c,c,z ) v (c,c,z ), (7) 0 0 where v(c,c,z) u(z/b,c,c, ). That s, the government has to select two 0 0 π complete bundles of labour tme (or gross ncome, equvalently) and consumpton n both perods, such that the more able person does not prefer the bundle assgned to the less able. (Agan we assume a-pror that the government wants to redstrbute ncome from the former to the latter type.) As before, a sngle-crossng condton s requred, whch has the same form as AM, but wth v (x,z,t ) { } max u(z b,c,c ; π ) c + N (c ) q x, c,c > 0, where N s the nverse of the net ncome functon N(qa) qa T(qa). AM has to hold for approprate T. Wth these preparatons we can formulate the planner's problem as max ρ v (c,c,z ) +ρ v (c,c,z ), (8) c,c,z 0 0 s. t. 0 0 v (c,c,z ) v (c,c,z ), c + c +π c +π c z + z G, (9) (20) 0 c,c,z 0, =,. (2) The correspondng frst-order condtons can be found n Appendx B. The soluton has the followng propertes: 9

21 Proposton 3: For type the optmum nonlnear tax on annuty payoffs exhbts a postve margnal tax rate equal to tax rate s undetermned. (q / π ) q. For type, the sgn of the margnal Proof: From the frst-order condtons for the ndvdual optmsaton problem max u(z /b,c,c ; π ), s.t.c = z T(z ) a and c = qa T(qa ) 0 0 one derves an expresson for the margnal tax rate T n terms of the margnal rate of substtuton as u/ c / c T = (q ) = (q ), =,, 0 0 q u/ c q / c (22) where the second equalty s mmedate from the defnton of v. On the other hand, from (B.4) and (B.5) n Appendx B we fnd that n the optmum / c /c 0 = π must hold, thus the frst part of the Proposton s proved. For type- ndvduals, (B.) and (B.2) tell us that n the optmum ρ / c υ v / c ρ υ π 0 0 = v / c v / c, (23) where / c, t = 0,, descrbes the margnal utlty of ndvdual n case of t mmckng,.e. optng for the type- bundle. In general, t cannot be concluded from (23), whether ( / c 0 )/( / c ) / π and therefore the margnal tax rate on annuty payoffs s undetermned. QED. One observes that the margnal tax rate for group wll be negatve, f the margnal rate of substtuton between present and future consumpton of ndvdual does not dffer much from that of ndvdual n case of mmckng, because then (23) mples that ( / c )/( / c ) s close to /π, whle the poolng rate of return q s smaller 0 20

22 than / π. On the other hand, f ndvdual values future consumpton more (n case of mmckng), because of her hgher lfe expectancy, then we have 0 0 ( / c )/( / c ) > ( / c )/( / c ), whch together wth (23) mples that ( / c )/( / c ) < / π. 7 Ths n turn, used n (22), tells us that the margnal tax 0 rate on annuty payoffs, for ndvdual may also be postve. It s mportant to notce the dfference of the result n Proposton 3 to the prevous one: whle the desrablty of a lnear tax on annutes essentally depends on the dfference n annuty demand of hgh- and low-rsk ndvduals, a further motve arses wth a nonlnear tax: the correcton of the rate of return n a poolng stuaton. Ths s more n lne wth the ntuton developed n Secton II: the loss (beneft) from poolng of the short-lvng (long-lvng) ndvdual, compared to ndvdually far payoff rates, gves rse to a dfferentated treatment by the tax system. It s obvous from the proof of Proposton 3 that f q was equal to the ndvdually far rates q, q, resp., then the margnal tax rate for ndvdual s zero, whle a dstorton - famlar from other Mrrlees-type models - arses for ndvdual. The correcton of ths market falure arsng from asymmetrc nformaton s specfc for annuty taxaton; n models nvestgatng the optmum nonlnear captal ncome tax, the margnal tax rate for the hgh-wage ndvdual s zero, whle for the low-wage ndvdual t s dstorted, except when preferences are weakly separable between consumpton and lesure (Ordover and Phelps 979). In fact, n the present model the payoff rate fxed on the annuty market does not drectly enter the planner's optmsaton problem, whch ams at determnng optmum second-best bundles. The margnal tax rate follows from a comparson wth the rate of return offered by the market. Such a correcton of the rate of return through the tax system becomes possble, f we follow the dea that n the retrement perod the authorty can ndeed dentfy ndvduals by ther types, because these are revealed when gross (and net) ncome s reported by the end of the workng perod. Yet, annuty demand s expressed durng ths perod already, when nether tax authorty Ths follows mmedately by rewrtng ( / c )/( / c ) > ( / c )/( / c ) as 0 0 ( / c ) ( / c ) > ( / c ) ( / c ) and by transformng equaton (23) to ( / c ) ( ρ υ ( / c ) ( / c ) ) ( / c ) ( ρ υ ( v / c ) ( / c ) ) = π. 2

23 nor nsurance frms can dstngush the types, therefore the latter are unable to offer ndvdually far rates. Fnally, t should be mentoned that n both models wth lnear or nonlnear taxaton of annuty payoffs the usual propertes concernng the optmum tax rates on labour ncome can be derved: It s zero for the more able ndvdual but postve for the less able. IV. Concludng comment Prvate lfe annutes are becomng a more wde-spread nstrument for old-age provson, as publc penson systems are expected to provde less support n the future. owever, t s well-known that the annuty market s affected by an adverseselecton problem, whch s a typcal obstacle to many nsurance markets. As a consequence, t provdes only less than far contracts for ndvduals wth low lfe expectancy. In addton, emprcal studes have found that lfe expectancy and ncome are postvely correlated. Therefore, t appears a natural queston to ask whether these facts should have an nfluence on the tax system, n partcular on the balancng of effcency and equty consderatons. Theoretcal studes on captal ncome taxaton have shown that n a varety of models such a tax cannot fulfl any further redstrbutve task, gven an optmally desgned tax on labour ncome. Intutvely, one mght be wllng to accept a smlar statement for the taxaton of annuty payoffs. On the other hand, ntuton also shows that such a tax falls on long-lvng ndvduals to a larger extent than on short-lvng and has, thus, a dfferent effect compared to a tax on ncome from labour or captal. Under the provson that lfe expectances and wages are postvely correlated, we were able to fnd clear results n Secton III.2 wthn the framework of optmum ncome taxaton: A lnear tax on annuty payoffs can be used for redstrbuton, f annuty demand ncreases wth lfe expectancy, whch s qute plausble. A nonlnear tax can be drectly employed to correct the dstorton of the rate of return caused by asymmetrc nformaton, rrespectve of demand. 22

24 Obvously, the basc queston underlyng the above results s, whether socety favours redstrbuton at all and, n partcular, how t values the welfare of groups wth dfferng lfe expectancy (and ncome). On ths, we presented some deas n Secton III., but of course, economc analyss cannot provde a fnal answer. Its man task s to pont out the dstrbutve consequences assocated wth the constructon of a tax system. 23

25 Appendx A The agrangan to the maxmzaton problem (2) - (5) reads ( ) ( ) =ρ v (x,z, τ ) +ρ v (x,z, τ) µ x + x z z σπ ( c +π c ) + G + +υ v (x,z, τ) v (x,z, τ), whch gves us the frst-order condtons: c ρ µ + µσπ υ = 0, (A.) x x x c ρ + µ + µσπ υ = 0, (A.2) z z z c ρ µ + µσπ + υ = 0, (A.3) x x x c ρ + µ + µσπ + υ = 0. (A.4) z z z Moreover, usng the Envelope Theorem we get σ c c S =ρ +ρ +µ ( π c +π c ) +µσπ ( +π ) + τ τ τ τ τ τ [] +υ υ, τ τ (A.5) where [] n the last term means that the dervatve of v s computed at (x,z,τ). By Roy's emma we have / τ= c(v/ x)(q/ τ ), where Q /(q( τ)). Usng ths and substtutng from (A.) for τ = σ = 0 µ = ρ υ (A.6) x x and from (A.3) (agan for τ = σ = 0) µ = ρ + υ (A.7) x x n turn n (A.5) we derve (note that, at τ = 0, σ/ τ =, Q / τ = / q ) 24

26 S =ρ π +ρ π + τ τ= 0 c( ) c( /q) x q x c [] x q x q +υ c ( π ) +υ ( π c ), (A.8) where c [] denotes consumpton n the second perod, whch the hgh-wage ndvdual would choose, f endowed wth gross and net ncome of the low-wage ndvdual. Approprate groupng and usng (A.6) and (A.7) agan gves us from (A.8) τ= 0 q q x q S v (c [] c ) =µ [c ( π ) + c ( π )] +υ. τ (A.9) Appendx B We wrte the agrangan functon for (8) - (2) as 0 0 ( v (c,c,z ) v (c,c,z ) ) ( ) =ρ v (c,c,z ) +ρ v (c,c,z ) µ c + c +π c +π c z z + G υ and get the frst-order condtons: ρ µ υ = 0, (B.) 0 0 c c ρ µπ υ = 0, (B.2) c c 0, (B.3) ρ +µ υ = 0 z z ρ µ+υ = 0, (B.4) 0 0 c c ρ µπ +υ = 0, (B.5) c c 0. (B.6) ρ +µ+υ = z z 25

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