The Efficiency of Unfunded Pension Schemes

Transcription

1 The Effiieny of Unfunded Pension Shemes Sefan Homburg Disussion Paper No. 53 ISSN This is a posprin of Homburg, S. (990) The Effiieny of Unfunded Pension Shemes. Journal of Insiuional and Theoreial Eonomis (JITE) 46, pp Absra: Publi pension shemes an be designed eiher as apial reserve sysems or as unfunded (or pay-as-you-go) shemes. In he lieraure i has been alleged ha unfunded shemes are inergeneraionally effiien in Pareo's sense. Here we show ha his holds only if onribuions o he sysem are levied as lump-sum axes. Bu in realiy, fla-rae inome axes are normally used insead and hen, an unfunded sheme indued disorions and an ompleely be abolished in finie ime wihou infliing damage upon any generaion. JEL-Classifiaion: H, H55

2 . Inroduion In an imporan arile in his Journal, Breyer [989] has onsidered he problem of he effiieny of unfunded (or pay-as-you-go: PAYG) pension shemes. He found ha hese shemes are inergeneraionally effiien in Pareo's sense even when he rae of ineres permanenly exeeds he growh rae. In suh a senario, an infiniy of generaions are made worse off and only he firs one is made beer off by he inroduion of PAYG. Neverheless, he unfunded pension sheme urns ou o be effiien for he following reason: In Breyer's model, onribuions o ha sysem are inrodued as lump-sum-axes and he pensions are lump-sum-ransfers. Now, applying he seond basi heorem of welfare eonomies, we know ha any lump-sum redisribuion of inomes enails an alloaion whih is differen bu also Pareo-effiien. Therefore, Breyer's resul is inuiively undersandable. The presen paper seeks o ake ino aoun he fa ha, in realiy, onribuions o PAYG are never raised as lump-sum paymens. Consider he US Soial Seuriy Sysem, for insane. Premium paymens are levied as a fla-rae ax on labor inomes whereas pension paymens are virually lump-sum a leas in he sense ha hey are independen of he preeding onribuions. Thus, a PAYG of he Amerian ype is basially a axransfer-sysem whih disribues inome beween generaions. We will show in he sequel ha suh a sysem is always ineffiien and ha, moreover, i an be onvered ino a apial-reserve sysem wihou infliing damage upon anyone.. The Basi Model Our framework is he sandard overlapping-generaions model wihou bequess whih has been widely used for analyzing he impas of publi pension shemes. We onsider he ase of a small open eonomy whih is haraerized by exogenous sequenes of wage raes w and ineres faors R:=+r. For onveniene, ineres faors and populaion are supposed o be onsan. The wage raes w, however, may vary in a perfely arbirary manner; hus he growh rae may be greaer or smaller han he ineres rae. In every period, a represenaive member of he younger generaion solves () max!,,l s..(i) (ii) U( s, w l ( ), R s, l p ), The assumpion of a small open eonomy is more favorable o a pay-as-you-go sysem sine, wih ineres and wage raes given from ouside, a derease in domesi savings will no hur laer generaions via depressed domesi wages. Breyer [989] analyzes he open as well as he losed eonomy and finds pay-as-you-go shemes o be effiien in boh ases.

3 3 where is onsumpion in he firs period, is onsumpion during he reiremen period, s denoes savings and l is he labor supply, he sum of labor and leisure ime having been normalized o one. U(.) is assumed o be srily monoonially inreasing, srily quasi-onave, and wie oninuously differeniable. The goods shall be normal. The variable is eiher a premium rae or an inome ax rae, and p + is a publi pension. Thus every household mus pay he amoun w l when young and obains a paymen p + when old. A pay-as-you-go sysem (PAYG) wih a onsan premium rae is simply a sequene (,p ), where boh and all p are srily posiive and where p w l for all This las equaion says ha he pension paymens whih are made o he preeding generaion mus equal urren onribuions; and he onribuion rae has no ime index beause i has been assumed onsan. Noe ha he hoie of a pariular PAYG generally affes he households' labor supplies; herefore, wage inome w l mus be evaluaed a he resuling perfe-foresigh equilibrium posiion whih, of ourse, need no be uniquely deermined. Heneforh, all variables whih are assoiaed wih some PAYG will be signed wih a bar ( ). I is now sraighforward o show ha every PAYG is ineffiien in he following sense: Generaion an be made beer off wihou any oher generaion being made worse off. This is done using a sysem I all apial reserve sysem um governmen deb (CRCD). Suh a sysem will be shown o be he opimal soluion o he following maximizaion problem of he sae: () max!, p,d s.. (.) (.) (.3) V(,p w l p p ) w p 0. l D, RD, Here V (, p + ) is he indire uiliy funion of generaion whih shall be maximized by planning an opimal poliy (, p +, D + ) in period. In iniial equilibrium, a pariular PAYG is in operaion, and he amoun w l is given o he elderly. The onsrain (.) requires ha his paymen may be finaned eiher by an inome ax w l or by governmen deb, D. The oal amoun wl D however, mus be equal o he iniial paymen w l so ha he preeding generaion is no made worse off. Assume ha a erain governmen deb D > 0 has been hosen in period. Then, aording o (.), generaion mus repay ha deb plus ineres before obaining a pension paymen p +. Pu differenly, he deb is no shifed ino he fuure bu is ompleely repaid by generaion so ha he subsequen generaions are no made worse off. Finally, (.3) demands p + 0, beause a negaive pension paymen would have o be inerpreed as a lump-sum-ax. Beause i is infeasible o levy lump-sum-axes, we exlude his possibiliy. The onsrains (.) and (.3) ogeher show ha governmen deb never exeeds he "implii deb" of PAYG: RD will no exeed p. Subraing R imes (.) from (.) yields he ombined onsrain of he above opimizaion problem:

4 4 (3) p R wl p R wl. This says ha he hoie of a pariular poliy (, p +, D ) may no hange he generaion's lifeime inome. As p equals imes he wage inome of he subsequen period, i is easy o see ha expression (3) is posiive whenever he growh rae exeeds he ineres rae. The generaion will hen be alled a winner of PAYG. In he more ineresing ase where R exeeds he growh faor w l /( w l ), expression (3) will beome negaive, and he generaion will be alled a loser. Proposiion: (i) If generaion is a winner, he only opimal poliy (,p,d ) is defined by 0, where p and D an be alulaed from he onsrains. (ii) If generaion is a loser, he only opimal poliy (,p,d ) is defined by p 0, where and D an be alulaed from he onsrains. (iii) If generaion is neiher a winner nor a loser, he only opimal poliy (,p,d ) is defined by 0 and p 0, where D an be alulaed from he onsrains. (iv) PAYG, i.e., p 0 and D 0 is opimal in neiher ase. Proof: From he household's ombined budge onsrain (4) R R( )wl p we obain he familiar Sluzky-equaion l l l (5) Rwl 0, p whih saes ha he ompensaed elasiiy of labor supply l wih respe o he premium rae is srily negaive. This follows immediaely from he sri quasi-onaviy of U(.). In order o solve he governmen's opimizaion problem () we alulae he derivaive of V (.) wih respe o : dv V V dp (6). d p d Now, from (3) we obain he derivaive dp + /d by applying he implii funion heorem: (7) dp d Combining (6) wih Roy s ideniy (8) R w l R w l R w l p V and subsiuing dp + /d from (7) gives (9) dv d R w l R w l V p R w l R w l R w l p V p

5 5 Finally, afer rearranging erms and employing he Sluzky-equaion (5) we obain dv V (0) l R w. d R w l p p As p is a lump-sum paymen, V /p + > 0 and l /p + < 0 (remember ha leisure, l, was assumed o be normal). Thus, for > 0, all erms in (0) are srily posiive exep (l /) whih is srily negaive. We onlude, herefore, ha dv /d is srily negaive for >0 and srily posiive for < 0. So = 0 is he globally unique opimum. This esablishes (i). If he generaion is a ne loser, however, = 0 implies p 0, whih is no possible. In his ase, mus be gradually inreased unil p 0. The orner soluion is a unique opimum beause dv /d < 0 for all > 0. This proves (ii). The laims (iii) and (iv) follow a one. Q.E.D. The inuiion behind his resul is very simple indeed. One has only o realize ha PAYG onsiss of an inome ax (he premium paymen w l ) and a lum-sum ransfer (he pension paymen p + ) Replaing PAYG by governmen deb enails a simulaneous reduion in he inome ax and he lump-sum ransfer. This is obviously benefiial o he household beause he inome ax disors he labor-leisure hoie and hene indues an exess burden. Seing (l /) = 0 in (0), i.e. assuming an exogenous labor supply, we obain he Breyer resul: Changes in neiher inrease nor redue he household's welfare, provided ha is lifeime inome remains unhanged. 3. Making all Generaions Beer off Subsequen o he parial analysis of he preeding seion, we now onsider he sequene of all generaions, V (.) denoing again generaion s uiliy level. All generaions, however, are assumed o have he same dire uiliy funion. We assume ha a PAYG is already in operaion and onsru an alernaive sysem whih makes every generaion beer off. In doing so, we deal only wih siuaions where he ineres rae exeeds he growh rae in iniial equilibrium so ha every generaion is a loser. This is he ase whih has been onsidered by mos auhors, inluding Breyer. The reader will noe, however, ha our resuls also hold for he reverse ase. This follows immediaely from saemen (i) in he above proposiion. An explii onsideraion of ha reverse ase would make he following noaion raher umbersome bu would no inrodue analyial diffiulies. From now on, W = w l denoes aggregae wage inome. Corollary : Assume ha, wih he ineres rae exeeding he growh rae, some speifi PAYG is already in operaion: 0 and p W. Then, here exis sequenes ( and (p, 0, 0,...) suh ha p W () W p /R and V () (, 0) V (,p ) for all.

6 6 Proof: Immediae from he proposiion, (ii). The sysem hus onsrued has already been alled CRCD. From he household's view, i is a pure apial reserve sysem (CR) beause no pension paymens are made. Bu on he aggregae level, suh a sysem implies a sequene of governmen debs whih are equal o he implii deb of PAYG. Therefore, he eonomy's ne wealh is he same as before. Noe ha he presenly living elderly are ompleely unaffeed by a ransiion from PAYG o CRCD beause hey ge he pension paymen p anyway. Bu all subsequen generaions are made beer off beause CRCD enails a reduion in he premium raes and hus diminishes he exess burden. 4. A Pareo-improving Conversion Poliy Among ohers, Breyer [989] has posed he quesion wheher i is possible o onver a PAYG ino a pure apial reserve sysem (wihou governmen deb), provided ha no generaion may be made worse off. His answer has been in he negaive even when every generaion (exep he firs) is hur by PAYG beause he ineres rae exeeds he growh rae. Conenraing on his ase also, we now wan o show ha, in an eonomy wih an endogenous labor supply, suh a Pareo-improving onversion poliy is indeed possible. The implii deb of PAYG is run down in finie ime and up o ha momen no generaion will be made worse off. Thereafer, all generaions are beer off. In he following, are he premium raes whih are used for suh a onversion poliy. I will be seen ha hese raes are greaer han, he premium raes assoiaed wih CRCD, bu smaller han he premium rae of PAYG. Observe ha, as labor supply is endogenous, oal wage inome W will generally vary if he onribuion rae hanges; herefore, he W in () and (3) are no neessarily he same. Corollary : Assume ha, wih he ineres rae exeeding he growh rae, some speifi PAYG is already in operaion: 0 and p W. Then here exis sequenes ( ), ( p ) (p, 0, 0,...) and ( ) > 0 suh ha (3) W W p /R W and (4) (, 0) V (, p ) for all...t. V Furhermore, here exiss T suh ha D 0 for all > T. Proof: The exisene of is readily inferred from orollary : If we an ax he generaions in a way ha V (,0) srily exeeds V (, p ) we an also ax hem a bi more heavily so ha V (,0) equals V(, p) As all dire uiliy funions are idenial, he also bounded away from zero. I remains o be shown ha he deb is repaid in finie ime. From (.), we have (5) p W, so ha, using (3) and p W D

7 (6) W /R W. 7 D Similarly, (7) R D W, D (8) D W 3 /R R W W and for arbirary T, T (9) D W /R R W T T W T T T W (0) DT R. R W T T As he ineres rae exeeds he growh rae, we neessarily have R W / WT. T (For a seady sae pah, we find W / WT / G and R > G, where G is he growh T faor. For oher pahs, G is he supremum of W / W. In eiher ase, (R /G).) And as > 0 he erms on he righ-hand side are bounded away from zero. Hene here exiss finie T suh ha he sum on he righ exeeds. Q.E.D. Again, he eonomi idea behind his proof is relaively simple. In he preeding seion we used he effiieny gains whih arue from leaving PAYG for improving all generaions' welfares. In he presen seion, he firs T generaions are made as well off as in PAYG, and he effiieny gains are used insead for repaying he governmen deb. I is hus possible o ompleely abolish PAYG in finie ime and o resore he eonomy's ne wealh o he level whih is assoiaed wih a pure apial reserve sysem. And during he ransiion, no one is made worse off. This opion is mos araive for hose generaions whih are living in he far fuure beause, from period T on, he individuals will no longer be axed. On he oher hand, he presenly living households among hem all urren voers will prefer CRCD sine his sysem makes hem beer off. Ye, onsidering he polar ases and as defined in he orollaries, we reognize a one ha here exis premium raes in beween whih make he firs generaions beer off and a he same ime allow he deb o be repaid in finie ime. Referene Breyer, Friedrih [989], "On he Inergeneraional Pareo Effiieny of Pay-as-you-go Finaned Pension Sysems," Journal of Insiuional and Theoreial Eonomis, 45, Dr. Sefan Homburg Fahbereih Wirshafs- und Sozialwissenshafen Universiä Dormund Posfah D-4600 Dormund 50 T