Quas-Hyperbolc Dscountng and Socal Securty Systems Mordecha E. Schwarz a and Eytan Sheshnsk b May 22, 26 Abstract Hyperbolc countng has become a common assumpton for modelng bounded ratonalty wth respect to ndvdual savngs decsons. We examne the effects of hyperbolc countng on the comparson of alternatve socal securty systems. We show that ths form of bounded ratonalty breaks the equvalence between funded and pay-as-you-go systems establshed n Sheshnsk and Wess (98. Intergeneratonal transfers wthn a PAYG economy are usually secured by the socal securty system and ndependent of longevty, whereas ths s not the case for the funded economy. The savngs level under hyperbolc countng s lower than under exponental countng (Labson et al. 996, but the rato between the savngs level under hyperbolc countng wthn a funded economy and a PAYG economy depen on the effectveness of the commtment devces. It s shown that f ndvduals are hyperbolc counters, then n a PAYG economy any change n the mandated level of ntergeneratonal transfers s neutralzed by ndvduals' voluntary bequests. Ths does not apply to a funded system. JEL Classfcaton: D64, G23, G28, H55 Key Wor: Hyperbolc Dscountng, Funded, Pay As You Go, Socal Securty, Penson. a Department of Economcs and Management, The Open Unversty of Israel, Gvat-Ram Campus, Jerusalem 994, Israel. Phone: 972-2-677-3338, Fax: 972-2-65-25 E-mal: morch@openu.ac.l. b Department of Economcs, Hebrew Unversty of Jerusalem, Mount Scopus, Jerusalem 995 Israel. Phone: 972-2-5883-44, Fax 972-2-588-3357, E-mal: mseytan@mscc.huj.ac.l
2. Introducton Tme nconsstency of ndvdual preferences and the noton of dynamc preferences or wth evolvng selves s a well-known and studed phenomenon (Pollak 968, Phelps and Pollak 968, Yaar and Peleg 972, Hammond 976, Thaler and Shefrn 98, Schellng 984. The prevalent tool for modelng tme nconsstency of ndvdual preferences n the last decades of the twenteth century was the hyperbolc countng functon (Ansle 992, Lowensten and Prelec 992, Labson et al. 996, Labson 2, and such modelng can be traced back to the md-twenteth century (Strotz 956, Chung and Herrnsten 96. The hyperbolc countng functon was appled to the study of undersavng and the sharp reducton n consumpton of old people (Labson 996, to the early retrement pattern of workers (Labson et al. 998, Damond and Kőszeg 22, and to job search behavor (Paserman and Della-Vgna 2. The hyperbolc or quas-hyperbolc countng hypothess (henceforth, HDH, as an emprcal fndng, was based on experments performed on humans and anmals that the researchers nterpreted as supportng ths hypothess (for a survey, see Labson et al. 998. However, as an emprcal fact, HDH s controversal (Rubnsten 2, 22, Read 2, Besharov and Coffey 23. From a theoretcal pont of vew, hyperbolc countng rases several problems. For nstance, tme nconsstency reflects rratonalty, or at least bounded ratonalty, especally n ts naïve verson. However, n certan crcumstances of uncertanty, hyperbolc countng can be reconcled wth ratonalty and does not necessarly generate tme nconsstency and reversal of preferences (Wetzman 998, Azfar 999, 22, Dasgupta and Maskn 22. Our purpose n ths paper s nether to decde on the emprcal controversy nor to fnd any further theoretcal justfcatons for assumng hyperbolc countng by ratonal ndvduals. Rather, takng HDH for granted and leavng asde the emprcal controversy, we explore ts mplcatons on optmal socal securty systems. The man result of ths paper s that the equvalence between an optmal pay-as-you-go ntergeneratonal transfers system (henceforth a PAYG system and an optmal funded We shall use the terms hyperbolc countng and quas-hyperbolc countng nterchangeably.
3 penson system, establshed n Sheshnsk and Wess (98, henceforth SW, does not hold n general, under hyperbolc countng. The model we analyze n ths paper s an extenson of the SW model. In order to enable analyss of the hyperbolc countng effect, we have added a thrd perod to the standard two pero model. We show that the equvalence between funded and PAYG penson systems, establshed n SW, hol only under exponental countng, (whch s a specal case of the hyperbolc countng. A socal securty system provdes perfect nsurance under PAYG structure, whereas n a funded economy bequests are random and depend on realzed longevty. The correlaton between bequests and longevty n a funded economy depen on the elastcty of the parents margnal utlty from bequests. A smlar result s obtaned for savngs. The savngs level s dentcal n the two economes only f effectve commtment devces exst n the PAYG economy. We also analyze the effect of government nterventon n prvate decsons (.e., ntroducng a compulsory penson scheme and show that under hyperbolc countng, such nterventon has dfferent consequences on the two types of economes. The theoretcal equvalence of PAYG and funded socal securty systems n varous theoretcal models was generally gnored n penson reforms carred out n western countres. The common drecton of all these reforms s a shft from a PAYG system to a funded system. However, the logc of ths shft and ts costs are controversal. It has been argued that one socal securty method has no advantages over the other (Orszag and Stgltz, 999. Our results mply that the ablty to rank these systems hol only under the lmtng assumpton of exponental countng. If people are hyperbolc counters then each penson method has ts own unque advantages and advantages. In the next secton we brefly descrbe the hyperbolc countng functon and the tme nconsstency that t causes n ndvdual decsons. Next, we present the famlar SW model modfed for the hyperbolc countng analyss usng a threeperod verson. The followng sectons nclude a number of propostons and a cusson about ther mplcatons for optmally-desgned penson systems. The fnal secton s a cusson and conclusons.
4 2. Quas-Hyperbolc Dscountng and Tme Inconsstency Assume an ndvdual who lves T pero ( T 3 and derves utlty from consumpton at each perod. The utlty functon of the ndvdual n perod t s u( c t and we make the standard assumptons regardng ths functon, namely ( ( u c >, u c <. Suppose that lfe-cycle utlty s a Von-Neuman t t Morgenstern, addtvely separable utlty functon. Namely, t s the sum of all counted one-perod utltes. The form of the countng functon and ts mplcatons are the core of our analyss. Assumng hyperbolc countng mples that the ndvdual lfe-cycle utlty s, ( { } T T t ( t = ( + β δ ( t= t V c u c u c t= Where < δ < and < β. A glance at ( reveals that standard exponental countng s actually a partcular case of hyperbolc countng, wth β =. The count factor s δ. When β <, the ndvdual assgns a relatvely hgher weght to current perod s consumpton than to that of future pero, as can readly be verfed by comparng the rato of the utltes between second and frst perod utlty, βδ, to the rato between any followng pero utlty for t, whch s δ. Equaton ( presents the optmzaton problem that an ndvdual faces at T t =. The soluton to ths problem beng a sequence, { ct }, whch correspon to * t= hs vew at ths pont n tme. However, at t =, for nstance, hs vew of the relatve weght of utlty of dfferent pero changes. In perod, the lfe-cycle utlty functon descrbng the ndvdual s preferences s (2 { } T T t ( t = ( + β δ ( t= t V c u c u c t= 2
5 and the optmal consumpton sequence from the frst perod s pont of vew s therefore { cˆ } T t t=. Tme nconsstency arses from the fact that frst perod s soluton mples u u * ( c * ( c = βδ and u perod s pont of vew mples u * ( ct + * ( ct u u = δ, t, whle the soluton from the second ( cˆ 2 ( cˆ = βδ and ( ˆt + ( cˆ u c u t = δ t 2. Thus, c * cˆ. In other wor, n each perod the ndvdual changes hs mnd regardng hs optmal lfe-cycle path of consumpton. The crtcal ssue n ths framework s the reversblty of decsons made n the past and the ndvdual s level of sophstcaton. If decsons were rreversble, a sophstcated consumer would consder ths when decdng on hs optmal level of consumpton at each decson node durng the lfe cycle. Irreversblty of past decsons seems to be a very mportant feature for socal securty systems, especally f ndvduals are tme-nconsstent. In a PAYG system, for example, f representatves of the old generatons control the ntergeneratonal transfers, nsuffcent savngs whle young mght result n smaller net bequests rather than smaller consumpton. Namely, f past decsons are reversble, future generatons wll have to pay the bll for nsuffcent old generaton savngs, whle old generatons wll contnue to smooth ther lfe-cycle consumpton. On the other hand, n a funded economy, past decsons seem less reversble. t t 3. The model In order to demonstrate the tme nconsstency, analyss of a hyperbolc counter s optmzaton problem requres at least a three-perod model. For the sake of convenence, we analyze the consumer s optmzaton problem under a funded system and under a PAYG system separately. In the next secton, we compare the results of the analyss of the standard model (wth exponental countng to the more general case of hyperbolc countng. Denote by c and c the consumpton of generaton ndvdual durng the frst and second pero, respectvely. Assume a three-perod overlappng-generatons model. In ths economy, all ndvduals of the same generaton have dentcal nstantaneous preferences represented by a utlty functon u( c, and future utltes
6 are counted usng hyperbolc countng. An ndvdual begns lfe wth an ntal wealth B nherted from hs parents, and derves utlty from bequeathng to G chldren. 2 Denote by the fracton of potental retrement perod that s actually realzed, ( generatons.. We assume that the trbuton of s the same for all Funded system In a funded economy the ndvdual has to decde durng the frst perod how much to consume, and how much to contrbute to a funded penson scheme and how much to save. So the optmzaton problem of a representatve hyperbolc counter wthn a funded economy n hs frst perod s, (3 sa, { ( βδ ( δ ( } V = max u c + E v c, + Gh B st.. c = w+ B s a Where β, δ are count factors, (, of consumpton utlty from bequests c gven, ( v, v, v v c s second perod utlty of the flow > < >, ( 2 B ( h, h h B s the ndvdual s > < whch occurs n the subsequent perod (three and hence counted by δ, a and s are penson contrbutons and savngs, respectvely and w s wages (obtaned only n the frst-perod and the same for all generatons and B s ntal endowment (bequests from the prevous generaton. However, due to the hyperbolc countng functon, V s not a contracton mappng and therefore has no fxed pont (Labson et al. 996. Solvng ths problem s possble by applyng a backwar nducton technque. Durng the thrd perod, the ndvdual has no decson to make. So we begn our analyss n the second perod n whch the consumer s optmzaton problem s: 2 The tmng of the chldren s brth on the ndvdual lfe-cycle s unmportant n ths model.
7 (4 c { ( + βδgh( B } max E v c, st.. Ra GB Rs c = + The frst order condton (F.O.C for an nteror soluton s: (5 ( ( ˆ E v c h B Wth solutons cˆ ˆ = c( s, a and B ˆ (,, ˆ, βδ = s a. Snce the objectve functon of the second perod s strctly concave, the solutons are unque, whch s also the case for all solutons n ths paper. 3 Insertng c ˆ and B ˆ nto (3 yel the hyperbolc consumer frst-perod optmzaton problem: (6 s, a { βδ ( ˆ δ ( ˆ } V = max u( c + E v c, + Gh B st.. c = w+ B s a The F.O.C are: (7 dv = u ( cˆ ( ( ˆ dcˆ ˆ ( ˆ + βδ E v ( c, δh B + δh B R = 3 In vew of our assumptons about the concavty of (, (, c, and also of h( B, the maxmzaton of (4 has a unque soluton. u c v c wth respect to c and
8 (8 dv da = u ( cˆ ( ( ˆ dcˆ ˆ ( ˆ R + βδ E v ( c, δh B + δh B = da Wth solutons sˆ, a ˆ and the correspondng c ˆ. Pay as You Go System The frst perod optmzaton problem of a hyperbolc counter n a PAYG economy s: (9 { ( βδ ( δ ( } V = max u c + E v c, + Gh B a st.. s c = w+ B s a Note that the contrbuton of the ndvdual to the PAYG system, a s gven whle the contrbuton of the next generaton, a, s endogenous. Ths reflects the assumpton that the "older" populaton determnes the taxes pad by workers. Agan, ths problem can be solved by usng backwar nducton. In hs second perod, the consumer s problem s: ( c, a { ( + βδ ( } max E v c, Gh B a st.. Ga GB Rs c = + whch yel the followng F.O.C: * * * ( E v ( c h ( B a, βδ =
9 * * (2 βδ E h ( B a = * * Denote the solutons by c ( s, a ( s and * ( B s. * * Now gong back to the frst perod, nsertng c ( s, a ( s and * ( yel: B s nto (9 (3 { ( βδ ( ( } * δ * * V = max u c + E v c, + Gh B a st.. s c = w+ B s a Whch has the frst order condton: (4 ( v( c, δh ( B a * * * * dc V * = u ( c + βδ E =. * s * * da + δ h ( B a R+ G Wth soluton * s and the correspondng c. * 3. Comparng the Two Systems under HDH In ths secton we show that assumng hyperbolc countng breaks the equvalence between funded and PAYG systems, establshed n Sheshnsk and Wess (98. Thus, the equvalence hol only n the specal case β =,.e. under exponental countng. Proposton
Bequests under a PAYG system are ndependent of, whle under a funded system bequests are random and dependent on. Proof: see appendx Proposton states that the equvalence of PAYG and funded systems, establshed by SW (98, s volated f ndvduals are hyperbolc counters, PAYG system solates both bequests and second perod consumptons from fluctuatons of longevty whle funded systems fals to do so. The ntuton of ths result stems from the well known fact that hyperbolc countng creates tme nconsstency whenever there s a tme gap between takng a decson and mplementng t and there are no commtment devces. In a funded economy, all decsons are made n the begnnng of the frst perod, but the executon of second perod consumpton and bequest s delayed one perod. Snce preferences are not tme consstent, ths delay opens the gate for reversals and correctons of past taken decsons. In a PAYG system, the decsons about second perod consumptons and bequests are all made smultaneously durng the second perod leavng no room for tme nconsstency caused by hyperbolc countng. When ndvduals count future utltes usng the conventonal exponental countng functon, no tme nconsstency arses and the two alternatve systems are equvalent. Snce by proposton bequests under a funded system wth hyperbolc counters are random and depend on, rsk averse ndvduals (regardng the well beng of ther descendants would try to lower the hazard rate over ther planed bequests by contrbutng enough to ther penson fund, ensurng that they wll not have to reduce bequest because nsuffcent penson benefts. Followng Kmball (99, we defne the relatve prudence coeffcent regardng welfare of future generatons as ( ˆ η B ˆ B h h ( ˆ B. We also defne ( ˆ ( ˆ ( ˆ ε B η B B ˆ B, the modfed coeffcent of relatve prudence, whch s Kmball's coeffcent multpled by the rato between the hazard rate and the sum under rsk. Proposton 2:
In a funded economy wth hyperbolc counters, (5 > Raˆ > a B cˆ < < ( ε ( ˆ ( b ε ( B ˆ Proof: See appendx. Proposton 2 states that as ndvduals are more prudent regardng the welfare of ther descendants, they wll nsure ther bequests by contrbutng more to penson fund, lowerng the probablty that they wll have to reduce planed bequest to fnance consumpton due to nsuffcent penson benefts. Proposton 3 provdes a necessary and suffcent condton for a hgher savng rate under a funded system. Proposton 3 Assumng hyperbolc countng (6 ( cˆ c * * * dc ˆ dc da sgn = sgn sgn = Proof: See appendx. Proposton 3 states that frst perod savngs of a hyperbolc ndvdual n a funded economy s hgher (lower than frst perod savngs of the same ndvdual n a PAYG economy f da * s negatve (postve. Ths dervatve reflects the exogenous nsttutonal arrangements of commtment devces wthn an economy. In the funded economy, the consumer decdes about both ŝ and â at the same tme (hs frst
2 perod, and there s no wthdrawal from that decson. In a PAYG economy, on the other hand, the consumer decdes about the decson about ( * * B a * s durng hs frst perod, whle postponng to the second perod. Therefore, the nsttutonal arrangements of commtment devces are crtcal n ths economy, and for ths very * da reason no wonder why we can not nfer sgn analytcally, as explaned n the proof of ths proposton, (see appendx. Put dfferently, a funded economy s equvalent to an economy wth effectve commtment devces, snce the consumer cannot reverse hs past decsons about savngs durng hs second perod. Thngs are qute dfferent n the PAYG system; the reversblty of past decsons depen on the poltcal system, and especally on the age trbuton of the rulng poltcans. If the poltcal system s ruled by old people, they may try to ncrease the PAYG penson benefts at the expense of young generatons. Sophstcated young ndvduals should take ths nto account. Namely, they should take nto account that ther frst perod plan regardng ( * * B a wll change when they reach ther second perod. If the nsttutonal arrangements wthn the economy establsh effectve commtment devces, then da * = and savngs levels wthn the two economes wll concde. As explaned n the proof for proposton 3 (see appendx, although s a farly plausble assumpton, we have no way of provng t. However, under ths assumpton t follows from (6 that a hyperbolc consumer n funded economy saves less than n a PAYG economy, although the funded system wth hyperbolc consumers provdes no nsurance for ntergeneratonal transfers. Ths means that f da * < da * <, the hyperbolc effect or the myopa s domnant. 4. Compulsory Penson Schemes Hyperbolc countng, sometmes referred to as myopa, s a common argument put forward by proponents of compulsory penson schemes n many economc and poltcal debates. Compulsory savngs means government nterventon n prvate
3 decsons, whch s n essence a volaton of ndvdual freedom. In democraces, such nterventon n prvate decsons s justfed f and only f negatve externaltes are proved. In the socal securty context, myopc low savngs rates lead to old age poverty whch has negatve externaltes on socety as a whole. Compulsory penson proponents clam that mandatory savngs s the perfect correctng tax on myopc or hyperbolc consumers to prevent ths negatve externalty. In ths secton we wll not go nto the poltcal, phlosophcal or moral aspects of bureaucratc nterventon n prvate decsons for the sake of the ctzens. Our man nterest here s to determne the economc consequences of such nterventon on the penson market, ether funded or PAYG. Government Interventon n the Funded Economy Suppose that the government sets a mandatory contrbuton for a penson fund of a > aˆ. Second perod optmzaton of the ndvdual becomes: (7 { ( + βδgh( B } max E v c, c st.. Ra GB Rs c = + The frst order condton s: (8 E v ( c h ( B. Implyng c = c ( s, a, B ( s, a,, βδ = Proposton 4
4 (A9 ( ( ( a c s, a > cˆ s, aˆ, c < cˆ ( (,, ˆ b B s a > B ( s, a, Proposton 4 states that compulsory addtonal savngs for pensons n the funded economy wll result n hgher second perod consumpton, hgher rates of savngs and larger bequests. The entre burden of the addtonal savngs s carred by frst perod consumpton. However, the randomness of bequests remans unchanged after government nterventon. In other wor, publc polcy wll not nsure ntergeneratonal transfers or second perod welfare by compulsory savngs to funded schemes, f people count future utltes usng hyperbolc countng functon. Proof: See Appendx. Government Interventon n a PAYG system Governmental nterventon n a PAYG system means settng the rate of ntergeneratonal transfers a from young to old generatons at each perod. Wthout loss of generalty, suppose that the government sets a compulsory ntergeneratonal * transfer of a > a. Proposton 5 Compulsory ntergeneratonal transfers n a PAYG economy cause bequests to be random, dependng on longevty. Proof: See Appendx. Corollary Governmental nterventon n a PAYG system reduces t to an effectvely funded system, wth the same equlbrum of consumpton and (random net ntergeneratonal transfers.
5 Proof: See Appendx. The ntuton here s qute clear. As mentoned above, the dfference between these two systems under hyperbolc countng stems from the fact that n a funded economy decsons regardng penson premums are taken durng the frst perod whle decson about penson benefts are taken durng the second, and ths tme gap s crucal when tme preferences are nconsstent. In a PAYG economy, on the other hand, both decsons are taken durng the same perod (the second. If the government sets a arbtrarly, t s no more a decson varable of the ndvduals and they have no decson to make about t n ther frst perod. They take both premums and benefts of the penson system as gven for the second perod, mplyng that ther optmzaton problem durng all pero s the same as n the funded economy. 5. Dscusson and Concludng Remarks The global trend durng the last decades of the twenteth century was a shft from PAYG DB penson systems to funded DC systems. The ratonale for ths shft s controversal (see Orszag and Stgltz 999. However, as Sheshnsk and Wess (98 show, assumng standard exponental countng provdes the postve queston of how ths shft affects the real varables of the economy wth a strct and clear answer. The two types of socal securty are equvalent and therefore, as long as the penson systems are optmally desgned, no real change s expected. In ths paper we show that f people assgn a hgher weght to current perod consumpton relatve to future pero as well as future generatons consumpton, ths answer s no longer vald. Whle a PAYG system operates wthn the economy very smlarly no matter what countng functon the ndvdual actually uses, provded that there are effectve commtment devces, the effect of the funded system and ts equvalence to the PAYG system depen on the countng functon and on ( Bˆ ε. In an economy wth hyperbolc counters and effectve commtment devces, a PAYG system mght be preferable to a funded system, because t neutralzes bequests from the randomness of longevty. However, ths neutralty s condtonal on
6 the refranng of the government from any attempt of nterventon n the penson market. Effectve governmental nterventon n a PAYG economy wth hyperbolc counters reduces t to the funded equlbrum, but wth random bequests and wthout the advantages of the funded system. As we emphaszed above, we have not attempted to solve the controversy surroundng hyperbolc countng, especally not the emprcal aspect. However, snce t s unreasonable to assume that the features of a utlty functon change by the shft from a PAYG system to a funded system; f future evdence shows that savngs and bequeathng patterns have changed because of the shft from one type of penson system to another, ths may also serve as an emprcal pece of evdence n favor of hyperbolc countng. 6. Appendx Proof of Proposton It follows from (2 that cov, = =, * * * * (A E h ( B a ( h ( B a Snce the optmal net bequest, ( * * B a B a, s monotone n and h <, t follows that * * must be ndependent of. Ths s a well-known result, snce Yaar (965 has shown that annutes, when chosen optmally, provde the consumer wth nsurance that solates hs economc well-beng from random shocks. The strkng fact here s that we do not have a condton lke (2 under the funded system, ndcatng that the funded system cannot nsure the well beng of the consumer, provded that the ndvdual s a hyperbolc counter. Formally, by subtractng (8 from (7 we obtan: ˆ ˆ (A2 ( ˆ dc dc δ R ˆ ( ˆ E v c h B E h B (, δ = (. da
7 Totally dfferentatng (5 wth respect to s and a gves: (A3 ( ˆ 2 βδ R ˆ ( ˆ βδ E v ˆ ( c, h B dc E h B + = G G ( ˆ βδ R ˆ ( ˆ βδ 2 2 E v ˆ ( c, h B dc E h B da + = G G Thus, (A4 ( dcˆ βδ R = ˆ E h B > G ( dcˆ βδ R = E h B > da G ˆ 2 βδ ˆ ˆ E v c, h B + < G. 2 Where ( ( It follows from (A4 that: dcˆ dcˆ βδ R E h B da G (A5 ( ˆ = ( From equaton (5 we have: ( ˆ, ( ˆ E v c δh B (A6 ( ˆ, ( ˆ ( ( ˆ = E v c βδh B + β δh B ( ˆ B = ( β δe h Insertng (A5 and (A6 nto (A2 we get:
8 β G (A7 ( ˆ ( ˆ ( ( ( ˆ β δ E h B E h B = E h B ( The rght-hand sde of (A7 s ( ˆ sde of (A7 yel cov ( ˆ h B, = cov h B,. Thus, settng β = on the left-hand. In other wor, β = reduces our model to the famlar exponental countng model of SW (98 and the equvalence between the two types of socal securty systems s restored. β <, than accordng to (A7 the other case n whch ( ˆ If cov h B, = s when ( ˆ E h B ( dcˆ dcˆ =, mplyng (by (A5 that = da, or da ˆ ˆ =. Namely, a full offset of savngs n one track as a result of ncreased savngs n the ˆ = possblty s excluded by proposton 2. QED. other track. However, E h ( B ( f and only f ε ( B ˆ =, but ths Proof of Proposton 2: Ra Suppose that c ˆ <. In ths case, as can be seen from (4, ˆ B depen negatvely on. Assume that ε ( B ˆ > d, or ( d ( h B h ( B h ( B ˆ = ˆ + ˆ >. Then, E h Bˆ > h Bˆ ( E =,, (A8 ( ( ( Therefore: ˆ > ˆ ( = (A9 E h ( B ( h ( B E ( Ra If, on the other hand, ˆ c >, then d namely ( d ( h B h ( B h ( B ˆ = ˆ + ˆ <, thus, ε ˆ <, B ncreases n. Assume ( B
9 ˆ < ˆ ( =. QED (a. (A E h ( B ( h ( B E ( It follows from (4 that the correlaton between Ra cˆ. Therefore, assumng that h ( B ˆ decreases n. If h ( B ˆ s constant n then h ( ˆ ( B ˆ ( E h B and depen on the sgn of s monotonc, B ether ncreases or B decreases n, therefore s never equal to zero, mplyng that f β <, the rght sde of Ra (A7 s also never equal to zero; ths means that ˆ c, mplyng that ( B ε ˆ. QED (b. Proof of Proposton 3: Frst perod savngs n the funded economy s determned by (7 and (8, and frst perod savngs n the PAYG economy s determned by (4, and a quck glance at these two equatons systems reveals that the only functonal dfference between them * * da da s G but E G = ; therefore, frst perod savngs n the funded economy dffers from frst perod savngs n the PAYG economy ff dc dc. By totally dfferentatng (A we obtan, * ˆ
2 (A E v c βδh B a dc βδ E h B a da G 2 * * * * * * (, + ( ( βδ R * * = E h ( B a G * * * * * E h ( B a ( dc E h ( B a da G + 2 As we already know (by proposton, ( * * B a ( * * as well as ( * * h B a h B a * * us to set E h ( B a ( R * * = E h ( B a ( G s ndependent of, therefore, are also ndependent of, and ths fact allows =. Thus, by the second equaton of (A, (A2 2 * * GE h ( B a * dc = * * da E h ( B a ( It also follows from (A that * * dc βδ da R = + G * * (A3 E h ( B a 2. G * * * Where = E v ( c, + βδh ( B a It follows from (A2 and (A3 that
2 (A4 dc da * and dc > * * * * E h ( B a ( < ε( B a < < > > da > < * * Notce that by (A4 and (A3 f da * = then dc dc =. Nevertheless, although * ˆ da * < s a plausble assumpton, we have no way to prove t. Therefore, the sgns of dc dc * * da and are unknown. Comparng (A3 to (A4, we obtan: (A5 * ˆ * dc dc da sgn sgn = Returnng to (7 and (4, t now follows that: (A6 ( cˆ c * * * dc ˆ dc da sgn = sgn sgn = and ths completes the proof of proposton 3. QED (a. Proof of Proposton 4 By pluggng (, c = c s a and ( h B nto (8 we get: (A7 dv da = u ( c dc R + βδ E ( v( c, δ h ( B + δh ( B = da a= a
22 dc Snce > da a = a equaton (8 mples ( ( ˆ h B < h B c (see (A4, t follows that (, ( ˆ v c < v c,, therefore holdng, thus ˆ B > B. The nevtable concluson s that < cˆ. Namely, frst perod consumpton bears the entre burden of addtonal compulsory savngs. Snce ε ( B unchanged after government nterventon. QED. stll hold, the randomness of bequests remans Proof of Proposton 5: Suppose that the government sets a = a and ths s no more a decson varable of ndvduals. In ths case the second perod optmzaton problem of the consumer s, (A8 c { ( + βδ ( } max E v c, Gh B a st.. GB Rs Ga c = + Frst order condton of ths problem s, (A9 E v ( c h ( B a, βδ = Implyng c ( s and B ( s. Notce that snce a s no more a decson varable of the consumer, we do not have here a condton lke (2, thus by ( bequests n ths PAYG economy wll be random, dependng negatvely on. QED Proof of Corollary 5 Gong back to the frst perod, nsertng c ( s, a and B ( s nto (9 yel:
23 (2 { ( βδ ( δ ( } V = max u c + E v c, + Gh B a s st.. c = w+ B s a Whch has the frst order condton: V dc * * = u ( c + βδ E ( v c, δh B a + δh ( B a R =. s (2 ( ( Wth soluton s and the correspondng c. Notce that equatons (A9 and (2 actually dentcal to equatons (5 and (7, respectvely, mplyng that under governmental nterventon both systems concde nto the same funded equlbrum wth random bequests. QED. References Azfar O., Ratonalzng Hyperbolc Dscountng Journal of Economc Behavor & Organzaton, 38 (999 245-252. Azfar O., Consstency of Choce and Non-consstent Dscountng, Center for Insttutonal Reform and Informal Sector, Workng Paper No. 257 (22. Besharov G. and Coffey B., Reconsderng the Expermental Evdence for Quas- Hyperbolc Dscountng, unpublshed (23. Dasgupta P. and Maskn E., Uncertanty, Watng Costs and Hyperbolc Dscountng, Mmeo, Faculty of Economcs, Unversty of Cambrdge (23. Damond P. and Kőszeg B., Quas-Hyperbolc Dscountng and Retrement, Journal of Publc Economcs, 87 (23 839-872. Hammond P., Changng Tastes and Coherent Dynamc Choce, Revew of Economcs Studes, 43 (976 59-73. Kmball M. S., Precautonary Savng In The Small and In The Large, Econometrca, Vol. 58 No. (99 pp. 53-73.
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