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1 htt:// Postrit This is the acceted versio of a aer ublished i Joural of Public Ecoomics. This aer has bee eer-reviewed but does ot iclude the fial ublisher roof-correctios or joural agiatio. Citatio for the origial ublished aer (versio of record): Arosso, T., Cialai, C., Löfgre, K. (2012) Geuie savig ad the social cost of taxatio. Joural of Public Ecoomics, 96(1-2): htt://dx.doi.org/ /j.jubeco Access to the ublished versio may require subscritio. N.B. Whe citig this work, cite the origial ublished aer. Permaet lik to this versio: htt://ur.kb.se/resolve?ur=ur:b:se:umu:diva-47547

2 1 Geuie Savig ad the Social Cost of Taxatio * Thomas Arosso, Catia Cialai ad Karl-Gustaf Löfgre Deartmet of Ecoomics Umeå Uiversity SE Umeå, Swede Setember 2011 Abstract Followig the 1987 reort by The World Commissio o Eviromet ad Develomet, the geuie savig has come to lay a key role i the cotext of sustaiable develomet, ad the World Bak regularly ublishes umbers for geuie savig o a atioal basis. However, these umbers are tyically calculated as if the tax system is o-distortioary. This aer resets a aalogue to geuie savig i a secod best ecoomy, where the govermet raises reveue by meas of distortioary taxatio. We show how the social cost of ublic debt, which deeds o the margial excess burde, ought to be reflected i the geuie savig. By resetig calculatios for Greece, Jaa, Portugal, U.K., U.S. ad OECDaverage, we also show that the umbers ublished by the World Bak are likely to be biased ad may eve give icorrect iformatio as to whether the ecoomy is locally sustaiable. JEL classificatio: D60, H21, I31, Q56. Keywords: Welfare chage, ivestmet, savig, taxatio. * The authors would like to thak two aoymous referees, Tomas Sjögre, ad Magus Wikström for helful commets ad suggestios. Research grats from the Swedish Research Coucil are also gratefully ackowledged.

3 2 1. Itroductio Sice the 1970s, a theory of welfare accoutig has gradually evolved. Oe of the basic ideas behid the research i welfare accoutig has bee to rovide a coheret framework for measurig chages i welfare i a dyamic ecoomy, as well as uderstadig how the curret rocedures for atioal accoutig ought to be modified with this articular objective i mid. 1 I this aer, we revisit the relatioshi betwee caital formatio ad the subsequet welfare chage by resetig a measure of geuie savig for a secod best ecoomy where the ublic reveue set o evirometal olicy is raised by distortioary taxes. We argue below that such a measure is ot oly iterestig from a theoretical oit of view; it has also bearig o statistics of relevace for evirometal olicy frequetly ublished by the World Bak. The geuie savig is a idicator of comrehesive et ivestmet, i.e. the value of the et ivestmet i all caital stocks of relevace for society. As such, geuie savig does ot oly reflect the social value of et ivestmet i hysical caital (the measure of et ivestmet used i covetioal atioal accoutig); it also reflects the social value of chages i other caital stocks, such as atural ad huma caital. The remarkable feature with geuie savig is that it costitutes a exact measure of welfare chage over a short time-iterval. 2 Followig the 1987 reort by The World Commissio o Eviromet ad Develomet, it has also come to lay a iterestig role as a idicator of sustaiable develomet. The World Commissio wrote that develomet is sustaiable if it meets the eeds of the reset without comromisig the ability of future geeratios to meet their ow eeds (Our Commo Future, age 54). Oe ossible iterretatio is that sustaiable develomet requires welfare to be o-decliig. 3 This suggests, i tur, that the geuie savig is a local idicator of sustaiable develomet, where the emhasis o the word local is due to that we are measurig the welfare chage over a short time-iterval. 4 1 The semial cotributio to the theory of welfare accoutig is Weitzma (1976), showig how a welfareequivalet measure of et atioal roduct ought to be defied if the resource allocatio is first best. Arosso (1998, 2008) aalyzes the corresodig welfare measuremet roblem i secod best ecoomies, where the ublic reveue is raised by distortioary taxes. See also the literature reviews by Weitzma (2003) ad Arosso, Löfgre ad Backlud (2004). 2 Although Weitzma (1976) did ot attemt to aalyze geuie savig, it shows u i the roof of his mai result, i.e. we eed Weitzma s welfare measure to relate the idicator of welfare chage to the geuie savig. Stadard refereces for geuie savig are Pearce ad Atkiso (1993) ad Hamilto (1994, 1996). 3 This defiitio is used i Arrow et al. (2003). 4 See also Asheim (1994) ad Pezzey (1993), who show that a ositive value of geuie savig does ot give ay iformatio as to whether the curret level of utility or cosumtio is sustaiable forever.

4 3 Aother iterretatio of sustaiable develomet is that the curret (istataeous) utility level must ot fall short of the maximum level that ca be sustaied forever, i which case o-ositive geuie savig idicates that the curret utility level faced by the reresetative cosumer exceeds the maximum sustaiable level (Pezzey ad Toma 2002, Pezzey 2004). As such, the geuie savig has become a imortat statistic uderlyig the evirometal olicy debate, ad the World Bak regularly ublishes umbers for geuie savig o a atioal basis for a large umber of coutries. 5 However, the aroriate rocedures for calculatig the geuie savig have ot received sufficiet attetio. I fact, the calculatios that we have see either assume that the resource allocatio is first best, or that the resource allocatio is subotimal i the sese that society has ot reached the best ossible outcome give its objective ad costraits (due to uiteralized market failures). 6 To our kowledge, there are o studies dealig with the measuremet of geuie savig (or a aalogue thereof) i ecoomies where the resource allocatio is secod best otimal; a sceario that will arise if restrictios faced by olicy makers revet them from imlemetig the first best resource allocatio. This ga i the literature is somewhat surrisig cosiderig that the reveue raised by the ublic sector i real world ecoomies tyically ecessitates distortioary taxes, which are associated with a excess burde that may affect both the sig ad magitude of the welfare chage that the ecoomy exerieces durig a short time-iterval. Arguably, the riciles for measurig geuie savig ought to be modified accordigly; at least if the welfare ecoomic foudatio is to be take seriously. Therefore, the urose of this aer is to reset a aalogue to geuie savig i a secod best ecoomy, where the govermet raises reveue through a distortioary tax (istead of a lum-sum tax). Our study is based o a model develoed by Chamley (1985), which is a extesio of the Ramsey model i the sese of addig a ublic sector ad assumig that the ublic reveue is raised by usig a liear, yet time-varyig, labor icome tax. We show that the margial excess burde of taxatio affects the secod best aalogue to geuie savig via the accumulatio of ublic assets. Fially, we exemlify by adjustig the World Bak umbers for geuie savig ad show that eglectig the social costs of taxatio (as the World Bak does) may give rise 5 See also Hamilto (2010) for a overview of research o geuie savig. 6 See, e.g., Arosso ad Löfgre (1998) ad Löfgre ad Li (forthcomig).

5 4 to biased estimates of geuie savig ad, i some cases, alter our coclusios as to whether the ecoomy is locally sustaiable. 2. The Model The model reseted below cotais cosumers, firms ad a govermet. We start by describig the decisio roblems faced by agets i the rivate sector ad the cotiue with the olicy roblem facig the govermet. Followig much earlier literature, the secod best roblem will be described as a Stackelberg game, where the govermet is actig leader ad the rivate agets are followers. Cosumers ad Firms The model develoed i this sectio largely resembles the Ramsey-tye models used i earlier literature o welfare accoutig with the modificatio that the ublic reveue is raised by a labor icome tax. 7, 8 Followig the covetio i earlier literature, we assume that the ecoomy is oulated by a fixed umber of idetical cosumers ormalized to oe. The refereces are described by a time-searable utility fuctio. The objective fuctio facig the cosumer is rereseted by the reset value of future utility, t U(0) u( c( t), z( t), q( t)) e dt 0, (1) where c is the cosumtio of a rivate good, z leisure ad q the quatity of a ublic good decided uo by the govermet, while the arameter deotes the utility discout rate (i.e. the margial rate of time referece). The ublic good is a state-variable ad may be thought of as ublic caital that leads to higher evirometal quality (e.g., eviromet-friedly ifrastructure, ublic arks, ublicly rovided carbo-siks, etc.). This is clearly a somewhat aïve descritio of evirometal quality; by focusig solely o the ublic sector cotributio to such quality, it leaves out a umber of vital relatioshis betwee roductio, cosumtio ad damages to the eviromet. Yet, this simlificatio is aalytically 7 Addig aother distortioary tax will ot affect the ricial fidigs below. See Chamley (1986) for a dyamic reresetative aget model with liear taxes o labor icome ad caital icome. 8 Arosso (2008) uses a similar model to derive a secod best aalogue to Weitzma s (1976) welfare measure (i.e. a secod best aalogue to the comrehesive et atioal roduct) whe ublic reveue is collected through distortioary taxes, as well as aalyzes the role of ublic goods i welfare accoutig.

6 5 coveiet ad is of o ractical imortace for the qualitative relatioshi betwee geuie savig ad tax distortios, which is the mai focus i this aer. As a cosequece, we abstract from other asects of evirometal quality. The determiatio of the ublic good is discussed below. Leisure is defied as a fixed time edowmet, l, less the hours of work, l. The istataeous utility fuctio, u( ), is icreasig i each argumet ad strictly cocave. The cosumer holds two assets; caital, k, ad govermet bods, b, which are assumed to be erfect substitutes. If we defie writte as a k b, the asset accumulatio equatio ca be a( t) r( t) a( t) w ( t) l( t) c( t) (2) with a(0) a0, where w is the margial wage rate defied as w ( t) w( t)[1 ( t)], i which w is the gross wage rate ad the tax rate. The variable r is the iterest rate. The rice of the rivate cosumtio good has bee ormalized to oe. The cosumer chooses his/her cosumtio of the rivate good, c, ad hours of work, l, at each istat to maximize the reset value of future utility subject to equatio (2), the iitial coditio, ad a No Pozi Game coditio (which is a restrictio o the reset value of the termial asset). The cosumer also treats the factor rices ad olicy variables at each oit i time as exogeous. By usig the first order coditios, oe ca write the demad for the rivate good ad labor suly as fuctios of the et-of-tax wage rate, the margial utility of wealth ad the ublic good, resectively, 9 c( t) c( w ( t), ( t), q( t)) (3) l( t) l( w ( t), ( t), q( t)). (4) The margial utility of wealth obeys, i tur, the differetial equatio 9 Note that the curret value Hamiltoia imlied by the cosumer s decisio roblem ca be writte as (if the time-idicator is suressed) J u( c, z, q) a where the margial utility of wealth i curret value terms aears as the costate variable attached to wealth. Equatios (3) ad (4) are derived from the first order coditios u ( c, z, q) 0 ad u ( c, z, q) w 0. c z

7 6 ( t) ( t) ( t) r( t). (5) Fially, by substitutig equatios (3) ad (4) ito the istataeous direct utility fuctio, we obtai the istataeous idirect utility fuctio defied coditioal o the margial utility of wealth v( t) v( w ( t), ( t), q( t)) u c( w ( t), ( t), q( t)), l l( w ( t), ( t), q( t)), q( t). (6) Turig to the roductio side, we assume that idetical cometitive firms use labor ad caital to roduce a homogeous good uder costat returs to scale ad ormalize the umber of firms to oe. The roductio fuctio is give by f ( l( t), k( t)), ad the firm obeys the first order coditios f ( l, k) w 0 l ad f ( l, k) r 0 k. The Govermet The social welfare fuctio coicides with the objective faced by the reresetative cosumer. By usig the coditioal idirect utility fuctio reseted i equatio (6), the social welfare fuctio at time 0 ca be writte as t V (0) v( w ( t), ( t), q( t)) e dt 0. (7) Turig to the state-variables faced by the govermet, the accumulatio equatio for the ublic good is assumed to take the followig form: q( t) g( t) q( t) (8) where gt () is the cotributio to the ublic good at time t ad the rate of dereciatio. We ca thik of g as reresetig ublic exediture o ivestmet i eviromet-friedly ifrastructure or abatemet of the atural eviromet. The govermet uses the icome tax to fiace the cotributios to the ublic good, although it does ot ecessarily balace the

8 7 budget at each istat. If we write the uit tax o labor as for govermet bods is writte as w ww, the equatio of motio b( t) r( t) b( t) g( t) [ w( t) w ( t)] l( t). (9) Fially, by combiig equatios (2), (9) ad the zero rofit coditio, we ca derive the resource costrait f ( l, k) wl rk 0, k( t) f ( l( t), k( t)) c( t) g( t). (10) To simlify the otatio, we assume that f () measures outut et of caital dereciatio, which meas that the left had side of equatio (10) reresets the et ivestmet i hysical caital. Equatio (10) meas that outut is used for rivate cosumtio as well as rivate ad ublic ivestmet. The decisio roblem facig the govermet will be to choose the tax rate (or et wage rate) ad cotributio to the ublic good at each istat to maximize the social welfare fuctio reseted i equatio (7) subject to the state-differetial equatios (5), (8), (9) ad (10), as well as subject to the first order coditios for the rivate cotrol variables give by equatios (3) ad (4), ad the first order coditios of the firm (which defie the gross wage rate ad iterest rate by the margial roduct of labor ad caital, resectively). The reaso as to why equatio (5) aears as a state-differetial equatio i the govermet s decisioroblem is that the equatio of motio for the rivate margial utility of wealth is art of the ecessary coditios faced by the cosumer ad, therefore, a costrait that the otimal tax ad exediture olicy must fulfill. 10 The curret value Hamiltoia associated with the ublic decisio roblem ca be writte as (suressig the time-idicator for otatioal coveiece) 10 The resource allocatio must also obey iitial coditios for k ad b as well as a No Pozi Game coditio for b. As oited out by Chamley (1985), the govermet does ot face ay exlicit costrait o the iitial rivate margial utility of wealth, (0).

9 8 H v( w,, q) k q b (11) where,, ad are the costate variables (measured i curret value utility) attached to the state variables i the decisio-roblem faced by the govermet, i.e. the stock of hysical caital, the evirometal ublic good, the stock of govermet bods ad the rivate margial utility of wealth, resectively. The first order coditios are reseted i the Aedix. Here, we use these coditios to derive a measure of welfare chage. 3. Measurig Geuie Savig The covetioal aroach to measurig geuie savig is to add the value of chages i evirometal ad/or atural caital stocks to the et ivestmet i hysical caital, as well as addig the value of et ivestmet i other caital goods such as huma caital. I our simle model, which abstracts from huma caital, this suggests that we should defie geuie savig by addig the value of et ivestmet i the evirometal ublic good to the value of et ivestmet i hysical caital, i.e. kq. 11 We show below that this rocedure gives a correct measure of welfare chage if the resource allocatio is first best, while it does ot give a correct measure of welfare chage i the secod best framework addressed here. Defie the otimal value fuctio at time t as follows: ( st) V ( t) u( c ( s), z ( s), q ( s)) e ds t (12) where c c( w,, q ) ad z l l( w,, q ) are defied by equatios (3) ad (4). We use the sueridex 0 to deote secod best otimal resource allocatio. By totally differetiatig the otimal value fuctio rereseted by equatio (12) with resect to time, we obtai a measure of welfare chage over a short time-iterval 11 Although ivestmet i huma caital would affect the exact form of the geuie savig measure, addig huma caital to the model would ot affect the qualitative results reseted below for how the riciles of measurig geuie savig ought to be modified i a secod best ecoomy by comariso with the corresodig riciles i the first best. The welfare measuremet roblem associated with huma caital is addressed by Arosso ad Löfgre (1996).

10 9 0 dv () t u c t z t q t V t dt ( ( ), ( ), ( )) ( ). (13) To exlore the relatioshi betwee the right had side of equatio (13) ad the measure of geuie savig suggested above, i.e. kq, ad to be able to relate our study to earlier comarable literature (see the itroductio), we begi by evaluatig the welfare chage measure i a first best resource allocatio. We will the cotiue with the secod best aalogue to geuie savig. Secial Case: Geuie Savig i the First Best I terms of our model, the first best resource allocatio costitutes a secial case where ( t) ( t) 0 ad ( t) ( t) for all t. Such a allocatio would follow if the labor icome tax were relaced by a lum-sum tax to fiace the cotributio to the evirometal ublic good at each istat. By usig that V( t) H( t) at the otimal resource allocatio, 12 ad if we use the sueridex * to deote the first best (to distiguish it from the secod best), we obtai the familiar result * dv () t * * * * ( t) k ( t) ( t) q ( t) dt. (14) If alied to the model set out above with a two-dimesioal caital cocet, the right had side of equatio (14) is the covetioal geuie savig measure. I our model, the geuie savig is give by the sum of the value of et ivestmet i hysical ad evirometal caital. This aroach to measure the geuie savig is also cosistet with the aroach take by the World Bak; let be that they use a broader caital cocet tha we do (that also icludes huma caital). Geuie Savig i the Secod Best Model Let us ow retur to the more geeral secod best model set out above. As we show i the Aedix, by alyig the same rocedure as above, we ca derive the followig result: 12 The otimal cotrol roblem is time-autoomous, excet for the time-deedece of the utility discout factor.

11 10 Proositio 1. The welfare chage measure for the secod best ecoomy is give by 0 dv () t ( t) k ( t) ( t) q ( t) ( t) b ( t) ( t) ( t) dt. (15) If we follow covetio ad defie geuie savig as ( t) k( t) ( t) q( t), the the right had side of equatio (15) is iterretable as a geeralized measure of geuie savig. The geeralizatio follows because the social laer faces two additioal state variables here (i additio to k ad q); amely, the stock of ublic debt, b, ad the rivate margial utility of wealth,. The costate variable () t attached to the govermet debt at time t is iterretable as the egative of the margial excess burde at time t; it reflects that icreased govermet debt at reset ecessitates higher distortioary taxes i the future. If ( t) 0, as oe would ormally exect, the ituitio is that ublic debt (asset) accumulatio gives rise to a social cost (beefit) due to the distortios geerated by the tax system. Therefore, ublic debt or asset accumulatio affects the geuie savig i the secod best (which it does ot i the first best where 0 i real cosumtio uits, ). As oited out by Chamley (1985), the margial excess burde measured MEB ( t) / ( t) (0) / (0), is costat over time alog the otimal ath. 13 Otherwise, it would be ossible for the govermet to reduce the overall welfare cost of taxatio by chagig its debt olicy. We will retur to the margial excess burde below. The welfare effect of chages i the rivate margial utility of wealth, i.e. the fourth term i equatio (15), is also due to the aearace of distortioary taxatio, although for aother reaso. The tax system distorts the labor suly ad rivate cosumtio ad, therefore, also the ath for the rivate margial utility of wealth, causig it to differ from the ath for the shadow rice of hysical caital, () t. The associated welfare cost of this discreacy is catured by the variable () t. 14 To uderstad why chages i the rivate margial utility of wealth affect the welfare chage measure, recall that the ublic decisio-roblem is 13 This is see by solvig the equatios of motio for () t ad () t subject to trasversality coditios.

12 11 formulated i terms of demad fuctios ad a idirect istataeous utility fuctio, which are defied coditioal o the rivate margial utility of wealth. If evaluated i the first best, H / 0 (i which case 0 ) because the rivate cost beefit rule for resectively, would i that case coicide with the corresodig social cost beefit rule, whereas H / rivate ad social cost beefit rules. is geerally ozero i the secod best due to discreacies betwee the c ad l, If we measure the covetioal geuie savig i cosumtio uits, we ca rewrite equatio (15) as follows: GS [ k q]/, 0 0 dv ( t) ( t) 0 ( t) GS ( t) MEB b ( t) ( t) 0 dt () t. (16) The exressio withi square brackets of equatio (16) is the geeralized measure of geuie savig exressed i cosumtio uits. Sice 0 ( t) 0, it follows that welfare icreases over the short time-iterval ( t, t dt) if, ad oly if, the exressio withi square brackets is ositive. Equatio (16) costitutes the basis for the alicatio below. 4. Alicatio I this sectio, we exemlify by calculatig how the margial value of ublic debt will modify the umbers for geuie savig ublished by the World Bak. Our startig oit is the World Bak measure of geuie savig, which is defied by subtractig atural resource deletio ad damages from carbo dioxide emissios from the et ivestmet i hysical caital ad the addig educatio exeditures (which serve as a roxy for ivestmet i huma caital; let be that the roxy is somewhat misleadig, as the coectio betwee such exediture ad the future earigs of the ivestors is uclear). To be able to adjust the curret umbers for geuie savig, we make two simlifyig assumtios: (i) the resource allocatio is secod best otimal i the sese discussed above, 14 I a simlified versio of this model without the ublic good, Chamley (1985) shows that the variable is equal to zero at time 0 (due to that there is o iitial coditio o the rivate margial utility of wealth i the secod best roblem), while it is egative i a steady state.

13 12 ad (ii) the iterest rate is costat ad equal to the rate of time referece, so ( t) 0 for all t, i which case the fourth term o the right had side of equatio (15) vaishes. The secod assumtio is eeded because there is o way to estimate the value of chages i the rivate margial utility of wealth, give the data to which we have access. 15 Therefore, we augmet the umbers for geuie savig ublished by the World Bak by the value of the chage i govermet debt defied as the ublic deficit times the margial excess burde measured i cosumtio uits, where the latter is based o estimates i emirical literature. 16 To be more secific, we will subtract db() t MEB dt from the umber for geuie savig ublished by the World Bak. We cosider three differet umbers for margial excess burde; 0.1, 0.3 ad 0.5, resectively, which are well i lie with although i the lower rage of the estimates summarized by Jacobs (2009). 17 The results are reseted i Figure 1, which cotais the umbers ublished by the World Bak as well as the umbers followig the adjustmet metioed above. Figure 1 HERE Figure 1 resets the umbers for geuie savig (GS) ublished by the World Bak, as well as our geeralized measure of geuie savig (GGS), for Greece, Jaa, Portugal, U.K., U.S. ad OECD-average, for the eriod All umbers o which the curves are based are give i hudreds of U.S. dollars i 2009 rices ad measured er caita. 19 We make three 15 This is, of course, a restrictive assumtio. It may serve as reasoable aroximatio if the ecoomy is relatively close to a statioary equilibrium. 16 See Jacobs (2009) for a recet comrehesive literature review. 17 We realize, of course, that the studies surveyed by Jacobs are tyically based o models differet from ours. The mea value of margial excess burde amog the studies i his survey is Data for the geuie savig origiates from the World Bak ad were obtaied from the World Develomet Idicators (collected i the srig of 2011) at htt://databak.worldbak.org/dd/home.do, whereas data for budget surluses ad deficits were collected from the OECD Ecoomic Outlook 87 data base. We used the GNP deflator (UN statistics) to covert the omial umbers for geuie savig ad budget surluses/deficits ito real umbers. 19 We assumed away oulatio growth i Sectio 2, because such growth is ot imortat for our qualitative uderstadig of how the riciles for measurig geuie savig i a secod best ecoomy differ from the corresodig riciles that aly i a first best resource allocatio. I ractice, oulatio growth adds comlicatios to welfare measuremet, sice chages i the oulatio affect the welfare chage betwee two subsequet eriods (deedig o the form of the objective fuctio). We abstract from these comlicatios here.

14 13 broad observatios from the figure. First, eglectig the accumulatio of ublic debt may lead to the wrog coclusio as to whether the ecoomy is locally sustaiable. This is the case for the U.S. i 2008, where the covetioal geuie savig is ositive, whereas the geeralized geuie savig is egative if based o the umbers 0.3 ad 0.5, resectively, for the margial excess burde. Similar fidigs aly for the U.K. i 2009, Greece i ad Portugal i Secod, with the highest umber for margial excess burde, i.e. 0.5, which is i lie with the emirical evidece referred to above, ublic debt accumulatio may have a cosiderable effect o the geeralized geuie savig. Third, ad erhas eve more imortat, sice the covetioal geuie savig ad the budget deficit move together to some extet (e.g., the et ivestmets i hysical caital tyically fall ad budget deficits tyically icrease durig recessios), the covetioal geuie savig measure may be a oor idicator as to whe the ecoomy is at the risk of becomig locally usustaiable, i.e. the sigal that this statistic is desiged to give may come several years after which the geeralized geuie savig has tured egative. It is ecessary to exercise cautio i the iterretatio of the results i Figure 1. Oe reaso is, of course, that the World Bak umbers are ucertai, ad it is ot always clear that the measure used by the World Bak covers all imortat asects of the covetioal geuie savigs measure - as it ought to be defied i a world without tax distortios - or that all comoets are measured i the best way ossible. The estimates of margial excess burde are also subject to ucertaity, ad the aroriate value to be used may also differ betwee coutries. However, to arrive at a accurate icture of the savigs behavior of society, our results suggest, evertheless, that the savigs by the ublic sector may be of ractical relevace whe determiig whether or ot the ecoomy is locally sustaiable. Aedix For the aalysis to be carried out below, it will be more coveiet to use the reset value Hamiltoia tha the curret value Hamiltoia. We will, therefore, reformulate equatio (11) i reset value terms, through multilyig by exlicitly. We have e t, ad also write out the costrait fuctios

15 14 H ( t) H ( w ( t), g( t), k( t), q( t), b( t), ( t), ( t), ( t), ( t), ( t), t) t v( w ( t), ( t), q( t)) e ( t)[ f ( l( t), k( t)) c( t) g( t)] ( t)[ g( t) q( t)] ( t)[ r( t) b( t) g( t) ( w( t) w ( t)) l( t)] ( t) ( t)[ r( t)] (A1) where the subidex attached to the Hamiltoia ad costate variables deotes reset value, the factor rices wt () ad rt () are defied by the margial roducts of labor ad caital, resectively, while c( t) c( w ( t), ( t), q( t)) ad l( t) l( w ( t), ( t), q( t)) accordig to equatios (3) ad (4). The first order coditios for the cotrol variables are (suressig the time idicator) H w 0 ad H g 0 (A2) while the equatios of motio for the costate variables become H k, H q, H b, ad H. (A3) Derivatio of Equatio (15) By totally differetiatig equatio (A1) with resect to time, we obtai dh H dw H dg H dk H dq H db H d dt w dt g dt k dt q dt b dt dt H d H d H d H d H dt dt dt dt t (A4) where H / t v( w,, q)ex( t), sice the direct effect of time is due to the utility discout factor. Now, H / dk / dt, H / dq / dt, H / db / dt ad

16 15 H / d/ dt accordig to the defiitio of the Hamiltoia. Therefore, by usig the first order coditios i equatios (A1) ad (A2), equatio (A4) reduces to read dh dt v( w,, q) e t. (A5) For otatioal coveiece, we have suressed the sueridex 0 for secod best i equatio (A5). Solvig equatio (A5) for H ( ) T gives T s H ( T) H ( t) v( w ( s), ( s), q( s)) e ds t. (A6) If T aroaches ifiity, ad by usig the trasversality coditio lim H ( T) 0 T, 20 we obtai s H ( t) v( w ( s), ( s), q( s)) e ds t. (A7) Fially, trasformig equatio (A7) to curret value, through multilyig both sides by e t, we obtai V( t) H( t), where t H( t) H ( t) e is the curret value Hamiltoia ad ( st) V ( t) v( w ( s), ( s), q( s)) e ds t the social welfare fuctio. 21 Substitutio of V( t) H( t) ito equatio (13) gives equatio (15). Refereces Arosso, T. (1998) Welfare Measuremet, Gree Accoutig ad Distortioary Taxes. Joural of Public Ecoomics 70, Arosso, T. (2008) Social Accoutig ad the Public Sector. Iteratioal Ecoomic Review 49, See Seierstad ad Sydsaeter (1987, Chater 4, age 245). 21 A result aalogous to equatio (A7) was also derived by Arosso (1998), although his study focuses o Hamiltoia-based welfare measures ad does ot address geuie savig.

17 16 Arosso, T. ad Löfgre, K-G. (1996) Social Accoutig ad Welfare Measuremet i a Growth Model with Huma Caital. Scadiavia Joural of Ecoomics 98, Arosso, T. ad Löfgre, K-G. (1998) Gree Accoutig: What Do We Kow ad What Do We Need To Kow? I Tieteberg, T. ad Folmer, H. (eds) Iteratioal Yearbook of Evirometal ad Resource Ecoomics 1998/1999. Edward Elgar Publishig Limited. Arosso, T., Löfgre, K-G. ad Backlud, K. (2004) Welfare Measuremet i Imerfect Markets: A Growth Theoretical Aroach. Edward Elgar Publishig Limited. Arrow, K. J., Dasguta, P. ad Mäler, K-G. (2003) Evaluatig Projects ad Assessig Sustaiable Develomet i Imerfect Ecoomies. Evirometal ad Resource Ecoomics 26, Asheim, G.B. (1994) Net Natioal Product as a Idicator of Sustaiability. Scadiavia Joural of Ecoomics 96, Chamley, C. (1985) Efficiet Taxatio i a Stylized Model of Itertemoral Geeral Equilibrium. Iteratioal Ecoomic Review 26, Chamley, C. (1986) Otimal Taxatio of Caital Icome i Geeral Equilibrium with Ifiite Lives. Ecoometrica 54, Hamilto, K. (1994) Gree Adjustmets to GDP. Resources Policy 20, Hamilto, K. (1996) Pollutio ad Pollutio Abatemet i the Natioal Accouts. Review of Icome ad Wealth 42, Hamilto, K. (2010) Geuie Savig, Social Welfare, ad Rules for Sustaiability. I Arosso, T. ad Löfgre, K-G. (eds) Hadbook of Evirometal Accoutig. Edward Elgar Publishig Limited. Jacobs, B. (2009) The Margial Cost of Public Fuds ad Otimal Secod-Best Policy Rules. Erasmus Uiversity. Mimeo. Löfgre, K-G. ad Li, C-Z. (forthcomig) Itroductio. I Löfgre, K-G. ad Li, C-Z. (eds) Gree Natioal Accoutig ad Sustaiability. Edward Elgar Publishig Limited. Pearce, D.W. ad Atkiso, G. (1993) Caital Theory ad the Measuremet of Sustaiable Develomet: A Idicator of Weak Sustaiability. Ecological Ecoomics 8, Pezzey, J. (1993) The Otimal Sustaiable Deletio of Noreewable Resources. Uiversity College, Lodo. Pezzey, J. (2004) Oe-Sided Sustaiability Tests with Ameities, ad Chages i Techology, Trade ad Poulatio. Joural of Evirometal Ecoomics ad Maagemet 48,

18 17 Pezzey, J. ad Toma, M. (2002) Progress ad Problems i the Ecoomics of Sustaiability." I Tieteberg, T. ad Folmer, H. (eds) Iteratioal Yearbook of Evirometal ad Resource Ecoomics 2002/2003, Chelteham: Edward Elgar Publishig Limited. Seierstad, A. ad Sydsaeter, K. (1987) Otimal Cotrol Theory with Ecoomic Alicatios. Amsterdam: North-Hollad. Weitzma, M.L. (1976) O the Welfare Sigificace of Natioal roduct i a Dyamic Ecoomy. Quarterly Joural of Ecoomics 90, Weitzma, M.L. (2003) Icome, Caital, ad the Maximum Pricile. Harvard Uiversity Press. World Commissio o Eviromet ad Develomet (1987) Our Commo Future. Oxford: Oxford Uiversity Press.

19 18 Figure 1 Geuie Savig ad Geeralized Geuie Savig Greece Portugal GS GGS(0.1) GGS(0.3) GGS(0.5) GS GGS(0.1) GGS(0.3) GGS(0.5) Jaa USA GS GGS(0.1) GGS(0.3) GGS(0.5) GS GGS(0.1) GGS(0.3) GGS(0.5) UK OECD GS GGS(0.1) GGS(0.3) GGS(0.5) GS GGS(0.1) GGS(0.3) GGS(0.5)