Efficient Subsidization of Human Capital Accumulation with Overlapping Generations and Endogenous Growth. Wolfram F. Richter and Christoph Braun

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1 Efficien Subsidizaion of uman Capial Accumulaion wih Overlapping eneraions and Endogenous rowh by Wolfram F. Richer and Chrisoph Braun T Dormund niversiy April 29 Firs Draf o be presened a he Conference in onour of Robin Boadway Queen s niversiy, ingson, Onario, 4-5 May 29 Absrac: This paper sudies second-bes efficien policies for human capial in an OL model wih endogenous growh. When young, individuals decide on educaion, saving and nonqualified labour. When old individuals supply qualified labour. rowh equilibria are inefficien in laissez-faire because governmen expendiures have o be financed by disorionary axes. The inefficiency is exacerbaed if selfish individuals exernalize he posiive effec of educaion on descendens produciviy. I is shown o be second bes o subsidize human capial accumulaion even relaive o he firs bes. This holds for selfish individuals a balanced growh and i equally holds for alruiss in he dynasy s firs generaion. eywords: OL model, endogenous growh, endogenous labour, educaion and saving, inergeneraional exernaliies, opimal axaion Address: Wolfram F. Richer and Chrisoph Braun, T Dormund niversiy, Deparmen of Economics, 4422 Dormund, ermany. Corresponding auhor: Wolfram.Richer@udormund.de, phone: , fax:

2 2. Inroducion The implicaions of adding endogenous educaion o he Ramsey problem of efficien axaion are by now well undersood. I is however irriaing o learn ha hey srongly depend on wheher he represenaive axpayer lives for finie or infinie periods. If he axpayer s planning horizon is infinie, he reason for employing disorionary linear axes urns ou o be weak. This poin has been made firs by Chamley (986) and Judd (985) and i exends o he model wih endogenous educaion as has been demonsraed by Bull (993), Jones, Manuelli, Rossi (993, 997), and Akeson, Chari, ehoe (999). Along a balanced growh pah no use should be made of disorionary axes. The problem one may however have wih his kind of resul is ha i is of lile help in idenifying differences beween human and nonhuman capial o be aken ino accoun by ax policy. The exisence of such differences is suggesed when solving he finie Ramsey ax problem. I hen becomes clear ha he efficiency of no axing saving is primarily a reflecion of he axpayer s ineremporal preference srucure. In paricular, savings should no be axed if he axpayer s uiliy is weakly separable beween consumpion and labour and homoheic in consumpion (Akinson and Sigliz, 972; Sandmo, 974). By way of conras, he design of efficien educaion policy is more a reflecion of he specific properies of he earnings funcion. If he earnings funcion is weakly separable in qualified labour supply and educaion and if he elasiciy wih respec o he laer is consan, hen he choice of educaion remains undisored in he second-bes opimum. In oher words, he reurn o educaion should equal is cos before axes and subsidies (Jacobs and Bovenberg, 28; Bovenberg and Jacobs, 25; Richer, 26). Furhermore, labour should be axed such ha qualified labour is disored less han nonqualified labour (Richer, 28). Sudying opimal ax design in he finie Ramsey problem is known o suffer from various shorcomings. The mos obvious ones are he focussing on a represenaive axpayer and he ignoring of heerogeneiy and informaional asymmery. Criical is also he ignoring of poenial reasons of capial marke or policy failure. The presen paper however ignores all such shorcomings. Is sole objecive is o conribue o he aemp of overcoming he saic naure of he finie Ramsey model. More specifically, he paper explores he implicaions for second-bes efficien policies when acknowledging he fac ha human capial accumulaion is a dynamic aciviy affecing he well-being of descenden generaions. If descenden generaions benefi by he human capial invesmens of preceding generaions, one would no necessarily expec non-disorionary educaion policy o be efficien. The differeniaion of

3 3 qualified and nonqualified labour shaping efficien axaion in he finie Ramsey problem raises addiional quesions when applied o a scenario in which qualified parens and nonqualified children coexis. ence i is no clear o wha exen he resuls of he saic Ramsey analysis survive in a dynamic framework wih overlapping generaions. The presen paper sudies second-bes efficien policies for educaion, labour and saving in a wo-period overlapping-generaions model wih endogenous growh. Individuals are assumed o supply nonqualified labour when young and qualified labour when old. They may be eiher perfec alruiss wih respec o descenden generaions or hey may behave selfishly. The implicaions of selfishness have been sudied before by Docquier, Paddison and Pesieau (26) for a framework in which he governmen is no consrained in he use of policy insrumens. The auhors show ha decenralizing he firs bes requires subsidizing educaion. The presen sudy goes beyond Docquier e al. (26) by endogenizing labour supply and by assuming ha he governmen can only employ linear axes and subsidies on labour and educaion. As i urns ou i is second bes along balanced growh o encourage educaion o such an exen ha human capial accumulaion is posiively disored relaive o he firs bes. This means ha he marginal social cos of human capial should exceed he marginal social reurn in he long-run second-bes opimum. This is a sriking resul. Sriking is he efficiency of disorion as such and even more sriking is he sign of he disorion. No surprising is he need o subsidize educaion relaive o he laissez-faire. This is so as he inergeneraional exernaliies of human capial invesmens have o be inernalized. A priori i is no obvious however why he accumulaion of human capial should be disored relaive o he firs bes. The sign of he disorion may wonder even more. Subsidizing educaion requires governmen revenue which in he model has o be raised by disorionary axes on labour and savings. Wih he inuiion of Lipsey and Lancaser (956/57) in mind one migh hypohesize ha i is second bes o provide insufficien incenives for human capial invesmen if labour has o be axed and if he level of comparison is he firs bes. The conrary is however rue. The reason is ha axes on labour have a negaive effec on educaion and growh and ha human capial policy has o compensae for his dynamic effec. More precisely, i will be demonsraed ha he srengh by which human capial accumulaion should be posiively disored in he long run increases in hree facors. One facor is he social marginal uiliy of income, he second facor will be called he dynamic cos of educaion and he hird facor is he gap beween he marginal reurn o capial and he rae of balanced growh. In oher words, he more binding he non-availabiliy of lump-sum axes is and he more cosly human capial accumulaion is and he more deficien growh is, he more should human capial accumulaion be encouraged

4 4 beyond wha appears o be firs bes. Addiionally i is shown o be second bes o disor qualified labour less han nonqualified labour. ence he resuls derived for he wo-period Ramsey ax model carry over o he overlapping-generaions model only in par. The srucural design of efficien labour axaion is preserved while he educaion policy differs srongly. Assuming alruisic individuals changes some conclusions bu no all. Alruiss inernalize he posiive effec ha educaion has on descendens produciviy. ence he need for governmen inervenion is reduced. owever, he second source of inefficiency modelled in his paper does no vanish. The second source is he need o employ disorionary axes for financing exogenous governmen expendiures. The implicaions for second-bes policy are shown o differ markedly beween he firs generaion and all descenden generaions. Wih respec o descenden generaions he following resuls are obained. Qualified and nonqualified labour should be axed uniformly and he accumulaion of human capial should no be disored. Such resuls srongly conras wih hose derived for he case of selfish individuals. Agains hese, he resuls obained for he firs generaion are more similar. The specific properies of opimal policy for he firs generaion depend on iniial values. Afer neuralizing he impac of iniializaion a case can however be made for subsidizing he human capial invesmen. The reason is he same encounered when individuals are assumed o be selfish. Taxing labour has a negaive effec on educaion and growh and human capial policy has o compensae for his dynamic effec. The unifying boom line for selfish and alruisic individuals is as follows. Alruism well reduces he need o subsidize educaion relaive o laissez-faire and alruism also implies ha descenden generaions should have non-disored incenives o accumulae human capial. The shor-run policy recommendaions for alruism however agree wih he long-run recommendaions for selfishness. Labour has o be axed and he resuling decrease in growh has o be compensaed by subsidizing human capial accumulaion relaive o he firs-bes. Wheher saving should be axed or no, primarily depends on assumpions made wih regard o consumpion preferences. The paper is srucured as follows. Secion 2 ses up he model of a wo-period overlappinggeneraions model wih endogenous growh. The firs-order condiions characerizing soluions of he planner s firs-bes maximizaion are derived. Secion 3 sudies he planner s problem when individuals behave selfishly and when only linear policy insrumens are available. Secion 4 sudies he same problem for individuals which are alruisic owards descenden generaions. Secion 5 summarizes.

5 5 2. The model and he planner s firs-bes problem Consider a represenaive individual of generaion. The individual is living for wo periods. Lifeime uiliy is given by C (, CL, L) where he argumens C, C L, L denoe consumpion and non-leisure in life-cycle periods zero and one. Non-leisure ime L is idenical wih second-period labour supply. By conras, only L is ime spen in he E marke while ime E is spen on educaion. The effec of educaion is o increase human capial and labour produciviy. is he sock of human capial inheried by generaion from he preceding one. is he sock buil up by generaion and effecive in s second life period. uman capial accumulaes according o he equaion [ E ( ) + δ ] =. ( μ β ) () ence i depreciaes a he rae of δ and i increases by he endogenous choice of E. The learning funcion ( ) E is assumed o display posiive bu diminishing reurns, >> wih ()=. The elasiciy of educaion, η E /, is hen posiive bu smaller han one. μβ is a Lagrange muliplier associaed wih he planner s problem we are abou o se up. There is a second sock variable o be inerpreed as (nonhuman) capial buil up by generaion in heir firs life period. I depreciaes a he rae of consan reurns o capial and effecive labour. The resource consrain is F (,( L E), L ) + ( δ ) δ. Producion displays = C + C + f E + + A ( β ) (2) where A denoes exogenous governmen expendiure. Obviously, human capial accumulaion is labour augmening. When aking parial derivaives use is made of he following shor-forms: F F, F F F, F (( L E) ) ( L ) I is suggesive o inerpre L E as nonqualified labour and L as qualified labour. The wo kinds of labour may be perfec or imperfec subsiues in producion. The reurn o

6 6 educaion is increased produciviy. Two kinds of cos are modelled. There is he cos of foregone earnings capured by F E and here are direc coss which for simpliciy sake are modelled as a linear funcion of learning ime and inheried human capial, fe. The planner maximizes β C (, CL, L ) = (3) in C, C L, L, E,, (=,,..) subjec o () and (2). The parameers,, L = L, = are exogenously given. The firs-order condiions are as follows: C =, = C + β, F C = L, F C = L, (4) F μ + δ = C/ C = C/ β C +, (5) = ( f + F ), (6) + β [ FL + F + ( L E ) + + f + E ] = + μ β [ + + δ ] μ + (7) Condiions (4) characerize efficien consumpion and labour choices. Condiion (5) characerizes efficien saving and efficien capial. Condiion (6) characerizes he efficien choice of E and (7) is he condiion characerizing he efficien choice of. Solving (6) for μ and insering ino (7) yields afer some sraighforward manipulaions he condiion characerizing he efficien accumulaion of human capial, + FL F + L + ( + F + f + ) E + f + F = [ F + δ ] [ + f + F + δ ] (8) The firs erm on he lef-hand side, FL, is he reurn o human capial accruing o generaion in he second period of life and he difference F + L + ( + F + f + E + ) is he f + F reurn accruing o individuals of he nex generaion in heir firs life period. + cos of human capial in period and f + F + + is he is he cos of human capial one period laer. The righ-hand side of (8) is he cos resuling from invesing in period insead of posponing

7 7 he invesmen o he nex period. When separaing erms indexed by from erms indexed by + he efficiency condiion (8) implies f + F FL [ F + δ ] = F δ L + ( F + + f + ) E + [ E + + ] < F + L + ( F + + f + ) E + [ ] <. η δ The firs inequaliy follows from E > E+ + while he second inequaliy relies on he regressiviy of he learning funcion, <. ence i is firs-bes ha generaion s cos, η + f + F [ F + δ ], exceeds generaion s reurn o human capial, FL. The difference is he posiive exernal effec on generaion + which has o be inernalized by firs-bes policy when individuals are selfish. Along a balanced growh pah L, and are consan while consumpion, oupu and L E boh ypes of capial all grow a he common rae of + δ g which, by assumpion, is sricly lower han he gross rae of reurn o capial, F + δ ( condiion of ransversaliy ). The marginal produciviies F, F (,) are homogeneous of degree zero. i A balanced growh pah requires marginal uiliies of consumpion Ci being consan in =,, for each,. This is guaraneed whenever he uiliy funcion is linear homogeneous in consumpion. In his case, (5) implies / β = F + δ > + δ g. (9) Evaluaed a a balanced growh pah, (8) simplifies o + ( +f)e = [ ( f + F FL F L F F δ ) ( δ ) ]. () The following analysis sudies he quesion of wheher i is second bes o provide or no o provide efficien incenives for human capial accumulaion. As we shall see, much depends on individual behaviour and he quesion of wheher individuals are perfec alruiss owards heir children or no. In he alruisic model also called he dynasy model individuals are assumed o maximize (3). In he oher case which is considered firs he represenaive individual is assumed o maximize own lifeime uiliy

8 8 C (, CL, L) () subjec o he own lifeime budge consrain. We sudy boh scenarios and we sar by analyzing efficien axaion in he sandard OL framework wih selfish individuals. 3. Opimal axaion in he sandard OL model wih selfish individuals The selfish individual represening generaion is assumed o maximize () in he six choice variables C, C L, L, E, π ω ( L E ) + π ω L + subjec o he lifeime budge consrain = π C + π+ C + πϕ E + ( π R+ π+ ) ( λ ) (2) = ( E ) [ ( E ) + δ ]. (3) In his opimizaion is reaed as an exogenous parameer. For each here are six firsorder condiions = π λ, C = π + λ, C ω C = L, ω C = L, (4) ω L C = ( ϕ + ω ) C, R + = π / π + (5) which can be used o subsiue for he five relaive prices ω, ω, ϕ, π / π, + R+ and he Lagrange muliplier λ. Afer subsiuing, he budge consrain (2) can be wrien as [ C i Ci + L i Li ] = L L / ( λ β ) (6) E which will ake he role of an Incenive (or Implemenaion) Consrain in he planner s second-bes problem. Because of L ( ϕ + ω) E = L C E, (7) he righ-hand side of (6) can be inerpreed as he privae value of he cos of educaion. This cos depends on various facors. As i urns ou, he dependence on E - measured by he marginal variaion in - is of paricular significance when characerizing second-bes

9 9 policy. Le us call he resuling marginal variaion he privae marginal cos of human capial accumulaion or, for shor, he dynamic cos of educaion. The formal definiion is C d MC [ d L L C E L ] = L C de ( + )[ d E E ] L = L C [ E " E ++ ]. (8) In order o guaranee ha C MC is posiive, he brackeed expression on he righ-hand side of (8) mus be posiive. Such posiiviy is ensured whenever ( ) E is isoelasic, i.e. if < η = E / = consan in. In his special case, E E " E ++ = δ η + δ >. In wha follows, C MC is assumed o be posiive. The planner maximizes (3) in C, C, L, L, E,, (=,,..) subjec o (6), (), and (2). The soluions are second bes in he sense ha hey have o fulfil he Incenive Consrain (6) in addiion o he firs-bes consrains () and (2). If lump-sum axes were available, he planner could ignore (6). Inclusion of (6) in he se of consrains implies ha he planner is resriced in he choice of policy insrumens. The resricion is however no an arbirary one. Quie o he conrary, implici in he derivaion of (6) is he assumpion ha he planner is no consrained in seing consumer prices ω, ω, ϕ, R+. This means in paricular ha labour income can be axed a differen raes over an individual s lifecycle. If such differeniaion is ruled ou by assumpion, he planner has o respec an addiional consrain which may have srong implicaions for he design of opimal axaion. See Erosa and ervais (22) for a discussion of his poin in an OL Model wihou endogenous educaion. To solve he planner s problem se W + λ { [ C i Ci + L i Li ] L L E / }. (9) When discussing he efficien axaion of saving, paricular focus is on uiliy funcions which are weakly separable beween consumpion and non-leisure, = ( V( C, C ), L, L ), (2)

10 wih a linear homogeneous nesed funcion V. Such uiliy funcions are known o have he aracive propery ha he privae and he social marginal raes of subsiuion in consumpion are equal. Remark: Assume weak separabiliy beween consumpion and non-leisure and assume linear homogeneiy of V. Then W C W = C C C. (2) The proof is sraighforward. For he sake of simpliciy he index is suppressed. W Ci = Ci Ci d dc {+ λ [ C j Cj + L j Lj] λ L L E / } j= i = + CjCi LjCi λ {+ [ Cj + Lj ] L j= Ci Ci LCi Ci E / } = + VV VLj VL λ {+ V + Lj L V j= V V E / } = consan in,. The firs quesion addressed is he one raised by efficien axaion of saving. The relevan firs-order condiions associaed wih he planner s problem are as follows: C, L WL : WC = = (22) F C, L WL : WC = + β = F (23) + By (2), + β [ F + δ ] = (24) : F + δ = β + W W = C = C C C. (22)

11 Proposiion : Assuming selfish behaviour, weak separabiliy beween consumpion and nonleisure and assuming linear homogeneiy of V, hen i is second-bes efficien no o disor saving. A sligh varian of Proposiion is obained when focussing on balanced growh. ence assume Li = Li (,) and E =E o be consan across ime and,, Ci (,) o grow a he common gross rae + δ = g, so ha = g, = g, Ci = g resuls. Assume furhermore ha uiliy is linear homogeneous in consumpion. W as defined by (9) is hen equally linear homogeneous in consumpion jus as and W while, W are funcions which are homogeneous of degree zero in consumpion. As a resul, he growh facor g cancels ou of condiion (22). Afer cancelling ou, he only variable carrying an Li Li C i Ci Ci index in W = W L C F is λ which herefore mus be consan along he balanced growh pah, λ = λ >. The same holds for =, C= C. The privae rae of reurn o saving, R + = π π + = C = β + β C hen equals he social rae of reurn o capial, F + δ. Proposiion 2: Assume selfish behaviour and o be linear homogeneous in consumpion. A a balanced growh pah i is second-bes efficien no o disor saving. Proposiions and 2 exend earlier resuls of Akinson and Sigliz (972), Sandmo (974), Akeson, Chari and ehoe (999) and ohers o he presen framework. Turn nex o human capial and educaion. We are going o prove ha i is efficien o encourage human capial invesmen along a balanced growh pah o such an exen ha he firs-bes level is exceeded. To show his, evaluae he firs-order condiions wih respec o E and : E : μ = L L λ [ + " ] E + ( f + F )

12 2 μ = f + F λ + " L L [+ E ] (25) λ L L : / λ β + L E 2 + L + + E + / + + β [ FL + F + ( L + E+ ) f E ] μ + β [ + + δ ] = μ (26) + Making use of (24) and (25) we obain [ FL + F + ( L+ E+ ) f E ] + f+ + F ( + + δ ) λ L + L+ + " E [ E ] ( + + δ ) f + F = [ λ + " L L (+ E E )][ F + ]. (27) δ Making use of (8), condiion (27) can be wrien in a more srucured form: [ FL + F + ( L + E + ) f E ] + + f+ + F+ + [ + λ + + C + MC + ] ( + + δ ) C f + F = [ λ C MC ][ F + ]. (28) C δ Seing Δ λ C allows one o resae (28): C λ + C MC ( F + δ ) C + MC + ( + δ ) + (29) + f + F Δ = f+ + F+ ( F + δ ) + ( + + δ ) FL [ F + ( L + E + ) f E ]. + + Comparing his condiion wih (8) shows ha Δ is he wedge by which human capial accumulaion should be efficienly disored. A posiive wedge sands for subsidizaion relaive o he firs bes. Condiion (29) invies o be evaluaed a a balanced growh pah. By

13 3 he reasons given in he proof of Proposiion 2, λ = λ >, =, =, C C MC = MC C C has o hold. Seing R= F + δ and evaluaing (29) a a balanced growh pah we end up wih Δ = λ Inerpre λ C C C MC (R g). (3) > as he social marginal uiliy of income measured in real erms. This facor is posiive if he Incenive Consrain is binding, λ >, which is he case if he non-availabiliy of lump-sum axes is a binding consrain. In his sense he facor measures he cos resuling from he non-availabiliy of lump-sum axes. C MC is he dynamic cos of educaion which is posiive by assumpion. Finally, R g is he growh gap. By he ransversaliy condiion (9) he gap mus be posiive as well. ence Δ is he produc of hree facors each of which is posiive. Proposiion 3: Assume selfish behaviour and o be linear homogeneous in consumpion. A a balanced growh pah i is second-bes efficien o subsidize human capial accumulaion relaive o he firs bes. The srengh of posiive disorion increases in (i) he cos resuling from he non-availabiliy of lump-sum axes, (ii) he dynamic cos of educaion, and (iii) he growh gap. This is a sriking resul for reasons explained before. A priori i is no clear ha human capial accumulaion should be disored along balanced growh while capial accumulaion should no be disored subjec o appropriaely chosen uiliy funcions. The sign of he efficien disorion is neiher obvious when aking he firs bes as he sandard of comparison. I is raher eviden and has been noed before ha he laissez-faire level of educaion is inefficien from he firs-bes perspecive. Wihou governmen inervenion selfish individuals exernalize he posiive effec of own educaion on descenden generaions welfare. The criical quesion herefore is in wha direcion second-bes policy should deviae from he firs bes if such a deviaion is efficien. Noe ha any revenue needed o subsidize he moneary cos of educaion has o be raised by disorionary labour axes. Wih he inuiion of Lipsey and Lancaser (956/57) in mind one could have hypohesized ha i is second bes o give negaive incenives for human capial accumulaion relaive o he firs bes if labour has o be

14 4 axed. The conrary is however rue. The inuiion is as follows. The ax on labour has a depressing effec on educaion. We speak of a depressing effec because i does no necessarily mean a saically inefficien choice of educaion. In saic analysis i may well be second-bes efficien o disor only labour bu no educaion (Jacobs and Bovenberg, 28; Bovenberg and Jacobs, 25; Richer, 26). Sill, he choice of educaion will be reduced when labour income is axed. In he saic second-bes opimum, educaion is reduced in he same proporion as nonqualified labour and even more han qualified labour (Richer, 28). In dynamic analysis his reducion slows down long-run growh. A dynamic inefficiency resuls he srengh of which increases in he facors lised by Proposiion 2. Educaion policy has o compensae for his effec. Wha can be said abou he efficien axaion of nonqualified labour relaive o qualified labour? We only analyze his quesion for balanced growh. Seing η = E / wihou assuming consancy of η, (9) can be wrien as W + λ { [ C i Ci + L i Li ] L L η /g}. (3) Noe ha he specificaion of W according o (3) is srucurally asymmeric in and. This indicaes ha qualified and nonqualified labour should be axed differenly. To be more specific evaluae (22) and (23). The former condiion saes WL + F =. (+ λ )[ L + F ] C W C L L = η λ { g L ( + ) LL F LC [ C + L ] i CiL i LiL F [ C i CiC + L i LiC] } = η λ { C i CiL + L LL +( ) L LL + b F C } g wih b η [ + L +( ) L LC ]/ C g C i CiC LC + = [ + +( η L FC a C i CiL L LL ) L LL b g L ] (32)

15 5 wih a λ /[+ λ + b λ ]. L Similarly, (23) W + Fg W = C (+ λ )[ L + Fg ] C = λ { η [ + ( + )] L L LL Fg LC g Fg [ C i CiC+ L i LiC] } [ C + L ] i CiL i LiL = η η λ { C i CiL + L LL + ( ) L L L + b Fg C g g L}. wih b η [ C i CiC + L LC + ( ) L LC ]/ g η η L+ Fg C = a [ C i CiL + L LL + ( ) L L L ( b + g ) L] (33) g C wih a λ /[+ λ + b λ ]. To make a clear case for differeniaed axaion of qualified and nonqualified labour, evaluae (32) and (33) for he uiliy specificaion V( C, C ) D( L) i (34) wih some linear homogenous funcion V. Then, = (,) and b i L + F = C λ " LD + (32 ) λ L+ Fg = C λ + λ [ " ( η ) LD g η g D ]. (33 ) C C Dividing (33 ) hrough by (32 ) and seing F τ, Fg τ L L F i = ( + τ ) ω, i i LD / D ν gives us " i i i i

16 6 τ τ = η η ( ) ν g g ν. (35) Inerpre he lef-hand side of (35) as he raio of second-bes opimal ax raes and ν i as he inverse of labour supply in period,. For η =, (35) is he familiar Inverse Elasiciy Rule. According o his rule wage axes should be se inversely proporional o he wage elasiciies of labour supplies / ν i. This rule is exended by (35) o cope for endogenous educaion. The effec of educaion is o reduce he ax on qualified labour relaive o he ax η on nonqualified labour. Jus noe ha η ( ) ν < ν. See Richer (28) who derives a g g similar rule for he saic framework. The deviaion from he Inverse Elasiciy Rule increases in η / g. ence he deviaion increases in he elasiciy of he learning funcion, η = E/, and in he share, / g = /( + δ ), ha newly formed human capial has in he periodic change in he sock of human capial. Proposiion 4: Assume selfish behaviour and o saisfy (34). On a balanced growh pah i is hen second-bes opimal o ax labour according o he Inverse Elasiciy Rule (35). The effec of endogenous educaion is o reduce he ax on qualified labour relaive o he ax on nonqualified labour. The reader may wan o learn how second-bes policy ranslaes ino ax and subsidy raes. owever, he finie-period Ramsey ax analysis only lends iself for an implici deerminaion of he policy raes. In he presen conex an explici deerminaion encouners even more difficulies han usual. Jus for he sake of illusraion, consider he special case in which he uiliy specificaion (34) and he Inverse Elasiciy Rule (35) hold. ence saving should remain unaxed. Denoe by σ he rae by which he moneary cos of educaion should be subsidized in second bes, f = ( + σ ) ϕ. The opimal se of raes τ, τ 2 and σ mus solve hree equaions. These are (35), he governmen budge consrain and he requiremen of subsidizing educaion efficienly. The laer means ha he exreme hand-sides of he following chain are equal: g FL + ( F + f )( E ) + Δ = F + f R R τ τ σ τ FL = [ F + f] + τ + σ (36)

17 7 The firs equaliy characerizes second-bes efficien educaion policy. I is derived from () afer correcing for he second-bes efficien wedge Δ. The second equaliy characerizes he opimal privae choice of educaion. I is derived from (5) and (4) afer seing F = ( + τ ) ω, f = ( + σ ) ϕ and afer making some sraighforward subsiuions. The i i i condiion suggess ha he need o subsidy he moneary cos of educaion and o ax nonqualified labour (relaive o qualified labour) increases boh in he second-bes wedge Δ and in he inergeneraional exernaliy of human capial accumulaion, g FL +( F + f )( E ). Noe however ha his parial analyical inerpreaion ignores he fac ha he efficien policy raes are joinly deermined by (35), (36) and he governmen budge consrain. 4. Opimal axaion in he OL model wih alruisic individuals The perfecly alruisic individual is assumed o maximize C (, CL, L)+ β + which by recursive subsiuion amouns o maximizing (3) in C, C L, L, E,, (=,,..) subjec o he human capial accumulaion consrain () and he dynasy s budge consrain, [ π+ ωl + πω( L E) ] = = [ πc + π+ C + πϕ E + ( π R+ π+ ) ]. ( λ ) (37) = The firs-order condiions are (=,,..): β = λπ, C β C λπ + =, ω C ω = L, C= L, (38) μ = ( ϕ + ω ) C, R + = π / π + (39) λπ + [ L ω + ω + ( L + E + ) ϕ + E + ] = The laer condiion implies β μ β + [ + + δ ] μ + (4) λ π [ ω L + ω ( L E ) ϕ E ] = = (4) + [ βμ β μ+ + ] =

18 8 = μ (39) ϕ + ω = C (4) Muliplying he budge consrain (37) hrough by λ and using (38), (39), and (4) o subsiue for λπ, λπ, ω, ω, wih + R+ in (37) yields he Incenive Consrain β C i Ci = B ( λ ) (42) = B {[ ( ) ] ϕ + ω ω L E } ϕe + C. Similarly, (38) and (39) can be used o subsiue for λπ+, ω+, ω, μ in (4) which leaves us wih (=,,..) L β [( L + E + ) + + L L ϕ + E + C+ ] { μ = (4) β [ + + δ ] μ + } = μ β μ + + = [ ϕ C L (39) ] β [ ϕ + C+ L+ ] + +. ( γ β ) (43) The planner maximizes (3) in C, C L, L, E,,, ϕ (=,,..) subjec o he Resource Consrain (2), he accumulaion consrain (), and he behavioural consrains (42) and (43). Noe however ha ϕ + (=,,..) only appears explicily in condiion (43). By conras, he planner s objecive funcion as well as he consrains (), (2), and (42) are independen of ϕ +. (43) can herefore be reaed as a relaionship defining he free variable ϕ +. ence he planner s problem is equivalen o he simplified version in which (3) is maximized in C, C L, L, E,, (=,,..), and ϕ subjec o (), (2), and (42). The same kind of soluion procedure has been used before by Akeson, Chari, and ehoe (999) and ohers before. We firs sudy hose firs-order condiions of he simplified planner s problem which are associaed wih variables which do no ener he Incenive Consrain (42) or which drop ou when making paricular assumpions. The opimizaion wih respec o hose variables is no affeced by (42) and should herefore remain undisored.

19 9 Proposiion 5: Assuming alruisic behaviour, hen i is second-bes efficien no o disor he accumulaion of human capial for all generaions (=,2,..) excep he firs. Proposiion 6: Assuming alruisic behaviour, weak separabiliy beween consumpion and non-leisure and assuming linear homogeneiy of V, hen i is second-bes efficien no o disor he accumulaion of capial for all generaions (=,2,...) excep he firs. Proposiion 7: Assuming alruisic behaviour, addiive separabiliy beween consumpion and non-leisure and assuming linear homogeneiy of V, hen i is second-bes efficien o ax qualified and nonqualified labour uniformly for all generaions (=,2,..) excep he firs. The proof of Proposiion 5 is raher sraighforward. Jus noe ha he variables E,, (=,2,..) do no ener he Incenive Consrain. Taking parial derivaives of he Lagrange funcion wih respec o hese variables and subsiuing for he Lagrange variables μ, yields he efficiency condiion (8) for =,2,... The proof of Proposiion 6 parallels he one of Proposiion and is herefore skipped. The proof of Proposiion 7 is as follows. Se W + λ C i Ci. Addiive separabiliy of implies W Li = Li (,). Taking parial derivaives of he Lagrange funcion wih respec o, L, L, yields (24) and W L + F = = + WL + β F (=,2,..). C C Taxes on labour are uniform if, and only if, F τ = τ F C = C β + C C (24) L / = F + (=,2,..) which is he condiion of efficien δ saving. The assumpions made for Proposiion 7 are such ha Proposiion 6 applies. ence saving should no be axed and a foriori labour should be axed uniformly (=,2,..). Wih regard o he lieraure Proposiion 6 can hardly surprise. This proposiion fully sands in he Chamley-Judd radiion. Proposiion 7 is less obvious and i even allows us o qualify he L

20 2 main resul of Erosa and ervais (22) saing ha i is generally opimal o differeniae labour axes across he individual lifecycle. The inuiive explanaion for his resul is ha labour supplied in he second life period is anoher commodiy as labour supplied in he firs period. owever Erosa and ervais assume selfish individuals and Proposiion 4 confirms heir resul. The qualificaion suggesed by Proposiion 7 is ha alruism removes he need o employ age-dependen labour axes for descenden generaions. Finally, Proposiion 5 is ineresing because i goes beyond he Chamley-Judd lieraure. I holds for arbirary uiliy funcions and wihou assuming balanced growh. I.e. Proposiion 5 is logically sronger han Proposiions 6 and 7. And i is also much sronger han he Educaion Efficiency Proposiion obained in saic Ramsey analysis. In he finie-period framework i is only second-bes efficien no o disor educaion if he earnings funcion is weakly separable in qualified labour supply and educaion and if he elasiciy wih respec o he laer is consan (Jacobs and Bovenberg, 28; Bovenberg and Jacobs, 25; Richer, 26). By conras, Proposiion 5 holds for funcions which need no be isoelasic. This reminds one of he Producion Efficiency Theorem of Diamond and Mirrlees (97). According o his heorem he allocaion of inermediae goods should no be disored in second bes given ha no pure profis accrue o he privae secor. This is jus wha holds in he presen model. Invesmen in human capial is modelled as an inermediae good in he sense ha i does no affec he Incenive Consrain for =,2,... Furhermore, no pure profi accrues o he privae secor. Jus noe ha because of consan reurns o scale in producion he only pure profi is income earned by he paren generaion of generaion zero. This income equals F ( L E) F. I does no show up in he dynasy s budge consrain (37). Because of Walras Law i necessarily accrues o he governmen budge. The Producion Efficiency Theorem is herefore applicable and Proposiion 5 can be considered o be a corollary. The governmen has o finance he exogenous cash flow of governmen expendiures A (=,,..). If he amoun of pure profi earned by he governmen is insufficien, disorionary axes have o be employed o balance he budge. In his case, he Incenive Consrain is binding and i canno be ruled ou ha i is efficien o disor he choice of educaion of generaion zero. This raises he quesion of how o design opimal human capial policy for generaion zero. As we are going o learn, he answer comes close o wha has been shown o be efficien in he world of selfish individuals. More precisely, a case is made for subsidizing he human capial invesmen of generaion zero. To show his we maximize (3) subjec o (), (2), (42), and (43). Taking parial derivaives of he Lagrange funcion yields he following resuls afer some sraighforward manipulaions have been made.

21 2 ϕ : γ = λ [ E / ] (44) : ϕ γ = γ [ E / ] (45) E : μ + F = (44) f λ ϕ + ω C " E [+ ] (46) E : μ f + F = (44),(45) λ " ϕ + ω E [ / ][+ E ] (47) C Making use of (44) (47) yields : f + F Δ = f + F ( F + δ ) ( ) + δ F L [ F ( L E ) f E ] (48) wih Δ λ C C MC ( F + δ ) λ C C MC [ E / ] ( + δ ) (49) ϕ + ω MC C " E [+ E ] (=,). (5) / The definiions of he symbols Δ and C MC are se such ha similariies wih (29) and (8) are sressed. Wihou making addiional assumpions, i is clearly difficul o sign Δ. In paricular, negaive values canno be ruled ou for appropriae choices of iniial values. To neuralize he impac of iniializaion assume C λ MC and C o be boh posiive and consan for =, and (5) F δ + > + δ. (52) This obviously implies posiiviy of Δ. Noe ha E E < / = η <.

22 22 Proposiion 8: Assume (5) and (52). Then i is second-bes efficien o subsidize he human capial invesmen of generaion zero. From a purely mahemaical poin of view Proposiion 8 comes close o a rivialiy. The assumpions (5) and (52) are so srong ha he resul is obvious. The rue significance of Proposiion 8 is however a concepual one. Proposiion 8 parallels Proposiion 3 and hus allows us o ell a unifying sory for selfish and alruisic individuals. Alruism well reduces he need o subsidize educaion relaive o laissez-faire. Alruism also implies ha second-bes ax policy for descenden generaions is more like firs-bes policy. The accumulaion of human capial should remain undisored and labour axes need no be differeniaed across he individual life-cycle. The shor-run policy recommendaions for alruism however fully resemble he long-run recommendaions for selfishness. Labour has o be axed and he resuling decline in growh has o be compensaed by subsidizing human capial accumulaion relaive o he firs-bes. Wheher saving should be axed is no a maer of selfishness or alruism. I primarily depends on assumpions made wih regard o consumpion preferences. 5. Summary The accumulaion of human capial may suffer from all sors of poenial inefficiencies. Mos of hem have simply been assumed away by he presen sudy. Such a procedure is, no doub, debaable. Criical is he ignoring of poenial reasons of capial marke or policy failure. Even more criical is he ignoring of individual heerogeneiy and informaional asymmery. Sill, he procedure is defended wih he objecive o sudy efficien axaion in Ramsey s radiion. More precisely, he paper aims a bridging he gap ha separaes he wo srands of Ramsey ax analyses which exis for he finie and he infinie planning horizons. Our knowledge of efficien human capial policy in Ramsey s radiion is largely shaped by incompaible resuls derived for saic and dynamic analysis. The resuls derived in dynamic analysis sugges ha educaion should no be disored in he long run jus as saving should no be disored in he long run. ence i seems as if no difference should be made beween human and nonhuman capial policies. By way of conras, saic analysis srongly suggess such differences. Wheher educaion should be disored or no appears o depend primarily on he quesion of how educaion affecs he individual s earning poenial. More precisely, only if he earnings funcion is weakly separable in qualified labour supply and educaion and if he elasiciy wih

23 23 respec o he laer is consan, should he choice of educaion no be disored by second-bes policy (Jacobs and Bovenberg, 28). On he oher hand, wheher saving should be disored or no appears o depend on he axpayer s consumpion preferences. More precisely, saving should no be axed if he axpayer s uiliy is weakly separable beween consumpion and labour/nonleisure and homoheic in consumpion (Akinson and Sigliz, 972). The model filling he gap beween finie and infinie Ramsey ax analysis is one wih overlapping generaions. The presen paper sudies human capial policy in such a model wih overlapping generaions and endogenous growh. There have been earlier aemps o do he same. In view of he presen sudy wo aemps deserve o be cied more han ohers. These are he papers by Akeson, Chari and ehoe (999) and Docquier, Paddison and Pesieau (26). The mos conspicuous differences o he presen sudy are he following ones. The focus of he presen sudy is on human capial accumulaion while he focus of Akeson e al. is on nonhuman capial. Their paper conains exensions o boh endogenous educaion and overlapping generaions bu i fails o inegrae he wo. The paper by Docquier e al. is one inegraing boh, endogenous educaion and overlapping generaions. owever, he paper is none on endogenous labour supply and second-bes axaion. The auhors assume he availabiliy of non-disorionary ax insrumens which he presen sudy does no. In a sense, he presen paper sars where Akeson e al. and Docquier e al. sop. I goes beyond Akeson e al. by inegraing endogenous educaion and overlapping generaions and i goes beyond Docquier e al. by endogenizing labour supply and by doing second-bes ax analysis. The presen paper sudies wo reasons of poenial allocaional inefficiency. One is he lacking availabiliy of non-disorionary ax insrumens. The oher is individual selfishness. Taxpayers are assumed o exernalize he posiive effec ha heir human capial invesmens have on he produciviy of descenden generaions. As sressed by Docquier e al. selfishness is he source of an inergeneraional exernaliy. I gives reason o subsidize human capial invesmens relaive o laissez-faire. Such subsidizaion however requires governmen revenues. In he framework sudied by Docquier e al. i is efficien o subsidize human capial accumulaion up o he firs-bes level where marginal social coss equal marginal social reurns. The resul assumes he availabiliy of non-disorionary ax insrumens. The key assumpion of he presen sudy however is ha no ax insrumens are available ha would allow he governmen o raise he revenue needed o subsidize educaion wihou creaing disorions. I hen urns ou o be efficien a a balanced growh pah o subsidize human capial accumulaion even beyond he firs-bes level. In oher words, i is efficien in he long run o combine posiive ax wedges in he labour marke wih a negaive ax wedge for human

24 24 capial invesmen. Firs-bes incenives for accumulaing human capial would be oo weak o compensae efficienly for he decline in long-run growh induced by labour axaion. The srengh of efficien posiive disorion is shown o increase in (i) he cos resuling from he non-availabiliy of lump-sum axes, (ii) he dynamic cos of educaion, and (iii) he growh gap. Furhermore, i urns ou o be efficien o ax labour such ha qualified labour is less disored han nonqualified labour. If axpayers are alruiss wih respec o descenden generaions, a clear reason for governmen inervenion disappears. The effec ha educaion has on descenden generaions produciviy is inernalized by alruiss. The only remaining inefficiency modelled in his paper is caused by he need o employ disorionary axes for financing exogenous governmen expendiures. As i urns ou all generaions excep he firs one should sill be given non-disored incenives for accumulaing human and nonhuman capial. Furhermore, labour should be axed uniformly across he individual life-cycle. The laer allows us o qualify he main resul of Erosa and ervais (22) who sress he need o employ age-dependen labour axes in second bes. Obviously, alruism removes he need. In view of he Chamley-Judd lieraure resuls suggesing non-disorionary axaion may no be oo surprising. Sriking is however he srengh of he resul concerning human capial accumulaion. While he oher resuls on non-disorionary axaion require special uiliy funcions, he resul on human capial accumulaion holds wihou any comparable qualificaion. One only has o assume ha no pure profi accrues o he privae secor. I is argued ha his resul on efficien educaion policy is bes inerpreed as a corollary o he Producion Efficiency Theorem of Diamond and Mirrlees (97). The resuls on non-disorionary axaion do no require removing all possible disorions. On he conrary, he labour supply of descenden generaions will be disored if he governmen has o finance exogenous governmen expendiures by relying on disorionary insrumens. The resuls on non-disorionary axaion do neiher exend o he dynasy s firs generaion indexed by zero in he presen paper. A more specific characerizaion of opimal policy for generaion zero is difficul as he specific feaures srongly depend on iniial values. Afer neuralizing he impac of iniializaion a case can however be made for subsidizing educaion relaive o he firs bes. The reason is he same idenified for he scenario wih selfish individuals. Taxing labour has a negaive effec on educaion and growh and educaion policy has o compensae for his dynamic effec. The unifying boom line for selfish and alruisic individuals is as follows. Alruism well reduces he need o subsidize educaion relaive o laissez-faire and alruism also implies ha descenden generaions should be given non-

25 25 disored incenives for accumulaing human capial. The shor-run policy recommendaions for alruism however agree wih he long-run recommendaions for selfishness. Labour has o be axed and he resuling decline in growh has o be compensaed by subsidizing educaion relaive o he firs-bes. Wheher saving should be axed is no a maer of selfishness or alruism. I primarily depends on assumpions made wih regard o consumpion preferences. References Akeson, A., V.V. Chari, and P.J. ehoe, 999, Taxing capial income: A bad idea. Federal Reserve Bank of Minneapolis Quarerly Review 23, 3 7. Akinson, A.B. and J.E. Sigliz, 972, The srucure of indirec axaion and economic efficiency. Journal of Public Economics, Bovenberg, A.L. and B. Jacobs, 25, Redisribuion and learning subsidies are Siamese wins. Journal of Public Economics 89, Bull, N., 993, When all he opimal dynamic axes are zero. Federal Reserve Board Working Paper 37. Chamley, C., 986, Opimal axaion of capial income in general equilibrium wih infinie lives. Economerica 54, Diamond, P. and J.A. Mirrlees, 97, Opimal axaion and public producion, I Producion efficiency, II Tax rules. American Economic Review 6, 8 27, Docquier, F., O. Paddison, and P. Pesieau, 26, Opimal accumulaion in an endogenous growh seing wih human capial. IZA Discussion Paper No. 28. Erosa, A. and M. ervais, 22, Opimal axaion in life-cycle economies. Journal of Economic Theory 5, Jacobs, B. and A.L. Bovenberg, 28, Opimal axaion of human capial and he earnings funcion. CESifo Working Paper No Jones, L.E., R.E. Manuelli, and P.E. Rossi, 993, Opimal axaion in models of endogenous growh. Journal of Poliical Economy, Jones, L.E., R.E. Manuelli, and P.E. Rossi, 997, On he opimal axaion of capial income. Journal of Economic Theory 73, Judd,.L., 985, Redisribuive axaion in a simple perfec foresigh model. Journal of Public Economics 28, Lipsey, R.., and. Lancaser, 956/57, The general heory of second bes. Review of Economic Sudies 24, -32. Ramsey, F.P., 927, A conribuion o he heory of axaion. Economic Journal 37, Richer, W.F., 27, Taxing human capial efficienly The double dividend of axing nonqualified labour more heavily han qualified labour. Ruhr Economic Papers 2. Richer, W.F., 28, Taxing educaion in Ramsey s radiion. CESifo Working Paper No Sandmo, A., 974, A noe on he srucure of opimal axaion. American Economic Review 64, 7-6.