OPTIMAL TIME-CONSISTENT FISCAL POLICY IN AN ENDOGENOUS GROWTH ECONOMY WITH PUBLIC CONSUMPTION AND CAPITAL

Transcription

1 OPTIMAL TIME-CONSISTENT FISCAL POLICY IN AN ENDOGENOUS GROWTH ECONOMY WITH PUBLIC CONSUMPTION AND CAPITAL Alfonso Novales Rafaela Pérez 2 Jesus Ruiz 3 This version: July 5, 204 ABSTRACT In an endogenous growh model where he fiscal auhoriy canno commi o policy decisions beyond he curren period, we explore he ime-consisen opimal choice for wo policy insrumens: he income ax rae and he spli of governmen spending beween uiliy bearing consumpion and producive services o firms. We show ha under he ime-consisen Markov policy he economy lacks any ransiional dynamics and here is local and global deerminacy of equilibrium. For empirically plausible parameer values we find ha he Markov-perfec policy implies a higher ax rae and a larger proporion of governmen spending allocaed o consumpion han hose chosen under a commimen consrain. As a resul, economic growh is slighly lower under he Markovperfec policy han under he Ramsey policy, wih growh under lump-sum axes being highes. The implicaion of our resuls is ha if he privae secor is aware of he governmen's inabiliy o pledge fuure policy decisions, hen he governmen should impose a slighly higher ax rae and devoe a higher share of public resources o consumpion, wih a relaively low cos in erms of growh. JEL classificaion: E6, E62, H2 Keywords: ime-consisency, Markov-perfec opimal policy, Ramsey opimal policy, endogenous growh, income ax rae, governmen spending composiion. Deparameno de Economía Cuaniaiva, Universidad Compluense de Madrid (Spain) and Insiuo Compluense de Análisis Económico (ICAE), anovales@ccee.ucm.es 2 Deparameno de Análisis Económico, Universidad Compluense de Madrid (Spain) and Insiuo Compluense de Análisis Económico (ICAE), rmperezs@ccee.ucm.es 3 Deparameno de Economía Cuaniaiva, Universidad Compluense de Madrid (Spain) and Insiuo Compluense de Análisis Económico (ICAE), jruizand@ccee.ucm.es The auhors would like o hank financial suppor from he Spanish Minisry of Science and Innovaion hrough gran ECO , he program of suppor o Research Groups from Universidad Compluense de Madrid and BSCH, he Xuna de Galicia hrough Gran 0PXIB30077PR and he Fundación Ramón Areces hrough is program of Research Grans in Economics.

2 . Inroducion The relevance of ime consisen policies sems from he fac ha he governmen has no incenive o change is policy once privae agens have made heir decisions condiional on he policy announcemen. Unforunaely, he difficuly in solving he ime-consisen Markov policy opimizaion problem has generally led academic research ino he characerizaion of he more limied Ramsey opimal policies. The laer assume commimen and are hence subjec o poenial deviaions by he governmen from he previously announced policy rule. The same echnical difficuly also explains ha mos research on ime-consisen opimal policies has been done in exogenous growh environmens. As main examples, Origueira (2006) and Klein, Krusell and Rios-Rull (2008) consider an sylized exogenous growh model, wih leisure and public consumpion in he uiliy funcion, o characerize he opimal ime-consisen ax policy under wo differen game designs. Klein, Krusell and Rios-Rull (2008) consider a game in which he governmen is a dominan player ha akes he opimal reacion of privae agens as given when deciding he opimal policy. Origueira (2006) compares he resuls obained under he srucure in Klein, Krusell and Rios-Rull wih hose from an alernaive design of he game in which he governmen and privae agens make heir respecive decisions simulaneously, characerizing he behavior of he economy along he ransiion o he opimal seady-sae. These auhors consider alernaive fiscal srucures, always wih a single insrumen: eiher a single ax levied on oal income, a single ax on capial income or a single ax on labor income. Marin (200) follows he same game srucure as Klein, Krusell and Rios-Rull (2008), exending he analysis o he simulaneous consideraion of differen ax raes for capial and labor income and solving for he opimal ime consisen choice for boh fiscal insrumens. A furher exogenous growh analysis is done by Azzimoni e al. (2009), who characerize he Markovian ax rae raised on oal income when used o finance public invesmen. However, for he analysis of opimal axaion i is essenial o overcome he wo limiaions menioned above, by describing how o characerize he opimal ime consisen fiscal policy under endogenous growh. Endogenous growh models no only allow for a more plausible represenaion of acual economies, bu also for explicily aking ino accoun he effec of fiscal policy on he rae of growh. This is crucial when analyzing he growh effecs of producive governmen spending, as in he seminal papers by Barro (990), 2

3 Cazzavillan (996) or Glomm and Ravikumar (997). Recenly, Jaimovich and Rebelo (203) propose an endogenous growh model wih heerogeneiy in enrepreneurial abiliy in suppor of he empirical evidence on he highly non-linear effecs of axaion on growh: when he ax raes are low or moderae, marginal increases in he ax rae have small growh impac whils, when ax raes are high, furher ax hikes have large, negaive impac on growh, due o he disincenives o invesmen and innovaion. The wo menioned exensions, ime-consisen policy in endogenous growh framework, have been considered by Malley e al. (2002), who characerize he Markov ax policy in an endogenous growh economy where he governmen raises ax revenues on oal income, using he proceeds o finance public consumpion and producion services. However, heir seup is sill resricive in wo aspecs: i) he spli of governmen spending beween consumpion and producion services is exogenously given, and ii) privae agens are supposed o have a logarihmic uiliy funcion and physical capial is supposed o fully depreciae each period. Under hese parameric resricions, he Ramsey policy is no subjec o a ime consisency problem and i coincides wih he Markov perfec soluion, a resul ha we show laer on. 4 In our analysis we dispose of hese wo addiional limiaions: Firs, we consider an economic environmen wih a CRRA uiliy funcion defined on privae and public consumpion, wih incomplee depreciaion of capial. Second, we incorporae an endogenously ime-varying spli of governmen spending beween public consumpion and producion services. We show ha a ime consisen opimal policy exiss and i is described by he opimal choice of boh, he income ax rae and he spli of public spending beween consumpion and producion aciviies. We prove ha he dynamics of he model can be characerized by he raio of producive services provided by he governmen over privae capial. Under he Markov soluion his raio is always on he Balanced Growh Pah. Addiionally, we numerically show ha here is no indeerminacy of equilibrium and hence, he Markov soluion lacks any ransiional dynamics. Under his more general economic framework, when comparing he opimal Markovperfec and Ramsey policies, we find ha: i) he income ax rae is higher under he ime consisen policy, since he Markov governmen canno inernalize he disorionary effecs of he curren ax on he level of invesmen underaken in previous periods (as in Origueira (2006), in a neoclassical growh framework), ii) he proporion of public resources devoed 4 Azzimoni e al. (2009) also show his resul for an exogenous growh economy. 3

4 o consumpion is higher under he Markov governmen han under he Ramsey governmen, since he former only commis o curren policies, hereby giving prioriy o curren consumpion, wih an immediae effec on uiliy, raher han o producion aciviies, whose effecs on welfare will mainly ake place in fuure periods, and iii) as a resul, economic growh is slighly lower under he Markov-perfec policy han under he Ramsey policy, wih he growh rae under lump-sum axes being he highes. The implicaion is ha a governmen ha is aware ha sociey knows is inabiliy o pledge fuure policy decisions should impose a slighly higher ax rae and devoe a higher share of public resources o consumpion, wih a relaively lower implied rae of growh. 2. The model economy We assume in wha follows ha populaion does no grow, and also ha he economy can be described by represenaive agens: firm, household and governmen. We furher assume full employmen. The represenaive firm maximizes profis subjec o a echnology ha produces he single consumpion commodiy. The sock of privae capial, K, ogeher wih producion services provided by he governmen, i p,, are used ogeher wih he labor force, L, as producion inpus in a echnology: Y BK ( L i ) p,. In line wih Barro (990) and Cazzavillan (996) we assume here a flow of public services of rival naure, and hence i is he quaniy of he public good assigned o each firm he relevan variable in he privae producion process. The represenaive firm pays rens rk wl o households for he use of privae capial and labor, solving each period he saic profi opimizaion problem: Max BK ( L i ) r K w L. K, L p, We assume ha labor is inelasically supplied by he household, and we normalize i o : L,. Markes for producion inpus are compeiive. A each poin in ime, opimaliy condiions imply ha inpu prices are equal o heir marginal produc: p, r B K / i, w ( ) B K / i i. p, p, 4

5 Apar from he producion services o firms, i p,, he governmen also provides consumpion services o households, g. We denoe by η he proporion of he proceeds from income axes ha are used each period o finance public consumpion services, g, he remaining ax revenues being used o pay for producion services, i p,. The governmen rk w g i, where budge consrain is, p g rk w, () ip, ( ) rk w. (2) From he governmen budge expendiure rules (), (2), and he opimaliy condiions for he compeiive firms we ge, i g BK i, p, p, ( ) BK ip,, so ha producion services are provided according o, while consumpion services are, / ip, ( ) B K (3) g ( K ;, ) B ( ) K, (4) / / / and equilibrium real ineres raes and real wages become, uiliy, / ( ;, ) ( ), r r K B (5) / ( ;, ) ( ) ( ). w w K B K (6) The represenaive household maximizes his/her life-ime discouned 0 Uc (, g), defined over privae and public consumpion, c, g, subjec o a fla ax rae τ on oal income. 5 They know he curren values of τ and η, and expec fuure governmens o follow policies ( K ) and ( K ). The ypical household solves he problem: c, k} k, K ; ; ; ; Max U c, g ( k, K ; ; ) (7) given k 0, and subjec o he budge consrain, 5 We also assume ha consumpion services provided by he governmen are of rival naure, so he argumen in he uiliy funcion is he per-capia level of he public good. 5

6 c k ( ) k ( ) w( K;, ) r( K;, ) k, (8) where k denoes he level of capial chosen by he household and K he economy-wide per capia sock of capial. The soluion leads o a consumpion funcion ( K ;, ) saisfying he Euler equaion, 6 U ( K ;, ), ( K ;, ) U ( K ;, ), ( K ;, ) c c / B ( )/ ( ). (9) Wih homogeneous households and firms, we have in equilibrium: K k, and all he variables in he model can be regarded eiher as aggregae per capia or in individual erms. Using (5), (6) and (8), he sock of capial k can be seen o evolve over ime according o: ( )/ / k ( ) k ( ) ( ) B k ( k ;, ). (0) Subsiuing (3) in he producion funcion, oupu is given by / ( ) y B k. As a consequence, in he compeiive equilibrium allocaion, i) he raio of producion services o oupu, i, / y is equal o ( ), an exension of he p framework in Barro (990), and ii) he raio of privae capial o oupu, k / y, is a funcion of ( ) and srucural parameers α and B. In line wih Barro (990), he consan reurns o scale in privae capial and public producion services, joinly wih he fac ha he complemenary public inpu expands in a parallel manner o he privae capial, guaranees he exisence of endogenous growh in our model economy. A poenial problem wih opimal policy design under endogenous growh is he possibiliy of having indeerminacy of equilibria. In ha siuaion, equilibrium rajecories are undeermined, being dependen on he iniial value of some conrol variables. Policy recommendaions emerging from economic models may hen lack any real meaning. Since his model shares some characerisics of he AK-family of models, i may also lack any ransiional dynamics, which would be very relevan for he characerizaion of opimal policy as well as for welfare evaluaion. 6 F Along he paper we denoe parial derivaives by Fv. v 6

7 Along he paper we argue 7 ha in he Markov and Ramsey soluions and, as well as he raios of privae and public consumpion o privae capial, c / k, g / k, are consan from he iniial period. This has wo implicaions: he model lacks any ransiional dynamics and here is no indeerminacy of equilibria, wih he economy being on he single balanced growh pah from he iniial period onwards. 3. Opimal policy 3. The ime-consisen opimal policy We use he same equilibrium concep as Klein, Krusell and Ríos-Rull (2008) and he governmen-moves-firs case in Origueira (2006). 8 We consider a dynamic game played by a sequence of governmens, each one of hem choosing curren period policies on he basis of he sae of he economy, defined by he sock of privae capial. Hence, each governmen chooses he curren ax rae τ and he proporion of revenues used o purchase public consumpion, η, before he household decides on consumpion and savings. When making opimal policy choices, he governmen knows he household decision rule ( k ;, ) ha describes he reacion of he household o policy decisions. Acing as a leader, he governmen chooses he curren ax rae and he spli of public resources aking as given he policies followed by fuure governmens and aking ino accoun ha reacion of he household o he policy choices, as follows: V k Max U k ;,, k ;, V ( k ) [P], } where k;, and k are given by (4) and (0). Vk and V k denoe he value funcion for he curren governmen and he coninuaion value funcion, respecively. Alhough hey will be reaed as differen funcions when characerizing opimaliy condiions in wha follows, in equilibrium he wo funcions will be he same. Proposiion. The ime consisen policy corresponding o he Markov equilibrium is he soluion o he se of Generalized Euler Equaions (GEE): 7 We have analyical proofs of hese resuls for he case of logarihmic uiliy and full depreciaion of capial, and numerical argumens for he general case. 8 Which is also used in Krusell and Rios-Rull (999), and Krusell, Quadrini and Rios-Rull (996). 7

8 Uc U g U c U g, (, ) k (, ) k and Uc U g Uc Uc k U U g g k, k,, k (, ) k ( ),, ( ) ( ) ( ) where ( )/ / Proof.- See Appendix. B. () (2) Equaion () is a condiion relaing he opimal choice of he wo policy insrumens a a given poin in ime, while equaion (2) characerizes he opimal ineremporal choice of income ax raes. From (0) we ge he size of he reducion in ime invesmen from an increase in axes is: k k / (, ) k. Hence, he lef hand side a () gives he change in uiliy produced by a ax increase, per uni of crowded-ou invesmen. This is wha Origueira (2006) calls oday s marginal value of axaion. By a similar argumen, he righ hand side a () is he change in uiliy from an increase in he share of resources devoed o public consumpion, per uni of crowded-ou invesmen. The opimal choices of he wo policy insrumens mus saisfy he equaliy beween hese wo marginal effecs on uiliy. The lef hand side a (2) is again he marginal change in uiliy per uni of crowded ou invesmen implied by a decrease in he ax rae. Lower axes a + simulae invesmen, and an addiional uni of capial a + has a direc effec on uiliy of U U hrough is effec on privae and public consumpion and an indirec effec c k g k hrough is impac on ime +2 capial sock, k k 2 (, ) k, which needs o be appropriaely discouned. The oal effec is given by he square bracke a he righ hand side of (2). I shows ha he change in uiliy per uni of crowded-ou invesmen a ime implied by a marginal change in he opimal ax rae mus be equal o he discouned change in uiliy resuling a ime +. 8

9 Definiion.- A Markov-Perfec equilibrium is a se of funcions ( k ;, ), ( k ), ( k ) and V( k ) such ha: i) Given governmen rules (3) and (4), ( k ;, ) solves he Euler equaion (9) subjec o he law of moion of he sock of capial (0), ii) ( k ), ( k ) saisfy condiions (3), (4), he law of moion of he sock of capial (0), as well as he Generalized Euler Equaions () and (2); and iii) V( k ) is he value funcion of governmen obained as a soluion o [P]: V( k ) U k ; ( k ), ( k ), k ; ( k ), ( k ) V( k ). 3.2 The Ramsey policy As usual, we define he benchmark Ramsey equilibrium as he soluion o an opimal-policy problem where he governmen can commi o fuure policies. The Ramsey opimal policy is hen he soluion o he problem of maximizing he ime aggregae, discouned uiliy of he household, subjec o he equilibrium condiions (4), (9) and (0) as consrains: Max c, g,,, k} 0 subjec o : g ( ) B k c c U c, g U U, k ( ) k (, ) k c. [P2] B where ( )/ /, ( ) ( ). In Appendix 2 we characerize he firs order condiions and he balanced growh pah for he Ramsey problem. The Ramsey policy akes ino accoun he opimal reacions of privae agens. However, i is ime inconsisen, since once privae agens adjus heir decisions o he announced economic policy i will be opimal for he governmen o change policy. Given he complexiy involved in characerizing opimal policy under lack of commimen, aenion has ofen been resriced o Ramsey policies, in spie of heir wellknown limiaion of assuming commimen on he par of he curren governmen on fuure periods. I is herefore imporan o evaluae o wha exen he Markov-perfec fiscal policy differs from he Ramsey policy in our seup. We will perform such analysis in Secion 6. 9

10 4. An analyical soluion: logarihmic uiliy and full depreciaion of privae capial We consider in his secion he special case of logarihmic preferences ha are separable in privae and public consumpion, U c, g lnc lng, ogeher wih full depreciaion of privae capial. The wo assumpions ogeher allow us o obain an analyical characerizaion of he ime consisen opimal fiscal policy ha we can compare wih he Ramsey soluion as well as wih he allocaion ha would be obained under lump-sum axes. Under his uiliy funcion, he compeiive equilibrium allocaion is characerized by he sysem: k (, ) k c, c (3) (, ) c Proposiion 2. Under full depreciaion of privae capial and a logarihmic uiliy funcion, he compeiive equilibrium allocaions are given by: k (, ) k, (4) c (, ) k. (5) Proof. Plugging in he previous sysem (3) a guess for he funcional form for he compeiive equilibrium allocaion as: k A k, i is easy o show ha A=. Expressions (4) and (5) for g, c allow us o compue he parial derivaives ha ener ino he Generalized Euler equaions ()-(2), o find an analyical soluion o he ime consisen opimal policy problem. The nex se of resuls shows ha under he Markov and Ramsey soluions he model lacks any ransiional dynamics and here is no indeerminacy of equilibria, wih he economy being on he single balanced growh pah from he iniial period onwards. Proposiion 3. Under full depreciaion of privae capial and a logarihmic uiliy funcion, separable in privae and public consumpion, he opimal ime-consisen fiscal policy saisfies: 0

11 Proof.- See Appendix 3. M M (6) Corollary.- Under he opimal Markov policy he raio of producive public services o privae capial is consan for all. i Proof.- Using (6) in (9), we ge: k ( ) B,. p, / / Proposiion 4.- Under full depreciaion of privae capial and a logarihmic uiliy funcion, separable in privae and public consumpion, i) There is no local indeerminacy of equilibria, ii) The economy lacks ransiional dynamics, iii) The opimal Markov policy is: M M M ( ),, (7) M ( ),. (8) ( ) Proof-. See Appendix 3. Noice ha he opimal spli of resources beween public consumpion and producive services is well defined, aking values beween 0 and α, while he opimal income ax rae is always beween -α and one. We now characerize he opimal allocaion of resources in erms of he raios of c g privae and public consumpion o he sock of privae capial:,. These raios k k mus remain consan along he balanced growh pah. Proposiion 5. The opimal allocaion of resources under he Markov-perfec opimal policy is given by: M 2 M k M M M ( ) / (, ) B, k M M c M M M ( ) / (, ) B, k

12 M M M M g M M M / (, ) B. M k Proof. Their expressions can readily be obained from (4), (4) and (5). The hree following corollaries can be readily shown from (7) and (8): Corollary 2. When public consumpion does no ener as an argumen ino he uiliy funcion (θ=0), he Markov-perfec opimal ax rae coincides wih ha in Barro (990):. In ha siuaion, public resources are fully devoed o producion. Corollary 3. The Markov-perfec opimal ax rae converges o he Barro ax as he discoun rae approaches, wih public resources again being fully devoed o producion. Corollary 4. i) The proporion of public resources devoed o public consumpion increases wih θ and α, and i decreases wih ρ; ii) he opimal ime consisen income ax increases wih θ, and i decreases wih α and ρ. As expeced, he proporion of public resources devoed o consumpion increases wih he relaive imporance of public consumpion in he uiliy funcion. I also increases wih he oupu elasiciy of privae capial. A more producive privae capial, relaive o public producion services, allows for a higher share of public resources being devoed o consumpion. Turning he argumen around, he more producive is he public inpu relaive o privae capial, he more ineresing i is o allocae public resources o producive aciviies raher han o consumpion. The share of public resources dedicaed o consumpion decreases for a larger. We hen end o value fuure consumpion almos as much as curren consumpion and i becomes more ineresing o shif resources o he fuure by increasing producive services. As public consumpion is more appreciaed by consumers for higher values of θ and lower values of ρ, i is appropriae o raise higher ax revenues o finance ha componen of public spending. On he conrary, a high elasiciy of privae capial, α, leads he privae secor o allocae more resources o invesmen, and axes can be lower. 2

13 4. Comparing Ramsey and Markov policies under logarihmic uiliy and full depreciaion of privae capial The following proposiion shows ha, for his special case, he Ramsey and Markov policies coincide. Proposiion 6. Under a logarihmic uiliy funcion and full depreciaion, he opimal Ramsey policy, becomes: R ( ), R ( ). ( ) Proof: See Appendix 3. The income ax and he proporion of public resources devoed o public consumpion under he Ramsey policy coincide wih he values obained under he imeconsisen policy, so he properies analyzed in Proposiion 4 and Corollaries 2 o 4 for he Markov-perfec opimal policy apply o he Ramsey policy as well. The equaliy of soluions arises because under a logarihmic uiliy and complee depreciaion of physical capial he Ramsey policy is ime consisen, a resul shown by Azzimoni e al. (2009) in a neoclassical growh model. 5. Opimal ime-consisen fiscal policy under CRRA preferences and incomplee depreciaion of privae capial The Generalized Euler condiions () and (2) should incorporae he consumpion decision rule of privae agens, which is characerized as he soluion o he Euler equaion (9) of he compeiive equilibrium. Unforunaely, i is no possible o find he analyical soluion o (9) in general, and ha precludes us from obaining an analyical characerizaion of he ransiion owards he balanced growh pah. ( ) c g Assuming a CRRA uiliy: Uc (, g), 0 compeiive equilibrium becomes, he Euler condiion of he c g c g B ( ) ( ) /, 3

14 So ha he relevan sysem of equaions for,,,,, wrien in raios, becomes: Euler condiion of he compeiive equilibrium: law of moion of physical capial: ( ) ( ) ( ) / B, (9) (, ) (20) he raio of public consumpion o capial, from (4): ( ), (2) / ( )/ / B and he wo Generalized Euler equaions: ( ),, ( ) ( ) ( ) (, ) ( ) k. k k (, ) (22) (23) Noe ha o obain (22) and (23) we have used C k ; C k ; G k ; G k, and he parial derivaives: ; ; ( ),, ; ;, ( ) ha emerge from (20) and (2). The righ-hand side a (9) involves values a ime + of policy variables,,, and raios of decision variables o he sock of privae capial,,. Each one of hese 4

15 mus be a funcion of he single sae variable in he economy, 9 he righ-hand side a (9) as a funcion Fk ( ). ( ) ( ) k, so ha we can hink of F( k ). (24) We can now characerize he opimal Markov policy in he more general se up considered in his Secion. We sar by showing ha he relaionship beween he wo policy insrumens is he same we found under logarihmic preferences and full depreciaion of privae capial. As shown in Appendix 4, he funcion Fk ( ) cancels ou in he Generalized Euler equaion () ha relaes boh policy choices and, as a consequence, i does no play any role in he characerizaion of he relaionship beween and. Proposiion 7.- The ime-consisen opimal choice of he wo policy insrumens saisfies: Proof.- See Appendix 4.,. (25) Again, opimal income ax raes will be above, whereas he opimal proporion of public resources devoed o consumpion will always be below α. We now argue ha his economy lacks any ransiional dynamics. Taking he resul from Proposiion 7 o (3) implies ha he raio i / k is consan, he oupu o capial raio ( / ) ( ) is also consan, and so is he real rae of ineres. Under he / ( )/ y k B Markov policy, his economy shares he same characerisics of sandard AK-models. These resuls also imply ha he raios, / k 0, / k 0. p are no funcions of k, so ha Hence, equaions (20), (2) and (25) allow us o wrie equaions (9) and (23) as a nonlinear dynamic sysem in, : ( ) ( ) ( ), (26) 9 They will be funcions of he sae k if here is ransiional dynamics. Laer on, we will show ha his economy lacks any ransiional dynamics so ha, parameers., and end up being jus funcions of srucural 5

16 where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) / B,, ( ) ( ),. ( ) / B (27) The nonlineariy of his sysem forces us o analyzing he poenial equilibrium local indeerminacy in his model numerically. The dynamic properies of his sysem can be analyzed hrough he wo eigenvalues of is linearized version. An eigenvalue below would sugges an indeerminaion of equilibrium since in ha case, we would need an iniial condiion for eiher or χ in order o compue he ime series for,. Any arbirary choice would yield a valid Markov equilibrium hen producing a siuaion of indeerminacy of equilibria. Tha would in urn generae ransiional dynamics, as he rajecory followed by he economy would depend on he iniial choice for eiher or χ. On he oher hand, having boh eigenvalues greaer han one would imply ha he only sable soluion is obained wih and χ being consan over ime:,,, wihou indeerminacy of equilibria. We have numerically compued such eigenvalues for wildly differen parameerizaions, obaining always boh eigenvalues above one, even for empirically implausible parameer values. Lacking an analyical proof, our numerical analysis suggess ha here is no indeerminacy of equilibria, wih,,, and hence,,,, he economy being a each poin in ime on is balanced growh pah. 6

17 5. Solving for he Markov equilibrium Since he opimal policy insrumens, and he consumpion o capial raio ( k;, ), are consan over ime, we can compue he Markov equilibrium along k he balanced growh pah. Evaluaing he Euler equaion (9) along he balanced growh pah and using (25), we ge: (, ) /( ( )). (28) Taking his expression o (20), we ge:, (, ) (, ) /( ( )). (29) wih parial derivaives and : ( ) ( ) (, ), ( ) (30) ( ) (, ). ( ) ( ) (3) Hence, equaion (27) becomes ( ) ( ) ( ) ( ) ( ) ( ). (32) M M M M Finally, he Markov equilibrium,,, is obained as he soluion o he sysem (25), (28), (29), and (32). The sysem can only be solved numerically and he following secion is devoed o analyze is properies. Under all parameerizaions considered, he sysem has been shown o have a single soluion, 0 suggesing ha he equilibrium is globally deermined. 0 When solving he nonlinear sysem of equaions, we have ried very differen ses of iniial condiions, always reaching he same soluion shown in he Tables. 7

18 6. Comparing he Ramsey and Markov soluions in he general case Le us now compare he Markov and Ramsey soluions beween hemselves, as well as wih he allocaion of resources ha would be achieved by a benevolen planner using lump-sum axes, which is characerized in Appendix 5. We will use g i P p, as a y P g measure of he size of he public secor in he planner soluion and we will use g i for he composiion of public expendiures. Boh of hem will be used in he graphs and ables we presen below. Le us now examine he values aken by he main variables in he economy along he balanced growh pah under he hree alernaive fiscal policies: i) he planner s policy under lump-sum axes, ii) he Ramsey policy and iii) he ime-consisen policy, all of hem under he more general seup, wih a CRRA uiliy funcion and incomplee depreciaion of privae capial. Unforunaely, our resuls are no readily comparable wih hose in he lieraure because numerical resuls are usually derived using a logarihmic, separable uiliy funcion, whereas our resuls correspond o general CRRA uiliy funcions, and also because of our consideraion of endogenous growh. The Markov equilibrium is obained as explained in secion 5.. As shown in Appendix 2, he soluion o he Ramsey problem [P2] is characerized by a sysem of 8 dynamic equaions in,,,,,,, ha allows us o compue he balanced growh 2 3 R R pah for he Ramsey policy (, ) as well as he implied allocaion of resources, R R R characerized by (,, ) and hree mulipliers,,, 2 3 p,. Tha sysem is made up only by conrol variables, wih no paricipaion of any sae variable. Hence, in he absence of local indeerminacy of equilibrium, he only possible soluion is ha conrol variables say on he balanced growh pah (BGP) from he iniial period, wih no ransiion. Under incomplee depreciaion of privae capial, he choice of parameer values: 0.4, 0.20, 0.99, 0.0, B , when generaing annual daa lead o sensible properies of he Markov soluion. Parameer values are sandard in he lieraure for annual daa excep for, which is chosen so ha he raio of public consumpion o privae For he parameerizaions used in he paper we have numerically checked he absence of local indeerminacy of equilibria. 8

19 consumpion for he Markov soluion is in line wih daa for he poswar US economy (g/c=0.25). For insance, for 2, we ge a raio of public o privae consumpion around ( )( ) 0.25, an annual growh rae =.5%, and a gross real ineres rae: /.03. The value chosen for is consisen wih a broad concep of capial ha includes boh physical and human componens, as i is commonly esablished in endogenous growh models wih public producion services and privae capial (see Cazzavillian, 996). As o he elasiciy of oupu wih respec o public capial, -α, we se a value which is in line wih previous lieraure: Azzimoni e al. (2009) akes a benchmark elasiciy of 0.25, bu he range of values varies significanly across auhors beween he 0.03 esimaed by Ebers (986), and he 0.39 esimaed by Aschauer (989). Figure shows values for he main variables in he economy under he hree equilibrium conceps as a funcion of he risk aversion parameer, σ. Over he whole range of values considered, he opimal income ax increases wih risk aversion. I always falls beween 20% and 30%, being higher under he Markov-perfec policy han under he imeinconsisen Ramsey policy. The proporion of public resources devoed o consumpion, relaive o producion, is also increasing in σ, saying beween 6% and 32%. I is also higher under he Markov-perfec soluion han under he Ramsey policy. Seady sae growh is slighly higher under he Ramsey policy. Growh raes are large for low values of he risk aversion parameer, bu hey become quie realisic for values of σ above.5. As a proporion of oupu, privae consumpion is higher under he Ramsey policy, while public consumpion is higher under he Markov policy. In erms of specific values, privae consumpion never exceeds 35% of oupu under eiher policy, while public consumpion remains below 0% of oupu, boh observaions below he levels observed in acual economies. However, he raio of public o privae consumpion is around 25%, as in observed daa. For he Markov and Ramsey soluions we could obain raios of public and privae consumpion o oupu similar o hose in acual daa, a he expense of geing income ax raes implausibly high. A planner wih access o lump-sum axes under commimen would devoe o consumpion an even higher proporion of public resources han he Markov and Ramsey soluions, and he growh rae would be considerably higher han under he alernaive soluions. Tha he income ax is higher under he Markov-perfec policy han under he Ramsey soluion is consisen wih he resul obained by Origueira (2006) in an exogenous 9

20 growh economy under inelasic labor supply. 2 This resul arises because he Markovian governmen canno inernalize he disorionary effecs of curren axaion on pas invesmen, while in he Ramsey soluion, he governmen akes fully ino accoun he negaive effec of he income ax on fuure invesmen. A similar argumen explains ha he Markov governmen devoes a higher proporion of public resources o consumpion, which has a direc impac on curren uiliy, o he expense of producive services, which would have a posiive effec mainly on fuure uiliy. Togeher wih a higher share of resources devoed o producion, a lower income ax rae leads o a higher growh rae under he Ramsey han under he Markov soluion. Figure 2 presens resuls for σ = 2, and values of he relaive weigh of public consumpion in he uiliy funcion, θ, beween 0.2 and.5, he remaining parameers being as in Figure. As expeced, public consumpion as a share of oal public spending increases wih θ. Qualiaive resuls say he same, wih he Markov-perfec policy imposing a higher income ax han he Ramsey policy and devoing a higher proporion of public resources o consumpion. The growh rae is again higher under he Ramsey han under he Markov policy. Table summarizes he resuls by displaying a single poin from Figure and Figure 2. Table 2 analyzes he effecs of a change in. The value of B has been chosen o guaranee posiive growh raes under he Markov and Ramsey soluions. Since he raio of producive services o oupu is he same for he hree soluions implies ha he produc ( ) and hence, he raio of privae capial o oupu, are also he same for he hree soluions under any parameerizaion. The common value of ( ) urns ou o be equal o he elasiciy of oupu wih respec o he public inpu, again an exension of he resul obained by Barro (990). The soluion under lump-sum axes leads o he larges public secor and devoes a lowes share of public resources o producion. Since axes are no disorionary under he planner s soluion, a larger proporion of resources exraced by he public secor can be made compaible wih a higher rae of growh. The comparison beween he wo panels in Table shows wha happens as public consumpion becomes more imporan in he uiliy funcion: while he raios of boh ypes of capial o oupu remain unchanged, he opimal ax rae increases, as i does he 2 Even hough he wo resuls are no sricly comparable, since one of hem refers o an exogenous growh economy and he oher o an endogenous growh economy. 20

21 proporion of public resources devoed o consumpion. These wo changes lead o a lower rae of growh. 3 Table. Values for he main variables under he hree soluion conceps. Effecs of a change in θ η (%) τ (%) γ (%) c/y(%) g/y(%) i p /y(%) k/y B = , σ = 2.00, B = , σ = 2.00, θ = 0.40, α = 0.80, θ =.00, α = 0.80, δ = 0.0, ρ = 0.99 δ = 0.0, ρ = 0.99 Planner Markov Ramsey Planner Markov Ramsey Table 2. Values for he main variables under he hree soluion conceps. Effecs of a change in α η (%) τ (%) γ (%) c/y(%) g/y(%) i p /y(%) k/y B = 0.658, σ = 2.00, B =0.658, σ = 2.00, θ = 0.40, α = 0.80, θ = 0.40, α = 0.70, δ = 0.0, ρ = 0.99 δ = 0.0, ρ = 0.99 Planner Markov Ramsey Planner Markov Ramsey g i P p, P g Noe o he ables: for he planner soluion and y g i p,. Table 2 shows ha an increase in he produciviy of he public inpu (lower ) leads o higher ax raes. The governmen hen deracs more aggregae resources from he economy and devoes a larger proporion of hem o producion. Because of he increase in he ax rae generaed by a lower parameer, he produciviy of privae capial and hence, he rae of growh, boh decrease. Raes of growh under he Ramsey and Markov policies in Tables and 2 are very similar. However, for many alernaive parameerizaions, hey may easily differ in close o one percen poin. 4 3 Raes of growh under he Ramsey and Markov policies in Tables and 2 are very similar. However, for many alernaive parameerizaions, hey may easily differ in close o one percen poin. 2

22 7. Welfare In his secion we compue he level of welfare ha would arise along he balanced growh pah under he ime consisen Markov policy and compare i wih he level of welfare ha would be obained under lump-sum axes. 5 As in Lucas (987), wha we compue is he consumpion compensaion (as a percenage of oupu) ha would be needed under he Markov rule o achieve he same level of welfare han under he resource allocaion of he planner wih non-disorionary axaion. Under a CRRA uiliy, welfare can be wrien, ( ) c i, g, i i i Wi, i P, M ( )( ) 0 i. Le {c,i, g,i }, wih i=p, M, be he opimal pah for privae and public consumpion for he planner s soluion (P) and he Markov soluion (M), respecively, ha is: ci, iki, ik0i i i, k0 gi, iki, ik0i i i, i P, M, R k0 where we have indicaed he normalizaion k 0 =. The consumpion compensaion needed for he Markov and Ramsey soluions o achieve he same level of welfare as under he planner s allocaion can be obained by solving he following equaion: 0 ( ) ( ) c, j g, j,, WP j M R ha is, and finally, ( ) P P j j, j M, R ( )( ) ( )( ) P j 4 For insance, for B 0.90, 2.00, 2.00, 0.85, 0.0, 0.96, growh raes under Markov and Ramsey policies become 2.76% and 3.43%. 5 We do no consider he level of welfare under he Ramsey soluion because of is ime-inconsisen naure. 22

23 ( )( ) j P P ( )( ) P j j, j M, R. (33) To ranslae his compensaion ino oupu unis, we have o compue c M R, which is he compensaion shown in Figure 3. y, j 00, j,, j As he risk aversion parameer changes beween and 5, he Markov consumpion compensaion falls from 45% o 3% of oupu. In paricular, for =2, he compensaion ha would be necessary o achieve he planner s welfare is around 8% of oupu. By and large, he decrease in consumpion compensaion is due o he decline in he value of he firs facor in (33). 6 The consumpion compensaion increases wih θ. For =2, he Markov consumpion compensaion increases from 6% o 23% of oupu. Again, his increase in he consumpion compensaion is mainly due o he firs facor in (33). 7 So, he difference in growh raes is he main deerminan of he welfare loss under he Markov soluion relaive o he planner s soluion, over and above he effecs of differences in he raios of privae or public consumpion o oupu. In boh cases, he Ramsey policy, if i could be mainained over ime, i would lead o a slighly bigger loss of welfare relaive o he planner policy. Boh policies are no sricly comparable, and he welfare comparisons could go eiher way. In Klein e al. (2008), he Ramsey policy leads o a bigger welfare loss when he economy is subjec o a oal income ax. In Origueira (2006) he same resul arises when he only source of revenue is a ax on labor income. 8 6 The firs facor, which depends on growh raes, falls from 7.3 for σ =., o.23 for σ =5. The second facor increases from 0.29 o 0.86, while he hird facor iniially increases from is saring value of.08 o.054, and i decreases afer ha o essenially is same iniial level. 7 The firs facor increases from.72 o 3.02 as θ changes from 0.2 o.5. The second facor gradually decreases from 0.70 o 0.54, and he hird facor shows a moderae increase, from.3 o These auhors do no repor welfare levels. We have used he seady-sae values hey provide o compue seady-sae welfare levels. Our saemens above are valid under he assumpion ha he policy rules in boh papers would posiion he economies on heir balanced growh pahs from he saring period. 23

24 8. Conclusions We have characerized he opimal Markov-perfec fiscal policy in an endogenous growh economy where he fiscal auhoriy canno commi o policy choices beyond he curren period. Tax revenues are used o finance public consumpion and public producion services, and we have considered wo policy variables: a single ax on oal income and he spli of public resources beween consumpion and producive services. Under logarihmic preferences and full depreciaion of privae capial, we can analyically characerize he opimal values of he wo policy variables. Wih ha paricular specificaion, we show ha he Markov-perfec policy coincides wih he opimal Ramsey policy ha would arise by imposing commimen. For he more general case of a CRRA uiliy funcion and less han perfec depreciaion of capial, we show he economy o be on is balanced growh pah from he iniial period onwards. In his case here is no closed form soluion, bu we compue numerical values for he Markov-perfec and he Ramsey opimal policies under parameer values calibraed o he US economy. We also explore he sensiiviy of he numerical soluions o he values of hree parameers: he ineremporal elasiciy of subsiuion of consumpion, he relaive weigh of public consumpion in agens uiliy funcion and he elasiciy of oupu wih respec o privae capial. For empirically plausible parameer values, he income ax is higher under he Markov policy han under he Ramsey soluion, and a higher proporion of public resources are devoed o consumpion. Consequenly, he growh rae is lower under he Markov policy han under he Ramsey soluion. The welfare loss of he Markov soluion relaive o he planner s allocaion is mainly deermined by he differences in growh raes, more han by differences in he raios of privae or public consumpion o oupu. The implicaion of our resuls is ha if he privae secor is aware of he governmen's inabiliy o pledge fuure policy decisions, hen he governmen should impose a slighly higher ax rae and devoe a higher share of public resources o consumpion, wih a relaively low cos in erms of growh. Considering a more complex ax srucure, as well as non-rivial ransiional dynamics in an endogenous growh model wih public deb, are lef as fuure exensions of his work. 24

25 References Ambler, S. and F. Pelgrin, (200): Time-Consisen Conrol in Nonlinear Models, Journal of Economic Dynamics and Conrol, 34, pp Aschauer, D.A. (989): Is public expendiure producive?, Journal of Moneary Economics, 23(2), Azzimoni, M., P.D. Sare and J. Soares (2009): Disorionary axes and public invesmen when governmen promises are no enforceable, Journal of Economic Dynamics and Conrol, 33, Barro, R.J. (990): Governmen spending in a simple model of economic growh, Journal of Poliical Economy, 98, S03-S25. Cazzavillan, G. (996): Public spending, endogenous growh and endogenous flucuaions, Journal of Economic Theory, 7, Ebers, R. (986), Esimaing he conribuion of urban public infrasrucure o regional growh, Working Paper 860, Federal Reserve Bank of Cleveland. Glomm, G. and B. Ravikumar (997): Producive governmen expendiures and long-run growh, Journal of Economic Dynamics and Conrol, 2, Jaimovich, N. and S. Rebelo (203): Non-linear effecs of axaion on growh, CQER WP 3-02, April 203. Klein, P., P. Krusell and J.V. Ríos-Rull (2008): Time-consisen Public Policy, Review of Economic Sudies, 75, Krusell, P., V. Quadrini and J.V. Ríos-Rull (996), Are Consumpion Taxes Really Beer Than Income Taxes? Journal of Moneary Economics, 37 (3), Krusell, P. and J.V. Ríos-Rull (999), On he Size of U.S. Governmen: Poliical Economy in he Neoclassical Growh Model, American Economic Review, 89 (5), Lucas, Rober E. Jr. (987), Models of Business Cycles, Blackwell, Oxford. Malley, J., A. Philippopoulos and G. Economides (2002), Tesing for ax smoohing in a general equilibrium model of growh, European Journal of Poliical Economy, 8, Marin, F.M. (200): Markov-perfec capial and labor axes, Journal of Economic Dynamics and Conrol, 34, Origueira, S. (2006): Markov-perfec opimal axaion, Review of Economic Dynamics, 9,

26 Appendix : Proof of Proposiion Firs order opimaliy condiions for he governmen s problem are: wih respec o τ: (, ) Uc 0 U g V k k, where: so ha: (, ) / ( ) ( ) ( ) B (, ) ( ), ( ) U U V (, ) ( ) k c g k wih respec o η: (, ) Uc 0 U g V k k, where: (, ) / ( ) ( ) B (, ), so ha: Uc (, ) U g V k k. The envelope condiion is: Vk U c k U c Ug k U g k k k k (, ) (, ) V k (, ), k k k k k which, afer using he firs order condiions derived above, i can be wrien as Vk U (, ). c k U g k V k k From he opimaliy condiions above we ge, 26

27 V V k k Uc U g k, Uc U g, k which leads o condiion (). Plugging he firs equaion ino he envelope condiion we ge, Uc U g Vk U (, ), c k U g k k k and, finally, we ge equaion (2):, Uc U g Uc Uc k U U g g k, k., k (, ) k,, Appendix 2: Opimal Ramsey policy under a CRRA uiliy funcion and incomplee depreciaion of privae capial The Ramsey opimal policy is he soluion o he uiliy maximizaion problem, subjec o he equilibrium condiions as consrains. Under he CRRA uiliy funcion, he Lagrangian for he Ramsey problem becomes: ( ) c g L (, ) k c k 0 / / 2 B k g ( ) ( ) 3 cg (, ) c g. Taking he derivaives wih respec o c, g, k,, o be equal o zero, we obain he opimaliy condiions for he Ramsey problem: ( ) ( ) ( ) c g 3 c g 3, c g (, ), ( ) ( ) c g 2 ( ) c g 3 3, (, ), / /, (, ) 2, B ( ) ( ) k 2 k 3c g 0,, 27

28 k k c g ( ) Transforming he mulipliers by:,,, and ( ) ( ) 3 k 2 2k 3 k k defining he rae of growh, and he raios of privae and public consumpion o k capial c g, k k, we can ge a sysem of equaions in saionary raios. Firs, from he global consrain of resources, we ge an expression for he growh rae: (, ). Whereas from he governmen budge consrain, we can wrie he raio of public o privae capial: / / B ( ). From he Euler equaion for he compeiive equilibrium: x x (, ), ( ) ( ) ( ) and from he se of opimaliy condiions above, we finally ge he sysem of equaions characerizing he opimal Ramsey policy represened in saionary raios: x x (, ), ( ) ( ) 3 3 ( ) ( ) x 2 ( ) 3 3 (, ), x ( ) / / (, ) 2 B ( ), ( ) 2 3 x 0, ( ) Along he balanced growh pah, he sysem of equaions for he Ramsey equilibrium becomes: ( ) (, ), (, ),. 28

29 / / B ( ), ( ) x 3 (, ), ( ) 2 x 3 (, ), ( ) ( ) / / 2B (, ) ( ), ( ) 2 3 x 0, ( ) Denoing by: ( ) F (, ), F (, ), and, ( ) we characerize he balanced growh pah of he Ramsey equilibrium by paricularizing he sysem of equaions above o: / / ( ) (, ), (, ), B ( ), /, ( ) 2, ( ) 3, / ( ) 3 2 0, ( ) 2 3 ( ) 0, a sysem of 8 equaions in,,,,,,, ha allows us o compue he balanced 2 3 R R growh pah for he Ramsey policy (, ) as well as he implied allocaion of resources, R R R characerized by (,, ). 29

30 Appendix 3.- Proofs of proposiions 3, 4 and 6 Proof of Proposiion 3: The problem solved by he governmen is: V( k ) Max ln ( k ;, ) ln ( k ;, ) V ( k ), where k (, ) k ( k ;, ), ( k ;, ) ( ) (, ) k, ( k;, ) (, ) k. The firs order condiions for his problem are: : ( ) ( ) V (, ) ( ) k 0, (A3.) k ( ) : ( ) V k (, ) 0 k, (A3.2) From (A3.) and (A3.2) we obain a relaionship beween he opimal values of he ax rae and he governmen spending spli in he Markov-perfec equilibrium: M. (A3.3) M Proof of Proposiion 4: i) To examine he dynamic properies of he Markov soluion, we consider he envelope condiion : V k (, ) ( ) ( ) k (, ) k k (, ) V k (, ), ( ) k k k k which, using condiions (A3.) and (A3.2), i can be wrien as, V ( ) V (, ). (A3.4) k k k Using (A3.2) and (A3.3) in (A3.4), we obain he dynamic equaion: 0, (A3.5) 30

31 where. The soluion o he difference equaion (A3.5) is unsable, since ( ) /. Hence he only sable soluion is ha says consan over ime, and he same applies o, ha is,,. ii) Since is consan, condiion (A3.3) implies ha τ is also consan. From he wo condiions: ( k;, ) ( ) (, ) k, ( k;, ) (, ) k, implying ha he raios and will also be consan. Togeher wih he absence of indeerminacy, his resul implies ha he economy lacks ransiional dynamics, being on he balanced growh pah from he iniial period on. iii) From (A3.5) we obain he value of : M ( ), ( ) and using (A3.3), we obain he Markov perfec opimal ax rae: 9 ( ) M. Proof of proposiion 6: Paricularizing he sysem of equaions for he balanced growh pah under he opimal Ramsey policy obained in Appendix 2 o he case of a logarihmic uiliy funcion (σ=) and full depreciaion (δ=), we obain: 9 Malley e al. (2002) obain a similar expression for he Markov perfec ax rae. 3

32 / B / B ( ) ( ), / / / / ( ) ( ) F, B ( ), B ( ) ( ), F B ( ) ( ), F /, F B ( )( ) ( ) F / B 3, ogeher wih: 2 / / ( ) 3 2 0, 2 3( ) 0.,, Subsiuing he expressions for he Lagrange mulipliers ino he las wo equaions gives us: 0, / ( )( ) / / B ( ) B ( ) ( ) 0. / B ( )( ) ( ) Finally leading o he sysem: 32

33 R R ( ) ( ) 0, R R ( ). ( )( ) R ( )( ) The firs equaion yields he Ramsey-opimal ax rae as a funcion of he srucural parameers α, θ, ρ, while he second equaion gives us he associaed opimal spli of public resources. I is easy o see ha he soluion o his sysem is given by, R ( ), R ( ). ( ) Appendix 4.- Opimal Markov policy Proof of Proposiion 7: Taking ino accoun he generalized Euler equaion (22): ( ), (A4.),, we need o compue he parial derivaives of,. To ha end, we differeniae in (24) o obain: wih respec o he wo policy variables ( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( ), d 0 ( ) so ha, ( ) ( ) ( ) ( ), d ( ), (A4.2) d ( ) ( ) ( ) and, similarly, we would obain: 33

34 d d As we can see, he produc ( ),. (A4.3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cancels ou again in boh parial derivaives. As a consequence, he characerizaion of ha produc ha we made a (24) as a funcion Fk ( ) of he sae of he economy does no play any role in he firs Generalized Euler equaion ha relaes he opimal choice of he wo policy variables,. Using now he parial derivaives, in he firs Generalized Euler equaion (A4.), we finally ge: ( ) ( ),, ( ) ( ),, ha can only hold if:,. Appendix 5. The planner s problem under lump-sum axes A planner wih access o lump-sum axes would allocae resources so as o maximize ime aggregae uiliy wih he global consrain of resources as is sole resricion, hereby solving he problem, subjec o: Max c, k, kp, g 0 c g ( ) k ( ) k c g k Bk k, p, p, leading o opimaliy condiions: ha defines he rae of growh P, and ( ) / c ( ) B ( ), c 34