On the Generalized Weitzman s Rule

Transcription

1 MPRA Munich Personal RePEc Archive On he Generalized Weizman s Rule Kazuo Mino Insiue of Economics, Kyoo Universiy March 2004 Online a hp://mpra.ub.uni-muenchen.de/16996/ MPRA Paper No , posed 29. Augus :40 UTC

2 On he Generalized Weizman s Rule Kazuo Mino March 2004 Absrac Weizman (1976) provides a foundaion for ne naional produc of a compeiive economy as he annuiy equivalen of he presen discouned value of maximized consumpion. This paper considers how Weizman s rule should be modified if he compeiive equilibrium is affeced by he presence of marke disorions. The paper firs examines he model wih exernal effecs of capial in which here are spillovers of knowledge. The paper also sudies he model wih policy inervenions where he policy maker seeks he second bes allocaion. The cenral concern of he paper is o elucidae he facors ha generae a divergence beween ne naional produc and he welfare equivalence of maximized consumpion. In discussing each model, he paper presens a ypical example ha has been widely discussed in he lieraure. JEL classificaion code: C43, D6, 047 Keywords: Weizman s rule, ne naional produc, exernaliies, policy disorions

3 1 Inroducion In his seminal conribuion, Weizman (1976) reveals he welfare significance of naional income in a dynamic conex. He shows ha in a compeiive economy, he curren level of ne naional produc (NNP) is a precise measure of he annuiy equivalen of he presen discouned value of maximized consumpion. This fundamenal observaion, which we refer o as Weizman s rule, means ha he curren level of naional income conains all informaion concerning he welfare of curren and fuure generaions. Furhermore, Weizman s finding plays a key role in considering he relevan issues of dynamic welfare economics such as susainabliliy of economic developmen and inergeneraional equiy. Weizman s (1976) conribuion led o a renewal of ineres in he welfare implicaion of naional income accouning. Since his fundamenal proposiion holds for he perfecly compeiive economy in which agens are endowed wih perfec foresigh and have a linear uiliy funcion, he subsequen sudies have invesigaed he income-welfare relaionship under more general circumsances han hose assumed in Weizman (1976). For example, Asheim (1997) and Pemberin and Ulph (2001) re-examine he concep of NNP when he uiliy funcion is concave so ha he consumpion rae of ineres is ime dependen, while Arronson and Löfgren (1995) and Weizman (1998) inroduce uncerainy ino he base model. In addiion, Weizman (1996) and Asheim (1997) explore he effec of exogenous echnical progress. Despie hose relevan exensions, here is sill a gap beween heory and realiy if we inend o apply Weizman s rule o inerpre he acual economic daa. In realiy a marke economy does no necessarily saisfy he ideal condiions for he compeiive economy. There are many causes ha generae marke failures. Therefore, once we ake ino consideraion he fac ha behavior of a decenralized economy may diverge from ha of a command economy, we have o examine alernaive marke environmens in exploring he relaionship beween naion s wellbeing and he concep of ne naional produc. The purpose of his paper is o reconsider Weizman s rule in he presence of marke disorions. Among many possible causes for marke failure, we focus on Marshallian exernaliies as well as on policy inervenion. The firs issue we deal wih is he modificaion of Weizman s rule for he economy wih knowledge spillovers. More specifically, we sudy a dynamic economy where capial socks generae knowledge exernaliies. This kind of exernal effec of 1

4 capial was inroduced by Arrow (1961) and Romer (1986), who provided one of he heoreical bases for recen developmen of growh economics. When capial socks have posiive exernaliies, i is easy o anicipae ha NNP underesimaes he annuiy equivalen of he presen discouned value of maximized consumpion. We derive a modified version of Weizman s rule in a general seing. We hen apply he modified rule o a simple model of endogenous growh and evaluae he difference beween he curren level of NNP and he curren wealh expressed by a discouned presen value of curren and fuure levels of consumpion. In he exising lieraure, Arronson and Löfgren (1996) also discuss he income-welfare relaionship in he conex of Lucas (1988) model of growh in which human capial is associaed wih exernal effecs. Our formulaion is more general han heir seing. Addiionally, in our specific examplewe conduc numerical experimens o evaluae he accuracy of NNP as a welfare measure. Oursecondopicisoinvesigaehowhebaseresulwouldbemodified if he governmen disors compeiive allocaion. In paricular, we consider he case where he benevolen governmen maximizes he privae agens welfare by using disorionary policy ools. When we deal wih he opimizing governmen, we sudy wo alernaive sraegies of he governmen: open-loop policy and feedback policy. In each case, NNP canno be a precise measure of welfare equivalence of maximized consumpion. Roughly speaking, he divergence comes from he fac ha he marginal value of capial form he privae perspecive is differen from he implici value of capial form he governmen perspecive. Weizman s rule sill holds if he value of ne invesmen is measured in erms of he implici prices of capial ha suppor he opimal condiions for he governmen s plan. However, NNP measured by he marke prices fails o evaluae he discouned presen value of maximized consumpion in a precise manner. Afer deriving general resuls, we explore a simple model of dynamic opimal axaion as a ypical example. The remainder of he paper is organized as follows. Secion 2 reviews Weizman s rule. Secion 3 reas models wih exernal effecs of capial, while Secion 4 sudies models wih policy inervenions. Secion 5 concludes he paper. 2

5 2 The Basic Resuls 2.1 Weizman s Rule Weizman s rule is esablished under he following assumpions: (i) preferences and producion echnology are saionary, (ii) he sociey s feliciy level is expressed as a discouned sum of uiliies wih a consan discoun rae, and (iii) he sociey aains he ineremporally opimum resources allocaion. Given hose assumpions, Weizman s rule may be expressed as Y () =u ()+p () z () =ρ e ρ(s ) u (s) ds, (1) where Y () is ne naional produc (NNP) in erms of uiliy a ime, u () is he uiliy level of consumpion, p () is vecor of invesmen price in erms uiliy, z () is vecor of invesmen and ρ (> 0) is a ime discoun rae. Poined ou by Kemp and Long (1982) and ohers, (1) is an expression of he Hamilon-Jacobi-Bellman equaion for he dynamic opimizaion problem solved by he command economy ha mimics he behavior of he corresponding decenralized economy. More generally, (1) is one of he conservaion laws ha can be derived by Noeher s heorem on invarian ransformaions in opimizing dynamical sysems: see Sao (Chaper 8, 1981) and Sao (1985) for deailed discussions on he income-wealh conservaion laws in opimal growh models. 1 In order o discuss Weizman s rule in a general seing, we follow he formulaion employed by Dasgupua and Mira (1999). Consider a planning economy in which here are n capial goods and m consumpion goods. Le us denoe he vecors of capial socks and consumpion goods by k R n + and c R m +, respecively. To avoid he index number problem, we assume ha he insananeous feliciy of he sociey, u, can be expressed as a cardinal uiliy funcion u = u (c), u : R n + R +, which ransforms he consumpion vecor ino he level of feliciy. 2 The echnology se is assumed o be saionary and i is defined as T R + R n R n +. The feasibiliy condiion requires ha (u, z, k) T. The echnology se is a closed, convex subse of R + R n R n + and 1 See also Sao and Ramachandran (Chaper 7, 1998) for exposiory discussion on he conservaion laws.. 2 Dasgupa and Mira (1999) call u he aggregae consumpion. To avoid confusion, we refer o u as uiliy. I is also o be noed ha u can be considered a money meric uiliy funcion. Weizman (Chaper 6, 2002) presens a deailed exposiion on he relaionship beween nominal naional income and NNP in erms of uiliy. 3

6 i saisfies he sandard regulariy condiions. 3 Now define he projecion on T in such a way ha S = {(z,k) :(u, z, k) T for some u}. Then we may define funcion v : S R + as follows: v (z,k) =max{u :(u, z, k) T }. (2) This funcion describes he maximum level of uiliy under given levels of invesmens and capial socks. We assume ha funcion v (z, k) is a leas wice coninuously differeniable and sricly concave on S. Under given seings, he dynamic opimizaion problem for he planner is formulaed in he following manner: subjec o max 0 e ρ v (z (),k()) d k () =z (), (3) (z (),k()) S, (4) and given iniial levels of capial socks, k 0. Leing p () be vecor of he capial sock prices (in erms of uiliy), he maximum principle yields he following condiions for an opimum: and he ransversaliy condiion: p () = v z (z (),k()), (5) ṗ () =ρp () v k (z (),k()), (6) lim e ρ p () k () =0. The opimal soluion saisfies (3), (4), (5), (6), ogeher wih (??) and he iniial condiion: k (0) = k 0. Suppose ha he above problem has he opimal soluion. The value funcion a ime is hen defined as ½ ¾ W (k ()) max e (s ) v (z (s),k(s)) ds : k (s) =z (s) and (z (s),k(s)) S. 3 See Assumpion 1 in Dasugupa and Mira(1999). 4

7 When W (k ()) is differeniable wih respec o k (), he Hamilon-Jaobi-Bellman equaion is given by ρw (k ()) = max {v (z (),k()) + W k (k ()) z ()}. (7) z() Since i holds ha W k (k ()) = p (), he righ hand side of (7) equals he maximized Hamilonian such ha M (k (),p()) max {v (z (),k()) + p () z ()}. z() The maximized Hamilonian shown above is equal o he sum of consumpion and ne invesmen (in erms of uiliy), so ha i represens ne naional produc of a closed economy. Denoe Y () M (k (),p()). (8) Then, noing ha v (z (),k()) expresses he opimum level of uiliy, (7) esablishes Weizman s rule (1). 4 I is worh emphasizing ha Weizman s rule may hold in a wide class of dynamic economies. Since he elemens in k () represen no only socks of physical capial bu also oher ypes of capial ha can be devoed o producion aciviies. Therefore, k () may include human capial, knowledge capial and non-renewable naural resources. In a similar vein, he echnology se would describe learning, educaion, research and developmen as well as regular producion aciviies. This means ha Weizman s rule is esablished in many ypes of endogenous growh models in which perpeual growh can be susained in he absence of exogenous echnical progress. 2.2 Exogenous Technical Change As poined ou above, since k () may include a wide variey of capial socks, endogenous echnical change can be incorporaed ino Weizman s rule, as far as i is fully capured by 4 I is worh noing ha our definiion of NNP is he maximized level of consumpion plus ne invesmen in erms of (cardinal) uiliy. As poined ou by Asheim (1997) and Pemberon and Ulph (2001), if here is a single consumpion good and he uiliy funcion is linear in consumpion, he consumpion rare of ineres rae is consan so ha he maximized Hamilonian expresses he convenional naional income in a closed, compeiive economy. However, if we assume ha he uiliy funcion is concave, he consumpion rae of ineres rae is ime dependen and hus he price index problem arises o define a proper concep of naional income. In his paper we avoid his issue by using concepual income and wealh evaluaed in erms of uiliy. 5

8 he planner. However, if here is exogenous echnical change, he base resul (1) should be modified. When exogenous echnical changes is anicipaed o occur, he producion echnology se is no saionary. As a resul, he maximum uiliy defined by (2) is expressed as u () = v (z (),k(),), where (u (),z(),k()) T (). In his case he maximized Hamilonian is wrien as M (k (),p(),) max {v (z (),k(),)+p() z ()} (9) z() and he value funcion is given by ½ W (k (),) max e ρ(s ) v (z (s),k(s),s) ds : o k (s) =z (s) and (z (s),k(s)) S (). The Hamilon-Jacobi-Bellman equaion (7) is now replaced wih ρw (k (),)=M (k (),p(),)+w (k (),), (10) where W W/. This shows a modified Weizman s rule wih exogenous echnical change. 5 By (8) he modified rule (10) is wrien as Y () W (k (),) + W (k (),) W (k (),) = ρ. Inuiively, his equaion is a non-arbirage condiion under which he ne rae of reurn o he curren level of wealh, Y/W, plus he anicipaed capial gain caused by exogenous echnical progress, W /W, equals he discoun rae. 6 In order o evaluae W (k (),), noice ha from (6), (8) and (9) heimederivaiveof 5 The original value funcion can be se as Ŵ (k (),) max e ρs v (z (s),k(s),s) ds, and hence e ρ W (k (),)=Ŵ (k (),). The Hamilon-Jacobi-Bellman equaion in erms of Ŵ (.) is Ŵ (k (),)=max z() {v (z (),k(),)+w k (k (),) z ()}. Using W (.), his equaion is wrien as (10). 6 Harwick and Long (1999) presen a generalized version of (10) for he sysem in which he discoun rae is ime dependen. 6

9 he ne naional produc (he maximized Hamilonian) can be wrien as Ẏ () =M k k ()+Mp ṗ ()+M = ρ [Y () u ()] + v (k (),p(),). (11) R Provided ha lim e ρ(s ) v s (z (s),k(s),s) ds exiss, (11) yields he following: 7 Y () =ρ Therefore, we obain e ρ(s ) u (s) ds W (k (),)= e ρ(s ) v s (z (s),k(s),s) ds. (12) e ρ(s ) v s (z (s),k(s),s) ds, which shows ha W (k (),) evaluaes he discouned presen value of curren and fuure consumpion changes due o he exogenous echnical progress. The modified rule (12) indicaes ha he maximum level of uiliy is given by ρ e ρ(s ) u (s) ds = Y ()+ e ρ(s ) v s (z (s),k(s),s) ds. (13) In general, echnical progress expands he echnology se T (), and hence he maximized insananeous consumpion v increases wih ime, ha is, v s (.) > 0 for all 0. Therefore, in he presence of exogenous echnical progress, NNP underesimaes he annuiy equivalen of susainable consumpion. Weizman (1997) derives a simple relaionship beween he average maximized uiliy defined byū () = ρ R e ρ(s ) ū () ds and he average NNP a given by Ȳ () =ρ R e ρ(s ) Y (s) ds. He finds: µ ū () = 1+ λ Ȳ (), (14) ρ γ where γ = dȳ d () /Ȳ () and λ = ρ R e (s ) v s v ds. Since γ and λ respecively denoe he average growh raes of income and echnology, he erm ρ γ λ Ȳ () in (14) demonsraes he growh effec of exogenous echnical change on he level of maximum uiliy a ime. 8 7 The general soluion of (11) is Y () =ρ e ρ(s ) c (s) ds Therefore, (12) requires ha lim e ρ( ) Y ( ) =0. 8 Pezzy (2002) modifies (14) by considering environmenal issues. e ρ(s ) v s (z (s),k(s),s) ds + e ρ( ) Y ( ). 7

10 2.3 The Command Opimum and he Marke Economy So far, he fundamenal resuls, (1) and (12), have been esablished for he command economy. In applying Weizman s rule o a marke economy, here are a leas wo issues o be noiced: one is a echnical problem and he oher is a pracical one. Firs, even hough we focus on he perfec-foresigh compeiive economy in which each marke is complee, he necessary condiions for susaining he compeiive equilibrium may diverge from hose for he command opimum. Dasugpa and Mira (1999) find ha a compeiive equilibrium saisfies (1) if and only if he following invesmen value ransversaliy condiion holds: lim e ρ p () k () =0, (15) which does no always coincide wih he sandard capial value ransversaliy condiion (??). Dasugpa and Mira (1999) provide a simple example in which a regular compeiive economy wih concave uiliy and producion funcions does no fulfill (15) so ha (1) fails o hold. Second and more imporanly, in he real world he decenralized economies may no aain he command opimum due o he presence of marke disorions. There are many causes ha make he marke economy diverge from he command economy. Among ohers, in wha follows we focus on wo ypes of disorions: Mashallian exernaliies and policy inervenions. As he model wih exogenous echnical progress suggess, i is easy o anicipae ha ne naional produc would diverge from he susainable consumpion in he presence of marke disorions. Unlike he case of exogenous echnical change, however, he difference beween NNP and he annuiy equivalen of he discouned presen value of maximized consumpion depends on he endogenous facors ha generae marke failures. By deriving modified versions of Weizman s rule, we consider wha facors may yield such a divergence. 3 Exernaliies In his secion, we rea models of compeiive economy in which capial socks generae exernal effecs. Since he compeiive economy canno inernalize exernaliies, ne naional produc evaluaed by he marke prices does no equal he welfare equivalence of maximized uiliy. We focus on he wedge producing he difference beween hem. 8

11 3.1 Weizman s Rule wih Exernal Effecs If he producion echnology of an individual firm is affeced by exernal effecs generaed by he aggregae capial in he economy a large, he privae echnology depends on he social level of capial denoed by k. Thus he echnological feasibiliy condiion is expressed as (u, z, k) T k. This implies ha he maximum level of uiliy can be shown by v = v z,k, k, where v is defined on S = z, k, k :(u, z, k) T k ª for some u. Here,weassumehasome elemens in k represen knowledge capial ha saisfy nonexcludabiliy and nonrivalry. According o Kehoe e al. (1992), he equilibrium condiions for a dynamic compeiive economy wih exernaliies can be characerized by he opimal soluion for a pseudo-planning problem in which he planner is assumed o ake he sequence of exernal effecs as given. In our seing, he compeiive economy coincides wih he opimal soluion for he following problem: max subjec o 0 e ρ v z (),k(), k () d k () =z (), z (),k(), k () S, ogeher wih he iniial level of capial: k (0) = k 0. When solving his problem, he planner akes an anicipaed sequence of exernal effecs, k ª (), as given. =0 The maximized Hamilonian for he above problem is given by M k (),p(), k () ª max v z (),k(), k () + p () z (). z() The opimizaion condiions under a given sequence of k () ª =0 are he following: p () = v z z (),k(), k (), (16) ṗ () =ρp () v k z (),k(), k (), (17) and he ransversaliy condiion: lim e ρ p () k () =0. Since he sequence of k (s) ª s= is exernal o he privae agens, he value funcion a ime can be wrien as a funcion of k () and : W (k (),) max e ρ(s ) v z (s),k(s), k (s) ds. 9

12 As well as in he case of exogenous echnical change, he Hamilon-Jacobi-Bellman equaion is: ρw (k (),)=M k (),p(), k () + W (k (),). (18) For analyical simpliciy, le us assume ha he number of agens is normalized o one. This assumpion means ha he consisency condiion requires ha k () =k () for all 0. (19) I is now easy o show he following: Proposiion 1 If consisency condiion (19) holds, in he presence of exernal effecs of capial, Weizman s rule (1) is modified as Y () =ρ e ρ(s ) u (s) ds e ρ(s ) v k (z (s),k(s),k(s)) z (s) ds. (20) Proof. Ne naional produc equals he maximized Hamilonian for he pseudo-planning problem, and herefore from (17) we have Ẏ () =M k k ()+Mp ṗ ()+M = ρ [Y () u ()] + v k (z (),k(),k()) z (). R Thus, assuming ha lim e ρ(s ) v k z (s),k(s), k () z (s) ds exiss, he soluion of he above differenial equaion is (20) Comparing (18) wih (20), we find: W (k (),)= e ρ(s ) v k (z (s),k(s),k(s)) z (s) ds. In oher words, he capial gain due o he presence of exernaliies equals he discouned presen value of ne invesmen evaluaed by he marginal exernal effec. Finally, (20) means ha ρ e ρ(s ) u (s) ds = Y ()+ e ρ(s ) v k (z (s),k(s),k(s)) z (s) ds. (21) Since NNP of he marke economy is defined as Y () u ()+p() z (), he righ hand side of (21) expresses NNP from he social perspecive. 10

13 3.2 An Example: Growh wih Exernal Increasing Reurns As a ypical example of he model wih capial exernaliies, we consider Romer s (1986) model of growh wih exernal increasing reurns. For noaional simpliciy, in his subsecion we drop from he variables unless i is useful o show i explicily. Suppose ha each firm produces a homogenous good by use of a single capial sock, k. The producion funcion of an individual firm is specified as y = f k, k, f k > 0, f kk < 0, f k > 0, where y is oupu of he firm and k denoes he capial sock in he economy a large. The presence of k in he producion funcion of each firm represens he spillover effec of knowledge capial. In he following, we normalize he number of firms o one. Thus each variable also expresses he aggregae one and he consisency condiion requires ha k = k holds in every momen. The marke equilibrium condiion for he final good is given by ³ z y = c + φ k + δk, (22) k where δ (0, 1) is he depreciaion rae of capial. Funcion φ (z/k) is assumed o be monoonically increasing and convex in z/k, which shows he presence of invesmen adjusmen coss. 9 The insananeous uiliy funcion of he represenaive consumer is specified as u (c) =c 1 σ /(1 σ) (σ (0, 1)). 10 We normalize he number of households o one as well. Hence, he pseudo-planning problem is o maximize e ρ 1 h f k, 1 σ k ³ z 1 σ δk φ ki d k 0 subjec o k = z, a given sequence of exernal effecs, k () ª k 0., and he iniial level of capial, 9 Following Romer s (1986) formulaion, capial accumulaion is deermined by z = k i = g k, k where i isneinvesmenspending. Funciong (i/k) is assumed o be sricly concave and monoonically increasing in i/k, which represens he presence of adjusmen coss of invesmen. Since i is expressed as i = g 1 (z/k) k, from he equilibrium condiion for he good marke we have ĉ + g 1 (z/k) k = f k, k. Denoing g 1 (z/k) φ (z/k), we obain (22). 10 Since we have assumed cardinal uiliy, σ is assumed o be less han 1 o keep u (ĉ) posiive. 11

14 The maximized Hamilonian for his pseudo-planning problem is M k, k, p ½ 1 h max f k, z() 1 σ k ³ z i ¾ 1 σ δk φ k + pz. k The firs-order condiion for opimal invesmen decision is h f k, k ³ z i ³ σ δk φ k φ 0 z = p. (23) k k By use of (23), we obain pf k (k, k) M k (k, k) = φ 0 (z/k). Considering he equilibrium condiion, k = k for all 0, he modified Weizman s rule (20) is hus given by he following: Y () =ρ e ρ(s ) c (s) ds ρ(s ) p (s) f k (k (s),k(s)) z (s) e φ 0 ds. (24) (z (s) /k (s)) Since p (s) > 0, f k (k, k) > 0 and φ 0 (z/k) > 0, he exernal effec evaluaed by he second erm in he righ hand side of (24) has a posiive value. Therefore, NNP is smaller han he welfare equivalence of maximized consumpion. Now consider a special case where f k, k is homogeneous of degree one in k and k, so ha he producion funcion becomes y = f (1, 1) k when k = k. In his case he producion echnology has an Ak propery. Denoing z/k = γ, from he opimizaion condiion, p follows ṗ = ρp v k (z, k, k) = p ρ f k (1, 1) δ + φ (γ) φ 0 (γ) γ φ 0. (25) (γ) Noe ha (23) is rewrien as [f (1, 1) δ φ (γ)] = pk 1 θ. In he balanced-growh equilibrium z/k = k/k says consan, so ha pk 1 θ is consan over ime as well. σ k/k = ṗ/p and hus from (25) we obain γ = 1 σ fk (1, 1) δ + φ (γ) φ 0 (γ) γ φ 0 (γ) This means ha ρ. (26) This deermines he balanced-growh rae. The righ-hand side of he above monoonically decreases wih γ, implying ha here exiss a unique rae of balanced growh. As is well known, he model wih an Ak echnology wih homoheic uiliy funcion does no involve ransiion dynamics and he economy always says on he balanced-growh pah. Noing ha 12

15 z (s) =γk (s) and k (s) =k () e γ(s ), we find ha he aggregae exernal effec over an infinie ime horizon can be expressed as W (k (),)= e ρ(s ) f k (1, 1) [f (1, 1) δ φ (γ)] σ γk (s) 1 σ ds = [f (1, 1) δ φ (γ)] σ f k (1, 1) γ k () 1 σ. ρ (1 σ)γ The above show ha he exernal effec on welfare is zero eiher if here is no exernaliy (f k =0)or if here is no growh (γ =0). On he oher hand, using (23), he curren level of NNP is given by Y () =v (z (),k(),k()) + p () z () = 1 1 σ [f (1, 1) δ φ (γ)]1 σ k () 1 σ +[f (1, 1) δ φ (γ)] σ φ 0 (γ) γk () 1 σ. Therefore, he raio of he average uiliy and he exernal effec, W (k (),)/ c () is W (k (),) ū () = W (k (),) Y ()+W (k (),) f k (1, 1) γ = (1 σ) 1 f (1, 1) δ φ (γ)+φ 0 (γ) γ [ρ (1 σ)γ]+f k (1, 1) γ. In order o presen numerical examples, suppose ha he producion and adjusmen coss funcions are respecively specified as Then we obain: y = Ak α k1 α (0 < α < 1), φ (γ) =γ β (β > 1). f (1, 1) = A, f k = αa, f k =(1 α) A, φ (γ) γφ 0 (γ) =(1 β) γ β. Given hose specificaions, (26) becomes γ = 1 αa δ +(1 β) γ β σ βγ β 1 Addiionally, W k/ū () is wrien as W k (k (),) ū () = ρ. (1 α) Aγ (1 σ) 1 [A δ +(β 1) γ β ][ρ (1 σ) γ]+(1 α) Aγ. (27) When here is no adjusmen cos for invesmen, we may se β = 1. If his is he case, γ = 1 σ (αa δ ρ) so ha (??) is reduced o W k (k (),k()) ū () = (1 α) Aγ (1 σ) 1 (A δ)[ρ (1 σ) γ]+(1 α) Aγ. (28) 13

16 Le us examine some numerical examples for he case where here are no adjusmen cos (β =1). 11 Firs, consider he following se of parameer values: α =0.4, δ =0.05, ρ =0.05, A =0.3, σ =0.8 Here he income share of capial, α, is 0.4. The magniudes of he depreciaion rae δ =0.05 and he discoun rae ρ = 0.05 are frequenly used in he calibraed real business cycle models. The value of σ is assumed o be less han one o make he uiliy posiive, bu i is no so small ha he elasiciy of ineremporal subsiuion in consumpion (1/σ) is close o one. The value of A is se o obain realisic levels of he balanced-growh rae γ. Given hose parameer values, he balanced-growh rae is γ =0.025 and W /ū () = This means ha he curren level of NNP expresses almos 93% of he susainable consumpion, so ha Y () is sill a good indicaor of welfare measure even in he presence of marke failure. However, he resul is sensiive o he parameer values, in paricular, he long-erm growh rae. Now se α =0.4 and A =0.35. Oher hings being equal, he balanced-growh rae is γ =0.0625, so ha he economy aains relaively high growh. In his case, W /ū () =0.1489, and hence he exernal effec shares 15% of he susainable consumpion. If α =0.4 and A =0.4, hen he balanced-growh rae is γ =0.075 and W /ū () = In he above examples, we se α =0.4. Alhough his is a plausible magniude for he income share of capial, he exernal effec, which equals 1 α =0.6, is oo high from he view poin of he exising sudies on esimaion of scale economies: see, for example, Basu and Fernald (1997). Noe ha if capial k involves human capial, α may ake a larger value. To examine he case where k is a composie of physical and human capial, le us assume ha α =0.8 and A =0.15. In his case γ =0.025 and we have W /ū () = Thus accuracy of NNP as a welfare measure increases wih α. If α =0.8 and A =0.25, hen γ =0.075 and W /ū () = Consequenly, when he income share of capial is high, he welfare effec of capial exernaliy is relaively small even if he economy grows a a rapid rae of 7.25% per year. Basedonhisformula(14), Weizman (1997) claims ha he presence of exogenous echnical progress may significanly elevae he susainable consumpion so ha NNP is much smaller 11 Unless β has an unusually large value, we obain he similar resuls as hose shown below even hough here are invesmen adjusmen coss. 14

17 han he welfare equivalence of maximized consumpion. For example, if he rae of echnical progress is λ =0.01, hegrowhraeisγ =0.025 and ρ =0.05, hen λ/ (ρ γ) =0.4. Therefore, even if he average growh rae is only 2.5% per year, W /Ȳ () =0.4 in he case of exogenous echnical progress. In our noaion his means ha W /ū () =0.4/1.4 = This relaively large value comes from he fac ha Weizman (1997) assumes ha he uiliy funcion is linear and capial does no depreciae. In fac, if σ =0and δ =0in our model, (28) becomes W k (k (),k()) ū () = (1 α) γ ρ γ +(1 α) γ When α =0.4, γ =0.025 and ρ =0.05, we have W k/ū = This example demonsraes ha preference and capial depreciaion would play relevan roles in evaluaing he welfare effec of exernal increasing reurns. 4 Policy Inervenion We now urn our aenion o he economy where he governmen disors he marke equilibrium. Since in a decenralized economy privae agens ake he governmen s acions as given, he governmen s inervenion plays he similar role as ha of exernal effecs discussed above. However, unlike he case of exernaliies, we should specify governmen s policy making process in order o evaluae he disoring effecs. In wha follows, we consider opimizaion as well as non-opimizing policy makers. 4.1 Weizman s Rule wih Policy Inervenion Suppose ha he governmen s policies affec consumpion and producion decisions of he privae agens. In general, such a siuaion may be described by assuming ha he level of maximum consumpion depends on he governmen s behavior. Le us denoe he vecor of policy variables by g () Γ R r,whereγ expresses a se of consrains on he policy-making decisions. Hence, he maximum uiliy under policy inervenion is shown by v () =v (z (),k(),g()). (29) 15

18 The privae agens ake he sequence of policy variables, {g ()} =0, as given. As a consequence, he pseudo-planning problem ha may characerize he compeiive equilibrium is as follows: subjec o max 0 e ρ(s ) v (z (s),k(s),g(s)) ds k () =z (), (z (),k(),g()) S, plus an anicipaed sequence of policies, {g ()} =0, and he iniial value of k (0). The poin is ha, as well as in models wih exernal effecs of capial, he privae agens selec heir opimal plan by aking he governmen s acions as exernal effecs. The necessary condiions for an opimum includes p () = v z (z (),k(),g()), (30) ṗ () =ρp () v k (z (),k(),g()). (31) From (30) he opimum invesmen a ime can be wrien as z () =z (k (),p(),g()). (32) Obviously, if he policy variables are kep consan over ime (g () =g for all 0), he fundamenal formula (1) sill holds, so ha NNP can fully capure he presen value of curren and fuure consumpion. Similarly, if he governmen selecs he sequence of {g ()} =0 wihou considering he privae secor s behavior and if privae agens perfecly anicipae {g ()} =0, he policy inervenion plays essenially he same role as ha of exogenous echnical change so ha we obain (12). On he oher hand, if he policy maker adops a feedback rule ha relaes g () o he curren levels of z () and k () in such a way ha g () =θ (z (),k()) ; θ : R n R n + R r, hen he effecs of policy disorion are similar o he disorion generaed by exernal effecs of capial. In paricular, if he policy variables depend on he level of capial socks alone, i.e. g () =θ (k ()), he role of policy disorion is exacly he same as ha of capial exernaliies. In his case, (20) becomes Y () = e ρ(s ) c (s) ds e ρ(s ) v g (k(s),z(s),g(s)) θ k (k (s)) z (s) ds. (33) 16

19 The second herm in he righ-hand-side of (33) shows he oal welfare effec of policy inervenion evaluaed by consumpion. 12 The above argumen has assumed ha he policy maker s behavior is exogenously specified. Now suppose ha he benevolen governmen chooses is policy variables in order o maximize he welfare of he privae secor. If his is he case, he privae agens selec heir invesmen and consumpion plans under a given sequence of policies deermined by he policy maker. On he oher hand, policy maker decides is opimal policies subjec o he opimizing behavior of he privae agens as well as o is own consrains such as he budge and resource consrains. Formally speaking, his kind of problem can be formulaed as a Sackelberg differenial game in which he policy maker plays a leader s role and he privae agens are followers. In general, he opimal sraegy of he policy maker may be presened eiher as open-loop policies or as he feedback policies. We will examine hose alernaive sraegies in urn. Firs assume ha he governmen adops an open-loop policy under which each policy variable depends on ime alone. By aking (31) and (32) ino accoun, he governmen maximizes subjec o 0 e ρ v (z (k (),p(),g()),k(),g()) d k () =z (k (),p(),g()), (34) ṗ () =ρp () v k (z (k (),p(),g()),k(),g()) (35) Namely, he policy maker deermines he opimal sequence of g () Γ in order o maximize a discouned sum of indirec uiliies of he represenaive agen under he consrains of dynamic behaviors of capial socks and he marke prices. 13 problem is given by H = v (z (k (),p(),g()),k(),g()) + q () z (k (),p(),g()) The curren value Hamilonian for his + ψ ()[ρp () v k (z (k (),p(),g()),k(),g())], where q () and ψ () respecively denoe he cosae variables for k () and p (). The open-loop soluion for an opimum should saisfy he following condiions: H g = v z z g + v g + q () z g ψ ()(v kg z g + v kg )=0, (36) 12 Noe ha θ (k ()) is a vecor in R r so ha θ k (k ()) denoes an r n marix. Hence, v gθ k z is a scalor. 13 The governmen s opimizaion is also subjec o he ransversaliy condiion for he households plan. 17

20 q () =ρq () v k v z z k q () z k + ψ ()(v kz z k + v kk ), (37) ψ () =ψ () v kz z p q () z p v z z p, (38) ψ (0) = 0, (39) lim e ρ q () k () =0, (40) plus (34) and (35). 14 Since he iniial value of p () is no predeermined, he corresponding cosae variable, ψ (), should saisfy he ransversaliy condiion a he ouse of he planning. Condiion (39) presens he ransversaliy condiion on ψ () a =0. 15 Examining he condiions displayed above, we find: Proposiion 2 Suppose ha he opimizing governmen adops an open-loop policy. Then a he iniial period ( =0) Weizman s rule (1) is modified as Y (0) = ρ 0 e ρ c () d +[p (0) q (0)] z (0), (41) where p (0) and q (0) respecively denoe he privae and social prices of capial ha saisfy v g (z (0),k(0),g(0)) + [q (0) p (0)] z g (0) = 0. (42) Proof. Since he governmen s problem is a sandard opimal conrol problem in which p () and k () are sae variables, he Hamilon-Bellman-Jacobi equaion is given by ρw (k (),p()) = max {v (z (k (),p(),g()),k(),g()) g() Γ + W k (k (),p()) z (k (),p(),g()) + W p (k (),p()) [ρp () v k (z (k (),p(),g()),k(),g())]}, where W (.) is he value funcion which saisfies W k (.) =q () and W p (.) =ψ (). Thus (39) means ha a =0we have ρw (k (0),p(0)) = max {v (z (k (0),p(0),g(0)),k(0),g(0)) g(0) Γ + W k (k (0),p(0)) z (k (0),p(0),g(0)). 14 In he necessary condiions for an opimum, z g,v gk and v zz are r n, n r and n n marices, respecively. 15 See, for example, Bryson and Ho (1975, pp.56-57). 18

21 This yields 0 e ρ c () d = c (0) + q (0) z (0). Thus Y (0) c (0) + p (0) z (0) and he above equaion presen (41). Addiionally, by use of (30), (36) and (39) we obain (42). In general, ψ () 6= 0for >0, so ha he behavior of p () affecs he privae agens decision. Thus he gap beween NNP and he welfare equivalence of maximized consumpion equals [q () p ()] z ()+ψ () ṗ () for >0. The firs erm in his expression, [q () p ()]z (), is he social value of invesmen ha is no capured by he marke evaluaion. The second erm, ψ () ṗ (), shows he capial gain from he social perspecive. Unless he economy is in he seady sae where z () =ṗ () =0, NNP generally diverges from he welfare equivalen wealh by reflecing hose wo erms. As is well known, he open-loop policy is generally ime inconsisen. This is because if he governmen reopimizes a period ˆ (> 0), he ranseversaliy condiion requires ha ψ ˆ =0. However, he value of ψ () deermined by he original program a =0does no necessarily saisfy ψ ˆ =0. Thus he original plan deermined a =0is subopimal if i is re-evaluaed a = ˆ > 0. This indicaes ha ime consisency requires ha he governmen akes a Markov feedback rule raher han an open-loop sraegy. Suppose ha opimal invesmen level seleced by he privae agens is given by z () =z (k (),g()). (43) In words, z () does no depends on he marke price p (). If his is he case, he governmen s opimizaion behavior is o maximize subjec o 0 e ρ v (z (k (),g()),k(),g()) d k = z (k (),g()). Since in his case moion of he forward-looking variable, p (), do no bind he governmen s planning, he opimizaion problem for he governmen is a sandard conrol problem. The Hamilon-Jacobi-Bellman equaion for his problem is given by ρw (k ()) = max g() Γ {v (z (k (),g()),k(),g()) + W k (k ()) z (k (),g())}. (44) 19

22 The firs-order condiion is v z (z (k (),g()),k(),g()) z g (k (),g()) + v g (z (k (g),g()),k(),g()) which gives he opimal feedback rule in such a way ha + W k (k ()) z g (k (),g()) = 0, g () =θ (k ()). (45) Observe ha, unlike (33), he above rule is derived by solving he opimizaion problem for he governmen. If he policy rule is given by he above, i is easy o see ha he modified version of (1) is as follows: Proposiion 3 If he governmen adops a feedback policy rule, he following holds for all 0: Y () =ρ where p () and q () saisfies Proof. e ρ(s ) u (s) ds +[q () p ()]z (), (46) v g (z (),k(), θ (k ())) + [q () p ()] z g () =0. (47) Since W k (k ()) = q (), equaion (44) is wrien as ρ e ρ(s ) u (s) ds = u ()+q () z (). Hence, using Y () =c ()+p () z (), we obain (44). In addiion, (30), (42) and W 0 (k ()) = q () gives (47). Consequenly, if he policy maker can ake a Markov feedback sraegy, he modified rule (41) ha is saisfied only a =0under he open-loop policy holds for all 0. As shown by (46), he difference beween NNP and he welfare equivalence of maximized consumpion reflecs he divergence beween he marke and social values of capial. Since sign of [q () p ()]z () canno be specified wihou imposing furher condiions, accuracy of NNP as an indicaor of welfare measure depends on he policyruleakenbyhegovernmen. Forexample, if boh k () and g () are scalars, (47) becomes q () p () = v g (z (k (),g()),k(),g()). z g (k (),g()) 20

23 Therefore, if policy inervenion has opposie impacs on consumpion and invesmen, ha is, eiher if v g < 0 and z g > 0 or if v g > 0 and z g < 0), hen NNP underesimaes (overesimaes) he welfare equivalen wealh according o ne invesmen z is posiive (negaive). Conversely, if v g and z g have he same sign, we obain he opposie oucome. 4.2 An Example: Opimal Income Taxaion As a ypical example, le us consider a simple model of dynamic opimal axaion. There is a single good and is producion funcion is given by y = k α, 0 < α < 1, (48) where y and k denoe per capia oupu and capial, repeiively. The governmen levies a proporional income ax on oupu o finance is spending. We assume ha he governmen does no rely on deb finance. Leing g (0, 1) be he rae of income ax and s be he per capia governmen s spending, he flow budge consrain for he governmen is s = gy. (49) The households feliciy depends on consumpion, c, and he public expendiure, s. The represenaive agen maximizes a discouned sum of uiliies U = 0 e ρ u (c, s) d, where he insananeous uiliy funcion is specified as u (c, s) = c1 σ 1 1 σ + s1 ξ 1 ;0< σ, ξ < 1. (50) 1 ξ Here, we assume ha he public spending has a posiive effec on he households welfare. The flow budge consrain for he household is he following: 16 (1 g) y = c + z + δk. (51) Therefore, he aggregae consumpion is defined as c = [(1 g) kα z δk] 1 σ 1 1 σ + (gkα ) 1 ξ 1 1 ξ v (z, k,g). 16 Observe ha in view of (49), (51) also shows ha he marke equilibrium condiion for he final good. 21

24 The household maximizes R 0 v (z, k, g) e ρ d subjec o k = z and an anicipaed sequence of {g (),s()} =0. Denoing he marke price of capial by p, he opimizaion condiions for he household include he following: c σ =[(1 g) k α z δk] σ = p, (52) ṗ = p ρ + δ α (1 g) k α 1, (53) k = z, (54) lim p () e ρ k () =0. (55) Firs, assume ha he governmen adops an open-loop sraegy. By aking is own budge consrain (49) ino accoun, he fiscal auhoriy conrols he rae of ax g ( [0, 1)) o maximize 0 " # e ρ p 1 1 σ 1 1 σ + (gkα ) 1 ξ 1 d 1 ξ subjec o (53), (54), he iniial condiion k (0) = k 0, and he households ransversaliy condiion (55). Se up he Hamilonian funcion for he governmen s problem: H = p1 1 σ 1 1 σ + (gkα ) 1 ξ 1 i + q h(1 g) k α p 1 σ δk + ψ ρ + δ α (1 g) k α 1. 1 ξ The opimum condiions are (53), (54) and he following: (gk α ) ξ = q αψ k, (56) q = q[ρ + δ α (1 g) k α 1 ] ψα (1 α)(1 g) k α 2 αgk α 1 (gk α ) ξ, (57) ψ = ρψ 1 σ p 1 σ, (58) ogeher wih he ransversaliy condiions: ψ (0) = 0 and lim e ρ q () k () =0. Combining (56) wih (57) gives q = q ρ + δ αk α 1 ψα (1 τ α) k α 2. (59) On he oher hand, (56) yields g = q αψ k 1 ξ k α. (60) 22

25 Subsiuing (60) ino (53), (54), (57) and (58), we obain a complee adynamic sysem wih respec o k,p, q and ψ. In his formulaion, he Hamilon-Jacobi-Bellman equaion yields ρw (k ()) = ū () =u ()+q () z ()+ψ () ṗ () (61) Ne naional produc measured by he marke prices is defined as Y () c () +p () z (). Thus (61) is rewrien as Y () =ρ e ρs u (s) ds [q () p ()] z () ψ () ṗ (). The difference beween NNP and he annuiy equivalen of he discouned presen value of consumpion is expressed as [q () p ()] z ()+ψ() ṗ () " =[q () p ()] k () α µ q () 1 # αψ () ξ p () 1 α δk () k () " + ψ () ρ + δ αk () α 1 + µ q () 1 # αψ () ξ k (). k () Since he iniial values of k () and ψ () are given, if here is a wo-dimensional sable saddle pah converging o he seady sae equilibrium, q () and p () are wrien as funcions of k () and ψ (). Therefore, on he converging saddle pah, he residual erm shown above can be evaluaed numerically by specifying parameer values involved in he model. Nex, examine he case in which he governmen can selec a feedback rule In order o obain a closed-form soluion, we assume ha he inverse of elasiciy of ineremporal subsiuion in consumpion, σ, is equal o α. 17 In his special case, from (51), (52) and (54) we obain he following: ṗ p = ρ + δ α (1 g) kα 1, (62) k k =(1 g) kα 1 1 δ. kp1/α (63) 17 This is one of he special case where he feedback policy rule can be analyically derived. Anoher well known example in which we can rea he feedback rule analyical is o assume ha σ =1so ha u (c, g) = log c + g ξ 1 1 / (1 ξ) :see Xie (1997) and Lansing (1999). Mino (2001) shows ha he similar resuls can be obained in a model wih labor-leisure choice. 23

26 Hence, leing x kp 1/α, (62) and (63) yield: ẋ x = ρ + µ 1 α 1 δ 1 x. This differenial equaion has a unique saionary soluion and i is oally unsable. Therefore, x exhibis a diverging behavior unless x says on he saionary poin a he iniial period. If x diverges from he saionary poin, eiher he ransversaliy condiion or he feasibiliy condiion will be ulimaely violaed. This implies ha x always keeps is seady sae level so ha i holds ha x = ρ +(1/α 1) δ for all 0. As a consequence, he following condiion is saisfied: where μ ρ +(1/α 1) δ. p =(μk) α, (64) Consequenly, using (64), he objecive of he policy maker is o maximize Z " # e ρ (μk) 1 α 1 + (gkα ) 1 ξ 1 d 1 α 1 ξ subjec o 0 k =(1 g) k α (μ + δ) k. The necessary condiions for an opimum involve he following condiions: (gk α ) ξ = q, (65) q = q[(ρ + μ + δ)q α (1 g) k a 1 ] (gk α ) ξ αgk α 1. From he above equaions and μ = ρ +(1/α 1) δ, we have µ q = q 2ρ + δ α 1 αk. (66) α Condiion (65) yields he opimal ax rae: g () =W 0 (k ()) 1 ξ k () α θ (k ()). (67) This gives he feedback conrol rule for g (). Using (67), he value funcion saisfies he following Hamilon-Jacobi-Bellman equaion: ρw (k ()) = (μk ())1 α 1 1 α + (W 0 (k ())) ξ 1 ξ 1 1 ξ + W 0 (k ()) h i k () α W 0 (k ()) 1 ξ (ρ + δ) k (). 24

27 For he purpose of obaining a closer look a he relaionship beween p () and q (), we furher assume ha θ = σ so ha ξ = α. Then we have q q ṗ µ 1 p = ρ + α 1 δ = μ. As a resul, i holds ha q () /p () =e μ [q (0) /p (0)], which yields: μ q (0) q () p () =p () e p (0) 1 This means ha if q (0) >p(0), i holds ha q () >p() for all 0. If q (0) <p(0), hen q () <p() for e μ <p(0) /q (0) and q () >p() for e μ >p(0) /q (0). Remembering ha p (0) = [μk 0 ] α and q (0) = W 0 (k 0 ), he sign of q () p () follows sign [q () p ()] = sign ½ 1 W 0 ¾ μ log (k 0 ) (μk 0 ) α. Suppose ha W 0 (k 0 ) < (μk 0 ) α. Furher, assume ha he iniial level of capial k 0 is less han he seady sae level of k and hus he economy converges o he seady sae wih posiive invesmen Given hose condiions, NNP overesimaes he annuiy equivalen of he presen discouned value of maximized consumpion for 0 <<(1/μ)log W 0 (k 0 ) / (μk 0 ) α. Afer ime passes he criical poin, NNP becomes smaller han he welfare equivalen wealh. In conras, if k 0 exceeds he seady-sae level of k so ha economy converges o he seady sae wih negaive invesmen, we obain he opposie resuls. I is o be poined ou ha he criical ime lengh,(1/μ)log W 0 (k 0 ) / (μk 0 ) α, may be relaively shor. For example, if α =0.4, ρ = δ =0.05, hen μ = Hence, when q (0) /p (0) = W 0 (k 0 ) / (μk 0 ) α =1.2, he criical ime is This shows ha NNP is over he welfare equivalen wealh for less han wo years if k 0 is smaller han he seady sae level of k. When q (0) /p (0) = 2.0, he criical ime is If capial involves human capial so ha α has a higher value such as 0.8, we have μ = This he swiching ime poin is if q (0) /p (0) = 2.0, whileiis2.91if q (0) /p (0) = Conclusion This paper has generalized Weizman s rule by inroducing marke disorions ino he basic framework. If a marke economy saisfies he ideal condiions so ha i mimics he behavior 25

28 of he corresponding command economy, Weizman s fundamenal resul ensures ha ne naional produc serves as an exac welfare measure. However, since marke economies generally fail o aain he command opimum because of he presence of various forms of disorions, here may exis a gap beween ne naional produc and he annuiy equivalen of he discouned presen value of maximized consumpion. In his paper, we have examined models wih marke disorions and have explored how he fundamenal rule is modified if disorions conaminae he basic relaionship beween naional income and welfare. I is o be noed ha when he marke economy does no aain he command opimum, he welfare significance of naional income depends upon he magniude of he conaminaion effec generaed by marke disorions. As some of he numerical examples in Secion 3.2 show, he divergence beween NNP and he discouned presen value of curren and fuure consumpion can be sufficienly small. In such a case, NNP is sill a good measure of welfare. If he divergence has a significanly large value, we should be careful in using NNP as a represenaion of he welfare equivalence wealh. When evaluaing he conaminaion effec, we should derive explici forms of modified Weizman s rules ha ake he disorionary effecs ino accoun. Needless o say, he modified rules presened in his paper are far from general. We have only ouched upon he issue by exploring specific examples. Theincome-welfare relaionship in alernaive marke environmens deserves furher invesigaion. 26

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