NATIONAL BANK OF BELGIUM

Transcription

1 NATIONAL BANK OF BELGIUM Conference 150th annversary WORKING PAPERS - RESEARCH SERIES GROWTH IN AN OPEN ECONOMY: SOME RECENT DEVELOPMENTS Stephen J. Turnovsky (*) The vews expressed n ths paper are those of the author and do not necessarly reflect the vews of the Natonal Bank of Belgum. (*) Unversty of Washngton, Seattle WA NBB WORKING PAPER No.5 - MAY

2 Edtoral Drector Jan Smets, Member of the Board of Drectors of the Natonal Bank of Belgum Statement of purpose: The purpose of these workng papers s to promote the crculaton of research results (Research Seres) and analytcal studes (Documents Seres) made wthn the Natonal Bank of Belgum or presented by outsde economsts n semnars, conferences and colloqua organsed by the Bank. The am s thereby to provde a platform for dscusson. The opnons are strctly those of the authors and do not necessarly reflect the vews of the Natonal Bank of Belgum. The Workng Papers are avalable on the webste of the Bank: Indvdual copes are also avalable on request to: NATIONAL BANK OF BELGIUM Documentaton Servce boulevard de Berlamont 14 B Brussels 2 NBB WORKING PAPER No.5 - MAY 2000

3 Imprnt: Responsblty accordng to the Belgan law: Jean Hlgers, Member of the Board of Drectors, Natonal Bank of Belgum. Copyrght Natonal Bank of Belgum Reproducton for educatonal and non-commercal purposes s permtted provded that the source s acknowledged. ISSN: X NBB WORKING PAPER No.5 - MAY

4 Edtoral On May 11-12, 2000 the Natonal Bank of Belgum hosted a Conference on "How to promote economc growth n the euro area?". A number of papers presented at the conference s made avalable to a broader audence n the Workng Papers seres of the Bank. Ths volume contans the frst of these papers. The other fve papers were ssued as Workng Papers Abstract Ths paper dscusses some of the recent developments n growth theory, dong so from the perspectve of a small open economy. After settng out a basc generc model, we show how t may yeld two of the key models that have played a promnent role n the recent lterature, the endogenous growth model and the non-scale growth model. We focus ntally on the former, emphaszng how the smplest such model leads to an equlbrum n whch the economy s always on ts balanced growth path. One aspect of the model s the mportance of fscal polcy as a determnant of the equlbrum growth rate, an aspect that s dscussed n detal. We also show how the endogenety or otherwse of the labor supply s crucal n determnng the equlbrum growth rate and ts responsveness to macroeconomc polcy. But transtonal dynamcs are an mportant aspect of the growth process and ndeed much research has been drected to determnng the speed wth whch the economy converges to ts balanced growth path. We dscuss alternatve ways that such transtonal dynamcs may be ntroduced. These nclude () restrcted access to the world captal market; () the ntroducton of government captal, and () the two-sector producton model, poneered by Lucas. In the orgnal analyss, the two captal goods relate to physcal and human captal and n the nternatonal context these naturally can be dentfed wth traded and nontraded captal, respectvely. Crtcsm of the endogenous growth model has led to the development of the nonscale growth model. Ths too s characterzed by transtonal dynamcs, whch are more flexble than those of the correspondng endogenous growth model. Ths model s much closer to the neoclasscal model; n partcular, the long-run growth rate s ndependent of macroeconomc polcy. However, snce such models are typcally assocated wth slow NBB WORKING PAPER No.5 - MAY

5 convergence speeds, polcy can nfluence the accumulaton of captal for extended perods of tme, leadng to sgnfcant long-run level effects. The dscusson seeks to emphasze the adaptablty of the models to a wde range of ssues. A fnal extenson addresses the mpact of volatllty on growth. Ths has been extensvely analyzed emprcally and a stochastc extenson of the endogenous growth model provdes a convenent framework wthn whch to nterpret ths research. 2 NBB WORKING PAPER No.5 - MAY 2000

6 TABLE OF CONTENTS: 1. INTRODUCTION Some Background Scope of ths Paper A CANONICAL MODEL OF A SMALL OPEN ECONOMY THE ENDOGENOUS GROWTH MODEL Inelastc Labor Supply Taxes, Growth, and Welfare Elastc Labor Supply Comparson wth Fxed Employment AK Model PRODUCTIVE GOVERNMENT EXPENDITURE The Barro-Model Optmal Fscal Polcy TRANSITIONAL DYNAMICS AND GROWTH Upward Slopng Supply Curve of Debt Publc and Prvate Captal TWO-SECTOR GROWTH MODEL WITH TRADED AND NONTRADED CAPITAL Outlne of the Model Determnaton of Macroeconomc Equlbrum NON-SCALE GROWTH MODEL Inelastc Labor Supply Elastc Labor Supply CONVERGENCE VOLATILITY AND GROWTH A Canoncal Model of a Stochastc Small Open Economy CONCLUDING COMMENTS...61 NBB WORKING PAPER No.5 - MAY

7 1. INTRODUCTION Economc growth s arguably the ssue of prmary concern to economc polcy makers n both developed and developng economes. Economc growth statstcs are among the most wdely publczed measures of economc performance and are always dscussed wth nterest. As a consequence growth theory has long occuped a central role n economcs. 1.1 Some Background Long-run growth was frst ntroduced by Solow (1956) and Swan (1956) nto the tradtonal neoclasscal macroeconomc model by ntroducng a growng populaton coupled wth a more effcent labor force. The drect consequence of ths approach s that the longrun growth rate n these models s ultmately ted to demographc factors, such as the growth rate of populaton, the structure of the labor force, and ts productvty growth (technologcal change), all of whch were typcally taken to be exogenously determned. Hence, the only polces that could contrbute to long-run growth were those that would ncrease the growth of populaton, and manpower tranng programs amed at ncreasng the effcency of the labor force. Conventonal macroeconomc polcy had no nfluence on long-run growth performance. Snce then, growth theory has evolved nto a volumnous lterature, n two dstnct generatons of models. The Solow-Swan model was the nspraton for a frst generaton of growth models durng the 1960s, whch, beng assocated wth exogenous sources of longrun growth, are now sometmes referred to as exogenous growth models. Research nterest n these models tapered off around 1970, as economsts turned ther attenton to other ssues, perceved as beng of more mmedate sgnfcance, such as nflaton, unemployment, and ol shocks, and the desgn of macroeconomc polces to deal wth them. Begnnng wth the semnal work of Romer (1986), there has been a resurgence of nterest n economc growth theory gvng rse to a second generaton of growth models. Ths revval of actvty has been motvated by several ssues, whch nclude: () an attempt to explan aspects of the data, not dscussed by the neoclasscal model; () a more satsfactory explanaton of nternatonal dfferences n economc growth rates; () a more central role for the accumulaton of knowledge; and (v) a larger role for the nstruments of macroeconomc polcy n explanng the growth process; see Romer (1994). These new models seek to explan the long-run growth rate as an endogenous equlbrum outcome of the behavor of ratonal optmzng agents, reflectng the structural characterstcs of the NBB WORKING PAPER No.5 - MAY

8 economy, such as technology and preferences, as well as macroeconomc polcy. For ths reason they have become known as endogenous growth models. The new growth theory s far-rangng. It has been analyzed n both closed economes, as well as n open economes. In fact, one of the characterstcs of the new growth theory s that t has more of an nternatonal orentaton; see e.g. Grossman and Helpman (1991). Ths may reflect the ncreased mportance of the nternatonal aspects n macroeconomcs n general. In comparson wth the frst generaton of growth models, the newer lterature places a greater emphass on emprcal ssues and the reconclaton of the theory wth the emprcal evdence. In ths respect a wdely debated ssue concerns the socalled convergence hypothess. The queston here s whether or not countres have a tendency to converge to a common per capta level of ncome. But new growth theory s also assocated wth mportant theoretcal advances as well, and one can dentfy two man strands of theoretcal lterature, emphaszng dfferent sources of economc growth. One class of models, closest to the neoclasscal growth model, stresses the accumulaton of prvate captal as the fundamental source of economc growth. Ths dffers n a fundamental way from the neoclasscal growth model n that t does not requre exogenous elements, such as a growng populaton to generate an equlbrum of ongong growth. Rather, the equlbrum growth s nternally generated, though n order to acheve that, certan restrctons relatng to homogenety must be mposed on the economc framework. Some of these restrctons are of a knfe-edge character and have been the source of crtcsm; see e.g. Solow (1994). In the smplest such model, n whch the only factor of producton s captal, the constant returns to scale condton mples that the producton functon must be lnear n physcal captal, beng of the functonal form Y = AK. For obvous reasons, ths technology has become known as the "AK model". As a matter of hstorcal record, explanaton of growth as an endogenous process n a one-sector model s not new. In fact t dates back to Harrod (1939). The equlbrum growth rate characterzng the AK model s essentally of the Harrod type, the only dfference beng that consumpton (or savngs) behavor s derved as part of an ntertemporal optmzaton, rather than beng posted drectly. These one-sector models assume (often only mplctly) a broad nterpretaton for captal, takng t to nclude both human, as well as nonhuman, captal; see Rebelo (1991). A drect extenson to ths basc model are two-sector nvestment based growth models, orgnally 2 NBB WORKING PAPER No.5 - MAY 2000

9 due to Lucas (1988), that dsaggregate prvate captal nto human and nonhuman captal; see also Mullgan and Sala--Martn (1993) and Bond, Wang, and Yp (1996). A second class of models, emphaszes the endogenous development of knowledge, or research and development, as the engne of growth. The basc contrbuton here s that of Romer (1990), whch develops a two-sector model of a closed economy, where new knowledge produced n one sector s used as an nput n the producton of fnal output. The knowledge sector has been extended n varous drectons by a number of authors; see e.g. Aghon and Howtt (1992) and more recently, Echer (1996). A related class of models deals wth nnovaton and the dffuson of knowledge across countres, and a comprehensve dscusson s provded by Barro and Sala--Martn (1995, Chapter 8). One s begnnng to see a merger evolvng between the old and the new growth theory. The new growth models are often characterzed by havng scale effects, meanng that varatons n the sze or scale of the economy, as measured by say populaton, affect the sze of the long-run growth rate. For example, the Romer (1990) model of research and development mples that a doublng of the populaton devoted to research wll double the growth rate. Whether the AK model s assocated wth scale effects depends upon whether there are producton externaltes that are lnked to the sze of the economy; see Barro and Sala--Martn (1995). By contrast, the neoclasscal Solow model has the property that the equlbrum growth rate s ndependent of the scale (sze) of the economy; t s therefore not subject to such scale effects. Emprcal evdence does not support the exstence of scale effects. For example, Jones (1995a) fnds that varatons n the level of research employment have exerted no nfluence on the long-run growth rates of the OECD economes. Backus, Kehoe, and Kehoe (1992) fnd no conclusve emprcal evdence of any relatonshp between US GDP growth and measures of scale. These emprcal observatons are begnnng to stmulate nterest n the development of non-scale models. Such models are hybrds n the sense that they share some of the characterstcs of the neoclasscal model, yet ther equlbrum s derved from ntertemporal optmzaton as n the new growth models 1. Jones (1995b) has proposed a specfc model, n whch the steady-state growth rate s determned by the growth rate of populaton, n conjuncton wth certan producton elastctes, n hs case pertanng to the knowledge producng sector. 1 Jones (1995a) referred to such models as sem-endogenous growth models. NBB WORKING PAPER No.5 - MAY

10 1.2 Scope of ths Paper It s beyond the scope of ths paper to present an exhaustve dscusson of growth theory. That s a specalzed topc n tself and for that the reader should refer to Grossman and Helpman (1991), Barro and Sala--Martn (1995), and Aghon and Howtt (1998) who provde comprehensve treatments of the subject from dfferent perspectves. Rather, the purpose of ths paper s to expost the nvestment-based growth models, but from an nternatonal perspectve. In followng ths approach, we try to cover the man developments, but from a vewpont that hopefully s of greater relevance for a small open economy such as Belgum. We begn our dscusson n Secton 2 by expostng a canoncal model of a small open economy, whch s suffcently general to encompass alternatve models. Sectons 3 and 4 then develop two alternatve versons of the AK growth model. Such models have been extensvely used to analyze the effects of fscal polcy on growth performance; see e.g. Barro (1990), Kng and Rebelo (1990), Jones and Manuell (1990), Rebelo (1991), Jones, Manuell, and Ross (1993), Ireland (1994) and Turnovsky (1996a). Most of these endogenous growth models have been developed for a closed economy, although some applcatons to an open economy are begnnng to develop; see Razn and Yuen (1992, 1996), Rebelo (1992), van der Ploeg (1996), Mno (1996), Turnovsky (1996b, 1996d, 1997c), van der Ploeg (1996), Baldwn and Forsld (1999, 2000). Secton 3 begns wth the smplest Romer (1986) model wth fxed labor supply and then modfes ths model to the case where labor s suppled elastcally. It emphaszes how gong from one assumpton to the other fundamentally changes the determnaton of the equlbrum growth rate and the mpact of fscal polcy. Secton 4 dscusses the Barro (1990) model where government expendture s productve and analyzes optmal fscal polcy n that settng. These ntal models all abstract from transtonal dynamcs, so that the economy s always on ts balanced growth path. Ths s an obvous lmtaton and Secton 5 dscusses n some detal two ways that transtonal dynamcs may be ntroduced. The frst s through restrcted access to world fnancal markets, n the form of ncreasng borrowng costs, lkely to be relevant for a small developng economy. The second s through the ntroducton of government captal, so that n contrast to the Barro model, government expendture mpacts producton as a stock, rather than as a flow. But transtonal 4 NBB WORKING PAPER No.5 - MAY 2000

11 dynamcs can also be ntroduced n other ways, and these are dscussed n the next two sectons. Secton 6, dscusses the case where the producton technology s augmented to two sectors, a traded and a nontraded sector, showng the nature of the dynamcs ths ntroduces. The two sector model, where the two sectors consst of physcal (nonhuman) and human captal, respectvely, was one of the orgnal models of endogenous growth poneered by Lucas (1988). Other authors to analyze the two-sector model nclude: Mullgan and Sala--Martn (1993), Devereux and Love (1994), and Bond, Wang and Yp (1996). Ths aspect s partcularly relevant for nternatonal economes. As we have already noted, the endogenous growth model has been subject to crtcsm along two lnes. Frst, t s often assocated wth scale effects meanng that longrun growth rates are lnked to the sze of the economy, a characterstc that s not supported by the emprcal evdence. Second, t holds only f strct knfe-edge condtons on the technology hold. In response to ths, we have seen the development of non-scale growth models, whch have the property that long-run growth rates are ndependent of the scale of the economy. Ths model s also assocated wth transtonal dynamcs and s dscussed n Secton 7. An mportant ssue that has receved promnent attenton n the recent lterature on growth theory concerns the noton of convergence. Ths ssue has two aspects: () whether or not economes converge to a common growth rate, and () the rate at whch the economy converges along ts transtonal growth path. Ths s dscussed n Secton 8. The speed of convergence s mportant, because even f polcy s unable to affect long-run growth rates, t can nevertheless have sgnfcant long-run level effects f the rates of convergence are suffcently slow, as much emprcal evdence suggests. The fnal ssue addressed s the relatonshp between volatlty and growth, dscussed n Secton 9. Ths ssue has been the subject of emprcal nvestgaton, and the growth model developed n ths paper provdes a natural analytcal framework for assessng ths work. Throughout the paper, our man objectve s to expost the structures of the varous models n ther basc form rather than analyzng any n detal. The models provde powerful analytcal tools that can be adapted to varous needs and crcumstances. One key ssue that dstngushes the endogenous growth model from the non-scale model s the mpact of polcy on the long-run equlbrum growth rate. Before embarkng further, we NBB WORKING PAPER No.5 - MAY

12 should acknowledge that the emprcal evdence pertanng to ths ssue s mxed. If one takes the evdence on non-scale growth models serously, and accepts that the long-run growth rate s determned as suggested by Jones (1995b), the scope for fscal polcy s lmted, although less so than n the Solow model. Indeed, emprcal evdence by Easterly and Rebelo (1993) and Stokey and Rebelo (1995) suggests that the effects of tax rates on long-run growth rates are nsgnfcant, or weak at best. Stokey and Rebelo argue that ther fndngs provde evdence aganst those models, such as AK models, that predct large growth effects from taxaton. In order for the predctons of these models to be consstent wth ther evdence, these growth effects would have to be largely offset by changes n other determnants of the long-run growth rate. But other studes such, as Grer and Tullock (1989), Barro (1991), and Barro and Lee (1994) obtan negatve relatonshps between growth and government consumpton expendture, whle Barro and Lee also fnd that government expendture on educaton has a postve effect on growth. Taken together, we do not vew the emprcal evdence as necessarly contradctng the ablty of fscal polcy to nfluence the growth rate. It may well be the case that a hgher ncome tax has a sgnfcant negatve effect on the growth rate, but that ths s roughly offset by a sgnfcant postve growth effect of the productve government expendture t may be fnancng, thus havng a small overall net effect. Indeed, the welfare-maxmzng rate of taxaton n the smple Barro (1990) model of productve government expendture concdes wth the growth-maxmzng tax rate, so that f the tax rate s n fact close to beng optmal there should be lttle effect on the growth rate, precsely as the emprcal evdence seems to suggest. But to understand ths relatonshp, t s mportant to develop a model n whch the varous components of fscal polcy are ntroduced explctly, and ther separate and possbly conflctng effects on the growth rate analyzed. It s n ths ven that we vew the AK model as provdng an nstructve framework for analyzng fscal polcy on growth. 6 NBB WORKING PAPER No.5 - MAY 2000

13 2. A CANONICAL MODEL OF A SMALL OPEN ECONOMY We begn by descrbng the generc structure of a small open economy whch consumes and produces a sngle traded commodty. There are N dentcal ndvduals, each of whom has an nfnte plannng horzon and possesses perfect foresght. Each agent s endowed wth a unt of tme that can be allocated ether to lesure, l, or to labor, (1 - l). Labor s fully employed so that total labor supply, equal to populaton, N, grows exponentally at the steady rate N = nn. Indvdual domestc output, Y, of the traded commodty s determned by the ndvdual's prvate captal stock, K, hs labor supply, (1 - l), and the aggregate captal stock K = NK 2. In order to accommodate growth under more general assumptons wth respect to returns to scale, we assume that the output of the ndvdual producer s determned by the Cobb-Douglas producton functon 3 : Y 1 σ σ η = α(1 l) K K 0 σ < 1, η 0 < > < (1a) Ths formulaton s akn to the earlest endogenous growth model of Romer (1986). The spllover receved by an ndvdual from the aggregate stock of captal can be motvated n varous ways. One s to nterpret K as knowledge captal, as Romer suggested. Another, s to assume N specfc nputs (subscrpted by ) wth aggregate K representng an ntra-ndustry spllover of knowledge 4. Each prvate factor of producton has postve but dmnshng margnal physcal productvty. To assure the exstence of a compettve equlbrum the producton functon exhbts constant returns to scale n the two prvate factors [Romer (1986)]. In contrast to the standard neoclasscal growth model, we do not nsst that the producton functon exhbts constant returns to scale, and ndeed total returns to scale are 1 + η, and are ncreasng or decreasng, accordng to whether the spllover from aggregate captal s postve or negatve. As we wll show n subsequent sectons, the producton functon s suffcently general to encompass a varety of models. For example, we wll show that the model s Snce all agents are dentcal, all aggregate quanttes are smply multples of the ndvdual quanttes, X = NX. Note that snce all agents allocate the same share of tme to work, there s no need to ntroduce the agent s subscrpt to l. When producton functons exhbt non-constant returns to scale n all factors, the exstence of a balanced growth equlbrum requres the producton functon to be Cobb-Douglas, as assumed n (1a); see Echer and Turnovsky (1999a). negatve exponent can be nterpreted as reflectng congeston, along the lnes of Barro and Sala--Martn (1992a). NBB WORKING PAPER No.5 - MAY

14 consstent wth long-run stable growth, provded returns to scale are approprately constraned. Ths contrasts wth models of endogenous growth and externaltes n whch exogenous populaton growth can be shown to lead to explosve growth rates; see Romer (1990). We should also pont out that the standard AK model emerges when σ + η = 1, n = 0, and the neoclasscal model corresponds to η = 0. Aggregate consumpton n the economy s denoted by C, so that the per capta consumpton of the ndvdual agent at tme t s C /N = C, yeldng the agent utlty over an nfnte tme horzon represented by the ntertemporal soelastc utlty functon: γ θ ρt ( 1 γ)( C l ) e dt; < γ < 1; θ > 0,1 > γ(1 + θ), > γθ Ω 1 0 (1b) where l /(1 γ) equals the ntertemporal elastcty of substtuton, and θ measures the substtutablty between consumpton and lesure n utlty 5. The remanng constrants on the coeffcents n (1b) are requred to ensure that the utlty functon s concave n the quanttes C and l. Agents accumulate physcal captal, wth expendture on a gven change n the captal stock, I, nvolvng adjustment (nstallaton) costs that we ncorporate n the quadratc (convex) functon Φ 2 ( I,K ) I + hi /2K = I ( 1 + hi /2K ) (1c) Ths equaton s an applcaton of the famlar Hayash (1982) cost of adjustment framework, where we assume that the adjustment costs are proportonal to the rate of nvestment per unt of nstalled captal (rather than ts level). The lnear homogenety of ths functon s necessary f a steady-state equlbrum havng ongong growth s to be sustaned Ths form of utlty functon s consstent wth the exstence of a balanced growth path; see Ladron et al (1997). The specfcaton n (1b) s the case of pure lesure; they also consder the case where utlty derved from lesure depends upon ts nteracton wth human captal. Many applcatons of the cost of adjustment n the Ramsey model assume that adjustment costs depend upon the absolute rate of nvestment, rather than ts rate relatve to the sze of the captal stock. They also often assume only that t s convex; the assumpton of a quadratc functon s made for convenence, smplfyng the soluton for the equlbrum growth rates n the endogenous growth model 8 NBB WORKING PAPER No.5 - MAY 2000

15 Convex adjustment costs are a standard feature of models of captal accumulaton n small open economes wth tradable captal facng a perfect world captal market, beng necessary for such models to gve rse to nondegenerate dynamcs; see Turnovsky (1997a). They are, however, less common n endogenous growth models of closed economes, whch typcally treat the accumulaton of captal as beng determned resdually; see e.g. Rebelo (1991), Barro (1990) 7. Adjustment costs turn out to have at least two mportant roles n ths model, partcularly n the basc AK verson of the model to be dscussed n Secton 3. Frst, they may preclude the exstence of a steady-state equlbrum growth path. Second they ntroduce an mportant flexblty nto the equlbratng process. In equlbrum, the after-tax rates of return on the two assets avalable to the economy, traded bonds and captal, must be equal. Gven the lnear technology, the margnal physcal product of captal s also constant, so that the equalty between these two after-tax rates of return n general constrans the feasble choce of tax rates. By contrast, the presence of adjustment costs ntroduces a varable shadow value of captal (the Tobn q), whch equlbrates the rates of return on these two assets, for any arbtrarly specfed tax rates. For smplcty we assume that the captal stock does not deprecate, so that the net rate of captal accumulaton s gven by: K = I nk (1d) In addton, agents have unrestrcted access to a world captal market, beng able to accumulate foregn bonds, B, whch pay an exogenously determned fxed rate of return, r. We shall assume that ncome from current producton s taxed at the rate τ y, ncome from bonds s taxed at the rate τ b, whle n addton, consumpton s taxed at the rate τ c. We shall llustrate the contrastng mplcatons of dfferent models by analyzng the purely dstortonary aspects of taxaton and assume that revenues from all taxes are rebated to the agent as lump sum transfers T. Thus the ndvdual agent's nstantaneous budget constrant s descrbed by: ( 1 τy ) Y + r( 1 τ ) [ b n] B ( 1 + τc ) C I[ 1 + ( h /2)( I /K )] T B = + (1e) 7 There are some exceptons; see Turnovsky (1996c) n a closed economy. Baldwn and Forsld (1999, 2000) emphasze the q-theoretc approach n an open economy. NBB WORKING PAPER No.5 - MAY

16 The agent's decsons are to choose hs rates of consumpton, C, lesure, l, nvestment, I, and asset accumulaton, B, K, to maxmze the ntertemporal utlty functon, (1a), subject to the accumulaton equatons, (1d) and (1e). The dscounted Hamltonan for ths optmzaton s: H e + q e ρt ρt 1 γ θ γ ρt ( C l ) + λe (1 τ )Y Φ (1 + τ )C + r(1 τ )B T B I nk K y c b where λ s the shadow value of wealth n the form of nternatonally traded bonds and q' s the shadow value of the agent s captal stock. Exposton of the model s smplfed by usng the shadow value of wealth as numerare. Consequently, q q' / λ can be nterpreted as beng the market prce of captal n terms of the (untary) prce of foregn bonds. The optmalty condtons wth respect to C, l, and I are respectvely C γ 1 θγ l = λ(1 + τ c ) (2a) θc γ θγ 1 l = λ(1 τ y )(1 σ)y (1 l) (2b) ( /K ) q 1 + h I = (2c) Equaton (2a) equates the margnal utlty of consumpton to the tax-adjusted shadow value of wealth, whle (2b) equates the margnal utlty of lesure to ts opportunty cost, the aftertax margnal physcal product of labor (real wage), valued at the shadow value of wealth. The thrd equaton equates the margnal cost of an addtonal unt of nvestment, whch s nclusve of the margnal nstallaton cost hi / K, to the market value of captal. Equaton (2c) may be solved to yeld the followng expresson for the rate of captal accumulaton: K /K = I /K n = ( q 1) /h n φ (3) 10 NBB WORKING PAPER No.5 - MAY 2000

17 Wth all agents beng dentcal, equaton (3) mples that the growth rate of the aggregate captal stock, s φ = φ + n, so that K K = K K + n = (q 1) h φ (3 ) Ths descrbes a "Tobn q" theory of nvestment, wth K < > 0 accordng to whether q < > 1. Optmzng wth respect to B and K mples the arbtrage relatonshps ρ λ λ = r( 1 τb) n 2 ( 1 τ ) σy /qk ) + q/q + ( q 1) / 2hq = r( 1 τ ) y b (4a) (4b) Equaton (4a) s the standard Keynes-Ramsey consumpton rule, equatng the margnal return on consumpton to the growth-adjusted after-tax rate of return on holdng a foregn bond. Lkewse (4b) equates the after-tax rate of return on domestc captal to the after-tax rate of return on the traded bond. The former comprses three components. The frst s the margnal after-tax output per unt of nstalled captal, (valued at the prce q), whle the second s the rate of captal gan. The thrd element reflects the fact that an addtonal beneft of a hgher captal stock s to reduce the nstallaton costs (whch depend upon I / K ) assocated wth new nvestment. Fnally, n order to ensure that the agent's ntertemporal budget constrant s met, the followng transversalty condtons must be mposed 8 : lm λb e t ρt = 0; lm λqk e t ρt = 0 (4c) The government n ths canoncal economy plays a lmted role. It leves ncome taxes on output and foregn nterest ncome, t taxes consumpton, and then rebates all tax revenues. In aggregate, these decsons are subject to the balanced budget condton τy Y + τbrb + τcc = T (5) 8 The transversalty condton on debt s equvalent to the natonal ntertemporal budget constrant. NBB WORKING PAPER No.5 - MAY

18 Aggregatng (1e) over the N ndvduals, and mposng (5) and (1d) leads to: B = Y + rb C I1 + [ ( h 2)( I K) ] (6) whch descrbes the country s current account. The model can thus be summarzed by the 5 optmalty condtons (2a) (2c), (4a), and (4b), together wth the current account relatonshp, (6). Many models assume that labor s suppled nelastcally, n whch case that the optmalty condton for labor, (2b), ceases to be operatve. 12 NBB WORKING PAPER No.5 - MAY 2000

19 3. THE ENDOGENOUS GROWTH MODEL The nvestment-based endogenous growth model has been the subject of ntensve research snce Most such models assume that labor s suppled nelastcally and, as we shall demonstrate, the endogenety or otherwse of labor s an mportant determnant of the equlbrum growth rate. The key feature of the endogenous growth model s that t s capable of generatng ongong growth n the absence of populaton growth;.e. n = 0. For ths to occur, the producton functon, (1a), must have constant returns to scale n the accumulatng factors, ndvdual and aggregate captal, that s, σ + η = 1 (7) Substtutng ths nto (1a), ths mples ndvdual and aggregate producton functons of the form Y = α η 1 η [(1 l)k ] K ; Y = α[ (1 l)n] η K (8) The ndvdual producton functon s thus constant returns to scale n prvate captal, K, and n labor, measured n terms of effcency unts (1-l)K. Summng over agents, the aggregate producton functon s thus lnear n the endogenously accumulatng captal stock. Note that as long as η 0 so that there s an aggregate externalty, the average (and margnal) productvty of captal depends upon the sze of the populaton. Increasng the populaton, holdng other technologcal characterstcs constant, ncreases the productvty of captal and the equlbrum growth rate. The economy s thus sad to have a scale effect ; see Jones (1995a). Such scale effects run counter to the emprcal evdence and have been a source of crtcsm of the AK growth model; see Backus, Kehoe and Kehoe (1992). These scale effects can be elmnated from the AK model f ether () there are no externaltes ( η = 0 ), or () f the ndvdual producton functon (1a) s modfed to 1 σ σ ( ) η Y = α(1 l) K K N NBB WORKING PAPER No.5 - MAY

20 so that the externalty depends upon the average, rather than the aggregate captal stock; see Mullgan and Sala--Martn (1993). Henceforth throughout ths secton, we shall normalze the sze of the populaton at N = 1 and thereby elmnate the ssue of scale effects. 3.1 Inelastc Labor Supply We begn wth the case where labor s suppled nelastcally,.e. l = l. Wth populaton normalzed, the ndvdual and aggregate producton functons are of the pure AK form: Y = AK ; Y = AK (9) where 1 σ A α(1 l) s a fxed constant. Wth the labor supply fxed, both the margnal and average productvty of captal are constant. The specfcaton of the technology, consstent wth ongong growth, s a very strong knfe-edge condton, one for whch the endogenous growth model has been crtczed; see Solow (1994) 9. To determne the macroeconomc equlbrum, we frst take the tme dfferental of (2a), and then combne the resultng equaton wth (4a), whch, wth fxed employment (and normalzed populaton), mples C r(1 τb) ρ = C 1 γ ψ (10) An mmedate consequence of (10) s that the equlbrum growth rate of domestc consumpton s proportonal to the dfference between the after-tax return on foregn bonds and the (domestc) rate of tme preference. From a polcy perspectve, t also mples that the consumpton growth rate vares nversely wth the tax on foregn nterest ncome, but s ndependent of all other tax rates. Solvng ths equaton mples that the level of consumpton at tme t s C ψt ( t) = C( 0) e (11) 9 Note that the technology (9) s dentcal to that of the orgnal Harrod-Domar model, to whch the AK model s a modern counterpart. It was Harrod, hmself, who orgnally referred to the knfe-edge characterstcs of hs model. 14 NBB WORKING PAPER No.5 - MAY 2000

21 where the ntal level of consumpton C(0) s yet to be determned. The crtcal determnant of the growth rate of captal s the relatve prce of nstalled captal, q, the path of whch s determned by the arbtrage condton (4b). To analyze ths further, we rewrte (4b) as the followng nonlnear dfferental equaton wth constant coeffcents: 2 q = 2h[r(1 τb )q A(1 τy )] (q 1) (12) In order for the captal stock domcled n the economy ultmately to follow a path of steady growth (or declne), the statonary soluton to ths equaton attaned when q = 0, must have (at least) one real soluton. Settng q = 0 n (12), mples that the steady-state value of q, ~ q say, must be a soluton to the quadratc equaton: A(1 (q 1) τy ) + 2h 2 = rq(1 τ b ) (13) A necessary and suffcent condton for the captal stock ultmately to converge to a steady growth path s that ths equaton have real roots, and by examnng ths equaton t can be seen that ths nvolves a tradeoff between the sze of the adjustment cost, h, and the productvty of captal, A. If, for a gven h, A s too large, the returns to captal domnate the returns to bonds, rrespectve of the prce of captal, so that no long-run balanced equlbrum exsts along whch the returns on the two assets are equalzed. Suppose that (13) has real roots whch we denote by q 1 (smaller) and q 2 (larger), respectvely. It s easly shown that the equlbrum pont that corresponds to the smaller equlbrum value, q 1, s unstable, whle the equlbrum that corresponds to the larger value, q 2, although locally stable, volates the transversalty condton (4c). The behavor of equlbrum q can thus be summarzed by: Proposton 1: The only soluton for q whch s consstent wth the transversalty condton s that q always be at the (unstable) steady-state soluton q 1, gven by the NBB WORKING PAPER No.5 - MAY

22 smaller root to (13). Consequently there are no transtonal dynamcs n the market prce of captal q. In response to any shock, q mmedately jumps to ts new equlbrum value. Correspondngly, domestcally domcled captal s always on ts steady growth path, growng at the rate = ( q 1 1) / h φ. The domestc government s assumed to mantan a contnuously balanced budget, n accordance wth (5), whle the domestc current account s gven by (6). Substtutng the expressons for I and K(t) from (3 ), and C(t) from (11) nto (6), and nvokng the transversalty condtons, we can determne () C(0) consstent wth the naton beng nternatonally solvent, and () the resultng tme path for traded bonds; see Turnovsky (1996b). Ths equaton, together wth (3 ) and (11), the soluton for q, and the ntal condton on C(0), comprse a closed form soluton descrbng the evoluton of the small open economy startng from gven ntal stocks of traded bonds B 0 and captal stock K 0. The followng general characterstcs of ths equlbrum can be observed. () Consumpton and tax-adjusted wealth, on the one hand, and physcal captal and output, on the other, are always on ther steady-state growth paths, growng at the rates ψ and φ respectvely 10. The former s drven by the dfference between the aftertax rate of return on foregn bonds and the domestc rate of tme preference. The latter by q, whch s determned by the technologcal condtons n the domestc economy, as represented by the (fxed) margnal physcal product of captal A, and adjustment costs h, relatve to the return on foregn assets. For the smple lnear producton functon, the rate of growth of captal also determnes the equlbrum growth of domestc output Y / Y. An mportant feature of ths equlbrum s that t can sustan dfferental growth rates of consumpton and domestc output. Ths s a consequence of the economy beng small and open. It contrasts wth the closed economy n whch, constraned by the growth of ts own resources, all real varables, ncludng consumpton and output, would ultmately have to grow at the same rate It turns out that consumpton s proportonal to the quantty B + qk + (Aτy qrτb) (r φ), whch we refer to as tax adjusted wealth. In the absence of taxes, t reduces to the usual measure of wealth, B + qk. We assume that the country s suffcently small so that t can mantan a growth rate that s unrelated to that n the rest of the world. Ultmately, ths requrement mposes a constrant on the growth rate of the domestc economy. If t grows faster than does the rest of the world, at some pont t wll cease to be small. Whle we do not attempt to resolve ths ssue here, we should note that the queston of convergence of nternatonal growth rates s an area of ntensve research actvty, whch we shall brefly dscuss n Secton 8 below. 16 NBB WORKING PAPER No.5 - MAY 2000

23 () Holdngs of traded bonds are subject to transtonal dynamcs, n the sense that ther growth rate B / B vares through tme. Asymptotcally the growth rate converges to max[ψ,φ] and whch t wll be depends crtcally upon the sze of the consumer rate of tme preference relatve to the rates of return on foregn nvestment opportuntes. In the case that domestc agents are relatvely patent, they choose to consume only a small fracton of ther wealth. The economy runs a current account surplus, thus generatng a postvely growng stock of foregn assets. The ncome from these assets then enables the small economy to sustan a long-run consumpton growth rate n excess of the growth rate of productve capacty. If agents are relatvely mpatent, the opposte apples. The economy accumulates an ever ncreasng foregn debt, the servcng of whch prevents t from beng able to mantan a consumpton growth rate equal to that of domestc producton. () Wth all taxes beng fully rebated and labor supply fxed, the consumpton tax s completely neutral. It has no effect on any aspect of the economc performance and acts lke a pure lump-sum tax. 3.2 Taxes, Growth, and Welfare Wth the neutralty of consumpton taxes, we can focus on the two forms of captal taxaton. Dfferentatng the soluton for q, (see (8)), together wth the defnton of the growth rate of captal φ, and consumpton ψ, leads to: Proposton 2: An ncrease n the tax on bond ncome ncreases the growth rate of captal and reduces the growth rate of consumpton; an ncrease n the tax on captal reduces the growth rate of captal, but leaves the growth rate of consumpton unaffected. Intutvely, an ncrease n the tax on bond ncome lowers the rate of return on bonds, thereby nducng nvestors to ncrease the proporton of captal n ther portfolos, rasng the prce of captal and nducng growth n captal. In addton, ths tax nduces agents to swtch from savng to consumpton, ncreasng the rato of consumpton to tax adjusted wealth. Ths slows down the rate of growth of consumpton. An ncrease n the tax on captal ncome generates the opposte portfolo response, lowerng the growth rate of captal. NBB WORKING PAPER No.5 - MAY

24 A key ssue concerns the effects of tax changes, on the level of welfare of the representatve agent, when consumpton follows ts optmal path. Ths s gven by the expresson: Ω = ψt γ ρ [ ] [ C(0) ] C(0)e e dt = 1 t 0 γ γ( ρ γψ) γ (14) Thus, the overall ntertemporal welfare effects of any polcy change depend upon ther effects on () the ntal consumpton level, and () the growth rate of consumpton. From (14) we may derve: Proposton 3: Startng from zero taxes, an ncrease n ether form of ncome tax leaves the overall level of welfare unchanged. However, the two taxes do have fundamentally dfferent effects on the tme profle of consumer welfare. Consder frst a change n the tax on captal τ y. Startng from zero taxes, t leaves the ntal tax-adjusted prce of captal, tax-adjusted wealth, and therefore the ntal consumpton level and ntal welfare, all unchanged. Moreover, snce the captal ncome tax has no effect on the growth rate of consumpton, the tme path of utlty and total overall dscounted welfare are unchanged as well. Startng from a zero tax stuaton, a tax on bond ncome also has no ntal mpact on tax-adjusted wealth. However, t rases the fracton of tax-adjusted wealth that s consumed, thereby ncreasng ntal consumpton and mprovng ntal welfare. But snce t also reduces the consumpton growth rate, these short-run gans come at the expense of longer-run losses. Indeed, these two effects can be shown to be exactly offsettng, so that startng from zero taxes, the mposton of a tax on bond ncome (wth full rebatng) has no effects on overall ntertemporal welfare. All t does s to redstrbute the tme path of consumer welfare. 3.3 Elastc Labor Supply The endogenous growth model we have been dscussng ncludes two nterdependent crtcal knfe-edge restrctons: () nelastc labor supply, and () fxed productvty of captal. The structure of the equlbrum changes fundamentally when the labor supply s endogenzed. Ths ntroduces two key changes. The frst s that the 18 NBB WORKING PAPER No.5 - MAY 2000

25 producton functon s modfed to (8), so that the productvty of captal now depends postvely upon the fracton of tme devoted to labor. Second, the fxed endowment of a unt of tme leads to the requrement that the steady-state allocaton of tme between labor and lesure must be constant. Ths latter condton provdes a lnk between the long-run rate of growth of consumpton and the rate of growth of output. Ths can be seen by dvdng the optmalty condton (2a) by (2b). On the left hand sde we see that wth the allocaton of tme remanng fnte, the margnal rate of substtuton between consumpton and lesure grows wth consumpton. On the rght hand sde we see that, gven the fxed labor supply, the wage rate grows wth output. For these condton to reman compatble over tme, the equlbrum consumpton-output rato must therefore reman bounded, beng gven by: C Y = C Y = l 1 σ 1 l θ (15) To obtaned the macroeconomc equlbrum we take the tme dervatves of: () the optmalty condton for consumpton, (2a), () the equlbrum consumpton-output rato, (15), and, () the producton functon, (8). Combnng the resultng equatons wth (8), (3 ), and (4b), the macroeconomc equlbrum can be expressed by the par of dfferental equatons n q and l: q = r(1 τ 2 (q 1) 1 σ b )q (1 τy ) α(1 l) 2h (16a) (1 γ)(q 1) l = 1 r(1 τb ) ρ F(l) (16b) h (1 γ) σ 1 γ(1 + θ) where F (l) + > 0. 1 l l The steady state to (16) s obtaned by settng q = l = 0 and s therefore characterzed by the relatve prce of captal, q, and the fracton of tme devoted to lesure, l, both beng constant. Lnearzng (16) around ts steady state, we can easly show that the two egenvalues to the lnearzed approxmaton are both postve; see Turnovsky (1999). Hence the only bounded equlbrum s one n whch both q and l adjust NBB WORKING PAPER No.5 - MAY

26 nstantaneously to ensure that the economy s always on ts balanced growth path (denoted by ~) namely 12 : ~ ~ q 1 r(1 τb) ρ ψ = = h 1 γ (1 τ y ~ ) α(1 l ) ~ q 1 σ ( ~ q 1) + 2h ~ q 2 = r(1 τ b ) (17a) (17b) where the transversalty condton now mples: ~ ~ q 1 ψ = < r h τ ( 1 ) b Equaton (17a) mples that the equlbrum s one n whch domestc output, captal, and consumpton all grow at a common rate determned by the dfference between the world rate of nterest and the domestc rate of tme preference, all multpled by the ntertemporal elastcty of substtuton. The form of the expresson s analogous to the equlbrum growth rate n the smplest AK model; see Barro (1990). The only dfference s that for the small open economy the (fxed) margnal physcal product of captal s replaced by the (gven) foregn nterest rate. Gven ths growth rate, (17a) determnes the equlbrum prce of captal, ~ q, whch wll ensure that domestc captal grows at ths equlbrum rate. Havng obtaned ~ q, (17b) then determnes the fracton of tme devoted to lesure (employment) such that the margnal physcal product of captal ensures that the rate of return on domestc captal equals the (gven) world rate of nterest. Hence n ths small open economy wth elastcally suppled labor, the growth rate of output and captal s ndependent of producton characterstcs such as the productvty parameter, α, and the 12 Ths local nstablty of the dynamc path depends n part upon our assumptons of a Cobb-Douglas producton functon and constant elastcty utlty functon, and justfes our focus on that equlbrum n the present analyss. For more general producton functons one cannot dsmss the possblty that the dynamcs has a stable egenvalue, gvng rse to potental problems of ndetermnate equlbra. In a model wth both physcal and nonhuman captal, Benhabb and Perl (1994), Ladrón-de-Guevara, Ortguera, and Santos (1997) show how the steady-state equlbrum may become ndetermnate. Other authors have emphaszed the exstence of externaltes as sources of ndetermnaces of equlbrum; see Benhabb and Farmer (1994). 20 NBB WORKING PAPER No.5 - MAY 2000

27 margnal cost of adjustment, h. Changes n these parameters are reflected n the ~ labor-lesure choce l 13. The vablty of the equlbrum requres that the naton's ntertemporal budget constrant, whch n ths growth context s B K ~ ~ ~ l (1 ) q 1 1 ~ ~ Y 2 0 σ + r ψ 1 l K 2h θ 0 = 0 (18) be met. The ntal value of ts foregn bonds plus the captalzed value of the current account surplus along the balanced growth path must sum to zero. Havng determned the equlbrum values of ~ l, ~ ~ q, and Y K, the ntertemporal constrant (18) determnes the combnaton of the ntal captal stock, K 0, and the ntal stock of foregn bonds, B 0, necessary for the equlbrum to be ntertemporally vable. Substtutng (18) nto the current account relatonshp, we fnd that the equlbrum stock of traded bonds accumulate at the common equlbrum growth ψ ~. If the nherted stocks of these assets volate (18) we assume that the approprate adjustment s attaned through ntal lump-sum taxaton (f necessary), of the form dt ~ 0 + db0 + qdk 0 = 0, whereby the prvate agent s forced to readjust hs portfolo to attan the ntertemporally vable rato consstent wth (18). Usng the government's balanced budget condton (5) one can determne the requred level of lump sum taxes at each pont n tme In order for the equlbrum to be vable, the mpled fracton of tme devoted to lesure must satsfy 0 < l < 1. Ths wll be so f and only f: τ γ + ρ < τ + h r(1 b)(1 2 ) 0 r(1 ) ( r(1 τ ) ρ) < (1 τ ) σα, a condton that s b b y 1 γ 2(1 γ) plausbly met; see Turnovsky (1999). ~ ψ Along the balanced growth path, ths s of the form: T(t) = (ak t 0 + bb0) e, where a, b are constant, easly derved from the balanced growth equlbrum. NBB WORKING PAPER No.5 - MAY

28 3.4 Comparson wth Fxed Employment AK Model Endogenzng labor fundamentally changes the macrodynamc equlbrum from ts determnaton n the pure AK model, descrbed n Secton 3.1, where the labor supply s fxed. The key to ths s the margnal rate of substtuton relatonshp, (15), whch mples that, because n equlbrum the allocaton of tme must be constant, the long-run consumpton-output rato must also be constant, forcng equlbrum consumpton and output to grow at the same rate. The dvergence n the equlbrum growth rates between consumpton and output, whch wth fxed labor supply could preval, s elmnated through the adjustment n the tme allocaton between work and lesure. Ths contrastng mechansm s reflected n the correspondng dynamcs of the external sector. As noted, wth fxed employment, these dfferental growth rates are sustaned by the transtonal dynamcs n the accumulaton of traded bonds from ther arbtrarly gven ntal stock, B 0, wth the ntal consumpton beng determned to ensure that nternatonal solvency s met. In the present case, however, the ntal consumpton level s determned by the optmalty condton (15). Now, however, the ntal stocks of assets B 0,K 0 must be chosen (through some ntal nterventon) to ensure that the economy remans nternatonally solvent. Ths seemngly modest change n economc structure has sgnfcant consequences. It mples that factors that under fxed employment are reflected n the equlbrum growth rate of output are now reflected n the allocaton of work tme. Thus, for example, whereas n the fxed employment AK model the growth rate of output depends upon the domestc producton parameters, α and h, t s now ndependent of these parameters, whch nstead nfluence l. Partcularly mportant dfferences arse wth respect to fscal polcy. From (17a) and (17b) we can summarze ther effects as follows: Proposton 4: An ncrease n the domestc ncome tax ncreases the tme devoted to work, that s reduces lesure. It has no effect on ether the growth rate of consumpton or output. An ncrease n the tax on bond ncome leads to a reducton n work and an ncrease n lesure. It also reduces the growth rates of both consumpton and output. 22 NBB WORKING PAPER No.5 - MAY 2000

29 The ntuton s as follows. Wth the equlbrum growth rate and the equlbrum prce of captal, ~ q, fxed, a hgher domestc ncome tax reduces the after-tax return to captal. In order to mantan equlbrum among rates of return, the productvty of captal must be ncreased. Ths s acheved by an ncrease n the fracton of tme devoted to labor, that s, by a declne n lesure. An ncrease n the tax on foregn bond ncome generates the opposte portfolo response. The lower return on foregn bonds requres a lower equlbrum return on domestc captal, whch s accomplshed by a reducton n the productvty of captal brought about by an ncrease n lesure. Both these responses contrast wth the fxed-employment open economy AK model, where each affects the equlbrum growth rate of output as summarzed n Proposton 2. On the other hand, the consumpton tax has no effect ether on the growth rate or on employment. Its only effect s on the consumpton-output rato, whch s reduced. In ths respect, the tax acts very much lke a lump-sum tax, as n the fxed employment AK model of the open economy 15. The fact that the growth rate s ndependent of most ncome taxes (except τ b ) offers an nterestng perspectve to the followng ssue. As we observed at the outset, one of mplcatons of the basc endogenous growth model, a feature that dstngushes t from the tradtonal neoclasscal model, s that ts equlbrum growth rate vares nversely wth dstortonary ncome taxes. The fact that emprcal evdence by Easterly and Rebelo (1993), Stokey and Rebelo (1995) and Jones (1995a, 1995b) does not support ths mplcaton, has been used as evdence aganst these endogenous growth models. Our results suggest some cauton mght be requred n reachng ths concluson. For small economes facng a perfect world captal market, the equlbrum growth rate s n fact ndependent of most tax rates. Instead, such economes respond to changes n tax rates through varatons n ther equlbrum labor-lesure choce. 15 As a further comparson, the effects of both the domestc ncome tax and the consumpton tax contrast sharply wth the analogous closed economy wth endogenous labor. In such an economy both lead to a reducton n the growth rate together wth an ncrease n lesure; see Turnovsky (2000). Wth the equlbrum growth rate fxed from the consumpton sde, ths tradeoff ceases to exst n the small open economy. NBB WORKING PAPER No.5 - MAY