Bringing Growth Theory Down to Earth by

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1 Bringing Growth Theory Down to Earth by Ramón López and Andrew Stoking WP Department of Agriultural and Reoure Eonomi The Univerity of Maryland, College Park

2 Draft lat revied: Marh 30, 2009 Bringing Growth Theory Down to Earth Ramón López Univerity of Maryland 3125 Symon Hall College Park, MD (301) Andrew Stoking Univerity of Maryland 2200 Symon Hall College Park, MD Abtrat: Expliitly aounting for ertain bai phyial law governing the earth etor dramatially enrihe our ability to explain a high degree of diverity in oberved pattern of eonomi growth. We provide a theoretial explanation of why ome ountrie have been able to utain a more or le ontant and poitive rate of eonomi growth for many deade while o many other have failed to do o. The analyi predit that ountrie that have an over abundane of phyial apital (a onept that i preiely defined in the text) may be unable to utain a poitive rate of eonomi growth over the long run. Too muh phyial apital may affet the dynami of the eonomy ultimately leading to tagnation. The plauibility of the growth model introdued here i demontrated by it ability to predit ome important tylized fat for whih tandard endogenou growth model generally annot aount. Keyword: endogenou growth theory, unbalaned growth, trutural hange, tagnation JEL: E22, Q01, O41 Correponding Author Fax: (301) Copyright 2008 by Ramón López All right reerved. Reader may make verbatim opie of thi doument for non-ommerial purpoe by any mean, provided that thi opyright notie appear on all uh opie.

3 Bringing Growth Theory Down to Earth I. Introdution Eonomi growth doe not take plae in a vauum but rather our in a natural world that i ubjet to it own phyial law. In thi paper we expliitly onider thi obviou fat by fully integrating the natural world into a growth model that allow for permanent and poitive eonomi growth over the long run a a poible but not inevitable outome. We how that expliitly aounting for bai phyial law governing the earth reoure, even at a minimum level of omplexity, reult in ome tartling differene with the tandard growth model, dramatially enrihing our ability to explain the high degree of diverity in oberved pattern of eonomi growth. The tandard endogenou growth model are either agnoti toward ertain important apet of new development pattern or they require ad-ho modifiation to explain thee new pattern. Our fou on pattern of eonomi growth a affeted by the earth etor tand in tark ontrat to muh of the environment/growth literature that foue mainly on the impliation of eonomi growth on the environment. A will be diued, the fundamental role played by the environment in affeting eonomi growth ha been alluded to by many author, partiularly in the o-alled reoure ure literature; however, thi i the firt time the environment ha been o intimately linked to deribe a wide array of growth pattern. An important ontribution of the model developed below i that it exhibit a tatedependent bifuration that allow u to explain ontrating growth pattern among ountrie evident today. That i, depending on the tate of aet endowment, the eonomy may endogenouly ahieve an eonomi growth rate that i ontant and poitive or dereaing toward ultimate eonomi tagnation. From the model we derive a well-defined interval within whih the initial aet ompoition mut fall for utainable and ontant eonomi growth to be feaible over the long run. If the aet ompoition i outide thi ritial interval, the eonomy will not be able to preerve a ontant (poitive) rate of eonomi growth and eventually may take a path that lead toward long run tagnation. Thi our beaue the endogenou dynami of the ytem aue the aet ompoition to diverge further and further away from the aet boundarie that allow for permanent eonomi growth, and thu prevent any autonomou Page 1

4 orretion that ould plae the eonomy bak into a teady growth path. Countrie that have an over abundane of phyial apital vi-à-vi human apital (a onept that i rigorouly defined below) may be unable to utain a poitive rate of eonomi growth over the long run. They may fall into what we all a apital ure. Some literature ha quetioned the ability of fator aumulation to explain the deep inome differential oberved aro ountrie (e.g., Aemoglu and Angrit, 2001; Eaterly and Levine, 2001; Prithet, 2001; Aemoglu and Dell, 2009). Empirial etimate uing year of hooling and related meaure a proxie for human apital have found that differene in uh indiator explain no more than 40% of ro-ountry variation (phyial apital explain muh le). In our analyi the nature of aet ompoition and the produtivity of human apital play a key role in long run growth, whih would ontradit the aid empirial evidene. However, reent advane in the meaurement of human apital have hown that uing year of hooling without aounting for quality differene may dramatially under-tate the true value of human apital. Hanuhek and Woemann (2008) diu everal new empirial finding and onviningly how that when quality indiator baed on international tandardized ognitive tet are ued to adjut for hooling year, human apital i a muh more powerful meaure for explaining ro-ountry differential in individual earning and inome growth. We provide a hypothei of why the aumulation of human apital an be mothered in eonomie where the aet ompoition i wrong or an fuel permanent poitive eonomi growth in ountrie where uh ompoition i within right level. Thi predition of the model that over abundane of phyial apital with repet to human apital may lead to tagnation, i.e., the apital ure annot be eaily teted with available empirial data. Conequently, we probe other predition of the model againt four well etablihed empirial tylized fat to demontrate the auray and relevane of the model. The fat that the model enhane our undertanding of thee tylized fat not only give redibility to the model, but alo upport it extenion to the finding about the apital ure. But before preenting the tylized fat, onider the following obervation. For muh of the 20 th Century, peritent eonomi growth wa mainly irumribed to a handful of ountrie aounting for le than 10% of world population: Wetern Europe, the USA and more reently Japan (heneforth, the North ). The ret of the world (the South ) wa not able to maintain a peritent rate of growth for prolonged period (Afria, Latin Ameria, and mot of Aia). At Page 2

5 the ame time the South wa the reeptale of vat natural reoure that appeared to provide an almot unbounded upply of primary ommoditie to atify the North demand and for whih the South itelf demanded little. Reently, however, the growth lub ha expanded into part of the South, partiularly China, India and everal other ountrie whih ontitute more than 40% of world population and onequently the demand for primary ommoditie from the South ha begun to rapidly inreae 1. The popular pre i ablaze in peulation of the growth impliation reulting from thi major hift in the growth lub over the lat two deade. The theoretial growth literature, however, ha been largely ilent on the impliation of thi growth lub expanion for the North ontinued growth ue. We now turn to four tylized fat that have been etablihed in the reent empirial literature but whih have not been omprehenively inorporated into exiting growth theory. Firt, eonomi growth in the North ha reulted in trutural hange away from apital-intenive and ommodity-intenive indutrie toward ervie etor, high tehnology etor, a well a other knowledge-intenive ativitie (Chenery, 1960; Kongamut et al., 2001; Aemoglu and Guierrieri, 2008) 2. Thi trutural hange ha been omplemented in the North with the inreaing importation of primary ommoditie and other apital-intenive indutrial good at a rate that outtrip growth (Ghertner and Fripp, 2007) 3. Seond, empirial evidene ha hown that the onventional Kaldorian aertion regarding the ontany of labor hare in GDP ha not held over the pat few deade (Poterba, 1998; Krueger, 1999; Aemoglu, 2003). In fat, a ro-ountry analyi of over 100 ountrie between 1972 and 1995 find an inreaing labor hare in the North and a dereaing labor hare in the South (Jayadev, 2007). 1 In the period , growth in the United State and the EU ountrie repreented 51% of world GDP growth while India and China with 35% of the world population aounted for le than 1% of world GDP growth. By ontrat, in the period , the United State and EU ontribution to world GDP growth dropped to 45% while the ontribution of China and India roe to 16%. And ine 2000, Brazil, Ruia, India, and China growth ha aounted for 22% of world GDP growth. (World Bank: World Development Indiator Databae). 2 Strutural hange ha alo been a onern in the literature on trade, growth and the environment (Copeland and Taylor, 2004). See alo Antweiler et al. (2001). 3 The other ide of the oin i the South whih ha inreaingly upplied the demand for ommoditie from the North. Thi ha aued the South to redue the ize of it produtive (non-ubitene) ervie etor vi-à-vi the ret of the eonomy. In China, for example, the ervie etor repreenting only 30% of GDP i onidered to be groly under-developed given it per apita GDP (Farrell and Grant, 2005). Page 3

6 Third, natural reoure wealth and eonomi growth appear to be loely onneted even if the natural reoure etor i a delining part of the eonomy. Thi tylized fat i validated by the influential and primarily empirial reoure ure literature (Sah and Warner, 1995, 2001; Mehlum, Moene and Torvik, 2006; Humphrey, Sah and Stiglitz, 2007), whih highlight the importane of natural reoure wealth a a fator affeting the rate of eonomi growth. A eond omponent of thi tylized fat i the nature of thi relationhip. While early work provided empirial evidene uggeting that reoure wealth and the rate of eonomi growth were inverely related, more reent tudie have hown that uh a relationhip may work in the oppoite diretion under ertain ondition (Barbier, 2005; Lederman and Maloney, 2008; Peretto, 2008). Conequently, the empirial evidene onerning the relationhip between natural reoure wealth and eonomi growth remain eluive. Fourth, depite rapid growth in the North whih demanded inreaingly larger volume of raw material, real ommodity prie have not trended upward. In fat, ommodity prie have been non-inreaing for mot of the 20 th entury (Page and Hewitt 2001; Zania, 2005; Kellard and Wohar 2006) 4. Some indiator ugget, however, that over the lat deade there may be a break of uh table prie trend. A a reult, there i inreaing interet in the onnetion between ommodity prie and eonomi growth. Our ontribution an be partly aeed with referene to the above tylized fat. Firt, with few exeption, growth model onider only one final good etor and thu are agnoti to trutural hange away from ommodity-intenive indutrie to ervie indutrie. Kongamut et al. (2001) i one uh exeption; however, thi paper aumed a partiular type of nonhomotheti preferene (that the onumer io-utility map hift away from primary ommoditie) to explain trutural hange. While thi aumption may be realiti, it i almot equivalent to impoing trutural hange and deviate from the tradition in general equilibrium and growth literature to aume neutral preferene. Baumol et.al. (1985) explain trutural hange by impoing the aumption that exogenou tehnial hange i biaed againt prodution of ommoditie and favor produtivity growth in the ret of the eonomy. Aemoglu and Guerrieri (2008) alo preent a model of unbalaned growth with exogenou tehnial hange 4 A notable exeption to thee downward trending ommodity prie i world timber prie whih have onitently inreaed over the pat 100 year. We will reonider thi important exeption in light of the aement of the growth model in Setion IX. Page 4

7 where the hifting diretion of growth i determined by the elatiity of ubtitution in onumption. López et.al. (2007) examine upply-indued trutural hange in the ontext of a bang-bang invetment model where all aet are produed by the ame prodution funtion; however, they do not integrate trutural hange a part of the broader framework implied by the four tylized fat diued earlier and do not aount for the role of the initial aet ompoition in affeting growth and trutural hange. We how that trutural hange in both output and produtive aet i an intrini onequene of eonomi growth even when preferene are entirely neutral and produtivity growth i endogenou and etor neutral. The growth model that we develop i notably not proportional uh that the phyial apital to human apital ratio (and the onumption to apital ratio) i perpetually hanging over the long run, even when the rate of eonomi growth may beome ontant. That i, we extend the onept of trutural hange to inlude hange in the ompoition of fator of prodution, not merely of output. While aet ratio hange over time we how that they follow ertain ytemati pattern that repliate peifi type of trutural hange oberved in growing eonomie. Continuou trutural hange both in term of output and fator of prodution i perfetly onitent with a ontant rate of eonomi growth. Seond, reolving the iue of hanging labor hare require a model that lead to differential rate of growth of human and phyial apital in the long run. Yet mot growth model predit ontant fator ratio over the long run. The fat that we extend the onept of trutural hange to inlude hange in aet ompoition allow u to etablih the ondition under whih the labor and apital hare an hange throughout the proe of eonomi growth. Thi repliate the eond tylized fat regarding fator hare diued above. Third, explanation of the reoure ure abound depite the general abene in formal growth model of a atifatory theory to delineate the ondition under whih a reoure endowment i aoiated with fater or lower eonomi growth. An exeption i Peretto (2008) who provide the mot rigorou theoretial treatment of the reoure ure yet available; however, thi work onider the welfare impliation reulting from a one-and-for-all exogenou hok on reoure availability and ignore the reoure dynami. We expliitly onider uh reoure dynami and endogenou hek of reoure wealth. We orroborate the firt tenet of the reoure ure finding; namely, that hange in reoure wealth and eonomi growth are loely linked in the intermediate run. We deribe ondition under whih Page 5

8 reoure wealth i poitively or negatively linked to eonomi growth over the intermediate run. The peed of eonomi growth may, however, be deoupled from natural reoure wealth over the long run when the rate of eonomi growth i driven primarily by the eonomy ability to reate and dieminate knowledge. However, while natural reoure do not affet the rate of eonomi growth over the long run, we how that long run reoure wealth doe affet the likelihood that an eonomy i able to utain eonomi growth or alternatively fall into a long run tagnation trap. That i, under ertain ondition reoure wealth a well a over abundane of phyial apital an be a ure for eonomi growth. Fourth, with repet to the ontany of ommodity prie, the model predit tability of real ommodity prie under ertain ondition likely to have prevailed throughout mot of the 20 th Century. The model allow apital invetment from the North to our the globe earhing for new ommodity oure where the marginal produt of apital i high (Caelli and Feyrer, 2007). Inreaing demand for ommoditie from the North lead to the development of new ommodity oure in the South, thu hifting the ommodity upply urve to the right along with the hifting demand urve (Deaton and Laroque, 2003). Key omponent of thi part of the model are the reognition of the vatne of unexploited ommodity reoure in the South, the relatively low demand for uh reoure in the South, and the almot omplete lak of regulation in the South that allow reoure extration without paying for the environmental ot aoiated with uh extration. Inluion of thee mehanim provide a framework to evaluate how reent hange in world growth pattern, in partiular the expanion of the growth lub, may affet the North in the future through potential new trend in ommodity prie. The earth or ommodity etor onidered in thi paper i imilar in nature to Brok and Taylor (2004) treatment in the environment/growth literature. It inlude prodution etor that rely either on natural reoure a oure of produtive input (e.g., timber, agriulture, fiherie, hydroeletri power), a ink of pollution (e.g., pipeline and other), or a both (e.g., oal mining, hemial proeing). A defined, thee etor aounted for roughly 14% of 2005 GDP in the United State 5. While we expliitly model the earth etor after renewable natural reoure, non-renewable reoure extration alo impoe heavy demand upon the renewable 5 Reoure or environment-dependent etor may inlude agriulture, energy, mining, utilitie, fiherie, wood and pulp, mineral, metal, and other (US Cenu: Statitial Abtrat of the United State: 2008) Page 6

9 reoure etor, and i thu onidered part of uh a etor. That i, we model non-renewable reoure extration through it effet on the urrounding renewable reoure bae 6. Thi approah aume that the limiting eonomi fator i the renewable reoure more o than the nonrenewable under-ground reerve 7. Our reearh trategy hart the following oure. We firt eluidate the intrini model mehani uing a mall open eonomy paradigm (open to trade in final good) whih doe not regulate the ue of the natural reoure (or ha no property right on them). Apart from allowing u to highlight the key qualitative nature of the model devoid of ompliation aoiated with endogenou ommodity prie and reoure regulation, we ugget that the mall open eonomy approah give the bai miro foundation of the theory muh like the theory of the firm and houehold provide the miro foundation for tati maroeonomi analye. The ue of a mall open eonomy a the bai unit of analyi i infrequent in growth modeling beaue the overwhelming majority of the growth literature aume a loed eonomy. The mall open eonomy paradigm i imple enough to allow analytial tratability of the model, but robut enough to reprodue the onventional Kaldorian tylized fat (Kaldor, 1961) a well a ome but not all of the tylized fat diued above. We next how that the preene of property right on the natural reoure ha little qualitative impat on the analyi. Finally, in the full model we expliitly inorporate endogenou world ommodity prie with property right in the North. Thi lat modeling effort allow u to repliate the remaining tylized fat that annot be analyzed in the ontext of a mall open eonomy with immobile apital. However, we how that the inherent logi of the analyi of the mall open eonomy remain intat in the ae of a large eonomy. 6 For example, non-renewable reoure extration affet water quality (mining, oil extration), oil and foret (mountain top removal for oal extration), all of whih are renewable reoure and are thu inluded in our earth etor. For reent work onidering nonrenewable reoure in a growth model ee for example, Brethger (2008), Pittel and Brethger (2008), and André and Smulder (2008). 7 Several reent analye ugget that with the exeption of a few ommoditie, there are no ign of arity of nonrenewable reoure, but that arity mainly affet the renewable reoure a a oure of eential ervie (Simpon, Toman, and Ayre (2005). The US ould, for example, dramatially inreae it oil prodution by expanding off-hore or Alakan prodution. The limiting fator i not the availability of under ground reerve but rather the teep environmental ot that uh expanion would entail. Page 7

10 II. The Model We onider three produtive etor: 1) a reoure etor (referred to hereafter a the ommodity etor) uing natural reoure and man-made input to produe a final good, 2) a eond final good etor that doe not ue natural reoure (referred to hereafter a the ervie etor), and 3) a knowledge etor that produe labor-augmenting human apital that benefit all three etor inluding the ommodity etor. There are three aet human apital or knowledge, phyial apital, and natural apital and the eonomy invet in enhaning human and phyial apital. Growth in human apital trigger labor-augmenting produtivity growth that benefit all three etor. Conumption. The repreentative onumer ha preferene defined over both the final ervie good ( x ) and the final ommodity good ( x ) in the following indiret utility funtion: (1) U( ; p) ε = ε 1 e( p,1) where ε i the invere of the elatiity of marginal utility and i fixed, = x + px i the level of total onumption expenditure in unit of the ervie good, e( p,1) i the onumer unit expenditure funtion, and p i the relative prie of the ommodity good relative to the ervie good 8. The peifiation ued in (1) impoe homotheti and trit onavity of onumer preferene (the latter ondition due to the fat thatε > 1) but the fat that we do not impoe any funtional form on e( p,1) mean that the analyi i generally onitent with any type of homotheti tritly onave preferene, inluding Cobb-Dougla, CES, or other more general preferene truture. Prodution. Firm ue raw labor ( l i ) and phyial apital ( k i ) to produe eah final good. Produtivity growth i repreented by human apital or knowledge ( h ). Prodution of the ommodity good alo require a natural reoure ( n ) a an additional fator of prodution. Firm produe the ervie and ommodity good aording to tandard neolaial prodution funtion: 8 The level of final good demand, x and x, an be determined through Roy Identity from the indiret utility funtion U(;p) one the optimal level of total real expenditure () i obtained from the olution to the enuing dynami optimization. ε 1 ε Page 8

11 y Ak hl α α = (2) ( ) 1 (3) ( ) 1 y n Dk hl θ β β = where y and y are output level of the ervie- and ommodity-baed good, repetively, 0< α < 1, 0< β < 1, 0 θ 1, and A and D are fixed parameter. We normalize human apital o that h 1. Inreaing h i equivalent to inreaing the effetive ize of the labor fore. The funtional form for prodution of the ommodity good add human effort to the tok of natural reoure to produe the final good. Human effort i repreented by the ompoite of man-made aet, Dk β ( hl ) 1 β, in equation (3). The parameter θ reflet the importane of the natural reoure in prodution of the ommodity good. Thi prodution funtion orrepond to the tandard peifiation ued for repreenting prodution of renewable natural reoure-baed ommoditie (Clark, 1990; Brander and Taylor, 1998). For mot of the analyi we will aume that β > α for developed ountrie where reoure-baed ativitie tend to be more phyial apital intenive while the non-reoure etor tend to be highly kill and labor intenive 9. By ontrat, developing ountrie may be haraterized by α > β, a the non-reoure etor tend to onit of indutrial prodution that i highly intenive in phyial apital rather than knowledge. In developing ountrie, mot of the population remain tied to a reoure etor produing mainly primary ommoditie that ue little apital but large amount of labor 10. Aet Aumulation and Market Equilibrium. The dynami of the renewable natural reoure tok are deribed by: θ β (4) ( ) φ ( ) 1 n= g n n Dk hl β 9 For example, in the USA for the period , the average employee ompenation a a % of etor-peifi GDP in the reoure-baed etor i 27-38% (agriulture, foretry, fihing, and hunting), 22-35% (mining), and 22-25% (utilitie); wherea, in non-reoure etor, labor aount for 41-54% (information), 54-56% (finane and inurane), 69-74% (profeional and buine ervie), and 79-80% (eduation and health are ervie) of the repetive etor-peifi GDP level (US BEA: Gro Dometi Produt by Indutry Aount, ). 10 Compare for example agriulture in the US whih relie on a vat array of farm mahinery and equipment with that in poor ountrie uh a India depending motly on human power. Page 9

12 The funtion g( n ) ummarize the phyial law governing intrini growth of the renewable natural reoure 11. The eond term repreent the redution of the natural reoure tok due to prodution of the ommodity good. Produing one unit of the ommodity good impoe demand upon the natural reoure; the parameter 0 < φ < 1 repreent the intenity of uh demand. Thu, the parameter θ and φ repreent the importane of the natural reoure a a fator of prodution and the environmental impat due to ommodity prodution, repetively. In fiherie or foret model (Gordon, 1954; Shaefer, 1957) the reoure tok i aumed to be uffiiently are to influene the produtivity of effort and thu θ = 1 under mot ondition 12. In other etor (mining, agriulture, energy) θ aquire intermediate value ( 0 < θ < 1) howing a leer dependene of prodution on the natural reoure. Finally, θ = 0 implie that the reoure i o abundant that output only depend on the level of effort, independent of the reoure 13. Human or knowledge apital growth i aumed to be ubjet to inreaing return a modeled by Lua (1988) and other (Barro and Sala-i-Martin, 2004): (5) h = Bhlr where B i a fixed parameter. The tok of phyial apital grow aording to: α (6) ( ) α θ β ( ) 1 1 β k = Ak hl + pn Dk hl Equation (6) alo repreent the budget ontraint of the eonomy, whih in the ontext of an open eonomy reflet the equilibrium in the urrent aount, i.e., the total value of dometi output (prodution of the ervie and ommodity good) i equal to the value of expenditure in onumption of the two good plu invetment. Direpanie between prodution and onumption of eah final good are filled by orreponding import and export. 11 When neeary for the analyi, we ue a logiti peifiation for g ( n) γ n[1 ( n / n)] parameter and n i the maximum arrying apaity of the natural ytem. = where γ > 0 i a fixed 12 For example, oal extration often deforet large area and even require the omplete detrution of the urrounding environment, but prodution of oal itelf i unaffeted by uh environment, whih implie a mall value for θ and a large value for φ. Similarly, oil prodution often aue evere demand on renewable reoure implying that θ=0 and a large φ. 13 For example, logging extration in the Amazon i likely to depend only on logging effort, not on the foret tok due to the heer immenity of the foret. Page 10

13 Labor and apital market are perfetly ompetitive and thu full employment of labor and apital prevail. We aume that the total labor fore L i fixed: (7) l + l + lr = L (8) k + k k = At thi point we need to make ertain aumption about the parameter value: Parameter Aumption A1. The elatiity of marginal utility i fixed and le than one:1 ε < 1. A2. Defining ρ > 0 a the pure time diount rate, the maximum rate of growth of human apital fall within the following range: ρ < BL < ρ /(1 1 ε ). With repet to aumption A1, Aghion and Howitt (1998) how that an elatiity of marginal utility greater than unity i implauible and implie odd behavior in the ontext of maroeonomi model. The firt inequality in A2 implie that invetment in human apital an be profitable. A we how below, thi i a neeary ondition for poitive growth to be at all feaible. In fat, a will be lear below BL i not only the maximum rate of growth of human apital but i alo equal to the rate of return to human apital in the long run. It will alo be lear below that the eond inequality effetively require that the long run rate of return to human aet be not too large to rowd out invetment in phyial apital under any ondition. The Soial Planner' Problem. Under the eond welfare theorem, the entral planner ondition for maximizing oial welfare are idential to the general equilibrium ondition ariing from a deentralized model of perfetly ompetitive firm and houehold independently maximizing utility and profit, repetively. The oial planner maximize the diounted utility from onumption aro all time: ρt V U p e dt, k, k, l, l, lr 0 (9) max ( ; ), where t denote time. Maximization of utility i ubjet to the ontraint hown in equation (2) -(8) repreenting the prodution funtion, aet growth equation and market learing ondition for labor and apital. Conitent with the interpretation of (9) a that of a ompetitive market eonomy we aume that the planner take p a given. In addition, the initial level of human Page 11

14 apital ( h 0 ), natural apital ( n 0 ), and phyial apital ( k 0 ), are aumed given and we have nonnegativity ontraint for onumption and prodution of the ervie and ommodity etor. Thi problem an be olved by maximizing the urrent value Hamiltonian where λ, μ, and η are the o-tate variable aoiated with phyial, human, and natural apital, repetively. 1 ; α 1 β 1 β r We onider two polar ae regarding property right on the natural reoure: 1) α θ β θ β (10) H = U ( p) + λ Ak ( hl ) + pn Dk ( hl ) + μbhl + η g ( n) φn Dk ( hl ) omplete open ae whih i equivalent to auming that η = 0 and 2) perfet property right on the reoure whih mean that η i optimally hoen. We how below that the eonomy ahieve a tationary level of the reoure tok even if the reoure i exploited under open ae. For impliity in preentation, we will aume initially that η = 0 but in Setion VIII we how that key qualitative reult do not hange in the property right enario where η > 0 i endogenou. The reulting firt order ondition for the maximization of the Hamiltonian under the aumption that η = 0 are derived below. The ontrol variable are the alloation of labor aro the three etor, apital aro the ommodity and ervie etor, and onumption. Interior v. Corner olution. We firt aume interior olution, e.g., that both final good etor produe poitive level of output and that the knowledge etor i ative ( l r > 0 and hene h > 0 ). Thi allow u to preent the firt order ondition a equalitie intead of their Kuhn-Tuker analog. A diued in the introdution, however, an interior olution with three produtive etor i not guaranteed and may not even be optimal to the oial planner or to a ompetitive market olution. In Setion V we will diu the ondition required for diverifiation. We will be partiularly onerned about the poibility that the non-negativity ontraint be binding for l r beaue permanent eonomi growth ritially hinge on h > 0 and hene on l r > The other orner olution ( l r phyial apital i greater than zero. = L) an be ruled out a long a the tok of 14 If l = 0 then the Kuhn-Tuker ondition for (13) beome w > ( μ / λ) B and equation (16) beome μ = μρ λwl. r Page 12

15 In addition to the equation of motion deribed in (4)-(6), the firt order ondition auming an interior olution are: (11) ( ; ) U p = λ θ β (12) ( 1 β) ( ) (1 α) ( ) p Dn k hl = A k hl w θ β (13) ( β) ( ) = ( μ λ ) p ndk hl B 1 α θ (14) α ( ) β ( ) 1 β 1 A k hl = p Dn k hl r The firt ondition tate that the marginal value of onumption i equal to the hadow prie of phyial apital. The next two ondition repreent the equalization of wage aro the three etor. And the fourth ondition repreent the equalization of the return to apital between the ervie and ommodity etor. We note that w i the wage rate per unit of human apital (the wage per unit of effetive labor time); then w= wh (with h 1) i the atual wage earning per work time, both meaured in term of the ervie good. Alo, r i the rental prie of phyial apital. The equation of motion for the o-tate variable are: α 1 ( ) (15) λ = λ ρ αak ( hl) (16) μ = μ( ρ BL) Finally, the tranverality ondition are: ρ (17) lim t ρt e μh= 0, lim e λk = 0 t Endogenou veru exogenou ommodity prie. We firt onider the mall open eonomy ae without property right on the natural reoure; that i, the ae where the eonomy i open to trade in good (but not in fator of prodution) o that the ommodity prie (p) i fixed and not affeted by the prodution and demand ondition of the eonomy. In Setion IX we preent the full model whih onider endogenou ommodity prie and property right. Whether or not p i endogenou, the nature of the firt order ondition of the optimization problem above are not affeted a long a the planner, like a ompetitive market eonomy, abtain from exeriing market power. In both ae, the firt order ondition are defined onditional on a partiular level of p. The only differene i that if the eonomy i large enough to affet prie the evolution over time of the olution will trigger a prie dynami t α Page 13

16 that will, in turn, feed bak into the olution over time. By ontrat, if the eonomy i mall no prie feedbak exit. III. Bai Equilibrium Condition If the eonomy produe both final output we an ombine equation (12) and (14) yielding: Lemma 1: The ratio of phyial apital to human apital-augmented labor in the ervie and ommodity etor are proportional to eah other: k β k k (18) = Ψ hl α hl hl ( 1 α) ( 1 β) Proof: By inpetion of equation (12) and (14). The apital to labor ratio in the ommodity etor i alway a ontant multiple of that in the ervie etor. If β > α ( α > β ) we have that 1 ( 1) Ψ> Ψ< whih mean that the required phyial apital to labor ratio in the ommodity etor i greater (le) than that in the ervie etor at all time, regardle of the level of prie and other parameter. Uing equation (18) in (14) allow u to olve for the equilibrium phyial apital to human apital ratio whih i exluively a funtion of the natural reoure tok: 1 1 α β α β α θ (19) k hl = ( ΧΨ ) ( pn() t ) where ( D( 1 β )) ( A( 1 α )) θ Χ. Heneforth we define Z ( pn ) k hl and θ Z ( pn ) k hl where Z i olved uing (19) in (18). Now we have the following Lemma: Lemma 2. (i) The phyial apital to human apital-augmented labor ratio in both final good etor are dereaing (inreaing) in the tok of natural reoure ( n ) if the ommodity etor i more (le) phyial apital intenive than the ervie etor, that i if β > α (α > β ). (ii) If β > α ( α > β ) then the wage rate per unit of human apital ( w ) i falling (riing) and the rental prie of phyial apital (r) i riing (falling) in n. Proof: By inpetion of Equation (12), (14), and (19). Intuitively, inreaing the natural reoure tok expand the ommodity etor and thu inreae demand for phyial apital and labor. If the ommodity etor i phyial apital intenive ( β > α ), thi expanion reate a greater demand for phyial apital than for human Page 14

17 apital whih lead to inipient exe demand for phyial apital and exe upply of labor. Thi, in turn, aue the prie of phyial apital to inreae and the wage rate to fall. Thi fator prie readjutment indue both final good etor to redue their apital-labor ratio, whih i what Lemma 2 predit. If β > α the lower wage rate indued by an inreae in the natural reoure tok reult in an expanion of the knowledge etor, whih i the mot labor intenive etor in the eonomy. Knowledge and natural reoure are omplement in thi ae. By ontrat, if α > β natural reoure and knowledge are ubtitute. IV. The Nature of Convergene Unlike mot growth model in the literature (e.g., Aemoglu and Guerrieri, 2008), the preent model doe not allow one to ditinguih between tranitional dynami and teady tate. The ytem i in perpetual tranition a it never reahe balaned growth often defined a the tate where aet ratio and aet to onumption ratio tabilize. The dynami of the ytem here, however, an till be eparated into two tage: (i) Stage 1 (the intermediate run ) in whih onumption grow at varying rate over time; and (ii) Stage 2 (the long run ) when the onumption growth rate beome ontant. In Stage 1 the natural reoure i endogenouly hanging over time until it beome ontant in Stage 2. The ontany of the natural reoure aue the onumption rate of growth to beome ontant a well. Below, we identify ondition under whih Stage 2 with poitive onumption growth i both feaible and utainable. Even in the long run (Stage 2) a growing eonomy i in perpetual evolution by adjuting aet ratio, onumption-to-aet ratio, a well a output ompoition. Depite the ontinual hange of the kh and k ratio, they are ubjet under ertain initial aet endowment ondition to well-defined boundarie to whih they may approah but never atually reah. We all thee boundarie infinite onvergene point (ICP). Thee boundarie are fundamentally different from the uual aymptoti onvergene point (ACP) ued in tandard multi-aet endogenou growth model. More formally, we define thee boundarie a follow: Definition. Conider two poitive variable, Nt () and M () t, then we define the ICP a ( N / M) lim[ N( t)/ M( t)], and the ACP a the more traditional N M ( N / M) lim[ N( t)/ M( t)] t. Page 15

18 Remark on Surrogate ICP. A real variable x() t f( N()/ t M()) t i aid to have a Surrogate ICP when the N / M ratio i at it ICP; that i, x = f[( N / M) ]. One way of illutrating the differene between ICP and ACP i to perform the following experiment: aume that by hane the above ratio are initially at ICP. We how below that the ytem will neearily move away from uh ondition even if no exogenou diturbane our. By ontrat, in the tandard growth model where the long run i haraterized by ACP, an initial ondition whih by hane oinide with ACP will be permanent unle the ytem i perturbed. In fat, exitene of ICP a natural boundarie provide the foundation for bifuration and tate dependene of the ytem. A we how below, diverified growth equilibrium i poible only if the initial aet ratio are on the orret ide of their repetive ICP. If the ratio are at or on the wrong ide of ICP, the eonomy onverge to an ACP whih i haraterized by eonomi tagnation. The two tage into whih we have laified the dynami of the ytem an be defined by their approah toward ACP (Stage 1) or ICP (Stage 2). Below we provide a deription of eah of thee phenomena. Stage 1: ACP onvergene There are two poible ACP equilibrium depending on whether an interior olution ( l r > 0) or a orner olution ( l r = 0 ) applie. We deribe both ae below in equene. Natural reoure: ACP onvergene. Given the dependene of the phyial apitallabor ratio on the reoure tok, we now haraterize the evolution of the natural reoure baed on any initial ondition ( n 0 ). With l > 0, we an ue equation (13). Logarithmially r differentiating it with repet to time and uing (15), (16), and (19) we obtain the equilibriumrequired rate of hange of natural apital or demand ide of natural apital (the hat denote growth rate) whih i the rate along the tranition path neeary to atify the market wage equilibrium ondition repreented by equation (13). (20) nt ˆ( ) = ( β α) ( θα)[ BL rn ( )] Sine rn ( ) i inreaing (dereaing) in n for β > α (α > β ), we have that the rate of growth of natural apital i delining in nt () for both β > α and α > β. Thi mean that the natural reoure onverge to a tationary value ( n ) at whih point rn ( ) = BLregardle of the apital Page 16

19 intenitie of the final good etor and the initial level of n 0. Similarly, the wage rate per unit of human apital onverge to a tationary value. Thi equilibrium i learly of the ACP type. The fat that the knowledge etor i ompetitive mean that the marginal produt of labor in the human apital etor i equal to the marginal produt of labor elewhere in the eonomy. That i, the relative (hadow) prie of knowledge i et ( w = μb λ) o that equation (13) hold at all time. In a growing eonomy λ and μ are both falling, albeit at different rate. Therefore the wage rate ha to adjut onomitantly to allow the equilibrium (13) to ontinuouly hold and thi adjutment mut orrepond to hange in the natural reoure tok. Finally, the adjutment of nt () fully determine the optimal ize of the ommodity etor and it orreponding alloation of labor. By equalizing the demand ide (equation (20)) of the natural reoure tok hange with it upply ide (equation (4)) a unique value for the ommodity labor requirement an be determined at all point in time: 1 β α θ θ β φdn [ Z( pn )] θα Where Z i defined in (19). From (21) it i lear that when n= n > 0 (and thu r = BL) then (21) hl = g( n) + n( r( n) BL) hl > 0. Uing (21) in (2) we determine the equilibrium prodution of the ommodity output: β α (22) y = [1/φ ] gn ( ) + nrn ( ( ) BL) θα Conumption Growth Rate: ACP Convergene. The dynami of nt ( ) alo determine the rate of growth of the eonomy. Differentiating (11) with repet to time and uing equation (15), (18), and (19) produe: (23) t ˆ( ) = ε [ rn ( ) ρ] Thu, an eonomy that i depleting it natural apital will experiene a delining rate of growth over time if β > α. A nt ( ) onverge toward it aymptoti tationary value, the rate of growth of the eonomy alo onverge to a ontant value. (24) ˆ () t ε [ r( n) ρ] ε [ BL ρ] = =, where the eond equality in (24) follow from (20) when nt ˆ( ) = 0. Aumption A.2 guarantee that long run onumption growth in thi diverified eonomy i poitive (i.e., ˆ > 0 onvergene i alo ACP. We formally define the ACP equilibrium in Propoition 1. ). Thi Page 17

20 Propoition 1 [On ACP Charaterization with Interior Solution]. Aume l r > 0, then: (i) The ACP equilibrium i deribed by the following ontant: 1) ( ) ( ) ( 1 α ) 2) 3) ( ) ( ) 1 β α θ α β n = BL/ A 1/ pχψ ; Z 1 α A 1 α =Ψ BL y = g( n ) φ ; ; Z ( hl ) 1 α A 1 α = BL gn ( )/ n = β φdz α 1 α θ θ ˆ = ε[ BL ρ] 1 α 1 α 4) w = A ( 1 α ) ; r = BL BL Ahievement of thee limit i ontingent on a natural reoure tok arrying apaity that i uffiiently large. (ii) The following additional ondition hold in the long run: k ˆ = ( hl ) =0; kˆ ˆ ˆ = h+ l ; ( wh ) = hˆ. Proof: See Appendix. We aume throughout the remaining analyi that 0 < n n. Thu, in ACP the eonomy ahieve a ontant rate of eonomi growth and a ontant rental prie of apital whih under ertain ondition an be upported over the long run. Thee reult are onitent with tylized fat originally identified by Kaldor (1961). While w reahe a tationary ACP value a well, the wage earning rate ( wh ) will ontinue to grow at the rate of human apital growth. Labor Alloation aro etor. Uing the fator market learing ondition (7) and (8) we an olve for atual level of effort ued in eah of the three etor in both Stage 15 : 1 (25) (i) hl = Ψh( L lr) ; ( ii ) r ( 1) Ψ 1 Z In addition, k k hl = hl + Ψ hl. Z hl i given by (21). From equation (25)(ii) it follow that an interior olution for l r (0< lr < L) require that (26) Z [ L+ ( Ψ 1)( hl ) / h] > k / h> Z ( Ψ 1)( hl ) / h 15 In deriving thee ondition we have ued the fator market learing ondition Zhl + Zhl = k and hl + hl + hlr = hl in onjuntion with Lemma 2. Page 18

21 We note that in a diverified eonomy, effort in the ommodity etor ( hl ) a defined in (21) i dependent only on n and hene, one n= n then hl = ( hl) a defined in Propoition 1. From (26) it follow that diverifiation require that the tok of apital be greater than a minimum level kmin Z ( Ψ 1)( hl ) (ontant in the long run one hl i at or in the neighborhood of ACP), and below a maximum level, k Z Lh+ k. We note that k min > 0 if max min Ψ> 1 whih happen if β > α, but k min < 0 if Ψ< 1 whih our when α > β. Stage 2: ICP Dynami: Corner v. Interior Solution Whether or not an eonomy an utain an interior olution with L> l r > 0 at all time depend on the dynami of the kh aet ratio and the bifuration boundarie. We define k k k min and, a hown in Propoition 2 below, we define the following ICP value for the repetive ratio, ( k) BLα ε( BL ρ ) = and( ) kh = ( Z B)[ BL ε ( BL ρ) ]. Aumption A2 guarantee that both ICP value are poitive. We an then ue the two equation in (25) together with (5), (6), (21) and (23) to derive: (27) k = k ( / k) rnk ( ) k ( ε 1 α)( BL rn ( )) (28) In addition we have: (29) ( ) ( ) (30) ( ) k h= B Z k h k h k ε + ε ερ min ( BL r( n) ) ( 1 Z Z) ( BL( 1) ) ( 1 α α) ( ) min ( ε 1 α)( ( )) k= k k + rnk k BL rn ( ) ( ) kh= BZ kh kh k BZ kmin h ( BL r( n) ) ( 1 Z Z) ( ( BL ) BL) ε + ε ρ From Propoition 1, when n n, Z Z, hl ( hl ) and rn ( ) rn ( ) BL = ; imilarly, k min i fixed when n n. Thu, in ACP the lat right-hand-ide term in (27) and (29) and the lat two right-hand-ide term in (28) and (30) all vanih. Propoition 2 provide a Page 19

22 formal proof of the derivation of equation (27)-(30), of the ICP for the kh and k ratio, and derive key impliation. Propoition 2 [On ICP Charaterization]. Aume the following initial ondition: n = n and k = k o ; h= ho, where k o and h o are arbitrarily poitive value. Define kmin Z ( Ψ 1)( hl ), then full and permanent diverifiation i poible (i.e., all etor output inluding the human apital etor are poitive at all future time) if and only if the initial phyial to human apital ratio atifie the following ondition: (i) If β > α : ( kh) < ko h < ( kh) + kmin h; (ii) If α > β : ( kh) + kmin h < ko h < ( kh). The ratio kh and k onverge toward but never o o reah non-negative and fixed ontant ( k/ h) and( k), repetively a long a k and h are finite. Proof: See Appendix. Propoition 2 provide the ondition under whih the eonomy an remain fully diverified indefinitely, i.e., l r > 0 with poitive prodution of both final good. Thi Propoition ombined with equation (26) alo define the boundarie for the two orner olution: 1) if k kmax, then there i no labor alloated to the prodution of human apital ( l r = 0 ); and 2) o o if k k min, all of the labor would be alloated to the human apital etor ( l r = L) and the two final good etor eae to be in operation. Note, however, that the ondition for l r < Li extraordinary weak beaue if l r = Lthen l = l = 0, whih imply that k min = 0. That i, a long a k > 0 we rule out thi orner olution. Hene we only fou on the poibility that the lower bound ontraint for l r i binding. If the ondition for l r > 0 are not met then the olution hange dramatially. The eonomy invet only in phyial apital ( k ) whih mean that the upport for rn ( ) i no longer et at BL. In fat, the eonomy ontinue inveting in k auing n to hange to a new ACP defined by rn ( ) = ρ, and ( ) ( ) ( 1 ) ( ) ( ) 1 β α θ α β n = ρ / Aα 1/ pχψ θ. From Lemma 2 it follow that ( n< n ) if β α ( α β ) n> n > >. From (23) it i evident that at thi point onumption growth beome zero and the eonomy tagnate. Propoition 2 deribe diverifiation boundarie Page 20

23 relevant to the tate variable ( khn),, that will determine whether or not l r an be poitive and remain poitive. We make the following additional remark about the reult in Propoition 2: Remark on Propoition 2. (a) Stable Diverifiation. If β > α then k min > 0. Thu, if the initial kh ratio i within the permanent diverifiation boundarie, it ontinuouly fall but never reahe it lower bound, ( kh), for finite k and h level. Whenα > β then k min < 0. Thu, if the initial k/ h ratio i within the permanent diverifiation boundarie, it ontinuouly rie but never reahe it upper bound, ( kh), for finite k and h level. (b) Untable Diverifiation. If the initial kh ratio i outide the diverifiation boundarie the eonomy annot utain diverifiation over time. If the kh ratio i above it repetive upper diverifiation boundary but below k max, the eonomy i able to remain diverified and grow for a period of time. However, phyial apital in the eonomy inexorably move toward k max, where the human apital etor i eventually rowded out, unable to ompete for labor with the ommodity etor. We refer to thi ituation a the apital ure. () Phyial Capital Intenive Corner Solution. If the tok of phyial apital k i initially above k max or move above it, the eonomy eventually tagnate a there i no longer invetment in human apital ( l = h = 0 ).The eonomy peialize in the final good, i unable to utain the natural reoure tok at r n whih eventually approahe a new equilibrium, n. The ondition tated by Propoition 2 onern the initial aet endowment k o and h o that reflet pat hitorial ondition whih annot be endogenouly hanged in the hort run. A tated in the Remark to Propoition 2, whether or not thee aet endowment atified the repetive ondition ha a dramati effet on the long run growth potential of the eonomy. Employment in Human Capital Setor: Surrogate ICP. Finally we onider the dynami of labor in the human apital etor auming that 2). Uing (5) and (23) we get that: (31) h = ε[ BL ρ] Blr n = n (i.e., the eonomy i in Stage Page 21

24 Uing / h ( / k )( k / h) it follow that h = k h+ k. Hene, from Propoition 2 we have that h 0 at ICP and we an olve (31) to obtain an expreion for the urrogate ICP for the level of employment in the human apital etor: (32) = ε( ρ) lr BL B whih i obviouly poitive in a growing eonomy (i.e., if BL urrogate nature of the ICP for l r i apparent by the fat that (27) and (28) in (31) yield: (33) k h ( k h) = Z ( lr lr) > ρ by Aumption A1). The l = L Z k h. Combining r (1/ )( / ) From Propoition 2 we have that if β > α (α > β ) the left-hand-ide of equation (33) mut be negative (poitive) for permanent eonomi diverifiation and thu the level of employment in the human apital etor mut be above (below) it repetive ICP, i.e., l r ( l r > l r ). In addition from (33) it follow that if the ondition of Propoition 2 hold, l r ontinuouly fall (inreae) over time beoming loer and loer to it lower (upper) boundary, l r, without ever reahing it boundary. Under thee diverified interior olution ondition, we an alo haraterize labor in the < l r other two etor. Given the ontany of ( ) hl and poitive alloation of labor to the human apital etor in a diverified eonomy, long run employment in the ommodity etor ( l ) mut ontinuouly fall in a growing eonomy. The fat that l r i alo falling (riing) mean that the ervie etor employment ( l ) mut ontinuouly expand (ontrat) over time toward it impliit urrogate ICP level when β > α (α > β ). Thi urrogate ICP i l = L l r (note that the urrogate ICP for employment level in the ommodity etor i zero; that i, l = 0 ). V. ICP Dynami and Bifuration In thi etion we diu the Stage 2 ondition under whih either a fully diverified interior olution or a peialized orner olution our. Figure 1 provide a graphial explanation of the poible Stage 2 aet endowment and the onequene for diverifiation Page 22

25 and long run growth pattern of the eonomy in the k Figure 2A and 2B provide a different perpetive in the ( k/ h) omparion of the ae when β > α and α > β. h pae (for the ae when β > α ). h pae whih permit a lear Region (I) in Figure 1 ( Speialization and Stagnation ) i defined by the KK boundary equivalent to the Phyial Capital Intenive Corner Solution preented in the Remark to Propoition 2 ( hl r = 0 ). From (26) it follow that the KK line i defined in the long run by k h= LZ or, equivalently by k = kmax. Thu, if the initial endowment of human apital ( 0 h ) i too low vi-à-vi the endowment of phyial apital ( k 0 ), the human apital etor may not operate auing human apital aumulation to be infeaible. Given that kh mut be inreaing when the human apital etor i tagnant, it i lear that an eonomy that enter Region (I) will remain in Region (I). Intuitively, in thi region human apital prodution i rowded out beaue it annot ompete with the final good etor. The opportunity ot of labor in prodution of the final good i too high relative to the human apital etor; equation (13) beome an inequality ( w > ( μ / λ) B) and the rate of return to human apital (equation (16)) eae to have the level BL a upport. Furthermore, the interior olution ACP for the natural reoure level ( n ) annot be utained in Region (I) beaue the eonomy i growing by inveting in phyial apital only and intead onverge to the orner olution ACP where rn ( ) = ρ, ˆ = 0 and the eonomy tagnate (when the tok of apital reahe a level k in Figure 1). Thu, an eonomy that i too rih in phyial apital and/or too poor in human apital may grow over the intermediate run on the bai of aumulating phyial apital only but it will not be able to avoid a tagnation trap over the long run. Region (II) in Figure 1 ( Temporary Diverifiation and Growth ) allow for untable diverifiation. Thi region i defined from below by the KK line and above by the KK and repreent an eonomy where the level of employment in the human apital etor i poitive but below it ICP (i.e., l r < l r ) for the ae where β > α. The KK line i given by k = ( k/ h) h+ k. So in Region (II) we have that k h ( k h). Initially, the eonomy i min fully diverified; the fator endowment allow all three etor to be in operation. However, by Propoition 2, the k h ratio in thi region i too high beaue it i above or at it ICP level Page 23