4 The Neoclassical model with Human Capital

Transcription

1 4 The Neoclassical model wih Human Capial Why doesn capial flow from rich counries o poor counries? [Rober Lucas Jr.] Learning Goals: - Undersand why he Solow model canno accoun for large cross-counry differences in per capia income - Acknowledge he poenial role of Human Capial in making he model more consisen wih he real world facs. - Acknowledge ha, even accouning for human capial, much of he crosscounry variance of per capia incomes remains o be explained. - Developmen accouning - Acknowledge he main achievemens as well as he main limiaions of he neoclassical growh model Inroducion We jus saw ha he Solow model is capable of describing very reasonably a wide range of real world facs of economic growh. No surprisingly, he Solow model rapidly became he workhorse model in he heory of economic growh and i sill is. During he las 30 years, however, various researchers have subjeced he model o furher and more demanding empirical scruiny. One direcion explored in his furher invesigaion relies on he fac ha he model specificaion implies no only he expeced signs of cerain parameers bu also heir approximae magniudes. In paricular, he model implies broad orders of magniude for he coefficiens linking per capia income o he savings rae and 18/02/2015 1

2 o he populaion growh rae. I happens however ha hese magniudes do no conform oo well wih he empirical evidence. The key parameer in his furher invesigaion is he elasiciy of oupu in respec o physical capial,. If, according o he model, here is perfec compeiion and facors of producion are paid heir marginal producs, hen he elasiciy of oupu in respec o physical capial should correspond o he share of capial in domesic income. As we will see, he empirical observaion ha he laer sands a around 30% o 40% makes he model incapable of describing he large differences in per capia incomes ha we observe in he real world. The conciliaion would require, for example, differences in savings raes as beween counries much larger han hose acually observed in realiy. This chaper addresses hese inconsisencies and explores one avenue ha was proposed o miigae he problem: he inroducion of a second reproducible facor in he producion funcion, Human Capial. As we will see, such an exension will help, bu i will no be enough o make he model fully consisen wih he real world facs. Secion 4.2 reviews he above menioned limiaions of he Solow model. Secion 4.3 inroduces he concep of Human Capial and proposes an augmened producion funcion, including human capial. Secion 4.4 shows a version of he neoclassical growh model exended wih Human Capial. Secion 4.5 urns o he empirical evidence o evaluae how far he exended version can go in accouning for cross-counry income differences. The conclusion is ha augmening he producion funcion wih human capial improves he predicabiliy of he model bu i does no allow i o fully explain he exising per capia income differences. The chaper concludes wih he need o have a beer undersanding of wha is behind he produciviy erm, and oulines he subsequen direcions of our search The Lucas Paradox Explaining cross-counry income differences 18/02/2015 2

3 In he Solow model, he seady-sae level of per capia income is given by: y 1 s 1 1 * A e [3.10] n In his secion, we invesigae wheher calibraing his expression wih sensible values for he main parameers makes i capable of explaining per capia income differences of magniudes observed in he real world. Firs, we need an esimae for. Because he model assumes perfec compeiion and rules ou marke failures (such as exernaliies, and public goods), i predics ha facors are paid according o heir marginal producs. This is refleced in equaions (2.11) and (2.12), which sae ha he shares of capial and of labour in naional income shall correspond o he respecive elasiciies in he producion funcion, and 1-. Empirically, we observe ha he shares of labour in naional incomes vary around 60%, depending on he counry. Given his, one may reasonably calibrae he Solow model seing he parameer equal o 1/3. Wih such calibraion, he elasiciy of income in respec o he savings rae, ( 1 in equaion 3.10) becomes equal o 0.5: ha is, a 1 percenage poin (p.p.) increase in he saving rae implies a 0.5p.p increase in per capia income. The inconsisency wih he empirical facs arises from ha fac ha his elasiciy is oo small o accoun for he large cross-counry per capia income differences we observe in he real world. To ge a sense of his, consider wo counries; say a Rich Counry (R), and a Developing Counry (L). From (3.10), and absracing from differences in A, he raio of per capia incomes in he seady sae will be: y y R L s s R L 1 n n L R 1 (4.1) In he following exercise, assume ha =1/3, =3% and =2% (he las corresponding o he rend growh rae of per capia GDP in he U.S.). 18/02/2015 3

4 s L Consider furher he following values for he remaining parameers: s 0. 2, 0.24, n R =1% and n L =3%. These values mach roughly hose of US and Tanzania beween 1960 and Wih hese parameer values, equaion (4.1) would predic per capia incomes in he rich counry and in he poor counry differing only by a facor of 1.05 (ha is, he difference beween he average income of a US ciizen and ha of a Tanzanian ciizen should be 5% only). Obviously his is oo lile: by 2000, per capia income in he US was 32 imes higher han ha of Tanzania. Now, as an exreme case, assume ha he poor counry had insead s L =1.5% and n L =5%. Clearly, hese assumpions are exreme, even for developing counries (no counry has susained a populaion growh rae as high as 5% and few save as lile as 1.5% of he respecive income). Sill, hese figures would imply a raio of per capia incomes of 4.7, only. Again, his is oo small o accoun for he observed differences in per capia incomes 1. The conclusion is ha, even using drasic assumpions concerning he differences in he saving rae and in he populaion growh rae, here is no way of generaing cross counry income gaps as large as we observe in realiy in he conex of he Solow model as i is. R Using he producion funcion By making use of equaion (3.10), exercise (4.1) implicily posulaes ha economies are in he respecive seady saes. Bu i could be ha he poor counry was sill on is way o he seady sae. In ha case, par of he poor counry income gap visa-vis he rich counry could be due o ha fac ha is income poenial (as deermined by n and s) was no ye fully maerialized. 1 If one used insead, which is also a reasonable assumpion, he raio of per capia incomes in he exreme scenario would rise o 7.9. This is sill oo low. 18/02/2015 4

5 An alernaive approach is o look a he producion funcion. This approach has he advanage of relying on a relaionship ha should hold each momen in ime, regardless wheher a counry is in he seady sae or engaged in ransiion dynamics. Evidence ha rich counries have more capial per worker han poor counries is provided in Figure 4.1. The figure crosses daa on per capia income and on capial per worker for 53 counries. The figure confirms ha capial per worker varies considerably across counries (by a facor of 100:1) and ha is posiively relaed o per capia income. The non-lineariy in he relaionship beween per capia income and capial per worker is also suggesive of diminishing reurns, as assumed in he Solow model 2. The quesion is wheher he relaionship beween he capial labour raio and per capia oupu in ha figure is consisen wih our guess for he capial-oupu elasiciy. The base-line producion funcion in he Solow model is: y A k [2.2] From Figure 4.1, le s pick a developing counry, say Peru. By 2000, per capia income in Peru was roughly 17.9% of he corresponding level in he Unied Saes. Could such difference be explained by he capial labour raio alone? To answer his quesion, le s assume again ha is equal o 1/3, and solve (2.2) for k, o find ou he level of capial per worker in Peru ha would be needed o mached he assumpions. The answer is k Tha is, for he income difference beween Peru and he US o be explained by he capial labour raio only, one would need a capial labour raio in Peru equivalen o 0.57% of ha in he Unied Saes. This is clearly unrealisic: acually, by 2000, he capial labour raio in Peru sood a abou 18% of he corresponding level in h US. 2 We have o be careful wih such inerpreaion: for insance, Hong Kong achieves a higher level of oupu per worker han Japan, wih a lower level of capial per worker. This reveals ha oher facors apar from capial per worker (in our formulaion, capured by parameer A) are driving cross-counry income differences. However, he discussion above absrac from he influence of his facor, so as o sress he limiaions of he original Solow model. 18/02/2015 5

6 Figure 4.1 GDP per worker and capial per worker, ,000 60,000 USA 50,000 IRL HKG BEL NOR GDP per worker 40,000 30,000 20,000 MUS MYS CHL MEX VEN PRT GBR GRE FIN JAP CHE EGY PAN 10,000 BOL PER 0,000 0,000 20,000 40,000 60,000 80, , , , , ,000 Capial per worker Source: Caselli and Feyrer (2007). The Figure crosses capial per worker and GDP per worker in he year The non-lineariy in he relaionship beween per capia income and capial per worker is suggesive of diminishing reurns, as assumed in he Solow model. Bu one may quesion wheher all hese counries are lying along he same producion funcion. Why doesn' capial flow from rich o poor counies? A hird way of formulaing he same problem was proposed by he Nobel Laureae Rober Lucas (1990): if, cross counry differences in per capia incomes were only relaed o differences in he raio of capial per worker, hen poor counries should have much higher ineres raes han rich counries. How much higher? To answer his quesion, Lucas solved equaion (2.2) for k, and used (2.12), o ge: 1 1 r A y. (4.2) Now consider a Rich counry, (R), and a Less Developed Counry, (L), operaing along he same producion funcion, (2.2) ha is, wih an equal A. In ha case, he raio of capial reurns in he wo counries should be equal o: 18/02/2015 6

7 r r L R y y R L 1 (4.3) As an example, Lucas considered he cases of USA and of India, which per capia incomes in 2000 differed by a facor of 15:1 3. Wih =0.4, he exponenial erm in equaion (4.3) equals 1.5. Thus, o explain a 15:1 gap in per capia incomes, he raio of capial reurns should be 15^1.5= 58. Tha is, he ineres rae in India would need o be 58 imes greaer han ha of he USA. In plain language, if capial were generaing a reurn of around 4% in he USA, hen he corresponding average reurn in India should be 233%. Clearly, his is wholly unrealisic, even admiing ha poorer counries have very high risk premia 4. Lucas poined ou ha such high ineres differenials canno hold in a world wih capial mobiliy. Thus, in face of such reurn differenials, capial goods should be flowing from rich counries o poor counries. Moreover one would expec almos no invesmen o occur in capial-abundan counries unil capial-labour raios - and hence ineres raes - were more or less equalised across he globe. This, in urn, would be good news for poor counries: if heir problem was lack of capial and if his was refleced in high ineres raes, hen capial should flow in and convergence of per capia incomes would be jus a maer of ime. Clearly his sory does no conform o oday s realiies: capial does no flow in huge amouns o poor counries, ineres differenials are nowhere near as large as 58 3 According o Maddison (2001), in 2000, per capia incomes in USA and in India in comparable unis (PPP) were, respecively, $ and $ Noe ha =0.4 is a generous assumpion. Wih =1/3 he raio of ineres raes would jump o 225! 18/02/2015 7

8 imes and here is no sysemaic endency for poor counries o grow faser han rich counries 5. Box 4.1 The reurn o physical capial in poor counries The quesion as o wheher he marginal produc of capial is higher or lower in poor counries han in rich counries has imporan policy implicaions: if one concludes ha he marginal produc of capial differs subsanially across he globe, his means ha marke fricions are prevening physical capial from being more efficienly allocaed across he globe. In ha case, here would be scope for expanding he world oupu by promoing a beer allocaion of capial beween rich counries and poor counries (for insance, hrough exernal aid). Caselli and Feyrer (2007) invesigaed wheher he low levels of capial per worker in poor counries are associaed wih higher marginal producs of capial or no. The auhors esimaed he marginal produc of capial for 53 counries, assuming an aggregae producion funcion exhibiing consan reurns o scale. However, he auhors accouned for a hird inpu o producion, labelled as naural capial, ha includes land and naural resources. Accouning for his hird inpu implies ha he share of capial in naional income () canno no be esimaed as one minus he share of labour on naional income. In he case of Unied Kingdom, for insance, he auhors esimae a share of labour on naional income equal o 75% and a share of physical capial equal o 18%, only. The 7% difference corresponds o he share of naural capial. In Bolivia, he share of labour is esimaed o be 67% while he share of physical capial is only 8%, which implies a 25% share for naural capial. Since he share of income rewarding naural capial ends 5 Lucas observed ha during he XVIII and XIX cenuries, a a ime when producion funcion was beer described as dependen on labour and land, labour acually moved from labour abundan counries (Europe) o labour scarce counries (New World). In he XX cenury, capial replaced land as he main facor in he producion funcion and became he poenially mobile one, a a ime when srong resricions on labour mobiliy were ereced. 18/02/2015 8

9 o be significan in counries specialized in agriculural and naural-resource indusries, ignoring hese would imply an overesimaion of he conribuion of capial o oupu. This is especially rue for poor counries. A second cavea poined ou by Caselli and Feyrer (2007) relaes o he relaive price of capial. If he price of oupu and he price of capial were he same as assumed in he Solow model hen in a world wih perfec capial mobiliy one should observe an equalizaion of marginal producs of capial across counries. However, in he real world, one uni of capial does no cos he same as one uni of oupu. In general he relaive price of capial is higher in poor counries han in rich counries, due o ariffs, ranspor coss and oher disorions. Since in poor counries, firms have o spend more unis of oupu o buy one uni of capial, hey need o achieve a higher produciviy on he invesed capial, all else consan. Formally, le p K be he relaive price of capial goods (in unis of Y). In ha case, when one uni of oupu is saved, he invesor will be able buy 1 pk unis of physical capial, only. Wih a marginal produc of physical capial equal o Y K Y K change in oupu obained by saving one uni of oupu will be Y K 1 18/02/ p K, he. Taking his ino accoun, Caselli and Feyrer correced he usual esimae of he Marginal Produc of Capial, dividing i by he relaive price of capial. Figure 4.2 compares he correced esimaes wih he naïve esimaes (ha is wihou price correcion and wihou accouning for he share of naural capial). As he figure reveals, he naïve esimaes poin o high and variable marginal producs of capial poor counries, whereas in rich counries he marginal produc of capial ends o be much lower. Tha being he ruh, i would suppor he idea ha capial has no being flowing from rich counries o poor counries due o some form of capial marke fricions. In ha case, a massive invesmen in poor counries would be he key for economic convergence and worldwide efficiency improvemen. Afer he correcions, however, he sory is differen. As shown in Figure 4.2, when correcions for he naural capial and for he relaive prices are implemened, he

10 marginal produc of capial is largely equalized across counries. In oher words, from he invesor s poin of view, he reurn o invesmen is no higher in poor counries han in rich counries. This evidence suggess ha he large variaion in capial per worker across counries canno be aribued o capial marke fricions: a reallocaion of capial across counries so as o exacly equal he marginal producs of capial would no bring a world significanly more equal han wha we observe oday. Figure 4.2 The marginal produc of capial (naïve and correced esimaes) 60% Correced by/k 50% Naive by/pk Marginal produc of capial 40% 30% 20% 10% 0% Real GDP per worker Source: Caselli and Feyrer (2007). The naïve esimaes correspond o he marginal produc of capial compued assuming a producion funcion wih capial and labour, only, and ignoring differences beween he price of oupu and he price of capial. The correced esimae accouns for he role of naural capial and for he exising cross-counry divergences beween prices of oupu and prices of capial. 18/02/

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12 caused capial reurns o converge across counris 6. So, in oday s world, he gains from furher capial marke inegraion are expeced o be small. So, how can we explain hese inconsisencies? The discussion above reveals ha capial accumulaion canno accoun for he large differences in per capia income we observe in he real world. If hese differences were accouned for by he differences in he availabiliy of capial per worker, only, hen he marginal produc of capial should be very high in poor counries and capial should flow from rich counries o poor counries. However, his is does no happen in our days: capial is no flowing massively from rich counries o poor counries and here is no evidence ha he marginal produc of capial is subsanially higher in poor counries han in rich counries. The conclusion is ha we can hardly explain he cross-counry variaion of per capia incomes wih differences in he level of capial per worker, only. In order o solve his puzzle, hree avenues can be explored: 1. One is o accoun for he role of oher inpus in producion, in paricular Human Capial. This avenue will be addressed in he remaining of his chaper. 2. A second one is o admi ha he role of capial in producion is larger han ha measured by he capial income share. Tha would be he case if capial was no correcly priced in he marke, say, because of a posiive exernaliy. This avenue will be addressed in Chaper The hird avenue is o focus on he facor ha has been silen so far and ha we loosely relaed o he sae of echnology. This avenue will be explored in pars II and III of his book. 6 Mello (2008). According o his auhor, he price-correced marginal produc of capial in poor and rich counries differed, on average, by 5.52 p.p. in he 1970s, 1.85 p.p in he 1980s and 0.37 p.p in he 1990s. 18/02/

13 4.3. Human capial Wha is human capial? Human capial is he erm used o coin he sock of knowledge and healh embodied in labour. The noion of human capial comes from he observaion ha people inves in knowledge and in healh (hrough schooling, on-he-job raining, exercise and healhcare) wih he aim o obain a reurn, jus like people inves in physical capial. Human capial is also similar o physical capial, in ha i depreciaes along ime. As an example, hink on he know-how needed o disinguish poisonous mushroom from he good ones: cerainly, in our days such knowledge is much less relevan han i was housands of years ago, when mos people lived in he woods. As for anoher example, hink on wha happens o your old knowledge each ime you need o adap o a new release of your compuer operaing sysem: you have o learn how o operae wih he new version, hrowing away par of he ime invesed in learning how o operae wih he old version. In general, as he ime goes by, par of he accumulaed knowledge ges oudaed and useless. Jus like physical capial, human capial erodes over ime. Noe however ha human capial is differen from physical capial in ha i is embodied in individuals. In conras o physical capial, here are physical limis in human capial accumulaion a he individual level. Laer, we ll see ha his propery makes inequaliy in human capial endowmens much more disrupive for economic developmen han inequaliy in physical capial. A similar reasoning holds for healh: healh depreciaes along ime (and a an increasing rae wih age!). Consequenly, coninuous invesmen in healh is required (exercise, safe nuriion, healh care), so as o preven healh capial from eroding oo fas. Noe ha he wo componens of human capial end o be correlaed o each oher: more educaed people end o be more aware of he advanages of a healhy nuriion and of exercise, so hey will end o be more healhy oo; by he same oken, healhier individuals, wih longer life expecancies, are likely o inves more in educaion, 18/02/

14 because hey will have longer payback periods. Thus, improving one dimension of human capial is expeced o deliver improvemens in he oher dimension oo. Human capial as an inpu o producion In order o accoun for he role of human capial in producion, le s consider a producion funcion where boh raw labour and human capial ener as inpus o producion: Y A K H N, wih <1 (4.4) 1 In (4.4), he new variable H sands for Human Capial (in he form of knowledge and healh) and all oher variables are defined as before. In ligh of his specificaion, producion needs bodies and brains and i is no possible o subsiue compleely bodies for brains or brains for bodies 7. This producion funcion exhibis CRS in all hese inpus: ha is, one would be able o duplicae Y if one could simulaneously duplicae he use of he hree facors, K, H and N. As in he basic Solow model, however, we will be consrained by he fac ha one inpu (N) evolves a exogenous rae. Hence, seing K and H o expand faser han N would deliver diminishing reurns. To see he link beween human capial and per capia income, le s wrie he producion funcion in he inensive form: y A k h, where h H N is defined and human capial per worker. Why does human capial impac on posiively on per capia oupu? 7 Laer, we will see formulaions where human capial and raw labour are merged ogeher ino a unique composie inpu called human capial. 18/02/

15 There are several reasons o believe ha human capial impacs posiively on oupu per worker: - Firs, more knowledgeable workers will be able o accomplish more complex asks wih a minimum oulay of ime. - Second, healhy and well nourished workers are expeced o have more physical and menal energy o learn and underake heir asks han sick and sarving workers 8. - Third, healh capial increases he amoun of healhy ime available for work, by reducing incapaciy, disabiliy and abseneeism. - Fourh, individuals wih longer life expecancy have incenive o work harder, because hey need o save more for heir longer reiremen period. All in all, hese various effecs imply ha a higher level of human capial should impac posiively on oupu and, by hen, on he average produc of labour. In addiion o he direc effec of human capial on y, human capial can also impac posiively on y via effecs ha are mediaed hrough he produciviy parameer, A. For insance, more skilled and educaed workers are more likely o adop new echnologies and o conribue hemselves o echnological change han less educaed people. These effecs run from H o A and hen o Y. For exposiional convenience, however, a his sage we absrac from he poenial causaliy from H o A (acually, we are absracing from any cross-counry differences in A). Human capial and he produciviy of physical capial As you may remember, whenever wo inpus are complemenary in producion, increasing he use of one inpu has a posiive impac on he marginal produc of he oher. To see his in erms of (4.1), le s ake he parial derivaive in respec o physical capial: 8 A classical sudy is Fogel (1991, 1994). The auhor analysed he relaionship beween work effor and caloric inakes in France and England since he 18 h cenury. He concluded ha in he 18 h cenury France, individuals in he boom 10 percenile of consumpion had daily caloric inakes ha were so low ha hey could no even have enough energy o work. In he cenuries ha followed, improvemens in nuriion impaced significanly on workers effor (some 30% among Briish workers, he esimaed), and also in he paricipaion rae (ha is, he fracion of working-age populaion ha is acually able o work). 18/02/

16 K Y y A k 1 k h (4.5) According o (4.5), he marginal produc of physical capial is a decreasing funcion of physical capial per worker (his is he law of diminishing reurns), bu an increasing funcion of human capial per worker (h). Thus, a higher level of human capial per worker has he poenial o miigae he diminishing reurns o physical capial. Figure 4.4 provides a graphical illusraion. The figure displays wo curves describing he marginal produc of capial (4.5) as a funcion of capial per worker, in wo differen counries which differ in erms of human capial per worker: a rich counry (upper case) and a poor counry (lower case). These funcions are downward sloping because of diminishing reurns: in any of hese counries, more capial per worker will ranslae ino a lower produciviy of capial (and hence, lower ineres raes), everyhing else consan. However, he fac ha he rich counry is endowed wih a higher level of capial per worker may preven he ineres rae from being lower han in he poor counry. In he figure, he marginal produc of capial in he rich counry is represened by poin C, which is roughly similar o ha of he poor counry (A). Hence, in his example, here would be no reason for capial o flow from he rich counry o he poor counry. Wheher in he real world adding human capial in he producion will be enough o solve he Lucas paradox is a differen sory: as we will see in he remaining of his chaper, he answer is no. 18/02/

17 Figure 4.4 Wha happens o he marginal produc of physical capial when human capial per worker increase? Y K r A C Y K rich B Y K poor k poor k rich k K / N Augmening he producion so as o include human capial implies ha he schedule describing he marginal produc of capial becomes parameric on he level of human capial per worker. Hence, a poor counry wih less physical and human capial per worker can end up wih exacly he same ineres rae as a counry wih more physical and human capial per worker The augmened Solow model (MRW) In his secion, we exend he Solow model by adding human capial o he producion funcion 9. The main assumpions As in he Solow model, le s assume ha he echnological parameer in (4.4) increases coninuously, a an exogenous rae g: 9 This model was firs proposed by Mankiw, Romer and Weil (1992). 18/02/

18 g A Ae [3.1] Rearranging, he producion funcion (4.4) in erms of efficiency labour, we ge: Y AK H L, (4.6) 1 where L N e and 1 g. Noe ha in his model, workers become more producive boh because of labour augmening echnological progress and because of invesmen in human capial. The criical disincion beween hese wo facors is ha he former is sricly exogenous, while he laer is produced and accumulaed, jus like physical capial. As in he Solow model, i is assumed ha one uni of oupu can be ransformed a no cos ino eiher one uni of human capial or ino one uni of physical capial 10. The resource consrain of he economy is given by: Y C I I, (4.7) H H where I refers o invesmen in Human capial and all oher variables are defined as in he Solow model. I is also assumed ha people inves consan fracions of heir incomes in human capial. Le s H be he fracion of income invesed in human capial and s he fracion of income invesed in physical capial. The dynamics of K and H are given, respecively, by: sy I K K (4.8) s H Y H I H H. (4.9) For simpliciy, he model assumes ha he socks of physical and of human capial depreciae a he same rae,. The depreciaion of human capial may be inerpreed as he erosion of knowledge (obsolescence, forgiveness) ne of he benefis from experience. 10 In alernaive, one could specify a second secor for his economy, specifically devoed o producion of human capial (like schools, universiies and hospials). This case is examined in he Chaper 5. 18/02/

19 The seady sae in he MRW model To solve he model in he simples possible manner, here is a helpful clue: in he seady sae, Physical Capial and Human Capial mus grow a he same rae. A his sage, he reason should be inuiive: like in he simple Solow model, diminishing reurns imply ha i will no worh having one inpu growing faser han he ohers: if, for insance, physical capial was growing faser han human capial, hen he reurn o physical capial would decrease relaive o ha of human capial, creaing he incenives for physical capial o expand less. condiion: Imposing K K H H in he wo equaions above, one obains a criical H K s H (4.10) s This condiion saes ha he raio of human o physical capial in he seady sae shall be proporional o he corresponding invesmen raes. Solving for H, and subsiuing in he producion funcion (4.4), one obains 11 : 1 Y (4.11) BK L where and sh B A s. (4.12) Noe ha, because we used (4.10), equaion (4.11) holds in he seady sae only. Given he similariies beween (4.11) and (2.1), you may guess ha he seady sae of his model will be very similar o ha of he Solow model. And his conjecure happens o be rue. Adaping equaion (3.10) for he parameers in (4.11), one obains: 11 In a foonoe o heir paper (foonoe 12), Mankiw e al. (1992) noed ha he properies of he model change dramaically in he case in which =1. In ha case, equaion (4.11) becomes linear in K, delivering endogenous growh. This alernaive case will be addressed in he nex chaper. 18/02/

20 y * B 1 1 s n 1 e (4.13) Subsiuing back and (4.12), one obains he seady sae level of per capia income in his augmened model: y * AsH s n 1 1 e (4.14) where g 1. Noe ha (3.10) is no more han a paricular case of (4.14), wih. Equaion (4.14) saes ha he seady sae level of income per capia depends posiively on he saving rae and negaively on he rae of populaion growh (as before). I also saes ha per capia income rises in he long run a a consan rae (as before). The novely here is ha he level of per capia income also depends on he share of income devoed o Human Capial Accumulaion, s H : an increase in s H gives rise o a level effec (oupu per capia will expand emporarily unil he new seady sae is reached). This is similar o wha happens wih invesmen in physical capial. A propery of his model is ha he wo saving raes impac on per capia income individually and ogeher, reinforcing each oher: for any given rae of human capial accumulaion, a higher saving rae leads o a higher level of per capia income in he seady sae, which in urn leads o a higher level of human capial. Hence, small differences in he saving raes may explain large differences in per capia income, allowing he model o fi much beer he real world facs. Facor income shares in he MRW model 18/02/

21 The producion funcion (4.4) applies o he economy as a whole. Assuming ha all firms in his economy are idenical, firm level profi maximizaion leads o he following aggregae demands for physical capial, human capial and raw labour 12 : Y K Y H Y r K Y r H (4.15) (4.16) Y N Y N 1 w As expeced, hese equaions imply ha facor income shares are equal o he corresponding elasiciies in he producion funcion (remember ha his is a direc consequence of CRS and no marke failures). Taking ogeher, hese equaions imply ha he oal share of labour in naional income is. Noe ha, alhough he producion funcion disinguishes wo ypes of labour inpu, in he real world, hese wo inpus are paid in he same wage bill. If, for insance, his means ha he labour share in income will be around 2/3, which accords o he empirical evidence. An arbirage (efficiency) condiion Absence of arbirage opporuniies implies ha invesmen in he wo ypes of capial shall be such ha heir marginal producs are equal. Combining he demands for physical and human capial (4.15) and (4.16), one obains: H K (4.17) 12 Noe ha, because in his model one uni of oupu can be ransformed a no cos in eiher one uni of physical capial or in one uni of human capial (equaion 4.7), he user coss of hese wo forms of capial are he same. 18/02/

22 This equaion implies ha capial and labour evolve proporionally, as assumed in (4.10) 13. For fuure reference, he efficiency condiion (4.17) will be aken as benchmark, namely o analyse deparures from he well funcioning economy case, in Par III of his book. Cross-counry income differences revisied To ge a sense on wheher augmening he neoclassical model wih human capial helps explain he observed cross-counry differences in per capia oupu, le s firs reconsider he rich counry - poor counry example of Secion 4.2. Using (4.14), he raio of per capia incomes in he seady sae becomes: y y R L s s R L 1 s s HR HL 1 n n L R 1 Comparing o (4.1), we see ha he exponenials are now higher: 1 1. This means ha he impac of each parameer in he seady sae level of per capia income is higher han in he simple Solow formulaion. To exemplify, le s consider a case wih =3% and =2%. This gives: y y R L s s R L s s HR HL n n L R Now, consider for insance, a case where s s 0. 2, n 0. 01, s s 0.10 and n In his case, he raio of per capia incomes beween he L HL L R HR R 13 I is worh noing ha his efficiency condiion is consisen wih he golden rule. As in he basic Solow model, he golden rule for capial accumulaion can be obained maximizing he seady sae level of per capia consumpion: * * * max c 1 s sr y, where y is given by (4.14). The soluion o his problem is s, sr as expeced: s= and s R. Using (4.10), we hen one obain (4.17). Noe, however, ha he golden rule is more resricive han (4.17), because i implies wih he level of savings. 18/02/

23 rich counry and he poor counry is 7.1:1. This is more han in he Solow model wih equivalen assumpions (1.92:1), bu sill - is half-way relaive o wha we need! In general, explaining real world cross-counry income differences using equaion (4.4) delivers beer resuls han using equaion (3.10) (see Box 4.2). Bu he empirical evidence also reveals ha a significan par of he cross-counry variaion of per capia incomes remains o be explained. In oher words: accouning for he role of Human Capial cerainly improves he explanaory power of he model; bu is no enough: definiely, one needs o depar from he assumpion ha A is equal across counries. Box 4.2. Empirical es on he MRW model seady sae Insead of calibraing equaion (4.14) o compare pairs of counries, one may use regression analysis o invesigae more sysemaically how well he model accouns for observed cross-counry income dispariies in he real world. This was done by MRW in heir paper. In order o run a linear regression, he auhors ook logs in equaion (4.14). The implied regression equaion is: ln y * i a 1 wih ln A 1 1 ln si lnni ln s Hi lnni ui a (4.18) Equaion (4.8) implicily assumes ha all counries are in heir seady saes or, more generally, ha deviaions from he seady sae are random. The random disurbance u i capures hese deviaions, as well as counry specific effecs deermining differences 18/02/

24 in he level of A, such as differences in resource endowmens or in climae 14. The Solow model corresponds o he paricular case in which. Table 4.1 describes he esimaion resuls obained by Mankiw e al (1992). The auhors used a sample of 98 counries for he year The proxies used were: he growh rae of he working-age populaion for n; he raio of real invesmen o GDP for s (period averages); he fracion of he eligible populaion enrolled in secondary educaion for s H ; he real GDP per working age person for y; he auhors also posulaed 0.05 for all counries. The esimaes of he Solow model (equaion 4.18 imposing are displayed in he firs column of able 4.1 (sandard errors in ialics). According o hese esimaes, he Solow model accouns for 59% of cross-counry variaion of per capia incomes. The esimaed coefficiens have he prediced signs and are significan. Noneheless, he implied value for is as high as 0.60 (using ). This is jus anoher incarnaion of he Lucas puzzle: in order for he observed differences in per capia incomes o be accouned for by differences in physical capial endowmens, one would need a conribuion of capial o producion much larger han he corresponding observed income shares. The coefficien on physical capial is biased upwards because human capial is omied from he regression equaion. 14 Noe ha he possibiliy of counry specific effecs leads o a poenial economeric problem: o he exen ha negaive counry specific effecs (e.g., poor resource endowmens) discourage capial accumulaion (as is likely), he error erm will be correlaed o he saving rae and herefore esimaes will be biased. This is one of he main objecions o he empirical implemenaion of his model. To ge around his problem, some auhors proposed panel daa esimaion, which allows for he conrol of counry ( fixed ) effecs (Islam, 1995, Caselli e al, 1996). 18/02/

25 Table 4.1. Esimaion of he Solow model and he Augmened Solow model Dependen variable: log GDP per working age person in 1985 Sample Observaions Non-oil counries 98 consan 6,87 7,86 0,12 0,14 ln s ln n 0,05 ln s H ln n 0,05 1,48 0,73 0,12 0,12 0,67 0,07 Implied bea 0,60 0,31 0,02 0,04 Implied alfa 0,28 R- Squared 0,59 0,78 0,03 S.E.E. 0,69 0,51 Source: Mankiw e al (1992), Tables I and II. Noes: Sandard errors are in ialics. The able displays he regression resuls of an equaion describing he seady sae level of per capial income, in ligh of he Solow model (firs column) and he MRW model (second column). In he firs column, he esimaed share of capial in producion in 0.6, much higher han he observed capial income shares. In he second column, he inclusion of human capial in esimaion brings he capial-oupu elasiciy o a level consisen wih he observed income disribuion. The resuls using he augmened model (equaion 4.18) are displayed in he second column of able 4.1. According o hese esimaes, his model explains 78% of he cross-counry variaion in per capia incomes. All coefficiens have he expeced signs and are highly significan. Moreover, he new esimae for (0.31) accords much more 18/02/

26 closely wih he observed facs. Taken ogeher, he evidence presened is more favourable o he augmened version of he neoclassical model han o is basic version 15. Condiional convergence In Figure 3.5, we showed ha here is no general endency for poor counries o grow faser han rich counries. In Secion 3.5, i was argued ha his finding is fully consisen wih he neoclassical growh model: he model allows counries o reach differen seady saes. The model also implies ha counries grow faser he farher hey are from heir seady saes. This propery of he neoclassical model is called condiional convergence. Empirically, he condiional convergence hypohesis may be esed by observing he relaionship beween growh and he iniial level of per capia income, afer he variables deermining he seady sae are conrolled for (remember equaion 3.14): if a significanly negaive parial associaion beween growh and iniial per capia income is found, his is aken as evidence of condiional convergence. 15 Mankiw e al (1992) also esimaed he model for a sample of OECD counries, only. The resuls wih he MRW specificaion were however disappoining: he R-squared was 0.28 and he coefficiens on invesmen and populaion growh were no significan. A naural explanaion is ha WWII caused significan deparures from he seady sae in his sub-sample. Since he regression model (4.18) does no accoun for ransiion dynamics, i canno isolae he fac ha, in hese economies, he invesmen raes and populaion growh raes have no ye deliver heir full impac on per capia income. 18/02/

27 Figure 4.5 Evidence of Condiional Convergence (98 counries) 1,5 Growh of GDP/adul Condiional on saving, populaion growh and human capial 1 0,5 0-0,5 y = -0,3x + 2,4747 R 2 = 0, ,5 6 6,5 7 7,5 8 8,5 9 9,5 10 GDP/adul 1960 (logs) Source: Mankiw e al, The figure displays he relaionship beween he growh rae of per capia GDP and he iniial level of per capia GDP, afer conrolling for he effecs of variables deermining he seady sae (saving rae, invesmen rae in human capial, and populaion growh rae). The negaive slope is suggesing of diminishing reurns and condiional convergence. Figure 4.5 shows he resuls of a es on condiional convergence performed by MRW using heir heoreical model - deails in he Box 4.3. The verical axes measures he growh rae of per capia income, afer conrolling for he effecs of: he saving rae; invesmen in human capial; and he populaion growh rae (e.g, using equaion 4.14). The horizonal axes measures he iniial per capia income. Clearly, he figure reveals a srong negaive associaion beween growh and iniial incomes, afer conrolling for he differen seady saes. This evidence srongly conrass o he finding in Figure 3.5, which uses he same daa, bu wihou conrolling for differences in he seady sae. The conclusion is ha he worldwide evidence is no supporive of absolue convergence, bu i is supporive of he neoclassical proposiion of condiional convergence. 18/02/

28 Box 4.3. Empirical implemenaion of he MRW model Condiional Convergence Esimaing equaion (4.18), one implicily assumes ha couries are already in heir seady saes. To overcome his limiaion, Mankiw e al (1992) also esimaed a version of he model accouning for he possibiliy of counries being engaged in ransiion dynamics. To undersand heir es, we refer o equaion (3.14), which describes he ransiion dynamics in he Solow model. This equaion also holds in he MRW, afer adaping he parameer measuring he speed of convergence, v, o he exisence of human capial: 1 n 16. Then, on can es for condiional convergence by invesigaing he sign and significance of parameer b (3.15), wihou forgeing o conrol for he deerminans of * he seady sae in he MRW model (ha is, replacing y by 4.18). Formally, he following empirical model is obained: ln y ln y0 ln y0 1 (4.19) where ln A 1 and e 1 ln s lnn ln s lnn i 1, n 1. H i Table 4.2 illusraes how he es on condiional convergence was implemened by Mankiw e al. (1992), for a worldwide sample and for he OECD economies. The hird 16 Noe ha in he MRW model is slighly differen from ha in he Solow model (Appendix 3.1). The laer shall be seen as a paricular case, wih. This means ha he inclusion of human capial generaes a larger role for ransiional dynamics han in he Solow model. I is also worh noing ha by imposing a common convergence parameer (while populaion growh raes differ across counries), here is a poenial bias (a discussion in Lee e. al., 1997). Aemps o address his limiaion include Evans (1997) and Arnold e al. (2007). 18/02/

29 column of Table 4.2 also shows he resuls of a similar esimaion, for a sample of 37 African counries 17. As shown in he able, in he hree samples, all coefficiens have signs in accordance o (4.19) and are saisically significan. The esimaed coefficiens of ln y 0 are negaive and significan a he 5% level. This means ha he condiional convergence hypohesis holds in he hree samples. The parial associaion beween he growh rae of per capia GDP and he iniial level of per capia GDP, as implied by he firs column of Table 4.2 is displayed in Figure 4.5. In respec o he parameer esimaes, Table 4.2 poin o some differences as beween he OECD and African counries. In paricular, he esimaed average speed of condiional convergence () is 1.7% for African counries and 2.1%, for OECD counries. This means ha he ime required o eliminae half of he iniial gap from heir seady saes in Africa is abou 42 years, which compares o 35 years in he OECD sample 18. Evenually, he lower speed of convergence in Africa is due o a lower abiliy o arac invesmen in physical capial, probably due o bad economic policies, poliical insabiliy, ec. (bu remember he model we are using is silen in respec o he effec of hese facors!). Table 4.2 Tess for condiional convergence 17 Murhy and Ukpolo (1999). These auhors used he same daase as Mankiw, Romer e Weil (1992) excep in ha heir measure of invesmen in human capial also includes primary school aainmen. 18 To assess how long i akes an economy o ge halfway o is balanced growh pah, he reader is referred o equaion (3.5), hough adjusing for he new definiion of in equaion (4.19) : he answer is e 0. 5, which solves for ln0.5. Noe ha, wih he inclusion of human capial, he parameer v is now lower, implying a slower convergence han in he Solow model (Box 3.5) 18/02/

30 Dependen variable: log difference GDP per working age person Sample Non-oil OECD Africa Observaions consan 2,46 3,55 2,16 0,48 0,63 0,65 ln y 0 ln s ln n 0,05 ln s H ln n 0,05-0,30-0,40-0,35 0,06 0,07 0,10 0,50 0,40 0,44 0,08 0,15 0,16 0,24 0,24 0,31 0,06 0,14 0,17 Implied bea 0,48 0,38 0,40 0,07 0,13 Implied alfa 0,23 0,23 0,28 0,05 0,11 Implied speed of convergence 0,014 0,021 0,017 0,00 0,00 R- Squared 0,46 0,66 0,41 S.E.E. 0,33 0,15 Source: Non-oil counries and OECD: Mankiw e al (1992), Table VI. African counries: Murhy and Ukpolo (1999), Table 1. Significan levels in ialic. The regression equaion is (4.19), describing he ransiion dynamics of he MRW model. The negaive and significan coefficien of iniial per capia income is suggesive of condiional convergence. The esimaed oupu-capial elasiciies are in accordance o he observed income shares. The implied speed of adjusmen o he seady sae is found o be lower in Africa han in OECD counries Produciviy maers! The model of MRW emphasizes he role of facor accumulaion as he explanaion for cross-counry income differences, disregarding he differences in 18/02/

31 echnology 19. Such approach is in line wih he radiional Solow model (which focus on physical capial accumulaion), and also wih he resuls of Alwin Young, ha he Eas Asian growh miracles were fuelled by old-syle facor accumulaion, raher han by echnological change (Box 3.5). This view ha capial accumulaion has such as prominen role has been challenged by many auhors. A seminal conribuion in his criicism was formulaed by Klenow and Rodriguez Claire (1997). These auhors firs poined ou ha he proxy used by MRW for human capial, he secondary school aainmen rae, is likely o overesimae he cross counry variaion of human capial. True, richer counries no only have more schooling bu also beer schooling han poor counries, so by he qualiy facor one would expec proxies based on educaion aainmen o underesimae he cross-counry dispariies of human capial. However, secondary school aainmen varies much more across counries han primary school aainmen. Because MRW did no include he primary school aainmen in heir proxy, he balance was likely o imply an overesimaion of he cross-counry differences in human capial socks. Using a more sophisicaed approach o esimae proxies for human capial, Klenow and Rodriguez Claire concluded ha he role of human capial in explaining cross counry differences in per capia incomes is subsanially lower han wha MRW made us believe. The following secion summarizes some of heir resuls. Developmen accouning As a saring poin, Klenow and Rodriguez Claire, considered a producion funcion equal o (4.6). They focused, however on a ransformaion of i, which you can easily obain, by dividing boh sides by Y and manipulaing a bi furher: 19 This paper akes Rober Solow seriously, wroe he auhors in he firs paragraph of heir seminal aricle (Mankiw, Romer and Weil, 1992, pp. 407). 18/02/

32 K H y A (4.20) Y Y As explained in Secion 3.6, re-wriing he producion funcion in such a way has he advanage of absracing from he capial accumulaion ha is induced by echnological change. Accordingly, he firs erm in he righ hand side measures he Harrod Neural level of echnology. Klenow and Rodriguez-Clare calibraed equaion (4.20) for 98 counries over he period , assuming and. Then, hey expressed all counry variables as percenage of he corresponding values in he Unied Saes (ha is, US=1.00) 20. Table 4.3 displays some of heir resuls. In he able, ake Tanzania, for insance. In 1985, per capia income in his counry was only 3% of ha in he Unied Saes. How much of ha difference could be explained by physical capial alone? In 1985, he K/Y raio in Tanzania was 59% of ha in he Unied Saes. If human capial played no role in he producion funcion (i.e, if ), hen he conribuion of he capial labour raio would be K Y Tha is, wih equal produciviy and no role for human capial, Tanzania should have a per capia income level corresponding o 79% of ha in he US! Now, le s include human capial in he producion funcion. In he able, we see ha he raio H/Y in Tanzania was 37% of he corresponding variable in he Unied Sae. Hence, he join conribuion of physical and human capial o relaive per capia income would be equal o: X. 20 Of course, such an exercise is plagued by he fac ha i ignores causal relaionships beween he differen variables. For insance, an increase in human capial may impac posiively on echnology adopion and hence on measured TFP. Any aemp o isolae he conribuion of he differen facors using descripive echniques is always a limied exercise. 18/02/

33 Thus, adding human capial improved he esimae, bu we are sill very far from realiy. In ligh of (4.20), he remaining difference can only be accouned for by he produciviy parameer: 1 1 A Thus, in he case of Tanzania, he main reason for per capia income o be so small relaive o ha of he Unied Saes should be produciviy, no human or physical capial accumulaion. Table 4.3 Developmen Accouning (1985, US=1.00) Oupu per Facor Conribuion Produciviy Working Age Person (Harrod Neural) Y/L K/Y H/Y X SOUTH AFRICA TANZANIA BRAZIL INDIA INDONESIA MALAYSIA FRANCE NETHERLANDS PORTUGAL TURKEY U.K PAPUA N.GUINEA Source: Klenow and Rodriguez-Clare (1997). The able displays he decomposiion (4.20) for each counry, as a percenage of he corresponding variables in he Unied Saes, assuming and. These figures poin o an imporan role of TFP differences in accouning for cross-counry dispariies of per capia income. Inspecing Table 4.3 for oher counries, we see ha wih he excepion of France - facor accumulaion alone does no fully accoun for per capia income 18/02/

34 differences vis-a-vis he Unied Saes: cross-counry differences in oal facor produciviy are a very imporan source of divergence. To look a he 98 counries in he sample a he same ime, we refer o Figure 4.6. The figure crosses he combined conribuion of physical and human capial (X) wih per capia incomes, wih all variables being measured as a percenage of he corresponding values in he Unied Saes. As he figure reveals, mos observaions in he figure fall on he righ hand side of he 45 degrees line, meaning ha, in general, facor accumulaion alone end o underesimae he observed income gaps. Figure 4.6 Developmen Accouning: conribuions of facor accumulaion versus per capia income (US=1.00) Per capia income (US=100) Facor conribuion: labour, human capial, physical capial (US=1.00) Source: Klenow and Rodriguez-Clare (1997). The figure compares he prediced per capia oupu differences oupu using equaion (4.10) and assuming equal TFP, wih he observed differences. The deviaions relaive o he 45º line measure he imporance of TFP. The echnique of calibraing a producion funcion o measure produciviy differences vis-a-vis a reference counry is know as developmen accouning. Developmen Accouning shares wih growh accouning he feaure ha i uses naional accouns daa o calibrae a producion funcion and disenangle he relaive conribuions 18/02/