Advanced Macroeconomics 5. PRODUCTIVE EXTERNALITIES AND ENDOGENOUS GROWTH

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1 PART III. ENDOGENOUS GROWTH 5. PRODUCTIVE EXTERNALITIES AND ENDOGENOUS GROWTH Alhough he Solow models sudied so far are quie successful in accouning for many imporan aspecs of economic growh, hey have one major limiaion: by reaing he rae of echnological change as exogenous, hey leave he economy's long-run (seady sae) growh rae unexplained. Solow models herefore belong o he class of so-called exogenous growh heories. In his par of he course we will ry o answer a very big quesion ha remains unanswered: how can we explain he rae of echnological change which is he source of long-run growh in income per capia? The search for an answer o his fundamenal quesion akes us o he modern heory of endogenous growh where he long-run rae of growh in GDP per person is ruly endogenous Page 277 of 1314

2 A model ha explains he long-run rae of growh in GDP per worker is one ha endogenizes he underlying rae of echnical change, ha is, makes his rae depend on basic model parameers. Hence, by an endogenous growh model we mean a model in which he long-run growh rae of echnology depends on basic model parameers such as he invesmen raes in physical and human capial, he populaion growh rae, or oher fundamenal characerisics of he economy. An endogenous growh model herefore allows an analysis of how economic policies ha affec hese basic parameers will affec long-run growh in income per capia. This is an imporan, and some will say he defining, feaure of endogenous growh models: srucural economic policy has implicaions for growh in oupu per capia in he long run. In his lecure we will sudy endogenous growh heory. The models o be presened can be divided ino wo caegories. In boh caegories here will be aggregae producion funcions involving a variable A ha describes echnology, bu here will be no assumpion of exogenous echnological progress such as A +1 = (1 + g)a, where g is exogenous Page 278 of 1314

3 One caegory conains models ha include an explici descripion of how echnological progress, A +1 A in period, is produced hrough a specific producion process ha requires inpus of is own. Since we hink of he producion of echnological progress as arising from research and developmen, we call such models R&D-based models of endogenous growh. We will no consider hem in his course. The oher caegory does no have an explici producion process for echnological improvemen, bu assumes ha he A of he individual firm depends posiively on he aggregae use of capial, or of oupu, because of so-called producive exernaliies. This implies ha he aggregae producion funcion, as opposed o ha of he individual firm, will have increasing reurns o scale. As we will see, his will resul in growh in GDP per worker in he long run wihou any exogenous echnological progress being assumed. The models in his caegory are referred o as endogenous growh models based on producive exernaliies, and hey are he subjec of his lecure Page 279 of 1314

4 A growh model wih producive exernaliies In Lecure 3 we explained why growh in income per worker had o vanish in he long run according o he basic Solow model. The explanaion was relaed o consan reurns o capial and labour and he associaed diminishing reurns o capial alone. Le us ake he explanaion once more in a way ha is well suied o our presen purposes. The producion funcion was Y = K α L 1-α, exhibiing consan reurns o K and L and diminishing reurns o K alone (since α < 1). Consequenly here were also diminishing reurns o capial per worker in he producion of oupu per worker: y = k α. Now, assume ha here is some growh in he labour force, say a 1 per cen per year. If capial also increases by 1 per cen per year, as in he seady sae of he basic Solow model, hen because of consan reurns o capial and labour, oupu will increase by 1 per cen per year. Hence oupu per worker will be consan Page 280 of 1314

5 How could here possibly be growh in oupu per worker? Wih he producion funcion of he basic Solow model, only if capial increases by more han 1 per cen per year. If capial increases a a given consan rae of more han 1 per cen per year, say a 2 per cen, hen each year oupu will also increase by more han 1 per cen and hence here will be growh in oupu per worker. Indeed, he (approximae) growh rae of capial per worker, g k, will be consan and equal o 1 per cen, and he (approximae) growh rae in oupu per worker, g y, will be g y = αg k. Hence, here will be a consan and posiive growh rae in oupu per worker. However, he formula g y = αg k already reveals he problem. Since α < 1, he growh rae of income per worker is smaller han he assumed growh rae of capial per worker. Therefore he coninued consan growh rae in capial per worker canno be susained by savings. Wha happens is ha as long as capial increases faser han labour here will be more and more capial per worker and his implies, due o diminishing reurns, ha addiional unis of capial per worker creae less and less addiional oupu per worker, and hence, less and less addiional savings per worker. As Page 281 of 1314

6 a consequence, growh in capial, and in GDP, per worker will have o cease in he long run. This reasoning suggess ha if here were increasing reurns o capial and labour, hen long-run growh in GDP per worker would be possible wihou exogenous echnological progress. If boh capial and labour were increasing a a rae of 1 per cen per year, hen, simply because of increasing reurns, oupu would be increasing by more han 1 per cen per year. And in his case growh would no have o cease in he long run, since i would be unnecessary o build up more and more capial per worker o susain growh and hence diminishing reurns would no be a problem. Increasing reurns o scale a he aggregae level herefore seems o be a poenial source of endogenous growh Page 282 of 1314

7 Consan reurns a he firm level and increasing reurns a he aggregae level One may hink ha he idea of an endogenous growh model suggesed by he above reasoning is simply he basic Solow model wih he producion funcion of he represenaive firm exhibiing increasing reurns, ha is, wih he wo exponens on capial and labour summing up o a number greaer han 1 raher han exacly 1. For wo reasons his is no an idea we will pursue. Firs, because of he replicaion argumen, we believe ha here should be close o consan reurns a he firm level o he inpus ha can be replicaed. Second, he idea of compeiive markes, involving price-aking behaviour of he individual firms, is no compaible wih increasing reurns a he firm level. Here is why. Under consan reurns o scale and given inpu prices, oal coss are proporional o oal oupu. The reason is simply ha an increase in oupu requires proporional Page 283 of 1314

8 increases in he inpus. Hence, under consan reurns average and marginal cos are consan and boh equal o he cos of producing one uni, Ĉ, say. Under increasing reurns, an increase in oupu requires less han proporional increases in he inpus, and herefore average and marginal coss will be decreasing in oupu. Wih a Cobb-Douglas producion funcion wih he exponens adding up o a number greaer han 1, he marginal cos, C ˆ( Y ), will be a decreasing funcion, and C ˆ( ) will go o 0 as Y goes o infiniy. Y If he firm akes he prices of inpus and oupus as given, o maximize profis i should wan o produce an infinie amoun of oupu. In oher words, profi maximizaion does no imply well-defined behaviour of he individual firm. Price-aking behaviour and perfec compeiion are no compaible wih increasing reurns a he firm level Page 284 of 1314

9 There is a way o keep he assumpion of consan reurns a he firm level and a he same ime have increasing reurns a he aggregae level. In our model we only have one represenaive profi-maximizing firm, bu his firm represens he aggregae behaviour of many firms each of which is small relaive o he whole economy. The firm herefore akes aggregae magniudes such as GDP or he aggregae use of capial as given, since i is oo small o have more han a negligible influence on aggregaes. In our model he aggregae use of capial has o be equal o he use of capial in he single represenaive firm, bu we should neverheless assume ha in making is individual decisions, he represenaive firm akes he aggregaes as given. We assume ha he individual producion funcion of he represenaive firm (in is role as a small individual firm) is: Y = (K d ) α (A L d ) 1-α, 0 < α < Page 285 of 1314

10 where he firm akes he labour produciviy variable A as given and here are consan reurns o he inpus of capial and labour, K d and L d. We assume furher ha he A of he individual firm depends posiively on he aggregae sock of capial K as expressed by he consan elasiciy funcion: A = K φ, φ The special case φ = 0 will bring us back o he basic Solow model, bu we will explain below why i may be reasonable o assume φ > 0. Since he individual firm has no influence on aggregae capial, i akes A as given. The aggregae producion resuls from insering (5.2) ino (5.1) and using he facs ha clearing of he inpu markes implies K d = K and L d = L, where L is oal labour supply in period : Page 286 of 1314

11 Y = K α (K φ L ) 1-α = K α+φ(1-α) L 1-α 5.3 When φ > 0, he aggregae producion funcion in (5.3) has increasing reurns because he sum of he exponens is 1 + φ (1 α) > 1. A doubling of boh aggregae inpus implies ha aggregae oupu is muliplied by he facor φ (1 α). Since he individual firm akes aggregae capial as given, he marginal producs enering he marginal produc equal o real renal rae condiions for opimal inpu demands should be hose ha appear when A is aken as given. Taking parial derivaives in (5.1), we herefore have: α 1 α d d K K r = α and w d α A d A L A L = ( 1 ) Page 287 of 1314

12 in which we can inser he marke-clearing condiions K d = K and L d = L as well as (5.2) o arrive a: α 1 α K K r = α and w ϕ α A ϕ K L K L = ( 1 ) 5.5 You will easily verify from (5.3) and (5.5) ha r K = αy and w L = (1 α)y. Hence, we have achieved wha we waned: we have consan reurns a he firm level (so perfec compeiion can be assumed), we have increasing reurns a he aggregae level, and our heory of he funcional income disribuion sill has he nice feaures of consan income shares and no pure profis, capial's share being α. For his whole consrucion he assumpion of a producive exernaliy from he aggregae use of capial o labour produciviy, Eq. (5.2), is crucial. Wha could be he moivaions for his? Page 288 of 1314

13 Empirically, he idea of increasing reurns a he aggregae level is no implausible. A survey of empirical esimaes of reurns o scale in aggregae producion funcions can be found, for example, in Sephanie Schmi-Grohe, Comparing Four Models of Aggregae Flucuaions due o Self-Fulfilling Expecaions, Journal of Economic Theory, 72, Esimaes of he sum of our exponens, 1 + φ(1 α), are sysemaically greaer han 1, poining o φ > 0. Furhermore hey vary widely across invesigaions, ranging from jus above 1 o levels way above 2. There is perhaps a endency ha mos esimaes (and he mos reliable ones) should be found in he (lower) range, A value for he sum of exponens, 1 + φ(1 α), around 1.3, say, corresponds (wih α around 1/3) o φ being around 0.45, while a value of 1.5 would mean ha φ should be This may give an indicaion of larges possible plausible values for φ. Taking he grea uncerainy of he esimaions ino accoun, however, one canno compleely exclude larger values for he sum of he exponens, for insance values close o 5/3 (1.67), The laer corresponds o a φ around 1, a possibiliy ha will be of imporance below Page 289 of 1314

14 For he heoreical moivaions he key phrase is learning by doing. The idea is ha he use of (addiional) capial in an individual firm will have a direc effec on producion as expressed by he individual producion funcion (5.1), bu in addiion i will have a posiive effec on he capabiliies of workers who gain new knowledge by working wih he new capial. The benefi from his effec does no only accrue o he firm which increases is capial sock, because oher firms can gain from i by looking over heir shoulders, and because in he longer run employees may move beween firms and hus bring heir acquired capabiliies wih hem o new employers. These feaures should explain why he addiional capabiliy effec spills over o firms in general and hence should be modelled as an exernaliy. Bu why should here be a capabiliy effec a all? Page 290 of 1314

15 Here he idea is ha one way workers ge more skilful and sophisicaed is hrough he insallaion of new capial in he firms, since new machinery is a carrier of new echnological knowledge. Thus, working wih new capial, learning by doing, workers become more sophisicaed. Taking his idea lierally, he capabiliy effec should arise from accumulaed gross invesmen raher han from he sock of capial, bu we may le he laer approximae all he gross invesmen underaken in he pas (in his connecion ignoring depreciaion). The imporan idea of producive exernaliies due o learning by doing comes from a famous aricle by he economis and Nobel Prize winner Kenneh J. Arrow, The Economic Implicaions of Learning by Doing, Review of Economic Sudies, 29, Arrow consrucs a model ha involves a disincion beween old and new capial and in which he idea of a capabiliy effec from he use of new capial is perhaps beer placed. The complee model Page 291 of 1314

16 The above explanaions have focused on he crucial and only new feaure of he growh model we are consrucing. In all oher respecs he model is jus like he basic Solow model. We can herefore wrie down he complee model. Equaion (5.6) below is he individual producion funcion of he represenaive firm aking A as given, bu wih he equilibrium condiions of he facor markes, K d = K and L d = L, insered. Equaion (5.7) saes he assumpion ha he labour produciviy variable, A, poenially (if φ > 0) depends on aggregae capial because of learning-by-doing producive exernaliies. The wo las equaions, (5.8) and (5.9), describe capial accumulaion and populaion growh, respecively, and are essenially unchanged from he basic Solow model: Y = (K ) α (A L ) 1-α, 0 < α < A = K φ, φ 0, 5.7 K +1 = sy + (1 δ)k, 0 < s < l, 0 < δ < L +1 = (1 + n)l, n > Page 292 of 1314

17 We have chosen no o resae he expressions for he real facor prices his ime, bu whenever needed hey can be aken from (5.5) above. From given iniial values, K 0 and L 0, of he sae variables, he model will deermine he evoluions of all he endogenous variables Y, K, L and A, and hence, using (5.5), of he real facor prices. Someimes i is assumed in similar models ha he exernal learning-by-doing effec on labour produciviy really arises from oal producion raher han from capial use. This would amoun o he formulaion, A = Y φ, raher han (5.7) above, he remaining model being he same. This alernaive model works qualiaively exacly as he one above. We could also have chosen o le he exernal effec from capial use (or producion) be affecing oal facor produciviy raher han labour-augmening produciviy, A, wihou any effec on our conclusions Page 293 of 1314

18 Semi-endogenous growh As menioned, in he special case of φ = 0, he model above is jus he basic Solow model. We will herefore assume φ > 0 in all ha follows, and he aggregae producion funcion (5.3) will hen have increasing reurns o K and L. Noe from (5.3) ha if φ < 1, hen here will be diminishing reurns o capial alone, since α + φ(1 α) will be smaller han 1, while φ = 1, he aggregae producion funcion will have consan reurns o K alone. Wheher here are diminishing reurns or consan reurns o capial alone urns ou o make an imporan difference. We will firs consider he case φ < l. The law of moion Assuming φ < 1, we can again analyse he model in erms of he echnology-adjused ~ variables, k k / A K /( A L ) and ~ y y / A Y /( A L ), bu noe ha A is no longer growing a an exogenous rae. Insead, he evoluion of A is endogenous and depends, hrough (5.2), on how aggregae capial evolves Page 294 of 1314

19 I follows sraighforwardly from he producion funcion (5.6) ha: ~ y k ~ = α 5.10 and (5.7) implies ha: ϕ + 1 K+ = 5.11 A K A 1 These wo equaions will be used in he following. From he definiion of k ~ we have Page 295 of 1314

20 Page 296 of 1314 ϕ ϕ = = = ~ ~ K K n L L K K K K L L A A K K k k 5.12 where we have used (5.11) and (5.9). Insering ha K +1 = sy + (1 δ)k, we ge: ( ) ϕ α ϕ ϕ δ δ δ = + + = + + = ) (1 ~ 1 1 ) (1 ~ ~ 1 1 ) (1 1 1 ~ ~ sk n k sy n K K sy n k k where we have used (5.10) for he laer equaliy. Muliplying on boh sides by k ~ gives he ransiion equaion:

21 ~ 1 ~ ~ 1 ( (1 )) 1+ n α 1 1 ϕ k + = k sk + δ 5.13 which we can also wrie in he form: ~ 1 ~ ~ 1/(1 ϕ ) 1 ϕ k + 1 = ( sk ( α + ϕ αϕ) /(1 ϕ) + (1 δ ) k ) n Noe ha since we have assumed φ < 1, he exponens in (5.14) are all well-defined and posiive. Also noe ha if we se φ = 0, boh of he equaions (5.13) and (5.14) become he ransiion equaion of he basic Solow model (wih B = 1). Convergence o seady sae Page 297 of 1314

22 The ransiion equaion has he form of a one-dimensional firs-order difference ~ equaion in k *. We will esablish properies of he ransiion equaion implying ha in he long run k ~ ~ converges o a specific value k *. From eiher of (5.13) or (5.14), i is clear ha he ransiion curve passes hrough (0, 0). From (5.14) i follows ha i is everywhere increasing. Furher, from (5.13) follows ha i ~ crosses he 45 line for exacly one posiive value k * of k ~ ~ : insering k = ~ + 1 k ino (5.13) and solving for k ~ gives 1/(1 α ) ~ ~ * s k = k = /(1 ϕ ) (1 + n) (1 δ ) verifying a unique posiive inersecion if (1 + n) 1/(1-φ) > 1 δ. The laer is realisically assumed (noe ha i is implied by he realisic assumpion, n + δ > 0) Page 298 of 1314

23 Finally, for convergence o 45 line a ~ * k from above, as shown in Figure 5.1. ~ * k i is imporan ha he ransiion curve inersecs he Page 299 of 1314

24 Figure 5.1: The ransiion diagram of he model of semi-endogenous growh Page 300 of 1314

25 To show his one can differeniae he ransiion funcion given in (5.13) wih respec o k ~, ~ inser k * for k ~ ~ o find he slope of he ransiion curve a k *, and hen derive he condiion for his slope being less han 1. You will find he condiion (1 + n) 1/(1-φ) > 1 δ, which is exacly he one we have jus assumed. The properies jus esablished and illusraed in Figure 5.1 imply ha in he long run k ~ ~ will converge o k * α. Consequenly, ~ ~ y = k will converge o he associaed: α /(1 α ) ~ * s y = /(1 ϕ ) (1 + n) (1 δ ) We have hus demonsraed long-run convergence of k ~ respecively. This defines he seady sae. ~ and y~ o k * and ~ y *, Page 301 of 1314

26 Semi-endogenous growh in seady sae Our conclusions so far seem reminiscen of hose we arrived a for he general Solow model. The auxiliary variables are defined in he same way, k ~ k / A and ~ y y / A, respecively, and we have found again ha hese variables converge o consan seady sae values. When hey are locked a hese consan values in seady sae, boh k and y mus grow a he same rae as A (oherwise k /A and y /A could no be consan). In he Solow model his sufficed for deermining he seady sae growh raes of k and y. Boh had o be equal o he exogenous growh rae, g, of A. This ime we do no have an exogenous growh rae of A, so o deermine he seady sae growh rae of y we mus deermine he endogenous growh rae of A in seady sae Page 302 of 1314

27 This is easy o do. Consider (5.12) above. In seady sae he lef-hand side, equal o 1. Therefore he righ-hand side mus also equal 1: ~ ~ k k + 1 /, is 1 1 ϕ K + 1 K+ 1 1/(1 ϕ ) 1 + n K = 1 K = (1 + n) Using (5.11) hen gives: A A + 1 ϕ /(1 ϕ ) + 1 ϕ /(1 ϕ ) = (1 + n) = (1 + n) 1 A A A g se 5.17 Hence, in seady sae he growh rae of A is g se. Noe ha g se is deermined endogenously and depends on model parameers Page 303 of 1314

28 In seady sae he growh raes of boh capial per worker, k, and GDP per worker, y, mus hen also be g se. Hence from (5.17), if he growh rae of he labour force is 0, hen he seady sae growh rae of GDP per capia is also 0. For he growh rae of GDP per capia o be posiive, a posiive populaion growh rae is required. To exploi he increasing reurns in he aggregae producion funcion an increasing labour force is required. The erm semi-endogenous growh refers o he fac ha we only have (endogenous) growh in GDP per worker if here is (exogenous) populaion growh. Noe ha in seady sae he inpus of capial and labour are no growing a he same raes. Labour inpu grows a rae n, bu since k K /L grows a he rae g se, capial mus be growing a approximaely he rae n + g se. Only if n = 0 are hese growh raes equal. Is he endogenous growh rae g se of GDP per person derived from his model of a realisic size for plausible parameer values? Annual populaion growh raes of around 0.5 per cen are ypical for Wesern counries and according o he empirical sudies Page 304 of 1314

29 menioned earlier, realisic values of φ could be up o As you will easily verify from (5.17), hese parameer values resul in seady sae growh raes, g se, in he range from 0.5 per cen o 1.5 per cen, which is no far-feched. Srucural policy for seady sae Le us now urn o he seady sae growh pah. From he definiion of y~ he seady sae growh pah of y mus be y * = ~* y A. The evoluion of A obeys A = K φ ϕ, so y * = ~* y K. Furhermore, when he economy is in seady sae here is a necessary link beween K and L. This follows because in seady sae he variable k ~, defined as K /(A L ) = K 1-φ /L, is ~ locked a he value k *, so in seady sae: K ~ ( k L ) * = 1/(1 ϕ ) Page 305 of 1314

30 Insering his expression for K ino he seady sae growh pah, per worker gives: y = y, for oupu * ~* ϕ K y * = ~ ~ ~ * * ϕ /(1 ϕ ) * α + [ ϕ /(1 ϕ )] ϕ /(1 ϕ ) [ ϕ /(1 ϕ ) y ( k L ) = ( k ) L0 (1 + n) α where we have used ha ~ ~ y = k, and L = (1 + n) L 0. We can now finally inser (5.15), and from (5.17) ha (1 + n) φ/(1-φ) = 1 + g se : ~ * k from ( α + [ ϕ /(1 ϕ )]) /(1 α ) * s ϕ /(1 ϕ ) y = L0 (1 + g se ) 1/(1 ϕ ) (1 + n) (1 δ ) 5.18 This gives he seady sae growh pah (y * ) for oupu per worker. The corresponding seady sae growh pah (c * ) for consumpion per worker is obained by muliplying by 1 s on boh sides of (5.18) Page 306 of 1314

31 Srucural economic policies for seady sae mus work o affec he posiions of hese pahs or he growh raes along hem (or boh). Wih respec o he posiions, he policy implicaions are somewha similar o hose in he basic Solow model. For example, a higher invesmen rae, s, shifs he growh pah for y upwards. Noe, however, ha he golden rule value for s, implying he highes possible posiion of (c ), is now α + φ(1 α), which is greaer han he α we have arrived a for he basic Solow model. The really new feaure is ha he seady sae growh raes of y and c depend endogenously on model parameers: boh are equal o g se = (1 + n) φ(1 φ) 1. Taking φ o be a given echnical parameer (no easily affeced by policy) he conclusion from his model's seady sae is ha, o promoe long-run growh in GDP and consumpion per capia, one should ry o promoe populaion growh. There are reasons why one would be cauious abou such a policy Page 307 of 1314

32 Empirics for semi-endogenous growh The model's seady sae predics ha a higher growh rae in he labour force, or he populaion, should give higher growh in GDP per capia. One way o es his predicion is o plo average annual growh raes in GDP per worker over some period agains average annual populaion growh raes over he same period across counries Page 308 of 1314

33 Figure 5.2: Average annual growh rae of real GDP per worker agains average annual growh rae in populaion, 55 counries Source: Penn World Table Page 309 of 1314

34 We find a raher clear negaive relaionship beween populaion growh and economic growh across counries in he period considered. The OLS-esimaed sraigh line has a significanly negaive slope. This does no necessarily mean ha he model of semiendogenous growh is wrong, however. Firs, i is no clear exacly wha kind of area he model covers, a region, a counry, or perhaps he (developed) world, since i is no easy o ell how wide he exernal effec of he model reaches. If he firms in one counry can look over he shoulders of firms in oher counries, hen perhaps he model should be considered o cover he world. I is herefore no clear if cross-counry evidence as repored in Figure 5.2 is relevan. Second, he model we consider has a seady sae and i has convergence o he seady sae in he long run. During he convergence period here will be ransiory growh in addiion o he underlying (endogenous) seady sae growh, and he ransiory growh depends negaively on populaion growh, jus as in he Solow models, and for he same reason: a decrease in he rae of populaion growh shifs he seady sae growh pah up bu reduces is slope (see Eq. (5.18)) Page 310 of 1314

35 Hence, during he ransiion o he new seady sae growh pah here will, for some ime, be a posiive ransiory conribuion o growh arising from he lower populaion growh. Furhermore, convergence in he model we consider may be quie slow due o he presence of he producive exernaliy. As φ comes close o 1, convergence will be very slow. Wha we have found in Figure 5.2 may be inerpreed as an expression of long-lasing ransiory growh ha should indeed depend negaively on populaion growh according o our model. These remarks seem o call for (even) longer run empirical evidence ha does no go across counries. Table 5.1 repors for each of he wo sub-periods and he average annual populaion growh rae and he average annual growh rae of GDP per capia. For all counries economic growh increased subsanially from he early subperiod o he laer one, while for all counries bu hree, populaion growh decreased, for many of he counries subsanially. If we consider quie long periods and say wihin each counry, he evidence shows lile sign ha growh in GDP per capia should be posiively relaed o populaion growh Page 311 of 1314

36 Sill we canno conclude ha he model is wrong. I could be ha convergence according o he model is so slow ha i really akes place over periods as long as 120 years. In ha case he general picure found in Table 5.1 could be compaible wih he model's ransiory growh. However, if convergence o seady sae is ha slow, hen he model's seady sae is no very descripive for periods of ineres, so we should consider he very slow convergence process iself as he model's main predicion. This is exacly he perspecive aken in he nex secion. We will apply a specific assumpion ha makes he convergence period no jus very long, bu infinie. We will hen consider he model's infinie, ouside seady sae behaviour as an approximaion of a very long-lasing convergence process. Table 5.1: Average annual growh raes in populaion and real GDP per capia in 17 indusrialized counries, Page 312 of 1314

37 Source: Angus Maddison, Monioring he World Economy , Paris, OECD, Page 313 of 1314

38 Endogenous growh Wha is he analyical expression for he rae of convergence for y~ according o our model wih φ < 1? I can be shown ha for he case where populaion growh is small, n 0, he rae is λ (1 α)(l φ)δ. This verifies he claim ha a large φ less han 1 implies very slow convergence. Below we will indeed se n = 0, since we now wan o invesigae he possibiliy ha over periods of ineres, such as cenuries, populaion growh is no he facor ha causes economic growh. The size of he labour force is hen consan, equal o L, say. The idea is o consider he growh model wih φ = l. This gives a zero rae of convergence and hence an everlasing convergence process. We consider he everlasing convergence resuling from φ = l o be of ineres because i approximaes he very long-lasing convergence ha would resul from a φ ha is large, bu sill less han uniy Page 314 of 1314

39 The AK model In he analysis in he previous secion he erm 1 φ appeared in many places, including in some denominaors, and several expressions would be meaningless for φ = 1. Hence we canno simply se φ = 1 in all he equaions above. For wha we did above, on he oher hand, i was no imporan wheher φ was smaller han or equal o 1, so he expressions for he real facor prices we found here and he associaed heory of income disribuion are also relevan when φ = 1. Wih L = L and φ = 1 we ge from (5.5) ha: r = r αl 1-α and w = (1 α)k /L α 5.19 so he real ineres rae, r δ, is consan and he wage rae, w, evolves proporionally o K, and hence o k K /L. In addiion o he above expressions for he real facor prices he model can be condensed o he wo equaions: Page 315 of 1314

40 Y = K L 1-α AK 5.20 K +1 = sy + (1 δ)k 5.21 where he firs equaion is he aggregae producion funcion resuling from (5.6) and (5.7), or direcly from (5.3), wih φ = 1, while he second equaion is he usual capial accumulaion equaion repeaed from (5.8). The model's las equaion, (5.9), has been replaced by L = L. Noe ha in (5.20) we have used he definiion A L 1-α. This is a bi of an abuse of noaion, bu i should cause no confusion if we denoe L 1-α by an A wihou a subscrip. We apply his noaion because he model we consider is so ofen referred o as he AK model. Growh according o he AK model Page 316 of 1314

41 I is easy o see ha he model above can resul in permanen growh in GDP per worker. Dividing boh sides of he producion funcion (5.20) by L, in order o ransform variables ino per worker erms, gives y = Ak. Dividing also boh sides of he capial accumulaion equaion (5.21) by L, and insering Ak for y hen gives he ransiion equaion: k +1 = (sa + 1 δ)k 5.22 Subracing k from boh sides gives he Solow equaion: k +1 k = sak δk 5.23 and hen dividing boh sides by k gives he modified Solow equaion: Page 317 of 1314

42 k k = sa g + 1 δ e 5.24 k The laer equaion direcly gives he consan and endogenous growh rae, g e, in capial per worker k and hence in capial, K, since k = K /L. Furhermore, since y is a consan imes k, he growh rae of oupu per worker is g e, and hence he growh rae of consumpion per worker, c = (1 s)y, mus be g e. Finally, he echnology variable A (no o be confused wih he consan A L 1-α ) mus increase a rae g e, since wih φ = 1, we have A = K. I follows ha according o he AK model, g e = sa δ is he common endogenous growh rae of all he variables we are ineresed in (we assume ha sa δ > 0). Hence, everlasing economic growh is possible in his model wihou an assumpion of exogenous echnological growh. To undersand his resul i may be useful o draw some well-known diagrams Page 318 of 1314

43 Figure 5.3 shows he ransiion diagram associaed wih (5.22), he Solow diagram associaed wih (5.23), and he modified Solow diagram associaed wih (5.24). The diagrams illusrae ha in his model here is no seady sae and k (and hence y and Y ) grow a a consan rae forever Page 319 of 1314

44 Page 320 of 1314

45 Figure 5.3: The ransiion diagram (op), he Solow diagram (middle), and he modified Solow diagram (boom) of he model of endogenous growh The curves, which are sraigh lines in Figure 5.3, would in Solow models be rue curves due o diminishing reurns o capial per worker (α < 1). The lineariy of he Page 321 of 1314

46 curves in he presen model reflecs ha here are no diminishing reurns o capial a he aggregae level. Raher here are consan reurns o capial per worker: y = Ak. The growh brake from he Solow models, diminishing reurns o capial, is simply no longer presen in he aggregae producion funcion. The source of growh is hus aggregae consan reurns o he reproducible facor, capial. Implicaions for srucural policy and he scale effec The main conclusion from he model is ha a higher savings (invesmen) rae, s, gives rise o a permanenly higher growh rae in GDP and consumpion per worker. This differs from our earlier conclusions since a higher s no longer jus gives a higher level of oupu per worker in he long run and a emporarily higher ransiory growh rae in he inermediae run, bu i resuls in a permanenly higher rae of growh in oupu per worker Page 322 of 1314

47 Since our model wih φ = l should be seen as an approximaion of he semi-endogenous growh model wih a large φ smaller han 1, we should remember ha sricly speaking he correc saemen is ha an increase in s gives very long-lasing ransiory growh in GDP per worker. The implicaions for economic policy are obvious: policies ha simulae savings now give a very long-lasing boos o growh. A decrease in δ has an effec similar o ha of an increased s, bu i may be more difficul o achieve hrough economic policy. More effecive aggregae invesmen should, however, lead o a lower rae of depreciaion, and a lower rae of depreciaion leads, in he model of endogenous growh, o a permanenly higher growh rae in GDP per worker. If, somehow, a governmen can ake acions ha make ne invesmen more effecive, i can perhaps ake advanage of his effec. This may be mos relevan for counries where he governmen is heavily involved in producion so ha a large par of invesmen is no driven by privae incenives, bu decided upon by governmen bureaucracies Page 323 of 1314

48 Hisorically his mehod of making invesmen decisions has ofen led o inefficien and rapidly worn ou (someimes even useless) invesmen. By relying more on privae, profimoivaed incenives in invesmen decisions, such a counry could probably aain higher prosperiy and growh in he long run, alhough here may be ransiion coss involved in changing from one ype of economic sysem o anoher. Policies for more effecive invesmen are no only relevan in connecion wih endogenous growh. In he Solow model a lower depreciaion rae resuls in a higher level of GDP per capia in he long run, which is also imporan. Noe he general endency ha parameer changes ha give a long-run level effec on GDP per worker in he Solow models give a long-run effec on he rae of change in GDP per worker in he model of endogenous growh. There is one odd feaure of he AK model which cass doub on any policy recommendaion derived from i. Remember ha A L 1-α. Hence he growh rae g e = sa δ Page 324 of 1314

49 is higher he larger he consan populaion size L is, and he growh rae would increase if populaion increased. This is he so-called scale effec, which is raher conroversial. Empirical evidence does no suppor he hypohesis ha larger counries should have larger long-run growh raes, bu again i is no clear wha geographical area he model covers. Perhaps i is he world, bu hen during he las 200 years he world populaion has increased rapidly while economic growh raes have no shown a similarly srong increase. I is possible o ge rid of he scale effec. Assume ha he producive exernaliy, which is he driving force of endogenous growh in his lecure, arises from capial per worker raher han from capial iself, so ha Eq. (5.7) in our model should be replaced by: K A = L ϕ Page 325 of 1314

50 You will easily verify ha wih φ = 1 he aggregae producion funcion becomes Y = K, ha is, he A (wihou subscrip ) is equal o 1 and independen of L. Consequenly he endogenous growh rae of he model will be g e = s δ, and he scale effec has been eliminaed. However, aggregae producion (no producion in he individual firm) is now compleely insensiive o labour inpu. The posiive effec on producion of a higher labour inpu, L, arising a he firm level is exacly offse by he negaive exernal effec of a lower K /L a he aggregae level (when φ = l). Eiher endogenous growh models have he unaracive scale effec or hey assume, unaracively, ha labour inpus are unproducive a he aggregae level. There is no way around his problem. Empirics for endogenous growh As menioned, a main predicion of he model of endogenous growh is he posiive influence of he savings or invesmen rae on he long-run growh rae of GDP per Page 326 of 1314

51 worker. Le us pu his predicion o a es by considering cross-counry evidence. Figure 5.4 plos average annual growh raes, g i, from 1960 o 2000 agains average invesmen raes, g i, over he same period across counries i Page 327 of 1314

52 Figure 5.4: Average annual growh rae of real GDP per worker agains average invesmen rae in physical capial, 90 counries Source: Penn World Table 6.1. The figure shows a quie good and igh posiive relaionship. This is a main reason why economiss ake he heories of endogenous growh seriously (sill remembering ha hese should be seen as approximaions of heories of very long-lasing ransiory growh). Ye he posiive relaionship does no prove ha he idea of endogenous growh is correc. As we have argued several imes, i is no clear wheher he model covers a counry or he world, so cross-counry evidence may no be he appropriae kind of empirics o sudy. Furhermore, in no way does Figure 5.4 conradic he radiional Solow models according o which an increase in he savings rae will imply a (higher) ransiory growh in excess of he underlying exogenous growh. Perhaps he figure jus shows ha counries wih a high average s i over he period considered ypically have experienced increases in s i and herefore have had relaively high ransiory growh raes. However, he nex secion will provide a reason for indeed viewing Figure 5.4 as indicaing endogenous growh Page 328 of 1314

53 Exogenous versus endogenous growh In his lecure we have sudied wo disinc and alernaive ypes of endogenous growh, semi-endogenous and endogenous growh, boh fundamenally caused by producive exernaliies. The presence of a posiive, bu no oo srong producive exernaliy, 0 < φ < 1, gave rise o a model of semi-endogenous growh. This model had convergence o a seady sae in he long-run, and herefore i had he convergence propery : growh is faser he furher below seady sae he economy is. Furhermore, in seady sae economic growh was explained by growh in he labour force, and he long-run growh rae of GDP per worker was higher he higher he rae of populaion growh. A srong producive exernaliy, φ = 1, resuled in a model of (genuinely) endogenous growh. This model had no seady sae and hence no convergence o one (or raher, in he appropriae model inerpreaion, very slow convergence o one). Therefore he model did Page 329 of 1314

54 no have he convergence propery. Growh in GDP per capia could occur wihou populaion growh, bu here was a scale effec: a larger labour force gave a higher rae of economic growh, and a fixed populaion growh would imply an exploding economic growh rae. Boh he model of semi-endogenous growh and he model of endogenous growh delivered explanaions of economic growh: he growh rae of GDP per worker depended on basic srucural model parameers. In he previous lecures we have sudied models of exogenous growh. All hese models had convergence o a seady sae and he convergence propery. In he long run he growh rae of GDP per worker was eiher independen of or negaively affeced by he populaion growh rae (he laer in he realisic presence of naural resources). In hese models permanen growh in GDP per worker was a reflecion of exogenous echnological growh, so he models did no really explain long-run economic growh Page 330 of 1314

55 The basic mechanism for endogenous growh in his lecure was producive exernaliies. The echnological growh arising from hese came as an uninended byproduc of economic aciviy. No agens were concerned wih deliberaely producing echnological progress. In he real world we know ha a lo of research and developmen aciviies inended o creae echnological progress are underaken in privae companies as well as a universiies. Therefore, perhaps he endogenous growh models in which echnological progress is he oucome of an explici and deliberae producion aciviy will be more convincing. We will no consider hem here. However, he simple exernaliy-based models of endogenous growh considered in his lecure share enough feaures wih he more advanced endogenous growh models o be represenaive for endogenous growh heory. Therefore we can already presen some main argumens from an ongoing and fascinaing debae among growh heoriss. The issue in his debae is wheher he economic growh we see in he real world is bes undersood by Page 331 of 1314

56 exogenous growh models or by endogenous growh models. Here are some main argumens from he debae. Explaining growh Everybody agrees ha i is good o have explanaions of long-run economic growh, and since he endogenous growh models deliver much more of an explanaion han exogenous growh models, his is cerainly an argumen in favour of endogenous growh models. However, i could be ha a he presen sage of knowledge he exising endogenous growh models were no convincing, so one would have o rely on exogenous growh models for undersanding growh, alhough he undersanding would be limied by echnological progress being unexplained. Wheher he growh explanaions and oher feaures of he endogenous growh models are sufficienly convincing is mainly an empirical maer, and some argumens presened below may be imporan for deciding his Page 332 of 1314

57 The knife-edge argumen Sricly speaking he model of (genuinely) endogenous growh only perains o a zero probabiliy, knife-edge case (φ = 1) which is unineresing. Endogenous growh models are someimes criicized on such grounds. However, as we have carefully argued, he model should righly be inerpreed as an approximaion of a wider case (φ slighly smaller han 1), which does no have zero probabiliy. On he face of i, his criicism is herefore no valid. The issue is really wheher he relevan parameer (here φ) can realisically be assumed o be so close o a limiing value (here 1) ha he model of endogenous growh can be seen as a good approximaion. This is again an empirical maer. For he exernaliy parameer φ considered in his lecure, empirical evidence seems o poin o posiive values of a mos up o 0.75, which speaks in favour of models of semi-endogenous growh. Populaion growh, economic growh and he scale effec Page 333 of 1314

58 If he knife-edge propery of he endogenous growh model does no represen a serious objecion o i, he scale effec does. The feaure ha an increasing populaion implies an increasing growh rae in GDP per capia is simply no realisic. The presence of he scale effec is he single mos imporan argumen agains models of (genuinely) endogenous growh. According o he model of semi-endogenous growh here should be a posiive relaionship beween he (consan) growh rae of GDP per capia and he (consan) populaion growh rae. Based on he evidence presened in Figure 5.2 and Table 5.1, one may have a hard ime finding his feaure convincing. There is, however, a counerargumen: wha Table 5.1 shows is rue, bu anoher fac is, according o his argumen, more impressive and more imporan in a very long-run perspecive. As we have seen, many counries in he Wes have experienced relaively consan average annual growh raes in GDP per capia of abou 2 per cen during he las 200 years. Over he several housands of years before ha, he average annual Page 334 of 1314

59 growh raes in GDP per capia mus have been close o 0 o fi wih he level of income per capia around year So, housands of years of almos no economic growh have been succeeded by 200 years of rapid economic growh. Populaion growh has behaved basically he same way, wih he world populaion being close o consan for hundreds of years up o he eigheenh cenury and hereafer increasing rapidly (for hese facs and an ineresing discussion of hem, we refer o Michael Kremer, Populaion Growh and Technological Change: One Million B.C. o 1990, Quarerly Journal of Economics, 108, 1993). This acually fis wih semi-endogenous growh. Sill, one may wonder if i is really he world's populaion growh ha has caused economic growh, or he oher way round. Furhermore, Table 5.1 shows ha he areas in he world ha have experienced he high (and even increasing) economic growh have had decreasing populaion growh. I is hard o believe ha populaion growh in Africa, India and China should have caused economic growh in Europe, USA and (oher) pars of Asia, while i seems more Page 335 of 1314

60 plausible ha economic growh in Europe and USA has (indirecly) conribued o reduce moraliy and creae populaion growh in Africa and pars of Asia. Therefore he empirical relaionship beween populaion growh and economic growh seems o be mosly in favour of he models of exogenous growh, which predic a negaive relaionship when naural resources are included in he model. Convergence We saw in Lecure 2 ha focusing on counries ha can be assumed o be srucurally similar, such as he OECD counries, here is a clear endency for he counries ha are iniially mos behind o grow he fases and hus cach up o he iniially riches. This paern is even more clear beween saes in he USA. We saw ha if one conrols for srucural differences beween counries in a way suggesed by exogenous growh models, hen even for he counries of he world, he iniially Page 336 of 1314

61 poores end o grow he fases. There is hus evidence in suppor of convergence among counries in he real world. In he world of models he convergence among counries arises naurally in models wih (an appropriae form of) convergence o a seady sae. As he seady sae is approached (from below), growh becomes slower and slower. Two differen counries wih similar characerisics will converge o he same seady sae, so he counry saring ou wih he lowes GDP per person will grow he fases. The empirically observed convergence beween counries is hus mos naurally explained by models of exogenous or semiendogenous growh. The endogenous growh model considered in his lecure, on he oher hand, did no have convergence o a seady sae and showed no endency for he iniial posiion o be imporan for subsequen growh. The observed convergence hus conradics he model and seems o speak agains endogenous growh. I mus be added, however, ha raher naural modificaions of he endogenous growh models can ake accoun of convergence, for Page 337 of 1314

62 example, by inroducing a gradual echnological ransmission beween counries ino he framework. The convergence beween counries observed in he real world does no deliver a decisive answer o wha kind of growh model does he bes job. Growh raes and invesmen raes The single empirical regulariy ha speaks mos srongly in favour of models of (genuinely) endogenous growh is he one illusraed in Figure 5.4: he clear posiive empirical relaionship beween invesmen raes and growh raes. The endogenous growh model predics ha a higher savings or invesmen rae should give a higher growh rae, and indeed high invesmen raes do go hand in hand wih high growh raes. In he models of exogenous or semi-endogenous growh a higher savings rae does no give a higher growh rae in he long run, bu i does so in he inermediae run due o ransiory growh. The posiive associaion beween invesmen raes and growh raes Page 338 of 1314

63 could hus be a resul of ransiory growh, and herefore i does no necessarily conradic he exogenous or semi-endogenous growh models. However, in models of exogenous growh a higher savings or invesmen rae, s, could no possibly give a higher growh rae in he echnological variable, A, exacly because his growh rae is exogenous. In he endogenous growh model a higher s will give a higher growh rae in A since he growh rae of A is sa δ, and in he semi-endogenous growh model a higher s gives a higher ransiory growh in A. I is hus of ineres o find ou wheher invesmen raes are also posiively relaed o growh raes in A. This is indeed an idea pursued in an aricle by he wo economiss Ben S. Bernanke and Refe S. Gurkaynak ( Is Growh Exogenous? Taking Mankiw, Romer, and Weil Seriously, NBER Working paper 8365, July 2001). One can ge an esimae of he average annual growh rae in he produciviy variable A, by growh accouning. In heir aricle, Bernanke and Gurkaynak firs do growh accouning for each of a large number of counries Page 339 of 1314

64 o ge esimaes of average annual growh raes of A. They hen plo hese raes agains average invesmen raes across counries. Using he daa of Bernanke and Gurkaynak he picure in Figure 5.5 emerges Page 340 of 1314

65 Figure 5.5: Average annual rae of labour-augmening echnological progress agains average invesmen rae in physical capial, 84 counries Sources: Penn World Table 6.1 and daa se for Bernanke and Gurkaynak (2001). There is clearly a posiive relaionship. Wha is shown in Figure 5.5 is he sronges empirical argumen in favour of endogenous, or perhaps semi-endogenous, growh models ha we know of. The debae from which we have ried o presen a few argumens is no seled. Raher i is ongoing. The empirical resuls linking invesmen raes o growh raes are sufficienly convincing o make he endogenous growh research programme, and he exising models, highly promising. Bu a serious challenge o endogenous growh heory, as i sands, comes from he empirical relaionship beween populaion growh raes and economic growh raes, which seems, if anyhing, o be a negaive one. In conras, semi-endogenous growh models predic a posiive effec of populaion growh on economic growh, while Page 341 of 1314

66 endogenous growh models even predic ha a consan posiive populaion growh rae should give an exploding growh rae in GDP per capia. Summary In endogenous growh heory he growh rae of echnology, which underlies he long-run growh in GDP per worker, is endogenous: i depends on basic behavioural model parameers and herefore on srucural policies ha affec hese parameers. According o he replicaion argumen, he individual firm's producion funcion should exhibi consan reurns o he inpus of capial and labour a any given echnological level. If here are producive exernaliies from he aggregae sock of capial (or from aggregae producion) o labour produciviy or oal facor produciviy in he individual firm, hen here can be consan reurns o capial and labour a he firm level and increasing reurns o capial and labour a he aggregae level. Wih producive exernaliies one can herefore mainain he convenien assumpion of perfec compeiion and he associaed heory of Page 342 of 1314