Obsolescence and modernization in the growth process

Transcription

1 Journl of Development Economics 77 (2005) Obsolescence nd moderniztion in the growth process Rouf Boucekkine, *, Fernndo del Río b, Omr Licndro c IRES nd CORE, Université Ctholique de Louvin, Plce Montesquieu, 3, B-348 Louvin-l-Neuve, Belgium b Universidde de Sntigo de Compostel, Spin c Europen University Institute, Itly, nd FEDEA, Spin Received My 2002; ccepted Mrch 2004 Abstrct In this pper, n endogenous growth model is built up incorporting Schumpeterin cretive destruction nd embodied technologicl progress. Under embodiment, long run growth is ffected by two opposite effects: (i) obsolescence costs dd to the user cost of cpitl, which hve negtive effect on reserch efforts; nd (ii) the moderniztion of cpitl increses the demnd for investment goods, rising the incentives to undertke reserch ctivities. Applied to the understnding of the growth enhncing role of both cpitl nd R&D subsidies, we conclude tht the positive effect of moderniztion generlly more thn compenstes the negtive effect of obsolescence. D 2004 Elsevier B.V. All rights reserved. JEL clssifiction: E22; E32; O40 Keywords: Schumpeterin growth; Cretive destruction; Embodiment; Obsolescence; Moderniztion. Introduction An importnt issue for growth nd development theory is the role of subsidies to both cpitl ccumultion nd R&D ctivities. In the neoclssicl growth frmework, * Corresponding uthor. Tel.: ; fx: E-mil ddresses: boucekkine@ires.ucl.c.be (R. Boucekkine)8 edelrio@usc.es (F. del Río)8 omr.licndro@iue.it (O. Licndro) /$ - see front mtter D 2004 Elsevier B.V. All rights reserved. doi:0.06/j.jdeveco

2 54 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 subsidizing cpitl hs no permnent impct on the growth rte, since investment only mtters in the short run while technologicl progress is the sole determinnt of per cpit growth in the long run. In contrst, fcilitting nd subsidizing R&D nd technology doption should foster long term growth. Moreover, in R&D bsed growth models, à l Romer (990), subsidies to reserch ctivities, which precisely drive technologicl progress, re effective in boosting growth. The dichotomy between cpitl ccumultion nd technologicl progress ws t the hert of the embodiment controversy in the 960s, s recently pointed out by Hercowitz (998). Supporters of the embodiment hypothesis rgued tht investment is the chnnel through which innovtions re implemented. Since investment plys moderniztion role under embodiment, it should be decisive determinnt of long run growth. In this pper, we clim tht the growth enhncing role of cpitl subsidies needs to be nlyzed in n endogenous growth model with embodied technicl chnge, nd we show tht the min implictions of the embodiment ssumption for fiscl policy cnnot be cptured if the moderniztion role of investment is neglected, which gives theoreticl support to the importnce of the embodied question. Howitt nd Aghion (998) show tht Romer s result is bised by the ssumption tht lbor is the sole input in the production of reserch. Indeed, if the R&D sector employs cpitl s n input, subsidizing cpitl is growth enhncing. Prdoxiclly, Howitt nd Aghion suggest tht if new technologies re embodied in new mchines, embodiment ctully hs the effect of wekening the result tht cpitl subsidy will ffect long run growth. The reson is tht replcement dds obsolescence costs to the user cost of cpitl, reducing the incentives to innovte. Nonetheless, the empiricl literture suggests tht the moderniztion of cpitl is growth enhncing. DeLong nd Summers (99) find tht countries with high growth rtes re precisely those with both lrge investment rtes nd fst decline rtes in the reltive price of equipment. These observtions cpture the moderniztion role nd the embodied nture of technologicl progress. Wolff (99), for smple of seven OECD countries, finds tht ctch-up in totl fctor productivity is highly correlted with cpitl ccumultion. He lso concludes tht embodiment plys centrl role in this reltionship s productivity growth is highly sensitive to the ge of the cpitl stock. Brdhn nd Prile (996) notice the significnt difference in the sving rte between Ltin Americ nd Est Asi, nd invoke the moderniztion role of investment to explin fst economic growth in Est Asi nd reltive stgntion in Ltin Americ. In this pper, we introduce cpitl nd embodied technicl progress in Schumpeterin growth model à l Aghion nd Howitt (992). The model we propose is n endogenous growth version of Greenwood et l. (997). As in Howitt nd Aghion (998), the user cost of cpitl is incresed by obsolescence costs. However, obsolescence costs re of different nture: in Howitt nd Aghion, the scrpping of mchines due to the replcement of obsolete technologies increses the user cost of cpitl; in our frmework, technologies re infinitely lived, but the investment-specific See Denison (964), Phelps (962) nd Solow (960).

3 R. Boucekkine et l. / Journl of Development Economics 77 (2005) nture of technicl progress permnently increses the productivity of the investment sector generting stble decline of investment prices, which dds cpitl losses to the user cost of cpitl. More importnt, in our frmework, reserch ctivities re exclusively ddressed to the improvement of productivity in the investment sector. It is the simplest wy to introduce embodied technicl chnge. For this reson, profitbility in the R&D sector depends crucilly on the demnd for investment goods. When technologicl progress is high, the demnd for investment goods is high too, which rises the incentives to undertke reserch ctivities. This is the moderniztion role of embodiment. When pplied to understnding subsidies to both cpitl nd R&D ctivities, we show tht the moderniztion effect more thn compenstes obsolescence costs, which contrdicts the Howitt nd Aghion clim tht embodiment reduces the incentives to innovte nd the efficcy of subsidies to cpitl. The endogenous growth model with embodied technicl chnge in this pper is close to the models by Krusell (998) nd Hsieh (200), which use n explicit R&D sector à l Romer (990), nd the Arrowin lerning-by-doing model designed by Brdhn nd Prile (996). 2 Firstly, our obsolescence mechnism is relted to investment-specific technicl chnge, s in Greenwood et l. (997) nd Krusell (998). The occurrence of n innovtion improves the productivity of the investment sector nd lowers the reltive price of investment goods. This reduction in the reltive price of investment genertes the sme type of cpitl losses s in Solow (960), reducing the incentives to invest nd innovte. In Brdhn nd Prile, the obsolescence mechnism is similr to the infltionry wge scheme first put forwrd by Solow et l. (966). Technologicl progress pushes wges upwrd, which ends up by exhusting the qusi-rents extrcted from the existing cpitl goods until they become obsolete nd re scrpped. A similr mechnism is highlighted in the work of Solow (960), s referred to by Hsieh. The equliztion of lbor mrginl productivities cross vintges reduces the lloction of lbor to old sectors, reducing the vlue of old cpitl vintges. The ssocited cpitl losses, the so-clled obsolescence costs, dd to the user cost of cpitl. This kind of mechnism tkes plce under vintge structure when some complementrity between cpitl nd lbor exists. Secondly, for Brdhn nd Prile, the moderniztion role of investment shows up in the negtive reltion between the growth rte nd the lifetime of mchines. 3 In our model, the moderniztion mechnism is different since the lifetime of cpitl goods is infinite. Actully, we identify moderniztion mechnism bsed on the investment to cpitl rtio, n inverse function of the verge ge of cpitl. This mechnism requires investment being determinnt of the profitbility of the R&D sector (which could be n doptionimittion sector in the cse of developing countries) to be effective. An increse in the rte of technicl progress hs multiplictive effects by rising the investment to cpitl rtio, which in turn ffects the rte of technologicl progress. A similr mechnism is t work in Krusell nd Hsieh. 2 See lso the LBD model with embodiment in Boucekkine et l. (2003). 3 A similr mechnism is t work in the second prt of Hsieh (200).

4 56 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 This pper is orgnized s follows. In Section 2, the proposed model is solved nd its min properties re stressed. In prticulr, it is compred with the exogenous growth model with embodied technicl chnge by Greenwood et l. (997), nd the R&D growth model with embodied technicl chnge by Krusell (998) nd Hsieh (200). Section 3 is devoted to the nlysis of the effects on growth of cpitl nd reserch subsidies. The results for the endogenous growth models with disembodied technicl chnge re pointed out. The min economic mechnisms, obsolescence nd moderniztion, re discussed. Sufficient conditions under which the moderniztion effect domintes the obsolescence effect re stted nd interpreted. Section 4 concludes. 2. The model The model in this pper is bsed on Aghion nd Howitt (992), nd it introduces embodied technicl chnge s in the work of Solow (960). There re two finl sectors, one producing non-durble good, nd nother producing n investment good. Technology in the non-durble sector is Cobb-Dougls on cpitl nd lbor, nd the non-durble good is llocted to consumption nd s n input in both the production of intermedite goods nd R&D ctivities. Technology in the investment goods sector is constnt elsticity of substitution on continuum of intermedite inputs. Both finl sectors re competitive. In the intermedite sector, continuum of differentited goods is produced under monopolistic competition. Technology in this sector only employs non-durble goods s inputs, nd benefits directly from innovtions developed in the R&D sector. Finlly, the R&D sector is competitive. For comprtive resons, we nlyze two lterntive technologies, one uses the nondurble good s n input nd the other uses lbor. The reduced form of the economy is very similr to the two sector exogenous growth model of Greenwood et l. (997), herefter GHK, where one sector employs cpitl nd lbor to produce non-durble good, nd the other sector produces n investment good using the non-durble good s the sole input. In our model, the trnsformtion of non-durble goods in investment goods requires n intermedite step, i.e. the production of intermedite inputs. In both models, technologicl progress benefits the investment sector, nd requires new investments to propgte over the whole economy; the so-clled embodied nture of technologicl progress. Howitt nd Aghion (998), herefter HA, combine the neoclssicl growth model à l Solow (956) nd their 992 cretive destruction model, nd shre some key properties with our model, in prticulr tht qulity improvements re the engine of growth nd profits in the intermedite sector re the min incentive to innovte. However nd differently from our model, HA ssume tht technicl progress is disembodied: A single finl goods sector produces both consumption nd investment goods, nd benefits from qulity improvements in the intermedite sector. As it is shown in the next, the different nture of technologicl progress is crucil to understnd the different economic mechnisms t work in these two models.

5 R. Boucekkine et l. / Journl of Development Economics 77 (2005) Growth under embodiment As in the stndrd optiml growth model, n infinitely lived representtive dynsty endowed with one unit of lbor mximizes intertemporl utility. The Euler eqution relted to the dynsty problem is ĊC t ¼ ð C t r r t qþ: ðþ As usul, the growth rte of per cpit consumption, C t, depends on the difference between the interest rte r t nd the dynsty s discount rte q, multiplied by the intertemporl elsticity of substitution (/r), which is supposed to be constnt nd strictly positive. Technology in the investment sector displys constnt returns to scle on continuum of intermedite inputs in the intervl [0,]: I t ¼ Z ; x dj jt ð2þ 0 where I t is per cpit investment, x jt is the per cpit mount of the intermedite good j used in the production of the investment good, nd ]0,[. The problem of the representtive firm in the investment sector is purely sttic. She tkes prices s given nd mximizes current profits subject to the technologicl constrint (2). The optiml demnd for the intermedite good j is: x jt ¼ pjt P t It ; ð3þ where P t is the price of the investment good nd p jt is the price of the jth intermedite good, both mesured in units of the non-durble good (which is tken s the numerire). The so-clled dtrue price indext ssocited with technology (2) is given by P t ¼ Z 0 p jt dj! : ð4þ Technology in the intermedite sector is liner in the sole input, the non-durble good. The mrginl productivity in the production of the jth intermedite good is q jt. The sector is under monopolistic competition nd ech intermedite good is produced by one firm only. Given the demnd function (3), the jth monopolist optimlly sets the price p jt ¼ : ð5þ q jt The mrkup / is constnt nd equl for ll monopolists, nd /q jt is the mrginl cost. From Eqs. (4) nd (5), we cn compute the price of the investment good P t ¼ Q t ; ð6þ

6 58 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 where Q t ¼ Z 0 q jt dj! ð7þ is qulity index of the inputs used in the production of the durble good. The price of the investment good is n inverse function of the verge qulity of intermedite inputs. In sttionry growth regime, Q t must be growing t positive constnt rte, implying tht the price of investment goods must be permnently declining. Notice lso tht the price of ny intermedite good reltive to the price of the investment good is p jt /P t = Q t /q jt. It depends on its reltive qulity only. More efficient intermedite goods re sold t smller prices. The per cpit mount of non-durble goods employed in the production of intermedite goods is given by X t ¼ Z 0 x jt q jt dj ¼ I t Q t : The equlity on the left hnd side of Eq. (8) cn be esily obtined from Eq. (3), ( p jt /P t )=(Q t /q jt ) nd Eq. (7). It is importnt to notice tht technologicl improvements in the intermediry sector lower the production costs of investment goods, since reduced mount of nondurble goods is required to produce one unit of investment goods. Technologicl progress is embodied in new mchines, implying tht new investments re needed to profit from the dvnces in technology. In the non-durble goods sector, technology is Cobb-Dougls, K t ( l t ), where K t is the per cpit stock of cpitl nd l t [0,] is the frction of the lbor endowment devoted to produce non-durble goods. The lw of motion of the per cpit cpitl stock is K t ¼ I t dk t ; ð9þ where dn0 is the deprecition rte. In this sector, the representtive firm tkes prices s given nd mximizes the discounted flow of profits subject to these technologicl constrints. Cpitl ccumultion is subsidized t the rte b K N0, so tht firms fce the net of subsidies interest rte r t b K. From the first order conditions of this problem, the mrginl productivity of cpitl must be equl to the corresponding user cost nd the mrginl productivy of lbor must be equl to the wge rte: K t ð l t Þ ¼ P t r t þ d ṖP t P t b K ð8þ ; ð0þ ð ÞKt ð l t Þ ¼ W t : ðþ As expected, chnges in the price of investment goods hve negtive effect on the user cost of cpitl. From Eq. (6), the decline on investment prices is equl to the growth rte of the qulity index, i.e. Ṗ t /P t = Q t/q t. Qulity improvements in the intermedite sector move the technologicl frontier up, reducing the future price of investment goods

7 R. Boucekkine et l. / Journl of Development Economics 77 (2005) nd cting s brke on cpitl ccumultion. This is the so-clled obsolescence cost relted to embodied technicl chnge. At equilibrium, per cpit production of the non-durble good is llocted to consumption, C t, nd s inputs in the production of both the intermedite sector, X t, nd the R&D sector, N t. All vribles re in per cpit terms. Formlly, C t þ X t þ N t ¼ K t ð l tþ : ð2þ 2... Embodied technicl chnge Combining Eqs. (), (6), nd (8) to (2), we get K t ¼ Q t K t ð l t Þ C t N t dkt ĊC t ¼ Qt 2 Kt ð l t Þ ðq þ d þ c C t r t b K Þ ; where c t u Q t/q t is the rte of embodied technicl chnge. This system is very close to the system representing the equilibrium of n optiml growth model with embodied technicl chnge, s in GHK. Along blnced growth pth with positive growth, it cn be esily shown tht the growth rte of consumption is equl to (/ )c, nd smller thn the growth rte of investment, equl to c/. Consequently, our model reproduces the min empiricl fcts relted to embodiment. Firstly, from Eq. (6) the reltive price of investment permnently declines t the rte of embodied technicl chnge c. Secondly, the investment to output rtio permnently increses. 4 The three min differences between our model nd GHK re the following. Firstly, the production of the finl good nd the time endowment my lso be llocted to R&D ctivities. Secondly, the obsolescence cost c t is endogenous. Its behvior is nlyzed in the next subsection. Finlly, monopolistic competition in the intermedite goods sector implies tht the mrginl productivity of cpitl is multiplied by, the inverse of the mrkup rte. Under exogenous growth, i.e. l t =0,N t =0,c t = cn0, nd perfect competition in the intermedite goods sector, these two equtions become equivlent to the dynmic system in GHK Schumpeterin R&D ctivities The rte of embodied technicl chnge is endogenized following Aghion nd Howitt (992). Let q jt = q j jt, where q j jt represents the qulity grd of the jth intermedite good t time t, qn/ being constnt. 5 j jt represents the number of qulity improvements of the 4 Bsed on the economic theory on index numbers, Licndro et l. (2002) find tht under embodied technicl progress the growth rte of output must be defined s in NIPA s methodology, i.e., liner combintion of the growth rte of both consumption nd investment, the weights being the corresponding nominl shres. Consequently, the growth rte of output is smller thn the growth rte of investment. 5 This condition sttes tht the difference in qulity between two successive innovtions should be sufficiently lrge, tht the current innovtion displces the previous one.

8 60 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 jth intermedite good chieved up to dte t. As usul in this literture, resercher discovering new qulity grde is supposed to hve the monopoly right to produce the good t the obtined qulity. Consequently, when new qulity improvement occurs, incumbents lose utomticlly their monopoly rentls. This feture genertes Schumpeterin process of cretive destruction. How does the cretive destruction process tke plce for the jth intermedite good? Let q j j be the leding qulity grde t time t. If resercher successfully introduces n innovtion t this time, the qulity grde increses to q jj+. The innovtion is ssumed to come out ccording to Poisson process, where g jj denotes the Poisson rrivl rte. We consider two lterntive technologies in the R&D sector: Technology A g jj ¼ kn jj /j j ; where n jj is the mount of non-durble goods devoted to reserch nd kn0 mesures the productivity of this sector. Technology B g jj ¼ kq t l jj /ðj j Þ; where l jj is the mount of lbor services devoted to reserch nd kq t N0 mesures the productivity of this sector. In both cses, the Poisson rrivl rte g is supposed to be decresing function of the reserch tsk, here cptured by j, representing the negtive effect of reserch complexity. More precisely, we set /j ð Þ ¼ q ðj þ Þ, implying / V(j)b0. This ssumption is consistent, s it is shown lter, with n equilibrium Poisson probbility ultimtely independent of the complexity of the reserch tsk. The expected vlue of n innovtion discovered t time t, V jj +,t, is equl to the expected flow of profits it genertes. The instntneous profits of the j j + innovtor, for ll time zzt until she will be displced by the j j +2 innovtor, re given by p jj þ ;z ¼ p jz q j x j þ jz ¼ I z q j j þ q j j þ : Q z The lst equlity comes fter substitution of p j from Eq. (5) nd x j from Eq. (3), using the condition p j /P = Q/q j. Given the embodied nture of technologicl progress, the benefits of R&D ccrue to the investment sector. Consequently, the instntneous profits of n innovtor depend on the demnd for investment goods only, in contrst to HA where technologicl progress is disembodied nd benefits lso the consumption sector. Under embodiment, the scope of R&D is restricted to the investment sector, enlrging the importnce of investment in the growth process. Then, V jj þ ;t ¼ Z l p jj t þ ;ze Z z t rs þ g jj þ ;s ds dz; ð3þ

9 R. Boucekkine et l. / Journl of Development Economics 77 (2005) where the two exponentil terms in the integrnd represent respectively the discount fctor nd the probbility of the qulity grde q jj+ still leding t time znt. Let us ssume tht the reserch sector is competitive nd reserch is subsidized t the rte b R. The rbitrge condition for strictly positive mount of resources spent in R&D ctivities stipultes tht the mrginl cost of reserch should be equl to the expected present vlue of profits, tht is: h t ¼ k/ðj j ÞV jj þ ;t: ð4þ where h t ¼ b R if technology A ð b R ÞW t Q t if technology B: ð5þ This rbitrge condition implies V jj þ ;t V jj ¼ ḣh t þ ;t h t, which yields by differentition of Eq. (3) V jj þ ;t ¼ q ðj j þ Þ r t þ g jj þ ḣh t h t Q t I t : ð6þ From Eqs. (4) nd (6), it turns out tht the Poisson rrivl rte g jj + does not depend on the complexity of the reserch tsk, tht is g jj +=g8 j. This mens tht qulity improvements cn occur for ll types of intermedite goods with the sme probbility, whtever the qulity grde is. This property of the model is entirely due to the specifiction of function /(j), s outlined by Brro nd Sl-i-Mrtin (995). The Poisson rrivl rte g j is ffected by j in two opposite wys. First, the monopoly profits ccruing to n innovtor increse with j, since its productivity depends directly on it. Secondly, by ssumption, the probbility of innovting decreses with the difficulty of the tsk, mesured by j. When the specifiction /j ð Þ ¼ q ðj þ Þ is dopted, the two effects exctly offset. As set out bove, t equilibrium Poisson rrivl rtes re equl for ll intermedite goods. By the Lw of Lrge Numbers, the verge growth rte of Q(t) is c t ¼ qg t ; ð7þ where q ¼ ðq Þ. From Eqs. (6) nd (7), the rbitrge condition (4) cn be written s h t ¼ k Q t I t r t þ c t q ḣh t h t : ð8þ Under technology A, the equilibrium lloction of non-durble inputs to the production of R&D ctivities is given by N t ¼ R 0 n j jt dj. Since the equilibrium Poisson rrivl rtes re the sme for ll intermedite goods, using Eq. (7), we get fter some trivil lgebr: N t ¼ c t k Q t ; ð9þ

10 62 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 where k ¼ k q. The equilibrium lloction of lbor to the production of R&D ctivities q is l t =0. Under technology B, the equilibrium lloction of lbor to the production of R&D ctivities is given by l t ¼ R 0 l j jt dj. Since the equilibrium Poisson rrivl rtes re the sme for ll intermedite goods, using Eq. (7), we get fter some trivil lgebr: l t ¼ c t k ; ð20þ The equilibrium lloction of non-durble inputs to the production of R&D ctivities is N t = Blnced growth pth equilibrium In order to chrcterize the equilibrium of this economy, some vrible chnges re introduced to eliminte trends. Concerning non-durble consumption nd inputs, z t ¼ Z t Q t for Z{C, X, N}. Concerning cpitl nd investment, z t ¼ Z t Q t for Z{I, K}. Implicit in this trnsformtion is tht consumption grows t smller rte thn both investment nd cpitl, direct impliction of embodied nture of technicl chnge. We define blnced growth pth (BGP) s n equilibrium pth long which detrended vribles re constnt. It is chrcterized by the following eqution system k ð lþ ¼ ðr þ d þ c b KÞ ðkþ h ¼ k km r þ c q ðaþ r ¼ r c þ q ðrþ m ¼ h ¼ c þ d : ðmþ ð b R Þ if technology A: ð b R Þð Þk ð lþ if technology B: ( 0 if technology A: l ¼ c if technology B: ðlþ k Eq. (K) is the optiml condition for cpitl nd it expresses the stedy stte vlue of cpitl intensity s function of the user cost of cpitl multiplied by the mrkup rte. It is derived from Eqs. (0) nd (6). Eq. (A) is the rbitrge condition in the R&D sector, Eq. (8), where m u i/k is n index of the moderniztion of cpitl. The lrger m is, the lrger the weight of new mchines in the stock of cpitl nd the lower the verge ge of it. Eq. (R) is the stndrd Fisher eqution showing how the interest rte depends on the growth ðwþ

11 rte of consumption long the BGP. It comes from the Euler Eq. (). We derive Eq. (M) from Eq. (9). It simply sttes tht, t the blnced growth pth, the investment to cpitl rtio must be equl to the deprecition rte plus the growth rte of cpitl. Eq. (W) defines the mrginl cost of R&D ctivities nd it is obtined from Eq. (5), fter substitution of W t from Eq. (). Eq. (L) reports the mount of lbor devoted to R&D ctivities. Before pursuing the nlysis, let us first estblish sufficient condition for existence nd uniqueness of blnced pth with positive growth. Proposition. If knk there exists unique solution to the system (K) (L) with cn0 nd k ¼ 8 >< >: ðd þ q b K Þ qð br Þ Cd ðq þ d b K Þqð b R Þ d if technology A if technology B where C ¼ ð Þ þ : Proof. To prove existence nd uniqueness of strictly positive solutions for c, some tedious lgebr is needed. Technology A Indeed fter successive substitutions from Eqs. (K), (R), (M), (W) nd (L) into (A), we cn write c s n implicit function of the prmeters of the problem: Ck b R ¼ r þ q c þ q c þ d r þ c þ d þ q b K uk A ðþ: c It is esy to check tht function K A (c) hs the following properties: (i) Ckd K A ðþ¼ 0, (ii) the limit of K ðqþd b K Þ q A is zero when c tends to infinity, (iii) K A is continuous nd strictly incresing in k nd (iv) there is t most one cn0 such tht KV A (c) =0. Technology B After successive substitutions from Eqs. (K), (R), (M), (W) nd (L) into (A), we cn write c s n implicit function of the sole prmeters of the problem: k c k c þ d b R ¼ r þ q c þ q r c þ þ d þ q b K ð2þ

12 64 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 Proposition sttes tht productivity in the reserch sector should be lrge enough for BGP with positive growth to be sustinble. Though this kind of conditions is very often required in endogenous growth models (even in the simplest ones, see Romer, 990), it is bsolutely needed in our frmework to dditionlly rule out multiplicity. 6 Positivity of k cn then be esily showed. Finlly, s in the stndrd growth model the condition qn ð rþ c is required to get bounded utility, nd it implies tht c nd i re strictly positive long the BGP On Krusell nd Hsieh It is well-known in the endogenous growth literture tht the different models of R&D hve very similr reduced forms. Of course, this is lso the cse of endogenous growth models with embodied technicl progress. For this reson, the reduced forms of Krusell (998) nd Hsieh (200) s models re very close to the system (K) (L) in this section. Consequently, our min results cn lso be generted using ny of these two models. 3. On the impct of subsidies 3.. Under disembodied technicl progress HA tke different view on technologicl progress, by imposing conditions tht mke it disembodied. The key ssumption is tht there is only one finl good sector producing simultneously the consumption nd the investment good. Technology in the finl sector is given by ð l t Þ R 0 A jtx jt dj, where s in the previous sections ( l t) is the frction of lbor llocted to production, A jt represents the stte of technology in sector j nd x jt is the per cpit mount of input j employed in the production of the finl good. The production of one unit of the intermediry input j requires A jt x jt units of mchines. Consistently, they define per cpit ggregte cpitl s K t u Z 0 A jt x jt dj ¼ A t x t ; where A t ¼ R 0 A jtdj, nd x jt = x t for ll j is direct impliction of symmetry in sectoril technologies. From the previous ssumptions, they get C t þ I t þ N t ¼ A t Kt ð l t Þ : ð24þ Given their definition of cpitl, technicl progress is disembodied, the reltive price of investment goods is unity nd both consumption nd investment grow t the sme rte long the blnced growth pth. Consequently, there re no obsolescence costs in the user ð23þ 6 Under kbk, multiple stedy stte equilibri re possible, s in Hsieh (200). Strtegic complementrities involved in the moderniztion mechnism re t the bsis of the multiplicity of stedy stte equilibri in R&D bsed growth models with embodied technicl chnge.

13 R. Boucekkine et l. / Journl of Development Economics 77 (2005) cost of cpitl nd the benefits of innovtion depend on totl production insted of investment. More precisely, Eqs. (K) nd (A) become k ð lþ ¼ ðr þ d b KÞ ðk D Þ h ¼ kð Þ k ð l r þ c ; ða D Þ q the other equtions remining identicl. Note tht Romer (990) s model genertes equtions very similr to Eqs. (K D ) nd (A D ), with the peculirity tht the term c/q is zero, since Shumpeterin obsolescence costs re nil. Nevertheless, the min difference between HA nd Romer is on the R&D technology. HA ssume tht R&D uses the finl good s input insted of the lbor endowment s in Romer. This ssumption is criticl, s stressed by HA, for the understnding of the role of cpitl subsidies on growth. When R&D uses lbor s input, fter substitution of Eq. (W) on Eq. (A D ), we obtin ð b R Þ ¼ k l r þ c : q Þ From Eqs. (R) nd (L), r nd l re functions of c implying tht the growth rte does not depend on b K. As sid before, this result does not depend on the ssumption tht growth is Shumpeterin, but on the extreme ssumption tht cpitl is not required for R&D ctivities. As long s R&D uses cpitl s production fctor, subsidy to the ccumultion of cpitl would be good for technologicl progress nd growth. Subsidies to cpitl ccumultion increse both cpitl per cpit nd output per cpit, ffecting the growth rte in two opposite wys. Firstly, n increse in finl production cuses the rise in profits to n innovtor, right hnd side of Eq. (A D ), nd secondly n increse in cpitl rises the mrginl product of lbor, rising R&D costs, Eq. (W). In the specil cse of disembodied technicl chnge nd lbor s the sole input in R&D, s in Romer (990), these two effects exctly cncel, becuse wges nd profits re proportionl to output per worker, s the previous eqution shows. This implies tht the growth rte does not depend on cpitl ccumultion Moderniztion nd obsolescence Under embodiment, the growth rte is ffected through two min economic mechnisms. The first is relted to the effect of obsolescence on the user cost of cpitl, the so-clled obsolescence costs, represented by the term c on the right hnd side of Eq. (K). It mitigtes ny positive effect on long run growth. The reson is strightforwrd. A rise in the rte of technicl chnge increses the user cost of cpitl by rising the decline rte of investment prices, which generte cpitl loses to cpitl owners. This increse in the user cost of cpitl reduces the demnd for cpitl, Eq. (K). Since the expected vlue of R&D is positively relted to the demnd for cpitl, Eq. (A), decrese in the lter reduces the intensity of R&D ctivities nd, consequently, the rte of technicl progress.

14 66 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 The second feture is relted to the crucil role of investment in the moderniztion of the cpitl stock. When R&D is stimulted by n exogenous prmeter chnge tht impinges on the reserch rbitrge condition (A), the resulting rise in productivity growth reinforces the stimulus by rising investment. This increse in investment comes bout becuse fster rte of innovtion implies moderniztion of cpitl, s cn be seen in Eq. (M). This rise in investment reinforces the stimulus to R&D becuse it increses the size of the mrket tht cn be cptured by successful innovtor. Consequently, when more resources re devoted to R&D, the ssocited increse in the rte of technicl progress hs multiplictive effects by rising the investment to cpitl rtio, which in turn ffects the incentives to innovte nd the rte of technologicl progress. This is the moderniztion mechnism tht the embodiment ssumption gives rise to. The eqution system (K) (L) is very similr to the system governing the blnced growth pth equilibrium in HA. There re however two min differences, both relted to the diverse nture of technologicl progress, s it cn be seen by compring Eqs. (K) (A) to (K D ) (A D ). Firstly, under embodied technicl chnge, the user cost of cpitl in the right hnd side of Eq. (K) involves obsolescence costs c, while it does not under disembodiment s it cn be seen in (K D ). Finlly, under embodiment the expected vlue of R&D, on the right hnd side of Eq. (A), depends on investment, while under disembodiement, on the right hnd side of Eq. (A D ), it depends on finl production. As sid in the previous section, subsidies to cpitl ccumultion ffect the growth rte by incresing both profits to innovtion nd R&D costs. Under embodiment, even if lbor is the sole input in R&D ctivities, the positive effect on profits more thn compenstes the negtive effect on costs, nd the net effect of the subsidy is positive. Therefore, under embodiment, subsidies to cpitl ccumultion nd innovtion re growth enhncing. This result is estblished in Proposition 2. Proposition 2. Under technologies A nd B, if cn0 t stedy stte, c increses when either b K or b R increses. Proof. Proposition 2 follows directly from Eqs. (2) nd (22), since K j (c), j = A, B, re incresing functions of b K nd the left hnd sides of Eqs. (2) nd (22) re decresing functions of b R Embodiment in Howitt nd Aghion (998) In order to nlyze the role of embodiment, Howitt nd Aghion, Section 4.4, introduce the following ssumption: When new technology is discovered in sector j, ll existing mchines become obsolete. 7 It implies, in the prticulr cse of technology A, tht t ech 7 More precisely, they ssume tht mchines ssocited to the old technology re instntneously scrpped. HA never discuss the optimlity of such rule. However, it is well-known in vintge cpitl theory tht the scrpping of old mchines is not necessrily immedite. See Brdhn nd Prile (996) nd Boucekkine et l. (997).

15 R. Boucekkine et l. / Journl of Development Economics 77 (2005) instnt frction kn of the stock of cpitl is destroyed. Consequently, the motion lw of cpitl tkes the following form K t ¼ X t ðd þ kn t ÞK t ; ð25þ where dn0 is the physicl deprecition rte. At stedy stte, Eqs. (24) nd (25) re very similr to the stndrd conditions in optiml growth models, with ( )Ȧ t /A t being the endogenous growth rte of disembodied technologicl progress nd d + kn the endogenous deprecition rte. This result is puzzling, becuse the uthors strt by ssuming tht technicl progress is embodied, but their model behves s if it were disembodied. In prticulr, it does not fit the empiricl regulrity tht the price of equipment is permnently declining with respect to the price of non-durble consumption, neither the observtion tht the equipment to output rtio is permnently incresing in rel terms. The only effect of embodiment is in the deprecition rte: it dds the obsolescence costs kn, directly ssocited to the scrpping of mchines embodying the replced technology. More importnt, HA s ssumptions hve strong implictions for the incentives to undertke reserch ctivities. The zero profit condition in the reserch sector cn be written s in Eq. (A D ). As in Section 2.4, the min difference with respect to Eq. (A) is tht in HA profits from R&D do not depend on investment, but on finl production. Given tht innovtion is ddressed to the production of finl goods, profits depend on finl goods demnd not on investment demnd. This kills the moderniztion effect. Additionlly, if the R&D sector uses lbor s sole input, the economy behves s if technologicl progress were disembodied nd subsidies to cpitl hve no effect on innovtion nd the growth rte. Let us propose the following vrible chnge to better understnd wht is going on in the Howitt nd Aghion model with embodied technicl chnge. We define J t ¼ Z A 0 it z it di! ; where z it = A it x it represents the number of mchines of type i opertive t time t (mesured in the sme unit s X t ). Given tht this technology is constnt elsticity of substitution, n improvement in the productivity of ny of these technologies would improve the productivity of the others. It mkes cler tht there is no need of relying on ny story bout intermediry goods to understnd the role of technicl progress in this economy: The relevnt ssumption is tht there is no perfect substitution between different vintges in terms of their bility to produce the nondurble good. Notice lso tht in HA, K t mesures (per cpit) ggregte cpitl in terms of production costs, but J t mesures (per cpit) ggregte cpitl in terms of their bility to produce the nondurble good.

16 68 R. Boucekkine et l. / Journl of Development Economics 77 (2005) 53 7 It is esy to show tht, t the symmetric equilibrium, J t = Q t K t, where Q t ¼ A t.it implies tht Eq. (24) becomes C t þ I t þ N t ¼ J t From Eq. (25), we get J d t ¼ Q t X ffl{zffl} t I t d þ kn t Q t Q t J t ð26þ ð27þ Under this definition of ggregte cpitl, the reltive price of investment goods, given by /Q t, is permnently declining due to embodied technicl progress, s in GHK. However, the lw of motion of efficient cpitl includes two dditionl terms: kn t, representing scrpping, nd Q t/q t, which represents the gins in productivity coming from the effects of time t innovtions on the whole cpitl stock (this gin is due to the complementrity of different types of mchine in the production of the nondurble good). The net deprecition rte of J t is d + kn t Q t/q t. Indeed, the negtive term Q t/q t cncels with the cpitl losses due to the permnent decline in investment prices, implying tht the user cost of cpitl only depends on d + kn t. Technologicl progress is driven by common source, R&D in the investment sector, nd it is embodied in new mchines. However, given the nture of nondurble goods technology, this common source lso genertes disembodied technicl progress s result of the complementrity of different cpitl goods in the production of the nondurble good. It is represented by the term Q t/q t J t in Eq. (27) On the impct of subsidies under embodiment The min concern of this section is to nlyze the suggestion by HA tht bincorporting embodiment ctully hs the prdoxicl effect of wekening the result tht cpitl subsidies ffect long run growth.q HA, Section 4.4, provide n intuitive explntion of the role of obsolescence in their frmework, by dding obsolescence costs to the user cost of cpitl, nd found tht cpitl intensity should be smller t stedy stte. Consequently, they rgue tht n increse in cpitl subsidies hs lower positive effect on growth if obsolescence costs re to be considered. This pper shows tht HA s rgument is incomplete, since it does not tke into ccount the moderniztion effect of investment ssocited to the embodiment hypothesis. When tken into ccount, the rise in the rte of technicl chnge due to cpitl subsidies yields indeed n increse in the investment to cpitl rtio, which gin stimultes reserch nd growth. This dditionl mechnism my well rule out the min conclusion of HA, nmely the negtive effect of embodiment on the efficiency of cpitl subsidies. Proposition 3 estblishes necessry nd sufficient conditions for the positive effect of moderniztion more thn compenste the negtive effect of obsolescence. Proposition 3. Under technology A, the moderniztion effect is lrger thn the obsolescence effect if nd only if r cnb K:

17 R. Boucekkine et l. / Journl of Development Economics 77 (2005) Proof. The stedy stte of c is implicitly defined by Eq. (2) s function of b K nd b R. Differentiting Eq. (2), fter some lgebr, we get dc db K ¼ r þ q r þ c q þ ð Þ r r ð Þ r þ ð Þ r fflfflfflffl{zfflfflfflffl} obsolescence effect ð Þ m fflfflfflfflfflffl{zfflfflfflfflfflffl} moderniztion effect ð28þ dc db R ¼ r þ q r þ c q ðkcþ ðr þ c q Þ m r r þ þ ð Þ r ð Þ r fflfflfflffl{zfflfflfflffl} obsolescence effect ð Þ m fflfflfflfflfflffl{zfflfflfflfflfflffl} moderniztion effect ð29þ where r = r + d + c b K is the user cost of cpitl. The moderniztion effect is higher thn the obsolescence effect if only if rnmzr cnb K: 5 A better understnding of Proposition 3 cn be chieved by bstrcting from the effects of c others thn those operting through the obsolescence nd the moderniztion mechnisms. From Eq. (K), n increse in the obsolescence cost rises the user cost of cpitl nd reduces the demnd for cpitl. The elsticity of the demnd for cpitl with respect to the obsolescence cost is proportionl to the weight of obsolescence costs in the dc c user cost of cpitl, i.e. dk k ¼ c r. In order for the moderniztion effect to fully compenste the obsolescence effect, the demnd for investment must remin unchnged fter the induced reduction in the demnd for cpitl. If not, from Eq. (A) the reduced incentives for R&D should lower the rte of technicl progress. Indeed, investment could be ccommodted by n increse in the investment to cpitl rtio, such tht (dk/k) =(dm/m). From Eq. (M), such reduction requires the following reltion dm m ¼ c m dc c to hold. Consequently, the positive effect of moderniztion fully compenstes the negtive effect of obsolescence costs if nd only if the user cost of cpitl r is equl to the investment to cpitl rtio m. Nevertheless, if the user cost of cpitl is lrger (smller) thn the investment to cpitl rtio, the moderniztion effect more (less) thn compenste the obsolescence effect. As stted t the end of Section 2.4, qnð r c is required to get bounded utility. This condition is equivlent to r cn0. Consequently, if cpitl subsidies re ner zero, the moderniztion effect lwys domintes, nd the positive effect of subsidies is lrger under embodiment thn under disembodied technicl chnge. Þ

18 70 R. Boucekkine et l. / Journl of Development Economics 77 (2005) Conclusions This pper introduces cpitl ccumultion nd embodied technicl progress in Schumpeterin cretive destruction model, following different strtegy thn Howitt nd Aghion (998). We show tht the embodied nture of technicl progress hs two min implictions. Firstly, the user cost of cpitl involves obsolescence costs, which ffects negtively reserch incentives. Secondly, the moderniztion of cpitl through investment rises the incentives to undertke R&D ctivities. The moderniztion of cpitl is shown to offset the growth losses due to obsolescence costs, which in prticulr improves the growth enhncing role of subsidies. The proposed model is consistent with the empiricl evidence on embodied technicl progress. Additionlly, it stresses the moderniztion role of investment pointed out by the recent empiricl literture on development economics. From theoreticl point of view, it reconciles the Schumpeterin pproch with the literture on growth with embodied technicl chnge, s in Greenwood et l. (997), by stressing the importnce of the investment-specific nture of technicl progress. Applied to the promotion of economic development, our results cn serve to dvocte the typicl fiscl nd trde policies ensuring the moderniztion of the cpitl stock. For exmple, cpitl subsidies re desirble under the strict condition tht the purchsed cpitl goods embody superior technologies. Though this policy is likely to generte reltively high obsolescence costs, the long term growth effects of the resulting moderniztion re likely to be high enough to compenste the ltter costs. Nturlly, such moderniztion should tke into ccount the necessrily limited cpcity of technologicl bsorption of the considered economies, of which the qulity of the vilble skills is key indictor. To properly ddress this issue, there is need to construct less stylized models including creful modelling of technologicl bsorption cpcity. This line of reserch is on top of our gend. Acknowledgements We would like to thnk Bruno Amble, Dvid de l Croix, Victor Ríos-Rull, nd two nonymous referees for their useful comments. Boucekkine cknowledges the finncil support of the Belgin French speking community (ARC 99/ nd ARC 03/08-235) nd of the Belgin Federl Government (PAI P5/0). Finlly, del Río nd Licndro cknowledge the finncil support of the Spnish Ministry of Sciences nd Technology (SEC ). References Aghion, P., Howitt, P., 992. A model of growth through cretive destruction. Econometric 60, Brdhn, P., Prile, R., 996. Endogenous Growth Theory in Vintge Cpitl Model. UC Berkeley, CIDER WP C Brro, R., Sl-i-Mrtin, X., 995. Economic Growth. McGrw Hill. Boucekkine, R., Germin, M., Licndro, O., 997. Replcement echoes in the vintge cpitl growth model. Journl of Economic Theory 74,

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