Beyond Balanced Growth : Some Further Results

Transcription

1 eyond alanced Growh : Some Furher Resuls by Dens Sec and Helmu Wagner Dscusson Paer o. 49 ay 27 Dskussonsberäge der Fakulä für Wrschafswssenschaf der FernUnversä n Hagen Herausgegeben vom Dekan der Fakulä Alle Reche legen be den Verfassern

2 Dens Sec Prof. Dr. Helmu Wagner har of acroeconomcs Dearmen of Economcs Unversy of Hagen Unversässr. 4, D-5884 Hagen, Germany Phone: / Fax: / E-mal: dens.sec@fernun-hagen.de E-mal: helmu.wagner@fernun-hagen.de h://

3 - - eyond alanced Growh : Some Furher Resuls Dens Sec and Helmu Wagner ay 27 Absrac ongsamu e al. 2 have demonsraed ha conrary o earler onon balanced growh of aggregaed varables and srucural change can be smulaneously generaed n a model of exogenous echnologcal rogress. Tha s, hey have resened a model ha s smulaneously conssen wh aldor s sylzed facs and sylzed facs of labor reallocaon beween secors. However, hey used n her model secoral roducon funcons ha dffer only by a mullcave consan. Thus her model s no conssen wh he emrcal fac of dfferen labor shares of ncome across secors. We generalze her model and show ha a model of exogenous growh can smulaneously be conssen wh aldor s sylzed facs, sylzed facs of labor reallocaon and dfferen labor shares of ncome across secors.e. comleely dfferen secoral roducon funcons. eywords: balanced growh, srucural change, aldor facs, labor reallocaon, labor shares of ncome. JEL odes: O4, O 4

4 Inroducon ongsamu e al. 2 have demonsraed ha conrary o earler onon balanced growh of aggregaed varables and srucural change can be smulaneously generaed n a model of exogenous echnologcal rogress. Therefore, her model s smulaneously conssen wh aldor s sylzed facs 2 and wh facs abou srucural change. Unforunaely, he auhors assumed secoral roducon funcons, whch dffer only by a consan roducvy arameer. Ths s an unrealsc assumon, because here s emrcal evdence ha he labor shares of ncome dffer across secors. 3 Therefore, ongsamu e al. 997 had nroduced secoral roducon funcons ha dffer comleely,.e. n every arameer. There hey roved ha n hs case srucural change s only conssen wh a consan real rae of reurn. Smulaneous balanced growh s no longer feasble. Thus no all aldor facs are sasfed. However, her model feaured only hree secors manufacurng, agrculure, servces. 4 The overall concluson ha can be drawn from hs research s ha balanced growh of aggregaed varables and srucural change canno be smulaneously unfed n hs model, as long as secoral roducon funcons are comleely dfferen. In oher words, he research by ongsamu e al. 997 and 2 conveys he mresson ha aldor s sylzed facs, srucural change facs and dfferen labor shares of ncome across secors canno be unfed n a neoclasscal model. Srucural change sands here for labor reallocaon beween secors such as manufacurng, agrculure and servces. 2 aldor s sylzed facs sae ha growh rae of er caa ouu, caal-o-ouu rao, shares of labor and caal n naonal ncome and rae of reurn o caal are nearly consan n he long run; and caal er worker grows over me. 3 See ongsamu e al. 997, ongsamu e al. 2 and 997 used a hree secor framework, because her am was o f he model o he emrcal facs of US-develomen durng he las cenury.

5 - 3 - Our am s o demonsrae ha hs s no rue. The reason why ongsamu e al. 2 and 997 were no successful n hs resec s ha hey dd no use enough secors o rove hs resul. We use he model resened by ongsamu e al. 2 for our roof. u, unlke ongsamu e al. 2, we assume ha here s an arbrary number of secors nsead of only hree secors n he economy and he secoral roducon funcons are comleely dfferen. We use he conce of a balanced growh ah of aggregaed varables. Such a growh ah feaures consan growh raes of aggregaed varables, a consan real rae of reurn o caal and consan relave rces. u, along hs growh ah, he growh raes of dsaggregaed varables such as secoral ouu ec. need no o be consan. We show ha a necessary condon for smulaneous balanced growh of aggregaed varables and srucural change s he exsence of a leas four secors n hs framework. Overall, he model ha s resened here s a more general verson of he model resened n ongsamu e al. 2, because he unrealsc assumon of nearly dencal secoral roducon funcons s no necessary. Thus, hs model s conssen a he same me wh hree emrcal fndngs: aldor s sylzed facs, srucural change facs, and dfferen labor shares of ncome across secors, whereas he verson of ongsamu e al. 2 s only conssen wh he frs wo emrcal fndngs. In he nex secon of hs aer we secfy he roducon secor and s effcency condons. Followng hs, we descrbe he household secor and solve s dynamc omzaon roblem by usng resuls from he roducon secor. In he fourh ar we look a he balanced growh ah of hs model and derve he necessary condons for s exsence. The ffh ar s abou srucural change along hs balanced growh ah. Fnally, we summarze our resuls.

6 Producon Secor Lke ongsamu e al. 2, we assume ha here are wo roducon facors: caal and labor. The oal amoun of labor avalable n he economy s exogenously gven and normalzed o one a every on of me. There s no oulaon growh. 5 The growh rae g of labor-augmenng echncal rogress s consan, exogenously gven and equal across secors. The ouu of he manufacurng secor can be used as caal and consumed, as well. The ouu of he oher secors can only be consumed. Tha s, only he manufacurng secor roduces caal. 6 Therefore, he ouu of he manufacurng secor has o be numérare. All caal and labor avalable has o be used n roducon. Unlke ongsamu e al. 2, we assume ha each secor roduces s ouu by a secor secfc obb-douglas roducon funcon and he number of secors n s arbrary. Thus he equaons descrbng he roducon sde of he economy are: - n, g 4 δ 5 5 Smlar resuls can be derved wh oulaon growh. 6 Ths assumon s emrcally reasonable a leas for he USA. See ongsamu e al. 2.

7 - 5 -, 6 7 The arameers are dfferen across secors,.e.,. reresens he fracon of caal labor devoed o secor ; denoes consumon of good ; s he relave rce of good exressed n manufacurng erms; δ denoes he economy-wde derecaon rae and s he me ndex. In order o faclae he calculaons, we follow ongsamu e al. 2 and 997 and elaborae he effcency condons n roducon: 7 Effcen allocaon across secors requres ha he margnal raes of echncal subsuon are equal across secors,.e.: 8 / / / / 8 oreover, also requres ha he margnal roducvy of labor s equal across secors, whch mles: / / 9 Prof-maxmzng roducers emloy caal and labor n such a manner ha he sum of he real rae of reurn on caal and he derecaon rae s equal o he margnal roducvy of caal, and he real wage rae s equal o he margnal roducvy of labor,.e.: 7 ecause he omzaon wh he Hamlonan yelds hese effcency condons anyway, faclaes he calculaons f we elaborae hese condons now and use hem n dynamc omzaon. 8 See ongsamu e al. 997 as well.

8 - 6 - r δ w For hs resuls n because of eq. and 7: 9 r δ w Aggregaed ouu n manufacurng erms s gven by because of eq. and 9: 2 A roof of equaon 2 s gven n Aendx A. 3. Preferences The reference srucure s he same as n ongsamu e al. 2. The only dfference s ha we generalze he uly funcon: he number of goods n s arbrary nsead of beng hree: n d e U,..., ρ σ σ where 9 The same resuls are obaned n ongsamu e al. 997,. 22.

9 - 7 - σ, ρ, > 3 4 The consans home roducon f nfny when can be nerreed as subssence levels f s osve or as he s negave of good because he margnal uly aroaches aroaches. defnes by how much he consumon of good conrbues o he uly of he household. These references are non-homohec,.e. he ncome elascy of demand dffers across goods, as long as no all. Ths can cause some srucural change. The reresenave household maxmzes s lfeme uly subec o s dynamc budge-resrcon, whch s gven by: δ E, 5 where he aggregaed consumon exendures are gven by: E 6 Accordng o ongsamu e al. 2 we assume 7 n order o ensure ha he labor share of he manufacurng secor says consan, whch s conssen wh emrcal facs. Ths omal conrol roblem can be solved by usng a Hamlonan. The ransversaly condon s gven by lm { }, where s he oeraor of he Hamlonan Ths resrcon can be obaned by addng o boh sdes of eq. 5. ecause of eq. 6 and 7, hs resuls n he dynamc resrcon above. Ths s conssen wh he emrcal facs, a leas for he develoed economes durng he las years; see e.g. ongsamu e al. 2.

10 - 8 - shadow rce of caal. The omal soluon s he same as n ongsamu e al. 2: 2, 8 σ ρ r 9 These resuls are roved n Aendx. We can now derve he consumon exendures E because of eq. 9, 4, 6 and 8: E 2 4. alanced Growh of Aggregaed Varables A balanced growh ah of aggregaed varables 3, whch s conssen wh aldor s sylzed facs of economc growh, requres r o be consan and g w w E E. 2 Ths soluon s only rue when relave rces are consan. We wll see ha relave rces are consan along he balanced growh ah ha we focus on. ongsamu e al. 997 acheve he same resuls, as well; see ongsamu e al. 997, A balanced growh ah n he radonal sense refers o a growh ah ha feaures consan growh raes of all varables. Thus, no srucural change akes lace along hs ye of growh ah. When we refer o a balanced growh ah of aggregaed varables n hs aer, we mean a growh ah ha feaures consan growh raes of aggregaed varables, consan relave rces and a consan real rae of reurn o caal, bu no necessarly consan growh raes of dsaggregaed varables, such as secoral ouu ec.

11 - 9 - Le us now assume ha s consan 4 and s consan as well. I can be seen a frs sgh ha n hs case a balanced growh ah of aggregaed varables whch s conssen wh aldor s sylzed facs exss when grows a rae g: r s consan eq. ; eq. 2, w eq. and E eq. 5 5 each grow a rae g. Thus, he aldor facs are sasfed. Addonally, relave rces are consan eq. 9 6 and grows a rae g eq. 5 and 7. The remanng ask s o elaborae he necessary condons ha ensure ha and are consan: For o be consan s necessary ha s consan eq. 3. I can be derved from equaons 2, 3 and 8, ha s gven by The exlc roof s n Aendx. 4 Ths assumon s necessary n order o faclae he soluon of dfferenal equaons, whch we wll need laer on. The assumon s conssen wh emrcal fndngs see ongsamu e al. 2. 5, δ and each grow a rae g. Thus, equaon 5 can only be fulflled n every on of me f E grows a rae g as well. 6 When s consan, s consan n all secors as well, because of eq I can be seen from eq. ha grows a rae g along he balanced growh ah of aggregaed varables, because we assumed ha s consan. Thus, because, δ and each grow a rae g along he balanced growh ah of aggregaed, eq. 5 can only be sasfed a any on of me f grows a rae g as well.

12 - - Thus, s consan, when s consan. We have already assumed ha s consan. Overall, he wo requremens ha are necessary for and o be consan, are: consan 23 consan 24 I can be roved by usng equaons, 6, 8, 8 and 22 ha along our balanced growh ah of aggregaed varables secoral labor shares are gven by: 8 Γ Γ ex g,, 25 where Γ σg ρ δ / 26 The exlc roof s n Aendx D. Wh hs resul, we can exress he equaons 23 and 24 as funcons of modelarameers: Γ Γ See also ongsamu e al. 997,. 23.

13 - - The queson ha now has o be answered s wheher hese wo requremens can be sasfed smulaneously. The answer s yes, bu here have o be a leas hree goods/secors wh a. If here are only wo secors wh a as n he aer resened by ongsamu e al. 997 and 2, he equaons 27 and 28 form a homogenous lnear-equaon sysem, ha s fully deermned.e. here are wo unknowns 9 and wo lnearly ndeenden equaons. Thus, here s only a rval soluon o hs sysem,.e. Γ,. I can be seen from equaons 25 and 26 ha n hs case has o be equal o zero n all secors, and no srucural change akes lace,.e. labor shares say consan. Of course, f secoral roducon funcons are dencal u o a consan.e.,,, he equaons 27 and 28 are lnearly deenden. Thus, here exs an nfne number of soluons of he sysem. Ths s he case n ongsamu e al. 2. Overall, when only wo secors have a, he equaons 27 and 28 can only be sasfed smulaneously, eher when here s no srucural change or when secoral roducon funcons are dencal u o a consan. If here are a leas hree secors wh a, he equaons 27 and 28 are a homogenous lnear-equaon sysem ha s no fully deermned,.e. he number of equaons s smaller han he number of unknowns wo equaons and a leas hree unknowns. Thus, here exs an nfne number of soluons of he sysem, even when 9 I can be seen from equaon 26 ha Γ s equal o zero when he corresondng. Thus, when here are only wo secors wh n he economy here are also only wo corresondng unknowns Γ n he equaons 27 and 28 he oher Γ s are equal o zero.

14 - 2 - he roducon funcon arameers dffer n all secors. Therefore, he equaons 27 and 28 can smulaneously be sasfed. 2 The remanng queson s: how can he wo condons 27 and 28 be nerreed? Equaon 27 s necessary, because we wan our model o have a consan, whch s an emrcal fac. As exlaned above, s consan when equaon 27 s sasfed. y usng equaons 9, 7, 26 and 27 can be roved ha f condon 28 s sasfed, s sasfed as well see Aendx E. An nerreaon of hs requremen has already been suggesed by ongsamu e al. 2: If we nerre he uly arameers as he household s nal endowmens of goods, hs condon saes ha he marke value of nal endowmens has o be equal o zero. 5. Srucural hange Wha abou srucural change? I can be seen from equaons 25 and 26 ha secors wh < > have ncreasng decreasng labor shares along he balanced growh ah of aggregaed varables. 2 The labor share of he manufacurng secor s consan, as menoned above. Thus, our balanced growh ah of aggregaed varables also feaures srucural change. The reason for srucural change s dfferen ncome elascy of demand across goods as n ongsamu e al. 997 and 2. The drecon of srucural change deends on he relaon beween demand growh and roducvy growh along he balanced growh ah of aggregaed varables. As long as demand 2 See eckl 22 for smlar condons n anoher model of balanced growh and srucural change. 2 See foonoe 3.

15 - 3 - grows a a rae hgher han g, he ncrease n roducvy due o echnologcal rogress s no suffcen o make suly kee ace wh demand growh. Thus, labor nu has o be ncreased n he corresondng secor, and vce versa. In he manufacurng secor, where demand grows a rae g here s no need o change he facor nus. How can he secors of hs economy be nerreed? As already menoned, he secor wh s he manufacurng secor, whch feaures no labor reallocaon. One secor wh > can be nerreed as he agrculure secor. As exlaned above, secors wh a osve have decreasng labor shares n our model. Thus, he labor share of he agrculure secor s decreasng n our model. Ths s conssen wh he emrcal facs of srucural change. 22 There are wo alernaves for he nerreaon of he remanng secors wh a : Eher, hey can be nerreed as subsecors of he servce secor. They should have ncreasng labor shares. 23 As saed above, hs requres < n hese secors. Or, f here are only wo remanng secors: One secor wh < can be nerreed as he servce secor ncreasng labor share. The oher secor can be nerreed as he ublc secor ha rovdes ublc servces ha corresond o governmen sendng. Wagner s law saes ha he rao of governmen sendng o oal ouu ncreases n he long run see for examle Oxley 994. The rao of secoral ouu o aggregae ouu s ncreasng n 22 For a revew of emrcal facs regardng srucural change see, e.g. uznes 976, ongsamu e al. 2 and 997, or ga/pssardes 24. These emrcal fndngs sae ha he labor share of he servce secor agrculural secor s ncreasng decreasng. Wh resec o he manufacurng secor, ongsamu e al. 997 and 2 sae ha s labor share can be regarded as consan n he las cenury n he develoed counres. Oher auhors e.g. ga/pssardes 24 sae ha he evoluon of he labor share of he manufacurng secor mgh be raher descrbed as hum-shaed n he longer run. 23 See foonoe 22.

16 - 4 - our model as long as he corresondng <. 24 Thus, we have o assume ha s negave n he secor ha s nerreed as he ublc secor. When hs assumon s made he rao of ublc secor ouu o aggregae ouu ncreases wh me n our model, corresondng o Wagner s law. Overall, he same srucural change dynamc as n ongsamu e al. 2 s feasble n hs model. 6. Résumé The work of ongsamu e al. 2 and 997 conveys he mresson ha balanced growh of aggregaed varables and srucural change canno be smulaneously generaed n a neoclasscal model of exogenous growh when he secoral roducon funcons dffer n all arameers. I seemed herefore ha aldor s sylzed facs, srucural change facs and dfferen secoral labor shares of ncome were no smulaneously feasble n such a model. We have roved n hs aer ha hs s no rue. The framework for hs roof s smlar o he framework of ongsamu e al. 2. The necessary condon for smulaneous balanced growh of aggregaed varables and srucural change s he exsence of a leas hree secors wh a. In hs case he arameer resrcons 27 and 28 can be sasfed smulaneously. These wo resrcons nvolve 6 24 The rao of secoral ouu o aggregaed ouu s gven by /. We know ha relave rces are consan and aggregae ouu grows a rae g along he balanced growh ah of aggregaed varables. y usng equaons 8 and 6 can be shown ha he growh rae of secoral ouu s gven by / g /, n our model. Thus, when <, he secoral ouu s growng a a rae hgher han g along he balanced growh ah of aggregaed varables. Therefore, he rao of secoral ouu o aggregae ouu / s ncreasng along he balanced growh ah of aggregaed varables f <.

17 - 5 - arameers. Therefore, hey wll robably no resrc he unversaly of he model. When hese wo resrcons are sasfed he economy s on a balanced growh ah, whch feaures smulaneously srucural change, every me when he aggregaed caal grows a rae g. Overall, we managed o ge he same aerns of srucural change as n ongsamu e al. 2 whou usng he unrealsc assumon of nearly dencal secoral roducon funcons. Thus, our model s conssen wh hree knds of emrcal fndngs: aldor s sylzed facs, srucural change facs and comleely dfferen secoral roducon funcons.e. dfferen labor ncome shares across secors.

18 - 6 - APPEDI A Inserng equaons 9 and no : resuls n: Solvng equaon 8 for and nserng no equaon 2 yelds: Inserng hs resul no he equaon above resuls n: q.e.d.

19 - 7 - APPEDI The Hamlonan for hs omzaon roblem s gven by: E u H δ where σ σ u. E / / are conrol varables and s he sae varable. The ransversaly condon s gven by { } lm. The well known omaly condons sae ha: n H,...,!.2 H ρ!.3 From he frs omaly condon.2 follows ha: n E u,...,! σ σ

20 - 8 -, σ.4 Inserng equaon 7 and 7 no equaon.4 yelds for : σ.5 Thus, follows from equaon.5: θ.6 Seng.4.5 yelds:, q.e.d..7 Solvng.7 for and nserng no equaon. resuls n: Thus, as long as rces are consan we have:.8 Inserng.8 n.6 yelds: θ.9 From he second omaly condon.3 follows ha: E ρ δ!. ecause of equaons 6 and 7 we know ha

21 - 9 - E. Thus, because of equaon and.: 25 δ r E.2 Inserng.2 no. resuls n: r ρ.3 Seng.9.3 yelds: σ ρ r q.e.d. 25 Remember: s a conrol varable.

22 - 2 - APPEDI y solvng equaon 8 for, nserng no equaon 2 and solvng for, we ge: Subsung n hs equaon by usng equaon 3 resuls n q.e.d.

23 - 2 - APPEDI D Solvng equaon for and subsung by usng equaon 6, gves, Dfferenang hs equaon wh resec o me resuls n: 26 g, The soluon o hs dfferenal equaon s, ex ex g gd D. From equaon 8 follows, ha on he balanced growh ah.e. when rces are consan and when grows a rae g: g g g, The soluon of hs dfferenal equaon s gven by g, ex D.2 ecause of equaon 6 follows from equaon, ha, D.3 ow we subsue n equaon as follows: by usng equaon 8 and r by usng equaon Thus we ge 26 Ths equaon s only rue along he balanced growh ah. 27 Remember ha grows a rae g along he balanced growh ah.

24 g, / δ ρ σ D.4 ow we subsue n equaon D. as follows: frs by usng equaon D.2, hen by usng equaon D.3, and fnally by usng equaon D.4. Thus we ge:, ex Γ Γ g, where g δ ρ σ Γ / q.e.d.

25 APPEDI E ecause of equaons 9 and 7, follows ha 28 y nserng equaon D.4 from Aendx D no hs equaon we ge because of equaon 26 and 27: Γ Γ Γ Γ g δ ρ σ / We can see now ha, when Γ. 28 Remember: s consan along he balanced growh ah.

26 References OGSAUT, P., REELO, A. and IE, D. 997, eyond alanced Growh, ER Workng Paer o OGSAUT, P., REELO, A. and IE, D. 2, eyond alanced Growh, Revew of Economc Sudes, 68, UZETS, S. 976, odern Economc Growh: Rae, Srucure and Sread ale Unversy Press, 7 h edon. EL, J. 22, Srucural hange and Generalzed alanced Growh, Journal of Economcs, 77, 3, GAI, R. L. and PISSARIDES. A. 24, alanced Growh wh Srucural hange London School of Economcs. OLE, L. 994, onegraon, ausaly and Wagner s Law: A Tes for ran 87-93, Scosh Journal of Polcal Economy, 4, 3,