A Neoclassical Growth Model for Environmental Expenditure of S-shaped Curve and Pollution of Inverted U-shaped Curve

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1 中国人民大学经济学院工作论文系列 School of Economcs, Renmn Unversty of Chna Workng aper Seres A Neoclasscal Growth Model for Envronmental Expendture of S-shaped Curve and olluton of Inverted U-shaped Curve # SERUC Workng aper no. 851E # Economc School, Renmn Unversty of Chna. Bejng, Chna. GUO Lu: doctor canddate of nternatonal economcs, E-mal: guolu@ruc.edu.cn. LU Yang: doctor canddate of nternatonal trade, E-mal: lusanmao@yahoo.com.cn. We are grateful to Wang Chuan and Yang Qjn for ther valuable suggestons on ths paper, and all errors are ours.

2 Abstract: In ths paper we tres to gve the nverted U-shaped and N-shaped EKC a theoretcal nterpretaton by usng neoclasscal growth model. We fnd that though the preference s homothetc and the envronmental expendture s constant return to scale, the nverted U-shaped EKC can stll be brthed naturally from neoclasscal growth model. In addton, the convergng envronmental expendture path s a S curve. We also fnd the steady state of polluton-economc growth s a saddle equlbrum whch offers an explanaton of N-shaped EKC by a departed path from steady state. Key Words: Neoclasscal Growth, EKC, Envronmental Expendture, olluton JEL Classfcaton: 1

3 A Neoclasscal Growth Model for Envronmental Expendture of S-shaped Curve and olluton of Inverted U-shaped Curve GUO Lu LU Yang 1. Introducton The typcal presupposton of neoclasscal economc growth theores mples that there s no polluton produced n producton, whch s controversal n real world. In recent day, the envronmental problems comng to focus markedly and envronmental restoratve cost rsng sharply make people begn to reconsder the envronmental problems n economc growth. The Envronmental Kuznets Curve (hereafter EKC) hypothess shows the nverted U-shaped relatonshp of economc growth and polluton brought nto world by producton. Grossman and Krueger (1991) frst tested the relatonshp between polluton emsson and economc growth; they found that some pollutants show an exact nverted U-shaped relatonshp between envronmental qualty and the ncome level. Economc growth ntally leads to ncreased of polluton, but when the polluton passes through the Turnng ont, the polluton would fall down. Ths phenomenon s called the Envronment Kuznets Curve (EKC). In Grossman and Krueger (1991) s paper, they use the scale effect (SE), abatement technque effect (ATE) and composton effects (CE) to explan the nverted U-shaped EKC. Many emprcal tests support the nverted U-shaped EKC, such as Grossman and Krueger(1991), anayotou (1993), Shafk(1994), Selde and Song(1994). In the mean tme, other emprcal tests found that the EKC s N-shaped n some pollutants (Selden and Song,1994; Grossman and Krueger,1995; anayoutou, ; Leb,3; Fredl and Getzner,3; L,7). Why the EKC shows nverted U-shaped n some pollutants and N-shaped n other pollutants smultaneously stll draws the attenton of economsts. There are two man drectons for envronmental economsts to amount for the EKC theoretcal background. Many researches are based on the abatement technque effect, whle that

4 of a few researches s based on the composton effect. Snce EKC exhbts the relatonshp between economc growth and polluton emsson, analyses naturally explan the EKC n the classc growth model framework by usng the steady state equlbrum and economc paths to show the relatonshp. Typcal ATE analyses regard the scale effect as brngng more polluton nto the economy, whereas ATE reduces polluton. Ths means that socal economc planners have to tradeoff between economc growth and envronmental expendture to maxmze the long term socal welfare. As one of the frst paper based on ATE, John and ecchenno(1994) setup a dynamc overlappng generaton model to explan the connecton between the economc growth and polluton emsson. In ther model, they obtan the optmal envronmental consumpton n compettve equlbrum, and fnd out that economy may have multple equlbra n steady state equlbrum: hgh polluton-hgh ncome and low polluton-hgh ncome. Though the multple equlbra phenomena explan the dfferent polluton level n dfferent economc growth level, the steady state equlbrum can t depct the essence of EKC exactly, because the EKC tell us a growth path of polluton wth the development of economy. Another lmtaton comes from the externalty of polluton n whch economy usually needs government actons to solve the externalty, but the analyss of the compettve equlbrum makes polluton s externalty neglected. To solve the problem, a successve paper by John and ecchenno (1995) manly ntroduced the government nto the model to treat wth the externalty of polluton. The paper as well as former one can not explan the EKC s dynamc characterstcs because all varables are analyzed n steady state equlbrum, n whch can not descrbe the exact shape of the EKC. The frst lterature descrbed the nverted U-shaped of EKC s Lopez (1994). Lopez explans the EKC not by steady state equlbrum but by the growth path. In the Lopez s model, envronment resources are supposed to be accurately prced by the market and are consdered as factors of producton n addton to labor nput and captal nput, frms maxmze proft and consumers maxmze utlty functons. Under condtons that preferences are non-homothetc and ncome elastcty of 3

5 envronmental nput s suffcent larger than one, t turns out that the relatonshp between the aggregate output and envronmental qualty depends on the comparson between the elastcty of substtuton between nputs (ESI) and consumer s coeffcent of relatve rsk averson (CRRA). When ESI s larger than the recprocal of CRRA, t corresponds to the up sde of EKC; on the contrary, t corresponds to the down sde of EKC. By the concluson, the nverted U-shaped EKC mples that CRRA should ncrease wth the economc growth. ractcally, the character of consumers for the normal commodty can t guarantee the nverted U-shaped EKC. Dnda (5) nherts Lopez(1994)presupposton and manly analyze the exstence of EKC by steady state equlbrum. Dnda (5) fnds that the polluton-output steady state s the saddle equlbrum, and the nverted U-shaped EKC actually shows the polluton growth path n dfferent phase space, whch mples that the economy wll not reach the steady state. Selden and Song (1995) s another nstructve paper focusng on ATE. They expand Forster (1973) model, n whch utlty s a functon of consumpton and polluton. Selden and Song (1995) thnk that envronmental qualty s not the factor of producton but a by-product: more products brng more polluton. In order to control polluton ncrease, the socal planer has to allocate the aggregate output to pay for envronmental expendture. The nstructve dea n the paper s that the authors analyze the EKC by usng polluton s growth path nstead of usng the steady state analyss. In order to gve the nverted U-shaped EKC a convenent descrpton, Selden and Song (1995) abandon economc ntal condtons of the Forster model, whch s nevtable for the model to get an nteror soluton. But f those ntal condtons are added nto the model, the EKC depcted by Selden and Song (1995) would be U-shaped. In addton, McConnell (1997) s paper s very smlar to Selden and Song s, whch cannot show the exact nverted U-shaped EKC by neglectng the ntal condtons. Egl and Steger (7) develop a dynamc model based on Andreon and Levnson (1) to nterpret the nverted U-shaped EKC by the ncreasng return to scale of ATE, though the assumpton of ncreasng return to scale n abatement technology s not so convncng. 4

6 Typcal composton-effect analyss s Lopez et al. (7). Accordng to Lopez model, they fnd that even though the economy does not have any technque effect on envronment, polluton wll automatcally decrease n the long-run growth. Gaeande et al. (1) s a paper focusng on how labor moblty affects the envronment qualty. In the model, they fnd that n spatal general equlbrum, there s an nverted U-shaped relatonshp between polluton and ncome. Though composton effect gves us an nterpretaton of EKC, many emprcal tests do not support the explanaton of composton effect. Stern (1996), Cole () fnd the consumpton structure n many developed countres does not change sgnfcantly, so the polluton problem n developed countres just transfers to developng countres by nternatonal trade and FDI. Ths means that polluton problem stll exsts n the large scope. What the EKC tells s the dynamc relatonshp between polluton and economc growth, and t s mpossble for steady states analyses to truly descrbe the dynamc relatonshp. Our paper, based on ATE, ntroduces fundamental economc ntal condtons nto neoclasscal growth model to nterpret the EKC. Thus, we nhert the Selden and Song (1995) analytcal method, explanng the dynamc relatonshp by path analyss, We fnd that the EKC s stll nverted U-shaped, even though envronmental technology exhbts constant return to scale, the J curve for envronmental expendture n Selden and Song (1995) s just part of our envronmental expendture curve that s S-shaped. Lke Dnda (5), our paper also fnds the polluton-output path s a saddle path, by whch we can explan the N-shaped EKC. The rest of paper s organzed as follows: a neoclasscal growth model s bult up on the bass of ATE n secton. We manly analyze the envronmental expendture path and fnd the envronmental expendture curve s S-shaped n secton 3. In secton 4, the nverted U-shaped EKC s dscussed. In secton 5, the N-shaped EKC s analyzed by polluton-output s saddle path. Secton 5 s the concluson.. Model There s a socal planner n a one-good closed economy wthout populaton 5

7 growth, and the socal planner should maxmze the representatve consumer s long term utlty as a functon of consumpton, C, and polluton, whch come nto beng by producton, consumpton and polluton are addtvely separable n the utlty functon. The planner has to allocate the aggregate output between consumpton and envronmental expendture for the consumer s utlty maxmzaton. Consumer s utlty, U(C, ), s concave, whch mples that: U >, U <, U <, U <.As Forster (1973) s defnton, polluton,, s an addtvely separable functon of captal stock, K, and envronmental expendture, E. KE (, ) s concave too, / K > / K > / E< / E >, and / E K =. To ensure the exstence of nteror soluton, we ntroduce economc ntal condtons nto model, LmU =, LmU =, c C C Lm = M <, Lm =. The socal planner E E E E maxmzes the consumer s long term utlty functon as follow: t W = e U( C, ( K, E), t) dt < ρ < 1 (1) ρ where ρ s rate of tme preference. The transton of captal stock s gven by C K = Y( K) C δ K E < δ < 1 () where δ s dscount rate of captal, Y( K) s the aggregate output of the socal whch s the functon of captal, K. C s the level of consumpton and E s the amount of envronmental expendture. Besdes the captal transton functon, the model has a polluton transton functon. There are three forces nfluencng the polluton transton: aggregate output, envronmental expendture and polluton per se, whch s shown as follow: = γy β αe γ >, α >, < β < 1 (3) The frst force n polluton transton functon tells us more aggregate output brngs more polluton, but margnal polluton ncrement decreases, ths force n the model can be thought as the scale effect defned n Grossman and Krueger (1991).. The second force to polluton change, envronmental expendture, s assumed to be constant returns to scale, so envronmental expendture s a lnear functon of 6

8 polluton change, α E s decrement of polluton due to envronmental expendture, for smplcty, we assume α = 1. The thrd force s polluton tself, As Forster (198), the stock of polluton s subject to exponental decay β, and thus we can wrte the polluton tself s also a lnear functon of polluton change. As above, the soluton of the problem n equaton (1) can be found by maxmzng a current value Hamlton functon. Stage 1, polluton can be absorbed completely by nature. Because of the absence of polluton, the center planner does not need to spend envronmental expendtures, whch means that =. At that stage, there s no polluton n economy, whch only exsts the consumpton, captal stock and output s dynamcs, ths s tradtonal Ramsey economc growth model. Stage, the polluton can not be absorbed fully n natural and no envronment expendture. As lmtaton of polluton-absorpton capacty of nature, when polluton can not be fully absorbed n nature, the polluton decrease the representatve consumer s utlty. In ths case, because low level output produce less polluton, the consumer wll prefer to consume more and has to put up wth more polluton, center planner, at ths tme, also do not allocate the envronmental fund Then equaton (3) wll become: = ηy β < β < 1, η > (4) There s no envronmental expendture n ths stage, the polluton functon wll be = ( K). We can analyze ths stage n a one perod. Based on utlty functon s separaton, we know when U U = d dk c / /, the utlty reach ts top. when polluton begns to affect the consumer, U s a small negatve value, whereas U c s relatvely large, therefore, U / U c s a great value, but at ths tme, due to polluton just emergng, d / dk s a very small value, whch means U / U > d / dk. At ths stage, the consumers would rather put up wth the c ncreasng polluton for ncreasng hs consumpton, and t s not necessary for the 7

9 central planner to pay the envronmental spendng. When the economy contnues to grow and contnuously declnes, U and d / dk ncrease, whch makes U c U / U = d / dk.when economy goes further, c U / U < d/ dk, the consumer c can no longer tolerate ncreasng polluton at the cost of relatve consumpton, so the center planner should start to allocaton a certan amount of envronmental spendng from aggregate output. Let K makes U / U = d / dk, when K < K, the center planner does c not allocate the envronmental expendture. The Hamlton operator wll be K H = U[ C, ( K)] + μ [ Y( K) C δk] + μ [ ηy( K) β( K)] (5) FOC: U c 1 1 = μ (6) H μ = = U μβ (7) H μ1 = = U + μ1[ Y( K) δ] + μ[ ηy ( K) β ] K K K (8) Transversalty condtons are μ ( ) = and μ ( ) =. 1 Stage 3, the optmal envronment expendture When the economy contnues to grow, makng the captal stock K larger than. Because of excessve polluton makes the consumer can not tolerate t, so the planner s supposed to spend envronmental expendture, snce the ntroducton of envronmental spendng, then the polluton dynamcs wll be equaton(3), the Hamlton Operator wll be as follow: H = U( C, ) + λ [ Y( K) C E δk] + λ [ γy β E] (9) 1 such that the followng FOC condtons should hold: U c = λ 1 (1) ( U = λ1+ λ β + 1) E E (11) H λ1 = = U + λ1[ Y( K) δ] + λ[ γy ( K) β ] K K K (1) 8

10 H λ = = U λβ Transversalty condtons are: (13) λ ( ) = (14) λ ( ) = (15) 1 The optmal consumpton and envronmental expendture chosen by the socal planner should satsfy the equatons of (1)-(15). In practce, the real consumpton and envronmental expendture may dverge from the optmal ones; however, we can thnk the departure s approxmate to the optmal one. In followng analyses, we assume that envronmental expendture, polluton and consumpton move along ther optmal tme paths. 3. The Growth ath of Envronmental Expendture Envronmental expendture s used to control the polluton wthn certan lmtaton. In a dynamc choce, envronmental expendture also has ts own optmal growth path. Combnng equaton (1) and (11), we can yeld: U / E λ λ ( β / E+ 1) = (16) 1 Based on equaton (16), we can easly get the path of envronmental expendtures. Takng dervatve of equaton (16) wth respect to tme t gves U( E) + U ( ) E+ U K = λ1+ λ ( β + 1) + λ ( β E) E E E K E E (17) In order to judge envronmental expendture, we need know the sgn of λ. We frstly suppose f λ >, then λ >, ths result s nconsstent wth the transversalty condton (15). Thus f and only fλ <, can be larger than, whch s consstent λ wth the transversalty condton (15). Rearrangng equaton (17),we get the envronmental expendture s dynamc growth path as below 1 E D N = (18) where D= U( / E) + ( U λβ )( / E ), N UcC λ ( β 1) U( )( = + + ) K. E E K From equaton (18), we can see the denomnator Uc C s negatve, ( )( U ) K E K 9

11 and λ d E are postve. uttng the equaton () nto equaton (18), we can get dk <, ths means E s concave functon on K. Because Y s convex functon on K, that dky ( ) d E s >, we can get dy dy <, by whch we know the change of envronmental expendture ( E ) has an up and down relatonshp wth the growth of economy. In the ntal stage of economc growth, a very few of aggregate output are allocated to envronmental expendture; Accordng to t, Lm E E = U ( / E ) s domnant n denomnator whch makes E s approxmated to zero. Consequently, wth the growth of envronmental expendture, the nfluence of and U ( / E ) n denomnator wll decrease and polluton level wll goes up, the reason s K >. In ths stage, U ( )( ) wll be a domnant one n the E K envronmental expendture transton, n whch the envronmental expendture wll ncrease contnuously. When economy keeps on gong and convergng to steady state (SSE), K, C and λ wll reach zero, and makes E. We can show ths process as Fg 1. The Fg 1 depcts the relatonshp of envronmental expendture wth the economc growth. When economy s n stage I shown n the zone of, the aggregate output and the envronmental expendture are both lttle, but the envronmental expendture wll sharply ncrease wth economc growth. When the economy reaches Y, the ncrement of envronmental expendture wll reach the top and the slope of envronmental expendture curve reaches maxmum. The envronmental expendture curve n stage I s the same as the envronmental expendture J curve of Selden and Song (1995). And economy goes nto stage II portrayed n the zone Y of Y Y n Fg 1, the ncrement of envronmental expendture wll drop and the envronmental expendture begns to slowly ncrease. 1

12 When economy grows to the pont ofy, the ncrement of envronmental expendture wll be zero, ths means that the economy enters nto the steady state, and n ths pont ofy, the envronmental expendture reaches ts maxmum. When economy enters nto stage III, the envronmental expendture wll not change any more. E E T (a) I II III O y y SSE y E E SSE E T I II III (b) O y y y Fg 1 Dagram of the optmal envronmental expendture movement 4. The Inverted U-Shaped EKC The EKC s a dynamc relatonshp between polluton and aggregate output; we can get the path of polluton by usng the path of E and K, and make use of the equaton (3) to get the dynamc relatonshp of EKC. Usng the condton = ( / K) K+ ( / E) E,polluton growth path wll be: β= γy ( / K) K ( / E) E E (19) 11

13 We have known that E s concave functon on Y and / E <, so we can get < Y, by whch we can get the nverted U-shaped EKC. The EKC s an up and down curve. In order to analyze the SE and ATE n EKC, we put = ( / K) K+ ( / E) E nto (13) and get the path of polluton. = ( / K) K + ( / E) D 1 N () From equaton (15), we can fnd an obvous dfference between the polluton path and envronmental expendture path, n the polluton path, SE not only nfluence the change of polluton but also ATE works on the polluton ( / E and / E, standng for ATE, affect the path of polluton). There are rough four stages to account for the growth of economy and polluton. When economy s n the ntal stage(stage I), government begns to allocate envronmental expendture nto economy, a lttle envronmental expendture wll reduce polluton sgnfcantly and make. Wth growth of economy, the envronmental expendture ncrease, whch makes ATE declne. On the other hand, the SE begns to force polluton level to ncrease (because / K s ncreasng). In ths stage defned by stage II, polluton wll ncrease. When economy enter nto stage III, / Ewll work much less than before and λ contnuously access to zero, whch makesu C+ U λβ< c. The ATE begns to domnate n ths stage. Economy goes ahead contnuously and enters nto steady state, whch s defned stage IV. Every varable does not change any longer n ths stage, whch means K C E and λ.all above analyses are shown n Fg 1

14 II O I y III y IV y Inverted U-shaped EKC T SSE I II III IV O y y y Fg Dagram of the optmal polluton movement The Fg depcts the relatonshp between polluton change and the economc growth. We can see that the relatonshp s nverted U-shaped. When economy s n ntal stage (stage I) shown n the pont of n Fg, the envronmental expendture also reduces rapdly the polluton brought n producton. The ATE domnate over the SE. When economy begns to grow and reaches the stage II (the zone of Y ), the ncrement of polluton wll ncrease remarkably wth the growth of economy, and the polluton level wll reach ts turn pont n ths stage, whch s shown by the T pont. In stage II, the SE domnates over ATE. When polluton passes through ts turn pont, the economy enters nto the stage III, the ATE wll be the predomnant effect and makes the polluton drop, whch s shown the rght sde of EKC. The economy contnuously 13

15 grows and enters nto the stage IV (steady state), the polluton and the aggregate output wll not change any more. 5. The Steady State Analyss and N-shaped EKC We have shown the paths of envronmental expendtures and the polluton and exstence of steady state equlbrum (SSE), moreover, the EKC s nverted U-shaped nterpreted by the path of polluton. In ths secton, we manly analyze the character of SSE, f the SSE of economy was a global equlbrum, the EKC, we nterprets, would be exactly nverted U-shaped; on the other hand, f the SSE of economy was a saddle equlbrum, the EKC may be N-shaped. The possble N-shaped EKC comes nto beng by shockng of some forces whch make the polluton path depart the SSE. Snce the SSE s saddle equlbrum, the departure wll move far away wth growth of economy. In order to analyze the SSE, we make lnear approxmaton of equaton () and (3): We frstly judge the SSE character of the captal stock and polluton. Takng dervatve of equaton () and (3) wth respect to K, and gves F K dk = = Y ( K) δ dk dk (1) F = = d () G K d = = γy ( K) β dk K (3) G d = = β d (4) Accordng to equatons (16)-(19), we can get the captal and polluton s steady state Jacob s matrx, whch s Y ( K) δ FK F K = = GK G γy ( K) β β K (5) Because of Jacob s determnant K <, a saddle equlbrum could be found n polluton-captal panel. We can also get the aggregate output transton equaton by puttng / K K = Y Y nto equaton (), the aggregate output transton equaton s Y = [ Y( K) C δ K E] Y K (6) Takng dervatve of equaton () and (1) wth respect to Y, and gves 14

16 F Y dy = = Y dy K (7) F dy = = d (8) G Y d = =γ dy (9) G d = = β d (3) matrx, As above, we can get the aggregate output and polluton s steady state Jacob s K FY F YK = = (31) GY G γ β Because of the Jacob s determnant K less than zero, path of polluton could converge at the pont of saddle equlbrum pont (SSE) n the polluton- aggregate output panel. The saddle equlbrum and polluton paths are drawn n Fg 3. III - = + SSE N-shaped EKC II IV Inverted U-shaped EKC I - + Y = Y Y Fg 3 hase dagram of the polluton From the secton of the nverted U-shaped EKC, we know that the polluton path s an up and down curve. When or after the nverted U-shaped EKC reaches the SSE, the polluton path may depart from the SSE for some reasons. The departed path has two phase spaces to go, one s n phase space I, the other s phase space IV. The departed polluton path n phase space I would return the SSE, whereas, the departed polluton path n phase space IV wll go far away and wouldn t back to SSE, whch makes the N-shaped EKC come nto beng. 15

17 6. Conclusons Our paper manly gve the nverted U-shaped and N-shaped EKC a theoretcal nterpretaton by usng neoclasscal growth model. Though ntal economc condtons have to be neglected for gettng the nverted U-shaped EKC n Selden and Song (1995), we fnd f those ntal economc condtons were added nto the neoclasscal growth model, the nverted U-shaped EKC can stll be brthed naturally. Unlke Egl and Steger (7), the ATE can stll be assumed constant returns to scale, and the nverted U-shaped EKC s also a reasonable path for polluton transton. The envronmental expendture path should reach SSE, whch makes the envronmental expendture J curve just part of the envronmental expendture path we get. Because SSE s a saddle equlbrum, some shocks make the polluton path depart the SSE and form the N-shaped EKC. Reference: Andreon, J., Levnson, A., 1. The smple analytcs of the Envronmental Kuznets Curve. Journal of ublc Economcs 8(), Barber, E., Introducton to the Envronmental Kuznets curve specal ssue. Envronment and Development Economcs (4), Beckerman, W., 199. Economc growth and the envronment: whose growth? Whose envronment? World Development, Brandoford, D.F., Schleckert, R., Shore, S.H.,. The Envronmental Kuznets Curve: explorng a fresh specfcaton. NBER.Workng paper 81. Brock, W.A. and M.S. Taylor, 4. The Green Solow Model, NBER Workng aper Seres, No Bulte, E.H., Van Soest, D.., 1. Envronmental degradaton n developng countres: households and the (reverse) envronmental Kuznets curve. Journal of Development Economcs 65, Cole, M.A., Ellott, R.J.R., Azhar, A.K.,. The determnants of trade n polluton ntensve ndustres: North-South evdence[j]. Unversty of Brmngham, UK. Mmeo. Cole, M.A., Rayner, A.J., Bates, J.M., The Envronmental Kuznets Curve: an emprcal analyss. Envronment and Development Economcs, Dasgupta, S., Laplante, B., Wang, H., Wheeler, D.,. Confrontng the Envronmental Kuznets Curve. Journal of Economc respectves 16(1), de Bruyn, S.M., Opschoor, J.B., Developments n the throughput-ncome relatonshp: theoretcal and emprcal observatons. Ecologcal Economcs, Dessus, S., Bussolo, M., Is there a trade-off between trade lberalzaton and polluton abatement? A computable general equlbrum assessment appled to Costa Rca. Journal of olcy Modelng (1), Dnda, S., 4. Envronmental Kuznets Curve Hypothess: A survey. Ecologcal Economcs 49, Dnda, S., 5. A theoretcal bass for the envronmental Kuznets curve. Ecologcal Economcs 53,

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