Positive and Negative Population Growth and Long - Run Trade Patterns: A Non - Scale Growth Model*

Transcription

1 43 The Interntionl Economy, Vol. 18, 2015 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model* Hiroki Sski Grdute School of Economics, Kyoto University Astrct This study uilds two - country, two - sector, non - scle growth model nd investigtes the reltionship etween trde ptterns nd the growth rte of per cpit rel consumption. Suppose tht oth countries hve different popultion growth rtes, tht t lest one country hs negtive popultion growth rte, nd tht the home country hs comprtive dvntge in mnufcturing t the initil point in time. Then, the following two ptterns re sustinle in the long run: (1) the home country diversifies (i.e., produces oth mnufctured nd griculturl goods), while the foreign country specilizes in griculture nd (2) the home country specilizes in griculture, while the foreign country diversifies. By compring the growth rte under utrky with tht under free trde, we find tht in the first cse, the foreign country increses its growth rte y engging in free trde, wheres the home country does so in the ltter cse. JEL Clssifiction: F10; F43; O11; O41 Key words: negtive popultion growth; trde ptterns; non - scle growth model 1. Introduction This study investigtes the reltionships etween positive s well s negtive popultion growth, trde ptterns, nd per cpit consumption growth. For this purpose, we present two - country, two - sector, two - fctor, non - scle growth model. In scle growth models, the long - run growth rte of per cpit income is proportionl to the scle of popultion, which seems to e counterfctul. On the contrry, non - scle growth models re chrcterized y long - run growth rte of per cpit income tht is proportionl to the growth rte of the popultion. Received 9 Decemer 2014, Accepted 19 April 2015, Relesed online in J - STAGE s dvnce puliction 20 June 2015 * I thnk Tkumi Nito nd the seminr prticipnts of the fourth Spring Meeting of the Jpn Society of Interntionl Economics held t Hosei University in 2014 s well s Noki Yoshihr nd the prticipnts of the 2014 Workshop of Anlyticl Politicl Economy in Jpn held t Hitotsushi University. I lso thnk n nonymous referee for useful suggestions, which improved the pper. For finncil support, I m grteful to Grnts - in - Aid for Scientific Reserch from the Jpn Society for the Promotion of Science (KAKENHI ). Grdute School of Economics, Kyoto University. Yoshid - Honmchi, Skyo - ku, Kyoto , Jpn. E - mil: sski@econ.kyoto - u.c.jp. Phone: (0)

2 44 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model The pioneering work on non - scle growth models y Jones (1995) removes the scle effects y presenting non - scle growth model in which the growth rte of output per cpit depends positively on the popultion growth rte rther thn on the size of the popultion. Tht is, the higher the popultion growth rte, the fster the country grows. 1) We use non - scle growth model in this study for two resons. First, doing so llows us to otin sustinle income per cpit growth, even when popultion growth is strictly positive. Second, we do not need to impose knife - edge conditions on the model prmeters. Previous studies hve nlyzed the reltionship etween trde ptterns nd growth. 2) Two good exmples re the works y Kneko (2003) nd Felermyr (2007). Kneko (2003) uilds two - country, two - sector, AK growth model nd endogenizes the terms of trde. 3) He finds tht if n utrkic country with lower growth rte thn its trde prtner hs comprtive dvntge in the consumption goods sector, then tht country cn nrrow or even reverse the growth gp y opening trde. Felermyr (2007) descries sitution in which cpitl - undnt north nd cpitl - scrce south trde with ech other. In his model, the trde pttern is endogenously determined, nd he nlyzes the sitution in which the north produces investment goods nd the south produces consumption goods. The production technology of investment goods is AK nd tht of consumption goods is decresing returns to scle. Along the lnced growth pth, the southern terms of trde re continuously improving such tht, even with decresing returns to scle, the south cn grow t the sme rte s the north. Therefore, the south cn eliminte the growth gp y opening trde. However, oth these uthors use scle growth models, which re suject to the forementioned prolems tht mke them somewht counterfctul. By contrst, Sski (2011, 2012) investigtes the reltionship etween trde ptterns nd growth rtes y using non - scle growth models. In prticulr, the model presented in Sski (2011) exmines the long - run trde pttern when oth countries popultion growth rtes re equl. Sski (2012) extends this model to include the cse in which the countries popultion growth rtes re different. 4) However, oth studies consider only positive popultion growth. To ridge this reserch gp, we consider cses in which popultion growth is oth negtive nd positive. Mny developed countries hve stgnnt popultion growth, nd even 1) For systemtic exposition of scle effects nd non - scle growth, see Jones (1999, 2005), Aghion nd Howitt (2005), nd Dinopoulos nd Sener (2007). For more sophisticted non - scle growth models, see Kortum (1997), Dinopoulos nd Thompson (1998), Peretto (1998), Segerstrom (1998), Young (1998), Howitt (1999), nd Dinopoulos nd Syropoulos (2007). 2) Wong nd Yip (1999), for exmple, present smll, open economy, two - sector model of endogenous growth including cpitl ccumultion nd lerning y doing to nlyze the reltionship etween economic growth, industriliztion, nd interntionl trde. 3) Kneko (2000) uilds growth model with humn cpitl ccumultion nd shows tht the reltionship etween the terms of trde nd growth depends on whether the country specilizes in the consumption goods sector or the investment goods sector. However, Kneko s (2000) study is sed on smll, open economy model, which mens the terms of trde re exogenous. 4) In this respect, Sski (2011) uilds non - scle growth, north - south economic development model nd shows tht oth countries grow t the sme rte long the lnced growth pth. However, their per cpit incomes grow t different rtes ecuse of the differences in popultion growth. Nevertheless, since the production pttern is fixed nd given exogenously in this model, we cnnot know whether the given trde pttern is sustinle over time.

3 45 H. Sski negtive growth rte in some cses. 5) However, existing economic growth theories typiclly ssume tht popultion growth is positive, principlly ecuse modeling negtive growth cn e complicted. 6) As Ferrr (2011) nd Christins (2011) show, incorporting negtive popultion growth into growth models is more complicted thn simply replcing positive vlue with negtive vlue. Specificlly, Christins (2011) demonstrtes the importnce of negtive popultion growth y using simple model. Consider Solow growth model with production function tht exhiits incresing, ut reltively smll returns to scle. 7) When the popultion growth rte is positive, per cpit income growth is positive nd incresing. By contrst, when the popultion growth rte is negtive, contrry to expecttions, per cpit income growth remins positive, ut is decresing. By using two - country model of interntionl trde, we investigte the reltionships etween the size of exogenously given popultion growth rtes, the sustinility of trde ptterns, nd the long - run growth rte of per cpit consumption in ech country, given tht specific trde pttern is sustinle. The min results re s follows. Suppose tht oth countries hve different popultion growth rtes, tht t lest one country hs negtive popultion growth rte, nd tht the home country hs comprtive dvntge in mnufcturing t n initil point of time. Then, the following two ptterns re sustinle in the long run: (1) the home country diversifies (i.e., it produces oth mnufctured nd griculturl goods), while the foreign country specilizes in griculture nd (2) the home country specilizes in griculture, while the foreign country diversifies. By compring the growth rte under utrky with tht under free trde, we find tht in the first (second) cse, the foreign (home) country increses its growth rte y engging in free trde. Moreover, suppose tht economic welfre is mesured y the level of per cpit rel consumption. Then, in country with positive popultion growth, long - run economic welfre is the sme under oth utrky nd free trde, while in country with negtive popultion growth, long - run economic welfre under free trde is higher thn tht under utrky. We mention the effect on economic growth of popultion ging, which leds to decline in popultion growth, nd even negtive growth in some cses. Nito nd Zho (2009) exmine how popultion ging ffects trde ptterns y formulting two - country, two - good, two - fctor, two - period - lived overlpping genertions model in which the two countries re identicl except for their exogenous rtes of popultion growth. In their model, good 1 is cpitl good tht is either invested or consumed, while good 2 is pure consumption good. Both goods re produced with constnt returns to scle production functions. They identify the ging (younger) country s the one with the lower (higher) exogenous rte of popultion growth nd find tht the low popultion growth ging country exports cpitl - intensive goods. In our model, s shown lter, the younger country with high popultion growth diversifies 5) For exmple, ccording to Jpn s Ministry of Internl Affirs nd Communictions, Jpn experienced its lrgest ever decline in popultion in Mrch ) Ritschel (1985) rgues tht in the stndrd Solow growth model, negtive svings rte is necessry for the existence of stedy - stte equilirium with negtive popultion growth. See lso Felderer (1988). 7) In the Solow model with constnt returns to scle production function, per cpit income growth is zero when popultion growth is positive, ut positive when popultion growth is negtive. For detils, see Christins (2011).

4 46 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model nd produces mnufctured goods, while the ging country with low popultion growth specilizes in griculture. Tht is, the younger country exports cpitl - intensive goods nd hence our result is contrry to tht of Nito nd Zho (2009). This difference lies in the specifiction of the production function of the mnufcturing (cpitl goods - producing) sector. While they use constnt returns to scle production function, we dopt n incresing returns to scle production function. This study investigtes possile trde ptterns in dynmic model. Here, we prticulrly mention the reltionship etween our model nd dynmic Heckscher - Ohlin models. Nishimur nd Shimomur (2002) incorporte sector - specific externlities into two - country, two - good (consumption nd investment goods), two - fctor (cpitl nd lor) dynmic Heckscher - Ohlin model. They show tht production functions exhiit decresing returns to scle t firm level ut constnt returns to scle t n industry level. Popultion growth is prohiited. Iws nd Nishimur (2014) lso incorporte sector - specific externlities into dynmic Heckscher - Ohlin model. Unlike Nishimur nd Shimomur (2002), however, they use consumption goods nd consumption - cum - investment goods. Further, lthough Doi, Nishimur, nd Shimomur (2007) incorporte sector - specific externlities, in contrst to the ove two studies, they use physicl cpitl nd humn cpitl, oth of which ccumulte over time. Accordingly, endogenous growth is otined. On the contrry, our study incorportes sector - specific externlity into dynmic Ricrdo - Viner model: the cpitl stock is used only in the mnufcturing sector. The production function of mnufcturing exhiits incresing returns to scle. Like Iws nd Nishimur (2014), we use consumption goods nd consumption - cum - investment goods. In ddition, like Doi, Nishimur, nd Shimomur (2007), we otin endogenous growth. However, in Doi, Nishimur, nd Shimomur s (2007) model, the reltive price (i.e., terms of trde) is constnt long the lnced growth pth, wheres in our model, it declines t constnt rte in the long run. The reminder of the pper is orgnized s follows. Section 2 presents our model nd determines the equilirium nd long - run growth rte of per cpit consumption under utrky. Section 3 investigtes the equilirium under free trde. Section 4 investigtes the long - run growth rte of per cpit consumption under free trde. In this section, we lso compre the growth rtes under free trde with those under utrky. Finlly, Section 5 concludes the pper. 2. The model Consider world tht contins two countries, termed Home nd Foreign herein. Both countries produce homogeneous mnufctured nd griculturl goods. The mnufctured good is used for oth consumption nd investment, wheres the griculturl good is used only for consumption. 2.1 Production Firms produce mnufctured goods, Xi M, with lor input, Li M, nd cpitl stock, Ki, nd produce griculturl goods, Xi A, with only lor input, Li A. Here, i =1 nd i =2 denote Home nd Foreign, respectively. Both countries hve the sme production functions, which

5 47 H. Sski re specified s follows: 8) Xi M = AiK i RLi M W 1-, where Ai = K i = K + i RLi M W 1-,0< < 1, 0 < < 1, + < 1, (2) Xi A = Li A. (3) Here, A i in eqution (1) represents n externlity ssocited with cpitl ccumultion, which cptures the lerning - y - doing effect introduced y Arrow (1962). By sustituting A i into eqution (1), we otin eqution (2), which shows tht mnufctured goods production hs incresing returns to scle, with corresponding to the extent of these incresing returns. Eqution (3) shows tht griculturl goods production hs constnt returns to scle. We further suppose tht lor supply is equl to the popultion nd thus tht the popultion is fully employed. Moreover, the popultion grows t constnt rte, n i, nd the initil popultion is unity in ech country: LiRt W= Li M Rt W+Li A Rt W= e nit, ni U 0. Note tht popultion growth cn e negtive. Let p i denote the price of mnufctured goods reltive to griculturl goods. Then, the profits of mnufcturing nd griculturl firms re given y ri M = pixi M -wili M -piriki nd ri A = Xi A -wili A, respectively, where w i denotes the wge pid to produce griculturl goods nd r i denotes the rentl rte of cpitl. From the profit - mximizing conditions, we otin the following reltions: (1) pi 2Xi M 2Li M = wi = 1, (4) 2Xi M = ri with Ai given. (5) 2Ki From eqution (4), we find tht the wge is unity s long s griculturl production is positive. We ssume Mrshllin externlity in deriving eqution (5); profit - mximizing firms regrd A i s exogenously given. Accordingly, firms do not internlize the effect of A i. 2.2 Consumption For simplifiction, we mke the clssicl ssumption tht wge income w i L i is entirely devoted to consumption, while cpitl income p i r i K i is entirely devoted to sving. 9) In the cnonicl one - sector Solow model, under the golden rule stedy stte in which per cpit consumption is mximized, consumption is equl to the totl rel wge nd totl cpitl income is sved nd invested. Hence, our ssumption hs some rtionlity nd cn e interpreted s 8) In this study, we need to find nlyticl solutions for the reltive price p nd mrginl rte of trnsformtion (MRT) of the production possiility frontier (PPF). For this purpose, we need to find nlyticl solutions for the endogenous vriles. Therefore, we use Co - Dougls production function. The sme holds for the utility function introduced lter. 9) The sme ssumption is used in Uzw (1961), which considers two - sector growth model, nd Krugmn (1981), which considers two - country, two - sector, North - South trde nd development model. If dynmic optimiztion is used, then the Euler eqution for consumption ppers nd the numer of differentil equtions increses, significntly complicting the nlysis. Therefore, n nlysis with dynmic optimiztion will e left for future reserch. In ddition, while consumption smoothing with dynmic optimiztion is stndrd tool in mcroeconomics, Mnkiw (2000) sttes tht, in relity, consumption ehvior devites from consumption smoothing.

6 48 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model simple rule of thum for consumers with dynmic optimiztion (Christins, 2008). We define rel consumption per cpit ci s ci = Ci/Li = RCi M W c RCi A W 1-c /Li, where C i denotes economy - wide rel consumption. In this cse, proportion, -c, of wge income is spent on Ci M nd the rest, 1-c, is spent on Ci A : pici M = cwili, (6) Ci A = R1-cWwiLi. (7) Moreover, the following reltionship etween rel investment, I i, nd sving holds: piii = piriki. From this eqution, we otin the rte of cpitl ccumultion: oki gki / Ki = ri. (8) Tht is, the rte of cpitl ccumultion is equl to the rentl rte of cpitl. A dot over vrile denotes the time derivtive of the vrile Re.g., o Ki / dki/dtw. 2.3 Equilirium under utrky nd per cpit consumption growth Under utrky, oth goods hve to e produced. The mrket - clering conditions re s follows: Xi M = Ci M +Ii nd Xi A = Ci A. Note tht wi = 1 under utrky. From the mrket - clering condition for mnufctured goods, we otin p i, which is used to derive ech sector s employment proportion: Li M /Li = c nd Li A /Li = 1-c. Therefore, under utrky, ech sector s employment proportion is constnt. Under utrky, the reltive price of mnufctured goods is given y RcLiW pi = R1- +. WKi (9) Equilirium with n i >0 First, we derive the lnced growth pth under utrky when the popultion growth rte is positive, tht is, ni > 0. Along the lnced growth pth, the rte of cpitl ccumultion is constnt nd equl to the rentl rte of cpitl, which is given in eqution (5) s ri = Ki +-1 RcLiW 1-. With ogki/gki = R+-1WgKi+ R1-Wni = 0, the lnced growth pth growth rtes of K i nd p i re, respectively, given y g * Ki = 1- ni > 0, (10) 1-- g * pi =- 1-- ni < 0, (11) where gx / ox/x denotes the growth rte of vrile x nd n sterisk denotes the lnced growth pth vlue. The rte of cpitl ccumultion is positive nd proportionl to popultion growth, while the reltive price of mnufctured goods is decresing t constnt rte. Consumption is defined s wges only nd hence the growth rte of per cpit rel consumption is equl to the growth rte of the rel wge: 10) gci = gwi-cg pi. (12) 10) In every cse in our model, the long - run growth rte of per cpit rel consumption is equl to tht of per cpit rel income. Therefore, we use the growth rte of per cpit rel consumption.

7 49 H. Sski Here, the rel wge is deflted y the consumer price index, p c i. 11) To otin gci, we must know gwi nd g pi. Note tht s long s griculturl goods re produced, the nominl wge is equl to unity, tht is, wi = 1, which mens tht gwi = 0. Accordingly, we otin the growth rte of per cpit consumption under utrky s follows: AT c gci = 1-- ni > 0, (13) where AT denotes utrky. Therefore, AT gci is incresing in n i. By considering the lnced growth pth growth rte of the cpitl stock, we introduce new vrile, nmely the scle - djusted cpitl stock, ki / Ki/L z i, where 1- z / > The dynmics of the scle - djusted cpitl stock re given y oki = c 1- k i + -zniki. (14) In the stedy stte, o ki = 0, from which we otin c ki * = 1-1 U Z 1--. (15) zni The stedy stte is stle ecuse d o ki/ki;ki=ki * =-ki * " R1--Wc 1- Rki * W +-2 +zni%< Equilirium with ni < 0 Next, we derive the long - run equilirium under utrky when the popultion growth rte is negtive, tht is, ni < 0. If ni < 0, from eqution (14), there never exists ki > 0 such tht oki = 0, nd we hve o ki > 0for ki > 0. Tht is, if the initil vlue of the cpitl stock is strictly greter thn zero, kir0 W> 0, then ki diverges to infinity. The growth rte of the scle - djusted cpitl stock is given y gki = c 1- ki +-1 -zni. (16) When ki "+3, from eqution (16), we hve lim gki =-zni > 0 (17) ki"+3 ecuse we ssume tht +-1 < 0. In this cse, the long - run growth rtes of the cpitl stock nd the reltive price of mnufctured goods re respectively given y g Ki g pi * = 0, (18) * = ni < 0. (19) We next exmine the long - run growth rte of per cpit consumption. Considering tht per cpit consumption is equl to the rel wge mesured in terms of the consumer price index, we hve AT gci =-cg pi =-cni > 0. (20) 11) Let p c denote the consumer price index tht is consistent with the expenditure minimiztion prolem of consumers. Then, pc = c -c R1-cW -R1-cW p c i, nd y definition, pcci = wi. Strictly speking, the consumer price index is given y c -c R1-cW -R1-cW p c i. However, we use p c i ecuse the constnt terms hve no effect on the results.

8 50 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model Therefore, AT gci is positive, even when ni < 0 nd is decresing in n i. 3. Equilirium under free trde In the following nlysis, we suppose tht t lest one country hs negtive popultion growth rte. 12) Accordingly, we consider the following five cses: n1 = n2 < 0 (Cse 1), n2 < 0 < n1 (Cse 2), n1 < 0 < n2 (Cse 3), n2 < n1 < 0 (Cse 4), nd n1 < n2 < 0 (Cse 5). Suppose tht Home nd Foreign engge in free trde t time zero. If K1R 0W> K2R0 W, then from eqution (9), p1r 0W< p2r0 W ecuse L1R 0W= L2R0 W= 1. Thus, if K1R 0W> K2R0 W, Home hs n initil comprtive dvntge (ICA) in mnufctured goods nd Foreign hs n ICA in griculturl goods. In the following nlysis, we ssume tht K1R 0W> K2R0 W, without loss of generlity. In this cse, we need to consider the following four trde ptterns: Pttern 1: Both countries produce oth goods; tht is, oth countries diversify. Pttern 2: Home diversifies nd Foreign completely specilizes in griculture. Pttern 3: Home completely specilizes in mnufcturing nd Foreign completely specilizes in griculture. Pttern 4: Home completely specilizes in mnufcturing nd Foreign diversifies. Whether Home hs n ICA in mnufcturing is unrelted to the rte of popultion growth, nd ccordingly we hve to consider ll cses, tht is, Cses 1, 2, 3, 4, nd 5 with regrd to popultion growth. Note tht even if Ptterns 1, 2, 3, nd 4 re possile t t = 0, it is not self - evident tht these trde ptterns re sustinle in the long run. Thus, we need to investigte the sustinility of these trde ptterns in detil. However, if neither country engges in free trde t time t = 0 ut oth do so fter some time psses, other trde ptterns re possile. 13) For exmple, if n2 < n1 < 0 (Cse 4), s shown elow, then we hve p1rt1 W= p2rt1 W t time t 1 nd p1r t W> p2r t W for t drt1,+3w ecuse ; g p1 ;<; g p2 ; even if p1r 0W< p2r0 W t t = 0. This mens tht Foreign hs dynmic comprtive dvntge (DCA) (Redding, 1999) in mnufcturing from t 1 onwrd when ; g p1 ;<; g p2 ; even if Foreign hs n ICA in griculture t t = 0. Accordingly, suppose tht Home nd Foreign engge in free trde t time t 1. Then, we need to consider the following three trde ptterns if ; g p1 ;<; g p2 ; under utrky. Pttern 5: Home diversifies nd Foreign completely specilizes in mnufcturing. Pttern 6: Home completely specilizes in griculture nd Foreign diversifies. Pttern 7: Home completely specilizes in griculture nd Foreign completely specilizes in mnufcturing. Figure 1 explins the difference etween n ICA nd DCA. The downwrd sloping curves show the time pths of p1r t W nd p2r t W under utrky. Here, we ssume tht 12) Sski (2012) exmines cses in which oth countries popultion growth rtes re positive. Moreover, if oth countries hve the sme positive popultion growth rte, we cn investigte the trnsitionl dynmics in detil y using phse digrms. For this, see Sski (2011). 13) The uthor is indeted to Tkumi Nito for pointing out these other trde ptterns.

9 51 H. Sski Figure 1: Time pths of the reltive price of mnufctured goods under utrky when g p1 < g p2 ; g p1 ;<; g p2 ; holds. At time t = 0, since K1R 0W> K2R0 W, nd hence p1r 0W< p2r0 W, Home hs n ICA in mnufcturing. If oth countries engge in free trde t t = 0, Ptterns 1, 2, 3, nd 4 re possile. However, if ; g p1 ;<; g p2 ;, then from t 1 onwrd, we hve p1r t W> p2r t W nd hence Foreign hs DCA in mnufcturing. If oth countries egin to trde t t =t 1, Ptterns 5, 6, nd 7 re possile. Here, we consider the conditions under which Foreign hs DCA in mnufcturing. When n1 = n2 < 0 (Cse 1), we hve g p1 = n1 = g p2 = n2. In this cse, we do not hve p1r t W> p2r t W for t > t1 nd hence Foreign cnnot hve DCA in mnufcturing. When n2 < 0 < n1 (Cse 2), we hve g p1 =- 1-- n1 < 0 nd g p2 = n2 < 0. In this cse, for Foreign to hve DCA in mnufcturing, we need ; g p1 ;-; g p2 ;= 1-- n1+n2 < 0. (21) This condition is fesile under the restriction n2 < 0 < n1. When n1 < 0 < n2 (Cse 3), we hve gp1 = n1 < 0 nd g p2 = 1-- n2 < 0. In this cse, for Foreign to hve DCA in mnufcturing, we need ; g p1 ;-; g p2 ;=-n1-1-- n2 < 0, n n2 > 0. (22) This condition is fesile under the restriction n1 < 0 < n2. When n1 < 0 <nd n2 < 0 (Cses 4 nd 5), we hve gp1 = n1 < 0 nd g p2 = n2 < 0. these cses, for Foreign to hve DCA in mnufcturing, we need ; g p1 ;-; g p2 ;= Rn2-n1W< 0. (23) From this, we need n2 < n1 < 0 for Foreign to hve DCA in mnufcturing. Accordingly, Foreign hs DCA in mnufcturing in Cse 4 ut not in Cse 5. Therefore, given tht Foreign hs DCA in mnufcturing, we focus on Cses 2, 3, nd 4 In

10 52 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model with regrd to the popultion growth rtes. As in ICA cses, even if Ptterns 5, 6, nd 7 re possile t t = t1, it is not self - evident whether these trde ptterns re possile in the long run. Hence, detiled nlysis is necessry. 3.1 Equilirium when oth countries diversify (ICA): Pttern 1 The mrket - clering conditions for oth goods re given y X 1 M +X 2 M = C 1 M +C 2 M +I1+I2, (24) X 1 A +X 2 A = C 1 A +C 2 A. (25) From these, we otin p = R1- " crl1+l2w%. (26) S W K K2 X Ech country s employment proportion of the mnufcturing sector, ii M, is given y i M M L 1 / 1 = L1 L2 cs1+ L1 X + K2 1+ S K1 X, i M M L 2 / 2 = L2 L1 cs1+ L2 X + K1 1+ S K2 X The rtes of cpitl ccumultion in oth countries re given y. (27) gk1 = K Ri M 1 L1W 1-, gk2 = K Ri M 2 L2W 1-. (28) First, if n1 = n2, so tht L1 = L2, 14) then, fter enough time hs pssed, we otin limt"+3 i 1M = 2c nd limt"+3 i 2 M = 0, where c < 1/2 is needed. Then, the mnufcturing employment proportion in Foreign pproches zero, nd Foreign symptoticlly completely specilizes in griculture. 15) Hence, Pttern 1 is not sustinle when n1 = n2. Second, if n1 > n2, then, fter enough time hs pssed, we otin limt"+3 i 1 M = c nd limt"+3 i 2 M = 0. In this cse, the mnufcturing employment proportion in Foreign lso pproches zero, nd Foreign symptoticlly completely specilizes in griculture. Hence, Pttern 1 is unsustinle when n1 > n2, too. Third, if n1 < n2, then, fter enough time hs pssed, we otin limt"+3 i 1M =+3 nd limt"+3 i 2 M = 0. In this cse, the mnufcturing employment proportion in Home exceeds unity, the mnufcturing employment proportion in Foreign pproches zero, nd Foreign symptoticlly completely specilizes in griculture. Hence, Pttern 1 is lso unsustinle when n1 < n2. By summrizing the ove discussions, we otin the following result. Result 1. Pttern 1, in which oth Home nd Foreign diversify, is unsustinle in the long 14) In ddition, if K1R0 W= K2R0 W, the mnufcturing employment proportion in ech country is given y i M i = c, which is constnt. Pttern 1 is only sustinle in this cse. However, the reltive prices in oth countries under utrky re equl nd therefore trde does not occur. 15) In our model, griculturl output pproches zero, ut it never disppers ecuse we ssume tht Foreign s cpitl stock is strictly positive. Therefore, the phrse symptoticlly completely specilizes in griculture is more pproprite. For more informtion, see Christins (2008).

11 53 H. Sski run in every cse. 3.2 Equilirium when Home diversifies nd Foreign specilizes in griculture (ICA): Pttern 2 The mrket - clering conditions for oth goods re given y X 1M = C M 1 +C M 2 +I1, (29) X A 1 +X A 2 = C A 1 +C A 2. (30) Hence, we otin " crl1+l2w% p = R1- +. WK1 The mnufcturing employment proportion in Home is given y (31) i M L2 1 = cs1+ X. L1 (32) First, if n1 = n2, the mnufcturing employment proportion in Home ecomes i M 1 = 2c. In this cse, we need c < 1/2 for Pttern 2 to hold. Second, if n1 > n2, we otin limt"+3 i M 1 = c. Here, the mnufcturing employment proportion of Home converges to c M nd thus Pttern 2 is sustinle. Third, if n1 < n2, then i 1 continues to increse, ecomes more thn unity, nd pproches infinity: limt"+3 i 1M =+3. In this cse, Pttern 2 is gin unsustinle. The growth rte of the cpitl stock is given y (33) gk1 = c 1- RL1+L2W 1- K We now exmine in detil the conditions under which Pttern 2 holds. Following Wong nd Yip (1999), we investigte whether the trde pttern is sustinle y compring the size of the terms of trde with the size of the MRT of the PPF t the corner point where country completely specilizes in mnufcturing (see Figure 2). In our model, the production function of the mnufcturing sector exhiits incresing returns to scle ut decresing returns to scle with respect to Li M. Accordingly, the PPF is concve towrd the origin, s shown in Figure 2. 16) The PPF t some point in time is the sme s tht of usul trde models. However, popultion growth nd cpitl ccumultion expnd or shrink the PPF s well s chnge its slope. Hence, trde pttern tht is possile t some point in time would e unsustinle in the long run nd dditionl conditions would e required. By using equtions (2) nd (3) with Li M +Li A = Li, we otin the size of the MRT of the PPF in country i s follows: dxi - A M = dxi K + i ii M " Li %1- R W 1- + R1-WK 1- i. By sustituting ii M = 1 into eqution (34), the MRT t the point where country i completely specilizes in mnufcturing is given y (34) 16) A similr PPF is otined in Wong nd Yip (1999), who use n incresing returns to scle production function in mnufcturing nd liner production function with respect to lor in griculture.

12 54 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model Figure 2: PPFs in Home nd Foreign L i r i = R1- WKi +. For Pttern 2 to e sustinle over time, we need oth p < r 1 nd p < r 2, which re given y " crl1+l2w% + < R1-WK 1 R1- L 1 WK1 + nd " crl1+l2w% + < R1-WK 1 R1- By considering the sizes of the popultion growth rtes nd investigting whether Pttern 2 is sustinle in the long run nd wht ech growth rte will e, we otin the following two results. 17) Result 2. Pttern 2 (Home diversifies, while Foreign symptoticlly completely specilizes in griculture) is sustinle in the long run in Cses 1 nd 2. Result 3. If Pttern 2 is sustinle in the long run, oth countries grow t the sme per cpit rte under free trde irrespective of whether their popultion growth is positive or negtive. The long - run growth rtes in Cse 1 re given y L 2 WK2 +. (35) (36) gk1 = 1- n1 > 0, 1-- g p =- 1-- n1 < 0, FT FT c gc1 = gc2 = 1-- n1 > 0. (37) (38) (39) 17) For the detil of the nlysis, see Appendix A.1.

13 55 H. Sski Note tht Cse 1 requires c # 1/2. Accordingly, oth countries growth rtes re incresing in n 1. The long - run growth rtes in Cse 2 re given y gk1 = gk1-zn1 = 0, g p = n1 < 0, FT FT = gc2 =-cn1 > 0. gc1 (40) (41) (42) Note tht Cse 2 requires 1-- re decresing in n 1. n1+n2 > 0. Accordingly, oth countries growth rtes 3.3 Equilirium when Home specilizes in mnufcturing nd Foreign specilizes in griculture (ICA): Pttern 3 The mrket - clering conditions for oth goods re given y X 1M = C 1 M +C 2 M +I1, X 2 A = C 1 A +C 2 A. With L 1 M = L1, ecuse of complete speciliztion in mnufcturing, we otin (43) (44) cl2 p = R1-WR1-c WK1 L 1 (45) The growth rte of the cpitl stock is given y gk1 = K L 1-1. (46) As with the cse of Pttern 2, y compring the terms of trde with the MRT of the PPF t the corner point where country completely specilizes in mnufcturing, we investigte whether Pttern 3 is sustinle in the long run. For Pttern 3 to e sustinle, we need r 1 < p < r 2, which is given y R1- L 1 + WK 1 < R1-W 1-c cl2 R + 1- WK1 L 1 < R1- L 2 + WK2 By exmining these conditions, we otin the following two results. 18). (47) Result 4. Pttern 3 (Home completely specilizes in mnufcturing, while Foreign symptoticlly completely specilizes in griculture) is sustinle in the long run in Cse 1. Result 5. If Pttern 3 is sustinle in the long run, oth countries grow t the sme per cpit rte under free trde. The long - run growth rtes re given y g p = n1 < 0, (48) FT =-cn1 > 0, (49) gc1 18) For the detil of the nlysis, see Appendix A.2.

14 56 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model FT gc2 =-cn1 > 0. (50) Note tht Cse 1 requires 1/2 < c. 3.4 Equilirium when Home specilizes in mnufcturing nd Foreign diversifies (ICA): Pttern 4 The mrket - clering conditions for oth goods re given y X 1 M +X 2M = C 1 M +C 2 M +I1+I2, (51) X 2 A = C 1 A +C 2 A. (52) From equtions (51) nd (52), we find tht the terms of trde stisfy the following eqution: R1-W p K2 = cl2-r1-wr1-cwpk + 1 L 1-1. (53) Here, p is implicitly nd uniquely determined from eqution (53) nd hence p is function of K 1, K 2, L 1, nd L2: p = prk1, K2, L1, L2W. 19) The growth rte of the cpitl stock in ech country is given y gk1 = K L , gk2 = R1-W p K2, where p is endogenously determined from eqution (53). The mnufcturing employment proportion in Foreign is given y (54) i 2 M = R1-W p K2 L2. In this cse, since nlyticl solutions re difficult to otin, we conduct numericl simultions. 20) The numericl simultion implies tht regrdless of whether n1 W n2, the mnufcturing employment proportion in Foreign tends to zero in finite time; tht is, i M 2 " 0. 21) Therefore, Pttern 4 is unsustinle in the long run. Result 6. Pttern 4 (Home completely specilizes in mnufcturing, while Foreign diversifies) is unsustinle in the long run in every cse. 3.5 Equilirium when Home diversifies nd Foreign specilizes in mnufcturing (DCA): Pttern 5 Pttern 5 is symmetricl to Pttern 4. Tht is, to otin Pttern 5, we interchnge suscripts 1 nd 2 in Pttern 4 to 2 nd 1 in Pttern 5, respectively. 22) (55) Result 7. Pttern 5 (Home diversifies nd Foreign specilizes in mnufcturing (DCA)) is 19) The left - hnd side of eqution (53) is n incresing function of p, wheres the right - hnd side is decresing function of p for the given vlues of K 1, K 2, L 1, nd L 2. By plotting oth functions, we find tht their intersection is unique nd tht it provides n instntneous equilirium vlue of p. 20) We use the following vlues for the prmeters nd initil cpitl stocks: = 0.3, = 0.2, c = 0.6, K1R0 W= 1.2, nd K2R0 W= 1. For the detil of the nlysis, see Appendix A.3. 21) For numericl simultions, see the Appendix, which is ville on request. 22) For the detil of the nlysis, see Appendix A.4.

15 57 H. Sski unsustinle in the long run in every cse. 3.6 Equilirium when Home specilizes in griculture nd Foreign diversifies (DCA): Pttern 6 Pttern 6 is symmetricl to Pttern 2. To otin Pttern 6, we interchnge suscripts 1 nd 2 in Pttern 2 to 2 nd 1 in Pttern 6, respectively. For Pttern 6 to e sustinle, we need n1 < n2. Thus, Cses 3 nd 5 re possile. However, Cse 5 contrdicts the condition tht Foreign hs DCA in mnufcturing nd hence we consider only Cse 3. 23) Result 8. Pttern 6 (Home specilizes in griculture nd Foreign diversifies (DCA)) is sustinle in the long run in Cse 3. Result 9. If Pttern 6 is sustinle, then oth countries grow t the sme per cpit rte. The long - run growth rtes re given y gk2 = 1- n2 > 0, 1-- (56) g p =- 1-- n 2 < 0, (57) FT FT c gc1 = gc2 = n > 0. (58) Note tht the following condition is dditionlly needed for Foreign to hve DCA in mnufcturing: n1+ n (59) > Equilirium when Home specilizes in griculture nd Foreign specilizes in mnufcturing (DCA): Pttern 7 Pttern 7 is symmetricl to Pttern 3. To otin Pttern 7, we interchnge suscripts 1 nd n2 < n1. Thus, Cses 2 nd 4 re possile. 24) However, y exmining the long - run sustinility conditions r 2 < p < r 1, we find tht Cses 2 nd 4 re oth unsustinle in the long run. 25) Result 10. Pttern 7 (Home specilizes in griculture nd Foreign specilizes in mnufcturing (DCA)) is unsustinle in the long run in every cse. 3.8 Summry By summrizing the ove results, we find tht the comintions of long - run sustinle trde ptterns nd popultion growth rtes re s follows: 23) For the detil of the nlysis, see Appendix A.5. 24) In Cse 2, the condition 1-- n1+n2 < 0 is lso necessry. 25) For the detil of the nlysis, see Appendix A.6.

16 58 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model 1. Pttern 2 nd Cses 1 nd 2. Home hs n ICA in mnufcturing. 2. Pttern 3 nd Cse 1. Home hs n ICA in mnufcturing. 3. Pttern 6 nd Cse 3. Foreign hs DCA in mnufcturing. If oth countries popultion growth rtes re negtive, long - run sustinle trde ptterns never exist unless their popultion growth rtes re the sme. If we consider generl cse in which oth countries popultion growth rtes re different, long - run sustinle trde ptterns exist only if one country tht ccumultes cpitl stock hs positive popultion growth rte nd the other country tht does not ccumulte cpitl stock hs negtive one. In our model, the difference in popultion growth etween the two countries considerly ffects the results for the following reson. Popultion growth determines the cpitl ccumultion rte, which in turn determines the rtes of chnge in the reltive price nd MRT of the PPF. According to the size of the popultion growth rtes, the reltionship etween the reltive price nd MRT determines the long - run trde pttern. Next, we provide n intuitive explntion s to whether ech trde pttern is sustinle in the long run. Pttern 1 is unsustinle in the long run in every cse for the following reson: for this trde pttern to lst, the employment shre of mnufcturing must rnge etween 0 nd 1. However, when the popultion growth rtes etween Home nd Foreign re different, the employment shre of mnufcturing of Foreign will e 0 within finite time nd hence Pttern 1 cnnot lst. Moreover, even if the popultion growth rtes re the sme, s long s the initil cpitl stocks re different, the employment shre of mnufcturing of Foreign will e zero within finite time. Accordingly, Pttern 1 is unsustinle in the long run. Pttern 2 is unsustinle in the long run when n1 < n2 ecuse demnd for mnufctured goods in Foreign grows fster thn supply from Home. As result, Home cnnot meet world demnd for mnufctured goods on its own. Pttern 3 is unsustinle in the long run when n1 > n2 ecuse demnd for griculturl goods in Home grows fster thn they cn e supplied y Foreign. Hence, Foreign cnnot meet world demnd for griculturl goods on its own. Pttern 4 is lso unsustinle in the long run in every cse. In this trde pttern, mnufctured goods re produced in oth countries, while griculturl goods re produced only in Foreign. When n1 $ n2, demnd for griculturl goods in Home grows fster thn the supply of griculturl goods in Foreign. Hence, Foreign cnnot meet world demnd for griculturl goods. Even when n1 < n2, the growth rte of world demnd for griculturl goods exceeds the growth rte of the supply of griculturl goods. Hence, Foreign cnnot meet world demnd for griculturl goods. Therefore, Pttern 4 is unsustinle in the long run. Finlly, ecuse Pttern 5 is symmetric to Pttern 4, Pttern 6 is symmetric to Pttern 2, nd Pttern 7 is symmetric to Pttern 3, similr rguments hold. 4. Comprisons of the growth rtes nd welfre under utrky nd free trde From the ove nlysis, we find tht Ptterns 2, 3, nd 6 re sustinle. In this section, we compre the growth rtes nd welfre under free trde with those under utrky. To egin with, with regrd to the growth rtes, we otin the following proposition: 26) 26) For the detil of the nlysis, see Appendix A.7.

17 59 H. Sski Proposition 1. Suppose tht oth countries popultion growth rtes re different, tht t lest one country hs negtive popultion growth rte, nd tht Home hs n ICA in mnufcturing. Then, the following two out of the seven possile trde ptterns discussed herein re sustinle in the long run: (1) Home diversifies, while Foreign specilizes in griculture nd (2) Home specilizes in griculture, while Foreign diversifies. By compring the growth rtes under free trde with those under utrky, we find tht in cse (1), Foreign s growth rte increses y opening trde t t = 0, while in cse (2), Home s growth rte increses y opening trde t time t = t1 > 0. Next, we compre welfre under utrky with tht under free trde. In our model, per cpit rel consumption is equl to the rel wge rte. Under utrky nd Ptterns 2 nd 6 (i.e., tht re sustinle), oth countries necessrily produce griculturl goods nd hence the nominl wge rte is given y wat w AT 1 = w AT 2 = w FT 1 = w FT 2 = 1. Accordingly, rel per cpit consumption under utrky nd tht under free trde re given y ci AT = 1/ Rpi AT W c nd ci FT = 1/ Rp FT W c, respectively. From this, we cn compre welfre under utrky with tht under free trde y compring the price level under utrky with tht under free trde. Note tht under free trde, oth countries price levels re equlized; consequently, we otin c FT FT 1 = c 2. Hence, under free trde, the welfre levels of oth countries re equlized. We investigte sitution in which Pttern 2 nd Cse 2 Rn2 < 0 < n1w hold. The price level of Home under utrky nd tht under free trde re given y R p AT cl1w 1 = R1- +, WK1 (60) R p FT cl1+l2w RcL1W = R1- + = WK1 R1- +. WK1 Since n2 < 0 in p FT, L 2 pproches zero, we hve L1+L2 " L1 in the long run. Moreover, AT g p 1 = g p FT. From this, we otin AT p1 = p FT in the long run. Therefore, the level of per cpit rel consumption in Home under utrky is equl to tht under free trde. In Foreign, the rte of chnge in the reltive price is different etween under utrky nd free trde. By sutrcting the rte of chnge in the reltive price under free trde from tht under utrky, we otin AT ; g p2 ;-; g FT p ;=-T (62) 1-- n1+n2y Here, the prentheses on the right - hnd side re positive ecuse of the sustinility condition. Accordingly, we hve ; g p2 ;<; g FT p ;. This finding mens tht the rte of chnge in the AT reltive price under free trde is lrger thn tht under utrky. Therefore, in Foreign, the level of per cpit rel consumption under free trde is higher thn tht under utrky. To sum up, when Pttern 2 nd Cse 2 hold, in Home, long - run welfre is equl under oth utrky nd free trde, wheres in Foreign, long - run welfre is higher under free trde thn under utrky. A sitution in which Pttern 6 nd Cse 3 hold is symmetric to the ove cse. Therefore, in Foreign, long - run welfre is equl under oth utrky nd free trde, wheres in Home, (61)

18 60 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model long - run welfre is higher under free trde thn under utrky. Summrizing the ove discussions, we otin the following proposition: Proposition 2. Suppose tht economic welfre is mesured y per cpit rel income. Then, in country with positive popultion growth, long - run economic welfre is the sme under oth utrky nd free trde, while in country with negtive popultion growth, long - run economic welfre under free trde is higher thn tht under utrky. Note tht the sum of the discounted present vlue of welfre t d" 0,+3% is uncertin, +3 j tht is, # c i Rt Wexp R -ttwdt ecuse totl welfre depends on when country switches 0 from utrky to free trde. 5. Conclusions In this study, we uilt two - country, two - sector, non - scle growth model nd investigted the reltionship etween trde ptterns nd per cpit consumption growth. In ddition, we considered oth negtive nd positive popultion growth. First, with regrd to trde ptterns, we otined the following results. If we confine ourselves to the sitution in which oth countries popultion growth rtes re different, t lest one country hs negtive popultion growth rte, nd Home hs n ICA in mnufcturing, the long - run sustinle comintions of popultion growth rtes nd trde ptterns re s follows: (1) Home with positive popultion growth produces oth goods, while Foreign with negtive popultion growth specilizes in griculture nd (2) Foreign with positive popultion growth produces oth goods, while Home with negtive popultion growth specilizes in griculture. With regrd to growth rtes, we otined the following results. In every cse, oth countries per cpit consumption growth rtes re identicl under free trde. By compring the growth rtes under free trde with those under utrky, we find tht in cse (1), Home s growth rte under free trde is the sme s tht under utrky, while Foreign s growth rte under free trde is higher thn tht under utrky, nd tht in cse (2), Foreign s growth rte under free trde is the sme s tht under utrky, while Home s growth rte under free trde is higher thn tht under utrky. These results imply tht in long - run sustinle cses, free trde increses one country s growth rte without decresing tht of the other. Therefore, free trde is desirle for oth countries. With regrd to welfre, we otin the following results. If we mesure economic welfre y the level of per cpit rel consumption, then, in country with positive popultion growth, long - run economic welfre is the sme under oth utrky nd free trde, while in country with negtive popultion growth, long - run economic welfre under free trde is higher thn tht under utrky. References Aghion, H. nd Howitt, P. (2005) Growth with qulity - improving innovtions: n integrted frmework, in P. Aghion nd S.N. Durluf (eds.) Hndook of Economic Growth, ch. 2, Amsterdm: North - Hol-

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20 62 Positive nd Negtive Popultion Growth nd Long - Run Trde Ptterns: A Non - Scle Growth Model Sski, H. (2011) Trde, non - scle growth, nd uneven development, Metroeconomic 62(4), pp Sski, H. (2011) Popultion growth nd north - south uneven development, Oxford Economic Ppers 63, pp Sski, H. (2012) Trde ptterns nd non - scle growth etween two countries, Kyoto University, Grdute School of Economics Reserch Project Center Discussion Pper Series, No. E Sski, H. (2014) Interntionl trde nd industriliztion with negtive popultion growth, Mcroeconomic Dynmics, doi: /S Segerstrom, P. (1998) Endogenous growth without scle effects, Americn Economic Review 88, pp Uzw, H. (1961) On two - sector model of economic growth, Review of Economic Studies 29(1), pp Wong, K. - Y. nd Yip, C.K. (1999) Industriliztion, economic growth, nd interntionl trde, Review of Interntionl Economics 7, pp Young, A. (1998) Growth without scle effects, Journl of Politicl Economy 106, pp A. Appendix A.1 Detiled nlysis of section 3.2 For Pttern 2 to e sustinle over time, we need p < r 1; tht is, from equtions (31) nd (35), " crl1+l2w% + < R1-WK 1 R1- L 1 WK1 + By rerrnging this condition, we otin. (A - 1) L2 < 1-c L1, e c Rn2-n1Wt 1-c <. c (A - 2) From this, if n1 = n2, Pttern 2 is sustinle if c < 1/2. If n1 > n2, Pttern 2 is sustinle in the long run irrespective of the size of c. By contrst, if n1 < n2, Pttern 2 is unsustinle in the long run. If n1 > 0 then from eqution (33), the growth rte of gk1 is given y ogk1 gk1 = R1- L1 WS L1+L2 L2 n1+ L1+L2 When n1 $ n2, fter enough time psses, this leds to n2 X+ R+-1WgK1. (A - 3) ogk1 gk1 = R1- Wn1+ R+-1WgK1. (A - 4) With ogk1/gk1 = 0, the growth rtes of the cpitl stock, terms of trde, nd per cpit consumption re respectively given y gk1 = 1- n1 > 0, 1-- g p =- 1-- n1 < 0, (A - 5) (A - 6)