Trade Patterns and Perpetual Youth in A Dynamic Small Open Economy

Transcription

1 Econ. J. of Hokkado Unv., Vol. 40 (2011), pp Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy Naoshge Kanamor n hs paper, examne he long-run specalzaon paerns ha arse n a small open economy usng a wo-secor growh model n whch households faced wh fne bu unceran lfemes underake neremporal opmzaon decsons. show ha n a small open economy wh a posve brh rae, mperfec specalzaon requres ha he subjecve dscoun rae be less han he neres rae. Ths fndng conrass sharply wh Sglz s (1970) fndng ha a counry mus be compleely specalzed f he subjecve dscoun rae dffers from he neres rae. n addon, show ha a seady sae equlbrum wh ncomplee specalzaon can be saddle-pon sable. JEL lassfcaon: E13, F11, F41 Keywords: Two-Secor Growh Model, Small Open Economy, Perpeual Youh Model 1. nroducon The purpose of hs paper s o examne he long-run rade paerns ha arse for a small counry by employng he wo-secor Blanchard (1985) perpeual youh model. Usng a dynamc Hecksher-Ohln model ha exends Onk and Uzawa (1965), Sglz (1970) esablshes he well-known resul ha a small open counry s perfecly specalzed n he capal (labor) -nensve secor f s more (less) paen. Under hs framework, when a more paen counry s mperfecly specalzed, faces he fac ha he neres rae, whch depends on world commody prces, s larger han he subjecve dscoun rae. Ths gap nduces he counry o accumulae capal sock. As he capal sock ncreases, he counry becomes capal abundan and evenually specalzes compleely n he capal-nensve secor. Thus, a counry s ncompleely specalzed n he long-run only when he counry s subjecve dscoun rae s nally equal o he neres rae. Snce a counry s subjecve dscoun rae s equal o he neres rae only by chance, mos counres mus be compleely specalzed. However, mos counres are ncompleely specalzed. One mehod for solvng hs conradcon s o nroduce endogenous me preferences. Karasawa and Yanase (2009), whch exend Sglz s (1970) model, sudy he rade paerns n a small open economy by employng endogenous me preferences ha depend would lke hank Professor aya for helpful commens. am also graeful o Yoshmasa Aok, Dasuke Amano and Kochro Sano for dscussons. ommens from an anonymous referee helped me o mprove he paper. Of course, all remanng errors are mne. Graduae School of Economcs and Busness Admmnsraon, Hokkado Unversy, Sapporo, , Japan. E-mal: nao-k@pop.econ.hokuda.ac.jp

2 30 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR boh on he level of ndvdual consumpon and on he level of average consumpon. They show ha a counry can be ncompleely specalzed n he long-run even f he neres rae s no equal o he counry s subjecve dscoun rae as an ncrease n he capal sock changes he counry s subjecve dscoun rae hrough a change n he level of consumpon 1). Anoher mehod s o use a perpeual youh model. Blanchard (1985) assumes ha all ndvduals face a posve probably of deah. Snce hey face an unceran lfemes, her asses are operaed by a lfe nsurance company n he annuy markes. He examnes he seady sae level of capal n a small open economy wh one secor and shows ha even f he counry s less paen, he seady sae level of capal may be posve. Ths resul occurs because aggregae savngs depend boh on he rao of consumpon o capal and on he gap beween he neres rae and he subjecve dscoun rae n he perpeual youh model. Recenly, Hamada, wasa and Kkuch (2009) employ he Blanchard (1985) perpeual youh model and analyze rade paerns n dynamc wo large counry model where each counry has a dfferen dscoun rae. They show ha even wh dfferen subjecve dscoun raes, boh counres can be ncompleely specalzed n he long-run. Noe ha hey assume ha capal s perfecly moble and ha he counres produce a radable good and a non-radable good. n hs paper, assume ha capal s mmoble beween counres and he economy can produce wo radable goods, such as n Slglz (1970), and focus on a small open economy. exend Blanchard (1985) o examne he long-run rade paerns ha arse for a small counry n a wo-secor perpeual youh model. show ha wh a posve brh rae mperfec specalzaon can occur for a small counry n he long-run only f he subjecve dscoun rae s less han he neres rae. Ths resul sems from he fac ha aggregae savngs depend on he rao of aggregae non-human wealh o aggregae consumpon n he perpeual youh model. Ths fndng conrass sharply wh Sglz s (1970) fndng ha f he subjecve dscoun rae dffers from he neres rae, he counry mus be compleely specalzed as aggregae savngs depend solely on he dfference beween he neres rae and he subjecve dscoun rae n an nfne horzon model. Furhermore, fnd ha a seady sae equlbrum wh ncompleely specalzaon can be saddle-pon sable. The paper s organzed as follows. n Secon 2, consder he opmzaon problems of frms and households. n Secon 3, analyze he long-run rade paerns and presen he conrbuons of hs paper. Secon 4 concludes. n Appendx, prove he wo Proposons. 2. The model 2.1 Frms consder a small open economy. Followng Onk and Uzawa (1965) and Sglz (1970), he counres produce wo goods, a pure nvesmen good (denoed by ) and a pure consumpon good (denoed by ) whch are produced usng 1) Nshmura and Shmomura (2002) presen he smlar resul by usng endogeneous me preferences.

3 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy 31 capal and labor. assume ha he facors (.e., capal and labor) are mmoble beween counres. However, suppose ha boh commodes are raded freely. denoe p() as he relave world prce for he consumpon good n erms of he nvesmen good and assume ha hs world prce s consan over me,.e., p () p a all mes. To smplfy my model, assume ha he deprecaon rae of capal s zero. The producon funcon n secor (, ) s Y F K(), L(), where K () s he capal used n secor and L () s he labor employed n secor a me. Ths funcon sasfes posve and dmnshng margnal producs for each npu, consan-reurns-o-scale, and he nada condons. Thus, can rewre he producon funcon n secor as Y F K (), L() f k () L(), where k() K()/ L() f k () s defned o equal F k (),1. f boh goods are produced n he counry, can derve he followng equaons from he prof maxmzaon of frms: s he rao of capal o labor n secor and r () f k() pf k (), K K w () f k() pf k (), L L where f j k(), r (), and w () are he margnal produc of npu j j K, L n secor, he renal rae on capal, and he wage rae, respecvely. Noe ha when he counry s ncompleely specalzed, all four varables, k (), k (), r (), w (), depend solely on he world prce, p. The correspondng value of varable x s denoed by xˆ( p ). n addon, assume ha he nvesmen goods secor s more capal nensve han he consumpon goods secor for all prces of he consumpon good, ha s, kˆ ( ) ˆ p k for all p. consder he value of naonal ncome per capa, y, whch s gven by F ( K y( p, K( ), L( )) max, ( ), L ( )) pf K, L L( ) ( K ( ), L ( )) s.. K () K() K (), L () L() L (), where K () and L () are aggregae capal and aggregae labor, respecvely. f he counry produces only he nvesmen good, y s f ( k ( )), whle f he counry produces only he consumpon good, y s pf ( k( )), where k () represens he aggregae capal sock per capa. f he counry produces boh goods, y s gven by

4 32 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR k () kˆ ( ) ˆ p k k( ) ˆ ( ) ˆ f k p pf k p k ( ) ˆ ( ) ˆ p k p k p k ˆ ( ) ˆ ( ) Wh kˆ ( ) ˆ p k, he counry s ncompleely specalzed f kˆ ( ) ( ) ˆ p k k. f k () kˆ, he counry s specalzed n secor, whle f kˆ ( p ) k ( ), he counry s specalzed n secor. Fnally, he margnal produc of capal (denoed by y ) and he slope of he k margnal produc of capal (denoed by y ) are kk. and pfk ( k( )) yk rˆ f ( k ( )) y kk K pf f KK KK f f f ( k( )) 0 0 ( k( )) 0 k () kˆ kˆ ( ) ( ) ˆ p k k, kˆ ( p ) k ( ) f f f k () kˆ kˆ ( ) ( ) ˆ p k k, (1) kˆ ( p ) k ( ) where f s he second dervave wh respec o capal n secor KK. 2.2 Households There s a large number of dencal consumers faced wh fne bu unceran lfemes. Each cohor s born a a consan rae and faces a consan probably of deah. The oal populaon a me s L () and he sze of he cohor born a me s L (). Thus, a cohor born a me s has a sze of s s Lse () e e a me, where L(0) 1. The oal populaon s obaned by negrang over he survvors of each cohor born a me : L( ) e e s ds e where n ( ) 0 s he growh rae of he oal populaon. Each consumer maxmzes he followng expeced lfeme uly a me : x ( )( ) c ( s, ) e d, n, log (2) where c ( s, ) and are ndvdual consumpon a me by a consumer born a me s and he subjecve dscoun rae, respecvely. assume ha he subjecve dscoun rae s larger han he growh rae of he oal populaon,.e., n. Each consumer supples one un of labor nelascally and receves he wage w () a me. She has zero non-human wealh when she s born,.e., a (,) s s 0. When she has a (, s ) a me s, she receves r( ) a ( s, ). follow Yaar (1965) and Blanchard (1985) n assumng he exsence of acuarally

5 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy 33 far annuy markes. Each consumer wll conrac wh an nsurance company whch gves her a premum of a (, s ) a each pon n me, as long as she lves. The flow budge consran for each consumer born a me s s The ransversaly condon s where rˆ (, ) r( ) d a (, s ) r a(, s ) w() pc(, s ). (3) r(, ) e lm a ( s, ) 0, (4) s he average neres rae beween mes and. negrang (3) and usng (4), can oban he neremporal budge consran: rˆ( p c ( s, ) e ~ d a ( s, ) w~ ( ),, ) ( ) (5) where rˆ(, ) ( ) w( ) w( ) e d s he presen value of wage ncome. Each consumer maxmzes expeced uly (2), subjec o he flow budge consran (3), nal endowmens a (,) s s 0, and he ransversaly condon (4), akng he me pahs of marke prces ( prces of he consumpon good, wages, and neres raes) as gven. Applyng he maxmal prncple, can oban he followng ndvdual Euler equaon: c (, s ) r c(, s ). (6) negrang (6) forward and usng he neremporal budge consran (5), can oban he ndvdual s consumpon funcon: c(,) s as, w() p. Aggregae consumpon a me, (), s obaned by negrang ndvdual consumpon n each cohor, weghed by he number of consumers alve n each cohor a me : s ( ) e c ( s, ) e ds. Smlarly, aggregae nonhuman wealh A() and aggregae wage ncome W () a me are gven by: s A( ) e a ( s, ) e ds, (7) s W ( ) e w~ ( ) e ds. (8)

6 34 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR Usng he above equaons, can oban he followng aggregae consumpon funcon: () A W () p. (9) Dfferenang equaons (7) and (8), he dynamcs of aggregae nonhuman wealh A() and ha of aggregae wage ncome W () can be wren as A () r() A() w() e n p(), (10) W () r () n W () we () n, (11) respecvely. Usng (9), (10) and (11), can oban he growh rae of aggregae consumpon: n A() () r n () p () Fnally, can show ha he growh rae of aggregae consumpon per capa s n a() c () r c () pc ()., (12) where c () and a () are aggregae consumpon per capa and aggregae nonhuman wealh per capa, respecvely. Ths equaon can be nerpreed as he aggregae Euler equaon. The dfference beween he aggregae Euler equaon (12) and he ndvdual Euler equaon (6) s he erm n a()/ pc(). Ths erm appears due o he presence of he brh rae (.e., n 0). Snce newborns wh zero asses ener he economy every perod, he growh rae of aggregae consumpon per capa s smaller han ha of ndvdual consumpon. Hence, he growh rae of aggregae savngs per capa dffers from ha of ndvdual savngs. Borrowng he ermnology used by Hejdra (2009) denoe he erm n a()/ pc() as he dsrbuonal effec. f he brh rae s zero, he aggregae Euler equaon s equal o he ndvdual Euler equaon and hus he dsrbuonal effec s zero. 3. Long-run Specalzaon Paerns n hs secon, consder he specalzaon paerns ha arse n a seady sae. Snce capal s nernaonally mmoble, non-human wealh per person s equal o he capal sock per person,.e., a () k (). The dynamcs of per capa capal can be wren as k () y () pc () nk (). (13)

7 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy 35 Usng equaon (12), r () yk () and a () k (), can oban he dynamcs of per capa consumpon as follows: n k() c () yk () c (). (14) p n a seady sae c k 0. ombnng (13) wh (14) yelds y, k, and y k n k y nk, (15) y are he seady sae levels of per capa ncome, per cap- k where a capal, and he margnal produc of capal, respecvely. Followng Karasawa and Yanase (2009), can nerpre he above equaon as he long-run capal marke clearng condon. The lef-hand sde of (15) mples he long-run capal demand funcon as frms demand a level of capal ha equaes he margnal produc of capal wh he neres rae. The rgh-hand sde of (15) represens he long-run capal supply funcon as he supply of capal depends on he sum of he dscoun rae,, and he dsrbuonal effec, nk / y nk. Fgure1: The long-run capal demand curve corresponds o y k, whle he long-run capal supply curve corresponds o n k / y nk. f he brh rae s zero, he long-run capal supply curve s. Fgure 1 provdes as llusraon of (15) followng Sglz (1970). The curve correspondng o he lef-hand sde of (15) s downward slopng f k kˆ or kˆ ( p ) k, and s fla f kˆ ( ) ˆ p k k. The curve correspondng o he rgh-hand sde of (15) s unambguously upward slopng (see Appendx A). have he followng Proposon. Proposon 1: Gven he world prce p, when he brh rae s posve (.e., n 0), he counry s ncompleely specalzed n he long-run f and only f

8 36 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR rˆ( ( )( n) kˆ ( )( ) ˆ n k rˆ(,, (16) pf ( kˆ () nkˆ f ( kˆ ) nkˆ s sasfed. When he brh rae s zero (.e., n 0), he counry s ncompleely specalzed n he long-run f and only f he subjecve dscoun rae s equal o he neres rae (.e., rˆ ). Proof: See Appendx A. onsder he nuon behnd Proposon 1. Suppose ha he rao of capal o labor n he counry, k, s kˆ ( p ). Ths counry produces only a consumpon good. f equaon (16) s sasfed, per capa capal ncreases as he consumers fnd ha he neres rae s larger han he subjecve dscoun rae and are wllng o accumulae more non-human wealh. However, snce newborns wh zero asses ener he economy every perod, he growh rae of aggregae savngs declnes. Ths effec corresponds o he dsrbuonal effec n Secon 2. As per capa capal ncreases, hs dsrbuonal effec ncreases. Hence, evenually he neres rae s equal o he sum of he subjecve dscoun rae and he dsrbuonal effec, and he capal sock per person sops rsng. As a resul, he counry s ncompleely specalzed n he long-run. n Fgure 1, he equlbrum s denoed by pon A and he rao of capal o labor n he counry s k. A However, f he brh rae s zero, he dsrbuonal effec vanshes. When he subjecve dscoun rae s less han he neres rae, r ˆ( per capa capal ncreases. Thus, he counry evenually specalzes n he capal-nensve secor, as poned ou by Sglz (1970). n Fgure 1, he equlbrum s gven by pon B and he rao of capal o labor n he counry s k. B Sglz (1970) fnds ha he counry s ncompleely specalzed only f he subjecve dscoun rae s equal o he neres rae. n conras, n my model, mperfec specalzaon of he counry requres ha he subjecve dscoun rae be less han he neres rae from equaon (16). The dfference sems from fac ha he presence of he brh rae modfes he aggregae Euler equaon. n oher words, n my model here exss a dsrbuonal effec. The above resuls are also argued by Hamada, wasa and Kkuch (2009). They employ he Blanchard (1985) perpeual youh model and analyze long-run rade paerns beween wo large counres where each counry has a dfferen subjecve dscoun rae. They show ha f he subjecve dscoun rae n he more paen counry s smaller han he neres rae, here can exs a unque seady sae equlbrum where boh counres are ncompleely specalzed. Ths resul sems from he dsrbuonal effec. Noe ha wh he perpeual youh model he dsrbuon effec appears regardless of wheher here are wo counres or one small open counry. Thus, her resuls are he same as he resuls derved n hs paper 2). However, hey assume ha capal s perfecly moved and ha one of he wo goods s a non-radable good, whle assume ha capal s 2) hank he referee for hs pon.

9 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy 37 perfecly mmoble and boh goods are radable goods and focus on a small open economy. n addon, Karasawa and Yanase (2009) show ha n a small open economy even f he subjecve dscoun rae s dfferen from he neres rae nally, here exss a seady sae where he counry s ncompleely specalzed. They do no employ he perpeual youh model bu assume endogenous me preferences ha rely on boh ndvdual consumpon and socal average consumpon. Thus, even f he subjecve dscoun rae s less han he neres rae nally, by modfyng he subjecve dscoun rae hrough he changes n he consumpon levels, he counry s subjecve dscoun rae s evenually equal o he neres rae. Fnally, fnd ha an equlbrum wh ncomplee specalzaon can be saddle-pon sable. More precsely, have he followng Proposon. Proposon 2: f rˆ rˆ n n s sasfed, hen he long-run equlbrum wh ncomplee specalzaon s saddle-pon sable. Proof: See Appendx B. When a posve brh rae and he subjecve dscoun rae s less han he neres rae, a long-run equlbrum wh ncomplee specalzaon can be saddlepon sable. The resul sems from he emergence of he dsrbuonal effec n he presence of a posve brh rae. 4. oncludng Remarks n hs paper, have examned long-run specalzaon paerns n a small open economy by employng he wo-secor Blanchard perpeual youh model. The man conrbuon of hs paper s o clarfy how he ncluson of fne lved agens nfluences he mplcaon of long-run specalzaon paerns. When he brh rae s posve, mperfec specalzaon can occur only f he subjecve dscoun rae s less han he neres rae. Ths resul sems from he fac ha he ndvdual Euler equaon dffers from he aggregae Euler equaon n he perpeual youh model. On he oher hand, Sglz (1970) fnds ha f he subjecve dscoun rae dffers from he neres rae, he counry s never ncompleely specalzed. Tha s, f he subjecve dscoun rae s less han he neres rae, he counry mus specalze n he capal nensve secor. Ths resul arses because he ndvdual Euler equaon has he same form as he aggregae Euler equaon n an nfne horzon model. When he brh rae s zero, my model reduces o ha of Sglz (1970). Furhermore, show ha he seady sae equlbrum wh ncomplee specalzaon can be saddle-pon sable when he brh rae s posve and he subjecve dscoun rae s less han he neres rae. The mos mporan and naural exenson s o allow for secor-specfc or economy wde producon exernales, as shown n Benhabb and Farmer (1994). n my paper he seady sae equlbrum can be he saddle-pon sable. However, he ncluson of exernales would creae a source for generang ndeermnacy. n addon, he equlbrum could be made ndeermnae by nroducng he desre

10 38 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR for socal saus (e.g., aya and Kanamor 2010), whch nduces prvae agens o care abou oher s capal levels or consumpon levels. The analyss n hs paper s a frs sep oward a more comprehensve analyss of sables n such sengs. Fnally, would be neresng o examne sably n a wo counry model. Ths exenson convers consan commody prces no varable prces. Hence, he sables of he equlbrum n a wo counry model may be dfferen from hose n a small open economy. Docoral Suden, Hokkado Unversy References Benhabb, J., Farmer, R.E.A., 1994, ndeermnacy and ncreasng reurns, Journal of Economc Theory 63 (1), Blanchard, O.J., 1985, Deb, defcs, and fne horzons, Journal of Polcal Economy 93 (2), aya, J., and Kanamor, N., 2010, onsumpon axaon, socal saus and ndeermnacy n models of endogenous growh wh elasc labor supply, Journal of Economcs 100 (2), Karasawa, Y., and Yanase, A., 2009, Endogenous me preference and consumpon exernales n a small open economy: mulple seady saes and ndeermnacy. Avalable a SSRN: hp://ssrn.com/absrac= Hamada, K., wasa, K., and Kkuch, T., 2009, Trade and capal movemens beween counres wh dfferen dscoun raes n a model of perpeual youh. Avalable from hp:// Hejdra, B., 2009, Foundaons of modern macro economcs second edon, Oxford Unversy Press. Nshmura, K., and Shmomura, K., 2002, ndeermnacy n a dynamc small open economy, Journal of Economc Dynamcs and onrol 27(2), Onk, H., and Uzawa, H., 1965, Paerns of rade and nvesmen n a dynamc model of nernaonal rade, Revew of Economcs Sudes 32 (1), Sglz, J.E., 1970, Facor prce equalzaon n a dynamc economy, Journal of Polcal Economy 78 (3), Yaar, M.E., 1965, Unceran lfeme, lfe nsurance, and he heory of he consumer, Revew of Economcs Sudes 32 (2),

11 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy 39 Appendx A: Proof of Proposon 1 defne he rgh-hand sde of (15) as ( k), ha s, ( k) n k/ ynk. Snce he per capa wages of workers, y y k, are posve, k n y ykk 2 ( k) k y nk 0. Thus, fnd ha ( k) s monooncally ncreasng n he rao of capal o labor. Noe ha f he brh rae s zero, ( k) s, whch s consan. have found ha from equaon (1) he lef-hand sde of (15) s decreasng n he rao of capal o labor when k kˆ c ( p ) or kˆ ( p ) k and s r ˆ( when kˆ ( ) ˆ c p k k. Frs, consder he brh rae s posve. suppose ha here exss a seady sae per capa capal ˆ ˆ k kc, k such ha ( k ) rˆ. Snce ( k) s monooncally ncreasng n he rao of capal o labor, boh ( kˆ ) rˆ( and ( kˆ ) rˆ( are sasfed, ha s, ( )( n) kˆ ( )( ) ˆ n k rˆ( rˆ(, pf ( kˆ ) nkˆ f ( kˆ ) nkˆ s sasfed. Suppose ha ( kˆ ) rˆ( and ( kˆ ( )) ˆ p r. There exss a seady sae per capa capal ˆ ( ), ˆ k ( k p k ) such ha ( k ) rˆ as ( k) s monooncally ncreasng n he rao of capal o labor. Nex, consder he brh rae s zero. suppose ha here exs a per capa capal ˆ ( ), ˆ k ( k p k ) such ha ( k ) rˆ. Snce ( k) s, rˆ s sasfed. Suppose ha rˆ, ( k ) rˆ s sasfed as ( k) s. Appendx B: Proof of Proposon 2 focus on he seady sae equlbrum wh ncomplee specalzaon. Usng equaons (13) and (14), ake a lnear approxmaon of he dynamc sysem around he seady sae values of per capa capal, k, and per person consumpon, c,: n k n p. c n nk c c p pc k pc k k Noe ha when he counry s ncompleely specalzed n he seady sae, he slope of he margnal produc of capal, y, s zero. The deermnan of hs marx (denoed by D ) can be kk wren as

12 40 Econ. J. of Hokkado Unv., Vol. 40 N. KANAMOR k k D n n n1. pc pc can oban ha f rˆ rˆ n n 1, (A) hen D s negave. Nex, he race of hs marx (denoed by T ) can be wren as k T 2n n. pc Snce n 0, T s always posve. Thus, f equaon (A) s sasfed, he seady sae equlbrum s saddle-pon sable.