JEL Classification: D50; D58; F10; F11

Transcription

1 Ttle: Fctor endowment commodty output reltonshps n three-fctor two-good generl equlbrum trde model: Further nlyss 1 By Yoshk Nkd, Kyoto Unversty, Fculty of Agrculture, Ktshrkw-owke-cho, Skyo, Kyoto, Jpn E-ml: nkd@ks.kyoto-u.c.p Abstrct: The poston of the EWS (economy-wde substtuton)-rto vector determnes the Rybczynsk sgn pttern, whch expresses the fctor endowment commodty output reltonshps, nd the Stolper- Smuelson sgn pttern, whch expresses the commodty prce fctor prce reltonshps n threefctor two-good generl equlbrum trde model (see Nkd (2016)). In ths rtcle, we show tht the EWS-rto vector exsts on the lne segment. Usng ths reltonshp, we develop method to estmte the poston of the EWS-rto vector. We derve suffcent condton for extreme fctors to be economy-wde complements, whch mples strong Rybczynsk result. Addtonlly, we derve suffcent condton for specfc Stolper-Smuelson sgn pttern to hold. We ssume fctor-ntensty rnkng s constnt. Ths rtcle provdes bss for further pplctons. Keywords: three-fctor two-good model; generl equlbrum; Rybczynsk result; EWS (economywde substtuton)-rto vector, Stolper-Smuelson sgn pttern. JEL Clssfcton: D50; D58; F10; F11 1. Introducton Btr nd Css (1976) (herenfter BC) wrote n rtcle on functonl reltons n three-fctor twogood neoclsscl model (or 3 x 2 model) nd clmed tht strong Rybczynsk result rses. However, ths s not the cse (see Nkd (2016)). Accordng to Suzuk (1983, p. 141), BC contended n Theorem 6 (p. 34) tht f commodty 1 s reltvely cptl ntensve nd commodty 2 s reltvely lbor ntensve, n ncrese n the supply of lbor ncreses the output of commodty 2 nd reduces the output of commodty 1. Ths s wht strong Rybczynsk result mples. Nkd (2016) defned the EWS-rto vector bsed on economy-wde substtuton (herenfter EWS) orgnlly defned by Jones nd Eston (1983) (herenfter JE) nd used t n n nlyss. Nkd (2016) concluded tht the poston of the EWS-rto vector determnes the Rybczynsk sgn pttern, whch expresses the fctor endowment commodty output reltonshps nd ts dul counterprt, the 1 An erler verson ws ttled, Economy-wde substtuton nd Rybczynsk sgn pttern n threefctor two-good model: Further nlyss. On ths, see 1

2 Stolper-Smuelson sgn pttern, whch expresses the commodty prce fctor prce reltonshps n 3 x 2 model of BC s orgnl type. Usng the EWS-rto vector, Nkd (2016) derved suffcent condton for strong Rybczynsk result to hold (or not to hold) n systemtc mnner. The followng result hs been estblshed (see Nkd (2016, p. 24)). Theorem 1. f the EWS-rto vector S, U exsts n qudrnt IV (or subregons P1 P3), n other words, f extreme fctors re economy-wde complements, strong Rybczynsk result holds necessrly. The followng questons rse. () How cn we estmte the poston of the EWS-rto vector? () Under wht condtons does the EWS-rto vector exst n qudrnt IV; n other words, re extreme fctors economy-wde complements? The purpose of ths pper s s follows. Frst, we show tht the EWS-rto vector exsts on the lne segment. Usng ths reltonshp, we develop method to estmte the poston of the EWS-rto vector. We derve suffcent condton for extreme fctors to be economy-wde complements. Addtonlly, we derve suffcent condton for specfc Stolper-Smuelson sgn pttern to hold. Ths rtcle wll provde the bss for further pplctons. For exmple, ths rtcle wll be useful n estmtng the Rybczynsk sgn pttern n some countres nd wll contrbute to nterntonl nd development economcs. 2 Smlr to Nkd (2016), some ppers re nterested n the role of complementrty. For exmple, Tkym (1982, p. 19) ssumed tht extreme fctors were ggregte complements. Suzuk (1987, Chpter 1, p ) ssumed tht extreme fctors were Allen complements n ech sector. Termch (1993) ssumed tht extreme fctors were ggregte complements. On the other hnd, Thompson (1985) ssumed tht ny two fctors were ggregte complements. 3 Other ppers lso ddress 2 For exmple, Nkd (2016b) ppled the results derved here to dt from Thlnd nd, n dong so, derved the fctor endowment commodty output reltonshp for Thlnd durng the perod To some extent, these results show how Chnese mmgrton ffected commodty output n Thlnd between 1920 nd Addtonlly, Suzuk (1983) ssumed tht cptl nd lnd (mddle fctor nd extreme fctor, respectvely) were perfect complements n ech sector, nd derved the mplctons usng Allenprtl elstctes of substtuton. JE ssumed tht two fctors were perfect complements nd derved the mplctons usng EWS (see JE (p )). However, Suzuk s proof s not plusble (see Nkd (2015)). JE s proof n subsectons 5.2.4, nd (p ) s questonble (see Nkd (2016, Appendx)). 2

3 complementrty, for exmple, Thompson (1995), Blss (2003), Eston (2008), Ide (2009), Bn (2007), nd Bn (2008). See lso Bn (2007b). In summry, some of these prevous studes re somewht complcted. We re uncertn whether ll of these studes re plusble. At lest, to the uthor s knowledge, none of the studes nlyze the condtons under whch extreme fctors re economy-wde complements. Secton 2 of ths study explns the model. In subsecton 2.1, we expln the bsc structure of the model. In subsecton 2.2, we ssume fctor-ntensty rnkng. In subsecton 2.3, we derve the mportnt reltonshp mong EWS-rtos nd drw the EWS-rto vector boundry. In secton 3, we estmte the poston of the EWS-rto vector under some ssumptons. In subsecton 3.1, we show tht the EWS-rto vector s on the lne segment AB (or the EWS-rto vector lne segment). In subsecton 3.2, we defne the fctor-prce-chnge rnkng. In subsecton 3.3, we derve the mportnt reltonshps mong some vrbles (e.g., H 0, H0 0 ). In subsecton 3.4, by nlyzng the Crtesn coordntes of pont A nd B, we develop method to estmte the poston of the EWSrto vector. Frst, we derve suffcent condton for the EWS-rto vector to exst n qudrnt IV, tht s, n ny subregon P1, P2, or P3. In ths cse, extreme fctors re economy-wde complements. If ths holds, strong Rybczynsk result holds, tht s, three of the Stolper-Smuelson sgn ptterns hold. Moreover, we derve suffcent condton for the EWS-rto vector to exst n ny subregon P1, P2, or P3. If ths holds, specfc Stolper-Smuelson sgn pttern holds. Fnlly, I show some mplctons. Secton 4 presents the conclusons, nd the Appendx derves the lner reltonshp between EWS-rtos. Secton 2 contns smlr content s Nkd (2016). 2. Model 2.1. Bsc structure of the model We ssume smlrly to BC (p.22-23). Tht s, we ssume s follows. Products nd fctors mrkets re perfectly compettve. Supply of ll fctors s perfectly nelstc. Producton functons re homogeneous of degree one nd strctly qus-concve. All fctors re not specfc nd perfectly moble between sectors, nd fctor prces re perfectly flexble. These two ensure the full employment of ll resources. The country s smll nd fces exogenously gven world prces, or the movement n reltve prce of commodty s exogenously determned. The movements n fctor endowments re exogenously determned. Full employment of fctors mples X V, T, K, L, (1) 3

4 where X denotes the mount produced of good (=l, 2); the requrement of nput per unt of output of good (or the nput-output coeffcent); V the supply of fctor ; T s the lnd, K cptl, nd L lbor. In perfectly compettve economy, the unt cost of producton of ech good must ust equl to ts prce. Hence, w p, 1,2, (2) where p s the prce of good ; w s the rewrd of fctor. BC (p.23) stted, Wth qus-concve nd lnerly homogeneous producton functons, ech nput-output coeffcent s ndependent of the scle of output nd s functon solely of nput prces: w, T, K, L, 1,2. (3) And they contnued, In prtculr, ech C [ n our expresson] s homogeneous of degree zero n ll nput prces. 4 Equtons (1)-(3) descrbe the producton sde of the model. These re equvlent to eqs (1)-(5) n BC. The set ncludes 11 equtons n 11 endogenous vrbles (X,, nd w ) nd fve exogenous vrbles (V nd p ). The smll-country ssumpton smplfes the demnd sde of the economy. Totlly dfferenttng eq. (1), we hve ( * X *) V *, T, K, L, (4) where n stersk denotes rte of chnge (e.g., X *=d X / X ), nd where λ s the proporton of the totl supply of fctor n sector (tht s, X / V ). Note tht 1. The mnmum-unt-cost equlbrum condton n ech sector mples Σ w d =0. Hence, we derve (see JE (p.73, eq. (9)), BC (p.24, note 5), * 0, 1, 2, (5) where θ s the dstrbutve shre of fctor n sector (tht s, θ = w / p ). Note tht Σ θ =l; d s the dfferentl of. Totlly dfferenttng eq. (2), we derve w * p *, 1, 2. (6) Subtrctng p * 1 from the both sdes of eq. (6), we hve 4 From the condton of cost mnmzton, we cn show tht s homogeneous of degree zero n ll nput prces (see Smuelson (1953, chpter 4, p. 68), Nkd (2016, p. 5)). 4

5 w * 0, 1 1 w * P, 2 1 P p * p * p / p *, w * w * p *, w w / p ; P s the rte of chnge n where the reltve prce of commodty; w 1 s the rel fctor prce mesured by the prce of good 1. where Totlly dfferentte eq. (3) to obtn * w * 0, T, K, L, 1, 2, (8) h h h log / log w. (9) h h h h σ h s the Allen-prtl elstctes of substtuton (herenfter AES) between the th nd the hth fctors n the th ndustry. For ddtonl defnton of these symbols, see Sto nd Kozum (1973, p.47-49), nd BC (p.24). AESs re symmetrc n the sense tht h h. (10) And ccordng to BC (p33), Gven the ssumpton tht producton functons re strctly qus-concve nd lnerly homogeneous, 0. (11) Snce s homogeneous of degree zero n nput prces, we hve h h h h h 0, T, K, L, 1, 2. (12) Eqs (8) nd (12) re equvlent to the expressons n BC (p.24, n. 6). See lso JE (p.74, eqs (12)- (13)). From these: * w *. (13) h h h1 Substtute eq. (13) n (4): where ( w * X *) g w * X * V *, T, K, L, (14) h h h1 h h h1 gh h,, h T, K, L. (15) Ths s the EWS (or the economy-wde substtuton ) between fctors nd h defned by JE (p.75). g h s the ggregte of h. JE (p.75) stted, Clerly, the substtuton terms n the two ndustres (7) 5

6 re lwys verged together. Wth ths n mnd we defne the term k to denote the economy-wde substtuton towrds or wy from the use of fctor when the kth fctor becomes more expensve, under the ssumpton tht ech ndustry's output s kept constnt:. Note tht hgh 0, T, K, L, (16) gh ( h / ) gh,, h T, K, L. (17) where nd re, respectvely, the shre of fctor, T, K, L, nd good, 1,2 n I, where I px = wv totl ncome. Tht s, px / I, wv /. On ths, see BC (p.25, eq. (16)). Hence, we obtn ( / ) (see JE (p.72, n. 9)). Note tht 1, g g h n generl. On eq. (17), see lso JE 1. g h s not symmetrc. Nmely,, (p.85). From (9), (11) nd (15), we cn show tht h h g 0. (18) From eqs (16) nd (18), we derve gkt gkl gkk 0, gtk gtl gtt 0, glk glt g LL 0. (19) From (19) nd (17), we cn esly show tht (,, ) LK LT KT,,,,,,,,,,,. (20) g g g = At most, one of EWSs ( glk, glt, g KT ) cn be negtve Fctor-ntensty rnkng In ths rtcle, we ssume T1 L1 K1, (21) T 2 L2 K 2. (22) L1 L2 Eq. (21) s, wht you cll, the fctor-ntensty rnkng (see JE (p69), see lso BC (p.26-27), Suzuk (1983, p.142). Ths mples tht sector 1 s reltvely lnd ntensve, nd sector 2 s reltvely cptl ntensve, nd tht lbor s the mddle fctor, nd lnd nd cptl re extreme fctors (see lso Ruffn (1981, p.180)). Eq. (22) s the fctor-ntensty rnkng for mddle fctor (see JE (p. 70)). It mples tht the mddle fctor s used reltvely ntensvely n sector 1. In the followng sectons, we show the nlyss under these ssumptons. Nkd (2016) lso ssumed eqs (21) nd (22) hold. Of course, even f we ssume L1 L2, we cn nlyze smlrly. 6

7 2.3. EWS-rto vector boundry In ths subsecton, we derve the mportnt reltonshp mong EWS-rtos. I show the EWSrto vector boundry. Ech s homogeneous of degree zero n ll nput prces (see eq. (3)). Recll eq. (11), 0. From these, Nkd (2016, eq. (35)) derved n mportnt reltonshp mong EWS s follows. gkk gtt gtk gkt L L = gktgtl gklgtk gklgtl = [ gkt( glt glk) glkglt ]( 0) T K. (23) From (19), glk glt g LL 0. Usng ths, trnsform eq. (23) to obtn g KT L glkglt g g K LK LT. (24) Trnsform eq. (24) to hve where L U ' K S ' L S ', f T>0; U ', f T<0, (25) S ' 1 K S ' 1 S, U S / T, U / T ( g / g, g / g ), (26) LK LT KT LT ( S, T, U) ( g, g, g ). (27) LK LT KT We cll (S, U ) the EWS-rto vector. Eq. (25) expresses the regon for the EWS-rto vector. Trnsform L S ' L L 1 U ' K S' 1 K K S' 1, (28) whch expresses the rectngulr hyperbol. We cll t the equton for the EWS-rto vector boundry. It psses on the orgn of O (0, 0). The symptotc lnes re S 1 U' L/ K,. We cn drw ths boundry n the fgure (see Fg. 1). S s wrtten long the horzontl xs, nd U long the vertcl xs. Ths boundry demrctes the boundry of the regon for the EWS-rto vector. Ths mples tht the EWS-rto vector s not so rbtrry, but exsts wthn ths bounds. The sgn pttern of the EWS-rto vector s, n ech qudrnt (see lso eq. (20)): 7

8 Qud. I: (S', U )=(+, +) (S, T, U) = (+, +, +); Qud. II: (S', U )=(-, +) (S, T, U) =(-, +, +); Qud. III: (S', U )= (-, -) (S, T, U) =(+, -, +); Qud. IV: (S', U )= (+, -) (S, T, U) =(+, +, -). (29) We my defne (for h), Fctors nd h re economy-wde substtutes, f g h >0; Fctors nd h re economy-wde complements, f g h <0. (30) 3. Estmtng the poston of the EWS-rto vector 3.1. EWS-rto vector lne segment In ths subsecton, we show tht the EWS-rto vector exsts on the lne segment. We hve shown tht the EWS-rto vector exsts on the strght lne, whch we cll the EWS-rto vector lne (see (A19) n Appendx). Recll eqs (A19) nd (25), tht s, U ' 1S ' b1, (31) L U ' K S ' L S ', f T>0; U ', f T<0, (32) S ' 1 K S ' 1 The EWS-rto vector stsfes these equtons. From eqs (31) nd (28), we cn mke system of equtons: U ' 1S ' b1, (33) L U ' K S ' S ' 1. (34) These re the equtons of the EWS-rto vector lne nd the EWS-rto vector boundry, respectvely. From eqs (33) nd (34), we obtn qudrtc equton n S. Solve ths to derve two solutons. The solutons re S ' W W TL KL, K 0 T 0 ' ' KT, (35) where we recll eq. (A20) nd (A10), tht s, Wh w * wh* ( w / wh)*, T, K, L, h, (36) h / h, T, K, L, h. (37) Hence, the Crtesn coordntes of the ntersecton pont re WTL L WLT ( wt * wl*) L ( wl * wt*) ( S', U') (, ) (, ), WKL K WKT ( wk * wl*) K ( wk * wt*) 8

9 ' '. (38) ' ' ( K0 KT, K0 ) T0 L0 We cll these ponts A nd B. In generl, the EWS-rto vector ( S', U ') exsts on the lne segment AB. We cll t the EWS-rto vector lne segment. Ths mples tht by nlyzng the Crtesn coordntes of Ponts A nd B, we cn estmte the poston of the EWS-rto vector Fctor-prce-chnge rnkng In ths subsecton, we show the reltonshp mong the fctor-prce chnges. Here, we ssume Recll eq. (7): P p p 1* 2* 0, (39) w * 0, 1 1 w * P, T, K, L. 2 1 (40) For ese of notton, defne tht ( X, Y, Z) ( wt1*, wk 1*, wl 1*) ( wt* p 1*, wk * p 1*, wl* p 1*). (41) These re the chnges n the rel fctor prce. These re not ndependent but re relted to ech other. Usng eq. (41), trnsform eq. (40) to derve T1 K1 X L1Z Y P Z. (42) T 2 K 2 L2 Solvng eq. (42), we hve X 1 K 2 K1 L1Z Y D 1 T 2 T1 P L2Z, (43) D. Hence, we hve where 1 T1 K 2 K1 T 2 1 D2 X K1P Z, D1 D1 1 D3 Y ( T 1P ) Z, nd D1 D1 Z Z, (44) where D2 K 2 L1 K1 L2, D3 T1 L2 T 2 L1. Eq. (21) mples ( D1, D2, D3) (,, ). (45) 9

10 Tret s f X nd Y were dependent vrbles nd Z were n ndependent vrble. Eq. (44) expresses the strght lnes n two dmensons. We cll these lnes, respectvely, Lnes X, Y, nd Z. Becuse we ssume eq. (39) (P > 0), the sgns of the grdent nd ntercept of Lnes X nd Y re, respectvely, D2 1 ( ), K1P ( ), for Lne X; D1 D1 D3 1 ( ), ( 1 ) ( ), D1 D T P for Lne Y. (46) 1 Hence, we cn drw Lnes X, Y, nd Z n the fgure (see Fg. 2). The Crtesn coordntes of the ntersecton of Lnes Y nd Z nd Lnes X nd Z re, respectvely, ( A T1 P, A T1 P ), ( B K1 P, B K1 P ). (47) Becuse eq. (22) holds, f we compre the grdent of Lnes X nd Y, we cn esly show tht D D 2 1 D3. (48) D1 Hence, Lnes X nd Y hve n ntersecton pont n qudrnt IV. Only four rnkngs re possble; they re, X Y Z, X Z Y, Z X Y, Z Y X. (49) Any of the four rnkngs re possble. We cll ths the fctor-prce-chnge rnkng. For exmple, we cn ssume X Z Y wt* wl* wk * (50) If eq. (50) holds, the chnge n rel rewrd for lbor s ntermedte (or moderte), nd the chnge n rel rewrd for lnd nd cptl re extreme Dervton of the mportnt reltonshps mong some vrbles (H < 0, H 0 < 0) In ths subsecton, we derve the mportnt reltonshps mong some vrbles. The unt cost of producton of good equls the prce of good (see eq. (2)): w p, 1,2. (51) Eq. (51) expresses the socost surfce (or IC) nd the soqunt surfce (or IQ) (see Fg. 3). Defne tht 10

11 w w, w' w w, OA, OB, AB OB OA=, T, K, L, 1,2, (52) where denotes the smll vrton. Vector w s vertcl to the socost surfce. Becuse producton functons re homogeneous of degree one nd strctly qus-concve, the soqunt surfce s convex to the orgn. The soqunt surfce touches the socost surfce t pont A. Tht pont s the equlbrum pont. (or the nput-output coeffcent) s determned by ths pont. We drw ths fgure by nlogy from the fgure of the socost lne nd the soqunt curve for the two-fctor cse. If the socost surfce chnges ts poston nd becomes IC, the equlbrum pont moves to pont B. Angles re the ngles between vectors w nd AB nd For ngles A, B, we obtn w ' nd BA, respectvely. A nd 0 A,0 B. (53) 2 2 Hence, the nner product of vectors stsfes w AB w w AB cos A >0, (54) w' BA ( w w)( ) w' BA cos B >0. (55) Summng up eqs (54) nd (55), we hve H 0, 1, 2, (56) where H w. (57) B H s the nner product of two vectors, w ( w) nd AB( ). From eq. (56), we hve 5 H 0, 1, 2. (58) Trnsformng eq. (57), we cn express n elstcty forms: H. (59) w * *p, 1, 2 From eq. (58), we derve 5 Eq. (58) s very smlr to the equton whch Smuelson (1983, Chpter 4, p. 78, eq. (82)) n derved, tht s, wv 0, where w s the prce of fctor ; 1 tht mnmze the totl cost, but the uthor derved ths equton dfferently. v s the combnton of fctors 11

12 Defne tht 1 H p H 0 0, 1, 2. (60) 1 H. (61) p We cll ths the ggregte of H / p. From eqs (60) nd (61), we cn show tht H 0 0. (62) Substtutng eq. (59) n (61), we hve H 1 w * *p p * w *. (63) 0 Recll eq. (A3), tht s, 0' *, T, K, L. Usng eq. (A3), rewrte (63) to obtn H 0 0' w *( 0). (64) Recll eq. (5)( * 0, 1, 2. ). From eqs (5) nd (A3), we cn show tht 0' 0. (65) Eq. (65) mples tht ( T 0', K 0', L0') (,, )(,, ), ( T 0', K 0', L0') (,, ),(,, ),(,, ),(,, ),(,, ),(,, ). (66) One or two of 0' must be negtve. We cll these sgn ptterns the sgns A, B, C, D, E, nd F, respectvely. From eq. (65), we derve L0' L ( T 0' T K 0' K). (67) Substtutng eq. (67) n (64), we derve H 0 ( wt * wl*) T 0' T ( wk * wl*) K 0' K <0. (68) Smlrly, we derve H 0 ( wt * wk*) T 0' T ( wl * wk*) L0' L <0, (69) H 0 ( wk * wt*) K 0' K ( wl* wt*) L0' L <0. (70) Eqs (68)-(70) re ust lke constrnts on some vrbles ( w * w h* nd 0' ). We show ther mplctons. For exmple, we ssume P p1* p2* 0, nd (71) X Z Y wt* wl* wk *. (72) 12

13 Ths ssumpton s plusble (see eq. (49)). From eq. (72), we derve ( wt * wl*, wk * wl*) (, ), ( wt * wk*, wl * wk*) (, ), ( wk * wt*, wl * wt*) (, ). (73) Substtutng eq. (73) n (68), (69), nd (70), we obtn ( T0', K0') (, ), ( T0', L0') (, ), ( K0', L0') (, ). (74) From eq. (74), we hve ( T 0', K 0', L0') (,, ),(,, ). (75) Hence, the sgns E nd F re mpossble. The sgns A, B, C, nd D re possble (see eq. (66)). Tht s, A B C D ( T 0', K 0', L0') (,, ),(,, ),(,, ),(,, ). (76) Here recll (5), tht s, * 0, 1, 2. From ths, we derve * ( * *). (77) L L T T K K Substtutng (77) n (60), we hve 1 H (w T* w L*) T * T (w K* w L*) K * K 0, p ( 78) Smlrly, we derve 1 H ( wt * wk*) T * T ( wl* wk*) L * L <0, p (79) 1 H ( wk * wt*) K * K ( wl * wt*) L * L <0. p (80) From these, smlrly to (76), we cn show tht the sgns A, B, C, nd D re possble for sector. Tht s, A B C D ( T*, K*, L*) (,, ),(,, ),(,, ),(,, ), 1.2. (81) The followng result hs been estblshed. Lemm 1: We ssume the fctor ntensty rnkng nd the chnge n the reltve prce of goods s follows. T1 L1 K1, L1> L2, (82) T 2 L2 K 2 13

14 P p p 1* 2* 0. (83) Ths mples tht only four fctor-prce-chnge rnkngs re possble, tht s, X Y Z, X Z Y, Z X Y, Z Y X. (84) And, further, f we ssume X Z Y wt* wl* wk * (85) holds, the sgns A, B, C, D re possble. Tht s, A B C D ( T 0', K 0', L0') (,, ),(,, ),(,, ),(,, ), (86) ( T*, K*, L*) (,, ),(,, ),(,, ),(,, ), 1.2, (87) where 0' s the ggregte of * (or nput-output-coeffcent-chnge) Estmtng the poston of the EWS-rto vector In ths subsecton, we ssume eqs (82), (83), nd (85) n Lemm 1 hold. We estmte the poston of the EWS-rto vector. In subsecton 3.4.1, by nlyzng the Crtesn coordntes of pont A nd B, we derve suffcent condton for the EWS-rto vector to exst n qudrnt IV. In ths cse, extreme fctors re economy-wde complements. If ths holds, strong Rybczynsk result holds, tht s, three Stolper-Smuelson sgn ptterns hold. Addtonlly,, n subsecton 3.4.2, we derve suffcent condton for the EWS-rto vector to exst n ny subregon P1, P2, or P3. If ths holds, specfc Stolper-Smuelson sgn pttern holds A suffcent condton for extreme fctors to be economy wde complements For pont A, we derve (see eq. (38)) W W (, W W TL L LT KL K KT ) = (, ). (88) Hence, pont A s n qudrnt IV. For pont B, f sgn A, B, C, nd D hold, we derve s follows, A B C D 0 0 ( K ' K ' KT, ) = (, ), (, ), (, ),(, ). (89) T0' L0' 14

15 Hence, pont B s n qudrnt III, II, IV, nd IV. Usng eqs (88) nd (89), we cn plot pont A nd B n the fgure. Here, we ssume the sgn C holds, tht s, ( T 0', K 0', L0') (,, ). (90) Both ponts A nd B re n qudrnt IV. Therefore, lne segment AB exsts n qudrnt IV. Hence, the EWS-rto vector ( S', U ') exsts n qudrnt IV. Recll eq. (68), tht s, H 0 (w T* w L*) T 0' T (w K* w L*) K 0' K <0. Usng eq. (68), we cn show whch pont s on the left-hnd sde, pont A or B. Trnsformng eq. (68), we hve ( wt * wl*) K 0' K ( wk * wl*) T 0' T H 0 ( wk * wl*) T 0' T{ } WTL K 0 ( w * *) ' K wl T 0' T{ KT} 0. (91) WKL T 0' From eqs (72) nd (90), we hve ( wk* wl*) ( ), T 0' ( ). (92) From eqs (91) nd (92), we derve W W Ths mples 0 ' ' TL K 0 KT >0. (93) KL T 0 W W ' ' TL K 0 KT. (94) KL T 0 Hence, pont A s on the left-hnd sde of pont B (see Fg. 1). We hve From eq. (94) nd Fg. 1, the EWS-rto vector ( S', U ') stsfes W 0 ' W ' ' TL S K 0 KT, KL T 0 L W LT K 0 0 U ' '. (95) K W KT L0' ( S', U') (, ) ( S, T, U) (,, ). (96) Ths mples tht fctors cptl nd lnd, extreme fctors, re economy-wde complements. In summry, the followng result hs been estblshed. Theorem 2. We ssume the fctor ntensty rnkng s follows. T1 L1 K1, T 2 L2 K 2 (97) L1 L2. (98) 15

16 The EWS-rto vector ( S', U ') exsts on the EWS-rto vector lne segment (or lne segment AB). Usng ths reltonshp, we cn estmte the poston of the EWS-rto vector. For exmple, f we ssume (from Lemm 1, these ssumptons re plusble enough) P 0, X Z Y w T* w L* w K*, ( T 0', K 0', L0') (,, ), (99) the Crtesn coordntes of ntersecton pont A nd B re, respectvely (see lso Fg. 1), W TL L W LT ( wt * wl*) L ( wl* wt*) (, ) (, ) = (, ), (100) WKL K WKT ( wk * wl*) K ( wk * wt*) 0 0 ( K ' K ' KT, ) = (, ). (101) T0' L0' Hence, both of ponts A nd B re n qudrnt IV. And, pont A s on the left-hnd sde of pont B. The EWS-rto vector re n qudrnt IV nd stsfes W 0 ' W ' ' TL S K 0 KT, KL T 0 L W LT K 0 0 U ' ', nd (102) K W KT L0' ( S', U') (, ) ( S, T, U) (,, ). (103) Ths mples tht cptl nd lnd, extreme fctors, re economy-wde complements. Eq. (99) mples the followng. The reltve prce of commodty prce ncreses. The rte of chnge n rel rewrd for lbor s ntermedte (or moderte), nd the rte of chnge n rel rewrd for lnd nd cptl re extreme. Both of the sgns of the ggregte of T * nd K * (or the rte of chnge n the nput output coeffcents of lnd nd cptl) re postve, nd the sgn of the ggregte of L * (or the rte of chnge n the nput output coeffcent of lbor) s negtve. Smlrly, f sgn D holds, both ponts A nd B re n qudrnt IV. We cn esly show tht pont A s on the rght-hnd sde of pont B by n nlyss smlr to the bove. However, we omt ths nlyss A suffcent condton for specfc Stolper-Smuelson sgn pttern to hold In ths subsecton, we ssume eq. (99) holds, hence, eqs (102) nd (103) hold. The followng result hs been estblshed (see Nkd (2016, Theorem 1)). Theorem 1. We ssume the fctor-ntensty rnkng s follows. 16

17 T1 L1 K1, (104) T 2 L2 K 2 L1 1L1L2. (105) L2 Further, f the EWS-rto vector S, U exsts n qudrnt IV (or subregons P1-P3), n other words, f extreme fctors re economy-wde complements, strong Rybczynsk result necessrly holds. In ths cse, the Rybczynsk sgn ptterns, s per Thompson s (1985, p. 619) termnology for subregons P1-P3 re, respectvely: s gn X*/ V* = P1 P2 P3. (106) The Stolper-Smuelson sgn ptterns for subregons P1-P3 re, respectvely: w* p* sgn[ ] P =. (107) By comprng the Crtesn coordntes of Ponts R L2 nd R L1 wth the Crtesn coordntes of Ponts A nd B, we cn show some exmples of suffcent condton for specfc Stolper-Smuelson sgn pttern to hold. We use eq. (102). The Crtesn coordntes of Ponts R L2 nd R L1 re, respectvely (Nkd (2016, eq. (71), Fg.1): S, U = (, 1 K1 K1 T1 L1, 1 L ), ( K2 K2 K T2 L2 L K ). (108) We derve the followng result. Proposton 1: We ssume ll the equtons n Theorem 2 hold. () Exmple 1: If the equton shown below holds, K 2 TL K 0 T 2 W S ' ' KT, (109) WKL T 0' both Ponts A nd B exst n subregon P1. Hence, the EWS-rto vector exsts n subregon P1. The suffcent condton for eq. (109) s, 17

18 K 2 T 2 W TL wl* p2* ( ) 0. (110) WKL () Exmple 2: If the equton shown below holds, W ', (111) W ' K1 TL K 0 K 2 S ' KT T1 KL T 0 T 2 both Ponts A nd B exst n subregon P2. Hence, the EWS-rto vector exsts n subregon P2. The suffcent condton for eq. (111) s the set of equtons shown below. W W TL KL K1 T1 K 2 wl* p2* ( ) 0, (112) T 2 WTL wl* p1* ( ) 0, nd (113) WKL '. (114) ' K 0 K K 2 T 0 T T 2 () Exmple 3: If the equton shown below holds, W 0 ' W ' ' TL K 0 K1 S KT, (115) KL T 0 T1 both Ponts A nd B exst n subregon P3. Hence, the EWS-rto vector exsts n subregon P3. The suffcent condton for eq. (115) s the set of equtons shown below. W W TL KL K1 wl* p1* ( ) 0, (116) T1 WTL 0, nd (117) WKL '. (118) ' K 0 K K1 T 0 T T1 We do not show the proof of ll the equtons bove. For exmple, we show the proof of eq. (110). Multplyng eq. (110) by W KL (<0), we hve K 2WKL WTLT 2 K 2( wk * wl*) ( wt * wl*) T 2 2 w * wl * 0. (119) Recll eq. (6)( w* p *. ). Usng eq. (6), trnsform (119) to derve p2* wl* 0 wl* p2* ( ) 0. (120) Apprently, from (85), (117) holds. We omt the proof of other equtons. 18

19 Recll eq. (65). From ths, we hve K0 ' K ( L0 ' L T 0' T). Substtutng ths n the left-hnd sde of eq. (114), we hve K 0' K L0' L T 0' T The left-hnd sde of (114) = L0' L = 1 T 0' T T 0' T T0 ' T * L L L ( L1 L1* L2 L2*) ( 1 1 * *) 2 1 L L L L L 1 1 TT * T ( T1T 1* T 2 T 2*) T ( T1 T1 1* T 2 T 2 *) 2. (121) Eq. (121) seems useful for pplcton. The left-hnd sde of (118) s equvlent to (121) Some mplctons We cn show some mplctons of the EWS-rto lne segment nlyss. We ssume ll the ssumptons shown n Theorem 2 hold. Hence, extreme fctors must be economy-wde complements. For exmple, Thompson nd Bn ssumed s follows. () Thompson (1995) ssumed tht producton functons re of Cobb-Dougls type nd n ll constnt CES type. () Bn (2007, 2008) ssumed tht producton functons re of the two-level CES type. See lso Bn (2007b). 6 However, re these studes plusble? 7 We cn show tht they re mplusble. 6 In hs model, three fctors re sklled lbor, cptl, nd unsklled lbor. Bn (2007) ssumed tht sklled lbor nd cptl could be [Allen-] complements n ech sector, nd he computed the vlues of AESs theoretclly. Bn (2007) ttempted to nlyze how commodty prces ffect reltve fctor prces. He descrbed the effects when he chnged fctor-ntensty rnkng. However, hs nlyss s somewht complcted, nd hs results re not cler. Ths s theoretcl study. Bn (2008, p. 4, Tble 1) showed tble clssfyng the results n Bn (2007) by fctor-ntensty rnkng. He clssfed the countres n the world nto 14 regons n totl nd computed the fctor-ntensty for ech re usng the GTAP verson 6 dtbse. Addtonlly, he ssumed 10 types of vlues for the elstctes of substtuton (equvlent to EWS) to smulte how commodty prces ffect the reltve fctor prces. Ths s n pplcton. Bn (2011, chpter 4, p ) summrzed Bn (2007) nd Bn (2008) nd modfed the studes. For hs results, see Bn (2011, p , Tble 4 1). On ths, see Nkd (2016). 7 Of course, n norml CGE (or computble generl equlbrum) nlyss, t s usul to ssume Cobb-Dougls type nd CES type. On ths, for exmple, see Bergmn (2005. P ). 19

20 We cn show tht t s not plusble to ssume s n (). It s becuse they do not llow ny two fctors to be Allen-complements. Moreover, we cn show tht t s not plusble to ssume s n (). I show the proof. Replce sklled lbor, cptl, nd unsklled lbor (S, K, L) n Bn (2007) wth lnd, cptl, nd lbor (T, K, L), respectvely. Bn (2007, 2008) chnged fctor-ntensty rnkng, but ths seems confusng. We ssume the fctor-ntensty rnkng s constnt s shown n (21) nd (22). And, we ssume T nd K could be [Allen-] complements. The vlues of AES re ( K, L, T ) ( c, c, T ), 1,2, (122) L T K K where c s constnt (>0); T T K >0, or K <0. Recll eqs (26), (15), nd (9), tht s, S, U S / T, U / T g / g, g / g, (123) LK LT KT LT g,,, h h, h T K L, (124) log / log w. (125) h h h h g h s the ggregte of h. Hence, we derve L K L K K T = (, ) L L S, U S / T, U / T L T L T (126) L K L K K KT T = (, ). (127) L L L T T L T T The EWS-rto vector contns AESs. Becuse nd re constnt, S nd T re constnt (>0). Hence, S s constnt (>0). K However, U cn chnge (>0, or <0). Tht s, f we chnge the vlue of T, the vlue of U lso chnges. The vlue of the EWS-rto vector s S, U ( d,?), (128) where d s constnt (>0). Hence, the EWS-rto vector s on the lne S d. (129) 20

21 However, s ths plusble? In ths rtcle, we hve shown tht the EWS-rto vector exsts on the lne segment AB (see (38)). For exmple, we ssume the ssumptons shown n Theorem 2 hold. Therefore, the EWSrto vector re n qudrnt IV nd stsfes (102) nd (103). The lne segment AB exts n qudrnt IV (see Fg. 1). And t mght () on the lne S () not on. d, or In cse of (), t ons only t one pont. Bn (2008) ssumed 10 types of vlues for the elstctes of substtuton (equvlent to EWS) to smulte how commodty prces ffect the reltve fctor prces. Only one vlue mght be plusble. Apprently, ths smulton seems questonble. In cse of (), the vlue of S, U on S d s not possble t ll. In summry, f we wnt to do plusble smulton, we should substtute the vlue of EWSrto vector, whch exsts on the lne segment AB. 4. Concluson In ths rtcle, we hve ssumed certn pttern of fctor-ntensty rnkng ncludng certn pttern of fctor-ntensty rnkng of the mddle fctor. In the Introducton, I posed the followng questons. () How cn we estmte the poston of the EWS-rto vector? () Under wht condtons does the EWS-rto vector exst n qudrnt IV; n other words, re extreme fctors economy-wde complements? We derve the results s follows. Answer to (): We hve shown tht the EWS-rto vector exsts on the lne segment AB (or the EWSrto vector lne segment). Ponts A nd B re the ntersecton ponts of the EWS-rto vector lne nd the EWS-rto vector boundry. Usng ths reltonshp, we hve developed method to estmte the poston of the EWS-rto vector. Tht s, wth the pproprte dt, we know the poston of pont A nd B, hence, we lso know the poston of the EWS-rto vector to some extent. Answer to (): Frst, we derve suffcent condton for the EWS-rto vector to exst n qudrnt IV (tht s, n subregon P1, P2, or P3). In ths cse, cptl nd lnd, extreme fctors, re economy-wde complements (see Theorem 2). If ths holds, from Theorem 1 n Nkd (2016), strong Rybczynsk result holds necessrly, tht s, three of the Stolper-Smuelson sgn ptterns hold. We cll ths rough estmte. 21

22 Addtonlly, we derve suffcent condton for the EWS-rto vector to exst n ny subregon P1, P2, or P3. If ths holds, specfc Stolper-Smuelson sgn pttern holds (see Proposton 1). If we use ths property, we cn conduct detled estmte. To derve the suffcent condton shown n (), we need dt on the chnge n some vrbles, whch requres dt for two tme ponts. Tht s, the sgn of the chnge n the reltve prce of commodty, the fctor-prce-chnge rnkng, nd the sgn of the ggregte of * (or the rte of chnge n the nput output coeffcent). On the other hnd, norml CGE (or computble generl equlbrum) nlyss only requres the dt for one tme-pont to estmte the vlue of bsc prmeters. To do detled estmte, we need more detled dt. Ths rtcle suggests the followng. In some cses, t s not plusble to ssume tht producton functons re Cobb-Dougls type, or ll-constnt CES type n ech sector, whch do not llow ny two fctors to be Allen-complements. Moreover, t s not plusble to ssume tht producton functons re the two-level CES type. Ths rtcle provdes bss for further pplctons. For exmple, ths rtcle contrbutes to the estmton of the Rybczynsk nd Stolper-Smuelson sgn pttern n some countres nd contrbutes to nterntonl nd development economcs. It s uncertn whether we cn reduce the rnge of the EWS-rto vector further. Appendx: Dervton of lner reltonshp between EWS-rtos: EWS-rto vector lne where Usng eq. (12), elmnte from eq. (13) to obtn (see BC (p. 33, note 6 n p. 24)) T* TK K( wk * wt*) TL L( wl* wt*), K* KL L( wl * wk*) KT T( w T* wk*), L* LT T( wt * wl*) LK K( wk * wl*), (A1) h h. Recll eq. (9), tht s, h log / log wh h h. eq. (A1) to derve where T* TK( wk * wt*) TL( wl * wt*), K* KL( wl * wk*) KT( w T* wk*), Usng ths, trnsform L* LT( wt * wl*) LK( wk * wl*), (A2) h h. Defne tht 0' *, T, K, L. (A3) Ths s the ggregte of * (or nput-output-coeffcent-chnge). Substtutng eq. (A2) n (A3), we derve, 22

23 K L, T T T 0' TT* T{ ( wk * wt*) ( wl * wt*)}, ' * { ( w * w *) ( w * w *)} K K K 0 K K K L L K T T K T K. (A4) L L L0' LL* L{ ( wt * wl*) ( wk * wl*)} Recll eq. (15), tht s, g,,,., h T K L Usng eq. (15), rewrte (A4) to h h hve T 0' gtk( wk * wt*) gtl( wl * wt*), (A5) K 0' gkl( wl * wk*) gkt( wt * wk*), (A6) L0' glt( wt *- wl*) glk( wk *- wl*). (A7) Recll eq. (17), tht s, gh ( h / ) gh. Usng ths, elmnte g TK,g TL,g KL from eqs (A5) nd (A6) to obtn T 0' KTgKT ( wk * wt*) LTgLT ( wl * wt*), (A8) K 0' LKgLK ( wl * wk*) gkt( w T* wk*), (A9) where h / h, h. (A10) In summry, from eqs (A8), (A9), nd (A7), we hve T 0' KTgKT ( wk * wt*) LTgLT ( wl * wt*), (A11) K 0' LKgLK ( wl * wk*) gkt( w T* wk*), (A12) L0' glt( wt * wl*) glk( wk * wl*). (A13) Multply eqs (A11), (A12), nd (A13) by glk T, glt K, gkt K, respectvely, nd tke the dfference to obtn T 0' glkt K 0' gltk ( wk * wt*)g 0, K 0' gltk L0 ' gktk ( wl * wk*)g 0, L0 ' gktk T 0' glkt ( wt * wl*)g 0, (A14), where G0 gktk ( glk glt) gltglkl ( 0). (A15) Eq. (A15) s derved from eq. (23). From eq. (A14), we hve G 0 ' g ' g ' g ' g ' g ' g.(a16) ( wk * wt*) w( L * wk *) w( T * wl *) T0 LK T 0K LT K 0 K LT K0 L KT K0 L KT0 K T LK T From eq. (A16), we hve ( T 0' glk T K 0' glt K)( wl * wk*) ( K 0' glt K L0' gkt K)( wk * wt*).(a17) 23

24 Dvde both sdes of eq. (A17) by g LT. Recll eqs (26), tht s, S U S T U T g g g g, /, / /, /. Usng these symbols, we hve LK LT KT LT ( T 0' S ' T K 0' K)( wl* wk*) ( K 0' K L0' U ' K)( wk * wt*) From eq. (A18), we derve where U ' 1. (A18) 1S ' b1, (A19) ' W ' W T0 T LK, b L0 K KT 1 K0' W LT, Wh w L0' W * w * h ( w / wh )*,, h T, K, L., (A20) h KT W h s the chnge n reltve fctor prce between fctors nd h. Eq. (A19) expresses the strght lne, whch we cll the EWS-rto vector lne. The EWS-rto vector ( S', U ') exsts on ths lne. Hence, U hs lner reltonshp wth S. References: Bn, H. (2007). Cptl-skll complementrty, fctor ntensty nd reltve fctor prces: model wth three fctors nd two goods. Kobe-gkun economc ppers 39, (n Jpnese). Bn, H. (2007b). Shhon-Gnou Roudou no Hoknse to Stolper-Smuelson Kok: 3 Yoso 2 Z Model n yoru Bunsek [Cptl-skll complementrty nd the Stolper-Smuelson effect: n nlyss by three-fctor two-commodty model]. Pper submtted for the Jpn Socety of Interntonl Economcs, The 66th Annul meetng /bnhkr-thess.pdf (n Jpnese). Bn, H. (2008). Cptl-skll complementrty nd the Stolper-Smuelson effect: n nlyss by three-fctor two-commodty model. Kobe-gkun economc ppers 40, 1-17 (n Jpnese). Bn, H. (2011) Gurobru Kez n Okeru Ouyou Ippn Knko Bunsek [Appled generl equlbrum n globl economy]. Koyo-shobo, Kyoto (n Jpnese). Btr, R.N. nd Css, F.R. (1976). A synthess of the Heckscher-Ohln nd the neoclsscl models of nterntonl trde. Journl of Interntonl Economcs, 6, Bergmn, L. (2005). CGE Modelng of Envronmentl Polcy nd Resource Mngement, n: K.-G. Mäler nd J.R. Vncent, eds., Hndbook of Envronmentl Economcs, Volume 3. Elsever, B.V., Amsterdm. Blss, C. (2003). A Specfc-Fctors Model wth Hstorcl Applcton. Revew of Interntonl Economcs, 11, Eston, S, T. (2008). Somethng new for somethng old: Reflectons on model wth three fctors nd two goods, n: S. Mrt, E.S.H. Yu, eds., Contemporry nd emergng ssues n trde theory nd polcy, Emerld group Ltd., Bngley. 24

25 Ide, T. (2009). The two commodtes nd three fctors model wth ncresng returns to scle technology: nother nterpretton of the Leontef prdox. Fukuok Unversty Revew of Economcs, 53, Jones, R.W. nd Eston, S.T. (1983). Fctor ntenstes nd fctor substtuton n generl equlbrum. Journl of Interntonl Economcs, 15, Nkd, Y. (2015) Comment on Suzuk's rebuttl of Btr nd Css. Avlble t Nkd, Y. (2016). Fctor endowment commodty output reltonshps n three-fctor two-good generl equlbrum trde model. Avlble t ReserchGte (forthcomng). Nkd, Y. (2016b) Dervng the fctor endowment commodty output reltonshp for Thlnd ( ) usng three-fctor two-good generl equlbrum trde model. Avlble t ReserchGte (forthcomng). Ruffn, R.J. (1981). Trde nd fctor movements wth three fctors nd two goods. Economcs Letters, 7, Smuelson, P.A. (1953). Foundtons of economc nlyss. Hrvrd Unversty Press. Smuelson, P.A. (1983). Foundtons of economc nlyss, enlrged edton. Hrvrd Unversty Press [Kez Bunsek no Kso, Zoho bn. Keso-shobo, Tokyo (trnslted by Sto Ryuzo n 1986 n Jpnese)]. Sto, R. nd Kozum, T. (1973). On the Elstctes of Substtuton nd Complementrty. Oxford Economc Ppers, 25, Suzuk, K. (1983). A synthess of the Heckscher-Ohln nd the neoclsscl models of nterntonl trde: comment. Journl of Interntonl Economcs, 14, Suzuk, K. (1987). Boek to Shgen Hbun [Trde nd Resource Allocton]. Yuhkku, Tokyo (n Jpnese). Tkym, A. (1982). On Theorems of Generl Compettve Equlbrum of Producton nd Trde A Survey of Some Recent Developments n the Theory of Interntonl Trde. Keo Economc Studes, 19, Termch, N. (1993). The Structure of Producton wth Two Goods nd Three Fctors. The bulletn of the Insttute for World Affrs nd Cultures, Kyoto Sngyo Unversty 13, (n Jpnese). Thompson, H. (1985). Complementrty n smple generl equlbrum producton model. Cndn Journl of Economcs, 17, Thompson, H. (1995). Fctor Intensty versus Fctor Substtuton n Specfed Generl Equlbrum Model. Journl of economc ntegrton, 10,

26 U'-xs qud. II qud. I O A B S'-xs qud. III qud. IV Fg.1 Illustrton of the EWS-rto boundry, EWS-rto vector lne nd lne segment AB 26

27 qud.ii Lne-X X,Y-xs 6 qud.i Lne-Z 4 2 Lne-Y Z-xs qud.iii qud.iv -2-4 Fg.2. Illustrton of Lnes X, Y, nd -6 Z (the chnge n rel fctor prce ) Note: (X, Y, Z)=(w T1 *, w K1 *, w L1 *) 27

28 k -xs Fg. 3. Isocost surfce nd Isoqunt surfce IC w w' A L -xs B IQ O IC' T -xs 28