7 Global Linkages and Economic Growth

7 Global Linkages and Economic Growth Y t = F(K t,e t L t ), (1) Y t C t = S t = sf(k t, E t L t ). (2) K t+1 K t = sf(k t, E t L t ) δk t, (3) Foundations of International Macroeconomics (297) Chapter 7

E t+1 = (1 + g)e t, (4) L t+1 = (1 + n)l t. (5) k e t = K t /E t L t. (6) K t+1 E t L t K t E t L t = sf(k t, E t L t ) E t L t δ K t E t L t (7) Foundations of International Macroeconomics (298) Chapter 7

K t+1 = K t+1 Et+1L t+1 = kt+1 e (1 + z) E t L t E t+1 L t+1 E t L t k e t+1 ke t = 1 1 + z [ sf (k e t ) (z + δ)k e t ], (8) sf ( k e ) = (z + δ) k e. (9) k e = ( ) 1 s 1 α. (10) z + δ Foundations of International Macroeconomics (299) Chapter 7

Y t L t = E t ( s ) α 1 α z + δ log Y t L t = log E 0 + t log(1 + g) + α 1 α log s α 1 α log(z + δ) log Y t L t = log E 0 + gt + α 1 α α 1 α log s log(n + g + δ). (11) Foundations of International Macroeconomics (300) Chapter 7

log Y t L t = 5.48 + 1.42 log s 1.97 log(n + g + δ), R 2 = 0.59. (1.59)(0.14) (0.56) (12) Y t = K α t H φ t (E t L t ) 1 α φ, y e t = (k e t )α (h e t )φ, (13) H t+1 H t = s h [K α t H φ t (E t L t ) 1 α φ ] δh t Foundations of International Macroeconomics (301) Chapter 7

h e t+1 he t = 1 1 + z {s h[(k e ) α (h e ) φ ] (z + δ)h e t }. (14) K t+1 K t = s k [K α t H φ t (E t L t ) 1 α φ ] δk t k e t+1 ke t = 1 1 + z {s k[(k e ) α (h e ) φ ] (z + δ)k e t }. (15) k e [ (s k ) 1 φ (s h ) φ z + δ ] 1 1 α φ, (16) Foundations of International Macroeconomics (302) Chapter 7

h e [ (s h ) 1 α (s k ) α z + δ ] 1 1 α φ. (17) log Y t L t = log E 0 + gt + log α 1 α φ log s k + α + φ (n + g + δ), 1 α φ φ 1 α φ log s h U t = L t s=t β s t (1 + n) s t u (c s ), (18) K t+1 = K t + F(K t,e t L t ) C t. Foundations of International Macroeconomics (303) Chapter 7

k t+1 k t = F(k t,e t ) c t 1+n nk t 1 + n, (19) u (c t ) = [1 + F K (k t+1, E t+1 )]βu (c t+1 ). (20) u (c) = c1 1 σ, (21) 1 1 σ c t+1 /c t = β σ [ 1 + F K (k t+1, E t+1 ) ] σ. (22) Foundations of International Macroeconomics (304) Chapter 7

β(1 + n)(1 + g) (σ 1)/σ < 1, (23) k e t+1 ke t = f(ke t ) ce t 1+z z 1+z ke t, (8 ) c e t+1 c e t = βσ [1 + f (k e t+1 )]σ 1 + g, (24) f ( k e ) = (1 + g)1/σ β 1. (25) Foundations of International Macroeconomics (305) Chapter 7

c e = f( k e ) z k e. (26) U t = L t s=t ( ) β s t (1 + n) s t u(c s ) 1 + n = L t s=t β s t u(c s ) (27) f ( k e ) = (1 + g)1/σ β/ (1 + n) 1 (28) Foundations of International Macroeconomics (306) Chapter 7

U v t = s=t β s t log(c v s ) (29) k v t+1 = (1 + r t)k v t + w t c v t, (30) c v t+1 c v t = (1 + r t+1 )β. (31) x t = x0 t + nx 1 t + n(1 + n)x 2 t +...+n(1 + n) t 1 x t t (1 + n) t. Foundations of International Macroeconomics (307) Chapter 7

k t+1 k t = f(k t) c t 1+n nk t 1 + n. (32) c 0 t+1 + nc1 t+1 +...+n(1 + n)t 1 c t t+1 (1 + n) t = (1 + r t+1 )βc t ( ) c t+1 = (1 + r t+1 )βc t n c t+1 c t+1 t+1. c t+1 c t+1 t+1 = (1 β)(1 + r t+1)k t+1. Foundations of International Macroeconomics (308) Chapter 7

c t+1 = (1 + r t+1 )βc t n(1 β)(1 + r t+1 )k t+1. c t+1 = [1 + f (k t+1 )][βc t n(1 β)k t+1 ]. (33) U t = s=t β s t log c n s, (34) h n t+1 = (1 + w s,t δ)h n t + w t c n t, (35) Foundations of International Macroeconomics (309) Chapter 7

c n t+1 c n t = (1 + w s,t+1 δ)β (36) w t = f(h t ) h t f (h t ), (37) w s,t = f (h t ), (38) h t+1 h t = f(h t ) δh t c t. (39) Foundations of International Macroeconomics (310) Chapter 7

c t+1 c t = [1 + f (h t+1 ) δ]β. (40) h = (f ) 1 ( 1 β β ) + δ. c = Ncn +Mc m N + M, (41) F(N h,n +M) F(N h,n) MF L (N h, N + M). Foundations of International Macroeconomics (311) Chapter 7

F(N h,n +M) F(N h,n) M >F L (N h, N + M). log ( Y1990 L 1990 ) log ( Y1950 L 1950 ) = 6.47 0.58 log (0.54) (0.06) ( Y1950 L 1950 ), R 2 = 0.83. log ( Y1979 N 1979 ) log ( Y1870 N 1870 ) = 8.46 1.00 log (0.09) ( Y1870 N 1870 ), R 2 = 0.88. k e = [ ] (1 τ) α 1/(1 α). (42) r w Foundations of International Macroeconomics (312) Chapter 7

k e t+1 ke t = s(ke t )α 1 + z z + δ 1 + z ke t, (43) k e = ( ) 1 s 1 α. z + δ k e t+1 k e = µ(k e t k e ), (44) Foundations of International Macroeconomics (313) Chapter 7

µ = [ 1 + sα( k e ) α 1 1 + z 1 + αz + (α 1)δ 1 + z ] z + δ 1 + z (45) µ = 1 + (1/3) [ (1.02)(1.01) 1 ] (2/3)(0.03) (1.02)(1.01) 0.96 Y t = K α t H φ t (E t L t ) 1 α φ, y e t = (k e t )α (h e t )φ. Foundations of International Macroeconomics (314) Chapter 7

r = α(k e t )α 1 (h e t )φ. k e t = αye t r. (46) y e t = χ(h e t )ν, (47) ν φ 1 α, Foundations of International Macroeconomics (315) Chapter 7

χ ( α r ) α 1 α. H t+1 H t + K t+1 K t + B t+1 B t = Y t + rb t C t δh t B t = K t. H t+1 H t =Y t rk t C t δh t Foundations of International Macroeconomics (316) Chapter 7

H t+1 H t = s(y t rk t ). h e t+1 he t = s (h e t )ν 1 + z z + δ 1 + z he t, (48) s s(1 α) ( α r ) α 1 α. h e = ( s z + δ ) 1/(1 ν), (49) Foundations of International Macroeconomics (317) Chapter 7

h e t+1 h e = µ (h e t h e ), (50) µ = [ 1 + s ν( h e ) ν 1 1 + z 1 + νz + (ν 1)δ 1 + z ] z + δ 1 + z (51) Y t = (K g t )φ K α (E t L t ) 1 α φ, Foundations of International Macroeconomics (318) Chapter 7

y t α k t (1 α) l t, y t = k α t. r d = f (k) δ = αk α 1 δ. (52) k t+1 + b t+1 = w t c y t, (53) Foundations of International Macroeconomics (319) Chapter 7

c o t+1 = (1 + rd t+1 )k t+1 + (1 + r)b t+1 (54) b t+1 ηw t. (55) U t = log(c y t ) + β log(co t+1 ) c y t + co t+1 1 + r d t+1 = w t ( r d t+1 r ) 1 + r d t+1 b t+1. (56) Foundations of International Macroeconomics (320) Chapter 7

s y t = w t c y t = βw t 1 + β ( r d t+1 r ) ηw t (1 + β) ( 1 + rt+1 d ). k t+1 = s y t b t+1 = [ ] β(1 + η) 1 + β + (1 + r)η (1 + β)(1 + rt+1 d ) w t (57) k t+1 = (1 α) [ β(1 + η) 1 + β ] (1 + r)η + (1 + β)αk α 1 t+1 k α t. (58) k d = [ ] 1 αβ(1 α)(1 + η) 1 α. (59) α(1 + β) η(1 α)(1 + r) Foundations of International Macroeconomics (321) Chapter 7

β w 1 + β + η w < k u, (60) βw 0 1 + β + ηw 0 k u, U t = s=t β s t c1 s 1 σ 1 1 σ, σ>0. (61) Foundations of International Macroeconomics (322) Chapter 7

( ct+1 ) 1 σ, (62) 1 + r t+1 = 1 β c t y t = Ak t, (63) r t+1 = A. (64) c t + i t = y t = Ak t, Foundations of International Macroeconomics (323) Chapter 7

c t+1 c t = [β(1 + A)] σ = 1 +ḡ. (65) i t = k t+1 k t =ḡk t = ḡ A y t, c t = A ḡ A y t. y j t =A ( k j t ) αk 1 α t, (66) Foundations of International Macroeconomics (324) Chapter 7

dy j dk j = αa ( k k j ) 1 α. (67) dy dk = αa = r. (68) c t+1 c t = [ β(1 + αa) ] σ = 1 +ḡ. y t =Ak t, Foundations of International Macroeconomics (325) Chapter 7

U t = E t { s=t β s t log C s }, (69) K t+1 = [ x t (1 + r t )+(1 x t )(1 + r) ] K t C t, (70) { } Ct 1 = (1 + r)βe t C t+1 (71) { 1 = βe t (1 + r t+1 ) C } t C t+1. (72) Foundations of International Macroeconomics (326) Chapter 7

C t = (1 β) [ x t (1 + r t )+(1 x t )(1 + r) ] K t (73) E t ( r t+1 ) r (1 + r)βcov t { Ct+1 C t } 1, r t+1 r (74) C t+1 C t 1 = β [ 1 + r + x( r t+1 r) ] 1, (75) E t ( r t+1 r) (1 + r)βcov t { β[1 + r + x( rt+1 r)] 1, r t+1 r } = x(1 + r)β 2 Var t ( r t +1 r). Foundations of International Macroeconomics (327) Chapter 7

x = E t ( r t+1 r) β 2 (1 + r)var t ( r t +1 r) (76) { } Ct+1 E t C t = [ Et ( r t+1 r) ] 2 β(1 + r)var t ( r t +1 r) + β(1 + r) (77) { C n } E t+1 t Ct n = [ Et ( r w t+1 r)] 2 β(1 + r)var t ( r w t +1 r) + β(1 + r) (78) y t y t 1 = a t a t 1 + α(k t k t 1 ) + (1 α)(l t l t 1 ) Foundations of International Macroeconomics (328) Chapter 7

Y t = L 1 α Y,t A t j=1 K α j,t, (79) Y = αl 1 α Y K j K α 1 j (80) αl 1 α Y Kj α 1 Kj =0 =. A t+1 A t =θa t L A,t, (81) Foundations of International Macroeconomics (329) Chapter 7

L = L A + L Y. (82) max {K j } L1 α Y A t j=1 A t Kj α j=1 p j K j, (83) p j = αl 1 α Y Kj α 1. (84) 1 α Y Kj α j = p jk j 1 + r K j = αl 1 + r K j. (85) Foundations of International Macroeconomics (330) Chapter 7

K = ( ) 1/(1 α) α 2 L Y, (86) 1 + r p = 1 + r α. (87) = p K 1 + r K = ( ) ( 1 α α 2 α 1 + r ) 1 1 α L Y (88) Foundations of International Macroeconomics (331) Chapter 7

p A = s=t (1 + r) s t = (1 + r) r. (89) ḡ = A t+1 A t A t = θ L A. (90) ( p A θal A ) L A = p A θa=(1 α)l α Y A K α = Y (91) L Y L Y = r θα, (92) Foundations of International Macroeconomics (332) Chapter 7

ḡ = θl r α. 1+ḡ= ( θl+ 1+α ) α 1+r α. (93) U t = s=t β s t c1 s 1 σ 1 1 σ, 1 + g = C t+1 C t = [(1 + r)β] σ, Foundations of International Macroeconomics (333) Chapter 7

1 + r = 1 β (1 + g)1 σ. (94) r = α(1 + θl β), (95) 1 + αβ ḡ = αβθl (1 β). (96) 1 + αβ θl> 1 β αβ Foundations of International Macroeconomics (334) Chapter 7

Y t = L 1 α Y,t A t K α t Y t = C t + A t+1 K t+1. ḡ plan = βθl (1 β). (97) ḡ = βθl (1 β), 1 + β Foundations of International Macroeconomics (335) Chapter 7

A t+1 A t = θa t L t, (98) Y t = A t L 1 α t. (99) C min = Y t L t, (100) A t = C min L α t. (101) Foundations of International Macroeconomics (336) Chapter 7

L α t+1 Lα t = θl 1+α t, L t+1 L t =(θl t +1) 1 α. (102) Y t = A t F(Z Y,t,L Y,t ), X t =A t F(Z X,t,L X,t ), L X +L Y = L, Foundations of International Macroeconomics (337) Chapter 7

Z X + Z Y = Z. A t+1 A t = θx t A t. U t =E t { s=t β s t log C s }, (103) Y t = A t K α t. Foundations of International Macroeconomics (338) Chapter 7

K t+1 = A t K α t C t. (104) 1 C t = βe t { } 1 + rt+1 C t+1. (105) r t+1 = αa t+1 K α 1 t+1 1, (106) 1 = βe t { αa t+1 K α 1 t+1 ( Ct C t+1 )}, (107) Foundations of International Macroeconomics (339) Chapter 7

C t = ωa t K α t (108) 1 A t K α t = βα K t+1, ω = 1 αβ. (109) K t+1 = αβa t K α t. Foundations of International Macroeconomics (340) Chapter 7

y t = χ 0 + αy t 1 + a t, (110) ȳ = χ 0 + ā 1 α. (111) ȳ = χ 0 + αȳ + ā y t ȳ = α(y t 1 ȳ) + (a t ā), (112) Foundations of International Macroeconomics (341) Chapter 7

y t ȳ = t s=1 α t s (a s ā) + (y 0 ȳ)α t. (113) Y t = A w t A t K t α, Y t = A w t A t ( K t ) α, α<1 K t+1 +K t+1 =Aw t [ At K α t +A t (K t )α] ( C t + C t ) (114) 1 C i t { ( )} = βe t αa w t+1 1 Aj t+1 (Kj t+1 )α 1 C i t+1 Foundations of International Macroeconomics (342) Chapter 7

C t = κ(1 αβ)y w t, (115) K t+1 = αβy w t, (116) E t { Y t+1 Yt+1 w } = K t+1 Kt+1 w. U t = E t { s=t β s t log C s }, Foundations of International Macroeconomics (343) Chapter 7

K t+1 K t = Y t C t. (117) Y t = K α t E t 1 α, (118) 1 C t = βe t { } 1 + rt+1 C t+1, (119) 1 + r t+1 =1+α ( Kt+1 E t+1 ) α 1, (120) Foundations of International Macroeconomics (344) Chapter 7

1 β = 1 + r, (121) ( ) 1 Ē 1 β K = 1 α, (122) βα Ȳ K = 1 β βα. (123) C Ȳ = 1. (124) Foundations of International Macroeconomics (345) Chapter 7

y t = αk t + (1 α)e t. (125) dk t+1 dk t = α + (1 α) ) 1 α (Ē dk t K ) α (Ē de t dc t K Foundations of International Macroeconomics (346) Chapter 7

dk t+1 K = 1 + α ) 1 α (Ē dk t K K + (1 α) (Ē K ) 1 α det Ē C K dc t C k t+1 = 1 β k t 1 β βα c t + (1 α)(1 β) e t βα (126) Foundations of International Macroeconomics (347) Chapter 7

1 C t 1 = βe t 1 { exp [ log ( )]} 1 + rt C t = β exp { E t 1 [log(1 + r t ) log(c t )] + 1 2 Var [log ( )]} 1 + rt C t E t 1 {log C t } log C t 1 = log β + 1 ( )] 1 [log 2 Var + rt + E t 1 log(1 + r t ) C t E t 1 {log C t log C} (log C t 1 log C) = E t 1 {log(1 + r t ) log(1 + r)}+χ 0 Foundations of International Macroeconomics (348) Chapter 7

E t 1 c t c t 1 = E t 1 r t + χ 0, E t 1 c t c t 1 = E t 1 r t. (127) 1 + r t =1+ r+α(α 1) ( K Ē ) α 1 ( dkt K de ) t Ē, r t = (1 α)(1 β) ( e t k t ). (128) Foundations of International Macroeconomics (349) Chapter 7

E t 1 c t c t 1 = (1 α)(1 β)(e t 1 e t k t ) (129) e t = ρe t 1 + ɛ t, (130) c t = a ck k t + a ce e t, (131) Foundations of International Macroeconomics (350) Chapter 7

[ 1 k t+1 = + β (1 β)a ck βα [ (1 α)(1 β) βα ] k t (1 β)a ] ce βα e t (132) a ck (k t+1 k t ) + a ce (E t e t+1 e t ) = (1 α)(1 β)(e t e t+1 k t+1 ) (133) Foundations of International Macroeconomics (351) Chapter 7

a ck [ 1 β β (1 β)a ] ck k t βα + a ck [ (1 α)(1 β) βα (1 β)a ] ce e t + a ce (ρ 1)e t βα = ρ(1 α)(1 β)e t (1 α)(1 β) [ 1 β (1 β)a ] ck k t βα (1 α)(1 β) [ (1 α)(1 β) βα (1 β)a ] ce e t (134) βα a 2 ck + [2α 1 + β(1 α)]a ck + α(1 α) = 0 (135) a ce = a ck(1 α) + (1 α) [ ρβα (1 α)(1 β) ] βα 1 β (ρ 1) [ a ck + (1 α)(1 β) ] Foundations of International Macroeconomics (352) Chapter 7

1 = βe t {[1 + α ( Et+1 ) 1 α ] ( Ct K t+1 C t+1 ) }, 1 = βe t ( ) E 1 α 1 + α t+1 K t+1 ( Ct C t+1 ). E t 1 c t c t 1 = (1 α)(1 β)(e t 1 e t k t ), E t 1 c t c t 1 =(1 α)(1 β)(e t 1 e t k t ) Foundations of International Macroeconomics (353) Chapter 7

E t 1 (e t k t ) = E t 1 (e t k t ). (136) K t+h K t = shf(k t, E t L t ) δhk t, (3 ) E t+h E t = ghe t, (4 ) L t+h L t = nhl t. (5 ) Foundations of International Macroeconomics (354) Chapter 7

k e t+h ke t h = sf (k e t ) (1 + nh)(1 + gh) (n + g + δ) + ngh (1 + nh)(1 + gh) ke t sf ( k e ) = [(n + g + δ) + ngh] k e. sf ( k e ) = (n + g + δ) k e, (9 ) K t = sf(k t, E t L t ) δk t, (3 ) Foundations of International Macroeconomics (355) Chapter 7

Ė t E t = g, (4 ) L t L t = n, (5 ) Ẋ t dx t dt X t+h X t = lim. h 0 h K t E t L t = sf(k t, E t L t ) E t L t δk t E t L t. Foundations of International Macroeconomics (356) Chapter 7

k e t = K t E t L t K t E t L t (Ėt + L ) t E t L t = K t E t L t k e t (g + n), k e t = sf (k e t ) (n + g + δ)ke t. E t = (1 + gh) t/h E 0, Foundations of International Macroeconomics (357) Chapter 7

lim E t = lim (1 + gh) t/h E 0 h 0 h 0 = lim n (1 + g n )tn E 0 = exp (gt) E 0 U t = s=t ( ) 1 (s t)/h u(c s )h, 1 + δh K t+h = K t + hf (K t ) hc t max {K s } s=t ( ) 1 (s t)/h [ Ks K s+h u 1 + δh h ] + F(K s ) h, K t given Foundations of International Macroeconomics (358) Chapter 7

u (C s ) = ( ) 1 [1 + hf (K ] s+h) u (C s+h) 1 + δh u (C s+h ) u (C s ) h = [ δ 1 + δh F ] (K s+h ) 1 + δh u (C s+h ) du (C s ) dc s dc s ds = u (C s )Ċ s = [ δ F (K s ) ] u (C s ). U t = t u(c s ) exp[ δ(s t)]ds Foundations of International Macroeconomics (359) Chapter 7

K s = F(K s ) C s. H(C s, K s, s) = u(c s ) + λ s [ F(Ks ) C s ], H C s = u (C s ) λ s = 0, λ s = δλ s H K s = λ s [δ F (K s )] λ s = u (C s )Ċ s = [δ F (K s )]u (C s ), Foundations of International Macroeconomics (360) Chapter 7

Y t = A t K α t L t 1 α, (137) U t = log(ct y ) + βe t log(ct+1 o ), (138) c o t+1 = (w t c y t )[x t+1(1 + r t+1 ) + (1 x t+1 )(1 + r t+1 )] (139) c y t = w t 1 + β. Foundations of International Macroeconomics (361) Chapter 7

s y t = βw t 1 + β. (140) K t+1 = L t s y t. k t+1 = βw t (1 + β)(1 + n). k t+1 = β(1 α)a tk α t (1 + β)(1 + n) (141) Foundations of International Macroeconomics (362) Chapter 7

k t+1 = log [ β(1 α) ] (1 + β)(1 + n) + αk t + a t (142) k t = y t a t α. y t = χ 0 + αy t 1 + a t, (143) χ 0 α log β(1 α) (1 + β)(1 + n). Foundations of International Macroeconomics (363) Chapter 7

Foundations of International Macroeconomics (364) Chapter 7 Table 7.1 Convergence in Output Per Capita, 1870 1979 Per Capita Growth in Per Capita 1870 Income Income (1870 1979, Country (1975 dollars) log difference 100) Australia 1,922 116 United Kingdom 1,214 145 Switzerland 1,118 174 Belgium 1,137 168 Netherlands 1,104 166 United States 1,038 207 Denmark 883 201 Canada 881 214 France 847 207 Austria 751 203 Italy 746 178 West Germany 731 223 Norway 665 228 Sweden 557 247 Finland 506 241 Japan 328 286 Sources: De Long (1988) and Maddison (1982).

Foundations of International Macroeconomics (365) Chapter 7 Table 7.2 Country The Once Rich Seven Per Capita Growth in Per Capita 1870 Income Income (1870 1979, (1975 dollars) log difference 100) New Zealand 981 157 Argentina 762 141 East Germany 741 199 Spain 728 176 Ireland 656 167 Portugal 637 150 Denmark 519 150 Source: De Long (1988).

Foundations of International Macroeconomics (366) Chapter 7 Table 7.3 Average Annual Total Factor Productivity Growth in East Asia and the G-7 Countries Annual Growth Country Period (percent) Hong Kong 1966 91 2.3 Singapore 1966 90 0.2 South Korea 1966 90 1.7 Taiwan 1966 90 2.1 Canada 1960 89 0.5 France 1960 89 1.5 Germany 1960 89 1.6 Italy 1960 89 2.0 Japan 1960 89 2.0 United Kingdom 1960 89 1.3 United States 1960 89 0.4 Source: Young (1995).

Foundations of International Macroeconomics (367) Chapter 7 Table 7.4 World Population Growth, 1,000,000 b.c. to 1990 Start of Population Population Growth Rate Period (millions) (percent per year) Major Calamities 1,000,000 0.125 0.0003 25,000 3.34 0.0020 5000 5 0.0562 2000 27 0.0873 500 100 0.1062 1 170 0.0559 200 190 0.0 400 190 0.0256 600 200 0.0477 800 220 0.0931 1000 265 0.1533 1200 360 0.0 Mongol invasions 1300 360 0.0282 Black Death 1400 350 0.2217 1600 545 0.1127 Thirty Years War, Ming dynasty fall 1700 610 0.3897 1800 900 0.5926 1900 1625 1.0125 1980 4450 1.8101