3 The Life Cycle, Tax Policy,

Transcription

1 3 The Life Cycle, Tax Policy, and the Current Account C 1 + I 1 + C 2 + I 2 = Y 1 T 1 + Y 2 T 2. (1) G 1 + G 2 = T 1 + T 2, (2) C 1 + I 1 + C 2 + I 2 = Y 1 G 1 + Y 2 G 2, Foundations of International Macroeconomics (85) Chapter 3

2 S p = Y T C, S g = T G. B p t+1 Bp t = Y t + r t B p t T t C t I t, (3) s=t ( ) 1 s t (C s + I s ) = ()B p t + s=t ( ) 1 s t (Y s T s ). (4) Foundations of International Macroeconomics (86) Chapter 3

3 B g t+1 Bg t = T t + r t B g t G t (5) s=t ( ) 1 s t G s = ()B g t + s=t ( ) 1 s t T s, (6) B = B p + B g, (7) Foundations of International Macroeconomics (87) Chapter 3

4 s=t ( ) 1 s t (C s + I s ) = () B t + s=t ( ) 1 s t (Y s G s ). (8) U ( c y t, co t+1) = log ( c y t ) + β log ( c o t+1 ). (9) ct y + co t+1 = yy t τ t y + yo t+1 τ t+1 o (10) c o t+1 = () βcy t. (11) Foundations of International Macroeconomics (88) Chapter 3

5 c y t = ( ) ( 1 yt y τ y 1 + β t + yo t+1 τ o t+1 ), (12) ( β ct+1 o = () 1 + β ) ( yt y τt y + yo t+1 ) τo t+1. (13) C t = c y t + c o t. (14) B g t+1 Bg t = τ y t + τ o t + rb g t G t. (15) Foundations of International Macroeconomics (89) Chapter 3

6 s=t ( ) 1 s t G s = ()Bt g + s=t ( ) 1 s t ( τ y s + τo s ). (16) C = [ 1 + ()β ](y y τ y + yo τ o ) 1 + β 1+r. G=rB g + τ y + τ o, Foundations of International Macroeconomics (90) Chapter 3

7 [ 1 + ()β C = ](y y + yo G rτ y + rb g ) 1 + β CA t = B t+1 B t = B p t+1 + Bg t+1 ( B p t + B g t ) = ( B p t+1 Bp t ) + ( B g t+1 B g t ). (17) S y t = B p t+1. (18) S o t = S y t 1 = Bp t. (19) Foundations of International Macroeconomics (91) Chapter 3

8 S p t = S y t + S o t = B p t+1 Bp t, (20) B t+1 = B p t+1 + Bg t+1 = Sy t + B g t+1. (21) S y t = y y t τ y t c y t = ( ) β [(y y 1 + β t τt y (22) ) ( y o t+1 τt+1)] o = B p t+1 S p t = B p t+1 Bp t = S y t S y t 1 = ( ) β [ ( y y t τ y ) ( t y o 1 + β t+1 τ o )] (23) t+1 Foundations of International Macroeconomics (92) Chapter 3

9 c o 0 = c o 0 + d 2. (24) c y 0 = c y β = c y β ( 1 r ) d 2 ( ) 1 d 2. (25) c o 0 + c y 0 (c o 0 + cy 0 ) = [1 + ] 1 d (1 + β)() 2 (26) Foundations of International Macroeconomics (93) Chapter 3

10 c o 1 = c o 1 β + () 1+β ( 1 r ) d 2 ( ) β d = c1 o β 2 (27) ( )( ) rd 2 = ( ) (d 2r+r 2 ) 1+r 2 c y t =c y t ( 1 1+β ) ( ) (d 2r+r 2 ) 1+r 2 (28) Foundations of International Macroeconomics (94) Chapter 3

11 c o t ( ) β = ct o 1 + β (2r + r 2 ) ( ) d. (29) 2 c o 1 + c y 1 (c o 1 + cy 1 ) = [ ( ) β 1 + β ( β ) ( )] (d 2r + r 2 ) 2 (30) CA 0 CA 0 = [ c o 0 +c y 0 (c o 0 + cy 0 )] = [ 1+ ] 1 d (1+β)() 2 (31) Foundations of International Macroeconomics (95) Chapter 3

12 CA 1 CA 1 = r ( CA 0 CA ) 0 [ c o 1 + c y 1 (c o 1 + cy 1 )] CA 1 CA 1 = ( β ) 1+β (1+r) ( ) d 2 (32) CA/Y = (T G)/Y, R 2 = (4.06)(0.33) u (c y t ) = ( t+1)βu (c o t+1 ). Foundations of International Macroeconomics (96) Chapter 3

13 u (c t ) = βe t { (t+1 )u (c t+1 ) } ( ) β CA 0 CA 0 = dy, 1 + β ( ) β CA 1 CA 1 = dy 1+β S p t =B p t+1 Bp t =S y t S y t 1 = ( ) β [ ( y y 1+β t τt y ) ( y o t+1 τ o t+1)] (23) Foundations of International Macroeconomics (97) Chapter 3

14 S p t = ( β ) 1 + β { ( yt y τt y ) ( y o t+1 τt+1 o ) }{{} (1+β)S y t /β [( yt 1 y τ t 1 y ) ( y o t τt o )] }{{} (1+β)S y t 1 /β }. y o t+1 = (1 + e)yy t, y y t+1 = (1 + g)yy t. Foundations of International Macroeconomics (98) Chapter 3

15 = β Y t 1+β S p t ( eg ) 2 + e + g. d(s p t /Y t) de = β 1+β [ g(2 + g) (2 + e + g) 2 ] < 0. d(s p t /Y t) dg = β 1+β [ e(2 + e) (2 + e + g) 2 ], S p t Y t = ( Nt N t 1 ) s y N t y y + N t 1 y o = ns y (1 + n)y y (33) + yo, Foundations of International Macroeconomics (99) Chapter 3

16 d ( S p /Y ) dn = sy ( y y + y o) [ (1 + n)y y + y o] 2 > 0. S p /Y = z. (1.3)(0.25) S/Y = z, R 2 = (1.90)(0.47) Y t = A t F(K t,l t )=A t K α t L1 α t, (34) Foundations of International Macroeconomics (100) Chapter 3

17 N t = (1 + n)n t 1. (35) r = A t F K (K t, L t ) = αa t k α 1 t, (36) w t = A t F L (K t, L t ) = (1 α)a t k α t, (37) K(r, A t, N t ) = L t k(r, A t ) = N t k(r, A t ) Foundations of International Macroeconomics (101) Chapter 3

18 ( ) 1/1 α αat K(r, A t, N t ) = N t k(r, A t ) = N t (38) r w t = (1 α)a t k(r, A t ) α = (1 α)a t ( αat r ) α/1 α (39) S y t = B p t+1 + K t+1 = B t+1 + K t+1. (40) s y t = (1 + n)(b t+1 + k t+1 ), (41) Foundations of International Macroeconomics (102) Chapter 3

19 b = sy 1 + n k. (42) S y t + S o t N t + N t 1 = ( ) 1 + n 2 + n s y + ( 1 ) 2 + n s o. K t+1 K t N t + N t 1 = (1 + n)n 2 + n k. A t+1 = (1 + g) 1 α A t, (43) Foundations of International Macroeconomics (103) Chapter 3

20 K Y = α r I Y = 1 α N t+1a 1 1 α N t A 1 t t+1 K Y K Y = (n + g + ng)α r. (44) c y t = w t 1 + β, co t+1 = ()βw t. (45) 1 + β Foundations of International Macroeconomics (104) Chapter 3

21 s y t = w t w t 1 + β = βw t 1 + β 1 = β(1 α)a 1 α t 1 + β ( α r ) α 1 α N t s y t + N t 1 s o t Y t = β(1 α) 1 + β = S Y [ 1 ] 1 (1 + n)(1 + g) (46) B Y = β(1 α) (1 + β)(1 + n)(1 + g) α r. Foundations of International Macroeconomics (105) Chapter 3

22 CA Y = S Y I Y = (n + g + ng)b Y. I/Y = S/Y, R 2 = (0.02)(0.07) I/Y = S/Y, R 2 = (0.02)(0.09) U = (1 + β) log(w) + β log() Foundations of International Macroeconomics (106) Chapter 3

23 du dr = 1 + β w ( ) dw dr + β, du dr (1 + β) = k + w β = β+ β 1+r = βr < 0 (47) kdr + βwdr (1 + β)() = kdr+ kdr 1+r = rkdr Foundations of International Macroeconomics (107) Chapter 3

24 rkdr () [ 1 + ( 1 ) + ( ) ] = kdr du dr = (1+β) k + w β = βk k + b + β kdr + (k + b)dr, s y t = β 1 + β ( wt τt y ). (48) Foundations of International Macroeconomics (108) Chapter 3

25 w t = (1 α)k α t. (49) αk α 1 t = αk α 1 t = r t = α(k w t ) α 1, (50) K t+1 + K t+1 = N ts y t + N t sy t. (51) L t = N t, L t = N t. Foundations of International Macroeconomics (109) Chapter 3

26 s y t = β(1 α) 1 + β (kw t ) α, K t+1 + K t+1 = β(1 α) 1 + β ( Nt + N t ) (k w t ) α K t+1 + K t+1 N t + N t = (1 + n) K t+1 + K t+1 N t+1 + N t+1 = (1 + n)k w t+1 k w t+1 = β(1 α) (1 + n)(1 + β) (kw t ) α (k w t ). (52) Foundations of International Macroeconomics (110) Chapter 3

27 k w = [ ] 1 β(1 α) 1 α. (1 + n)(1 + β) α( k w ) α 1 = r= α(1 + n)(1 + β). (53) β(1 α) B g t+1 = ( t)b g t + N t τ y t, Foundations of International Macroeconomics (111) Chapter 3

28 d = Bg t+1 N t+1 = ( t ) N t d N t τt y N t+1 N t+1 = ( t) d τt y 1 + n τ y t = (r t n) d. (54) s y t = β 1 + β [ wt (r t n) d ]. s y t = β 1 + β { (1 α) ( k w t ) α [ α ( k w t ) α 1 n ] d} (55) Foundations of International Macroeconomics (112) Chapter 3

29 K t+1 + K t+1 Bg t+1 = N ts y t + N t sy t. kt+1 w = β{ (1 α)(kt w ) α x [ α(kt w ) α 1 n ] d } (1 + n)(1 + β) x d ( kt w, d ) x N t N t + N t U t = u(c t ) + βu t+1, (56) Foundations of International Macroeconomics (113) Chapter 3

30 ()H t + Y t T t = C t + H t+1, (57) H t+1 0. (58) s=t ( ) 1 s t C s = ()H t + s=t ( ) 1 s t (Y s T s ), (59) Foundations of International Macroeconomics (114) Chapter 3

31 U t+1 = u(c t+1 ) + βu t+2. U t = u(c t ) + βu(c t+1 ) + β 2 U t+2. U t = s=t β s t u(c s ) + lim s βs t U s. U t = s=t β s t u(c s ) (60) Foundations of International Macroeconomics (115) Chapter 3

32 U p a 1 > 0, U c a 1 0. U v t = s=t β s t log(c v s ), (61) s=t ( ) 1 s t cs v = ()bp,v t + s=t ( ) 1 s t ( y v s τv s ) (62) Foundations of International Macroeconomics (116) Chapter 3

33 b p,v v = 0. (63) c v t = (1 β) [ ()b p,v t + s=t ( ) 1 s t ( y v s τv s ) ] (64) c t = c0 t + nc 1 t + n(1 + n)c 2 t + +n(1 + n) t 1 c t t (1 + n) t (65) c t = (1 β) [ ()b p t + s=t ( ) 1 s t (y s τ s )] (66) Foundations of International Macroeconomics (117) Chapter 3

34 b p,v t+1 = ()bp,v t + y v t τ v t c v t, b p,0 t+1 + nbp,1 t+1 + +n(1 + n)t 1 b p,t t+1 (1 + n) t = ()b p t + y t τ t c t, (1 + n) [ ] b p,0 t+1 + nbp,1 t+1 + +n(1 + n)t 1 b p,t t+1 + n(1 + n)t b p,t+1 t+1 (1 + n) t+1 = (1 + n)b p t+1 Foundations of International Macroeconomics (118) Chapter 3

35 b p t+1 = ()bp t + y t τ t c t 1 + n (67) b p t+1 = [ ()β 1 + n ] b p t y t τ t (1 β) ( s t(ys 1 s=t 1+r) τ s ) n (68) b t+1 = [ ] ()β 1 + n b t + [ ] ()β 1 r(1 + n) ȳ. (69) Foundations of International Macroeconomics (119) Chapter 3

36 b = [ ()β 1 (1 + n) ()β ] ȳ r. (70) b t+1 = ()b t + y t c t 1 + n. c = (r n) b + ȳ. (71) y s = { ȳ (s = t), ȳ (s >t). (72) Foundations of International Macroeconomics (120) Chapter 3

37 [ ] ()β b t+1 = 1 + n + ȳ (1 β) b [ ȳ + s=t+1 ( 1 1+r) s t ȳ 1 + n ] = [ ] (1+ r)β 1+ n b + [ ] (1+ r)β 1 r(1+ n) ȳ + β 1 + n (ȳ ȳ) = b + β 1 + n (ȳ ȳ) > b, y t+1 = (1 + g)y t, Foundations of International Macroeconomics (121) Chapter 3

38 c t = (1 β) [ ()b t + ( ) ] y t r g b t+1 = [ ] ()β 1 + n b t + [ ] ()β (1 + g) (1 + n)(r g) y t. b t+1 y t+1 = [ ()β (1 + n)(1 + g) ] bt y t [ ] ()β (1 + g) + (1 + n)(1 + g)(r g) (73) b t+1 = b t 1 + n, Foundations of International Macroeconomics (122) Chapter 3

39 bt+1 g = () bg t + τ t g t, (74) 1 + n τ s = (r n) d + g s. (75) s=t ( ) 1 s t τ s = s=t ( ) 1 s t (r n) d + s=t ( ) 1 s t g s Foundations of International Macroeconomics (123) Chapter 3

40 c t = (1 β) [ ()(b t + d) + s=t ( ) 1 s t (y s g s ) ()(r n) r d ] = (1 β) [ () ( b t + n d r ) + s=t ( ) 1 s t (y s g s )]. Y = F(K,EL), (76) E t+1 = (1 + g)e t, E t+1 L t+1 = (1 + n)e t (1 + g)l t = (1 + z)e t L t, Foundations of International Macroeconomics (124) Chapter 3

41 1 + z (1 + n)(1 + g). y e = F(K/EL,1) f(k e ). K t+1 K t =F(K t,e t L t ) C t, k e t+1 = ke t +f(k e t ) ce t 1+z, Foundations of International Macroeconomics (125) Chapter 3

42 c e = f( k e ) z k e. (77) d c e d k e = 0 f ( k e ) = r=z. (78) B g t+1 = (1+r)t D. Foundations of International Macroeconomics (126) Chapter 3

43 p t+1 p t =, rk Y > zk Y. Foundations of International Macroeconomics (127) Chapter 3

44 Foundations of International Macroeconomics (128) Chapter 3 Table 3.1 Growth and Saving in the Seven Largest Industrial Countries, Period GNP Growth Net Private Saving Rate Rate Source: Guiso, Jappelli, and Terlizzese (1992). Growth rates are a simple average of period average growth rates for Canada, France, Italy, West Germany, Japan, the United Kingdom, and the United States, expressed in percent per year. Saving rates are a simple average of period average ratios of inflation-adjusted net private saving to net national product, expressed in percent.

45 Foundations of International Macroeconomics (129) Chapter 3 Table 3.2 Government Saving in the Main Industrial Countries Country 1960s 1970s 1980s Canada France n.a West Germany Italy Japan United Kingdom United States Source: Shafer, Elmeskov, and Tease (1992). Government budget surpluses are expressed as a percent of GNP.