拉格朗日泛函理论与经济学应用 赵晓军王小华 北京大学经济学院工作论文 编号 : C-2013-013 2013 年 12 月 12 日 版权归作者所有
拉格朗日泛函理论与经济学应用 Lagrange Functional and It s Applications in Economics 赵晓军北京大学经济学院 Xiaojun ZHAO School of Economics, Peking University * 王小华中央财经大学中国经济与管理研究院 Xiaohua Wang CEMA, Central University of Finance and Economics * 通讯作者 : 赵晓军, 北京大学经济学院, 通讯地址 : 北京大学经济学院 318 室, 邮编 100871 Email:zhaoxiaojun@pku.edu.cn Corresponding author: Xiaojun Zhao, School of Economics, Peking University, Beijing, 100871, China. E-mail: zhaoxiaojun@gsm.pku.edu.cn.
拉格朗日泛函理论与经济学应用 Lagrange Functional and It s Applications in Economics 摘要 : 庞特里亚金留给人类的最大精神财富是最优控制原理, 而最优控制原理的核心是变分方法, 将最优化问题与变分法联系起来的就是拉格朗日泛函 由于拉格朗日泛函方法是传统拉格朗日函数的推广, 与普通优化方法不同的是所有的相关概念都将建立在函数空间 本文首先建立拉格朗日泛函的概念, 接着介绍泛函极值的求解, 通过变分的方法得到泛函的 `` 导数 (Frechet 导数 ) 最后, 本文将列举动态经济学的各类实例优化问题, 应用拉格朗日泛函进行求解 关键词 : 拉格朗日泛函 ;Frechet 导数 ; 变分 ;Rieze 表现定理 中图分类码 :F019.2;F062.5;F062.6 Abstract: The mental Legacy from Pontryajin is the principle of optimal control, while the core of the the principle of optimal control is the variation, and the Lagrange functional if the bridge between the optimization problem and the variation method. Since Lagrange functional is an extension of Lagrange functions, the corresponding conception should be based on the functional space. This paper begin with the construction of Lagrange functional, then derive the solution method to find the optimal points of certain functional, we reach the differential of functional (named as Frechet differential) through the variation method. Finally, we cite a lot of economic examples to apply the Lagrange functional method. Keywords: Lagrange Functional, Frechet Differentiation, Variation, Rieze Representative Theorem Chinese Literature Classification: F019.2;F062.5;F062.6
1 1 1 0 Frechet 1.1 Frechet (X, X )(Y, Y )NXY ΛXY ΛxFrechetA N Λ(x ) Λ(x) = A(x x) + o ( x x X ), x U(x). (1) U(x)xXAΛxFrechetδ x Λx XΛΛFrechet Y = RΛXΛ 1.1. Λ : X RXRx x U(x )Λ(x) Λ(x )Λ(x) Λ(x ) U(x ) δ x Λ = 0. (2). 1.1. X = C[0, 1[0,1ΛΛ(x) = f(x(1/2))frr δ x Λ(η) = f (x(1/2))η(1/2), η X. δ x Λ2.1δ x Λ = 0 ηf (x(1/2)) = 0 f (x(1/2))η(1/2) = 0, η X. 1.2. X(Ω, F, µ)frrλ(x) = fxdµ δ x Λ(η) = f (x)ηdµ, η X. δ x Λ = 0f (x) = 0 1.3. X(Ω, F, F t, µ)f ( t F = ) F t f t R RRΛ(x) = E f t (x t, y) y F ft (x t, y) δ xt Λ(η) = ηdµ, η F t. x t δ xt Λ = 0E t ( ft(x t,y) x t ) = 0
1 2 ΛXΛ 1.1. ΛXx, y Xλ [0, 1 Λ(λx + (1 λ)y) ( )λλ(x) + (1 λ)λ(y), Λx yλ (0, 1)Λ 1.2. ΛXx, y X Λ(x) Λ(y) ( )δ x Λ(x y).. X = R n 1.2 N T ΩN T = {1,..., T }Ω(Ω, F, µ) F = {F t : t N T }F F t µω M t = {x : Ω R, xx F t }, t N T, (3) M t x = max ω Ω x(ω) (M t, )Banach P1: max x t+1 M t,u t M t E [ T β t 1 f(x t, u t ), (4) x t+1 = g(x t, u t, ε t, t), t = 1,..., T. x 1 (5) x t u t β (0, 1)ε t M t f, g (x, u)p1 [ T T L (x, u, z) = E β t 1 f(x t, u t ) + β t 1 z t, g(x t, u t, ε t, t) x t+1, (6) z t M t, x, y = E(xy) = xydµ
1 3 1.3. (x t+1, u t ), t = 1,..., T P1zt M t, t = 1,..., T (x t+1, u t, zt ) L (x, u, z)( Saddle Point)ε > 0 L (x, u, z) L (x, u, z ) L (x, u, z ), (7) x B(x, ε), u B(u, ε)z B(z, ε)b(x, ε), B(u, ε)b(z, ε)x, u z ε. (x t+1, u t )P1x t+1 = g(x t, u t, ε t, t) [ T L (x, u, z) = L (x, u, z ) = E β t 1 f(x t, u t ), (8) (2.7) T T T M = M t M t, N = M t. t=0 M 0 T Λ 1 : Λ 1 (x, u) = E[ β t 1 f(x t, u t ) (9) Λ 2 : Λ 2 (x, u) t+1 = g(x t, u t, ε t, t) x t+1, t = 1,..., T 1. (10) f, gx, uλ 1 : M RΛ 2 : M N z N L (x, u, z) = Λ 1 (x, u) + z, Λ 2 (x, u). (11) f, g(x, u)λ 1 Λ 2 (x, u)frechetδ x,u Λ 1M RRiezeη Mξ M δ x,u Λ 1(ξ) = η, ξ. (12) δ x,u Λ 2MNAANM ξ Mζ N ζ, δ x,u Λ 2(ξ) = A(ζ), ξ. (13) Torres(1990)1573η Im(A)z NA( z ) = η L (x, u, z ) = Λ 1 (x, u) + z, Λ 2 (x, u). (14) m = (x, u) MΛ 1 1.2 Λ 1 (m ) Λ 1 (m) δ m Λ 1 (m m) = η, m m, (15) A( z ) = η η η, m m = A( z ), m m = z, δ m Λ 2 (m m), (16) z, Λ 2 (m) m1.2 z, Λ 2 (m ) z, Λ 2 (m) z, δ m Λ 2 (m m). (17)
2 RAMSEY 4 Λ 2 (m ) = 0 z, Λ 2 (m) z, δ m Λ 2 (m m), (18) (15)(16) Λ 1 (m ) Λ 1 (m) z, Λ 2 (m) (19) Λ 2 (m ) = 0(14) L (x, u, z ) L (x, u, z ). (20) 1.3FrechetP1δ x,u L (x, u, z) = 0 δ xt+1 L (x, u, z) = 0, t = 1,..., T. (21) (21) δ ut L (x, u, z) = 0, t = 1,..., T. (22) (z t β(f x (t + 1) + z t+1 g x (t + 1))) ηdµ = 0, η M t E t z t = βe t [f x (t + 1) + z t+1 g x (t + 1). (23) (22) f u (t) + z t g u (t) = 0. (24) (23)(24)P1 2 Ramsey RamseyHamiltionian Ramsey 2.1 Ramsey Ramsey P2: max c t,k t 0 e ρt u(c t )dt, (25)
2 RAMSEY 5 k t = f(k t ) c t, k 0 (26) M[0, ) c t, k t, u(c t ), f(k t ) M L cfrechet L L = δ c L = δ k L = 0 0 L (c, k, λ) = δ c L (ξ) = 0 0 e ρt [u(c t ) + λ t (f(k t ) c t k t ). (27) e ρt (u t λ t )ξ t dt, ξ M. (28) e ρt [u(c t ) λc t + λ t f(k t ) + ( λ ρλ t )k t + lim t e ρt λ t k t λ 0 k 0. (29) δ k L (η) = 0 e ρt (λ t f + λ ρλ t )η t dt, η M. (30) u = λ, (31) λ = λ t (ρ f ). (32) Hamiltionian Ramsey 2.2 ta t R t tx t w 0N t N t πx t Poisson tc t u(c t )t u(c t ) wx t 0 < β < 1 P2 : max c t,x t E β t (u(c t ) wx t ), (33) 0 R t+1 = R t c t + N t, 0 c t R t, A t+1 = A t x t, 0 x t A t, N t P (πx t ), R 0 = R, A 0 = A. c t, R t, N t x t, A t F t = {c 0,, c t }R t+1 F t L = E + β t [u(c t ) wx t + λ 1 t (R t c t + N t R t+1 ) + λ 2 t (R t c t ) + λ 3 t R t 0 β t [µ 1 t (A t x t A t+1 ) + µ 2 t (A t x t ) + µ 3 t A t. 0
3 6 Kuhn-Tuck L c t, R t, A t Frechet u = λ 1 t + λ 2 t, (34) λ 1 t = βe t [λ 1 t+1 + λ 2 t+1 + λ 3 t+1, (35) µ 1 t = β(µ 1 t+1 + µ 2 t+1 + µ 3 t+1). (36) δ xt L δ xt E[N t E[N t = πx t δ xt E[N t = π µ 1 t + µ 2 t = πe[λ 1 t w. (37) (P2 ) 2.1. (P2 )x t, c t {0, A t, R t }.. λ 2 t = µ 2 t = λ 3 t = µ 3 t = 0, λµ u = λ, λ t = βe t [λ t+1, µ t = πe[λ t w µ t = βµ t+1. π(e[λ t βe[λ t+1 ) = (1 β)w. (38) λ t = βe t [λ t+1 E[λ t = βe[λ t+1 (1 β)w = 0, (39) (P2 ) 3 Rogerson(1985a,1985b) Rogerson(1985a,1985b)
3 7 3.1 Rogerson(1985a,1985b) u(c, l)cll = y/θyθ 1 χ(θ j ) χ(θ j ) = E[χ(θ) = j χ(θ)dµ. (40) µθθt θ t (ω) (Ω, F, µ){f t } T Θ T θ t (ω) µθ T µθ t (ω)θθ = (θ 1,..., θ T ) K t Y t = y t (θ)dµt Q t = F (K t, Y t ) [ T T P3 : max β t 1 u(c t (θ j ), y t (θ j )/θ j t ) = max E β t 1 u(c t (θ), y t (θ)/θ t ), (41) c t,y t,k t+1 c t,y t,k t+1 j K t+1 = F (K t, y t (θ)dµ) + (1 δ)k t c t (θ)dµ, K 1 (42) P3 u c (t) = β(f k (t) + 1 δ)e t (u c (t + 1)), (43) u l (t) = F y (t)u c (t). θ t (44) P3 (c t, y t )θ U(θ) = T β t 1 u(c t (θ), yt (θ)/θ t ), U(θ)[W, W W < W 0 < W Pr(U(θ) < W 0 ) > 0W 0 P3 T β t 1 u(c t (θ), y t (θ)/θ t ) W 0, θ Θ T,
3 8 P3P4 [ T T P4 : max β t 1 u(c t (θ j ), y t (θ j )/θ j t ) = max E β t 1 u(c t (θ), y t (θ)/θ t ), (45) c t,y t,k t+1 c t,y t,k t+1 j K t+1 = F (K t, y t (θ)dµ) + (1 δ)k t c t (θ)dµ, (46) T β t 1 u(c t (θ), y t (θ)/θ t ) W 0, θ Θ T, K 1 (47) P4P3P4θ Θ T P4 (46)W 0 P4P3 P3P4 P4 M t = {uu F t }, t = 1,..., T. (48) (c t, y t, K t ) t T P4t < T ξ t, η t, ξ t+1, η t+1 c t, y t, c t+1, y t+1 χ t t 1. tt + 1 2. t + 2K t+2 χ t ξ t, η t F t ξ t+1, η t+1 F t+1 tt + 1 R t = χ t + F (Kt, yt (θ)dµ) F (Kt, (yt (θ) + η t )dµ) + ξ t dµ. (49) Kt+2t + 1c t+1yt+1 F (Kt+1 + χ t, (yt+1(θ) + η t+1 )dµ) + (1 δ)χ t ξ t+1 dµ = F (Kt+1, y t+1(θ)dµ), (50) P4 P4(Dual): min R t (51) χ t,ξ t,η t,ξ t+1,η t+1 u(c t (θ) + ξ t, (y t (θ) + η t )/θ t ) + βu(c t+1(θ) + ξ t+1, (y t+1(θ) + η t+1 )/θ t+1 ) = u(c t (θ), y t (θ)/θ t ) + βu(c t+1(θ), y t+1(θ)/θ t+1 ), (52) F (Kt+1 + χ t, (yt+1(θ) + η t+1 )dµ) + (1 δ)χ t ξ t+1 dµ = F (Kt+1, y t+1(θ)dµ). (53) Zhao and Gong(2010d)
3 9 3.1. P4c t, y t, K t (0,0,0,0,0)P4(Dual) P4(Dual)z t+1 M t+1, λ R L = R t + λ, F (Kt+1 + χ t, (yt+1(θ) + η t+1 )dµ) (54) + λ, (1 δ)χ t ξ t+1 dµ 1.3Frechet + z t+1, u(c t (θ) + ξ t, (y t (θ) + η t )/θ t ) + z t+1, βu(c t+1(θ) + ξ t+1, (y t+1(θ) + η t+1 )/θ t+1 ). 1 δ χt L = δ ξt L = δ ηt L = δ ξt+1 L = δ ηt+1 L = 0, Frechet(χ t, ξ t, η t, ξ t+1, η t+1 )(0,0,0,0,0) 1 λ = F k (t) + 1 δ, (55) η u c (t) dµ + z t+1, η = 0, η M t, (56) λ η 1 β u c (t + 1) dµ + z t+1, η 1 = 0, η 1 M t+1, (57) F Y (t) ξθ t /u y (t)dµ + z t+1, ξ = 0, ξ M t, (58) λ θ t+1 F Y (t + 1) β u y (t + 1) ξ 1dµ + z t+1, ξ 1 = 0, ξ 1 M t+1, (59) M t M t+1 (57)(58) 1 u c (t) = 1 β(f k (t) + 1 δ) E t ( ) 1, (60) u c (t + 1) Golosovet al.(2003)(59) u y (t)/θ t = F Y (t)u c (t). (61) (60)(61) P4zhao and Gong(2010d) P4(47) T = P5 K t+1 = F (K t, P5 : max E c t,y t,k t+1 [ β t 1 u(c t (θ), y t (θ)/θ t ), (62) y t (θ)dµ) + (1 δ)k t c t (θ)dµ, K 1 (63)
3 10 β τ t 1 u(c τ (θ), y τ (θ)/θ τ ) β τ t u(c τ (θ), y τ (θ)/θ τ ), θ Θ ω. 2 (64) τ=t+1 τ=t (χ t, ξ t, η t, ξ t+1, η t+1 ) 1. tt + 1 2. t + 2K t+2 P4P4 P5(Dual): min R t (65) χ t,ξ t,η t,ξ t+1,η t+1 u(c t (θ) + ξ t, (y t (θ) + η t )/θ t ) = u(c t (θ), y t (θ)/θ t ), (66) u(c t+1(θ) + ξ t+1, (y t+1(θ) + η t+1 )/θ t+1 ) = u(c t+1(θ), y t+1(θ)/θ t+1 ), (67) F (Kt+1 + χ t, (yt+1(θ) + η t+1 )dµ) + (1 δ)χ t ξ t+1 dµ = F (Kt+1, y t+1(θ)dµ). (68) (66)(67)tt + 1(68)t + 2K t+2 (3.1) 3.2. P5c t, y t, K t (0,0,0,0,0)P5(Dual) (3.2)(3.1)P5(Dual)P4(Dual) λ R, z t M t z t+1 M t+1 L = R t + λ, F (Kt+1 + χ t, (yt+1(θ) + η t+1 )dµ) (69) + λ, (1 δ)χ t ξ t+1 dµ + z t, u(c t (θ) + ξ t, (y t (θ) + η t )/θ t ) + z t+1, u(c t+1(θ) + ξ t+1, (y t+1(θ) + η t+1 )/θ t+1 ). (54)χ t, ξ t, η t, ξ t+1, η t+1 Frechet 1 λ = F k (t) + 1 δ, (70) η u c (t) dµ + z t, η = 0, η M t, (71) η 1 λ u c (t + 1) dµ + z t+1, η 1 = 0, η 1 M t+1, (72) F Y (t) ξθ t /u y (t)dµ + z t, ξ = 0, ξ M t, (73) θ t+1 λf Y (t + 1) u y (t + 1) ξ 1dµ + z t+1, ξ 1 = 0, ξ 1 M t+1, (74) 2 Θ ω Θ
3 11 (71)(73)(72)(74) u y (t)/θ t = F Y (t)u c (t), u y (t + 1)/θ t+1 = F Y (t + 1)u c (t + 1). (75) P5P3 (73)η 1 = z t /z t+1 η 2 z t η 2 λ z t+1 u c (t + 1) dµ + z t, η 2 = 0, η 2 M t+1. (76) η 2 = η(72) ( ) 1 u c (t) = 1 F k (t) + 1 δ E zt 1 t. (77) z t+1 u c (t + 1) (60)P4P5 P5 3 P4 P5 P5 3.2 ABABA ABGolosovat.al.(2003)Rogerson(1985a,1985b) AB Mirrlees(1971,1976)Diamond and Mirrlees(1986) [0,1j [0, 1 tθ j t θ j t j{θ t } 1 t T (Ω, F, µ)f t F F = T F t tθ(x, u)f(x, u, θ, t)x u [ T T β t 1 f(x j t, u j t, θ j t, t) = max E θ β t 1 f(x t, u t, θ t, t). (78) x t,u t max x j t,uj t j [0,1 E θ {θ t } 1 t T β (0, 1) x t+1 = g(x t, Λ(u t, θ t ), t); t = 1,..., T. (79) 3 P5z tz t+1 P4z t+1
3 12 Λ(u t, θ t ) = (Λ 1 (u t, θ t ),..., Λ m (u t, θ t ))u t θ t (IC;Incentive Compatible) σ : Θ T Θ T Σ σθ Θ T T W (σ, θ) = β t 1 f(x t (σ(θ)), u t (σ(θ)), θ t, t), (80) σ σ (θ) = θθ Θ T IC W (σ, θ) W (σ, θ) θ Θ T, σ Σ. (81) (78)(79)(81)P6 [ T P6: max E θ β t 1 f(x t, u t, θ t, t), (82) x t,u t (79)(81)x 1 x t Ru t R n 3.3 P6 f(x, u, θ, t) = f 1 (x, u 1, t) + f 2 (x, u 1, t). (83) u 1 = (u 2,..., u n ) R n 1 (x t, u t ) : t = 1,..., T P6 M t tχ t ξ t ξ t+1 u 1,tu 1,t+1 1. tt + 1 2. t + 2K t+2 R t = χ t + g(x t, Λ(u t, θ t ), t) g(x t, Λ(u 1,t + ξ t, u 1,t, θ t ), t), (84) P6P6(Dual) P6(Dual): min R t, (85) χ t,ξ t,ξ t+1 g(x t+1 + χ t, Λ(u 1,t+1 + ξ t+1, u 1,t, θ t+1 ), t + 1) = g (t + 1), (86) f 1 (x t, u 1(t) + ξ t, t) + βf 1 (x t+1 + χ t, u 1,t+1 + ξ t+1, t + 1) = f 1 (t) + βf 1 (t + 1). (87) g (t + 1), f 1 (t)f 1 (t + 1)
3 13 3.3. P6x t, u t (0,0,0)P6(Dual). χ t, ξ t, ξ t+1 R t < 0 u 1(1),..., u 1(t 1), u 1,t + ξ t, u 1,t+1 + ξ t+1,..., u 1(T ), x 1,..., x t, x t+1,..., x T. u 1(79) IC(81)θσWnew(σ, θ) W opt (σ, θ)(87) Wnew(σ, θ) = W opt (σ, θ), (88) W opt (σ, θ) W opt (σ, θ) σ Σ, θ Θ T. (89) W opt (σ, θ) = f 1 (x t (σ(θ)), u 1(t)(σ(θ)), t) + βf 1 (x t+1(σ(θ)), u 1,t+1(σ(θ)), t + 1) + W, W W opt (σ, θ)w σ, θw (σ, θ) W+(σ, θ) = f 1 (x t (σ(θ)), u 1,t(σ(θ)) + ξ t, t) + βf 1 (x t+1(σ(θ)) + χ t, u 1,t+1(σ(θ)) + ξ t+1, t + 1), (87) W+(σ, θ) = f 1 (x t (σ(θ)), u 1,t(σ(θ)), t) + βf 1 (x t+1(σ(θ)), u 1,t+1(σ(θ)), t + 1) Wnew(σ, θ) = W +(σ, θ) + W (σ, θ) = W opt (σ, θ), (90) (88)-(90) Wnew(σ, θ) Wnew(σ, θ) σ Σ, θ Θ T. (91) (87) R t < 0 x t, u t 1. 3.13.23.3 3.3fu 1 θ (90)fkθ kk fcyθ f(c, y, θ) = u(c) v(y/θ),
3 14 λ R, z t+1 M t+1 L = R t λg(x t+1 + χ t, Λ(u 1,t+1 + ξ t+1, u 1,t, θ t+1 ), t + 1) (92) + z t+1, f 1 (x t, u 1(t) + ξ t, t) + βf 1 (x t+1 + χ t, u 1,t+1 + ξ t+1, t + 1). FrechetL χ t, ξ t, ξ t+1 1 λg x (t + 1) + β z t+1, f 11 (t + 1) = 0, (93) m g Λj (t)δ u1 Λ j (t), η + z t+1, f 12 (t)η = 0 η M t, (94) m λ g Λj (t + 1)δ u1 Λ j (t + 1), φ + β z t+1, f 12 (t + 1)φ = 0 φ M t+1. (95) f 1i f 1 i(93)(95) 1 λ = g x (t + 1) m. (96) g Λj (t + 1)δ u1 Λ j (t + 1), f 11 (t + 1)/f 12 (t + 1) (94)η = η /f 12 (t)(95)φ = φ /f 12 (t + 1) M t M t+1 m g Λj (t) f 12 (t) δ u 1 Λ j (t), η + z t+1, η = 0 η M t, (97) m g Λj (t + 1) λ f 12 (t + 1) δ u 1 Λ j (t + 1), φ + β z t+1, φ = 0 φ M t+1. (98) m g Λj (t) f 12 (t) δ u 1 Λ j (t), η = λ m β g Λj (t + 1) f 12 (t + 1) δ u 1 Λ j (t + 1), η η M t. (99) w j t δ u1 Λ j (t)(rieze Representative) m g Λj (t) f 12 (t) wj t ηdµ = λ β m m g Λj (t + 1) f 12 (t + 1) wj t+1 ηdµ η M t, g Λj (t) f 12 (t) wj t = λ [ β E gλj (t + 1) t f 12 (t + 1) wj t+1. (100) (96)λGolosovet al.(2003)(100) Golosovet al.(2003) P6
3 15 3.4 P6IC ICIC T T β t 1 f(x t (θ), u t (θ), θ t, t) β t 1 f(x t (θ ), u t (θ ), θ t, t) θ, θ Θ T. (101) λ t R(79)λ(θ, θ) : Θ T Θ T R(101) λ(θ, θ) F F [ T T L = E θ β t 1 f(x t, u t, θ t, t) + β t 1 λ t (g(x t, Λ(u t, θ t ), t) x t+1 ) (102) [ T + E θ,θ λ(θ, θ) β t 1 (f(x t (θ), u t (θ), θ t, t) f(x t (θ ), u t (θ ), θ t, t)). L x, ufrechet 3.4. P6 { } λ t = β λ t+1 g x (t + 1) + E[(1 + E θ [λ(θ, θ))f x (t + 1) E[λ(θ, θ)f x (t + 1, θ t+1) (103) m λ t g Λj (t)w ij (t) = (1 + E t [E θ λ(θ, θ))f u i(t) + E t [E θ [λ(θ, θ )f u i(t, θ t). (104) f x (t), g x (t)tf x (t, θ t) = f x (x t (θ), u t (θ), θ t, t) w ij (t)δ u iλ j (t) ff 3.1. f(x t (θ), u t (θ), θ t, t) = f(x t (θ), u t (θ), θ t, t)v(θ t, θ t)v(θ t, θ t ) = 1 (103)(104) λ(θ, θ t ) = 1 + E θ [λ(θ, θ) + λ(θ, θ )v(θ t, θ t), λ t = β[λ t+1 g x (t + 1) + βe[λ(θ, θ t )f x (t + 1), (105) m λ t φ i = g Λj (t)w ij (t) = E t [λ(θ, θ t )f u i(t). (106) f u i m g Λj w ij ψ i = g x f x f u i m g Λj w ij. λ t = β[λ t+1 E[ψ i (t + 1), (107) λ t = E t [λ(θ, θ t )φ i (t). (108)
3 16 3.5. 1P6 φ i (t) = β E t+1[λ(θ, θ t+1 ) φ i (t + 1)E[ψ i (t + 1), (109) E t [λ(θ, θ t ) φ i = φ j, i, j = 1,..., n. (110) (109)u i (110) u i u j φ i (111)Atkinson and Stiglitz(1976,1980) AtkinsonStiglitz1 4 3.2. f(x t (θ), u t (θ), θ t, t) = f 1 (x t (θ), u 1 t (θ), t) + f 2 (u 2 t (θ),..., u n t (θ), θ t, t)w(θ t, θ t), w(θ t, θ t ) = 1λ(θ) = 1 + E θ [[λ(θ, θ) [λ(θ, θ ) 3.6. 2P6 1 φ 1 (t) = βe[ψ 1(t + 1)E t [ 1, (111) φ 1 (t + 1) φ 1 (t) = E t[λ(θ, θ t ) φ i (t), i = 2,..., n, (112) E t [λ(θ) φ i = φ j, i, j = 2,..., n. (113). 23.4 λ t = βλ t+1 g x (t + 1) + βe[λ(θ)f x (t + 1), λ t = E t [λ(θ)φ 1 (t), λ t = E t [λ(θ, θ t )φ i (t), i = 2,..., n. λ t R E t [λ(θ)φ 1 (t)e t [1/φ 1 (t) = E t [λ(θ). λ t = βλ t+1 E[ψ 1 (t + 1), λ t+1 R E t+1 [λ(θ)φ 1 (t + 1)E t [1/φ 1 (t + 1) = E t [λ(θ). (112)(113)(114) 4 12
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