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1 / 34

Li-Yao,, Li-Yao The Life-Cycle Effects of House Price Changes (Li-Yao ),,,, ( ) 1 ω ( ) ω 1 γ Ct Ht t T βt U(C t, H t, N t) = N t N t t T βt N t 1 γ H t : t C t : t β : ω : γ : W. Li, R. Yao, The life-cycle effects of house price changes. Journal of Money, Credit and Banking, 39, 1375 1409, 2007. 2 / 34

Li-Yao, 1 2 3 ( ) 4 5 (, ) 6 3 / 34

,, Li-Yao Li-Yao,, 4 / 34

T = {1,..., t max }, J x tj : t j ( :1, :0) y tj : t j ( :1, :0) z tj : t j ( :1, :0) c t : t s t : t x tj y tj j J x tj + j J y tj 1 t T 5 / 34

u tj : t j ( :1, :0) v tj : t j ( :1, :0) m t : t u tj v tj j J u tj + j J v tj = 1 t T 6 / 34

g t : t h j : j r : r : π : u 0j : t = 0 j v 0j : t = 0 j j J u 0j + j J v 0j = 1 7 / 34

maximize subject to β t U c t, t T j J ( ) h j utj + v uj (1) (1 + r)s t 1 + g t c t s t (1 + r )m t 1 + m t π j J h j v tj j J h j x tj + j J h j z tj = 0 t T (2) m t h j u tj j J t T (3) z tj u (t 1)j t T, j J (4) z tj u (t 1)j + x tj + y tj 1 j J j J t T, j J (5) z tj x tj + y tj t T, j J (6) j J j J x tj + y tj 1 t T (7) j J j J 8 / 34

u tj x tj t T, j J (8) u tj u (t 1)j j J y tj j J x tj t T, j J (9) v tj y tj t T, j J (10) v tj v (t 1)j x tj y tj j J j J t T, j J (11) u tj + v tj = 1 j J j J t T (12) m tmax = 0 (13) c t, s t, m t 0 t T (14) x tj, y tj, u tj, v tj, z tj {0, 1} t T, j J (15) 9 / 34

t T βt U(C t, H t, N t ) = t T βt N t ( Ct N t ) 1 ω ( ) ω Ht 1 γ, t T, 1 10 1 10 2, 1 1 2 2 N t 1 γ 10 / 34

, 2 1 2,, 1, 2 2, 1, 1 11 / 34

: Numerical Optimizer(18.1.0), 12 / 34

T 10 J 4 s 0 1 g 1,..., g 10 4 1 h 1 3 2 h 2 5 3 h 3 7 4 h 4 9 r 0.030 r 0.050 π 0.075 β 0.800 m 10 0 13 / 34

: 0.355 ( : -0.794) v0_1 Figure:,, x1_4 = 1 u1_4 x2_4 = 1 u2_4 x3_4 = 1 u3_4 u4_4 x5_4 = 1 u6_4 u7_4 u8_4 x9_4 = 1 u9_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 5.030 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 14 / 34

: 0.355 ( : -0.794) v0_4 Figure:,, x1_4 = 1 u6_4 u1_4 x2_4 = 1, z2_4 = 1 u7_4 u2_4 u8_4 u3_4 u9_4 u4_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 5.030 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 15 / 34

: 0.330 ( : -0.750) u0_1 x1_4 = 1 x6_4 = 1, z6_4 = 1 u1_4 u6_4 u7_4 u2_4 x8_4 = 1, z8_4 = 1 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u9_4 u4_4 y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 8.030 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 16 / 34

: 0.278 ( : -0.715) u0_4 x1_4 = 1 u6_4 u1_4 x2_4 = 1, z2_4 = 1 u7_4 u2_4 x8_4 = 1, z8_4 = 1 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 x4_4 = 1, z4_4 = 1 u9_4 u4_4 y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 14.030 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 17 / 34

u0_1 x1_4 = 1 x6_4 = 1, z6_4 = 1 u1_4 u6_4 u7_4 u2_4 x8_4 = 1, z8_4 = 1 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u9_4 u4_4, y10_4 = 1, z10_4 = 1 v10_4 18 / 34

(1 + r)s t 1 + g t c t s t (1 + r )m t 1 + m t π j J h j v tj (1 + }{{} δ ) j J h j x tj + j J h j z tj = 0 δ : 19 / 34

T 10 J 4 s 0 1 g 1,..., g 10 4 1 h 1 3 2 h 2 5 3 h 3 7 4 h 4 9 r 0.030 r 0.050 π 0.075 β 0.800 δ 0.030 m 10 0 20 / 34

: 0.358 ( : -0.801) v0_1 x1_4 = 1 Figure:,, u1_4 u2_4 u3_4 u4_4 u6_4 u7_4 u8_4 u9_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 4.760 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 21 / 34

: 0.358 ( : -0.801) v0_4 x1_4 = 1 Figure:,, u1_4 u2_4 u3_4 u4_4 u6_4 u7_4 u8_4 u9_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 4.760 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 22 / 34

: 0.332 ( : -0.753) u0_1 Figure:,, x1_4 = 1, z1_1 = 1 u6_4 u1_4 u7_4 u2_4 u8_4 u3_4 u9_4 u4_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 7.760 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 23 / 34

: 0.278 ( : -0.715) u0_4 Figure:,, u1_4 u2_4 u3_4 u4_4 u6_4 u7_4 u8_4 u9_4 y10_4 = 1, z10_4 = 1 v10_4 t c t s t m t 1 14.030 0 9.000 2 3.550 0 9.000 3 3.550 0 9.000 4 3.550 0 9.000 5 3.550 0 9.000 6 3.550 0 9.000 7 3.550 0 9.000 8 3.550 0 9.000 9 3.550 0 9.000 10 2.875 0 0 24 / 34

2 u0_1 x1_4 = 1, z1_1 = 1 u1_4 u2_4 u3_4 u4_4 u6_4 u7_4 u8_4 u9_4 y10_4 = 1, z10_4 = 1, v10_4 25 / 34

, 1 10 m t m t 1 0.1 j J h j u tj + M j J x tj M : M 26 / 34

T 10 J 4 s 0 1 g 1,..., g 10 4 1 h 1 3 2 h 2 5 3 h 3 7 4 h 4 9 r 0.030 r 0.050 π 0.075 β 0.800 δ 0.030 m 10 0 27 / 34

: 0.364 ( : -0.856) v0_1 x1_4 = 1 u6_4 u1_4 x7_4 = 1, z7_4 = 1 u2_4 u7_4 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u4_4 u9_4 x5_4 = 1, z5_4 = 1y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 4.760 0 9.000 2 2.650 0 8.100 3 4.225 0 9.000 4 2.650 0 8.100 5 4.225 0 9.000 6 2.650 0 8.100 7 4.225 0 9.000 8 2.650 0 8.100 9 4.225 0 8.100 10 2.875 0 0 28 / 34

: 0.364 ( : -0.856) v0_4 x1_4 = 1 u6_4 u1_4 x7_4 = 1, z7_4 = 1 u2_4 u7_4 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u4_4 u9_4 x5_4 = 1, z5_4 = 1y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 4.760 0 9.000 2 2.650 0 8.100 3 4.225 0 9.000 4 2.650 0 8.100 5 4.225 0 9.000 6 2.650 0 8.100 7 4.225 0 9.000 8 2.650 0 8.1000 9 4.225 0 9.000 10 2.875 0 0 29 / 34

: 0.338 ( : -0.808) u0_1 x1_4 = 1, z1_1 = 1 u6_4 u1_4 x7_4 = 1, z7_4 = 1 u2_4 u7_4 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u4_4 u9_4 x5_4 = 1, z5_4 = 1y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 7.760 0 9.000 2 2.650 0 8.100 3 4.225 0 9.000 4 2.650 0 8.100 5 4.225 0 9.000 6 2.650 0 8.100 7 4.225 0 9.000 8 2.650 0 8.100 9 4.225 0 9.000 10 2.875 0 0 30 / 34

: 0.287 ( : -0.771) u0_4 x1_4 = 1, z1_4 = 1 u6_4 u1_4 x7_4 = 1, z7_4 = 1 u2_4 u7_4 x3_4 = 1, z3_4 = 1 u8_4 u3_4 x9_4 = 1, z9_4 = 1 u4_4 u9_4 x5_4 = 1, z5_4 = 1y10_4 = 1, z10_4 = 1 v10_4 Figure:,, t c t s t m t 1 13.760 0 9.000 2 2.650 0 8.100 3 4.225 0 9.000 4 2.650 0 8.100 5 4.225 0 9.000 6 2.650 0 8.100 7 4.225 0 9.000 8 2.650 0 8.100 9 4.225 0 9.000 10 2.875 0 0 31 / 34

3,,,,,,, 32 / 34

Li,Yao 2,, t = 1 t = 9,,,,, 33 / 34

, ( ) 70 34 / 34

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