COE-RES Discussion Paper Series Center of Excellence Project The Normative Evaluation and Social Choice of Contemporary Economic Systems Graduate School of Economics and Institute of Economic Research Hitotsubashi University COE/RES Discussion Paper Series, No.130 September 2005 Naka 2-1, Kunitachi, Tokyo 186-8603, Japan Phone: +81-42-580-9076 Fax: +81-42-580-9102 URL: http://www.econ.hit-u.ac.jp/~coe-res/index.htm E-mail: coe-res@econ.hit-u.ac.jp http://wakame.econ.hit-u.ac.jp/~koho/1intro/coe/index.htm
COE 2005 9 JEL Classification Numbers: E62, H21 Keywords: 1 E-mail: tyamada@econ.hit-u.ac.jp. 21 COE 1
1 Aiyagari (1994) Huggett (1996) Huggett (1996) 2 1 2 ( ) Imrohoroglu et al. (1995) 1 ( ) ( ) 1 (2005) 2
2 Aiyagari (1994) Huggett (1996) Imrohoroglu et al. (1995) Kreps-Porteus 2 3 4 5 2 2.1 T Huggett (1996) Kreps-Porteus t =1,...,T µ t 1 ( P T t=1 µ t =1) 3
T s t (0, 1) t t +1 s t t t +1 t +1 µ t+1 µ t+1 = s t µ t R (T >R) R +1 1 R 1 {η t } R t=1 t wη t e t E t 3 wη t e t 1 e t 2 η t 3 w e t wη t t>r η t =0 e t e e 0 π (e 0 e) π Π (e) a t A R + a t 0 (a 1 =0) T Tr (pay-as-you-go) τ ( ) wb b R + (replacement rate) 2 2 4
3 " # ρ 1 ρ X γ U t ( ) = cρ t + s t β π (e 0 e) U γ t+1 ( ), for all t =1,...,T. (1) e 0 E c t t β T U T +1 ( ) =0 (elasticity 1 of intertemporal substitution; EIS) 1 ρ 1 γ γ < 1 ρ < 1 ρ 6= 0 ρ = γ (1) CRRA 2.2 t η t e t K N a t s t ((1 s t ) µ t ) µ t (1 s t ) a t+1 Y K N Y = K α N 1 α w r (dynamic programming) t (t, a, e) =(a, e) X = A E V t (x) t x def x t =1,...,T 3 Kreps and Porteus (1978) Epstein and Zin (1989) 5
V t (x t )=max c,a cρ t + s tβ X e 0 t+1 E π e 0 t+1 e t V γ t+1 (x t+1) ρ γ 1 ρ (2) subject to c t + a t+1 (1 + r) a t +(1 τ) wη t e t + Tr c t + a t+1 (1 + r) a t + wb + Tr ( ) ( ) a 1 =0,a t 0, a T +1 =0ande 1 given. c t (a, e) a t (a, e) V T +1 ( ) =0 2.3 π (e 0 e) x Q t (, ) e 0 (X, B (X), Φ t ) B (X) Borel σ-field Φ t (B) B B (X) B B (X) ( ) 0 Φ 1 0 1 Q t : X B(X) [0, 1] Q t (x, B) = X π (e 0 e) if a t (x) B 0else e 0 B,forallt =1,...,T. x t B Φ 1 t {Φ t } T t=1 Z Φ t+1 (B) = X Q t (x, B) dφ t, ( B B (X)), t =1,...,T (3) (3) K = TX Z µ t t=1 X k t dφ t,n = RX Z µ t η t t=1 X e t dφ t 6
2.4 1 V t : A E R a t : A E R + c t : A E R + (K, N) (w,r ) (τ,b) Tr Φ t (i) (w, r) V t (, ) (2) c t (a, e) a t (a, e) V t (, ) c t (, ) a t (, ) B(X) (ii) w r r = F (K, N) K δ, w = F (K, N). N δ (0, 1) (iii) TX Z TX Z TX Z µ t c t ( ) dφ t + µ t a t+1 ( ) dφ t Y +(1 δ) µ t a t ( ) dφ t, t=1 K = t=1 TX Z µ t t=1 X a t dφ t, N = RX Z µ t η t t=1 X t=1 edφ t. (iv) Z Φ t+1 (B) = X Q t (x, B) dφ t, ( B B (X)), t =1,...,T. (v) RX X TX τ µ t Π (e t ) η t e t = µ t b. (4) t=1 e t E t=r+1 (vi) Tr = TX t=1 Z µ t (1 α t ) (1 + r) a t+1 dφ t. X (i) (ii) (iii) (iv) (v) (vi) (iii) 2 1 7
4 3 2 3.1 1 1 Hayashi and Prescott (2001) 1984 1989 Cobb=Douglas α 0.362 δ =0.089 β Hayashi and Prescott (2001) 0.976 (2005) ( ) β =0.971 982 β =0.976 Kreps-Porteus (1) 2 0 RBC 1 5 EIS {0.2, 0.8}(ρ { 4.0, 0.25}) Epstein and Zin (1991) EIS 0.8 0.2 γ 0 γ =0 20 (t =1) 85 (i.e., T =66) 65 (i.e., R =46) {s t } T t=1 (2002) 20 65 66 85 22.57% 2000 20.24% 2% {η t } R t=1 4 5 EIS Gouvenen (2005) 8
14 [ ] 12+[ ] 50 3.2 y t =loge t AR(1) y t = θy t 1 + ε t (5) ε t N 0, σε 2 t θ σε 2 t Ohtake and Saito (1998) σε 2 t Abe and Yamada (2005) 1 σε 2 t 6 [ 1: ] θ 0.95 Abe and Yamada (2005) θ 1 θ =1 0.95 θ 1 Tauchen (1986) (5) 15 4 4.1 1 (EIS) 0.2 0.8 (τ =5%, 10%, 15%, 20%) (4) b 6 Abe and Yamada (2005) Storesletten et al.(2004) 9
5% 18% 15% 55% ( 1) 15% 55% 13.58%(+ ) 59.3% 7 0.2 τ 5% 20% 0.638 0.705 1 0.638 0.650 ( ) 7 (2005) 10
8 EIS 20 2 τ 10% 20% τ EIS 1 [ 2a, 2b: ] 8 11
4.2 3 τ =0.1 0.2 20 80 5 (1994) Aiyagari (1994) 40 [ 3: ] τ 2 EIS 60 τ =0.2 2 12
4.3 9 İmrohoroğlu et al. (1995) Ω ({c t })= X e E Π (e) V 1 (k 1,e). (6) (6) 20 1 (6) β 1 (6) 10 9 10 İmrohoroğlu et al. (1995) 13
5 A : 11 3 1. t 2. 3. 11 Fortran 90/95 IMSL 14
1. a 1,...,a Iª r 1 w ˆV t a i,e j ; r, w " ρ X γ =max c,a cρ t + βα t π (e 0 e) ˆV γ t+1 a 0 t+1,e 0 t+1 ; r, w # e E 1 ρ, t = T,...,1, j ˆV t,e 100 200 t a i e j a t a i,e j 2. a 1,...,a Lª L 6000 Aiyagari (1994) Huggett (1996) [1] Abe, N. and T.Yamada (2005) Non-Linearity in Household Income Process and Consumption Inequality over the Life Cycle, work in progress, Hitotsubashi University and Bank of Japan. [2] Aiyagari, S.R. (1994): Uninsured Idiosyncratic Risk and Aggregate Saving Quarterly Journal of Economics, 109, 659-684. [3] Epstein, L.G. and S.E. Zin (1989): Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework Econometrica, 57, 937-969. [4] Epstein, L.G. and S.E. Zin (1991): Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis Journal of Political Economy, 99, 263-286. [5] Gouvenen, F. (2005): Reconciling Conflicting Evidence on the Elasticity of Intertemporal Substitution: AMacroeconomicPerspective forthcoming in Journal of Monetary Economics. [6] Hayashi, F. and E.C. Prescott (2002): The 1990s in Japan: A Lost Decade Review of Economic Dynamics, 5, 206-235. [7] Huggett, M. (1996): Wealth Distribution in Life-Cycle Economies Journal of Monetary Economics, 38, 469-494. 15
[8] İmrohoroğlu, A., S. İmrohoroğlu and D.H. Joines (1995): A Life Cycle Analysis of Social Security Economic Theory, 6, 83-114. [9] Kreps, D.M. and E.L. Porteus (1978): Temporal Resolution of Uncertainty and Dynamic Choice Theory Econometrica, 46, 185-200. [10] Ohtake, F. and M. Saito (1998): Population Aging and Consumption Inequality in Japan Review of Income and Wealth, 44, 361-381. [11] Storesletten, K., C.I. Telmer and A. Yaron (2004): Cyclical Dynamics in Idiosyncratic Labor-Market Risk Journal of Political Economy, 112, 695-717. [12] Tauchen, G. (1986): Finite State Markov-Chain Approximations to Univariate and Vector Autoregressions Economics Letters, 20, 177-181. [13] (2005) 56 3 pp.248-265 [14] (2005) [15] (2005) 3 [16] (1994) pp.59-78. [17] (2002) 13(2001) 62(2050) : 63(2051) 112(2100) ( 14 1 ) 16
EIS 0.2 0.8 τ 5% 10% 15% 20% 5% 10% 15% 20% 0.638 0.672 0.694 0.705 0.638 0.646 0.650 0.650 2.19 4.58 6.90 9.06 3.05 3.76 4.38 4.92 2.11 1.55 1.21 0.99 1.88 1.71 1.59 1.49 (%) 28.48 23.36 19.85 17.38 26.52 25.02 23.85 22.90 K/Y 3.26 2.68 2.29 2.02 3.03 2.86 2.73 2.62 (b) 0.18 0.37 0.55 0.73 0.18 0.37 0.55 0.73 (Ω ({c t })) 0.16 0.14 0.12 0.11 4.40e-07 4.20e-07 3.99e-07 3.78e-07 ( 1 C/Y C Y 1: 17