COE-RES Discussion Paper Series Center of Excellence Project The Normative Evaluation and Social Choice of Contemporary Economic Systems Graduate Scho

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1 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 , Japan Phone: Fax: URL:

2 COE JEL Classification Numbers: E62, H21 Keywords: 1 tyamada@econ.hit-u.ac.jp. 21 COE 1

3 1 Aiyagari (1994) Huggett (1996) Huggett (1996) ( ) Imrohoroglu et al. (1995) 1 ( ) ( ) 1 (2005) 2

4 2 Aiyagari (1994) Huggett (1996) Imrohoroglu et al. (1995) Kreps-Porteus T Huggett (1996) Kreps-Porteus t =1,...,T µ t 1 ( P T t=1 µ t =1) 3

5 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

6 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

7 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 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

8 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

9 Hayashi and Prescott (2001) Cobb=Douglas α δ =0.089 β Hayashi and Prescott (2001) (2005) ( ) β = 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 γ 0 γ =0 20 (t =1) 85 (i.e., T =66) 65 (i.e., R =46) {s t } T t=1 (2002) % % 2% {η t } R t=1 4 5 EIS Gouvenen (2005) 8

10 14 [ ] 12+[ ] 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 = θ 1 Tauchen (1986) (5) (EIS) (τ =5%, 10%, 15%, 20%) (4) b 6 Abe and Yamada (2005) Storesletten et al.(2004) 9

11 5% 18% 15% 55% ( 1) 15% 55% 13.58%(+ ) 59.3% τ 5% 20% ( ) 7 (2005) 10

12 8 EIS 20 2 τ 10% 20% τ EIS 1 [ 2a, 2b: ] 8 11

13 4.2 3 τ = (1994) Aiyagari (1994) 40 [ 3: ] τ 2 EIS 60 τ =

14 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) İmrohoroğlu et al. (1995) 13

15 5 A : t Fortran 90/95 IMSL 14

16 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 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, [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, [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, [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, [7] Huggett, M. (1996): Wealth Distribution in Life-Cycle Economies Journal of Monetary Economics, 38,

17 [8] İmrohoroğlu, A., S. İmrohoroğlu and D.H. Joines (1995): A Life Cycle Analysis of Social Security Economic Theory, 6, [9] Kreps, D.M. and E.L. Porteus (1978): Temporal Resolution of Uncertainty and Dynamic Choice Theory Econometrica, 46, [10] Ohtake, F. and M. Saito (1998): Population Aging and Consumption Inequality in Japan Review of Income and Wealth, 44, [11] Storesletten, K., C.I. Telmer and A. Yaron (2004): Cyclical Dynamics in Idiosyncratic Labor-Market Risk Journal of Political Economy, 112, [12] Tauchen, G. (1986): Finite State Markov-Chain Approximations to Univariate and Vector Autoregressions Economics Letters, 20, [13] (2005) 56 3 pp [14] (2005) [15] (2005) 3 [16] (1994) pp [17] (2002) 13(2001) 62(2050) : 63(2051) 112(2100) ( 14 1 ) 16

18 EIS τ 5% 10% 15% 20% 5% 10% 15% 20% (%) K/Y (b) (Ω ({c t })) e e e e-07 ( 1 C/Y C Y 1: 17

Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue Date Type Technical Report Text Version publisher URL

Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue 2012-06 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/23085 Right Hitotsubashi University Repository