2017 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 exante (The Overlapping-Generations Models:OLG) OLG OLG Al
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1 2017 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 exante (The Overlapping-Generations Models:OLG) OLG OLG Allais (1947) 2 Samuelson(1958) 3 OLG Solow Ramsey 1 Bewley, T. (1977): The permanent income hypothesis: A theoretical formulation, Journal of Economic Theory, 16(2), Bewley, T. (1983): A difficulty with the optimum quantity of money, Econometrica,51(5), Allais, M. [1947], Economie et Intret, Imprimerie Nationale, Paris.( ) 3 Samuelson, Paul A An Exact Consumption-Loan Model of Interest With or Without the Social Contrivance of Money. Journal of Politica Economy, 66(6):
2 OLG Samuelson Diamond (1965) 4 OLG Diamond Barro (1974) 5 Diamond Ramsey OLG Ramsey Ramsey Solow (g>r!) Ramsey OLG Gale (1973) 6 Azariadis (1981) 7, Kehoe and Levin (1985) 8 Spear et al. (1990) 9, Geanakoplos (1987) 10 Farmer (1999) OLG OLG OLG Auerbach and Kotlikoff (1987) 13 4 Diamond, Peter National Debt in a Neoclassical Growth Model. American Economic Review, 55(5): Barro, Robert Are Government Bonds Net Wealth? Journal of Political Economy, 81(6): Gale, David Pure Exchange Equilibrium of Dynamic Economic Models. Journal of Economic Theory, 6(1): Azariadis, Costas, (1981), Self-fulfilling prophecies, Journal of Economic Theory, 25, issue 3, p Kehoe, T. J. and D. K. Levine Comparative Statics and Perfect Foresights in Infinite Horizon Models, Econometrica 53: Spear, Stephen E. & Srivastava, Sanjay, & Woodford, Michael, (1990) Indeterminacy of stationary equilibrium in stochastic overlapping generations models, Journal of Economic Theory, Elsevier, vol. 50(2), pages , April. 10 Geanakoplos, John The Overlapping Generations Model of General Equilibrium. In The New Palgrave Dictionary of Money and Finance, vol. 1, ed. Peter Newman, Murray Milgate, and John Eatwell, Palgrave Macmillan. 11 Roger Farmer (1999) Macroeconomics of Self-fulfilling Prophecies, second edition, MIT Press 12 Feng and Hoelle (2017) Zhigang Feng & Matthew Hoelle, (2017. Indeterminacy in stochastic overlapping generations models: real effects in the long run, Economic Theory, Springer; vol. 63(2), pages , February. 13 Alan J. Auerbach, Laurence J. Kotlikoff (1987) Dynamic Fiscal Policy, Cambridge 2
3 (1996) 14 Acemoglu Ljungqvist and Sargent 1990 McCandless and Wallace (1993) 15 Azariadis (1993) Ramsey RBC Romer Acemoglu, Ljuongqvist and Sargent Blanchard and Fisher (1989) Sargent (1989) OLG ( )) OLG 21 ( ) OLG OLG Auerbach and Kotlikoff (1987) OLG RBC DSGE Ramsey OLG University Press pp George T. McCandless, Neil Wallace (1998) Introduction to Dynamic Macroeconomic Theory, Harvard Univesity Press. 3
4 G5 young G4 young old G3 young old G2 young old G1 young old G0 old OLG 16 2 (2 ) 2 OLG OLG 17 I id i I N I 1 id (G0),G1,G2,G3,G4,... G0 1 ( ) G ( ) G OLG Nishiyama and Smetters, (2007) Does Social Security Privatization Produce Efficiency Gains?, The Quarterly Journal of Economics, vol. 122(4), pages Storesletten, Telmer, and Yaron (2007) Asset Price with Idiosyncartic Risk and Overlapping Generations, Review of Economic Dynamics, 10, 4, ( ) 4
5 t Max c t t + c t t+1 s.t. b t p t ( ω 0 c t t), p t+1 c t t+1 p t+1 ω 1 + b t. c i j i j ω0, ω 1 p t t b t ( G0) Max U ( c 0 ) 1 ( s.t. p 1 c 0 1 ω 1) b Samuelson Case i > 0 ( ω 0, ω 1) = (1, 0) i = 0 G0 b 0 Gale (1973) Samuelson 18 1(p t = 1) (b t = 0) (Autarky) G0 c 0 1 = 0 G1 1 G1 1 G2, G3, G4,... G1 18 Samuelson (1958) 5
6 G1 G0 G2 G1 G0 G0 1/2 Samuelson OLG ( ) OLG (Fiat Money) ( ) fiat money G0 b 0 1/2 R Samuelson Sameuelson 2.2 Classical Case i > 0 ( ω 0, ω 1) = (0, 1) i = 0 G0 1 Gale(1973) Samuelson i = 0 1/2 G1 G2 i > 0 (0, 1) (1/2, 1/2) G0 6
7 2.3 OLG? N I (x, p) x i p x i px i > px i x 1 p n x i,n < i n Samuelson OLG OLG Debreu(1959) T heory of V alue Samuelson (1958) Debreu indeterminacy (determinate) (indeterminate) 7
8 locally unique locally unique locally indeterminate local uniqueness regular economy ( ) ( ) regular economy 19 (measure) OLG indeterminacy Geanakoplos (1987) ( ) unique unique OLG Geanakoplos(1987) OLG ( ) L L L L-1 2L 2L-1 20 locally unique indeterminate Geanakoplos (1987) OLG 19 Mas-Colell (1995) p Feng and Hoelloe (2015) Indeterminacy in Stochastic Overlapping Generations Models: Real Effects in the Long Run 8
9 2.5 [0,1] Samuelson indeterminacy? Kotolikoff Good Behavior Balasko-Shell t (1 + r s ) = +, t=1s=1 Krueger and Kubler (2006) 21 Ljungqvist and Sargent (2012) Geanakoplos Indeterminacy OLG Geanakoplos Farmer Keohe 21 Dirk Krueger and Felix Kubler (2006) Pareto-Improving Social Security Reform when Financial Markets are Incomplete!? American Economic Review, vol. 96, no. 3, June, (pp ) 9
10 3 Diamond (1965) : OLG Samuelson Diamond (1965) U ( c y t, c o ) t+1 = u (c y t ) + βu ( c o t+1), β > 0, u > 0, u < 0, L t = (1 + n) t L 0, Y t = F (K t, L t ). (100%) 22 Intensive form k = K/L Solow 1 + r t = f (k t ), w t = f (k t ) k t f (k t ), where f (k) F (k, 1). Max u (c y t ) + βu ( c o ) t+1 s.t. c y t + s t w t, c o t+1 (1 + r t+1 ) s t. Time Line (1) (2) ( )
11 (3) (4) (5) (6) u (c y t ) = β (1 + r t+1 ) u ( c o t+1). s t = s (w t, r t+1 ). c y t = w t s t c o t+1 = (1 + r t+1 ) s t 0 = β (1 + r t+1 ) u ((1 + r t+1 ) s t ) u (w t s t ). ds t u y = dw t β (1 + r t+1 ) 2 u o + u y. 0 < ds t dw t < 1. 1 ds t dr t+1 = βuy + β (1 + r t+1 ) s t u y β (1 + r t+1 ) 2 u o + u y K t+1 = S t = L t s t, K t+1 = L t s (w t, r t+1 ). Intensive form k t+1 = s (w t, r t+1 ). 1 + n k t+1 = s (f (k t) k t f (k t ), f (k t+1 )). 1 + n 11
12 k = s (f (k ) k f (k ), f (k )). 1 + n U ( c y t, c o ) t+1 = ln (c y t ) + β ln ( c o t+1). 1 c y t = β (1 + r t+1 ) 1 c o t+1. c y t + s t = w t s t = w t c o t+1/ (β (1 + r t+1 )) c o t+1 = (1 + r t+1 ) s t s t = w t (1 + r t+1 ) s t / (β (1 + r t+1 )) = w t s t /β ( ) β + 1 s t (1 + 1/β) = s t = w t β ( ) β s t = w t. 1 + β Solow k t+1 = f (k t) k t f ( ) (k t ) β. 1 + n 1 + β f (k t ) = k α t k t+1 = β (1 α) (1 + n) (1 + β) ka t. s = β (1 α) / (1 + β) Solow 12
13 CRRA u (c y t ) + βu ( ( ) c o ) c y(1 σ) o(1 σ) t+1 = t 1 c t β, 1 σ 1 σ c o t+1 c y t = (β (1 + r t+1 )) 1/σ, c o t+1 = c y t (β (1 + r t+1 )) 1/σ. c o t+1 = (1 + r t+1 ) s t = c y t (β (1 + r t+1 )) 1/σ, c y t + s t = w t. s t = c y t β 1/σ (1 + r t+1 ) (1 σ)/σ = w t c y t w t = c y t (1 + β 1/σ (1 + r t+1 ) (1 σ)/σ) ( c y t = w t 1 + β 1/σ (1 + r t+1 ) (1 σ)/σ) 1 s t = w t c y t w t = w t 1 + β 1/σ (1 + r t+1 ) (1 σ)/σ = w tβ 1/σ (1 + r t+1 ) (1 σ)/σ 1 + β 1/σ (1 + r t+1 ) (1 σ)/σ s t = w t β 1/σ (1 + r t+1 ) (1 σ)/σ + 1. CRRA σ 1 σ = 1 σ > 1 σ < 1 13
14 k t+1 = ( β 1/σ ( 1 + αk α 1 t+1 (1 α) kt α ) (1 σ)/σ ) + 1. (1 + n) k (1 α) k α = ( ). β 1/σ (1 + αk α 1 ) (1 σ)/σ + 1 (1 + n) CRRA Unique σ > 1 (! ) 3.1 CES 23 k t+1 = s (f (k t) k t f (k t ), f (k t+1 )) 1 + n 0 = s (f (k t ) k t f (k t ), f (k t+1 )) + (1 + n) k t+1 dk t+1 = s w (f kf f ) dk t 1 + n s r f = s wkf 1 + n s r f. s w > 0 s r E Solow A B B A 23 OLG Woodford Azariadis Farmer
15 K(t+1) E 15
16 3.2 Diamond k = ( ) 1 β (1 α) 1 α, (1 + n) (1 + β) ( ) f (k t ) (1 + n) k t+1 = c y t + co t 1 + n, 1 c t f (k t ) = c t + (1 + n) k t+1. f ( k gold) = 1 + n. αk gold α 1 = 1 + n k gold α 1 = 1 + n α ( α k gold = 1 + n ) 1 1 α k k k gold = ( ) 1 β (1 α) 1 α (1 + β) α k > k gold when β (1 α) > (1 + β) α β (1 + β) > α (1 α), β = 0.8, α = 0.3 k > k gold r < n (dynamic inefficient) 16
17 Ramsey dynamic inefficient OLG ( n )?? OLG
2015 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 (The Overlapping-Generations Models:OLG) OLG OLG Allais (1
2015 : 27 6 13 1 (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 (The Overlapping-Generations Models:OLG) OLG OLG Allais (1947) 2 Samuelson(1958) 3 OLG Solow Ramsey Samuelson
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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
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