2015 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 (The Overlapping-Generations Models:OLG) OLG OLG Allais (1

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1 2015 : (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 Diamond (1965) 4 OLG Diamond 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 1958 An Exact Consumption-Loan Model of Interest With or Without the Social Contrivance of Money Journal of Political Economy, 66(6): Diamond, Peter 1965 National Debt in a Neoclassical Growth Model American Economic Review, 55(5):

2 Barro (1974) 5 Diamond Ramsey OLG Ramsey Ramsey Solow (g>r!) Ramsey OLG Gale (1973) 6 Kehoe and Levin (1985) 7 Azariadis (1993) 8, Geanakoplos (1987) 9 Farmer (2002) 10 OLG OLG Auerbach and Kotlikoff (1987) 11 (1996) 12 5 Barro, Robert 1974 Are Government Bonds Net Wealth? Journal of Political Economy, 81(6): Gale, David 1973 Pure Exchange EquilibPhilippe Weil 133 rium of Dynamic Economic Models Journal of Economic Theory, 6(1): Kehoe, T J and D K Levine 1985 Comparative Statics and Perfect Foresight in Infinite Horizon Models, Econometrica 53: Costas Azariadid, (1993) Intertemporal Macroeconomics, Wiley-Blackwell 9 Geanakoplos, John 1987 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 10 Roger Farmer (2002) Macroeconomics of Self-fulfilling Prophecies, second edition, MIT Press 11 Alan J Auerbach, Laurence J Kotlikoff (1987) Dynamic Fiscal Policy, Cambridge University Press pp1-31 2

3 Acemoglu Ljungqvist and Sargent 1990 McCandless and Wallace (1993) 13 Azariadis (1993) Ramsey RBC Romer Acemoglu, Liyongqist and Sargent Blanchard and Fisher (1989) Sargent (1989) OLG ( )) OLG 21 () OLG OLG Auerbach and Kotlikoff (1987) OLG RBC DSGE Ramsey OLG OLG George T McCandless, Neil Wallace (1998) Introduction to Dynamic Macroeconomic Theorym Harvard Univesity Press 14 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 2 (2 ) 2 OLG OLG 15 I id i I N I id,1,2,3,4, G0 1 ( ) G1 1 2 ( ) G2 2 4 t 15 ( ) 4

5 Max c t t + c t t+1 st 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 ( st p 1 c 0 1 ω 1) b 0 21 Samuelson Case i > 0 ( ω 0, ω 1) = (1, 0) i = 0 G0 b 0 Gale (1973) Samuelson 16 1(p t = 1) (b t = 0) (Autarky) G0 c 0 1 = 0 G1 1 G1 1 G2, G3, G4, G1 G1 G0 G2 G1 G0 16 Samuelson (1958) 5

6 G0 1/2 Samuelson OLG ( ) OLG (Fiat Money) ( ) fiat money G0 b 0 1/2 R Samuelson Sameuslon 22 Classical Case i > 0 ( ω 0, ω 1) = (0, 1) i = 0 G0 1 Gale Samuelson i=0 1/2 G1 G2 i > 0 (0, 1) (1/2, 1/2) G0 6

7 23 OLG? N I (x, p) x i p x i px i > px i x 1 p n x i,n < i n Samuelson 1 24 OLG OLG Debreu Theory of Value Samuelson (1958) Debreu indeterminacy locally unique locally unique locally indeterminate local uniqueness regular economy ( ) ( 7

8 ) regular economy 17 (measure) OLG indeterminacy Geanakoplos (1987) ( ) unique unique OLG Geanakoplos(1987) OLG ( ) L L L L-1 2L 2L-1 18 locally unique indeterminate [0,1] 17 Mas-Colell (1995) p Feng and Hoelloe (2015) Indeterminacy in Stochastic Overlapping Generations Models: Real Effects in the Long Run 19 8

9 Samuelson Fiat Money indeterminacy? Kotolikoff Good Behavior Balasko-Shell t (1 + r s ) = +, t=1s=1 Krueger and Kuber (2006) Ljungqvist and Sargent (2012) Geanakoplos Indterminacy OLG Geanakoplos Farmer Keohe 3 Diamond (1965) : OLG Samelson 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, 9

10 Y t = F (K t, L t ) 20 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 st c y t + s t w t, c o t+1 (1 + r t+1 ) s t Time Line (1) (2) ( ) (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 ) K t+1 = S t = L t s t,

11 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 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 11

12 s = β (1 α) / (1 + β) Solow 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 = (w t s t ) β 1/σ (1 + r t+1 ) (1 σ)/σ, s t ( 1 + β 1/σ (1 + r t+1 ) (1 σ)/σ) = w t β 1/σ (1 + r t+1 ) (1 σ)/σ s t = s t = w tβ 1/σ (1 + r t+1 ) (1 σ)/σ 1 + β 1/σ (1 + r t+1 ) (1 σ)/σ w t β /σ (1 + r t+1 ) (1 σ)/σ + 1 CRRA σ 1 σ = 1 σ > 1 σ < 1 12

13 k t+1 = ( β /σ ( 1 + αk α 1 t+1 (1 α) kt α ) (1 σ)/σ ) + 1 (1 + n) k = (1 α) k α ( ) β /σ (1 + αk α 1 ) (1 σ)/σ + 1 (1 + n) CRRA Unique σ > 1 (! ) 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) = 1 + n k gold = ( α ) 1 1 α 1 + n 21 OLG Woodford Azariadis Farmer

14 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 = 08, α = 03 k > k gold r<n (dynamic inefficient) Ramsey dynamic inefficient OLG ( n ) 14

2017 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 exante (The Overlapping-Generations Models:OLG) OLG OLG Al

2017 : (heterogenous) Heterogeneous homogeneous Heterogenous agent model Bewley 1 exante (The Overlapping-Generations Models:OLG) OLG OLG Al 2017 : 29 5 24 1 (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