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1 中央大学経済研究所年報第 45 号 (2014)pp 金融政策とタイム ラグ ( 浅田 松本 ) 143 非線形経済理論研究会 In this paper, we study how the time lag that is related to the monetary policy influences on dynamic stability of the macroeconomic system by using a dynamic Keynesian model. In our model, initial conditions of all endogenous variables are fixed unlike the mainstream New Keynesian Dynamic model that allows for the jump variables that ensure the convergence to the steady state even if the unstable roots exist. It is shown that the existence of the time lag has the destabilizing effect in the sense that the stability region of the parameter space of the monetary policy shrinks ) 1) Asada (ed.)2014general Introduction

2 144 中央大学経済研究所年報第 45 号 Friedman Kalecki1935Frisch and Holme1935James and Belz Steindl1952 Johansen1959Lange1969 Phillips Asada and Yoshida2001 Yoshida and Asada2007Matsumoto and Szidarovsky2013 Asada ChiarellaFlaschel and Franke2010 Asada2010Asada2010 Keynes1936 2) Galί2008 Woodford2003 jump variables 3 3) (1) (2) 1 2) Asada ) Asada2010Asada2014 Asada ChiarellaFlaschel and Franke2010

3 2014 金融政策とタイム ラグ ( 浅田 松本 ) 145 (3) 2 Asada ) IS Y = Y (R) = F (r, π e ) 1 Y r π e R = r π e F r = Y < 0, F π = Y > 0, Y = dy e dr M P = L(Y, r, πe ) 2 M P L(Y, r, π e ) L Y > 0, L r < 0, L π e 0 4)

4 146 中央大学経済研究所年報第 45 号 π = ε(y Ȳ ) + πe 3 π = P /P π e Ȳ ṙ(t) = α [π(t) π] + β ˆY (t) Ȳ, π e (t) = γ {θ [ π π e (t)] + (1 θ) [π(t) π e (t)]} 4 π α 0, β 0, γ > 0, ε > 0, 1 θ 0 IS-LM r 4 1 r ) α β 4 2 forward-looking expectation formation backward-looking expectation formation θ 2 θ 1 θ = 1 2 π e (t) = γ [ π π e (t)] π e (t) = π + (π e 0 π)e γt π 0 e π0 e < π π π0 e > π θ = 0 2 π e (t) = γ [π(t) π e (t)]

5 2014 金融政策とタイム ラグ ( 浅田 松本 ) 147 π e (t) = Z t 0 γe γ(t τ) π(τ)dτ t t π(τ), τ < t r π e 2 ṙ(t) = (αε + β)f (r(t), π e (t)) + απ e (t) (α π + (α + β)ȳ ), π e (t) = εγ(1 θ)f (r(t), π e (t)) γθπ e (t) + (γθ π (1 θ)ȳ ) 5 t r(t) π e (t) F (r(t), π e (t)) 2 α β (r, π e ) ṙ(t) = π e (t) = 0 F (r, π e ) = Ȳ π e = π = π (αε + β)y J α (αε + β)y 1 A 6 (1 θ)εγy γ [θ + (1 θ)εy ] J λi = 0 I 2 λ 2 rjλ + DetJ = 0 7

6 148 中央大学経済研究所年報第 45 号 DetJ DetJ = γ(αε + βθ)( Y ) > 0 8 rj rj = (αε + β)y γ[θ + (1 θ)εy ] 0 9 5) rj = λ 1 + λ 2, DetJ = λ 1λ 2 8 rj < 0 Re[λ i] < 0 θ = θ = 0 F B rj F = (αε + β)y γ < 0, rj B = [(α γ)ε + β] Y 0 rj B rj F rj B 9 rj(θ) = θ rj F + (1 θ) rj B 10 θ d rj(θ) dθ = γ[1 + ε( Y )] < rj B < 0 θ [0, 1] rj < 0 rj B 2 rj B < 0 (i) α γ, (ii) α < γ, β > γε αε (i)(ii) (α, β) α 5) Gandolfo2009

7 2014 金融政策とタイム ラグ ( 浅田 松本 ) 149 β α γ β γε α β β > γε αε rj B > 0 11θ 0 (0, 1) θ 0 = B rj rj rj F rj B 13 θ < θ 0 rj > 0, θ > θ 0 rj < 0 13 θ 0 = Y γ(1 + ε( Y )) (αε + β) + ε( Y ) 1 + ε( Y ) 14 β = αε + εγ k(θ 0) 15 k(θ 0) k(θ 0) = γ(1 + ε( Y )) Y θ 0, k (θ 0) > 0 16 rj B = 0 k(θ 0) = 0 rj B > 0 θ 0 > 0 15 εγ k(θ 0) = 0 θ (α, β)

8 150 中央大学経済研究所年報第 45 号 θ max 0 = ε( Y ) 1 + ε( Y ) < 1 θ 0 > θ max 0 15 β < 0 k(θ 0) θ 0 [0, θ max 0 ] rj B > 0 α < γ, αε +β < γε β > αε + εγ k(θ 0) = rj < 0 β = αε + εγ k(θ 0) = rj = 0 17 β < αε + εγ k(θ 0) = rj > γε αε > β > αε + εγ k(θ 0) rj B > 0, rj < 0 (r, π e ) (α, β) 1 (i) α γ, (ii) α γβ > αε + γε, (iii) αε + γε > β > αε + εγ k(θ 0) 14 (αε + β, θ) 2-2 6) (α, β) (αge + β, θ) α, β, θ 6 λ 1,2 = rj ± p ( rj) 2 4DetJ 2 DetJ > 0 rj = 0 λ 1,2 9 α β 6) Asada2010

9 2014 金融政策とタイム ラグ ( 浅田 松本 ) (αε + β, θ) d rj dα = εy < 0, d rj dβ = Y < (β = αε + (γε k(θ 0))) α β 2 (H1) (H2) αβ rj 2 5 α β 11 d rj dθ < 0 (αε + β, θ) 2-2 θ 2 5 (r, π e ) αβ θ

10 152 中央大学経済研究所年報第 45 号 3. r r 2 1 Fixed ime Delay 1 Continuously Distributed ime Delay heterogenous r r e (t) = Z t 0 1 t τ e r(τ)dτ 18 t t t 185 r(t) r e (t) ṙ(t) = (αε + β)f (r e (t), π e (t)) + απ e (t) (α π + (α + β)ȳ ), π e (t) = εγ(1 θ)f (r e (t), π e (t)) γθπ e (t) + (γθ π (1 θ)ȳ ) 18 3 ṙ(t) = (αε + β)f (r e (t), π e (t)) + απ e (t) (α π + (α + β)ȳ ), π e (t) = εγ(1 θ)f (r e (t), π e (t)) γθπ e (t) + (γθ π (1 θ)ȳ ), ṙ e (t) = 1 [r(t) re (t)] 19

11 2014 金融政策とタイム ラグ ( 浅田 松本 ) 153 (r, π e, r e ) F (r e, π e ) = Ȳ, r = r e, π e = π 19 0 J D = 0 (αε + β)( Y ) + α (αε + β)y 0 γε(1 θ)( Y ) γθ γε(1 θ)y 1 0 J D λi = 0 3 λ 3 + a 1λ 2 + a 2λ + a 3 = C A a i(i = 1, 2, 3) a 1 = 1 + γ [θ + (1 θ)εy ], a 2 = γθ + ( Y )[αε + β γ(1 θ)ε], a 3 = γ( Y )(αε + βθ) > ) a 1 > 0, a 2 > 0, a 3 > 0, a 1a 2 a 3 > 0 23 θ = 1θ = 0 0 < θ < θ = 1 θ = 1 22 a i(i = 1, 2, 3) a F 1 = 1 + γ > 0, a F 2 = γ + (αε + β)( Y ) > 0, a F 3 = γ(αε + β)( Y ) > ) a 1 > 0 a 2 > 0 3

12 154 中央大学経済研究所年報第 45 号 a F 1 a F 2 a F 3 = γ(1 + γ ) + (αε + β)( Y ) 23 > θ = 0 θ = 0 22 a i(i = 1, 2, 3) a B 1 = 1 + γεy, a B 2 = [(α γ)ε + β] ( Y ), a B 3 = αεγ( Y ) > 0 26 a B 1 a B a B 2 a B 1 a B 2 a B 3 = ( Y ) > 0 «1 + γεy [β f(α)] 27 «γ f(α) = γε αε 1 1/ + γεy 28 a B 1 a B 2 a B γεy > 0 2β > αε + γε 1 f(α) 18 r γ

13 2014 金融政策とタイム ラグ ( 浅田 松本 ) γ 1 γ 1 «1 γ + γεy γ < 0 1 < 1/ + γεy f(a) > αε + γε 3-1(A) 1 > γ 1 > [1 + ω( Y )]γ > γ ω( Y ) > 0 df(α) dα «= ε γ 1 < ε 1/ + γεy 3-1(B) 1 = [1 + ω( Y )]γ = f(α) = γε 1/ γ β = f(α) γε 1/ > γ γε

14 156 中央大学経済研究所年報第 45 号 3-1 (α, β) α β β > f(α) 3 3 : 0 < θ < 1 0 < θ < 1 22 a i(i = 1, 2, 3) 2426 θ a i(θ) = θa F i + (1 θ)a B i, i = 1, 2, g(θ) = a 1(θ)a 2(θ) a 3(θ) g(θ) = θ(a F 1 a F 2 a F 3 ) + (1 θ) a B 1 a B 2 a B 3 + (1 θ)θ(a F 1 a B 1 )(a B 2 a F 2 ) 30 (a F 1 a B 1 )(a B 2 a F 2 ) = γ2 [1 + ε( Y )] 2 > 0 a F i > 0 a F 1 a F 2 a F 3 > a B i > 0

15 2014 金融政策とタイム ラグ ( 浅田 松本 ) 157 β > f(α) a B 1 a B 2 a B 3 > 0 a i(θ) > 0 g(θ) > 0 0 < θ < 1 5 θ = 0 θ = 1 (0 < θ < 1) 1 2 β < f(α) 30 F = a F 1 a F 2 a F 3 > 0, B = a B 1 a B 2 a B 3 < 0, Ω = (a F 1 a B 1 )(a B 2 a F 2 ) > 0 g(θ) g(θ) = θ F + (1 θ) B + (1 θ)θω 31 θ g(0) = B < 0, g(1) = F > 0, g (0) = F B > 0 θ (0, 1) g( θ) = 0 θ (0, 1) θ ( F B + Ω) B Ω = ( F ) 2 2 F ( B Ω) + ( B + Ω) 2 > 0 g(θ) = 0 2 θ = F B + Ω p ( F B + Ω) B Ω 2Ω 0 < θ < θ g(θ) < 0, θ < θ < 1 g(θ) > 0 a B i > 0 a i > 0 g(θ)

16 158 中央大学経済研究所年報第 45 号 6 a B 1 a B 2 a B 3 < 0 19 θ > θ θ < θ g(θ) = a 1(θ)a 2(θ) a 3(θ) g( θ) = 0 a 3( θ) = a 1( θ)a 2( θ) ˆλ + a1( θ) ˆλ 2 + a 2( θ) = q λ 1,2 = ±i a 2( θ), λ 3 = a 1( θ) < «`3λ 2 dλ + 2ā 1λ + ā 2 d = λ 2 dā 1 d + λ dā2 d + dā3 d 33 ā i = a i( θ) 22 dā 1 d = 1, dā 2 2 d = ā2, dā 3 d = ā3 λ 1 = i ā 2 3λ ā 1λ 1 + ā 2 = 2(ā 2 iā 1 ā2) 33 dλ/d» " ` dλ 1 ā2 ā2 # Re = Re ā 3 iā 2 d 2(ā 2 iā 1 ā2) Re» dλ = d ā 2 2(ā 2 > ā2) 2

17 2014 金融政策とタイム ラグ ( 浅田 松本 ) Gandolfo2009p.481 (H1) (H2) 2 7 a B 1 a B 2 a B 3 < (C), , https:// ko131018a.htm/ P.Krugman, P.2013 PHP 2009

18 160 中央大学経済研究所年報第 45 号 Asada,. (2010), Central Banking and Deflationary Depression : A Japanese Perspective, in Cappello, M. and Rizzo, C. (eds.) Central Banking and Globalization, Nova Science Publishers : New York, pp Asada,. (ed.) (2014), he Development of Economics in Japan : From the Inter-war Period to the 2000s, Routledge : London. Asada,. (2014), Macrodynamics of Deflationary Depression : A Japanese Perspective, in Asada(ed.) pp Asada,., Chiarella, C., Flaschel, P. and Franke, R. (2010), Monetary Macrodynamics, Routledge: London. Asada,. and Yoshida, H. (2001), Stability, Instability, and Complex Behavior in Macrodynamic Models with Policy Lag, Discrete Dynamics in Nature and Society, Vol. 5, pp Friedman, M. (1948), A Monetary and Fiscal Framework for Economic Stability, American Economic Review, Vol. 38, pp Frisch, R. and Holme, H. (1935), he Characteristic Solutions of a Mixed Difference and Differential Equation Occuring in Economic Dynamics, Econometrica Vol. 3, pp Galί, J. (2008), Monetary Policy, Inflation, and the Business Cycles : A New Keynesian Framework. Princeton : Princeton University Press. Gandolfo, G. (2009), Economic Dynamics, Fourth Edition, Berlin : Springer. James, R. W. and Belz, M. H. (1938), he Significance of the Characteristic Solutions of Mixed Difference and Differential Equations, Econometrica Vol. 6, pp Johansen, L. (1959), Substitution versus Fixed Production Coefficients in the heory of Economic Growth : A Synthesis, Econometrica Vol. 27, pp Kalecki, M. (1935), A Macro-dynamic heory of Business Cycles, Econometrica Vol. 3, pp Keynes, J. M. (1936), he General heory of Employment, Interest and Money, London : Macmillan Lange, O. (1969), heory of Reproduction and Accumulation, Oxford : Pergamon Press. Matsumoto, A. and Szidarovsky, F. (2013), Dynamics in Delay IS-LM Model with ax Collections, IERCU # 208, Institute of Economic Research, Chuo Universityhttp:// economic/publication/discussion/pdf/discussno208.pdf. Phillips, A. W. (1957), Stabilization Policy and the ime-form of Lagged Responses, Economic Journal Vol. 67, pp Steindl, J. (1952), Maturity and Stagnation in American Capitalism, Oxford : Oxford University Press. Woodford, M. (2003), Interest and Prices : Foundations of a heory of Monetary Policy, Princeton : Princeton University Press. Yoshida, H. and Asada,. (2007), Dynamic Analysis of Policy Lag in a Keynes-Goodwin Model : Stability, Instability, Cycles and Chaos, Journal of Economic Behavior and Organization Vol. 62, pp

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