p *2 DSGEDynamic Stochastic General Equilibrium New Keynesian *2 2

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1 nabe@ier.hit-u.ac.jp Jorgenson Tobin q : Hayashi s Theorem : Jordan : 1 investment *1 *1 93SNA 1

2 p *2 DSGEDynamic Stochastic General Equilibrium New Keynesian *2 2

3 2 * GDP GDP GDP GDP 6 *3 NEEDS web GDP 3

4 3 3.1 IS-LM *4 F K, L K: L Inada *5 lim F K =, lim F L = 1 k 0 L 0 lim F K = 0, lim F L = 0 2 K L F K > 0, F L > 0 3 F KK < 0, F LL < 0 4 λ > 0, λf K, L = F λk, λl 5 Y = Const K a L 1 a 0 < α < 1 CES Constant Elasticity of Substitution? CES Y = A a bk ψ + 1 a 1 bl ψ 1/ψ 6 0 < a, b < 1, ψ < 1 Iso Quant *4 Hall, Robert E., and Dale Jorgenson 1967: Tax Policy and Investment Behavior, American Economic Review, 57 Jorgenson * Derivative Condition Uzawa 1961 Inada Condition 4

5 dl dk iso = F/ K 7 F/ L = ab ψ K ψ 1 / 1 a 1 b ψ L ψ 1 8 ab ψ 1 ψ L = 1 a 1 b ψ 9 K 1% %? Slope L/K [ slope L/K L/K ] 1 10 slope [ ab ψ ψ L L 1 a 1 b ψ ] 1+ψ 1 L = 1 ψ ψ 1 a 1 b K K ab ψ 11 K 1 = 12 1 ψ 1/1 ψ ψ 1 Y = Const K a L 1 a ψ 1 CES Inada CES F K = A a bk ψ + 1 a 1 bl ψ 1 ψ/ψ ψ 1 ab bk 0 < ψ < 1 lim K F K > 0 lim K 0 F K = ψ < 0 lim K F K = 0 lim K 0 F K < CES Inada lim k F k = 0 *6 13 *6 CES Translog CES K, L, Y K 10 PC K missing variables 5

6 3.2 Jorgenson *7 r Y w 1 Minimize wl + rk 14 s.t. F K, L = Y 15,F K, L = AK a L 1 a K = λ ay /r 16 K r *8 K I dk dt Jorgenson 1963 *9 16 Kt 1 K = t 1 λ 18 λ 0 < λ < 1 1 Kt Kt λ 1 ln ln 1 = λ ln Kt ln 1 19 λ 1 K t 16 *7 Tobin q 1984 *8 λ *9 D.W. Jorgenson 1963 Capital Theory and Investment Behavior, American Economic Review, Vol. 53, p

7 16 λ λ Time to build 3.3 Tobin q Tobin Jorgenson Tobin q q + q = 20 q 1 * 10 q 1 q 1 * 11 q q + q = 21 * 12 *10 *11 irreversible investment irreversibility Dixit and Pindyck BOJ Working Paper06-J-9 *12 q 7

8 q q * 13 q q q Tobin q Tobin q 4 Jorgenson Tobin q PC *

9 irreversibility * 14 * r = r r/ = r t 1 lim n r n nt r nt = n 1 + r n nt 25 m = n r 26 *14 Perko 1991 *15 9

10 * r nt = m rt 27 n m lim n r n nt = lim m rt 28 m m e e lim m 29 m m t 1 lim n r n nt = e rt 30 r r rt 0 t t s=0 r s ds 31 [ t ] exp r s ds s=0 32 r t s=0 rds = rt t π * 17 I t It φ I t 34 t It π I t φ I t 35 *16 r *17 10

11 π > 0, π < 0 I t φ > 0 36 φ 0 = φ It + φ > 0 38 lim x 0 xφ x = 0 39 φ φ φ φ I φ φ t I t φ Romer2012 Lucas Uzawa t I t I t = +1 40? 11

12 r t r t+1 r t 2 [ It 1 π I t φ I t + π +1 I t+1 φ 1 + r t+1 It+1 +1 I t+1 ] 41 2 s [ ] 1 It+s π +s I t+s φ I t+s 1 + r t+m +s s=1 m=0 42 s=0 s [ ] 1 It+s π +s I t+s φ I t+s 1 + r +s 43 I t = +1 for all t 44 K 0, K 1, K 2, K 3,..., +1,... 2 Policy Function Max s=0 s [ ] 1 It+s π +s I t+s φ I t+s 1 + r +s 45 s=0 s.t. I t = +1 for all t 46 λ t s [ ] 1 It+s L = π +s I t+s φ I t+s + λ t+s I t+s +s+1 + +s r +s 12 s=0

13 q q t+s = 1 + r s λ t+s 48 L = s=0 s [ ] 1 It+s π +s I t+s φ I t+s + q t+s I t+s +s+1 + +s 1 + r +s 49 s=0 I t 1 φ I t φ + q t = 0, 50 q t = 1 + φ + I t φ 51 q q t+1 = q t+1 q t 2 π +1 It+1 + φ + q t r q t = π +1 It+1 + φ = rq t+1 q t+1 r q t * 18 t+1 t+1 q t q t+1 > q t+1 q T T T+1 T q T 1 q T K T = r *18 13

14 t 1 lim q t = 0 55 t 1 + r Transversality Condition * s=0 s [ ] 1 It+s π +s I t+s φ I t+s 1 + r +s 56 e e π rt It I t φ I t dt 57 t=0 d dt = I t 58 * 20 It H = π I t φ I t + q t I t 59 q t = 1 + φ + I t φ 60 q t rq t = H 61 2 π It + φ = rq t q 62 t *19 Arrow and Kurz1970 *

15 lim t e rt q t = T 0 e π rt It I t φ I t dt 64 d dt = I t 65 L = T 0 K T e rt 0 T e π rt It I t φ I t dt + e rt q t I t K t dt + λk T e rt 0 q t λ I t K t K t T 0 e rt q t dt = [ e rt ] T T q t 0 d e rt q t dt dt = e rt q T K T q 0 K 0 0 T 0 d e rt q t dt dt L = T 0 e π rt It I t φ T T I t dt + e rt q t I t dt e rt q T K T 0 d e rt q t + q 0 K 0 + dt + λk T e rt 0 dt T L = e π rt It I t φ I t + q t I t rq t q t dt 0 + λk T e rt e rt q T K T + q 0 K 0 π + It 2 φ rq t q t = 0 15

16 2 rq t q t = π It + φ I t 1 φ I t φ + q t = Tobin Q φ I/K I/K q t = 1 + φ + I t φ 66 2φ + It φ 67 I/K q I t = h q t, h > 0, h 1 = 0 68 q q q 1 I/K q 1 h q I/K q 1 q Tobin q I/K q q q I/K q I/K q q Tobin q q q q Hayashi1982 q q

17 3. 3 q q 2 F K, L q t = rq t F 2 It φ 69 q t = q t + K t q t 70 2 It = K φ 71 rq F + I 1 + φ + I t φ 72 = Krq + I 1 + φ F K, L + wl 73 t cashflow = π t = F K, L I 1 + φ wl 74 q t = Krq π t 75 x t = q t x t = rx t π t 76 rx t e rt xt rx t e rt = e rt π t 77 t=0 de rt x t dt = e rt π t 78 de rt x t dt = e rt π t dt 79 dt t=0 lim t e rt x t x 0 = e rt π t dt 80 t=0 K 0 q 0 = t=0 e rt π t dt 81 q 0 = t=0 e rt π t dt K

18 Tobin q q q q q 1982 q q q * 21 5 = h q t 83 2 q t = rq t π It φ 84 lim t e rt q t = h q t > 0 q t > 1 86 q 1 q 2 q t > 0 rq t > π It + φ 87 *21 q q q 18

19 1 q q K q K 2 q t = 0 = 0 q K q K = 0 q=1 q q q t = 0 q saddle path Phase Diagram saddle path f, g = f K, q 88 19

20 q, K Taylor q t = g K, q 89 = f q, K + f k q, K K K + fq q, K q q 90 q t = g q, K + g k q, K K K + gq q, K q q 91 q, K f q, K = g q, K = 0 K q fk q, K = g k q, K f q q, K g q q, K K K q q 92 K = 0, q = 0 h q = r = π K, q = 1, 1 φ + φ + = 1 I K φ 2φ K q 0 K = 2φ π r K K q q λ 1, λ 2 P 1, P 2 P = P 1, P 2 * 22 0 K 2φ λ1 0 π = P P 1 94 r 0 λ 2 93 K q P 1 K q = P = λ1 0 P 1 K K 0 λ 2 q q λ1 0 P 1 K K 0 λ 2 q q z1 z 2 P 1 K K q q 97 z 1 z 2 = P 1 K q 98 *

21 z 1 z 2 = λ1 0 0 λ 2 z1 z 2 99 z 1t = e λ1t z z 2t = e λ 2t z P11 P P = 12 P 21 P K = P 11 e λ 1t z 10 + P 12 e λ 2t z q t q = P 21 e λ 1t z 10 + P 22 e λ 2t z λ 1, λ 2 K q P z K q λ 1 < 0 < λ 2 P 12 z 20 P 22 z 20 z 20 = 0 P 1 K K q q 2 = K q saddle path saddle P 1 P 11 P = 12 P 21 P 22 saddle path P 21 K K + P 22 q q = λ 1 λ 2 = π K 2φ < saddle * 23 *23 21

22 saddle saddle saddle saddle saddle q?q 84 q saddle path saddle path saddle? 5.2 saddle path q K π 3 q t = 0 q 1? q q 22

23 2 q q q saddle path q q 1 1? q K q q q q saddle path 6 Fabio-Cesare Bagliano and Giuseppe Bertola 2008 Models for Dynamic Macroeconomics, Oxford University Press. 23

24 Romer Barro and Salai-I-Martin Adda and Copper q Hayashi, F., Tobin s Marginal q and Average q: A Neoclassical Interpretation, Econometrica, 50, 1 January, Hayashi & Inoue Hayashi, F., and T. Inoue, The Relation between Firm Growth and Q with Multiple Captial Goods: Theory and Evidence from Panel Data on Japanese Firms, Econometrica, 59, 3, Tobin q Hoshi, T., A. Kashayp, and D. Sharfstein, Corporate Structure, Liquidity and Investment: Evidence from Japanese Industrial Groups, Quarterly Journal of Economics, February, irreversible investment Bagliano and Bertola Dixit, A., and R. Pindyck, 1994 Investment under Uncertaintiy, Princeton, NJ: Princenton University Press. Cabellero Handbook of Macroeconomics Caballero, R.J., 1999, Aggergate Investment, in Taylor and Woodford ed, Handbook of Macroeconomics, 1B, North-Holland. Perko, L., 1991 Differential Equations and Dynamical Systems, Springer-Verlag. 24

25 A.K. Dixit, 1990, Optimization in Economic Theory 2nd edition, Oxford University Press Arrow, K.J., and M. Kurz, 1970, Public Investment, the Rate of Return, and Optimal Fiscal Policy, Baltimore, Johns Hopkins Press. appendix R.J.Barro, and X. Sala-I-Martin, 2003 Economic Growth, second edition, New York: MacGraw-Hill. 25

Jorgenson F, L : L: Inada lim F =, lim F L = k L lim F =, lim F L = 2 L F >, F L > 3 F <, F LL < 4 λ >, λf, L = F λ, λl 5 Y = Const a L a < α < CES? C

Jorgenson F, L : L: Inada lim F =, lim F L = k L lim F =, lim F L = 2 L F >, F L > 3 F <, F LL < 4 λ >, λf, L = F λ, λl 5 Y = Const a L a < α < CES? C 27 nabe@ier.hit-u.ac.jp 27 4 3 Jorgenson Tobin q : Hayashi s Theorem Jordan Saddle Path. GDP % GDP 2. 3. 4.. Tobin q 2 2. Jorgenson F, L : L: Inada lim F =, lim F L = k L lim F =, lim F L = 2 L F >, F

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