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

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1 Jorgenson Tobin q : Hayashi s Theorem Jordan Saddle Path. GDP % GDP Tobin q 2 2.

2 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? CES Y = A a b ψ + a bl ψ /ψ 6 < a, b <, ψ < / ψ ψ Y = Const a L a CES Inada CES F = A a b ψ + a bl ψ ψ/ψ ψ ab b 7 < ψ < lim F > lim F = ψ < lim F = lim F < CES Inada lim k F k = * * CES Translog CES, L, Y PC missing variables 2

3 2.2 Jorgenson *2 r Y w Minimize wl + r 8 s.t.f, L = Y 9,F, L = A a L a = ay /r r? I d dt Jorgenson 963 t = t λ 2 λ < λ < t t λ ln ln = λ ln t ln 3 λ t *2 Tobin q 984 3

4 Time to build 2.3 Tobin q Tobin Jorgenson Tobin q q + q = 4 q *3 q q *4 q q + q = 5 *5 *3 *4 irreversible investment irreversibility Dixit and Pindyck BOJ Working Paper6-J-9 *5 q 4

5 q q *6 q q q Tobin q Tobin q 3 Jorgenson Tobin q irreversibility *

6 *7 * r = + r 6 r/2 = + r t lim n + r n nt 8 + r nt = n + r n nt 9 m = n r 2 *9 + r nt = + m rt 2 n m *7 Perko 99 *8 *9 r 6

7 lim n + r n nt = lim + m rt 22 m m e e lim + m 23 m m t lim n + r n nt = e rt 24 r r rt t t s= r s ds 25 [ e R t ] t s= rsds = exp r s ds s= 26 r t s= r s ds = rt t π I t It φ I t 28 t It π I t φ I t 29 π >, π < I t φ > 3 φ = 3 7

8 2φ It + φ > 32 lim x xφ x = 33 Romer24 Lucas Uzawa 3..4 t I t I t = + 34? r t 8

9 + r t+ r t 2 [ It π I t φ I t + π + I t+ φ + r t+ It+ + I t+ ] 35 2 s [ ] It+s π +s I t+s φ I t+s + r t+m +s s= m= 36 s= s [ ] It+s π +s I t+s φ I t+s + r +s 37 I t = + for all t 38 2 Policy Function Max s= s [ ] It+s π +s I t+s φ I t+s + r +s 39 s.t. I t = + for all t 4 λ t L = s= s [ ] It+s π +s I t+s φ I t+s + r +s q + λ t+s I t+s +s+ + +s 4 s= q t+s = + r s λ t 42 L = s= s [ ] It+s π +s I t+s φ I t+s + q t+s I t+s +s+ + +s + r +s 43 9

10 s= I t φ I t φ + q t = 44 q t = + φ + I t φ 45 q q t+ = q t+ q t 2 π + It+ + φ + q t+ + r q t = π + It+ + φ = rq t+ q t+ r q t * t+ t+ q t q t+ > q t+ q T T T+ T q T q T T = 48 + r t lim q t = 49 t + r Transversality Condition * * * Arrow and urz97

11 3..5 s= s [ ] It+s π +s I t+s φ I t+s + r +s 5 e e π rt It I t φ I t dt 5 t= d dt = I t 52 * 2 It H = π I t φ I t + q t I t 53 q t = + φ + I t φ 54 q t rq t = H 55 2 π It + φ = rq t q 56 t lim t e rt q t = 57 *2 995

12 3..6 T e π rt It I t φ I t dt 58 d dt = I t 59 L = T T e rt T e π rt It I t φ I t dt + e rt q t I t t dt + λ T e rt q t λ I t t t T e rt q t dt = [ e rt ] T T q t e rt q t t dt = e rt q T T q T e rt q t t dt L = L = T + q + T e π rt It I t φ T e rt q t t dt + λ T e rt e rt π I t φ It + λ T e rt e rt q T T + q π + It T I t dt + e rt q t I t dt e rt q T T I t + q t I t rq t q t dt 2 φ rq t q t = 2 rq t q t = π It + φ 2

13 I t φ I t φ + q t = 3..7 Tobin Q φ I/ I/ q t = + φ + I t φ 6 2φ + It φ 6 I/ q I t = h q t, h >, h = 62 q q q I/ q h q I/ q q Tobin q I/ q q q I/ q I/ q q Tobin q q q q 982 q q q q 2 F,L 3

14 q t = rq t F 2 It φ 63 q t = q t + t q t 64 2 It = φ 65 rq F + I + φ + I t φ 66 = rq + I + φ F, L + wl 67 t cashflow = π t = F, L I + φ wl 68 q t = rq π t 69 x t = q t x t = rx t π t 7 rx t e rt xt rx t e rt = e rt π t 7 de rt x t dt = e rt π t 72 t= de rt x t dt = e rt π t dt 73 dt t= lim t e rt x t x = e rt π t dt 74 t= q = t= e rt π t dt 75 q = t= e rt π t dt 76 Tobin q q q q q 4

15 982 q q q * 3 4 = h q t 77 2 q t = rq t π It φ 78 lim t e rt q t = h q t > q t > 8 q q 2 q t > rq t > π It + φ 8 q q q q t = = q q = q= q q q t = q *3 q q q 5

16 saddle path 4. 2 f, g = f, q 82 q, Taylor q t = g, q 83 = f q, + f k q, + fq q, q q 84 q = g q, + g k q, + gq q, q q 85 q, q fk q, = g k q, f q q, g q q, q q 86 =, q = h q = r = π, q =, φ + φ + I φ = 2φ q = 2φ π r q q λ, λ 2 P, P 2 P = P, P 2 * 4 2φ λ π = P P 88 r λ 2 87 q λ = P λ 2 P q q 89 *

17 P q λ = λ 2 P q q 9 z z 2 P q q 9 z z 2 = P q 92 z z 2 λ = λ 2 z z 2 93 z t = e λt z 94 z 2t = e λ 2t z 2 95 P P P = 2 P 2 P = P e λt z + P 2 e λ2t z 2 97 q t q = P 2 e λt z + P 22 e λ2t z 2 98 λ, λ 2 q P z q λ < < λ 2 P 2 z 2 P 22 z 2 z 2 = P q q 2 = 99 q saddle path saddle P P P = 2 P 2 P 22 saddle path P 2 + P 22 q q = 7

18 λ λ 2 = π 2φ < saddle * 5 saddle saddle saddle saddle saddle q?q 78 q saddle path saddle path saddle? 4.2 saddle path π 2 q t = q? q q q q q saddle path q q 5 Adda and Copper23 *5 8

19 q Hayashi, F., Tobin s Marginal q and Average q: A Neoclassical Interpretation, Econometrica, 5, January, 982. Hyashi & 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, 99. Tobin q Hoshi, T., A. ashayp and D. Sharfstein, Corporate Structure, Liquidity and Investment: Evidence from Japanese Industrial Groups, Quarterly Journal of Economics, February, 99. irreversible investment Dixit, A., and R. Pindyck, 994 Investment under Uncertaintiy, Princeton, NJ: Princenton University Press. Cabellero Handbook of Macroeconomics Caballero, R.J., 999, Aggergate Investment, in Taylor and Woodford ed, Handbook of Macroeconomics, B, North-Holland. Perko, L., 99 Differential Equations and Dynamical Systems, Springer-Verlag Arrow,.J., and M. urz, 97, Public Investment, the Rate of Return, and Optimal Fiscal Policy, Baltimore, Johns Hopkins Press. appendix Barro, J.R., and X. Sala-I-Martin., 995 Economic Growth, New York; NY, McGraw Hill. 9

20 q q = = 2 q q = = q q = =

Keynes 1936 Tobin 1 Hicks 1937 Keynes IS LM 2 IS LM 3 IS LM Keynes Keynes animal spirits

43 2000 17 28 Masahiro NAKANO 2.1 Keynes 2.2 3.1 3.2 43 2000 Keynes 1936 Tobin 1 Hicks 1937 Keynes IS LM 2 IS LM 3 IS LM Keynes Keynes animal spirits Keynes IS LM 1990 Keynes 1980 90 2 1 43 2000 Keynes