Eaton 1987 Roldos Eaton Roldos Galor and Lin Shimomura 1993 Nakanishi Turnovsky 1997, Chap. 4

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1 Eaton 987 Roldos 99 Eaton 987 Galor and Lin 997 Roldos 99 Shimomura 993 Nakanishi 000 Turnovsky 997, Chap. 4

2 i i ODA 5 ii i heterogeneous capital joint production iii Ikemoto 969 Jones 97 Mussa 974 Neary 978 Mayer 974

3 9 indivisibility i i i i K i L i y i I i K i L i z i F i 4 z i = F i (K i, L i ). z i x i y i y i y i x i η i xi gi η i 5 I i = g i (x i - η i ). i p i w r i δ i t 6 max i 0 + [pi {F i (K i, L i )- x i } - wl i ] e -rt dt, {L i, x i } s.t. K = g i (x i - η i ) - δ i K i. L i x i F 5 g i : R + R (i) 0 x i η i g i ( ) = 0 (ii) x i > η i g i ( ) > 0 g' i ( ) > 0 g'' i ( ) < 0 (iii) lim Xi + g i ( ) = + lim Xi +0 g' i ( ) = + lim Xi + g' i ( ) = 0 6 K i L i x i t p i w r

4 0 5 F i L i / K i ω i w/p i l i L i K i = l i (ω i ). l i ω i F i F i (K i, K i l i (ω i )) = K i F i (, l i (ω i )) f i (ω i ) F i (, l i (ω i )), h i (ω i ) f i (ω i ) - ω i l i (ω), f i h i ω i µi λi pi / µi λ i λ i λ i ψ i φ i ψ i (λ i ) min {x 0 g i (x i - η i ) = λ i x i } x i = φ i (λ i ) g' i (x i - η i ) = λ i. ψ i λ i φ i λ i I i i K i λ i ω i I ~ i ψ 0 if K i < i (λ i ), f i (ω i ) I i = I ~ ψ i (λ i ) i (K i, λ i, ω i ) g i (K i l i (ω i ) - η i ) if f i (ω i ) K i φ i (λ i ) g i (φ i (λ i ) - η i ) if f <K i. i (ω i ) φ i (λ i ) f i (ω i ) I ~ i K i λ i ω i

5 p i w r δ i λ i λ * i λ i = λ * δ i i + r. h i (ω i ) 004 K * i K * i K * i g i (φ i (λ * i) - η i ). δ i K c i K c i K * i K c i ψ i (λ * i). f i (ω i ) K * i K c i ω i r δ i ω i r δ i λ * i K * i K c i ω i w p i h i ω i ω i λ * i φ i λ i g i x i = φ i (λ i ) λ * i K * i ψ i λ i f ω i λ * i K c i r r λ * i K * i λ * i K c i r K c i δ i λ * i δ i λ * i K * i δ i K * i K c i

6 5 p p /p ω ω w/p ω ω / p L > 0 K l (ω / p) + K l (ω) = L. K (i =, ) p L ω ω = ω (K, K, p, L). ω K i (i =, ) p L ω p λ * i I ~ i K i = I i - δ i K i K = I ( ~ K δ +r ω, h ) - δ K E (K, K, p, L, δ, r), (ω (K,K,p,L)/p) p K = I ( ~ K δ +r, h ω ) - δ K E (K, K, p, L, δ, r), (ω (K,K,p,L)) (K, K ) K K K i E i (K, K, p, L, δ i, r) = 0. i K

7 3 K - K K (K, K) K = K * g ( ( ) φ δ + r i) - η h (ω (K,K,p,L)/p) δ G (K, K, p, L, δ, r). K K G (K, K, p, L, δ, r) G K K p δ r 7 G K K p δ r K K K - K I ~ i K i E i I ~ i K c i (K, K ) E i K = K = δ ψ ( + r h i (ω ) (K,K,p,L)/p) f (ω (K,K,p,L)/p) δ ψ ( + r ) h i (ω (K,K,p,L) f (ω (K,K,p,L)/p) D (K, K, p, L, δ, r), D (K, K, p, L, δ, r). G i D D K i δ i r (i =,) D p D p (K, K ) K - K K K D i D i >, i =,. K i K j ( j i ) D K i K i Ki K i 7 004

8 4 5 K i K K - K K K G G K K D D K K G D G D κ (K, K ) G D K κ G D K κ G D K κ G D K D D β β 0 8 β i i (i =, ) 0 γ ζ ζ K > 0 K > 0 κ 0 (K 0, K 0 ) D D K - K D γ D D γξ D γξ ξ γξ 0 κ 0 D γ D κ 0 D γξ D γξ 8 9

9 5 K D G β ζ G ζ D ξ γ 0 ξ β K β β κ 0 ξ γξ 0 (basin of attraction) D γ D D γξ D γξ ξ γξ 0 p r δ i p K K p ω ω ω ω/p K = 0 ω ω p K = 0 ω ω/p p K = 0 K = 0 p G D G D 0 0

10 6 5 K * i K c i p (i) K β (ii) K β (iii) K (iv) K p ' K K K D' D G G' β ' G G' D' D ξ γ' γ 0 ξ β K z (z, z ) β K =0 β K p

11 7 K K κ * (K *, K * ) p κ *' (K * ', K * ' ) K * < K * ' K * > K * ' κ *'' (K *, K * ') κ * κ *' κ * κ *'' κ *'' κ *' * K * ' p κ * z z κ * κ *'' K * K z z κ *'' κ *' K * ' K * K * ' z z z z r r λ * i K K K K 3 i β i r K i I i i r 3 K K

12 8 5 K D' D G G' b' b a' a a'' G' G D x g D' g' 0 3 x b b' K δ K 3 K K β β ' K ''

13 9 ODA 4 κ K ξ Q β K Q D' D G β κ κ' γ κ'' G D ξ 0 ξ K 4 β K

14 30 5 K κ 4 D 'ξ K D ξ 4 κ' Q K κ' K ξ ODA K ODA κ' K ODA ODA ODA κ' κ'' 4 κ''

15 3 K K Eaton, J., 987, A Dynamic specific-factors model of international trade, Review of Economic Studies LIV, Galor, O. and S. Lin, 997, Dynamic foundations for the factor endowment model of international trade, in: Jensen, B.S. and K-Y. Wong (eds.), Dynamics Economic Growth, and International Trade, University of Michigan Press.

16 3 5 Ikemoto, K., 969, Specific factors of production and the comparative cost theory, Kobe University Economic Review, 3-3. Jones, R.W., 97, A three-factor model in theory, trade, and history, in: J.N. Bhagwati, R.W. Jones, R.A. Mundell, and J. Vanek (eds.), Trade, Balance of Payments and Growth: Papers in International Economics in Honor of Charles P. Kindleberger, North-Holland. Markusen, J.R. and R. Manning, 993, Long-run production frontiers for the Jones specific-factors model with optimal capital accumulation, in: W.J. Ethier, E. Helpman, and J.P. Neary (eds.), Theory, policy and dynamics in international trade: Essays in honor of Ronald W. Jones, Cambridge University Press. Mayer, W., 974, Short-run and long-run equilibrium for a small open economy, Journal of Political Economy 8, No.5, Mussa, M., 974, Tariffs and the distribution of income: The importance of factor specificity, substitutability, and intensity in the short and long run, Journal of Political Economy 8, No.6, Nakanishi, N., 000, An overlapping-generations model of international trade with a durable consumption good, The International Economy 6, -38 (published by the Japan Society of International Economics)., 004,, 89, Neary, P.J., 978, Short-run capital specificity and the pure theory of international trade, Economic Journal 88, Negishi, T., 968, Protection of the infant industry and dynamic internal economies, Economic Record 44, Roldos, J.E., 99, A dynamic specific-factors model with money, Canadian Journal of Economics XXV, No. 3, Shimomura, K., 993, Durable consumption goods and the pattern of international trade, in: N.V. Long and H. Herberg (eds.), Trade, welfare, and economic policies: Essays in honor of Murray C. Kemp, The University of Michigan Press. Turnovsky, S.J., 997, International Macroeconomic Dynamics, MIT Press.

17 33 A DYNAMIC SPECIFIC-FACTORS MODEL OF PRODUCTION WITH THE INDIVISIBILITY IN INVESTMENT ACTIVITIES NORITSUGU NAKANISHI We develop a two-commodity, three-factor, dynamic specific-factors model of production, which is applicable to many development-related international trade issues. Two specific factors are treated as heterogeneous capitals, each of which is accumulated within the corresponding industry. We assume that there is a minimum requirement of factor inputs to attain a positive gross investment-the investment activity exhibits a sort of indivisibility. We show that there are three kinds of long-run production equilibria: (i) the diversification equilibrium at which both industries produce positive amounts of the commodities; (ii) the specialization equilibrium at which only one of the industries survives and the other industry vanishes; and (iii) the vanishing equilibrium at which both industries are completely eliminated from the market. Which equilibrium will be attained in the long run depends upon the initial endowments of the specific factors, the commodity prices, and the interest rate on the internationally traded bond. Some implications of our model for policy interventions are also discussed.

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