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Optimal growth in the Robinson-Shinkai-Leontief model: the case of capital-intensive consumption goods

  • Liuchun Deng , Minako Fujio EMAIL logo and M. Ali Khan
Published/Copyright: May 30, 2019

Abstract

We study the two-sector Robinson-Shinkai-Leontief (RSL) model of discrete-time optimal economic growth for the case of capital-intensive consumption goods. We frame the model in the context of Nishimura’s oeuvre, and more specifically, relate it to its neoclassical cousin: the Uzawa-Srinivasan continuous-time version studied in Haque, W. 1970. “Sceptical Notes on Uzawa’s ‘Optimal Growth in a Two-Sector Model of Capital Accumulation’, and a Precise Characterization of the Optimal Path.” The Review of Economic Studies 37: 377–394. We take Fujio’s identification of the marginal rate of transformation ζ of capital goods today into capital goods tomorrow (Fujio, M. 2006. Optimal Transition Dynamics in the Leontief Two-Sector Growth Model. Ph.D. thesis, The Johns Hopkins University), and using a mix of the Bellman-Blackwell methods of dynamic programming and the value-loss approach of Brock-Mitra, pin down the optimal policy for a specific subset of the parameter space. We discern a bifurcation pattern with respect to the discount factor that echoes the results of Benhabib, J., and K. Nishimura. 1979. “The Hopf Bifurcation and the Structure and Stability of Closed Orbits with Heterogeneous Steady States in Multi-Sector Models of Optimal Growth.” Journal of Economic Theory 21: 421–444 for the neoclassical model, and Khan, M. A., and T. Mitra. 2005. “On Choice of Technique in the Robinson-Solow-Srinivasan Model.” International Journal of Economic Theory 1: 83–110 for the RSS model. We also study the optimal policy correspondence for a general parameter set.

JEL Classification: C60; D90; O21

References

Barna, T. 1957. “Review of ‘Essays in the Theory of Economic Growth’ by Joan Robinson.” Economic Journal 67: 490–493.10.2307/2227364Search in Google Scholar

Benhabib, J., ed. 1992. Cycles and Chaos in Economic Equilibrium. Princeton, NJ: Princeton University Press.10.1515/9780691225210Search in Google Scholar

Benhabib, J., and K. Nishimura. 1979. “The Hopf Bifurcation and the Structure and Stability of Closed Orbits with Heterogeneous Steady States in Multi-Sector Models of Optimal Growth.” Journal of Economic Theory 21: 421–444.10.1016/0022-0531(79)90050-4Search in Google Scholar

Benhabib, J., and K. Nishimura. 1985. “Competitive Equilibrium Cycles.” Journal of Economic Theory 35: 284–306.10.1007/978-3-642-22397-6_4Search in Google Scholar

Benhabib, J., and K. Nishimura. 1989. “Stochastic Equilibrium Oscillations.” International Economic Review 30: 85.10.2307/j.ctv19fvxt1.16Search in Google Scholar

Benhabib, J., K. Nishimura, and A. Venditti. 2002. “Indeterminacy and Cycles in Two-Sector Discrete-Time Model.” Economic Theory 20: 217–235.10.1007/s001990100217Search in Google Scholar

Boldrin, M. 1989. “Paths of Optimal Accumulation in Two-Sector Models.” In Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity, eds. William Barnett, John Geweke, and Karl Shell, 231–252. Cambridge: Cambridge University Press.Search in Google Scholar

Boldrin, M., and R. J. Deneckere. 1990. “Sources of Complex Dynamics in Two-Sector Growth Models.” Journal of Economic Dynamics and Control 14: 627–653.10.2307/j.ctv19fvxt1.13Search in Google Scholar

Boldrin, M., and L. Montrucchio. 1986. “On the Indeterminacy of Capital Accumulation Paths.” Journal of Economic theory 40: 26–39.10.1016/0022-0531(86)90005-0Search in Google Scholar

Burmeister, E., and A. R. Dobell. 1970. Mathematical Theories of Economic Growth. New York: MacMillan.Search in Google Scholar

Chen, B.-L., K. Nishimura, and K. Shimomura. 2004. “An Uzawa-Oniki-Uzawa Dynamic Two-Country Model of International Trade: Back to Heckscher and Ohlin.” mimeo, Kobe University.Search in Google Scholar

Deneckere, R. 1988. An Improved Sufficiency Condition for Complex Dynamics in Optimal Growth Models. Evanston, IL: mimeo, Northwestern University.Search in Google Scholar

Deneckere, R., and S. Pelikan. 1986. “Competitive Chaos.” Journal of Economic Theory 40: 13–25.10.1016/0022-0531(86)90004-9Search in Google Scholar

Dixit, A. K. 1976. The Theory of Equilibrium Growth. Oxford: Oxford University Press.Search in Google Scholar

Fujio, M. 2005. “The Leontief Two-Sector Model and Undiscounted Optimal Growth with Irreversible Investment: The Case of Labor-Intensive Consumption Goods.” Journal of Economics 86: 145–159.10.1007/s00712-005-0143-9Search in Google Scholar

Fujio, M. 2006. Optimal Transition Dynamics in the Leontief Two-Sector Growth Model. Ph.D. thesis, Baltimore, MD: The Johns Hopkins University.Search in Google Scholar

Fujio, M. 2008. “Undiscounted Optimal Growth in a Leontief Two-Sector Model with Circulating Capital: The Case of a Capital-Intensive Consumption Good.” Journal of Economic Behavior & Organization 66: 420–436.10.1016/j.jebo.2006.01.008Search in Google Scholar

Fujio, M. 2009. “Optimal Transition Dynamics in the Leontief Two-Sector Growth Model with Durable Capital: The Case of Capital Intensive Consumption Goods.” The Japanese Economic Review 60: 490–511.10.1111/j.1468-5876.2008.00461.xSearch in Google Scholar

Fujio, M., and M. A. Khan. 2006. “Ronald W. Jones and Two-Sector Growth: Ramsey Optimality in the RSS and Leontief Cases.” Asia-Pacific Journal of Accounting & Economics 13: 87–110.10.1080/16081625.2006.9720774Search in Google Scholar

Hamberg, D. 1963. “Review of ‘Essays in the Theory of Economic Growth’ by Joan Robinson.” American Economic Review 53: 1109–1114.Search in Google Scholar

Haque, W. 1970. “Sceptical Notes on Uzawa’s ‘Optimal Growth in a Two-Sector Model of Capital Accumulation’, and a Precise Characterization of the Optimal Path.” The Review of Economic Studies 37: 377–394.10.2307/2296727Search in Google Scholar

Khan, M. A., and T. Mitra. 2005. “On Choice of Technique in the Robinson-Solow-Srinivasan Model.” International Journal of Economic Theory 1: 83–110.10.1111/j.1742-7363.2005.00007.xSearch in Google Scholar

Khan, M. A., and T. Mitra. 2006. “Discounted Optimal Growth in the Two-Sector RSS Model: A Geometric Investigation.” Advances in Mathematical Economics 8: 349–381.10.1007/4-431-30899-7_14Search in Google Scholar

Khan, M. A., and T. Mitra. 2007. “Optimal Growth Under Discounting in the Two-Sector Robinson-Solow-Srinivasan Model: A Dynamic Programming Approach†.” Journal of Difference Equations and Applications 13: 151–168.10.1080/10236190601069069Search in Google Scholar

Khan, M. A., and T. Mitra. 2012. “Impatience and Dynamic Optimal Behavior: A Bifurcation Analysis of the Robinson–Solow–Srinivasan Model.” Nonlinear Analysis: Theory, Methods & Applications 75: 1400–1418.10.1016/j.na.2011.05.053Search in Google Scholar

Khan, M. A., and T. Mitra. 2013. “Optimal Growth in a Two-Sector RSS Model with Discounting: A Further Geometric Investigation.” Advances in Mathematical Economics 17: 1–33.10.1007/978-4-431-54324-4_2Search in Google Scholar

Lancaster, K. 1960. “Mrs. Robinson’s Dynamics.” Economica 27: 63.10.2307/2551428Search in Google Scholar

Le Van, C., and R. Dana. 2003. Dynamic Programming in Economics. Dordrecht: Kluwer Academic Publishers.Search in Google Scholar

Little, I. M. D. 1957. “Classical Growth.” Oxford Economic Papers 9: 152–177.10.1093/oxfordjournals.oep.a042277Search in Google Scholar

Magee, S. P. 1973. “Factor Market Distortions, Productions, Production, and Trade: A Survery.” Oxford Economic Papers 25: 1–43.10.1093/oxfordjournals.oep.a041239Search in Google Scholar

Magee, S. P., W. A. Brock, L. Young. 1989. Black Hole Tariffs and Endogenous Policy Theory. Cambridge Books, Cambridge University Press, No. 9780521377003.Search in Google Scholar

Majumdar, M., and T. Mitra. 1994. “Periodic and Chaotic Programs of Optimal Intertemporal Allocation in an Aggregative Model with Wealth Effects.” Economic Theory 4: 649–676.10.1007/978-3-662-04060-7_3Search in Google Scholar

Majumdar, M., T. Mitra, and K. Nishimura. 2000. Optimization and Chaos. Vol. 11, Berlin: Springer.10.1007/978-3-662-04060-7Search in Google Scholar

Malinvaud, E. 1961. “The Analogy between Atemporal and Intertemporal Theories of Resource Allocation.” The Review of Economic Studies 28: 143–160.10.1007/978-1-349-15492-0_15Search in Google Scholar

May, R. M. 1976. “Simple Mathematical Models with very Complicated Dynamics.” Nature 261: 459–467.10.1038/261459a0Search in Google Scholar

McKenzie, L. W. 1986. “Optimal Economic Growth, Turnpike Theorems and Comparative Dynamics.” In Handbook of Mathematical Economics, eds. K. J. Arrow, and M. Intrilligator, Vol. 3, 1281–1355, New York: North-Holland.10.1016/S1573-4382(86)03008-4Search in Google Scholar

McKenzie, L. W. 2005. Classical General Equilibrium Theory. Cambridge, Mass: The MIT press.Search in Google Scholar

Morishima, M. 1965. “The Multi-Sectoral Theory of Economic Growth.” In Mathematical Optimization in Economics, ed. B. de Finetti, 81–163. Fondazione CIME, Berlin: Springer-Verlag.Search in Google Scholar

Morishima, M. 1969. Theory of Economic Growth. OUP Catalogue, Oxford University Press, No. 9780198281641.10.1093/0198281641.001.0001Search in Google Scholar

Nishimura, K., and M. Yano. 1994. “Optimal Chaos, Nonlinearity and Feasibility Conditions.” Economic Theory 4: 689–704.10.1007/978-3-662-04060-7_4Search in Google Scholar

Nishimura, K., and M. Yano. 1995. “Nonlinear Dynamics and Chaos in Optimal Growth: An Example.” Econometrica 63: 981–1001.10.2307/2171805Search in Google Scholar

Nishimura, K., and M. Yano. 1996. “Chaotic Solutions in Dynamic Linear Programming.” Chaos, Solitons & Fractals 7: 1941–1953.10.1007/978-3-642-22397-6_7Search in Google Scholar

Oswald, A. J. 1985. “The Economic Theory of Trade Unions: An Introductory Survey.” The Scandinavian Journal of Economics 87: 160.10.1007/978-1-349-08596-5_2Search in Google Scholar

Ramsey, F. P. 1928. “A Mathematical Theory of Saving.” The Economic Journal 38: 543–559.10.2307/2224098Search in Google Scholar

Robinson, J. 1956. The Accumulation of Capital. London: Macmillan, reprinted by Palgrave Macmillan in 2013.Search in Google Scholar

Robinson, J. 1962. Essays in the Theory of Economic Growth. London: Macmillan.10.1007/978-1-349-00626-7Search in Google Scholar

Robinson, J. 1970. “Review of ‘The Neoclassical Theory of Production and Distribution’ by C. E. Ferguson.” Economic Journal 80: 336–339.10.2307/2230122Search in Google Scholar

Robinson, J. 1971. Economic Heresies: Some Old-Fashioned Questions in Economic Theory. New York: Basic Books.10.1007/978-1-349-81576-0Search in Google Scholar

Robinson, J. 1978. Contributions to Modern Economics. New York: Academic Press.Search in Google Scholar

Samuelson, P. A. 1958. “An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money.” Journal of Political Economy 66: 467–482.10.1086/258100Search in Google Scholar

Shinkai, Y. 1960. “On Equilibrium Growth of Capital and Labor.” International Economic Review 1: 107.10.2307/2525290Search in Google Scholar

Solow, R. 2000. Growth Theory, An Exposition. Oxford: Oxford University Press.Search in Google Scholar

Solow, R. M. 1962. “Comment.” The Review of Economic Studies 29: 255–257.10.1007/978-1-349-03196-2_44Search in Google Scholar

Solow, R. M. 1970. “Growth Theory. an Exposition.” In Growth theory. An exposition. Oxford: Clarendon Press.Search in Google Scholar

Solow, R. M. 1999. “Chapter 9 Neoclassical Growth Theory.” In Handbook of Macroeconomics, Elsevier, 637–667.10.1016/S1574-0048(99)01012-5Search in Google Scholar

Srinivasan, T. N. 1964. “Optimal Savings in a Two-Sector Model of Growth.” Econometrica 32: 358.10.2307/1913041Search in Google Scholar

Stiglitz, J. E., and H. Uzawa. 1969. Readings in the Modern Theory of Economic Growth. Cambridge: MIT Press.Search in Google Scholar

Uzawa, H. 1964. “Optimal Growth in a Two-Sector Model of Capital Accumulation.” The Review of Economic Studies 31: 1.10.1017/CBO9780511664496.019Search in Google Scholar

Worswick, G. D. N. 1959. “Mrs. Robinson on Simple Accumulation. A Comment with Algebra.” Oxford Economic Papers 11: 125–142.10.1093/oxfordjournals.oep.a040819Search in Google Scholar

Worswick, G. D. N. 1963. “Review of ‘Essays in the Theory of Economic Growth’ by Joan Robinson.” Economic Journal 73: 295–297.10.2307/2228542Search in Google Scholar


Article Note

An earlier version of the results reported here was presented in the special session in honor of Professor Kazuo Nishimura at the 26th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics in Keio University on March 19, 2018. The authors wish to acknowledge the inspiration received over the years from the pioneering writings of Kazuo Nishimura; Khan should also like to acknowledge with gratitude the privilege of a friendship dating to the first meeting in Atlantic City in 1976. The authors gratefully acknowledge the encouragement and stimulus into our thinking provided by Jess Benhabib, Hülya Eraslan, Tapan Mitra, and Makoto Yano.



Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2019-0032).


Published Online: 2019-05-30

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