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Acyclicity and Dynamic Stability: Generalizations and Applications

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  • Michele Boldrin
  • Luigi Montrucchio

Abstract

We study the asymptotic stability of infinite horizon concave programming problems. Turnpike theorems for this class of models generally have to assume a low level of discounting. By generalizing our precedent work we provide a one-parameter family of verifiable conditions that guarantee convergence of the optimal paths to a stationary state. We call this property theta-acyclicity. In the one-dimensional case we show that supermodulatity implies our property but not viceversa. In the multidimensional case supermodularity has no relevant implications for the asymptotic behavior of optimal paths. We apply theta-acyclicity to a pair of models which study firms' dynamic behavior as based on adjustment costs. The first is the familiar model of competitive equilibrium in an industry in the presence of adjustment costs. IN the second case firms act strategically and we study the dynamic evolution implied by the closed-loop Nash equilibria. In both instances our criteria apply and allow us to obtain stability results that are much more general than those already existing in the literature.

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Bibliographic Info

Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 980.

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Date of creation: Nov 1987
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Handle: RePEc:nwu:cmsems:980

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Postal: Center for Mathematical Studies in Economics and Management Science, Northwestern University, 580 Jacobs Center, 2001 Sheridan Road, Evanston, IL 60208-2014
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Web page: http://www.kellogg.northwestern.edu/research/math/
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  1. Chaim Fershtman & Eitan Muller, 1984. "Turnpike Properties of Capital Accumulation Games," Discussion Papers 604, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. DasGupta, S., 1985. "A local analysis of stability and regularity of stationary states in discrete symmetric optimal capital accumulation models," Journal of Economic Theory, Elsevier, vol. 36(2), pages 302-318, August.
  3. Uzawa, H, 1969. "Time Preference and the Penrose Effect in a Two-Class Model of Economic Growth," Journal of Political Economy, University of Chicago Press, vol. 77(4), pages 628-52, Part II, .
  4. Robert E. Lucas & Jr., 1967. "Adjustment Costs and the Theory of Supply," Journal of Political Economy, University of Chicago Press, vol. 75, pages 321.
  5. Boldrin, Michele & Montrucchio, Luigi, 1986. "On the indeterminacy of capital accumulation paths," Journal of Economic Theory, Elsevier, vol. 40(1), pages 26-39, October.
  6. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-80, January.
  7. L. W. McKenzie, 2010. "Turnpike Theory," Levine's Working Paper Archive 1389, David K. Levine.
  8. Araujo, A & Scheinkman, Jose A, 1977. "Smoothness, Comparative Dynamics, and the Turnpike Property," Econometrica, Econometric Society, vol. 45(3), pages 601-20, April.
  9. Dechert, Dee, 1978. "Optimal control problems from second-order difference equations," Journal of Economic Theory, Elsevier, vol. 19(1), pages 50-63, October.
  10. Fumio Hayashi, 1981. "Tobin's Marginal q and Average a : A Neoclassical Interpretation," Discussion Papers 457, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  11. Magill, Michael J P & Scheinkman, Jose A, 1979. "Stability of Regular Equilibria and the Correspondence Principle for Symmetric Variational Problems," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 20(2), pages 297-315, June.
  12. Boldrin, Michele & Montrucchio, Luigi, 1988. "Acyclicity and Stability of Intertemporal Optimization Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 29(1), pages 137-46, February.
  13. Fershtman, Chaim & Muller, Eitan, 1984. "Capital accumulation games of infinite duration," Journal of Economic Theory, Elsevier, vol. 33(2), pages 322-339, August.
  14. Mortensen, Dale T, 1973. "Generalized Costs of Adjustment and Dynamic Factor Demand Theory," Econometrica, Econometric Society, vol. 41(4), pages 657-65, July.
  15. William A. Brock & Jose A. Scheinkman, 1974. "Global Asymptotic Stability of Optimal Control Systems with Applications to the Theory of Economic Growth," Discussion Papers 85, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  16. Treadway, Arthur B, 1971. "The Rational Multivariate Flexible Accelerator," Econometrica, Econometric Society, vol. 39(5), pages 845-55, September.
  17. Alexandre Scheinkman, Jose, 1976. "On optimal steady states of n-sector growth models when utility is discounted," Journal of Economic Theory, Elsevier, vol. 12(1), pages 11-30, February.
  18. Flaherty, M Therese, 1980. "Industry Structure and Cost-Reducing Investment," Econometrica, Econometric Society, vol. 48(5), pages 1187-1209, July.
  19. Lucas, Robert E, Jr & Prescott, Edward C, 1971. "Investment Under Uncertainty," Econometrica, Econometric Society, vol. 39(5), pages 659-81, September.
  20. McKenzie, Lionel W., 1979. "Optimal Economic Growth and Turnpike Theorems," Working Papers 267, California Institute of Technology, Division of the Humanities and Social Sciences.
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Cited by:
  1. Marcus Hagedorn, 2007. "Optimal Ramsey Tax Cycles," IEW - Working Papers 354, Institute for Empirical Research in Economics - University of Zurich.

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