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The Nature of the Steady State in Models of Optimal Growth Under Uncertainty

Author

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  • Mitra, Tapan

    (Cornell U)

  • Montrucchio, Luigi

    (U of Turin)

  • Privileggi, Fabio

    (Universita degli Studi del Piemonte Orientale)

Abstract

We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent alpha. Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1 - p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters alpha and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever alpha 1/2. Singularity with respect to Lebesgue measure also appears for values alpha, p such that alpha p[superscript p] (1 - p) [superscript (1 - p)] and 1/3

Suggested Citation

  • Mitra, Tapan & Montrucchio, Luigi & Privileggi, Fabio, 2001. "The Nature of the Steady State in Models of Optimal Growth Under Uncertainty," Working Papers 01-04, Cornell University, Center for Analytic Economics.
  • Handle: RePEc:ecl:corcae:01-04
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    File URL: https://cae.economics.cornell.edu/nsmgim2.pdf
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    References listed on IDEAS

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    1. Donaldson, John B. & Mehra, Rajnish, 1983. "Stochastic growth with correlated production shocks," Journal of Economic Theory, Elsevier, vol. 29(2), pages 282-312, April.
    2. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-1406, November.
    3. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
    4. Mirman, Leonard J., 1973. "The steady state behavior of a class of one sector growth models with uncertain technology," Journal of Economic Theory, Elsevier, vol. 6(3), pages 219-242, June.
    5. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
    6. Brock, W. A. & Majumdar, M., 1978. "Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted," Journal of Economic Theory, Elsevier, vol. 18(2), pages 225-243, August.
    7. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
    8. L. Montrucchio & F. Privileggi, 1999. "Fractal steady states instochastic optimal control models," Annals of Operations Research, Springer, vol. 88(0), pages 183-197, January.
    9. Mirman, Leonard J, 1972. "On the Existence of Steady State Measures for One Sector Growth Models with Uncertain Technology," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(2), pages 271-286, June.
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    Citations

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    Cited by:

    1. Takashi Kamihigashi & John Stachurski, 2014. "Interlinkage between Real Exchange rate and Current Account Behaviors: Evidence from India," Working Papers 2014-86, Department of Research, Ipag Business School.
    2. Mitra, Tapan & Privileggi, Fabio, 2003. "Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty," Working Papers 03-09, Cornell University, Center for Analytic Economics.
    3. La Torre, Davide & Marsiglio, Simone & Privileggi, Fabio, 2011. "Fractals and Self-Similarity in Economics: the Case of a Stochastic Two-Sector Growth Model," POLIS Working Papers 157, Institute of Public Policy and Public Choice - POLIS.
    4. Guido Cozzi & Fabio Privileggi, 2009. "The fractal nature of inequality in a fast growing world: new version," Working Papers 2009_30, Business School - Economics, University of Glasgow.
    5. Santanu Roy & Itzhak Zilcha, 2012. "Stochastic growth with short-run prediction of shocks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(3), pages 539-580, November.
    6. Shilei Wang, 2012. "Iterated Function Systems with Economic Applications," Papers 1209.4849, arXiv.org.
    7. Gardini, Laura & Hommes, Cars & Tramontana, Fabio & de Vilder, Robin, 2009. "Forward and backward dynamics in implicitly defined overlapping generations models," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 110-129, August.
    8. repec:ipg:wpaper:2014-086 is not listed on IDEAS
    9. Mitra, Tapan & Privileggi, Fabio, 2005. "Cantor Type Attractors in Stochastic Growth Models," POLIS Working Papers 43, Institute of Public Policy and Public Choice - POLIS.
    10. Kamihigashi, Takashi & Stachurski, John, 2016. "Seeking ergodicity in dynamic economies," Journal of Economic Theory, Elsevier, vol. 163(C), pages 900-924.
    11. La Torre, Davide & Marsiglio, Simone & Privileggi, Fabio, 2018. "Fractal Attractors in Economic Growth Models with Random Pollution Externalities," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201801, University of Turin.
    12. Kamihigashi, Takashi, 2007. "Stochastic optimal growth with bounded or unbounded utility and with bounded or unbounded shocks," Journal of Mathematical Economics, Elsevier, vol. 43(3-4), pages 477-500, April.
    13. Shilei Wang, 2015. "The Iterative Nature of a Class of Economic Dynamics," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 9(3), pages 155-168, December.
    14. Davide La Torre & Simone, Marsiglio & Mendivil, Franklin & Privileggi, Fabio, 2015. "Self-Similar Measures in Multi-Sector Endogenous Growth Models," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201509, University of Turin.
    15. La Torre, Davide & Marsiglio, Simone & Mendivil, Franklin & Privileggi, Fabio, 2016. "Fractal Attractors and Singular Invariant Measures in Two-Sector Growth Models with Random Factor Shares," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201620, University of Turin.
    16. Olson, Lars J. & Roy, Santanu, 2005. "Theory of Stochastic Optimal Economic Growth," Working Papers 28601, University of Maryland, Department of Agricultural and Resource Economics.
    17. Guido Cozzi & Fabio Privileggi, 2002. "Wealth Polarization and Pulverization in Fractal Societies," ICER Working Papers - Applied Mathematics Series 39-2002, ICER - International Centre for Economic Research.

    More about this item

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

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