# The nature of the steady state in models of optimal growth under uncertainty

## Author Info

• Tapan Mitra

()

• Luigi Montrucchio

()

• Fabio Privileggi

()

## 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 $\alpha < 1/2$ . More delicate is the case $\alpha > 1/2$ . Singularity with respect to Lebesgue measure also appears for values $\alpha ,p$ such that $\alpha < p^{p}\left( 1-p\right)^{\left( 1-p\right) }$ . For $\alpha > p^{p}\left( 1-p\right) ^{\left( 1-p\right) }$ and $1/3\leq p\leq 2/3,$ Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs $\left( \alpha ,p\right)$ . Copyright Springer-Verlag Berlin/Heidelberg 2003

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File URL: http://hdl.handle.net/10.1007/s00199-002-0340-5

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## Bibliographic Info

Article provided by Springer in its journal Economic Theory.

Volume (Year): 23 (2003)
Issue (Month): 1 (December)
Pages: 39-71

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Handle: RePEc:spr:joecth:v:23:y:2003:i:1:p:39-71

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## Related research

Keywords: Stochastic optimal growth; Iterated Function System; Singular and absolutely continuous invariant distribution.;

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Find related papers by JEL classification:
• C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
• O41 - Economic Development, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

## References

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1. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
2. 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.
3. 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 271-86, June.
4. Donaldson, John B. & Mehra, Rajnish, 1983. "Stochastic growth with correlated production shocks," Journal of Economic Theory, Elsevier, vol. 29(2), pages 282-312, April.
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. 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.
7. 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.
8. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-406, November.
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## Citations

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Cited by:
1. Takashi Kamihigashi & John Stachurski, 2014. "Seeking Ergodicity in Dynamic Economies," Working Papers 2014-086, Department of Research, Ipag Business School.
2. 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.
3. Shilei Wang, 2012. "Iterated Function Systems with Economic Applications," Papers 1209.4849, arXiv.org.
4. 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.
5. Takashi Kamihigashi, 2005. "Stochastic Optimal Growth with Bounded or Unbounded Utility and with Bounded or Unbounded Shocks," Discussion Paper Series 176, Research Institute for Economics & Business Administration, Kobe University.
6. Mitra, Tapan & Privileggi, Fabio, 2005. "Cantor Type Attractors in Stochastic Growth Models," POLIS Working Papers 43, Institute of Public Policy and Public Choice - POLIS.
7. 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.
8. Olson, Lars J. & Roy, Santanu, 2005. "Theory of Stochastic Optimal Economic Growth," Working Papers 28601, University of Maryland, Department of Agricultural and Resource Economics.
9. Santanu Roy & Itzhak Zilcha, 2012. "Stochastic growth with short-run prediction of shocks," Economic Theory, Springer, vol. 51(3), pages 539-580, November.
10. 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.

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