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Stochastic Optimal Growth through State-Dependent Probabilities

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Abstract
We extend the classical discrete time stochastic one-sector optimal growth model with logarithmic utility and Cobb-Douglas production ´a-la Brock and Mirman (1972) to allow probabilities to be state-dependent. In this setting the probability of occurrence of a given shock depends on the capital stock, thus, as the economy accumulates more capital, the probability of occurrence of different shocks changes over time. We explicitly determine the optimal policy and its relation with state-dependent probabilities both in the cen- tralized and decentralized frameworks, focusing on two alternative scenarios in which the probability function, assumed to take a logarithmic form, is either decreasing or increasing with capital. We show that state-dependent probabilities introduce a wedge between the centralized and decentralized solutions, as individual agents do not internalize the effects of capital accumulation on the probability of shocks realization. In particular, when- ever the probability is decreasing (increasing) in the capital stock the probability of the most (least) favorable shock increases, leading the decentralized economy to underinvest (overinvest) in capital accumulation, resulting in the long run into a steady state capital distribution characterized by a leftward (rightward) shifted support. We also show how the features of state-dependent probabilities affect the spread and shape of such a steady state distribution, which tends to be more skewed (more evenly spread) whenever the probability decreases (increases) with capital.

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  • La Torre, Davide & Marsiglio,Simone & Mendivil, Franklin & Privileggi, Fabio, 2023. "Stochastic Optimal Growth through State-Dependent Probabilities," Department of Economics and Statistics Cognetti de Martiis. Working Papers 202312, University of Turin.
    • Handle: RePEc:uto:dipeco:202312
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    References listed on IDEAS

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    1. Mitra, Tapan & Privileggi, Fabio, 2006. "Cantor type attractors in stochastic growth models," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 626-637.
    2. John G. Fernald & J. Christina Wang, 2016. "Why Has the Cyclicality of Productivity Changed? What Does It Mean?," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 465-496, October.
    3. William A. Brock & Leonard J. Mirman, 2001. "Optimal Economic Growth And Uncertainty: The Discounted Case," Chapters, in: W. D. Dechert (ed.), Growth Theory, Nonlinear Dynamics and Economic Modelling, chapter 1, pages 3-37, Edward Elgar Publishing.
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    5. Torre, Davide La & Marsiglio, Simone & Mendivil, Franklin & Privileggi, Fabio, 2019. "A stochastic economic growth model with health capital and state-dependent probabilities," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 81-93.
    6. Dirk Bethmann, 2007. "A Closed-form Solution of the Uzawa-Lucas Model of Endogenous Growth," Journal of Economics, Springer, vol. 90(1), pages 87-107, January.
    7. Mitra, Tapan & Privileggi, Fabio, 2009. "On Lipschitz continuity of the iterated function system in a stochastic optimal growth model," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 185-198, January.
    8. Tapan Mitra & Luigi Montrucchio & Fabio Privileggi, 2003. "The nature of the steady state in models of optimal growth under uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 23(1), pages 39-71, December.
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