Stochastic Optimal Growth through State-Dependent Probabilities

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.