Stochastic optimal growth under state-dependent probabilities

We analyze the role of state dependent probabilities in a stochastic growth model, where the probability of occurrence of a given shock depends on the capital stock and thus changes over time with capital accumulation. We explicitly determine the optimal policy and its relation with state-dependent probabilities both in the centralized and decentralized frameworks, focusing on two alternative scenarios in which the probability function 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, whenever 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 in 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.