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

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 α. 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 α and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever α < 1/2. More delicate is the case α > 1/2. Singularity with respect to Lebesgue measure also appears for values α, p such that α < p^P (1 - p)^(1-p). For α > p^p (1 - p)^(1-p) and 1/3 ≤ p ≤ 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 (α, p).