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 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. Singularity with respect to Lebesgue measure also appears for values alpha, p such that alpha p[superscript p] (1 - p) [superscript (1 - p)] and 1/3
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