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 <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\alpha $</EquationSource> </InlineEquation>. Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\alpha $</EquationSource> </InlineEquation> and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\alpha < 1/2$</EquationSource> </InlineEquation>. More delicate is the case <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\alpha > 1/2$</EquationSource> </InlineEquation>. Singularity with respect to Lebesgue measure also appears for values <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$\alpha ,p$</EquationSource> </InlineEquation> such that <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$\alpha < p^{p}\left( 1-p\right)^{\left( 1-p\right) }$</EquationSource> </InlineEquation>. For <InlineEquation ID="Equ7"> <EquationSource Format="TEX">$\alpha > p^{p}\left( 1-p\right) ^{\left( 1-p\right) }$</EquationSource> </InlineEquation> and <InlineEquation ID="Equ8"> <EquationSource Format="TEX">$1/3\leq p\leq 2/3,$</EquationSource> </InlineEquation> 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 <InlineEquation ID="Equ9"> <EquationSource Format="TEX">$\left( \alpha ,p\right) $</EquationSource> </InlineEquation>. Copyright Springer-Verlag Berlin/Heidelberg 2003