In this article the authors study properties of the optimal policy function in a one-sector stochastic optimal control growth model with iso-elastic utility function \(u(c) = c^{1-\beta}/(1-\beta)\) and a production function \(f(x,r) = rh(x)\), where \(h\) is of Cobb-Douglas type, i.e. \(h(x)= x^{1-\alpha}/(1-\alpha)\), and \(r\) is random, taking value \(q \in (0,1)\) with probability \(p\) and \(1\) else. The value \(q\) is representing a downward production shock. The authors state, that there is no closed form analytic expression for the optimal policy function \(g :\mathbb{R}_+ \to \mathbb {R}_+\). For any output level \(y\) the value \(g(y)\) is interpreted as the optimal consumption in the current period and the value \([y-g(y)]\) as optimal input for the production in the next period. The output \(h[y-g(y)]\) in the following period, when \(r\) takes the value \(1\), is denoted by \(G(y)\). If \(r\) takes the value \(q\) then the output obtained in the following period is \(H(y) := qG(y)\). Iterating over the periods the authors obtain a stochastic process \(y_{t+1}=r_{t+1}G(y_t)\), which alternatively can be interpreted as an iterative function system \(\{H,G,p,1-p\}\). It follows from the standard theory of Markov processes, that the distribution \(\mu_t\) of optimal outputs at time \(t\) converges weakly to the unique invariant distribution \(\mu\). The authors state that standard theory of iterated function systems implies that the support \(A\) of the invariant distribution of \(\mu\) satisfies \(G(A) \cup H(A) = A\) and hence that \(A\) is a self similar set of cantor type. The main result in the article ( Proposition 5 ) states that under specific conditions on the parameters of the model, the set \(A\) is of Lebesgue measure zero and therefore that \(\mu\) is singular with respect to the Lebesgue measure. The authors indicate, that the methods in the article might also be applicable in the case of more general utility functions.