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

Abstract

Summary. 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 . Production is a¤ected 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 and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever < 1=2. More delicate is the case > 1=2. Singularity with respect to Lebesgue

Self-similar measures in multi-sector endogenous growth models

“…Note that the one-sector stochastic optimal growth model discussed in Mitra et al (2003) exhibits the same optimal dynamics as in (15). Indeed, besides the different constant σ as in (12), the dynamics described by (11) is the same as the optimal dynamics of capital in the one-sector growth model; hence, also the no-overlap condition 0 < α < 1/2 yielding a support which is a Cantor set is exactly the same.…”

“…Especially the one-dimensional 2-maps IFS (λx, λx + (1 − λ) ; p, (1 − p)), with 0 < λ < 1 and 0 < p < 1, characterized by the same contraction factor λ in both maps, has received much attention since the first half of the twentieth century, as its invariant measure µ * is the same as that of the Erdös series ∞ s=0 ±λ s [it being understood that the minus sign is taken with probability p and the plus sign with probability (1 − p)] translated over the interval [0, 1] (see Mitra et al, 2003). For p = 1/2 the topic is known as the study of "symmetric infinite Bernoulli convolutions"; an exhaustive survey on the whole history of this subject can be found in .…”

“…To the best of our knowledge, the contribution by Peres and Solomyak (1998), established for a generic family of contracting similitudes on the real line-including the general case 1/2 ≤ λ < 1 and 0 < p < 1 for the 2-maps IFS above-is still the most advanced available in the literature and proves useful for our purposes in the next Section 3. We summarize below their results, together with the simpler cases discussed before, on the basis of the analysis in Mitra et al (2003), so to provide an overall picture of the state of the art.…”

“…The traditional one-sector growth model with Cobb-Douglas production and logarithmic utility has been extensively studied later. Mitra et al (2003) show that its optimal path converges to a singular measure supported on a Cantor set, characterizing singularity versus absolute continuity of the invariant probability in terms of the parameters' values. Mitra and Privileggi (2004, 2006) further generalize the model and provide also an estimate of the Lipschitz constant for the maps of the optimal policy defined by an IFS.…”

Estimating the stationary distribution of a Markov chain

“…Some of the best known stochastic dynamic models in economics -both descriptive and normativefall into this category. We note that the literature on the "inverse optimal problem" identifies conditions under which a given IFS is "generated" by a stochastic dynamic programming model (see Mitra, Montrucchio and Privileggi (2001) and the list of references). This line of research owes much to the pioneering efforts of Mordecai Kurz (1969).…”