On Stochastic Models of Economic Growth

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In recent years there has been a growing interest in two main areas of stochastic economic growth models: the normative area of intertemporal optimization under uncertainty, and the positive area of the behavior of a growing economic system in the presence of stochastic elements. This paper is concerned with thle second area; we are interested in the concept of a steady state for economic systems. A steady state in stochastic models means that we have ever-ch;anging outcomes over time with stationary probability distributions. In the deterministic sense a steady state never actually exists; therefore, a stochastic model might be more suitable. In Section 2 of this paper we introduce a useful theorem in probability which enables us to prove the existence of limiting stationary distributions for some well-known dynamic economic models. The deterministic version of these models has been extensively discussed in the literature. In Section 3 we apply this theorem to a one-sector economic growth Model. Stochastic generalizations of the concept of steady state equilibrium for olnesector models of economic growth have been discussed in recent papers by Dar and the authors [1974], Brock and Mirman [1972, 1973], Mirman [1972, 1973] and Radner [1971]. While most of the literature in the area confined itself to optimal growth under uncertainty, Radner [1971] and Mirman [1972, 1973] posed the problem for positive theory in which an aggregate savings function for the economy is exogeneously given. Obviously, many of the results obtained for positive theory models can also be used to characterize optimal policies. We analyze a positive economic model of growth Under ulncertainty for which we prove that there exists al unique stationary limiting probability distribution of the capitcal-labor ratio. While OLur economic model is sim1 ilar to the one considered by Mirman [1972, 1973], we provide a rather different (and, we believe, simpler) method of proof for which many of the unpleasant assumptions appearing in Mirman's work are not necessary. In particular, we do not require (as in Mirman [1972, 1973]) that isoquants of the aggregate production function do not cross as the uncertainty parameter varies, or that consumption should not equal income. We believe that the approach used in this paper might eventually spawn useful new ideas in the investigation of stochastic economic models.