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Almost sure convergence to zero in stochastic growth models.

Article Excerpt
Abstract This paper considers the resource constraint commonly used in stochastic one-sector growth models. Shocks are not required to be i.i.d. It is shown that any feasible path converges to zero exponentially fast almost surely under a certain condition. In the case of multiplicative shocks, the condition means that the shocks are sufficiently volatile. Convergence is faster the larger their volatility, and the smaller the maximum average product of capital.

Keywords Stochastic growth * Inada condition * Convergence to zero * Extinction

JEL Classification Numbers: C61 * C62 * E30 * O41

1 Introduction

In a seminal paper, Brock and Mirman (1972) showed that the optimal paths of a stochastic one-sector growth model converge to a unique nontrivial stationary distribution. While various cases are known to which their theorem can be extended, (1) it does not seem to be well understood when the theorem fails. Most of the extensions of the Brock-Mirman theorem assume that the production function satisfies the Inada condition at zero, i.e., that the marginal product of capital tends to infinity as to capital tends to zero. (2)

Although the Inada condition at zero is widely used in economics, the only justification for its use seems to be mathematical convenience. (3) In fact it is known to have the rather unrealistic implication that each unit of capital must be capable of producing an arbitrarily large amount of output with a sufficient amount of labor (e.g., Fare and Primont 2002).

In this paper, we consider the resource constraint commonly used in stochastic one-sector growth models, focusing on the case in which the Inada condition at zero is not satisfied. Our framework encompasses stochastic endogenous growth models as well as stochastic overlapping generation models. To accommodate nonconcave production functions, we assume that the maximum (stochastic) average product of capital is always finite, which is equivalent to the violation of the Inada condition at zero in the concave case. Under this assumption we show that any feasible path converges to zero exponentially fast almost surely if there is a negative upper bound on the long-run sample average of the logarithm of the maximum average product of capital. In the case of multiplicative shocks, this general condition means that the shocks are sufficiently volatile. Convergence is faster the larger their volatility, and the smaller the maximum (deterministic) average product of capital.

To our knowledge, this relationship between almost sure convergence to zero and the volatility of shocks has not been documented in the stochastic growth literature, though technically similar results have recently been obtained independently by Mitra and Roy (2003) (MR) and Nishimura et al. (2004) (NRS). Both MR and NRS study optimal stochastic growth...

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