Optimal Savings under Uncertainty1

There has been extensive discussion in the literature of optimal savings behaviour under certainty in the context of infinite time horizon. To our knowledge the extension of the results under certainty to a situation of uncertainty has been attempted by Mirrlees [1] and Phelps [2]. Mirrlees considers a one-commodity neoclassical model 'with two factors of production, labour and capital; constant returns to scale in production; exponential labour force growth, and Harrod-neutral technical change. Uncertainty is introduced in the model form of a Wiener process over time for the logarithm of the Harrod-neutral technical change. The maximand is the expected value of the integral of discounted future per capita utilities. Mirrlees establishes a set of conditions characterizing an optimal consumption policy as a function of capital stock per unit of labour and the level of technology. These conditions correspond to the Euler equations and the transversality condition characterizing the optimal accumulation path under certainty. For the case of a constant (but negative) elasticity utility function and a Cobb-Douglas production function, Mirrlees shows that optimal savings can increase with increasing uncertainty, at least for some set of values of the capital-labour ratio and the level of technology. Phelps considers a pure capital model where an individual at any moment of time has the option of either consuming his wealth and current wage income or investing part of it. The return to investment is uncertain but the probability distribution of returns is assumed to be independently and identically distributed over time. The objective is again to maximize the expected value of the sum of discounted future utility over a finite horizon. Phelps takes the limit, as the horizon extends to infinity, of the finite-horizon optimal policy and discusses the behaviour of the limit policy as the " riskiness " of return to capital increases. The utility function is one for which the elasticity of marginal utility is constant. It is shown that the limit policy results in lower (higher) consumption for any given level of wealth as riskiness increases if the elasticity of marginal utility exceeds (falls short of) unity in absolute value. In the in-between case of unitary elasticity, consumption policy is invariant with respect to riskiness. Phelps somehow feels the last result to be " odd"". Our purpose is first to re-examine a slightly simplified version of Phelps's model (we assume wage income to be zero) in the context of an infinite horizon, and to derive a set of sufficient conditions characterizing an optimal policy provided one exists. These conditions will be derived on the assumption of a strictly concave utility function, a class which includes, but is wider than, the class of constant elasticity utility functions. Second, we show that the limit policy obtained by Phelps is indeed the optimal policy for an infinite horizon for the class of constant elasticity utility functions. The set of sufficient conditions we derive is quite similar to the set derived by Mirrlees for his more general (and hence complicated) continuous time model. We believe that our derivation is of independent interest in that it yields all the Mirrlees-Phelps results in a considerably more simple and intuitively appealing way, though our model is not as rich as Mirrlees' because of its omission of diminishing returns to capital. In the last section we extend our model to include dynamic portfolio choice and discuss some possible implications about portfolio choice of change of the risk parameters of one of the assets.