Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

Abstract

This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.MSC (2010): 91B62 93E20.1 Surprisingly, many of these works require an a priori uniform upper bound, usually the constant 1, for consumption processes {c t } t≥0 . This is implicitly suggested in the problem formulation of [7], and explicitly stated as 0 ≤ c t ≤ 1 in [10] and [6]. While this uniform upper bound provides technical conveniences, it can not be fully justified economically in continuous time. After all, for each t ≥ 0, c t represents the consumption ratio per unit of time instantly at time t, which does not admit any natural upper bound. This is in contrast to the discrete-time setting where the upper bound 1 can be easily justified. Morimoto [8,9] consider general, unbounded consumption processes, but not without a cost. There, the production function in the Ramsey model is required to have finite first derivatives, along a boundary of its domain. This particularly rules out the standard Cobb-Douglas production function, commonly used in economic modeling.In other words, a tradeoff exists between the viscosity solutions approach in [10, 6] and Banach's fixed-point argument in [8,9]. The former accommodates the classical Cobb-Douglas production function, but is limited to uniformly bounded consumption processes; the latter allows for general consumptions, but fails to cover the Cobb-Douglas production function. We aim to resolve this tradeoff: this paper considers both unbounded consumption processes and the Cobb-Douglas production function, in the stochastic Ramsey problem. The goal is to characterize the associated value function V , as well as a (possibly unbounded) optimal consumption processĉ.The upfront challenge of our studies is the non-standard stochastic differential equation (SDE) of the state process X, which represents capital per capita; see (2.8) below. On the one hand, the Cobb-Douglas production function renders the drift coefficient of X non-Lipschitz (see Section 5.1 for...