This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to previous studies, we allow for general consumption processes, without any a priori boundedness constraint. The associated value function is characterized as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. An optimal consumption process, expressed in terms of the value function , is in a feedback form of the state process. The characterization of the value function relies on constructing a suitable sequence of approximating functions and employing viscosity solutions techniques. The derivation of the optimal consumption process involves a mixture of probabilistic arguments concerning the explosion and pathwise uniqueness of the controlled state process .
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 .
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previous
prior
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The associated value function is characterized
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
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An optimal consumption process, expressed in terms
This characterization involves a condition on the limiting behavior
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, is in a feedback form of the state process. The characterization of the value function relies on constructing a suitable sequence of approximating functions and employing viscosity solutions techniques. The derivation of the optimal consumption process involves a mixture of probabilistic arguments concerning the explosion and pathwise uniqueness of the controlled state process
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