A stochastic Ramsey model is studied with the Cobb-Douglas production function maximizing the expected discounted utility of consumption. We transformed the Hamilton-Jacobi-Bellman (HJB) equation associated with the stochastic Ramsey model so as to transform the dimension of the state space by changing the variables. By the viscosity solution method, we established the existence of viscosity solution of the transformed Hamilton-Jacobi-Bellman equation associated with this model. Finally, the optimal consumption policy is derived from the optimality conditions in the HJB equation. 1. Introduction In financial decision-making problems, Merton’s [1, 2] papers seemed to be pioneering works. In his seminal work, Merton [2] showed how a stochastic differential for the labor supply determined the stochastic processes for the short-term interest rate and analyzed the effects of different uncertainties on the capital-to-labor ratio. The existence and uniqueness of solutions to the state equation of the Ramsay problem [2] is not yet available. In this study, we turned to Merton’s [2] original problem that is revisited considering the growth model for the Cobb-Douglas production function in the finite horizon. Let us define the following quantities: = , ？=？capital stock at time , ？=？labor supply at time , ？=？constant rate of depreciation, , ？=？consumption rate at time , ？=？totality of consumption rate per labor; ？=？ with and is a constant, production function producing the commodity for the capital stock and the labor supply , =？rate of labor growth (nonzero constant), ？=？non-zero constant coefficients, =？discount rate , ？= utility function for the consumption rate , =？one-dimensional standard Brownian motion on a complete probability space endowed with the natural filtration generated by . Let us assume that is a consumption policy per capita such that is nonnegative , a progressively measurable process, and we denote by the set of all consumption policies per capita. The utility function is assumed to have the following properties: Following Merton [2], we make the following assumption on the Cobb-Douglas production function : We are concerned with the economic growth model to maximize the expected discount utility of consumption per labor with a horizon over the class subject to the capital stock , and the labor supply is governed by the stochastic differential equation This optimal consumption problem has been studied by Merton [2], Kamien and Schwartz [3], Koo [4], Morimoto and Kawaguchi [5], Morimoto [6], and Zeldes [7]. Recently, this kind of problem is
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