An optimal consumption problem is studied in a growth model for the Cobb-Douglas production function in a finite horizon. The problem is transferred into a stochastic Ramsey problem so as to reduce the dimension of the state space. The corresponding state equation is a stochastic differential equation with inherently non-Lipschitz coefficients, whose unique solvability is established. The unique existence of the classical solution of the Hamilton-Jacobi-Bellman equation associated with the original problem is proved, and a synthesis of the optimal consumption policy is presented in the feedback form.