An optimal consumption problem is studied in a growth model for the Cobb-Douglas production function in a finite horizon. The problem is transfered into a stochastic Ramsey problem so as to reduce the dimension of the state space. The corresponding state equation is a stochastic differential The second author is supported in part by a start-up fund at University of Oxford, and RGC Earmarked Grants CUHK418605 and CUHK418606. 1