We consider the Merton consumption problem on a finite time horizon to optimize the discounted expected power utility of consumption and terminal wealth in risk-averse cases. The returns and volatilities of the assets are random and affected by some economic factors, modeled as a diffusion process. The problem becomes a standard stochastic control problem. We derive the Hamilton-Jacobi-Bellman (HJB) equation and study its solutions. Under general conditions we construct a suitable subsolution-supersolution pair. We prove the existence and uniqueness of solution for this HJB equation. Finally, we show the verification theorem.