This study compares the Poisson process and Brownian motion based models in derivations of Keynes-Ramsey rules and expected consumption growth rates. These two models can also be used to solve for a simple case with constant investment opportunities in the classic consumption-investment problem. A numerical example is shown to provide details of (1) how investors allocate their portions in assets; (2) how consumption growth related to Jump process differs from that of Wiener process; and (3) how changes in interest rates affect consumption growth.