This thesis considers an investor who can distribute wealth between two assets , one with deterministic rate of growth (eg. bank deposit account) , the other with growth governed by a Brownian motion with drift (eg. equity share) . Transfers between these holdings incur proportional transaction costs . The investor may consume continuously and costlessly from the bank , and requires a consumption and investment strategy which maximises total discounted utility of consumption over an infinite horizon .For a large class of utility functions , it is proved that maximum utility is achieved by a strategy-which confines the investor's portfolio to a certain wedge-shaped region in the portfolio plane, by minimal trading, and allows consumption at a rate depending at any time on the current state of the portfolio .Moreover it is shown that the optimal strategy does not lead to bankruptcy in finite time.The problem is treated rigorously as a reflected diffusion process, and the existence and uniqueness of the optimally-controlled wealth process are demonstrated.