We consider an optimal investment problem with intermediate consumption and random endowment, in an incomplete semimartingale model of the financial market. We establish the key assertions of the utility maximization theory, assuming that both primal and dual value functions are finite in the interiors of their domains and that the random endowment at maturity can be dominated by the terminal value of a self‐financing wealth process. In order to facilitate the verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.