  In this paper, we study the problem of expected utility maximization of an
agent who, in addition to an initial capital, receives random endowments at
maturity. Contrary to previous studies, we treat as the variables of the
optimization problem not only the initial capital but also the number of units
of the random endowments. We show that this approach leads to a dual problem,
whose solution is always attained in the space of random variables. In
particular, this technique does not require the use of finitely additive
measures and the related assumption that the endowments are bounded.
