  In this paper, we consider a num\'eraire-based utility maximization problem
under constant proportional transaction costs and random endowment. Assuming
that the agent cannot short sell assets and is endowed with a strictly positive
contingent claim, a primal optimizer of this utility maximization problem
exists. Moreover, we observe that the original market with transaction costs
can be replaced by a frictionless shadow market that yields the same
optimality. On the other hand, we present an example to show that in some case
when these constraints are relaxed, the existence of shadow prices is still
warranted.
