Before revision:  By using the Kuhn-Tucker theorem , some ideas from convex analysis and the notion of aggregate state price density  , we provide a solution to this problem , in the setting of both complete and incomplete markets (with random endowments).
After revision:  By using the Kuhn-Tucker theorem and the notion of aggregate state price density , we provide a solution to this problem , in the setting of both complete and incomplete markets (with random endowments).