On the objective of firms under uncertainty with stock markets.

In a multi-period, multi-commodity economy with stock markets, we try to extend the work of Drèze (1974) to define the behaviour of the firms. We exhibit first order necessary conditions for a constrained Pareto optimal allocation. The financial constraints lead to non-collinear supporting spot prices for the consumers at each node. Nevertheless, the firms are satisfying a first order necessary condition for profit maximization with respect to a price computed as the Drèze's prices. These prices are also consistent in the sense that the present value of the firms computed with the personal prices of the stockholders and with the Drèze's prices coincide when short sales are allowed. We also show that these conditions are simpler if we consider an allocation at which each consumer maximizes his preferences, when they are smooth. This allows us to give a formal definition for the objective of the firms, which extend the Drèze's criterion. We also discuss different definitions of constrained feasibility and we provide the related necessary conditions, which do not differ for the production sector.