We investigate the nonatomic assignment model introduced by Gretsky, Ostroy and Zame (1992) under general conditions on the economic agents and utility functions. This extension allows considering various function spaces (price processes and risk curves) as models for the market. We show that the assignment problem based on “inequality” constraints is equivalent to a corresponding transportation problem under “equality” constraints. This equivalence leads to a general duality theorem in this setting. In order to model some external regulations on the market requiring at least a certain minimum level of activity on the part of the agents, we then introduce a modified assignment model with constraints on the agents. We establish duality theorems for this modified assignment problem and existence results for optimal solutions.