In this article, we consider the general multimodal traffic equilibrium model with elastic demands. The link travel costs may depend upon the entire load pattern and the travel demands associated with each origin-destination pair and mode may depend upon travel costs associated with every origin-destination pair and every mode of transportation. For this model, we define the concepts of user-optimality and equilibrium. We establish, by means of a constructive proof motivated by the theory of variational inequalities, the existence of a unique equilibrium under appropriate monotonicity conditions. We then show that the existence proof induces an algorithm for the computation of equilibrium traffic patterns. The algorithm proceeds by iteration, each step of which amounts to computing the equilibrium pattern for a single modal linear traffic equilibrium problem with elastic demands. We derive estimates for the speed of convergence.