Private ownership economies with externalities and existence of competitive equilibria: a differentiable approach

We consider a general equilibrium model of a private ownership economy with consumption and production externalities. Utility functions and production technologies may be affected by the consumption and production activities of all other agents in the economy. We use homotopy techniques to show that the set of competitive equilibria is non-empty and compact. Fixing the externalities, the assumptions on utility functions and production technologies are standard in a differentiable framework. Competitive equilibria are written in terms of first order conditions associated with agents' behavior and market clearing conditions, following the seminal paper of Smale (J Math Econ 1:1-14, 1974). The work of adapting the homotopy approach to economies with externalities on the production side is non-trivial and it requires some ingenious adjustments, because the production technologies are not required to be convex with respect to the consumption and production activities of all agents.