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

Abstract

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 (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.

Externalities in private ownership production economies with possibility functions. An existence result

“…Our main result states that for all initial endowments which satisfy classical survival conditions, the set of competitive equilibria with consumptions and prices strictly positive is non‐empty and compact. Following the seminal work by Smale (1974), and the recent contributions made by del Mercato (2006) and del Mercato and Platino (2017), we prove our result using an homotopy argument and the topological degree modulo 2 . As shown by del Mercato and Platino (2017), due to the fact that the production sets are not required to be convex, one needs to provide a “piece‐wise homotopy” which makes the proof of our result nontrivial.…”

“…The following theorem is a consequence of the homotopy invariance of the topological degree. Following recent contributions by del Mercato (2006), Bonnisseau and del Mercato (2008), and del Mercato and Platino (2017), our homotopy approach is based on the degree modulo 2, hereafter “”.…”

“…del Mercato (2006), Balasko (2015), Ericson and Kung (2015), and del Mercato and Platino (2017) use a differentiable approach to general equilibrium analysis. In Ericson and Kung (2015), individual preferences and production technology also depend on the price system, and the consumption set coincides with the positive orthant of the commodity space.…”

“…In Balasko (2015), wealth‐dependent preferences are taken into account. del Mercato and Platino (2017) consider private ownership economies with externalities and standard consumption sets. So, our paper extends the latter one to the case of private ownership economies in which the consumption sets of the agents are described by possibility functions.…”

Wealth concerns and equilibrium

“…With preferences that can be represented by separable utility functions that are the sum of a direct utility for the consumption of goods and of another function that represents the impact of externalities as in Dufwenberg et al (2011), equilibrium prices and allocations are those of the economy without externalities defined by the utility functions for goods and the classical existence and regularity properties directly apply to that setup. Without separability, the existence of equilibrium for economies that are convex once externalities are fixed has been proved, but only recently, by del Mercato (2006), del Mercato and Platino (2011), and Ericson and Kungy (2012). Still without separability, regular economies fail to be generic despite an equal number of independent equilibrium equations and unknowns as follows from an example of Bonnisseau and del Mercato (2010).…”