Solving equilibrium problems in economies with financial markets, home production, and retention

TL;DR

This paper introduces a novel method for computing equilibria in complex economies with financial markets and home production by transforming the problem into a maxinf-optimization, simplifying calculations and handling market incompleteness.

Contribution

It develops a new maxinf-optimization approach incorporating non-arbitrage conditions, enabling equilibrium computation without directly pricing financial contracts.

Findings

Equilibrium prices can be found via maxinf-optimization.

The approach handles incomplete financial markets effectively.

Numerical examples demonstrate the method's efficiency.

Abstract

We propose a new methodology to compute equilibria for general equilibrium problems on exchange economies with real financial markets, home-production, and retention. We demonstrate that equilibrium prices can be determined by solving a related maxinf-optimization problem. We incorporate the non-arbitrage condition for financial markets into the equilibrium formulation and establish the equivalence between solutions to both problems. This reduces the complexity of the original by eliminating the need to directly compute financial contract prices, allowing us to calculate equilibria even in cases of incomplete financial markets. We also introduce a Walrasian bifunction that captures the imbalances and show that maxinf-points of this function correspond to equilibrium points. Moreover, we demonstrate that every equilibrium point can be approximated by a limit of maxinf points for a family of perturbed problems, by relying on the notion of lopsided convergence. Finally, we propose an augmented Walrasian algorithm and present numerical examples to illustrate the effectiveness of this approach. Our methodology allows for efficient calculation of equilibria in a variety of exchange economies and has potential applications in finance and economics.