Existence of an equilibrium with limited participation

TL;DR

This paper proves the existence and uniqueness of a Radner equilibrium in a limited participation economy with exponential preferences, using quadratic BSDEs, extending previous models to more general dividend streams.

Contribution

It establishes the global existence and uniqueness of equilibrium in a limited participation setting with exponential preferences, generalizing prior models and providing explicit examples.

Findings

Unique solution to coupled quadratic BSDEs established

Existence of Radner equilibrium with limited participation proven

Model extended to include general dividend streams

Abstract

A limited participation economy models the real-world phenomenon that some economic agents have access to more of the financial market than others. We prove the global existence of a Radner equilibrium with limited participation, where the agents have exponential preferences and derive utility from both running consumption and terminal wealth. Our analysis centers around the existence and uniqueness of a solution to a coupled system of quadratic backward stochastic differential equations (BSDEs). We prove that the BSDE system has a unique $\mathcal{S}^\infty\times\text{bmo}$ solution. We define a candidate equilibrium in terms of the BSDE solution and prove through a verification argument that the candidate is a Radner equilibrium with limited participation. This work generalizes the model of Basak and Cuoco (1998) to allow for a stock with a general dividend stream and agents with exponential preferences. We also provide an explicit example.