# Existence theory for a time-dependent mean field games model of household wealth

## TL;DR

This paper establishes the existence and uniqueness of solutions for a relaxed mean field games model of household wealth, and identifies conditions under which the original model has no solutions.

## Contribution

It introduces a relaxed version of a macroeconomic mean field game model, proving existence, uniqueness, and nonexistence results for solutions under small time horizons.

## Key findings

- Existence and uniqueness of solutions for the relaxed problem.
- Conditions where the original problem has no solutions.
- Demonstration of nonexistence of solutions in certain cases.

## Abstract

We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This system together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou, Buera, Lasry, Lions, and Moll. We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are in fact not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.