Existence and Uniqueness of Multi-Agent Equilibrium in a Stochastic, Dynamic Consumption/Investment Model

Abstract

We consider an economy in which a set of agents own productive assets which provide commodity dividend streams, and the agents also receive individual commodity income streams, over a finite time horizon. The agents can buy and sell the commodity at a certain spot price and buy and sell their shares of the productive assets. The proceeds can be invested in financial assets whose prices are modelled as semimartingales. Each agent's objective is to choose a commodity consumption process and to manage his portfolio so as to maximize the expected utility of his consumption, subject to having nonnegative wealth at the terminal time. We derive the optimal agent consumption and investment decision processes when the prices of the productive assets and commodity spot prices are specified. We prove the existence and uniqueness of an “equilibrium” commodity spot price process and productive asset prices. When the agents solve their individual optimization problems using the equilibrium prices, all of the commodity is exactly consumed as it is received, all of the productive assets are exactly owned and the financial markets are in zero net supply.