Forward utility and market adjustments in relative investment-consumption games of many players

Abstract

We study a portfolio management problem featuring many-player and mean field competition, investment and consumption, and relative performance concerns under the forward performance processes (FPP) framework. We focus on agents using power (CRRA) type FPPs for their investment-consumption optimization problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin "market-risk relative consumption preference". The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle her risk-tolerance and elasticity of intertemporal substitution (EIS) just like Epstein-Zin preferences under recursive utility framework and unlike the classical utility theory one. This, in turn, allows a finer analysis on the agent's consumption "income" and "substitution" regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent's perception of consumption in a non-trivial manner. We provide numerical illustrations of our results.