Martingale Consumption

TL;DR

This paper introduces martingale consumption as a natural consumption pattern aligned with investment strategies, providing explicit solutions in certain models and analyzing its properties compared to classical utility-based strategies.

Contribution

It proposes a novel consumption approach that avoids utility optimization, derives explicit solutions in deterministic models, and explores its properties in stochastic settings.

Findings

Explicit solutions in deterministic models

Uniqueness in general models with random coefficients

Comparison with classical CRRA strategies

Abstract

We propose martingale consumption as a natural, desirable consumption pattern for any given (proportional) investment strategy. The idea is to always adjust current consumption so as to achieve level expected future consumption under the arbitrarily chosen investment strategy. This approach avoids the formulation of an optimization objective based on preferences towards risk, intertemporal consumption, habit formation etc. We identify general explicit solutions in deterministic-coefficient models. In the general case with random coefficients we establish uniqueness, but the question of existence of a solution is unsettled. With the interest rate as the only random factor we derive a PDE for the wealth-to-consumption factor as a function of the state variables, which, however, is non-linear and without known closed-form solutions. We briefly consider the discrete-time case and obtain similar results. Throughout, we compare with well-known optimal strategies for classical CRRA investors with time-additive utility of consumption and find that under suitable time preferences they may in certain cases achieve martingale consumption simultaneously.