A greedy algorithm for habit formation under multiplicative utility

ArXiv ID: 2305.04748 “View on arXiv”

Authors: Unknown

Abstract

We consider the problem of optimizing lifetime consumption under a habit formation model, both with and without an exogenous pension. Unlike much of the existing literature, we apply a power utility to the ratio of consumption to habit, rather than to their difference. The martingale/duality method becomes intractable in this setting, so we develop a greedy version of this method that is solvable using Monte Carlo simulation. We investigate the behaviour of the greedy solution, and explore what parameter values make the greedy solution a good approximation to the optimal one.

Keywords: Habit Formation, Lifetime Consumption, Martingale/Duality Method, Monte Carlo Simulation, Power Utility, Macro-finance (Consumption)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematical, featuring advanced stochastic calculus (Brownian motion, Itô’s formula, PDEs), martingale methods, and complex Lagrange multiplier analysis, which drives up the math complexity score. However, the empirical rigor is low because it focuses on theoretical model derivation and approximation analysis using Monte Carlo simulation, with no mention of real-world data, backtesting, or implementation details for trading strategies.

  flowchart TD
    A["Goal: Optimize lifetime consumption<br>with habit formation"] --> B["Method: Greedy Algorithm<br>via Martingale/Duality method"]
    B --> C["Inputs: Power Utility on<br>Consumption/Habit Ratio"]
    C --> D["Computational Process:<br>Monte Carlo Simulation"]
    D --> E["Outcome: Optimal Consumption<br>Policy Function"]
    E --> F["Findings: Greedy solution approximates<br>optimal under specific parameters"]