Bifurcation in optimal retirement

ArXiv ID: 2506.02155 “View on arXiv”

Authors: Bushra Shehnam Ashraf, Thomas S. Salisbury

Abstract

We study optimal consumption and retirement using a Cobb-Douglas utility and a simple model in which an interesting bifurcation arises. With high wealth, individuals plan to retire. With low wealth they plan to never retire. At a critical level of initial wealth they may choose to defer this decision, leading to a continuum of wealth trajectories with identical utilities.

Keywords: Cobb-Douglas Utility, Optimal Consumption, Retirement Planning, Bifurcation Analysis, Life-Cycle Models, Wealth Management

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper features dense mathematical derivations including HJB equations, ODEs, and free boundary problems, placing it in the high math range. It includes numerical implementation details (Matlab, finite-difference schemes) and calibration based on real-world data, indicating substantial empirical work, though not full backtesting.

  flowchart TD
    A["Research Goal:<br>How does initial wealth affect<br>optimal retirement decisions?"] --> B["Methodology:<br>Cobb-Douglas Utility &<br>Dynamic Programming"]
    B --> C["Data/Inputs:<br>Initial Wealth Levels<br>High vs. Low vs. Critical"]
    C --> D["Computational Process:<br>Optimize Consumption & Retirement<br>under HJB Constraints"]
    D --> E{"Outcome Analysis"}
    E --> F["Findings: Bifurcation Behavior"]
    F --> G["High Wealth → Retire Early"]
    F --> H["Low Wealth → Never Retire"]
    F --> I["Critical Wealth →<br>Indeterminate/Continuum of Paths"]