Paper: arXiv 2309.12588

Authors: Zhou Yang, Junkee Jeon

Abstract

In this paper, we undertake an investigation into the utility maximization problem faced by an economic agent who possesses the option to switch jobs, within a scenario featuring the presence of a mandatory retirement date. The agent needs to consider not only optimal consumption and investment but also the decision regarding optimal job-switching. Therefore, the utility maximization encompasses features of both optimal switching and stochastic control within a finite horizon. To address this challenge, we employ a dual-martingale approach to derive the dual problem defined as a finite-horizon pure optimal switching problem. By applying a theory of the double obstacle problem with non-standard arguments, we examine the analytical properties of the system of parabolic variational inequalities arising from the optimal switching problem, including those of its two free boundaries. Based on these analytical properties, we establish a duality theorem and characterize the optimal job-switching strategy in terms of time-varying wealth boundaries. Furthermore, we derive integral equation representations satisfied by the optimal strategies and provide numerical results based on these representations.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using dual-martingale methods and parabolic variational inequalities with double obstacle problems to model optimal job-switching and utility maximization. While mathematically advanced and novel in its formulation, it lacks empirical validation using real-world market or labor data, relying primarily on theoretical characterizations and numerical representations.

Research Flowchart

  flowchart TD
    A[Goal: Finite-Horizon Utility Maximization<br>with Optimal Job-Switching & Retirement] --> B[Inputs: Asset Dynamics, Utility Functions,<br>Wage Profiles & Fixed Retirement Date]
    B --> C[Formulate Mixed Stochastic Control<br>& Optimal Switching Problem]
    C --> D[Apply Dual-Martingale Approach<br>to derive Dual Optimal Switching Problem]
    D --> E[Analyze Parabolic Variational Inequalities<br>via Double Obstacle Problem Theory]
    E --> F[Establish Duality Theorem & Characterize<br>Free Boundaries for Job-Switching]
    F --> G[Outcomes: Integral Representations<br>& Optimal Wealth-Boundary Strategies]