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Finite-Horizon Optimal Consumption and Investment with Time-Varying Job-Switching Costs

Gugyum Ha, Junkee Jeon, Jihoon Ok

Abstract

In this paper, we study the finite-horizon problem of an economic agent's optimal consumption, investment, and job-switching decisions. The key new feature of our model is that the job-switching cost is time-varying. This extension leads to a novel mathematical characterization: the agent's dual problem reduces to a parabolic double obstacle problem with time-dependent upper and lower obstacles. By employing rigorous PDE theory, we establish not only the existence and uniqueness of the solution to this double obstacle problem, but also the smoothness of the two free boundaries that emerge from it. Building on these results, we characterize the agent's optimal consumption, portfolio, and job-switching strategies.

Finite-Horizon Optimal Consumption and Investment with Time-Varying Job-Switching Costs

Abstract

In this paper, we study the finite-horizon problem of an economic agent's optimal consumption, investment, and job-switching decisions. The key new feature of our model is that the job-switching cost is time-varying. This extension leads to a novel mathematical characterization: the agent's dual problem reduces to a parabolic double obstacle problem with time-dependent upper and lower obstacles. By employing rigorous PDE theory, we establish not only the existence and uniqueness of the solution to this double obstacle problem, but also the smoothness of the two free boundaries that emerge from it. Building on these results, we characterize the agent's optimal consumption, portfolio, and job-switching strategies.
Paper Structure (10 sections, 14 theorems, 131 equations)

This paper contains 10 sections, 14 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Derivation of Finite-Horizon Optimal Switching Problem
  4. A Parabolic Double Obstacle Problem with Time-varying Obstacles
  5. Existence and Uniqueness
  6. Free boundary analysis
  7. Existence and properties of the free boundaries
  8. Estimate involving time and spatial derivative
  9. Regularity of the free boundary
  10. Optimal Strategies

Key Result

Theorem 3.1

(ZJ2025) If $\mathcal{Q} \in W^{1,2}_{p,\mathrm{loc}}(\Omega_T) \cap C(\overline{\Omega_T})$ is the unique strong solution of the obstacle problem obs:prev for any $p \ge 1$, then there exists a unique strong solution $(P_0, P_1)$ with $P_0, P_1 \in W^{1,2}_{p,\mathrm{loc}}(\Omega_T) \cap C(\overlin

Theorems & Definitions (25)

  • Theorem 3.1
  • Remark 4.1
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • Lemma 4.4
  • Proof 4
  • ...and 15 more