Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal retirement in presence of stochastic labor income: a free boundary approach in an incomplete market

Daniele Marazzina

TL;DR

The paper investigates optimal retirement under stochastic, non-hedgeable labor income in an incomplete market using a free boundary approach. It develops a duality framework with a state-price density and convex conjugates to transform the problem into a dual shadow-price control, culminating in a PDE that characterizes the retirement boundary. Post-retirement decisions reduce to a Merton-type problem with an analytically described indirect utility, while the overall solution requires solving the dual PDE numerically. The framework clarifies how wage risk and market incompleteness reshape the retirement decision and associated consumption, investment, and leisure strategies, and it generalizes the complete-market case as a special instance.

Abstract

In this work, we address the optimal retirement problem in the presence of a stochastic wage, formulated as a free boundary problem. Specifically, we explore an incomplete market setting where the wage cannot be perfectly hedged through investments in the risk-free and risky assets that characterize the financial market.

Optimal retirement in presence of stochastic labor income: a free boundary approach in an incomplete market

TL;DR

The paper investigates optimal retirement under stochastic, non-hedgeable labor income in an incomplete market using a free boundary approach. It develops a duality framework with a state-price density and convex conjugates to transform the problem into a dual shadow-price control, culminating in a PDE that characterizes the retirement boundary. Post-retirement decisions reduce to a Merton-type problem with an analytically described indirect utility, while the overall solution requires solving the dual PDE numerically. The framework clarifies how wage risk and market incompleteness reshape the retirement decision and associated consumption, investment, and leisure strategies, and it generalizes the complete-market case as a special instance.

Abstract

In this work, we address the optimal retirement problem in the presence of a stochastic wage, formulated as a free boundary problem. Specifically, we explore an incomplete market setting where the wage cannot be perfectly hedged through investments in the risk-free and risky assets that characterize the financial market.
Paper Structure (6 sections, 3 theorems, 59 equations)

This paper contains 6 sections, 3 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. The Optimal Consumption-Investment Problem
  3. A dual approach
  4. The Bellman equation and the free boundary problem
  5. Correlated Wage
  6. Conclusions

Key Result

Proposition 3.1

Let If $z\geq\widetilde{z}$, then where $\widehat{c}=\frac{\alpha}{1-\alpha}Y\widehat{l}\ \ \textrm{and} \ \widehat{l}=z^{-\frac{1}{\gamma}} \left(\frac{\alpha}{1-\alpha}Y\right)^{\frac{\alpha(1-\gamma)-1}{\gamma}}.$ If $z<\widetilde{z}$, then where $\widehat{c}=\left(zL^{(\alpha-1)(1-\gamma)}\right)^{\frac{1}{\alpha(1-\gamma)-1}}.$

Theorems & Definitions (6)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof