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Habit Formation, Labor Supply, and the Dynamics of Retirement and Annuitization

Crisent Birungi, Cody Hyndman

Abstract

The decision to annuitize wealth in retirement planning has become increasingly complex due to rising longevity risk and changing retirement patterns, including increased labor force participation at older ages. While an extensive literature studies consumption, labor, and annuitization decisions, these elements are typically examined in isolation. This paper develops a unified stochastic control and optimal stopping framework in which habit formation and endogenous labor supply shape retirement and annuitization decisions under age-dependent mortality. We derive optimal consumption, labor, portfolio, and annuitization policies in a continuous-time lifecycle model. The solution is characterized via dynamic programming and a Hamilton-Jacobi-Bellman variational inequality. Our results reveal a rich sequence of retirement dynamics. When wealth is low relative to habit, labor is supplied defensively to protect consumption standards. As wealth increases, agents enter a work-to-retire phase in which labor is supplied at its maximum level to accelerate access to retirement. Human capital acts as a stabilizing asset, justifying a more aggressive pre-retirement investment portfolio, followed by abrupt de-risking upon annuitization. Subjective mortality beliefs are a key determinant in shaping retirement dynamics. Agents with pessimistic longevity beliefs rationally perceive annuities as unattractive, leading them to avoid or delay annuitization. This framework provides a behavior-based explanation for low annuity demand and offers guidance for retirement planning jointly linking labor supply, portfolio choice, and the timing of annuitization.

Habit Formation, Labor Supply, and the Dynamics of Retirement and Annuitization

Abstract

The decision to annuitize wealth in retirement planning has become increasingly complex due to rising longevity risk and changing retirement patterns, including increased labor force participation at older ages. While an extensive literature studies consumption, labor, and annuitization decisions, these elements are typically examined in isolation. This paper develops a unified stochastic control and optimal stopping framework in which habit formation and endogenous labor supply shape retirement and annuitization decisions under age-dependent mortality. We derive optimal consumption, labor, portfolio, and annuitization policies in a continuous-time lifecycle model. The solution is characterized via dynamic programming and a Hamilton-Jacobi-Bellman variational inequality. Our results reveal a rich sequence of retirement dynamics. When wealth is low relative to habit, labor is supplied defensively to protect consumption standards. As wealth increases, agents enter a work-to-retire phase in which labor is supplied at its maximum level to accelerate access to retirement. Human capital acts as a stabilizing asset, justifying a more aggressive pre-retirement investment portfolio, followed by abrupt de-risking upon annuitization. Subjective mortality beliefs are a key determinant in shaping retirement dynamics. Agents with pessimistic longevity beliefs rationally perceive annuities as unattractive, leading them to avoid or delay annuitization. This framework provides a behavior-based explanation for low annuity demand and offers guidance for retirement planning jointly linking labor supply, portfolio choice, and the timing of annuitization.
Paper Structure (37 sections, 9 theorems, 67 equations, 9 figures, 1 table)

This paper contains 37 sections, 9 theorems, 67 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Economic background
  3. The Financial Market
  4. Problem Formulation
  5. State Processes: Wealth and Habit
  6. Deterministic force of mortality
  7. Admissible Strategies and the Optimization Problem
  8. Leisure process
  9. Dynamic Programming and the HJB Variational Inequality
  10. Dynamic Programming Principle
  11. The Hamilton-Jacobi-Bellman Variational Inequality
  12. Dimensionality Reduction
  13. Numerical Results, Implementation, and Discussion
  14. Policy Characterization
  15. Optimal consumption to habit ratio ($\kappa_{t}^*$)
  16. ...and 22 more sections

Key Result

Proposition 3.1

The value function $V(t, x, z)$ satisfies the following dynamic programming equation for any small time interval $\Delta t > 0$

Figures (9)

  • Figure 1: Optimal Policy: Case I (Interior labor solution).
  • Figure 2: Optimal Policy: Case II (Corner labor solution).
  • Figure 3: Optimal consumption-to-habit surface as a function of age and the wealth-to-habit ratio.
  • Figure 4: Optimal labor supply surface as a function of age and the wealth-to-habit ratio.
  • Figure 5: Optimal portfolio weight surface as a function of age and the wealth-to-habit ratio.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Proposition 3.1: Dynamic Programming Principle
  • proof
  • Lemma 3.2: SDE for the Wealth-to-Habit Ratio
  • proof
  • Remark 3.1: Annuitization Value Function
  • Definition 3.1
  • Definition 3.2
  • Remark 3.2: Stationary condition of the value function
  • Remark 3.3: Structure of the optimal strategy
  • ...and 17 more