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Wage Rigidity and Retirement in Optimal Portfolio Choice

Sara Biagini, Enrico Biffis, Fausto Gozzi, Margherita Zanella

Abstract

We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing constraints and a finite retirement date. Similarly to arXiv:2002.00201, wages evolve in a path-dependent way, but the presence of a finite retirement time leads to a novel, two-stage infinite dimensional stochastic optimal control problem with explicit optimal controls in feedback form. This is possible as we find an explicit solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of parabolic type. The identification of the optimal feedbacks is delicate due to the presence of time-dependent state constraints, which appear to be new in the infinite dimensional stochastic control literature. The explicit solution allows us to study the properties of optimal strategies and discuss their implications for portfolio choice. As opposed to models with Markovian dynamics, path dependency can now modulate the hedging demand arising from the implicit holding of risky assets in human capital, leading to richer asset allocation predictions consistent with wage rigidity and the agents learning about their earning potential.

Wage Rigidity and Retirement in Optimal Portfolio Choice

Abstract

We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing constraints and a finite retirement date. Similarly to arXiv:2002.00201, wages evolve in a path-dependent way, but the presence of a finite retirement time leads to a novel, two-stage infinite dimensional stochastic optimal control problem with explicit optimal controls in feedback form. This is possible as we find an explicit solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of parabolic type. The identification of the optimal feedbacks is delicate due to the presence of time-dependent state constraints, which appear to be new in the infinite dimensional stochastic control literature. The explicit solution allows us to study the properties of optimal strategies and discuss their implications for portfolio choice. As opposed to models with Markovian dynamics, path dependency can now modulate the hedging demand arising from the implicit holding of risky assets in human capital, leading to richer asset allocation predictions consistent with wage rigidity and the agents learning about their earning potential.

Paper Structure

This paper contains 16 sections, 15 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Assumptions and problem statement
  3. Rewriting the state equation and the constraint
  4. Reformulation of the SDDE for labor income.
  5. A formula for human capital
  6. Solving the optimization problem: preliminary work
  7. Statement of Problem \ref{['p0']} rewritten
  8. Solving Problem \ref{['pb1']}: solution strategy
  9. The evolution of admissible paths
  10. Solving the infinite horizon Problem \ref{['pb2']}.
  11. Solving Problem \ref{['pb3']}: a finite horizon problem
  12. The HJB equation and its explicit solution
  13. The fundamental identity
  14. Verification Theorem and optimal feedbacks
  15. Stating and interpreting of the main result
  16. ...and 1 more sections

Key Result

Proposition 3.1

For every $t\ge 0$ and $x=(x_0,x_1)\in M_2$ the infinite dimensional equation INFINITE_DIMENSIONAL_STATE_EQUATION has a unique mild (and weak) solution with almost surely continuous trajectories, denoted by $X^x(\cdot;t)=(X_0^x(\cdot;t),X_1^x(\cdot;t))$. Moreover the equation for $y$ in (DYNAMICS_WE

Theorems & Definitions (42)

  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • lemma 3.3
  • Proposition 3.4
  • Corollary 3.5
  • Remark 3.6
  • Remark 3.7
  • Definition 4.1
  • Remark 4.2
  • ...and 32 more