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Optimal consumption under adjustment costs with respect to multiple reference levels

Yijie Huang, Kaixin Yan, Qinyi Zhang

TL;DR

The paper advances the continuous-time consumption-portfolio framework by incorporating adjustment costs tied to two reference levels—the past running maximum and minimum of consumption—and derives a complete, closed-form solution via a dual transform. The authors obtain a region-wise, five-part piecewise feedback policy for the optimal consumption and investment controls, and prove a verification theorem to validate optimality. Key insights include monotone, convex/concave behavior of the boundary wealth curves, asymptotic convergence to Merton’s problem as adjustment costs vanish, and explicit impact of adjustment costs on risk-taking and long-run consumption patterns. Numerically, the model yields smoother consumption paths than standard models and reveals how proactive versus reactive adjustments to reference levels shape optimal decisions under realistic adjustment frictions.

Abstract

This paper studies a type of consumption preference where some adjustment costs are incured whenever the past spending maximum and the past spending minimum records are updated. This preference can capture the adverse effects of the historical consumption high and low values on the agent's consumption performance, thereby matching with some empirically observed smooth consumption patterns. By employing the dual transform, the smooth-fit conditions and the super-contact conditions, we obtain the closed-form solution of the dual PDE problem, and can characterize the optimal investment and consumption controls in the piecewise feedback form. We provide the rigorous proof of the verification theorem and compensate the theoretical findings with some numerical examples and financial implications.

Optimal consumption under adjustment costs with respect to multiple reference levels

TL;DR

The paper advances the continuous-time consumption-portfolio framework by incorporating adjustment costs tied to two reference levels—the past running maximum and minimum of consumption—and derives a complete, closed-form solution via a dual transform. The authors obtain a region-wise, five-part piecewise feedback policy for the optimal consumption and investment controls, and prove a verification theorem to validate optimality. Key insights include monotone, convex/concave behavior of the boundary wealth curves, asymptotic convergence to Merton’s problem as adjustment costs vanish, and explicit impact of adjustment costs on risk-taking and long-run consumption patterns. Numerically, the model yields smoother consumption paths than standard models and reveals how proactive versus reactive adjustments to reference levels shape optimal decisions under realistic adjustment frictions.

Abstract

This paper studies a type of consumption preference where some adjustment costs are incured whenever the past spending maximum and the past spending minimum records are updated. This preference can capture the adverse effects of the historical consumption high and low values on the agent's consumption performance, thereby matching with some empirically observed smooth consumption patterns. By employing the dual transform, the smooth-fit conditions and the super-contact conditions, we obtain the closed-form solution of the dual PDE problem, and can characterize the optimal investment and consumption controls in the piecewise feedback form. We provide the rigorous proof of the verification theorem and compensate the theoretical findings with some numerical examples and financial implications.

Paper Structure

This paper contains 13 sections, 16 theorems, 130 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Market Model and Problem Formulation
  3. Heuristic Derivation of the Solution
  4. Quantitative Properties of the Optimal Control
  5. Numerical Examples
  6. Proofs
  7. Proof of Proposition \ref{['prop: solu_dual_value']}
  8. Proof of Theorem \ref{['thm: verification']} and Corollary \ref{['cor: main_res']}
  9. Proof of Lemma \ref{['lemma: v_convex']}
  10. Proof of Lemma \ref{['lemma: control_Lipschitz']}
  11. Proof of Theorem \ref{['thm: sensitivity']}
  12. Proof of Proposition \ref{['prop:asymptotic']}
  13. Proof of Proposition \ref{['prop:portfolio']}

Key Result

Proposition 3.1

Under the boundary conditions eq: dual_boundary_adjust1-eq: dual_boundary_adjust2, homogeneous property eq: dual_homo, and smooth-fit conditions at points $y = h_1^{-\gamma}$ and $y = h_2^{-\gamma}$, the dual value function $v(y,h_1,h_2)$ is given by where $\bar{h}_1(y):= (y/z_\alpha)^{-1/\gamma}$, $\underline{h}_2(y):= (y/z_\beta)^{-1/\gamma}$, $m_1\in (1,+\infty)$, $m_2\in(-\infty, \min(\gamma^

Figures (5)

  • Figure 1: Boundary curves
  • Figure 2: Sensitivity of $\alpha$
  • Figure 3: Sensitivity of $\beta$
  • Figure 4: Sensitivity of $\mu$
  • Figure 5: Expectation and Variance of Consumption Process

Theorems & Definitions (22)

  • Proposition 3.1
  • Theorem 3.1: Verification Theorem
  • Lemma 3.1
  • Corollary 3.1
  • Proposition 4.1
  • Theorem 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Corollary 4.1
  • proof
  • ...and 12 more