Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon

Abstract

We study an optimal consumption, leisure, and investment problem over a finite horizon in a continuous-time financial market with partial borrowing constraints. The agent derives utility from consumption and leisure, with preferences represented by a Cobb–Douglas utility function. The agent allocates time between work and leisure, earning wage income based on working hours. A key feature of our model is a partial borrowing constraint that limits the agent’s debt capacity to a fraction of the present value of their maximum future labor income. We employ the dual-martingale approach to derive the optimal consumption, leisure, and investment strategies. The problem reduces to solving a variational inequality with a free boundary, which we analyze using analytical and numerical methods. We provide an integral equation representation of the free boundary and solve it numerically via a recursive integration method. Our results highlight the impact of the borrowing constraint on the agent’s optimal decisions and the interplay between labor supply, consumption, and portfolio choice.

1. Introduction

The problem of determining optimal consumption and investment strategies has been a cornerstone in financial economics ever since the influential work of Merton [1,2], who introduced a continuous-time approach to portfolio selection. Building upon this foundation, numerous studies have extended the classical framework to incorporate various market frictions and real-world constraints. Among these, the integration of labor income, leisure choices, and borrowing restrictions has emerged as a critical topic, particularly when individuals make financial decisions over a finite time horizon.

Labor supply decisions are inherently linked to consumption and saving behaviors. Individuals must decide how to allocate their time between earning labor income and enjoying leisure, with wages playing a pivotal role in wealth accumulation. This labor–leisure choice introduces complexity beyond the traditional consumption-investment problem, as both financial and temporal resources must be jointly managed. Borrowing constraints further complicate this decision-making process by limiting individuals’ ability to smooth consumption over time through debt.

In practice, borrowing restrictions are often partial, meaning that individuals can only borrow up to a fraction of their expected future labor income. Such constraints arise from credit market imperfections and institutional lending limits. They significantly influence consumption, labor supply, and portfolio allocation decisions throughout an individual’s working life.

Existing research has provided valuable insights into optimal consumption and investment under borrowing constraints. El Karoui and Jeanblanc-Picqué [3] explored optimal consumption and portfolio strategies in the presence of labor income and liquidity constraints, utilizing techniques from American option valuation. Jeon and Shin [4] addressed a finite-horizon consumption and investment problem with a negative wealth constraint, deriving an integral equation representation of the free boundary using Mellin transform techniques. Lee et al. [5] examined the interaction between borrowing constraints, labor supply flexibility, and portfolio selection in an infinite-horizon setting with Cobb–Douglas preferences. More recently, Jeon et al. [6] investigated the effects of reversible retirement and borrowing constraints on human capital and portfolio choice, emphasizing the interplay between labor supply flexibility and borrowing limits.

Despite these advancements, no prior work has comprehensively analyzed the joint optimization of consumption, leisure, and portfolio selection under partial borrowing constraints within a finite time horizon. This paper aims to fill this gap by extending the classical Merton model to incorporate labor–leisure choice and a borrowing limit linked to the present value of future labor income. The agent derives utility from both consumption and leisure, modeled through a Cobb–Douglas utility function that captures their relative importance.

Our approach employs the dual-martingale method to characterize the optimal consumption, labor supply, and investment policies. The problem is formulated as a stochastic control problem subject to a borrowing limit, leading to a variational inequality with a free boundary condition. A key result is the derivation of an integral equation representing the free boundary, which can be efficiently and accurately solved using the Recursive Integration Method (RIM) developed by Huang et al. [7]. This numerical solution facilitates the computation of optimal consumption, labor supply, and investment strategies.

The findings of this study contribute to the literature on optimal consumption and labor supply by elucidating how borrowing constraints influence individuals’ financial and time allocation decisions. The results underscore the importance of accounting for labor supply flexibility and borrowing restrictions when devising optimal wealth management strategies.

The remainder of this paper is structured as follows. Section 2 describes the model and the agent’s optimization problem. Section 3 presents the dual-martingale approach and the formulation of the variational inequality with a free boundary and derives the integral equation for the free boundary. Section 4 reports numerical results and discusses the economic implications of borrowing constraints. Section 5 provides policy implications. Finally, Section 6 concludes with a summary and future research directions.

2. Model

We consider a continuous-time financial market that is both complete and frictionless, and we analyze the problem over a finite time horizon $[0,T]$ for a given $T>0$.

The market features two tradable assets: a risk-free bond and a risky stock. Their price dynamics are described by the following stochastic differential equations:

$$\left\{\begin{array}{c}d{S}_{0,t}=r{S}_{0,t}dt,\hfill \\ d{S}_{1,t}=\mu S_{1,t}dt+\sigma S_{1,t}d{B}_t,\hfill \end{array}\right.$$

(1)

where $S_{0,t}$ and $S_{1,t}$ represent the prices of the bond and the stock at time $t\ge 0$, respectively. The constant $r>0$ denotes the risk-free interest rate, $\mu \ne r$ is the stock’s drift (expected return), and $\sigma >0$ is its volatility. The process ${\left\{B_t\right\}}_{t\ge 0}$ is a standard Brownian motion defined on a filtered probability space $(\Omega ,\mathbb{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$. The filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ is the natural filtration generated by $B_t$, augmented to satisfy the usual conditions; that is, ${\mathcal{F}}_t=\sigma (B_s:0\le s\le t)\vee \mathcal{N}$, where $\mathcal{N}$ is the collection of $\mathbb{P}$-null sets.

We assume that the agent’s risk preference is represented by a time-separable Cobb–Douglas utility function that depends on the consumption rate $c_t$ and the leisure rate $l_t$. For constants $0<\alpha <1$ and $0<\gamma \ne 1$, the utility function takes the form

$$u(c_t,l_t)={\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{c_t^{\alpha }{l}_t^{1-\alpha }}{1-\gamma }}\right)}^{1-\gamma }={\displaystyle \frac{c_t^{1-{\gamma }_1}{l}_t^{{\gamma }_1-\gamma }}{1-{\gamma }_1}},\mathrm{where}{\gamma }_1:=1-\alpha (1-\gamma ).$$

(2)

Here, $\alpha$ reflects the relative weight on consumption, and $\gamma$ represents the coefficient of relative risk aversion.

Alternative Utility Specifications: CES Utility Function

The Cobb–Douglas utility function in Equation (2) assumes a unit elasticity of substitution between consumption and leisure. This assumption simplifies analysis but may not fully capture real-world behaviors, where the elasticity of substitution may vary.

A more general alternative is the Constant Elasticity of Substitution (CES) utility function, given by

$$u(c,l)={\left[\alpha c^{\rho }+(1-\alpha )l^{\rho }\right]}^{{\displaystyle \frac{1}{\rho }}},$$

(3)

where $\rho$ governs the elasticity of substitution:

$$\zeta ={\displaystyle \frac{1}{1-\rho }}.$$

(4)

-

When $\rho =0$, the CES function reduces to Cobb–Douglas utility.

-

When $\rho >0$, consumption and leisure are more substitutable.

-

When $\rho <0$, consumption and leisure are less substitutable.

Implications for Our Model

If CES preferences were used, the following would be true:

1.

The agent’s optimal choices of consumption and leisure would adjust differently based on $\zeta$.

2.

Borrowing constraints would interact with labor supply more dynamically.

3.

The variational inequality solution would require a modified boundary condition.

For future research, extending the model to allow for CES preferences could provide deeper insights into constrained intertemporal decisions.

The agent earns income from labor and can freely adjust their working and leisure time within certain limits. Let the total available time in each period be $\overline{L}$, which can be allocated between leisure and work. If the agent enjoys $l_t$ hours of leisure at time t, the remaining time devoted to work is $\overline{L}-l_t$. The corresponding wage income is given by

$$\mathcal{I}\left(l_t\right)=\omega (\overline{L}-l_t),$$

where $\omega >0$ is a fixed wage rate per unit of working time. We assume that the leisure choice is subject to the constraint

$$0\le l_t\le L\le \overline{L},\forall t\ge 0.$$

This condition implies that leisure hours $l_t$ cannot exceed a maximum level L, while the agent is required to work at least $\overline{L}-L$ hours. The base-level labor income from this minimum work requirement is $\omega (\overline{L}-L)$, and any reduction in leisure below L represents extra working hours, generating additional income.

Given an initial wealth endowment $x>0$, let ${\pi }_t$ represent the amount of wealth invested in the risky asset at time t. The agent’s wealth process ${\left\{X_t^{c,\pi ,l}\right\}}_{t\ge 0}$, under a chosen consumption–leisure–investment strategy $(c_t,l_t,{\pi }_t)$, evolves according to the stochastic differential equation:

$$d{X}_t^{c,\pi ,l}=\left[r{X}_t^{c,\pi ,l}+(\mu -r){\pi }_t-c_t+\omega (\overline{L}-l_t)\right]dt+\sigma {\pi }_{t}d{B}_t,$$

(5)

with initial condition $X_0^{c,\pi ,l}=x$.

In this setting, we impose a partial borrowing constraint that allows the agent to borrow against a fraction $\eta \in [0,1)$ of the present value of their maximum potential future labor income. This restriction ensures that the agent can only use a limited portion of their anticipated wage earnings as collateral for immediate consumption or investment. Mathematically, this constraint is expressed as follows:

$$X_t^{c,\pi ,l}\ge -\eta {\mathbb{E}}_t\left[{\int }_t^T{e}^{-r(s-t)}\omega \overline{L}ds\right]=-\eta \omega \overline{L}{\displaystyle \frac{1-e^{-r(T-t)}}{r}},\forall t\in [0,T].$$

(6)

This requirement implies that the agent’s wealth can fall below zero but only up to a fraction $\eta$ of the expected present value of their remaining maximum labor income.

In other words, their total financial resources—comprising investment returns, labor income, and savings, along with the limited borrowing capacity against future labor income—must be sufficient to cover consumption expenses at all times. The partial borrowing constraint plays a crucial role in shaping the agent’s optimal consumption, labor supply, and investment decisions, as it restricts excessive borrowing while still allowing some degree of leverage based on future earnings potential. This constraint balances the agent’s desire to smooth consumption with the need to maintain financial feasibility over time.

We now formulate the agent’s utility maximization problem as follows:

Problem 1.

Given an initial wealth level $x>0$, the agent seeks to maximize their expected lifetime utility:

$$V\left(x\right):=\underset{(c,l,\pi )\in \mathcal{A}\left(x\right)}{sup}\mathbb{E}\left[{\int }_0^T{e}^{-\beta t}u(c_t,l_t)dt\right],$$

(7)

where $\beta >0$ is the subjective discount rate, and $\mathcal{A}\left(x\right)$ denotes the set of all admissible consumption, leisure, and investment strategies satisfying the following conditions:

(i) 

$c_t\ge 0$, $0\le l_t\le L\le \overline{L}$, and the processes $c_t,l_t,{\pi }_t$ are progressively measurable with respect to the filtration $\mathcal{F}$, such that

$${\int }_0^t{c}_{s}ds<\infty a.s.,{\int }_0^t{\pi }_s^{2}ds<\infty a.s.,{\int }_0^t{l}_{s}ds<\infty a.s.,\forall t\in [0,T].$$

(ii) 

The agent’s wealth process $X_t^{c,\pi ,l}$ associated with $(c,l,\pi )$, evolving according to (5), adheres to the partial borrowing constraint (6).

3. Optimization Problem

We use the dual-martingale method developed by El Karoui and Jeanblanc-Picqué [3] to address Problem 1. To proceed, we first introduce the stochastic discount factor (SDF) defined as

$${\xi }_t:=e^{-(r+{\displaystyle \frac{1}{2}}{\theta }^2)t-\theta B_t},\mathrm{with}\theta :={\displaystyle \frac{\mu -r}{\sigma }},$$

(8)

where $\theta$ is referred to as the market price of risk.

By El Karoui and Jeanblanc-Picqué [3], we deduce that for any $(c,\pi ,l)\in \mathcal{A}\left(x\right),$

$$\underset{D\in \mathcal{NI}(0,T)}{sup}\mathbb{E}\left[{\int }_0^T{D}_t{\xi }_t\left(c_t-\omega (\overline{L}(1-\eta )-l_t)\right)dt\right]\le x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}},$$

(9)

where $\mathcal{NI}(s_1,s_2)$ is the collection of all $\mathcal{F}$-adapted, non-negative, non-increasing, right-continuous processes ${\left\{h_t\right\}}_{t=s_1}^{s_2}$ with left limits (RCLL), such that $h_{s_1-}=1$.

By utilizing the static budget constraint (9) for Problem 1, we define the following Lagrangian $\mathfrak{L}$ for Problem 1:

$$\begin{array}{cc}\hfill \mathfrak{L}:=& \mathbb{E}\left[{\int }_0^T{e}^{-\beta t}u(c_t,l_t)dt\right]\hfill \\ & +\lambda \left(x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}-\underset{D\in \mathcal{NI}(0,T)}{sup}\mathbb{E}\left[{\int }_0^T{D}_t{\xi }_t\left(c_t-\omega (\overline{L}(1-\eta )-l_t)\right)dt\right]\right)\hfill \\ \hfill =& \underset{D\in \mathcal{NI}(0,T)}{inf}\left\{\mathbb{E}\left[{\int }_0^T{e}^{-\beta t}\left(u(c_t,l_t)-{\Lambda }_t^{\lambda }{D}_t(c_t+\omega l_t)+\omega \overline{L}(1-\eta ){\Lambda }_t^{\lambda }{D}_t\right)dt\right]\right\}\hfill \\ & +\lambda \left(x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}\right),\hfill \end{array}$$

(10)

where ${\Lambda }_t^{\lambda }:=\lambda e^{\beta t}{\xi }_t.$

Let us consider the dual-conjugate utility $\tilde{u}\left(\lambda \right)$ for $u(c_t,l_t)$ given by

$$\begin{array}{cc}\hfill \tilde{u}\left(\lambda \right):=& \underset{c>0,;0\le l_t\le L}{sup}\left(u(c_t,l_t)-\lambda (c_t+\omega l_t)+\omega \overline{L}(1-\eta )\lambda \right)\hfill \\ \hfill =& \left[{\displaystyle \frac{\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\lambda }^{-{\displaystyle \frac{1-\gamma }{\gamma }}}+\overline{L}(1-\eta )\lambda \omega \right]{\mathbf{1}}_{\{\tilde{\lambda }<\lambda \}}\hfill \\ \hfill +& \left[{\displaystyle \frac{{\gamma }_1}{1-{\gamma }_1}}{L}^{{\displaystyle \frac{{\gamma }_1-\gamma }{{\gamma }_1}}}{\lambda }^{-{\displaystyle \frac{1-{\gamma }_1}{{\gamma }_1}}}+(\overline{L}(1-\eta )-L)\lambda \omega \right]{\mathbf{1}}_{\{0<\lambda \le \tilde{\lambda }\}},\hfill \end{array}$$

(11)

where

$$\tilde{\lambda }:={\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\gamma }_1}{L}^{-\gamma }$$

(12)

and we have used the first-order and second-order conditions in (11).

Moreover, the second equality holds if and only if $c_t=\widehat{c}\left({\Lambda }_t^{\lambda }{D}_t\right)$ and $l_t=\widehat{l}\left({\Lambda }_t^{\lambda }{D}_t\right),$ where

$$\begin{array}{cc}\hfill \widehat{c}\left(\lambda \right):=& \left\{\begin{array}{cc}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\lambda }^{-{\displaystyle \frac{1}{\gamma }}}\hfill & \mathrm{if}\lambda >\tilde{\lambda },\hfill \\ L^{{\displaystyle \frac{{\gamma }_1-\gamma }{{\gamma }_1}}}{\lambda }^{-{\displaystyle \frac{1}{{\gamma }_1}}}\hfill & \mathrm{if}0<\lambda \le \tilde{\lambda },\hfill \end{array}\right.\hfill \\ \hfill \widehat{l}\left(\lambda \right):=& \left\{\begin{array}{cc}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1}{\gamma }}}{\lambda }^{-{\displaystyle \frac{1}{\gamma }}}\hfill & \mathrm{if}\lambda >\tilde{\lambda },\hfill \\ L\hfill & \mathrm{if}0<\lambda \le \tilde{\lambda }.\hfill \end{array}\right.\hfill \end{array}$$

(13)

Hence, it follows from (10) and (11) that for any admissible $(c,\pi ,l)\in \mathcal{A}\left(x\right)$

$$\begin{array}{c}\hfill \mathbb{E}\left[{\int }_0^T{e}^{-\beta t}u(c_t,l_t)dt\right]\le \mathfrak{L}\le \underset{D\in \mathcal{NI}(0,T)}{inf}\mathbb{E}\left[{\int }_0^T{e}^{-\beta t}\tilde{u}\left({\Lambda }_t^{\lambda }{D}_t\right)dt\right]+\lambda \left(x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}\right).\end{array}$$

(14)

Thus, we can define the following dual problem:

Problem 2.

For given $\lambda >0,$ we consider the following singular control problem:

$$J\left(\lambda \right)=\underset{D\in \mathcal{NI}(0,T)}{inf}\mathbb{E}\left[{\int }_0^T{e}^{-\beta t}\tilde{u}\left({\Lambda }_t^{\lambda }{D}_t\right)dt\right].$$

(15)

Let us consider the dynamic form $\mathcal{J}(t,\lambda )$ for Problem 2 as follows:

$$\mathcal{J}(t,{\Lambda }_t):=\underset{D\in \mathcal{NI}(t,T)}{inf}\mathbb{E}\left[{\int }_t^T{e}^{-\beta (s-t)}\tilde{u}\left({\Lambda }_s^{\lambda }{D}_s\right)ds\mid {\mathcal{F}}_t\right].$$

(16)

By utilizing the dynamic programming principle, $\mathcal{J}(t,\lambda )$ satisfies the following Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint: on the domain ${\Omega }_T:=\{(t,\lambda )\mid 0\le t<T,0<\lambda <\infty \}$

$$min\left\{{\partial }_t\mathcal{J}(t,\lambda )+\mathcal{L}J(t,\lambda )+\tilde{u}\left(\lambda \right),-{\partial }_{\lambda }\mathcal{J}(t,\lambda )\right\}=0$$

(17)

with $\mathcal{J}(T,\lambda )=0,$ where the differential operator $\mathcal{L}$ is given by

$$\mathcal{L}:={\displaystyle \frac{{\theta }^2}{2}}{\lambda }^2{\partial }_{\lambda \lambda }+(\beta -r)\lambda {\partial }_{\lambda }-\beta .$$

In terms of the HJB Equation (17), we define the two regions, the non-jump region $\mathbf{NR}$, and the jump region $\mathbf{JR}$ as follows:

$$\mathbf{NR}:=\{(t,\lambda )\in {\Omega }_T\mid {\partial }_{\lambda }\mathcal{J}(t,\lambda )<0\}\mathrm{and}\mathbf{JR}:=\{(t,\lambda )\in {\Omega }_T\mid {\partial }_{\lambda }\mathcal{J}(t,\lambda )=0\}.$$

(18)

The function $\mathcal{J}(t,\lambda )$ that satisfies the HJB Equation (17) is of class $C^{1,2}\left({\Omega }_T\right)$ and strictly convex in $(t,\lambda )\in \mathbf{NR}$. In this paper, the main focus is not on proving the analytical properties of the solution to the HJB equation, but rather on finding an analytical representation of the optimal strategies that satisfy the equation. Therefore, detailed proofs are omitted.

Let us denote $\mathcal{X}(t,\lambda )$ by

$$\mathcal{X}(t,\lambda )=-{\partial }_{\lambda }\mathcal{J}(t,\lambda ).$$

(19)

Then, it is easy to see that $\mathcal{X}(t,\lambda )$ satisfies the following variational inequality (VI): on the domain ${\Omega }_T$

$$max\left\{{\partial }_t\mathcal{X}(t,\lambda )+\widehat{\mathcal{L}}\mathcal{X}(t,\lambda )+\psi \left(\lambda \right),-\mathcal{X}(t,\lambda )\right\}=0$$

(20)

with $\mathcal{X}(T,\lambda )=0,$ where the operator $\widehat{L}$ and the function $\psi \left(\lambda \right)$ are given by

$$\widehat{L}:={\displaystyle \frac{{\theta }^2}{2}}{\lambda }^2{\partial }_{\lambda \lambda }+(\beta -r+{\theta }^2)\lambda {\partial }_{\lambda }-r,$$

(21)

and

$$\begin{array}{cc}\hfill \psi \left(\lambda \right):=-{\tilde{u}}^{\prime }\left(\lambda \right)=& \left[{\displaystyle \frac{1-\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\lambda }^{-{\displaystyle \frac{1}{\gamma }}}-\overline{L}(1-\eta )\omega \right]{\mathbf{1}}_{\{\tilde{\lambda }<\lambda \}}\hfill \\ \hfill +& \left[L^{{\displaystyle \frac{{\gamma }_1-\gamma }{{\gamma }_1}}}{\lambda }^{-{\displaystyle \frac{1}{{\gamma }_1}}}-(\overline{L}(1-\eta )-L)\omega \right]{\mathbf{1}}_{\{0<\lambda \le \tilde{\lambda }\}}.\hfill \end{array}$$

(22)

Since both the terminal condition and the obstacle of the variational inequality (VI) (20) are zero, and $\psi \left(\lambda \right)$ is strictly decreasing in $\lambda >0$, the comparison principle for VI implies that

$${\partial }_t\mathcal{X}(t,\lambda )\ge 0\mathrm{and}{\partial }_{\lambda }\mathcal{X}(t,\lambda )\le 0.$$

(23)

Moreover, by the standard theory for VIs (see Friedman [8]), we can easily obtain that there exists a unique solution $\mathcal{X}\in W_{\mathrm{p},\mathrm{loc}}^{1,2}\left({\Omega }_T\right)\cap C^1\left({\overline{\Omega }}_T\right).$

Due to the property ${\partial }_{\lambda }\mathcal{X}\le 0$, there exists a free boundary $z\left(t\right)$ such that

$$\mathbf{NR}=\{(t,\lambda )\in {\Omega }_T\mid 0<\lambda <z\left(t\right)\}\mathrm{and}\mathbf{JR}=\{(t,\lambda )\in {\Omega }_T\mid \lambda \ge z\left(t\right)\}.$$

(24)

(see Figure 1).

Furthermore, utilizing the method in Friedman [9], the free boundary $z(·)$ is smooth and strictly decreasing in $t\in [0,T)$, and

$$\underset{t\to T^{-}}{lim}z\left(t\right)=\overline{z},$$

(25)

where $\overline{z}>0$ is the unique solution to $\psi \left(\overline{z}\right)=0.$

Then, in the region $\mathbf{NR}$, $\mathcal{X}(t,\lambda )$ satisfies the following partial differential equation (PDE):

$${\partial }_t\mathcal{X}(t,\lambda )+\widehat{\mathcal{L}}\mathcal{X}(t,\lambda )+\psi \left(\lambda \right)=0,\mathrm{for}(t,\lambda )\in \mathbf{NR}.$$

(26)

Clearly, in the region $\mathbf{JR}$, since $\mathcal{X}(t,\lambda )=0$, we have

$${\partial }_t\mathcal{X}(t,\lambda )+\widehat{\mathcal{L}}\mathcal{X}(t,\lambda )=0,\mathrm{for}(t,\lambda )\in \mathbf{JR}.$$

(27)

Combining the above results, we obtain

$${\partial }_t\mathcal{X}(t,\lambda )+\widehat{\mathcal{L}}\mathcal{X}(t,\lambda )+\psi \left(\lambda \right){\mathbf{1}}_{\{0<\lambda <z(t\left)\right\}}=0,\mathrm{with}\mathcal{X}(T,\lambda )=0.$$

(28)

Let us define an equivalent martingale measure $\mathbb{Q}$ by

$${\displaystyle \frac{d\mathbb{Q}}{d\mathbb{P}}}=e^{-{\displaystyle \frac{1}{2}}{\theta }^{2}T-\theta B_T},$$

(29)

so that $B_t^{\mathbb{Q}}:=B_t+\theta t$ for $t\in [0,T]$ is a standard Brownian motion under the measure $\mathbb{Q}$.

It is easy to check that

$$d{\Lambda }_t^{\lambda }=(\beta -r+{\theta }^2){\Lambda }_t^{\lambda }dt-\theta {\Lambda }_t^{\lambda }d{B}_t^{\mathbb{Q}}.$$

(30)

Hence, applying Itô’s lemma to $e^{-r(s-t)}\mathcal{X}(s,{\Lambda }_s^{t,\lambda })$ yields that

$$d\left(e^{-r(s-t)}\mathcal{X}(s,{\Lambda }_s^{t,\lambda })\right)=e^{-r(s-t)}\left({\partial }_s\mathcal{X}(s,{\Lambda }_s^{t,\lambda })+\widehat{\mathcal{L}}\mathcal{X}(s,{\Lambda }_s^{t,\lambda })\right)ds-e^{-r(s-t)}\theta {\partial }_{\lambda }\mathcal{X}(s,{\Lambda }_s^{t,\lambda })d{B}_s^{\mathbb{Q}},$$

(31)

or equivalently,

$$\begin{array}{cc}\hfill \mathcal{X}(t,\lambda )=& \mathcal{X}(T,{\Lambda }_T^{t,\lambda })-{\int }_t^T{e}^{-r(s-t)}\left({\partial }_s\mathcal{X}(s,{\Lambda }_s^{t,\lambda })+\widehat{\mathcal{L}}\mathcal{X}(s,{\Lambda }_s^{t,\lambda })\right)ds\hfill \\ & +\theta {\int }_t^T{e}^{-r(s-t)}{\partial }_{\lambda }\mathcal{X}(s,{\Lambda }_s^{t,\lambda })d{B}_s^{\mathbb{Q}},\hfill \end{array}$$

(32)

where ${\Lambda }_s^{t,\lambda }:=\lambda e^{\beta (s-t)}{\xi }_s/{\xi }_t$ with ${\Lambda }_t^{t,\lambda }=\lambda .$

By taking the conditional expectation ${\mathbb{E}}^{\mathbb{Q}}[·\mid {\mathcal{F}}_t]$ in the both-side of the Equation (32), it follows from (28) that

$$\begin{array}{c}\hfill \mathcal{X}(t,\lambda )={\int }_t^T{e}^{-r(s-t)}{\mathbb{E}}^{\mathbb{Q}}\left[\psi \left({\Lambda }_s^{t,\lambda }\right){\mathbf{1}}_{\{0<{\Lambda }_s^{t,\lambda }<z\left(s\right)\}}\right]ds.\end{array}$$

(33)

Applying (A.7) and (A.8) in Jeon and Oh [10] to the dynamics of ${\Lambda }_t^{\lambda }$ in (30), we have the integral equation representation for $\mathcal{X}(t,\lambda )$ as follows:

$$\begin{array}{cc}\hfill \mathcal{X}(t,\lambda )=& L^{{\displaystyle \frac{{\gamma }_1-\gamma }{{\gamma }_1}}}{\lambda }^{-{\displaystyle \frac{1}{{\gamma }_1}}}{\int }_t^T{e}^{-K_1(s-t)}\mathcal{N}\left(-d_{1-{\displaystyle \frac{1}{{\gamma }_1}}}(s-t,{\displaystyle \frac{\lambda }{z\left(s\right)\wedge \tilde{\lambda }}})\right)ds\hfill \\ \hfill & -(\overline{L}(1-\eta )-L)\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(-d_1(s-t,{\displaystyle \frac{\lambda }{z\left(s\right)\wedge \tilde{\lambda }}})\right)ds\hfill \\ \hfill & +{\displaystyle \frac{1-\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\lambda }^{-{\displaystyle \frac{1}{\gamma }}}{\int }_t^T{e}^{-K(s-t)}\mathcal{N}\left(d_{1-{\displaystyle \frac{1}{\gamma }}}(s-t,{\displaystyle \frac{\lambda }{\tilde{\lambda }}})\right)ds\hfill \\ \hfill & -{\displaystyle \frac{1-\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\lambda }^{-{\displaystyle \frac{1}{\gamma }}}{\int }_t^T{e}^{-K(s-t)}\mathcal{N}\left(d_{1-{\displaystyle \frac{1}{\gamma }}}(s-t,{\displaystyle \frac{\lambda }{z\left(s\right)\vee \tilde{\lambda }}})\right)ds\hfill \\ \hfill & -\overline{L}(1-\eta )\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(d_1(s-t,{\displaystyle \frac{\lambda }{\tilde{\lambda }}})\right)ds\hfill \\ \hfill & +\overline{L}(1-\eta )\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(d_1(s-t,{\displaystyle \frac{\lambda }{z\left(s\right)\vee \tilde{\lambda }}})\right)ds,\hfill \end{array}$$

(34)

where $\mathcal{N}(·)$ is the standard normal distribution function,

$$d_{\nu }(t,\lambda ):={\displaystyle \frac{log\lambda +(\beta -r-\frac{1}{2}{\theta }^2+\nu {\theta }^2)t}{\theta \sqrt{t}}},$$

(35)

and

$$K:=r+{\displaystyle \frac{\beta -r}{\gamma }}+{\displaystyle \frac{\gamma -1}{{\gamma }^2}}{\displaystyle \frac{{\theta }^2}{2}}\mathrm{and}{K}_1:=r+{\displaystyle \frac{\beta -r}{{\gamma }_1}}+{\displaystyle \frac{{\gamma }_1-1}{{\gamma }_1^2}}{\displaystyle \frac{{\theta }^2}{2}}.$$

(36)

By using the continuity of $\mathcal{X}(t,\lambda )$ at $\lambda =z\left(t\right),$ we have the following integral equation for the free boundary $z\left(t\right)$:

$$\begin{array}{cc}\hfill 0=& L^{{\displaystyle \frac{{\gamma }_1-\gamma }{{\gamma }_1}}}{\left(z\left(t\right)\right)}^{-{\displaystyle \frac{1}{{\gamma }_1}}}{\int }_t^T{e}^{-K_1(s-t)}\mathcal{N}\left(-d_{1-{\displaystyle \frac{1}{{\gamma }_1}}}(s-t,{\displaystyle \frac{z\left(t\right)}{z\left(s\right)\wedge \tilde{\lambda }}})\right)ds\hfill \\ \hfill & -(\overline{L}(1-\eta )-L)\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(-d_1(s-t,{\displaystyle \frac{z\left(t\right)}{z\left(s\right)\wedge \tilde{\lambda }}})\right)ds\hfill \\ \hfill & +{\displaystyle \frac{1-\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\left(z\left(t\right)\right)}^{-{\displaystyle \frac{1}{\gamma }}}{\int }_t^T{e}^{-K(s-t)}\mathcal{N}\left(d_{1-{\displaystyle \frac{1}{\gamma }}}(s-t,{\displaystyle \frac{z\left(t\right)}{\tilde{\lambda }}})\right)ds\hfill \\ \hfill & -{\displaystyle \frac{1-\gamma }{1-{\gamma }_1}}{\left({\displaystyle \frac{{\gamma }_1-\gamma }{\omega (1-{\gamma }_1)}}\right)}^{{\displaystyle \frac{{\gamma }_1-\gamma }{\gamma }}}{\left(z\left(t\right)\right)}^{-{\displaystyle \frac{1}{\gamma }}}{\int }_t^T{e}^{-K(s-t)}\mathcal{N}\left(d_{1-{\displaystyle \frac{1}{\gamma }}}(s-t,{\displaystyle \frac{z\left(t\right)}{z\left(s\right)\vee \tilde{\lambda }}})\right)ds\hfill \\ \hfill & -\overline{L}(1-\eta )\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(d_1(s-t,{\displaystyle \frac{z\left(t\right)}{\tilde{\lambda }}})\right)ds\hfill \\ \hfill & +\overline{L}(1-\eta )\omega {\int }_t^T{e}^{-r(s-t)}\mathcal{N}\left(d_1(s-t,{\displaystyle \frac{z\left(t\right)}{z\left(s\right)\vee \tilde{\lambda }}})\right)ds,\hfill \end{array}$$

(37)

Let us denote $\Pi (t,\lambda )$ by

$$\Pi (t,\lambda ):={\displaystyle \frac{\theta }{\sigma }}\lambda {\partial }_{\lambda \lambda }\mathcal{J}(t,\lambda )=-{\displaystyle \frac{\theta }{\sigma }}\lambda {\partial }_{\lambda }\mathcal{X}(t,\lambda ).$$

(38)

Then, We establish the duality theorem and optimal strategy in the following theorem.

Theorem 1.

Let $x>0$ be given. Then, the following duality relationship holds:

$$V\left(x\right)=\underset{\lambda >0}{inf}\left[J\left(\lambda \right)+\lambda \left(x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}\right)\right]=J\left({\lambda }^{*}\right)+{\lambda }^{*}\left(x+\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}\right),$$

(39)

where ${\lambda }^{*}\in (0,z\left(0\right))$ is the unique solution to $x=-J^{\prime }\left({\lambda }^{*}\right)-\eta \omega \overline{L}{\displaystyle \frac{1-e^{-rT}}{r}}.$

Moreover, the optimal strategies $(c^{*},{\pi }^{*},l^{*})$ is characterized as

$$c_t^{*}=\widehat{c}\left({\Lambda }_t^{*}{D}_t^{*}\right),l_t^{*}=\widehat{l}\left({\Lambda }_t^{*}{D}_t^{*}\right),and{\pi }_t^{*}=\Pi (t,{\Lambda }_t^{*}{D}_t^{*}),$$

(40)

where

$${\Lambda }_t^{*}={\lambda }^{*}{e}^{\beta t}{\xi }_{t}and{D}_t^{*}=min\left\{1,\underset{0\le s\le t}{sup}{\displaystyle \frac{z\left(s\right)}{{\Lambda }_s^{*}}}\right\}$$

The wealth $X_t^{*}=X_t^{c^{*},{\pi }^{*},l^{*}}$ corresponding to the strategy $(c^{*},{\pi }^{*},l^{*})$ is given by

$$X_t^{*}=\mathcal{X}(t,{\Lambda }_t^{*}{D}_t^{*}).$$

Proof. 

See El Karoui and Jeanblanc-Picqué [3]. □

4. Numerical Results

We numerically solve the integral Equation (37), which characterizes the free boundary $z\left(t\right)$, by employing the recursive integration method (RIM) developed by Huang et al. [7]. This approach allows us to efficiently and accurately obtain the numerical values of the free boundary $z\left(t\right)$ over the interval $[0,T]$. Using these values, we can easily compute the optimal wealth $\mathcal{X}(t,\lambda )$ and the portfolio $\Pi (t,\lambda )$ by applying the simple trapezoidal rule.

To provide numerical results for the optimal strategy, we use the following parameters as the baseline:

$$\begin{array}{cc}\hfill T& =10,\beta =0.06,r=0.02,\mu =0.1,\sigma =0.3,L=1,\overline{L}=1.2,\hfill \\ \hfill \alpha & =0.5,\omega =1,\gamma =3,\eta =0.3.\hfill \end{array}$$

Figure 2 illustrates the optimal consumption, leisure, and portfolio policies as functions of wealth $X_t$, under different levels of the borrowing constraint parameter $\eta \in \{0,0.3,0.7\}$. The parameter $\eta$ determines the fraction of the present value of future labor income that can be used as collateral for borrowing. A larger $\eta$ allows the agent to borrow more, whereas a smaller $\eta$ represents a stricter borrowing constraint.

Figure 2a shows that the optimal consumption $c_t$ is increasing in wealth $X_t$ for all levels of $\eta$. However, the slope and level of the consumption curve depend on the borrowing constraint. When $\eta =0$, meaning borrowing is not allowed, consumption is considerably lower when wealth is negative, as the agent cannot borrow to smooth consumption. With a moderate borrowing capacity ($\eta =0.3$), the agent can sustain a higher level of consumption even when wealth is negative. When borrowing is more flexible ($\eta =0.7$), consumption is significantly higher in the negative wealth region, as the agent can leverage future labor income to maintain current consumption. Borrowing constraints restrict an agent’s ability to smooth consumption over time. When borrowing is severely limited ($\eta =0$), the agent is forced to reduce consumption when wealth is low. In contrast, a more relaxed borrowing constraint ($\eta =0.7$) allows the agent to maintain relatively smooth consumption by borrowing against future labor income, even when current wealth is negative.

Figure 2b depicts the relationship between wealth and leisure $l_t$. Leisure increases with wealth in all cases, reflecting the fact that wealthier individuals can afford to reduce labor supply and enjoy more leisure. However, borrowing constraints significantly influence leisure choices when wealth is low. When borrowing is not allowed ($\eta =0$), the agent allocates more time to work and less to leisure in the low-wealth region, as labor income is the only means to finance consumption. As borrowing capacity increases ($\eta =0.3$ and $\eta =0.7$), the agent can reduce working hours and enjoy more leisure even when wealth is low, as they can rely on borrowing to supplement consumption. Labor supply serves as a buffer to ensure consumption when borrowing is restricted. With a strict borrowing constraint ($\eta =0$), the agent must work more when wealth is low to finance consumption. A higher borrowing capacity ($\eta =0.7$) allows the agent to reduce labor supply and enjoy more leisure, even when wealth is low, because borrowing provides an alternative source of funds.

Figure 2c shows the optimal portfolio allocation ${\pi }_t$ with respect to wealth. The agent’s investment in the risky asset increases as wealth rises, consistent with standard portfolio choice models. However, borrowing constraints affect risk-taking behavior when wealth is low. When borrowing is not allowed ($\eta =0$), the agent invests conservatively in the low-wealth region, as losses would be difficult to recover without borrowing capacity. With moderate borrowing capacity ($\eta =0.3$), the agent invests more aggressively even when wealth is low. When borrowing is highly flexible ($\eta =0.7$), the agent is willing to hold a larger risky position even in the negative wealth region, as future labor income serves as a cushion against potential losses. Borrowing constraints amplify the agent’s aversion to financial risk when wealth is low. Without borrowing capacity ($\eta =0$), the agent must adopt a conservative investment approach to avoid being trapped in a situation where consumption cannot be financed. When borrowing is easier ($\eta =0.7$), the agent can afford to take more investment risk because future labor income can serve as a backstop in the event of portfolio losses.

Figure 3 illustrates the optimal consumption, leisure, and portfolio policies as functions of wealth $X_t$, under different time horizons $T\in \{5,10,20\}$. The planning horizon T represents the length of the agent’s decision-making period, with a longer T indicating that the agent considers consumption, labor, and investment decisions over a more extended future.

Figure 3a shows that the optimal consumption $c_t$ increases with wealth $X_t$ across all time horizons. However, when the time horizon is short ($T=5$), the agent consumes more aggressively even with low wealth, as there is less time to benefit from future labor income and portfolio growth. When the time horizon is long ($T=20$), the agent is more cautious, particularly when wealth is low, as they prioritize maintaining resources over an extended period. When the time horizon is short, the agent has limited time to accumulate wealth or recover from negative shocks. This encourages higher consumption in the short term, as future labor income and investment returns play a smaller role. Conversely, with a longer horizon, the agent places more value on future consumption and adopts a more conservative spending pattern when wealth is low.

Figure 3b illustrates the relationship between wealth and leisure $l_t$. Leisure increases with wealth in all cases. However, for shorter horizons ($T=5$), the agent works more (i.e., leisure is lower) when wealth is low, as immediate labor income is crucial to finance consumption over the brief remaining period. For longer horizons ($T=20$), the agent allocates more time to leisure, even when wealth is low because they have a longer period over which to smooth consumption and income. With a shorter time horizon, the agent must prioritize earning income over enjoying leisure, especially when wealth is low, as the remaining time to generate returns from investments is limited. When the horizon is longer, the agent can rely on future labor income and investment returns over an extended period, reducing the immediate need to work excessively.

Figure 3c shows the optimal portfolio allocation ${\pi }_t$ with respect to wealth. The investment in the risky asset increases with wealth in all cases. However, for shorter horizons ($T=5$), the agent invests more conservatively across all wealth levels. As the horizon increases ($T=20$), the agent becomes more willing to take on investment risk, especially when wealth is positive, as they have a longer period to recover from potential losses and benefit from compounding returns. A shorter time horizon reduces the agent’s risk tolerance, as there is little time to recover from investment losses. With a longer horizon, the agent can afford to take on more risk, as potential losses can be offset by future gains over a more extended period.

Economic Interpretation of Borrowing Constraints

Our numerical results illustrate key insights into how borrowing constraints influence financial decisions:

• When borrowing is restricted ($\eta =0$), agents increase labor supply to compensate, reducing leisure and consumption.
• As borrowing constraints relax ($\eta =0.7$), agents allocate more time to leisure and maintain higher consumption levels.

These findings align with empirical observations that credit-constrained individuals tend to work more and exhibit more conservative spending habits.

5. Policy Implications

Our findings suggest several policy insights:

1. The impact of borrowing constraints on labor supply: Stricter borrowing constraints force individuals to work more and delay consumption. Expanding credit access could help smooth consumption patterns.

2. Implications for social safety nets: government programs such as unemployment benefits can mitigate the need for excessive labor supply when credit is constrained.

3. Retirement planning and credit access: Limited borrowing opportunities may delay retirement and increase labor supply among older individuals. Policymakers should design pension systems considering credit constraints.

4. Interest rate and credit market regulations: since borrowing constraints are affected by lending policies, regulatory frameworks should balance debt capacity with financial stability.

6. Concluding Remarks

This paper presents a comprehensive analysis of the optimal consumption, labor–leisure, and portfolio choice problem under partial borrowing constraints in a finite-horizon setting. By integrating the labor–leisure decision into the classical consumption-investment framework and deriving a tractable numerical approach to solve the resulting free boundary problem, we provide practical insights into how individuals adjust their financial and labor supply decisions when borrowing against future income is limited. Our findings highlight the crucial role of borrowing constraints in shaping optimal strategies and suggest potential extensions to incorporate other real-world factors such as income uncertainty, retirement options, or taxation policies.