An optimal portfolio, consumption-leisure and retirement choice problem with CES utility: a dynamic programming approach
- Ho-Seok Lee^{1} and
- Yong Hyun Shin^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-015-0841-y
© Lee and Shin 2015
Received: 12 August 2015
Accepted: 24 September 2015
Published: 6 October 2015
Abstract
In this paper, we study an optimal portfolio, consumption-leisure and retirement choice problem for an infinitely lived economic agent with a CES utility function. Using the dynamic programming method, we obtain the value function and optimal investment, consumption, leisure, and retirement strategies in analytic form. Numerically we observe that the threshold retirement wealth level is an increasing function with respect to the elasticity of substitution.
Keywords
MSC
1 Introduction
We consider an optimal portfolio and consumption-leisure choice problem of an infinitely lived economic agent with a voluntary retirement option and a CES utility function of consumption and leisure. The agent can choose her labor supply flexibly above a certain minimum level in accordance with the trade-off between utility from leisure and labor income. The economic agent receives labor income proportional to the amount of labor supplied before retirement and enjoys full leisure after retirement at the cost of labor income.
Bodie et al. [1] investigated the influence of the labor supply flexibility on portfolio and consumption choice under the lifetime portfolio and consumption choice model of Merton [2, 3]. Farhi and Panageas [4] studied the optimal portfolio, consumption and retirement choice problem of an economic agent with a Cobb-Douglas utility function and a binomial leisure rate process (\(l_{1}\) before retirement and l̄ after retirement) using the martingale method. Shin [5] solved the problem using the dynamic programming method, and Koo et al. [6] extended the model by allowing the leisure rate process to be chosen flexibly, also using the dynamic programming method. Borrowing constraints have significant effects on an economic agent’s optimal portfolio, consumption and retirement choice, and this is well documented in the literature, for example, Dybvig and Liu [7]. Barucci and Marazzina [8] also considered borrowing constraints with stochastic labor income in solving optimal consumption, investment, labor supply and retirement choice problem through a duality approach. Choi et al. [9] extended Farhi and Panageas [4] by employing a general CES utility function, which has the Cobb-Douglas utility function as a special case, and by allowing a continuum between labor and leisure. In Choi et al. [9], the martingale method is used to provide an analytic form for the value function and optimal strategies.
We revisit the optimization problem of Choi et al. [9] to give some methodological contributions and supply some numerical results which were not considered therein. We use the dynamic programming approach based on Karatzas et al. [10] to derive the closed-form solutions including the value function and optimal strategies. Since our optimization problem utilizes CES utility, it is important to investigate the effect of the elasticity of substitution between consumption and leisure on the optimal policies. Numerically we observe that the threshold retirement wealth level is an increasing function with respect to the elasticity of substitution. This is due to the fact that an economic agent with a large elasticity of substitution between consumption and leisure may consume more and enjoy higher utility than an economic agent with a small elasticity of substitution when the optimal leisure rate reaches the maximum leisure rate allowed while working. We also show that our results converge to those with Cobb-Douglas utility in Koo et al. [6] as the elasticity of substitution goes to 1.
The rest of the paper is organized as follows. Section 2 sets up the economic model. In Section 3, we solve the optimization problem using the dynamic programming method. Section 4 provides numerical examples with a limiting case, and Section 5 concludes.
2 The economy
In the financial market, we assume that the agent has investment opportunities given by a riskless asset with a constant interest rate \(r>0\), and one risky asset \(S_{t}\) whose price evolves according to the following stochastic differential equation: \(dS_{t}/S_{t}=\mu \,dt+\sigma \,dB_{t}\), where μ is a constant mean rate of return, σ is a constant volatility, and \(B_{t}\) is a standard Brownian motion on a probability space \((\Omega,\mathcal{F}, \mathbb{P})\). \(\{ \mathcal {F}_{t} \}_{t \geq0}\) is the \(\mathbb{P}\)-augmentation of the filtration generated by the standard Brownian motion \(\{B_{t} \} _{t \geq0}\).
3 The optimization problem
Remark 3.1
Now we provide the following assumption that holds throughout the paper and guarantees the optimization problem (3.3) will be well defined.
Assumption 3.1
We postulate that there exists a threshold wealth level x̄ (see, for similar conjecture and validation, Choi and Shim [11] or Dybvig and Liu [12]) corresponding to the optimal retirement time \(\tau^{*}\) such that the value function \(V(x)\) satisfies HJB equation (3.4) for \(x<\bar{x}\) and \(V(x)=U(x)\) in (3.2) for \(x\geq \bar{x}\).
Theorem 3.1
Assume that a strictly increasing function \(v(x)\in C^{1}(-w\bar{L}/r, \infty)\) solves HJB equation (3.4) for \(x<\bar{x}\) and \(v(x)=U(x)\) in (3.2) for \(x\geq\bar{x}\), where x̄ is determined by the smooth-pasting (or continuous differentiability) condition at \(x=\bar{x}\). Also assume that \(v(x)\in C^{2}(-w\bar{L}/r, \infty)\setminus\{\bar{x}\}\). Then \(v(x)\geq J(x; \mathbf {c},\boldsymbol {\pi}, \mathbf {l}, \tau)\) for any admissible control \((\mathbf {c},\boldsymbol {\pi}, \mathbf {l}, \tau)\in\mathcal{A}(x)\).
Proof
Theorem 3.2
Proof
After retirement, we derive the value function \(V(x)=U(x)=\bar{J}(\bar {C}(x))\) from Karatzas et al. [10], where a function \(\bar {X}(\cdot)\) and its inverse function \(\bar{C}(\cdot)\) are given such that the agent’s wealth \(x=\bar{X}(c)\) and the optimal consumption \(c=\bar{C}(x)\), and \(\bar{X}(c)\) and \(\bar{J}(c)\) are defined in (3.7).
FOCs (3.10) and (3.12) give us the following optimal strategies.
Theorem 3.3
Remark 3.2
The optimal strategies \((\mathbf {c}^{*},\mathbf {l}^{*},\boldsymbol {\pi}^{*},\tau^{*})\) in Theorem 3.3 are the same as those in Theorem 5.3 from Choi et al. [9]. Although it is difficult to obtain an explicit transformation between them, simple but tedious calculations show that they are equivalent.
4 Numerical examples and Cobb-Douglas utility
In this section, we present numerical examples and a limiting case \(\rho \rightarrow0\), that is, the elasticity of substitution goes to 1. In this case, our CES utility function becomes a Cobb-Douglas utility function.
5 Conclusion
With the dynamic programming method, we have provided some methodological contributions to the previous research, Choi et al. [9] which solved an optimal portfolio, consumption-leisure and retirement choice problem for an infinitely lived economic agent with a CES utility function. Through some illustrative numerical examples, we see that the threshold retirement wealth level increases with the elasticity of substitution. The solutions to a limiting case when the elasticity of substitution goes to 1 become those of the optimization problem with a Cobb-Douglas utility function.
Declarations
Acknowledgements
The corresponding author (YH Shin) gratefully acknowledges the support of Sookmyung Women’s University Research Grants 2013 (1-1303-0272).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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