Optimal Retirement Time and Consumption with the Variation in Habitual Persistence

We consider the wealth process of an individual who participates in the social endowment insurance. Before retirement, the individual receives wage as the labor income and contributes part of the income as the premium of the social insurance. After retirement, the individual receives benefit from the social insurance. Meanwhile, investment is allowed both before and after retirement. The individual dynamically chooses the asset allocation and consumption policies, as well as the retirement time to maximize the overall utility of the consumption. First, let $B=\{B_{t},\ t\geq 0\}$ be a Brownian motion on complete filtered probability $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, the filtration $\mathbb{F}\!=\!\{\mathcal{F}_{t}:\ t\geq 0\}$ satisfies the usual conditions and is generated by the Brownian motion $B$, i.e., $\mathcal{F}_{t}=\sigma\{B_{u},\ 0\leq u\leq t\}$, $t\geq 0$. For simplicity, we assume that the financial market consists of one risk-free asset and one risky asset. The price of the risk-free asset $S_{0}=\{S_{0}(t),\ t\geq 0\}$ is given by

and the price of the risky asset $S_{1}=\{S_{1}(t),\ t\geq 0\}$ follows the stochastic differential equation (abbr. SDE):

where $r$ is the risk-free interest rate, $\mu$ and $\sigma$ are the expected return and the volatility of the risky asset, respectively. $s_{0}$ and $s_{1}$ are positive constants. The wage process $W=\{W(t),\ t\geq 0\}$ satisfies the following SDE:

where $\alpha(t)$ and $\beta(t)$ are the expected growth rate and the volatility of the wage at time $t$. Next, we establish the settings of the social insurance rules and the retirement time. The individual chooses his/her optimal retirement time $\tau$, which lies within $[\tau_{min},\tau_{max}]$. $\tau_{min}$ and $\tau_{max}$ are minimal and maximal retirement time required by the government. In addition, we assume that the optimal retirement time is a pre-commitment policy, that is, the individual chooses the optimal time to retire at the initial time rather than choosing it by constantly observing the follow-up situations. The settings are consistent with the ones in Hey and Panaccione (2011) and Chen, Hentschel and Xu (2018). Before retirement, the individual mandatorily participates in the social endowment insurance, and has the obligation to contribute $k$ proportion of the wage as the insurance premium. After retirement, the individual has the right to receive the benefit from the social insurance. The amount of the benefit depends on the retirement time. Particularly, the individual receives the amount of $g(\tau)De^{\xi t}$ as the benefit at time $t$, where $D$ is the benefit at time $0$, $\xi$ is the growth rate of the benefit, which is designed to maintain the purchasing power, and $g(\tau)$ is a penalty variable for early retirement. We assume $g(\tau)=e^{-\zeta(\tau_{st}-\tau)^{+}}$, where $\tau_{st}$ is the statutory retirement time and $\zeta>0$ is the elastic parameter. As such, $g(\tau)$ is increasing with respect to $\tau$ before $\tau_{st}$ and invariant after $\tau_{st}$. For early retirement $(\tau<\tau_{st})$, the individual receives discounted benefit and is encouraged to work to the statutory retirement age. For the wage process, we assume $\alpha(t)=\alpha 1_{\{t\leq\tau_{min}\}}$ and $\beta(t)=\beta 1_{\{t\leq\tau_{min}\}}$, where $\alpha$ and $\beta$ are positive constants. It is a realistic setting that the wage stops rising after a certain time. The last, we establish the individual’s wealth process. Assume that $T$ is the maximal survival time. The dynamics of the wealth $X=\{X_{t},\ 0\leq t\leq T\}$ is determined by three control variables: the wealth allocated to the risky asset $\pi=\{\pi_{t},\ 0\leq t\leq T\}$, the consumption level $C=\{C_{t},0\leq t\leq T\}$ and the retirement time $\tau$. It is natural to assume that $C:[0,T]\times\Omega\to[0,\infty)$ and $\pi:[0,T]\times\Omega\to[0,\infty)$ are $\mathbb{F}$-progressively measurable and satisfy the integrability condition $\int_{0}^{T}(C_{t}+\pi_{t}^{2})dt<\infty$ almost surely. Besides, $\tau$ is a pre-commitment control variable, as such, we treat it as a given parameter in the first step of optimization. The individual’s wealth process satisfies the following SDEs:
When $0\leq t\leq\tau$,

and when $\tau\leq t\leq T$,

In the next subsection, we will establish the individual’s optimization objective and the admissible domain of the control variables.

In this subsection, we establish the S-shaped utility function of the individual and the variations in the level and the sensitivity of the habitual consumption after retirement, which are better depictions of the reality. First, we set up the habitual consumption level of the individual. The habitual behavior was originally studied by Pollak (1970) and Ryder and  Heal (1973), which establish the criterion of evaluating consumption, and the habitual level is measured by the weighted average of the past consumption. Composing the S-shaped utility, only the consumption exceeding the habitual level produces positive utility. The habitual level $h=\{h_{t},\ 0\leq t\leq T\}$ is defined by

where $\psi(t)$ and $\widetilde{\eta(t)}>0$ are nonnegative and bounded parameters varying with respect to time $t$. $\psi(t)=\widetilde{\eta(t)}$ usually holds. Under this assumption, the habitual consumption level is the arithmetic average of the past consumption, as discussed in Bilsen, Laeven and Nijman (2020). Particularly, when the actual consumption equals to the habitual consumption, the habitual level remains unchanged. Next, we establish the two important variations in the habitual consumption after retirement. Briefly, both the level and the sensitivity of the habitual consumption decline at the time of retirement. The modifications of $\{\psi(t),\ 0\leq t\leq T\}$ and $\{\widetilde{\eta(t)},\ 0\leq t\leq T\}$ are based on two observations. The former observation is that the individual has more leisure time and is able to make more household productions. As such, the individual feels equally satisfied when the consumption level declines. Because only the difference between the actual consumption and the habitual consumption produces utility, this observation could be modeled as the shrinking of habitual consumption level after retirement. The latter observation is that the retired individual’s habitual level is sluggish to change over time. The consumption after retirement is influenced by the wealth constraints and the daily routines. It is unusual that one sudden extremely high (low) consumption could change the habitual consumption level a lot. Thus, the sensitivity of the variation in the habitual consumption level also decreases after retirement. Based on the above discussions, we assume $h_{\tau}=lh_{\tau^{-}}$, $0<l<1$, which refers to the decline of the habitual consumption level after retirement. Furthermore, we assume $\psi(t)=\psi$, $\widetilde{\eta(t)}=\eta$ when $0\leq t<\tau$; $\psi(t)=m\psi$, $\widetilde{\eta(t)}=m\eta$ when $\tau\leq t\leq T$, where $\psi$ and $\eta$ are constants. And $0<m<1$ refers to the decline of the sensitivity of the habitual consumption level. As such, the habitual consumption level $h=\{h_{t},\ 0\leq t\leq T\}$ is given by

In order to obtain the concise representation of $h_{t}$, we introduce the dirac function $\delta(x)$, which is a generalized function defined by

and

By substituting $\eta(s)$ with $\widetilde{\eta(s)}-\ln(l)\delta(s-\tau)$ in (2.4), $h$ is transformed into

As such, $h$ has the following concise form:

In addition, $\delta(\cdot)$ in (2.5) can also be an indicator to distinguish the pre-retirement and post-retirement scenarios. The last, we establish the S-shaped utility function of the individual. As discussed above, only the difference between the actual consumption and the habitual consumption produces utility. Besides, we assume that the actual consumption below the habitual level is allowed. As such, we need a well defined utility function to express positive and negative utilities synthetically, and we naturally choose the S-shaped utility function. The S-shaped utility function is represented as the compositions of two CRRA utilities:

where $\gamma$ refers to the relative risk aversion of the individual, and $\kappa$ is the loss aversion parameter. The second term in (2.7) defines the negative utility when the individual’s actual consumption is below the habitual level. In this circumstance, the individual suffers from the gap of inadequate consumption, as such, it produces negative utility. $\kappa>1$ represents the psychological phenomenon that the suffering of loss is greater than the happiness of equal gain. Therefore, we assume that the retirement time $\tau$ is given in the first step. The individual’s objective is to maximize the overall utility of the consumption:

where $\rho$ refers to the rate of time preference along with the mortality rate. Actually, the integral in (2.8) is divided into two parts. Before retirement, the individual earns wage and contributes to the social endowment insurance. After retirement, the individual receives benefit. At the same time, the level and the sensitivity of the habitual consumption both decline in this period. In the second step, we try to establish $\arg\max\limits_{\tau}\{V(\tau)\}$ satisfying

which is the solution of the static optimization problem. Furthermore, we study the restrictions on the admissible domain of the control variables. Although we set up the non-addictive consumption, the actual consumption should be above some threshold to maintain the minimal standard of living. We assume that there exists $L\geq 0$ such that $h_{s}-C_{s}\leq L$ holds for $s\in[0,T]$. Particularly, when $L=0$, the actual consumption below the habitual consumption is not allowed and it depicts the addictive consumption model. In addition, the individual may initiate some loans to consume the labor capital in advance. And he/she is required to repay the loans at the maximal survival time. As such, the triple control $\mathfrak{a}\triangleq(\tau,\pi,C)$ is called admissible if the individual’s wealth $X_{T}^{\mathfrak{a}}$, under the $\mathfrak{a}$, remains nonnegative at time $T$, i.e.,

almost surely. We denote the family of admissible triple controls $\mathfrak{a}\triangleq(\tau,\pi,C)$ by $\mathcal{A}$. Our goal is to solve the individual’s optimization problem (2.8)-(2.9) within the admissible domain $\mathcal{A}$. The optimal asset allocation and consumption policies will be established correspondingly. In addition, the optimal retirement time will be obtained by the pre-commitment optimization after the value function $V(\tau)$ being established.

Because the actual and habitual consumption processes are both stochastic, we usually explore $\mathrm{E}[C^{*}_{t}]$ and $\mathrm{E}[h^{*}_{t}]$ to study the evolutions of the two processes over time. However, the traditional explored method is not accurate, especially under the S-shaped utility and the long time horizon. There is a very small probability that the amounts of the actual and habitual consumption are magnificent because of excellent wealth appreciation or radical investment policies. In this circumstance, the magnificent consumption levels of some tracks will greatly raise the level of the expectations $\mathrm{E}[C^{*}_{t}]$ and $\mathrm{E}[h^{*}_{t}]$ over the long time horizon. Thus, the traditional method loses its representativeness. We naturally explore the certainty equivalence of the actual and habitual consumption processes to study the optimal policies. Under this perspective, the process $\hat{C}$ and its corresponding habitual process $\hat{h}$ satisfying $u_{t}(\hat{C}_{t}-\hat{h}_{t})=\mathrm{E}[u_{t}(C^{*}_{t}-h^{*}_{t})]$ are defined as the certainty equivalence of the consumption processes. That is, when the individual’s deterministic actual and habitual consumption amounts are $\{\hat{C}_{t},\ t\in[0,T]\}$ and $\{\hat{h}_{t},\ t\in[0,T]\}$, he/she obtains the same utility as the one under the stochastic consumption processes. Particularly, $\hat{c}_{t}=\hat{C}_{t}-\hat{h}_{t}$ is the excess consumption under certainty equivalence. It directly measures the magnitude of the utility obtained at time $t$. In the next subsection, we study the evolution of the optimal consumption over time under the certainty equivalence perspective.

In this subsection, we first set up the reasonable parameter settings according to the real data in the financial market, the labor market and the social insurance market. For the financial market, the risk-free interest rate is $r=0.02$, and the expected return and the volatility of the risky asset are $\mu=0.08$, $\sigma=0.4$, respectively. As such, the market price of risk is $\theta=0.15$. For the rules of the labor market, we assume that the parameters of the individual’s wage income are $\alpha=0.028$, $\beta=0.02$ and $W_{0}=10$. The individual begins to work at the age of $25$ and his/her maximal survival age is $100$. Besides, the individual can choose the actual retirement age within $[50,80]$ and the statutory retirement age is $65$. Retiring at the statutory retirement age, the individual can obtain the full benefit. As such, we have $t_{0}=0$, $T=75$, $\tau_{min}=25$, $\tau_{max}=55$ and $\tau_{st}=40$. Furthermore, the parameters related to the social insurance are given as follows. The contribution rate is $k=0.2$ and the full benefit is $D=6$ at time $0$. The elastic parameter in the penalty variable is $\zeta=0.015$ and the growth rate of the benefit is $\xi=0.018$. In general, if the individual retires at the minimal retirement age, he/she will obtain nearly $70\%$ of the full benefit. For the settings of the habitual consumption level, we assume that $\psi=\eta$ usually holds. As such, the habitual level is the arithmetic average of the past consumption levels. And it is realistic to assume that $\eta=\psi=0.05$. After retirement, both the level and the sensitivity of the habitual consumption decline, and we have $l=0.6$ and $m=0.3$. These two parameters are originally used to depict the changes of the habitual consumption after retirement. Thus, we choose the logical values and change their values to study the impacts on the optimal consumption and retirement policies. Besides, we assume that the initial habitual level is $h_{0}=6$ and the minimal consumption constraint requires that $L=0.5$. For the parameters in the S-shaped utility function, $\rho=0.04$, $\gamma=0.8$ and $\kappa=2.25$ are realistic settings. Then, based on the baseline model, we exhibit the certainty equivalence of the optimal consumption $\hat{C}$ and the habitual consumption $\hat{h}$. First, we assume that the retirement time is given at the statutory retirement time. That is, $\tau=\tau_{st}=40$. Figure 1 shows the evolution of the optimal consumption and the habitual consumption over time. We observe a sharp decline in the actual and habitual consumption levels at retirement. This is consistent with the empirical evidence observed in the “retirement consumption puzzle”. Because the incomes of the baseline model are relatively abundant, the actual consumption is higher than the habitual consumption, and this leads to the spiral rise of the two consumption levels. However, the growth rates of the actual and the habitual consumption levels decline after retirement, which is caused by the reduced sensitivity of the habitual formulation. In addition, we find that the initial consumption is higher than the wage income. In this circumstance, the individual initiates some loans to improve the early consumption level and obtains higher utility.

In Figure 2 and Figure 3, we select three typical retirement times $\tau=35$, $40$, $45$ to study its influence on the excess consumption level $\hat{c}$ and the optimal consumption level $\hat{C}$. Theoretically, the optimal excess consumption level $\hat{c}$ is balanced between two effects. The first is the wealth effect. If the individual obtains more incomes, he/she will naturally consume more. The second is the habitual effect. Larger excess consumption rises the habitual consumption level even more and leads to more pressure on the follow-up consumption. This prevents the individual from consuming too much in the early time. If the habitual effect weakens, time preference effect will be the dominance. That is, the early consumption produces higher utility due to the human nature of impatience. In this circumstance, the individual will consume more at the moment, and vice verse. Thus, we observe a sharp rise in the excess consumption level after retirement because of the reduced sensitivity of the habitual level and the weakened habitual effect.

In Figure 2, the excess consumption levels are both low before and after retirement in the $\tau=35$ case due to the lower total incomes. Interestingly, the excess consumption is lower before retirement and higher after retirement in the $\tau=45$ case than in the $\tau=40$ case. Longer working time also means using the larger habitual consumption benchmark for a longer time. Thus, the habitual effect strengthens. The individual decreases the former consumption and increases the latter consumption to balance the two effects. Furthermore, although the excess consumption rises a lot at retirement, it cannot offset the sharp decline in the habitual level. Thus, we observe a decline in the actual consumption $\hat{C}$ at retirement in Figure 3. In Figure 4, we study the optimal asset allocation policies over time to ensure the integrity of the problem. As usual, we explore the expectation of the amount allocated to the risky asset. We observe a downward curve due to the life cycle phenomenon. Thus, the special variation assumption in habitual persistence after retirement does not affect the trend of investment. Interestingly, there is an unsmooth rise around the time $\tau_{min}=25$. Before the time $\tau_{min}$, the individual’s wage income is fluctuating. However, after the time $\tau_{min}$, the individual receives static wage and benefit incomes. In addition, the random sources of the wage and the risky investment are the same. As such, the former risky investment should be smaller to avoid bearing too much risk. The latter risky investment can be larger as the risk bearing ability is improved. The result could also be drawn from Eq. (A.6).

In this subsection, we study the impacts of the exogenous parameters on the certainty equivalence of the optimal consumption $\hat{C}$ under the assumption that the individual retires at the statutory retirement time. In Figure 5, we study the impacts of total wealth $A$ (the discounted present value of the incomes) on the optimal consumption $\hat{C}$. In fact, the value of $A$ is determined by a family of parameters, such as $W_{0}$, $D$, $r$, $\mu$, $\sigma$, $k$, $\xi$ and $\zeta$, etc. The result confirms the positive correlation between the total wealth and the optimal consumption. When the total wealth $A$ is relatively large, the trends of the optimal consumption are almost the same. Interestingly, we observe that the percentage of decline in consumption at retirement is negatively correlated with the total wealth. This result is consistent with the empirical evidence that the consumption drop is negatively correlated with income replacement rate after retirement (cf. Schwerdt (2005)). However, when $A$ is relatively small, the optimal consumption shows a declining trend. In the case of $A=200$, which could be interpreted as $W_{0}=4$ and $D=2$, the total incomes are not enough to support the consumption above the habitual level. Thus, the actual consumption is lower than the habitual consumption. This leads to the spiral decline of the both consumption levels and produces negative utilities.

In Figure 6, we study the impacts of the habitual parameters $\psi$ and $\eta$ on the optimal consumption $\hat{C}$. In order to make the model well defined, we assume that $\psi=\eta$. As discussed in Subsection 4.2, smaller $\psi$ and $\eta$ represent lower sensitivity of the habitual level. Because the habitual effect weakens, the time preference effect becomes the dominance. As such, the individual increases former consumption and reduces latter consumption. In the extreme case $\psi=\eta=0$, we observe a completely declining trend of the optimal consumption. In this circumstance, the habitual consumption level is not affected by the former consumption. Thus, larger consumption at the moment will produce higher utility according to the time preference effect.

In Figure 7, we study the impacts of the shrinking habitual proportion $l$ on the optimal consumption $\hat{C}$. $l$ measures the magnitude of the decline in the level of habitual consumption after retirement. Different $l$ leads to the redistribution of the consumption levels before and after retirement. When $l$ is smaller, the lower consumption makes the individual feel the same satisfactory as before retirement. Thus, he/she can consume more before retirement. Although the consumption level after retirement is reduced, the excess consumption is still considerable because of the sharp decline of the habitual effect.

In Figure 8, we study the impacts of the shrinking sensitivity parameter $m$ on the optimal consumption $\hat{C}$. Because $m$ only influences the sensitivity of the habitual level after retirement, its variation hardly has impacts on the consumption levels before retirement. When $m$ is smaller, the habitual effect weakens. In this circumstance, the individual prefers to consume more in the early time because of the time preference effect. Similar to the result in Figure 6, the optimal consumption exhibits a declining trend after retirement in the extreme case $m=0$.

In Figure 9, we study the impacts of the initial habitual consumption level $h_{0}$ on the optimal consumption $\hat{C}$. The result shows that the lower initial consumption increases the admissible domain of the follow-up consumption. And the inertia of the larger excess consumption in the former time leads to the higher rise of the latter consumption. On the contrary, when $h_{0}$ is large, the actual and habitual consumption levels in the latter time are relatively small because of the small excess consumption in the former time.

In this subsection, we treat the optimal retirement time as a pre-commitment control variable and establish the optimal retirement time to maximize the individual’s value function. Then, we establish an analytical method, which can be used to analyze the impacts of the parameters on the optimal retirement time quantitatively. Before establishing the optimal retirement time, we first analyze the two impacts of the retirement time on the overall utility. The first is the wealth effect. The wealth effect can be accurately estimated by the function $A(\tau)$. In Figure 10, we observe that $A(\tau)$ is an increasing function with respect to $\tau$. From this perspective, the later an individual retires, the more wealth and higher utility he/she can obtain. The second is the habitual effect. The habitual level increases with the extension of the working time according to the baseline model. From this perspective, the later an individual retires, he/she suffers from higher habitual consumption level and obtains lower utility. In Figure 11, we modify the parameter settings to preserve the validation of $A=500$, as such, the influence of the wealth effect is excluded. We observe a negative relationship between the retirement time and the overall utility simply based on the habitual effect. Overall, the optimal retirement time is the balance between the two effects.

Next, we study the impacts of the retirement time on the overall utility based on the two effects and establish the optimal retirement time numerically. In Figure 12, the individual’s utility function first increases and then decreases with respect to the extension of retirement time. In the former time, the wealth effect dominates the habitual effect. And the habitual effect becomes the dominance in the latter time. Interestingly, the optimal retirement time is around $\tau_{st}=40$, which is exactly the statutory retirement age of $65$, according to the baseline model.

The last, the above discussions on the two effects are helpful to study the impacts of the exogenous parameters on the optimal retirement time. Besides, we establish a quantitative analytical method as follows. For any exogenous parameter $y$, we have the denotation that $\tau^{*}(y)\triangleq\arg\sup\limits_{\tau}\left\{V(y,\tau)\right\}$. As such, $V_{\tau}(y,\tau^{*}(y))=0$ is valid. Differentiating with respect to $y$ at both sides of this equality, $\frac{d\tau^{*}(y)}{dy}=-\frac{V_{y\tau}(y,\tau^{*}(y))}{V_{\tau\tau}(y,\tau^{*}(y))}$, where $V_{\tau\tau}(y,\tau^{*}(y))<0$. As such, $\frac{d\tau^{*}(y)}{dy}$ has the same sign with $V_{y\tau}\left((y,\tau^{*}(y)\right)$. Furthermore, the second-order derivative can be approximated by

Even if $V$ is not smooth, judging the sign of (4.1) is still useful. The sign is determined by comparing the size of the two values $V(y+\delta y,\tau^{*}(y)+\delta\tau)-V(y,\tau^{*}(y)+\delta\tau)$ and $V(y+\delta y,\tau^{*}(y))-V(y,\tau^{*}(y)))$, i.e., we can check whether the disturbance of $y$ will produce higher utility in the case of earlier retirement or later retirement. Remarkably, we only exhibit the parameters of high relevance and sensitivity in Table 1. Table 1 shows the relationship between the exogenous parameters and the optimal retirement time. The upward arrow indicates that the larger parameter leads to later retirement, and vice verse.

Echoing the discussions on the wealth effect and the habitual effect, $W_{0}$, $\alpha$, $k$, $D$ and $\xi$ are the parameters reflecting wealth effect, and $\psi$, $m$, $l$ and $h_{0}$ are the parameters reflecting habitual effect. Naturally, larger initial level $W_{0}$ and wage growth rate $\alpha$ will lead to more wealth if the retirement is postponed. On the contrary, larger $k$ leads to more contribution during the working period. And larger $D$ and $\xi$ lead to more benefit during the retirement period. Thus, these parameters make the individual retire early. For the habitual effect, larger $\psi$ and $m$ reflect higher sensitivity of the habitual level. As such, earlier retirement is required to prevent the habitual level from rising too high and reducing overall utility. Besides, if the shrinking habitual proportion $l$ is larger, the utility improvement by the shrinking habitual level after retirement is less effective. Thus, later retirement better balances the wealth effect and the habitual effect in this circumstance. Naturally, early retirement is required to control the habitual level when the initial habitual consumption $h_{0}$ is large.