We study consumption behavior, retirement decisions, and endogenous growth within a dynamic equilibrium when individuals have present-biased preferences. Compared to individual with exponential preferences, individual with hyperbolic preferences will choose to retire early for present-biased preferences but to delay retirement for the initial time preference rate. We extend the benchmark equilibrium model to age-dependent survival law and solve numerically the equilibrium effects. It shows that, at the same age, the consumption-capital ratio may have slightly positive effect on increasing life expectancy before retirement but has a significantly positive effect on it after retirement.
Most studies in psychology and experimental economics confirm that (quasi-)hyperbolic discounting provides a better answer for future utility than exponential discounting (see e.g., Diamond and Köszegi [
In this paper, we extend time-inconsistent consumption problems in the context of endogenous growth and retirement choice following Blanchard-Yarris model (as depicted in Blanchard [
To show these, we combine Prettner and Canning’s [
In addition, we extend our model to age-dependent survival law which is adapted from Boucekkine et al. [
Our findings provide a possible theoretical explanation for government intervention in individuals’ retirement decisions and increasing life expectancy. These help to align theory with the intuition of demographic structure and social schemes in retirement decisions.
The present paper is organized as follows. Section
In this section, we first introduce briefly the basic structure of Blanchard-Yarris model with time-inconsistent preferences. That is, the mortality rate is constant (and age-independent) before retirement (How unrealistic is the assumption of a constant
In this part, we derive models with hyperbolic preferences to capture optimal consumption-savings and retirement decisions with constant mortality rate. We follow the assumptions in Yaari [
Individuals enter the labor market as adults at time
The Hamiltonian for this problem is
The following are the first-order conditions:
These conditions yield the following:
Similar to Strulik [
Since lifetime consumption expenditures are equal to lifetime income, we can calculate consumption from a reparametrization of the budget constraint as follows:
For endogenous retirement the situation is more complex. We denote the retirement ages (or working ages) by
Substituting for the initial consumption-wage ratio (
In Figure
Difference in the retirement for varying speed of declining impatience and initial discount rates under hyperbolic preferences.
Difference in the retirement for varying discount rates under exponential discounting.
Figures
In this part, we derive expressions for aggregate capital accumulation and aggregate consumption expenditures similar to the overlapping generations’ framework of Blanchard [
Consider all cohorts that are alive at a certain instant
Similar to Prettner and Canning [
In particular, let
This part of the neoclassical economic growth model strictly follows Solow [
In order to set up the long-run equilibrium, we assume that aggregate income, or economic GDP, consists of wage income and capital income. We also assume that the economy grows at the constant rate
Next, we can obtain the following propositions.
An increase in longevity raises aggregate consumption expenditures.
Since the function
The intuition for this finding is that as longevity increases, individuals pay more fees in physically and spiritually (such as health care) to meet their own needs.
(i) The rate of economic growth is declining in the mortality rate; (ii) the rate of economic growth is increasing in the speed of declining impatience but is decreasing in the initial time preference rate.
By the partial differential derivation method, we can get the following inequalities:
The intuition behind this finding is straightforward. First, in a society where mortality is declining, the population is growing, resulting in additional domestic demand, which has a positive impact on the economy. Second, according to Laibson [
In this section, we compare the dynamic behaviors of optimal retirement under exponential and hyperbolic discounting. To illustrate this, recall the stream of lifetime utility from Bloom et al. [
We first illustrate the effects of the discount rate and retirement age on individuals’ unwillingness to work with numerical examples, i.e.,
Effects of discount rate and retirement on unwillingness to work.
| | | | | |
| |||||
| 45 | 44 | 43 | 42 | 41 |
| |||||
| 0.7191 | 0.7423 | 0.7904 | 0.8571 | 0.9397 |
Introducing a format of the statistical curve fitted by multidimensions least square, we assume the following measurement of unwillingness to work under exponential discounting:
Next, we study that the retirement age for hyperbolic and exponential preferences is equivalent under a plausible condition. In order to assess this, we extend the assumption following Strulik [
Motivated by (
In Figure
Effects of retirement age on the speed of declining impatience with
Difference between hyperbolic and exponential preferences in retirement age given the equivalent present value.
We summarize the above findings in the following.
Given equivalent present value (
Next, we show that the main results are robust to the effects of generalizing the instantaneous utility function, disutility of work, the mortality process, the interest rate, the discount rate and economic growth. In addition, we assess the sensitivity of our central result with respect to changes in the speed of declining impatience. Results in Table
Effects of the speed of declining impatience on the optimal duration of working life.
| 0.1 | | | | | | | | 6 |
| |||||||||
| 43.97 | 44.00 | 44.06 | 44.16 | 44.28 | 44.44 | 44.62 | 44.83 | 45.06 |
It is worth noting that when individuals have the same degree of unwillingness to work, the higher discount rate leads to longer working hours. However, when the degree of the unwillingness to work depends on the time preference and retirement, the situation will change; that is, the longer the working time to produce negative effects is driven by the individual choosing to retire early. We measure the degree of the unwillingness to work based on time preference and retirement age. The higher the discount rate is, the earlier the individual chooses to retire. For example, for two jobs with the same remuneration but with different working hours, the individual is clearly willing to take less time with the same incomes of work.
In the previous model, we implicitly assume that the mortality rate is age-independent, which is clearly unrealistic after retirement. In this section, we relax this assumption and consider age-dependent survival law. We use a survival law adapted from Boucekkine et al. [
The set of individuals are alive in
Under age-dependent survival law, an individual’s maximize lifetime utility is
It states that consumption expenditure growth has no effect on age-dependent mortality rate. However, when an individual chooses to work, the lower limit of the ratio of wages to consumption is reduced (e.g.,
Next, we consider an individual’ retirement age based on age-dependent mortality rate. Similarly, we can calculate consumption as follows:
In Figure
Partial equilibrium effects for different
For a low value of
Based on age-dependent mortality rate, consider all cohorts that are alive at a certain instant
Furthermore, we have
Therefore, along a balanced growth path, we can write the system as
Next, we describe the response of an economy’s consumption to capital ratio in life expectancy. Solving the system (
In Figure
Consumption to capital ratio effects of different
The dashed line displays the effect of a decrease in parameter
In this paper, we extend time-inconsistent consumption problems in the context of endogenous growth and retirement choice. In continuous-time models, individuals with hyperbolic preferences will choose to retire early in the short discount rate but to delay retirement in the long discount rate.
A major extension in the theoretical model is the introduction of endogenous growth within a dynamic equilibrium based upon Boucekkine et al. [
Another major extension is the numerical assessment of time inconsistency and age-dependent equilibrium model. They are clearly important in practice, and they change retirement decisions with hyperbolic preferences considerably. We find that, under the assumption of constant unwillingness to work, the higher the rate of time preference is, the later individuals choose to retire regardless of quasihyperbolic or exponential preferences. Considering the unwillingness to work is a function of the speed of declining impatience, the discount rate, and the retirement age, we find that individuals with strong future-biased preferences prefer to delayed retirement, but a high initial discount rate will cause early retirement. Consider age-dependent mortality rate, numerically, we give some interesting findings, such as how the consumption to capital ratio has effect on increasing life expectancy before or after retirement. These insights may have implications for individuals’ increasing life expectancy and capital-consumption allocations based policy schemes and their effects on retirement decisions.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This paper is supported by the National Natural Science Foundation of China (Grant Nos. 71521061, 71790593) and Hunan Provincial Graduate Innovation Fund of China (No. CX2016B076).