Mathematical Problems in Engineering

Volume 2019, Article ID 8681471, 9 pages

https://doi.org/10.1155/2019/8681471

## Time-Inconsistent Preferences, Retirement, and Increasing Life Expectancy

Business School of Hunan University, Changsha, Hunan, China

Correspondence should be addressed to Guangbing Li; nc.ude.usc@il_gnibgnaug

Received 13 August 2018; Revised 22 December 2018; Accepted 30 December 2018; Published 10 January 2019

Academic Editor: Sebastian Anita

Copyright © 2019 Shou Chen and Guangbing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], Zou et al. [2], Holmes [3], and Findley and Caliendo [4]). Some prominent economic topics are delayed saving for retirement and endogenous economic growth. For instance, Bloom et al. [5] find that increased longevity raises aggregate savings rates in countries with universal pension coverage and retirement incentives. Strulik [6] concludes that present-biased preferences, providing the same present value of a constant infinite income stream, are harmless for economic growth. Naturally, this raises questions such as how hyperbolic preferences and time-inconsistent behavior do matter in retirement decisions and how the effects of increasing life expectancy is on aggregation consumption to capital ratio before and after retirement. Most of the classic quasihyperbolic discounting model is beginning with a discrete form model developed by Laibson [7]. For example, earlier selves think that the deciding self tends to choose early retirement and may save less to induce delayed retirement in Diamond and Köszegi [1]; that is, quasihyperbolic discounting can cause dynamic inconsistency. Retirement plans are never time-inconsistent with such preferences in Holmes [3].

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 [8], the survival rate is age-independent). We show that, given the equivalent present value, the retirement is later (earlier) under hyperbolic discounting than under exponential discounting for the initial time preference rate (present-biased preferences). It is so that individuals will choose early retirement under certain circumstances while they do not care about future goals. The lower the quasihyperbolic discount rate is, the later individuals will retire since individuals’ savings will be enough to sustain consumption. We also show that the faster the speed of declining impatience is, the older the retirement age is as for hyperbolic discounting increasing the consumption rate.

To show these, we combine Prettner and Canning’s [9] endogenous retirement and economic growth model with Strulik’s [6] hyperbolic preferences. Firstly, present-biased preferences allow us to use continuous time model to capture dynamics in savings and retirement decisions. Secondly, we extend the standard model of endogenous growth for aggregate capital accumulation and aggregate consumption expenditure. Thirdly, we explore the effects of changes in the mortality rate on the consumption-capital ratio and in mortality rate, the interest rate, and the speed of declining impatience on economic growth rate. Finally, we recalibrate our model using US data and compare the dynamic behaviors of retirement decisions of exponential and hyperbolic preferences.

In addition, we extend our model to age-dependent survival law which is adapted from Boucekkine et al. [10, 11] and Azomahou et al. [12]. In particular, the assumption of age-dependent mortality rate taken in the more realistic modeling is absolutely crucial. Numerically, we find that age-dependent mortality rate leads individual to work longer. What is more, in case of aggregation over cohorts, the consumption to capital ratio has slight effect on age-dependent mortality rate before retirement. But this ratio has a positive effect on the mortality rate after retirement. People are willing to pay for higher consumption expenditures as age increases. Besides, at the same age, the longer individuals’ life expectancy is, the lower this ratio is.

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 2 considers the benchmark model with time-inconsistent preferences merging Blanchard and Yarris structures. Section 3 compares the dynamic behaviors of optimal retirement under exponential and hyperbolic discounting. In Section 4, the model is considered with more realistic demographics, that is, with age-dependent survival probabilities. Section 5 concludes the paper.

#### 2. The Benchmark Model with Time-Inconsistent Preferences

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 ? Evidence on mortality rates suggests low and approximately constant probabilities at working ages, see Blanchard [8] for example.). We introduce continuous-time consumption models by merging Prettner and Canning [9] with Strulik [6] and then discuss hyperbolic discounting of Laibson [7].

##### 2.1. Individuals with Constant Mortality Rate

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 [13]; i.e., individuals start their working life without capital holdings and there are no bequests.

Individuals enter the labor market as adults at time and maximize lifetime utilitysubject to the budget constraintwhere represents the mortality rate; , , , and denote consumption, capital, wage rate, and the interest rate, respectively; is a scaling parameter measuring individuals’ unwillingness to work; and is an indicator function with a value of when working and zero when retired. According to Frederick and O’Donoghue [14] and Strulik [6], we consider hyperbolic discounting. The discount function in (1) is given by hyperbolic discounting containing exponential case; that is, . Throughout the paper, the present-biased preferences refer to the short-term discount rate, and the initial time preference rate is the long-term discount rate, according to Laibson [7].)where controls the present bias (The smaller the parameter is, the stronger the present-biased preferences is, and the lower the speed of declining impatience is.) and controls the instantaneous discount rate (or the initial time preference rate) at next instant in time. Generally, the quasihyperbolic discount rate at discount time to time is defined as

The Hamiltonian for this problem is

The following are the first-order conditions:

These conditions yield the following:It states that consumption expenditure growth is positive if and only if the interest rate exceeds the initial discount rate (it is also similar to the standard neoclassical growth model with an infinite lifetime horizon if ; see Ramsey [15], for example). Intuitively, individuals prefer to work as long as the additional utility of working longer is able to compensate them for their disutility of delaying retirement, given that the consumption while working is higher than retirement.

Similar to Strulik [6], we obtain individuals’ optimal consumption aswhere and denotes individuals’ retirement date.

Since lifetime consumption expenditures are equal to lifetime income, we can calculate consumption from a reparametrization of the budget constraint as follows:Integrating and using , which follows from the individuals’ Euler equation, and , which follows from denoting wage growth by , we arrive at an expression for the fraction of consumption expenditures to wages at the beginning of working lifeIntuitively, this expression tells us that if individuals want to consume more or save less in relation to initial income, they will choose to delay retirement. Following Bloom et al. [5], we also introduce the parameter restriction , which ensures a finite present value of lifetime wage income.

For endogenous retirement the situation is more complex. We denote the retirement ages (or working ages) by and rewrite the retirement equation as follows:which we can be simplify to

Substituting for the initial consumption-wage ratio (10) into (12), we have

In Figure 1, we sketch for varying and for different speed of declining impatience and discount rate scenarios. To compare this with exponential discounting, Figure 2 depicts for varying with the exponential discount rate according to Prettner and Canning [9]. We obtain the parameter values in the numerical analysis which fit the actual values for the United States directly from Economagic [16] and World-Bank [17]. That is, , , , and . (Following Prettner and Canning [9], we reconstruct averages for 2010-2014 to obtain business–cycle adjusted values for banks’ prime loan rate (), discount window primary credit (), per capita gross domestic product (GDP) growth (), and life expectancy at a birth of 78.7 years. Furthermore, we recalibrate and in (21) to obtain 78.7 years for life expectancy and a working life duration of 45 years, which corresponds to these figures for the US in 2012 according to OECD [18].)