The decision about how much to save for retirement is likely to be dependent on when an individual plans to be retired, and vice versa. Yet, the established literature on hyperbolic discounting and life-cycle saving behavior has for the most part abstracted from choice over retirement. Two notable exceptions are Diamond and Koszegi [1] and an important follow-up study by Holmes [2], which demonstrates that time-inconsistent retirement timing is impossible when saving behavior is explicitly modeled in a stylized three-period setting. In this paper, we build upon the framework of Diamond and Koszegi [1] and Holmes [2] by generalizing the assumptions about initial income and assets. We show analytically and via simple numerical examples that time-inconsistent retirement can exist in a three-period life-cycle model of consumption and saving.
Research findings from psychology have been used to gain insight into many of the important questions that are typically studied by economists. A prominent example is that of hyperbolic discounting, in which a sizable body of research has documented that hyperbolic discount functions provide a better fit to choice data relative to the exponential discount function.1,2 A hyperbolic discount function is characterized by a discount rate that declines in the delay. And as demonstrated in the influential study of Strotz [
Excessive debt and delayed saving for retirement are some of the most important economic applications of hyperbolic discounting.3 Yet, the established literature on hyperbolic discounting and life-cycle saving behavior has generally abstracted from choice over labor supply. Although understandable given the additional complexity that might exist as a result of having multiple margins of time inconsistency, it is conceivable that ignoring labor supply decisions could lead to skewed predictions about the effect of hyperbolic discounting on saving outcomes. This is due to the possibility that saving and labor supply decisions are determined in tandem. Indeed, the decision about how much to save for retirement is likely to be dependent on when an individual plans to be retired (the extensive labor supply decision), and vice versa.
The study by Diamond and Kőszegi [
An individual lives for three periods and acquires utility from consumption and from leisure. The utility acquired from consumption each period is, where is consumption in period. Leisure in period 1 and period 3 is exogenously imposed, namely and. The representative individual has choice over leisure in period 2 such that, where is the period utility of leisure (the cost of working). From the perspective of the first period, the intertemporal utility function of the individual is
where and are the short-term and longterm discount factors. From the perspective of the second period, the intertemporal utility function is
Note that if, then (1) is not consistent with (2), meaning that the marginal rate of substitution between and is from the perspective of period 1, yet it is from the perspective of period 2. We assume that the individual is naive about his time-inconsistent preferences. This means that in period 1 the individual selects an allocation of consumption and leisure that he believes will be followed in the current and in future periods in order to maximize (1). Yet, the individual will update his choices in period 2 such that (2) is maximized.
We generalize the setup of Diamond and Kőszegi [
We assume a zero interest rate, but we do not impose any restrictions on S1. Yet, and are necessary to entertain the possibility of retirement in period 2.
With the superscript on the choice variables denoting the period of planning, the individual plans to consume
with and, given the individual’s period-1 intention of his period-2 leisure choice,. These intentions imply period-1 savings,
The individual will choose in period 1 an intention of his period-2 leisure, , in order to maximize his well-being from the perspective of period 1. Therefore, he will plan to be working during period 2 if
Otherwise, he will intend to be retired during period 2.
Given, which is dependent on whether the individual intended to work or to be retired during period 2 from the perspective of period 1, the individual will select the consumption allocations from the perspective of period 2
and. Note that these allocations are both dependent on the individual’s choice to actually work or to be retired during period 2, meaning that the individual will also choose to maximize his intertemporal utility from the perspective of period 2. Therefore, the representative individual will choose to actually work during period 2 if
where is the threshold level of saving that is required to finance retirement during periods 2 and 3.
We first study whether or not the possibility can exist for the individual to initially plan on working in period 2 from the perspective of period 1, and then reverse his original plan by actually retiring when period 2 arrives. The normal retirement intention will occur if, but the individual will actually choose to be retired when period 2 arrives if. The latter of these two inequalities is equivalent to
where is the lower-bound value for period-1 cash on hand that would enable savings from period 1 to be high enough to finance retirement during period 2, even though the individual had intended on normal retirement from the perspective of period 1. Holmes [
This indicates that time-inconsistent retirement timing can exist for the special case of unit labor income and a zero initial savings account balance, if long-term patience is entertained.6,7 However, we are primarily interested in examining parameterizations with. From (7) and (10),
as, meaning thatmust exist if.
This highlights the existence of time-inconsistent early retirement when initial assets are non-zero and/or if. Yet, time-inconsistent retirement can also exist for larger values of, such as, and which yields.
We now examine whether or not it is possible for the individual to intend to be retired during period 2 from the perspective of period 1, and then reverse his original intention by delaying retirement and actually working during period 2. An early retirement intention will occur if, but the individual will actually end up working during period 2 if. This implies
where is the upper-bound value for period-1 cash on hand that would lead to insufficient savings in period 1 such that the individual cannot finance retirement when period 2 arrives, even though period-2 retirement was the intention from the perspective of period 1. We formally state our finding for this case.
Proposition. The following sequence of retirement timing is impossible: 1) The individual initially plans to be retired in period 2 from the perspective of period 1; and then, 2) The individual actually chooses to work during period 2.
Proof. The ratio of (7) to (12) can be mathematically arranged as
where
Given, the first multiplicative term in (13) is greater than 1 if. The second term is also greater than 1 on account that is increasing in for and given
under the same conditions on. Thus, for all in the parameter space, meaning that (12) can never be satisfied. ■
The retirement decision is one of the most important choices that an individual can make during his lifetime, since the timing of retirement determines the life-cycle budget constraint to a large degree. Holmes [
Findley and Caliendo [
We acknowledge and thank Nick Guo, Frank Caliendo, and two anonymous referees for helpful comments and suggestions.