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 [
, which can consist of current labor income and financial wealth.5 The individual earns unit income in period 2 if he chooses to actually work. Therefore, the individual’s constraints in periods 1 and 2 are respectively
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 [
given
, meaning that the retirement decision must be time-consistent with unit labor income in period 1 and with
. We find parameters 
with 
such that
.
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 that
must 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. ■