GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.206

Saving and Retirement Behavior under Quasi-Hyperbolic
Discounting

Lin ZhangAugust 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Saving and Retirement Behavior under Quasi-Hyperbolic
Discounting∗

Lin Zhang †Graduate School of Economics, Osaka University, Japan

Abstract
This paper investigates saving and retirement behavior in a quasi-hyperbolic discounting modelà la Laibson (1997) by incorporating endogenous labor supply. This behavior underquasi-hyperbolic discounting is characterized by: (i) comparing those with long-run optimal
behavior, obtained under exponential discounting; and by (ii) comparing the behavior of
sophisticated consumers with those of naïve consumers. Quasi-hyperbolic discounters, eithernaïve or sophisticated, definitely under-save and at the same time, if the wage rate is sufficiently
low, retire earlier than long-run optimizers would. Consistent with empirical studies, therefore,
under-saving and early-retirement can arise simultaneously.

Keywords: Saving, Retirement, Quasi-Hyperbolic discounting

I am very grateful to Shinsuke Ikeda for his invaluable guidance through my research.
I am thankful to Kouichi Futagami, Tatsuro Iwaisako and Yoshiyasu Ono for theircomments and advice. I also thank to Shoko Morimoto for her helpful suggestions. Allerrors remain mine.
† Email address: kge017zl@mail2.econ.osaka-u.ac.jp∗

11. Introduction
The quasi-hyperbolic discounting model is widely applied to study people’s behavior of saving for
retirement, for its good approximation to hyperbolic discounting’s accurate description oftime-inconsistent impatience – discounting the near future much more heavily than the distant future
for the same length of time period, which is inspired by experimental research and common
intuitions. On the other hand, time-consistent time preference can be described by exponential
discounting under which the marginal rate of substitution of consumption between any two points oftime depends only on how far apart the points are. Intuitively, the quasi-hyperbolic discounters do
not save as much as the exponential discounters, driven by the stronger impatience to consume
immediately. And as a result of this, they cannot accumulate enough wealth to support themselves
after they stop working, have to delay the retirement age, and work for a longer time than the
exponential discounters. This interplay between saving level and retirement age underquasi-hyperbolic discounting has been discussed by Laibson et al. (1998). On the other hand, retiring
earlier is also desirable for the quasi-hyperbolic discounters; in order to realize it, they have to giveup some current consumption and save at a higher level (Diamond and Koszegi (2003)). However,
either under-saving or early-retirement can only be predicted separately by the previous studies. The
aim of paper is to investigate that people could under-save and simultaneously early-retire underquasi-hyperbolic discounting.
Being widely observed, quasi-hyperbolic discounters’ saving and retirement behavior isempirically investigated, even though they have not been jointly examined so far. Eisenhauer and
Ventura (2006) find that wealth accumulation is negatively related to hyperbolic discounting in Italy
and hyperbolic discounters are less likely to utilize commitment devices to control their choices.
Fang and Silverman (2007) show that never-married mothers in USA fail to remain sufficient labor
supply under quasi-hyperbolic discounting. By relating wealth level and labor supply to thediscounting styles, under-saving jointly determined with early-retirement could be indirectly
revealed. In this context, this theoretical analysis helps to explain this phenomenon which has notbeen jointly predicted by previous theoretical studies.
A series of empirical studies describe the indications of under-saving or early-retirement.
Attanasio (1993) discovers that aggregate personal saving has declined in the USA since 1980s byaccessing the CEX data set with the Bureau of Labor Statistic tapes of the Consumer Expenditure
Survey. Bosworth et al. (1991) show that the American private saving rate has declined steadily overthe last twenty years. Gendell (2001) shows that the average age at retirement declined in the 1990s.
Gruber and Wise (2002) demonstrate that workers have been leaving the labor force at younger andyounger age in recent years. These empirical studies jointly show that in the recent 20 years
American household saving has tended to be less, and at the same time people have retired earlier.
2

The result of this paper could help to explain these empirical phenomena.
To demonstrate the results in this paper, a three-period quasi-hyperbolic discounting model
incorporated with endogenous labor supply is employed. The saving level is determined in the first
period and the retirement age in the second period. This setup is different from Diamond andKoszegi (2003) in that the labor supply chosen in the second period can be from 0 to 1, while it is “0or 1” in Diamond and Koszegi (2003). Following O’Donoghue and Robin (1999), two types ofconsumers are distinguished: (i) people who can foresee their self-control problems are defined as
sophisticated consumers; (ii) people who cannot recognize such problems are defined as naïve
consumers. Furthermore, the instantaneous utility function is specified as the form of constantabsolute risk aversion (CARA hereafter) in order to facilitate calculation, calibration and comparisonof results.
I compare the saving and retirement behavior of different types of individuals in two aspects.
Firstly, the saving level and retirement age under quasi-hyperbolic discounting are compared withthose under exponential discounting. Following O’Donoghue and Robin (1999), the outcome oftime-consistent preference is considered as best in the long run. In this context, the definition ofunder-saving and early-retirement is derived accordingly. Secondly, I compare the saving levels andretirement ages of the sophisticated consumers with those of the naïve consumers. By following
Salanie and Treich (2006), I denote the present bias parameter separately according to different
effects in each period and relate the present bias parameter in the second period to self-control
problems.
The results found are as follows. (i) When the wage rate in the second period is sufficiently low,
people with quasi-hyperbolic discounting, either sophisticated or naïve, will save less and jointly
retire earlier than those with exponential discounting. This reveals the phenomenon of under-saving
and early-retirement which is not predicted by the previous studies. (ii) Quasi-hyperbolic discountersdefinitely under-save. However, consistent with Diamond and Koszegi (2003), consumers underquasi-hyperbolic discounting may either early-retire or late-retire. (iii) Naïveté enhances
under-saving and delays retirement age, and the difference in saving level and retirement age
between the two types of discounters is shown to be monotonic in a present bias parameter.

It is thepresent bias parameter in the second period which represents the self-control problem thatdifferentiates the sophisticated consumers from the naïve ones.
The novelty of this paper is mainly proposed in the following aspects: the specification of constantabsolute risk aversion instantaneous utility function is made to obtain closed-form solutions to theutility maximization problem with the present bias, and thereby conduct comparative dynamics withrespect to the present bias parameter; and the difference in the behavior of the sophisticatedconsumers and the naïve consumers is proved to be monotonic in the present bias parameter. Thus,
the present bias parameter is a key factor that differentiates not only quasi-hyperbolic discounters3

from exponential discounter but also sophisticated consumers from naïve consumers.
The remainder of this paper is organized as follows. Section 2 introduces the theoretical modeland the solutions. Section3 is devoted to the results of naïve consumers’ saving and retirement
behavior. Section 4 discusses the solutions to the sophisticated consumers’ problem. Section 5compares sophisticated consumers with naïve consumers in their saving level and retirement age andinvestigates the effect of present bias parameter. Section 6 discusses the results got from theforegoing sections. Section 7 makes the concluding remarks.

2. The Model
To describe saving and retirement behavior of quasi-hyperbolic discounters, I extend thequasi-hyperbolic discounting model, à la Laibson (1997), by incorporating endogenous labor supply.
The quasi-hyperbolic discounting model enables us to approximate the hyperbolic discounting
preference in a tractable form.
Consider consumers, who live three periods, where the length of each period equals one. In thefirst period, people have no choice but to supply one unit of labor in-elastically at wage rate w1, andat the same time, they decide the level of consumption c1 and saving s1.
Consumers get retired in the second period. Retirement decision is made by choosing period-2
working hours l between 0 and 1 for wage income w2 l . Following Frogneux (2009), consumers
have to endure the cost function of working e (l), which is assumed to be convex. After they retire inperiod 2, they do not receive income any more. In period 3, they just consume the amounts they
saved before the retirement.
I follow Laibson (1997) in specifying the consumers’ life time utility function with the present
bias as the quasi-hyperbolic discounting model:3−tU t = Et u (ct ) + β ∑ δ τ u (ct +τ ) − ( βδ ) 2−t e(l ), t = 1,2
τ =1



(1)

where 0<β<1 and 0<δ<1. I assume the utility function to be concave. The βcaptures the present
bias whereas δ is representing the long-run discount factor. A smaller β implies a stronger
present bias. When βequals 1, (1) reduces to the case of exponential discounting.
Specifically, from (1), the utility in period 1 is:

U 1 = u (c1 ) + βδu (c2 ) + βδ 2u (c3 ) − βδe(l )whereas the utility in period 2 is given by:
4(2)

U 2 = u (c2 ) + βδu (c3 ) − e(l )(3)When present bias βis less than 1, the marginal rates of substitution between c2 and c3 in period 1
differs from that in period 2. It causes time-inconsistency if this problem is solved forwardly without
incorporating the future shifting of the inter-temporal marginal rate of substitution. The consumers
who do not notice this problem and solve the optimal problem forwardly are defined as being naïve.
On the other hand, the consumers who regard the optimal problem as a game between selves indifferent periods and behave following the sub-game perfect Nash equilibrium solution by solving
the problem backwardly are defined as being sophisticated. When there is no present bias (β=1) andhence no self-control problem, the solution of the naïve consumers coincides with that of thesophisticated consumers.
In order to simplify the calculation and thereby to give numerical examples later, I assume theinstantaneous utility function is of constant absolute risk aversion (CARA) form.

Assumption 1 The instantaneous utility function in each period is:

u (ct ) = −e − Act

(A>0)

(4)

This specification enables us to obtain closed-form solutions to the utility maximization problem
with the present bias, and thereby conduct comparative dynamics with respect to β. Out of thesame consideration, the assumption on the effort cost in the second period is made:e(l ) = e Bl − 1

(B>0)

(5)

In what follows, the problem is solved in alternative cases to compare the solutions from two
viewpoints. Firstly, I compare solutions under the present bias (β<1) and those under exponential
discounting (β=1). As pointed out by O’Donoghue and Robin (1999), with quasi-hyperbolic
discounting, solutions that would be obtained under exponential discounting gives those of thelong-run optimizer who maximizes utility at the point in time just before he starts to consume. The
comparison with solutions of exponential discounters is necessary to evaluate normatively the effectof the present bias. Specifically, I use the following terminology to characterize saving andretirement behavior normatively:
Definition: Consumers are said to under-save when their saving is smaller than that under
exponential discounting. They are also said to get retired early when their retirement age is earlier
than that under exponential discounting.

5Secondly, I compare naïve consumers’ solutions with sophisticated consumers’. By considering asingle consumer as consisting of several inter-temporal selves in each period, these two types ofconsumers are included according to whether they are aware of their self-control problems. This
enables me to consider the impacts of the level of sophistication.

3. Naïve Consumers
3.1 Solutions to naïve consumers problem
To start with, consider the case in which the consumers are naïve. By this assumption, theconsumers cannot foresee their future self-control problem and the corresponding present-bias, they
believe that they will carry out whatever plan they formulate today. I could derive their optimizing
behavior by solving the utility maximization problems in periods 1 and 2 consecutively.
Period 1 Problem
Letting R be the gross rate of interest (R>1), the consumers’ problem in period 1 is given by:

Max U 1 = u1 (c1 ) + βδu 2 (c 2 ) + βδ 2 u 3 (c3 ) − βδe(l N )
N

N

N

(6)

c1 ,c2 ,c3 ,ls. t. c3N

= R 2 w1 + Rw2 l N − R 2 c1 − Rc 2
N

N

(7)

The first order condition is:

u1 ' (c1 ) = βδ 2 R 2 u3 ' (c3 )(8)u 2 ' (c2 ) = δRu3 ' (c3 )(9)e' (l N ) = δRw2u3 ' (c3 )

(10)

N

N

N

N

N

Conditions (10) through (13) jointly determine the period 1 optimal solution for (c1N, c2 N, c3 N,
and l N). The consumers in period 1 actually consume the amount that equals the optimal solution for
c1 N, where it is obtained form (7) through (10).

c1 =N

ABR 2 w1 − (2 B + BR + ARw2 ) log(δR) − ( B + BR + ARw2 ) log βA( B + BR + BR 2 + ARw2 )+ARw2 log w2 + ARw2 log( A / B)
A( B + BR + BR 2 + ARw2 )6(11)

s1 = w1 − c1NN

= w1 −N(12)

ABR 2 w1 − (2 B + BR + ARw2 ) log(δR) − ( B + BR + ARw2 ) log βA( B + BR + BR 2 + ARw2 )Although they also plan to carry out the period-1 solution for (c2 N, c3 N, and l N), they will change
the behavioral decisions in period 2, due to the present bias.
Period 2 Problem
Given the value of c1 N and hence of savings s1 N =w1-c1 N, which were determined in period 1, theself in period 2 re-solves the optimization problem with respect to (c2 N, c3 N, and l N).

Max U 2 = u 2 (c2 ) + βδu 3 (c3 ) − e(l N )(13)= R 2 s1 + Rw2l N − Rc2

(14)

N

N

l , c2s. t. c3

N

N

N

The first order condition is:

u 2 ' (c2 ) = βδRu3 ' (c3 )(15)e' (l N ) = βw2δRu3 ' (c3 )

(16)

N

N

N

The period-2 optimal solution for (c2 N, c3 N, and l N) is determined by (14)-(16). In the presence ofthe present bias (β<1), note that the period-2 optimal (c2 N,c3 N, and l N) differ from that of theperiod-1 optimal solution obtained form (7)-(10).
From (14) to (16) the solutions to the naïve consumers’ problem under Assumption 1 are obtained
as follows:lN =1[−( B + BR + ARw2 ) AR 2 w1
( B + BR + ARw2 )( B + BR 2 + BR + ARw2 )(17)+ ( B + BR + ARw2 − BR 3 − AR 3 w2 ) log β + ( B + BR + ARw2 )(1 + R 2 + R ) log w2+ (1 − R)( B + BR 2 + ARw2 ) log(δR ) + ( B + BR + ARw2 )(1 + R + R 2 ) log(( A / B)]7

3.2 The effects of the present bias
The impacts of present-bias on saving and retirement behavior of the naïve consumers areinvestigated here. In order to examine how much the present-bias affects the saving level andretirement age, I differentiate(20) and (21) respect to β to obtain:

B + BR + ARw2
∂c1
<0
=−
∂β
A( B + BR + ARw2 + BR 2 ) β

(18)

∂s1
∂c
B + BR + ARw2=− 1 =>0∂β∂βA( B + BR + ARw2 + BR 2 ) β(19)

B + BR + ARw2 − BR 3 − AR 3 w2
∂l N
=
∂β
A(1 + R + Rw2 )(1 + R + Rw2 + R 2 ) β

(20)

N

N

Equation (23) implies thatN∂l N
> 0 if and only if ( B + BR + ARw2 − BR 3 − AR 3 w2 ) > 0 .
∂β

From the partial derivatives above, the naïve quasi-hyperbolic discounters always save less thanthe exponential discounters. Under the condition above, the retirement age of the naïve
quasi-hyperbolic discounters is lower than that of the exponential discounters.

Proposition 1 Suppose that consumers are naïve. Then, in the setting that is specified in theprevious section, a stronger present bias (i.e. a smaller β) leads to: (i) smaller saving definitely; and1(ii) a earlier retirement age if and only if w2 < B ⋅− 1.
A R ( R + 1)( R − 1)
Proposition 1 implies that naïve consumers with β<1 definitely under-save in that their saving
levels are definitely lower than the long-run optimal level, i.e. the saving level when β=1. Similarly,naïve consumers with β <1 get retired earlier that long-run optimizer would do when
w2 <

B
1
⋅
− 1 . w2 is the marginal benefit got from working in period 2, while
A R ( R + 1)( R − 1)B1⋅
− 1 is regarded as the marginal cost. When the marginal benefit is sufficiently
A R ( R + 1)( R − 1)low, consumers will decide to retire early rather than late. Because even though they work longer,
the income from working cannot cover their lives. In this circumstance of naïve consumers,
under-saving and early-retirement arise, which is not consistent with the previous studies. In the later
section I will show this is also the case when the consumers are sophisticated. These results are8

summarized as follows.

Corollary 1 Suppose that consumers are naïve. Then when w2 <

B
1
⋅
−1 ,
A R( R + 1)( R − 1)

quasi-hyperbolic discounters under-save and jointly retire early.

N

It is straightforward to prove that there is a critical value R ∈ [1,+∞) which satisfies
1
N

R (R

N

+ 1)( RN

− 1)

− 1 = 0 . Therefore, it is reasonable to conclude that the area of under-saving

and early-retirement exists. See Figure 1 which demonstrates a special case of A=B. In other cases,
the boundary line which separates early-retirement and late-retirement only deviates from the specialcase, but still guarantees the existence of early-retirement. The line in Figure 1 is the boundary
between early-retirement and late-retirement. In the area below the boundary line early-retirementNarises. The critical value R ensures the existence of this area.

4. Sophisticated Consumers
4.1 Solutions to sophisticated consumers problem
When people are sophisticated and incapable of commitment, their optimal behavior is obtainedby solving backwardly the sub game-perfect equilibrium played by different inter-temporal selves.
Period 2 Problem
In period 3, the consumers just consume their wealth accumulated in periods 1 and 2. In period 2,
they choose the consumption level in this period c2 and decide when to retire (l) with a given saving
level s1 which is determined in period 1. The maximization problem faced by self 2 is:

MaxU 2 = u 2 (c2 ) + βδu3 (c3 ) − e(l S )(21)

s. t. c3 = R 2 s1 + Rw2 l S − Rc2

(22)

S

S

l , c2SS

S

By solving the optimal problem, the first order conditions in period 2 are obtained as:

u 2 ' (c2 ) = βδRu3 ' (c3 )
S

S

9

(23)

e' (l S ) = βw2δRu3 ' (c3 )S(24)Period 1 Problem
In the Period 1, the consumption level c1 is determined. The maximization problem faced by self 1
is:

MaxU 1 = u1 (c1 ) + βδu 2 (c2 ( s1 )) + βδ 2 u3 (c3 ( s1 )) − βδe(l S ( s1 ))
S

S

S

S

S

S

(25)

c1

s. t. c3 ( s1 ) = R 2 w1 + Rw2 l ( s1 ) − R 2 c1 − Rc2 ( s1 )
S

S

S

S

S

S

(26)

And the first order condition in the first period is:

u1 (c1 )' = βδ 2u3 (c3 )' [−
S

S

∂c3
∂c2
∂l S
−
R
−
(
)]
β
w
2
S
S
S
∂c1
∂c1
∂c1
S

S

(27)

The solutions to the sophisticated consumers’ problem under Assumption 1are given as follows:

c1 =
S

1
[ ABR 2 w1 + ARw2 log w2 − B log βA( B + BR + BR + ARw2 )2(28)− (2 B + ARw2 + BR) log(δR) + ARW2 log( A / B)
− ( B + ARw2 + BR) log( B + β SC BR + β SC Aw2 R)+ ( B + ARw2 + BR) log( B + BR + Aw2 R)]
s1 = w1 − c1S

= w1 −S(29)

1[ ABR 2 w1 ARw2 log w2 − B log βA( B + BR + BR + ARw2 )2− (2 B + ARw2 + BR) log(δR) + ARW2 log( A / B)
− ( B + ARw2 + BR) log( B + βBR + βAw2 R)+ ( B + ARw2 + BR) log( B + BR + Aw2 R)]
1[(1 + R2 + R) log w2 + log β
lS =2( B + BR + BR + ARw2 )+ (1 − R 2 ) log(δR) − AR 2 w1 + (1 + R + R 2 ) log( A / B)
− R 2 log( B + βBR + βAw2 R) + R 2 log( B + BR + ARw2 )]
10

(30)

4.2 The effects of the present bias
Similarly, I differentiate saving level and retirement age with respect to the present bias parameterβ to compare the behavior under quasi-hyperbolic discounting with those under exponential
discounting.

∂s1
B 2 + 2 βB 2 R + 2 βABRw2 + 2 βABR 2 w2 + βB 2 R 2 + βA 2 R 2 w2=>0∂β
βA[ B + βBR + βARw2 )][ B + BR + ARw2 + BR 2 ](31)∂l S B + βBR + βARw2 − βBR 3 − βAR 3 w2=
Aβ (1 + R + Rw2 )(1 + R + Rw2 + R 2 )
∂β

(32)

2

S

The derivative of saving level respect to the present bias parameter β is definitely positive. It
implies that the saving level of the sophisticated quasi-hyperbolic discounters, whose present bias
parameter β is less than 1, is definitely lower than that of the exponential discounters. Meanwhile,the derivative of retirement age respect to β can be either positive or negative.
Equation (48) implies that the necessary and sufficient condition of

∂l S
>0 is
∂β

( B + βBR + βARw2 − βBR 3 − βAR 3 w2 ) > 0 , which is similar to the condition of the naïve
consumers. When β is close to 1 enough, this condition is the same as the one in the naïve

1
consumers case. Therefore, if and only if w2 < B ⋅ [− 1] , the sophisticatedA β R ( R + 1)( R − 1)
quasi-hyperbolic discounter retire earlier than the exponential discounters do.

Proposition 2 Suppose that consumers are sophisticated. Then, in the setting that is specified in
the previous section, a stronger present bias (i.e., a smaller β) leads to: (i) less saving definitely;and (ii) a earlier retirement if and only if w2 <B
1⋅[− 1] .
A βR( R + 1)( R − 1)

It is necessary to discuss whether the condition for∂l S>0 really exists. Since w2>0, the right∂β

hand side of this condition is supposed to be positive. After calculation, it can be obtained that for1

S

R>1, there exists a critical value R ∈ [1,+∞) which makes

the condition for

S

S

S

β R ( R + 1)( R − 1)

− 1 =0 and

S
S
∂l S
>0 can be satisfied for R ∈ [1, R ] . We can also notice that R is the
∂β

11

S

decreasing function of β,i.e., a smaller βleads to a larger R . And it is easier to meet thecondition for∂l S>0. To some extent, consumers with a higher level of present bias will retire∂βmore easily. This phenomenon could be obtained by comparing Figure 2 which demonstrates a
special case of A=B. Otherwise, the boundary line which separates early-retirement andlate-retirement only rotates around the special case, but still guarantees the existence ofearly-retirement.
So far from the general and specified aspect, I have formed the view that present bias can bothlead to early-retirement and late-retirement.
Integrating the results above, if and only if w2 <B
1⋅[− 1] , the sophisticated
A βR( R + 1)( R − 1)

quasi-hyperbolic discounters both under-save and retire early.

Corollary 2. Suppose that consumers are sophisticated. Then, quasi-hyperbolic discounters
under-save and jointly retire early when w2 <B
1⋅[− 1] .
A βR( R + 1)( R − 1)

When the marginal benefit of working in period 2 is sufficiently low, people choose to retire early
even though they under-save in period 1. This phenomenon is verified here in the case of thesophisticated consumers.

5. A static comparison: sophisticated consumers versus naïve consumers
The foregoing analysis investigates the saving level and retirement age of quasi-hyperbolic
discounters in comparison with the exponential discounters. And then the following comparison I
will make is between sophisticated consumers and naïve consumers. As naïve consumers, theycannot foresee their future self-control problem and corresponding present bias just as the results
have been obtained above.
The consumption levels could be compared by subtraction:c1 − c1 =S

N

1[( BR + ARw2 ) log βA( B + BR + BR + ARw2 )2+ ( B + ARw2 + BR) log( B + BR + ARw2 )− ( B + ARw2 + BR) log( B + βBR + βARw2 )]

12

(33)

Since c1 − c1 | β →0 = −∞ and c1 − c1 | β →1 = 0 ; c1 − c1 can be shown to be monotonic in β,
S

N

S

N

S

N

which is proved in the Appendix. Therefore, c1 < c1 . On the other hand, the saving level ofSN

sophisticated consumers is higher than that of naïve consumers.
By the same way, the retirement age could be compared as follows:lS − lN =1[ R 3 ( B + Aw2 ) log β2( B + BR + ARw2 )( B + BR + BR + ARw2 )(34)+ ( B + ARw2 + BR) log( B + BR + Aw2 R)− ( B + ARw2 + BR) log( B + βBR + βAw2 R)]
Sincel S − l N |β →0 = −∞ , l S − l N | β →1 = 0 and l S − l N is monotonic in β, l S < l N . See

Figure 3. The retirement age of sophisticated consumers is lower than that of naïve consumers.
Even though either sophisticated consumers or naïve consumers behave under-saving, the saving
levels of them diverge. In virtue of realizing their self-control problem, they make use ofcommitment devices to commit the saving behavior. Without the sophistication, the level ofunder-saving of naïve consumers is aggravated. Therefore, sophisticated consumers can work for ashorter time than the naïve one in the second period for they could enjoy the saving got from the first
period.

Proposition 5 The saving level of the sophisticated consumers is higher than that of the naïve
consumers. The retirement age of the sophisticated consumers is lower than that of the naïve
consumers.
In order to consider the implications of the present bias parameter β, I follow Salanie and Treich(2006) in denoting β separately by its different effects in each period: the β that makes thediscounting factor δ lower （0＜β＜1）in period 1, which is denoted by β1 ; and the part of βthat causes different marginal rate of substitution between c2 and c3 , which is denoted as β2. Moreexplicitly, by using β1 and β2, the utility function in periods 1 and 2, (2) and (3), can be rewritten
asrespectively.

U 1 = u (c1 ) + β1δu (c2 ) + β1δ 2u (c3 ) − β1δe(l ) ,(35)

U 2 = u (c2 ) + β 2δu (c3 ) − e(l ) ,(36)

Therefore13c1 − c1 =S

N

1[( BR + ARw2 ) log β 2A( B + BR + BR + ARw2 )2(37)+ ( B + ARw2 + BR) log( B + BR + ARw2 )− ( B + ARw2 + BR) log( B + β 2 BR + βARw2 )]lS −lN =1[ R 3 ( B + Aw2 ) log β 2( B + BR + ARw2 )( B + BR 2 + BR + ARw2 )(38)+ ( B + ARw2 + BR) log( B + BR + Aw2 R)− ( B + ARw2 + BR) log( B + β 2 BR + βAw2 R)]
By denoting the present bias parameter β separately, the effects of βin differentiate thesophisticated consumers from the naïve consumers are clearly indicated. Both the differences ofsaving level and retirement age are only affected by β in period 2, i.e. β2, but not β1. Therefore,
it is the present bias parameter in period 2 differentiates the sophisticated consumers from the naïve
ones and causes the self-control problem.

Proposition 6 The divergence of saving level and retirement age between sophisticated consumers
and naïve consumers is affected only by the present bias parameter in the second period.

6. Discussions
So far how the saving level and retirement age are affected by the discounting style has been
investigated by taking partial of saving level and labor supply with respect to present bias parameter.
And it is found that both the saving level and labor supply decrease with a decreasing present bias
parameter β when the wage rate in the second period is sufficiently low, implying thatquasi-hyperbolic discounters may under-save and jointly early-retire as contrasted with exponential
discounters. In fact, quasi-hyperbolic discounters definitely under-save regardless of the condition.
While the retirement behavior depends on the relationship of wage rate and gross interest rate. The
wage rate w2 is the marginal benefit of working in the second period, and the right side of thecondition could be considered as the marginal cost of it. When the marginal benefit is lower than themarginal cost, it is reasonable for quasi-hyperbolic discounters to retire early in the second period
since the longer they work the more they suffer from working.
Although this result is applicable to both sophisticated consumers and naïve consumers, themarginal conditions for early-retirement of them are not exactly the same. For naïve consumers, thecondition for early-retirement is w2 <B
1⋅
− 1 , while for sophisticated consumers
A R( R + 1)( R − 1)14

it is w2 <B
1⋅[− 1] . The present bias parameter β differentiates the conditions.
A β R ( R + 1)( R − 1)

With exogenous wage rate w2 and gross interest rate R, the condition for early-retirement ofsophisticated consumers is easier to be achieved than that of naïve ones. In this context,
sophisticated people are more like to retire early.
With the abnormity of under-saving and early-retirement, quasi-hyperbolic discounters suffer fromdouble lost in the welfare than exponential discounters. Since a decreasing present bias parameterβ leads to less saving and earlier retirement age, the quasi-hyperbolic discounters who are withsmaller βsuffer even more.
On the other hand, I find that in this model there is no evidence of over-saving which has beenobtained by Diamond and Koszegi (2003). This is consistent with the result of Laibson (1997).
However, the possibility of both early-retirement and late-retirement coincides with the results of
Diamond and Koszegi (2003). In Laibson et al. (1998) as well as Diamond and Koszegi (2003),
consumers have to face a tradeoff between consuming more and retiring early, although they areboth desirable. In this paper, consumers also face this tradeoff, but when the wage rate of working is
too low, even the income received from retiring later cannot support their lives.
A series of empirical studies describe the indication of under-saving or early-retirement. Attanasio(1993) discovers that aggregate personal saving declined in the USA since 1980s accessing to the
CEX data set with the Bureau of Labor Statistic tapes of the consumer Expenditure Survey.
Bosworth, Burtless, and Sablehaus (1991) show that the American private saving rate has declined
steadily over the last twenty years. Gendell (2001) shows that the average age at retirement declinedin the 1990s. Gruber and Wise (2002) demonstrate that workers are leaving the labor force atyounger and younger age in recent years. These empirical studies jointly show that in the recent 20
years American household saving has tended to be less, and at the same time people got retired
earlier. The result of this paper could help to explain these empirical phenomena.
During analyzing the quasi-hyperbolic discounters, two types of consumers are consideredaccording to whether they are aware of their time-inconsistent preferences. With different level ofsophistication, their saving level and retirement age diverge from each others’. It is showed that thesaving level of naïve consumers is less than that of the sophisticated ones and the retirement age ofnaïve consumers is higher than that of the sophisticated ones. Naïveté enhances under-saving andtherefore naïve consumers have to delay retirement age or else there will be no enough amounts tobe consumed after stopping working.
By taking partially derivative, it is also found that the difference of consumption level andretirement age between sophisticated consumers and naïve consumers is monotonically increasing in
the present bias parameter β. With the same level of present bias, the divergences of consumption
level and retirement age between sophisticated consumers and naïve consumers are larger when15

present bias parameter β increases from 0 to 1. When β equals to 1, they converge to the
exponential discounting case.
The separate denotation of present bias parameter allows for investigating the effect of them ineach period. As Proposition 6 discusses, it is only the present bias in the second parameter that
determine the difference of saving level and retirement age between sophisticated consumers andnaïve consumers. The present bias parameter in the first period does not influence the divergence ofconsumers with different level of sophistication, only discounting the utilities in future.

7. Conclusion
In this study, saving and retirement behavior under quasi-hyperbolic discounting is investigated bycomparing the case of quasi-hyperbolic discounting with that of exponential discounting. In thequasi-hyperbolic discounting case, both under-saving and early-retirement can exist simultaneously.
Meanwhile, the behavior of the sophisticated consumers is compared with that of the naïve
consumers. The self-control effect causes a deviation in the behavior of sophisticated consumers
from that of the naïve consumers.
The wage rate and the gross interest rate appear to be the condition of early-retirement. However,they are exogenously determined in this paper. It would be interesting to extend this study in a
general equilibrium model with endogenous wage rates and/or interest rates.

16

References
Attanasio, O.. 1993. A cohort analysis of saving behavior by U.S. households. NBER working paper
4454.
Bosworth, B., Burtless, G. and Sablehaus, J., 1991. The decline in saving: evidence from household
surveys. Brookings Papers On Economic Activity, 183-256.
Diamond, P. and Koszegi, B., 2003. Quasi-hyperbolic discounting and retirement. Journal of Public
Economics, 87 (2003) 1839–1872.
Eisenhauer, J. and Ventura, L. (2006). The prevalence of hyperbolic discounting: some European
evidence, Applied Economics, 2006, 38, 1223–1234.
Laibson, D., 1997. Golden eggs and hyperbolic discounting. Quarterly Journal of Economics 112,
443–477.
Laibson, D. Repetto, A. and Tobacman, J., 1998. Self-control and saving for retirement. Brookings
papers on economic activity, volume 1998, issue 1, 91-172.
Fang, H. and Silverman, D., 2007. Time Time-inconsistency and welfare program participation:evidence from NLSY. NBER Working Paper 13375.
Frogneux. V. 2009. Retirement Decision and Quasi-hyperbolic Discounting. Research report of
FUNDP.
Gendell, M., 2001. Retirement age declines again in 1990s. Monthly Labor Review, 12-21.
Gruber, J. and Wise, D., 2002. Social security programs and retirement around the world: micro
estimation. NBER working paper 9407.
O’Donoghue. T and Robin. M. 1999. Doing it now or later. The American Economic Review, Vol. 89,
No. 1 (Mar., 1999), pp. 103-124
Salanie, F. and Treich, N.. 2006, Over-savings and hyperbolic discounting. European Economic
Review 50, 1557–1570.

17

Figure 1Naïve Consumers: the case of A=B

Figure 2

Sophisticated Consumers: the case of A=B

18

Figure 3

19

Appendix
A.1 The monotonic property of c1 S − c1 N and l S − l N(1 − β ) BR( B + Aw2 )∂ (c1 − c1 )
=
>0 ,
2
∂β
β ( B + BR + ARw2 + BR)[ B + βR( B + Aw2 )]
S

N

so

c1 − c1solS − lNS

N

ismonotonically increasing in β.

(1 − β ) BR( B + Aw2 )∂ (l S − l N )
=
>0
2
∂β
β ( B + BR + ARw2 + BR)( B + ARw2 + BR)
monotonically increasing in β.

20,

is