GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 266

Saving and retirement behavior
under quasi-hyperbolic discounting

Lin Zhang

May 2012

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Saving and retirement
hyperbolic discounting

behavior

under

quasi-Lin Zhang

Abstract This paper investigates the saving and retirement behavior of consumers using
a quasi-hyperbolic discounting model with endogenous labor supply. The optimal
behavior, which is obtained under exponential discounting, is compared with the
behavior of naive and sophisticated consumers. If and only if the present-biased
marginal utility of future consumption decreases with stronger present bias, the quasihyperbolic discounter, whether naive or sophisticated, under-saves and retires earlycompared with an exponential discounter. In other words, quasi-hyperbolic discounting
explains why, under-savers might also be early retirees. This is consistent with previous
empirical studies. Further, two functional forms are employed as examples of thisnecessary and sufficient condition. And I show that under logarithmic utility, a wage tax
and an interest subsidy can improve consumer welfare.

Keywords: Saving · Retirement · Quasi-Hyperbolic discounting · Sophisticated ·Naive

JEL Classification: D91 · E21 · J26

1 Introduction
The quasi-hyperbolic discounting model has been widely applied to study how people
save for retirement, because of its good approximation to hyperbolic discounting’s
accurate description of time-inconsistent impatience, namely discounting the near futuremuch more heavily than the distant future for the same length of time. This research
approach has been inspired by experimental research as well as by common intuitions.
By contrast, time-consistent preference can be described by exponential discounting
under which the marginal rate of the substitution of consumption between any two
points of time depends only on the distance between these. Intuitively, the quasihyperbolic discounter does not save as much as the exponential discounter does, owing

L. Zhang Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043
Japan
Email: kge017zl@mail2.econ.osaka-u.ac.jp

1

to the stronger impatience to consume immediately. Consequently, he or she cannot
accumulate enough wealth to support him-or herself after stopping working, has todelay retirement, and thus works for a longer time compared with the exponential
discounter. This trade-off relation between saving level and retirement age under quasihyperbolic discounting has been discussed by Laibson et al. (1998). Retiring early isalso desirable for the quasi-hyperbolic discounter; however, in order to realize this, heor she has to forego a proportion of current consumption and saves at a higher level(Diamond and Koszegi, 2003). Given that either under-saving or early retirement has
only been predicted separately by previous studies, the aim of this paper is to investigate
whether the consumers can under-save and still retire early under quasi-hyperbolic
discounting.
Although the saving and retirement behavior of quasi-hyperbolic discounter has been
empirically investigated, these behavioral traits have not been jointly examined thus far.
Eisenhauer and Ventura (2006) find that wealth accumulation is negatively related to
hyperbolic discounting in Italy and that hyperbolic discounters are less likely to utilize
commitment devices to control their choices. Fang and Silverman (2007) show that unmarried mothers in the US fail to maintain sufficient labor supply under quasihyperbolic discounting. By relating wealth level and labor supply to discounting styles,
under-saving jointly determined with early retirement can be indirectly shown. In thiscontext, the present analysis helps explain this as-yet underexplored phenomenon.
A number of empirical studies discuss resent trends in under-saving or early
retirement. For example, Attanasio (1993) uses a CEX data set provided the Bureau ofLabor Statistic tapes of the Consumer Expenditure Survey to show that aggregate
personal saving has declined in the US since the 1980s. Bosworth et al. (1991) showthat the private saving rate in the US has declined steadily over past 20 years. Gendell(2001) shows that the average age at retirement declined in the 1990s, while Gruber andWise (2002) demonstrate that workers have been leaving the labor force at younger andyounger ages in recent years. These empirical studies imply that over the past 20 years
American household saving has declined, even though people have been retiring earlier.
The findings in this paper could help explain these seemingly paradoxical empiricalphenomena.
This paper employs a three-period quasi-hyperbolic discounting model withendogenous labor supply. Saving level is determined in the first period and retirement
age in the second period. This setup is different from that presented by Diamond andKoszegi (2003) in that the labor supply chosen in the second period is continuous.
Following O’Donoghue and Robin (1999), two types of consumer can be distinguished:(i) those who can foresee self-control problems (defined as “sophisticated consumers”)
and (ii) those unaware of such problems (defined as “naive consumers”).
Saving level and retirement age under quasi-hyperbolic discounting are comparedwith those under exponential discounting. Following O’Donoghue and Robin (1999),
the outcome of time-consistent preference is considered to be optimal from the long-run
perspective, and the definition of under-saving and early retirement is derived
accordingly.
Furthermore, how a wage tax and an interest subsidy affect consumer welfare isinvestigated. This paper thus extends the model by including government policy, whichaims to optimize a consumer’s long-run well-being.
This paper makes four principal findings. First, if and only if the present-biased2

marginal utility decreases as present bias strengthens, consumers under quasi-hyperbolic
discounting, whether sophisticated or naive, save less and retire earlier than those under
exponential discounting. This result shows the co-existence of under-saving and early
retirement. Second, two examples of utility function form (i.e., constant absolute risk
aversion (CARA) and logarithmic) are found to show the establishment of thisnecessary and sufficient condition. With these two functional forms, the consumer
definitely under-saves; however, consistent with the finding of Diamond and Koszegi(2003), consumers under quasi-hyperbolic discounting may either retire either earlier orlater.
Third, a necessary and sufficient condition is provided under which the government
improves consumer welfare by imposing a proportional wage tax. Interest subsidy
definitely improves consumer welfare. Fourth, because the discounting effect of thepresent bias parameter can increase or decrease labor supply, the long-run Phillips curve
relation may display a trade-off or stagflation between unemployment and inflation.
This paper makes the following original contributions to the body of knowledge inthis area. A necessary and sufficient condition of the general form for under-saving andearly retirement is derived, while two examples of utility function form are used toobtain closed-form solutions to the utility maximization problem with present bias,
thereby conducting comparative dynamics with respect to the present bias parameter.
Furthermore, a proportional wage tax and an interest subsidy are included to extend themodel by discussing how they affect consumer welfare when the government’s budget
constraint is satisfied.
The remainder of this paper is organized as follows. Section 2 introduces thetheoretical model. Section 3 presents the saving and retirement behavior of naïve
consumers. Section 4 discusses the solutions to the problem of sophisticated consumers.
Section 5 analyzes how representative consumer welfare is affected by a wage tax andan interest subsidy. Section 6 discusses on the results of this analysis and explores how
these results work upon the long-run Phillips curve relation. Section 7 concludes.

2 The Model
To describe the saving and retirement behavior of quasi-hyperbolic discounters, thequasi-hyperbolic discounting model of Laibson (1997) is extended by incorporating
endogenous labor supply. This model enables us to approximate present-biased
preferences in a tractable form.
Consider a representative consumer who lives for three periods. In the first period, heor she has to supply labor in-elastically. He or she retires in the second period.
Retirement is decided by choosing period-2 working hours l. Before retirement, he or
she receives wage at a rate of w . Following Frogneux (2009), she has to suffer thedisutility of working e(l). After he or she retires in period 2, there is no further income.
In period 3, he or she only consumes the amount saved before retirement. Variable ct
denotes the consumption level in each period and s1 is the saving level in the first
period ( s1  w  c1 ).
Following Laibson (1997), the representative consumer’s lifetime utility function
with present bias is

3

3t

U t  u (ct )      u (ct  )  (  ) 2t e(l ), t  1,2

(1)

 1

where 0<  <1 and 0<  <1.  captures present bias or diminishing impatience,
whereas  represents the long-run discount factor. A smaller  implies a stronger
present bias. Therefore,  could be referred to as the present bias parameter. When equals 1, (1) reduces to the case of exponential discounting.
In the last period, the consumer exhausts all the remaining savings and faces theinter-temporal budget constraint:c3  R 2 w  R 2 c1  Rwl  Rc2(2)

where R is the gross interest rate.
I assume the follow:
ASSUMPTION 1 The instantaneous utility function u () is concave in consumption
ct and the working disutility function e() is convex in labor supply l .
In the following sections, functional-form assumptions are made to obtain analyticalresults and to show that the propositions are robust to change in functional form. Two
specific functional forms are employed as examples and they both meet Assumption 1.
Example 1: Constant Absolute Risk Aversion Utility Function
CARA forms the utility function intou(ct )  e  Act(A>0),(3)and assume that the disutility function of working ise(l )  e Bl  1(B>0).

(4)

Example 2: Logarithmic Utility Function
The utility and disutility functions are in the forms:u(ct )  ln ct ,e(l )   ln(l ) .

(5)
(6)

In the following sections, consumption level and retirement age are compared
between the cases under quasi-hyperbolic discounting (  <1) and under exponential
discounting (  =1). As pointed out by O’Donoghue and Robin (1999), the behavior of
exponential discounter is considered to be the optimal outcome from the long-run
perspective, because their preference is time-consistent. This comparison is necessary toevaluate normatively the effect of present bias. Specifically, I use the following
terminology to characterize saving and retirement behavior normatively:

4

DEFINITION 1 A consumer is said to under-save when his or her saving is lower thanthat under exponential discounting. He or she is also said to retire too early when his or
her retirement is earlier than that under exponential discounting.
When the present bias parameter  is less than 1, the marginal rates of substitutionof c2 for c3 (and l for c3) in period 1 differ from those in period 2. Time-inconsistency
arises if this change in the marginal rates of substitution is not incorporated into theagent’s consumption plan. A representative consumer who is unaware of this problem
and optimistically believes that he or she will carry out what he or she plans now isdefined as being naive (denoted with a superscript N). By contrast, a representative
consumer who regards the life time utility maximizing problem as a game between
different periods and behaves following the sub-game perfect Nash equilibrium solution
by solving the problem backwardly is defined as being sophisticated (denoted with asuperscript S). Under logarithmic utility, the solution for naive consumers coincides
with that for sophisticated consumers.

3 Naïve Consumers
Consider that a consumer is naïve, that is to say, the consumer cannot foresee his or her
present bias and the corresponding time inconsistency and believes that he or she will
carry out whatever he or she plans today. His or her optimizing consumption levels andlabor supply are derived by solving the utility maximization problems in period 1 and 2
consecutively.
In period 1, the naive consumer consumes c1N and optimistically plans the~consumption levels c~2N and c~3N as well as labor supply l in terms of:NNu' (c1 )   2 R 2u' (c~3 ) ,NNu' (c~2 )  Ru ' (c~3 ) ,~Ne' (l N )  Rwu ' (c~3 ) .

(7)
(8)
(9)

The inter-temporal budget constraint (2), and (7), (8), and (9) jointly determine theconsumption levels and labor supply for the lifetime utility maximization problem inperiod 1. However, time inconsistency will arise by neglecting the change in themarginal rates of substitution of c2 for c3 (and l for c3) in period 2. In order to separate
the actual consumption level and labor supply from those planned in period 1, those are~
optimistically planned are denoted as c~2N , c~3N , and l . As a result, only c1N isrealized in period 1.
When period 2 comes and given c1N , the consumer re-solves the lifetime utility
maximization problem in terms of:u' (c2 )  Ru ' (c3 ) ,(10)e' (l )  R w u' (c3 ) .

(11)

N

N

N

N

5

Consumption levels c2 N and c3 N, and labor supply l N are determined by (2), (10), and(11). In the presence of present bias (  <1), c2 N, c3 N, and l N deviate from the plan madein period 1. By differentiating consumption level and labor supply with respect to  ,the effect of present bias on the saving and labor supply of quasi-hyperbolic discounters
is investigated.
LEMMA 1 Suppose that the representative consumer is naïve. Then, in the setting
specified in the previous section, stronger present bias (i.e., a smaller  ) leads to lessd [ u ' (c~3N (  ))]saving if and only if
 0.
dThe proof of Lemma 1 is shown in the Appendix. Lemma 1 demonstrates thenecessary and sufficient condition for under-saving by naive consumers. Notice thatu ' (ctN (  )) is the marginal utility of consumption in period t, which can ultimately be
expressed as a function of the present bias parameter  , while u' (ctN ( )) represents
marginal utility discounted by the short-run discounting factor or marginal utility in thenext period. This paper follows the expression “weighted marginal utility” in line with
Pareto (1909), while u ' (ct ) is referred to as the “present-biased marginal utility of
consumption”.
In the three-period model, u ' (ct ) captures the degree to which a consumer isconcerned about his or her consumption in the future. Therefore, in Lemma 1
u' (c~3N ( )) represents the present-biased marginal utility of period-3 consumption in
the naive consumer’s plan, which affects how much c1N is consumed. For the naive
consumer making plans in period 1, a smaller  means stronger present bias for theutility in the future. If in the naive consumer’s plan, a smaller present-biased marginalutility of period-3 consumption follows a smaller  , this implies that he or she is lessconcerned about future consumption than might an exponential discounter, because
present bias makes he or her so desirous of the current consumption. Then she does notsave much for future in period 1.
Saving level is determined in period 1, and thus it is based on the naive consumer’splans. However, these plans are too optimistic for a quasi-hyperbolic discounter andthus they will be modified in period 2.
LEMMA 2 Suppose that the representative consumer is naïve. Then, in the setting
specified in the previous section, stronger present bias (i.e., a smaller  ) leads to and [ u ' (c3N (  ))] 0.
earlier retirement if and only ifd
The proof of Lemma 2 is similar to that of Lemma 1. Notice that Lemma 2 concerns
the present-biased marginal utility of actual period-3 consumption c3N , which isdetermined in period 2. In period 2, the naive consumer modifies his or her consumption
level and labor supply because he or she aims to maximize the lifetime utility from theperspective of the present period. The consumer in period 2 is the victim of his or her

6

under-saving in period 1 and has to work longer for period-3. However, he or she isreluctant to work and faces a trade-off between enjoying the current leisure or working
longer for future. When a smaller present-biased marginal utility of actual period-3
consumption c3N follows stronger present bias, the unwillingness to work dominates
and the consumer chooses to retire earlier, because he or she is still concerned less about
his or her future (period-3) consumption.
PROPOSITION 1 Suppose that the representative consumer is naïve. Then, if and onlyd [ u ' (c~3N (  ))]d [ u ' (c3N (  ))]if
 0 and 0 , the quasi-hyperbolic discounter underddsaves and retires too early.
Proposition 1 implies a condition that makes saving and working less attractive to theconsumer. When the naive consumer is less concerned about planned and actual period3 consumption less, he or she chooses to save less and retire earlier than might an
exponential discounter. In other words, the consumer can raise period-3 consumption byeither increasing saving in period 1 or labor supply in period 2, and when he or she isless concerned about period-3 consumption in both periods, saving and labor supply areat a lower level. Therefore, under-saving can co-exist with early retirement, which isconsistent with empirical studies.
However, there remains the question of whether this necessary and sufficientcondition for under-saving and early retirement exists and whether is robust to a specificform of utility function. Two examples of utility function are included to show the coexistence of under-saving and early retirement.
COROLLARY 1 Under CARA utility, the naive consumer definitely under-saves; ifB1 1] , he or she retires too early.
and only if w   [A R( R  1)( R  1)
Therefore, for a naive consumer with CARA utility under-saving can co-exist withearlier-retirement.
COROLLARY 2 Under logarithmic utility, the naive consumer definitely under-saves;
if and only if  2 2 (  2)  2 she retires too early.
The logarithmic utility function form thus allows the co-existence of under-saving
and early retirement.

4. Sophisticated Consumers
When the representative consumer is sophisticated, his or her optimal behavior is
obtained by solving the sub-game perfect equilibrium backwardly, which is carried out
by different inter-temporal selves.
First the sophisticated consumer anticipates his or her optimal behavior in period 2
S
and period 3 with given c1 :7u' (c2 )  Ru ' (c3 ) ,(12)e' (l S )  wRu ' (c3 ) .

(13)

S

S

S

In combination with (2), he or she foresees how much will be consumed in periods 2
and 3 as well as how long to work for a given saving level. This representative
consumer is sophisticated in that he or she integrates this anticipation into the lifetime
utility maximization problem in period 1

u ' (c1 )   2u ' (c3 )[R 2  (1   )
S

S

c3
],
S
c1
S

(14)

S

and determines the consumption level c1 . In this case, time inconsistency is absent.
Similarly, I differentiate saving level and labor supply with respect to the present bias
parameter  to compare the behavior under quasi-hyperbolic discounting with that
under exponential discounting.
LEMMA 3 Suppose that the representative consumer is sophisticated. Then, in thesetting specified in the previous section, stronger present bias (i.e., a smaller  ) leads tod{[ [ R 2  (1   )less saving if and only ifdc3]u ' (c3S (  ))}
S
c1
S

 0.

The proof of Lemma 3 is similar to that of Lemma 1. The only difference is that thenecessary and sufficient condition in Lemma 3 for under-saving involves the
anticipation of the next two periods. The present-biased marginal utility can thus beSc3 (  )2
] . Theapplied here, with an additional sophistication fraction [ R  (1   )S
c1
sophisticated consumer foresees that increasing period-3 consumption c3S follows
increasing saving level, and therefore he or she integrates this insight into consumption
strategy in period 1 in order to induce an adverse effect.
Similarly to the naive consumer, if the present-biased marginal utility of period-3
S
consumption c3 with sophistication fraction decreases with stronger present bias, therepresentative consumer saves less than might the exponential discounter because he or
she is too desires of current consumption and is less concerned about his or her future
consumption.
LEMMA 4 Suppose that the representative consumer is sophisticated. Then, in thesetting specified in the previous section, stronger present bias (i.e., a smaller  ) leads
to an earlier retirement if and only if

d [ u ' (c3S (  ))] 0.
dThe necessary and sufficient condition for early retirement is also similar to that of8the naive consumer. However, it should be noted that the labor supply levels of these
two types of consumer are not necessarily identical.
The sophisticated consumer foresees his or her unwillingness to work in period 2 andincorporates it into peiod-1 plan. Meanwhile, he or she is also desires of consumption
now. As a result, he or she has to sacrifices the consumption in period 3.
PROPOSITION 2 Suppose that the representative consumer is sophisticated. Then, ifScd{[ [ R 2  (1   ) 3S ]u ' (c3S (  ))}d [ u ' (c3S (  ))]c1
and only if
 0 and 0 , theddquasi-hyperbolic discounter under-saves and retires too early.
Proposition 2 implies that for sophisticated consumers, if the present-biased marginalS
utility of period-3 consumption c3 with (or without) the sophistication fraction
decreases with stronger present bias, under-saving co-exists with early retirement. In
this context, the consumer finds that more consumption now and less work in next
period are attractive and eventually decides to save less and retire earlier than an
exponential discounter because he or she cares his- or herself in period 3 less.
Two specific utility function forms are thus employed to show that the co-existenceof under-saving and early retirement is allowed.
COROLLARY 3 Under CARA utility, the sophisticated consumer definitely underB1saves; if and only if w2   [ 1] , he or she retires too early.
A R( R  1)( R  1)
COROLLARY 4 Under logarithmic utility, the sophisticated consumer definitely
under-saves; if and only if  2 2 (  2)  2 , he or she retires too early.

5. Government Policy
This section investigates the welfare effect of a proportional wage tax and an interest
subsidy. In this partial equilibrium model, wage rate and gross interest rate areexogenously given, while interest makes saving more advantageous and a wage tax
decreases income. The government’s policy would thus considerably adjust
consumption levels and labor supply.
Here, ex-ante welfare is evaluated at the point when the consumer is born. I evaluate
the consumer’s well-being from the long-run perspective, namely under exponential
discounting:W  u(c1 )  u(c2 )   2u(c3 )  e(l ) .

(15)

Further, the utility function is specialized to logarithmic utility. This functional form
assumption makes the model tractable and widely applicable.
The sophisticated consumer’s behavior is different from that of the naive one in thatincorporating foresight leads to time inconsistency disappearing. However, underlogarithmic utility the naive consumer’s consumption level and labor supply coincides

9with that of the sophisticated consumer. Hence, government policy has an identicaleffect on these two types of consumers, whether a proportional wage tax or an interest
subsidy.
5.1 Proportional Wage Tax
Assume that the government imposes a proportional wage tax and gives a lump-sumtransfer M in period 2. The wage tax rate is t ( 0  t  1 ). Therefore, the consumer’s
budget constrain is(16)
c3  R 2 w  R 2 c1  Rwl (1  t )  Rc2  RM .
The budget constrain of the government is satisfied:M  twl .

(17)

Accordingly, all consumers modify their first-order conditions regarding the
proportional wage tax, whether naive or sophisticated. In period 2, labor is elasticallysupplied, while the proportional wage tax affects labor supply through two channels.
First, it reduces income from working and hence decreases labor supply. Second, therepresentative consumer has to maintain consumption levels by increasing his or her
labor supply. By contrast, labor is in-elastically supplied in period 1 and a lump-sumtransfer would entirely counteract the effect caused by the wage tax in period 1, which
has a negligible effect on welfare.
The government aims to optimize consumer welfare by selecting the tax rate properly.
This effect is investigated by differentiating welfare with respect to the tax rate in theneighborhood of t  0 . Proposition 3 provides the necessary and sufficient condition
for the government to improve consumer welfare by imposing wage tax.
PROPOSITION 3 In the setting of this model with logarithmic utility, for both naive
and sophisticated consumer, the optimal proportional wage tax is positive if and only if2
 2.
 (2   )The proof of Proposition 3 is provided in the Appendix.
Under logarithmic utility, the wage tax leads to a decrease in labor supply. Recalling2
from Corollary 2 and Corollary 4, we can conclude that when   2the (2   )consumer retires later than the exponential discounter does. A positive wage tax
mitigates later retirement and thus enhances consumer welfare.
5.2 Interest Subsidy
This sub-section considers the case in which the government subsidizes interest inperiod 2 at a rate r ( 0  r  1 ) and therefore a lump-sum transfer M occurs. The
corresponding budget constraint of the consumer reads:c3  (1  r ) R 2 w  (1  r ) R 2 c1  Rwl  Rc2  RM .

(18)

10

The government’s budget constraint meetsM  rR(w  c1 ) .

(19)

In this case, the government optimizes the consumer’s long-run well-being bychoosing the interest subsidy rate r . Proposition 4 states that the interest subsidy
improves consumer welfare, whether consumers are naive or sophisticated.
PROPOSITION 4 In the setting of this model with logarithmic utility, for both naive
and sophisticated consumers, the optimal interest subsidy is positive.
The proof of Proposition 4 is in the Appendix. Under logarithmic utility, interest
subsidy increases saving level in period 1. Recall from Corollary 1 can Corollary 3,
under logarithmic utility the consumer definitely under-saves. A positive interest
subsidy mitigates under-saving and thus improves consumer welfare.

6. Discussion
This study investigates how the saving level and labor supply are affected bydiscounting style by differentiating with respect to the present bias parameter. It findsthat under-saving can co-exist with early retirement and how the present-bias parameter affects saving level and labor supply depends on whether a smaller present-biased
marginal utility of period-3 consumption c3 (or with the sophistication fraction)
follows stronger present bias.
On one hand, the necessary and sufficient condition for under-saving has been shown.
On the other hand, over-saving is also possible, in line with Diamond and Koszegi(2003) and Salanie and Treich (2006). As long as the present-biased marginal utility of
period-3 consumption c3 (or with sophistication fraction) increases with stronger
present bias, over-saving by quasi-hyperbolic discounting consumer arises. In thiscircumstance, such consumers are more concerned about their future consumption morecompared with the exponential discounters and they find that saving at a higher levelcan make up for the consumption in period 3.
Further, the possibility of both early retirement and late retirement coincides with theresults presented by Diamond and Koszegi (2003). In Laibson et al. (1998) andDiamond and Koszegi (2003), consumers face a trade-off between consuming more or
retiring early, although both options are desirable. In this paper, the representative
consumer also faces this trade-off, but when the benefit of working is not sufficient,even the income received from retiring later cannot support his or her lifestyle.
Note that the concavity of the instantaneous utility function and the convexity of thedisutility function matter when they determine the necessary and sufficient conditions
for under-saving and early retirement. In particular, the convex disutility function ofworking describes a consumer who is exceedingly reluctant to work in period 2.
6.1 The Effect on the Long-run Phillips Curve
The foregoing sections investigate the saving level and labor supply of a representative
quasi-hyperbolic discounter in comparison with an exponential discounter. This section

11

investigates how these results affect the long-run Phillips curve relationship.
According to Graham and Snower (2008), inflation can affect the long-run Phillips
curve by three channels:(i)the employment cycling effect,(ii)
the labor smoothing effect,(iii) the discounting effect.
As inflation increases, (i) labor demand reduces, (ii) labor supply reduces, and (iii)
labor supply increases. The third effect dominates and thus there is a negative relation
between inflation and unemployment in the long run.
However, in addition to this long-run trade-off relation between inflation andunemployment, a positive relation is found, namely stagflation. The results found in theforegoing sections support the conclusions of Graham and Snower (2008).
According to Lemma 2 and Lemma 4, whether the present bias of a consumer underquasi-hyperbolic discounting leads to more labor supply or less depends on whether thepresent-biased marginal utility of period-3 consumption c3 increases with a stronger
present bias parameter  . Therefore, the third channel that affects the long-run Phillips
curve includes two aspects.
When later retirement arises, the discounting effect increases labor supply. Moreover,because this effect dominates other two effects, there is a long-run trade-off relation
between inflation and unemployment. However, when stronger present bias leads toearly retirement and hence the discounting effect decreases labor supply, as well as theemployment cycling effect and labor smoothing effect, this generates a positive relation
between long-run inflation and unemployment, which is stagflation.

7. Conclusion
This paper compares quasi-hyperbolic discounting with exponential discounting to
explore saving and retirement behavior. In the quasi-hyperbolic discounting case, undersaving can co-exists with early retirement. However, this depends on whether thepresent-biased marginal utility of period-3 consumption decreases with stronger present
bias. The consumer under quasi-hyperbolic discounting with the CARA (or logarithmic)
utility function form definitely under-saves, but he or she can still earlier than might theexponential discounter.
Furthermore, By investigating the effects of a proportional wage tax and an interest
subsidy, this paper shows that such policies can improve representative consumer
welfare.
Because quasi-hyperbolic discounting can increase or decrease labor supply, the longrun Phillips curve relation can be expressed as a trade-off or stagflation between
unemployment and inflation.

Acknowledgments:
I am very grateful to Shinsuke Ikeda for his invaluable guidance
throughout my research. I am thankful to David Flath, Koichi Futagami, Tatsuro
Iwaisako and Yoshiyasu Ono for their comments and advices. I thank Shoko Morimoto

12

for her suggestions. I am also deeply indebted to the anonymous referee and editor-inchief Giacomo Corneo for their very helpful comments. All remaining errors are mine.

Appendix
A.1 Proof of Lemma 1
LEMMA 1 Suppose that the representative consumer is naïve. Then, in the setting
specified in the previous section, stronger present bias (i.e., a smaller  ) leads to lessd [ u ' (c~3N (  ))]saving if and only if 0.
dProof: For a consumer under quasi-hyperbolic discounting, the consumption and labor
supply plan he or she makes in period 1 is too optimistic. The actual consumption levels
and labor supply in period 2 and 3 deviate from what he or she plans in period 1. How
much is consumed in period 1 is based onNNu' (c1 )   2 R 2u' (c~3 ) .

(A.1)

Taking the total differential of (A.1) with respect to  leads toN~N
du ' (c1 ) dc1N
2 2 d [ u ' (c3 (  ))]



R

.
N
d
d
dc1

(A.2)

According to Assumption 1, the concavity of the instantaneous utility functionNdu ' (c1 )
ds1N
N
N

0
implies.
Because,has the same sign as
s

w

c
1
1
N
d
dc1
d [ u ' (c~3N (  ))].
d
A.2 Proof of Proposition 3
PROPOSITION 3 In the setting of this model with logarithmic utility, for both naive
and sophisticated consumers, the optimal proportional wage tax is positive if and only if2
 2.
 (2   )Proof: Because the naive consumer’s consumption levels and labor supply coincides
with that of the sophisticated one’s, I only consider the naive consumer here. For thenaive consumer under a proportional wage tax, the first-order conditions imply that in13

period 1:NNu' (c1 )   2 R 2u' (c~3 ) ,NNu' (c~2 )  Ru ' (c~3 ) ,~Ne' (l N )  Rw(1  t )u' (c~3 ) ;

(A.3)
(A.4)
(A.5)

Further in period 2:

u' (c2 )  Ru ' (c3 ) ,
N

N

(A.6)

e' (l N )  Rw(1  t )u' (c3 ) .
N

(A.7)

The first order conditions (A.3), (A.4), (A.5), (A.6), and (A.7) and the consumer’sNNN
budget constraint (16) jointly determine the consumption levels c1 , c2 , and c3 aswell as labor supply lN.

I take the derivative of welfare with respect to the wage tax rate t in theneighborhood of t  0 in order to find the optimal wage tax policy:
N
N
N
dl N
dW
N dc1
N dc2
2
N dc3
N
 u ' (c1 ) u ' (c2 )  u ' (c3 ) e' (l )dt t 0
dt t 0
dt t 0
dt t 0
dt

.
t 0

(A.8)
Substituting these first-order conditions and taking the derivatives with respect to thewage tax rate t leads to:dW 2 3 (  2) Rw(1   )
u ' (c3N )[2   2 (2   )] .
2
2 2
dt t 0 (2   ) (1  2   )

(A.9)

2
dW
 0 implies   2.
dt t 0 (2   )Therefore the optimal wage tax of the naive consumer is positive if and only if2
 2.
 (2   )A.3 Proof of Proposition 4
PROPOSITION 4 In the setting of this model with logarithmic utility, for both naive
and sophisticated consumer, the optimal interest subsidy is positive.
Proof: Because the naive consumer’s consumption levels and labor supply coincides
with that of the sophisticated one’s, I only consider the naive consumer here. The firstorder conditions of the naive consumer in period 1:

14

N
N
u' (c1 )   2 (1  r ) R 2u' (c~3 ) ,NNu' (c~2 )  Ru ' (c~3 ) ,~Ne' (l N )  Rwu ' (c~3 ) ;(A.10)(A.11)(A.12)

those in period 2:

u' (c2 )  Ru ' (c3 ) ,(A.13)e' (l )  Rwu ' (c3 ) .

(A.14)N

N

N

N

The first-order conditions (A.10), (A.11), (A.12), (A.13), and (A.14) and theNNconsumer’s budget constraint (18) jointly determine the consumption levels c1 , c2 ,NNand c3 as well as labor supply l .
I take the derivative of welfare with respect to the interest subsidy rate r in theneighborhood of r  0 in order to find the optimal interest subsidy policy:
N
N
N
dl N
dW
N dc1
N dc2
2
N dc3
N
 u ' (c1 ) u ' (c2 )  u ' (c3 ) e' (l )dr r 0
dr r 0
dr r 0
dr r 0dr,r 0(A.15)
Substituting these first-order conditions and taking the derivatives with respect to theinterest subsidy rate r leads to:dW
3   (2   )
,dr r 0 1  2   2

(A.16)

which is definitely positive because 0    1 and 0    1 .
Therefore, the optimal interest subsidy is positive.

15

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 fromhousehold surveys. Brookings Papers on Economic Activity, 183–256.
Diamond P and Koszegi B (2003) Quasi-hyperbolic discounting and retirement. Journalof Public Economics, 87: 1839–1872.
Eisenhauer J and Ventura L (2006). The prevalence of hyperbolic discounting: some
European evidence. Applied Economics, 38: 1223–1234.
Fang H, and Silverman D (2007) 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.
Graham L and Snower D J (2008) Hyperbolic discounting and the Phillips curve.
Journal of Money, Credit and Banking, 40(2–3):427–448
Gruber J and Wise D (2002) Social security programs and retirement around the world:
micro estimation. NBER working paper 9407.
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, 1: 91–172.
O’Donoghue T and Robin M (1999) Doing it now or later. The American Economic
Review, 89(1): 103–124.
Pareto V (1909) Manuel d'Bconomie Politique, Paris, 159.
Salanie F and Treich N (2006) Over-savings and hyperbolic discounting. European
Economic Review 50: 1557–1570.

16