GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 293

Mandatory Social Security Regime, Consumption and Retirement
Behavior of Quasi-Hyperbolic Discounters

Lin ZhangJanuary 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Mandatory Social Security Regime, Consumption and Retirement
Behavior of Quasi-Hyperbolic Discounters∗

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

Abstract
This paper proposes a mandatory social security contribution regime in order to adjust theconsumption and retirement age of quasi-hyperbolic discounters to the optimal level in thelong-run perspective which is under exponential discounting. Within this mandatory pension
contribution regime, this paper ascertains that the behavior of some generations can be adjusted
to the optimal level while other generations cannot have both consumption and retirement age
adjusted to the optimal level. With behavior adjusted to the optimal level, the adjusted
generations’ welfare is improved. However, the un-adjusted generations’ welfare is deteriorated.

Keywords: Social security system, Consumption, Retirement, Quasi-hyperbolic discounting

JEL Classification Numbers E21 H55 D91∗

I am very grateful to Shinsuke Ikeda for his invaluable guidance through my research. I am

thankful to the members of Ikeda Seminar for their comments and advices. All errors remain mine.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan
Email address: kge017zl@mail2.econ.osaka-u.ac.jp
11. Introduction
Quasi-hyperbolic discounters, with strong present bias, tend to consume at a higher level than the
exponential discounters do, which is demonstrated by theoretical studies. And in order to support
themselves after they stop working, quasi-hyperbolic discounters have to work for a longer time than
exponential discounters. Since exponential discounters’ preference is time-consistent, their decisionsof consumption level and retirement age are regarded as the optimal outcome in the long-run
perspective. Therefore, compared with exponential discounters, the present bias causes deviations ofconsumption level and retirement age of quasi-hyperbolic discounters from exponential discounters.
On the other hand, social security system, particularly pension system, reallocates consumption
level across a consumer’s life and is regarded as one of commitment devices to help individuals
whose time preference is inconsistent commit their future behavior. Blake (2004) shows that pension
promotes greater saving and encourages earlier retirement. It is of great interest to ask how amandatory social security regime adjusts quasi-hyperbolic discounters’ behavior which is considered
to be irrational in the long-run perspective. This paper investigates that whether social security
system adjusts consumption level or retirement age of quasi-hyperbolic discounters, and to what
extent it can adjust them.
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 of2

time-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.1 The behavior in pension system under hyperbolic discounting (or quasi-hyperbolic
discounting) has been included in theoretical studies. Schwarz and Sheshinski (2007) examine theeffects of hyperbolic discounting on the comparison of alternative social security systems, and findthat intergenerational transfers within a pay-as-you-go economy are usually secured by the social
security system and independent of longevity, whereas this is not the case for the funded economy.
Previous studies of pension systems focus on the effects of pension system on the wealthaccumulation and consumption after retirement, not including policy target. In these previous studies,social security system did not act initiatively. However, the pension system in this paper aims tomodify the behavior of hyperbolic discounters’ which is considered as irrational in the long run. This
paper is intended to incorporate social security system as a policy instrument to adjust theconsumption level and retirement age of hyperbolic discounters. The novelty of this paper includes:i) investigating the effects of mandatory pension system on the adjustment of consumption level andretirement age of quasi-hyperbolic discounters; ii) ascertain not only the welfare effects but also the
behavior effects of pension adjustment.
An overlapping-generation model of quasi-hyperbolic discounters with endogenous labor supply

1

Hereafter, the expressions “quasi-hyperbolic discounting” and “hyperbolic discounting” will bereferred interchangeably.
3

is employed. A pay-as-you-go pension system is incorporated to adjust the over-consumption andlate-retirement of quasi-hyperbolic discounters to the level that is obtained under exponential
discounting and regarded as optimal in the long-run perspective. The pay-as-you-go social security
system is characterized by intergenerational redistribution with benefits paid directly from current
workers' contributions and employed here to show how it works among generations withpresent-bias.
By applying mandatory pension system to the adjustment of consumption level and retirement ageof quasi-hyperbolic discounters in a small open economy, it is ascertained that as two policy target
the consumption level and retirement age cannot be adjusted to the optimal level at the same time.
For a second-best choice, only several generations’ behavior can be adjusted, while othergeneration’s welfare will be hurt. However, in order to lead to a Pareto Improvement there is path ofcontribution rates along which the welfare of each generation could be increased.
The remainder of this paper is organized as follows. Section 2 introduces the theoretical model.
Section 3 is devoted to the solutions. Section 4 discusses the adjustment effects of mandatory social
security system. Section 5 discusses the results got from the foregoing sections. Section 6 makes theconcluding remarks.

2. The Model
Consider a small open economy populated by overlapping generations of quasi-hyperbolic

4

discounters who live for three ages with a population growth rate n. In this economy, thethree-period model with endogenous retirement decision is employed. Besides, a pay-as-you-gosocial security system is incorporated into the model economy, in which the generation ofindividuals who are old enough, that is to say at age 3 of the whole life, receive a lump-sum pension
benefit b. The time horizon is infinite.
Let’s consider a generation who is born in period t-1 (hereafter generation t-1), and assume thepopulation of this generation is unit. Therefore, for the generation who are born at period t-1, at age
1 they work and contribute to the social security system at some contribution rate kt −1 ; and at age 2,
they can choose when to retire with 0 ≤ lt −1 ≤ 1and contribute to social security system at a certain
rate k t ; and at age 3, they retire and receive a lump-sum pension benefit bt −1 . The contribution
rates of social security system vary over each period, but stay the same in one period. Due to thefeature of pay-as-you-go, in each time period, the pension benefits of a generation of individuals arecollected from the contributions of generation of individuals who are at age 1 and working at age 2.
Hence for the generation t-1, they receive a lump-sum pension benefit at period t+1,

bt −1 = kt +1 w1 (1 + n) 2 + kt +1 w2 (1 + n)l .

(1)

Quasi-hyperbolic discounting (Laibson, 1997) forms the discounted utility function into
3− jU t , j = u (ct , j ) + β ∑[δ τ u (ct , j +τ )] − ( βδ ) 2− j e(lt , 2 ),
τ =1

5

j = 1,2 ,(2)where 0 < β < 1 	
 denotes the present bias parameter and 0 < δ < 1 . e(l ) denotes the cost ofworking at age 2, including endogenous labor supply (Frogneux, 2009). When present bias βis lessthan 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.
Assume the consumers to be constant relative risk aversion with the inverse of the elasticity of
substitution equal to 1, which simplifies the utility function to be in the form of natural logarithm,

u (ct , j ) = ln(ct , j ) .

(3)

Similarly, the cost function of working at age 2 is assumed to bee(lt ) = ln(lt ) .

(4)

The budget constraint faced by the consumers is

ct ,3 = R 2 (w1 − ct ,1 − kt w1 ) + Rw2 (1 − kt +1 )lt − Rct , 2 + bt ,

where

(5)

w1 and w2 are the wage rates of working at each age, and R>1 is the gross interest rate.

Characterized by their strong impatience for future utility, quasi-hyperbolic discounters areverified to consumer more than exponential discounters do, and retire later to support themselves
after they stop working. It is the present bias that causes the deviation of consumption level andretirement age. Social security system is known for its reallocation effect on individual’s
6consumption and it is of great interest to investigate whether the consumption level and retirement
age can be adjusted by it.

3. The solutions
Following O'donoghue & Robin (1999), the sophisticates and the naïfs are considered. The former
are capable of realizing their self-control problem and incorporating it into the future plan while thelater are unaware of their present-bias and behave time-inconsistently. However, the set-up of modelin the foregoing section makes the consumption and labor-supply of both types of individuals
coincide, that is to say that the naïfs just happen to correctly predict their consumption level andretirement age at age 1. Therefore, it is only necessary to consider the sophisticates (or the naïfs).
For the sophisticates, they fully predict the dynamic inconsistency of their own and solve theinter-temporal utility maximizing problem backwardly.
Let’s consider a generation who are born at period t-1(generation t-1). Since the sophisticates are
considered, they can incorporate the dynamic inconsistency into plan, and behave according to it. To
start with planning the optimal consumption level at each age and optimal retirement age, they begin
with solve the utility maximizing problem at age 2. And assume that social security system covers
every generation and every generation incorporates it the utility maximizing problem.

Max U t −1, 2 = ut −1, 2 (ct −1, 2 ) + βδut −1,3 (ct −1,3 ) − e(lt −1 )s.t. ct −1,3 = R 2 ( w1 − ct −1,1 − kt −1w1 ) + Rw2 (1 − kt )lt −1 − Rct −1, 2 + bt −1
7

(6)
(7)

bt −1 = kt +1 w1 (1 + n) 2 + kt +1 w2 (1 + n)lt −1(8)

The first order condition at age 2 isut −1, 2 ' = βδ Ru t −1,3 ' ,et −1 ' = βδw2 [ R(1 − kt ) + kt +1 (1 + n)]ut −1,3 ' .

(9)
(10)

Similarly, the first order condition at age 1 is obtained by incorporating the first order condition atage 2,

ut −1,1 '+βδ 2ut −1,3 '{βR∂ct −1, 2 / ∂ct −1,1 + ∂ct −1,3 / ∂ct −1,1
− βw2 [ R(1 − k t −1 ) + k t (1 + n)]∂lt −1 / ∂ct −1,1} = 0 .

(11)

The first order conditions show that pay-as-you-go social security system does have effects on
consumption through the endogenous labor supply.
And the optimal consumption levels and retirement age of generation t-1 are determined as[ R 2 (1 − kt −1 ) + kt +1 (1 + n) 2 ]w1,
R 2 (1 + βδ 2 )[ R 2 (1 − kt −1 ) + kt +1 (1 + n) 2 ]δw1
H,
ct −1, 2 =R(1 + βδ 2 )[ R 2 (1 − kt −1 ) + kt +1 (1 + n) 2 ]βδ 2 w1
H,
ct −1,3 =(1 + βδ 2 )H

ct −1,1 =

lt −1

H

[ R 2 (1 − kt −1 ) + kt +1 (1 + n) 2 ]δw1.
=
[ R(1 − kt ) + kt +1 (1 + n)](1 + βδ 2 ) w2

(12)
(13)
(14)

(15)

The generation t-1 chooses when to retire at age 2, which is period t. In the same period, they have8

to contribute to the social security system at rate of kt. And kt only has effects on the labor supply.
The consumption level at each age is merely independent of the contribute rate of social security atage 1 and 3, those are period t-1 and period t+1. Consistent with intuition, consumption level
decreases with the contribution rate which is in the period generation t-1 is at age 2, and increases
with the contribution rate which is in the period generation t-1 is not working. It is noticeable thatthe consumption level at each age is independent of the contribution rate at age 2, even thoughconsumers contribute to the social security system at age 2 when they are working.

4. A mandatory social security system
4.1 The adjusted generations
In the pay-as-you-go social security system, the consumption level and labor supply of generation
t-1 are influenced by contributing to social security at rate kt-1 and kt in each period and receive alump-sum pension benefit whose amount is independent of the contribution rate in period t+1. And it
is obtained that for generation t-1of quasi-hyperbolic discounters in the case without social security
system which is denoted as “NH”,

ct −1,1

lt −1

HN= w1 /(1 + βδ 2 ) ,(16)= Rδw1 /(1 + βδ 2 )w2 .

(17)

HN

Following O’Donoghue & Robin (1999), the exponential discounters’ consumption level andretirement age are considered as the optimal outcome from the long-run perspective, for their9preference is time consistent. And in this model, these are denoted as “E”,

c1 = w1 /(1 + δ 2 ) ,(18)l E = Rδw1 /(1 + δ 2 )w2 .

(19)

E

It is straightforward that the consumption level and retirement age of quasi-hyperbolic discounters
without social security system are higher than those of exponential discounters, since 0 < β < 1 .
These have been demonstrated as over-consumption and late-retirement of quasi-hyperbolic
discounters.
In this paper, social security system is employed to modify the consumption level and labor
supply of quasi-hyperbolic discounters, who are verified to over-consume or late-retire, to the level
under exponential discounting without social security:

H

E

(20)

lt −1 = l E .

(21)

ct −1,1 = c1andH

The adjustment effect of social security system for generation t-1 causes

kt = kt −1 R /(1 + n) − (1 − β )δ 2 R /(1 + δ 2 )(1 + n) ,(22)

kt +1 = kt R /(1 + n) .

(23)

Consider first-best optimal conditions that the consumption level and labor supply of every
generation are adjusted. And similarly for generation t,
10

kt +1 = kt R /(1 + n) − (1 − β )δ 2 R /(1 + δ 2 )(1 + n) ,(24)

kt +2 = kt +1 R /(1 + n) .

(25)

However, the first-best optimal conditions cannot be realized since they contradict to each other.
In this context, the consumption level and labor supply of quasi-hyperbolic discounters cannot be
adjusted to be the same level as exponential discounters for every generation.

Proposition 1 The consumption level and retirement age of all generations cannot be adjusted to the
optimal levels at the same time.
Consider second-best optimal conditions that social security system adjusts not every generation’s,
but every other one generation’s consumption level and labor supply, for generation t+1,

kt + 2 = kt +1 R /(1 + n) − (1 − β )δ 2 R /(1 + δ 2 )(1 + n) ,(26)

kt +3 = kt +2 R /((1 + n) .

(27)

By considering the conditions for generation t-1 and t+1 simultaneously, a dynamic path ofcontribution rate of social security system can be achieved. In this context, the consumption level
and retirement age of every other generation can be regulated.
Figure 1 to Figure 3 illustrate how contribute rate to social security in each period dynamicallyrelates to each other’s. Before moving to the analysis of figures, relative interest ratio

R /(1 + n) is

introduced, which is defined as the ratio of gross interest rate to the speed of population growth.

11Definition 1 The ratio of gross interest rate to the speed of population growth

R /(1 + n) is defined

as relative interest ratio.
Based on the simplified assumptions that the population of generation t-1 is unit and the
population growth rate is n, these figures illustrate the tendency of the path of contribution ratequalitatively rather than quantitatively.
When R>1+n, that is relative interest ratio is larger than 1, shown in Fig. 1, the contribute rate ofsocial security grows on a path of divergence. R>1+n implies that the gross interest rate exceeds the
speed of population growth. And in order to adjust every other generation’s behavior, thecontribution rate of social security has to grow on a path of divergence. However, there is a steadypoint on which every other generation’s behavior can be adjusted to the optimal level, and when1

R /(1 + n) > [(1 + δ 2 ) /(1 + βδ 2 )]2 , the steady point of contribution rate is between 0 and 1. This isa steady state of contribution rate when the population grows sufficiently slow, which implies that inthis steady state the social security system meets the pension benefit and adjust every othergeneration’s consumption level and retirement age.

12

kt(kt+1)

R>1+n

0

Kt(Kt+2)
Kt-1(Kt+1)

0
Kt-1(kt+1)

Fig. 1

The contribution rate of pension system: the case of R>1+n

Fig. 2 shows that when R<1+n, i.e. relative interest ratio is less than 1, the contribution rate ofsocial security converges. R<1+n implies that the speed of population growth exceeds the grossinterest rate. When population grows rapidly, benefitting from a large population in working age, thewhole population doesn’t have a growing burden of pension paid for the retired people. However,the steady point to which the contribution rate of social security converges is definitely negative,
which implies people receive from rather than contribute to the social security system when they are
working.

13

kt(kt+2)

R<1+n

0

Kt(Kt+2)
Kt-1(Kt+1)0
kt-1(kt+1)

Fig. 2The contribution rate of pension system: the case of R<1+nFig.3 shows that when R=1+n, i.e. relative interest ratio equals to 1, the contribution rate of social
security system proportionally decreases and there is no steady point. R=1+n implies that the grossinterest rate equals to the speed of population growth. In this context, people don’t have a growing
burden of contribution to social security system either.
Proposition 2 In pay-as-you-go pension system, in order to adjust the consumption and retirement
behavior of some generations to the optimal level, the path of contribution rate to this systemdepends on the relative interest rate R /(1 + n) .
Proposition 2 implies that the path of contribution rate to pension system depends on therelationship between the gross interest rate and the speed of population growth, i.e. the relative
14

interest rate. Intuitively, in a pay-as-you-go pension system, the pension benefits of the retired
generations are paid by the younger working generations. Therefore, it is strongly dependent on the
population structure.

4.2 The un-adjusted generations
The behavior of the generations who have been adjusted by the social security system has been
investigated; however, it is unknown what the behavior of un-adjusted generations is. The
consumption level and retirement age of the un-adjusted generations are discussed in thissub-section.
Definition 2 The generations whose consumption level and retirement age are adjusted to the
optimal levels by the mandatory pay-as-you-go pension system are referred as the adjusted
generations; the other generations whose behavior is not adjusted are referred as the un-adjusted
generations.

15

kt(kt+2)

R=1+n

0

Kt(Kt+2)
Kt-1(Kt+1)0
kt-1(kt+1)

Fig. 3The contribution rate of pension system: the case of R=1+n

For generation t, whose consumption level and labor supply aren’t adjusted,

ct ,1

H

H

lt =

[ R 2 (1 − k t ) + k t + 2 (1 + n) 2 ]w1,=
R 2 (1 + βδ 2 )(28)[ R 2 (1 − kt ) + kt +2 (1 + n) 2 ]δw1.
[ R(1 − kt +1 ) + kt +2 (1 + n)](1 + βδ 2 ) w2

(29)

It is has been assumed that every generation incorporates the social security system into thelife-time utility maximizing problem even though their behavior are not adjusted by the system.
Therefore, the social security system still has indirect effects on the un-adjusted generations. And it
is can be calculated that how the consumption level and retirement age are affected by the social

16

security system by making subtractions.
For the consumption level,w1δ 2 (1 − β )(1 + n)
< 0,
R(1 + βδ 2 )(1 + δ 2 )w δ 2 (1 − β )(R − 1 − n)HE.
ct ,1 − c1 = 1
R(1 + δ 2 )(1 + βδ 2 )H

ct ,1 − ct ,1

NH

=−

(30)
(31)

As the result of subtraction shows, the consumption level of un-adjusted generation t is definitely
lower than the case without social security system. In this context, the social security system does
correct the over-consumption of quasi-hyperbolic discounters to some degree even if it does not
adjust the consumption to the optimal level.
By comparing with the optimal consumption level, it can be found that whether the consumption
level is corrected depends on the relationship between gross interest rate and the speed of population
growth.
When R>1+n, ct ,1

HE> c1 , the generation t still over-consume. However, they still consume lessthan the case without social security system. When R<1+n,HE

ct ,1 < c1 , the generation t

under-consume. The social security system over-corrects the consumption level and results in that it
is under the proper limit. When R=1+n, ct ,1

HE= c1 , the generation t consume at an appropriate

level. In this case, the consumption of un-adjusted generation is corrected to the optimal level eventhough the social security level does not intend to do so.
Proposition 3 For the un-adjusted generations, they always consume less than the case without17

social security system; and when relative interest ratio R /(1 + n) > (<) 1, they consume more (less)than the optimal level.
For the retirement age,Hlt − lt

lt

H

NH

=

δ 3 w1 (1 − β )(R − 1 − n)],(1 + βδ 2 ) 2 w2

δw1[ R(1 + βδ 2 ) + ( R − 1 − n)(1 + δ 2 )].
−l =w2 (1 + βδ 2 )(1 + δ 2 )
E

(32)

(33)

The results of subtractions show that the correction extent of un-adjusted generation’s behavior
depends on the relationship between gross interest rate and the speed of population growth, as wellas the present bias parameter βand long-run discounter factorδ.
Compared with the case without social security system, when R=1+n it distinguishes between thatgeneration t within social security system retire earlier or later than those without it. And R=1+n
implies that social security system has no effects on the retirement age of quasi-hyperbolic
discounters within it, since they work for the same time as those without the system.
The comparison of retirement age between generation t within social security system andexponential discounters implies that whether

R /(1 + n) > 1/[1 + (1 + βδ 2 ) /(1 + δ 2 )] distinguishes

that whether generation t retire later than the exponential discounters do or not.
Proposition 4 For the un-adjusted generations, when R /(1 + n) > (<) 1/[1 + (1 + βδ 2 ) /(1 + δ 2 )],they retire later (earlier) than the optimal level; when R /(1 + n) > (<) 1, they retire later (earlier)18

than the case without social security system.

5. Discussions
5.1 The adjustment effects
Thus far the conditions distinguishing how social security system corrects quasi-hyperbolic
discounters’ consumption level and retirement age has been demonstrated separately. And acomprehensive analysis is made in this section.
As it has been shown, the behavior of generation t-i (i = ± (2k+1), k=0, 1, 2 … ) it adjusted to the
optimal level by the social security system. However, the extent to which the behavior of generationt-i (i= ± 2k, k=0, 1, 2 … ) is corrected depends on several conditions. The conditions are divided into
3 regions by the relationship of the gross interest rate and the speed of population growth. When
R>1+n, the contribution rate of social security system grows on a divergence path. And in thiscontext, the un-adjusted generations still consume more than the optimal level even though theirconsumption is corrected by the pension system to a lower level. Meanwhile, they have to retire later
than both the optimal level and those without the pension system. In the region of

1/[1 + (1 + βδ 2 ) /(1 + δ 2 )] < R /(1 + n) < 1, where the contribution rate converges, the un-adjusted
generation t-i (i= ± 2k, k=0, 1, 2 … ) consume even less than the optimal level. And at the same time,they retire later than the optimal level but earlier than those without pension system. When

1/[1 + (1 + βδ 2 ) /(1 + δ 2 )] > R /(1 + n) , the un-adjusted generation t-i (i= ± 2k, k=0, 1, 2 … ) still19

consumer at a low level, and in the mean time they retire earlier than both the optimal level andthose without the pension system.
Let’s notice 2 critical points where R=1+n and 1/[1 + (1 + βδ 2 ) /(1 + δ 2 )] = R /(1 + n) . When
R=1+n, the gross interest rate coincides with the speed of population growth, and the un-adjusted
generations consume at the optimal level. However, they retire later than the optimal level and thepension system has no effect on retirement age. When 1/[1 + (1 + βδ 2 ) /(1 + δ 2 )] = R /(1 + n) , theun-adjusted generations consume at a lower level than the optimal one, but retirement age is at the
optimal level. And in this context, they retire earlier than those without social security system. To
conclude, the consumption level and retirement age cannot be corrected to the optimal level at thesame time. The policy makers have to confront with a tradeoff between the optimal consumption
level and the optimal labor supply.
The critical value which distinguishes whether the un-adjusted generations retire at the optimal
age deals with the time preference parameters: the long-run discount factor δ	 and the present bias
parameter β. A smaller β which implies stronger present bias increases the region where theun-adjusted generations get retired earlier than the optimal level. In this context, the time preference
parameters do not only have effects on determining the labor supply level but also on the degree ofthe un-adjusted generations’ retirement age being corrected.

5.2 The welfare effects

20This sub-section discusses welfare consequences of each generation after the mandatory contribution
rate of social security system. From a long-run perspective, variations on life discounted utility after
compulsory pension contribution rate regime are investigated. The behavior of exponential
discounters is regarded as rational since their time-consistent preferences. And therefore compulsory
pension contribution regime is designed in the foregoing section to adjust the consumption andretirement age to the optimal level which is under exponential discounting. It has been ascertainedthat for some generations consumption and retirement age can be adjusted to the optimal level while
for other generations the optimal adjustment cannot be realized at the same time. And it is of interest
to investigate what effects does the compulsory pension contribution regime have on eachgeneration’s welfare.
Following O'Donoghue & Robin (2006), ax-ante welfare with exponential discounting isevaluated. Welfare is considered to be discounted utility from one’s point of view that is at the
beginning of each generation in a long-run perspective. For example, the welfare of generation t isdefined as

Vt = u1 + δ (u 2 − e) + δ 2 u3 .

(34)

By making subtraction, let’s compare welfare level between quasi-hyperbolic discounters withinmandatory pension contribution regime and those without it.
For generations whose behavior is adjusted to the optimal level, say generation t-i (i = ± (2k+1),21

k=0, 1, 2 … ), the welfare difference is

Vt −Hi − Vt −NHi = (1 + δ 2 ) ln[(1 + βδ 2 ) /(1 + δ 2 )] − δ 2 ln β (i = ± (2k+1), k=0, 1, 2 … ).
(35)
And

it

is

provablethat

Vt −Hi − Vt −NHi > 0 , ∀β ∈ (0,1) . This welfare improvement

straightforwardly holds since the optimal consumption level and retirement age are considered to be
those under exponential discounting which is represents the long-run perspective,
For generations whose behavior is not adjusted to the optimal level, i.e. generation t-i (i= ± 2k,k=0, 1, 2 … ), the welfare difference is2
Vt −Hi − Vt −NHi = (1 + δ ) ln[1 −δ 2 (1 − β )(1 + n)
] + δ ln[(1 + βδ 2 ) /(1 + δ 2 )]2R(1 + δ )(i= ± 2k, k=0, 1, 2 … ),(36)

which is definitely negative. This implies that the mandatory pension contribution regime cause
welfare deterioration to the un-adjusted generations.

Proposition 5 After employing the mandatory pay-as-you-go pension system, the welfare of theadjusted generations is improved; however, the welfare of the un-adjusted generations isdeteriorated.
So far it has been ascertained that under mandatory pension contribution regime proposed by thispaper for some generations the consumption and retirement age can be adjusted to the optimal level

22while other generations have to suffer from welfare deterioration caused by this regime even though
their behavior is adjusted to some extent. In the behavioral context, the policy maker has to confront
with a tradeoff between the optimal consumption level and the optimal labor supply. And in thewelfare context, the policy maker has to consider the fairness – which generations’ welfare to besacrificed.
However, it is still available to find out a mandatory pension regime to raise the welfare level ofeach generation. Consider a certain generation i of hyperbolic discounters, pension system causes
Pareto improvement of welfare implies

U iH − U iNH ≥ 0 .

(37)

A sufficient but not necessary condition of this improvement is

R 2 (1 − ki ) + ki +2 (1 + n) 2 ≥ R 2(38)

R(1 − ki +1 ) + ki +2 (1 + n) ≥ R ,(39)

ki + 2 ≥ [ R /(1 + n)] 2 ki

(40)

and k i + 2 ≥ [ R /(1 + n)]k i +1 .

(41)

and
which imply that

This sufficient but not necessary condition implies that in order to cause Pareto improvement thecontribution rate of pension should at least keep proportionally to ratio of gross interest rate over
population growth rate, which is referred as relative interest ratio.

23

When the relative interest ratio

R /(1 + n) is larger than 1, which implies that the gross interest

rate is higher than the population growth speed, the population grows at a relatively slow rate. In thiscontext, the boundary condition of leading a Pareto improvement on contribution rate to pension
system converging. While on the other hand, when the relative interest ratio is less than 1, theboundary condition is diverging. They both depend on the relative interest ratio.

6. Conclusions
This paper proposes a mandatory pay-as-you-go social security contribution regime in order toadjust the consumption and retirement age of quasi-hyperbolic discounters to the optimal level in thelong-run perspective which is under exponential discounting. Within this mandatory pension
contribution regime, this paper ascertains that behavior of all generations cannot be adjusted to the
optimal levels at the same time. With behavior adjusted to the optimal level, the adjusted generations’welfare is improved. However, the un-adjusted generations’ welfare is deteriorated.
There are some points of this paper that need to be further developed. This model is based on thespecific set up of a small open economy in which the gross interest rate is exogenously given. Whatis more, by incorporating the mandatory pay-as-you-go pension system, the adjusted and un-adjusted
generations are separated. With different welfare conditions, the fairness is out of the considerationof this paper.

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References
Blake. D. (2004) “The impact of wealth on consumption and retirement behaviour in the UK”,
Applied Financial Economics, 2004, 14, pp. 555–576
Frogneux, V. (2009) “Retirement Decision and Quasi-hyperbolic Discounting”, Research report of
FUNDP.
Laibson, D. (1997) “Golden eggs and hyperbolic discounting”, Quarterly Journal of Economics 112,
pp. 443–477.
O’Donoghue, T and M. Robin (1999) “Doing it now or later”, American Economic Review, Vol. 89,
No. 1 , pp. 103-124―― and ―― (2006) “Optimal Sin Tax”, Journal of Public Economics, 90 (2006) pp. 1825–1849.
Schwarz. M. E and E. Sheshinski (2007) “Quasi-hyperbolic discounting and social security systems”,
European Economic Review, 51 (2007) pp. 1247–1262

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