GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 292

Funded Pension Scheme and Time Preference in an
Endowment Economy

Lin ZhangJanuary 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Funded Pension Scheme and Time Preference in an
Endowment Economy

Lin Zhang†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka,
Osaka 560-0043, Japan
Abstract
This paper incorporates the three-period overlapping generations model with a funded pension
scheme and quasi-hyperbolic discounting in an endowment economy, it and discusses thebehavior of each generation under the general equilibrium condition. For saving instruments,
there is a substitution relation between the funded pension scheme and investments. Theequilibrium gross return rate on asset holdings decreases with a higher pension contribution rateand stronger present bias. The funded pension scheme deteriorates consumer welfare.

Keywords: Overlapping Generations Model, General Equilibrium, Funded Pension Scheme,
Quasi-Hyperbolic Discounting

JEL Classification: D91, E21, H55†Tel.: +81 06 6850 6111; fax: +81 06 6850 5205
Email address: kge017zl@mail2.econ.osaka-u.ac.jp
11.

Introduction
A mandatory pension contribution scheme provides a savings commitment device for people who

prefer immediate consumption to saving for retirement. In the case of a funded pension scheme,
people’s pension benefits depend on their contributions and the return rate. The aim of this paper is
to investigate the endogenous gross return rate under general equilibrium conditions, and examine
how it is affected by the pension contribution rate and the time preference parameter.
People with strong present bias saves at a low level (Laibson et al., 1998) and social security
system provide them with compulsory savings for retirement. There are a number of studies
discussing the relation between time preference, labor supply, and social security systems in a partial
equilibrium. For example, Schwarz and Sheshinski (2007) examine the effects of hyperbolic
discounting on the comparison of alternative social security systems. Diamond and Koszegi (2003)
show that quasi-hyperbolic-discounting consumers may retire either earlier or later than exponential
discounters, with different consumption patterns. Zhang (2012) finds that quasi-hyperbolic
discounting explains why under-savers might also be early retirers. Therefore, it is of interest to ask
how time preference affects consumer behavior through a funded pension system in a generalequilibrium.
The main contributions are as follows. First, this paper extends the model of Diamond andKoszegi (2003) by employing endogenous labor supply and incorporating mandatory funded pension
scheme. This mandatory pension scheme as a commitment device has effects on consumption level
and labor supply, and hence consumer welfare. Second, this paper derives a general equilibrium
model of overlapping generations in an endowment economy. Third, this paper incorporates
quasi-hyperbolic discounting (Laibson, 1997) to investigate the effect of time preference on theendogenous gross return rate. Fourth, the welfare effect of the funded pension scheme is discussed
under the general equilibrium condition.
2

The remainder of this paper is organized as follows. Section 2 introduces the model. Section 3
derives the general equilibrium condition and offers certain propositions. Section 4 concludes.

2.

The ModelConsider an economy populated by overlapping generations living for three periods (young,middle-aged, and old). They supply labor in-elastically when they are young and elastically when
they are middle-aged. Before retirement, they receive wages at a rate of w . In the last period of
their life, they do not work but exhaust all remaining resources. I assume zero population growth andthat the population of each generation is one. A funded pension scheme is employed in this economy.
It levies a pension contribution at a rate of k ∈ [0,1] when the consumer is working, and when she
is old, gives a pension benefit bt to the generation born in period t (hereafter “generation t ”).
The time horizon is infinite.
I assume that there is only one type of goods that are unstorable. Consumers can invest them in (orborrow them from) the bank and hold assets. Variable ct ,i (i = 1, 2, 3 here and hereafter) denotes
the consumption level of generation t in each period and st ,i denotes the net flows (a positive
one indicates outflows and a negative one indicates inflows) of assets. Variable lt denotes the labor
supply in her middle age. The inter-temporal budget constraint of generation t isct ,3 = Rt +1 Rt + 2 [ w(1 − k ) − ct ,1 ] + Rt + 2 [ w(1 − k )lt − ct , 2 ] + bt ,(1)where Rt denotes the gross return rate on asset holdings in period t . The government imposes abalanced funded pension scheme as follows:bt = Rt +1 Rt +2 kw + Rt +2 kwlt .
Consumer preference is given by3(2)3− jU t , j = u (ct , j ) + β ∑[δ τ u (ct , j +τ )] − ( βδ ) 2− j e(lt ),j = 1,2 ,(3)

τ =1where e(lt ) denotes the disutility of working in middle age. 0 < β ≤ 1 represents the short-run
discount factor, and 0 < δ ≤ 1 , the long-run discount factor.
The preference given by (3) includes two types of consumers: exponential discounters ( β = 1 )and quasi-hyperbolic discounters (0 < β < 1) . Under quasi-hyperbolic discounting, the short-run
discount factor

β captures the consumer’s strong impatience for immediate utility. Therefore, italso can be referred to as the present-bias parameter. The behavior of an exponential discounter isconsidered to be the optimal outcome from the long-run perspective, because her preference istime-consistent (O’Donoghue and Rabin, 1999).
Furthermore, I make functional form assumptions by stipulating thatu (ct , j ) = ln ct , j ,(4)and e(lt ) = ln lt .

(5)

Different responses follow different degrees of sophistication during dynamic decision making(O’Donoghue and Rabin, 1999). However, under logarithmic utility, the consumption level and labor
supply of the naive consumer coincides with that of the sophisticated one. Therefore, differentiating
between the two types of consumers is not necessary. The functional form assumption of logarithmic
utility enables us to focus on time preference parameters rather than the degree of sophistication.
Budget constraint (1), preference (2), and functional form assumptions (4) and (5), jointly
determine the consumption level and labor supply of generation t .

w

ct ,1 =,

(1 + βδ 2 )δwRt +1,ct , 2 =(1 + βδ 2 )ct ,3 =βδ 2 wRt +1 Rt +2,(1 + βδ 2 )
4(6)
(7)
(8)

and lt =3.

δRt +1.
(1 + βδ 2 )(9)General Equilibrium in the Endowment Economy

3.1 Equilibrium Gross Return Rate on Assets Holdings

I assume the consumers are homogeneous in time preference. In this endowment economy,
consumers have to invest their savings in the bank given the nature of goods (unstorable), and hence,
the gross return rate on asset holdings Rt is endogenously determined. The general equilibrium
condition in period t + 1 reads as

st −1,3 + st , 2 + st +1,1 = 0 .
Becausest +1,1 = w(1 − k ) − ct +1,1,st , 2 = ( Rt +1 − 1) st ,1 + w(1 − k )lt − ct , 2(10),andst −1,3 = ( Rt +1 − 1)[ Rt st −1,1 + w(1 − k )lt −1 − ct −1, 2 ] + bt −1 − ct −1,3 , substituting consumption level andlabor supply leads to the equilibrium gross return rate:βδ 2 − (1 + δ + βδ 2 )k.
Rt =δ (1 − k )(11)The endogenous gross return rate is time-invariant ( Rt = R ). Further, equation (11) implies that ifand only if 0 ≤ k <βδ 2, the gross return rate on assets is positive.
1 + δ + βδ 2Proposition 1 discusses how equilibrium in an endowment economy relates to the funded pension
scheme.

Proposition 1 In an endowment economy with an endogenous gross return rate on asset holdings,
equilibrium exists if and only if 0 ≤ k ≤βδ 2( R ≥ 0 ), and the gross return rate decreases
1 + βδ 25

with a higher pension contribution rate rises (dR< 0 ).
dk

Proposition 1 states that a sufficiently low pension contribution rate leads to a non-negative gross
return rate on asset holdings and equilibrium among generations in each period.
In this endowment economy, consumers cannot store their savings. They may save for retirement
by investing savings in a bank and receiving repayments, or making contributions to the fundedpension system and receiving pension benefits. In other words, there is a substitution relation
between investments and the funded pension scheme. Therefore, the gross return rate on asset
holding decreases with higher pension contribution rates.
When the gross return rate on assets is positive, each young-generation consumer invests hersavings in the bank and receives repayments in her middle-aged and old period. For each period, theamount invested by the young generation provides the repayments received by the middle-aged andold generations.
When the gross return rate on assets is zero, each generation invests savings when she is young
and receives repayments in her middle age. The pension benefit covers her consumption when she isold. The net flow of the old generation is zero and only the middle-aged generation gets repaid. In
these contexts, the equilibrium in this endowment economy maintains both intra- andinter-generational balances.
In the absence of a funded pension scheme, i.e., k = 0 , instrument in banks and their repayments
serve as the sole saving instrument. The old generations receive no pension benefit but to rely on theremaining assets. Therefore, the strong desire of young generation to save for old period pushes up
the gross return rate to a high level.
As the pension contribution rate increases, the funded pension system becomes another saving
instrument. Old generations receive not only repayments from the bank, but also pension benefits

6from the funded pension system. Therefore, the decreasing demand for bank investments among
young generations drags the gross return rate down.
In the extreme case of k =βδ 2, the gross return rate equals to zero. The pension
1 + δ + βδ 2contribution rate is so high that the pension benefits itself can cover the old generations’consumption, and the young generations are no compelled to save. As the pension contribution rateincreases beyondβδ 2, the equilibrium is broken.
1 + δ + βδ 2Notice that Proposition 1 implies that the equilibrium gross return rate is less than 1. In otherwords, young generations invest their savings in the bank even though they have to bear a loss intheir asset holdings.

3.2 Time Preference

Consumer preference given by (3) implies different time preferences: exponential discounting( β = 1 ) or quasi-hyperbolic discounting (0 < β < 1) . A lower present-bias parameterβdescribes stronger impatience prompting immediate consumption, and leads to a lower level ofsaving. Proposition 1 implies that the existence of equilibrium in this endowment economy is related
to time-preference parameters. Proposition 2 states the manner in which the gross return rate relates
to time preference.

Proposition 2 In an endowment economy with an endogenous gross return rate on asset holdings,
the gross return rate decreases with stronger present bias (i.e., a lowerβ ).

Stronger present bias leads to lower levels of saving, lower demand for investment by young
generations, and hence, a lower gross return rate on asset holdings. Furthermore, a lower
present-bias parameter implies that realizing equilibrium in the endowment economy is more7

difficult.

3.3 Welfare Effect of Funded Pension System

This sub-section investigates the welfare effect of a funded pension system. In this generalequilibrium model of an endowment economy, the gross return rate is endogenously determined,
while the equilibrium depends on the pension contribution rate. The funded pension system canadjust consumer behavior by controlling the pension contribution rate.
Here, ex-ante welfare is evaluated at the point of birth of the consumer. I evaluate the consumer’swell-being from a long-run perspective, namely under exponential discounting:Wt = u(ct ,1 ) + δu(ct , 2 ) + δ 2u(ct ,3 ) − δe(lt ) .

(12)

The government aims to optimize consumer welfare by selecting the optimal pension contribution
rate. This welfare effect is investigated by differentiating welfare with respect to the pension
contribution rate. Proposition 3 shows how the funded pension system affects consumer welfare.

Proposition 3 In an endowment economy with an endogenous gross return rate on asset holdings, afunded pension scheme is welfare-deteriorating when the gross return rate is positive.

The proof of Proposition 3 is in the Appendix.
The funded pension scheme affects welfare through the endogenous gross return rate. Recallingfrom Proposition 1, an increase in the pension contribution rate leads to a decrease in the gross returnrate. On one hand, a decrease in gross return rate implies lower consumption level in middle-aged
and old period because of lower return on assets. On the other hand, a decrease in the gross return
rate leads to lower pension benefits owing to lower labor supply in the middle age. Therefore, thelower gross return rate caused by higher pension contribution rate decreases the efficiency of savings

8investments by young generations and therefore, leaves consumers worse off.

4.

Concluding Remarks
By incorporating the overlapping generations model under quasi-hyperbolic discounting with afunded pension scheme, this paper shows that the equilibrium condition depends on the pension
contribution rate and time preference parameters. The endogenous gross return rate decreases with ahigher pension contribution rate and a lower present-bias parameter. Consumer welfare decreases as
the pension contribution rate increases.

Appendix

A. Proof of Proposition 3
Proof. I take the derivative of welfare with respect to the pension contribution rate k in order toinvestigate the welfare effect:

dcdc
dcdWt
dl
= u ' (ct ,1 ) t ,1 + δu ' (ct , 2 ) t , 2 + δ 2 u ' (ct ,3 ) t ,3 − δe' (lt ) t .
dkdk
dkdkdk(A.1)

Substituting first-order conditions, consumption level and labor supply leads to:dWt 2βδ 4 w
dR=u ' (ct ,3 ) R ,2
dk 1 + βδ
dkwhere(A.2)

dR< 0 because of Proposition 1. Therefore, for a positive gross return rate we have
dkdWt
< 0 . Thus, the funded pension scheme is welfare-deteriorating.
dk9

Acknowledgement

I am grateful to Shinsuke Ikeda for his guidance through my research. This research is supportedby Grant-in-Aid for JSPS Fellows. All errors remain mine.

10References

Diamond, P., Koszegi. B., 2003. Quasi-hyperbolic discounting and retirement. Journal of Public
Economics 87:1839–1872
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., Rabin, M., 1999. Doing it now or later. The American Economic Review, 89(1):103–124.
Schwarz, M. E., Sheshinski, E., 2007. Quasi-hyperbolic discounting and social security systems.
European Economic Review, 51: 1247–1262.
Zhang, L., 2012. Saving and retirement behavior under quasi-hyperbolic discounting. Journal ofEconomics, forthcoming, DOI: 10.1007/s00712-012-0302-8.

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