ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E22-1

Social Security, Economic Growth, and Social Welfare
in an Overlapping Generations Modelwith Aggregate and Idiosyncratic ShocksbyToshiki Tamai

March 2022

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Social Security, Economic Growth, and Social Welfare
in an Overlapping Generations Modelwith Aggregate and Idiosyncratic Shocks

Toshiki Tamai*

Abstract
This paper presents a study of the growth and welfare effects of four public pension
systems under aggregate and idiosyncratic shocks. The equilibrium growth rate obtained
under the pay-as-you-go pension system is lower than the growth rate achieved under
the funded pension systems because the unfunded pension system hinders capital
accumulation. However, pay-as-you-go with additional benefits for saving enhances
capital accumulation by incentivizing people to save. Particularly, the equilibrium
growth rate under the modified unfunded pension system exceeds that under the fundedpension system if the degree of relative risk aversion is sufficiently small. With regard
to social welfare, within the Rawlsian welfare function, pay-as-you-go without saving
credit is superior to the fully funded system if people are highly risk-averse. By contrast,
if they have low risk aversion, then pay-as-you-go with saving credit is preferable.
Considering the Benthamite welfare function, these results hold if the low-income
classes have thick population. This finding implies that the demographic structure ofincome classes is important to ascertain the optimal extent of social security.
Keywords: Public Pension; Risk; Economic Growth
JEL Classifications: H55; O41*

Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan. Tel: +81-52-789-2384.
E-mail address: tamai@soec.nagoya-u.ac.jp. This work was supported by JSPS KAKENHI Grant Numbers 18H00865, 20H01492,
and the Kampo Foundation.

1

1. Introduction
Economically developed countries provide their own social security programs to safeguard
incomes of elderly persons during old age in cases of insufficient savings. Social security
programs in most such countries are based on pay-as-you-go system, whereas a few countries
entirely or partly manage their social security systems by funded systems. 1 The retirement
benefits in pay-as-you-go systems are financed by contributions levied from people from current
working generations. Because they do not depend directly on pensioners’ paid contributions, payas-you-go systems have intergenerational redistribution effects. By contrast, a fully funded systemprovides retirement benefits that are perfectly related to the pensioners’ earnings and contributions.
Therefore such a system has no intergenerational redistribution effects.
Against this background, social security has been studied as a core issue in the fields of publicfinance, public economics, and macroeconomics.2 Specifically addressing matters of savings andold age income, earlier studies of the literature have demonstrated that social security impedes
capital accumulation and induces early retirement (e.g., Feldstein, 1974, 1977; Kotlikoff, 1979).
Findings show that a fully funded system is preferable over a pay-as-you-go system in a
dynamically efficient economy, with the former being superior to the latter when considering
economic growth. However, the importance of social security is based on the presence of
uncertainty because it affects sharing risks and the optimality of allocation under various shocks
and improves social welfare (e.g., Enders and Lapan, 1982; Gordon and Varian, 1988; Thøgersen,
1998; Demange and Laroque, 1999; Wagener, 2004; Gottardi and Kubler, 2011).3
Recent studies of the literature describing social security treat issues of idiosyncratic shocks
and the heterogeneity of individuals to elucidate the redistributive effects of social security (e.g.,
Conesa and Krueger 1999; Harenberg and Ludwig, 2015, 2019; Bagchi, 2019).4 Idiosyncratic
shocks affect personal earnings and generate income differences. Therefore, it is necessary toanalyze effects of social security not only on intergenerational inequality but also on
intragenerational inequality to consider idiosyncratic shocks. In particular, one study related toours is an elaborate research developed by Harenberg and Ludwig (2015). They show that payas-you-go social security can give partial insurance against idiosyncratic and aggregate risks when1

For example, the pension system in Singapore, Central Provident Fund (CPF), is a social security system funded both by employers’
and employees’ contributions. Superannuation in Australia and the premium pension in Sweden are funded parts of the earningsrelated pension, although these two countries have pay-as-you-go pension systems which work as the intragenerational redistribution
devices.
2
Issues related to social security have been discussed widely, for example, intergenerational risk sharing (e.g., Smith, 1982; Bohn,
2001; Gollier, 2008), adverse selection (e.g., Abel, 1986), and optimal social security (e.g., Samuelson, 1975; Sheshinski and Weiss,
1981).
3
Hauenschild (2002) developed a general equilibrium model with stochastic production and social security to examine the existence,uniqueness, and stability of stochastic equilibrium.
4
Bagchi (2019) specifically examines the presence of differential mortality. Regarding this, Kelly (2021) demonstrates that theassumption of mortality homogeneity biases the equilibrium growth rate and welfare analysis. Harenberg and Ludwig (2015, 2019)
treat the aggregate and idiosyncratic earning shocks.

2markets are incomplete. Their studies clarify insurance against risks and crowding-out of capital
because of distortionary tax as key determinants of the welfare effects of social security.
In reality, social security programs are specifically operated depending on each country’s
economic circumstances, even though most are based on pay-as-you-go systems. For instance,some countries provide fringe benefits to encourage saving for retirement (e.g., tax deductions,credits, and allowances).5 The Savings Credit of social security in the United Kingdom (UK) wasan extra payment for people who had saved up money for retirement. These facts naturally castsome questions on the negative effects of social security on capital accumulation if it is operated
as a pay-as-you-go system with additive benefits to induce retirement saving.
Furthermore, two views of public pensions exist in relation to its redistributive effects within
pay-as-you-go systems: Beveridgean and Bismarckian schemes.6 The former has a weak link
between individuals’ contributions and pension benefits, which exhibits large intragenerational
redistribution (e.g., Australia, Ireland, Netherlands, and UK). By contrast, the latter shows lessintergenerational redistribution because the individuals’ contributions are linked tightly to theirretirement benefits (e.g., France, Germany, and Italy). Using an overlapping generations modelwith three income classes of low, medium, and high income individuals, Conde-Ruiz and Profeta(2007) examined which scheme is chosen under majority voting. 7 Without uncertainty, they
showed that low-income individuals prefer a small Beveridgean system, whereas middle-income
individuals favor a large Bismarckian system. High-income individuals wish for a fully fundedsystem. Therefore, depending on the density of income classes, either scheme could be chosen
politically.
Integrating findings obtained from earlier studies, one finds that the analysis of the effects ofsocial security on economic growth and intragenerational and intergenerational redistribution isfundamentally important to evaluate welfare effects of social security under aggregate andidiosyncratic shocks.8 To address this, we develop a simple overlapping generations model withsocial security under the economic-wide productivity and labor supply shocks. We consider social
security programs of four types: The fully funded, modified funded, pay-as-you-go (unfunded),and modified unfunded systems, which are related to existing social security programs. For
instance, the modified unfunded system in our model is a pay-as-you-go system with Saving
Credit in UK, which are fringe benefits that are anticipated as stimulating incentives to savemoney for old-age consumption. The modified funded system is an intragenerational risk-sharing
5

Savings Credit is one type of fringe benefit to attract people to save more. According to the report by European Commission’s highlevel group of experts on pensions, some countries (e.g., Germany, Croatia, and Italy) apply tax exemption and incentives to encourage
personal savings. The tax deduction functions similarly to saving credit here.
6
Disney (2004) provides details of Beveridgean and Bismarckian schemes.
7
Numerous studies have addressed this issue (e.g., Casamatta et al., 2000; Cremer and Pestieau, 2000; Cremer et al., 2007; Glasso
and Profeta, 2007).
8
Empirical studies have elucidated a significant relation between economic growth and shocks by postulating risks (e.g., Kormendi
and Meguire, 1985; Grier and Tullock, 1989; Ramey and Ramey, 1995; Furceri and Karras, 2007; Imbs, 2007, Alouini and Hubert,2019). Emphasizing income risks by productivity shocks is important for the examining social security effects.

3system: the social security fund manages the pooled revenues of social security taxes and pays
pension benefits from the pooled funds. Therefore, it differs from personal savings. It is fully
funded based on personal pension accounts.
First, we demonstrate that the equilibrium growth rate under a modified unfunded system
might be larger than that under a fully funded system, depending on the relative risk aversion. A
savings credit gives people an incentive to save more. With low relative risk aversion, people
might prefer saving because it makes their future returns become large. They also benefit fromlarge returns with risks. For that reason, a modified unfunded pension is superior in terms ofgrowth-enhancement. However, with high relative risk aversion, people wish to avoid risky
behavior. The growth rate under the modified unfunded system becomes the second-lowest after
pay-as-you-go.
Second, we derive the welfare effects of different pension systems. Under the Rawlsian
welfare function, the social welfare level under the fully funded pension system is the lowest
because it provides no benefit for the poorest people. By contrast, the modified funded system
will improve social welfare, although its economic growth rate does not differ from that achieved
under the fully funded system. Furthermore, a fully compulsory pension managed by the modified
funded system is expected to have the highest welfare level because the poorest people obtainconsumption opportunities without risks. The pay-as-you-go system with optimal interior social
security tax rate generates a higher welfare level than a fully funded system. Furthermore, payas-you-go is superior to modified unfunded with fringe benefits at the welfare level, except forsmall relative risk aversion.
Third, numerical analyses confirm the theoretical findings and quantitative implications. The
results provide illustrative examples of theoretical findings. Particularly if people are highly riskaverse, they will choose pay-as-you-go irrespective of the type of social welfare function anddistribution of labor endowments. Under the Benthamite welfare function, the distribution oflabor endowments affects the optimal social security tax under pay-as-you-go. The thickpopulation of the low-income class tends to increase the optimal tax rate.
The remainder of this paper is organized as follows. Section 2 presents a description of thebasic setup of our model. Section 3 examines equilibrium properties of economies with the fundedsystem and also provides equilibrium analyses of economies with the unfunded system. Section
4 investigates the relation between risk, economic growth, and social welfare under different
pension systems. Furthermore, it provides a discussion about policy implications of our study
based on the related literature by particularly addressing the viewpoints of the welfare states.
Finally, Section 6 concludes the paper.

4

2. The model
Consider a closed economy with a homogeneous good. The economy is in discrete time, the timeis indexed by subscript 𝑡. firms exist as a continuum and the production technology is assumed
to be the specification presented by Turnovsky (2000) and Kenc (2004), which includes external
effects of capital on labor productivity.9 The production function of each firm is
𝑦! = 𝑋! 𝑘! ,(1)

where 𝑦! denotes the firm output at period-t, 𝑋! stands for firm productivity at period t, and 𝑘!
represents capital input at period t. We assume that the capital is fully depreciated during one
period.
Total factor productivity 𝑋! is a probabilistic variable with probabilistic density function
𝑓(𝑥! ) ; it is independent and identically distributed over time. Each firm faces different
productivity shocks even though they are generated by identical stochastic processes. Therefore,
investors also face idiosyncratic shocks if security is insufficient to cover the risks. That is, weconsider the incompleteness of asset markets, as analyzed by Harenberg and Ludwig (2015). On
the other hand, the total factor productivity shocks also affect macroeconomic activity.
Based on the production function (1), we do not treat labor input explicitly for analytical
simplicity, although labor is also an input in the production process. As shown in Turnovsky(2000) and Kenc (2004), the workers receive rewards for labor depending on the relative
contribution to the average worker. Assuming capital and labor shares are constant over time, we
have
𝑟! 𝑘! = 𝛼𝑦! ,𝑤! = (1 − 𝛼)𝑦! ,where 𝑟! denotes the interest rate, 𝑤! signifies the wage payment of each firm, and 𝛼
represents the capital distribution rate ( 0 < 𝛼 < 1 ). Inserting Equation (1) into the above
equations yields
1 + 𝑟! = 𝛼𝑋! ,(2)

𝑤! = (1 − 𝛼)𝑋! 𝑘! .

(3)

Individuals live two periods: young and old. We consider a stationary population, similarly toHarenberg and Ludwig (2015). Hence, the population of each generation is normalized to unity.
During the young period, the generation born at period t supply 𝑍! unit of labor, where 𝑍!

9

They assume a positive externality of aggregate capital developed by Romer (1986). Bruce and Turnovsky (2013) examine therelation between social security, economic growth, and welfare under lifetime uncertainty using a continuous-time overlapping
generations model with the production function similar to ours. By assuming that the production externality arises from the interaction
between the aggregate capital-labor ratio, 𝑦! = 𝑋! 𝑘!" (𝐾! /𝐿! )#$" 𝑙!#$" , where 𝐾! is the aggregate capital stock, 𝑙! is the firm’s labor
input, and 𝐿! is the aggregate labor input. As it is seen later, the predetermined stock level of capital has no uncertainty. In equilibrium,
firm’s capital input level coincides with mean of the capital-labor latio. This provides the microeconomic foundation, leading toEquations (1), (2), and (3).

5

represents a probabilistic variable with probabilistic density function 𝑔(𝑧! ) that is independently
and identically distributed over time. The young generation receives rewards for labor, depending,
on ℎ! , which is ℎ! ≡ 𝑍! /𝑍!̅ , and 𝑍!̅ is defined as 𝑍!̅ ≡ ∫ 𝑧! 𝑔(𝑧! )𝑑𝑧! . Therefore, ℎ! can be-interpreted as worker’s productivity, which is also a probabilistic variable following 𝑔(𝑧! ). Labor
income 𝑤! ℎ! is allocated to purchasing private goods, paying social security tax, and savings.
In the old period, individuals retire and live on savings and public pension benefits. Then, thebudget equations for period-t generation in the two periods are.

.
where 𝑐!

𝑐! + 𝑠! + 𝜏! = 𝑤! ℎ! ,

(4)

1
𝑐!/0
= (1 + 𝑟!/0 )𝑠! + 𝑏!/0 ,(5)

denotes private consumption in the young period, 𝑠! represents saving, 𝜏! stands for1
the social security tax, 𝑐!/0
expresses private consumption in the old period, and 𝑏!/0 stands

for the pension benefit.
The lifetime utility function for the period t generation is
023.
1 )023C𝑐 D−11 (𝑐!/0−1
𝑈! = !
+for 𝜃 > 0, 𝜃 ≠ 1,
1−𝜃
1+𝜌
1−𝜃
.

𝑈! = log 𝑐! +1
1
log 𝑐!/0for 𝜃 = 1,
1+𝜌where 𝑈! represents the utility level, 𝜌 is the discount rate (𝜌 > 0), and 𝜃 denotes the relativerisk aversion (𝜃 > 0). Each individual chooses the level of saving to maximize the expected
lifetime utility subject to Equations (4) and (5). With pension system (𝜏! ≥ 0 and 𝑏!/0 ≥ 0), thefirst-order condition becomes𝑑𝐸[𝑈! ]
1= −(𝑤! ℎ! − 𝑠! − 𝜏! )23 +𝐸P𝑅!/0 (𝑅!/0 𝑠! + 𝑏!/0 )23 R = 0,
𝑑𝑠!
1+𝜌

(6)

where 𝑅!/0 ≡ 1 + 𝑟!/0 . Depending on the risks, the social security taxes, and the retirement
benefits, Equation (6) gives the individual saving function.
The pension system is operated publicly. We consider pension systems of four types: fully
funded (FF), modified funded (MF), pay-as-you-go (PG), and modified unfunded (MU) systems
that represent the existing pension schemes. Regardless of the operation methods, the aggregate
tax revenue and retirement benefits in period t are
𝑇! = T 𝜏! 𝑓(𝑥! )𝑔(𝑧! )𝑑𝑥! 𝑑𝑧! ,(7)

𝐵! = T 𝑏! 𝑓(𝑥! )𝑔(𝑧! )𝑑𝑥! 𝑑𝑧! .

(8)

To consider the capital market equilibrium condition, one must set up details of the pension
system. Depending on the pension systems, the budget of public pension is varied: For fundedpension systems (FF and MF), the social security tax revenue in period 𝑡 must be equal to the6aggregate retirement benefits in the next period. Because we have 𝑇! = 𝐵!/0 for FF and MF,
under the funded pension system, the capital market equilibrium condition becomes
𝐾!/0 = 𝑆! + 𝑇! = T 𝑠! 𝑓(𝑥! )𝑔(𝑧! )𝑑𝑥! 𝑑𝑧! + T 𝜏! 𝑓(𝑥! )𝑔(𝑧! )𝑑𝑥! 𝑑𝑧! ,(9)where 𝑆! denotes the aggregate saving.
On the other hand, unfunded pension systems based on a pay-as-you-go principle require thatthe tax revenue in period 𝑡 is equal to the aggregate retirement benefits in period 𝑡. Therefore,
𝑇! = 𝐵! holds. The capital market equilibrium condition under the unfunded pension system is
𝐾!/0 = 𝑆! = T 𝑠! 𝑓(𝑥! )𝑔(𝑧! )𝑑𝑥! 𝑑𝑧! .

(10)

Going forward, we assume that 𝑋! and 𝑍! respectively follow lognormal distributions, suchthat𝑓(𝑥! ) =𝑔(𝑧! ) =1√2𝜋𝜎4 𝑥!
1√2𝜋𝜎6 𝑧!

exp _−(log 𝑥! − 𝜇4 )5a,
2𝜎45

exp _−(log 𝑧! − 𝜇6 )5a.
2𝜎65Lognormality of shocks is used widely in the literature on risk.10
Finally, the labor market equilibrium condition requires that the demand for labor is equal to
the labor supply. Consequently, we have,

𝐿! = c 𝑧! 𝑔(𝑧! ) 𝑑𝑧! = exp _𝜇6 +-𝜎65
a = 𝑍̅.
2

In those equations, 𝑍̅ is the aggregate labor supply, which is equal to the mean of labor supply
by the normalization of population. Capital and labor market equilibrium conditions derive theresource constraint of this economy. With production technology (1) and (9) or (10), the aggregate
economic variables of 𝐾! , 𝑌! , 𝐵! , and 𝑆! grow at the identical (equilibrium) growth rate.
Hereinafter, we specifically examine the case in which the equilibrium growth rate is positive.

3. Social security programs
This section explains details of four pension systems and characterizes the equilibrium with social
security programs. In particular, Subsections 3.1 and 3.2 specifically examine funded pension
systems. Subsections 3.3 and 3.4 present examination of unfunded pension systems.

10

Harenberg and Ludwig (2015) considered two aggregate shocks such that a productivity shock a shock to the unit user costs ofcapital and one idiosyncratic shock that is related to old-age wage. They also assume that the shocks follow a stochastic processes
with log-normal distribution.

7

3.1. Fully funded pension
Fully funded pensions can have two patterns of expected returns for individuals: risky and riskfree. Given that private investment (saving) involves a risky return on investment, individuals will
choose different options with regard to risks.
Fully funded pension with announcement of the expected returns (as a benchmark: BM). In
Singapore, the Central Provident Funds (CPF) provide a publicly operated fully funded pension.
The CPF ensures a rate of return of at least 2.5% for insured persons. The point of this type is thatthe fund announces the expected returns of the pension. The pension is risk free for insured
persons, although there is heterogeneity of their earnings.
Formally, if the government releases the expected returns of public pension as 𝑅e!/0 , then thebudget of the public pension must satisfy
𝑏!/0 = 𝑅e!/0 𝜏! .
Accounting for the portfolio selection between private saving and public pension, Equations (4)–(6) yield 𝑠! = 0 and the following equation (see Appendix A):
𝜏! =1
𝑤 ℎ ≡ 𝜉89 𝑤! ℎ! ,1 + 𝛽𝐴7 ! !

(11)

where
2!

𝛼023 "
𝜃−1𝜎45
𝛽≡_a ,𝜂 ≡, log 𝐴 ≡ 𝜇4 + .
1+𝜌𝜃2
Using Equations (9) and (11) with 𝑠! = 0, the equilibrium growth rate under fully funded pension
with risk-free return is𝛾89 ≡(1 − 𝛼)𝐴− 1.
1 + 𝛽𝐴7(12)

Fully funded pension with announcement of the distribution of the expected returns (as a usual
fully funded pension: FF). Superannuation in Australia is one kind of compulsory savings. It isbased on defined contribution pension system. The defined contribution pensions such as
Superannuation, 401K, and others are equivalent to personal savings if the risks are not coveredby the pension programs. One can suppose that the government announces that the return of the
public pension depends on the productivity 𝑋! with the probabilistic density 𝑓(𝑥! ). Here, thepublic pension does not differ from private savings because no disparity exists among the returns:
𝑏!/0 = 𝑅!/0 𝜏! .
If individuals can optimally choose their private and public (pension) savings, then from
Equations (4)–(6), the total saving function is obtained as (see Appendix A)
𝑠! + 𝜏! =1
𝑤 ℎ ≡ 𝜉:: 𝑤! ℎ! ,1 + 𝛽𝐴7 𝜆023 ! !

8

(13)

wherelog 𝜆 =

𝜎45.
2Equations (9) and (13) lead to the equilibrium growth rate under a fully funded pension:𝛾:: ≡(1 − 𝛼)𝐴− 1.
1 + 𝛽𝐴7 𝜆023

(14)

Suppose that 𝐴 is sufficiently large to ensure a positive rate of economic growth.11
The relation between growth rates under fully funded pension. Comparison between Equations(12) and (14) leads to𝛾:: ⋛ 𝛾89 ⇔ 𝜃 ⋛ 1.
When 𝜃 = 1, no difference exists between risky and risk-free returns. The saving rates are
identical (i.e., 𝜉89 = 𝜉:: ) because the income and substitution effects of the interest rate changeon the youth consumption exactly offset. Alternatively, the income (substitution) effect dominates023
the substitution (income) effect if 𝜃 > 1 (𝜃 < 1). Given that 𝐸P𝑅!/0
R takes a smaller (larger)
023
value than 𝑅e!/0
for 𝜃 > 1 (𝜃 < 1), 𝜉:: is larger/(smaller) than 𝜉89 : the total effect of theinterest rate change on youth consumption under risky returns is smaller (larger) than that underrisk-free returns when 𝜃 > 1 (𝜃 < 1). Therefore, the growth rate under the FF-regime is higher(lower) than that under the BM regime if 𝜃 > 1 (𝜃 < 1).

3.2. Modified funded pension
In any case, the fully funded pension systems presented in the preceding subsection ensure
actuarial fairness, but the equilibrium growth rates differ depending on the relative risk aversion.
However, the government might wish to use a funded pension to share economic risks during thesame generations. In other words, one can consider the modified funded pension (MF), by which
individuals pay the social security tax amount depending on their income and receive the same
benefits from the reserve. Therefore, even when such a pension is fundamentally funded, no direct
link exists between individuals’ contribution and their retirement benefits. Then, we have𝑏!/0 = 𝑅!/0 𝜙𝑤
o! ℎe! ,where 𝜙 denotes the social security tax rate on income under the MF regime (0 ≤ 𝜙 ≤ 1),,ℎe! = 𝑍̅ 20 c 𝑧! 𝑔(𝑧! )𝑑𝑧! = 1,-,𝑤
o! = (1 − 𝛼)𝑘e! c 𝑥! 𝑓(𝑥! )𝑑𝑥! .
-11This assumption covers all cases of economic growth rates under the three pension systems considered herein.

9

The idea of the MF system could be found in the Japanese modified funded pension system.
In Japan, the public pension program started as a fully funded system. However, facing inflation
after the World War II, it was amended to a system similar to MF in the sense that no direct link
between individuals’ contribution and retirement benefits has existed since 1948.12 With partly
defined benefits, the pension system had intragenerational redistribution effects. By facing theshortage of a reserve for public pension, in reality, Japanese pension programs operated on a payas-you-go principle at present after changing the program repeatedly. However, it is worthwhile
to investigate its intergenerational redistribution effects as another benchmark to fully fundedsystems.
;
Individuals predict the pension benefit in the next period as 𝑏!/0
= 𝛼𝜙𝑋!/0 𝑤
o! because theaverage value of labor income and productivity depends on the stochastic process. Using;
Equations (4)–(6), 𝑏!/0
, and 𝑤
o ! , the savings function under the MF pension becomes (Appendix

A)
𝑠! =

(1 − 𝜙)𝑤! ℎ! − 𝛽𝐴7 𝜆023 𝜙𝑤o!
1 + 𝛽𝐴7 𝜆023.

The individual saving function is decreasing in both the social security tax rate and benefits.
This result is based on Equations (4)–(6). The increased tax rate decreases the disposable income
and increases the marginal utility of the young period for the given saving. To maintain equalityof marginal utilities, the individual reduces saving. For the given saving, the increased pension
benefit reduces the marginal utility of the old period consumption. To equalize the marginalutilities of the young and old periods, savings must be decreased with the pension benefit. The
saving effect of the increased tax rate is explained as the combined effects of the increased tax
rate and benefit.
The average saving function under the MF pension is
1 − C1 + 𝛽𝐴7 𝜆023 D𝜙𝑠̅! = qr𝑤
o ! ≡ 𝜉9: 𝑤
o! .
1 + 𝛽𝐴7 𝜆023

(15)

The intuition of Equation (15) is fundamentally identical to that of the individual saving function.
Using Equations (9) and (15) with the pension budget, we obtain the economic growth rate underthe modified pension system as
𝛾9: ≡(1 − 𝛼)𝐴− 1.
1 + 𝛽𝐴7 𝜆023

Equation (16) is identical to Equation (14). Thereby, we obtain the following.
Lemma 1. 𝛾:: = 𝛾9: .

12

Yoshihara and Hata (2016) provide a historical review of the Japanese public pension system.

10

(16)

At the aggregate level, the MF pension does not differ from fully funded pension because thegovernment appropriates the social security tax revenue for investment in the capital market andthe reserve for paying pension benefits. However, the MF pension has redistributive effects within
the same generation at an individual level. From the viewpoint of social security types, fully
funded systems such as BM and FF correspond to the Bismarckian scheme, whereas the MU
system corresponds to a Beveridgean scheme.

3.3. Pay-as-you-go pension
Based on the pay-as-you-go principle (PG), the social security tax and benefits in period t must
be balanced within the existing generations. Then, we have𝑇! = 𝐵! .
In many countries, a public pension is aimed at ensuring a certain level of pension replacement
rate. Then we assume that it is fixed over time. Consequently, the pension benefit is𝑏!/0 = 𝜓𝑤o!/0 ℎe!/0 ,where 𝜓 denotes the fixed replacement rate (0 < 𝜓 < 1),,ℎe!/0 = 𝑍̅ 20 c 𝑧! 𝑔(𝑧! )𝑑𝑧! = 1,-,𝑤
o!/0 = (1 − 𝛼)𝑘e!/0 c 𝑥!/0 𝑓(𝑥!/0 )𝑑𝑥!/0 .
-As stated by Disney (2004), public pension programs are classified to Beveridgean andBismarckian schemes. With the fixed replacement rate, the pay-as-you-go pension represents asignificant departure from actuarial fairness. The pay-as-you-go pension system described in thissection corresponds to the Beveridgean scheme, similar to those found in Australia, Ireland, andthe UK. Alternatively, we can consider the Bismarckian scheme by which the individuals’
contributions link to their retirement benefits. To examine the intragenerational redistribution
effects of public pension, we specifically examine Beveridgean schemes such as the UK pension
system.
With PG, the capital market equilibrium condition is (10). In the average term, we have;𝑘e!/0 = 𝑠̅! . Individuals anticipate the future pension benefit as 𝑏!/0
= (1 − 𝛼)𝜓𝑋!/0 𝑘e!/0
because they regard the average capital stock as given and the productivity generated from the;stochastic process. Then, the saving function under PG is derived from Equations (4)–(6), 𝑏!/0,
and 𝑘e!/0 = 𝑠̅! (Appendix B):

𝑠! =(1 − 𝜓)𝑤! ℎ! − 𝛽𝜒𝜓𝐴7 𝜆023 𝑠̅!
,
1 + 𝛽𝐴7 𝜆023

11

where1−𝛼𝜒≡uv.
𝛼The economic meaning of the saving function under PG is identical to the saving function undera MF pension. Given that the replacement rate is fixed over time, a rise in the replacement rate
has the combined effect of the increased tax rate and benefit. This result is parallel to the negative
effect of public pension on capital accumulation in the deterministic and stochastic models (e.g.,
Feldstein, 1974; Hauenschild, 2002; Hillebrand, 2012). In our model, sustainable growth isgenerated by capital accumulation. Decreasing capital accumulation will negatively affecteconomic growth (e.g., Yakita, 2001). To see this, we consider aggregate capital accumulation
through aggregate saving.
Using the individual saving function, the average saving function under unfunded pension is(Appendix B)
𝑠̅! = w1−𝜓x𝑤
o ≡ 𝜉<= 𝑤o! .
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023 !

(17)

The PG social security program has two negative effects on saving through the distortionary effectof income tax and saving adverse effect of retirement benefits. Particularly, the latter effect isaffected by the risks and relative risk aversion. The effect of the interest factor under risks appears
in the deflator of Equation (17) as well as Equation (14), although it is strengthen by the presenceof the saving adverse effect in case of PG. Furthermore, larger 𝜃 engenders the strong saving
adverse effect. The relative risk aversion determines the sensitivity of economic growth change
in response to risks. The average saving rate decreases with the fixed replacement rate:𝜕𝜉<=
1 + (1 + 𝜒)𝛽𝐴7 𝜆023=−
< 0.
[1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023 ]5𝜕𝜓The economic intuition follows the result obtained for individual saving.
Using Equations (10) and (17), we obtain the economic growth rate under PG, such that𝛾<= ≡(1 − 𝜓)(1 − 𝛼)𝐴− 1.
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023

(18)

Partial derivation of Equation (18) with respect to 𝜙 yields(1 − 𝛼)P𝐴 + (1 + 𝜒)𝛽𝐴0/7 𝜆023 R
𝜕𝛾<=
=−
< 0.
[1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023 ]5𝜕𝜓Because the capital accumulation depends on saving, the replacement rate influences the
economic growth rate through saving. As demonstrated, an increase in the replacement rate has anegative effect on saving because of the distortionary effect of income tax and saving adverse
effect of retirement benefits. Therefore, the economic growth rate is negatively associated withthe replacement rate. Alongside this result and Lemma 1, comparing Equations (13) and (18), one

12obtains
Lemma 2. 𝛾<= ≤ 𝛾:: for 𝜓 ≥ 0.
Irrespective of the value of 𝜃, the economic growth rate under PG pension is dominated bythat under the fully funded or MF pensions. When 𝜓 = 0, 𝛾<= = 𝛾:: = 𝛾9: holds. In principle,
the pension benefit decreases private saving. Furthermore, the pension benefits of PG derive fromthe social security tax of the next generation, which is not used in the capital market. The growthrate under PG is decreasing in the replacement rate. This result is the same as that of the model
without risk. However, the risks quantitatively affect savings and economic growth.

3.4. Pay-as-you-go pension with additive benefits to stimulate saving incentives
Earlier, we considered the PG pension designed in a conventional manner. As demonstrated inearlier studies and ours, PG discourages saving for retirement. Some countries have adopted
fringe benefits to encourage saving for retirement (e.g., tax deduction, credit, and allowance). For
instance, Savings Credit of the public pension in the United Kingdom was an extra payment forpeople who had saved up money for retirement. To address such a pension system in reality, we
consider the modified unfunded pension (MU), which gives individuals incentive for saving.
Given that savings credit is positively associated with private saving relative to average saving,
the pension benefit is formalized as𝑠!
𝑏!/0 = w𝜋 + (1 − 𝜋) x 𝜓𝑤
o !/0 ℎe!/0 ,𝑠̅!
where (1 − 𝜋) denotes the parameter related to savings credit ( 0 < 𝜋 < 1 ). This can beinterpreted as one way to introduce actuarial fairness. When 𝜋 = 1, the modified pay-as-you-go
pension coincides with the standard pay-as-you-go system follows the Beveridgean manner. In
contrast, the modified pay-as-you-go system corresponds to unfunded but fund-like pension withthe returns as the economic growth rate if 𝜋 = 0 . Therefore, MU such as 𝜋 = 0 is one
representative of the Bismarckian scheme.13 The pension budget is the same as 𝑇! = 𝐵! at theaggregate level.
Equations (4)–(6) engender the following (Appendix B):
𝑠! =(1 − 𝜓)𝑤! ℎ! − 𝜔𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝑠̅!
,
1 + (1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023

13

From this viewpoint, 𝜋 can be interpreted as the Bismarckian factor, which is formulated by Conde-Ruiz and Profeta (2007).
However, our specification is more specified to the existing pension program.

13!

where 𝜔 ≡ (1 + (1 − 𝜋)𝜒𝜓)2" . 0 < 𝜔 < 1 holds. The average saving function becomes(Appendix B)
1−𝜓𝑠̅! = wx𝑤
o ≡ 𝜉9? 𝑤
o! .
1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 !

(19)

By the same manner of interpreting Equation (19), the MU system has two negative effects on
saving: the distortionary effect of income tax and saving adverse effect of retirement benefits. The
most attractive point is that the saving credit affects the saving adverse effect of retirement
benefits; it will weaken the saving adverse effect. Using Equations (10) and (19), the economic
growth rate under MU pension system is𝛾9? ≡(1 − 𝜓)(1 − 𝛼)𝐴− 1.
1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023

(20)

We now characterize the effects of the unfunded pension with saving-induced fringe benefits
on economic growth rate. Partial differentiation of Equation (20) regarding 𝜓 provides
𝜕𝜔
P1 + (1 + 𝜒)𝜔𝛽𝐴7 𝜆023 R + (1 − 𝜓)(1 + 𝜒𝜓)𝛽𝐴7 𝜆023 𝜕𝜓
∂𝛾9?
(1 − 𝛼)𝐴,=−[1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 ]5
∂𝜓
where
!

(1 + (1 − 𝜋)𝜒𝜓)2"#! (1 − 𝜋)𝜒
𝜕𝜔
=−
< 0.
𝜕𝜓
𝜃
Unfunded pension with saving-induced fringe benefits might enhance economic growth because
the growth effect of a rise in 𝜓 through a change in 𝜔 is positive.
When 𝜓 = 0, the marginal growth effect is(1 − 𝜋)𝜒𝛽𝐴7 𝜆023
P1 + (1 + 𝜒)𝛽𝐴7 𝜆023 R −∂𝛾9?
𝜃(1 − 𝛼)𝐴.
|=−(1 + 𝛽𝐴7 𝜆023 )5∂𝜓 @AFor a small 𝜃, a rise in 𝜓 increases the growth rate at 𝜓 = 0. Contrarily, the effect of 𝜓 on
growth rate at 𝜓 = 1 is negative, as
∂𝛾9?
1
|
=−
< 0.
∂𝜓 @A01 + (1 + 𝜒)𝜔𝛽𝐴7 𝜆023
Therefore, the social security tax and economic growth rate have a hump-shaped (monotonicallydecreasing) relation if 𝜃 is sufficiently small (large). The reported values of 𝜃 by empiricalstudies are varied over the range of 0.2–10. The most widely accepted value of 𝜃 would be
between 1 and 3 (Gandelman and Hernandez-Murillo, 2015). Gandelman and Hernandez-Murillo(2015) found that some economically developed countries have values smaller than 0.5 (e.g.,

14

Ireland, Japan, Korea, and the Netherlands).14 Sufficiently small values of 𝜃 would be plausible.
We next consider the effects of a change in 𝜋 on economic growth. The partial derivative of
Equation (20) regarding 𝜋 yields(1 − 𝜓)(1 − 𝛼)𝜒𝜓𝛽𝐴0/7 𝜆023 𝜕𝜔
∂𝛾9?
=−
< 0.
[1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 ]5 𝜕𝜋∂𝜋The equation above demonstrates that small (large) saving credits engender a low (high)economic growth rate.
Equations (13) and (20) show that the properties of equilibrium growth rate, and Lemmas 1
and 2 provide the following (Appendix C provides the proof):
Proposition 1. (i) When 𝜃 is sufficiently small, there exists 𝜓} such that 0 < 𝜓} < 1 and𝛾9? = 𝛾:: . Then, 𝛾<= < 𝛾:: < 𝛾9? if 𝜓 < 𝜓} while 𝛾<= < 𝛾9? ≤ 𝛾:: if 𝜓 ≥ 𝜓}. (ii) When
𝜃 is sufficiently large, 𝛾<= < 𝛾9? < 𝛾:: holds for 𝜓 > 0.
Small 𝜃 denotes that the consumptions between youth and old age are more substitutable.
The income shocks have less effect on consumption-saving choice than large q. Given that othereconomic conditions are unchanged, saving under small 𝜃 is less than under large 𝜃. Therefore,
the growth rate under fully funded pension will be larger than under PG because the latter induces
people to consume more. For small 𝜃, unfunded pension with saving-induced fringe benefits
might accelerate economic growth by stimulating saving compared to a fully funded pension. For
small saving under small 𝜃 , strengthening saving incentive will positively affect economic
growth. The MU pension system increases the economic growth rate over the level under full
funding within the appropriate values of 𝜓.

4. Growth and welfare effects of risks through public pension
A tradeoff between economic growth and social justice is a central issue of public economics. If
no transitional dynamics or no heterogeneity of economic agents with the production function (1)exists, then the maximizing growth rate can be consistent with attaining social justice. However,this will not be true with income shocks. To consider the tradeoff between economic growth andsocial justice, this section presents examination of the growth and welfare effects of income risks
through different public pension systems.

14

The values of 𝜃 are 0.35 in Ireland, 0.44 in Japan, 0.27 in Korea, and 0.10 in the Netherlands. 𝜃 equals 2 is rejected at the 10
percent level for all countries and 𝜃 = 1 is also rejected at the 10 percent level for Korea.

15

4.1. Risks and economic growth
Following conventional methods, risks should be measured as variances. Parameters related torisks are 𝜇D and 𝜎D (𝑖 = 𝑥, 𝑧). Both 𝜇D and 𝜎D affect the mean and variance of 𝑋! or 𝑍! . The
distribution of labor ability only influences individual saving and does not aggregate saving. The
variance 𝑉[𝑋! ] is zero if 𝜎4 = 0. Herein, we specifically examine the comparative statics of 𝜎4 .
Taking a logarithmic differentiation of Equations (13) and (15) for 𝜎4 , the elasticities of thegrowth factors in the funded pension systems for 𝜎4 become𝜖:: ≡(𝜃 − 1)5𝜎4 𝜕 log(1 + 𝛾:: )𝛽𝐴7 𝜆023= q1 −_ar 𝜎45 ,𝜕𝜎4𝜃
1 + 𝛽𝐴7 𝜆023

𝜖9: ≡𝜎4 𝜕 log(1 + 𝛾9: )= 𝜖:: .
𝜕𝜎4

Similarly, the logarithmic differentiations of Equations (18) and (20) for 𝜎4 lead to𝜖<= ≡(𝜃 − 1)5 (1 + 𝜒𝜓)𝛽𝐴7 𝜆023
𝜎4 𝜕 log(1 + 𝛾<= )= •1 −
qr‚ 𝜎45 ,𝜕𝜎4𝜃
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023

𝜖9? ≡(𝜃 − 1)5 (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023
𝜎4 𝜕 log(1 + 𝛾9? )= •1 −
qr‚ 𝜎45 .
𝜕𝜎4𝜃
1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023

We now consider the marginal effects of increased risk on the growth effects from 𝜎4 .
Comparison between the elasticities results in the following.
Proposition 2. (i) (1 + 𝜒𝜓)𝜔 > 1 ⇔ 𝜖9? < 𝜖<= < 𝜖:: , while (ii) (1 + 𝜒𝜓)𝜔 < 1 ⇔ 𝜖<= <𝜖:: < 𝜖9? .
A rise in 𝜎4 increases not only the average productivity but also the income risks. The overall
effect on economic growth is varied, depending on the schemes and preference parameters.
Specifically examining 𝜎4 = 0, we have
(1 + 𝜒𝜓)𝜔 ⋛ 1 ⇔ 𝜃 ⋛log(1 + (1 − 𝜋)𝜒𝜓)≡ 𝜃},
log(1 + 𝜒𝜓)

where 𝜃} < 1. For example, we consider 𝜃 > 𝜃}. We obtain 𝜖9? < 𝜖<= < 𝜖:: from Proposition
2. Presuming that 𝜖:: < 0 holds, then |𝜖9? | > |𝜖<= | > |𝜖:: | > 0 because of 𝜖9? < 𝜖<= <𝜖:: < 0. The unfunded pension with saving credits exhibits the highest sensitivity of economic
growth to increased risk. However, if we assume 𝜖9? > 0 and |𝜖:: | > |𝜖<= | > |𝜖9? | > 0
holds, then the fully funded pension shows the highest sensitivity of economic growth to increased
risk. The relation between risk and economic growth can be elucidated using numericalsimulations.

16

4.2. Optimal pension system
Using Equations (4), (5), and (14), the indirect utility function under a fully funded pension withrisky returns is023

𝑈!:: =

C(1 − 𝜓:: )𝑤! ℎ! D
1−𝜃

−1

+

1 𝐸P(𝑅!/0 𝜓:: 𝑤! ℎ! )023 R − 1
.
1+𝜌
1−𝜃

Similarly to deriving 𝑊!:: , we obtain the indirect utility functions under MF pension such that023[(1 − 𝜙)𝑤! ℎ! + 𝜙𝑤
o! ]𝛽𝐴7 𝜆023…
†
1 + 𝛽𝐴7 𝜆023
𝑈!9: =1−𝜃

−1023(1 − 𝜙)𝑤! ℎ! + 𝜙𝑤o!
𝐸 qu𝑅!/0 v r − 1
1 + 𝛽𝐴7 𝜆023
1
+
,
1+𝜌
1−𝜃
023

(1 − 𝜓)𝛽𝐴7 𝜆023 𝑤! ℎ! + 𝛽𝜒𝜓𝐴7 𝜆023 𝑠̅!
wx
1 + 𝛽𝜆023 𝐴7
𝑈!<= =
1−𝜃

−1
023

1
+
1+𝜌

(1 − 𝜓)𝑤! ℎ! − 𝛽𝜒𝜓𝐴7 𝜆023 𝑠̅!
𝐸 ‡u
𝑅!/0 + 𝜓𝑤
o!/0 ℎe!/0 v
1 + 𝛽𝐴7 𝜆023

ˆ−1
,

1−𝜃
023

[(1 − 𝜓)𝑤! ℎ! ](1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023 + 𝜔𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝑠̅!
…
†
1 + (1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023
𝑈!9? =
1−𝜃

−1

023
𝑠!
e!/0 Œ x − 1
(1
𝐸
w‰𝑅
𝑠
+
Š𝜋
+
−
𝜋)
‹
𝜓𝑤
o
ℎ
!/0
!
!/0
1
𝑠̅!
+
.
1+𝜌
1−𝜃

We assume that the social welfare function takes the form of the Rawlsian welfare function.
Formally, the social welfare function under pension scheme i (𝑖 = 𝐹𝐹, 𝑀𝐹, 𝑃𝐺, 𝑀𝑈) is defined as,

1

𝑊!D ≡ ‘ u
v
1+𝛿
EA!

E

min 𝑈ED ,where 𝛿 denotes the social discount rate. 𝛾D < 𝛿 (𝑖 = 𝐹𝐹, 𝑀𝐹, 𝑃𝐺, 𝑀𝑈) is necessary to ensurethat the social welfare is bounded. Other welfare criteria can be presumed, such as the Benthamite
welfare function:,
1 E
D
𝑉! ≡ ‘ uv T 𝑈ED 𝑓(𝑥E )𝑔(𝑧E )𝑑𝑥E 𝑑𝑧E .
1+𝛿
EA!

However, deriving analytical results other than the Rawlsian welfare function is complicated.
Therefore, we specifically examine the Rawlsian welfare function, although the Benthamite social

17

welfare function will be analyzed numerically.
The welfare effect on 𝑊!9: consists of effects on indirect utility and economic growth. As
shown earlier, a rise in 𝜙 has no effect on economic growth. Hence, the welfare effect of achange in 𝜙 is
sgn _

𝜕𝑊!9:
𝜕 min 𝑈!9:a = sgn _
a
𝜕𝜙
𝜕𝜙
023

023

𝛽𝐴7 𝜆023 𝑤o!
= 𝜙 23 —_a
1 + 𝛽𝐴7 𝜆023

+

023
1
𝑅0/!
𝑤
o!
𝐸 ‡_
a
1+𝜌
1 + 𝛽𝐴7 𝜆023

ˆ˜ > 0.

The optimal social security tax rate on income under the MF regime is equal to its upper limit.
When 𝜙 = 0, 𝑊!9: = 𝑊!:: holds. We obtain 𝑊!9: ≥ 𝑊!:: for 𝜙 ≥ 0. Within the fundedpension systems, these results lead to the following.
Proposition 3. For 𝜙 > 0, 𝑊-9: is larger than 𝑊-:: . The optimal social security tax rate of
MF pension is 𝜙 ∗ = 1.
We turn to the analysis of the welfare effects of the unfunded pension systems. Within theunfunded pension systems, the social security tax rates affect the economic growth rates.
Therefore, the increased pension replacement rate influences social welfare through short-termand long-term effects: the former involves intergenerational redistribution and tax burden effects;
the latter is based on the negative growth effects of public pension. As mentioned in the previous
sections, the relative risk aversion affects the negative growth effect because saving adverse effectof retirement benefits are weakened or strengthened depending on 𝜃. Therefore, the welfare
effects of the unfunded social security programs and the optimal system are varies on 𝜃 .
Considering these effects, we obtain the following proposition (Appendix D provides the proof):
Proposition 4. There exists an interior optimal social security tax rate of PG pension, 𝜓 ∗ . If 𝜃
is sufficiently small, there might exist an interior optimal saving credit rate 𝜋 ∗ . Contrarily, PG is
optimal if 𝜃 is sufficiently large. In either case, 𝑊-9? is equal to or greater than 𝑊-<= for the
optimal saving credit rate.
People with no income do not benefit from economic growth at present because they cannot
consume and save. They are willing to receive pension benefits. Then, governments must careabout the negative growth effects of PG because future pension benefits depend on the next young
generations’ income levels (ultimately economic growth rate). Therefore, the social security tax
rate is optimally selected as an interior solution irrespective of the parameter values. However,18

the optimal tax rate of social security depends on 𝜃, 𝜎4 , and so on.
With regard to the optimal saving credit rate, a government might choose either an optimal
interior credit rate or no saving credit (i.e., PG pension system). If 𝜋 = 0, then it means no
pension for the poorest people. Therefore, the government has an incentive to increase 𝜋 anddecrease saving credit because the current effect on the poorest people includes the redistributive
effect of PG if 𝜋 > 0. For 𝜋 > 0, the poorest person’s utility depends on the economic growthrate. The governments aim not to decrease the economic growth rate too much and might keep ithigh to set 𝜋 < 1. This is true if 𝜃 is sufficiently low. However, if 𝜃 is sufficiently large, thenno saving credit (i.e., 𝜋 = 1) is preferable because high growth and risks with large saving credit
negatively affect utility level.

4.3. Numerical analysis
We set the parameters as 𝛼 = 0.3 and 𝜌 = 0.5.15 Following the estimated result presented
by Gandelman and Hernandez-Murillo (2015), the key parameter 𝜃 is considered at threedifferent values: 0.5 , 1 , and 1.5 . 16 Furthermore, 𝜇6 = 0 , and 𝜎6 = (0.5, 1, 3, 5, 10) .
Microeconomic data for firms and households are necessary to identify parameters related to
idiosyncratic productivity shocks. However, our specified model requires specified data and mustovercome a lack of the data. This numerical analysis aims at providing certain examples to show
theoretical predictions in the preceding sections. To avoid difficulties of data treatment, we
specifically examine the case in which idiosyncratic shock parameters correspond to
macroeconomic shocks parameters one to one. Therefore, the economy consists of representative
economic agents with identical probability density functions. Although incompleteness of ourestimation exists, it is apparently sufficient for giving the parameters which generate realistic
equilibrium values.
The parameters related to productivity shocks should be calculated using observed economic
variables. Using Equations (1) and (16), the relation between productivity and observableeconomic variables satisfies
𝑋!/0 =

𝑌! 𝑌!/0
Growth factor
×
=
.
𝑆!
𝑌!
Investment-to-output ratio

Using the US data of GDP and investment from 1947Q1 to 2020Q4, we calculated the value of

15

The capital share lies between 0.3 and 0.5. Therefore, the value around 𝛼 = 0.3 is used frequently for numerical analysis. For
instance, Imrohoroglu et al. (1998) set 𝛼 = 0.36 to calibrate the effects of individual retirement accounts on capital accumulation.
Based on estimation of Gandelman and Hernandez-Murillo (2015), the density is quite low, out of the range of 0.5–1.5. Therefore,
we specifically examine these three values as representatives.

16

19

𝑋 from 1947Q1 through 2020Q4.17
The distribution parameters during all data periods are estimated as (𝜇4 , 𝜎4 ) = (2.025,0.187).
The Anderson–Darling statistic is 3.199 (p-value: 0.022). Consequently, the estimated parameteris statistically not significant. This non-significance might derive from large shocks and structuralchanges in the distribution parameters. We divide the time periods into four: first Period (1950Q1–
1975Q4), Second Period (1976Q1–1997Q4), Third Period (1998Q1–2008Q4), and Fourth Period(2009Q1–2020Q4). Estimation of the First Period yields (𝜇4 , 𝜎4 ) = (2.200,0.092), where theAnderson–Darling statistic is 0.282 (p-value: 0.951). These parameter values are significant at
the 5% level, whereas no other period was found to be significant at the 5% level.18 Based on ourestimated results, we set the productivity shock parameters as 𝜇4 = 2 and 𝜎4 =(0, 0.25, 0.5, 0.75, 1).
Economic growth, public pension, and risks. Actually, 𝜓 = 0.2 and 𝜋 = 0.9 are assumed
for calculating the equilibrium growth rate under the PG and MU systems. Tables 1 and 2 present
the equilibrium growth rates. In Table 1, a comparison between benchmark and fully funded/MF
gives the numerical example of the results from Section 3. Table 2 demonstrates that the growthrate under MU is larger than that under PG for each case (Proposition 1). Comparison between
the results Tables 1 and 2 indicates that a fully funded/MF pension generates the highest growthrate. Therefore, within plausible values of parameters, fully funded/MF is the best way to boost
economic growth.
The MU system stimulates economic growth through saving credit, but its effect is too weak
to overweigh that of the funded system within the value of 0.5 ≤ 𝜃 ≤ 1.5 for 𝛼 = 0.3. However,when 𝛼 = 0.2 and 𝜃 = 0.5, the growth rate under MU is larger than that under FF/MF for 0 <𝜓 < 0.292. For small value of 𝛼, the MU system generates economic growth rate higher thanthe FF/MF system within the plausible rage of the relative risk aversion. The recent trend for
declining labor share may weaken the advantage of MU to FF/MF regarding economic growth.
Social welfare, public pension, and risks. With regard to economic growth, funded pension
systems are superior to unfunded pension systems. However, under the Rawlsian welfare function,
the fully funded pension system brings about the lowest welfare level for the poorest workers
because no income derives no saving for consumption in retirement. Therefore, we should
quantitatively verify the order of welfare levels and the optimal social security. The labor supply
shocks do not affect the optimal tax rates under the Rawlsian welfare function.
We set 𝛿 = 1 as the social discount rate. Table 3 demonstrates that the optimal social security
tax rate is equal to unity under the MF system because the poorest workers benefit from a public

17

U.S. Bureau of Economic Analysis, Real Gross Domestic Product [GDPC1] and Real Gross Private Domestic Investment[GPDIC1], retrieved from FRED, Federal Reserve Bank of St. Louis.
The estimated parameter values are (2.029, 0.079) for the 2nd Period, (1.793, 0.039) for the 3rd, and (1.819, 0.123) for the 4th. The
Anderson–Darling statistics are 0.589 (p = 0.658) for the 2nd Period, 0.520 (p = 0.726) for the 3rd, and 2.289 (p = 0.064) for the 4th.

18

20

pension with no tax burden (Proposition 3). By contrast, PG outcomes indicate that an optimal
interior social security tax rate exists. For 𝜃 = 0.5, the tax rate range of 4% and 15%. For 𝜃 = 1
and 𝜃 = 1.5, the tax rates are around 20%. Observation of PG outcomes indicates that larger 𝜎4engenders a smaller tax rate. An increase in 𝜎4 raises the expected productivity and its volatility.
It increases the interest and economic growth rates. These effects diminish the need for raisingsocial security tax rate by increasing pension benefits.
In Table 3, MU can be superior to PG for 𝜃 = 0.5, although PG is preferable (i.e., 𝜋 ∗ = 1)
for 𝜃 = 1 and 𝜃 = 1.5. When 𝜃 = 0.5, the optimal percentage of saving credit is 78.9% –86.3% for the domain of 𝜎4 (Proposition 4). Therefore, the unfunded system with saving credit
is theoretically justified in certain cases. For small 𝜃, high growth and interest rates by increased
risks improve welfare. The poorest households seek to benefit from increasing interest andeconomic growth rates. Therefore, the saving credit is optimally selected.
Regarding the welfare level, Table 3 demonstrates that MF generates the highest welfare level.
When 𝜃 = 1 and 𝜃 = 1.5, no saving credit (𝜋 = 1) is selected; PG is in second place. However,if 𝜃 = 0.5, then the order is reversely changed. The MU system is a better choice by the intuitionof Proposition 4.
Social welfare, public pension, and population density of income class. Based on computations,any class of specified welfare function gives optimal social security tax rate and its welfare level.
To verify the robustness of our guess related to the optimal social security tax rate, we consider
the Benthamite welfare function 𝑉!D . Parameter 𝜎4 is fixed at 𝜎4 = 0.5, whereas 𝜎6 is variedin 𝜎6 = (0.5, 1, 3, 5, 10). The density of each income class determines the optimal social security
tax rate. Small 𝜎6 indicates that the income distribution is not widely spread, whereas large 𝜎6
indicates that the low-income people have a thick population.
Table 4 presents the calibrated results. The outcomes under the Benthamite welfare function
exhibit similarities to those obtained under the Rawlsian welfare function: social security tax rates
and the orders of welfare levels. However, when 𝜃 = 0.5 and 𝜎6 = 0.5, welfare under the MU
system is greater than that under MF. The table demonstrates that the optimal social security tax
rate is increasing in 𝜎6 . High density of low-income class engenders high tax rate. Furthermore,high relative risk aversion tends to increase social security tax rate similarly to that in Table 3.

4.4. Discussion
This subsection provides discussion of the policy implications of our study for existing WelfareStates and further analyses for endogenous fertility.
Policy implication for Existing Welfare States. Our theoretical and numerical analyses reveal

21

that an unfunded pension is preferred to a fully funded pension if the government cares aboutpoor people. Rich people naturally prefer to buy private annuities, which are identical to a fully
funded pension. However, for various political reasons, the governments in the real world must
devote attention to low-income people. Therefore, the government operates the public pension ona pay-as-you-go principle.
Our results also indicate that the modified funded pension system has some advantages over a
pay-as-you-go system and fully funded pension systems in response to risks. Although providing
a higher welfare level than others, this system loses the link between contributions and retirement
benefits. In reality, it might be difficult to obtain public agreement for the modified funded system.
Consequently, we conclude that the unfunded pension system might be a better choice in an
economy with risks, which explains why unfunded pension systems are mainstream to provide
social security programs.
Within unfunded pension systems lies the issue of intragenerational redistribution. As
described earlier, real social security programs are classified as Beveridgean and Bismarckian
pension systems. Using the political economic approach, some studies have tackled issues ofwhich of them is politically chosen (e.g., Casamatta et al., 2000; Cremer and Pestieau, 2000;
Conde-Ruiz and Profeta, 2007; Cremer et al., 2007; Glasso and Profeta, 2007). They
demonstrated that the Beveridgean system is politically supported by low-income people,
although Bismarckian systems are favored by high-income people, and demonstrated that which
system arises depending on the retirement timing, demographics, and so on.19
In our study, qualitative and quantitative analyses demonstrate that the Beveridgean systemsuch as PG is preferred if people tend to avoid risks strongly, whereas the Bismarckian systemsuch as MU is favored when people tend to weakly avoid risks. In fact, this result is similar tothat described earlier in the literature on Beveridgean and Bismarckian pension systems because
the relative risk aversion can be interpreted as the relative inequality aversion. However, ournumerical results imply a difference of welfare implications in response to risks of different types.
The degree of saving credit, 𝜋, decreases with the variance of labor productivity directly
related to earning, 𝜎6 ; a large variance of the worker’s productivity engenders the Beveridgean.
Furthermore, whether a small or large pension system is better depends on the labor-related risk
and social welfare function. For given the firm’s productivity, large inequality (i.e., large value of𝜎6 ) meets a large pay-as-you-go pension. However, different patterns are visible for the firms’
productivity shocks. A large variance of the firm’s productivity, 𝜎4 , engenders small pay-as-yougo pensions. Because the firm’s productivity is linked to aggregate productivity, which determines
the equilibrium economic growth rate, a large value of 𝜎4 generates a high growth rate with high19

Kaganovich and Zilcha (2012) examined the fiscal sustainability of social security including the public education funding. They
demonstrate that that the fully funded social security system generates political support for a higher education funding and therefore
a higher economic growth rate than the pay-as-you-go system.

22

risks. It requires reduction of a distortionary effect of a tax. The optimal social security tax rate
decreases with 𝜎4 .
These results imply that the desirable pension systems are faced with different levels of risk
and relative risk aversion. In reality, almost all economically developed countries have adopted
pay-as-you-go pension systems between Beveridgean and Bismarckian schemes. Based on our
analyses, if people are strongly risk or inequality averse, the society tends to prefer large
intragenerational redistribution within pay-as-you-go pension systems. The Bismarckian factor
averages in 1988–2008 are 0.05 for Australia, 0563 for Germany, 0.341 for Ireland, 0.307 for the
Netherlands, and 0.127 for UK (Rivera-Rozo et al., 2018), whereas the corresponding values of
the relative risk aversion are 1.17 for Australia, 0.77 for Germany, 0.35 for Ireland, 0.1 forNetherland, and 1.03 for United Kingdom (Gandelman and Hernandez-Murillo, 2015). Thesetendencies support our result that large 𝜃 engenders large 𝜋. Therefore, our model gives aplausible explanation about the existing Welfare States.20
Further analyses. As described in this paper, we treat labor supply shocks to consider
heterogeneity of workers’ earnings. Because the population of the economy is stationary, there isno demographic change. However, if the variance of exogenous labor supply is changed, then theaggregate labor supply is also changed. Such a shock can be interpreted as a shock for the working
population. Therefore, our results are applicable to explain the effects of exogenous change inworking populations on equilibrium outcomes through social security programs.
Demographic change can be generated by changes in economic circumstances such as income.
Therefore, the demographic shocks are not independent of labor supply and are one determinantof fertility. Considering endogenous determination of the numbers of children, certain studies
show that social security systems affect fertility rates (e.g., Cigno, 1993; Zhang, 1995; van Grozen
et al., 2003; Sinn, 2004). Although this issue is fundamentally important to analyze thedemographic shocks in relation to fertility and mortality rates, we can guess the feedback effects
of social security on fertility using the theoretical findings presented in earlier studies of theliterature.
Numerous researchers have examined the relation between population aging, social security,and economic growth using overlapping generations model with endogenous fertility (e.g.,
Pecchenino and Pollard, 1997; Yakita, 2001; Hirazawa et al., 2010; Ono, 2017), which enables us
to analyze population aging and its economic effect on (or through) social security. However,separate from fertility choice, demographic shocks have not been specifically examined in
endogenous growth models. The different demography in each generation causes asymmetriclabor supply and intragenerational and intergenerational income inequality. Furthermore,
20

Rivera-Rozo et al. (2018) found that the Bismarckian factor is affected by cultural factors (e.g., individualism) as same as economic
factors. Therefore, our model cannot all the cases of the existing Welfare States. For example, the Bismarckian factor and relative risk
aversion in United States are 0.489 and 1.39 respectively. To explain this, we must incorporate additional factors into our basic model.

23

productivity shocks are significant with regard to economic growth. Our theoretical findings fill
the gap in the research.
When the number of children is formulated as one of consumption goods and normal goods,
a decrease in income decreases the number of children. With large relative risk aversion, an
increase in the variance of labor productivity will decrease the number of children. It negatively
affects the revenue of social security taxes under pay-as-you-go pension systems and therefore
might decrease social welfare. In contrast, the firms’ productivity risks will have different effects
on fertility. An increase in variance of the firms’ productivity increases the economy-wide
productivity and therefore raises average wages and interest rates with large volatilities of them.
Depending on the relative risk aversion, these increases in wages, interest rates, and theirvolatilities affect the expected utility in the second period. However, the basic mechanism
revealed by our study will provide an analytical basis for the extensions of endogenous fertility.

5. Conclusion
This paper presented an examination of the relation between public pensions, economic growth,and social welfare in an overlapping generations model with income shocks. We considered four
pension systems for comparison: FF, MF, PG, and MU. Particularly, MF and MU arecharacteristics of this study. The former provides a risk-pooling function, which involves income
redistribution effects within the same generations, whereas the latter gives people incentives forsaving by fringe benefit.
First, we analyzed the relation between economic growth and public pension. The equilibrium
growth rate under the fully funded system coincides with that under modified pensions. Actually,
PG generates the lowest growth. The results obtained for the fully funded and PG regime
complement those of earlier studies. The main contribution of this study is the role of fringe
benefits in providing saving incentives, indicating that the equilibrium growth rate under MU
might be higher than that under full funding, depending on the relative risk aversion. Because
private saving behavior is affected by risks and fringe benefit, they can stimulate saving andeconomic growth.
Next, we investigated the relation between social welfare and public pension. Specificallyexamining the Rawlsian welfare function, the social welfare level under fully funded pension is
the lowest. Fully funded does not differ from private pension. The poorest people lacking income
can neither save for retirement nor consume any goods. The MF system will improve social
welfare as compared with fully funded, even though their economic growth rates coincide.
Furthermore, a full compulsory pension by the MF system provides the highest welfare level24within the MF system because the poorest people obtain consumption opportunities without risks.
PG has its optimal social security tax rate. Its welfare level is always higher than that under full
funding and is superior to MU with fringe benefits.
We also conducted quantitative analysis with plausible values of parameters to elucidate thequalitative results and to provide realistic examples. Results demonstrate that the growth rate
under fully funded is the highest. With highly relative risk aversion, the welfare under PG
overweighs those under MU (i.e., PG system with saving credit) and fully funded, irrespective ofthe types of social welfare functions and distribution of labor endowments. Furthermore, using
the Benthamite welfare function, the optimal social security tax under PG increases with the
population of low-income class. These results elucidate the effects of social security on economic
growth and welfare through productivity and demographic shocks. The calculated tax rates
complement those in the study by De Menil et al. (2016), although the setting differs. This
complementary nature implies that the demographic structure (especially income class) strongly
influences optimal social security tax rates.
Lastly, we consider the future directions for research in this area. The Rawlsian social welfare
function provides a clear view of the redistributive policy. However, other criteria must be
examined. In relation to the optimal social security, democratic determination of social security
policy will provide new insights. Furthermore, incorporating endogenous fertility into the model
proves to be interesting. Fundamentally, these extensions represent more realistic economic
situations. Through our analyses, we can infer some conclusions of extensions. For example,
social needs for redistribution will be weakened by the substitution of a political determination ofsocial security tax rate for the Rawlsian welfare function. Such an extension can be expected todecrease the optimal social security tax rate and to increase economic growth rate. Therefore, ourstudy provides an analytical basis of these extensive analyses.

25Appendix
A. Saving functions under the funded pension systems
Equation (11). If the fully funded pension has risk-free return, then individuals prefer to purchase
only the public pension. The first-order condition is(𝑤! − 𝑠̅! )23 =

1
𝛼023
𝜎65
023 23
𝐸P𝑅e!/0R𝑠̅! ⇔ (𝑤! − 𝜏! )23 =exp Ÿ(1 − 𝜃) _𝜇6 + a 𝜏!23 .
1+𝜌
1+𝜌
2

Solving this equation for saving yields
𝑠! =

1
1−𝜃
𝜎5
1 + 𝛽 exp _− ‰ 𝜃 Œ u𝜇6 + 26 va

𝑤! .

Using definitions for 𝛽, 𝜂, and 𝐴, we obtain Equation (10).
Equation (13). Inserting Equations (4) and (5) into the expected utility function yields
𝐸[𝑈! ] =(𝑤! ℎ! − 𝑠! − 𝜏! )023 − 1
1 𝐸P(𝑅!/0 𝑠! + 𝑏!/0 )023 R − 1
+
.
1−𝜃
1+𝜌
1−𝜃

When 𝜏! = 𝑏!/0 = 0, the first-order condition of the utility maximization is𝑑𝐸[𝑈! ]
1
023 23= −(𝑤! ℎ! − 𝑠! )23 +𝐸P𝑅!/0R𝑠! = 0.
𝑑𝑠!
1+𝜌The first-order equation is rewritten as(𝑤! ℎ! − 𝑠! )23 =(1 − 𝜃)5 𝜎45 23
𝛼023
exp Ÿ(1 − 𝜃)𝜇4 +
𝑠! .
1+𝜌
2

Solving the equation above with respect to 𝑠! , we obtain
𝑠! =

1
𝑤ℎ.
1 + 𝛽𝐴7 𝜆023 ! !

With social security tax and benefit (𝜏! > 0 and 𝑏!/0 > 0), the same equation of saving must
hold if𝜏! ≤1𝑤ℎ.
1 + 𝛽𝐴7 𝜆023 ! !

(A1)

Therefore, Equation (13) is obtained if (A1) holds.
Equation (15). Under the MF pension, the first-order condition of the utility maximization is𝑑𝐸[𝑈! ]
1023 (𝑠= −[(1 − 𝜙)𝑤! ℎ! − 𝑠! ]23 +𝐸P𝑅!/0o! )23 R = 0.
! + 𝜙𝑤𝑑𝑠!
1+𝜌
This equation becomes[(1 − 𝜙)𝑤! ℎ! − 𝑠! ]23 =(1 − 𝜃)5 𝜎45
𝛼023(𝑠! + 𝜙𝑤
exp Ÿ(1 − 𝜃)𝜇4 +o ! )23 = 0.
1+𝜌
2

26

Solving the equation above for saving yields(1 − 𝜙)𝑤! ℎ! − 𝛽𝐴7 𝜆023 𝜙𝑤
o!
.
7
023
1 + 𝛽𝐴 𝜆𝑠! =Taking average of the individual saving function, we arrive at Equation (15).

B. Saving functions under unfunded pension systems
Equation (17). Using the budget equation of the pension system and Equation (3), we obtain𝑏!/0 = (1 − 𝛼)𝜓𝑋!/0 𝑠̅! . With this equation, the first-order condition is𝑑𝐸[𝑈! ]1= −[(1 − 𝜓)𝑤! ℎ! − 𝑠! ]23 +𝐸P𝑅!/0 (𝑅!/0 𝑠! + (1 − 𝛼)𝜓𝑋!/0 𝑠̅! )23 R = 0.
𝑑𝑠!
1+𝜌The equation above becomes−[(1 − 𝜓)𝑤! ℎ! − 𝑠! ]23 +𝛼
023
𝐸P𝑋!/0R(𝛼𝑠! + (1 − 𝛼)𝜓𝑠̅! )23 = 0.
1+𝜌Therefore, we have!

(1 − 𝜃)𝜇4 (1 − 𝜃)5 𝜎45
𝛼 #"−[(1 − 𝜓)𝑤! ℎ! − 𝑠! ] + u−a [𝛼𝑠! + (1 − 𝛼)𝜓𝑠̅! ] = 0.
v exp _−1+𝜌𝜃
2𝜃Solving the equation above for individual saving leads to𝑠! =(1 − 𝜓)𝑤! ℎ! − 𝛽𝜒𝜓𝜆023 𝐴7 𝑠̅!
.
1 + 𝛽𝜆023 𝐴7

Using the individual saving, the average saving function is1−𝜓𝑠̅! = w
x𝑤
o .
1 + (1 + 𝜒𝜓)𝛽𝜆023 𝐴7 !
Equation (19). With fringe benefits (to give individuals saving incentive), the first-ordercondition of the utility maximization is[(1 − 𝜓)𝑤! ℎ! − 𝑠! ]23 =

1
𝐸¡
1+𝜌

𝑅!/0 + (1 − 𝛼)(1 − 𝜋)𝜓𝑋!/0
¢.
3
𝑠!
(1
(1
‰𝑅!/0 𝑠! + Š𝜋 + − 𝜋) 𝑠̅ ‹ − 𝛼)𝜓𝑋!/0 𝑠̅! Œ!

This equation is rewritten as[(1 − 𝜓)𝑤! ℎ! − 𝑠! ]23 =23𝑠𝛼023 (1 + (1 − 𝜋)𝜒𝜓)𝐴023 𝜆(320)3 ‰𝑠! + Š𝜋 + (1 − 𝜋) 𝑠̅! ‹ 𝜒𝜓𝑠̅! Œ
!

1+𝜌

.

Consequently, we obtain!

𝑠! =(1 − 𝜓)𝑤! ℎ! − (1 + (1 − 𝜋)𝜒𝜓)2" 𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝑠̅!
"#!

1 + (1 + (1 − 𝜋)𝜒𝜓) " 𝛽𝐴7 𝜆023
27=

(1 − 𝜓)𝑤! ℎ! − 𝜔𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝑠̅!
.
1 + (1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023

Using the individual saving function, the average saving function is(1 − 𝜓)𝑤
o!
𝑠̅! =
.
!
1 + (1 + (1 − 𝜋)𝜒𝜓)2" 𝛽𝐴7 𝜆023 (1 + 𝜒𝜓)

C. Proof of Proposition 1
Lemmas 1 and 2 show 𝛾<= ≤ 𝛾:: = 𝛾9: for 𝜓 ≥ 0. Using Equations (13) and (20), we obtain
𝛾9? = 𝛾:: ⇔ 𝜓} =(1 − 𝜔£ )𝛽𝐴7 𝜆023,
1 + (1 + 𝜔£𝜒)𝛽𝐴7 𝜆023

where 𝜔£ ≡ 𝜔C𝜓}D. Therefore, we have𝛾9? ⋛ 𝛾:: ⇔(1 − 𝜓)(1 − 𝛼)𝐴(1 − 𝛼)𝐴⋛⇔ 𝜓 ⋚ 𝜓}.
1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 1 + 𝛽𝐴7 𝜆023

Depending on 𝜃, 𝜓} can be larger or smaller than unity. 𝛾9? = 𝛾:: (𝛾9? < 𝛾:: ) holds if
𝜓 = 0 (𝜓 = 1). Consequently, a rise in 𝜙 must have a positive growth effect under the MU
pension at 𝜙 = 0. If 𝜃 is sufficiently small/(large), we obtain
∂𝛾9?
∂𝛾9?
|
> 0_
|
< 0a.
∂𝜓 @A∂𝜓 @AActually, 𝛾9? > 𝛾:: holds if 𝜃 and 𝜓 are sufficiently small. Furthermore, we have
∂𝛾9?
|
< 0.
∂𝜓 @A0
The results obtained for the growth effect of 𝜓 above indicate that 𝜓} exists as 0 < 𝜓} < 1 if
𝜃 is sufficiently small. Then, we arrive at 𝛾9? > 𝛾:: for 0 < 𝜓 < 𝜓} and a sufficiently small
𝜃. Contrarily, if 𝜃 is sufficiently large, then 𝜓} < 0 holds. Consequently, 𝛾9? < 𝛾:: . We have
𝛾9? > 𝛾<= because of
∂𝛾9?
< 0.
∂𝜋

D. Proof of Proposition 4
One key determinant of social welfare level under PG is
023

023

𝛽𝜒𝜓𝐴7 𝜆023 𝑠̅!
uv
1 + 𝛽𝜆023 𝐴7<=
min 𝑈! =
1−𝜃

−1

1
+
1+𝜌

𝐸 ‡u𝜓𝑤o !/0 −

Using this equation, we obtain28

𝛽𝜒𝜓𝐴7 𝜆023 𝑠̅!
𝑅 v
1 + 𝛽𝐴7 𝜆023 !/0
1−𝜃

ˆ−1
.

𝑈 <=
023
𝜕 min u ! v
𝛽𝜒𝐴7 𝜆023
𝜕𝜉<=
𝑤!
=q
r
u𝜉<= + 𝜓v (𝜓𝜉<= )23
023
7
𝜕𝜓
𝜕𝜓
1 + 𝛽𝜆 𝐴
1
1
𝑐!/0
+
𝐸 ‡_
a
1+𝜌
𝑤!

23

_1 + 𝛾<= + 𝜓

𝜕𝛾<= 𝛽𝜒𝐴7 𝜆023 𝜉<=𝛽𝜒𝜓𝐴7 𝜆023 𝜕𝜉<=
−
𝑅
−
𝑅 aˆ.
!/0
𝜕𝜓
1 + 𝛽𝐴7 𝜆023
1 + 𝛽𝐴7 𝜆023 𝜕𝜓 !/0

For the corner values of 𝜓, the above partial derivative becomes𝑈 <=𝜕 min u 𝑤! v
!

𝜕𝜓

023

𝛽𝜒𝐴7 𝜆023=qr
1 + 𝛽𝜆023 𝐴7

§

023

023
𝜉<=
lim 𝜓 23 +
@→-

1
1
𝑐!/0
𝐸 ‡_
a
1+𝜌
𝑤!

ˆ = +∞, (A2)

@A-

𝑈 <=
𝜕 min u 𝑤! v
!

𝜕𝜓

023

𝛽𝜒𝐴7 𝜆023=qr
1 + 𝛽𝜆023 𝐴7

§

𝜕𝜉<=
23
lim 𝜉<=@→- 𝜕𝜓 @→lim

@A0
1
1
𝑐!/0
+
𝐸 ‡_
a
1+𝜌
𝑤!

23

𝜕𝛾<=
𝛽𝜒𝐴7 𝜆023
𝜕𝜉<=
−
lim
𝑅!/0 aˆ = −∞.
7023@→- 𝜕𝜓
@→- 𝜕𝜓
1 + 𝛽𝐴 𝜆_ lim

These results indicate that there exists the value of 𝜓© satisfying
𝑈 <=𝜕 min u 𝑤! v
𝜕𝜓

!

§

= 0.

(A3)

M
@A@

Using (A2), (A3) and 𝜕𝛾<= /𝜕𝜓 < 0 lead to
𝜕𝑊!<=
𝜕𝑊!<=
ª
> 0 andª< 0.
𝜕𝜓 @A𝜕𝜓 @A@M
Furthermore, there exists 𝜓 ∗ satisfying 0 < 𝜓 ∗ < 1 and
𝜕𝑊!<=
ª
= 0.
𝜕𝜓 @A@∗
If the pension is managed by the MU system, then the indirect utility function for generation 𝑡
becomes023𝜔𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝑠̅!
x
1 + (1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023
9?
min 𝑈! =
1−𝜃
w

−1

023
o!/0 ℎe!/0 D ‹ − 1
1 𝐸 ŠC𝜋𝜓𝑤
+
.
1+𝜌
1−𝜃

The welfare effect of a rise in 𝜋 includes its effect on the poorest people at generation 𝑡 and its
cumulative effect of income growth. Such a growth effect is captured as#

1 + 𝛾! " 1 + 𝛿
' =
,
!"
1+𝛿
𝛿 − 𝛾!
"$%

29

𝜕
1+𝛿
1 + 𝛿 𝜕𝛾𝑀𝑈
"
'=
< 0.
(𝛿 − 𝛾)( 𝜕𝜋𝜕𝜋 𝛿 − 𝛾&'With regard to the effect on the poorest people at generation 𝑡 for 𝜋 = 0, we have thefollowing.
𝑈!9?
𝜕 min u 𝑤
v
1
!
lim ¡
¢=
lim 𝐸P𝜓023 [𝜋(1 + 𝛾9? )]23 (1 + 𝛾9? )RP→𝜕𝜋
1 + 𝜌 P→𝜔𝛽𝜋𝜒𝜓𝐴7 𝜆023 𝜉9?
+ lim qr
P→- 1 + (1 + (1 − 𝜋)𝜒𝜓)𝜔𝛽𝐴7 𝜆023

23

𝜔𝛽𝜒𝜓𝐴7 𝜆023 𝜉9? P1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 R•‚[1 + (1 + 𝜒𝜓)𝜔𝛽𝐴7 𝜆023 ]5= +∞,This equation suggests that an increase in 𝜋 improves social welfare because it dominates thenegative welfare effect by decreasing the economic growth rate. Furthermore, we obtain thefollowing equation for 𝜋 = 1:
𝑈 9?
𝜕 min u ! v
𝑤!
lim ¡
¢
P→0
𝜕𝜋
023

𝛽𝜒𝜓𝐴7 𝜆023 𝜉9?
=qr
1 + 𝛽𝐴7 𝜆023

•1 + q+(𝜃 − 1)𝛽𝐴7 𝜆023 𝜒𝜓
1
+
r ‚
𝜃
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023
1 + 𝛽𝐴7 𝜆023

1𝜕𝛾9?
lim 𝐸 w𝜓023 [𝜋(1 + 𝛾9? )]23 u1 + 𝛾9? + 𝜋
vx . (A4)
1 + 𝜌 P→𝜕𝜋

The following equations hold:1 𝜕𝜔 1𝜒𝜓
1 𝜕𝜔𝜒𝜓=, lim wx=
, lim 𝜔 = 1,𝜔 𝜕𝜋 𝜃 1 + (1 − 𝜋)𝜒𝜓 P→0 𝜔 𝜕𝜋
𝜃 P→0
(1 + 𝜒𝜓)𝛽𝐴7 𝜆023 𝜒𝜓
1𝜕(1 + 𝛾9? )
lim q
r=−
.
P→0 1 + 𝛾9?
𝜕𝜋
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023 𝜃
The first term on the right-hand side of Equation (A4) has an ambiguous sign, whereas thesecond term is negative.
(𝜃 − 1)𝛽𝐴7 𝜆023
1
1 + 𝛽𝐴7 𝜆023
+
⋛
0
⇔
𝜃
⋛
1
−
.
[1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023 ]𝛽𝐴7 𝜆023
1 + (1 + 𝜒𝜓)𝛽𝐴7 𝜆023
1 + 𝛽𝐴7 𝜆023
If θ is sufficiently large, then Equation (A4) becomes positive; the negative growth effect isdecreased. Therefore, 𝜋 = 1 is chosen. However, we obtain 0 < 𝜋 < 1 if 𝜃 is sufficiently
small as Equation (A4) becomes negative.

30

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33

Tables
Table 1. Equilibrium growth rates within the funded system
SystemBenchmark

Fully funded/Modified funded𝜎4𝜃 = 0.5

𝜃=1

𝜃 = 1.5

𝜃 = 0.5

𝜃=1

𝜃 = 1.5

0

1.567

1.069

0.910

1.567

1.069

0.910

0.25

1.690

1.135

0.957

1.669

1.135

0.977

0.5

2.092

1.344

1.107

2.000

1.344

1.192

0.75

2.880

2.741

1.382

2.641

1.741

1.605

1

4.278

2.411

1.825

3.763

2.411

2.316

Table 2. Equilibrium growth rates within the unfunded system
SystemPay-as-you-go

Modified unfunded𝜎4𝜃 = 0.5

𝜃=1

𝜃 = 1.5

𝜃 = 0.5

𝜃=1

𝜃 = 1.5

0

0.663

0.293

0.180

0.754

0.334

0.206

0.25

0.732

0.334

0.222

0.826

0.376

0.249

0.5

0.955

0.465

0.357

1.060

0.512

0.387

0.75

1.390

0.713

0.617

1.514

0.767

0.651

1

2.159

1.132

1.064

2.314

1.199

1.108

34

Table 3. Optimal tax rates and social welfare levels (Rawlsian welfare function)
Modified funded𝜃 = 0.5
𝜎4

𝜙∗

𝜃=1
𝑊 9:

𝜙∗

𝜃 = 1.5
𝑊 9:

𝜙∗

𝑊 9:

0.00

25.551

4.053

3.916

0.25

27.825

4.261

3.941

0.50

100%

36.491

100%

4.886

100%

4.015

0.75

62.430

5.928

4.131

1.00

230.72

7.386

4.280

Pay-as-you-go𝜃 = 0.5
∗

𝜃=1
𝑊

<=

𝜓

∗

𝜃 = 1.5
𝑊

<=

𝜓

∗

𝑊 <=

𝜎4

𝜓

0.00

14.5%

0.451

−4.66922.1%−3.270

0.25

14.0%

0.827

−4.46122.0%−3.178

0.50

12.3%

2.166

−3.836

21.7%−2.911

0.75

9.1%

5.538

−2.79421.3%−2.4961.00

3.4%

18.982−1.336

20.7%−1.971

21.3%

Modified unfunded𝜃 = 0.5

𝜃=1

𝜎4

𝜋∗

𝜓∗

𝑊 9?

0.00

86.3%

16.9%

0.470

0.25

86.0%

16.4%

0.847

0.50

85.0%

14.7%

2.190

0.75

82.9%

11.2%

5.571

1.00

78.9%

4.4%

19.022

𝜋∗

100%

𝜓∗

21.3%

35

𝜃 = 1.5
𝑊 9?

𝜓∗

𝑊 9?

−4.66922.1%−3.270−4.46122.0%−3.178

21.7%−2.911−2.79421.3%−2.496−1.336

20.7%−1.971−3.836𝜋∗100%Table 4. Optimal tax rates and social welfare levels (Benthamite welfare function)
Modified funded𝜃 = 0.5

𝜃=1

𝜃 = 1.5

𝜙∗

𝑊 9:

𝜙∗

𝑊 9:

𝜙∗

𝑊 9:

100%

36.491

100%

4.886

100%

4.153

<=

𝜓

∗

𝑊 <=

𝜎6
0.50
1.00
3.00
5.00
10.0

Pay-as-you-go𝜃 = 0.5
∗

𝜃=1
𝑊

<=

𝜓

∗

𝜃 = 1.5

𝜎6

𝜓

𝑊

0.50

0.0%

35.162

0.0%

4.469

0.0%

3.906

1.00

0.0%

31.420

0.0%

3.219

10.5%

3.246

3.00

3.0%

8.922

17.3%

−2.246

24.9%

0.778

5.00

10.9%

2.831

20.9%

−3.682

26.8%

0.060

10.0

12.3%

2.166

21.3%

−3.83626.9%−0.017

Modified unfunded𝜃 = 0.5
𝜎6

𝜋

∗

𝜓

0.50

0.0%

1.00

∗

𝜃=1
9?

𝜋

∗

𝜓

8.6%

37.018

3.4%

0.0%

0.0%

8.6%

33.109

8.9%

3.00

29.5%

9.7%

9.327

5.00

79.9%

13.8%

10.0

85.0%

14.7%

𝜃 = 1.5
𝜓∗

𝑊 9?

4.469

0.0%

3.906

0.0%

3.219

10.5%

3.246

100%

17.3%

−2.246

24.9%

0.778

2.871

100%

20.9%

−3.682

26.8%

0.060

2.190

100%

21.3%

−3.83626.9%−0.017

𝑊

36

∗

𝑊

9?

𝜋

∗

100%