PAYG Public Pension and Economic Growth in An OLG Economy with Human Capital Accumulation

PAYG Public Pension and Economic Growth in An OLG Economy with Human Capital Accumulation

Hiroyuki HASHIMOTO ^# September, 2020

ABSTRACT

This paper shows the negative relationship between population aging and economic growth by developing an endogenous growth model of overlapping generations, where human capital accumulation is the engine of income growth. The effect of PAYG public pension system on economic growth is examied. We show that the PAYG public pension system could be a growth-enhancing social contrivance, and government can run the PAYG public pension system to offset the negative effect of population aging on economic growth.

Keywords: Public Pensipon System; Human capital; Economic growth JEL classification: H22, J13, J22, J24, J26, O41

# University of Hyogo8-2-1 Gakuennishi-machi Nishi-ku Kobe Hyogo, 6512197 JAPAN

E-mail hiro@mba.u-hyogo.ac.jp

1. Introduction

This paper develops a model to displays negative relationship between population

aging and economic growth and examines the effect of PAYG public pension on

economic growth in long run. Ever since Romer (1986), while many types of growth

models have been developed to clarify underlying mechanisms in economic growth,

empirical studies have tested the theoretical assertions and found empirical evidence on

growth phenomena. Along the strand of this literature, especially stimulated by

Lucas(1988), human capital has been recognized as one of substantial factors to sustain

economic growth. Various types of growth models with human capital accumulation

have developed. For example, Becker, Murphy and Tamura (1990) demonstrate the

existence of poverty trap. As for the study on the interaction between population aging

and social security system, most of theoretical models that have been ever developed are

of overlapping generations. Kaganovich and Zilcha (1999), Grozen et al (2003) study

importance of how government finances payments of public pension. In order to study a

role of pay-as-you-go public pension in a tractable overlapping generations model with

endogenous growth setting, we construct an overlapping-generations model with two

types of capital, physical and human capital, in which human capital is the engine of

growth by incorporating different two types of workers in production. Our model is

basically a modified version of Hashimoto et al (1997) that is very similar to Lucas

model (1988) of endogenous growth with human capital accumulation. Moreover as

employed in Cipriani (2014), we take the child-rearing cost into account in order to

study how the cost affects economic growth, although the number of children is

exogenously given. In such a model we demonstrate that pay-as-you-go public pension

system may have growth-enhancing effect. This result shows the possibility of

pay-as-you-go public pension to offset the negative effect of population aging.

The rest of this paper is organized as follows. Section 2 develops a model and Section3 considers how pay-as-you-go public pension system works in the model and what it could bring about. Section 4 includes summary and concluding remarks.

2. The Model

We consider a small open economy which exists over infinite number of periods. ¹ The economy is composed of individuals who live for three periods without uncertainty on lifetime and many competitive firms that produce a single good. The production requires physical capital and two types of workers with human capital. We label the generation that was born at period t as “generation t”. Individuals in the same generation are assumed to be identical and each has n children. The number of children n is assumed to be exogenously determined. We assume that N _{t + 1} = + ( 1 n N ) _t holds, where N t is the size of generation t. In both the first and second period of life, each individual is endowed with one unit of time. In the first period of life, each individual is young worker and allocates the productive time between work and accumulation in human capital which is the engine of growth in this economy. In the second period of life, each is old worker and supplies the labor force inelastically to labor market. In the third period of life, each is retired. The members of the different two generations engage in production in any period t.

1 The assumption needs for the determinacy of equilibrium, even though it does not ensure the

determinacy all by itself. In other words, the closed-economy version of the following model

exhibits the determinacy of equilibrium.

2.1 Human Capital Accumulation

Only the young govern investment in human capital. Suppose that each young individual of generation t is endowed with human capital, h _t ^y . We assume that it is accumulated according to

o

h t _{+ 1} ⁼ H ( ) u t h t ^y , (1)

where u _t ∈[ , ] 0 1 is the part of time devoted to human capital accumulation, h _t ^o ₊₁ is the level of human capital of old worker of generation t at period t+1, and H(⋅) is assumed to be increasing, concave and continuously twice differentiable function such that

H(0)=1 and = +∞

→ ' ( )

lim

0 t

u H u

t

hold. Since one remarkable aspect as for human capital accumulation is a social activity as stated in his seminal work of Lucas (1988), we simply assume that each young individual at period t inherits a portion of average level of human capital of old individuals at period t:

y

h t = ( 1 − δ ) h _t ^o : 0≤δ≤1, (2) where h _t ^o is the average level of human capital of old individuals at period t and initial level h ₀ ^o is given. ² Individuals between neighboring generations complete “the inheritance” expressed in (2) without intention, therefore without costly activities. Thus the inheritance mechanism is intergenerational externality. The parameter δ stands for Lucasian intergenerational externality, which could be interpreted as the extent of inheritance of human capital. ³ For simplifying the following analysis, we set δ to be zero. Note that the consistency condition h _t ^o = h _t ^o is required in equilibrium, since individuals are identical within generation. Thus ^h t

y = h _t ^o must hold in equilibrium.

2 Lucas (1988, p19) says: …the initial level each new member begins with is proportional to (not equal to!) the level already attained by older members of the family….

3 If δ is unity, it means the absence of intergenerational externality.

2.2 Individuals

The utility of each individual of generation t, U _t , depends on consumption at the second and the third period of life, c _t+1 ² and c _t+2 ³ . For simplifying the analysis, we assume that consumption does not occur in the first period of life. We give each individual a log-linear utility function

3 2 1 2

1 ln

ln ₊ + ⁻ ₊

= _{t t}

t c c

U ρ , ρ>1 (3) where ρ represents the subjective discount factor on the third-period utility, ln c _t ³ _{+ 2} . Each young individual with ^h t

y allocates u _t units of time for accumulating own human capital, while supplying 1− u _t units of time to firm through competitive labor market in order to earn income w _t ^y ( 1− u h _t ) _t ^y . Since individuals are assumed not to need consumption in the first period of life, they save all of wage income in that period and earn the going rate of return R _{t + 1} . Each old individual with h _t ^o ₊₁ supplies one unit of time to firm inelastically to earn income w h _t ^o _{+ 1 t} ^o _{+ 1} , and receives the saving with accrued interest, R _{t + 1} w _t ^y ( 1 − u _t ) h _t ^y . Although the number of children n is exogenous in this model, we assume that raising children is costly activity, and the parents bear whole of the child-rearing cost as employed in Cipriani (2014). The amount of resources for parents to raise each child is assumed to be a fixed proportion to income earned in the first period of life, qw _t ^y ( 1 − u _t ) ^h t

y . ⁴ Since the parents have n children, the child-rearing cost is written as qw _t ^y ( 1 − u _t ) ^h t

y n . Each old individual consumes a part of wealth and saves the rest s _{t + 1} to earn the going rate of return R _{t + 2} . Each retired individual spends

4 It is interesting to study the case that the cost is also proportional to income in the

second period of life. It could be interpreted as the case that the grand-parents bear a

part of the child-rearing cost. It is worthy for further investigation.

all of the saving with accrued interest R _{t + 2} s _{t + 1} on the third-period consumption. Thus the budget constraints of each individual of generation t are written as

c _t+1 ² = R _{t +1} w _t ^y ( 1 − u _t ) ^h t

y − qw _t ^y ( 1 − u _t ) ^h t

y n + w _t ^o ₊₁ h _t ^o ₊₁ − s _t+1 and (4) c _t+2 ³ = R _t+2 s _t+1 . (5) An individual born at period t is to maximize (3) subject to (4), (5) and (1). It is straightforward to obtain the first-order conditions for the choice of s t+1 and u t to be optimal. The conditions are

c _t ³ ₊₂

c _t ² ₊₁ = R _{t +2}

ρ

(6)

( ) o ( ) t

t y t

t H u

w qn w

R − = ′

+ +

1

1 1 , (7)

together with the budget constraints. Equation (6) means that the marginal loss of the second-period utility by saving one unit of income must equal the marginal gain of the third-period utility by receiving R _{t +2} units of income. The LHS of (7) represents the marginal opportunity cost of allocating additional time for accumulation in human capital and the RHS indicates the marginal benefit of such time allocation. Thus equation (7) displays that each individual allocates one unit of endowed time so as to maximize his lifetime income equating the marginal benefit with the marginal cost. The optimal decision of each individual is characterized by (7) and

s _{t+1 =} ^{R t+1 (1 − u t )(1 − qn)w t}

y + w _t+1 ^o H(u _t )

[ ] ^h t

y

1+ ρ (8) Eq.(8) comes from (6) and the budget constraints. Note that u t in (8) satisfies (7), in other words, the numerator in the RHS of (8) is the maximized lifetime income.

Thus an individual’s decision making is of step-wise. The first decision is to maximize

the lifetime income by choosing the optimal time allocation. The second one is to

choose the optimal saving with the maximized lifetime income.

2.3 Firms

In each period, firm is assumed to be profit-maximizer in competitive markets.

Assuming constant-returns-to-scale technology in each firm, the production function at the aggregate level at period t is represented as the following technology:

Y _t = AK _t ^α [ (1 − u _t )H _t ^y ] ^β [ ] ^{H t o γ , α + β + γ =1,} (9)

where A is technological parameter, K _t denotes aggregate physical capital stock, H _t ^y is aggregate human capital stock of young workers given by N h _{t t} ^y , and H _t ^o is aggregate human capital stock of old workers given by N h _{t −1 t} ^o , at period t, respectively.

Assuming that physical capital fully depreciates at one period and using consistency condition h _t ^y = _{h t} ^o , the profit maximization gives the following first-order conditions:

R _t = α A k _t h _t ^o



  

 

α −1

(1 − u _t ) ^β n ^{α +β −1} , (10)

w _t ^y = β A k _t h _t ^o



  

 

α

(1 − u _t ) ^{β −1} n ^{α +β −1} , and (11)

w _t ^o = (1 − α − β )A k _t h _t ^o



  

 

α

(1 − u _t ) ^β n ^{α +β} , （12）

where _{k t} is defined as

t t

N

K , R _t stands for the rental price of capital, and w _t ^y and w _t ^o are wage rates to be spent for young and old workers, at period t, respectively.

These conditions require that firms must employ productive factors in order to equate

the marginal product with the market price they face in the competitive markets.

2.4 Equilibrium

The equilibrium condition in the capital market is given as

y t t

y t t t t

t N s N w u qn h

K _{+ 1} = _{− 1} + ( 1 − )( 1 − ) , (13) where N _{t − 1} and N _t stand for the number of the old and the young workers in period t, respectively. The LHS of (1) is the capital stock at period t+1, while the RHS displays aggregate saving in period t.

Since h _t ^y = _{h t} ^o holds in equilibrium, it is useful to denote h _t instead of h _t ^y or _{h t} ^o in equilibrium. Using (1) and (2), We obtain the following equation.

t t

t H u h

h _{+ 1} = ( ) . (14) A competitive equilibrium must satisfy the equation of motion for human capital (14) and the capital-market-clearing condition (13), together with households’ and firms’

optimality conditions, (7), (8), (10), (11), (12), consistency condition, ^o

t y

t h

h = ≡ h _t , and _{r = r t} , at every period. Substituting all of these conditions into (13) and (14) generates the following dynamical system about the motion of k _t , h _t and u _t

1

) 1

) ( 1

( ) 1 (

+

= +

 

 





−

−

−

t t t t

t

h u k h H

k n

R qn

β α

β ,

) , ( ) 1

1 ( ) 1 (

) ( ) (

1 ) 1

( )

1 ( )

1 (

1 1

1 2 1

1 2 1

t t

t t t t

t t

t t t

t

u H h qn k R n

u H u H h Rn k u

h H qn k h R

n k

 

 



−  +

+

 

 



 



 



−  + −

 

 



− 

= +

−

−

−

− +

+

α ρ β α

β α α

ρ β

and

t t

t H u h

h _{+ 1} = ( ) ,

where

k 0 and

h 0 are given as initial value. Though, on the face of it, the equilibrium dynamics is of complicated nonlinear difference equations, introducing a new variable, X _t ≡ k h _{t t} , can reduce the system to a simpler two-dimensional nonlinear first-order difference equations.

) ,

1 ( t t

t X u

X ₊ = φ (15) )

1 ( t

t u

u ₊ = ϕ , (16) where X ₀ ≡ k ₀ h ₀ ⁱ s given, and (15) satisfies ₁ ( ) ¹

) 1

( ) 1

( ₋

+ − −

= − _{t t}

t X H u

n R qn

X α β

β and

(16) satisfies

) (

) ( ) 1 )(

1 (

) 1

( 1

) 1 )(

1 (

) 1

( ) (

)

( ^{2 2}

1 1

t t t

t

u H

u H qn qn

u H

u

H ′

− +

− + −

− + −

− +

−

= −

′ ⁺ ₊ αβ ρ

β α α

β α ρ

αβ

β

α . It is

hereafter assumed that the equilibrium is determinate. ⁵ We focus the equilibrium on the balanced growth path in the following analysis. The balanced growth path is such that both _{X t} and _{u t} take constant value. Though the whole system is depicted by (15) and (16), it is useful to employ (16) and (10) instead of (15) to characterize the balanced growth path. Thus equilibrium balanced growth path is characterized by

n qn u R

H −

− ⋅

= − 1

) 1 (

' α β

β (17)

γ β

α α ⁻ − + ⁻

= AX ¹ ( 1 u ) ( 1 n )

R . (18)

The assumptions imposed on H（・）ensures the existence and the uniqueness of the balanced growth path. The production function (9) implies that the growth rate of per-worker income, _{y t + 1 / y t} , is H (u ) on the balanced growth path. This shows that the driving force of this economy is the accumulation of human capital and an increse in u means enhancing growth on the balanced growth path.

5 ( )

1

) 1 )(

1 ) (

( _t qn H u _t

u

H α β

ρ αβ

−

−

−

> +

′ is assumed to ensure the determinacy of equilibrium.

2.5 Effect of Population Aging and Child-rearing Cost on Economic Growth

Using (17), we immediately derive

) (

) 1

( ²

u H

n R dn

du

′′

−

−

−

= α β

β

. (19) Since the sign of (19) is strongly positive, we obtain the following result.

Result 1 Population aging due to a fall in the number of children is harmful for growth.

The intuition for Result 1 is clear. A decrease in the number of children n means implies a rise in the ratio of the old worker to the young worker,

t t

N N _{− 1}

for any t on the balanced growth path. A rise in the old-young ratio means a rise in relative scarcity of the young worker in the labor market. The scarcity leads to a rise in the young worker’s wage rate that also stands for the opportunity cost of human capital accumulation.

Moreover, a fall in the number of children means directly a decrease in the child-rearing cost implying a rise in the young workers’ net wage rate, w _t ^y ( 1 − qn ) h _t ^y . This also push up the opportunity cost of human capital accumulation. Summing up the both channels, a rise in the opportunity cost makes individuals to allocate their time less to human capital accumulation. Thus population aging is harmful for growth with discouraged human capital accumulation.

Using (17) again, we get

) (

) 1

( u H

R dq

du

′′

−

−

−

= α β

β

. (20) The sign of (20) is strictly positive, and we get

Result 2 An increase in child-rearing cost captured by a rise in q enhances economic

growth.

The result is seemed to be surprising, but the intuition is simple and reasonable as follows. A rise in the child-rearing cost means directly a decrease in the young workers’

net wage rate. This means a decrease in the opportunity cost of human capital accumulation. A fall in the opportunity cost makes individuals to allocate their time more to human capital accumulation. Thus a rise in child-rearing cost induces the economy to engage with faster growth.

3. PAY-AS-YOU-GO Public Pension

We will examine how the introduction of pay-as-you-go public pension system affects economic growth. Let a _t ^y be the contributions of a young worker at period t,

o

a t be that of an old worker at period t, and b _t be the benefit received by a retired individual at period t. Then pay-as-you-go public pension system is such that

o t y t

t n a na

b = ² + holds at any period. We consider two cases as for the way government finances the social security scheme. One is such that they are financed in a manner of lump-sum, another is such that they are proportional to wage income. Since no changes occur in firms’ optimality conditions and the equation of motion of human capital in both cases, thus possible changes appears in both two conditions for individuals’

optimality and the good-market-clearing condition. While welfare analysis need the full

information described in the three conditions, in order to explore how pay-as-you-go

public pension system affects the economic growth, it is sufficient to use one of two

equations for individuals’ optimality, i.e., the optimality condition as for time allocation

between work and human capital accumulation, because the optimality equation

determines the economic growth rate. ⁶

3.1 CASE 1 a _t ^y and a ^o _{t + 1} are financed in a manner of lump-sum The counterpart of (7) is obtained as

R _t+2 ( 1 − qn ) ^{w t y}

w _t ^o ₊₁ = ′ H u ( ) _t （21）

Comparing (3-7) with (21), we find that nothing affects the decision making on time allocation. The intuition is that lump-sum finance does not affect wage profile between the first and the second periods of one’s life; the slope of wage profile is closely related to the speed of economic growth. Summarizing the above, we get

Proposition 1 Suppose that a government runs a pay-as-you-go public pension system which finances the contributions in a manner of lump-sum tax. Then such a pay-as-you-go public pension system has no effect on the rate of economic growth.

3.2 CASE 2 a _t ^y and a ^o _{t + 1} are proportional to wage income

Let a _t ^y be θ ^y w _t ^y ( 1 − u _t ) h _t ^y and a _t ^o _{+ 1} be θ ^o w _t ^o _{+ 1} h _t ^o , where 0 ≦ θ ^{i ≦} 1. The individual’s first-order condition as for time allocation is rewritten as

ˆ ) ( 1 '

1

1

2 t

o t y o t y

t qn w w H u

R ₊ = ₊

−

−

− θ

θ （22）

Comparing (7) with (22), we find that the LHS is different between them. It seems to bring about remarkable effect on the rate of economic growth immediately. However we postulate that even in this case if a government set the rate of contribution such that

6 Since welfare analysis is much complicated, we conduct the following analysis

focusing on growth effect, apart from welfare effect. Welfare analysis is worthy for

future study.

o

y qn θ

θ = ( 1 − ) holds, the pay-as-you-go public pension system has again no impacts on time-allocating decision making and thus on the rate of economic growth.

Proposition 2 Suppose that a government runs a pay-as-you-go public pension system which finances the contributions in such a way that the contributions of workers are proportional to their wage incomes. Then such a pay-as-you-go public pension system has no effect on economic growth, if the rate of contribution of the young workers and that of the old workers are set such that θ ^y = ( 1 − qn ) θ ^o holds.

If the government imposes a heavier contribution on the young worker, the young decides to reallocate time more in human capital accumulation due to the relative decrease in opportunity cost of human capital accumulation. Thus the pay-as-you-go public pension system has growth-enhancing effect. We obtained the following.

Proposition 3 Suppose that a government runs a pay-as-you-go public pension system which finances the contributions in such a way that the contributions of workers are proportional to their wage incomes. Then such a pay-as-you-go public pension system accelerates the human capital accumulation and promotes the economic growth, if the government imposes heavier tax on the young worker’s wage income in the sense that

o

y qn θ

θ > ( 1 − ) holds.

The intuition for the growth-enhancing pay-as-you-go public pension system with wage-proportional contributions is as follows. Heavier contribution for the young, changing the wage profile, means lower opportunity cost of working at the first period.

This wage-profile effect induces the young to allocate their given time more to human

capital accumulation, leading the economy to attain faster growth.

Proposition 3 indicates the possibility of PAYG public pension system in order to offset the negative effect due to population aging.

4. CONCLUDING REMARKS AND COMMENTS

We have explored the effect of a pay-as-you-go public pension system on the rate of

economic growth in an endogenously growing small open economy with

three-period-lived non-altruistic individuals, where the accumulation of human capital is

the engine of economic growth. We have shown that the introduction of a pay-as-you-go

public pension system has potential of enhancing economic growth in an appropriate

manner. The pay-as-you-go public pension system has growth-enhancing effect when it

is financed with a tax on wage income and imposes appropriately a heavier tax on the

wage income of young workers relative to that of the old workers, while when the

contributions from workers of both the two generations are financed in a manner of

lump-sum tax, it turns out that the pay-as-you-go public pension system has no such

effect. These results depend on the structure of the young’s decision making as for time

allocation. We now have an important policy implication that a government who is

running a pay-as-you-go public pension system must pay much attention to the behavior

of individuals who engages in work that contributes to the current production and the

alternative activities that contribute to the future production i.e., economic growth. The

result that the growth rate depends on the contribution ratio has an interesting

theoretical implication in the literature of economic growth and size of government

activity. For example, Barro (1990) deals with it in an endogenous growth model with

public expenditure, showing that the size of government, i.e., rate of national tax burden

is one of key determinants of economic growth rate. Our result as in Proposition 3

implies that the structure of tax burden may rather important other than the total size of

government expenditure. It is also important to give welfare analysis for the effect of

pay-as-you-go public pension. Moreover we can treat these issues in models of

endogenous fertility setting that might be a straightforward extension of the model

presented in this paper. We also postulate that the formulation of child-rearing cost must

be careful in the sense that child-rearing cost itself could change opportunity cost of

human capital accumulation. It is interesting to examine other types of child-rearing

cost. All of these issues are worthy of further investigation.

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