Population aging, economic growth, and the social transmission of human capital: An analysis with an overlapping generations model

Economic Modelling 50 (2015) 138–147

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Population aging, economic growth, and the social transmission of human capital: An analysis with an overlapping generations model☆ Ki-Hong Choi a, Sungwhee Shin b,⁎ a b

National Pension Research Institute, Dosanro 128, Kangnam-gu, Seoul 138-811, Republic of Korea Department of Economics, University of Seoul, Seoulsiripdaero 163, Dongdaemun-gu, Seoul 130-743, Republic of Korea

a r t i c l e

i n f o

Article history: Accepted 30 May 2015 Available online xxxx Keywords: Overlapping generations model Population aging Human capital Economic growth

a b s t r a c t We develop a computable overlapping generations model in which the accumulation of human capital is endogenous. The model is similar to that of Fougere et al. (2009) and Sadahiro and Shimasawa (2002). It is, however, distinguished from them in several aspects regarding the individual utility function and the intergenerational transmission of human capital. We use the model to explore the effect of population aging on economic growth in Korea. The simulation shows that population aging causes a decrease in labor supply growth and an increase in capital stock growth, thus yielding capital deepening. The aging of the population may significantly undermine growth potential. However, the result is sensitive to the manner of intergenerational transmission of human capital. Our study shows that the mode of social transmission of human capital is quite important for longterm growth of the economy. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Korea faces a rapidly aging population. The total fertility rate was 1.23 in the year 2010, which is much lower than the populationsustaining rate of 2.1. As a result, the population is expected to decrease after 2030. The mortality rate is decreasing rapidly and the population aging is accelerating. According to population forecasts from the Korea Statistical Office, Korea will become an aged society in which the share of the population aged more than 64 is greater than 14% in 2017. It will become a post-aged society in which this share is greater than 20% in 2026. This population contraction and aging have emerged as a potential threat to economic growth. First of all, there would be a significant shortage of labor supply. Thus firms should find a way to procure appropriate human resources. The shortage of labor might become a factor hindering investments by firms. Moreover, the fiscal burden would become heavier as the share of the elderly becomes greater. As the labor supply decreases, investments by firms are impaired and tax rates increase, meaning that the economic growth might be significantly hampered. The purposes of this paper are twofold. The first goal is to develop a deterministic computable overlapping generations (OLG) model in

☆ We are grateful to the participants in the session of the Summer Conference of Korea Econometrics Society. In particular, we are deeply grateful to Yongjin Kim, journal editor S. Mallick, and two reviewers for their valuable comments. We are also grateful to Chang Hee Kim for her research assistance. All remaining errors are our own. ⁎ Corresponding author. Tel.: +82 2 6490 2059. E-mail addresses: [email protected] (K.-H. Choi), [email protected] (S. Shin).

http://dx.doi.org/10.1016/j.econmod.2015.05.015 0264-9993/© 2015 Elsevier B.V. All rights reserved.

which the element of human capital is formed endogenously. The next task is to apply this model to the Korean economy. With regard to the OLG model, a basic formulation that incorporates endogenous human capital was initiated and applied to the population aging problem in OECD countries by Fougere and Merette (1999). Fougere et al. (2009) revised the OLG model and applied it to the Canadian economy. Sadahiro and Shimasawa (2002) and Shimasawa (2007) also developed computable OLG models with endogenously determined human capital and applied them to the Japanese economy. Kim (2011) developed an OLG model similar to that of Sadahiro and Shimasawa (2002) and applied it to the population aging problem in Korea. Our model has some characteristics that distinguish it from the above approaches. First, we use a version employing a household utility function proposed by Kimball and Shapiro (2008). Kimball and Shapiro's (KS) utility function has advantages in that it is more consistent with the empirical findings related to labor supply and household consumption. The KS utility function also does not require the level of human capital to affect the utility of household directly. Second, we introduce a slightly modified functional form of human capital transmission among generations. It turns out that this slight modification has a pronounced effect on long-run economic growth. The simulation results show that population aging has a significant effect on economic growth. It reduces the annual growth rate of percapita GDP by around 0.5 percentage points in the year 2100 under the baseline scenario in comparison with the fixed population scenario. The simulation result, however, reveals high sensitivity to the functional form of human capital transmission among generations. For example, in an alternative scenario for the social transmission of human capital,

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147

Fig. 1. Trend of total population.

139

Fig. 3. The change in population age of structure.

annual growth rate of per-capital GDP can be higher than that under the fixed population scenario. This indicates the importance of the human capital transmission and suggests the need to scrutinize the model of social transmission of human capital. The simulation results also show that the decrease in labor supply due to population aging raises the wage rate; this in turn promotes the accumulation of human and physical capital. The increase in human and physical capital significantly offsets the adverse effect of population aging on economic growth. Our paper consists of five sections. Section 2 presents the main features of population projections for 2010–2100, which are based on the Future Population Projection published by the Korea Statistical Office in 2011. Section 3 presents the model and Section 4 describes the simulation results, while Section 5 provides some concluding remarks.

and then it decreases a little to 0.89 in 2083 (Table 1). After that, the ratio increases to 0.96 in 2100 (Fig. 2). The age structure of the population changes radically over the period 2010–2100 as shown by Fig. 3 and Tables 2 and 3. 3. The model There are two economic agents in this formulation: the household and the firm. The representative household consumes goods and supplies the production factors, i.e., labor and capital, to a representative firm. The firm supplies the goods and demands the production factors. There are three markets: those for goods, labor, and capital. We consider the dynamic general equilibrium of these markets.

2. Population aging

3.1. The household

We use the population projections for 2010–2100 made by the Korea Actuarial Projection Committee of the National Pension Service which extended the forecast for 2010–2060 implemented by the National Statistical Office. According to the projection, the Korean population will increase from 50.0 million in 2010 to 52.2 million in 2030. The population then decreases to 48.1 million in 2050 and to 28.2 million in 2100 (Fig. 1). Note that the ratio of the elderly aged more than 64 to the young aged 20–64 increases monotonically from 0.18 in 2010 to 0.93 in 2066

The household consumes goods (cj), provides labor (nj), and invests time into education (zj) for members of age j. The utility obtained from the consumption of goods, provision of labor, and investment into education is represented by the following utility function, as proposed by Kimball and Shapiro (2008).

Table 1 Total population and the population ratio of the old to the young.

2010 2030 2050 2100

1þ1=η 1−1=γ " 1þ1=θ !#1=γ cj nj zj
 A j þ B j ð1−γ Þ þ u c j; n j; z j ¼ 1−1=γ 1 þ 1=η 1 þ 1=θ

Table 2 The change in the population age structure.

Total population (million)

The old/the young ((65+)/(20–64))

50.0 52.2 48.1 28.2

0.18 0.41 0.77 0.96

Source: Korea Actuarial Projection Committee of National Pension Service.

20–29 30–39 40–49 50–59 60–69 70–79 80+

2010

2050

2100

7,001,498 8,128,301 8,507,226 6,676,317 4,683,420 2,645,334 960,612

4,330,485 4,514,951 4,709,943 6,296,936 6,904,047 7,403,251 6,893,653

2,189,683 2,591,042 2,893,839 3,103,390 3,634,806 4,331,275 5,694,228

2010

2050

2100

0.18 0.21 0.22 0.17 0.12 0.07 0.02 1.00

0.11 0.11 0.11 0.15 0.17 0.18 0.17 1.00

0.09 0.11 0.12 0.13 0.15 0.18 0.23 1.00

Table 3 The share of the population by age ranges.

Fig. 2. The ratio of the old to the young.

20–29 30–39 40–49 50–59 60–69 70–79 80+ 20+

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The lifetime utility of household is the discounted sum of both the period utilities and the discounted utility from bequests (b). The lifetime utility is represented by the following function: T X

"

1−1=γ

βj

j¼1

cj

1−1=γ

1þ1=η

A j þ B j ð1−γÞ

nj

1 þ 1=η

1þ1=θ

þ

zj

!#1=γ

1−1=γ

þ βT κ

1 þ 1=θ

b 1−1=γ ð1Þ

where βj = ψj/(1 + ρ) j is the individual discount factor for the utility at age j, Ψj is the survival probability until the age j, and ρ is the time preference rate. The survival probability is estimated from the national life table from the National Statistical Office. The parameter 1 − γ reflects the substitutability between consumption and labor. The parameter γ reflects the inter-temporal elasticity of substitution as can be seen in the Appendix A (Eq. (A8)). The parameter η represents the elasticity of the labor supply with respect to wage rates. The parameter θ represents the elasticity of education with respect to wage rates. The parameter κ represents the weight on bequests. The lifespan is 61 and is denoted by T. The age index j = 1 corresponds to the real age 20 and j = 61 corresponds to the real age 80. The accumulation of human capital (reflecting labor productivity) of a household depends on its investment into education. The evolution of the human capital of a household is described by the following equations. h j ¼ h; j ¼1 h j ¼ h j−1

 1 þ nzqj−1 ; j ¼ 2; ⋯; T 1 þ m j−1

ð2Þ

The parameter mj represents the depreciation of human capital. The parameters n and q represent the coefficients of human capital formation in response to investment into education. The level of human capital (labor productivity) affects the wage income earned by the household. We denote the wage rate per efficiency unit at age j by wj. Then the wage rate incorporating the human capital is hjwj. The income consists of both the labor income and transfer income 1 j (sj). Denoting the market discount factor d j ¼ ∐k¼1 for the inð1 þ r k Þ come at age j, the budget constraint of the household is as follows: T X

d j c j þ dT b≤

j¼1

T X


 d j h jw jn j þ s j

ð3Þ

j¼1

empirical findings on household behavior. According to Browning and Crossley (2001), the elasticity of the long term labor supply is equal to zero and the consumption profile across the ages is hump-shaped. Fougere and Merette (1999), Sadahiro and Shimasawa (2002), Shimasawa (2007), and Kim (2011) use a CRRA (constant relative risk aversion) type utility functions (see Table 4 below). Note that education itself increases the utility in Sadahiro and Shimasawa (2002) and Kim (2011). In contrast with this, Fougere et al. (2009) and this paper suppose that the process of education itself causes disutility to the individual. 1−1=γ

1 xj Fougere et al. (2009) use a utility function ∑ β j 1−1=γ

where

j

1/ε + α(hjlj)1− 1/ε]1/(1 − 1/ε), γ is the intertemporal elasticity xj = [c1− j of substitution and ε is the elasticity of substitution between consumption and leisure. This function is the same as the one used by Auerbach and Kotlikoff (1987) except that the human capital hj at age j is multiplied by the leisure lj in the equation expressing xj. This means that the increase in human capital proportionately increases the effective amount of leisure. This assumption leads to the consumption profile by the age not being hump-shaped, but monotonically increasing when the interest rate is larger than the time preference rate. In the original formulation of Auerbach and Kotlikoff (1987), in which human capital is not multiplied by the leisure, the consumption profile satisfies the following equation:

c jþ1 ¼ cj

1 þ α ε wjþ1 1−ε 1 þ α ε wj 1−ε

!ε−γ

β jþ1 d j β j d jþ1

!γ ð5Þ

where wj * is the opportunity cost of leisure at age j and is equal to hjwj + vj, where vj is related to the Lagrange multiplier for the constraint on the leisure to be less than 1 (see Auerbach and Kotlikoff (1987), p. 32). In this equation, the terms of human capital hj and hj + 1 are located in the denominator and the numerator respectively. This creates a hump-shaped consumption profile. However, when the human capital hj is multiplied by the leisure in 1−1=ε

the period utility function, it is equivalent to changing α into αh j . This therefore makes hj and hj + 1 cancel out except for the case where vj is not zero. This leads to the consumption profile not being humpshaped, as we can observe from the following equation derived from the first order conditions: 2 !γ 1−ε 3ε−γ ε−1
β jþ1 d j c jþ1 41 þ α ε h jþ1 h jþ1 w jþ1 þ ν jþ1 5 ¼ 1−ε ε−1
cj β j d jþ1 1 þ αε h h jw j þ ν j

ð6Þ

j

Thus, the representative household faces the following utility maximization problem:

max

T X j¼1

s:t:

T X j¼1

1−1=γ

βj

cj

"

1−1=γ

d j c j þ dT b≤

1þ1=η

A j þ B j ð1−γÞ T X

nj

1 þ 1=η

!#1=γ

1þ1=θ

þ

zj

1 þ 1=θ

1−1=γ

þ βT κ

b 1−1=γ


 d j h jw jn j þ s j

j¼1

ð4Þ Our formulation is distinguished from the other OLG models in three aspects. First, we introduced the survival probability Ψj until the age j and incorporated it in the discount factor βj. Since we investigate the effect of aging on economic growth, it seems important to incorporate the effect of the decreasing mortality rate on the individual household's decision making on savings and labor supply. We therefore estimated and projected the survival probabilities for the years 1970–2100 using the life table maintained by the Korea National Statistical Office. Second, the household utility function is of the form suggested by Kimball and Shapiro (2008). This utility function is consistent with the

None of the utility functions used in the studies mentioned above satisfies either the zero long-term elasticity of labor supply or the hump-shaped consumption profile. The KS utility function satisfies both of these properties. Moreover, the computing time required is quite short. We used the GAUSS program for simulation. Using an IBM PC with core i3-530 (2.93 GHz), it took only 2–3 minutes to obtain the simulation results. The third aspect that distinguishes our model from the others is related to the formation of social human capital. In each period t, 61 overlapping generations (aged from 1 to 61) are living together. Each household is identified by two indices. One is period t and the other is the age j. The literature generally assumes that the initial level of human capital of a new generation is determined by a scaled arithmetic mean of the combined human capital of all the generations in the previous year: ht,1 = π∑Tj = 1 ht − 1, j where ht,1 denotes the human capital of the generation of age j = 1 in the year t and π is a constant. This implies that the population share of each generation does not affect the transmission of human capital to new generations. We argue that this is not the case. For instance, as the population share of older generations becomes larger, the new generation will be affected more by these

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147

141

Table 4 The utility function and the accumulation function of human capital in the related literature.

Fougere and Merette (1999)

Period utility function   1−1=γ 1−1=γ 1 þ κb j 1−1=γ c j

Formation of human capital h i 1 htþ1; jþ1 ¼ ht; j 1þm þ n  zqt; j T

ht;1 ¼ π∑ ht−1; j Sadahiro and Shimasawa (2002)

1 1−1=γ



1−1=γ

cj

1−1=γ

þ αz j

j¼1



ht + 1, j + 1 = (1 − δ)ht, j + B(m kt)ϕ(ht, jzt, j) T

ht;1 ¼ π∑ ht−1; j 

j¼1

 −1

Bouzahzah, De la Croix, and Docquier (2002)

1 1−1=γ

Shimasawa (2007)

1−1=γ 1 1−1=γ c j

ht + j,j + 1 = θ(1 + ξzqt,1)ht,1 ht,1 = ht − 1,1(1 + ξzqt − 1,1) h i 1 htþ1; jþ1 ¼ ht; j 1þm þ n  zqt; j j

1−1=γ 1 , 1−1=γ x j

ht;1 ¼ π∑ ∑ hk; k¼1 j¼1   1 htþ1; jþ1 ¼ ht; j 1þm þ n  zqt; j þ ; expt; j j

1−1=γ

cj

t−1 t−1

Fougere et al. (2009)

Kim (2011)

where 1/ε xj = [c1− + α(hjlj)1− 1/ε]1/(1 − 1/ε) j h
1−1=γ i 1−1=γ 1−1=γ 1 þ ξαz j þ α h jl j 1−1=γ c j

T

ht;1 ¼ π∑ ht−1; j j¼1

ht + 1, j + 1 = (1 − δ)ht, j + B(m kt)ϕ(ht, jzt, j) T

ht;1 ¼ π∑ ht−1; j j¼1

older generations than by the younger generations. We therefor suppose that the initial level of human capital of each generation is determined partly by the simple arithmetic mean and partly by the weighted arithmetic mean of the aggregated human capitals of all the generations existing in the previous year. The economic rationale of this formulation is as follows. Human capital consists of knowledge and know-how. People obtain this knowledge and know-how not only through schooling, but also through selflearning activities such as reading books and through communication and consultation with other people. The school curriculum defines the basic level of human capital. The self-learning activities described are related to the acquisition of more specific and advanced knowledge. We can consider that a book contains knowledge from its author's generation. Since knowledge has the property of non-rivalry, the human capital obtained through books by a representative cohort may be modeled as the simple arithmetic mean of the knowledge in the population. Interactions with other people, such as communication and consultation, transmit knowledge and know-how among a community. Assuming random matching with other people, the transmission of the knowledge though communication and consultation is affected by the age distribution of the population.

3.3. The firm The representative firm produces goods with the labor and capital supplied by the household. The production technology is represented by the Cobb–Douglas function of labor and capital: Y t ¼ AK βt LE1−β t

where Kt denotes the capital stock at period t, LEt denotes the effective labor supply in efficiency units, and Yt denotes the output net of depreciation. Denoting the capital to labor ratio by kt, we can represent the production function by a simple form as follows: β

yt ¼ f ðkt Þ ¼ Akt yt ¼

Yt ; LEt

kt ¼

ð9Þ Kt LEt

ð10Þ

The firm maximizes profit, with the first order conditions for profit maximization being as follows: 0

β−1

r t ¼ f ðkt Þ ¼ Aβkt

3.2. The evolution of social human capital

0

Let hat denote the weighted average of human capital at period t where the weight is the average of both the share of each age in the population and the inverse of the lifespan (which is T = 61). Then:  61  X pt; j =pt þ 1=61 ht; j hat ¼ 2 j¼1

ð8Þ

ð7Þ

where ht,j and pt,j denote the human capital and the population of the household aged j in the year t respectively, while pt denotes the population in the year t.   The initial human capital ht;1 of each household at period t is assumed to equal hat − 1. This is the channel that determines the evolution of economy-wide human capital (productivity), which reflects technological progress. 61 pt; j We also consider the case where hat ¼ ∑ j¼1 h and hat ¼ pt t; j 61 1 ∑ j¼1 ht; j . The simulation result shows high sensitivity to the form 61 of this function (see Subsection 4.3).

wt ¼ f ðkt Þ−f ðkt Þkt ¼

ð11Þ β Að1−βÞkt

ð12Þ

2.3. General equilibrium In the general equilibrium, all the three markets, that is, the goods market, the labor market, and the capital market are in equilibrium. The equilibrium condition for the capital market and the labor market reduces to the above profit maximization conditions of the firm. The intra-temporal equilibrium condition for the goods market is that the savings are equal to the investments. This condition is implicitly used in the identification of aggregate capital stock with the aggregate assets of households where household assets increases with the inflow of household savings. The inter-temporal equilibrium in the goods market is ensured by discounting with interest rates. We denote the initial period (period 1) price of the goods at period t by qt. Then the initial period price of the goods in period t + 1 is equal to the initial period price of the qt . goods in period t discounted by the interest rate at period t: qtþ1 ¼ ð1þr tÞ  1 ,t= Thus, letting q1 = 1, we obtain the following: qtþ1 ¼ t ∏s¼1 ð1 þ rs Þ 1,2,…. This relationship is derived from the condition preventing intertemporal arbitrage.

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The variable nt,j denotes the labor supply of the household with age j in period t and pt, j represents the number of households of age j in period t. The supply of labor in efficiency units at period t is the sum of effective labor supplies of overlapping generations in that period. It is represented by the following equation: LEt ¼

61 X

ð13Þ

ht; j nt; j pt; j

j¼1

The capital stock at period t is equal to the weighted sum of household assets at, j: Kt ¼

61 X

ð14Þ

at; j pt; j

Fig. 4. Income and consumption profile of the individual whose age is 20 in the year 2010.

j¼1

The household asset at,j evolves according to the following equation: atþ1; jþ1 ¼ at; j ð1 þ r t Þ þ yt; j −ct; j ;

at;1 ¼ s

ð15Þ

where s denotes subsidies, which are equal to the bequests received from parents. 4. Simulations 4.1. Calibration of parameters The household parameters were chosen to generate characteristics similar to those recorded in the study of Korean household behavior in Park and Jun (2009). We focused on the following characteristics. First, the income profile is hump-shaped and has a peak in the age of early 50's. Second, the consumption profile is hump-shaped and has a peak in age in the late 50's (Fig. 4). We assume that the values of Aj and Bj vary with age in household utility function. The values of the main parameters are shown in the following table (Table 5). The parameters in the human capital function of the household are as follows. The depreciation parameter starts at 0.05 and increases by 3% as age increases. The parameters n and q related to the production efficiency of education are set at n = 0.1, q = 0.1. These values produce a hump-shaped labor productivity profile of households. Bouzahzah et al. (2002) used a value of q equal to 0.1. In contrast to that, Fougere and Merette (1999) and Fougere et al. (2009) set the value of q at 0.7. The capital share β in the total product net of depreciation is assumed to be 0.3. This is higher than the 0.25 that was used in Miles (1999) and Altig et al. (2001), but is lower than the 0.33 used in Chun (1998). The labor income share that appears in the National Account was around 60% during 2000–2010, but this underestimates the labor share since the income of the self-employed is not counted as labor income, but as capital income.

The initial improvement in labor productivity was assumed to be 2% during the years 1950–2010. This figure was calibrated for our model to produce an average annual growth rate of GDP around 3% during the period 2010–2013. 4.2. Simulation results We consider three scenarios. One is the baseline scenario in which population aging occurs in line with the Statistical Office forecasts. Another is the case of fixed time allocations for education while population aging progresses. In this case, the lifetime profile of the time invested into education is fixed at the same level for all generations as the education time profile across the generations existing at the baseline year 2010. The third is the case of fixed populations, with the population in each year remaining the same as in the year of 2010. 4.2.1. The change of aggregate work time The aggregate work time of the economy decreases as the population decreases, and the rate of decrease becomes greater as population aging progresses in the baseline scenario. In the scenario of fixed education time, the changing rate of aggregate work time shows almost the same pattern as that in the baseline scenario. In the scenario of fixed population, the aggregate work time remains almost constant (Fig. 5). 4.2.2. The growth of labor productivity and the effective labor supply The growth rate of aggregate labor productivity in the baseline scenario is lower than that in the fixed population scenario. This seems to be due to the increases in the population older the 60. The elderly of age greater than 60 have less human capital than the young. As the elderly population increases, the growth rate of aggregate human capital decreases.

Table 5 Values of household parameters. Household parameters

Values

ρ γ η ε κ Aj Bj

0.001 0.25 0.3 1.6 30 Varying with agea Varying with ageb

a The value of Aj increases from 1 at age 20 (j = 1) by 0.04 every year until the age 53 and stays constant for six years and then decreases by 0.06 every year. b The value of Bj decreases from 1 at age 20 by the rate of 3% every year until the age 53 and stays constant for six years and then increases by the rate of 5% every year.

Fig. 5. The Changing Rate of Aggregate Work Time.

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147 Table 6 The growth rate of labor productivity.

Baseline scenario Fixed education time Fixed population

143

Table 8 The growth rate of capital stock.

2011

2020

2030

2040

2050

2060

2100

1.38% 1.33% 1.98%

1.20% 1.02% 1.98%

1.10% 0.79% 1.99%

1.15% 0.71% 2.00%

1.20% 0.71% 2.00%

1.28% 0.82% 1.99%

1.37% 0.94% 1.98%

Baseline scenario Fixed education time Fixed population

2011

2020

2030

2040

2050

2060

2100

4.42% 4.52% 2.83%

3.72% 3.83% 2.56%

3.15% 3.18% 2.26%

3.29% 3.21% 1.98%

3.41% 3.24% 1.83%

2.83% 2.62% 1.83%

1.44% 1.14% 1.92%

Fig. 8. The Growth Rate of Capital Stock. Fig. 6. The Growth Rate of Labor Productivity.

The growth rate of labor productivity (Table 6, Fig. 6) in the baseline scenario is higher than that in the fixed education scenario. This can be explained by the increased education levels in response to higher wage rates in the baseline scenario. Accordingly, the growth the rate of aggregate effective labor supply decreases (Table 7, Fig. 7) as the population aging process advances. Under the fixed education scenario, the decrease would be larger, and the rate of change would reach around −0.5% in the year 2060. 4.2.3. The growth of capital stock The profile of the growth rate of the capital stock in the baseline scenario has a hump in the 2040s and the growth rate is larger than that in the fixed population scenario until the year 2079 (Table 8, Fig. 8). This is due to the increase in the population share of the people aged from around 50 to 70 whose asset size is large. As time passes, the population share of people aged more than 70 whose asset size is small becomes greater. The growth rate of capital in the baseline scenario therefore

becomes lower than that in the fixed population scenario from the year 2080. A similar pattern appears with respect to the fixed education scenario. 4.2.4. The growth rate of wages and the interest rate As the supply of labor decreases, the wage rate increases. The growth rate of wages in the baseline scenario shows a humped-shape across the time period 2010–2100. It is larger than that in the fixed population scenario, where the growth rate of wages approaches zero as time goes by. The growth rate of wages is higher in the fixed education scenario than in the baseline scenario (Fig. 9).

Table 7 The growth of the effective labor supply. 2011

2020

2030

2040

2050

2060

2100

Baseline scenario 2.76% 1.66% 0.75% 0.49% 0.29% 0.02% 0.05% Fixed education time 2.71% 1.47% 0.42% 0.03% −0.25% −0.48% −0.38% Fixed population 1.96% 1.96% 1.96% 1.98% 1.99% 1.99% 1.98% Fig. 9. The Growth Rate of Wages.

Fig. 7. The Growth of the Effective Labor Supply.

Fig. 10. The interest rate.

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Table 9 The growth rate of GDP.

Baseline scenario Fixed education time Fixed population

Table 11 The growth rate of population which is older than 19 (20+). 2011

2020

2030

2040

2050

2060

2100

3.25% 3.25% 2.22%

2.27% 2.17% 2.14%

1.46% 1.24% 2.05%

1.32% 0.97% 1.98%

1.22% 0.78% 1.94%

0.86% 0.44% 1.94%

0.47% 0.08% 1.96%

Population growth rate

2011

2020

2030

2040

2050

2060

2100

−0.16%

0.73%

0.09%

−0.18%

−0.62%

−1.05%

−0.98%

to around 1.5% in the year 2100, which is lower than the roughly 2.0% obtained in the fixed population scenario. It is generally lower in the baseline scenario than in the fixed population scenario. (Table 11). 4.3. The functional forms for social transmission of human capital

Fig. 11. The growth rate of GDP.

Table 10 The growth rate of per-capita GDP.

Baseline scenario Fixed education time Fixed population

2011

2020

2030

2040

2050

2060

2100

1.72% 1.71% 2.22%

1.54% 1.44% 2.14%

1.37% 1.15% 2.05%

1.51% 1.15% 1.98%

1.85% 1.42% 1.94%

1.93% 1.51% 1.94%

1.46% 1.07% 1.96%

The interest rate decreases as the population aging progresses, because the capital stock grows faster than the labor supply. The rate of decrease is lower in the baseline scenario than in the fixed education scenario (Fig. 10).

We consider two alternative functional forms of social human capital transmission and compare the simulation results with those in the baseline scenario. The first supposes that the initial human capital of each generation is equal to the simple arithmetic mean of the human capital of all the generations existing in the previous period. The other supposes that the initial human capital of each generation is equal to the weighted average, where the weight is given by the population share of each generation. As can be seen in Figs. 13–17 below, the growth rate of labor productivity is higher in the case of the simple arithmetic mean than in the baseline scenario, while it is lower in the case of the weighted average. This difference between the two cases is transmitted into different profiles of effective labor supply and also of GDP and per-capita GDP. 4.4. Comparison of the results with related studies Auerbach and Kotlikoff (1987) pointed out that population aging is accompanied by capital deepening. An important part of the economy's capital stock is the accumulated assets for retirement. Thus the rise in the ratio of old to young leads to an increase in the capital labor ratio.

4.2.5. The GDP and per-capita GDP As population aging progresses, the growth rate of the effective labor supply decreases while the growth rate of capital stock is seen to be generally decreasing with a hump in the 2040s. Thus the GDP growth rate decreases as time passes with a small hump in the 2040s. The GDP growth rate in the baseline scenario is lower than that in the fixed population scenario from the year 2022 (Table 9; Fig. 11). Due to the increase in the population aged from 20 to 80 until 2034, the growth rate of GDP in the baseline scenario during the period 2010–2021 is larger than that in the fixed population scenario. The growth rate of per-capita GDP shows a hump-shaped profile across the years (Table 10; Fig. 12). The growth rate reaches a peak of around 1.9% in the period from 2053 to 2066. This seems to be mainly due to the rapid growth of capital stock during this period. It decreases

Fig. 13. The growth rate of labor productivity.

Fig. 12. The growth rate of per-capita GDP.

Fig. 14. The Growth Rate of Effective Labor Supply.

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147

Fig. 15. The growth rate of capital stock.

Fig. 16. The growth rate of GDP.

As a result, wages rise and interest rates fall gradually. The welfare level in the new steady state increases in comparison with the initial steady state by around 7%. On the other hand, Kotlikoff et al. (2007) show that capital weakening is possible in the case of the U.S. due to the increase in effective labor supply. The formation of capital does not keep up with the combined growth of the population and labor productivity in the U.S. As a result, interest rates rise gradually. The reason for these developments lies in the rapid increases of tax rates in order to maintain the fiscal balance. The per-capita GNP triples during the period from 2000–2100. Hviding and Merette (1998) show that population aging brings forth capital deepening and a fall of interest rates in seven OECD countries, using an OLG model with exogenous technological progress. The per-capita GDP falls for all seven OECD countries in spite of the capital deepening. This might appear puzzling; however, it is possible, since the value for labor used in the calculation of the capital labor ratio is the effective labor supply, which reflects the age profiles of both work hours and labor productivity. Since both the work hours and the productivity of the old are lower than that of the young, both the total work hours and the total labor productivity decrease as population aging progresses.

Fig. 17. The growth rate of per-capita GDP.

145

Fougere and Merette (1999) show that the long-run effects of population aging are significantly altered when the formation of human capital is introduced in the model. The population aging increases human capital investment which raises the effective labor supply; this in turn stimulates economic growth. This leads to an increase in the per-capita GDP. The reduction in the national savings rate is less pronounced and the real return on capital declines less markedly. Fougere et al. (2009) refined the model in Fougere and Merette (1999) by incorporating an endogenous labor supply. They applied the model to the Canadian economy and show that population aging leads to capital deepening and to a substantial increase in real wages. This results in the middle-aged workers being more skilled and working more. As a result, the long-term impact of population aging is smoother. The fall of real output per-capita in comparison with the initial steady state between 2015 and 2050 is smaller than that in the case of exogenous investments into education. Sadahiro and Shimasawa (2002) show that a declining population brings forth a positive GDP growth rate via the enhanced labor productivity achieved through the accumulation of human capital. Our findings are in line with those of the preceding studies. Our result shows capital deepening and rising wage rates, and a fall of interest rates as population aging progresses. In comparison with the case of fixed education time, the capital deepening is moderated slightly in the case of endogenous human capital formation. This is reflected in the trend of interest rate in Fig. 10. The growth rate of per-capita GDP decreases due to population aging. The per-capita GDP grows at a rate of around 1.5% in the year 2100 which is lower than the growth rate of around 2.0% obtained under the scenario of fixed population. This is due to the supposition that human capital accumulation is partly affected by the weighted average of the human capital profile by age. If we suppose that human capital accumulation is affected solely by the simple average of human capital profile by age as in Sadahiro and Shimasawa (2002) or Fougere et al. (2009), the speed of human capital accumulation accelerates. As a result, the growth rate of per-capita GDP becomes higher than that under the scenario of fixed population. Again, we may observe that the mechanism of human capital accumulation plays a key role in economic growth. 5. Concluding remarks We constructed a deterministic computable OLG model in which the human capital is endogenously determined. Using this model, we investigated the impact of population aging on the growth of labor supply, capital stock, and economic growth. The simulation shows that population aging decreases the growth rate of the labor supply remarkably. This causes the wage rate to rise. This, in turn, fosters investment into human and physical capital. The growth of human capital, however, is lower than that under the fixed population scenario. In addition, the increase in the growth rate of physical capital partially offsets the decrease in labor supply growth. As a result, the growth rate of GDP decreases from 3.3% in 2011 to 0.5% in 2100, while the growth rate of per-capita GDP reaches a peak at around 1.9% in the period of the 2050s and 2060s and then decreases to around 1.5% in the year 2100. The decrease in GDP and per-capita GDP would have been more severe if there were no change in the investment into human capital. Without the increase in human capital investment, the growth rate of per-capita GDP would stay at around a mere 1.1% while that of GDP would remain below 0.1% in the year 2100. The aging of the population is thus expected to affect the Korean economy greatly. It reduces the growth potential of per-capita GDP by around 25% according to our simulation. We therefore need to keep alert to the trend of population aging and consider appropriate measures in response to this phenomenon.

146

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147

Our simulation results, however, are sensitive to the forms of the social transmission of human capital. In particular, the growth rate of per-capita GDP becomes higher in the case of simple average formulation due to the amplified accumulation of human capital. This emphasizes the importance of human capital transmission among generations and the necessity for further scrutiny of this issue. Our model provides an alternative to the growth accounting approach that is often used in projecting long-run economic growth. The present study shows the possibilities of this alternative and might be regarded as a first step toward that direction. In order to improve the reality of the model for the projection of long-run growth, we need to incorporate the public sector and the foreign sector for a fully-fledged model. These remain as further research topics.

"

1−1=γ

βj

j¼1 T X

cj

1−1=γ

d j c j þ dT b≤

s:t:

1þ1=η

j¼1

T X

!#1=γ

1þ1=θ

nj

A j þ B j ð1−γÞ

1 þ 1=η

þ

zj

1 þ 1=θ

ðA7Þ

j

On the other hand, from the Eq. (A1) for the period t and t + 1, we obtain the following. 2

1þ1=η

!3

1þ1=θ

n jþ1

z jþ1

6A jþ1 þ B jþ1 ð1−γ Þ þ 1 þ 1=η 1 þ 1=θ c jþ1 6 6 ¼6 1þ1=η 1þ1=θ ! 6 cj nj zj 4 þ A j þ B j ð1−γ Þ 1 þ 1=η 1 þ 1=θ

β j d jþ1 β jþ1 d j

n jþ1 ¼ nj

The lifetime optimization problem of a household is as follows. T X

1þ1=η 1þ1=θ !# nj zj Aj h jw j cj ¼ þ þ ð1−γ Þ 1=η Bj 1 þ 1=η 1 þ 1=θ n

7 !γ 7 β 7 jþ1 d j 7 7 β j d jþ1 5

ðA8Þ

Using Eq. (A6), we can rewrite the term within the bracket and obtain the following.

Appendix A. The algorithm solving the household problem

max

"

!γη 

B j h jþ1 w jþ1 B jþ1 h j w j

η ðA9Þ

1−1=γ

þ βT κ

b 1−1=γ

Using Eq. (A6), Eq. (A2) can be simplified as follows, 


 d j h jw jn j þ s j

Bj h jw j

λd j ¼ β j

j¼1

1=γ

1

ðA10Þ

nγη j

nqzq−1

The Lagrangian is as follows.

L¼

T X

1−1=γ

βj

cj

"

1þ1=η

nj

A j þ B j ð1−γ Þ

1−1=γ 1 þ 1=η 2 3 T X
 4 d j h j w j n j þ s j −c j −dT b5 þλ

1þ1=θ

þ

j¼1

zj

!#1=γ

1 þ 1=θ

1−1=γ

þ βT κ

b 1−1=γ

k i From Eqs. (A1) and (A3), using the relationship ∂h ¼ m þnz q hk ∂zi i i   nqzqi ∂hk zi or ∂z hk ¼ m þnzq which shows the independence of (∂hk/∂zj)(1/hk) i

i

1þ1=η

A j þ B j ð1−γ Þ

j¼1

E j≡

The first order condition is as follows: " 1þ1=η 1þ1=θ !#1=γ nj zj ∂L −1=γ þ ¼ β jc j A j þ B j ð1−γÞ −λd j ¼ 0 1 þ 1=η 1 þ 1=θ ∂c j

ðA1Þ

h jw j

1þ1=η

1þ1=θ

nj zj ∂L 1−1=γ þ ¼ −β j c j A j þ B j ð1−γÞ 1 þ 1=η 1 þ 1=θ ∂z j T X ∂hk þλ dk wk nk ¼ 0 ∂z j k¼ jþ1

!#1=γ−1

1þ1=η

nj

1 þ 1=η

1þ1=θ

þ

zj

1 þ 1=θ

1 þ 1=θ

1=θ

¼

B jc jd jz j Ej

;

ðA11Þ

1=η

Ej

⇒z j ¼

E jn j

!θ

d jh jw j

ðA12Þ

On the other hand, from Eq. (A8), we obtain the following.

d jþ1 β jþ1 ¼ dj βj



 1þ1=η 31=γ 2 1þ1=θ n jþ1 z jþ1 −1=γ A jþ1 þ B jþ1 ð1−γÞ 1þ1=η þ 1þ1=θ 7 6 c jþ1 6  1þ1=η  7 1þ1=θ 5 4 nj zj cj A j þ B j ð1−γÞ 1þ1=η þ 1þ1=θ

ðA13Þ

Using Eq. (A11), we can rewrite the bracket term and obtain the following.

ðA4Þ

z jþ1 ¼ zj

!γθ  θ β j d jþ1 B j E jþ1 d j β jþ1 d j B jþ1 E j d jþ1 !γθ   θ ð1−γÞθ βj B j E jþ1 dj ¼ B jþ1 E j β jþ1 d jþ1

ðA5Þ

ðA14Þ

The bequest is determined by the following equation derived from the first order condition Eq. (A4);  b¼

A j þ B j ð1−γ Þ

!

ðA3Þ

From Eqs. (A1) and (A2), we obtain the following equations.

!

B jc jd jz j

1=θ

T
 ∂L X d j h j w j n j þ s j −c j −dT b ¼ 0 ¼ ∂λ j¼1

þ

1=θ

¼

B jz j

∂L −1=γ −λdT ¼ 0 ¼ βT κb ∂b

1 þ 1=η

1þ1=θ

zj

From Eqs. (A6) and (A11), we obtain the following. 1=η

" 1þ1=η 1þ1=θ !#1=γ−1 nj zj ∂L 1=η 1−1=γ þ ¼ −β j c j A j þ B j ð1−γ Þ B jn j 1 þ 1=η 1 þ 1=θ ∂n j þ λd j h j w j ¼ 0 ðA2Þ

nj

T ∂hk 1 X dk hk wk nk ∂z j hk k¼tþ1

B jc jn j

"

i

from the index k, we obtain the following equation.

βT λdT

γ

¼ κγ

hT wT −η n BT T

ðA15Þ

1=η

¼

B jc jn j

h jw j

ðA6Þ

Therefore, if {Zj} is given, then {hj} is determined. Now, if n1 is given, nj is determined by Eq. (A9). If nj is given, then, from Eqs. (A7) and

K.-H. Choi, S. Shin / Economic Modelling 50 (2015) 138–147

(A12), Cj,Zj is determined. The bequest b is determined by Eq. (A15). The value of n1 is determined by the budget constraint Eq. (A5). References Altig, D., Auerbach, A.J., Kotlikoff, L.J., Smetters, K., Walliser, J., 2001. Simulating fundamental tax reform in the US. Am. Econ. Rev. 91, 574–595. Auerbach, A.J., Kotlikoff, L.J., 1987. Dynamic Fiscal Policy. Cambridge University Press, Cambridge. Bouzahzah, M., De la Croix, D., Docquier, F., 2002. Policy reforms and growth in computable OLG economies. J. Econ. Dyn. Control. 26, 2093–2113. Browning, M., Crossley, T.F., 2001. The life-cycle model of consumption and saving. J. Econ. Perspect. 15, 3–22. Chun, Y., 1998. Inter-generational and Intra-generational Redistribution Effect of the Reform Recommendations of National Pension System. Korea Institute of Public Finance (Written in Korean). Fougere, M., Merette, M., 1999. Population ageing and economic growth in seven OECD countries. Econ. Model. 16, 411–427. Fougere, M., Harvey, S., Mercenier, J., Merette, M., 2009. Population ageing, time allocation and human capital: a general equilibrium analysis for Canada. Econ. Model. 26, 30–39.

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Hviding, K., Merette, M., 1998. Macroeconomics Effects of Pension Reforms in the Context of Ageing: OLG Simulations foro Seven OECD Countries. OECD Woring paper no. 201. Kim, K.-H., 2011. Comparison of employment policies aimed to deal with aging population with respect to their effect of improving growth rate. Econ. Anal. 17 (4), 52–98 (Written in Korean). Kimball, M.S., Shapiro, M.D., 2008. Labor supply: are the income and substitution effects both large or both small? NBER working paper Kotlikoff, L.J., Smetters, K., Walliser, J., 2007. Mitigating America's demographic dilemma by pre-funding social security. J. Monet. Econ. 54, 247–266. Miles, D., 1999. Modelling the impact of demographic change on the economy. Econ. J. 109, 1–37. Park, M.-H., Jun, B., 2009. The Effect of National Pension Reform on Households' Savings and Labor Supply. Korea Institute of Public Finance (written in Korean). Sadahiro, A., Shimasawa, M., 2002. The computable overlapping generations model with an endogenous growth mechanism. Econ. Model. 20, 1–24. Shimasawa, M., 2007. Population ageing, policy reforms and economic growth in Japan: a computable OLG model with endogenous growth,. Econ. Bull. 3 (49), 1–11.