Silviase/jgov_v1.4_full_ann huggingface dataset:
## 様式2

## 政策ごとの決算との対応について(個別表)【一般会計】

（所管）内閣府（組織）內閣本府等（除＜警察庁、宮內庁、公正取引委員會、金融庁、消費者庁）

(单位:千円)

政策評価体系	組織	項	事項	虛出予算額	前年度綜越額	予備費使用額	流用等增△減額	予算決定後移替增△減額	虛出予算現額	支出減虛出額	翌年度綜越額	差引額	備考
1 公文書等管理政策	2,214,815	121,667	0	0	0	2,336,282	2,330,335	0	0	5,947
適正な公文書管理の実施	內閣本府	公文書等管理政策	公文書等管理政策に必要な経費	16,268	0	0	0	0	16,268	14,270	0	1,997
0	0	0	0	0	0	0	0	0
內閣本府	独立行政法人国立公文書館運營費	独立行政法人国立公文書館運營費交付金に必要な経費	1,944,919	0	0	0	0	1,944,919	1,944,919	0	0
0	0	0	0	0	0	0	0	0
內閣本府	独立行政法人国立公文書館施設整備費	独立行政法人国立公文書館施設整備に必要な経費	253,428	121,667	0	0	0	375,095	371,145	0	3,950
2 政府廠報·広誌	4,062,485	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	4,062,485	4,028,982	0	33,502
政府廠報·広誌による政府施策の理整、協力の促進	內閣本府	政府廠報	政府廠報の実施等に必要な経費	4,062,485	0	0	0	0	4,062,485	4,028,982	0	33,502
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
3 遺業化学兵器廠業処理事業の推進	遺業化学兵器廠業処理事業	內閣本府	遺業化学兵器廠業処理事業に必要な経費	遺業化学兵器廠業処理事業に必要な経費	20,770,993	2,694,570	0	0	0	23,465,563	20,334,343	2,170,171	961,048
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
4 原子力貿害対策	12,871,875	0	0	0	0	0	12,871,875	0	0	0
0	0	0	0	0	0	0	0	0	0
原子力貿害対策の充実·強化	內閣本府	原子力貿害対策	原子力貿害対策に必要な経費	12,871,875	0	0	0	0	12,871,875	0	0	0
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
5 経済財政政策	3,623,435	0	0	0	0	0	3,623,435	3,449,815	0	173,619
0	0	0	0	0	0	0	0	0
経済財政政策の推進	內閣本府	経済財政政策	経済財政政策の企画立案等に必要な経費	3,623,435	0	0	0	0	3,623,435	3,449,815	0	173,619
0	0	0	0	0	0	0	0	0
6 地域活性化政策	1,410,568,045	0	0	0	0	0	1,411,081,224	1,062,762	1,398,806,960	11,211,501
0	0	0	0	0	0	0	0	0
地域活性化の推進	內閣本府	地域活性化政策	地域活性化政策の推進に必要な経費	1,946,750	513,179	0	0	0	2,459,929	1,062,762	806,960	590,206
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
## 様式3

## 政策ごとの決算との対応について(個別表)【特別会計】

（所管）内閣府東日本大震災復興特別会計（除く警察庁、金融庁、消費者庁）

(单位: 千円)

政策評価体系	勘定	項	事項	歲出予算額	前年度線越額	予備費使用額	予算総則の規定による経費增額	流用等增△減額	歲出予算現額	支出済歳出額	翌年度線越額	差引額	備考
1 公文書等管理政策	47,527	0	0	0	0	47,527	47,527	0	0
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
適正な公文書管理の実施	東日本大震災復興独立行政法人国立公文書館運營費交付金必要な経費	東日本大震災復興に係る独立行政法人国立公文書館運營費交付金必要な経費	47,527	0	0	0	0	47,527	47,527	0	0
2 政府広報·広聴	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
250,828	0	0	0	0	250,828	250,565	0	262
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
政府広報·広聴による政府施策の理解、協力の促進	地域活性化等復興政策費	政府広報の実施等に必要な経費	250,828	0	0	0	0	250,828	250,565	0	262
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
4 原子力災害対策	2,740,273	0	0	0	0	2,740,273	60,852	1,843,303	836,116
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
原子力災害対策の充実·強化	原子力の安全確保に必要な経費	原子力の安全確保に必要な経費	2,740,273	0	0	0	0	2,740,273	60,852	1,843,303	836,116
5 経済財政政策	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
1,180,526	0	0	0	0	1,180,526	744,424	0	436,101
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
経済財政政策の推進	地域活性化等復興政策費	経済財政政策に関する調査等に必要な経費	1,163,480	0	0	0	0	1,163,480	735,996	0	427,483
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
経済財政政策費	経済財政政策に関する調査等に必要な経費	17,046	0	0	0	0	17,046	8,427	0	8,618
6 地域活性化政策	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
150,000	0	0	0	0	150,000	63,080	0	86,919
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
地域活性化の推進	地域活性化政策費	地域活性化政策の推進に必要な経費	150,000	0	0	0	0	150,000	63,080	0	86,919
10 防災政策	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
646,612	0	0	0	0	646,612	330,745	31,820	284,046
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
防災政策の推進	防災政策費	防災基本政策に必要な経費	646,612	0	0	0	0	646,612	330,745	31,820	284,046
14 男女共同参画社会の形成の促進	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
87,128	0	0	0	0	87,128	73,785	0	13,342
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
男女共同参画社会の形成の促進	地域活性化等復興政策費	男女共同参画社会の形成の促進に必要な経費	76,672	0	0	0	0	76,672	66,002	0	10,669
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
男女共同参画社会の形成の促進に必要な経費	10,456	0	0	0	0	10,456	7,783	0	2,672
21 科学に関する重要事項の審議等	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
33,255	0	0	0	0	33,255	20,536	0	12,718
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
科学に関する重要事項の審議及び研究の運締	日本学術会議	科学に関する重要事項の審議等に必要な経費	33,255	0	0	0	0	33,255	20,536	0	12,718
(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數	(0) ∅内數
合計	5,136,149	0	0	0	0	5,136,149	1,591,517	1,875,123	1,669,507

（注）1. 政策評価の対象となる計数及び政策評価の対象外の計数で政策に関連付けられるものを掲記している。

2. 下段<>外書きは、複数の政策にまたがる計数(例:施設整備費等)及び複数の政策にまたがると整理できる計数であり、総額の「内数」で掲記し、

3. 東日本大震災復興特別会計については、内閣府省所管分のみ掲記している。
ESRI Discussion Paper Series No.96

# Population ageing, policy reforms and endogenous growth in Japan: a computable overlapping generations approach

by

Manabu SHIMASAWA

April 2004

Economic and Social Research Institute

Cabinet Office

Tokyo, Japan
# Population ageing, policy reforms and endogenous growth in Japan: a computable overlapping generations approach

by

Manabu SHIMASAWA $ ^{*} $

Visiting Fellow, Economic and Social Research Institute, Cabinet Office; Akita

Keizaihoka University

April, 2004
## (Abstract)

We have developed a multi-period computable endogenous growth overlapping generations model generated by the accumulation of human capital. To study whether ageing itself and policy reform that aims to cope with aging make any quantitative impact through human capital formation on the Japanese economy, we simulate two policy change scenarios and compare the results of those with endogenous growth to those with exogenous growth. Three main results are obtained: (1) the human capital investment is important to assure sustained positive economic growth; (2) policy changes, pension reform, and fiscal consolidation promote human capital accumulation and thus boost economic growth; and (3) the traditional exogenous growth OLG model underestimates the effect of policy reform, comparing the endogenous growth of the OLG model. The robustness of these results is guaranteed from the sensitivity tests.

JEL classification : C62; C63; C68; E27; E66; J11; J24; O41; O53;

keywords: Overlapping generations; human capital; endogenous growth; population ageing
### 1. Introduction

Japan is now, like most developed countries, experiencing the ageing of its population. Moreover, ageing in Japan is expected to progress further at a serious pace. As a result, Japan seems to be the eminent ageing society in the world at the beginning of this century.

As is well known, economic growth is mainly determined by physical capital, labor force, and technical progress. Especially, technical progress is the most critical factor for sustained economic growth from the viewpoint of standard growth theory. Moreover, Romer (1986, 1990) and Lucas (1988) pointed out that technical change has a positive relation to human capital, which also has a positive relation to population levels. And if so, ageing reduces the working age population, and thus technical change declines. Consequently, ageing lowers the economic growth rate through the following three channels: labor supply, capital supply, and technical progress via human capital. Meanwhile, Rebelo (1991) studied the effects of policy changes on economic growth under an endogenous growth framework. He found that policy changes have cumulative effects on economic growth in the case of endogenous growth, unlike that of exogenous growth.

A number of papers written after the seminal study by Auerbach and Kotlikoff (1987) have examined the impacts of ageing and policy changes by using computable general equilibrium models with the overlapping generations. They found outcomes such as a sharp reduction in the national savings rate and economic welfare in the long run. These studies, however, were based on the exogenous growth model initiated by Solow (1956) and Swan (1956), and disregarded interrelations that might exist between population change and technical progress, and between policy reform and economic growth. In this respect, those studies seemed to be incomplete. The few exceptions are the papers of Fougère and Mérette (1999), Bouzahzah, De la Croix and Docquier (2002), and Sadahiro and Shimasawa (2003), which endogenize the rate of labor productivity growth.

We develop a more realistic endogenous growth OLG model by allocating time to education for accumulating human capital. And we analyze numerically the impacts of ageing on the Japanese economy without a priori assumptions about the relation between population change and technical progress.
Moreover, we also conduct some alternative simulations to study the impacts of the policy changes.

More closely related to our contribution is a recent paper by Bouzahzah, De la Croix and Docquier (2002). The authors study the effects of demographic change on the economy by using a computable general equilibrium model with overlapping generations of agents and an endogenous growth specification à la Uzawa-Lucas. While similar to our approach, there are important differences, and three are worth noting: (i) as agents live for 60 periods in our model, one period in the model is approximately equivalent to one year of the real world. Thus we are succeed in modeling realistic population dynamics capable of capturing complicated patterns of "baby boom and bust" along the transition path; (ii) we don't assume that the starting point of the simulation is in a steady state--thus the economic variables, e.g. individuals' asset profiles, capital-labor ratio, behave more realistically; and (iii) we take a more careful calibration of the model to actual fiscal/public pension conditions and institutions. Thus we can compare the simulation results with the actual economy appropriately.

The rest of the paper is organized as follows. Section 2 overviews the population projection and the public pension program in Japan. Section 3 depicts the model. In Section 4 we present the calibration, the scenarios, and results. Finally, Section 5 concludes, summarizes the paper, and indicates some policy implications.

### 2. Population Projection and Public Pension Program in Japan

First, following the National Institute of Population and Social Security Research (2003), we overview a population projection for Japan. Though the total population of Japan was 126.93 million in 2000, it is expected to gradually increase until its peak of 127.74 million in 2006, then turn to decrease, to about 100.6 million in 2050. The population decline rate is 0.5 percent (annual rate) on the average during these periods (2006/2050). As for other developed countries, their populations begin to decrease until 2030 at the latest in all countries except for the United States. This projection shows that Japan will soon enter into the process of population decline. The old-age dependency ratio increases from the current 25.5 percent to the 50.0 percent.
range in 2030, then eventually up to 66.5 percent in 2050. The births per thousand is expected to decline from 9.4 permillage in 2001 to 8.0 permil in 2013, and it continues to decline, reaching 7.0 permil in 2035 and falls up to 6.7 permil in 2050.

Next, we overview the Japanese public pension system, which was established in 1941. That system, however, is inferior to present one regarding the range of the targeted people. This system, namely, didn’t include self-employed people and agricultural workers. In 1959, the National Pension Law was enacted and was enforced on a full scale in April 1961. As a result, every citizen age 20 and over was covered by some public pension programs that ensured the payment of pensions for their old age.

In 1985, a fundamental reform was carried out mainly to ensure fair benefits and burdens among different pension programs. A basic pension program was introduced specifically to improve the employee pension programs divided among occupations. The previous employee pensions were positioned as providing additional benefits on top of this basic pension program.

The Japanese public pension system thus came to consist of a two-tier pension program: the first being the “basic pension program” (national pension program), and the second tier is the “employee pension program.” They are mandatory, financed through contributions, and essentially pay-as-you-go systems (see Figure 1) $ ^{1} $ . The age of eligibility will gradually rise to 65 by 2025 (for males) and 2030 (for females).

The first tier pension program includes all the citizens age 20 and over.

The second tier pension program includes only employees. Contributions are shared between employees and employers (in 2003 13.58 percent of the standard monthly wages in total), and benefits are earnings based in this second tier pension program (the average gross replacement rate about 60 percent in 2003).

The ratio of the social security benefit expenditure to GDP is about 7.6 percent in 2001. And the basic pension program, the first tier, received 17.6905 trillion yen, about 3.5 percent of GDP, from the central government as a subsidy in 2001. Each public pension program has a reserve to implement the smooth
and stable operation of it. These funds amount to 238.0092 trillion yen, or about 47.4 percent of GDP in same fiscal year.

### 3. The Model Structure

In this section, we present the endogenous growth OLG model, consisting of five sectors: households, human capital, firms, government, and public pensions. There is a representative individual for each generation in the households sector. Each individual at age 21 maximizes his/her intertemporal utility function with consumption. The representative firm maximizes its profits under the production function. The model is a one-country closed model, where not only the goods market but also factor markets are perfectly competitive. Details of each sector follow.

#### 3.1. Household Sector

The overlapping generations model used in this paper is based on the life cycle theory of consumptions/savings behavior. We consider an economy in which every person lives for a fixed number of periods. Each generation enters the labor market at age 21 ( $ 1^{st} $ period), retires at age 64 ( $ 44^{th} $ period), is granted a pension at age 65 ( $ 45^{th} $ period), and dies at age 80 ( $ 60^{th} $ period). These are rational, forward-looking agents. His/her utility function may be specified thus $ ^{2} $ :

$$ U_{i}=\sum_{j=l}^{60}\left(\frac{1}{1+\rho}\right)^{j-l}\frac{\left[c_{i,j}\right]^{I-\gamma}}{1-\gamma} $$

where i refers to the i th generation, j refers to the j th period of life, $ \rho $ is the pure rate of time preference, and $ \gamma $ is the reverse of the elasticity of intertemporal substitution. The arguments of the utility function are the consumption per period $ (c_{i,j}) $ .

His/her intertemporal budget equation may be described as follows:

$$ \begin{aligned}\sum_{j=1}^{44}PDV_{i,j}(I-\tau w_{t}-\tau p_{t})w_{t}h_{i,j}(I-e_{i,j})\end{aligned}\begin{aligned}=\sum_{j=1}^{60}PDV_{i,j}(I+\tau c_{t})c_{i,j}+\sum_{j=45}^{60}PDV_{i,j}p_{i,j}\end{aligned} $$

where PDV refers to the factor of the present discounted value, $ w_{t} $ is the wage
rate at time t, $ h_{i,j} $ is the human capital stock of generation i at age j, $ e_{i,j} $ measures the time invested in education of generation i at age j, $ \tau w_{t} $ is the labor income tax rate at time t, $ \tau p_{t} $ is the public pension contribution rate at time t, $ \tau c_{t} $ is the consumption tax rate at time t, and $ p_{i,j} $ stands for pension benefit of generation i at age j. Each generation maximizes his/her utility function (Equation (1)) under a budget constraint (Equation (2)). With the maximization procedure, the following Euler equations can be solved, concerning consumption per period.

$$ c_{i,j}=\left\{\frac{1+r_{t}\left(1-\tau r_{t}\right)}{1+\rho}\right\}^{\frac{l}{\gamma}}\left\{\frac{1+\tau c_{t-l}}{1+\tau c_{t}}\right\}^{\frac{l}{\gamma}}c_{i,j-l}\quad,\quad C_{t}=\sum_{j=l}^{60}N_{t,j}c_{i,j} $$

where $ N_{t,j} $ measures the number of the people of age j at period t, and $ C_{t} $ is the aggregated consumption at time t.

Maximizing with respect to the educational investment gives the following result:

$$ e_{i,j}=\left\langle\frac{\theta l_{i,j+1}\left(1-\tau w_{t+1}-\tau p_{t+1}\right)w_{t+1}}{\left\{1+\left(1-\tau r_{t+1}\right)r_{t+1}\right\}\left\{\left(1-\tau w_{t}-\tau p_{t}\right)w_{t}\right\}}\right\}^{\frac{1}{1-\theta}} $$

where $ I_{i,j} $ stands for the time allocated to labor activity of generation i at age j, and $ \theta $ measures the elasticity of human capital production with respect to the

fraction of time allocated to education, $ \theta\in(0,1) $ This equation shows that

the educational investment increases with the discounted level of future net wages, but decreases with the current net wage, which means an opportunity cost. We can obtain the following physical wealth accumulation equation:

$$ a_{i,j}=a_{i,j-l}\left\{l+r_{t}\left(l-\tau r_{t}\right)\right\}+(l-\tau w_{t})w_{t}h_{i,j}(l-e_{i,j})-(l+\tau c_{t})c_{i,j}\\ ,P A_{t}=\sum_{j=l}^{60}N_{t,j}a_{i,j}(\equiv K_{t}^{S}) $$

where $ a_{i,j} $ is physical wealth asset of generation i at age j, $ r_{t} $ is the interest rate at time t, $ \tau r_{t} $ is the tax rate on interest income at time t, and $ PA_{t} $ is the aggregated private asset at period t.

Each generation optimally allocates his/her total time (normalized as unity) into educational activity $ (e_{t,j}) $ and labor activity $ (I_{t,j}) $ . Therefore, effective labor supply for each period is defined as

$$ L_{e,t}^{S}\equiv\sum_{j=l}^{44}N_{t,j}h_{t,j}\left(I-e_{t,j}\right) $$
#### 3.2. Human Capital Sector

This sector’s formula is be largely attributable to Fougère and Mérette (1999) and Sadahiro and Shimasawa (2003).

First, the accumulation of human capital is:

$$ h_{i,j+l}=\frac{h_{i,j}}{1+\delta_{h}(j)}+\varPsi e_{i,j}^{\theta}h_{i,j} $$

where $ \delta_{h}(j) $ is the exogenous human capital depreciation rate and $ \psi $ is a scaling factor. The depreciation rate is a function of age and has been calibrated to replicate a realistic Japanese earnings profile $ ^{3} $ . This allows us to compare, as we do in Section 4, the results of the endogenous growth OLG version with that of the exogenous growth OLG version.

Equation (7) indicates that new human capital is produced by only existing human capital and education time. No other inputs are required. And the equation also expresses that the human capital stock increases most markedly where all available time is allocated to schooling investment (e=1). Conversely, the human capital decreases by the order of depreciation where he/she invests nothing for schooling activity (e=0).

The initial human capital level of the new generation is assumed to include a certain percentage of the previous generation’s accumulated human capital.

$$ h_{i,l}=\pi\sum_{k=l}^{i-l}\sum_{g=l}^{i-l}h_{k,g} $$

The parameter n is calibrated to replicate the same effective labor productivity level at 2001 in Japan. This equation models the basic educational institution and plays a role in transmitting to the newcomer at time t an initial human capital stock that is equivalent to the fraction of the human capital accumulated by its previous generations as a kind of social bequest.

Aggregate human capital is defined as:

$$ H_{t}=\sum_{j=l}^{44}h_{i,j}N_{t,j} $$

#### 3.3. Firm Sector

The input/output structure is represented by the Cobb-Douglas production function with constant return to scale. The firm decides the demand for
physical capital ( $ K^{D} $ ) and effective labor ( $ L^{D} $ ) to maximize profits with the given factor prices of wage and rent, which are determined in the perfect competitive markets.

$$ Y_{t}=A K_{t}^{\alpha}L_{e,t}^{I-\alpha}\quad,\quad K_{t}=K_{t}^{D}\quad,\quad L_{e,t}=L_{e,t}^{D} $$

$$ K_{t}=I_{t}+\big(1-\delta\big)K_{t-I} $$

$$ r_{t}=\alpha A K_{t}^{\alpha-l}L_{e,t}^{l-\alpha}\mathrm{-}\delta\mathrm{~,~}w_{t}=\left(l\mathrm{-}\alpha\right)A K_{t}^{\alpha}L_{e,t}^{-\alpha} $$

where Y is output, a stands for capital income share, A is a scale parameter, K is the physical capital stock, and $ L_{e} $ is the effective labor.

#### 3.4. Government Sector

The government sector issues bonds and collects three types of taxes as its revenue; wage tax, consumption tax and capital tax. And government expenditure is restricted to subsidy to pension sector, public goods expenditure, and interest payments on the public debt. The government budget constraint in each period may be written as

$$ T_{t}=\tau w_{t}w_{t}L_{e,t}+\tau c_{t}C_{t}+\tau r_{t}r_{t}K_{t}\quad,\quad G_{t}=g Y_{t} $$

$$ D_{_{t+l}}=G_{_{t}}+SUBP_{_{t}}+(I+r_{_{t}})D_{_{t}}-T_{_{t}} $$

where $ G_{t} $ stands government expenditure at time t, $ T_{t} $ denotes tax revenue at time t, $ D_{t} $ denotes public debt at time t, $ SUBP_{t} $ is the subsidy to pension sector at time t, and g is a fraction of GDP.

As the government decides the tax rate according to the following intertemporal budget constraints, the budget does not have to balance for each period. Here, the wage tax rate is endogenously determined according to the difference of government revenues and government expenditure.

$$ D_{t}+\sum_{i=0}^{\infty}\left(G_{t+i}+SUBP_{t+i}\right)/\prod_{j=0}^{i}R_{t+j}=\sum_{i=0}^{\infty}T_{t+i}/\prod_{j=0}^{i}R_{t+j} $$

where $ R \equiv I/I + r_{t} $

#### 3.5. Public Pension Sector

The pension sector grants a pension to the retirement generations while the pension contribution is collected from the working generations.

$$ B_{t}=\sum_{j=I}^{r e t}N_{i,j}\tau p_{t}w_{t}h_{i,j}l_{i,j} $$

where B stands for the aggregated pension contribution.
As we saw in Section 2, pension benefit consists of basic pension (first tier) and employee pension (second tier) in Japan. Thus we represent the pension benefit as

$$ p_{i,j}=p b_{i,j}+p e_{i,j} $$

$$ p e_{i,j}=\beta\frac{1}{r e t}\sum_{j=1}^{r e t}w_{t}h_{i,j}l_{i,j} $$

$$ P_{t}=\sum_{j=ret+1}^{60}N_{i,j}p_{i,j} $$

where pb is the benefit of the basic pension, pe is the benefit of the employee pension, $ \beta $ denotes replacement rate, ret stands for retirement age, and P is the aggregated pension benefit.

Here, the budget constraint of the pension sector can be shown as follows

$$ F_{_{t+l}}=(I+(I-\tau r)r_{_{t}})F_{_{t}}+SUBP_{_{t}}+B_{_{t}}-P_{_{t}} $$

where $ F_{t} $ represents a reserve of the public pension at time t.

The subsidy to the basic pension program from the government can be shown as follows.

$$ S U B P_{t}=\pmb{\xi}_{t}\sum_{j=r e t+1}^{60}N_{i,j}p b_{i,j} $$

where $ \xi $ is a government subsidy rate on basic pension at time t.

The pension contribution rate is endogenously determined to keep this budget constraint (19).

#### 3.6. Equilibrium Condition

To close the model structure, the following two market-equilibrium conditions must be hold. The first condition is the equilibrium in the financial market.

$$ K_{_{t}}+D_{_{t}}=PA_{_{t}}+F_{_{t}} $$

The second condition is the equilibrium in the goods market.

$$ Y_{t}=C_{t}+G_{t}+SUBP_{t}+I_{t} $$

In the model simulation, private investment $ (I) $ is determined by using this equilibrium condition.

### 4. Simulation Results