Discussion Paper No. 700

FROM DUTY TO RIGHT:
THE ROLE OF PUBLIC EDUCATION

IN THE TRANSITION TO
AGING SOCIETIESYoshiaki Sugimoto
Masao Nakagawa

October 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

From Duty to Right: The Role of Public Education in theTransition to Aging Societies
Yoshiaki Sugimoto and Masao Nakagawa
October 9, 2007

Abstract
This paper argues that currently advanced, aging economies experienced a qualitative change
in the role of public education during the process of industrialization. In the early phases of
the Industrial Revolution, public education was regarded as a duty that regulated child laborand thereby discouraged childbirth. As these economies developed and the population aged,
younger generations came to view public education as a right, whereas older generations desirousof other public services became more politically in‡uential. The eventual policy bias in favorof the elderly placed a heavier education burden on the young, inducing them to have fewer
children. This vicious cycle between population aging and the undersupply of public education
may have decelerated the growth of advanced economies in the last few decades.

Keywords: Compulsory Education; Generational Con‡ict; Fertility; Growth.
JEL Classi…cation: D70; H50, J10; J20; O40.

Sugimoto: Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan (e-mail: sugimoto@iser.osaka-u.ac.jp); Nakagawa: Graduate School of Economics, Osaka University, 6-1
Mihogaoka, Ibaraki, Osaka 567-0047, Japan (e-mail: mnakaga@iser.osaka-u.ac.jp). The authors are grateful to
Mayumi Kawasugi and Akira Nakamura for excellent research assistance, and Charles Y. Horioka, Shinsuke Ikeda,
Fumio Ohtake, Yoshiyasu Ono, Shizuka Sekita, Elmer Sterken, Shinji Takenaka, and seminar participants at Kobe
University; Kyoto University; and the Osaka University Forum 2007 in Groningen, Netherlands, for their useful comments and encouragement. This research is supported by a JSPS Grant-in-Aid for Scienti…c Research (No. 18830036).

1

1

Introduction

This study hypothesizes that the origin of population aging, pervasive in developed economies,
dates back to the educational reforms during the era of the Industrial Revolution. We argue thatthe qualitative e¤ect on fertility of education policy changed with the individual stances on public
education. In the early phases of the Industrial Revolution, investment in human capital was notdesirable for low-income households needing child labor. Compulsory education thus worked asa ‘duty’ that regulated child labor and thereby discouraged childbirth. By contrast, in the later
phases of the Industrial Revolution, younger generations came to regard education investment as
desirable. That is, public education was desired as a ‘right’. The undersupply of public educationin the postwar period placed the …nancial burden of child education on households, and thereby
discouraged childbirth.
A question to be raised is how public schooling arose and which factors have a¤ected subsequent
education policies. To tackle this question, the present article highlights demographic factors as thedeterminants of social spending, for currently advanced economies over the modern period. Until
the early 19th century, population expansion gradually raised the demand for a public education
system by promoting skill- (age-) biased technological progress. In the postwar period, on the other
hand, older generations, who have increased their population share and political representation,
have sought national resources for their own well-being.1
In fact, one of the postwar aging societies, Japan, has improved the relative well-being of theelderly in some respects.2 Figure 1 plots the trends in population aging (the dotted line) and theallocation of public spending between the young and the old (the solid line). Noting that old-age
and education expenditures are both in per capita terms, one …nds that the public service allocation
1

In the U.S., for example, the American Association of Retired Persons (AARP) and the National Council on
Aging (NCOA) increased their political power in the 1960s (Longman, 1987, p. 234). According to a recent article of
The Economist (2005), the AARP has by far the largest members (35 millions) among the politically active groups
in the country.
2
There is some evidence for other countries. Pampel (1994) conducts econometric analysis by using the 1959–1986
annual data for 18 advanced industrial democracies. The result shows that, in the absence of class-based corporatism
and strong leftist parties, a large aged population tends to exacerbate age inequality in public spending in favor of theelderly. Preston (1984) shows that the aged American society reduced poverty among the elderly by the expansionof social security bene…ts in contrast to raising poverty among children. Poterba (1997, 1998), using the U.S. panel
data over the 1960–1990 period, …nds that an increase in the fraction of the elderly in a jurisdiction is associatedwith a signi…cant reduction in per-child education spending. For earlier periods, the regression results of Lindert(1994), based on the 1880–1930 panel of 21 (mostly currently advanced) countries, indicate that aged countries
tend to have large public spending on social transfers including pension and health subsidies, and low primary- andsecondary-school enrollment rates.

2

is shifting over the period in favor of the elderly. Such a trend would be attributed in large part toincreased political pressures from retiring workers and pensioners.
These arguments assert that demographic changes and public policies have interacted witheach other in the process of industrialization. In particular, developed economies may experiencea vicious cycle between social aging and the undersupply of public education, possibly leading toa population decline and a slowdown in economic growth.3 This spiral can therefore be one of thepotential forces for the postwar convergence across advanced economies. The main objective of thispaper is to develop a theoretical framework to understand the socioeconomic structures of suchphenomena and to consider its policy implications.
The growth theory developed features …ve properties deserving special attention. First, parents
face a trade-o¤ between the quantity and quality of children, as shown in standard models of fertility
and education such as Becker et al. (1990) and Galor and Weil (2000).4 Second, public and private
education are substitutes for each other in terms of skill acquisition. Those two properties together
generate a potential link between education policy and fertility: in response to the undersupplyof public education, for instance, parents may spend more on private education by having fewer
children. Third, public education compulsorily takes away the time children have for paid work.
Hence, their forgone wages equal the opportunity costs of child education. Fourth, technologicalprogress is assumed to be skill- (age-) biased. This improves the productivity of adult workers overthat of children, and thus makes sending children to school more bene…cial than having them work.5
Finally, older generations prefer social services that directly and immediately bene…t themselves,
such as pensions and social security, over public education. This leads to age-cohort di¤erences in
the willingness to enlarged education policy.6
3

The undersupply of public education does not necessarily mean a decline in public expenditures for education
per pupil. Indeed, it means that the provision of public education becomes insu¢ cient for the given level of economic
development.
4This paper extends the uni…ed growth theory developed by Galor and Weil (2000), who explore the transition fromMalthusian stagnation to modern growth through the demographic transition. In contrast to their representativeagent model, this article takes a political economy approach in analyzing the demographic trend since the earlyIndustrial Revolution. The extended model also allows for the other four properties discussed.
5
A similar mechanism is proposed by Hazan and Berdugo (2002). However, their single-agent model makes no
distinction between public education and private education.
6
Using the American National Election Survey for 1988, Vinovskis (1993, pp. 202–209) …nds that additional federal
assistance for public schools is supported by 77.1 % of interviewees aged through 18–29, and only by 46.9 % of those
aged 70 and above. Furthermore, his multiple classi…cation analysis based on the same data reveals that, after
controlling for the e¤ects of other factors (such as sex, race, education, family income and so on), there remains thetendency of the elderly to oppose federal funding for public schools.

3This paper also establishes a more solid foundation for the recent and growing literature on
demographic change and macroeconomics. It appears that no established theory has examined theinterrelationship between public education, private education and fertility, despite its importancefor the assessment of education policy. As discussed below, while there are three related seminal
theories, these can be interpreted as partial theories from the perspective of this research, in thatnone encompasses the rise and fall of education policy over the long term.
First, Galor and Moav (2006) demonstrate the mechanism of the birth of public schooling on
the course of industrialization. Unlike this research, however, their growth theory analyzes themonotonic evolution of education policy, excluding the possibility of its aggravation in maturedevelopment stages. Their model also assumes away demographic changes so as to highlight thepro…t rates of capitalists as a key determinant in their policy preferences. The current study
takes the opposite approach by omitting physical capital to highlight the demographic factors.
Furthermore, in their scenario, educational reform occurs without any political con‡ict— in sharp
contrast to this analysis, which emphasizes compulsory education as a duty for the poor.7
Second, Doepke and Zilibotti (2005) explore the mechanism of child labor legislation. By noting
the interaction between the number of children per family and policy preferences, they show thepossibility of multiple equilibria such that child labor may or may not be abolished. Whereas
their attention is directed at underdeveloped stages, the present study builds on a longer-term
perspective, and encompasses more development stages where population aging provokes politicalconfrontation between generations.
Third and …nally, Holtz-Eakin et al. (2004) develop a growth model where the fraction of theelderly population is a prime determinant of government spending on education.8 They demonstratethat a fertility decline alters the policy preferences of the median voter in favor of the elderly.
However, unlike the present article, their analysis leaves unexplored an opposing causality between
population aging and education policy. This is because their model abstracts from the familystrategies of child rearing by assuming exogenous fertility and no private education.
The remainder of the paper is organized as follows. Section 2 presents historical evidence
7This di¤erence is largely owing to the fact that the present article highlights the role of compulsory education
as a child labor regulation. As evidenced in Section 2, early factory legislation required education for child workers
while limiting their work hours.
8
Similar demographic approaches are employed in Kemnitz (1999), Pecchenino and Utendorf (1999) and Gradstein
and Kaganovich (2004). All of these, however, assume exogenous population growth.

4supporting the central argument of the paper. Section 3 describes the basic structure of the modeland then solves the optimization problems. This section also derives the political equilibrium that
determines the allocation of public spending between generations. Section 4, as the main part ofthe paper, conducts a dynamic analysis to demonstrate the evolution of the economy. Further,
it indicates the possibility for a slowdown in economic growth in the developed stage. Section 5
concludes the discussion and addresses some possible modi…cations and extensions to the research.
The description of the data used and the proofs of the technical results are provided in the Appendix.

2

Historical EvidenceThis section presents historical and empirical evidence supporting the central thesis of the paper.
The focus here is placed on the experiences of advanced economies— Japan, the United States, andWestern Europe— over the last two centuries. Consistent with the theory developed below, it showsthat industrialization forces triggered the decisive shift of individuals’attitudes toward child laborand education.

2.1

Public Education as a Duty

The notion of compulsory education as a duty has been embraced by the poor, who live from handto mouth and need the earnings of their children. In the early stages of their industrial revolutions,these countries employed children, and their respective child labor regulations and educational
reforms were not embraced by the lower class. As a result of compromise, early educational reforms
were designed to provide special classes for working children. Consistent with this observation, thegrowth theory developed here encompasses the coexistence of child labor and public schooling.
2.1.1

Japan

It is widely recognized that the Japanese industrial revolution took place after the Meiji Restorationin 1868. Ohkawa and Rosovsky (1965, p. 66; 1978, p. 142) argue that modern economic growthbegan in the mid-1880s. Similarly, Rostow (1978, p. 425) dates the years 1885–1905 as the periodof the take-o¤.
Japan’s modern educational reform was launched with the proclamation of the Educational System Order in 1872, which aimed for egalitarian education by establishing school districts throughout

5the country.9 The law made four-year elementary education compulsory, along with the charging
of school fees (Taira, 1978, p. 196). An educational system in the early stages of development suchas this placed a heavy …nancial burden on the lower classes at that time. As described by Taira(ibid.), “To poor farmers, compulsory education appeared as an encouragement to the children
to loaf in school when they could be helping on the farm. In some poorer parts of Japan there
were a number of riots against compulsory education, in which hundreds of school buildings weredestroyed.”In fact, statistical records indicate that not many children were sent to primary school in theearly years of industrialization. The classroom attendance rate was approximately 20.44% in 1873
and 31.24% in 1890, and exceeded 50% for the …rst time in 1900 (Japan National Commission for
UNESCO, 1966, p. 64).10 Poor classroom attendance might have even been observed in not only
day schools, but also night schools, which allowed children to work during the day. A public survey
reports that among 4043 pupils registered in primary night schools (run by the city of Tokyo) in
1927, about 74% dropped out before graduation (Ishii, 1992, p. 165).
The low attendance rates were a result of the economic dependence of poor households on childlabor. According to the Annual Report of Factory Inspection, even in 1926 nearly 20% of workers
in textile mills were aged 16 or less (Ayusawa, 1966, pp. 186–188). Saito (1996, pp. 87–89) argues
that, while child employment in paid work was not as extensive as in Britain, Japanese children
were mostly expected to support their parents in the …elds or around the house.
In order to encourage education for child workers, some forms of half time education wereintroduced in the late 19th century. In 1875, the city of Tokyo permitted night classes opened byprivate elementary schools, and in 1876 permitted night classes by public elementary schools (Ishii,
1992, p. 25). In 1894, the Ministry of Education ordered local governments to develop special
curriculums for children who could not attend regular classes, and subsequently night classes,
Sunday schools, and special classes for children involved in the care of babies were opened by state9

See the webpage of the Ministry of Education:
http://www.mext.go.jp/b_menu/hakusho/html/hpbz198103/index.html
where the history of Japan’s modern educational system is documented comprehensively. According to Dore (1964,
p. 176), although there were many small schools for reading, writing and arithmetic before the Meiji Restoration,
they were typically run privately with no subsidies from the government.
10
While the commission also reports the rate of school attendance, they believe that the rate of classroom attendance
provides more accurate information about actual attendance: the former is de…ned as “an o¢ cial ratio of the numberof children attending school to the total number of the school-age children,”whereas the latter is “the ratio of average
daily attendance to the total number of the school-age children (ibid., p. 64–65).”6

schools and sometimes by factories (Saito, 1996, pp. 83–85).11
2.1.2

The United States

As is well known, there were regional di¤erences in the timing of industrialization in the UnitedStates. The economic take-o¤ of New England occurred in the period 1815–1850, whereas that of
the American North in 1843–1870 (Rostow, 1978, p. 392). In the north-eastern region, the shareof child workers employed in manufacturing was 23.1% in 1820 (Goldin and Sokolo¤, 1982, p. 748).
It is estimated that in 1832 children amounted to about 40% of factory workers in New England(Weiner, 1991, p. 142). The proportion of child labor would have been much higher in agriculture,
a dominant sector in the early 19th century.12 At the national level, about one-sixth of children
between the ages 10 to 15 were gainfully employed even in 1880 (Sanderson, 1974, p. 297; Weiner,
1991, p. 145). Given these …gures, one can imagine that earlier years would have seen higher ratiosof children in the labor force.
These records indicate that one of the principal obstacles to universal education was how toprovide education for those working children. Hence, early educational reforms were presumably
a compromise between the need for child labor and that for child education. In the late 1830s,
industrial states passed laws which required factory children to go to school three months every
year (Church, 1976, p. 59). In 1836, Massachusetts legislated that (at least) three months of school
attendance was required, in advance, for the employment of children under the age of 15 in manufacturing (Weiner, 1991, pp. 142–143). Under the Massachusetts’ labor-attendance law in 1842,
it was the duty of the local school committees to secure the school attendance of children, while
all factory children under the age of 12 were prohibited from working more than 10 hours a day(Ensign, 1969, p. 49).
2.1.3

BritainChild labor played an important role even in early 19th century Britain, one of the most advanced
economies at that time. In 1816, workers under the age of 18 accounted for 51.2% of workers
11

A careful investigation by Osada (1995) identi…es the locations of about 300 childminders’ schools that were
established in 36 prefectures between 1876 and 1930 (pp. 211–215), and the typical student in the subsample heanalyzed were girls between the ages of 11 and 14 (pp. 325–334).
12
In 1820, the proportion of workers employed in agriculture was 73% in New England, and 74% in the Middle
Atlantic (Goldin and Sokolo¤, 1982, p. 748).

7

in the British cotton industry, and 60.2% in the Scottish ‡ax industry (Nardinelli, 1990, p. 109).
The Census of England and Wales records that at least 36.6% of boys aged 10–14 were working
in 1851 (Cunningham, 1996, p. 42). The earnings of children were an important income sourceof the working class. In the period from 1817 to 1839, records show that the children of mining,
factory, and outworking workers earned, respectively, 23.9%, 28.2%, and 27.5% of household income(Horrell and Humphries 1995, p. 491).
These …gures indicate parents’hesitation in sending their children to school at that time. For
instance, the records of the Mitcham National School show that during the years 1830–1939, theaverage length of stay of pupils was 34.3 months, and 57% of boys left school to start work (MadocJones, 1977, pp. 45, 47). Even in 1889, the London School Board issued as many as 96,450 initial
notices to parents failing to send their children to school (Rubinstein, 1977, p. 245).13 Furthermore,
there is a view that the London poor, estimated to be 30.7% of the population in 1891, opposed
education for their children, in part because those poor needed the earnings of their children(ibid., pp. 235–236). It is conjectured from these …gures that the …rst half of the 19th century saw
widespread, intense opposition to compulsory education.14
In these circumstances, early factory legislation compromised between regulating child laborand promoting child education. The Factories Act of 1833 imposed two-hour schooling on six daysa week on working children aged 9 to 11 in the main textile industries (Silver 1977, p. 141). The
Factory Act of 1844 enacted the half-time system, which restricted the work time of children tohalf a day, either before or after the dinner hour (Hutchins and Harrison, 1926, pp. 76, 85). As
stated by Silver (1977, p. 141), “The half-time system, foreshadowed in the early Factory Acts ande¤ective from 1845, began as a strategy for combating excessive child labour and became, in the
1850s and 1860s, an educational theory.”
2.1.4

FranceAs in Britain’s case, the progress of French educational reform was gradual. The education law in1833 made primary education universally available (Weissbach, 1989, p. 3). Nevertheless, that was
not fully made use of by the poorer classes, who needed child labor for their daily life. In 1840,
13

Hurt (1979, p. 155) introduces two court cases, from The Times in November 1875, in which the London School
Board was involved. They show that compulsory school attendance was a big burden on the poor.
14
See Hopkins (1994, p. 142) for supporting episodes in the mid-19th century Black Country.

8more than a quarter of the children aged 6 to 12 in France did not attend school in part because
their labor was needed at home (ibid.).15
Evidence shows that children were working for their families out of necessity, at least before
the mid-19th century. According to a study published in 1840, it cost 960 francs a year for a fourmember family to live in Melhouse, whereas a laborer or dyer in the local cotton industry could
earn at most 450 francs a year (Heywood, 1988, p. 108). As for handloom weavers in Rouen, theannual cost of living, 912 francs, was greater than the standard parental income, 861 francs (ibid.,
pp. 108–109). In the mid-1840s, several School Inspectors found that school attendance in rural
areas declined when bread became expensive (ibid., p. 88).
Given such economic importance of children for households, it is not surprising that France
began its educational reform while accepting child employment. Evening classes were held by themunicipality in Mulhouse in the early 1830s, and in Lille in the early 1840s, although they were
not very successful (ibid., pp. 244–245). The child labor law passed in 1841 provided a certain levelof education for children aged 12 or below, while regulating the minimum age of employment to
8 (Ogg and Sharp, 1929, p. 382). Faced with working class opposition, this …rst child labor law
was ine¤ective, and the child labor law passed in 1874 was more e¤ectively enforced by a special
inspectorate (Nardinelli, 1990, pp. 126–127, 137).
2.1.5

Germany

In Prussia, the …rst school edict was issued as the General-Landschul-Geglement of 1763, withthe aim of providing compulsory education for all children aged 5 to 13 (Melton, 1988, p. 174).
However, such an early attempt was unsuccessful. The four schools in Berlin, Kloster Berg, Stettin
and Königsberg “produced far too few graduates to meet the goals of the reform (ibid., p. 175–
176).”One of the reasons for the failure was the need for child labor for the poor. Indeed, summer
school was impossible to run because rural families, especially in summer, needed their children to
be working (ibid., p. 176).
Similarly, 40 out of about 100 petty schools in Bremen were closed between 1788 and 1810,
during which children found new work opportunities in tobacco processing (Cunningham 1995,
p. 103). This indicates that a large proportion of households in Bremen gave priority to child labor15

In 1840, 756,464 boys were in communal primary schools in winter, whereas the number attending declined to463,464 in summer, possibly a season when their labor was especially needed at home (ibid.).

9

over child education at that time, and probably, in earlier periods as well. During the period 1800–
1846, children under the age of 14 accounted for between 10 and 20% of factory workers (Lee, 1978,
p. 466). In particular, until the 1850s children were of great assistance in textile factories, where
their hands were suitable for picking up the threads (ibid.).
Under these circumstances, early educational reform encountered di¢ culties in its enforcement.
The Prussian law of 1839 obliged children under the age of 16 to attend school …ve hours a day,while limiting their work time to 10 hours (Ogg and Sharp, 1929, p. 389). However, the local
authorities (including police, teachers and clergymen) were unwilling and unable to enforce the law
and, accordingly, the schooling time was cut to three hours a day in 1853 with limited enforcement(ibid., p. 390). One can imagine from these events that the local authorities had sympathy for poor
children who were working for their families. Similar results were observed in other German statessuch as Bavaria, Baden, Württemberg and Hesse: they failed to restrict child labor and to spread
public schooling (ibid.).

2.2

Public Education as a Right

More recently, compulsory education is widely perceived to be essential for securing the minimum
living standard in advanced countries. In fact, children in OECD countries attend on average 11.9
years of schooling (OECD, 2006, p. 41), exceeding the compulsory years in most cases.16 This fact,
along with the evidence given in Section 2.1, suggests that the notion of compulsory education for
their citizens had shifted from a duty to a right.17
Such a shift in individual attitudes toward education became widespread around the turn of the20th century, although the timing varied across countries. In Japan, the aforementioned classroom
attendance rate at primary level rose to 89.63% in 1913 (Japan National Commission for UNESCO,
1966, p. 64). This implies that most school-age children remained at primary schools after being
enrolled. In the United States, 32.3% of children aged 14 to 17 were enrolled in secondary schools
in 1920, a sharp increase from 6.7% in 1890 (Church, 1976, p. 289). In Secondary Education forAll, published in 1922, the British Labor Party suggests the need for post-elementary education for
16

See the webpage of the Japanese Ministry of Education:
http://www.mext.go.jp/b_menu/shingi/chukyo/chukyo3/gijiroku/001/04052101/009/005.htm
17This point has been emphasized by many scholars. See Shibuya (1988, p. 167) for Japan’s case, and Weiner(1991) for a more comprehensive analysis.

10working-class children (Hopkins, 1994, p. 321).18
This hypothesis is supported by Nardinelli (1990), who investigates the historical evidence for
Britain, France, Germany, Japan, and the U.S. He argues that “As working-class incomes rose,parents kept children out of the labor force until later and later ages. The production of welleducated children became a realistic and desirable alternative for the working-class family in thelate nineteenth and twentieth centuries. The desire of parents to improve the quality of theirchildren would have been su¢ cient, in the absence of child labor laws, to remove children fromfactories in the long run (p. 149).” The minor role of child labor laws on school levels is also foundby Landes and Solmon’s (1972) regression analysis for the U.S. states in the late 19th century.
Also, in Prussian textile factories, the number of child workers dropped by about 30% for theperiod 1846–1853, when child labor legislation was not e¤ective (Lee, 1978, p. 467).
While several socioeconomic forces were at work in the transformation, two forces are emphasized as of particular importance in this paper: the rise in the net return on education investment,
and the improvement in living standards. The …rst force motivated parents to educate their children
if they could a¤ord it.19 By 1880, Félix Pécaut, apostle of progressive education in France, reported
that families began to notice the advantage of obtaining a school certi…cate for …nding jobs, inducing
parents to permit longer school attendance by their children (Weber, 1976, pp. 73, 328). Likewise,high school education in the United States began to be regarded as a means to secure white-collar
occupations emerging at the end of the 19th century (Church, 1976, p. 290). In Britain at that time,
secondary and university education were increasing their importance in achieving or maintaining
social status (Glass and Grebenik 1965, p. 117). On the cost side, technological progress reduced
the forgone wages of children attending school. For example, in the aforementioned German textile
factories, the signi…cant role of child labor was replaced by improved machines in the late 1840s(Lee, 1978, p. 467).
The second force, improved living standards, did more than reduce child labor: it alteredparents’ myopic views on child rearing, and thereby induced them to care about the future wellbeing of their children. In the United States, the aforementioned spread of secondary education18

In 1950, the share of secondary school pupils in the 10–19 age group reached 33.7% in England and Wales (Floraet al., 1983, pp. 626).
19
The underlying assumption here is that imperfect credit markets prevent households from borrowing school fees.
This credit-market imperfection appears to be plausible in under- and semi-developed stages.

11was in part because of rapid, widespread growth in per capita income after 1900, which would
have allowed more families to keep sending their children to school without relying on their forgone
wages (Church, 1976, pp. 289–290). Similarly in France, fewer children were in the labor force inthe second half of the 19th century, during which real wages kept rising (Heywood, 1988, p. 109;
Nardinelli, 1990, p. 146). In Britain, the average real wage over the period 1900–1909 increased toalmost twice the level of the period 1850–1859 (Polland, 1978, p. 171). The improved living standard
was also con…rmed by looking at the number of prosecutions. By 1910, it dropped to 36,823 fromthe peak level of 92,274 in 1883 and, furthermore, truancy was the second most common casein roughly the same period (Hurt, 1979, p. 203).20 Finally, in Japan, real GDP per capita morethan doubled between 1890–1990 (Maddison, 2001, p. 206). Taira (1971, p. 374) points out that
accelerated Japanese growth in this period lightened the burden on households while the rate ofclassroom attendance increased by about 20%.

3The Model

Consider a closed overlapping-generations economy operating over an in…nite discrete-time horizon.
In perfectly competitive environments, producers generate a single homogeneous …nal good byemploying individuals of all generations. Growth in output per worker is driven by education
investment and population growth.
In every period, the government levies an income tax on working adults in order to provide
education for children and an old-age related service. The allocation of tax revenues between thetwo services is determined by the adult and old generations, whose political power varies with theage composition of the population. Compulsory schooling works as a child labor regulation: thisa¤ects households’child-rearing strategies.

3.1

Producers

The …nal good is produced in two independent sectors, A and B. In sector A, competitive …rms
employ adult workers as a single factor of production. Letting YtA be the aggregate level of output
20

Although not mentioned, this is probably the record for England and Wales because these two countries arefocused on throughout the book.

12

produced by this sector in period t (0), the production function is written as
YtA = At Ht ;where At > 0 is the level of technology in period t and Ht is the aggregate amount of e¢ ciency labor
employed in period t. For simplicity, the price of the …nal good is normalized to unity. Standard
pro…t maximization reveals that the wage rate per e¢ ciency labor in period t, denoted as wt ; equals
the marginal labor productivity, At .
On the other hand, sector B employs children and old individuals by using a less advanced
technology. Letting YtB be the aggregate level of sector B’s output in period t, the production
function is
YtB = wLt ;where w > 0 is the stationary level of technology and Lt is the aggregate amount of raw labor
employed in period t: As a result of pro…t maximization, w equals the competitive wage rate per
raw labor in sector B:

3.2

Households

A new generation, consisting of a continuum of individuals, is born at the beginning of every
period and lives for three periods. This means that there are three generations in society at eachpoint in time. All individuals have perfect foresight and behave as price takers. Adult individuals
make three choices in child rearing: how many children to bear, how much education to give thesechildren, and how much their children will work.21 The political system is controlled by threeinterest groups: the adult rich, the adult poor, and the old. While adult individuals di¤er in theirskill levels, old individuals are homogeneous in this respect.
3.2.1

The Environment

Consider the life stream of an individual who is born in period t

1 to be a member of group i(= R; P ): The individual has one unit of time in each of the three life periods. In the …rst period(childhood), the individual works and/or receives education, while consuming a time fraction of
21

The omission of capital markets would not a¤ect households’ child rearing strategies. Considering the limitedaccess to capital markets in the early years of the Industrial Revolution, presumably most households at that time
were unable to borrow education costs.

13

her/his single parent. The work time of the child depends on the levels of private and public
education. There is no decision made by the child.
In the second period (adulthood or parenthood), the individual is obliged to serve
time for the government, whereunits of2 (0; 1) is exogenously given and stationary over time. Thus

the time left to the individual is 1

T; which is allocated between working and child rearing.

The individual works in sector A by supplying e¢ ciency labor, which is the product of her/his
work time and skill level hit . Meanwhile, the individual raises nit units of children by spending a…xed time fraction> 0 per child, and possibly further invests eit units of time per child for skill

acquisition. Each of the children supplies lti units of labor in sector B to earn wlti for her/his parent.
These wage incomes are spent for their consumption in this period, cit . It follows that the budget
constraint of the parent is
cit
where zti

zti [T( + eit! it )nit ];wt hit denotes parental potential income, and ! it(1)
wlti =zti is the child labor income

in terms of the parent’s work time. It is assumed that ! it is not high enough to fully cover the
noneducation cost of child rearing; i.e., ! it <8t:22 Human capital hit depreciates completely by

the end of adulthood.
In the third period (elderhood), the individual becomes unskilled, and earns w by inelasticallysupplying one unit of labor in Sector B. The obtained income is spent for her/his own consumption
c3t+1 , and there is no intergenerational transfers. The individual is exempted from taxation andreceives an old-aged related public service for free of charge. It follows that the budget constraintof the old individual is c3t+1w:

Preferences in adulthood are de…ned over the level of consumption and the quantity of her/hischildren. Preferences in elderhood are de…ned over three economic factors: her/his consumption,
the received amount of public service, and the average quality of her/his children. The lifetime
utility function of an adult individual i (an adult member of group i) in period t is given by
uit = (1where

2 (0; 1);) ln cit +2 (0; 1);ln nit +> 0; andit [(1) ln min( c3t+1 ; xt+1 ) +ln hit+1 ];(2)

i denotes the income-dependent discount rate taking at22This assumption negates child labor income as an incentive for childbirth, and excludes the feasibility of anin…nite number of children to be born. See Footnote 25.

14

discrete formit =(zti )

8< 0:

for zti < z ;> 0 for zti

(3)

z :

A plausible interpretation of (3) is that low-income adults live from hand to mouth, and do notcare about the future.23 Following Galor and Weil (2000), it is assumed for simplicity that theweight on the quantity of children equals the weight on the quality of children; i.e.,
3.2.2

=

:

The Production of Child Labor and Human Capital

Let eGt denote the time the government spends for the education of each child in period t. Then,the labor supply of a child i (a child of group i) in period t, denoted as lti ; is given by a continuousfunction

8< > 0 if ei + eG < e^;ttlti = l(eit + eG)t
: = 0 if ei + eG e^;twhere e^ >

0:24

(4)

t

Namely, the work time is left to the child if and only if less than e^ units oftime is invested in her/his education. Furthermore, the function satis…es l(0) = 1; l0 (e) < 0 andl00 (e)0 8e 2 [0; e^): Thus child labor supply is inversely related to the level of education, and no

education investment allows the child one unit of labor supply. In this sense, compulsory schoolingplays the role of a child labor regulation.
It follows that the labor income of a child i in period t is, in terms of her/his parent’s worktime,
! it =

wl(eit + eG
t )
i
zt

i
!(eit ; eGt ; zt ):(5)

In light of (1), this shows that the forgone wages of children are the only costs of public education
incurred by households. Thus, in terms of cost, public education is superior to private education.
The skill level of an adult individual i in period t + 1 is determined by the production function
hit+1 = ai h(eit ; eG
t );

(6)

where ai (aR > aP = 1) denotes the stationary learning ability of group i. This formulation meansthat only teaching time enhances the human capital children will acquire. The function h( ) is23
If instead it = > 0 8zti 0, parents might invest in private education while having their children work. As will
become apparent, (3) is useful to exclude this possibility and simpli…es the dynamic analysis without changing thecentral argument of the paper.
24 i
et + eGt is the total time spent for the education of a child i in period t; rather than the time the child spends foreducation. For this reason, the critical level e^ does not have to be the total time the child has (i.e. one unit of time).

15

continuous and is de…ned on R2+ ; featuring three properties. First, a noneducated adult of dynasty
P obtains one unit of human capital; that is, h(0; 0) = 1: Second, the function is increasing andstrictly concave with respect to each argument. Third, and as mentioned in the introduction,
private and public education are substitutes. These properties are expressed as hj ( ) > 0; hjj ( ) <0 and h12 ( ) < 0 for (eit ; eG
t )
3.2.3

0 and j = 1; 2.

Optimization

By taking market prices and government policies as given, an adult individual i in period t makes
a decision of childbirth and private education so as to maximize her/his utility (2). Since thisdecision has no e¤ect on the wage income w and thus consumption when old, the optimal choice isfnit ; eit g = arg max(1) ln[T( + eitii!(eit ; eGt ; zt ))nt ] +ln nit + (zti ) ln h(eit ; eG
t );

(7)

0; zti > w= .25

subject to the budget constraint (1) and (nit ; eit )The objective function is strictly concave with respect to nit . The …rst-order condition for nit
reveals that the net cost of child rearing is( + eit! it )nit = T;(8)

implying that nit > 0: The relationship between nit and eit in (8) indicates that the individual facesa trade-o¤ between the quantity and the quality of children. Equally important, an increase in therelative wage of child labor, ! it ; induces her/him to have more children.
Optimization with respect to eit is straightforward for individuals who cannot a¤ord to careabout the future: in light of (3), (zti ) = 0 and thus eit = 0 if zti < z : If zti

z and thus (zti ) = ;on the other hand, optimization is more complicated because, as shown below, multiple local optima
may exist in this case.
The situation in the second case above is shown by Figure 2. On the lower panel is the (partiallyindirect) utility function obtained from substituting (8) into (7).26 The solid curve there indicatesthe utility levels based on a given pair eGteit = e^0 and z min

z : Note that the curve is kinked ateGt ; where no work time is left to a child i in period t: In fact, there may be two locally25
The restriction on zti is necessary to assure the existence of the optimal pair (nit ; eit ): If this condition is violated,int can be in…nitely large by choosing eit = 0; for example.
26
i
h i
G
The slope of the indirect utility function is depends on the signs of the functions Dl (eit ; eGt ; zt ) and D (et ; et ) in

the upper panel. See Appendix 1 for details.

16optimal levels of private education, respectively, below and above the kink point: the lower level
allows children to work, whereas the higher level does not.27
In what follows, it is assumed that the marginal productivity of private education, h1 (eit ; eGt );is large enough to satisfy
h1 (^e; e^)( + e^) > h(^e; e^):
Lemma 1 If (A1) is satis…ed and zti(A1)z ; a locally optimal level of private education is given byGGa continuous single-valued function (eG^ andt ) such that (et ) + et > estrict inequality if

0

(eG
t )

0 8eG
t

0; with(eGt ) > 0:

Proof. See Appendix 1.
According to the …rst property of( ) above, choosing the level(eGt ) for private education

leaves no time for children to work. The second property means that parents not using child labor
aim to substitute private education for reduced public education.
The global optimality of the level (eGt ) depends on the relative productivity of child workers
to their parents. A rise in parental potential income makes child labor less productive, and thereby
reduces the opportunity cost of schooling. This transition is illustrated by Figure 2. As z i increases
to z high and further, the utility curve in the lower panel shifts downward only on the interval [0; e^),where child labor is used, and ultimately the level (eGt ) becomes globally optimal. This result isformally given by Lemma 2 below.
Lemma 2 If (A1) is satis…ed and zti

z ; there exists a level of potential income, z^; aboveG
which (eGt ) is the globally optimal level of private education for any et

0:

Proof. See Appendix 1.
Lemmas 1–2 are summarized as follows: Parents do not rely on child labor and invest in private
education, provided that their potential incomes are greater than z^ as well as than z . In order tosimplify the dynamic analysis below, it is assumed that parents do not use child labor if they are
wealthy enough to care about the future skills of their children (i.e.
z^ < z :27

i =t). That is to say,
(A2)

There is the possibility of a corner solution. For example, suppose that eit = eGt = 0: Substituting (8) into (7) yields
ln + (1) ln(1) + ln T
ln(
w=zti ): In this case, utility goes to in…nity as w=zti approaches . On theiiother hand, if eit > eGt = 0; then (8) shows that nt and thus utility remain …nite when w=zt approaches . Hence aiiG
corner solution et = 0 may exist depending on the values of zt and et :17This condition assures that no parents choose a level of private education on the range (0; e^), wherechildren have time to work. In other words, it excludes the case where parents privately educate
their children while sending them to work.
It follows that the optimal decision on private education is summarized as
8< 0
if w= < zti < z ;i)eit = e(eG;ztt
: (eG ) 0 if z i z ;tGwhere (eG^ andt ) + et > e

0

(eG
t )

0 8eG
t

(9)

t

0; with equality only if (eGt ) = 0: Substituting (9)into (8) and noting (5) yield thatinit = n(eGt ; zt )

where

T

8
<
:

T
i > 1
!(0;eGt ;zt )
T
>0
+ (eG
t )

if w= < zti < z ;if zti(10)z ;by assumption.28 Note that the choice of childbirth hinges on education policy aswell as the level of parental potential income, and their e¤ects change qualitatively depending on
the three cases below.
First, if eG^ and w= < zti < z ; an adult individual i in period t has her/his children workt < einstead of educating them privately. Hence, an increase in eGt works as a child labor regulation,
and an increase in the parent’s potential income zti reduces the relative wage of children ! it . Bothi
G iof them discourage childbirth, meaning that n1 (eGt ; zt ) < 0 and n2 (et ; zt ) < 0:Second, if eG^ and w= < zti < z ; the parent does not invest in private education, while thet >egovernment regulates child labor completely. Under these environments, an increase in either eGtor zti has no e¤ect on the relative wage of children, no e¤ect on the decision of private education,i
G i
and thus no e¤ect on the incentive of childbirth. That is, n1 (eGt ; zt ) = n2 (et ; zt ) = 0:

Third and …nally, if zti

z ; the parent invests in private education with no use of child labor,iiregardless of education policy. An increase in eGt induces the parent to reduce et and to increase nt ,because of the substitutability between public and private education, and through the quantity–
quality trade-o¤ in child rearing. Furthermore, as in the second case, no child labor income impliesithat nit is independent of zti . It is now clear that n1 (eGt ; zt )28i
0 and n2 (eGt ; zt ) = 0 in this case.

This assumption requires the weight parameter on childbirth, ; to be su¢ ciently larger than the noneducation
cost per child, : Without this condition, even a family relying on child labor may not have more than one child— acase not supported by historical evidence.

18

3.3

The Government

In period t, the government imposes a tax, in the form of time, on Nt units of adult workers. Eachof them is obliged to serve

2 (0; 1) units of time for public services. Thus the government obtains

Nt units of time in period t, allocating them between education for children and the old-age related
service. These two public services are free of charge for all bene…ciaries. All adult individuals havethe same ability in the production of the public services.
Let ntNt+1 =Nt denote the average number of children per adult in period t: Each publicschool class consists of nt units of students and one teacher; that is to say, each adult individual
equally teaches nt units of children at the same time.29 Thusis the maximum spending on public

education per child, and the budget constraint facing the government is eGt 2 [0; ]:Let xt be the level of public service each old individual receives in period t. For simplicity, xt
is assumed to be the time allocated for this service; that isxt =where nt 1

(

eG
t )Nt
Nt 1

=(

eG
t )nt 1x(eGt ; nt 1 );(11)

Nt =Nt 1 denotes the child/adult ratio in period t 1 (or the adult/old ratio in periodt). The e¤ects on xt of education policy and of fertility is straightforward: A rise in eGt shifts thegovernment spending from the elderly to the young, leading to a reduction in xt . A rise in nt 1
increases the number of taxpayers per old individual, and thereby increases xt :3.4

The Political System

In every period, adult individuals, together with the old, decide the budget allocated to young
generations. The decision on childbirth is made in advance by perfectly forecasting the politicaldecision.30 As will become apparent, the three interest groups, the adult rich, the adult poor, andthe old, have di¤erent political powers.
First, consider the level of public education desirable for an adult individual i in period t;
denoted as eGit . As implied by (11), education policy in this period has no direct e¤ect on theold-age related service xt+1 : It thus follows from (2) thateGit = arg max(10 eGt) ln[T( + eitiii
i G!(eit ; eGt ; zt ))nt ] + (zt ) ln h(et ; et );

29

(12)

More generally, one may assume that "nt units of children are enrolled in each class, where " > 0 is a constant.
As will be disccussed in Footnote 32, the timing of childbirth changes the reason for poor families to oppose
educational reform.
30

19

subject to eit

0; nit > 0 and zti > w= .

Next, consider the level of public education desirable for the elderly in period t; eG3t . Substituting
the budget constraint c3t = w and (11) into (2), one …nds thatGeG3t = arg max min[ w; x(et ; nt 1 )];0 eGtwhere x(eGt ; nt 1 ) = (eG~t such thatt )nt 1 : Note that there exists a unique value e

w =

x(~
et ; nt 1 ); and the qualitative e¤ect of education policy changes at this critical level. Increasing eG~t has no e¤ect on the elderly, whereas the e¤ect becomes negative on the range above
t up to e
e~t :The elderly, when indi¤erent, are assumed to accept the policy proposed by younger generations.
Then the desirable level of public education for old individuals in period t; denoted as eG3t ; is
8
< eG2
e~t ;
if eG2
t
t
(13)
eG3
=
t
: max(~
e ; 0) if eG2 > e~ ;
t

t

t

where

e~t =
Note that e~(nt 1 ) <w

e~(nt 1 ):

nt 1

(14)

and e~0 (nt 1 ) > 0 8nt 1 > 0: The …rst property means that the maximum

spending on public education,; is large enough to provoke opposition by the elderly. This isstraightforward because such an extreme policy leaves no budget for the old-age related service xt :The second property implies that the elderly can accept higher levels of public education as therelative number of taxpayers increases.
It is now ready to discuss how education policy is determined in the presence of potential
con‡ict. Based on the evidence in Section 2, the political system considered here incorporates two
ideas. First, all interest groups have some political in‡uence regardless of their voting status.31
Second, the political power of adult individuals in period t increases with the adult/old population
ratio in period t; nt 1 . More speci…cally, the equilibrium level of public education is a weighted
average such that
GR
eG
+ (1
t = (nt 1 )[ et

where

)eGP
t ] + [12 (0; 1) and (0) = 0; limn!1 (n) = 1 and

31

0(nt 1 )]eG3t ;(nt 1 ) > 0 8nt 1

(15)

0:

Flora et al. (1983) show that su¤rage was limited in Western Europe in the …rst half of the 19th century.
Nevertheless, as evidenced in Section 2.1, administrators at that time encountered practical di¢ culties in enforcing
school attendance.

20

3.5

Population Growth and Technological Changei
Let Nt+1
denote the number of adult members i in period t + 1: Then, noting that there is no

within-group heterogeneity, one …nds that
i
Nt+1
=

Z Nti
s=0

nit ds = nit Nti :The average fertility rate in period t, nt , is de…ned asP
N R + Nt+1
Nt+1
= t+1
= q t nRt + (1Nt
Nt

nt
where qt

qt )nPt ;(16)

NtR =Nt is the share of group R in the adult generation in period t: Note that nt is thechild/adult ratio in period t, and also the adult/old ratio in period t + 1:
Then it follows from (8) that the aggregate level of e¢ ciency labor employed by Sector A inperiod t + 1 is
Ht+1 =

X[T( + eit+1

i! it+1 )nit+1 ]hit+1 Nt+1

i=R;P

= (1

R
)T Nt [qt nR
t ht+1 + (1qt )nPt hPt+1 ]:(17)

Suppose that Ht+1 is the single determinant of technological progress between periods t and t + 1:
That is
gt+1

At+1 At= g(Ht+1 );At

where g( ) is a single-valued continuous function such that g(0) = 0 and g 0 (H) > 0 8H

(18)
0:

This formulation implies that there is no depreciation in the level of technology (or knowledge)
and At is nondecreasing over time. Equally important, it incorporates the well-known idea thatlarger populations tend to create more ideas (cf. Kuznets [1960]; Kremer [1993]; Galor and Weil
[2000]). In the presence of such a scale e¤ect, education investment is not necessary for economic
growth: as long as a population exists, the productivity of adult workers improves over time. By
contrast, there is no scale e¤ect and no technological progress in sector B. This type of age-biased
technological progress decreases the relative productivity of child labor over time.

4

The Evolution of the Economy

This section demonstrates that the dynamic model presented above generates the development
process consistent with the historical experience in Section 2. The economy goes through three21

stages of development through which individuals’ views toward public education change. The
resulting progress in educational reform is followed by a steady decline in the birthrate.
Stage I is the premature development stage without skill investment. Because of low living
standards, adult workers are not inclined to educate their children at the sacri…ce of child labor
income. As a result, neither private nor public investment in education occurs, and population
expansion is the single driving force of technological progress. It is shown that a small degree ofthe scale e¤ect leads to the acceleration of population growth.
Stage II is marked by the emergence of public education. Skill-biased technological progress
gradually raises the relative productivity of skilled workers and makes them wealthy enough to careabout the future performance of their children. These environmental changes induce them to abandon child labor, triggering the educational reform movement. However, they encounter opposition
from the less skilled poor, who still need child labor income for consumption. These con‡icting
policy stances limit the provision of public education and, accordingly, its growth-enhancing effects. On other hand, the legislation of compulsory schooling discourages childbirth for the poorby regulating the work time of their children.
Stage III encompasses population aging and generational con‡ict over public policies. In thisdeveloped stage, all adult individuals are wealthy enough to care about the future skills of their children, and the relative productivities of child workers are su¢ ciently low. Now even the less skilledadults come to regard the receipt of public education as a right and there is no political con‡ict
within the adult generation. Accordingly, educational reform makes further progress. Moreover,
the onset of universal investment in education, both by households and the government, decelerates
population growth.
The rise of education policy, however, is followed by a drop in later stages. As society ages,
old generations demanding other public services become more politically in‡uential. The risingpressures of the old cause a policy bias in their favor, and thereby lay a …nancial burden on young
households. To make up for the undersupply of public education (as a right), they spend more on
private education by having fewer children. This chain reaction generates a vicious cycle betweensocial aging and the deterioration of public education, and this spiral may cause population decline
and a slowdown in economic growth in the long run.

22

4.1

Stage I: Economic Growth Driven by Population Expansion

Stage I is the initial development stage that occurs before period t ; when ztR exceeds z for the …rst
time (namely, ztR < z 8t < t and ztRz for t = t ): In this stage, no parents are wealthy enoughto care about the future of their children. Consequently, neither private nor public education
investment occurs, and technological progress is driven solely by population expansion.
4.1.1

Education, Child Labor, and Fertility

Sector A in period 0 consists of adult workers who have only innate skills. Since their potential
incomes are z0R = aR A0 and z0P = A0 ; there is between-group inequality. As shown by (6) and (9),a rise in zti has a nonnegative e¤ect on private investment in child education, eit ; and then on theacquired skill level, hit+1 = ai h(eit ; eGt ): Therefore, the persistent ability gap between adult workers
results in
ztR > ztP
Thus it follows from (3) that ztP < ztR < z and

8t

0:

(19)

R =
t

P = 0 8t 2 [0; t ): Hence in Stage I, adulttindividuals do not gain utility from the future skills of their children. In this situation, as shown byGi(9), no one invests in private education regardless of education policy; i.e., eit = e(eGt ; zt ) = 0 8et

0:Using these results for (12), the desirable education policy for the parental generation in StageI iseGi
t = arg maxf10ii!(0; eGt ; zt )]nt g = 0:

[

eG
t

In other words, parents do not desire compulsory education as sending their children to schooltakes away the opportunity for them to work. Educational reform would merely diminish theirchild labor income ! it ; consumption cit ; and thus utility uit , because their choice on childbirth ismade in advance. These predicted adverse e¤ects prompt the adult generation to oppose theintroduction of public education.32 Then it follows from (13) and (15) that there is no generationalcon‡ict and eGt = 0 8t 2 [0; t ):32

The reason for their opposition changes depending on the timing of childbirth. If parents have children after theP
education policy was determined, nP)T ztPt is adjusted so that (8) holds. Then it follows that consumption is ct = (1regardless of education policy. In this case, it is the negative e¤ect on childbirth, rather than on consumption, thatinduces the opposition of group P against educational reform.

23

As a result of no schooling, all children spend their entire time on working in sector B: They
grow up to have only innate skills, hit = ai h(0; 0) = ai , and potential incomes zti = ai At : Thus inStage I, technological progress is the single force pushing up the level of zti , and At grows towardz =aR .
Then it follows from (5) and (10) that households’birthrates in Stage I are
nPt =

T
> nR
t =
w=At

T
> 1;
w=(aR At )(20)

where w= < At < z =aR : Note that parents of group P are more fertile than those of group R; as
child labor is more valuable for the formers than the latters. Equally important, nit decreases with
At because technological progress in Sector A reduces the relative productivity of children, working
in Sector B:
4.1.2

The Dynamical System

PRP
As shown above, nRt and nt depend only on At whereas ht+1 and ht+1 are constant for t 2 [0; t ).

Thus the resulting level of e¢ ciency labor, Ht+1 in (17), depends on At , Nt and qt :Using these results for (16) and (18), the evolution of the economy in Stage I, where At < z =aR ;is governed by the …rst-order, three-dimensional autonomous system:
At+1 = [1 + g(Ht+1 )]At
Nt+1 = nt Nt
qt+1 =

nR
t
nt

qt

GI (At ; Nt ; qt )At ;nI (At ; qt )Nt ;(21)mI (At ; qt )qt ;where the initial set (A0 ; N0 ; q0 ) satis…es33
w< A0 <z;aR

N0 > 0;0 < q0 < q max :(A3)

The three state variables of the dynamical system evolve in the following ways. First, thegrowth rate of technology gt+1 has a positive lower bound, and thus the level of technology At
grows without converging to a certain level.34 Second, nt > 1 and thus the total population Nt
keeps expanding. Third, nRt < nt and thus the share of group R in the adult generation, qt ; strictly
monotonically declines over time.
33

The upper limit q max 2 (0; 1] is de…ned by the proof of Equation (31) in Appendix 1.
As follows from (17) and (20), Ht+1 in (21) has a positive lower bound such that Ht+1 > (1(1
)T N0 .
34

24

)T Nt+1 >Accordingly, the parental potential income ztR = aR At grows strictly monotonically over time,
ultimately exceeding the critical level z . This means that there is no steady state in Stage I and
the economy inevitably departs from there. As for the average fertility rate, nt may or may notdecrease over time because of two opposing forces at work: On the one hand, technological progress
discourages childbirth by making child labor less productive compared to adult labor. On the other
hand, the more fertile group, the adult poor, increases its share over time (note that nI (At ; qt )is decreasing in qt ): The lemma below suggests that the average fertility rate increases when slow
technological progress makes the former negative force negligible.
Lemma 3 Under (A1)–(A3), nt > nt 1 for t 2 (0; t ) if At is su¢ ciently small.
Proof. See Appendix 1.
The possibility of population growth acceleration in this initial stage of industrialization appears
to be historically plausible. According to Maddison (2001, p. 242), the average growth rate of
population reached a peak during the period 1820–1870 in the U.S., 1870–1913 in Germany andthe U.K., and 1913–1950 in Japan.

4.2

Stage II: Educational Reform with Class Con‡ict

The economy develops in Stage II on the time interval [t ; t0 ); where t0 is the critical period in which
ztP exceeds z for the …rst time (i.e., ztP < z 8t < t0 and ztPz for t = t0 ). In this stage, membersof group R are wealthy enough to care about the future skills of their children, whereas this is
not the case for group P . Furthermore, the relative productivity of child workers is low enoughfor group R to abandon child labor. In these circumstances, educational reform is promoted by
the rich class, who face the opposition of the poor. Compulsory schooling works as a child labor
regulation that discourages childbirth in group P:
4.2.1

Education and Child LaborIn the initial period of Stage II, ztR = aR At

z : Since the level of technology At is, according to(18), nondecreasing over time, it follows thatztR = At hR
t

R
t =

z ;

As for group P; on the other hand, ztP < z and thus
25

;

8t

t :

P = 0 in Stage II.
t(22)

Under these circumstances, (9) reveals that individuals’decisions of private education are eR
t =
P
G
(eG
t ) and et = 0 8et

G
0: In fact, since (eG^ 8eGt ) + et > et0; (4) shows that ltR = 0 8eGt

0.

In other words, whether the government regulates child labor or not, adult members of group R do
not send their children to work. By contrast, those of group P are still myopic and rely on childlabor income, which depends on the level of compulsory education eGt :Using these results for (12), the desirable policy for adult individuals in Stage II areGeGR= arg max h(eRtt ; et ) = ;
0 eG
t

eGP
t

= arg maxf10

PP!(0; eGt ; zt )]nt g = 0:

[

eG
t

These results indicate the existence of between-group con‡ict (class con‡ict) over education policy:
Unlike group P; keeping children at school is no longer a burden for group R. The shift in theirpolicy preferences is caused by ztR exceeding z and z^: Given such a high potential income, parentsof group R take into account the future skills of their children (i.e.

R =t). The parents also …ndthat sending their children to work, rather than to school, does not compensate for the future loss.
The two adult groups facing such class con‡ict make a compromise at
In light of the historical evidence, suppose that the compromise level
reform. In particular,, according to (15).

does not lead to a drastic

is not large enough to completely regulate child labor, and not enough toprovoke the opposition by the elderly. That is to say,35< min(^
e; e~t )

8t

0:

(A4)

It thus follows from (15) that the resulting education policy iseGt =8t 2 [t ; t0 );

< e^

(23)

This outcome is interpreted as the half-time system mentioned in Section 2.1, in that children ofgroup P engage in both work and compulsory education.
4.2.2

Fertility

Substituting the results obtained above into (10), households’birthrates in Stage II are
nR
t =
35

In light of (10) and (14),that nmin nit 8t 0:T+ (< e~t 8t

)
0 if

R 1;
<

nPt =

T
> 1;
!(0; ; ztP )w=nmin ; where nmin

26

(24)

T =[ + (0)] is a lower bound suchwhere ztP = At h(0;): This shows that nR
t remains constant over Stage II, whereas the higher rate

nPt depends on the level of technology, At :Note that introducing compulsory education promotes group R’s childbearing by supporting
their child education; i.e., n(; ztR ) > n(0; ztR ): In this sense, public education works as a childcare

service, rather than as a child labor regulation, for the rich class. By contrast, it discourages group
P ’s childbearing by decreasing their child labor income; i.e., n(; ztP ) < n(0; ztP ): In light of (16);their overall e¤ect on average fertility depends on the relative share between the two groups. If
qt is su¢ ciently small, the former positive e¤ect on group R is negligible andqt n(; ztR ) + (1qt )n(; ztP ) < qt n(0; ztR ) + (1qt )n(0; ztP ):Now one obtains the proposition below.
Proposition 1 If (A1) is satis…ed and qt is su¢ ciently small, the provision of compulsory education in period t 2 [t ; t0 ) decreases the average fertility rate in the same period.
Thus, compulsory education is likely to delay population growth in the developing stage wherechild labor is needed by the masses. The initial value q0 is important for this result to hold because,
as will become apparent, qt strictly monotonically declines over Stages I–II.
4.2.3

The Dynamical System

RP
0As shown above, nPt depends only on At whereas nRt ; ht+1 and ht+1 are all constant for t 2 [t ; t ).

Thus it follows from (17) that the resulting level of e¢ ciency labor Ht+1 depends on At , Nt and qt :Using these results for (16) and (18), the evolution of the economy in Stage II, where z =aR <At < z =h(0;); is governed by the …rst-order, three-dimensional autonomous system:
At+1 = [1 + g(Ht+1 )]At
Nt+1 = nt Nt
qt+1 =

nR
t
nt

qt

where the initial condition (At ; Nt ; qt )GII (At ; Nt ; qt )At ;nII (At ; qt )Nt ;(25)mII (At ; qt )qt ;0 is determined by (21).

The dynamical system above characterizes the evolution of the three state variables as follows.
First, as in Stage I, the growth rate of technology gt+1 has a positive lower bound, and thus At
27

grows without converging to a certain level.36 Second, unlike in Stage I, the total population Nt
may not grow monotonically because, in light of (24), nt R 1. As long as the share of group R issu¢ ciently small, however, nt is greater than unity and thus the total population keeps growing.
Third, nRt < nt and thus qt+1 < qt ; namely, the share of group R in the parental generation declines
over time.
Under these circumstances, both ztR and ztP strictly monotonically grow over time and ztP ultimately reaches the critical level z (ztP0 = At0 h(0;)

z ). This assures the economy’s departurefrom Stage II. Furthermore, as in Stage I, the time trend of average fertility is generally ambiguous because of two opposing e¤ects. While the birthrate of group P declines over time, group R;
which is less fertile, reduces its share. As asserted below, the former downward pressure becomesdominant and decelerates population growth when the share of group R is su¢ ciently small.
Lemma 4 Under (A3), nt+1 < nt for t 2 [t ; t0 ) if qt is su¢ ciently small.
Proof. See Appendix 1.

4.3

Stage III: Social Aging and the Rise of Generational Con‡ict

Stage III, which begins in period t0 ; is symbolized by the rise and fall of education policy. In
this stage, all adult individuals are wealthy enough to care about the future performance of theirchildren. In addition, the relative wages of child workers are su¢ ciently low for even less skilledparents to abandon child labor. As a result, adult generations unanimously support extensive
education policy. Meanwhile, population aging gradually enhances the political power of the elderly,
who demand the old-age related public service. The resulting undersupply of public education laysan education burden on young families, and thereby discourages childbirth. These results well …t
the postwar evidence of social aging paralleled by generational inequality in public services.
4.3.1

Private Education, Child Labor, and Education Policy

Consider a period in which ztP(3) and (9) show that

R =
t

P =
t

z : Noting (19), this implies that ztR > z . In this circumstance,PG
G
and eRt = et = (et ) 8et

G
0: Since (eG^ 8eGt ) + et > et0;any parents no longer send their children to work regardless of the degree of child labor regulation
36

As follows from (17) and (24), Ht+1 in (25) has a positive lower bound such that Ht+1 > (1(1
)T NtP .

28

P)T Nt+1>(i.e., ltR = ltP = 0 for any eGt0). In this sense public schooling does not play a role of child labor

regulation.
Therefore, it follows from (12) that the policy stance of an adult individual i in this period isi GeGit = arg max h(et ; et ) = :0 eGtThis shows that public education is bene…cial and viewed as a right for all adult individuals, andno political con‡ict arises among them. The shift in the policy preferences of group P is caused by
the improvement in parental potential income. First, ztP exceeding z allows parents of the groupto care about the future performance of their children. Second, ztP exceeding z^ reduces the relativelabor productivity of their children.
The adult generation’s positive attitude to public education triggers generational con‡ict, because

exceeds the upper limit for the elderly, e~t in (14). In this case, as follows from (13) and(A4), the policy stance of the elderly is eG3= e~t >t

: These results and (15) reveal that theresulting education policy is
eG
t =

t

+ (1

et >
t )~

:

(26)

It follows that the parental potential income of group P in the subsequent period isPGzt+1= At+1 h( (eGt ); et ) > At0 h(0;where At+1

At

P > z if z P
At0 : Therefore, zt+1t

) = ztP0z ;z : Because of the persistent inequality shownby (19), it is now clear thatztR > ztP

z ;

R
t =

Thus the results obtained above hold for any t

P
t =

;

8t

(27)

t0 : In particular, note that the level of public

education in Stage III is always higher than in Stages I–II, where eG
t
4.3.2

t0 :

:

The Evolution of Fertility and Education Policy

As follows from (10), (16) and (27), the average fertility rate in Stage III isPnR
t = nt = nt =

T
;
+ (eG
t )

(28)

G
where (eGt ) is decreasing in et . The birthrate of group P in Stage III is lower than in the previous

stages, as the group no longer relies on child labor.
29GR = eGP =Now consider how eGt is determined in period t; in which ett

and the adult/old

ratio nt 1 > 0 is exogenously given. Since (13) reveals that eG3~(nt 1 )] in this case, thet = max[0; eeducation policy based on the political system (15) is given by a single-valued function such thateG (nt 1 ) = (nt 1 ) + [1where 0 < eG (nt 1 ) <(nt 1 )] max[0; e~(nt 1 )];(29)

and eG (nt 1 ) is strictly monotonically increasing in nt 1 > 0: Notethat a decline in nt 1 enhances the relative political power of the old generation in period t, andaccordingly reduces eGt :Substituting (29) into (28) reveals that the evolution of the average fertility in Stage III isgoverned by a …rst-order, autonomous system
nt =

T
+

(eG (nt 1 ))

nIII (nt 1 );(30)

where nIII (nt 1 ) is monotonically increasing in nt 1 > 0 and the initial value nt0 1 is given by(25).37
In addition to the positive correlation between nt and nt 1 ; the dynamical system (30) features
two important properties. First, as shown by Appendix 1,
nt0 = nIII (nt0 1 ) < nt0 1 :(31)This fertility decline re‡ects the switch of group P from the reliance on child labor to the educationof children. Second, because 0 < eG (nt 1 ) <8nt 1 > 0; the function nIII (nt 1 ) is bounded insuch a way thatT
< nIII (nt 1 ) <
+ (0)

T
+ ( )

8nt 1 > 0:(32)

These properties are depicted by Figure 3. The initial value of Stage III, nt0 = nIII (nt0 1 );is below the 45 degree line. Equally important, the function is monotonically increasing and has
a positive lower bound. These ensure the existence of a nontrivial steady-state equilibrium nt =nIII (nt ) > 0; towards which nt monotonically declines over Stage III.38 Since the corresponding
path of eGt is characterized by (26) and (29), one obtains the following proposition.
37
III
As shown by Lemma 1, 0 (eG
0 8eG0; with strict inequality if (eG(nt 1 ) is strictlyt )tt ) > 0: Therefore, nmonotonically increasing in nt 1 > 0 if ( ) > 0:38
The properties of nIII (nt 1 ) shown above do not ensure the uniqueness of the steady-state equilibrium. In thecase of multiple equilibria, nt converges to the largest steady-state level on the interval (0; nt0 1 ):30

Proposition 2 (Population Aging and Political Bias) Under (A1)–(A4);(a) ntnt 1 and

< eG
t+1

eG
t 8t

t0 ;

G
(b) nt converges to a nontrivial steady-state equilibrium where nt = nIII (nt ) > 0 and eG
t+1 = et :Proposition 2 asserts a vicious cycle between population aging and a deterioration in education
policy. As the old/adult ratio increases, the old generation becomes more politically in‡uential in
the allocation of public resources. The resulting budget cut in public education raises education
costs on young families, and thereby discourages childbirth. The level of public education in StageIII is nevertheless higher than in Stages I–II, where eGt:

There are a few remarks on this result. First, as mentioned in Footnote 3, the proposition does
not mean a nominal decline in aggregate spending on public education. eGt should be viewed as
the real education expenditure in terms of the time of adult workers. Second, private education is
stimulated by a reduction in eGt because, in this stage, child education is bene…cial for the adultgeneration. Such a substitutional relationship between private and public education does not exist
in underdeveloped stages, where households rely on child labor. Third and …nally, the steady-state
fertility rate, nt = nIII (nt ); may be smaller than unity, depending on the structural parameters of
the economy.39
The sequence fnt 1 g1
t=t0 determines the population dynamics given by
Nt+1 = nIII (nt 1 )Nt ;(33)

where the initial condition Nt0 is obtained from (25). Population keeps either shrinking or expanding
in the long run, depending on the steady-state fertility rate.
4.3.3

Technological ProgressPRPG
The results above demonstrate that nRt ; nt ; ht+1 and ht+1 depend only on et in period tt0 .

PFurthermore, because nR
t = nt the group ratio is stationary over Stage III; i.e., qt+1 = qt0

q

8t
39

t0 ; where qt0 is given by (25). It then follows from (17) that the aggregate level of e¢ ciency
Since (eG^t ) > eeGt in (28),nt <where eGt 2 (T+ e^ eG
t

8t

t0 ;

; ) as shown by (26). This shows that nt < 1 if the di¤erence e^

31

is su¢ ciently large, for example.

labor Ht+1 becomes
Ht+1 = (1)T Nt nt [qhR
t+1 + (1q)hPt+1 ]III G
Gwhere H1III (eGt ; Nt ) > 0 and H2 (et ; Nt ) > 0 8(et ; Nt )H III (eGt ; Nt );(34)

0. The …rst property is because a rise

i i
40
in eG
t increases nt ht+1 , and the second property re‡ects the scale e¤ect.

Substituting (34) into (18), the evolution of technology in Stage III is
At+1 = [1 + g(Ht+1 )]AtGIII (eGt ; Nt )At ;(35)

1where the initial value At0 is obtained from (25), and the sequence feGt ; Nt gt=t0 is fully determinedby (29), (30) and (33). Since eGt >for all t

t0 ; the growth rate of technology in this stage

is higher than in the previous stages, as long as population keeps growing. However, such rapid
growth may not be sustainable in the long run, depending on the demographic trend.
Proposition 3 (Economic Growth Slowdown) If (A1)–(A4) are satis…ed and nt = nIII (nt ) <1; the growth rate of technology, gt ; converges toward zero in the long run.
Proof. Proposition 2 and (33) reveal that under the conditions above, the adult population Nt
declines towards zero over Stage III. Since (35) shows that g(0) = 0 if Nt = 0; gt converges toward
zero in the long run.
The ultimate population extinction implied by this proposition does not appear to be quite
implausible, considering the current fertility rates of industrial economies.41 Nevertheless, thelong-run result needs to be interpreted carefully since it is based on the political system (26).
Rather than o¤ering an accurate prediction, the proposition alerts the possibility of economic
growth slowdown in the presence of political bias: If public policies excessively re‡ect the opinionsof the elderly, aging economies will underinvest in young generations, who will be the main labor
force in the future.
In order to avoid such a miserable consequence, the government needs to be impartial between
generations. By allocating a fair budget to public education, the government is able to lighten
40

Gi
Substituting (zti ) = ; eit = (eGt ) and (28) into (7) yields the indirect utility function of (et ; zt ): Then theGenvelope theorem reveals that the indirect utility function is strictly increasing in et : Moreover, as implied by (8);consumption cit is constant for any eG= ; one …nds that nit hit+1 increases with eGt : Thus recallingt .
41
For example, a French demographer Bourgeois-Pichat estimates that the population of European countries would
die out within 300 years if their fertility rates were and remained the West German level (Johnson et al., 1989).

32the households’burden on education, (eGt ); and thereby encourage childbirth. Equally important,these e¤ects can be augmented by improving the e¢ ciency of education policy.42 Therefore, witha high level of quantity and quality, public education works as a substitute for private education,
and as an e¤ective means of preventing population decline.

5

Concluding Remarks

This research has unveiled the crucial role played by public education in the transition to aging
societies since the early Industrial Revolution. In the early phases of Industrial Revolution, skill investment is undesirable for the masses, and compulsory education worked as a child labor regulationthat discouraged childbirth. On the other hand, in the later phases of the Industrial Revolution,
public education began to be viewed as a desirable service that assists households’ child rearing.
Nevertheless, it appears that public education may have been undersupplied in the postwar aging
countries.
The theory developed in this paper demonstrates that advanced economies may have poor
growth performance through a vicious cycle between population aging and the deterioration of
education policy. In order to make an escape from this cycle and sustain national pension systems,
policymakers should aim to prevent the allocation of national resources from being biased to aparticular age cohort. Furthermore, they need to carefully examine whether the budgets allocatedto public education are being used e¢ ciently. Without their dedicated e¤orts, public education
would not be an e¤ective substitute for private education, and thus fail to encourage childbirth.
The theoretical foundation behind these arguments is that education policy stimulates individuals’ incentives for childbirth as well as for skill investment. By taking this two-sided e¤ect into
account, one …nds that education policy may be more productive than other policies, such as childbene…ts, for the purpose of enhancing human capital at the national level. Future research should
be directed at investigating this potential superiority of public education.
42

To see this, consider public education investment in e¢ ciency units, denoted as t eG
t measures how
t ; whereGe¢ ciently eGt is invested: for instance, it depends on the allocation of et between facilities and teachers. In this casethe level of private education in Stage III is eit = ( t eG
t ):

33

Appendix 1

Technical Discussions

The Local Optimality Conditions for Private Education
This section discusses the case that zti

z and thus (zti ) => 0 in (3). Substituting (8) into

the objective function (7) yields the (partially) indirect utility function that depends on eit ; eGt andzti : The sign of the …rst derivative of this function with respect to eit is determined by the sign of8< Dl (ei ; eG ; z i ) if ei + eG < e^;
t t
t
t
t
i
D(eit ; eG
;
z
)
t
t
: Dh (ei ; eG )iif e + eG > e^;
t

t

t

t

[1

i
i G
! 1 (eit ; eGt ; zt )]h(et ; et );

where

i
Dl (eit ; eGt ; zt )Dh (eit ; eGt )ih1 (eit ; eGt )[ + eti!(eit ; eGt ; zt )]ih1 (eit ; eGt )( + et )h(eit ; eGt ):i
For analytical convenience, the domain of Dl (eit ; eGt ; zt ) is de…ned as
l

i
3
i
G
f(eit ; eGt ; zt ) 2 R+ : et + ete^ and zti

z g:

Noting the properties of !( ) in (5) and h( ) in (6), one obtainsi
D1l (eit ; eGt ; zt ) < 0;i
h i G
l i G ilimzti !1 Dl (eit ; eGt ; zt ) = D (et ; et ) > D (et ; et ; zt );ifor (eit ; eGt ; zt ) 28(eit ; eG
t )

(36)

l . On the other hand, D h (ei ; eG ) is de…ned on R2 and has properties such that,
t t
+

0;
D1h (eit ; eGt ) < 0;D2h (eit ; eGt ) < 0;lim Dh (eit ; eGt ) < 0:

eit !1(37)

In order to prove the last property in (37), note thath(eit ; eG
@
t )
> 1:
i
i
@et h1 (et ; eGt )Thus it follows thatlim

eit !1( + eit )h(eit ; eG
t )
< 0:
i
h1 (et ; eGt )The properties with respect to eit are depicted by the upper panel of Figure 2. The two solid
i
h i G
G
i
low . They
curves indicate Dl (eit ; eGt ; zt ) and D (et ; et ); respectively, for a given et and zt = z

are both negatively-sloped and are disconnected at eit = e^34eGt ; where the wage function !( ) is

kinked. The corresponding utility level is shown by the solid curve in the lower panel. The curveis continuous and kinked at eit = e^

i G i
eG
t : Furthermore, its slope is positive where D(et ; et ; z ) > 0;i
and is negative where D(eit ; eGt ; z ) < 0: These properties establish the following lemma.

Lemma 5 For an adult individual with zti

z , a level of private education, eit ; is at least locally

i
i
i G i
optimal if (a) D(eit ; eGt ; zt ) = 0 and et > 0 or (b) D(et ; et ; zt )0 and eit = 0:

Proof of Lemma 1
(a) 0

eG
t

e^: Since (A1) is equivalent to Dh (^e; e^) > 0; the properties of Dh ( ) in (37) reveal that
Dh (^
e

G
eG
t ; et ) > 08eG^]:t 2 [0; eUsing (37) again, one …nds that there exists a unique value eit > e^

h i G
eG
t such that D (et ; et ) = 0:

Thus as depicted by Figure 2, the curve Dh (eit ; eGt ) crosses the horizontal line at a unique point
above e^eGt :(b) eG^: The results in case (a), along with (37), imply that there exists a unique eitt >ethat Dh (eit ; eGt )0 such0; with equality if eit > 0:

Cases (a)–(b) show that there exists a single-valued function (eGt )GDh ( (eGt ); et )0;0 such that

G
(eG
^;
t ) + et > e

Gwhere the equality holds if (eGt ) > 0: In light of Lemma 5, (et ) is the optimal level of privateih i G
education at least on the interval where D(eit ; eGt ; zt ) = D (et ; et ).

Furthermore, applying the Implicit Function Theorem and noting (37), one obtains
0

(eG
t )=

where eit = (eGt ) > 0: Lastly,

0

(eG
t )

i
h12 (eit ; eG
h2 (eit ; eGt )( + et )
t )
< 0;
i
G
h11 (et ; et )( + eit )0 if (eGt ) = 0.

Proof of Lemma 2
(a) 0

eG
t

e^: Using (36)–(37) with (A1),ih i G
lim Dl (eit ; eGt ; zt ) = D (et ; et ) > 0

zti !18eit 2 [0; e^

eG
t ]:

i
This result is shown by Figure 2. As zti increases to z high and further, the curve Dl (eit ; eGt ; zt ) in

the upper panel shifts upward, approaching the upper bound Dh (eit ; eGt ). Correspondingly, the35

(partially indirect) utility function in the lower panel has a strictly positive slope on [0; e^ eGt ). On
the other hand, the proof of Lemma 1 shows that (eGt ) is the optimal level of private education atleast on the interval (^eeGt ; 1). Since the utility function is continuous on [0; 1) in the diagram,one …nds that (eGt ) is globally optimal.
(b) eG^: The proof of Lemma 1 shows that (eGt > et ) is the optimal level of private educationon the interval [0; 1): This directly proves the global optimality of (eGt ).

Proof of Lemma 3
Substituting qt+1 = nR
nt obtained from (16),t qt =nt into the expression of nt+1
!
!
nPt+1
nR
t+1
R
1 q t nt +1 (1 qt )nPt ;nt+1 nt =nt
nt
where qt

q0 < 1 for t 2 [0; t ] because of (21). Furthermore, in light of (16) and (20),
nit =
nt
nit+1

for t 2 [0; t

= qt

T
;
w=ai At
w=ai At+1+ (1w=aR At1); where aR > aP = 1 and At+1

qt )w=ai At+1;
w=At

At > w= because of (18) and (A3). Therefore,as At+1 (and thus At ) approaches w= from above, nPt goes to in…nity and nPt+1 =nt becomes greaterRthan unity, while nRt and nt+1 =nt remain …nite. In this case nt+1

nt goes to in…nity.

Proof of Lemma 4
Note that Ht+1 > (1P in (17). Furthermore, (20) and (24) imply that N P = nP N P > N P)T Nt+1ttt
t+1

for t 2 [0; t0 ): Then it follows from (18) that the growth rate of technology has a lower bound suchthat
gt+1 > g min
where N0P = (1g((1)T N0P ) > 0

t 2 [0; t0 );q0 )N0 > 0 under (A3). Using this result with (24) and (28) reveals that,

8t 2 [t ; t0 );nPt+1 <

wl(T)=[(1 + g min )At h(0;)]< nPt ;

where, in light of (25); At and nPt are independent of qt+1 and qt (> qt+1 ): On the other hand, (16)shows that nPt approaches nt as qt goes to zero. Thus, the relationship nPt+1 < nPt above implies
that nt+1 < nt if qt (and thus qt+1 ) is su¢ ciently small in period t 2 [t ; t0 ):36

Proof of Equation (31)
As follows from (16); (24) and (32), a su¢ cient condition for nt0 < nt0 1 isT
+ ( )where ! min

!(0;

qt0 1

T
+ (

)

+ (1

T
;
! min

qt0 1 ); z ) < !(0;; ztP0 1 ) because ztP < z 8t 2 [0; t0 ). Solving the expression

q

! min
+ ( )

above for qt0 1 yields

where ( )

(

t0

1

1

1

1

q max ;

) and thus q max 2 (0; 1]: Note that the condition qt0 1(A3) because, as shown in Section 4, qt+1

Appendix 2

! min
+ ( )

qt 8t

q max is satis…ed under0.

Data Sources

Figure 1 (Japan)
Old-age related bene…ts (Koreisha Kankei Kyufuhi ) are obtained from the National Instituteof Population and Social Security Research (2007, p. 102).
The number and the percentage of the population aged 65 and above are published by theJapanese Statistics Bureau. The data for 1973–2000 and 2001–2004, respectively, are obtained
from the following websites:
http://www.stat.go.jp/data/jinsui/wagakuni/index.htm
http://www.stat.go.jp/data/jinsui/2004np/index.htm#05k16-b
Public expenditures for national schools are recorded on the special account budgets (revised)of …scal years 1973–2003 as “Kokuritsu Daigaku”, and on the General Account budget (revised) of 2004 as “Kokuritsu Daigaku Hojin Uneihi ”.43 The whole data are published onlineby the Ministry of Finance Japan. Besides, the data for 1973–2000 are compiled in the electric…le “Kokuritsu Gakko Tokubetsu Kaikei (98)”by the Research Institute for Higher Educationof Hiroshima University. Their respective URLs are:http://www1.mof.go.jp/data/index.htm
http://rihe.hiroshima-u.ac.jp/data_list.php?%20-%2035
43

The di¤erence in the budget sources is due to the incorporation of national universities in 2004.

37

The number of students enrolled in national universities and graduate schools is obtained
from the …le “Zaigakushasu (2)”available at the Hiroshima University’s website listed above.

References
Ayusawa, Iwao, F., A History of Labor in Modern Japan, Honolulu: East-West Center Press, 1966.
Becker, Gary S., Kevin M. Murphy, and Robert Tamura, “Human Capital, Fertility, and Economic Growth,”
Journal of Political Economy, October 1990, 98 (5.2), S12–S37.
Church, Robert L., Education in the United States, New York: The Free Press, 1976.
Cunningham, Hugh, Children and Childhood in Western Society Since 1500, New York: Longman, 1995.
, “Combating Child Labour: The British Experience,” in Hugh Cunningham and Pier Paolo Viazzo,
eds., Child Labour in Historical Perspective 1800-1985: Case Studies From Europe, Japan and Colombia,
Florence, Italy: UNICEF, 1996, pp. 41–55.
Doepke, Matthias and Fabrizio Zilibotti, “The Macroeconomics of Child Labor Regulation,”American Economic Review, December 2005, 95 (5), 1492–1524.
Dore, Ronald P., “Education: Japan,”in Robert E. Ward and Dankwart A. Rustow, eds., Political Modernization in Japan and Turkey, Princeton, NJ: Princeton University Press, 1964, pp. 176–204.
Economist, “The AARP: Still the Biggest Bruiser?,” February 5th 2005, 374 (8412), 40.
Ensign, Forest Chester, Compulsory School Attendance and Child Labor, New York: Arno Press, 1969.
Flora, Peter, Jens Alber, Richard Eichenberg, Jurgen Kohl, Franz Kraus, Winfried Pfenning, and Kurt
Seebohm, State, Economy, and Society in Western Europe 1815–1975, Vol. I, Frankfurt: Campus
Verlag, 1983.
Galor, Oded and David N. Weil, “Population, Technology, and Growth: From Malthusian Stagnation to the
Demographic Transition and Beyond,” American Economic Review, September 2000, 90 (4), 806–828.
and Omer Moav, “Das Human-Kapital: A Theory of the Demise of the Class Structure,” Review of
Economic Studies, January 2006, 73 (1), 85–117.
Glass, D. V. and E. Grebenik, “World Population, 1800–1950,”in H. J. Habakkuk and M. Postan, eds., The
Cambridge Economic History of Europe, Vol. VI (Part 1), Cambridge: Cambridge University Press,
1965, chapter II, pp. 56–138.
Goldin, Claudia and Kenneth Sokolo¤, “Women, Children, and Industrialization in the Early Republic:
Evidence from the Manufacturing Censuses,” Journal of Economic History, December 1982, 42 (4),741–774.
Gradstein, Mark and Michael Kaganovich, “Aging Population and Education Finance,” Journal of Public
Economics, December 2004, 88 (12), 2469–2485.
Hazan, Moshe and Binyamin Berdugo, “Child Labour, Fertility, and Economic Growth,”Economic Journal,
October 2002, 112 (482), 810–828.
Heywood, Colin, Childhood in Nineteenth-Century France, Cambridge: Cambridge University Press, 1988.
Holtz-Eakin, Douglas, Mary E. Lovely and Mehmet S. Tosun, “Generational Con‡ict, Fisical Policy, andEconomic Growth,” Journal of Macroeconomics, March 2004, 26 (1), 1–23.
Hopkins, Eric, Childhood Transformed: Working-Class Children in Nineteenth-Century England, Manchester:Manchester University Press, 1994.

38

Horrell, Sara and Jane Humphries, “The Exploitation of Little Children: Child Labor and the Family
Economy in the Industrial Revolution,” Explorations in Economic History, October 1995, 32 (4), 485–
516.
Hurt, John S., Elementary Schooling and the Working Classes 1860-1918, London: Routledge & Kegan Pual,
1979.
Hutchins, B. L. and A. Harrison, A History of Factory Legislation, third ed., Oxon: Frank Cass, 1926.
Ishii, Shoji, Kindai no Jido Rodo to Yakan Shogakko (Child Labor and Night Schools in the Modern Era),
Tokyo: Akashi Shoten, 1992.
Japanese National Commission for UNESCO, The Role of Education in the Social and Economic Development, Tokyo: Institute for Democratic Education, 1966.
Johnson, Paul, Christoph Conrad, and David Thomson, “Introduction,”in Paul Johnson, Christoph Conrad,and David Thomson, eds., Workers Versus Pensioners: Intergenerational Justice in an Ageing World,
Manchester: Manchester University Press, 1989, chapter 1, pp. 1–16.
Kemnitz, Alexander, “Demographic Structure and the Political Economy of Education Subsidies,” Public
Choice, December 1999, 10 (3-4), 235–249.
Kremer, Michael, “Population Growth and Technological Change: One Million B.C. To 1990,” Quarterly
Journal of Economics, August 1993, 108 (3), 681–716.
Kuznets, Simon, “Population Change and Aggregate Output,” in “Demographic and Economic Change in
Developed Countries,” Princeton: Princeton University Press, 1960, pp. 324–351.
Landes, William M. and Lewis C. Solmon, “Compulsory Schooling Legislation: An Economic Analysis of
Law and Social Change in the Nineteenth Century,”Journal of Economic History, March 1972, 32 (1),54–91.
Lee, J. J., “Labour in German Indusrialization,” in Peter Mathias and M. M. Postan, eds., The Cambridge Economic History of Europe, Vol. VII (Part 1), Cambridge: Cambridge University Press, 1978,
chapter IX, pp. 442–491.
Lindert, Peter H., “The Rise of Social Spending, 1880–1930,” Explorations in Economic History, January
1994, 31 (1), 1–37.
Longman, Phillip, Born to Pay: The New Politics of Aging in America, Boston: Houghton Mi- in Company,
1987.
Maddison, Angus, The World Economy: A Millennial Perspective, Paris: the OECD Development Centre,
2001.
Madoc-Jones, Beryl, “Patterns of Attendance and Their Social Signi…cance: Mitcham National School 1830–
39,” in Phillip McCann, ed., Popular Education and Socialization in the Nineteenth Century, London:
Methuen, 1977, chapter 2, pp. 41–66.
Melton, James Van Horn, Absolutism and the Eighteenth-Century Origins of Compulsory Schooling in Prussia and Austria, Cambridge: Cambridge University Press, 1988.
Nardinelli, Clark, Child Labor and the Industrial Revolution, Bloomington: Indiana University Press, 1990.
National Institute of Population and Social Security Research, Shakai Hosho Tokei Nempo: Heisei 18 Nen
Ban (Annual Report on Social Security Statistics: 2006 Edition), Tokyo: Hoken, 2007.
Ogg, Frederic Austin and Walter Rice Sharp, Economic Development of Modern Europe, New York: Macmillan, 1929.
Ohkawa, Kazushi and Henry Rosovsky, “A Century of Japanese Economic Growth,” in William W. Lockwood, ed., The State and Economic Enterprise in Japan, Princeton: Princeton University Press, 1965,
chapter II, pp. 47–92.

39

and Henry Rsovsky, “Capital Formation in Japan,” in Peter Mathias and M. M. Postan, eds., The
Cambridge Economic History of Europe, Vol. VII (Part 2), Cambridge: Cambridge University Press,
1978, chapter III, pp. 134–165.
Organisation for Economic Co-Operation and Development (OECD), Education at a Glance: OECD Indicators 2006, Paris: OECD Publishing, 2006.
Osada, Mitsuo, Komori Gakko no Jisshoteki Kenkyu (Empirical Studies on Childminders’ Schools), Tokyo:
Waseda University Press, 1995.
Pampel, Fred C., “Population Aging, Class Context, and Age Inequality in Public Spending,” American
Journal of Sociology, July 1994, 100 (1), 153–195.
Pecchenino, Rowena A. and Kelvin R. Utendorf, “Social Security, Social Welfare and the Aging Population,”
Journal of Population Economics, November 1999, 12 (4), 607–623.
Pollard, Sidney, “Labour in Great Britain,” in Peter Mathias and M. M. Postan, eds., The Cambridge
Economic History of Europe, Vol. VII (Part 1), Cambridge: Cambridge University Press, 1978, chapter
III, pp. 97–179.
Poterba, James M., “Demographic Structure and the Political Economy of Public Education,” Journal of
Policy Analysis and Management, Winter 1997, 16 (1), 48–66.
, “Demographic Change, Intergenerational Linkages, and Public Education,” American Economic Review (Papers and Proceedings), May 1998, 88 (2), 315–320.
Preston, Samuel, “Children and the Elderly in the United States,” Demography, November 1984, 21 (4),435–457.
Rostow, Walt Whitman, The World Economy: History and Prospect, London: Macmillan, 1978.
Rubinstein, David, “Socialization and the London School Board 1870–1904: Aims, Methods and Public
Opinion,” in Phillip McCann, ed., Popular Education and Socialization in the Nineteenth Century,
London: Methuen, 1977, chapter 9, pp. 231–264.
Saito, Osamu, “Children’s Work, Industrialism and the Family Economy in Japan, 1872-1926,” in Hung
Cunningham and Pier Paolo Viazzo, eds., Child Labour in Historical Perspective 1800-1985: Case
Studies From Europe, Japan and Colombia, Florence, Italy: UNICEF, 1996, pp. 73–87.
Sanderson, Allen R., “Child-Labor Legislation and the Labor Force Participation of Children,” Journal of
Economic History, March 1974, 34 (1), 297–299.
Shibuya, Hideaki, “Hi Gakko Seido ni yoru "Gimu Kyoiku" no Hosho (The Securing of "Compulsory Education" by Non-School Systems),”in Miyao Mano and Toshiaki Kuwahara, eds., Kyoiku Ken to Kyoiku
Seido (Right to Education and Educational System, Tokyo: Daiichi Hoki, 1988, chapter 6, pp. 162–190.
Silver, Harold, “Ideology and the Factory Child: Attitudes to Half-Time Education,”in Phillip McCann, ed.,Popular Education and Socialization in the Nineteenth Century, London: Methuen, 1977, chapter 6,
pp. 141–166.
Taira, Koji, “Education and Literacy in Meiji Japan: An Interpretation,”Explorations in Economic History,
Summer 1971, 8 (4), 371–394.
, “Factory Labour and the Industrial Revolution in Japan,” in Peter Mathias and M. M. Postan, eds.,
The Cambridge Economic History of Europe, Vol. VII (Part 2), Cambridge: Cambridge University
Press, 1978, chapter IV, pp. 166–214.
Vinovskis, Maris, “An Historical Perspective on Support for Schooling by Di¤erent Age Cohorts,”in Vern L.
Bengtson and W. Andrew Achenbaum, eds., The Changing Contract Across Generations, New York:
Aldine de Gruyter, 1993.
Weber, Eugen, Peasants into Frenchmen: The Modernization of Rural France, 1870–1914, Stanford: Stanford University Press, 1976.

40

Weiner, Myron, The Child and the State in India, Princeton: Princeton University Press, 1991.
Weissbach, Lee Shai, Child Labor Reform in Nineteenth-Century France, Baton Rouge: Louisiana State
University Press, 1989.

41

Figure 1.

Population Aging and Generational Bias in Public Spending: Japan, 1973–2004

Notes: OLD AGE = old-age related bene…ts per person aged 65 and above. UNIVERSITY =public expenditures for national schools (mostly university) per student. OLD SHARE = the shareof the population aged 65 and above.
Sources: See Appendix 2.

42

Figure 2.

The Determination of Private Education for

i =
t

>0

Notes: The diagram depicts the decision of private education by an adult individual who cares
about the utility in elderhood. There may exist multiple local optima, depending on the levels ofiipublic education eGt and parental potential income zt . Given a su¢ ciently large amount of zt ; theindividual spends (eGt ) > 0 units of time per child on education.

43

Figure 3.

The Evolution of Fertility in Stage III

Notes: The diagram shows that fertility declines monotonically over Stage III, converging to a
nontrivial stationary-state equilibrium. The long-run fertility rate may or may not be lower thanunity (the critical level for stationary population), depending on how private education reacts toundersupplied public education.

44