Economic Growth with Locked-in Childbirth: From Under- toOver-Investment in Education
Masao Nakagawaa and Yoshiaki Sugimotob,yaInstitute of Social and Economic Research, Osaka University, Japanb

Faculty of Economics, Kagawa University, Japan

November 4, 2011

Abstract
This research argues that a quantitative change in the irreversible fertility decisions triggers the transition from under- to over-investment in education in the growth process. In theearly stage of economic development, parents place greater importance on the quantity than
the quality of children. The resulting large sunk cost of child rearing makes una§ordable education investment for children who are unexpectedly competent. Hence, a population control
policy that reduces the sunk cost advances the growth process and prevents the emergence of apoverty trap. In the later stage, by contrast, the quantity of children is locked into a small size,
leading to over-investment in education. An expansion of child care subsidies will promote theaccumulation of aggregate human capital and improve the welfare of future generations.
Keywords: Over-investment, Under-investment; Education; Fertility; Growth.
JEL ClassiÖcation: D10; J13; J24; O15.

The authors are grateful to Shin-ichi Fukuda, Seung-gyu Sim, and seminar participants at the University of Tokyo
for their useful comments and encouragement.
yCorresponding author. 2-1 Saiwai-cho, Takamatsu 760-8523, Japan. Tel.: +81 87 832 1836; fax: +81 87 832
1820. E-mail address: sugimoto@ec.kagawa-u.ac.jp (Y. Sugimoto).

11

Introduction

It is widely recognized that investment in human capital is one of the steady means to improve thewelfare of individuals and of an economy as a whole. Despite the recognition, under-investment ineducation is widespread over developing countries. A large fraction of households cannot a§ord theexpenses for schooling in the presence of borrowing constraints. On the other hand, the recognition
brought the issue of over-investment in higher education on developed countries. Students of thesecountries do not necessarily reward the Önancial aid from their parents.1 Using the data on UK
graduates between 1985 to 1990, an empirical analysis by Chevalier (2003) calculated that the wage
loss of over-educated workers was amount to 22% to 26%.2
A plausible conjecture from these facts is that the developed economies went through a transition
from under- to over-investment in education in the growth process. It appears, however, that no
single theory has shed light on the underlying mechanism of the phenomenon. Existing theories
are only partially satisfactory in this respect. The literature on inequality and growth, whichhas áourished since the 1990s, asserts the possibility of under-investment in human capital in thepresence of capital market imperfections (cf. Galor and Zeira, 1993; Moav, 2002; Mookherjee andRay, 2003). By contrast, the other theoretical literature argues that information imperfections,along with market imperfections, may induce precautionary reactions of individuals toward overinvestment in human capital (cf. Gould et al., 2001; Aiyagari et al., 2002).3 None of these studies
encompass the aforementioned transition.
Motivated by these observations, this research develops a theory to analyze the optimality ofprivate education, which varies with the process of economic development. It argues that the optimality is determined through the dynamic interaction between education investment and fertility
decisions in the growth process. Although imperfect, parents adjust the quantity of children they
intend to raise in order to prepare for future education expenses.4 The resulting accumulation
1

For the 2009 academic year in Japan, for instance, no less than 39.3 percent of high school dropouts was due tothe failure of adaptation to school life and study (the Japanese Ministry of Education, Culture, Sports, Science andTechnology. http://www.mext.go.jp/b_menu/houdou/22/12/1300746.htm).
2
In this line of empirical research is Sicherman (1991), who relies on the PSID (Panel Study of Income Dynamics)
data for the late 1970s.
3
Gould et al., (2001) considers the eroding e§ect of technological progress, which is biased and random acrosssectors, on human capital. Aiyagari et al. (2002) highlight the lack of insurance markets for ability as well as that ofloan markets.
4
Goldstein et al. (2003, p. 487, Table 2) compare mean personal ideal family size and mean personal expectedfamily size for young women by using the Eurobarometer 2001 survey. They report that the former measure is smaller2

of human capital alters technological progress, the return on education investment, and fertility
decisions of the next generation. This research also aim to o§er some policy implications for welfare
improvement, by examining policy reforms that a§ect the incentive of childbirth.
The growth model presented later features four key elements. First, the educational attainmentof children, a determinant of human capital, is the fruit of their own e§orts and parental support.
Second, there may arise conáict over private education policy within households, because of thedi§erence in their motivations.5 Children make e§orts to acquire skills at the cost of leisure time,
whereas parents face the trade-o§ between the quantity and the quality of their children.6 Second,childbirth is the irreversible investment in the quantity of children, as proposed by Fraser (2001) andDoepke and Zilibotti (2005).7 Once determined, the number of children to raise is not adjustablein either directions and hence is interpreted as a sunk cost of child rearing. Third, childbirthis accompanied by idiosyncratic, unexpected ability shocks on children. While these shocks may
induce parents to revise their initial education plan, their ex-post reactions may be constrained by
the irreversibility of childbirth.
Figure 1 gives insight into the lock-in e§ects of childbirth on education investment, which
are generated by the second and the third elements above.8 Panel (a) shows optimization by ahousehold whose children are, in fact, more competent than expected. At the time of childbirth,the households does not observe the true ability of children and its ex-ante (before the observationof true ability) optimal choice occurs at (n; eP ); where the ex-ante indi§erence curve is tangent tothan the latter by 0.2 to 0.4 points in major European countries (p. 486). A similar pattern applies to the UnitedStates (Hagewen and Morgan, 2005, p. 509, Figure 1). These disparities are consistent with this paperís view thatthe budget constraint is binding in child rearing; i.e., parents can raise funds for desirable education at the cost of
the family size.
5
There is a strand of literature analyzing child-rearing strategies and parent-child conáict in various contexts.
Weinberg (2001) develops a static agency model in which an altruistic parent motivates her child to make e§orts bya pecuniary means. The author Önds that in the presence of the subsistence level of consumption, the e§ort level
increases with parental income only at low income levels. Akabayashi (2006) develops a dynamic theory that explainschild maltreatment by a parent who imperfectly observes the accumulation of the childís human capital. The toughlove model of Bhatt and Ogaki (2008) shows that parents leave little transfers to impatient children so that poor
consumption in childhood makes them more patient.
6
See Becker and Lewis (1973) for the formulation of the quantity-quality trade-o§ faced by parents.
7
In relation to schooling, a recent study by de la Croix and Doepke (2009) focused on the lock-in e§ect of fertility
decisions on individualsí voting preferences, in accounting for the di§erences in public education systems acrosscountries. The assumption of perfect irreversibility would be relaxed by dividing the period of childbirth into two sothat unexpected ability shock occurs between them. This approach is taken by Iyigun (2000) for di§erent research
objectives from the present paper. The author develops a growth model with no uncertainty (i.e., no lock-in e§ect ofchildbirth) and demonstrates that the timing of childbearing is delayed by the accumulation of human capital.
8
The model presented in Figure 1 is not identical to the one introduced in Section 2. In particular, education
investment is a discrete choice in the latter model. Figure 1 nonetheless conveys the essence of the lock-in e§ects ofchildbirth that are analyzed later.

3

Figure 1.

The Lock-in E§ects of Childbirth on Education.

the budget line including the broken line. The quantity of children, n , is locked into a large levelin prospect of small education investment indicated by eP :In the diagram, ìPerfect foresightî indicates the point at which the ex-post indi§erence curve(i.e., the indi§erence curve under perfect foresight) is tangent to the original budget line. However,this point is beyond the actual budget constraint, which is kinked at n
 : That is, because thehousehold observes the true ability after childbirth, it is not possible to reduce the quantity ofchildren to achieve any higher education level than eP : The ex-post optimal decision is therefore
to carry out the initial education plan, eP , with no change in the family size. The opposite case is
explained by panel (b), in which children are in fact less competent than expected.
Taking the two types of lock-in e§ects into account, the theory developed below demonstrates thefollowing scenario of economic development. In the early stage of development, where technologicalprogress is sluggish, households place greater importance on the quantity than the quality of children
at the time of childbirth. The resulting sunk cost of child rearing, dominant in the household budget,
makes una§ordable education investment for children who are unexpectedly competent. Althoughthese children Önd skill acquisition advantageous, borrowing constraints prevent them from making
education loans.9 In this situation, a policy reform that discourages childbirth mitigates underinvestment in education and advances technological progress.
In the later stage of development, by contrast, households place more importance on the quality
9

Given this result, one may interpret locked-in childbirth as an implicit assumption in the aforementioned literature
on inequality, human capital, and growth.

4

of children at the time of childbirth. Now that the quantity of children is locked into a smaller
size in the household budgets, the burden of education expenses is not as heavy as before. As aresult, households invest in education as planned unless their children are signiÖcantly incompetent.
The problem here is that children receiving parental support are not necessarily inclined to engage
in skill acquisition. Under the circumstance, an expansion of child care subsidies at the cost of
the elderlyís welfare, will mitigate over-investment in education with no adverse e§ect on humancapital accumulation. It will also enrich the government by accelerating growth in the taxpaying
population, leaving the possibility of welfare improvement of future generations.
The rest of the present paper consists of the following sections. Section 2 describes the structureof the baseline model and considers optimal fertility decisions and individualsí attitudes toward
education. The last part of the section reveals the determination of macroeconomic variables.
Section 3 demonstrates the transition from under- to over-investment in education as a result of
fertility decline induced by exogenous technological progress. Section 4 incorporates endogenous
technological progress, which permits the interaction between fertility and education, and theninvestigates policy reforms that a§ect the incentive for childbirth. The proofs of mathematicalresults are given in the Appendix.

2

The Model

The economy has an overlapping-generations structure and operates over an inÖnite discrete time
horizon t  0.10 One single homogeneous good is produced in one sector by using human capital.
The formation of human capital is the fruit of the e§orts by children as well as education investmentby their parents. As a result of parent-child conáict over private education policy, either under- orover-investment in education arises in the growth process.

2.1

Firms

In perfectly competitive environments, producers generate a single homogeneous good by employing
human capital (i.e., e¢ciency unit of labor) with a linear technology. The level of output per worker
10This is an extension of the model developed by Galor and Weil (2000), who explore the mechanism underlying
the demographic transition in the long-term growth process.

5

in period t, denoted as yt ; is determined through the production function
yt = At Ht =Nt ;(1)

where At , Ht , and Nt are the technology level, the employed amount of aggregate human capital,
and the size of the working population, respectively, in period t:
For the sake of simplicity, the price of the Önal good is normalized to unity. As a result of
proÖt maximization by competitive producers, who are price takers, Ht maximizes the aggregate
proÖt At Ht  wt Ht , where wt is the market wage rate per unit of human capital in period t. In thecompetitive labor markets considered herein, wt is adjusted so that the resulting proÖt is neither
negative nor inÖnity large. It follows thatwt = At :Thus, the wage rate wt increases proportionally with the technology level At :2.2

Households

A new generation is born at the beginning of each period and lives for two periods. Generation t,born in period t  1, consists of a continuum of individuals existing on the interval [0; Nt ].
2.2.1

Environment

Consider the lifetime of an individual i 2 [0; Nt ] of generation t: In the Örst period (childhood),i ; and may also engage in skill acquisition. In the second period
the individual enjoys leisure, lt1

(adulthood or parenthood), the individual acquires hit e¢ciency units of labor and allocates thembetween child rearing and working. The individual raises nit units of identical children by spending( + eit ) e¢ciency units of labor per child, where  > 0 and eit  0 are the Öxed and the education
cost, respectively.11 The remaining labor is supplied to producers to earn wages, that are used up
for consumption, cit : It follows that
cit = wt [hit  ( + eit )nit ]:(2)

Utility of an individual i of generation t, uit , depends on leisure in childhood, consumption in
adulthood, and the quantity and quality of his/her children. Each of these children, indexed by
11

Parents may either train their children on their own or hire a teacher from the outside by paying wt eit :

6

j(i)

j(i) 2 [0; Nt+1 ]; acquires ht+1 e¢ciency units of labor in period t + 1: Taking these into account,the utility function is given by
n
h
io
j(i)
i
uit = (1  ) ln lt1+  (1  ) ln cit +  ln nit ht+1 ;(3)

where (; ) 2 (0; 1)  (0; 1).
2.2.2

Production of Human Capital

While children may di§er in educational attainment and innate ability across households, there is
no heterogeneities among siblings. The quantity of e¢ciency units of labor obtained by a child j(i),born from a parent i in period t, is determined according to the production functionj(i)ht+1 = h(it ; ait ; gt+1 ):(4)

where ait  0 and it  0 denote the levels of his/her ability and education attainment, respectively,
and gt+1 is the growth rate of technology between periods t and t + 1:12 One of the key assumptionsof the model is that it does not necessarily coincide with the level of parental education investment,
eit , which may be wasted by children.
The function h is assumed to satisfy three key properties. First, educational attainment has adiscrete impact on the formation of human capital. Children become either educated (skilled) or
uneducated (unskilled) labor, depending on whether their educational attainment reach a thresholdlevel  > 0. That is, 8(ait ; gt+1 ) 2 R2+ ;13h(it ; ait ; gt+1 ) =8< h(0; ai ; gt+1 ) > 0t: h(; ai ; gtt+1 ) > 0if it < ;if it  :(5)Second, a higher ability level makes skill acquisition more gainful (i.e., ability-education complementarity).14 That is, a (ait ; gt+1 ) > 0;8(ait ; gt+1 ) 2 R++  R+ ;(0; gt+1 )  1 andlimait !1 (ait ; gt+1 ) = 1;

12

(6)8gt+1  0:

As for the notation of eit and ait ; note that there is no need to use the superscript j(i) instead of i; because
children raised by a same parent are identical.
13
Throughout the present paper, fx (x; y) denotes the partial derivative of a function f with respect x:
14
The condition (0; gt+1 )  1; which is not indispensable, means that education investment does not enhance
human capital of children who are signiÖcantly incompetent. Because education investment is costly, these children
are raised as unskilled labor. On the ground that this is a minor case under the su¢cient condition, any uneducatedworker is simply referred to as unskilled labor throughout the present paper.

7

where the function  gives the relative skill level for given ability and the growth rate of technology:h(; ait ; gt+1 ):h(0; ait ; gt+1 )(ait ; gt+1 ) (7)Third, the acceleration of technological progress makes skilled labor more productive thanunskilled labor (i.e., skill-biased technological progress). That is, g (ait ; gt+1 ) > 0;8(ait ; gt+1 ) 2 R+  R++ ;(1; 0)  1 andlimgt+1 !1 (1; gt+1 ) = 1:

(8)

As will become evident, the conditions in the second line of Eqs. (6) and (8), which can be relaxedat the cost of exposition, ensure the existence of some critical values for education decisions.
2.2.3

Optimization by Parents

Optimization by parents, who are price takers with perfect foresight, is divided into two steps. At
the time of childbirth (ex-ante optimization), parents plan for future education investment in the
belief that their newborn children have average ability. After childbirth (ex-post optimization),parents unexpectedly Önd the true ability level of their children and thus may be inclined to alter
their initial plans.
Ex-Ante Optimization: Childbirth and Education Planning An individual i of generationt (a parent i in period t) decides the quantity of children, nit ; and the planned level of education
investment, ept : This decision making builds on the belief that these children will have averageability, whose level is normalized to unity, and fully make use of parental education support (i.e.,
ait = 1 and ept = it ). Then, in light of Eqs. (2), (3), and (4),fnit ; ept g = arg max (1  ) ln[hit  ( + ept )nit ] +  ln[nit h(ept ; 1; gt+1 )] ;(9)

subject to (nit ; ept )  0: The Örst-order condition for nit yields
nit =


hi :
 + ept t(10)

As will become apparent, the education planning determines the quantity of children, which in turnbinds actual education investment.
Substituting Eq. (10) into Eq. (9) reveals that
ept = arg max

h(ept ; 1; gt+1 ); + ept8

subject to ept  0: In view of Eq. (5), ept takes either 0 and ; and there is a critical level of gt+1 ,denoted as g~ > 0, for which parents are indi§erent between them; that is,(1; g~) = d:where d  ( + )=: By assuming for simplicity that the lower education level is chosen at g~,8< 0 if gt+1  g~;p
et =:  if g
> g~:

(11)

t+1

p

 e (gt+1 ):(12)Hence, the planned level of education investment is identical among the members of each generation.
Ex-Post Optimization: Education Investment

After having babies all at once, the adult in-dividual i in period t unexpectedly observes the ability level of his/her children, ait . In reconsidering
the education plan, two important assumptions are imposed. First, regardless of the level of observed ability, the individual continues to believe that these children will fully make use of parental
education support (i.e., eit = it ). Second, the predetermined fertility choice is unadjustable. At
this timing, it is not possible to change the quantity of children in either direction. Namely, theindividual takes nit in Eq. (10) as given in deciding the actual level of education investment, eit .
Thus in light of Eqs. (9) and (12),
eit




= arg max (1  ) ln 1  ( + eit ) arg max V (eit ; ait ; gt+1 );




i i
+  ln h(et ; at ; gt+1 ) + ept(13)

subject to eit  0:
Note that the term =( + ept ) in Eq. (13) indicates the burden of the locked-in fertility decision
on ex-post optimization. If  is so small that ( + )=  1, for example, any households choosing
ept = 0 beforehand cannot a§ord eit =  to secure a positive amount of consumption. Such a case isbeyond the scope of the present paper and thus is excluded by assuming
d < 1;
where d  ( + )= > 1:
9(14)

Now two points deserve special attention regarding ex-post optimization in Eq. (13). First,
the degree of the lock-in e§ect of childbirth varies with the growth rate of technology through thechange in the education plan, ept . Second, there is no income e§ect on the actual level of education
investment, eit . This is because a rise in hit proportionally increases the quantity of children, nit ;and thus has no impact on the budget constraint.
Let a~t be a critical ability level for which parents in period t are indi§erent between ex-post
education decisions. That is,
V (0; a~t ; gt+1 ) = V (; a~t ; gt+1 ):(15)Then, as will become apparent, a~t is given by a single-valued, noncontinuous, decreasing functiona~(gt+1 ) such that
eit

=

8
< 0
: 

if ait  a~(gt+1 );if ait > a~(gt+1 ): e(ait ; gt+1 ):(16)

where ait is the ability level of children raised by a parent i in period t: Thus, unlike in the ex-antecase, the ex-post education decision is heterogeneous across households of each generation.
A Benchmark Case: Perfect Foresight Environment

As a benchmark case, suppose ten-tatively that parents have perfect foresight into childrenís ability, while they believe that theirchildren will not waste parental education support (i.e., eit = it ). Using Eqs. (9) and (10), thepreferred (and Örst-best) level of education investment for a parent i in period t; denoted as ei
t ; isei
t = arg maxih(eit ; at ; gt+1 ): + eitj(i)iNote that ei
t is optimal for maximizing nt ht+1 ; which is the aggregate amount of human capital

produced by a parent i in period t: As will become clear, there is no reason that ei
t generally
coincides with the actual level of education, eit .
Noting Eq. (5) implying that ei
, let at be a critical ability level for whicht takes either 0 and 
parents in period t are indi§erent between the two education decisions. That is, by using Eq. (7),(at ; gt+1 ) = d > 1;

10

(17)

Figure 2.

The Determination of the Critical Ability Levels for Education.

where d  (+)= as deÖned earlier. This condition means that the relative advantage of education
investment is equal to the relative disadvantage in the quantity of children.
Figure 2 is useful to understand the determination of at . The line indicating the value d lies
between the top and the third horizontal lines because15
1   1  =d
1
1
ln
< ln d <
ln
:

1

1  d(18)

It follows that there is a unique ability level for which Eq. (17) is satisÖed for any gt+1  0:
Furthermore, an increase in gt+1 shifts the function (ait ; gt+1 ) upward, raising the critical ability
level. These are the intuition about Lemma 1.
Lemma 1 There exists a single-valued function at = a (gt+1 ) > 0 that satisÖes Eq. (17) for any
gt+1  0: Furthermore,(a) a0 (gt+1 ) < 0 8gt+1 > 0;(b) a (~g ) = 1:15
Eq. (10) implies that given gt+1 > 0 and ept = 0; an adult individual i in period t obtains more utility by choosingi
nt =  hit than choosing nit = +hi . This result along with Eq. (9) reveals the Örst inequality in Eq. (18). On thee tother hand, Eq. (10) implies that given gt+1 > 0 and ept = e; an adult individual i in period t obtains more utility
by choosing nit = +hi than by choosing nit =  hit . Using this result for Eq. (9) reveals the second inequality ine t

Eq. (18).

11



Proof. See the Appendix.
Recalling that the function  in Eq. (17) is strictly increasing in the ability level,8< 0 if ai  a (gt+1 );
t
ei
=
t
:  if ai > a (g ):
t

(19)

t+1

The properties of a (gt+1 ) presented in Lemma 1 are interpreted as follows. First, the existenceof a (gt+1 ) > 0 implies that in the absence of uncertainty, there are both skilled and unskilledworkers in the economy regardless of the growth rate of technology. Second, the negative reactionof a (gt+1 ) implies that education investment ei
t is promoted by the acceleration of technologicalprogress, which is assumed to be skill-biased. Finally, the result a (~g ) = 1 means that whether they
have perfect foresight or not, if gt+1 = g~, parents whose children have average ability are indi§erent
between the education decisions [cf. Eq. (11)]. This is straightforward because these parents indeedreceive no unexpected ability shock.
2.2.4

Optimization by Children

Faced with a trade-o§ between leisure and skill acquisition, children do not necessarily come up to
the expectations of their parents. They enjoy leisure if the parental education support is burdensome
on them. If it is insu¢cient for them, on the other hand, the absence of capital markets prevents
anyone from taking out education loans to cover the shortage. Given one unit of time, children
determine the optimal time allocation with the accurate information on their own abilities.
Consider a child j(i) in period t; who is born from a parent i. The leisure time left for the child,
j(i)

lt

, is
j(i)

lt

= 1  sit ;where  2 (0; 1) by assumption and sit 2 [0; 1] is the e§ort level or, more precisely, the childís time
devoted to skill acquisition.16 The condition  < 1 implies that a process of skill acquisition isperceived in part as leisure. On the other hand, achieving a certain education level requires not
only parental support but also the childís own e§orts. That is,it = sit eit :

(20)

16
The restriction on  ensures the existence of two corner solutions, sit = 1 and sit = 0: As is evident from Eq. (21),no one chooses sit = 0 if  = 0; whereas no one chooses sit = 1 if  = 1:

12

j(i)

Given Eqs. (2) and (10), the child knows that his/her consumption in adulthood, ct+1 ; has aj(i)linear relationship with his/her skills, ht+1 : Thus it follows from Eqs. (3) and (4) that the skillacquisition time chosen by the child, sit ; is
sit = arg max (1  ) ln(1  sit ) +  ln h(sit eit ; ait ; gt+1 ) ;(21)

subject to sit 2 [0; 1]. Because children are identical within households, one may index their optimal
choices by the superscript i rather than j(i).
Recalling Eq. (16) showing that eit is either 0 or , optimization by a child j(i) in period t isdivided into two cases. First, if eit = 0; then sit = 0. Because  is the minimum requirement ofeducational attainment to become skilled labor, it is wastefulness for the child to devote his/hertime, which is at most one, to schooling if parental support is less than . Second, if eit = ; thensit = 0 or 1: The child may engage in skill acquisition depending on his/her ability level and thegrowth rate of technology. Unless parental support is more than , it is necessary for the childsacriÖce his/her entire time to become skilled labor.
Let a^t be a critical ability level for which the child given eit =  is indi§erent between working
as skilled labor and as unskilled labor. That is, in view of Eq. (7), ln (^at ; gt+1 ) = (1  ) ln

1
> 0;
1

(22)

meaning that the relative advantage of working as skilled labor is just equal to the relative disadvantage in leisure. Figure 2 illustrates the determination of a^t graphically. It shows that there
is a unique ability level, indicated by a^(gt+1 ); for which Eq. (22) is satisÖed for any gt+1  0:
Furthermore, an increase in gt+1 shifts the function (ait ; gt+1 ) upward, raising the critical ability
level. These are the intuition about Lemma 2.
Lemma 2 There exists a single-valued function a^t = a^(gt+1 ) > 0 that satisÖes Eq. (22) for any
gt+1  0: Furthermore, a^0 (gt+1 ) < 0 8gt+1 > 0:
Proof. The lemma is proven in a similar way to Lemma 1.



Now suppose that children choose to work as unskilled labor when they are indi§erent. It
follows that the skill acquisition time chosen by a child of a parent i in period t is, for eit =  and13

for all gt+1  0;
sit =

8
< 0

if ait  a^(gt+1 );

: 1

(23)

if ait > a^(gt+1 );noting that the function  in Eq. (22) is strictly increasing in the ability level. As mentioned above,sit = 0 if eit = 0, regardless of the growth rate of technology.
Figure 2 provides the case that  1 ln(1  ) is in the neighbor of ln d, which occurs if andonly ifa^(gt+1 )  a (gt+1 )8gt+1  0;

(A1)

noting Eqs. (17) and (22). In comparison of Eqs. (19) and (23), this condition means that given
perfect foresight, parents would take similar stances toward education policy to their children, whofully observe their own abilities. In other words, parent-child conáict barely exists in the absenceof asymmetric information on ability. The analysis below focuses on this case, in order to exclude
some possible but irrelevant scenarios.17
Figure 3 draws the function a^(gt+1 ) using the results of Lemma 1. The negative reaction ofa^(gt+1 ) implies that more children are induced to acquire skills as skill-biased technological progress
accelerates.

2.3

Education Attainment

Educational attainment it reaches  only if both parents and their children take positive stances toprivate education. Combining the results of Eqs. (16) and (23) for Eq. (20), the level of educational
attainment by a child j(i) in period t is8
< 0 if ai  max[~a(gt+1 ); a^(gt+1 )];tit =:  if ai > max[~a(g ); a^(g )]:
t+1

t



t+1

(ait ; gt+1 ):(24)

As depicted by Figure 3, a^t = a^(gt+1 ) may or may not be greater than a~t = a~(gt+1 ), depending on
the level of gt+1 . When a^t < a~t ; for instance, children born with ait 2 (^at ; a~t ] Önd skill acquisition
17

The explicit condition for Eq. (A1) is
1   1  =d
1
1
1
1
ln
<
ln
<
ln
:

1

1

1  dNoting Eq. (18), the condition above holds if  1
ln(1  )  ln d.
14advantageous whereas their parents are not inclined to support them. That is, the borrowing
constraint is binding on them, and education investment is insu¢cient from the viewpoint of theyoung generation as a whole. Throughout the present paper, this case is referred to as under investment in education. Conversely, over -investment occurs when a~t < a^t : Children born withait 2 (~
at ; a^t ] waste education investment they receive, growing up with unskilled labor.
Comparing Eqs. (9) and (21) reveals the mechanism through which the relative position of a~t toa~t may change over time. In deciding their education policies, adult individuals take into accountthe quantity of children they intend to raise, whereas these children do not care about how many
siblings they have. As a result, only the formers are constrained by the irreversibility of childbirth.
Fertility decline in the growth process may therefore reverse the relationship between a~t and a~t .

2.4

Macroeconomic Variables

Suppose that ability levels are continuously distributed on R+ according to a stationary probability
distribution function (PDF). They are independent of each other not only within generations but
also across generations. In other words, ability is not inherited within dynasties, although it isidentical among children born from a same household.
Let G(at ) denote the corresponding cumulative distribution function, where at denotes an ability
level observed in period t: Substituting Eq. (24) into Eq. (4), the average level of e¢ciency unitsof labor in period t, denoted as ht , is
ht =
=

Z Nt
Z0 1

hit di=Nth((at1 ; gt ); at1 ; gt )dG(at1 ):

(25)

0

Then, it follows from Eq. (10) that the ratio of the child population to the adult population inperiod t, denoted as nt , is
nt 

Nt+1
=
Nt

Z Nt
0

nit di=Nt =ht : + ep (gt+1 )(26)

It follows that the evolution of the working population is
Nt+1 = nt Nt =

t
Q

j=0

15

nj N 0 ;(27)

where the initial size N0 is historically given. Then, in light of Eqs. (12), (13), and (16), theaggregate level of human capital employed by Örms in period t, Ht ; isHt = N t ht





Z 1

1
+
e(at ; gt+1 )dG(at ) ; + ep (gt+1 )0where the negative term is, as a whole, interpreted as the time cost of child rearing per worker.
These results show that all aggregate variablesónt ; Nt ; ht ; Ht ; and yt in Eq. (1)óare determined
for all t  0 once the initial size of the working population, N0 ; and the dynamic path of the growthrate of technology, fgt g1
t=0 , are given.

3

Exogenous Technological ProgressThis section demonstrates the transition from under- to over-investment in education in the processof economic development driven by exogenous technological progress. It focuses on the dynamic
path on which the growth rate of technology gt eventually exceeds the critical level g~ and remains
above g~ afterwards, although it does not necessarily increase monotonically. Then, there is a critical
period t~ after which gt exceeds g~ for the Örst time; i.e., gt  g~ 8t  t~ and gt > g~ 8t > t~. The growth
process is then divided into the following two stages.
Stage I (0  t < t~): This underdevelopment stage involves under -investment in education. Because the quantity of children is locked into a large level in the household budget, parents
whose children are unexpectedly competent cannot a§ord education investment. These children have no access to capital markets to make education loans.
Stage II (t  t~): This developed stage is characterized by over -investment in education. Now thatthe quantity of children is locked into a small level in the household budget, even parents
whose children are unexpectedly incompetent have spare resources for education investment.

3.1

Stage I: Under-Investment in Education

In Stage I, where 0  t < t~ and 0 < gt+1  g~; all households concentrate their resources on thequantity, rather than the quality, of children before the true ability level is observed. That is, theirplanned level of education and fertility decisions are
ept = 0 and

16

nit =

 i
h:
 t

(28)

It follows from Eq. (13) that for any gt+1 2 [0; g~];8
< (1  ) ln(1  ) +  ln h(0; ai ; gt+1 )ti iV (et ; at ; gt+1 ) =: (1  ) ln(1  d) +  ln h(; ai ; g )tt+1

if eit = 0;if eit = ;(29)

where 0 < 1  d < 1   from Eq. (14). The di§erence between 1   and 1  d, indicating theadverse e§ect of revising the initial education plan on consumption, is a result from the irreversibilityof childbirth. The di§erence would not exist if the quantity of children was adjustable at the timeof education investment.
It follows from Eqs. (7) and (29) that in Stage I, the critical ability level a~t in Eq. (15) satisÖes
 ln (~at ; gt+1 ) = (1  ) ln

1
> 0:
1  d(30)This condition means that the relative advantage of education investment is equal to the relative
disadvantage in consumption. Figure 2 illustrates the determination of a~t in Stage I. It showsthat there is a unique ability level, indicated by a
~I (gt+1 ); for which Eq. (30) is satisÖed for any
gt+1  0: An increase in gt+1 shifts the function (ait ; gt+1 ) upward, raising the critical ability level.
Furthermore, a~I (gt+1 ) > a (gt+1 ) as follows from the parameter relationship in Eq. (18). These are
the intuition about Lemma 3.
Lemma 3 There exists a single-valued function a~t = a~I (gt+1 ) that satisÖes Eq. (30) for any gt+1 
0: Furthermore,(a) a~I0 (gt+1 ) < 0 8gt+1 > 0;(b) a
~I (gt+1 ) > 1 8gt+1 2 [0; g~];(c) a
~I (gt+1 ) > a (gt+1 ) 8gt+1  0:

Proof. See the Appendix.

Thus in Stage I, where gt+1 2 [0; g~]; the preferred education level for a parent i in period t is8
< 0 if ai  a~I (gt+1 );
t
eit =
(31)
:  if ai > a~I (g );
tt+1

recalling that the function  in Eq. (30) is strictly increasing in the ability level. Figure 3 graphically
represents the properties of a
~I (gt+1 ) based on the results of Lemma 3. The solid line portion

indicates that a~t is given by a
~I (gt+1 ) only on the interval [0; g~]. Because a~I (gt+1 ) is decreasing
17

Figure 3. The Relationship Between the Growth Rate of Technology and the Critical Ability
Levels for Education.

in gt+1 , education investment is promoted by the acceleration of technological progress, which isassumed to be skill-biased.
Furthermore, a~I (gt+1 ) is greater than 1 on [0; g~], implying that in Stage I, parents whose children
have average ability do not invest in education. This result is explained by noting Eq. (28). In
prospect of no education investment, the quantity of children in Stage I is locked into a large levelthat magniÖes the education expenses per household. In order to o§set the adverse e§ect andinduce education investment, the ability level needs to be su¢ciently higher than the expectedlevel at childbirth (i.e., ait  a~I (gt+1 ) > 1).
The diagram depicts a~I (gt+1 ) above a (gt+1 ), which represents the critical ability level for theperfect-foresighted parents considered in Section 2.2.3. Given the ability information in advance,
parents would adjust childbirth so as to achieve their preferred level of education investment. Thatis, unlike imperfect-foresighted parents, their education decisions would not be constrained by theirreversibility of childbirth. This is why a (gt+1 ) is lower than a~I (gt+1 ) for any g+1  0:18
Under Eq. (A1), the gap between a~I (t+1 ) and a (gt+1 ) implies parent-child conáict over education policy, because children behave as if they were the aforementioned perfect foresighted parents.
This is the intuition of Proposition 1 below.
Note that a~I (gt+1 ) is greater than a (gt+1 ) for all gt+1  0: Technically speaking, this is because a~I (gt+1 ) is
derived, regardless of gt+1 ; from the utility comparison under the child-rearing plan in Eq. (28).
18

18

Proposition 1 (Under-Investment in Education) Under Eq. (A1),a~I (gt+1 ) > a^(gt+1 )8gt+1  0:

Proof. See the Appendix.

In Figure 3, adult individuals whose children have ait 2 (^at ; a~t ) do not invest in education
against the demand from their children. In such a circumstance, childbirth-discouraging policy,
which locks the quantity of children into a smaller level, mitigates under-investment in educationand may thereby prevent the emergence of a poverty trap in Stage I. This possibility will be
examined later in Section 4.2.

3.2

Stage II: Over-Investment in Education

In Stage II, where t  t~ and gt+1 > g~; all households prepare for future education by adjusting thequantity of children they intend to raise. That is, their planned level of education and corresponding
fertility decisions are
ept =  and

nit =

 i
h:
 +  t(32)

It follows from Eq. (13) that for any gt+1  g~;8< (1  ) ln(1  =d) +  ln h(0; ai ; gt+1 )tV (eit ; ait ; gt+1 ) =: (1  ) ln(1  ) +  ln h(; ai ; g )tt+1

if eit = 0;if eit = ;(33)

where 1=d > 1 from Eq. (14). The di§erence between 1 and 1=d indicates the adversee§ect of revising the initial education plan on consumption. Because the quantity of children isunadjustable at the time of education investment, the remaining income after choosing no educationis spent entirely on consumption.
It follows from Eqs. (7) and (29) that in Stage II, the critical ability level a~t in Eq. (15) satisÖes
 ln (~at ; gt+1 ) = (1  ) ln

1  =d
> 0:
1

(34)

This condition means that the relative advantage of education investment is equal to the relative
disadvantage in consumption. Figure 2 illustrates the determination of a~t in Stage I. It showsthat there is a unique ability level, indicated by a
~II (gt+1 ); for which Eq. (30) is satisÖed for any
gt+1  0: An increase in gt+1 shifts the function (ait ; gt+1 ) upward, raising the critical ability level.
Furthermore, a~II (gt+1 ) < a (gt+1 ) as follows from the parameter relationship in Eq. (18). These
are the intuition about Lemma 3.
19

Lemma 4 There exists a single-valued function a~t = a~II (gt+1 ) > 0 that satisÖes Eq. (34) for any
gt+1  0: Furthermore,(a) a~II0 (gt+1 ) < 0 8gt+1 > 0;(b) a
~II (gt+1 ) < 1 8gt+1  g~;(c) a~II (gt+1 ) < a (gt+1 ) 8gt+1  0:

Proof. See the Appendix.

Thus in Stage II, where gt+1 > g~; the preferred education level for a parent i in period t is8
< 0 if ai  a~II (gt+1 );
t
eit =
(35)
:  if ai > a~II (g );tt+1

recalling that the function  in Eq. (34) is strictly increasing in the ability level. Figure 3 graphically
represents the properties of a
~II (gt+1 ) from Lemma 4. The solid line portion indicates that a~II (gt+1 )is equal to a~t only on the interval (~g ; 1). Like the other functions in the diagram, the negative
slope of a
~II (gt+1 ) shows that education investment is promoted by the acceleration of skill-biased
technological progress.
The property that a~II (gt+1 ) is smaller than 1 on (~g ; 1) implies that in Stage II, parents whosechildren have average ability invest in education. This result is explained by noting Eq. (32).
In prospect of education investment, the quantity of children in this stage is locked into a small
level that lowers the sunk cost of child rearing. Parents are consequently inclined to follow theirinitial education plans, unless their children are signiÖcantly incompetent as opposed to the initialexpectation (i.e., ait  a~II (gt+1 ) < 1).19
The diagram depicts a~II (gt+1 ) below a (gt+1 ), which represents the critical ability level forthe perfect foresighted parents considered in Section 2.2.3. Unlike imperfect-foresighted parents
considered herein, they would adjust the quantity of their children in accordance with the abilityof their children, so that the child-rearing cost would not become unexpectedly small. This is whya (gt+1 ) is higher than a~II (gt+1 ) for any g+1  0:20
As shown by Lemma 4(b), this is also true for the case gt+1 = g~, in which parents are indi§erent between ept = 0
and ept = . The reason is that a~II (gt+1 ) is the critical level for education investment when nit is locked into a smallilevel, + ht :20
Moreover, this result and Lemma 3(c) show that a~(gt+1 ) given a positive value for any gt+1  0. Namely, bothskilled and unskilled labor exist at any stage of economic development (recall that ability levels are distributed on
R+ ).
19

20

In fact, the gap between a~II (gt+1 ) and a (gt+1 ) implies parent-child conáict over private education policy, because children behave as if they were the aforementioned perfect foresighted parents.
This is the intuition of Proposition 2 below.
Proposition 2 (Over-Investment in Education) Under Eq. (A1),a~II (gt+1 ) < a^(gt+1 )8gt+1  0:

Proof. See the Appendix..

Thus in Stage II, there exists the ability range (~at ; a^t ), on which children waste education support
provided by their parents. The following section extends the baseline model to an endogenous
growth model by assuming that these children do not contribute to technological innovation.

4

An Extension: Endogenous Growth with Public Policy

This section endogenizes technological progress and investigates the resulting evolution of the economy. It demonstrates the possibility of multiple steady-state equilibria and then examines how thegovernmentís policy allows the economy to develop beyond a a poverty trap.

4.1

Technological Progress

In the economy considered herein, technological progress is driven by skilled workers, who arewilling, not forced, to engage in skill acquisition in childhood. SpeciÖcally,
gt+1 = g(t );(36)

where t 2 (0; 1) is the fraction of skilled workers in generation t:21 g(t ) is a continuous function
characterized by the following properties. First, no new technology is created without skilledworkers; i.e., limt !0 g(t ) = 0. Second, in line with Galor and Moav (2000), technological progressis accelerated by an increase in the fraction of skilled workers; i.e., g 0 (t ) > 0 8t 2 (0; 1). Third,
technological progress is bounded above; i.e., limt !1 g(t ) < 1.
Members of generation t  1 face the same probability of drawing a certain ability level at1
according to the cumulative distribution function G(at1 ): In light of Eq. (24), the probability that
21This formulation implies that the two important factors of technological progress are o§set by each other: thescale e§ect of population and the Öshing out e§ect (i.e., reaching a higher technology level increases the di¢culty inÖnding new ideas). See for example Jones (1998, Ch. 5) and Weil (2009, Ch. 9)21any member of generation t  1 becomes unskilled labor in period t is pt  G(max[~a(gt ); a^(gt )]),
where
max[~
a(gt ); a^(gt )] =8< a~I (gt )
: a
^(g )
t

if gt 2 [0; g~];if gt > g~;noting Propositions 1ñ2. It then follows from Eq. (26) thatt =Z Nt10(1  pt )nit1 di=Nt= (1  pt )nt1 Nt1 =Nt= 1  G(max[~a(gt ); a^(gt )]);(37)

where G0 (at1 ) > 0 8at1 > 0:
Substituting Eq. (37) into Eq. (36), the evolution of gt is governed by the dynamical system8< g(1  G(~
aI (gt )))  I (gt ) if gt 2 [0; g~];gt+1 =: g(1  G(^a(g )))  II (g ) if g > g~:ttt (gt );(38)

where g0 is historically given. Figure 4 represents the dynamical system characterized by four key
properties. First, I (0) > 0 because even in the prospect of no technology growth, households whosechildren are su¢ciently competent Önd education investment desirable; i.e., a~I (0) < 1. Second,I0 (gt ) and II0 (gt ) > 0 8gt > 0 because the acceleration of skill-biased technological progress
attracts both parents and children toward education; i.e., a~0 (gt ) < 0 and a^0 (gt ) < 0 8gt > 0. Third,I (gt ) < II (gt ) 8gt  0 because parental education investment is constrained by the quantity ofchildren, which is locked into a large level (cf. Eq. (28), whereas childrenís education decisions arenot; i.e., a~I (gt ) > a^(gt ) 8gt  0: Forth, (gt ) is discontinuous at gt = g~; above which the economy
enters Stage II. Lastly, limgt !1 (gt ) < 1 because the function g(t ) is bounded above.
In the diagram, gl and gh indicate potential steady-state levels of gt , for which I (g l ) = gl andII (
g h ) = gh ; respectively.22 Suppose that the initial condition is given by
0  g0 < gl :

(A2)

22
While the existence of gh and gl is guaranteed by the properties of I (gt ) and II (gt ), their uniqueness isgenerally ambiguous because these functions may not be concave. To avoid confusion, these values are deÖned asgl  minf
g  0 j I (g ) = gg and gh  minf
g  0 j II (g ) = gg, implying that gl < gh :22

Figure 4.

The Evolution of the Growth Rate of Technology.

Note that steady-state equilibrium gt = (gt ) occurs at either gl or gh ; yet may not at both ofthem. Multiple steady-state equilibria occur if and only ifI (~g )  g~ < II (~
g ):

(A3)

This multiplicity condition holds depending on the quantitative properties of the underlying functions , G, a
~I , and a^ as well as on the level of g~. In particular, the gap between I (~g ) and II (~g ) isdue in part to the deviation of a~I (~g ) from a^(~g ) or, in other words, the prevalence of under-investment
in education at the end of Stage I.
Note that both of the steady-state equilibria are locally stable. In particular, one of them, which
occurs at gl ; works as a poverty trap in Stage I. Given the initial condition in Eq. (A2), the growthrate of technology increases monotonically over time and converges to the lower steady-state levelgl in Stage I. In this circumstance, only a small fraction of households are inclined to invest in
education (i.e., a~I (gt ) is substantially large), and technological progress becomes stagnant before
gt reaches the threshold level g~:
gt converges toward gh if and only if g0 > g~ or, equivalently, if and only if the economy starts
out with Stage II. This implies that, under Eq. (A3), there is no mechanism that permits theendogenous transition from Stage I to Stage II. The following section examines how public policy
can bring the economy out of the poverty trap to the higher steady-state level.

23

4.2

Public Policy

Now suppose that individuals live three periods (childhood, adulthood, and elderhood), whereas
no change is made on the production of Önal output. The economy has a central government thatlevies an income tax to provide child care for young parents and a public service for the aged. It
is shown that a population control policy that discourages childbirth prevents the emergence of apoverty trap. Furthermore, an expansion of child care subsidies in developed stages will promote
the accumulation of aggregate human capital and improve the welfare of future generations.
4.2.1

The Government

In every period t; the government levies a tax on potential income wt hit with a constant rate
 2 (0; 1). The resulting tax revenue, which is amount to  wt ht Nt ; is allocated between generations
living in the period. On the one side, each adult individual receives wt ( 0  )nt as child casesubsidies, where  0 > 0 and  > 0 are the pre- and the post-subsidy cost of child rearing per child,respectively, in terms of e¢ciency units of labor. On the other side, each old individual receives xt
units of the elderly-related public service. Under the balanced budget condition, the governmentísbudget constraint is, ht = ( 0  )nt +xt;
wt nt1(39)

where nt1 = Nt =Nt1 is the ratio of the taxpaying population to the aged population in period
t: Therefore, a rise in wt nt1 allows, ceteris paribus, the government to increase either  or xt inperiod t:
4.2.2

Households

Consider the life of an individual i of generation t (a parent i in period t). As for the Örst and second
periods of life, the environment is the same as before, except that the individual in the second period
receives child care in exchange for the tax payment. In the third period, the individual retires andmerely consumes the public service for the elderly, xt+1 . Then it follows from Eq. (2) that thebudget constraint faced by the individual is replaced with
cit = wt [(1   )hit  ( + eit )nit ]:24

In view of Eq. (3), the utility function is extended to
n
h
io
j(i)
i
uit = (1  ) ln lt1+  (1  ) ln cit +  ln nit ht+1
+ ln xt+1 :Individuals take not only the wage rate but also public policy as given. At the time of childbirth,they believe as before that their children will be born with average ability and fully make use ofparental education support (i.e., ait = 1 and ept = it ). Then the quantity of children raised by aparent i in period t isnit =

(1   )hit ; + ept

where the only deviation from the baseline model is that nit now depends on the tax rate  : Becauseof the multiplicative relationship between (1   ) and hit , introducing the income tax has no e§ect
on parentsí attitudes toward education represented by ept and eit . These results and Eq. (26) revealthat the ratio of the taxpaying population to the aged population, nt ; isnt =

(1   )ht ; + ept(40)

where ept and ht are respectively given by Eqs. (12) and(25).

4.3

Analysis

This section explores the implications of policy reforms that mitigate either under- or over-investment
in education. The following results, most of which are apparent from Figure 2, reveal how a change
in child care subsidies a§ects individualsí attitudes toward education.
Lemma 5 Let a^(gt+1 ), a
~I (gt+1 ), and a~II (gt+1 ) be expressed as a^(gt+1 ; d), a~I (gt+1 ; d), and a~II (gt+1 ; d),respectively, where d  1 + = > 1. Then,(a) a^d (gt+1 ; d) = 0 8gt+1  0;(b) a
~Id (gt+1 ; d) > 0 and a~IId (gt+1 ; d) > 0 8gt+1  0;(c) limd!1 [^a(gt+1 ; d)  a~I (gt+1 ; d)]  0 8gt+1  0;(d) @~
g =@d > 0.


Proof. See the Appendix.

25

Figure 5.

The Dynamical System for a Large :Given the results of Lemma 5 and , consider an increase in  (i.e., shift in the government budget
from child care subsidies to the elderly-related service) during Stage I. The parameter change leads
to the downward shift of a
~I (gt+1 ) in Figure 3, with no e§ect on a^(gt+1 ): The reduction in theinterval between a^(gt+1 ) and a~I (gt+1 ); on which children demand parental education investment,
implies the mitigation of under-investment in education and the increase in the fraction of skilledlabor in Stage I.23 This result is straightforward. A rise in  discourages childbirth and thereby
weakens its lock-in e§ect on education investment. By contrast, child care subsidies have no impact
on the education decisions of children, because they do not take into account the household budgetthat depends on the family size.
The rise in  also brings about the downward shift of a
~II (gt+1 ) in Figure 3 for the same reason
as a~I (gt+1 ): The reduction in the interval between a~II (gt+1 ) and a^(gt+1 ); on which children waste
parental education investment, implies the mitigation of over-investment in education in Stage II
with no e§ect on the fraction of skilled labor.
Figure 5 depicts the resulting e§ect of the parameter change on the dynamical system underEq. (A3). Increasing  shifts up the function I (gt ) toward II (gt ) as a result of the increase in
the fraction of skilled labor for given gt . The change in  also has no e§ect on II (gt ) and an
adverse e§ect on the critical level g~, which divides Stages I and II. These structural changes in the23

Note that Lemma 5(c) implies the violation of Eq. (A1). The second inequality in Footnote 17 does not holdif  is su¢ciently large (and thus d is su¢ciently close to 1). Nevertheless, Eq. (A1) is kept satisÖed unless therelationship between a^(gt ) and a~I (gt ), which reáects the relationship between I (gt ) and II (gt ), is reversed.

26

dynamical system advance technological progress in Stage I. Furthermore, if  is su¢ciently small,the poverty trap dissipates as depicted in the diagram; that is,24
I (~g ) > g~:(A3í)

In this case, gt converges toward the steady-state level in Stage II, gh , regardless of the initialcondition.
Now two remarks deserve special attention. First, the government does not need to change permanently in order to keep the economy away from the poverty trap. It may decrease  backto the original level once the growth rate of technology exceeds the threshold level g~: Second, it
is not clear whether the economy in the poverty trap is dynamically ine¢cient, because of theambiguous overall e§ect of the policy reform on the government budget in Eq. (??). While theacceleration of technological progress boosts the wage rate wt ; it is to some extent countervailed
by the deceleration of growth in the taxpaying population (i.e., a decrease in nt1 ).
These results are summarized below.
Proposition 3 (Population Control Policy in Stage I) Under Eqs. (A1) and (A2), consider
a policy reform that shifts the government budget during Stage I, from child care subsidies to theelderly-related service.
(a) The reform mitigates under-investment in education and advances technological progress inStage I.
(b) If Eq. (A3) is initially satisÖed, the reform may prevent the economy from being trapped inStage I, depending on its extent.
Next, consider a temporary decrease in  (i.e., shift in the government budget from the elderlyrelated service to child care subsidies) after the economy enters Stage II. The parameter change
brings about the upward shifts of a
~I (gt+1 ) and a~II (gt+1 ). The reduction in the interval betweena~II (gt+1 ) and a^(gt+1 ); on which children waste parental education investment, implies the mitigationof over-investment in education in Stage II with no change in the fraction of skilled labor, t ; andin average human capital, ht . This result is intuitive. As follows from Eq. (40), a reduction in 
24

Lemma 5(c) is of particular importance for the dissipation of the poverty trap. It ensures the existence of a smallvalue of d > 1 for which I (gt ) coincides with II (gt ) for given gt . This fact, along with the property I (0) > 0 andLemma 5(d), reveals that I (~g ) is greater than g~ if d is su¢ciently small.

27

encourages childbirth and thereby raises the sunk cost of child rearing that constrains education
investment.
The transition from Figures 5 to 4 illustrates the resulting e§ect of the policy reform on the
dynamical system. A decline in  shifts down the function I (gt ) away from II (gt ) and also
increases the critical level g~. Although this may lead to the re-emergence of the poverty trap,
technological progress is intact as long as the increase in g~ is moderate.
Because the wage rate and average human capital are una§ected by the policy reform, accelerated growth in the taxpaying population enlarges the government budget in the period after thepolicy reform.25 The increment of the tax revenue can be allocated to child care subsidies in the
period without curtailing the budget for the elderly-related service. Therefore, it is possible toimprove the welfare of future generations by gradually raising child care subsidies over Stage II (tothe extent that Eq. (A1) is satisÖed). These results are summarized below.
Proposition 4 (Child Care Subsidies in Stage II) Under Eqs. (A1), (A30 ), and (A2), consider a policy reform that gradually shift the government budget during Stage II, from the elderlyrelated service to child care subsidies. The reform mitigates over-investment in education andimproves the welfare of future generations with no harm on technological progress.
As a Önal remark, it is notable that the proposition is silent about Pareto improvement. In
the absence of capital markets, no compensation is possible for the squeezed elderly in the initialreform period.

5

Concluding Remarks

This research has elucidated the role of irreversible childbirth in economic growth. In the stageof underdevelopment, unexpected ability shocks on newborn children induce some households torevise their education plans upward. Their ex-post reactions, however, may be constrained by thepredetermined family size, which squeezes the household budget as a sunk cost. Fertility decline
in the growth process therefore lessens the sunk cost and promotes education investment, easing
borrowing constraints on children who aim to raise the money for education. In the more developed25

By using Eq. (39), this means that nt1 increases whereas ht and wt , both of which vary with technologicalprogress, are unchanged by the policy reform.

28

stage, the irreversibility constraint is binding in the opposite direction. That is, households areunable to have children additionally even if they do not invest in education against the initial plan.
For the second best, households tend to invest their resources on the existing children, leading toover-investment in education.
These results derive two notable policy implications. First, a policy reform that discourages
childbirth lessens the sunk cost of child rearing and promotes education investment in the underdeveloped stage. Technological progress fueled by the increased share of skilled workers ultimately
alters householdsí (ex-ante) stances toward education and brings about fertility decline. Second,raising child care subsidies in the developed stage will mitigate over-investment in education andaccelerate growth in the working population. The resulting expansion of the government budget
improves the welfare of future generations at the cost of the elderly-related service in the initialreform period.
While the central thesis of the present research is intuitive, the developed theory builds on anumber of simplifying assumptions. A more general theory would allow for the following aspects.
First, parental human capital or ability would be one of the key factors in the formation of the
expectation of childrenís ability. In this case, ability shocks after childbirth would not be totally
unexpected. Second, parents would more or less make a coordination of private education policy
with their children, which would moderate under- and over-investment in education. Third, fertility
adjustment to prepare for future education expenses should be bounded below (that is, the minimum
number of children is 1) because in reality investment in the quantity of children is a discrete choice.
Unbounded fertility adjustment would make over-investment in education more likely to occur inthe developed stage. Lastly, it is desirable to examine the possibility of Pareto improvementin the presence of capital markets, through which the government can transfer resources acrossnonadjoining generations. These issues should be addressed in future research.

Appendix: Technical Discussions
Proof of Lemma 1.

The properties of (at ; gt+1 ) with respect to ait , given by Eq. (6), ensure theexistence of a unique value at > 0 such that (at ; gt+1 ) = d for any gt+1  0: This implies theÖrst statement of Lemma 1. It then follows from Eq. (11) that at = a (gt+1 ) = 1 if gt+1 = g~.
29

Furthermore, the Implicit Function Theorem reveals that a0 (gt+1 ) =  g (at ; gt+1 )= a (at ; gt+1 ) <

0 8gt+1 > 0; by noting Eqs. (6) and (8).
Proof of Lemma 3.

The Örst statement and result (a) of the lemma are proven in a similar way

to Lemma 1.
(b) Since, in view of Eq. (30), (ait ; gt+1 ) increases with ait , the result holds ifln (1; gt+1 ) <

1
1
ln

1  d

8gt+1 2 [0; g~]:Noting Eqs. (8) and (11), one Önds that 8gt+1 2 [0; g~];(1; gt+1 )  (1; g~)= d:These results and Eq. (18) prove Lemma 3(b).
(c) Noting Eq. (6), the result holds if, 8gt+1  0;ln (a (gt+1 ); gt+1 ) < ln (~aI (gt+1 ); gt+1 ):In light of Eq. (18) and Lemma 1, 8gt+1  0;ln (a (gt+1 ); gt+1 ) <

1
1
ln
:

1  dThen, the result follows from the Örst statement of Lemma 3, which shows that 8gt+1  0;ln (~aI (gt+1 ); gt+1 ) =

1
1
ln
;

1  d(41)
Proof of Proposition 1. Noting Eq. (6), the result holds if, 8gt+1 2 [0; g~];ln (^a(gt+1 ); gt+1 ) < ln (~aI (gt+1 ); gt+1 ):Lemma 2 shows that 8gt+1  0;ln (^a(gt+1 ); gt+1 ) =

1
1
ln
:

1

(42)

Thus noting Eq. (41), the proposition holds under Eq. (A1), which is explicitly expressed in Footnote


17
30

Proof of Lemma 4.

The Örst statement and result (a) of the lemma are proven in a similar way

to Lemma 1.
(b) Since, in view of Eq. (30), (ait ; gt+1 ) is increasing in ait , the result holds ifln (1; gt+1 ) >1   1  =d
ln

1

8gt+1  g~:

Noting Eqs. (8) and (11), one Önds that 8gt+1  g~;(1; gt+1 )  (1; g~)= d:These results and Eq. (18) prove Lemma 4(b).
(c) Noting Eq. (6), the result holds if, 8gt+1  0;ln (a (gt+1 ); gt+1 ) > ln (~aII (gt+1 ); gt+1 ):In light of Eq. (18) and Lemma 1, 8gt+1  0;ln (a (gt+1 ); gt+1 ) >1   1  =d
ln
:

1

On the other hand, Lemma 4 shows that 8gt+1  0;ln (~aII (gt+1 ); gt+1 ) =1   1  =d
ln
:

1

(43)


The result therefore follows.
Proof of Proposition 2. Noting Eq. (6), the result holds if, 8gt+1 2 [0; g~];ln (~aII (gt+1 ); gt+1 ) < ln (^a(gt+1 ); gt+1 ):Comparing Eqs. (42) and (43) reveals that the proposition holds under Eq. (A1), which is explicitly

expressed in Footnote 17.
Proof of Lemma 5. (a) The result is obtained by noting that Eq. (22) is una§ected by d:(b) A rise in d increases the term on the right side of Eq. (30) and that of Eq. (34). Then the
results are obtained by applying the Implicit Function Theorem as in the proof of Lemma 1.
(c) If d is su¢ciently close to 1,
1
1
1
1
ln

ln
:

1

1  d31

Then, the result is obtained by utilizing the proof of Proposition 1.
(d) A rise in d increases the term on the right side of Eq. (11). Then the result is obtained byapplying the Implicit Function Theorem as in the proof of Lemma 1.

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33