ISSN (Print) 0473-453X

Discussion Paper No. 1071

ISSN (Online) 2435-0982

THE EDUCATION GENDER GAP
AND THE DEMOGRAPHIC TRANSITION
IN DEVELOPING COUNTRIES

Nguyen Thang Dao
Julio DávilaAngela Greulich

Revised August 2020
December 2019

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

The Education Gender Gap and the Demographic Transition in
Developing Countries*
Nguyen Thang Dao



Julio DávilaAngela GreulichAugust 3, 2020

Abstract
This paper explores, theoretically and empirically, the role of the declining gender gap in education in the
demographic transition and the emergence of modern economic growth. Specically, the paper develops amodel in the tradition of the unied growth theory that captures and interconnects the key empirical featuresof the demographic transition, the decline in gender gap in education, and the transition to sustained growthacross less-developed economies. The mechanism on which the model relies comprises several interplaying
components.

First, technological progress reduces housework time through the creation and diusion oflabor-saving home appliances, which frees women's time for childrearing, resulting in an initial increasein fertility, as well as in labor-force participation.

Second, due to the possibly higher female labor-force

participation as housework time decreases, households invest relatively more in their daughters' education,
given its higher return following the initial imbalance.

This improves gender equality in education andincreases the opportunity cost of childrearing, which leads to a subsequent decrease in fertility. Third andnally, the decrease in the education gender gap through higher investment in daughters' education increases
average human capital, thus accelerating technological progress in turn. This reinforcing loop results in thetransition to a new fertility regime and accelerated economic growth. We provide the empirical conrmationof the model's predictions using data from developing countries in the late 20th and early 21st centuries.

Keywords: Unied growth model; gender inequality; demographic transition; developing countries.
JEL Classication: J11, J13, J16, O11, O40.

* We thank the editors of the journal (Oded Galor and Madeline Zavodny), and four anonymous referees for providing helpful comments andsuggestions. We also thank David de la Croix, Cuong Le Van, Sascha O. Becker, and David N. Weil for helpful discussions and encouragements.
Earlier versions of this paper were presented at several seminars and conferences at the Universite catholique de Louvain, MCC Berlin, Vienna
Institute of Advanced Studies, Catholic University of Lisbon, University of Warwick, University of Göttingen, and Osaka University. We havebeneted from the comments and suggestions of participants. Funding from an MIS Ulysse research grant from the Belgian F.R.S.-FNRS is
acknowledged. The rst author would like to acknowledge the supports from the Japan Society for the Promotion of Science (JSPS) KAKENHI
Grant-in-Aid for Scientic Research JP15H05728 and from the International Joint Research Promotion Program (Osaka University). The
scientic responsibility is ours.

 Corresponding author. Institute of Social and Economic Research, Osaka University, 6 - 1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan;

Leibniz Institute of Freshwater Ecology and Inland Fisheries, Müggelseedamm 310, 12578 Berlin, Germany; and Mercator Research Instituteon Global Commons and Climate Change, Germany. Emails: daonguyen@iser.osaka-u.ac.jp and dao@igb-berlin.de.
School of Sciences and Humanities, Nazarbayev University, email: julio.davila@nu.edu.kz; and CES, Université Paris 1 Panthéon Sorbonne,
106-112 Boulevard de l'Hôpital, 75647 Paris, email: julio.davila@univ-paris1.fr.

 OSC, Sciences Po Paris, 27 rue St. Guillaume, 75007 Paris, France. Email: angela.greulich@sciencespo.fr.

1

1

1

INTRODUCTION

Introduction

This paper explores, both theoretically and empirically, the role of the declining education gender gap
in the demographic transition and the emergence of modern economic growth.

The paper develops

a model in the tradition of unied growth theory that captures and interconnects the key empiricalfeatures of the demographic transition, the decline in gender gap in education, and the transition tosustained growth across less-developed economies.

In order to do so, and to explain some stylized

facts from developing countries in the late 20th and early 21st centuries, we provide a new mechanism.
Specically, we show how the reduction of housework time due to technological progress operates
through its interaction with households' education and fertility choices aecting the transition to alower fertility and sustained growth, the reduction of earnings and human capital gender gaps, and theincrease of female labor market participation.
Indeed, the generalization of home appliances in the 20th century reduced women's houseworkburden by decreasing housework time, freeing women's time for childrearing and leading to an initial
increase in fertility, as well as in labor market participation. In turn, foreseeing higher labor market
participation, parents invested relatively more in their daughters' education to increase their income,as well as to improve their bargaining position in the marriage market and as insurance against divorce.
This improved gender equality in education. The subsequent increasing opportunity cost of childrearing
then led to a decrease in fertility.

In addition, the improvement in gender equality in education

accelerated technological progress as it raised the average human capital in the economy, leading to thecreation and diusion of additional time-saving home appliances. The resulting reinforcing feedback
loop thus generated a fertility transition along with accelerated economic growth.
This paper combines and goes beyond Galor and Weil (1996, 2000), Greenwood et al. (2005a, b) andSoares and Falcao (2008), thus complementing the literature on 19th century demographic transitions1in the West (see Galor and Weil 2000, Galor and Moav 2002, Hansen and Prescott 2002, Doepke
and Tertilt 2009, Cervellati and Sunde 2015, and recently Dao 2016, among others). It then seeks to
explain the 20th century fertility transition of developing countries. Indeed, compared to that of theWest, the demographic transitions in Asia, North Africa, and Latin America started around a century
later at much lower

per capita

GDP, with fertility falling faster and from higher rates, prompting thesearch for alternative explanations.

Consistent with unied growth theory (Galor 2011), our model

highlights the role of technological progress and its impact on fertility and human capital formation in
the demographic transition. However, unlike unied growth theory, in which the child quantity-quality
tradeo is induced by the gradual increase in the rate of technological progress, as was the case in
the current developed economies, we focus on the experience of less-developed societies, in which thedemographic transition was inuenced by other important impacts of technology. To explain this, ourmodel instead proposes that the developing countries' transitions hinge specically on the spread of
1 A demographic transition refers to the passing from a high fertility and high mortality regime including child mortality to a low
fertility and low mortality one, generating an inverted U-curve for

net fertility along the way. For exposition purposes, we do not address
net fertility. Child mortality can be included in the model without

mortality throughout the paper, but fertility here can be interpreted aschanging its results signicantly.

This is consistent with the related literature, where the term demographic transition is used instead of

fertility transition to refer implicitly to the potential extension/power of the models. In addition, as pointed out in Herzer

et al. (2012), thedecline in fertility triggered by falling mortality is not essential to explain the secular decline in population growth experienced in the 20th
century. For a more comprehensive discussion on this, see Doepke (2005) and Galor (2012).

2

2 STYLIZED FACTS AND RELATED LITERATURE
home appliances, and its eects on households' fertility and education choices (see Greenwood andSeshadri (2005) and Greenwoodet al. (2005a) for the US, and Mayers and Sathaye (1989) for theirrapid spread in developing countries).2 The fall in fertility in developed countries took place instead,almost a century earlier, in the absence of a comparable creation and diusion of home appliances, which
was therefore understandably overlooked by the literature. Our model focuses precisely on the role ofinnovation in home appliances and their diusion to explain the demographic transition of developing
countries in the late 20th and early 21st centuries.3

Compared to that of developed countries, theliterature on the demographic transition in developing countries is limited. The two cases dier mainlyin the timings of the fall in fertility that are linked to the literature on comparative development, basedon geographical factors (Dao 2016) and trade between industrial and less-developed countries (Galorand Mountford 2006, 2008).
The rest of paper is organised as follows.

Section 2 reviews the literature.

Section 3 introduces

the model and its equilibrium dynamics. Section 4 analyses the evolution of the economy. Section 5analyses its development to explain the stylized facts and discusses some extensions of the model. The
empirical investigation is presented in Section 6, and Section 7 concludes the paper.

2

Stylized facts and related literature

The mechanism in this paper explains why, in the late 20th and early 21st centuries, developing
countries have experienced a transition exhibiting the stylized facts illustrated in Figures 1, 2 and 3

a negative correlation between fertility and female-to-male education ratio a positive correlation
between per capita income and female-to-male education ratio declines over time in the education andlabor income gender gaps a fertility transition as sustained economic growth takes o an increaseover time in the female formal labor market participation an eventual reversal in education acrossgenders
the demographictransition in unied growth theory technology and changing role of women
below:

;

;

;

;

; and

.

Our paper integrates two strands of economic growth and development literature:, and. In the next subsections,

we review these related literature.

2.1 Demographic transition from unied growth theory perspective
Demographic transitions have taken place across dierent regions of the globe since the rst half of
2 The share of households in Bangkok with refrigerators increased from 26% to 62%, and from 63% to 95% in Taiwan in the period 19751984. The generalization of home appliances also shows up in the increase in residential electricity consumption per capita, from 17 kWh to
110 kWh per capita in the period 1970-1986 in Thailand; from 25 kWh to 248 kWh in South Korea; from 75 kWh to 190 kWh in Mexico; andfrom 90 kWh to 261 kWh in Brazil (Mayers and Sathaye 1989).
The impact of technological progress on the economic role of women was even anticipated very early in the 20th century, as shown by Thomas

The housewife of the future will be neither a slave to servants nor herself a drudge. She will give less attention to thehome, because the home will need less; she will be rather a domestic engineer than a domestic laborer, with the greatest of all handmaidens,
electricity, at her service. This and other mechanical forces will so revolutionize the woman's world that a large portion of the aggregate of
woman's energy will be conserved for use in broader, more constructive elds , Good Housekeeping Magazine, LV, no. 4 (October 1912, p.
436). Quoted in Greenwood et al. (2005a).

A. Edison's words: 

3 Recently, Bucci and Prettner (2020) also advance an endogenous growth model to consider the relationship between innovation anddemographic change. However, dierent from ours, Bucci and Prettner (2020) focus on the impact of population growth, through its eects on
aggregate human capital accumulation, on research and development activities in production sectors (not in household sector) to account for
the reversal between fertiltiy and innovation-driven productivity growth. In addition, Bucci and Prettner (2020) do not consider the genderdimension of the problem while it is the main focus of our paper.

32.1 Demographic transition from unied growth theory perspective
2 STYLIZED FACTS AND RELATED LITERATURE

Cross-country plots of fertility and per capita income (in US dollars) against gender equality in education (1970 and2000). Each dot represents one country. The list of countries includes: Algeria, Argentina, Bangladesh, Belize, Benin, Bolivia,
Botswana, Brazil, Burundi, Cameroon, Central African Republic, Chile, China, Colombia, Congo. Dem. Rep., Congo. Rep., Costa
Rica, Cote d'Ivoire, Cuba, Ecuador, Egypt. Arab Rep., El Salvador, Gambia, Ghana, Guatemala, Honduras, India, Indonesia, Iran.
Islamic Rep., Iraq, Jamaica, Kenya, Liberia, Malawi, Malaysia, Mali, Mauritania, Morocco, Nepal, Nicaragua, Niger, Pakistan,
Papua New Guinea, Paraguay, Peru, Philippines, Rwanda, Senegal, Sierra Leone, South Africa, Sri Lanka, Sudan, Thailand, Togo,
Tunisia, Turkey, Uruguay, Zambia, Zimbabwe. Source: WB WDI and Barro & Lee (2013).
Figure 1:

the 19th century.

First in the West, and then across the developing world in the 20th century.

It

has both fostered the accumulation of production factors (physical and human capital), technologicalprogress, and labor productivity, etc. as well as having been fostered itself by these very same factors
and outcomes along the development process. Thus, the interactions between a demographic transition
and all these factors have been studied extensively in the framework of unied growth models.
Indeed, the phenomenon of a demographic transition is one of the key features of the unied growththeory rst advanced by Galor and Weil (2000) in their explanation of the development process fromMalthusian stagnation to modern growth. Specically, unied growth theory Galor and Weil 2000,
Galor and Moav 2002, Hansen and Prescott 2002, Cervellati and Sunde 2013, Dao 2016, among others suggests that the technologically-driven increase in demand for human capital, and its eects
on the children quality-quantity trade-o plays the central role in the demographic transition and,more generally, the transition to modern economic growth. This theory has been conrmed empiricallyby Becker and Woessman (2009), Becker et al. (2010, 2011), and more recently Fernihough (2017).
Thus, the rise in demand for human capital and the decline in gender gap have been documented as4

2.1 Demographic transition from unied growth theory perspective
2 STYLIZED FACTS AND RELATED LITERATURE

Figure 2:

Fertility across countries (1960 - 2015). Source: World Bank (2020).

the major causes of demographic transition.4 Galor and Weil (1996) advanced an alternative unied
growth model showing that the decline in the wage gender gap resulting from physical capital being
more complementary to women's labor than to men's plays a central role in generating a decline infertility too.
In this tradition, our model contributes a novel mechanism to explain, in particular, the demographictransition experienced by developing countries since the second half of the 20th century. Specically,
we focus on the role played by innovations in, and diusion of, time-saving technologies for houseworkin the decline in gender gap, as well as the feedback of the latter into technological progress. In a waythat is complementary to the traditional unied growth theory, we therefore put the gender gap at thecenter of the analysis. Indeed, complementing the approach in Galor and Weil (1996) that considers
the decline in the wage gender gap to be due to the accumulation of physical capital, we show thattechnological progress also plays a crucial role in reducing the human capital gender gap, improving
thus income equality across genders. Alternatively, Soares and Falcao (2008) suggest that the narrowingof the wage gender gap and the increase in female labor supply follow from a demographic transition
initially triggered by improvements in life expectancy and a decline in fertility, which complements
rather than contradicts the mechanism put forward in our paper.
4 Other factors such as the increase of income per capita (Becker 1960, Becker and Lewis 1973), the decrease of infant and child mortality(Kalemli-Ozcan (2002, 2003), Tamura 2006, Tamura et al. 2016, Tamura and Simon 2017), andthe old-age security (Neher 1971, Caldwell

1976, Boldrin and Jones 2002) have been showed to be inconsistent, on their own, with the data, and therefore are considered as reinforcing
minor factors rather than main factors in explaining the demographic transition. For a more comprehensive overview, see Galor (2011).

5

2.2 Technology and the changing role of women

Figure 3:

2 STYLIZED FACTS AND RELATED LITERATURE

Female labor force participation in non-agricultural sector. Source: WB WDI.

2.2 Technology and the changing role of women
In addition to inserting itself in the child quantity-quality tradeo tradition (Becker 1960, Becker andLewis 1973, Galor and Weil 2000, Tamura 2006, Hazan and Zoabi 2015a), our paper relates to theliterature on technology and the changing role of women in the household and society (Galor and Weil
1996, Greenwood and Seshadri 2005, Greenwood et al. 2005b, Soares and Falcao 2008, Bloom et al.
2009, Doepke and Tertilt 2009, De la Croix and Vander Donckt 2010, Duo 2012), as well as thedemographic transition (Galor and Weil 2000, Hansen and Prescott 2002, Doepke 2004, Cervellati andSunde 2015, among others).
Greenwoodet al.

(2005b) study the link between home appliances and the baby boom in developed

countries after World War II. Home appliances have reduced the marginal cost of childrearing higher
in developed countries and have thus increased fertility as the depressing eect of rising incomes ismore than oset.5 We focus instead on the reduction of the education gender gap due to the rebalancing
across genders of education investment resulting from the diusion of time-saving appliances, whichshapes the demographic transition in developing countries.
Doepke (2004) shows (in a unied growth theory framework) that the introduction of child labor
regulations accounts for the dierent timings and speeds of the demographic transition across countries.
Hazan and Berdugo (2002) consider child labor too, and argue that during the development process,technological progress enlarges the wage dierential between parental and child labor, thereby inducing
5 Using US data at county level, Bailey and Collins (2011) have identied a negative coecient of the home technology adoption rate on

et al. (2005b). But
et al. (2015) point to a misinterpretation by Bailey and Collins (2011), who ignore that poorer

fertility, while a baby boom implies a positive one. This result is usually interpreted as evidence contradicting Greenwood
in a paper reply to these nding, Greenwood

households adopted electricity and home appliances later than richer ones, while the latter intrinsically have lower fertility rates.

6

2.2 Technology and the changing role of women

2 STYLIZED FACTS AND RELATED LITERATURE

a substitution of child education for child labor, which in turn reduces fertility. In an open economy
model, the authors suggest that a compulsory schooling policy and an intergenerational transfer to theelders would be Pareto-improving and would accelerate development. While not contradicting these
papers, our approach is dierent in not focusing child labor. We pay instead particular attention to theeect of creating and diusing home appliances on education investment in children and on fertility
during the development process.
Regarding women's rights and evolution of human capital, Doepke and Tertilt (2009) argue that an
expansion of wives' legal rights increases human capital investment for children. This, in turn, helps
children to nd quality spouses, just like fathers wish for their children. Thus, when voting on women's
rights, men hold conicting goals, between the diminished rights they want for their own wives andthe expanded rights they wish for other women. Men's weight on these goals shifts over time with theincreasing value of human capital as technological progress takes place, shaping the evolution of humancapital and demographic transition.
The model by De la Croix and Vander Donckt (2010) is based on intra-household bargaining toanalyse the impact of gender inequality on economic outcomes for least developed countries. It points tothe increasing survival probabilities of females and infants, making women more likely to be active in themarket and leading to an increase in female education as a key means to escape poverty. Furthermore,higher female education increases women's bargaining power within the household, decreasing fertility
and improving the quality of children, and so fostering economic growth.6
Our paper could have followed the literature of women's empowerment and development more closely(Duo 2012, Doepke and Tertilt 2019, Hazan et al. 2019). But we do not engage in this interesting
debate here. In our view, this literature does not contradict, but rather complements, our paper. Theinterdependence between women's empowerment and development is a naturally complex issue that no
single model captures, the exact nature of the link between women's empowerment and development
remaining elusive.
Finally, as for the empirical literature on gender equality and development, most papers study theeects of gender inequality in education or human capital on economic growth: e.g. Barro and Lee(1994), Dollar and Gatti (1999), Klasen (2002), and Klasen and Lamanna (2009), among others. Morerecently, Klasing and Milionis (2020) show the empirical link between improvements in life expectancy
and the closing of the education gap across genders as the global spread of Western medicine against
infectious diseases led to relatively larger gains in female life expectancy. These results complement ours
to explain the improvement of gender equality in education in the 20th century, since the model can beextended to capture the eect of an increase in life expectancy on fertility and education investment,in order to explain the empirical evidence in Klasing and Milionis (2020). Regarding fertility, humancapital and development, Vogl (2016) shows empirically that before 1960 children from larger families
obtained more education because they had richer and more educated parents, while this pattern had
reversed by the end of the 20th century. These results again also do not contradict ours, even if ourmodel does not take into account heterogeneity across households.
6 On women's rights and marital bargaining power see also Basu (2006), Fernandez (2009), Doepke et al. (2012), Doepke and Tertilt (2019).

7

3 THE MODEL

3

The ModelThis section sets up a model to explain the demographic transition in developing countries. Accordingly,
it does not take into account the markets for childcare services and housework aid that contributed toshape fertility patterns in developed countries throughout the second half of the 20th century. Instead,
it focuses on the introduction and diusion of home appliances and their impact on households' decisions
on education investment, female labor force participation, and fertility. The economy consists, at any
periodt, of a number Lt of identical 2-people and 2-gender (male and female) households.

Eachmember of a household lives for two periods childhood and adulthood and is endowed with one
unit of time when adult. Adults (i) allocate their time between supplying labor, doing housework, andchildrearing, (ii) decide how much to invest in their children's human capital, and (iii) consume theremainder of their income. As in Becker (1985), in order to capture gender gaps in earnings and humancapital, childrearing and housework are assumed to be carried out by the woman, in accordance withcross-cultural evidence.7 Regarding childrearing, each household's child also a 2-people, 2-genderhousehold requires ρ units of parents time, while the human capital of each member of a child
household is a power θ ∈ (0, 1) of the parent household's education investment, et for a son and ẽt
for a daughter. As for housework time, it is assumed to be decreasing with technological progress, sothat the fraction of time φt ∈ [0, 1) devoted by period t's adult household to housework is a function
φt = φ(at ) of the technological level at satisfying the following assumption8.

Assumption 1. φ0 (a) < 0 and lim φ(a) = 0
a→+∞

For the sake of simplicity, we do not assume that childcare time ρ decreases with technology.9 In themodel, ρ should be interpreted as the time for pregnancy, initial care, and breastfeeding being closely
linked to the birth.

As for the housework time φ(a), it could alternatively be made to depend on

the size of the household. However, indeveloping countries

alongside their parents (see Webbink et al.

children usually participate in housework2012) so that fertility has opposite eects on women'shousework burden. For the sake of simplicity, we assume that these eects oset each other. Finally,
we model housework as unavoidable and hence as not being a choice variable. That ts well into thecase of developing countries in which the outsourcing of household chores is almost nil.

3.1 Households

potential

ct , the number of their household-children nt ,10 and theof the latter through its education investment in human capital et for sons and ẽt for daughters

Households choose consumption

incomesolving

7 Specially in developing economies, conventions still make women responsible for childrearing and housework, while the labor market formostly physically demanding tasks leads households to specialise in a division of labor that sees almost only men participating in wage labor.
In Galor and Weil (1996), the opportunity cost of raising children is higher for men than women. Thus, the woman takes care of children until
her time constraint is binding, when the man gets involved. However, Galor and Weil (1996) restrict the weight of children in preferences sothat men do not participate in childrearing, devoting instead all their time to wage labor. Anyway, letting men do housework and childcare
does not change the qualitative analysis as long as it is assumed they do less than women, which seems uncontroversial.

8 Greenwood et al.

(2005a) argue that technological progress plays a key role in households' allocation of time. Indeed, the appearanceof a mass market for ready made clothes (in the 19th century) and the generalization of appliances washing machines, vacuum cleaners,
refrigerators (in the mid-20th century), etc. as well as frozen foods and ready meals (in the mid-late 20th century) freed a considerable
amount of time from housework.

9 This eect, that would indeed foster fertility, can be introduced without changing the qualitative results, in particular, the explanation of

the hump-shaped fertility pattern. On the increasing part of the hump, this eect reinforces our mechanism, while on the decreasing part it isdominated by the reduction in fertility through an increased gender equality in education for reasonable parameters and functional forms.

10 We assume that the gender birth ratio (male over female) is 1 which is close to the natural gender ratio. Accordingly, a household-childis assumed to consist of a son and a daughter.

83.1 Households

3 THE MODELmax

ct ,nt >0; et ,ẽt ≥0(1 − γ) ln ct + γ ln nt wt+1 [eθt + (1 − φt+1 )ẽθt ]

(1)subject to
ct + nt (et + ẽt ) ≤ wt eθt−1 + (1 − φt − ρnt )ẽθt−1(2)

φt + ρnt ≤ 1(3)given the wage rates wt , wt+1 and the time φt , φt+1 needed for housework, at t and t + 1, as well as
the education investments et−1 and ẽt−1 made by the household's parents.11 The parameter γ ∈ (0, 1)
captures the household's degree of altruism.
A household adult at

t is thus assumed to perfectly foresee the fraction of time φt+1 that its

household-children will devote to housework, since this is determined by its own average human capitalthrough technological progress.12 But the adult will not foresee what results follow from its children's

potentialρnt+1 in the objective. Thus, since households derive utility from theirθθ
income wt+1 [et + (1 − φt+1 )ẽt ], parents always invest in their daughters' education

choices, hence the absence ofchildren's

regardless of the latter's choice on childrearing and housework time.13

It follows from the necessary rst order conditions of the problem (derived in the Appendix) that,since by Assumption 1 housework time φt = φ(at ) decreases with the technological level at , the time
constraint is not binding, so that women supply labor to the market, whenever at is beyondh γ(1 − θ) i1−θ 
a =φ
1−
1 − γ(1 − θ)∗−1

The following assumption guarantees that a

Assumption 2. φ(0) = φ̄ > 1 −hγ(1−θ)
1−γ(1−θ)

i1−θ

∗

> 0.

, for some a > 0.

> φ(a)(4)

14

Assumption 2 requires (for very low technologies) housework time to be so high as to keep womenaway from the labor market. Without this assumption, the time constraint would never be binding,women would always participate in the labor market regardless of the level of technology, and fertility
would always be decreasing in the latter. Nevertheless, in the early stages of development, fertility is
typically observed to increase with the level of technology, and women participate in the labor market
only when the return to labor is high enough. Assumption 2 allows chosen fertility to be expressed asnt = 1−φ(at )if at ≤ a∗

if at ≥ a∗ρhiθγ(1−θ)
θ−1 + 1 − φ(a )
[1
−
φ(a
)]
t
t
ρ

(5)

11 The main results of our model do not change if we assume that the input for education investment is parents' time instead of income, while
this alternative modelling choice requires additional assumptions about who the father or the mother is responsible for it, or what is theparents' division of labor in educating children. At any rate, regardless of this, the education investments in our model would correspond, in thealternative approach, to the opportunity cost of educating children. All in all, the gender gap in education and its properties are determinedby equations (11) and (12) whose qualitative behaviour does not depend on this modelling choice. Ours is, at any rate, in accordance withthe fact that the importance of formal education for the labor market increasingly exceeds that of education involving parents' time.

12 Indeed, φt+1 has been assumed to be determined by the level of technology at+1 , which itself is determined by the education investments
et and ẽt chosen for this household by its parent household (see equation (13) below), all of which is known to it.

13 That is to say, a daughter's education has a value for her parents, independently of the use that she makes of it. This is consistent with

Becker and Tomes (1979) and Galor and Weil (2000), and captures other aspects excluded from the model, like the use of education as hedging
device against the risk of a daughter's divorce: see Chiappori and Weiss (2007).

14 This necessarily requires the condition 1 −hi1−θγ(1−θ)> 0 since φ is decreasing and non-negative for all a1−γ(1−θ)must holds. This last condition obviously holds for an altruism factor γ < 1/2, conventionally assumed in the literature:i.e. γ(1 − θ) < 1/2
e.g. Galor and Weil

(1996), Doepke (2004), and Lagerlöf (2003, 2006). We will restrict the value of θ in such a way that this condition holds.

93.2 Production

3 THE MODEL

Comparative statics for education investments and fertility in (16), (17), and (18) with respect to thewage rate wt and parameters ρ, γ deliver∂et ∂ẽt ∂et ∂ẽt,;,>0
∂wt ∂wt ∂ρ ∂ρ∂et ∂ẽt  > 0 if φt + ρnt = 1,∂γ ∂γ  = 0 if φt + ρnt < 1

and

and

∂nt
∂nt
= 0,
<0
∂wt
∂ρ

∂nt  = 0 if φt + ρnt = 1∂γ  > 0 if φt + ρnt < 1

Thus, when the woman's time constraint is not binding, a rise in the wage rate wt generates income
and substitution eects on fertility such that the former raises fertility while the latter, which reects
an increase in the opportunity cost of childrearing, reduces it. From the homotheticity of preferences,
the two eects oset each other. A rise in wt translates therefore into a rise in education investments
for children. As for the impact of the time cost of childrearing, an increase in ρ reduces the fertility of
the household, that trades child quantity for child quality, increasing the education investments.
When women's time constraint is binding, a higher weight γ for children in preferences makes parents invest more in their children's education. When the time constraint of the woman is not binding,a higher γ increases fertility without changing education investments, as it generates two oppositeeects on child quality. On the one hand, it makes parents allocate more resources to children, bothfor child quantity and quality. On the other hand, it increases the marginal costs of education investments through the increase in fertility nt . The two eects cancel out, leaving education investments
unchanged.
Finally, equations (16) and (17) show that when the woman's time constraint is not binding, children's education depends on the income of the mother only, since an increase in the father's income has
two opposing eects on children's education that cancel each other out: the income eect increases thenumber of children, but the increase in the number of children increases the marginal cost of education
investment for all children.

3.2 Production
Output yt is linear in the eective units of labor supplied,15 so that
yt = At eθt−1 + (1 − φt − ρnt )ẽθt−1(6)where At > 0 is the productivity per eective unit of labor, and hence the return to human capitaland the wage rate too (i.e. wt = At ). Productivity is assumed to be a function whereAt = A(at ) of the0level of technology at , with A(a) > 0 and A (a) > 0 for all a > 0, as well as

lim A(a) = Ā. In turn,a→+∞

15 For the sake of simplicity, we do not take into account physical capital. However, the main results are robust when introducing physicalcapital and assuming diminishing returns for all production factors.

Indeed, with physical capital and 3-period living households working

and saving in the second period and consuming in the third the rst period being childhood savings are lent to rms and their return isconsumed in the third period, while education investments and fertility choices remain the same. Also, we ignore consumption when young.
Nonetheless, with a log-linear utility the saving rate is constant and fertility is independent of the interest rate, and consumption when young
could be incorporated into the logarithm utility function, altering the dynamics of the model by just a multiplicative constant.

10

3.3 Equilibria

3 THE MODEL

technology is assumed to evolve according toat+1 = 1 + g

 eθ

θ 
t−1 + ẽt−1

2


at

(7)

where the rate of technological progress between t and t+1 is determined by the average human capitalas a function of education investments of the generation educated at t − 1 and working at t, withg(·) > 0, g 0 (·) > 0.16

We consider the impact of

average(rather thanaggregate) human capital on

technological progress, not taking into account the population size. For the impact of population sizeon technological progress see for example Galor and Weil (2000), Galor (2005), and Dao and Dávila
(2013), according to whom population growth fostered technological progress in the very early stagesof development, before education takes place.

Beyond some technological level, education plays theessential role and the marginal eect of population size on technological progress is negligible. In the20th century, population in the developing world was already big and the level of technological high,while education investment was growing. For the sake of simplicity, we ignore the impact of population
size on technological progress, without loss of generality for the qualitative results of the model.

3.3 Equilibria
At equilibrium, households maximize their utility according to their budget constraints, while output
matches consumption and education investment, in every period t, given the dynamics for technology.

A competitive equilibrium is thus characterized given initial conditions e−1 ,ẽ−1 , a0  by areduced dynamics in the education investments for sons and daughters et and ẽt , along with the levelof technology at+1 ,

et =


γθ


 1−γ+γθ




ẽt =

θ
1−θ

ρ
1

1+[1−φ(at+1 )] 1−θ


γθ


 1−γ+γθ




θ
1−θ

ρ

 A(at )eθ1
t−1(1−φ(at )) 1+[1−φ(at+1 )] 1−θ

A(at )ẽθt−1

ρ

1(1−φ(at )) 1+[1−φ(at+1 )] θ−1ρ
1

1+[1−φ(at+1 )] θ−1

 A(at )eθt−1

A(at )ẽθt−1 eθ + ẽθ 
t−1
at+1 = 1 + g t−1
at
2

if

at ≤ a∗if

at ≥ a∗if

at ≤ a∗

if

∗

(8)

(9)

at ≥ a(10)

given e−1 , ẽ−1 , a0 > 0.
16 The impact of human capital on technological progress might depend on the eective participation of workers in production, so that gwould depend on1 θ(e+ [1 − φ(at ) − ρnt ]ẽθt−1 ) instead. This would only slow down the transition to sustained growth, but not change it
2 t−1

qualitatively (the proof is available upon request). It can nonetheless be argued that women's human capital impacts technological progress
through childrearing via the children's cognitive abilities, capacity of adaptation to new knowledge, and health, which in turn impact their
human capital. The function g(.) describing the dynamics for technology follows the literature of unied growth theory, e.g. Galor and Weil
(2000), Galor and Moav (2002), Diebolt and Perrin (2013 a, b).

11

4 THE EVOLUTION OF THE ECONOMY TOWARDS ITS LONG RUN REGIME

4The Evolution of the Economy towards its Long Run Regime

Subsections 4.1 and 4.2 study the dynamics for gender inequality in education, fertility, female labor
supply as they interact with technology. Subsection 4.3 addresses the convergence to sustained growth.

4.1 Gender inequality in education
Gender inequality in education is gauged by the female-to-male education investment ratio

µt =

ẽt−1
et−1

(11)

That is to say, from (8) and (9), lagged one period, it follows that

θ

1

µt =

1 + [1 − φ(at )] 1−θ
1

1 + [1 − φ(at )] θ−1= [1 − φ(at )]11−θ≡ µ(at ) < 1[1 − φ(at )] 1−θ φ0 (at )with µ (at ) =
>0
θ−1
0

(12)

so that: (i) there is an education bias against women when µt < 1; (ii) a higher µt means more equality
across genders in education, with complete equality at µt = 1; and (iii) gender equality in education
improves with technological progress. And, under Assumption 1 and given the technology dynamics in(10), the education gender gap eventually vanishes, i.e.

Figure 4:lim µ(at ) = 1.

at →+∞

Gender inequality in education against level of technology.

Thus, parents' education choice is biased against daughters (i.e. sons receive more education investment than daughters) because women allocate a time to childrearing and housework for which they do
not earn any income.17 Moreover, gender equality in education increases with the level of technology
since a higher level of technology reduces housework time, so that women can increase their labormarket participation, leading parents to invest more in their daughters' education in order to increase
their labor productivity.

As a result, inequality in education decreases with technological progress.

17 In section 5 and in the Appendix we introduce heterogeneity across genders in non-cognitive skills to generate a reversal of this bias ineducation investment.

124.2 Fertility and female labor supply
4 THE EVOLUTION OF THE ECONOMY TOWARDS ITS LONG RUN REGIME
In turn, the narrowing of the education gender gap speeds up technological progress, in a reinforcing
feedback loop, as established in Lemma 1 next.

Whenever the rate of technological progress increases with the average human capital, a better
equality in education across genders (i.e. a higher µ < 1) implies a faster technological progress (i.e.
a higher growth rate g for a ), for any given level of aggregate human capital
Lemma 1.

ttt

et + ẽt =




γθ
ρ
A(at )eθt−11−γ+γθ 1−φ(at )if ρnt + φ(at ) = 1θρA(at )ẽθt−1
1−θ

if ρnt + φ(at ) < 1(13)according to (8) and (9).
Proof. See Appendix A1.
Intuitively, Lemma 1 follows from the diminishing returns to education investment captured byθ ∈ (0, 1), since a reallocation of education investments towards gender equality in education increases
average human capital, and hence the rate of technological progress that boosts income growth.18

4.2 Fertility and female labor supply
To see how technology impacts fertility and female labor supply, we rewrite the fertility function 1 [1 − φ(at )] ≡ n1 (at )if at ≤ a∗ρhint = n(at ) ≡θ γ(1−θ) [1 − φ(at )] θ−1+ 1 − φ(at ) ≡ n2 (at ) if at ≥ a∗
ρ(14)

and dene the women's fraction of time allocated to labor market participation as follows

L(at ) = 1 − ρn(at ) − φ(at )(15)

The following proposition states the responses of fertility and female labor force participation to achange in the level of technology.

In the economy as set out above:(i) When the level of technology is suciently low, in particular a < a , then(i.a) fertility increases in technological level,(i.b) women spend their full time rearing children and doing housework.
(ii) When the level of technology surpasses the threshold a and the return on education does notdecrease too quickly, in particular θ ∈ [ , 1), then(ii.a) fertility decreases with the level of technology, and converges to a constant level,(ii.b) female labor force participation increases with the level of technology, and converges toan upper bound.
Proposition 1.

∗

∗

1
2

19

18 Lemma 1 is also consistent with the empirical results in Klasen (2002) showing a positive eect of gender equality in education on economic

growth. Klasen (2002) shows that gender inequality in education aects economic growth directly by lowering the average level of humancapital, and indirectly through its impact on population and investment.

19 This is just a sucient condition to guarantee fertility decreasing with technological progress when the level of technology is high enough.

13

4.2 Fertility and female labor supply
4 THE EVOLUTION OF THE ECONOMY TOWARDS ITS LONG RUN REGIME
Proof. Indeed,1n01 (at ) = − φ0 (at ) > 0ρandγ(1 − θ)n02 (at ) =



ρ


1
θ
[1 − φ(at )] θ−1 − 1 φ0 (at ) < 0
1−θNote that n1 (at ) starts below n2 (at ) at the origin, sinceiθ
1 − φ̄
γ(1 − θ) hθ−1
n1 (0) =+ 1 − φ̄ = n2 (0)<(1 − φ̄)ρρ∗while n1 (at ) converges to a higher limit than n2 (at ), overtaking n2 (at ) at a , as at increases

lim n2 (at ) =at →+∞

2γ(1 − θ)1< = lim n1 (at )ρρ at →+∞

And

L(at ) =0

L (at ) =




0




if at ≤ a∗0

 1 − ρn2 (at ) − φ(at ) if at ≥ a∗

if at < a∗

 −ρn0 (at ) − φ0 (at ) > 0 if at > a∗2

andlim L(at ) = 1 − 2γ(1 − θ).

at →+∞The eects of technological progress on fertility and female labor force participation are illustrated
in Figure 5.
Thus, while at low enough levels of technology women's time is entirely devoted to childrearing and∗housework, as at increases (but stays below a ) the time freed from housework shifts to childrearing sothat, during the early stages of development, technological progress increases fertility.20 As at surpassesa∗ , women enter the labor market and carry out housework, childrearing only part-time. Technologicalprogress shrinks the human capital gender gap in (12) because of the higher return to female educationand the additional time for female labor market participation. In turn, the higher female labor market
participation and the decrease in the education gender gap reduce the earnings gender gap also. The
increase in women's earnings raises, in turn, the opportunity cost of childrearing. Thus, as the cost ofraising children increases, households substitute the quality of children to quantity.21
20 Some studies in the literature provide empirical evidence on the positive correlation between income/productivity and fertility in earlystages of development supporting this Malthusian feature (see Crafts and Mills (2009) for England in the 16th-18th centuries, Lagerlöf (2015)
for Sweden in the 18th-19th centuries, and Ashraf and Galor (2011) for the period 1-1500 CE). Several theories also predict such a positive
correlation in early stages of development (see e.g. Malthus 1798, Galor and Weil 2000, Galor and Moav 2002).

21 The model abstracts from child mortality to explain the demographic transition.

Indeed, there has been a lack of consensus in theliterature about the role of child mortality in explaining demographic transition. Kalemli-Ozcan (2002, 2003), Tamura (2006), Tamura et al.
(2016), Tamura and Simon (2017), and recently Cuberes and Tamura (2019) are in favor of the view that falls in child mortality may induce
declines in fertility. Using the altruistic parents model of Barro and Becker (1989), Doepke (2005) provides results suggesting that the fall in
net fertility during the 19th century in the industrialized countries was driven by factors other than the fall in child mortality, pointing outthat the decline in child mortality has been dismissed as a cause of the decline in fertility, because of the inconsistency of this link. Galor(2012) shares this view, showing that the fall in fertility in Western Europe started nearly a century after the decline in child mortality. Herzer

et al. (2012) show a positive correlation between child mortality and fertility in the 20th century, but not a strong enough relationship to

explain the secular decline of population growth. Murphy (2015) has shown empirically that child mortality had no impact on fertility during
the demographic transition in France, while female education seems to have played a role. Recently, Blanc (2019) provides an interesting
discussion on the determinants of the demographic transition in France.

14

4.3

Long run convergence

Figure 5:

4 THE EVOLUTION OF THE ECONOMY TOWARDS ITS LONG RUN REGIME

Fertility (upper graph) and female labor-force participation (lower graph) against technology

Finally, note that while the model predicts a female labor supply which is constant at zero when
fertility is increasing, this is consistent with the reports in Goldin (1995) of a sharp increase in female
clerical employment in the US during period 1890-1950, as the reported labor supply increase is associated with a decline in fertility during that period, i.e. when the technological level exceeded threshold

a∗ .

4.3

Long run convergence

The long run behavior of the economy in Section 3 is characterized in the following proposition.

The overlapping generations economy converges from any initial conditions a , e ,ẽ > 0 to a stationary regime of sustained growth characterized by:(i) a growth rate of technology ḡ,(ii) high education investments ē and complete equality across genders,(iii) a low fertility rate n̄.
ProofProposition 2.

0

−1

−1

. Firstly, Proposition 2 hinges next on the following Lemma 2 whose proof is in Appendix A2.

15

4.3

Long run convergence

Lemma 2.

For xαt+1 = mt xt

4 THE EVOLUTION OF THE ECONOMY TOWARDS ITS LONG RUN REGIME

with 0 < α < 1, x > 0, and lim m = m > 0, it holds lim x = m
0

t→+∞

t

t→+∞

t

1
1−α

.

et > 0 or ẽt > 0 for all t, the level of technology at increases
unboundedly over time. Therefore, since lim φ(a) = 0 is decreasing and lim A(a) = Ā is increasing,Thus, according to (10), since

a→+∞

a→+∞

it follows from (8), (9) and Lemma 2 that technology will progress asymptotically at a constant rate 1
θ ρ 1−θḡ = g(ē ) where ē = lim ẽt = lim et =
Ā
t→+∞
t→+∞
1−θ2


θ

Finally, from (14), the fertility will converge at a decreasing rate towards the constant raten̄ = lim n2 (at ) = 2t→+∞γ(1 − θ).
ρ2

Proposition 2 is consistent with the modern regime of sustained growth characterized by unbounded
economic growth, a low and decreasing fertility rate, and high human capital. It allows the processexperienced by developing countries in the 20th century to be accounted for, and which was characterized by: (i) an initial increase in per capita income and fertility; (ii) a demographic transition after
which fertility fell while per capita income kept growing; and (iii) a nal regime of sustained growthwith fertility converging to a low level and accelerated per capita income growth. During the transition
through these three stages, gender equality in education and income, as well as female labor force
participation, increased steadily.

∗
Indeed, consider an economy at an early stage of development, with a low level of technology a0 < aand low human capital across genders. The low technological level prevents women from participatingin the labor market in two ways. First, directly, as a large fraction of women's time is required forhousework. Second, indirectly, as the low level of technology biases the education investments against
daughters,22 making the opportunity cost of childrearing too low, so that households choose a highfertility rather than female labor market participation.

In turn, as at this early stage, householdincome is very low due to the low human capital and low technological level households' education
investments are very small too. Average human capital, which drives technological progress, is very low
for both male and female children, and the growth rate of technology therefore also remains low. In
addition, at this early stage of development, since housework requires a very large fraction of women'stime, the remaining time for childrearing is very small, so that the population growth rate is initiallylow. Nevertheless, since technology improves even when the average human capital is small, albeit very
slowly, the technological level eventually increases, so that housework time decreases, as long as it stays∗
low enough (i.e. for at < a ). This frees time for childrearing, while the cost of childrearing is still
low, so that the fertility rate increases. As the technological level at increases, fertility increases and∗∗reaches a maximum when at reaches a . When it increases even further (i.e. for at > a ), women startand increase their labor market participation due to two eects. First, technological progress reduces
housework time. Second, it improves the return to human capital, increasing women's earnings, and22 Households anticipate the large fraction of time that daughters will have to devote to housework, due to the low level of technology. Since
men's supply of labor to the market is inelastic, while women have to spend a fraction of their time for housework, households allocate mostof their education investment to their sons, making the education of their ospring very unequal across genders.

16

5 DISCUSSION
thus raising the opportunity cost of childrearing. Fertility therefore decreases as the level of technology
increases. It follows also from (8) and (9) that the total education investment increases with the level of∗technology when at > a , so that households substitute quality to quantity in their fertility choice. The
increase in the level of technology makes households invest relatively more in daughters' education as
the return to female education is higher and, as a consequence, gender equality in education improves,accelerating technological progress. This reinforcing feedback loop makes the economy enter a regimeof sustained economic growth.

5

Discussion

Our model captures key aspects of the development process experienced by many developing countries
during the late 20th end early 21st centuries, but it does not address them all. Related to this process
there is, for instance, the U-shaped relation between female labor force participation and income,the U-shaped relation between women's education and fertility, and the reversal in education acrossgenders that we discuss next. Other aspects not captured by the model like the link between women's
empowerment and development (Duo 2012, Doepke and Tertilt 2019, and Hazan et al. 2019, among
others) are worth integrating into future research.
As for the U-shaped relationship between female labor force participation and income, the model
predicts that female labor force participation never decreases with the level of technology, and increases∗
when the latter is high enough, specically for at > a , which is consistent with Galor and Weil (1996).
Nevertheless, Goldin (1995) and Mammen and Paxson (2000) nd an empirical U-shaped relation
between female labor force participation and per capita income, across countries and stages of economic
development. Goldin (1995) argues that, during early stages of development, most households live inrural areas where childrearing and agricultural work are easily compatible, and women are often unpaid
workers on their family farms.

As incomes increase, driven by technological progress going hand inhand with urbanization, female labor force participation rates fall because of an income eect and afall in the demand for female labor in agriculture.
Our model is actually silent about women's non-market activities during the early stages of development, considering only formal labor market participation. Galor and Weil (1996) discuss variationsof their model that generate a U-shaped relationship. One of them consists of considering that womencan produce marketable goods at home, as well as taking care of childrearing, which can be incorporated in the current model to replicate the U-shaped relationship in Goldin (1995). Indeed, if we
assume that home production is compatible with childrearing and housework, and the time for home
production φ̃(at ) is a decreasing function of the level of technology so that women spend a large fractionof time doing housework and home production during the early stages of development when the levelof technology is low, then the low fertility rate in this period is combined with a high rate of female
participation in the labor force. As the level of technology increases reducing housework timehome production timeand

 fertility increases. When technology reaches a high enough level, increasing

the return to female education, women enter the paid labor force, generating a persistent increase infemale labor market participation rates.

17

6 EMPIRICAL ANALYSIS
Looking at the current U-shaped relationship between women's education and fertility in the US,
Hazan and Zoabi (2015a) recently used the emergence of markets for childcare and housework aid to
explain the current U-shaped relationship between women's education and fertility in the US. Nonetheless, while these factors matter for developed countries, where increases in female education and labor
force participation go hand in hand with outsourcing housework and childrearing to cheap immigrantlabor and kindergartens, these factors play much less of a role in developing countries. Thus, Hazan
and Zoabi (2015a) do not contradict the stylized fact of a negative correlation between fertility and thefemale-to-male education ratio addressed in this paper.
As for the reversal in education across genders, Goldin

et al.

(2006) show that by 2002 women's

college enrollment rates exceeded men's in 15 out of 17 OECD countries, compared to 4 out of 17 in the
mid-1980s. Interestingly, the reversal in education across genders has not only taken place in developed
countries but also in many developing ones (Becker

et al.

2010). This phenomenon has generated a lotof research recently, and several factors, causes, and mechanisms have been proposed to explain it.23 In
the Appendix, we present a modication of our model incorporating a dierence in non-cognitive skills
across genders to generate a reversal in the education gap within our mechanism. The dierence in
non-cognitive skills may refer to, for example, the degree of self-motivation and seriousness in study, thedislike of homework exercises, and the like. A poorly non-cognitively skilled person nds school moredicult and hence more costly (Becker

et al.
et alon college costs across genders in Goldin2010). The eects of heterogeneity in non-cognitive skills. (2006) and Becker

et al.

(2010) can be reinterpreted as

the fact that investing in education for an average daughter is more ecient than that for an averageson.

6

Empirical Analysis

In this section, we illustrate the correlation which has existed within developing countries during thesecond half of the 20th century and the early 21st century between technical progress on the one handand fertility, woman's education and woman's employment on the other hand. Our theoretical model
suggests that the impact of technological progress on fertility, the gender gap in education and female
formal labor force participation is transmitted through the creation and diusion of labor-saving home
appliances. More specically, our model-based hypotheses can be stated, in the context of developing
countries throughout the 20th century, as follows:

Fertility has an inverted U-shaped response to the increasing creation and diusion of laborsaving home appliances.
 H2 The increasing creation and diusion of labor-saving home appliances reduces the gender gap
in education investment.

 H1:

:

23 An incomplete list includes, to name but a few: the cost of college education (Goldin et al.

2006 and Becker et al. 2010), the rise of
et al. 2006 and Chiappori et al. 2009), discrimination in labor market (Gosling 2003, Chiappori et al. 2009, Hazan andZoabi 2015b), technological progress in the household sector (Chiappori et al. 2009), a short supply of men in the marriage market (Iyigun
divorce rates (Goldin

and Walsh 2007), dierences in the anticipated dispersion of future wages across genders (Charles and Louh 2003), and the independence offertility on the gender composition of the children when the return to human capital is suciently high (Hazan and Zoabi 2015b).

18

6 EMPIRICAL ANALYSISThe increasing creation and diusion of labor-saving home appliances increases the female
formal labor-force participation.

 H3:

A challenge for the empirical investigation is the unavailability of data about the diusion and uses oflabor-saving home appliances in the household sector for emerging and developing countries. Hitherto
existing empirical studies on the impact of home appliances on gender-related issues such as fertility
and economic participation are mostly based on developed countries. Cavalcanti and Tavares (2008),
for example, show by using data for OECD countries between 1980 and 1999, that declining prices andthe greater availability of home appliances have a positive impact on female labor force participation.
To control for endogeneity, they use the manufactured price index and the terms of trade adjustment
as instrumental variables for the price index of home appliances. However, as data on the price anddiusion of home appliances is not available for a large number of less-developed countries and time
periods, we have to nd plausible proxies.
In order to proxy the diusion of labor-saving home appliances in less-developed countries, we use
three dierent measures:
1. The density of telephone subscriptions (measure available for 149 emerging and developing coun-tries; yearly observations for the time period 1960 to 2015; N = 5846; data source: World BankWorld Development Indicators)
2. The percentage of the population with access to electricity (measure available for 150 emerging

and developing countries; yearly observations for the time period 1990 to 2014; N = 3652; data
source: World Bank World Development Indicators)
3. The percentage of households with a refrigerator (measure available for 67 emerging and developing

countries; highly unbalanced panel covering the time period 1990-2015; N= 276; data source:

Demographic Health Surveys)
We investigate the within-country variations between fertility and the diusion of technology by modeling Total Fertility Rates (TFRs) as a function of every single one of these three variables which proxy
the diusion of time-saving home appliances.
Our empirical setup is as follows: We rst start with a simple model estimating TFRs as a functionof the technology proxy, its squared term and country-xed eects.

T F Ri,t = α + αi + β1 log(technology proxy) + β2 log2 (technology proxy) + εi,twhere T F Ri,t is the total fertility rate of country i in period t; and αi is a dummy variable for countryi.
We use the log of our technology variables in order to capture proportional rather than linearchanges.

Country-xed eects are included in the regression analysis in order to implicitly control

for country-specic variables that are constant over time. With this simple setup, we estimate thus
the overall

within-country

variation between fertility and our technology proxies, which is tested forconcavity. We then illustrate the estimated general within-country correlation pattern by plotting ourreal observations against the within-country variation that we have estimated by this simple model.

19

6 EMPIRICAL ANALYSIS
In a second step, we add time-xed eects and a series of control variables in order to test therobustness of the estimated within-country correlation:

T F Ri,t = α + αt + αi + β1 log(technology proxy) + β2 log2 (technology proxy) +Xγj cji,t + εi,tjjwhere αt is a dummy variable for year, ci,t s are country- and year-specic control variables.
Year-xed eects capture the inuence of aggregate time trends.

Our control variables lter out

technical advancement from other forms of progress, in economic, social, and institutional terms. This
helps us interpret our technology variables as proxies of technical progress in the eld of home appliances. The control variables are the log of GDP per capita and its square, population density, exports in% of GDP and the under-ve mortality rate. They are taken from the World Bank World Development
Indicators.
All regressions are run with cluster robust standard errors (cluster at country level). The estimation
results with the standard errors are reported in Tables 1 and 2.
When using telephone lines and electricity as proxies for time-saving home appliances, we focus ondeveloping and emerging countries by dropping advanced countries, tax heaven, very small countries
with a population size smaller than 30,000, and oil exporting Arab countries from the World Bank's
data base.24 The DHS data on the diusion of refrigerators is available for emerging and developing
countries only.
Access to electricity and the diusion of refrigerators are certainly more plausible proxies for thediusion of time-saving home appliances than the density of telephone subscriptions. But these measures contain important technical limitations for our empirical investigation: the observations are only
available for a relatively short time period (from 1990 on), they cover a lower number of countries
and are, in the case of refrigerator diusion, very unevenly distributed over time (data on refrigerator
diusion is not available on a yearly basis in the DHS but is only observed every three to six years
depending on the country). When using the density of telephone subscriptions, we dispose of almost
6,000 observations, while the number of observations is reduced by 40% when using access to electricity,and reduced to 5% only when using refrigerator diusion.
In the following, regression results and illustrations of the within-country correlation patterns between fertility and technology are presented for all three technology proxies.

However, due to thetechnical superiority of the rst proxy in terms of country and time coverage, our further empiricalinvestigations presented in this article focus on telephone density as proxy for time-saving home appli24 List of dropped countries:

Advanced countries (GDP per capita in PPP higher than 30.000 USD in 2017): Australia, Austria, Belgium, Canada, Czech
Republic, Finland, Denmark, Estonia, France, Germany, Iceland, Israel, Ireland, Italy, Japan, Lithuania, Luxembourg, Netherlands, New
Zealand, Norway, Portugal, Slovakia, Slovenia, South Korea, Spain, Sweden, Switzerland, UK, USA;

Fiscal paradises, oil exporting Arab countries, and very small countries: American Samoa, Andorra, Aruba, Bahamas, Bahrain,
Bermuda, British Virgin Islands, Brunei Darussalam, Channel Islands, Cayman Islands, Cyprus, Faroe Islands, Gibraltar, Guam, Greenland,Hong Kong, Isle of Man, Kuwait, Liechtenstein, Macao, Malta, Marshall Islands, Monaco, Nauru, Northern Mariana Islands, Oman, Palau,
Panama, Qatar, San Marino, Saudi Arabia, Singapore, St. Martin (French part), Trinidad and Tobago, Turks and Caicos Islands, Tuvalu,
United Arab Emirates, Virgin Islands (U.S.).
We also run alternative specications which: (1) focus only on countries and time periods for which GDP per capita is smaller than 20,000(constant 2010 USD); (2) exclude only the advanced countries listed in footnote 24; and (3) exclude only the scal paradises, oil exporting
Arab countries and very small listed in footnote 24. The results are consistent with those presented in this section. Results are available on
request.

20

6 EMPIRICAL ANALYSIS
ances. This concerns the empirical investigation of the within-country correlation between technology
diusion on the one side and the gender gap in education and female formal employment on the otherside. We abstain from estimating correlation patterns between the other two technology proxies andthe gender gap in education, respectively female formal employment, as these combinations of variables
lead to fragmented and highly unbalanced data: data on female formal employment is only available
from 1990 on and only for a subset of emerging and developing countries, and data on the gender gap
in education is only available on a quinquennial basis.
Data on female formal labor-force participation comes from the World Bank World Development
Indicators, covering the time period 1990  2015 (yearly observations). Data on education comes fromthe Barro and Lee Database, and observations are available from 1950 on, on a ve-year basis (Barroand Lee, 2013). A summary of the panel data used for our empirical investigations can be found in
Appendix A4. This appendix also provides a correlation matrix of all regressors used in our empiricalanalysis.
Telephone subscriptions certainly do not directly reect the diusion of labor-saving home appliances. However, as several studies have shown that the appearance of xed telephone lines in households
was observed soon after the arrival of other home appliances in the 20th century, even in the developing world, we assume that the penetration of telephone lines can be used as a plausible proxy forthe diusion of labor-saving home appliances. Studies by Paroush (1965), McFall (1969), Hebden andPickering (1974), Kasulis et al. (1979), Dicken et al. (1983) or more recently Dholakia and Banerjee(2012), for example, order consumer acquisition priorities for home appliances during the second halfof the 20th century. They show an absence of telephone lines in the priority list of home appliances
while other time-saving appliances such as refrigerators, clothes washers, kitchens, ovens, etc.

werelisted. Dholakia and Banerjee (2012) point out the late penetration of telephone lines for the case ofIndia. Such a pattern of acquisition priorities for home appliances is also observed in other developed
countries at earlier time periods. Bowden and Oer (1994) point out the ranking of diusion rate of
selected home appliances in the US and Britain since 1920s, in which the penetration of telephonelines is ranked after those of refrigerators, electric irons, clothes washers, and vacuum cleaners. This
time delay suggests that the observed penetration of telephone lines in households in the 20th century
reects the diusion of many other time-saving home appliances a couple of years before.

Further-more, the evolution of telephone densities is found to be relatively homogenous across developing andemerging countries, while other proxies of technical advancements such as electricity consumption or
air transport vary markedly even at similar income levels (World Bank 2008). This is also the casefor internet diusion, for which Andres et al. (2010) nd a signicant disparity within the group of
emerging and developing countries. In addition, data on internet diusion is only available from the
1990s on, and in most emerging and developing countries, internet has only been spreading since theearly 2000s. Besides, the fact that in many developing countries, internet access develops independentlyof the infrastructure that is necessary to use household appliances, has led us to abstain from using
internet diusion as a proxy for time-saving home appliances.
For our empirical illustrations using telephone density as proxy for time-saving home appliances, wegroup together xed and mobile telephone lines. We furthermore use the variable number of xed and21

6.1 The eect of home appliances on fertility

6 EMPIRICAL ANALYSIS

mobile telephone lines per 100 inhabitants with a 10-year forward gap as exogenous variable. Thismeans, telephone density is observed 10 years later than fertility, which is due to the above mentioned
priority order of owning home appliances. In concrete terms, in order to proxy the correlation of TFRs
and labor-saving home appliances at time t, we use TFRs at time t as endogenous variable and thedensity of telephone lines at time t + 10 as exogenous variable. The density of telephone lines at timet + 10 is supposed to reect the access to labor-saving home appliances 10 years earlier, at the time
when fertility outcomes are observed.25

6.1 The eect of home appliances on fertility
The rst two columns of Table 1 present regression results of TFRs estimated as a function of the tenyear forward gap of log(telephone lines) and its square. Model 1 estimates the overall within-country
variation, whereas Model 2 also includes time-xed eects and further control variables.
Model 1 shows a signicantly negative coecient of the square of the log(telephone lines) variable,
implying inverted U-shaped pattern between TFRs and technology diusion. This means that within
the emerging and developing countries in our sample, initial increases in the diusion of technology
come hand in hand with increases in TFRs, while from a certain technology level on, further increases
in technology diusion come hand in hand with decreases in TFRs. Note, however, that the estimatedcoecient of the level of log(telephone lines) is signicantly negative, which implies that the dominating pattern of the estimated within-country variation is the negative correlation between technology
diusion and fertility.
Figure 6 plots the overall within-country correlation between TFRs and the ten-year forward gapof log(telephone lines) that has been estimated by Model 1 against real observations (pooled data ofcross-country and time series). Figure 6 illustrates a concave pattern between telephone lines and TFRs
by conrming a maximum in the estimated correlation: at low technology levels, TFRs increase withan increasing diusion of technology, while at higher technology levels, fertility decreases with further
increases in the diusion of technology. This estimation result is consistent with our theoretical prediction of an inverted U-shaped pattern of fertility in response to the increase in the level of technology
as depicted in the upper graph of Figure 5. Figure 6 also shows that the dominating pattern in theestimated overall within-country variation is a negative correlation between fertility and technology
diusion, as the estimated turning point of the correlation pattern is situated at a relatively low levelof technology diusion. The maximum fertility level of 5.75 is reached a level of log(telephone lines) of-1.75, which corresponds to 0.17 telephone lines per 100 inhabitants. Around 6% of the observations
are in the interval of a positive correlation between TFRs and telephone penetration following model
1.26
25 Our theory works with an overlapping generations model in which each period lasts for 20 - 25 years, corresponding to the reproductive
period of women or childhood period of children.

Consistent with our theory, the proxy for technological progress should cover a certain

period of time, and not only a point in time when we observe fertility. That is to say, fertility observed in year 1970 should not only dependon the contemporary diusion of home appliances but also on the expected diusion of home appliances in the following years. Given thetime delay between the penetration of time-saving home appliances and the penetration of telephone lines, we choose a 10-year forward gap
for the variable 'telephone lines' when estimating its correlation with fertility. Results are robust when using a 5-year and a 15-year forward
gap instead of the 10-year forward gap. Results are available on request.

26 The inverse U-shape is conrmed when using a reduced panel with observations until the year 1990 only.

This robustness check wasapplied due to the fact that since the 1990s, telephone lines and other home appliances appear almost simultaneously in households.

226.1 The eect of home appliances on fertility

6 EMPIRICAL ANALYSIS

Table 1: Total fertility rates as a function of technology diusion.

23

6.1 The eect of home appliances on fertility

6 EMPIRICAL ANALYSIS

Figure 6: Fertility and density of telephone lines. Data: WB WDI, 148 countries, 1960-2015 (N = 5592).

The estimation results obtained by Model 1 and illustrated by Figure 6 can be quantied as follows:
Increasing the log(telephone lines) by 1 unit from -2.8 (second percentile) to -1.8 (6th percentile) (i.e.
from 0.1606 to 0.164 xed and mobile telephone lines per 100 inhabitants) implies that TFRs increaseby 0.13 children per women. Increasing the log(telephone lines) by 1 unit from 1.8 (50th percentile) to
2.8 (66th percentile) (i.e. from 6 to 17 xed and mobile telephone lines per 100 inhabitants) implies
that TFRs decrease by 0.42 children.27
Model 2 in Table 1 shows that the estimated coecient of the square of log(telephone lines) stays
signicantly negative when controlling for aggregate time trends, GDP per capita, population densityand exports.

This conrms the robustness of our nding of a concave within-country correlation

pattern between fertility and technology diusion. Note also that in Model 2, the estimated coecientof log(telephone lines) is now smaller and positive. This suggests that once we include controls for timetrends and variables that lter out technical advancement from other forms of progress, the estimatedinverse U-shaped correlation pattern between fertility and telephone lines is no longer dominated by adeclining branch. Model 2 suggests furthermore that within the developing and emerging countries ofour sample, fertility decreases come hand in hand with increases in GDP per capita (10% signicancelevel) as well as with increases in population density.28
27 Countries that are associated with the mentioned percentiles in our data set are for example: Benin in 1960 (second percentile), Tanzania
in 1975 (sixth percentile), Hungary in 1980 (50th percentile) and Armenia in 1990 (66th percentile).

28 We also tested for the indirect link that exists between telephone density and fertility due to the relative increase in female education.

We therefore rst estimated the ratio of female to male education as a function of telephone density (while controlling for population density
and child mortality), and then estimated fertility in a second step, as a function of the estimated gender ratio in education, while we include
population density and child mortality as control variables.

Our results suggest that an access of parents to time-saving home appliances246.1 The eect of home appliances on fertility

6 EMPIRICAL ANALYSISThe third and fourth column of Table 1 present regression results of TFRs estimated as a functionof log(% of population with access to electricity) and its square. Model 3 estimates the overall withincountry variation, whereas in Model 4, time-xed eects and controls are included.

Both Models 3

and 4 conrm a concave within-country variation between TFRs and our second proxy for time-saving
home appliances, namely access to electricity.

Figure 7: Fertility and access to electricity. Data: WB WDI, 150 countries, 1990-2014 (N = 3561).

Figure 7 illustrates the estimated overall within-country correlation based on Model 3. The correlation pattern is again inverted U-shaped, suggesting that fertility rst increases with access to technology
at lower technological levels and then decreases with the further diusion of technology. Figure 7 also
plots the real observations, illustrating the much lower number of observations in comparison to Figure
6. This lower number is due to the late appearance of the time series data on access to electricity, which
explains the accumulation of observations at relatively high electricity levels (a log of `electricity' of
4.6 corresponds to 100% of electricity coverage). Indeed, in the 1990s, most countries even the least
developed ones had already experienced their peak of fertility and many more developed countries
have already completed their demographic transitions with stable and low fertility rates. A comparisonof Models 3 and 4 reveals -similar to what we have found when using telephone lines as a proxy fortime-saving home appliances- that the declining branch of the inverse U-shaped correlation pattern
between fertility and access to electricity is somewhat dominating, while this is no longer the caseonce we control for time-xed eects, GDP per capita, exports, population density and child mortality:
leads to relative increases in their daughters' education, which later on leads to decreases in TFRs within developing and emerging countries.
Subsequent impacts are taken into account by allowing for a certain time delay between the observed technology level, the education and thefertility outcomes. Results are available upon request.

25

6.2 The eect of home appliances on the gender gap in education

6 EMPIRICAL ANALYSIS

While the coecient of the square of our electricity variable is signicantly negative in both models,Model 4 shows that the coecient of electricity itself is not signicant in comparison to Model 3.
Models 5 and 6 of Table 1 show regression results of TFRs estimated as a function of log(% ofhouseholds with a refrigerator) and its square. Model 5 is the simple model with country-xed eects
only, while Model 6 also contains time-xed eects and control variables. The number of observations
used here is extremely reduced due to the limited time and country coverage of the DHS. Model 5
suggests a concave overall within-country correlation between TFRs and our third proxy for timesaving home appliances, i.e. the di usion of refrigerators. The concavity is, however, only signicanton a 10% level, and disappears once we add time-xed eects and controls to the estimation model(Model 6). Figure 8 illustrates that the estimated overall within-country correlation pattern between
fertility and refrigerator diusion is slightly concave, albeit with a more dominant negative pattern incomparison to Figures 6 and 7. The weak within-country estimation results are likely to be caused by
the fact that data on refrigerator diusion in the DHS is not available for every year and observations
are very unevenly distributed over time. On average, each country has only 3.6 observed time periods inthe DHS. For comparison, the average number of observations per country is 38 in Model 1 (telephonediusion) and 25 in Model 3 (access to electricity).

Figure 8: Fertility and refrigerator diusion. Data: DHS, 76 countries, 1990-2015 (N = 276).

6.2 The eect of home appliances on the gender gap in education
Models 7 and 8 in Table 2 present regression results of the gender gap in education, estimated asa function of log(telephone lines) and its square.

The gender gap in education is measured by the26

6.2 The eect of home appliances on the gender gap in education

6 EMPIRICAL ANALYSIS

female-to-male ratio of average years of total schooling for ages 20-24.
Access to telephone lines, which gures here as a proxy for access to time-saving home appliances
one decade before, allows parents to send their children, and especially girls, to school, when they arearound ten to fteen years old. The telephone density that we observe at the time when the children
are aged 20-24 gives us thus information about the parental access to time-saving home appliances 10
years earlier, i.e. when their children were 10 to 15 years old. Thus, in this setting, we do not need aforward gap for telephone lines when empirically investigating the correlation between telephone lines
and the gender gap in education.

Table 2: The gender gap in education and female formal labor force participation as a function of technology diusion.

27

6.2 The eect of home appliances on the gender gap in education

6 EMPIRICAL ANALYSIS

Figure 9: Female/male years of schooling (ages 20 to 24) and density of telephone lines. Data: WB WDI and Barro &Lee (2013), 100 countries, 1960-2010; 5-year intervals (N = 963).

Model 7 contains country-xed eects and therewith estimates the overall within-country variation,
whereas Model 8 also includes time-xed eects and control variables (GDP per capita and its square,
exports, population density, child mortality, and TFRs).
Both models suggest a concave within-country correlation between telephone lines and the genderratio in education, as the estimated coecient of the square of log(telephone lines) is signicantly
negative. Figure 9 illustrates the estimated overall within-country variation as estimated by Model 7,
and contrasts it to real observations. The gure shows a concave pattern between telephone lines andthe gender ratio in education with a dominating rising branch: the female to male ratio in education
increases with an initial increase in telephone density, and then stagnates at higher levels of access totelephone lines. The ratio actually converges around 1 (i.e. gender equality in education) once a certain
level of technological advancement is achieved. This is in line with our theoretical prediction about the
improvement of gender equality in education in response to an increase in the level of technology, as
depicted in Figure 4. Further controls, which conrm the robustness of our ndings are available on
request. They include alternative measures of education (for example the female to male ratio of years
of total schooling for ages 20-24, with the female to male ratio of years of total schooling for ages 15+)and dierent time gaps for telephone lines.

28

6.3 The eect of home appliances on female formal labor-force participation

6 EMPIRICAL ANALYSIS

6.3 The eect of home appliances on female formal labor-force participation
Regression results of the female formal labor-force participation estimated as a function of log(telephonelines) and its square can be found in the last two columns of Table 2. Model 9 only contains countryxed eects while model 10 also contains time-xed eects and further controls.
For this empirical investigation, we prefer formal employment over general employment measures
including informal employment and subsistence activities, as female formal employment has the technical advantage that it evolves uniformly more or less during the process of economic development(and therewith over time), whereas women's general labor market participation is known to follow a
U-shaped pattern (i.e. decrease rst with initial stages of economic development, see feminization Uhypothesis by Goldin (1995)).

Figure 10: The percentage of women in wage employment in the nonagricultural sector and density of telephone lines.
Data: WB WDI, 142 countries, 1990-2013 (N = 1248).

Formal female employment is measured by the share of women in wage-employment in the nonagricultural sector, for which data is available for the 1990s onwards only. In order to keep the data
set as large as possible, we abstain here from using forward gaps of the exogenous variable `telephonelines'.
Here, only the full model (Model 10) suggest a slightly concave within-country correlation between
telephone lines and the female formal market participation, as the estimated coecient of the square29

8 APPENDIX
of log(telephone lines) is signicantly negative. Model 9, which does not include further controls andtherewith estimates the overall within-country pattern, rather suggests a positive correlation, which isillustrated by Figure 10.
It seems thus that overall within developing and emerging countries, increases in access to telephonelines are positively correlated with increases in female formal wage employment for initial stages oftechnological advancement, while levels of formal female wage employment stagnate at relatively highlevels of technological advancement. This is consistent with our theoretical prediction of an increasein female labor force participation, in response to an increase in the level of technology, as depicted in
the lower graph of Figure 5.

7

Concluding remarks

This paper has introduced a model in the tradition of unied growth theory to uncover a new mechanism
linking the demographic transition and the onset of sustained growth to the decline in the education
gender gap.

It has argued that the impact of technological progress on the eciency of householdproduction triggered an increase in female labor force participation, narrowing the education gender
gap and triggering a fertility decline. In turn, the subsequent increase in the average human capital
fosters further technological progress creating a feedback loop able to trigger sustained growth. Alongthe way, fertility increases rst, as women's time is freed up, enabling additional child-rearing, while it
eventually decreases, after increased labor force participation raises education investment in daughtersand hence their opportunity cost of childrearing.

The model predictions have been tested against

evidence from emerging and developing countries having undergone a demographic transition, and it
has helped to explain some stylized facts characterizing the development process.
Specically, the paper sheds light on the transition from stagnation to sustained growth, through ademographic transition experienced by developing countries in the late 20th and early 21st centuries.
The paper also shows that technological progress may increase female labor market participation,
not only by freeing women's time from housework due to the appearance of time-saving householdsector products, but also by leading households to reduce fertility due to an increase in the returnof women's human capital, which increases the opportunity cost of raising children.

In addition,technological progress makes households substitute quality to quantity in their fertility choice. Finally,
the paper provides empirical evidence to validate the main predictions of the model about the eectsof technological progress, through the creation and spread of labor-saving home appliances, on fertility
and education investment across genders, as well as on formal female labor-force participation.

8

Appendix

A0. Household's rst-order conditions
The rst-order conditions of the household's problem are30

8 APPENDIX



Uct









1

0





0





0











 t 
θ








 Un 
 = λ1  ρwt ẽt−1 + et + ẽt  + λ2  0  + λ3  0  + λ4  ρ 









 Ut 
nt
 0 
 0 
 −1 


 e 
t
0
−1
0
nt
Uẽ
along with the constraints and the slackness conditions, for multipliers
marginal utilities.

λi ≥ 0, where Ujt are theAccordingly, households choose education investments to equalize the marginalutilities of their sons' and daughters' human capitals, as well as fertility and consumption, in order toequalize the marginal rate of substitution of consumption and fertility with the marginal cost of thelatter. Thus, depending on whether the time constraint is binding, the household's choice on educationand fertility is

et =


γθ

 1−γ+γθ



ẽt =

ρ
1

(1−φt )[1+(1−φt+1 ) 1−θ ]ρθθ
1 wt ẽt−11−θ
1−θ
1+(1−φt+1 )γθ 1−γ+γθρ
1(1−φt )[1+(1−φt+1 ) θ−1 ]ρθθ
1 wt ẽt−11−θ
θ−11+(1−φt+1 )
nt = minwt eθt−1 if φt + ρnt = 1(16)

if φt + ρnt < 1
wt eθt−1 if φt + ρnt = 1(17)

if φt + ρnt < 1
1 − φt γ(1 − θ) eθt−1,+ 1 − φtρρẽθt−1(18)

so that the fertility in the binding and non-binding cases coincide for some φt , i.e.

γ(1 − θ) eθt−1
1 − φt+ 1 − φt =
θ
ρ
ρ
ẽt−1

(19)

but, since from (16) and (17) it holds in any period t − 1

#θ
"
1
θ
eθt−1
1 + (1 − φt ) θ−1= (1 − φt ) θ−1 ,
=
1
θ
ẽt−1
1 + (1 − φt ) 1−θ(20)then (19) implies that the time constraint will be binding wheneverγ(1 − θ)φt = 1 −1 − γ(1 − θ)

1−θ
(21)

for which the binding and non-binding consumption and education choices coincide.

A1. Proof of Lemma 1
Let us denote Et−1 = et−1 + ẽt−1 , which is determined in period t − 1 as in (13). The growth rate oftechnological progress between periods t and t + 1 is

 θ



et−1 + ẽθt−1(Et−1 − ẽt−1 )θ + ẽθt−1
gt = g=g
2
2
31

8 APPENDIX
so that∂gt= g0∂ẽt−1(Et−1 − ẽt−1 )θ + ẽθt−12θ  θ−1ẽt−1 − (Et−1 − ẽt−1 )θ−1
2

0
Since g (.) > 0 and θ ∈ (0, 1) then ∂gt /∂ẽt−1 > (=)(<) 0⇔ ẽt−1 < (=)(>) Et−1 /2, which implies

the result stated in Lemma 1.

A2. Proof of Lemma 2
Since

lim mt = m > 0 then ∀ε ∈ (0, m), ∃T such that ∀t ≥ T , we havet→+∞m − ε ≤ mt ≤ m + εDeneX0 = Q0 = Z0 = xT
and

Xt+1 = mT +t Xtα ,Qt+1 = (m + ε)Qαt ,Zt+1 = (m − ε)Ztα

We know that1

lim Qt = (a + ε) 1−α1lim Zt = (a − ε) 1−α

and

t→+∞

t→+∞

We also have

Z1 = (m − ε)X0α ≤ X1 = mT X0α ≤ (m + ε)X0α = Q1
Z2 = (m − ε)Z1α ≤ (m − ε)X1α ≤ X2 = mT +1 X1α ≤ (m + ε)X1α ≤ (m + ε)Qα1 = Q2.
.
.
and so on, by induction we have Zt ≤ Xt ≤ Qt , ∀t. Hence,(m − ε)11−α= lim Zt ≤ limt→+∞
inf Xt

T →+∞

t≥T







1
≤ lim supXt ≤ lim Qt = (m + ε) 1−α
T →+∞

t≥T

t→+∞

That is to say(m − ε)

1
1−α

≤ lim

T →+∞

inf Xtt≥T
1≤ lim supXt ≤ (m + ε) 1−α , ∀ε ∈ (0, m)
T →+∞

t≥T

Hence,lim (m − ε)

ε→0+

1
1−α


≤ lim

T →+∞


inf Xtt≥T
1≤ lim supXt ≤ lim+ (m + ε) 1−α
T →+∞

32

t≥T

ε→0

8 APPENDIX
i.e.

m

1
1−α


≤ lim

T →+∞


inf Xt
1≤ lim supXt ≤ m 1−α
T →+∞

t≥T

t≥T

which implies
1

lim Xt = m 1−α ,

i.e.

t→+∞

1

lim xt = m 1−α .

t→+∞

A3. Reversal in the education gap across genders
Following Goldin

et al. (2006) and Becker

et al.

(2010) on heterogeneity in non-cognitive skills acrossgenders, we can modify the human capital formation function for women to be ψ̃ e˜tθwith ψ̃ > 1, while

the human capital formation function for men remains unchanged. The parameter ψ̃ gauges the nonAlso, let a man's income with education et−1 becognitive skills advantage of an average daughter.

wt (ψ + eθt−1 ) in period t, where ψ > 0 captures the gender discrimination because of men's physicalstrength and no career interruptions linked to paternity. This allows capturing the fact that, with thesame levels of education, men usually receive higher wages than women.
The household's optimization problem then becomesmax

ct ,nt >0, et ,ẽt ≥0(1 − γ) ln ct + γ lnnt wt+1 [ψ + eθt + (1 − φt+1 )ψ̃ẽθt ]subject to
ct + nt (et + ẽt ) ≤ wt ψ + eθt−1 + (1 − φt − ρnt )ψ̃ẽθt−1
φt + ρnt ≤ 1θnote that the potential incomes of sons wt+1 (ψ+et ) consists of two parts, the potential basic, unskilledθlabor income wt+1 ψ and the potential skilled labor income wt+1 et . At the solution to the optimization
problem above, the gender inequality in education µt is determined by

µt =

1
i 1−θ
ẽt−1 h= (1 − φ(at ))ψ̃≡ µ̃(at )et−1

with, from Assumption 1,µ̃0 (a) > 0

and1

lim µ̃(a) = ψ̃ 1−θ > 1

a→+∞

(22)

The following assumption further guarantees that for low levels of technology, the education investment
in daughters is lower than in sons.

.

Assumption 3. (1 − φ̄)ψ̃ < 1Thus, the following lemma states that the reversal in education investment across genders, for ahigh level of technology follows straightforwardly from Assumption 3 and the properties of µ̃(at ) in(22).

Under assumptions 1 and 3, there exists a unique level of technology ã > 0 at which µ̃(ã) = 1
so that complete equality in education investment across genders holds. Moreover, for all a ∈ (0, ã) it
holds µ̃(a) < 1, and for all a > ã, µ̃(a) > 1.

Lemma 3.

33

8 APPENDIX

Figure 11:

Reversal in education investment across genders.

In other words, during the development process, and along with the technological progress, theeducation investment in daughters increases relative to that in sons, and a reversal in the education
investment gap across genders occurs with a suciently high level of technology. The reasoning hereis quite intuitive.

Technological progress makes parents invest relatively more in their daughters'

education when the marginal return to education investment in daughters is greater than in sons. So
whenever the technology surpasses a tipping point ã above which the potential income of the daughter
is greater than the potential skilled labor income of the son, the parents will optimally invest more in
the education for their daughter than for their son, in order to equalize the marginal utility from anincrease in the education of either child. Thus, the daughter's education eventually exceeds the son's.29

A4. Correlation matrix and summary of data on next pages

29 Note that, what we observe from the data is the reversal in education achievement eθ and ψ̃ẽθ for sons and daughters respect

tively rather than education investment.

i.e.

lim µ̃(a) =h(1 − φ(a))ψ̃

i

1
1−θ

1

= ψ̃ 1−θ > 1. Such a reversal, however, can occur for the education achievement even if there is
h
i 1
1−θ
not any in education investment, i.e. whenlim (1 − φ(a))ψ̃= 1. Indeed, as proved in Proposition 2, e = lim et = lim ẽt = ẽ.
a→+∞

lim

Assumption 1 and ψ̃ > 1 are sucient for a reversal in education investment across genders,

a→+∞

a→+∞

As a consequence, eventually eθ

t→+∞

t→+∞

< ψ̃ẽθ , so that the education achievement of the daughter exceeds that of the son even if the daughter's

education investment does not. That is to say, when the level of technology is suciently high the education achievement will reverse acrossgenders because of both (i) the improvement in gender equality in education investment and (ii) the better eciency of female education
investment.

34

8 APPENDIX

35

8 APPENDIX

36

9 REFERENCES

Funding: This study was funded by research grant MIS Ulysse from the Belgian F.R.S.-FNRS(grant number F. 6007.09).

Conict of Interest: The authors declare that they have no conict of interest.

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