Gender Inequality, Endogenous Cultural Norms and Economic Development∗ Victor HILLER† LEM, Université Paris II Panthéon-Assas May 25, 2010

Abstract This article explores the joint dynamics between gender inequality and cultural norms along the process of development. Cultural norms about gender roles are shaped by the relative women labor supply and thus depend on the gender gap in education. In turn, norms influence the relative education provided to boys and girls. We show that, this two way causality between the inegalitarian nature of norms and the educational gender gap might explain the emergence of high gender inequality and low development traps. The model also allows us to replicate a U-shaped relationship between the relative status of women and the economic development.

JEL Codes: J16, O11, Z10. Keywords: gender inequality, gender roles, cultural norms, economic development, inequality traps. ∗ This paper has been partly written during my Ph.D. at the University of Paris 1 Panthéon-Sorbonne. I would like to thank Thomas Baudin, Cecilia Garcia-Peñalosa, Fabio Mariani, Natacha Raffin, Bertrand Wigniolle, participants of JGI 2008 at Aix, CEPR conference "Models of Cultural Dynamics and Diversity" at Paris, ADRES Doctoral Days 2009 at UCL, RES 2009 Conference at the University of Surray, PET 2009 conference at Galway and seminar participants at Paris 1 University, University of Göteborg and University of Valencia for useful comments. † Adress: Université Paris II Panthéon Assas - LEM, 5-7 Avenue Vavin, 75006 Paris, France. E-mail: [email protected]. Tel: +33-1-55-42-50-26. Fax: +33-1-55-42-50-24.

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1

Introduction

The reduction of gender inequality has become a crucial issue for international organizations and policy makers. As an emblematic illustration, promoting gender equality has been identified as the third Millennium Summit Development goal. In this context, a growing literature focuses on the sources of gender discrepancies in developing countries (Dollar and Gatti 1999; Morrisson and Jütting 2004 and 2005). Those empirical works underline the role played by institutional, cultural and social factors beyond the sole economic determinants. As a direct implication of these findings, the economic development may reduce gender inequalities only if it comes with some cultural and social changes. Should it be otherwise, the economic development would translate into a rise of opportunities only for men and then, a wider gender gap. This conclusion may be related to older theories dealing with the evolution of women’s status in the course of development. Already, at the beginning of seventies, Esther Boserup claimed that first stages of development may be characterized by a deterioration in the relative status of women until that inegalitarian cultural and social barriers have been overcome. Goldin (1995) emphasizes a similar point when highlighting a U-shaped relationship between female labor force participation and the industrialization process. She argues that the existence of social stigma against women working in manual occupations may be responsible for the decreasing part of the U. Up to now, the possibility of a loss of status for women during first stages of economic development and, all the more, the role played by cultural norms in this pattern, has been rarely theoretically investigated.1 The present article aspires to fill this gap by providing a theoretical framework in which endogenous cultural norms play a key role in explaining the non-monotonic relationship between gender inequality and economic development. 1. Most of the literature relating long-term economic development and gender inequalities focuses on the reduction of the gender gap (Lagerlöf 2003; Greenwood et al. 2005; Soares and Falcão 2008). The work of Galor and Weil (1996) constitutes a noticeable exception. Indeed, the authors replicate a U-shaped relationship between women labor supply and the industrialization process.

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We consider an overlapping generations economy constituted of men and women, working both at home and on the market. The economy is characterized by the existence of cultural norms that define gender stereotypes and assign specific roles to men and women. We consider that the inegalitarian nature of cultural norms is a crucial determinant of parental investments in children’s human capital. For instance, if the norm is highly discriminatory2 , parents hold the view that women should specialize in home production and give a low value to the education of their daughter. Moreover, cultural norms about gender roles are endogenous. In particular, they evolve according to the observation of the relative female labor force participation. If women’s labor supply increases, stereotypes with regards to their role within the society change, and norms become less discriminatory. These mechanisms allow us to model a two-way causality between norms and the educational gender gap. On the one hand, gender differences in education determine women’s labor participation which, in turn, shapes individuals’ beliefs about their role within the society. On the other hand, the resulting new norms affect the relative education of boys and girls. During pre-industrial times, individuals do not receive any education and provide unskilled labor force. Since such activities require physical strength, men are more productive than women and supply a higher amount of labor. Consequently, social norms favor parental investment in the education of boys. Men’s education fosters development through human capital accumulation. However, this accumulation has a perverse effect: it increases the productivity gap between men and women, which induces a fall in the relative labor supply of women and less egalitarian views about gender roles. In the end, the initial stage of development gives rise to two opposite forces that shape incentives 2. By discriminatory norm, we mean a norm that favors boys and then discriminates against girls. In the same spirit, following the terminology proposed by Farré and Vella (2007), the term traditional reflects the view that women should specialize in home production and men in market.

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to educate girls. First, a positive income effect that fosters education spending for both children. Second, a negative cultural effect delaying parental investment in girls’ human capital. The long-term situation reached by one economy depends on the relative strength of these two effects. In particular, we show that the mechanism of mutual reinforcement between the inegalitarian nature of the norm and the educational gender gap may generate multiple equilibria. In that configuration, initial conditions, in terms of both development level and social beliefs about gender roles, crucially matter. If initial conditions are not too inegalitarian, the positive income effect overcomes the negative cultural effect. In that case, the accumulation of men’s human capital enables the economy to reach a sufficiently high level of development, girls are also educated and the gender gap in labor productivity is progressively filled. Then, the economy ends up with a high-output equilibrium featured by the equality between men and women. Along the convergence path, women labor participation and, more broadly, the relative status of women follows a U-shaped pattern. Indeed, as mentioned above, the beginning of the development process widens the gender gap. However, as soon as women are educated too, the productivity gap between genders decreases and women labor supply rises. Nevertheless, if an economy is initially characterized by too inegalitarian cultural norms, the initial rise of income cannot offset the negative cultural effect. Consequently, the economy ends up in a high gender inequality / low development trap. The way we introduce cultural norms and their evolutions constitutes a cornerstone of this paper. We do not focus on the micro-process behind the emergence of gender stereotypes, like in Vendrik (2003) or Breen and Garcia-Peñalosa (2002) for instance.3 Instead, cultural norms are embodied in households preferences and are shaped by the 3. Vendrik (2003) employs the Theory of Social Custom of Akerlof (1980). Alternatively, in Breen and Garcia-Peñalosa (2002), inegalitarian norms take the form of inegalitarian beliefs about the respective ability of men and women, these beliefs being updated through Bayesian learning.

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observation of average behavior within the population. However, our own article shares common features with these works. In particular, norms evolutions influence and are influenced by the observation of career choices of men and women, leading to a potential persistence of gender inequality. Moreover, our simplified framework, even though more restrictive, is much more tractable and allows to model, in a simple and convenient fashion, the intergenerational persistence of the earning gap between men and women. Thus, we are able to characterize the joint dynamics between gender roles, gender inequality and economic development in a fully analytical way. Our technique of modeling social norms as a parameter of preferences shaped by the evolution of an aggregate variable common in the literature (Lindbeck et al. 1999 and 2003 or Palivos 2001). Hazan and Maoz (2002) apply this reduced-form approach to the analysis of the joint relationship between female labor force participation and social stigma against working women. Nevertheless, our paper allows us to account for both the development process and the evolution of the gender wage gap besides female labor supply and cultural norms dynamics. The present article may also be related to a recent paper by Kirchsteiger and Sebald (2010). The authors consider that education a parent has received during childhood determines his willingness to invest into human capital of his own child. In our framework, in which gender matters, the human capital received by a wife, compared to her husband, shapes the relative weight given by the couple to the education of their daughter vs. their son. This intergenerational transmission of the earning gap being conveyed through decisions of participation to the labor market. Our research also relates to the literature dealing with the relationship between gender inequality and long-run economic development. In this strand of the literature, we depart from the seminal contribution of Galor and Weil (1996) by focusing on the accumulation of human capital rather than physical capital. Moreover, Galor and Weil do not consider the

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potential role played by cultural factors. Lagerlöf (2003) introduces the idea that the educational gap between girls and boys is derived from the existence of inegalitarian cultural norms. Our approach differs from Lagerlöf’s one by considering the process of cultural evolution as endogenous. It may generate gender inequality traps and non-monotonic evolutions of the gender gap, which are absent from Lagerlöf’s model. Recently, de la Croix and Vander Donckt (2010) consider an alternative way to take into account the impact of cultural norms on gender inequality. They assume that the relative bargaining power of women within the household is partly determined by cultural factors. Hence, de la Croix and Vander Donckt (2010) make a step behind by considering a non-unitary representation of the decision making between spouses. However, they do not deal with the endogeneity of cultural norms. The rest of the paper is organized as follows. Section 2 exposes some empirical insights that motivate our research. The model is set up in Section 3. Section 4 exposes the dynamics and the long-term properties generated by the model. Section 5 discusses the robustness of the results. Finally, Section 6 concludes.

2 2.1

Empirical background Evolution of women’s status in the course of development

The thesis according to which women’s status may decline during first stages of economic development finds its roots in Boserup’s (1970) influential work. The author argues that a sufficient level of income has to be reached in order to experience both economic development and a reduction of gender inequality.4 One of the most striking empirical illustration 4. Consistently with our own story, Boserup argues that: "In primitive communities, the difference in productivity between men and women labor is not very large. Although most men have advantage of superior physical strength, at this stage neither men nor women can benefit from specialization. The gap in productivity between the two sexes widens considerably at the stage when boys gets some systematic training in schools or in workshops, while girls continue to be taught only by mothers. At a later stage, when girls go to school, the gap is reduced." (p. 201)

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of this non-monotonic pattern is the U-shaped relationship between female labor participation and development level, supported by both cross-country analysis5 and longitudinal data.6 Along the same lines, Latican et al. (1996), in a study apply to Bangladesh, Philippines and South Korea, conclude that gender inequalities, at elementary school, first increase and then decrease as the economy industrialize. Similar to us, the authors stress out the role played by the inegalitarian allocation of resources between girls and boys within poor households. In our paper, the loss of status for women, during first stages of development, goes hand in hand with more inegalitarian social beliefs about gender roles. This idea may comply with the rise of "separate spheres" ideology, observed at the beginning of the industrialization process in Europe and the United States. Indeed, such kind of beliefs put an increasing emphasis on the role of women at home vs. the role of men on the market (see, the historical study of Abrams (2002) for the European case).

2.2

Impact of gender inequalities on the economic development

Our analysis suggests that gender inequalities may impede economic development. This can be supported by Klasen (1999) and Dollar and Gatti (1999) who find that gender inequalities in education display a direct and negative impact on economic growth. Knowles et al. (2002) estimate a neoclassical growth model including male and female education, and conclude that the educational gender gap is a barrier to economic development. In our setting, this negative causal link stems from two channels. First, there exists a direct volume effect: if one half of the population has a limited access to schooling, the aggregate level of education is mechanically lowered. In this line, Morrisson and Murtin (2009) note that the gap between enrollment rates of boys and girls in Sub-Saharan Africa 5. See Durand (1975), Goldin (1995), Cagatay and Olzer (1995) or Mammen and Paxson (2000). 6. See Goldin (1995) for United States or Mammen and Paxson (2000) for evidences from Thailand and India.

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and Southern Asia is largely responsible for the backwardness of education in these countries. Second, an indirect effect passes through the existence decreasing returns to education expenditure. Using micro data, Psacharopoulos (1994) confirms the relevance of this channel.7

2.3

Cultural norms and the gender inequality trap

In the World Development Report (WDR) 2006, the World Bank emphasizes that economic and cultural inequalities may reinforce each other, leading to gender inequality traps. Besides, some empirical studies assess the impact of cultural factors on gender inequality. In cross country analysis, Dollar and Gatti (1999), Inglehart and Norris (2003), Morrisson and Jütting (2005) or Self and Grabowski (2009) show that religious variables or the existence of inegalitarian cultural norms significantly depress the relative status of women. In our set-up, the relationship between inegalitarian cultural norms and gender inequality is derived from educational choices. Few empirical works directly investigate the link between gender role attitudes and investment in human capital. Nevertheless, focusing on developing countries, sociological studies commonly claim that cultural factors, as patriarchal norms, may explain the girls’ limited school participation (Greenhalgh 1985; Davison and Kanyuka 1992; Hyde 1993). In addition, Dollar and Gatti (1999) provide some empirical evidences that cultural variables, like religious preferences, partly explain gender differences in educational achievement. Finally, since in our model social norms depend on women’s participation, there might exist a direct link between this variable and investment in girls’ education. This would be consistent with findings of Anderson et al. (2003) on Malaysian’s data or Rosenzweig and Schultz (1982) on Indian’s data. 7. Gender discrepancies may affect output levels in several other ways. For instance, the reduction of gender inequality may reduce fertility and infant mortality, it may also improve children health and human capital (see Behrman and Deloakitar 1988; Schultz 1988; Subbarao and Raney 1995). Taking into account these alternative channels would reinforce our results.

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3

The model

Consider a two-sex overlapping generations economy, where each individual lives for two periods. During childhood, both girls and boys may receive education. During adulthood, both women and men work to produce two goods: the domestic good, produced at home and the manufactured good, produced on the market. Individuals’ productivity on the market depends on parental investment in education. Two parents (one man and one woman)8 compose a couple which has two children (one boy and one girl). Women and men display identical preferences: couples are assumed to have joint consumption and a joint utility function. In addition, since there is no heterogeneity within each generation, we can assume that men and women are randomly matched on the marriage market.

3.1

Production

Each adult is endowed with one unit of time that can be shared between housework and work on the market.

3.1.1

The domestic good

Housework enables the production of a domestic good. The amount of time devoted to housework by an individual j ∈ {f, m} at time t is denoted ltj . This housework time exhibits decreasing returns: a quantity of time ltj implies the production of (ltj )γ units of domestic good, with γ ∈ (0, 1). This assumption captures fatigue effects associated with the production process. Then, the total quantity of domestic good produced by a couple at time t (denoted Dt ) equals: Dt = (ltf )γ + (ltm )γ . 8. Respectively indexed by m for male and f for female.

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(1)

This implies that men and women are able to produce the same amount of domestic good if they provide the same quantity of time.9

3.1.2

The manufactured good

When individuals do not work at home, they work in the market, supplying (1−lti ) units of time as an input for the production of a manufactured good. This good is produced according to a constant return to scale technology, which requires human capital. Nevertheless, physical strength may also affect the productivity of unskilled workers. In particular, we consider that individuals who have received no education, have only access to manual activities requiring physical strength. Furthermore, and following the argument proposed by Galor and Weil (1996), it is assumed that a differential of strength endowment between men and women exists. The latter translates into a loss of productivity for women in unskilled physical tasks. However, since education enables to have access to less manual jobs, when educational attainment of women rises, this productivity loss reduces. In our model, we consider that workers have access to skilled occupations as soon as they receive a positive amount of education. Then, women do no longer suffer the loss of productivity associated with the physical strength gap since they are educated. This restriction is made for ease of presentation. The results would have been identical if we had assumed that this loss progressively reduces throughout the process of women’s human capital accumulation. Formally, the market income per unit of time, for men and women respectively, writes 9. The production function of household good proposed by Fernandez et al. (2004) displays the same kind of properties. Considering a CES production function (as in Albanesi and Olivetti 2007 and 2009) would not affect qualitatively the main results of the model.

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as:

wtm = h(em t ),

(2a)

wtf = h(eft ) − δ(eft )s,

(2b)

where h(ejt ) denotes the human capital of an individual j when adult, it is positively related to the level of education (ejt ) received during childhood, with h(0) > 0. The function δ(eft ) takes the value 1 when women receive no education and 0 otherwise:

δ(eft ) =

    1 if eft = 0

.

(3)

   0 if ef > 0 t

Finally, s measures the productivity loss for uneducated women, working in manual activities. Obviously, s is assumed to be not too high, to ensure that the income of an unskilled woman remains positive: s < h(0). Let us define the variable yt as the total potential income of a couple at date t: yt = wtf + wtm .10 The latter variable is crucial for our analysis since it is considered as our index of economic development.

3.2

Preferences and budget constraint

Preferences of a couple at date t are defined over consumption of domestic and manufactured goods (denoted Dt and Ct respectively) as well as the level of human capital reached by their children. Preferences are represented by the following utility function:

i h )) . Ut = µ ln Ct + (1 − µ) ln Dt + β θt ln(h(eft+1 )) + ln(h(em t+1

(4)

10. yt is a potential income because it is the income per unit of time. However, since parents devote some of their time to housework, they only earn a share of yt on the market.

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The parameter µ ∈ (0, 1) measures the relative importance attached to the consumption of the market good vs. the domestic good; while β > 0 represents the weight agents give to the future human capital of their children. Our key assumption lies in the existence of cultural norms about gender roles that determine the relative weight given by parents to the education of their daughter compared to the education of their son. The parameter θt ∈ [0, 1] measures the discriminatory nature of these norms at date t.11 When θt is low, the norm is highly discriminatory: parents hold the view that women should work at home and therefore they value less the education of their daughter. Conversely, if θt is close to one, the norm is relatively egalitarian and the weight attached to the education of girls is higher.12 The variable θt is endogenous since it is shaped by the relative labor force participation of men and women within the economy as we will see in section 4.2. However, θt is taken as given by parents when they decide on their education expenditure. In particular, they do not take into account the impact of their labor supply decisions on the evolution of cultural norms. Indeed, since the number of couples is large, each atomistic individual’s decision has a negligible impact on the aggregate labor supply and then on the evolution of the norm. Couples allocate their earnings between consumption of the manufactured good and educational spending. Then, couple’s budget constraint can be expressed as:

  (1 − ltf )wtf + (1 − ltm )wtm = Ct + τ eft+1 + em t+1 ,

(5)

11. To comply with empirical regularities that women had never been more educated than men during first stages of human capital accumulation, we focus on the case θ0 < 1. It implies that initially parents favor boys. As it will be clear later, under this condition, θt is lower than one for all date t. Thus, the f analysis is restricted to this later case implying em t ≥ et for all t. 12. Let us note that, qualitative results of the model would be unaffected under the more general specification of the utility function: i h Ut = µ ln Ct + (1 − µ) ln Dt + β θtf ln(h(eft+1 )) + θtm ln(h(em t+1 )) , with θtm + θtf = 1. The choice to normalize θtm to one is made for ease of simplicity.

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τ > 0 being the relative cost of one unit of education and ejt+1 the quantity of education given to a child j. Each couple chooses Ct , Dt , ltf , ltm , eft+1 and em t+1 in order to maximize its utility (4) subject to the budget constraint (5) and the technological constraint on the domestic production (1).

3.3

labor supply behaviors

Optimization with respect to the housework time of men and women (ltm and ltf ), yields the following first-order conditions:

(lf )γ−1 µwtf ≤ t , γ(1 − µ)Ct Dt

(6a)

µwtm (lm )γ−1 = t . γ(1 − µ)Ct Dt

(6b)

Our analysis is restricted to the interior solution for both spouses, meaning that men and women work at home as well as on the market: ltm ∈ (0, 1) and ltf ∈ (0, 1). In that case, (6a) and (6b) hold with equality.13 Combining (6a) and (6b) yield the following relationship between the optimal values of ltm and ltf : ltm ltf

!1−γ

=

wtf . wtm

(7)

This equation describes the optimal share of housework between men and women, such that the relative productivity of women at home (left hand side) equals their relative 13. Since the marginal productivity of housework equals infinity (positive) as ltj tends towards zero, ltf and ltm are always strictly positive. Moreover, ltm is always strictly lower than one. Indeed, ltm = 1 would imply that ltf = 1, since wtf ≤ wtm , and then Ct = 0. This case can not exist thanks to the properties of the utility function. In particular, the marginal utility of Ct equals infinity when Ct tends towards zero. Hence, the unique corner solution that we should consider is ltf = 1. However, we choose to abstract from this specific case since it will heavily complexify the ensuing analysis. Moreover, the WDR 2006 (p.53) documents that men and women perform the same time on market and non-market activities, both in developed and developing countries.

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opportunity cost of housework (right hand side). Consequently, the higher the relative productivity of men on the labor market, the larger the relative time spent by women in domestic production. The optimal share between work at home and on the market being determined for the two spouses, the following section focuses on the optimal education choices.

3.4

Educational choices

f Let us now consider optimization with respect to the education of children (em t+1 and et+1 ),

which yields the two following first-order conditions: h′ (eft+1 )

≤

h(eft+1 ) h′ (em t+1 ) m h(et+1 )

τµ , βθt Ct

(8a)

τµ , βCt

(8b)

≤

where (8a) (respectively (8b)) holds with equality if the optimal value of eft+1 (em t+1 ) is strictly positive. From now on, production of human capital is assumed to take place according to the following function: h(ejt ) = (c + aejt )α .

(9)

The parameter c > 0 prevents human capital from being zero even if parents do not invest in education, as in de la Croix and Doepke (2003, 2004). Moreover, the parameter α ∈ (0, 1) is a measure of the elasticity of human capital with respect to schooling and a > 0 accounts for the productivity of educational expenditure. Due to the existence of a basic level of human capital, parents can choose to provide no education to one or both of their children. In particular, three different regimes may be identified: the interior regime in which both boys and girls are educated, the gender 14

inequality regime in which only boys receive education and the poverty regime in which neither boys nor girls are educated. In the following sections we characterize each one of the three regimes.

3.4.1

The interior regime

The optimal choices of education are interior solutions if both (8a) and (8b) hold with equality. In that case, the ratio (8a) over (8b) yields:

c + aem t+1 c+

aeft+1

=

1 . θt

(10)

It measures the educational gap between men and women and is a decreasing function of θt . Indeed, a rise of θt induces an increase in the weight given to girls’ education in parental preferences. The optimal levels of education for girls and boys are deduced using the budget constraint (5) and FOCs (6a), (6b), (8a) and (8b):

eft+1 =

aαβθt yt − τ c[µ + γ(1 − µ) + αβ(1 − θt )] , aτ [µ + γ(1 − µ) + αβ(1 + θt )]

(11a)

aαβyt − τ c[µ + γ(1 − µ) − αβ(1 − θt )] . aτ [µ + γ(1 − µ) + αβ(1 + θt )]

(11b)

em t+1 =

Obviously, in this regime, the education of both children increases with parental income (yt ). Furthermore, the education provided to girls rises when the norm becomes more egalitarian (increase in θt ), while the education of boys reduces with θt through a substitution effect.

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3.4.2

The gender inequality regime

When (8a) holds with inequality while (8b) holds with equality, parents choose to provide no education to their daughter (eft+1 = 0) and a positive amount to their son. From (5), (6a) and (6b), the optimal level of em t+1 in that regime writes as:

em t+1 =

aαβyt − τ c[µ + γ(1 − µ)] , aτ [µ + γ(1 − µ) + αβ]

(12)

and is increasing in the parental income yt . It is worth noticing that it does not depend on θt since girls receive no education. We can now derive the set of all pairs (θt , yt ) for which parents do not provide education to their daughter. Let us define, the boundary formed by all pairs (θt , yt ) such that yt = y˜(θt ) with:   τ c µ + γ(1 − µ) + αβ(1 − θt ) y˜(θt ) ≡ . a αβθt

(13)

From (8a), if yt < y˜(θt ) then eft+1 = 0. This boundary is a downward slopping curve in the plan (θt , yt ).

3.4.3

The poverty regime

Finally, if both (8a) and (8b) hold with inequality then eft+1 = em t+1 = 0, children do not receive any education. It is the case if:

yt <

τ c[µ + γ(1 − µ)] = y˜(1). aαβ

(14)

This threshold is decreasing in a. Indeed, when the productivity of educational expenditures is high, it becomes optimal to educate children for relatively low income levels. In the same vein, y˜(1) is increasing in τ . Finally, a rise in c allows for an increase in the

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yt

y˜(θt )

em t+1 > 0 eft+1 > 0 em t+1 > 0 eft+1 = 0 y˜(1)

em t+1 = 0 eft+1 = 0 0

1

θt

Fig. 1. Corner solutions for education

basic level of human capital, then it cuts down the incentives to educate children: y˜(1) is increasing in c.

3.4.4

Educational regions in the plan (θt , yt )

To sum up, the plan (θt , yt ) can be divided into three regions, as depicted in Figure 1. If yt is too low i.e. lower than y˜(1), the economy belongs to the poverty regime. For intermediary values of income i.e. yt ∈ [˜ y (1), y˜(θt )), the economy is in the gender inequality regime. Finally, for a high enough levels of development i.e. yt ≥ y˜(θt ), the economy belongs to the interior regime. For a given value of yt , the size of the gender inequality regime reduces with θt (˜ y (θt ) is decreasing in θt ). Indeed, if θt is low, the weight given to the education of girls in parental preferences is small. Thus, the level of development that is required for women to be educated is high.

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Dynamics

Using optimal microeconomic choices, we are able to analyze the dynamics of the economy.

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4.1

Evolution of income

As developed in section 3.4, parental investments in education (eft+1 and em t+1 ) depend on the potential income yt , the level of social norms θt and parameters of the model. Since human capital is the unique factor of production, the value of the potential income in t + 1 f f m is fully determined by eft+1 and em t+1 : yt+1 = h(et+1 ) + h(et+1 ) − δ(et+1 )s. Hence, we obtain

the expression of yt+1 as a function of yt and θt : yt+1 = ρ(yt , θt ). From (9) and (11a)-(16), we derive:     2cα − s if yt < y˜(1)     ρ(yt , θt ) = cα − s + χ(0)(ayt + τ c)α if yt ∈ [˜ y (1), y˜(θt )) ,        χ(θt )(1 + θα )(ayt + 2τ c)α if yt ≥ y˜(θt ) t

(15)

where χ(θt ) is defined as follows:

χ(θt ) ≡



αβ τ [µ + γ(1 − µ) + αβ(1 + θt )]

α

.

(16)

Equation (15) provides us the conditional dynamics of yt , which describe the income time path conditionally to the level of the norm. The dynamics are characterized by the three regimes described above. In the poverty regime, there is no education spending so that yt remains constant. In the gender inequality regime, the rise in yt is driven by the increase in boys’ education. In the interior regime, education of both boys and girls triggers income growth. Consequently, in this regime, yt+1 depends on θt . Indeed, norms determine the share of education between men and women that, in turn, influences the aggregate productivity of educational spending. Let us underline that, thanks to the existence of decreasing returns to education expenditure, when θt is given, the income monotonously converges towards a stationary value both in the gender inequality regime and the interior

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regime.

4.2

Evolution of gender roles

Although it is taken as given by parents, θt is endogenous. In particular, we consider that norms are partly transmitted across generations14 but are also shaped by the observation of men and women labor supply behaviors.15 Formally, θt is a function of θt−1 , due to the intergenerational transmission of gender role attitudes, as well as the ratio ltm /ltf (i.e. the relative time spent by men to work at home). Indeed, if parents observe a change in women’s participation to the labor market, they may revise their conceptions of gender stereotypes. For instance, when ltm /ltf drops, the role of women is more associated with housework, then parents are less prone to educate their daughter. In order to obtain closed-form solutions, we choose the following functional form, describing the evolution of the norm:  κ θt = σθt−1 + (1 − σ) ltm /ltf .

(17)

The parameter σ ∈ (0, 1) measures the strength of habits. The higher σ, the more persistent is the gender bias. The parameter κ ∈ (0, 1) accounts for the sensitivity of norms to changes in relative labor supply.16 Plugging (7) into (17), we obtain the following expression: θt+1 = σθt + (1 − σ)

h(eft+1 ) − δ(eft+1 )s h(em t+1 )

!

κ 1−γ

.

(18)

14. Fernandez et al. (2004) or Fernandez (2006 or 2007) provide evidences on the intergenerational transmission of cultural attitudes regarding the role of women within the society. 15. As previously mentioned, atomistic individuals’ choices have no significant impact on the aggregate labor force participation. However, since there is no heterogeneity, neither among men nor among women, aggregate labor supplies equal individual ones. 16. Let us note that, expression (17) joint with condition (10) imply that an increase in the relative labor supply of men may induce a fall of the relative investment in girls education. This result is, for instance, backed up by the findings of Anderson et al. (2003) on Malaysian’s data or Rosenzweig and Schultz (1982) on Indian’s data.

19

Hence, from (9) and (11a)-(16), θt+1 can be expressed as a function of θt and yt , θt+1 = η(yt , θt ) with 
κ   cα −s 1−γ  σθ + (1 − σ) α t  c    i κ h 1−γ cα −s η(yt , θt ) = σθ + (1 − σ) t χ(0)(ayt +τ c)α     ακ    σθt + (1 − σ)θ 1−γ t

if yt < y˜(1) if yt ∈ [˜ y (1), y˜(θt )) .

(19)

if yt ≥ y˜(θt )

Similar to the income dynamics, we obtain the conditional dynamics of θt , describing how the norm evolves for a given yt . This dynamical behavior is crucially affected by the relative female labor force participation. In the poverty regime, the relative labor supply of women is exogenous. Men supply more labor than women since they are able to earn a higher wage on the market. In the gender inequality regime, the relative labor supply of women falls with yt . Indeed, the rise of parental income induces more education only for men. This implies a growing productivity gap and consequently a reduction of the relative female labor participation. In the interior regime, for the same level of education, men and women receive the same wage. Hence, the relative labor supply of women only relies on the educational gap, which is fully determined by the social norm θt (see equation (10)). In the poverty regime and the gender inequality regime, θt monotonously converges towards a stationary value, since σ ∈ (0, 1). In order to ensure that it is also the case in the interior regime, we assume that the following parametric restriction is satisfied:

ακ < 1 − γ.

(20)

It implies that the sensibility of norms to the gender gap in participation should be not too large.

20

4.3

The dynamical system

All endogenous variables of the model may be expressed as a function of the income (yt ) and gender roles attitudes (θt ). Then, the sequence {yt , θt }+∞ t=0 determines the dynamic path of the economy. It is described by the two-dimensional, first-order dynamical system given by equations (15) and (19):     yt+1 = ρ(yt , θt )

(21)

,

   θt+1 = η(yt , θt )

where y0 and θ0 are given. An intuitive representation of the dynamics requires the derivation of its phase diagram. The first step is to characterize the yy and θθ loci where yt and θt are respectively in a steady-state.

4.3.1

The yy locus

Let yy be the locus of all pairs (yt , θt ) such that yt is constant: yy ≡ {(yt , θt ) : yt+1 = yt }. The features of this locus depend on parameter configurations. In particular, let us define the two following thresholds on a, the productivity of educational spending:

a ˜≡

τ c[µ + γ(1 − µ)] αβ(2cα − s)

and

a ¯≡

τ c[µ + γ(1 − µ)] . 2αβcα

(22)

The yy locus can be characterized for all values of a. All the results presented in this section are proven in Appendix A.1. If a < a ¯, the yy locus is an horizontal line, yt = 2cα − s, located in the poverty regime. If a ∈ [¯ a, a ˜), the yy locus is constant and equal to yt = 2cα − s in the poverty regime; and is an increasing and concave function of θt , denoted yiyy (θt ), in the interior regime.17 This 17. yrl denotes the value of the income on the locus l ∈ {yy, θθ} in the regime r ∈ {gi, i} (gi refers to the gender inequality regime and i the interior regime).

21

locus is depicted in Figure 2(a). Finally, if a ≥ a ˜, the yy locus is an horizontal line denoted yy by ygi , belonging to the gender inequality regime, while it is represented by the function

yiyy (θt ) inside the interior regime. This last case is drawn in Figure 2(b). yt

yt

y˜(θt )

y˜(θt )

yiyy (θt )

yy ygi

y˜(1) 2c − s

yy

yiyy (θt )

yy

yy

y˜(1) yy

α

0

1

θt

0

(a) a ¯≤a
1

θt

(b) a ≥ a ˜ Fig. 2. The yy locus

In the poverty and gender inequality regimes, girls do not receive any education. Therefore, the stationary value of income is independent of the level of the norm. Conversely, in the interior regime, both boys and girls are educated. A rise in θt induces an increase in the overall weight attached to education by parents, thus mechanically implying a higher investment in education. It follows that the stationary value of parental income (yiyy (θt )) is an increasing function of θt . The positive relationship between yiyy (θt ) and θt is also an indirect consequence of the concavity of the human capital production function. In fact, when θt raises the educational gap reduces, meaning that education expenditures are somewhat transferred from boys (whose marginal productivity of education is relatively low, since they already have a relatively large human capital endowment) to girls (whose marginal productivity is relatively high). Therefore, the aggregate productivity is enhanced as well as the stationary level of human capital.18 18. Due to the presence of this second channel, model’s results hold under the alternative specification of the utility function mentioned in footnote 12.

22

4.3.2

The θθ locus

Let θθ be the locus of all pairs (yt , θt ) such that θt is constant: θθ ≡ {(yt , θt ) : θt+1 = θt }. This locus is depicted in Figure 3 (see Appendix A.2 for a technical analysis). The level of yt

y˜(θt )

θθ

θθ

θθ ygi (θt )

y˜(1)

0

cα −s cα

κ
1−γ

1

θt

Fig. 3. The θθ locus

yt affects the shape of this locus through its effect on the relative labor supply of men and women. In the poverty regime, this relative labor supply is exogenous and independent of yt , so that the θθ locus consists in a vertical line. It is worth noticing that, since unskilled men earn more than unskilled women, men always supply more labor than women in this regime. Hence, the stationary norm favors boys (its value is lower than one). In the gender inequality regime, a higher yt means more education for boys, while girls remain uneducated. Then, the relative productivity of men on the labor market, and consequently their relative labor supply, grows with yt . Hence, there exists a negative relationship between the stationary value of θt and the level of income yt . In the interior regime, boys and girls are both educated so that the initial productivity gap, due to physical strength differences, vanishes. Then, the stationary level of θt corresponds to the egalitarian situation, and the θθ locus consists in the vertical line where θt = 1.

23

4.4

Steady-state equilibria

¯ The steady-state equilibria of the dynamical system (21) are defined as those pairs {¯ y , θ} such that the yy and the θθ loci intersect. Based on the properties of both loci (Figures 2 and 3), we can claim the following: Proposition 1 Under condition (20), a threshold a ˆ (higher than a ˜) exists, such that the dynamical system (21) exhibits the following properties: i if a < a ¯, there exists a unique globally stable equilibrium that belongs to the poverty regime; ii if a ∈ [¯ a, a ˜), there exist two locally stable equilibria, one located in the poverty regime and the other in the interior regime; iii if a ∈ [˜ a, a ˆ), there exist two locally stable equilibria, one located in the gender inequality regime and the other in the interior regime; iv if a ≥ a ˆ, there exists a unique globally stable equilibrium that belongs to the interior regime.

Proof. See Appendix A.3 The four possible configurations listed in Proposition 1 are depicted in Figure 4.19 For a < a ¯, the productivity of educational spending is low. Then, a large level of income is required to start educating children (˜ y (1) is high). In that case, the dynamical system exhibits a unique and stable equilibrium in the poverty regime. We refer to this as a poverty trap. In this trap, norms about gender roles are fully shaped by the physical strength differential between men and women. 19. The yy and θθ loci are depicted in bold lines. The motion arrows show how state variables evolve off the loci.

24

yt

yt

y˜(θt )

θθ

y˜(θt )

θθ

yy

θθ

y˜(1)

0

1

yy b

θt

0

1

θt

(b) a ¯≤a
(a) a < a ¯ yt

y˜(θt )

θθ

y˜(θt )

θθ yy

yy

b

b

θθ

yy

b

y˜(1) yy b

yt

θθ

yy

θθ

b

y˜(1)

y˜(1)

0

1

θt

0

(c) a ˜≤a
1

θt

(d) a ≥ a ˆ Fig. 4. Phase diagrams

For a ∈ [¯ a, a ˜), the poverty trap coexists with an equilibrium placed in the interior regime, namely the high-output equilibrium. The later is characterized by the equality between men and women (θt = 1). Since there are multiple equilibria, initial conditions determine long-term situations. In particular, a country initially too poor and/or too inegalitarian will fall into the poverty trap while a richer/less egalitarian country will converge towards the high-output equilibrium. For a ∈ [˜ a, a ˆ), the poverty trap disappears whereas a high-output equilibrium coexists with an equilibrium that belongs to the gender inequality regime. The later is called inegalitarian equilibrium since only men receive education. Consequently, this equilibrium

25

is characterized by a more inegalitarian norm compared to the poverty trap. Indeed, initial differences in physical strength are reinforced by the growing educational gender gap. This configuration is more deeply discussed in section 4.5. For a ≥ a ˆ, thanks to sufficiently high returns to education, the high-output equilibrium is unique and globally stable.

4.5

Social norms and the gender inequality trap

As mentioned in the Introduction, the World Bank highlights the existence of gender inequality traps and emphasizes the role played by cultural norms in the persistence of such kind of traps. Our model allows for a theoretical assessment of this phenomenon. In fact, as illustrated in Figure 5, multiple equilibria may arise and initial conditions do matter: if initially the norm is too inegalitarian, the economy is caught into a trap where the level of development is low and gender inequalities are high (the inegalitarian equilibrium). However, if the norm is more egalitarian, the economy may end up with the high-output equilibrium.20 yt

y˜(θt )

θθ yy b

θθ yy b

y˜(1) 0

1

θt

Fig. 5. Multiple equilibria

The emergence of gender inequality traps stems from the complementarity between 20. As discussed in Sections 4.1 and 4.2, our two state variables evolve monotonically over time when they belong to the gender inequality regime or the interior regime. Thus, the trajectories implied by the phase diagram approximate the actual dynamic path, despite the fact that the dynamical system is discrete.

26

the inegalitarian nature of the norm and the educational gender gap. Since, at early stages of development, the norm is inegalitarian, parents only invest in boys’ education. It strengthens initial gender inequalities and the norm becomes even more discriminatory. Consequently, it becomes harder to escape from the gender inequality regime. The idea that cultural factors may explain the long-term persistence of gender inequality has been backed-up by some empirical studies. In particular, Inglehart and Norris (2003) point out that societies based on religious norms are most likely to exhibit a strong gender gap and a low acceptance of gender equality. In the same spirit, Dollar and Gatti (1999) conclude that gender differences in educational achievement partly result from religious preferences. In a recent article, Self and Grabowski (2009) study the relative impact of malleable institutions (like the rule of law or the government effectiveness) and nonmalleable institutions (as cultural and religious practices) on gender inequalities. They find that both kinds of institutions have an impact on the level of equality between genders, and conclude that policy reform alone is not likely to achieve gender equality: cultural changes are also important. We share this conclusion but, since we consider cultural norms as endogenous, we believe that cultural practices may also be malleable. In compliance with Dollar and Gatti’s results, the model also shows that the inegalitarian equilibrium is characterized by relatively low development levels. Indeed, the education of women would foster the accumulation of human capital and then would generate further economic developments. The reduction of educational costs as well as the increase in the productivity of education, are obviously relevant means to escape from the gender inequality trap. However, let us remark that these changes may also have negative effects on gender equality. Indeed, since at the inegalitarian equilibrium only boys receive education, a better access to school or a greater productivity of skills will benefit only to men and then increase the gender pro-

27

ductivity gap. This effect is highlighted in a recent empirical study by Oostendorp (2009). The latter shows that, in developing countries, FDI net inflows targeted towards high-skill occupations have a widening impact on the gender wage gap. Indeed, women being less educated, they have few accesses to such kind of occupations and thus benefit less from the positive productivity shock generated by FDI inflows. In the model, an increase in the productivity of education will also benefit to women if and only if it is large enough to destabilize the inegalitarian equilibrium (see Proposition 1). The fact that, the trap emerges through the reinforcing effect of endogenous cultural norms allows us to underline original means to escape from it. In particular, we consider that cultural norms are shaped by the observation of labor supply behaviors within the local population. Accordingly, it is reasonable to think that information about men and women behaviors outside the local population may change gender role attitudes. Several ethnographic and anthropological studies, focusing on Indian’s villages, suggest that television is an important channel of information about lifestyle outside the village (Jensen and Oster 2009 and references therein). For instance, and central to our concern here, many female characters in series diffused by cable television work outside the home. In a recent paper Jensen and Oster (2009) empirically estimate the impact of the introduction of cable television on the status of women in rural India. They conclude to an improvement of women’s relative status and, in particular, to a lower son preference. In our framework, the spread of such kind of program may instill new beliefs, according to which the role of women is more easily associated with work on the market and then translate in an exogenous increase of θt . If this increase is high enough, the gender inequality trap may be destabilized, thus inducing a convergence towards the high-output equilibrium.

28

4.6

The full transition and the U-shaped pattern of the relative status of women

This section focuses on the full transition path from the poverty equilibrium to the highoutput one. Let us consider an economy starting from the situation represented in Figure 4(a): in such a case, it remains trapped in the low equilibrium. In fact, parents do not engage in educational expenditure and individuals are only endowed with their basic human capital. Since women earn less than men on the labor market, their labor supply is lower and the norm is inegalitarian (θt < 1). Let us now consider a sufficiently high productivity shock on a (or a permanent technological progress) such that the relevant dynamical system is now described by Figure 6. In this case, the high-output equilibrium is globally stable and the economy converges towards it. Along the convergence path, the economy passes successively through the gender inequality regime and the interior regime. At early stages of development, only boys receive education: the educational gender gap grows. Later on, the accumulation of men’s human capital allows the economy to step out from the gender inequality regime and parents start educating their daughter. The educational gender gap is progressively filled and the economy finally reaches the high-output equilibrium. Thus, yt

y˜(θt ) b

yy

θθ

θθ

b

y˜(1) 0

1

θt

Fig. 6. The full transition from the poverty trap to the high-output equilibrium

thanks to multiple development regimes, our model is able to replicate the non-monotonic 29

relationship between women labor supply and economic development, as stressed out by Boserup (1970) among others (see Section 2.1). As discussed in Introduction, one of the most striking illustration of this pattern is the U-shaped relationship existing between the female labor force participation and the economic development. Consistently with the mechanisms proposed in our model, Mammen and Paxson (2000) insist on the crucial role played by social norms and education to explain this relationship (see quotation in Section 2.1). In the model, inegalitarian social norms imply that, at early stages of development, only men are educated. It follows that the relative opportunities of men on the labor market rise and women labor supply falls. However, as soon as the economy reached a sufficiently high level of income, girls are educated too and the productivity gap between genders is progressively filled. Consequently, female labor force participation increases.21 The model leads to similar conclusions from a cross-country perspective. In fact, its dynamics allow for potentially three steady-states, each one associated with a development stage (see Proposition 1). The poverty trap, characterized by a low level of development, is located on the decreasing part of the U pattern. At the inegalitarian equilibrium, the level of development is higher and women labor supply is lower (at the bottom of the U curve). Finally, the high-output equilibrium corresponds to the right portion of the U.

5

Discussion

This section discusses the assumption according to which there exists an initial gender wage gap that disappears as soon as women receive education. The existence of a gender 21. Let us underline that, the initial increase of women’s housework boosts the relative productivity of men’s housework. This effect could induce a fall in men’s labor participation during first development stages. However, it should be offset by the sharp increase in the relative productivity of men on the market. In later stages of development (rising part of the U), a slight reduction of men’s labor supply may occur due to the decrease of their relative productivity on the market.

30

difference for low development levels explains why the norm initially favors boys and why gender equality is not achieved in pre-industrial societies. Indeed, without any differences between men and women, the norm becomes progressively fully egalitarian in an economy trapped in the poverty regime. The reduction of this initial productivity gap throughout the industrialization process seems to be a reasonable assumption. It captures the fact that the spread of the service sector, and the underlying increase in skills requirement, makes the role played by physical strength in workers’ productivity less and less important. From a theoretical perspective, the disappearance of this initial productivity gap implies that the high-output equilibrium is more egalitarian than the poverty trap. Indeed, in this two situations men and women receive the same human capital endowments and potential gender disparities only stem from biological factors or discriminatory policies. Finally, let us underline that, the hypothesis according to which the gap vanishes as soon as eft > 0 is made for ease of presentation. In particular, would have obtained similar results if we had considered a progressive fall of the gap. In the model, the existence of this initial gender gap is justified by a physical strength differential between men and women. Nevertheless, some other stories may reinforce the physical strength argument. The two following compelling explanations are successively addressed: (i) the existence of some kinds of discrimination against women working at manual labor, and (ii) the role played by technological improvements in the fall of women’s comparative advantage in housework. Social barriers against women as manual workers. The fact that women may suffer from a stigma when they choose manual works is well documented by Goldin (1995). In addition, she argues that education gives access for women to white-collar jobs in which there is no more stigma. In her discussion about Indian and Pakistan cases, Boserup (1970)

31

also argues that Public opinion makes a sharp distinction between work in home industries and ‘literate work’ which are regarded as respectable occupations, and factory work which is not regarded as respectable for women. (p. 103) Such a stigma, reducing women’s incentives to work in unskilled occupations would play a similar role than physical strength differences. The evolution of relative productivity at home. In the present article, we have chosen to model the initial gender gap as an advantage for men on the market. A symmetric story, yielding similar results, would assume that the gap comes from an ‘advantage’ for women to work at home. In line with Albanesi and Olivetti (2007) we could have considered that an infant good is produced, exclusively using wife’s time and that this time of production is decreasing with some technological innovations.22 Under the usual assumption that the technological level is rising in the level of human capital, the incompressible time devoted by women to the production of the infant good would decrease during the process of development and the initial gap be partially filled.

6

Conclusion

This article proposes a theory of the relationship between gender inequality and economic development in which cultural norms about gender roles are endogenous. On the one hand, those cultural norms determine the education sharing between boys and girls; on the other hand, they are shaped by the relative female labor supply and then, by the gender educational gap. This two-way causality between the inegalitarian nature of the norm and the gender gap in education allows us to obtain some new and empirically relevant results. 22. The time of production of the infant good embodies feeding time, pregnancy, childbirth and recovery. Technological progresses which reduce this time may be regarded as progress in medical knowledge and obstetric practices or the use of infant feeding instead of breast feeding.

32

First of all, we are able to replicate the U-shaped relationship between the relative status of women and the development process. During first stages of development, cultural norms are inegalitarian and men are favored in the process of human capital accumulation: gender inequality grows. Then, if the economy reaches a sufficiently high development level, parents start educating girls and the gender gap is progressively filled. In addition, the model assesses the role played by cultural norms in the emergence of high gender inequality and low development traps. Indeed, an economy could never reach the necessary income level that would enable the education of girls. In that case, the economy remains trapped in a situation where only men are educated and where the aggregate level of human capital is consequently low. A natural extension of the model would be to introduce fertility decisions. As the share of housework time between men and women in the present framework, the fertility level may depend on the gender wage gap (Galor and Weil 1996). Since, fertility is time consuming for women, qualitative results of the model would be slightly affected. However, such an extension would allow to replicate the demographic transition pattern.

A

Appendices

A.1

Analysis of the yy locus

Poverty regime If yt < y˜(1), neither boys nor girls are educated, then yt+1 = yt = 2cα −s. As a consequence, the yy locus consists in the horizontal line: yt = 2cα − s. This stationary locus exists in the poverty regime only if y˜(1) > 2cα − s. Using (14), this condition holds if: a<

τ c[µ + γ(1 − µ)] =a ˜. αβ(2cα − s)

33

(A.1)

Gender inequality regime From (15), the locus of stationary value of yt corresponds to the solution of the following equation: yt = cα − s + χ(0)(ayt + τ c)α ≡ ρgi (yt ).

(A.2)

Since ρgi (.) is increasing and concave and ρgi (0) > 0, this solution is unique and denoted yy yy by ygi . Hence, the yy locus consists in the horizontal line: yt = ygi . yy This stationary locus exists in the gender inequality regime only if ygi ≥ y˜(1). The yy value of a which ensures the equality between y˜(1) and ygi is the solution of the equation

y˜(1) = ρgi (˜ y (1)) with ρgi (˜ y (1)) = 2cα − s. It directly follows that this value of a is a ˜, yy defined in (A.1). Since ygi is increasing in a and y˜(1) decreasing in a, the yy locus exists

in the gender inequality regime only for a ≥ a ˜.

Interior regime From (15), the yy locus corresponds to the solution of the equation:

yt = χ(θt )(1 + θtα )(ayt + 2τ c)α ≡ ρi (yt , θt ).

(A.3)

Since ρi is increasing and concave in yt and ρi (0, θt ) > 0, this solution is unique, denoted yiyy (θt ). Moreover, yiyy (θt ) is increasing and concave in θt . To prove this statement, let us note that (A.3) can be rewritten as:

f (yt ) ≡

yt = χ(θt )(1 + θtα ) ≡ g(θt ). (ayt + 2τ c)α

34

(A.4)

The differentiation of (A.4) with respect to yt and θt yields:

dyt g ′ (θt ) = ′ . dθt f (yt )

(A.5)

It is straightforward that f ′ (yt ) > 0. Then:

sign



dyt dθt



= sign{g ′ (θt )}

(A.6)

and  (µ + γ(1 − µ))θtα−1 + αβ(θtα−1 − 1) , g (θt ) = αχ(θt ) µ + γ(1 − µ) + αβ(1 + θt ) ′



(A.7)

which is positive since θt ∈ [0, 1]. Moreover, g ′ (θt ) decreases in θt . Hence, in the plan (yt , θt ), the yy locus is described by the upward sloping curve yiyy (θt ). To ensure that this stationary locus exists in the interior regime, it is sufficient to state that yiyy (1) ≥ y˜(1). The value of a which ensures the equality between y˜(1) and yiyy (1) is the solution of the equation y˜(1) = ρi (˜ y (1), 1) with ρi (˜ y (1), 1) = 2cα . This solution is:

a=

τ c[µ + γ(1 − µ)] ≡a ¯. 2αβcα

(A.8)

Since yiyy (θt ) is increasing in a and y˜(1) decreasing in a, the yy locus exists in the interior regime only for a ≥ a ¯. Finally, let us underline that a ¯
A.2

Analysis of the θθ locus

Poverty regime From (19), it is straightforward that, in the plan (yt , θt ), the θθ is a vertical where:

θt =



cα − s cα 35



κ 1−γ

.

(A.9)

Gender inequality regime From (19), the evolution of θt in this regime is given by:

∆θt = (1 − σ)

(

cα − s χ(0)(ayt + τ c)α



κ 1−γ

)

.

(A.10)

τc . a

(A.11)

− θt

It follows, the set of values of yt such that ∆θt = 0: 1

θθ ygi (θt )

=

τ (cα − s) α [µ + γ(1 − µ) + β] 1−γ ακ

−

aαβθt

θθ (θ ) = θθ ([(cα −s)/cα ]κ/(1−γ) ) = y ˜(1) and limθt →0 ygi It is decreasing and convex in θt , with ygi t

+∞. θθ (θ ) is From (19), the level of θt at the point of intersection between y˜(θt ) and ygi t

obtained by resolving the equation:

cα − s θt = χ(0)(a˜ y (θt ) + τ c)α 



κ 1−γ

=



cα − s cα



κ 1−γ

ακ

θt1−γ .

(A.12)

θθ (θ ) crosses y ˜(θt ) only once in θˆ such that: We can state that ygi t

θˆ =



cα − s cα



κ 1−γ−ακ

∈ (0, 1).

(A.13)

Interior regime From (19), it is straightforward that, in the plan (yt , θt ), the θθ locus is a where:

θt = 1.

36

(A.14)

A.3

Proof of Proposition 1

This proof is in three steps. First, Lemma 1 and 2 discuss the stability of the yy and θθ loci. Then, the threshold a ˆ is defined and it is shown that if a ∈ [˜ a, a ˆ], the two loci cross only once in the gender inequality regime while if a > a ˆ they do not cross in the gender inequality regime. Finally, Proposition 1 is proven. Lemma 1 Stability of the yy locus. For a given value of θt : i if yt < y˜(1), yt+1 equals to 2cα − s; yy ii if yt ∈ [˜ y (1), y˜(θt )), for yt < (>)ygi , ∆yt > (<)0;

iii if yt ≥ y˜(θt ), for yt < (>)yiyy (θt ), ∆yt > (<)0. Proof. i. Directly derived from (15); ii. In this configuration, yt+1 = ρgi (yt ) (defined in yy (A.2)) and ygi is the solution of the equation yt = ρgi (yt ). From the concavity of ρgi (yt ) yy and since ρgi (0) > 0, yt+1 > (<)yt if yt < (>)ygi ; iii. In this configuration, yt+1 = ρi (yt , θt )

(defined in (A.3)) and yiyy (θt ) is the solution of the equation yt = ρi (yt , θt ). Since ρi (yt , θt ) is concave with respect to yt and ρi (0, θt ) > 0, yt+1 > (<)yt if yt < (>)yiyy (θt ). Lemma 2 Stability of the θθ locus. For a given value of yt : i if yt < y˜(1), for θt < (>)[(cα − s)/cα ]κ/(1−γ) , ∆θt > (<)0; θθ (θ ), ∆θ > (<)0; ii if yt ∈ [˜ y (1), y˜(θt )), for yt < (>)ygi t t

iii if yt ≥ y˜(θt ), for θt < 1, ∆θt > 0. Proof. i. Directly derived from (19); ii. In this configuration, the expression of ∆θt is given θθ (θ ). It follows by (A.10), it is a decreasing function of yt ; Moreover, ∆θt = 0 for yt = ygi t ακ

θθ (θ ). iii. In this configuration, ∆θ = (1 − σ)θ [θ 1−γ that ∆θt > (<)0 if yt < (>)ygi t t t t

Under condition (20), ακ/(1 − γ) < 1, hence ∆θt > 0 if θt < 1. 37

−1

− θt ].

yy θθ (θ ) cross at the same Let us now define a ˆ as the value of a such that y˜(θt ), ygi and ygi t θθ (θ ) and y point. The value of θt at the crossing point between ygi ˜(θt ) is θˆ (defined in t yy and y˜(θt ) (A.13)), it is independent of a. The value of θt at the crossing point between ygi

is solution of the equation:

y˜(θt ) = cα − s + χ(0)(a˜ y (θt ) + τ c)α = 2cα − s,

(A.15)

which yields after some computations:

τc αβ[θt (cα − s) + θt1−α cα ] = . a µ + γ(1 − µ) + αβ(1 − θt )

(A.16)

yy ˜ It follows that the value of θt , denoted θ(a), such that ygi = y˜(θt ) is decreasing in a.

˜ ˜ Moreover, lima→0 θ(a) = +∞ and lima→+∞ θ(a) = 0. Consequently, there exists a unique yy θθ (θ ) cross at the same point. This value, denoted a ˆ value of a such that y˜(θt ), ygi and ygi t

˜ = θ. ˆ If a ∈ [˜ ˜ > θ, ˆ thus y yy and y θθ (θt ) cross once is solution of the equation θ(a) a, a ˆ), θ(a) gi gi in the gender inequality regime (configuration depicted in Figure 4c). Conversely, if a ≥ a ˆ, ˜ ˆ thus y yy and y θθ (θt ) do not cross in the gender inequality regime (configuration θ(a) ≤ θ, gi gi depicted in Figure 4d). Using those results, Proposition 1 is now proven: i For a < a ¯, the yy locus consists only in an horizontal line, that belongs to the poverty regime. Since the θθ locus, in this regime, consists in a vertical line, it is straightforward that the two loci cross only once and that this intersection is located in the poverty regime. The result of global stability directly comes from Lemmas 1 and 2. ii For a ∈ [¯ a, a ˜), the yy locus is composed by an horizontal line in the poverty regime

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and the increasing function yiyy (θt ) in the interior regime. In the poverty regime, the θθ locus consists in a vertical line and then crosses the yy locus once. In the interior regime, the θθ locus is described by θt = 1 and then also crosses the yy locus once. The results of local stability of these two equilibria directly come from Lemmas 1 and 2. iii For a ∈ [˜ a, a ˆ), as discussed above, the θθ locus and the yy locus cross once in the interior regime. In addition, since a ≥ a ˜, the yy locus no longer belong to the poverty regime. However, since a < a ˆ, the two loci cross once in the gender inequality regime. The results of local stability of these two equilibria directly come from Lemmas 1 and 2. iv For a ≥ a ˆ, the two loci do not intersect in the gender inequality regime nor in the poverty regime and cross only once in the interior regime. Hence, the unique steadystate is in this regime. The result of global stability directly comes from Lemmas 1 and 2.

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