Impact of Gender Empowerment on Gender-based Wage Inequality and Unemployment: A Theoretical Analysis *

Abstract

The article develops a three-sector competitive general equilibrium model with an agricultural sector, a manufacturing sector and a non-traded service sector and three factors: male unskilled labour, female unskilled labour and skilled labour. The article shows how sociocultural barriers impact female labour force participation and male–female wage disparity in the presence of male unemployment in the labour market in a developing economy. The analysis finds a rise in gender wage inequality and a decrease in male unskilled labour unemployment when the female labour force rises due to a fall in sociocultural barriers. Finally, we show that the Gini coefficient of income inequality of unskilled labour and gender-based wage inequality move in opposite directions due to the rise in female labour participation in the workforce and the fall in the male unemployment rate.

JEL: D50, J31, J61

Keywords

Gender-based wage inequality female labour supply general equilibrium social liberalisation unemployment Gini coefficient

I. Introduction

Gender inequality is one of the vibrant issues in lower-middle income countries. For a better socio-economic set-up, gender inequality needs to be reduced. Women are generally underpaid, according to the literature, and that is the main reason for gender-based wage inequality (e.g. Blau & Kahn, 1994; Cotton, 1988; Neumark, 1988; Reimer & Schrodere, 2006). According to Kindom and Unni (2001), women in the workforce experience significant wage discrimination.

The total labour force can be divided into two parts: skilled and unskilled. This unskilled labour force can be further subdivided into male and female labour forces. As women are busy with homemaking activities, they cannot take part in the labour market in most of the lower-middle income countries. Women have played a more significant role in unpaid work like child care, cooking, cleaning and other activities that require household function than men. Most of the time, their contribution to society is unaccounted. When male income is not sufficient for a family, females have to take part in the labour market. In 2011–2012, 35.3 per cent of all rural females and 46.1 per cent of all urban females in India were reported to be attending to domestic duties. In rural India, women’s participation in domestic duties increased from 51.8% to 59.7% in 2004–2005 to 2011–2012 (Ghai, 2018) and at the same time, women’s participation in urban areas increased from 45.6% to 48% (Ghai, 2021) in domestic duties. All over the world, women provide more unpaid care than men do. In numerical terms, globally, women perform 76.4% of the total amount of unpaid work. Women devote 3.2 times as much time as men do to unpaid caregiving on average (Charmes, 2019). Another recent study by Lakshman (2017) found that women in many countries are criticised because when they work in paid work, they ignore the unpaid household work. Women from low-income households who work are often employed in hazardous, poorly paid jobs while also carrying a heavy load of unpaid caregiving duties (Sengupta & Sachdeva, 2017). Both their mental and physical health are negatively impacted by this dual strain.

There are various empirical studies to show the effect of different globalisation-related parameters on gender-based wage inequality.¹ But, apart from globalisation, gender empowerment also has a significant impact on gender-based wage inequality. Gender empowerment helps to develop a positive social attitude towards women participating in various economic activities. This gender empowerment refers to releasing women from the socio-economic constraints of dependency. Thus, the participation of women in the labour force is variable. Rahman and Islam (2013), Klasen and Pieters (2012) and Goldin (1994) empirically show this variability of women’s participation in the labour force. They find a U-shaped relationship between female labour participation and income. With the rise in revenue, women initially withdraw themselves from the labour force, but after a particular growth stage, female labour participation rises with income. At the initial stage, female labour forces are primarily unskilled and illiterate and participate only in home-based activities. However, after a particular growth stage, they become skilled and perform skilled jobs like male workers. Sanghi et al. (2015) discuss the formative transformation of the economy that has a multi-aspect effect on FLFP.

There is theoretical literature in the field of gender-based wage inequality. Chaudhuri et al. (2019), Mukhopadhyay (2018), Poddar and Chaudhuri (2016) and Mukhopadhyay and Chaudhuri (2013) are some of them. All of the mentioned papers have developed three-sector general equilibrium models. Chaudhuri et al. (2019) consider a non-traded service sector that provides service to skilled families. The paper analyses the impact of foreign direct investment (FDI) and credit market reform on gender-based wage inequality. Mukhopadhyay (2018) analyses the effects of trade liberalisation on gender-based wage inequality and female workforce participation by allowing for the interaction between changes in relative wages, intra-household bargaining strength and social norms. Mukhopadhyay and Chaudhuri (2013) consider the female labour-based export sector to examine the impact of different economic liberalisation policies on gender-based wage inequality. Poddar and Chaudhuri (2016) consider factor market distortions by introducing efficient wages for male and female labour and analysing the impact of economic liberalisation on gender-based wage inequality. However, none of the papers presents unemployment in their paper.² Like gender-based wage inequality, unemployment is a crucial problem in lower-middle income countries. Also, the above paper considers a supply-based economy with female labour supply as given. We consider the female labour supply as an endogenous variable determined optimally by maximising the welfare of unskilled families. Also, none of the papers analyses the impact of gender empowerment on gender-based wage inequality and unemployment of unskilled labour.

We introduce unemployment through the Harris and Todaro (1970) framework, where the agricultural sector offers a flexible wage, but the manufacturing sector provides a fixed wage. The wage rate in the manufacturing sector is higher than in the farming sector. Workers are migrating from the agricultural industry to the manufacturing sector searching for jobs. Due to fixed wages, the manufacturing sector can only provide jobs to some migrated labour, and they remain unemployed. Migration equilibrium is where the expected wage in the manufacturing sector is equal to the actual wage of the agricultural industry. In literature, many competitive general theoretical models use the concept of Harris–Todaro’s (1970) framework to explain unemployment, which includes Chaudhuri (2004), Chaudhuri (2008), Chaudhuri and Banerjee (2010), etc.

In our model, gender empowerment helps to develop a positive social attitude towards women participating in various economic activities. This gender empowerment refers to releasing women from the socio-economic constraints of dependency. Women’s empowerment in India is a significant condition for the overall development of society. We develop a static competitive general equilibrium model of a small open economy with two traded sectors and a non-traded final sector. We examine the consequences of an exogenous increase in the female labour supply due to gender empowerment on the gender-based wage inequality and unemployment of unskilled labour.

We develop a three-sector competitive general equilibrium model with a traditional agricultural sector, advanced manufacturing sector and a non-traded service sector and three factors: male unskilled labour, female unskilled labour and skilled labour. The paper shows how sociocultural barriers impact FLFP and male–female wage disparity in the presence of male unemployment in the labour market in a developing economy. The analysis finds a rise in gender wage inequality and a decrease in male unskilled labour unemployment when the female labour force rises due to a fall in sociocultural barriers. Also, we show that the Gini coefficient income inequality of unskilled labour and gender-based wage inequality move in opposite directions due to the rise in female labour participation in the workforce. Gini coefficient as a measure of inequality among male and female unskilled labour was not introduced earlier.

In this article, the source of the disparity of wages between male and female unskilled labour is the difference in their productivity. Wage disparity of labour arises due to market discrimination, where there are different payment rules for males and females.³ In this article, we have not considered this market discrimination angle but the productivity angle. The productivity of low-skilled jobs depends on the physical strength of workers. And men generally possess more physical strength than women.

On the other hand, for highly skilled jobs, differences in productivity arise due to differences in training, education and experience. Though training, education and experience may vary across genders, the percentage of female labour engaged in low-skilled sector(s), having the maximum discrimination, is significantly larger than that in the high-skilled industry(s) where the pay gap is relatively low. Due to the above reason, we have considered wage differences only for male and female unskilled labour and not for male and female skilled labour. Also, the wage difference between skilled male–female and unskilled male–female labour will complicate our analysis. The same treatment is also done in Chaudhuri et al. (2019).

The rest of the article is organised as follows. Section II describes the model and in Section III, we examine the consequences of an exogenous increase in female labour supply due to gender empowerment on the gender-based wage inequality and unemployment of unskilled labour. Concluding remarks are given in Section IV.

II. The Model

We consider a small open developing economy with two traded good sectors and a non-traded final good sector. Sector 1 is the agricultural sector and sector 2 is the manufacturing sector. The non-traded unskilled female labour-specific sector is denoted by sector N. This N sector provides domestic services to the well-off families in the economy. Housemate services and babysitting are some services provided by this unskilled female labour belonging to the poor section of the society. Sector 1 or the agricultural sector uses unskilled male and female labour, and it is complementary between male labour and female labour.⁴ Sector 2 or the manufacturing sector uses male unskilled labour and skilled labour. Unskilled wage rates in sector 1 and N and skilled wage rate in the manufacturing sector are fully flexible. Wage rate of unskilled male labour in the manufacturing sector is constant. So there exists unemployment of unskilled male labour in the manufacturing sector explained by Harris–Todaro (1970) framework.⁵

We can use the following notation:

$a_{i j} =$ Technological co-efficient, i = M, F, S and J = 1, 2, N

$P_{i} =$ Price of the i^th good, i = 1, 2, N

$W_{i} =$ Wage rate of unskilled male and female labour, i = M, F

$W_{S} =$ Wage rate of skilled male or female labour

$X_{i} =$ Output level of the i^th sector, i = 1, 2, N

$L =$ Number of homogenous unskilled working families

$\bar{L} =$ Unskilled male population

$L_{F} =$ Unskilled female population engaged in market-based wage-earning activity

$\bar{S}$ = Number of homogeneous skilled families

$λ_{j i} =$ Proportion of the j^th input employed in the i^th sector, i = 1, 2, N; j = M, F, S

$S_{j k}^{i} =$ The degree of substitution between factors j and k in the i^th sector, i = 1, 2, N

$θ_{j i} =$ Distributive share of the j^th input in the i^th sector, i = 1, 2, N

U = Each unskilled household utility

V = Each skilled household utility

$1 - l_{F}$ = Proportion of female unskilled members engaged in market-based wage-earning activity

$α, δ =$ The share of physical consumption of the commodity 1, 2 respectively for unskilled workers

$γ =$ Household’s valuation of female work within homestead

$β^{1}, β^{2}, β^{N}$ = The share of the consumption of the commodity 1, 2, N respectively for the skilled labour

We have considered that there are L numbers of homogeneous unskilled working families consisting of one male labour and one female labour each. Each female member has two options. They can spend one unit of time in either wage-earning activities in the market economy or non-market activities within the household. Male members may be employed in the agriculture sector or manufacturing sector. Some of the male members are unemployed. This unemployment is explained by Harris–Todaro (1970) model. Each household maximised their utility, which is a positive function of their consumption of physical commodities and a negative function of market-based activity of female households. Thus, each household’s problem is stated as follows.

Max U = B x_{1}^{α} x_{2}^{δ} {l_{F}}^{γ}; α + δ + γ = 1

(1)

Subject to, P_{1} x_{1} + P_{2} x_{2} = W_{M} (1 - v) + W_{F} (1 - l_{F})

(2)

Equation (1) represents the utility function of a single decision-making family and it obeys all standard properties. $α + δ$ stands for the share of physical consumption of two goods in a household’s total spending and $γ$ household’s valuation of female work within homestead.

Some female members are engaged in market-based wage-earning activity, (1 – $l_{F})$ and some in unpaid household work within the homestead, $l_{F}$ . So average income of unskilled female member is $W_{F} (1 - l_{F})$ . Due to imperfection in the male labour market, there is some unemployed male labour in the economy. Employed male labours are getting the expected wage rate $W_{M}$ .⁶ Unemployed male labours are not getting any wage. $v$ denotes the rate of unemployment of unskilled labour. So the expected or average income of the male unskilled labour is $W_{M} (1 - v)$ . Therefore, the average family earnings of unskilled households is $W_{M} (1 - v) + (1 - l_{F}) W_{F} .$ This also represents budget constraint of unskilled households.

The optimisation exercise yields the following female labour supply by each household. The following female labour supply function is derived from Equations (1) and (2).⁷

1 - l_{F}^{*} = α + δ - \frac{W_{M} (1 - v) γ}{W_{F}}

(3)

From Equation (3), $(1 - l_{F}^{*})$ varies inversely with the male wage rate $W_{M}$ . This relationship indicates that with an increase in the male wage rate, family income increases, and thus the household chooses less female work. Second, $(1 - l_{F}^{*})$ varies directly with the female wage rate. Third, a decrease in $γ$ raises female labour supply participation, indicating a more liberal attitude of the families and the society. Lastly, $(1 - l_{F}^{*})$ varies positively with $v$ . Past financial crises suggest that men’s employment is more affected than women and male-dominated sectors like construction and manufacturing of the economy are affected more, while women’s working hours increased (Alon et al., 2020; Hupkau & Petrongolo, 2020).

L_{F} = L (1 - l_{F}^{*}) (W_{M}, W_{F}, γ, v)

(4)

In this model, we consider sector N as a non-traded service sector, and in order to close the model, we need a market clearing condition for the non-traded commodity. Here, we assume that $\bar{S}$ number of homogeneous skilled families, and each family consists of one male and one female skilled worker and their skilled wage rate is $W_{S}$ . Each skilled family consumes traded commodities and non-traded services.⁸

Skilled labour maximised the utility function:

Max V = C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N}}; [β_{1} + β_{2} + β_{N} = 1]

(5)

S ubject to, P_{1} X_{1}^{1} + P_{2} X_{2}^{1} + P_{N} X_{N}^{1} = 2 W_{S}

(6)

The optimisation exercise gives the following equation;⁹ which is also the demand function of non-traded services provided by female unskilled labour.

X_{N}^{D} = X_{N}^{D} (P_{N}, W_{S})

(7)

where, $X_{N}^{D}$ represents demand for non-traded services.

Finally, the market clearing equation of non-traded services is given by the following equation:

X_{N} = X_{N}^{D}

(8)

The General Equilibrium Analysis

The general equilibrium is represented by the following set of equations which is based on Jones (1965, 1971).

W_{M} a_{M 1} + W_{F} a_{F 1} = P_{1}

(9)

W_{M}^{*} a_{M 2} + W_{S} a_{S 2} = P_{2}

(10)

W_{F} a_{F N} = P_{N}

(11)

\frac{W_{M}^{*} a_{M 2} X_{2}}{a_{M 2} X_{2} + v \bar{L}} = W_{M}

(12)

a_{M 1} X_{1} + a_{M 2} X_{2} + v \bar{L} = \bar{L}

(13)

a_{F 1} X_{1} + a_{F N} X_{N} = L_{F} (W_{F}, W_{M}, γ, v)

(14)

a_{S 2} X_{2} = 2 \bar{S}

(15)

Here, Equations (9)–(11) represent the profit-maximising conditions of competitive firms in sectors 1, 2 and N, respectively. Equation (12) implies the Harris–Todaro migration equilibrium condition. Here, $a_{M 2} X_{2}$ is the total employment of labour in sector 2, $(a_{M 2} X_{2} + v \bar{L})$ is the total labour supply in sector 2, and $\frac{a_{M 2} X_{2}}{a_{M 2} X_{2} + v \bar{L}}$ is the probability of getting a job in sector 2. Thus, the left-hand side of Equation (12) is the expected earning of the unskilled labour in sector 2, and the right-hand side of Equation (12), that is, $W_{M}$ is the wage rate of the unskilled labour in sector 1. Until these two sides become equal, migration of labour continues. Equations (13)–(15) represent the full-employment equilibrium conditions in skilled labour and unskilled male and female labour, respectively.

In this model, $P_{1} and P_{2}$ are exogenously given, due to small open-economy assumptions. There are nine endogenous variables in the model— $L_{F}, X_{N}, W_{M}, W_{F}, W_{S}, P_{N}, v, X_{1}, X_{2}$ with nine independent equations. The production structure does not possess the decomposition property.

The working of the model is described as follows. From Equation (10), we determine $W_{S}$ for the given $W_{M}^{*}$ . From Equation (11), we solve for $W_{F}$ as a function of $P_{N}$ . Then, from Equation (9), we obtain $W_{M}$ as a function of $P_{N}$ . Now, from Equations (12)–(13) and (15), we solve for $v, X_{1} and X_{2}$ as a function of $P_{N}$ . Finally, from demand–supply Equation (8), we can solve for the equilibrium value of $P_{N}$ .

III. Comparative Static Exercise

In this section of our model, we examine the consequences of an exogenous increase in female labour supply due to gender empowerment on the gender-based wage inequality and unemployment of unskilled labour. As we consider a small open economy, the prices of all the traded sectors are given, and we do not consider any change in prices in this section. Thus, $P_{1} = P_{2} = 0$ .

Taking the total differentiation of Equations (8), (9) and (11)–(12) and of the basic model, we obtain the following results:

\hat{W_{M}} = - \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}}

(16)

\hat{W_{F}} = \frac{\hat{P_{N}}}{θ_{F N}}

(17)

\hat{v} = \hat{L_{U}} = \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{1}{v} + 1) \hat{P_{N}}

(18)

E_{γ} \hat{γ} = (e_{P_{N}} - A) \hat{P_{N}}

(19)

From the above equations, we can say that as $e_{P_{N}} < 0, E_{γ} < 0 and A > 0$ then as $γ$ falls, $P_{N}$ decreases. Hence, $W_{M}$ increases and $W_{F}$ decreases and vice versa. Moreover, v decreases as $γ$ falls.

Impact of Gender Empowerment on Wage Inequality

Gender empowerment helps to develop a positive social attitude towards women participating in various economic activities. In our model, it is reflected by a fall in $γ$ . Given that female labour supply is negatively related to $γ$ ; implies fall in $γ$ raises $L_{F}$ .¹⁰ That indicates an increase in female labour supply causes $W_{F}$ to fall. Female labour is mobile between sector 1 and sector N, thus both the sectors expand in terms of output. An expansion of output in sector N causes $P_{N}$ to fall which raises the male wage rate. On the other hand, as sector 1 expands, it demands more unskilled male labour. This excess demand for unskilled male labour causes $W_{M}$ to rise. Thus, male–female wage gap rises, and we have the following proposition.

Proposition 1. With gender empowerment in terms of an exogenous increase in female labour supply; gender-based wage inequality worsens.

However, from the demand side effect as sector 1 expands, it demands for more unskilled male workers, and this excess demand for unskilled males causes reverse migration from sector 2 to sector 1. So the unemployment of unskilled male labour falls. Hence, an exogenous increase (decreases) in female labour supply lowers (raises) the unemployment of unskilled male labour. From this result, we can draw the following proposition.

Proposition 2. An exogenous increase in female labour supply due to gender empowerment lowers the unemployment of unskilled labour.

So gender empowerment, which raises female labour supply in the economy, lowers the return of existing female labour and the economy. Now, as female and male labour are complementary in the agriculture sector, so the demand for male labour increases with an increase in the supply of female labour. And gender-based wage inequality rises and unemployment of male labour falls.

Gini Coefficient

We consider the Gini coefficient of wage income distribution as a measure of wage income inequality of unskilled families; and this Gini coefficient, denoted by G, is obtained as follows.¹¹

G = \frac{(L v + L l_{f}) \{L (1 - v) + L (1 - l_{f}) Δ\} + L (1 - v) L (1 - l_{f}) (Δ - 1)}{(2 L - 1) \{L (1 - v) + L (1 - l_{f}) Δ\}}

(20)

where

Δ = \frac{W_{M}}{W_{F}}

If there is no unemployment of male unskilled labour then, that is, if $v = 0$ , and also all female unskilled labour is working, that is, if $l_{f} = 0$ then, from Equation (20), we have:

G = \frac{L^{2} (Δ - 1)}{(2 L - 1) (L + L Δ)}

and, from this expression, we have:

\frac{d G}{d Δ} = \frac{2 L}{(2 L - 1) {(1 + Δ)}^{2}} > 0

So the Gini coefficient varies positively with the gender-based wage inequality with full employment of unskilled labour. So with full employment, we may use gender-based wage inequality as a measure of income equality among unskilled labour, but this should be replaced by Gini coefficient with the existence of unemployment.

Using Equation (20), we obtain:¹²

\hat{G} = B \hat{v} + C {\hat{l}}_{f} + E \hat{Δ}

(24)

where,

B = \frac{v [L^{3} {(1 - l_{f})}^{2} Δ + 2 L^{3} (1 - v) (1 - l_{f}) Δ + L^{3} {(1 - v)}^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0

C = \frac{l_{f} [L^{3} {(1 - v_{U})}^{2} + 2 L^{3} (1 - v_{U}) (1 - l_{f}) Δ + L^{3} {(1 - l_{f})}^{2} Δ^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0

E = \frac{Δ [L^{2} (1 - v) (1 - l_{f}) \{L (1 - v) + L (1 - l_{f})\}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0

and $\hat{v}$ , ${\hat{l}}_{f}$ and $\hat{Δ}$ are given by Equations (3), (16), (17) and (18), respectively. Equation (21) implies that the relative rate of change in the degree of income inequality of the working class is explained not only by the relative rate of change in gender-based wage inequality but also by relative rates of change in unemployment rate of unskilled male labour and female unskilled labour participation rate in workforce. Here $B$ , $C$ and $E$ represent elasticities of Gini coefficient of wage income distribution with respect to the unemployment rate in the unskilled labour market, unskilled female labour participation in the labour market and skilled–unskilled relative wage, respectively.

In a full employment model, $v = l_{f} = 0$ ; and these imply that $B = C = 0$ . Also,

E = \frac{2 Δ L^{3}}{G (2 L - 1) {(L + L Δ)}^{2}}

So, in this case, change in inequality is explained by the change in relative wage only.

Now, when $v \neq 0$ and $l_{f} \neq 0$ , then change in Gini coefficient also depends on the change in the unemployment rate of male unskilled labour and female unskilled labour participation rate in the workforce. Due to a rise in gender empowerment, male–female wage inequality increases, but male unemployment falls and female participation rate increases, that is, $l_{f}$ falls. So, with an increase in female participation and fall in male unemployment, the Gini coefficient moves in the opposite direction of wage inequality. So in the absence of full employment, Gini coefficient of unskilled labour and gender-based wage inequality may move in opposite directions under certain conditions.

In this model, due to gender empowerment, female unskilled labour participation rate rises. That raises the effective supply of female labour and thus their wages fall. As male unskilled labour and female unskilled labour are complementary in the agricultural sector, demand for unskilled labour increases and gender-based wage inequality rises. But in the absence of full employment, ratio of male–female wages does not represent a clear picture of income inequality. Due to gender empowerment, female participation in the workforce rises, and male unemployment rate falls. So those who are getting earlier zero wages, now they are able to earn positive income. To get a clear idea about income inequality among unskilled labour, we thus have to use other measures of income inequality. Here, we introduce the Gini coefficient as a measure of income inequality and see that the increase in female participation and fall in male unskilled unemployment improves the Gini coefficient. So, gender empowerment may improve the income inequality among unskilled labour, if these other two forces dominate the rise in gender-based wage inequality.

IV. Conclusion

This article has made a modest attempt to describe the societal movement from conservatism to modernism towards women’s participation in the labour market by developing a static three-sector competitive general equilibrium model of a small open economy where an unskilled working family’s optimising behaviour determines the family supply function of female labour. It has been discovered that the availability of female labour for families depends not only on male and female wages but also on how society views the contribution of women to the economy (social norms).

In this work, we have studied how sociocultural barriers affect FLFP and male–female wage disparity in the presence of male unemployment in the labour market within developing economies. We obtain some interesting results that impact two different manners on gender-based wage inequality and unemployment of unskilled labour. With gender empowerment in terms of an exogenous increase in female labour supply, gender-based wage inequality worsens, but the unemployment of unskilled male labour falls. Finally, we have shown that the Gini coefficient of income inequality among unskilled labour and gender-based wage inequality may move in opposite directions due to the rise in female unskilled labour participation and the fall in male unemployment rate.

However, our model only introduces some essential aspects of reality. We have not considered capital in our model. We also fail to show the effect of gender empowerment on skilled–unskilled wage inequality. We plan to do further research in the future, attempting to remove the above-mentioned significant problems.

Appendix A

Derivation of Equations (4) and (8)

Max U = B x_{1}^{α} x_{2}^{δ} {l_{F}}^{γ}

(A1)

Subject to, P_{1} x_{1} + P_{2} x_{2} = W_{M} (1 - v) + W_{F} (1 - l_{F})

(A2)

Using the Lagrange method, we get:

\begin{matrix} z = B x_{1}^{α} x_{2}^{δ} {l_{F}}^{γ} + λ [W_{M} (1 - v) + (1 - l_{F}) W_{F} - (P_{1} x_{1} + P_{2} x_{2})] \\ \frac{\partial z}{\partial x_{1}} = \frac{1}{P_{1}} α B x_{1}^{α - 1} x_{2}^{δ} {l_{F}}^{γ} = λ \end{matrix}

(A3)

\frac{\partial z}{\partial x_{2}} = \frac{1}{P_{2}} δ B x_{1}^{α} x_{2}^{δ - 1} {l_{F}}^{γ} = λ

(A4)

\frac{\partial z}{\partial l_{F}} = \frac{1}{W_{F}} γ B x_{1}^{α} x_{2}^{δ} {l_{F}}^{γ - 1} = λ

(A5)

\frac{\partial z}{\partial λ} = W_{M} (1 - v) + (1 - l_{F}) W_{F} - P_{1} x_{1} - P_{2} x_{2} = 0

(A6)

From Equation (A6), we have:

(1 - l_{F}) = \frac{P_{1} x_{1} + P_{2} x_{2} - W_{M} (1 - v)}{W_{F}}

(A7)

From Equations (A3) and (A4), we get:

P_{1} x_{1} = \frac{α}{δ} P_{2} x_{2}

From Equations (A4) and (A5), we have:

P_{2} x_{2} = \frac{δ}{γ} W_{F} l_{F}

From Equation (A7), we have:

\begin{matrix} \Rightarrow ({1 - l}_{F}) = \frac{P_{1} x_{1} + P_{2} x_{2} - W_{M} (1 - v)}{W_{F}} \\ \Rightarrow ({1 - l}_{F}) = \frac{\frac{α}{δ} P_{2} x_{2} + P_{2} x_{2} - W_{M} (1 - v)}{W_{F}} \\ \begin{matrix} \begin{matrix} \Rightarrow ({1 - l}_{F}) = \frac{(\frac{α + δ}{δ}) P_{2} x_{2} - W_{M} (1 - v)}{W_{F}} \\ \Rightarrow ({1 - l}_{F}) = \frac{(\frac{α + δ}{δ}) \frac{δ}{γ} W_{F} l_{F} - W_{M} (1 - v)}{W_{F}} \\ \begin{matrix} \Rightarrow ({1 - l}_{F}) = \frac{(1 - l_{F}) W_{F} (α + δ)}{γ W_{F}} - \frac{W_{M} (1 - v)}{W_{F}} \\ \Rightarrow ({1 - l}_{F}) + \frac{α + δ}{γ} ({1 - l}_{F}) = \frac{α + δ}{γ} - \frac{W_{M} (1 - v)}{W_{F}} \\ \Rightarrow \frac{(α + γ + δ)}{γ} ({1 - l}_{F}) = \frac{α + δ}{γ} - \frac{W_{M} (1 - v)}{W_{F}} \end{matrix} \end{matrix} \\ \begin{matrix} \Rightarrow \frac{1}{γ} ({1 - l}_{F}) = \frac{α + δ}{γ} - \frac{W_{M} (1 - v)}{W_{F}} & (A s, α + δ + γ = 1) \end{matrix} \\ \Rightarrow ({1 - l}_{F}) = α + δ - \frac{W_{M} (1 - v) γ}{W_{F}} \end{matrix} \end{matrix}

(A8)

\Rightarrow L_{F} = L (1 - l_{F}) (W_{M}, W_{F}, γ, v)

(A9)

Equations (A8) and (A9) are the same as Equations (3) and (4) in the body of the article.

Appendix B

Derivation of Equation (7)

Skilled labour maximised the utility function.

Max V = C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N}}

(B1)

Subject to, P_{1} X_{1}^{1} + P_{2} X_{2}^{1} + P_{N} X_{N}^{1} = 2 W_{S}

(B2)

Using the Lagrange method, we get:

\begin{matrix} Z = C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N}} + λ [2 W_{S} - (P_{1} X_{1}^{1} + P_{2} X_{2}^{1} + P_{N} X_{N}^{1})] \\ \frac{\partial Z}{\partial X_{1}^{1}} = β_{1} C {(X_{1}^{1})}^{β_{1} - 1} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N}} - λ P_{1} = 0 \\ \Rightarrow \frac{1}{P_{1}} β_{1} C {(X_{1}^{1})}^{β_{1} - 1} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N}} = λ \end{matrix}

(B3)

\begin{matrix} \frac{\partial Z}{\partial X_{2}^{1}} = β_{2} C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2} - 1} {(X_{N}^{1})}^{β_{N}} - λ P_{2} = 0 \\ \Rightarrow \frac{1}{P_{2}} β_{2} C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2} - 1} {(X_{N}^{1})}^{β_{N}} = λ \end{matrix}

(B4)

\begin{matrix} \frac{\partial Z}{\partial X_{N}^{1}} = β_{N} C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N} - 1} - λ P_{N} = 0 \\ \Rightarrow \frac{1}{P_{N}} β_{N} C {(X_{1}^{1})}^{β_{1}} {(X_{2}^{1})}^{β_{2}} {(X_{N}^{1})}^{β_{N} - 1} = λ \end{matrix}

(B5)

\begin{matrix} \frac{\partial z}{\partial λ} = [2 W_{S} - (P_{1} X_{1}^{1} + P_{2} X_{2}^{1} + P_{N} X_{N}^{1})] = 0 \\ \Rightarrow X_{N}^{1} = \frac{2 W_{S} - P_{1} X_{1}^{1} - P_{2} X_{2}^{1}}{P_{N}} \end{matrix}

(B6)

From Equations (B3) and (B4), we get:

P_{1} X_{1}^{1} = \frac{β_{1}}{β_{2}} P_{2} X_{2}^{1}

From Equations (B4) and (B5), we have:

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} P_{2} X_{2}^{1} = \frac{β_{2}}{β_{N}} P_{N} X_{N}^{1} \\ \Rightarrow X_{N}^{1} = \frac{2 W_{S} - P_{1} X_{1}^{1} - P_{2} X_{2}^{1}}{P_{N}} \end{matrix} \\ {\Rightarrow X}_{N}^{1} = \frac{2 W_{S} - [\frac{β_{1}}{β_{2}} P_{2} X_{2}^{1} + P_{2} X_{2}^{1}]}{P_{N}} \\ {\Rightarrow X}_{N}^{1} = \frac{2 W_{S} - [\frac{β_{1}}{β_{2}} + 1] P_{2} X_{2}^{1}}{P_{N}} \end{matrix} \\ {\Rightarrow X}_{N}^{1} = \frac{2 W_{S} - [\frac{β_{1} + β_{2}}{β_{2}}] \frac{β_{2}}{β_{N}} P_{N} X_{N}^{1}}{P_{N}} \\ {\Rightarrow X}_{N}^{1} = \frac{2 W_{S}}{P_{N}} - \frac{β_{1} + β_{2}}{β_{N} P_{N}} P_{N} X_{N}^{1} + \end{matrix} \\ {\Rightarrow X}_{N}^{1} = \frac{2 W_{S}}{P_{N}} - \frac{β_{1} + β_{2}}{β_{N}} X_{N}^{1} \\ {\Rightarrow X}_{N}^{1} + \frac{β_{1} + β_{2}}{β_{N}} X_{N}^{1} = \frac{2 W_{S}}{P_{N}} \end{matrix} \\ \Rightarrow (β_{N} + β_{1} + β_{2}) \frac{X_{N}^{1}}{β_{N}} = \frac{2 W_{S}}{P_{N}} \\ \begin{matrix} \Rightarrow \frac{X_{N}^{1}}{β_{N}} = \frac{2 W_{S}}{P_{N}} & (A s, β_{N} + β_{1} + β_{2} = 1) \end{matrix} \end{matrix} \\ \Rightarrow X_{N}^{1} = \frac{2 W_{S}}{P_{N}} β_{N} \\ X_{N}^{D} = X_{N}^{D} (P_{N}, W_{S}) \end{matrix}

(B7)

Equation (B7) is the same as Equation (7) in the body of the article.

Appendix C

Derivation of Equations (16), (17), (18) and (19)

From Equation (10) with given prices, we get:

\begin{matrix} \hat{W_{M}^{*}} θ_{M 2} + \hat{W_{S}} θ_{S 2} = 0 \\ \begin{matrix} \Rightarrow \hat{W_{S}} θ_{S 2} = 0 & (as W_{M}^{*} is constant) \end{matrix} \\ \Rightarrow \hat{W_{S}} = 0 \end{matrix}

(C1)

From Equation (11), we have:

\begin{matrix} \hat{W_{F}} θ_{F N} = \hat{P_{N}} \\ \Rightarrow \hat{W_{F}} = \frac{\hat{P_{N}}}{θ_{F N}} \end{matrix}

(C2)

Equation (C2) is the same as Equation (17) in the body of the article.

From Equation (9), we get:

\begin{matrix} {\hat{W}}_{M} θ_{M 1} + \hat{W_{F}} θ_{F 1} = 0 \\ \Rightarrow \hat{W_{M}} = - \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}} \end{matrix}

(C3)

Equation (C3) is the same as Equation (16) in the body of the article.

From Equation (12), we get:

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} W_{M}^{*} a_{M 2} X_{2} = W_{M} (a_{M 2} X_{2} + L_{U}) \\ \Rightarrow \hat{W_{M}^{*}} + \hat{a_{M 2}} + \hat{X_{2}} = \hat{W_{M}} + \hat{(a_{M 2} X_{2} + L_{U})} \\ \Rightarrow \hat{W_{M}^{*}} + \hat{a_{M 2}} + \hat{X_{2}} = \frac{d (a_{M 2} X_{2} + L_{U})}{(a_{M 2} X_{2} + L_{U})} + \hat{W_{M}} \end{matrix} \\ \Rightarrow \hat{a_{M 2}} + \hat{X_{2}} = \frac{a_{M 2} X_{2} (\hat{a_{M 2} X_{2})} d L_{U}}{(a_{M 2} X_{2} + L_{U})} + \hat{W_{M}} \\ \Rightarrow \hat{a_{M 2}} + \hat{X_{2}} = \frac{\frac{a_{M 2} X_{2}}{L} (\hat{a_{M 2} X_{2})} + \frac{L_{U} \hat{L_{U}}}{L}}{\frac{a_{M 2} X_{2}}{L} + \frac{L_{U}}{L}} + \hat{W_{M}} \end{matrix} \\ \Rightarrow \hat{a_{M 2}} + \hat{X_{2}} = \frac{λ_{M 2} (\hat{a_{M 2}} + \hat{X_{2}}) + \frac{L_{U}}{L} \hat{L_{U}}}{λ_{M 2} + \frac{L_{U}}{L}} + \hat{W_{M}} \\ {\Rightarrow λ}_{M 2} (\hat{a_{M 2}} + \hat{X_{2}}) + \frac{L_{U}}{L} (\hat{a_{M 2}} + \hat{X_{2}}) = λ_{M 2} (\hat{a_{M 2}} + \hat{X_{2}}) + \frac{L_{U}}{L} \hat{L_{U}} + \hat{W_{M}} (λ_{M 2} + \frac{L_{U}}{L}) \end{matrix} \\ \Rightarrow \frac{L_{U}}{L} (\hat{a_{M 2}} + \hat{X_{2}}) = \frac{L_{U}}{L} \hat{L_{U}} + \hat{W_{M}} (λ_{M 2} + \frac{L_{U}}{L}) \\ \begin{matrix} \Rightarrow \frac{L_{U}}{L} (\hat{X_{2}} - \hat{L_{U}}) = \hat{W_{M}} (λ_{M 2} + \frac{L_{U}}{L}) \\ \Rightarrow (\hat{X_{2}} - \hat{L_{U}}) = \hat{W_{M}} (λ_{M 2} \frac{L}{L_{U}} + 1) \\ \begin{matrix} \Rightarrow \hat{L_{U}} = \hat{X_{2}} - \hat{W_{M}} (λ_{M 2} \frac{L}{L_{U}} + 1) \\ \Rightarrow \hat{L_{U}} = \hat{X_{2}} + \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \end{matrix} \end{matrix} \end{matrix}

(C4)

From Equation (15), we have:

\begin{matrix} λ_{S 2} (\hat{a_{S 2}} + \hat{X_{2}}) = 0 \\ \Rightarrow \hat{X_{2}} = - \hat{a_{S 2}} \\ \begin{matrix} \Rightarrow \hat{X_{2}} = - (ε_{S S}^{2} \hat{W_{S}} + ε_{S M}^{2} \hat{W_{M}^{*}}) \\ \Rightarrow \hat{X_{2}} = 0 \end{matrix} \end{matrix}

(C5)

Putting the value of $\hat{X_{2}}$ in Equation (C4), we get:

\hat{L_{U}} = \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} = \hat{v}

(C6)

Equation (C6) is the same as Equation (18) in the body of the article.

From Equation (13), we get:

\begin{matrix} a_{M 1} X_{1} + a_{M 2} X_{2} + L_{U} = \bar{L} \\ {\Rightarrow λ}_{M 1} \hat{X_{1}} + λ_{M 2} \hat{X_{2}} + λ_{M 1} \hat{a_{M 1}} + λ_{M 2} \hat{a_{M 2}} = - \frac{L_{U} \hat{L_{U}}}{(L - L_{U})} \\ \begin{matrix} {\Rightarrow λ}_{M 1} \hat{X_{1}} + λ_{M 1} (ε_{M M}^{1} \hat{W_{M}} + ε_{M F}^{1} \hat{W_{F}}) = - \frac{L_{U} \hat{L_{U}}}{(L - L_{U})} \\ \Rightarrow \hat{X_{1}} = - \frac{L_{U} \hat{L_{U}}}{λ_{M 1} (L - L_{U})} - ε_{M M}^{1} \hat{W_{M}} - ε_{M F}^{1} \hat{W_{F}} \\ \Rightarrow \hat{X_{1}} = - \frac{L_{U}}{λ_{M 1} (L - L_{U})} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} + ε_{M M}^{1} \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}} - ε_{M F}^{1} \frac{\hat{P_{N}}}{θ_{F N}} \end{matrix} \end{matrix}

(C7)

From Equation (14), we have:

\begin{matrix} λ_{F 1} ({\hat{a}}_{F 1} + {\hat{X}}_{1}) + λ_{F N} ({\hat{a}}_{F N} + {\hat{X}}_{N}) = E_{F} {\hat{W}}_{F} + E_{M} {\hat{W}}_{M} + E_{γ} \hat{γ} + E_{U} {\hat{L}}_{U} \\ {\Rightarrow λ}_{F 1} (ε_{F M}^{1} {\hat{W}}_{M} + ε_{F F}^{1} {\hat{W}}_{F}) + λ_{F 1} {\hat{X}}_{1} + λ_{F N} (ε_{F F}^{N} {\hat{W}}_{F}) + λ_{F N} {\hat{X}}_{N} = E_{F} {\hat{W}}_{F} + E_{M} {\hat{W}}_{M} + E_{γ} \hat{γ} + E_{U} {\hat{L}}_{U} \\ \begin{matrix} {\Rightarrow λ}_{F N} {\hat{X}}_{N} = E_{F} {\hat{W}}_{F} + E_{M} {\hat{W}}_{M} + E_{γ} \hat{γ} + E_{U} {\hat{L}}_{U} - λ_{F 1} ε_{F M}^{1} {\hat{W}}_{M} - λ_{F 1} ε_{F F}^{1} {\hat{W}}_{F} - λ_{F 1} {\hat{X}}_{1} - λ_{F N} ε_{F F}^{N} {\hat{W}}_{F} \\ \Rightarrow {\hat{X}}_{N} = \frac{E_{F}}{λ_{F N}} {\hat{W}}_{F} + \frac{E_{M}}{λ_{F N}} {\hat{W}}_{M} + \frac{E_{γ}}{λ_{F N}} \hat{γ} + \frac{E_{U}}{λ_{F N}} {\hat{L}}_{U} - \frac{λ_{F 1}}{λ_{F N}} ε_{F M}^{1} {\hat{W}}_{M} - \frac{λ_{F 1}}{λ_{F N}} ε_{F F}^{1} {\hat{W}}_{F} - \frac{λ_{F 1}}{λ_{F N}} {\hat{X}}_{1} - ε_{F F}^{N} {\hat{W}}_{F} \\ \begin{matrix} \Rightarrow {\hat{X}}_{N} = (\frac{E_{F}}{λ_{F N}} - ε_{F F}^{N} - \frac{λ_{F 1}}{λ_{F N}} ε_{F F}^{1}) {\hat{W}}_{F} + (\frac{E_{M}}{λ_{F N}} - \frac{λ_{F 1}}{λ_{F N}} ε_{F M}^{1}) {\hat{W}}_{M} + \frac{E_{γ}}{λ_{F N}} \hat{γ} + \frac{E_{U}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} + \frac{λ_{F 1}}{λ_{F N}} \frac{L_{U}}{λ_{M 1} (L - L_{U})} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} - \frac{λ_{F 1}}{λ_{F N}} ε_{M M}^{1} \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}} + \frac{λ_{F 1}}{λ_{F N}} ε_{M F}^{1} \frac{\hat{P_{N}}}{θ_{F N}} \\ \Rightarrow {\hat{X}}_{N} = (\frac{E_{F}}{λ_{F N}} - ε_{F F}^{N} - \frac{λ_{F 1}}{λ_{F N}} ε_{F F}^{1}) \frac{{\hat{P}}_{N}}{θ_{F N}} - (\frac{E_{M}}{λ_{F N}} - \frac{λ_{F 1}}{λ_{F N}} ε_{F M}^{1}) \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} {\hat{P}}_{N} + \frac{E_{γ}}{λ_{F N}} \hat{γ} + \frac{E_{U}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1 θ_{F N}}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} + \frac{{L_{U} θ_{F 1} λ}_{F 1}}{λ_{F N} λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} - ε_{M M}^{1} \frac{λ_{F 1}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1}} \frac{\hat{P_{N}}}{θ_{F N}} + \frac{λ_{F 1}}{λ_{F N} θ_{F N}} ε_{M F}^{1} \hat{P_{N}} \\ \begin{matrix} \Rightarrow {\hat{X}}_{N} = [\frac{1}{θ_{F N}} (\frac{E_{F}}{λ_{F N}} - ε_{F F}^{N} - \frac{λ_{F 1}}{λ_{F N}} ε_{F F}^{1}) - \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} (\frac{E_{M}}{λ_{F N}} - \frac{λ_{F 1}}{λ_{F N}} ε_{F M}^{1}) + \frac{E_{U}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1 θ_{F N}}} (λ_{M 2} \frac{L}{L_{U}} + 1) + \frac{{L_{U} θ_{F 1} λ}_{F 1}}{λ_{F N} λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) - ε_{M M}^{1} \frac{λ_{F 1}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{λ_{F 1}}{λ_{F N} θ_{F N}} ε_{M F}^{1}] \hat{P_{N}} + \frac{E_{γ}}{λ_{F N}} \hat{γ} \\ \Rightarrow {\hat{X}}_{N} = A \hat{P_{N}} + \frac{E_{γ}}{λ_{F N}} \hat{γ} \end{matrix} \end{matrix} \end{matrix} \end{matrix}

(C8)

where, $A > 0$ .

A = [\frac{1}{θ_{F N}} (\frac{E_{F}}{λ_{F N}} - ε_{F F}^{N} - \frac{λ_{F 1}}{λ_{F N}} ε_{F F}^{1}) - \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} (\frac{E_{M}}{λ_{F N}} - \frac{λ_{F 1}}{λ_{F N}} ε_{F M}^{1}) + \frac{E_{U}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1 θ_{F N}}} (λ_{M 2} \frac{L}{L_{U}} + 1) + \frac{{L_{U} θ_{F 1} λ}_{F 1}}{λ_{F N} λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) - ε_{M M}^{1} \frac{λ_{F 1}}{λ_{F N}} \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{λ_{F 1}}{λ_{F N} θ_{F N}} ε_{M F}^{1}]

Now from Equations (C1), (C8) and (8), we have:

\begin{matrix} {\hat{X}}_{N} = e_{W_{S}} {\hat{W}}_{S} + e_{P_{N}} {\hat{P}}_{N} \\ \Rightarrow A \hat{P_{N}} + \frac{E_{γ}}{λ_{F N}} \hat{γ} = e_{W_{S}} {\hat{W}}_{S} + e_{P_{N}} {\hat{P}}_{N} \\ \begin{matrix} \Rightarrow \frac{E_{γ}}{λ_{F N}} \hat{γ} = e_{P_{N}} {\hat{P}}_{N} - A {\hat{P}}_{N} \\ \Rightarrow E_{γ} \hat{γ} = λ_{F N} (e_{P_{N}} - A) {\hat{P}}_{N} \end{matrix} \end{matrix}

(C9)

Equation (C9) is the same as Equation (19) in the body of the article.

Here, $e_{P_{N}} < 0, E_{γ} < 0$ .

From Equation (4), we get:

L_{F} = L . l_{F}^{*} (W_{F}, W_{M}, γ, v)

\begin{matrix} \Rightarrow \hat{L_{F}} = E_{F} {\hat{W}}_{F} + E_{M} \hat{W_{M}} + E_{γ} \hat{γ} + E_{U} \hat{v} \\ \Rightarrow \hat{L_{F}} = E_{F} \frac{\hat{P_{N}}}{θ_{F N}} - E_{M} \frac{θ_{F 1} \hat{P_{N}}}{θ_{M 1} θ_{F M}} + E_{γ} \hat{γ} + E_{U} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) \hat{P_{N}} \\ \Rightarrow \hat{L_{F}} = [\frac{E_{F}}{θ_{F N}} - \frac{E_{M} θ_{F 1}}{θ_{M 1} θ_{F M}} + \frac{E_{U} θ_{F 1}}{θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)] \hat{P_{N}} + E_{γ} \hat{γ} \end{matrix}

(C10)

Putting the value of $E_{γ} \hat{γ}$ in Equation (C8), we get:

\begin{matrix} {\hat{X}}_{N} = A \hat{P_{N}} + \frac{E_{γ}}{λ_{F N}} \hat{γ} \\ \Rightarrow {\hat{X}}_{N} = \frac{λ_{F N} A \hat{P_{N}} + E_{γ} \hat{γ}}{λ_{F N}} \\ \begin{matrix} \Rightarrow {\hat{X}}_{N} = \frac{λ_{F N} A \hat{P_{N}} + e_{P_{N}} {\hat{P}}_{N} - A \hat{P_{N}}}{λ_{F N}} \\ \Rightarrow {\hat{X}}_{N} = \frac{{(λ}_{F N} - 1) A \hat{P_{N}} + e_{P_{N}} {\hat{P}}_{N}}{λ_{F N}} \end{matrix} \end{matrix}

(C11)

Here,

(λ_{F N} - 1) < 0, A > 0, e_{P_{N}} < 0

From Equation (C10), we get:

\begin{matrix} \hat{L_{F}} = (E_{F} \frac{1}{θ_{F N}} - E_{M} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} + E_{U} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)) \hat{P_{N}} - A \hat{P_{N}} + e_{P_{N}} \hat{P_{N}} \\ \Rightarrow \hat{L_{F}} = (E_{F} \frac{1}{θ_{F N}} - E_{M} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} + E_{U} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)) \hat{P_{N}} - \frac{1}{λ_{F N}} (\frac{E_{F}}{θ_{F N}} - \frac{E_{M} θ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{E_{U} θ_{F 1}}{θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)) \hat{P_{N}} - \frac{1}{λ_{F N}} (\frac{1}{θ_{F N}} (- λ_{F N} ε_{F F}^{N} - λ_{F 1} ε_{F F}^{1}) + \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} λ_{F 1} ε_{F M}^{1} + \frac{L_{U} θ_{F 1} λ_{F 1}}{λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) - ε_{M M}^{1} \frac{θ_{F 1} λ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{λ_{F 1}}{θ_{F N}} ε_{M F}^{1}) \hat{P_{N}} + e_{P_{N}} \hat{P_{N}} \\ \Rightarrow \hat{L_{F}} = (1 - \frac{1}{λ_{F N}}) (E_{F} \frac{1}{θ_{F N}} - E_{M} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} + E_{U} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)) \hat{P_{N}} - \frac{1}{λ_{F N}} (\frac{1}{θ_{F N}} (- λ_{F N} ε_{F F}^{N} - λ_{F 1} ε_{F F}^{1}) + \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} λ_{F 1} ε_{F M}^{1} + \frac{L_{U} θ_{F 1} λ_{F 1}}{λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) - ε_{M M}^{1} \frac{θ_{F 1} λ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{λ_{F 1}}{θ_{F N}} ε_{M F}^{1}) \hat{P_{N}} + e_{P_{N}} \hat{P_{N}} \end{matrix}

(C12)

Here, $(1 - \frac{1}{λ_{F N}}) < 0,$

(E_{F} \frac{1}{θ_{F N}} - E_{M} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} + E_{U} \frac{θ_{F 1}}{θ_{M 1}} \frac{1}{θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1)) > 0

\frac{1}{λ_{F N}} (\frac{1}{θ_{F N}} (- λ_{F N} ε_{F F}^{N} - λ_{F 1} ε_{F F}^{1}) + \frac{θ_{F 1}}{θ_{M 1} θ_{F N}} λ_{F 1} ε_{F M}^{1} + \frac{L_{U} θ_{F 1} λ_{F 1}}{λ_{M 1} (L - L_{U}) θ_{M 1} θ_{F N}} (λ_{M 2} \frac{L}{L_{U}} + 1) - ε_{M M}^{1} \frac{θ_{F 1} λ_{F 1}}{θ_{M 1} θ_{F N}} + \frac{λ_{F 1}}{θ_{F N}} ε_{M F}^{1}) > 0

and $e_{P_{N}} > 0$ .

So, from Equations (C9) and (C12), total female labour force is a negative function of social liberalisation.

When,

γ ↓ \Rightarrow P_{N} ↓ \Rightarrow W_{F} ↓ \Rightarrow W_{M} ↑ \Rightarrow W_{S} and X_{2} unchange \Rightarrow L_{U} ↓ \Rightarrow X_{1} ↑ \Rightarrow X_{N} ↑ \Rightarrow L_{F} ↑

Appendix D

Derivation of Equations (20) and (21)

G = \frac{\bar{Δ}}{2 μ}

(D1)

where

\bar{Δ} = \frac{1}{n (n - 1)} \sum_{i = 1}^{n} \sum_{j = 1}^{n} |x_{i} - x_{j}|

and

$μ =$ Mean income.

Here,

\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1}^{n} |x_{i} - x_{j}| = 2 [(L v + L l_{f}) L (1 - v) W_{F} + (L v + L l_{f}) S (1 - l_{f}) W_{M} \\ + L (1 - v) L (1 - l_{f}) (W_{M} - W_{F})] \\ \begin{matrix} \underset{}{\Rightarrow} (L + L) (L + L - 1) \bar{∆} = \\ 2 [(L v + L l_{f}) \{L (1 - v) W_{M} + L (1 - l_{f}) W_{M}\} + L L (1 - v) (1 - l_{f}) (W_{M} - W_{F})] \end{matrix} \end{matrix}

(D2)

and

μ = \frac{[L (1 - v) W_{F} + L (1 - l_{f}) W_{M}]}{(N + S)}

(D3)

Using Equations (D1), (D2) and (D3), we have:

G \begin{matrix} = \frac{(L v + L l_{f}) \{L (1 - v) W_{U} + L (1 - l_{f}) W_{M}\} + L L (1 - v) (1 - l_{f}) (W_{M} - W_{F})}{(N + S - 1) N (1 - v) W_{F} + S (1 - l_{f}) W_{M}} \\ \underset{}{\Rightarrow} G = \frac{(L v + L l_{f}) \{L (1 - v) + L (1 - l_{f}) \frac{W_{M}}{W_{F}}\} + L L (1 - v) (1 - l_{f}) (\frac{W_{M}}{W_{F}} - 1)}{(L + L - 1) \{L (1 - v) + L (1 - l_{f}) \frac{W_{M}}{W_{F}}\}} \\ \underset{}{\Rightarrow} G = \frac{(L v + L l_{f}) \{L (1 - v) + L (1 - l_{f}) ∆\} + L (1 - v) L (1 - l_{f}) (∆ - 1)}{(2 L - 1) \{L (1 - v) + L (1 - l_{f}) ∆\}} \end{matrix}

(D4)

where

Δ = \frac{W_{M}}{W_{F}}

Equation (D4) is the same as Equation (20) in the body of the article.

Now, differentiating Equation (B.24), we have:

\begin{matrix} (L + L - 1) \{L (1 - v) + L (1 - l_{f}) ∆\} d G - G (L + L - 1) L d v - G (L + L - 1) L ∆ d l_{f} \\ G (L + L - 1) S (1 - l_{f}) d ∆ = [- (L v + L l_{f}) L + \{L (1 - v) + L (1 - l_{f}) ∆\} L - L L (1 - l_{f}) (∆ - 1)] d v + [- (L v + L l_{f}) S ∆ + \{L (1 - v) + L (1 - l_{f}) ∆\} L - L L (1 - v) (∆ - 1)] d l_{f} + [(L v + L l_{f}) L (1 - v_{S}) + L L (1 - v) (1 - l_{f})] d ∆ \end{matrix}

(D5)

Now, using Equations (D4) and (D5), we obtain:

\begin{matrix} (L + L - 1) {\{L (1 - v) + L (1 - l_{f}) ∆\}}^{2} d G = [\{- L^{2} v - L S l_{f} + L^{2} (1 - v) + L L (1 - l_{f}) ∆ - L L (1 - v_{S}) (∆ - 1)\} \{L (1 - v) + L (1 - l_{f}) ∆\} + \{L^{2} v + L L l_{f}\} \{L (1 - v) + L (1 - l_{f}) ∆\} + L^{2} L (1 - v) (1 - l_{f}) (∆ - 1)] d v + [\{- L L ∆ v - L^{2} ∆ l_{f} + L L (1 - v) + L^{2} (1 - l_{f}) ∆ - L L (1 - v) (∆ - 1)\} \{L (1 - v) + L (1 - l_{f}) ∆\} + \{L L ∆ v + L^{2} ∆ l_{f}\} \{L (1 - v) + L (1 - l_{f}) ∆\} + L L^{2} (1 - v) (1 - l_{f}) ∆ (∆ - 1)] d l_{f} + [\{(L v + L l_{f}) L (1 - l_{f}) + L L (1 - v) (1 - l_{f})\} \{L (1 - v) + L (1 - l_{f}) ∆\} - (L v + L l_{f}) \{L (1 - v) + L (1 - l_{f}) ∆\} L (1 - v_{S}) - L L^{2} (1 - v) {(1 - l_{f})}^{2} (∆ - 1)] d ∆ \\ \begin{matrix} \underset{}{\Rightarrow} d G = \frac{[L^{3} {(1 - l_{f})}^{2} ∆ + 2 L^{3} (1 - v) (1 - l_{f}) ∆ + L^{3} {(1 - v)}^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) ∆\}}^{2}} d v_{U} + \frac{[L^{3} {(1 - v_{U})}^{2} + 2 L^{3} (1 - v_{U}) (1 - l_{f}) ∆ + L^{3} {(1 - l_{f})}^{2} ∆^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) ∆\}}^{2}} d v + \frac{[L^{2} (1 - v) (1 - l_{f}) \{L (1 - v) + L (1 - l_{f})\}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) ∆\}}^{2}} d ∆ \\ \underset{}{\Rightarrow} \hat{G} = B \hat{v} + C {\hat{l}}_{f} + E \hat{∆} \end{matrix} \end{matrix}

(D6)

where,

\begin{matrix} B = \frac{v [L^{3} {(1 - l_{f})}^{2} Δ + 2 L^{3} (1 - v) (1 - l_{f}) Δ + L^{3} {(1 - v)}^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0 \\ C = \frac{l_{f} [L^{3} {(1 - v_{U})}^{2} + 2 L^{3} (1 - v_{U}) (1 - l_{f}) Δ + L^{3} {(1 - l_{f})}^{2} Δ^{2}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0 \\ E = \frac{Δ [L^{2} (1 - v) (1 - l_{f}) \{L (1 - v) + L (1 - l_{f})\}]}{G (2 L - 1) {\{L (1 - v) + L (1 - l_{f}) Δ\}}^{2}} > 0 \end{matrix}

Equation (D5) is the same as Equation (21) in the body of the article.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Notes

ORCID iD

Priya Brata Dutta

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