Macroeconomic costs of gender gaps: the case of Mexico

1.1 Setup description

The economy we consider has two sectors (market and household) that produce a homogeneous good, as well as a continuum of agents, indexed by their skill level x, who own one unit of time. Talent here should be interpreted more broadly than in Lucas (1978)[26] Lucas Jr., R. E., 1978. On the Size Distribution of Business Firms. The Bell Journal of Economics 9(2), 508-523.
or Cuberes and Teignier (2016)[11] Cuberes, D., and Teignier, M., 2016. “Aggregate Costs of Gender Gaps in the Labor Market: A Quantitative Estimate.” Journal of Human Capital, vol. 10, no. 1.
since now it not only affects the entrepreneurs’ profits, but also the workers’ earnings.²In what follows we will refer to an entrepreneur as someone who works as either an employer or a self-employed. We assume the economy is closed, with an exogenous workforce of size 𝑃. Skill-adjusted labor and capital are supplied to firms by the consumers, in exchange for a wage rate per unit of skill, 𝑤, and a capital rental rate, 𝑟, respectively. These inputs are then combined by firms to produce a unique, homogeneous consumption good. The stock of capital has a steady-state value and, hence, its marginal product is equal to the depreciation rate plus the intertemporal discount factor.

Men choose to become either firm worker in the market sector, who earn the equilibrium wage rate 𝑤 times their skill level 𝑥, or entrepreneurs, who earn the profits generated by the firm they manage in the market sector. Women can also become workers or entrepreneurs, but they also have the option of producing in the household sector. As in Lucas (1978)[26] Lucas Jr., R. E., 1978. On the Size Distribution of Business Firms. The Bell Journal of Economics 9(2), 508-523.
and Buera and Shin (2011)[5] Buera, F. J., Kaboski, J. P., Shin, Y., 2011. “Finance and Development: A Tale of Two Sectors.” American Economic Review 101 (5), 1964.
, the production function of an employer is given by

where $x$ denotes the talent or productivity level of the employer, $n\left(x\right)$ is the units of skill-adjusted labor hired by the employer, $k\left(x\right)$ is the units of capital rented by the employer, and $y\left(x\right)$ represents the units of output produced. The parameter $\eta \in \left(\mathrm{0,1}\right)$ measures the span of control of entrepreneurs and, since it is smaller than one, entrepreneurial technology involves an element of diminishing returns. Since the price of the homogeneous good is normalized to one, employers’ profits are equal to $\pi \left(x\right)=y\left(x\right)-rk\left(x\right)-wn\left(x\right)$ .

On the other hand, an agent with talent $x$ who chooses to become self-employed in the market sector operates a technology given by

where $\tilde{k}\left(x\right)$ denotes the units of capital used and $\tilde{y}\left(x\right)$ the units of output produced. $\tilde{n}\left(x\right)=x$ are the skill-adjusted labor units that the self-employed agents work in his or her own firm.³The consumption good produced by the self-employed and the capital they use is the same as the one in the employers’ problem. However, it is convenient to denote them y and k to clarify the exposition. The parameter $\tau$ , which is calibrated to match the aggregate share of self-employed workers, captures the fact that self-employed agents must spend some time on management tasks. Self-employed profits are equal to $\tilde{\pi }\left(x\right)=\tilde{y}\left(x\right)-r\tilde{k}\left(x\right)$ .

Finally, women can also produce in the household sector, operating the following technology:

where $k_h$ denotes the units of capital rented to produce in the household sector and $n_h$ the units of time allocated to the household sector. Note that this production function can be seen as the perfect substitute of the one in equation (1), with the productivity parameters $A$ and $B$ being independent of the agent talent. Women choose $k_h$ and $n_h$ in order to maximize their total earnings, which are given by their market-sector plus their household sector earnings.⁴Arguably this is a unitary approach to the problem in the sense that a household in this model is effectively composed of only one person who can either be a man or a woman. A more realistic but complicated approach would recognize the importance of intra-household decisions as in Chiappori (1997). We leave this promising avenue for further research. Specifically, when the opportunity cost of time is their market wage $wx$ , women choose to allocate their unit of time in the household sector when $\frac{A}{B}<\frac{r}{wx}$ , and they choose to allocate it to the market otherwise.⁵As explained in Appendix A, depending on the parameter values, women choosing to work at home may still want to rent some capital because their time endowment is limited. At the same time, there may be a group of women who allocate part of their time to the household sector and part of their time to the market sector. Under this household production function, changes in the home technology parameter $A$ (which can be interpreted as an increase in the availability of home appliances or the consumer durable goods revolution mentioned in Greenwood et al., 2005[20] Greenwood, J., A. Seshadri, and M. Yorukoglu, 2005. “Engines of Liberation. Review of Economic Studies.” 72: 109-33.
) lead to a rise of female labor participation, as in the model by Greenwood et al. (2005)[20] Greenwood, J., A. Seshadri, and M. Yorukoglu, 2005. “Engines of Liberation. Review of Economic Studies.” 72: 109-33.
which is empirically assessed by Cavalcanti and Tavares (2008)[6] Cavalcanti, T., and Tavares, J., 2008. “Assessing the Engines of Liberation: Home Appliances and Female Labor Force Participation.” The Review of Economics and Statistics, 90(1): 81-88.
.

1.2 Frictionless Equilibrium

In equilibrium, employers choose the units of labor and capital they hire in order to maximize their current profits, denoted by ${\pi }_e$ ; while self-employed workers choose the units of capital to rent in order to maximize their profits, denoted by ${\pi }_s$ . Market workers earn labor compensation equal to $wx$ . Women also choose the units of capital to rent for the household-sector production and the fraction of their time they want to allocate to this sector. If they choose to become full-time household workers, they earn an income denoted by ${\pi }_h^{00}$ , while if they choose to become part-time household workers, they earn an income denoted by ${\pi }_h^{01}$ , which includes market-sector earnings plus household-sector earnings.

The first plot of Figure 2 displays the payoff of the three market occupations at each talent level and shows the optimal occupational choices in equilibrium for men. Men with the highest skill level (those with talent above $z_2$ )become employers, whereas those with intermediate skill levels become self-employed. Finally, men with a level of talent lower than $z_1$ become market workers. The second plot of Figure 2 displays the slightly more complicated occupational map for women. As it was the case for men, women with talent above $z_2$ become employers, whereas those with talent between $z_1$ and $z_2$ choose to be self-employed. Women work in the market sector if their talent is between $z_0^f$ and $z_1$ . Women with talent below $z_0^f$ allocate their time to the household sector production, either part time (between $z_{00}^f$ and $z_0^f$ ) or full time (below $z_{00}^f$ ).⁶To be precise, h00 and h01 are defined here as the household production profits by household workers relative to market workers, who may also choose to engage in household production but using only capital.

In this economy, aggregate (market) production is the sum of output by male employers and male self-employed, as well as output by female employers and female self-employed:

where $\Gamma \left(x\right)$ denotes the talent cumulative density function, which, again, it is assumed to be the same for men and women. The first term inside the bracket represents the production by male and female employers, whereas the second is the corresponding term for the self-employed.

Total production in the economy, $Y_T$ , is the sum of market output ( $Y$ ) and household output, $Y_h$ .

$Y_h$ is equal to household production by full-time household workers, $y_h^{00}$ , plus household production by part-time household workers, $y_h^{01}$ , plus household production by female market workers, $y_h^1$ (who use some capital in the household sector in order to produce there):

1.3 Introducing gender gaps into the framework

The model assumes that women are identical to men in terms of their innate skills, but they face exogenous constraints in their market-sector occupational choice. These frictions may reflect discrimination, or other demand factors, but they might also reflect differences in optimal choices of women, or other supply factors. In this sense, our estimated effects should be interpreted as the result of all the factors that make women behave differently than men in the labor market, including the presence of social norms.

The first constraint we impose is that females face a probability $\mu$ of being “allowed” to be an employer and a probability $1-\mu$ of being excluded from employership.⁷Again, this constraint may represent either demand barriers, supply choices, or a combination of the two. Out of the group of women not allowed to be employers, some have the possibility of becoming self-employed while the rest are also excluded from self-employment. In particular, women excluded from employership have a probability ${\mu }_o$ of being allowed to be self-employed and a probability $\left(1-{\mu }_o\right)$ of not being allowed to be self-employed.

As a result, a fraction $\left(1-\mu \right)\left(1-{\mu }_o\right)$ of women is shut out from entrepreneurship, i.e. both employership and self-employment can only become workers. Appendix B shows a graphical representation of the occupational choice of women taking the just described constraints into account.⁸Note that, in this setup, we are not allowing for the possibility of women being excluded from self-employment but not from employership, since we think that whichever are the barriers women face to become self-employed, they should apply even more strongly to become an employer. In terms of the parameters of the model, if =1, then the value of o does not affect the occupational choices of women. Finally, the third fraction we introduce is that only a fraction $\lambda$ of women is allowed to participate in the labor market, while a fraction $\left(1-\lambda \right)$ of randomly selected women is excluded from all the possible occupations in the labor market.⁹We say that women excluded from the labor force are randomly selected because the talent of these excluded women is drawn from the same distribution as the rest of the population. In this setup, women who do not participate in the formal labor market become full-time workers in the household sector and, hence, the estimated aggregate income loss due to the $\lambda$ gender gap depends on the difference between the market participants earnings and the household-sector earnings.

The effects of the entrepreneurship gaps, $\mu$ and ${\mu }_o$ , are illustrated in Figure 3 for the case without part-time workers. When some women are excluded from entrepreneurship, the supply of market workers increases, leading to a fall in the wage rate and a rise in the employers’ profit function. This makes both $z_1$ and $z_2$ fall, implying a lower average talent of entrepreneurs and a lower firm productivity. The capital stock adjusts downwards to keep its marginal product equal to the depreciation rate plus the intertemporal discount factor. Moreover, in the case of women, there is a rise in $z_0^f$ , implying that the number of workers in the market-sector falls and the number of workers in the household sector rises. As a result of all these effects, the market-sector output decreases. If part-time work was also considered, the fall in wages would lead to a rise of both $z_{00}^f$ and $z_0^f$ , implying also a fall in female market labor.

The effects of the labor force participation gap, $\lambda$ , are more straightforward. When some women get excluded from the market sector, they become household-sector workers, leading to a fall in the market-sector labor and a rise in the home-sector labor.¹⁰Admittedly, in the case of the Mexican economy, a significant fraction of women not working in the formal labor market do it in the informal one. We could reinterpret our household sector as including not only the output generated at home but also the one in the informal sector. We leave for future research the addition of a well-defined informal sector into the framework. As before, the capital stock adjusts downwards to keep its marginal product equal to the depreciation rate plus the intertemporal discount factor. These effects clearly reduce total output from the market sector, but they are likely to slightly increase output per worker because the household-sector capital demand falls and, thus, the market sector capital-to-labor ratio increases.

2.1 Data

Data on employment status and working hours is obtained from the National Survey of Occupation and Employment (ENOE) made public by the National Statistics Institute (INEGI). The National Statistics Institute (INEGI) compiles satellite accounts on non-remunerated household work. The data show that this type of work amounted in 2017 to 5.1 billion pesos (0.25 billion US dollars), or about 23% of Mexico’s GDP. The estimation of this satellite account is based on two inputs: 1) A measure of time spent on unpaid work, which is approximated through the number of hours of unpaid work and the identification of the individuals who perform it (both indicators being taken from the National Time Use Survey); and 2) The cost per hour spent on unpaid care and domestic work, estimated from the National Occupation and Employment Survey, providing gross values from average earnings by economic activity, according to the North American Industry Classification System (NAICS).

The activities included for this estimation are those household’s activities defined as productive, if can be delegated to somebody else or provide a product or service that can be exchanged in the market, like provision of food, cleaning and maintenance of a dwelling, cleaning and care of clothes and shoes, shopping and household management, care and support, community services and volunteer work.¹¹Due to the very nature of the non-remunerated activities, some degree of measurement error should be assumed.

2.2 Talent Distribution and Model Parametrization

To simulate the model, we use a Pareto function for the talent distribution, as in Lucas (1978)[26] Lucas Jr., R. E., 1978. On the Size Distribution of Business Firms. The Bell Journal of Economics 9(2), 508-523.
and Buera et al. (2011)[5] Buera, F. J., Kaboski, J. P., Shin, Y., 2011. “Finance and Development: A Tale of Two Sectors.” American Economic Review 101 (5), 1964.
. The cumulative distribution of talent is, hence, given by

where $\rho , B>0$ .

The values used for the model parameters are shown in Table 1. The parameter $B$ of the talent distribution is normalized to 1, while the parameter $\eta$ is set to 0.79 as in Buera and Shin (2011[5] Buera, F. J., Kaboski, J. P., Shin, Y., 2011. “Finance and Development: A Tale of Two Sectors.” American Economic Review 101 (5), 1964.
).¹² Buera and Shin (2011) choose to match the top five percent income share in the U.S., which is 30%. This is a reasonable approximation given that the top earners are entrepreneurs both in the model and the U.S. data. The capital-output elasticity parameter $\alpha$ is set to 0.114 in order to match the 30% capital income share observed in the data.¹³Entrepreneurs’ profits are considered capital income, thus we set αη+1- equal to 30%. The parameters $\left(\rho , \tau , A, B\right)$ are estimated to match four different moments of the Mexican data. First, the fraction of employers in the labor force (which is 4.4%), second, the fraction of self-employed workers in the labor force (which is 22.3%), third, the household sector productivity relative to the market-sector one (which is 0.26), and, fourth, the gap between the share of female part-time workers and the share of male part-time workers (which is 12.4%).Compared to the United States case, we observe that Mexico has a slightly larger share of employers (4.4% vs. 3.6%), a much larger share of self-employed workers (22.3% vs 6.5%) and a similar relative household productivity and part-time gap.

The values of the country-specific gender gaps $\left(\mu ,{\mu }_o,\lambda \right)$ are computed to simultaneously match the female-to-male ratio of employers, self-employed workers, and labor market participation in each country. After matching these moments, we obtain that the value of the employership gender gap, $1-\mu$ , is 0.6 (very similar to the U.S. one), while the self-employment gender gap, $\left(1-\mu \right)\left(1-{\mu }_o\right)$ is equal to 0.08 (compared to 0.41 in the U.S.), and the labor force gender gap, $1-\lambda$ , is 0.44 (compared to 0.14 in the U.S.).

2.3 Numerical Results

The numerical results for Mexico are summarized in Table 2, which shows that gender gaps lead to a fall in total output (market plus household) is much larger in Mexico than in the United States (22% vs. 12.7%). In Mexico, there is an almost five-fold rise in household sector production due to the presence of gender gaps (487% in Mexico vs. 6.5%) which only compensates partially the fact that the fall in market output is much larger in Mexico than in the United States (26.5% vs. 17.3%). The effects of the entrepreneurship gender gaps on market output, however, are greater in the United States, the reason being that the fall in female market sector hours due to the entrepreneurship gender gaps is significantly smaller in Mexico (0.11% vs 11.9%).

Comparing we also the results with the ones for Southern Europe in Cuberes and Teignier (2018)[12] Cuberes, D. & Teignier, M. (2018). Macroeconomic costs of gender gaps in a model with entrepreneurship and household production. The B.E. Journal of Macroeconomics, 18(1).
we find that the fall in total output due all gender gaps is 16%, much smaller than in Mexico. The rise in household output is of about 4% only, but the fall in market output is of 20.5%, 6 percentage points lower than in the case of Mexico.

Beyond the mechanical effect of labor force participation gap, the intuition behind the loss in aggregate output is as follows. When a woman with high management skills happens to be "banned'' from becoming an employer, this will result in a decrease of the demand of workers and possibly an increase in the supply of workers, which leads to a reduction of the equilibrium wage rate (as well as the capital rental rate for similar reasons). The model then implies that a less skilled agent will now find it profitable to become an employer and will take her position as manager of the firm. As a result, the firm will be less productive and due to the nature of the span-of-control technology, also smaller. In equilibrium, aggregate productivity, wages, profits, and output will be lower because of this restriction.