# Gender Inequality— Elasticities of Labor Supply

#### REFERENCE: The Quiet Revolution that Transformed Women's Employment, Education, and Family»

The relative economic position of women and men is still quite unequal, yet has undergone great changes over the course of the last century. Economic inequality between women and men has many dimensions, including the following.There is, first, inequality of pay for the same occupation, maybe due to discrimination, which we will discuss more in the next chapter. There are, second, differences in the distribution of men and women across occupations, maybe due to social norms and aspirations and due to the workings of the educational system. And there are unequal intra-household divisions of labor. Traditional divisions of roles would often require women to be primarily in charge of unpaid reproductive labor — housekeeping, taking care of children, the elderly, and the sick, etc. — while men would be in charge of paid work in the labor market.

While this description of a traditional model might be correct in some "typical" sense, there are great differences across social classes and over time, shaped by market forces, social provisions by the state, changing social norms, and other factors. These differences and this historical evolution are the subject of Goldin (2006). Claudia Goldin focuses on the changing prevalence of women's participation in the labor market, and the changing career trajectories of women. She structures her historical description in terms of two key elasticities of women's labor supply, the income elasticity and the substitution elasticity.

## 1. Elasticities of labor supply

Economists like to express causal effects and other relationships in terms of elasticities. Elasticities are unit-less magnitudes. Suppose $$L$$ is a function of $$Y$$. Elasticities answer questions such as "By what percentage does $$L$$ increase (or decrease) if $$Y$$ increases by 1%?" The elasticity $$\epsilon$$ is formally defined as $$\epsilon = \frac{\partial \log L}{\partial \log Y } =\frac{\partial L}{\partial Y} \cdot \frac{Y}{L}.$$

Let $$L$$ be the labor supply of a woman, that is the amount of hours or weeks of paid labor. Let $$w$$ be the hourly wage that she receives or would receive in the market, and let $$Y$$ be household income, which includes partners' earnings and unearned income (from capital ownership and other sources).

Wages $$w$$ have greatly increased over time, and vary across social classes / levels of education. These changes of wages are an important explanatory factor for changing patterns of women's participation in wage labor. We can think of the effect of a change in wages on labor supply as being composed of two parts.

The first part is due to an effect of incomes. As households get richer, they are able to afford more. To the extent that it is considered desirable that (married) women do not work for wages, as per traditional role models (as prevalent early in the 20th century in the United States), an increase in incomes might lead to a decrease in women's labor force participation, both across social classes and over time. Richer households can afford women's staying home more easily. The second part is due to an effect of relative prices. When wages are higher, then the return to paid work relative to unpaid work is higher, creating an incentive to switch from the latter to the former. An increase in wages would suggest higher labor force participation for women, leading to increased labor force participation over time, going in the opposite direction of the income effect.

This decomposition into two parts can be made formal using two elasticities. The first is the income elasticity. It measures the percentage change of labor supply for a 1% change of (household) income, $$\epsilon = \frac{\partial \log L}{\partial \log Y }.$$ Total household income $$Y$$ depends not only on own-earnings, but also on partners' earnings. If we assume that household decisions are made jointly then both these sources of income affect women's labor supply decisions in the same manner. If that is the case then we can learn about $$\epsilon$$ by looking at the effect of partners' earnings on $$L$$, since partner's earnings do not themselves affect the incentives (relative prices) of women's labor supply.

The second effect is measured by the substitution elasticity $$\eta^s$$. We can only measure $$\eta^s$$ indirectly. $$\eta^s$$ is the effect of women's wages on their labor supply, after we subtracted the income effect. Put differently, increasing women's wages $$w$$ has a total effect $$\eta$$, which is the sum of substitution and income effect:

###### (1)
$$$$\eta = \frac{\partial \log L}{\partial \log w}= \frac{\partial \log L}{\partial \log Y } \frac{\partial \log Y}{\partial \log w}+ \eta^s$$$$ To calculate the effect $$\alpha$$ of wages on household income,
###### (2)
$$$$\alpha = \frac{\partial \log Y}{\partial \log w},$$$$ we just need to do some accounting, which we can do once we know household income and women's earnings. A 1% increase in $$w$$ leads to a 1% increase of earnings $$w L$$, and we get $$\alpha = \frac{w L}{Y}.$$ We can finally define $$\eta^s$$ by
###### (3)
$$$$\eta^s = \eta - \alpha \cdot \epsilon.$$$$

## 2. Decomposing changes

Goldin (2006) uses this decomposition to make sense of changing patterns in women's labor force participation since the late 19th century. For any given year, we can attempt to learn about $$\epsilon$$, the income elasticity, by comparing households with different partners' earnings, but similar wages for women. Formally, we might regress $$\log L$$ on $$\log Y$$, controlling for $$\log w$$:

###### (4)
$$$$\log L_i = \beta_0 + \epsilon \cdot \log Y_i + \beta_1 \cdot \log w_i + U_i.$$$$ Variation in $$Y_i$$ given $$w_i$$ comes from partners' earnings. We can learn about $$\eta$$, the total elasticity of labor force participation with respect to wages, by regressing $$\log L$$ on $$\log W$$,
###### (5)
$$$$\log L_i = \gamma_0 + \eta \cdot \log w_i + V_i. \label{eq:totalelasticity}$$$$

There are issues in estimating these elasticities due to possible endogeneity, that is due to correlation between $$(Y_i, w_i)$$ and $$(U_i, V_i)$$. We will ignore these for now, and assume that we got correct estimates of $$\epsilon$$ and $$\eta$$. We can finally learn about $$\alpha$$ by simply calculating how much women would earn when working full-time.

Plugging in the decomposition of $$\eta$$ into equation (5), we get

###### (6)
$$$$\log L_i = \gamma_0 + (\eta^s + \alpha \cdot \epsilon) \cdot \log w_i + V_i.$$$$

This equation allows us to interpret changes of labor supply over time and differences across social classes. First, labor supply might increase over time as $$\gamma_0$$ increases. This is an outward shift of women's labor supply, which implies that women would work more for any given wage level. Second, labor supply might shift over time as wages $$w_i$$ increase. This effect might go either way, depending on the sign of $$\eta = \eta^s + \alpha \epsilon$$. If negative income effects, $$\epsilon <0$$, dominate, then $$\eta <0$$, and increasing wages lead to a reduction of labor supply. If substitution effects, $$\eta^s >0$$, dominate, then $$\eta >0$$ and increasing wages lead to an increase of labor supply. Third, Labor supply might shift because of a change of these elasticities. According to Goldin (2006), the total elasticity $$\eta$$ used to be negative but increased to become positive at some point in history, as income elasticities increased (became less negative) and substitution elasticities increased (became more positive). Possible explanations for these changing elasticities include changes in the workplace, as new technologies and occupations became available (office work as opposed to factory work), changes in the legal environment (married women used to be barred from many occupations), changes in household technology (laundromats), public provision of care (public kindergartens; at least in European welfare states), and changes in social norms.

The net effect of these changes was an increase of labor force participation of working-age women from below 20% in the US around 1900 (and in fact close to zero for married white women) to almost 80% today. This increase was initially driven by an outward shift in labor supply (increase in $$\gamma$$), followed by increases in $$\eta$$ and subsequently a positive effect of increasing wages. More recent important changes since the 1980s are not quite captured by labor force participation. In tandem with women overtaking men in educational attainment, the nature of women's occupations, in particular for well-off women, has changed from being merely a source of supplementary income with little advancement over time to involving long-term careers.

## 3. Some critical remarks

Celebratory descriptions of this changing role of women in the labor market encounter some criticism in particular in feminist discussions, see for instance Fraser (2013). We shall briefly mention two.

First, descriptions such as the one of the "quiet revolution" of women's careers since the 1980s Goldin (2006) focus on college-educated women. Additional emphasis is put on those with professional and advanced degrees (lawyers, doctors, managers, academics...). These descriptions neglect the quite different historical changes for women at the low end of the wage distribution (in service and care occupations, in particular) in recent decades, facing stagnating low wages in a time of eroding social provisions. In addition to the focus on privileged women among all women, the consideration of labor supply differences by gender alone also neglects important heterogeneity. If one cares about inequality in general, then a focus on inequality solely along the dimension of gender might obscure other inequalities.

Second, the massive increase of women's participation in paid labor is the flip-side of an increased marketization of all spheres of life, including social spheres such as care of children and the elderly traditionally outside the reach of markets. Fraser (2013) argues that we should aim for a third alternative beyond (i) a traditional division of roles with women in charge of unpaid care-work and dependent on men's wage incomes, but also beyond (ii) a complete marketization of all spheres of life with its consequences for inequality, uncertainty, and erosion of social bonds. Such a third alternative would involve an equal role of men and women in care work organized in ways outside the anonymous market.