Measuring the Effect of Discrimination

1. Measuring the Effect of Discrimination The Earnings Function Approach

2. You may recall from your math background the equation of a line being written in the form y = mx + b, where y values depend on values of x, b is the y intercept (the value of y when x = 0), and m is the slope of the line (which means m is how much y changes when x changes by one unit.) Here the slope is a positive number , but can be negative (which means the line would slope downward, from left to right.) y m 1 x

3. In the math class all the points of interest are one the line. You can pick an x and solve for a y. I have shown two examples in the graph. Once values for m and b are picked, you can solve for specific values. For example, say m = 2 and b = 7. Then y = 2x + 7. If x = 3, y = 13. If x = 12, y = 31. y m 1 x

4. In statistics we have a similar graph, but there are some differences. Here the points represent subjects, which are often people in economics. The points are the values on two variables, like years of schooling and wage. A statistical technique called regression is used to put a line into the graph that “best” fits the data. Once we have the line we are back in our math class in many respects. Let’s not worry about how the regression stuff works, let’s just use it (you will see regression in a stats class.) y x

5. In the context of labor economics the y variable is often the wage, or the natural log of the wage, written ln w. The x variable could be years of schooling and we would write s. The equation could be written as ln w = b0 + b1s. Here you notice the intercept is written first and is b sub zero instead of just b and the slope is b sub 1 instead of m. So, in stats the intercept and slope are written as b’s with subscripts, or a Greek letter like alpha, beta, gamma and the like. Also in stats, if we think more than one “x” variable affects “y” we just add it in with another slope. The graph we had before becomes less useful, but the idea is similar. Say we have ln w = b0 + b1s + b2X +b3X2, where X is years of work experience.

6. The regression technique will give numerical values for the b’s. Say b sub one on the variable s is 0.1. This means that one more year of schooling should make the ln w go up by 0.1. Folks who are really smart and play around with this stuff know this means the wage then goes up by about 10%. When the “y” variable is measured in natural logs, the influence of a unit change in “x” is approximately 100 times the slope value – so here we had 100(0.1) = 10 or 10%. Digress – perpetuity. If you have P today and you can earn i then by the end of the year you have P + Pi. If you spend Pi you still have P and it can earn for another year and you would have P +Pi. You could spend Pi and start all over again. So, Pi is an annuity you could spend forever. So A = Pi

7. Say the only cost of going to school is the wage you give up. But you do it in the hopes of increasing your future wage. If you are young enough you basically are making a perpetuity annuity for yourself. P = w, the wage you give up and A = wb1, is the percentage increase in the wage you would get by one more year of schooling, so wi = wb1 and thus i = b1. In other words the slope is also interpreted as the rate of return to schooling. Now, if we want to see if wages are different for males and females, then our first inclination might be to add a variable known as a dummy variable. Values of the dummy variable could be 0 for males and 1 for females. Thus the coefficient would show how the log of the wage is changed as we switch from males to females. If the slope is negative then that would indicate that females get lower wages, controlling for years of schooling and work experience.

8. The problem with the approach mentioned is that maybe discrimination works its way through many variables. So, the solution here is to estimate a separate equation using only male data and estimate an equation using only female data. Say we did so and received the following: For males ln w = 0.5 + 0.1s + 0.08X – 0.003X2, and For females ln w = 0.5 + 0.13s + 0.05X – 0.003X2. Now, if you take the average values of s and X and plug them into the equations you get the average ln w for each. I will denote the average here by putting a subscript of m for males and f for females. Then let’s take the difference between the two. Let’s see this on the next page.

9. We get ln wm – ln wf = 0.1sm – 0.13sf + 0.08Xm – 0.05Xf. Now, lets assume the average value of s for men is 13.7 and for women is 13.5 and the average value of X for men is 23 and for women is 13. The average ln wage gap is then 0.1(13.7) – 0.13(13.5) + 0.08(23) – 0.05(13) = 0.805 Now, when you look at the top line you the coefficients are different for the male and female equations and from the example the averages are different. By some algebraic chicanery we can decompose the top line to show how much of the 0.805 is due to differences in averages and how much is due to differences in coefficients. The trick is to add and subtract from the above a term that takes the male coefficient and multiplies by the female mean for each of s and X – so we have 0.1sf – 0.1sf + 0.08Xf – 0.08Xf added to get (on the next page).

10. We get ln wm – ln wf = 0.1sm – 0.13sf + 0.08Xm – 0.05Xf + 0.1sf – 0.1sf + 0.08Xf – 0.08Xf. (rearranging) = 0.1(sm – sf) + 0.08(Xm-Xf) + sf(0.1 – 0.13) + Xf(0.08 – 0.05), So the first part shows the differences because of differences in means and the second part shows the differences due to differences in coefficients. = 0.1(13.7 – 13.5) + 0.08(23 -13) + 13.5(0.1 – 0.13) + 13(0.08 – 0.05) = 0.02 + 0.8 + -0.405 + 0.39 = 0.805 Notice here that men on average have more schooling and more experience. Although this may be for discriminatory reasons, at least at the firm level the differences in wages seem justified due

11. to the differences in averages. Of the total difference of 0.805 this part accounts for 0.82, which is actually more than the total. The differences in coefficients is called by the authors as unjustified discrimination because it means males and females with similar schooling and experience are treated differently, and this accounts for -0.015 of the difference.