Suppose there are two groups in society, W and B. The productivity of a person j in group i (W or B) denoted by P^{j}_{i}. Let the average of P among group i be represented by M_{i}. Let the variance of P be represented by V_{W}=V_{B}=V. If productivity is perfectly observed, and if labor markets are competitive, then the wage to individual j is simply w^{j} = P^{j}. Group membership has no bearing on wages.
Now suppose productivity can be measured only with error. In particular, suppose employers only observe a "test score" which is equal to actual productivity plus an independent noise. Assume that for group W (the majority group) the noise is e^{W}, while for group B (the minority group) the noise is e^{B}. These noises are i.i.d. across individuals. For i=W,B, we have
Make the assumption that Var[e^{W}] < Var[e^{B}]. Why?
What is the wage paid to individual j in Group i? In a competitive market,
If M_{W} = M_{B}, then the wage distribution for W is more dispersed than that for B, but they will have the same average level.
However, the fact that wages are less sensitive to productivity for group B discourages investment in human capital. Let
In this model, forcing employers to treat people in the two groups equally (i.e., forcing β_{W} to be equal to β_{B}) could improve net social surplus.