Suppose there are two groups in society, W and B. The productivity of a person j in group i (W or B) denoted by Pji. Let the average of P among group i be represented by Mi. Let the variance of P be represented by VW=VB=V. If productivity is perfectly observed, and if labor markets are competitive, then the wage to individual j is simply wj = Pj. 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 eW, while for group B (the minority group) the noise is eB. These noises are i.i.d. across individuals. For i=W,B, we have
Make the assumption that Var[eW] < Var[eB]. Why?
What is the wage paid to individual j in Group i? In a competitive market,
If MW = MB, 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 βW to be equal to βB) could improve net social surplus.