Can the signaling game serve as a model of statistical discrimination in hiring?

Some scholars argue that Spence's signaling game with an index may serve as a model of statistical discrimination in hiring processes. This would then explain that both mean education level and mean wage are greater for men than for women in industrialized societies. To examine the validity of this conjecture, I formulated a generalized version of the game. In this version, I assumed that the educational level is a signal of productivity while the gender is an index of productivity. I then followed the refinement procedure of Perfect Bayesian Equilibria to eliminate unreasonable outcomes. My analysis reveals that an anomaly is derived from the separating equilibrium that survives the Intuitive Criterion in the procedure: the mean wage for men would be equivalent to that for women. As the employer is assumed to know that the educational cost for women is greater than that for men, he or she would believe that women with a lower level of education have the same productivity as men with a higher level of education. Therefore, the employer would offer the same wage for the men and the women. I also examined other classes of equilibria and alternative assumptions.