To understand this equation it is useful to compare it to the simplified basis case in which union coverage rates, wage gaps, and variance gaps are all constant across skill groups (i.e., u(c) = u; Д*(с) = Д^с) = AJ. In this situation the effect of unions on the variance of wages reduces to the two-sector formula presented by Freeman (1980):
Relative to this basis case, variation in either the union coverage rate u(c) or the union wage effect Aw(c) introduces two additional factors into the overall wage dispersion effect. The first is a necessarily positive variance component that arises if the union wage gain ( u(c)Aw(c)) varies across groups. The second is a covariance term that may be positive or negative, depending on whether the union wage gain is larger or smaller for higher- or lower-wage workers.

Unobserved Heterogeneity

The preceding formulas have to be modified slightly if union and nonunion workers in a given skill category would earn different wages even in the absence of unions (i.e. if there is unobserved heterogeneity between union and nonunion workers in a given skill group). As before, assume that workers are classified into skill categories on the basis of observable characteristics, and suppose that where the first term is the true union wage effect and the second is the difference in the mean of unobserved heterogeneity between union and nonunion workers in the group. The latter term introduces a component of variance that would exist even in the absence of unions between workers who are currently unionized and those who are not. Taking account of this component, the difference in the variance of wages in the presence of unions and in the counterfactual situation in which all workers are paid according to the nonunion wage structure is:

Only the last term of this equation, which reflects the gap in mean wages between union and nonunion workers with the same observed skills in the presence and absence of unions, differs from equation (5).

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