Bils (1987) observes that in many industries, a higher wage is paid for overtime hours (i.e., , hours in excess of 40 hours per week). He thus proposes to quantify the extent to which the marginal wage rises as firms ask their employees to work longer hours, by measuring the extent to which the average number of overtime hours per employee, V, rises with increases in the total number of hours worked per employee H, and then assuming that W(H) = +pV(H)], where w0 is the straight-time wage and p is the overtime premium (0.5 according to the U.S. statutory requirement).
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For example, he finds that when average hours per employee rise from 40 hours per week to 41 hours, the average number of overtime hours worked per employee rises by nearly 0.4 hours, while when they rise from 41 to 42 hours per week, overtime hours rise by another 0.5 hours. This increase in the fraction of hours that are overtime hours as average hours increase means not only that the marginal wage exceeds the average wage, but that the ratio of the marginal wage to the average wage rises as hours increase.
Assuming p = .5, Bils finds that an increase in average hours from 40 to 41 hours increases the average wage by about 0.5%, but increases the marginal wage by 4.6%. On average, he finds that the factor uj in (2.11) has an elasticity of 1.4 with respect to variations in average hours.24 Thus a log-linear approximation to (2.11) is again of the form (2.9), where in Bils’ work H refers to fluctuations in average hours per worker,25 and 6 = -1.4.