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.

Since average hours worked in U.S. manufacturing are strongly procyclical, taking into account this factor makes the implied markup significantly more countercyclical. Indeed, when Bils regresses his constructed markup series (using (2.9)) on a measure of cyclical employment,26 he finds that markups decline, on average, by 0.33% for each one-percent increase in employment. Of this cyclical variation, a 0.12% decline is implied by the increase in the labor share (which is mildly procyclical in his sample), while the remaining 0.21% decline comes from the increase in the ratio of the marginal wage to the average wage.

One may question whether the statutory premium paid for overtime hours represents a true cost to the firm; some argue, for example, that the opportunity to work overtime is in fact dispensed as a reward for exemplary behavior at other times. Bils answers this objection by pointing out that if one assumes that because of sophisticated implicit contracts, the true cost to the firm is proportional to the worker’s disutility of working v(H), then one might well obtain estimates of the degree of procyclical movement in the ratio of the marginal wage to the average that are as large as those obtained using his method.

Under the assumption suggested above about the steady-state level of hours, the coefficient b in (2.9) would in that case equal —v,,/H*v’. or — chw, where €hw is now the Frisch (or intertemporal) elasticity of labor supply by a wage-taking household in a competitive spot market. A value of b less negative than Bils’ value of -1.4 would then be obtained only if one assumed preferences implying an elasticity of labor supply greater than 0.7, whereas many microeconomic studies of labor supply estimate a lower elasticity than that.