PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 8

In the data, however, employment variations and variations in total person-hours are not the same, even if they are highly correlated at business-cycle frequencies. This leads us to suppose that firms can vary both employment N and hours per employee h, with output given by F(K, zhN), and that costs of adjusting employment in period t are given by KtNt(f){Nt/Nt-i). If, however, there are no costs of adjusting hours, and wage costs are linear in the number of person-hours hired Nh, firms will have no need ever to change their number of employees (which is clearly not the case). If, then, one is not to assume costs of adjusting hours per employee, one needs to assume some other motive for smoothing hours per employee, such as the sort of non-linear wage schedule discussed above. We thus assume that a firm’s wage costs are equal to W(h)N, where W(h) is an increasing, convex function as above.

One can then again compute the marginal cost of increased output at some date, assuming that it is achieved through an increase in employment at that date only, holding fixed the number of hours per employee h at all dates, as well as other inputs. One again obtains (2.12), except that the definition of Q in (2.13) must be modified to replace 7я by 7/v, the growth rate of employment, throughout. (In the modified (2.13), w now refers to the average wage, W(h)/h.) Correspondingly, (2.15) is unchanged.

Thus one obtains, as in the simpler case above, a correction to (2.5) that results in the implied markup series being more countercyclical (since employment is strongly procyclical,
just as with the total labor input).
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Alternatively, one could compute the marginal cost of increased output, assuming that it is achieved solely through an increase in hours per employee, with no change in employment or in other inputs. In this case, one obtains again (2.11), but with H everywhere replaced by h in the first factor on the right-hand side. There is no contradiction between these two conclusions. For the right-hand sides of (2.11) and (2.12) should be equal at all times; cost-minimization requires that
which implies that П = w. Condition (2.17) is in fact the Euler equation that Bils (1987) estimates in his “second method” of determining the cyclicality of the marginal wage; he uses data on employment and hours variations to estimate the parameters of this equation, including the parameters of the wage schedule W(h).33 An equivalent method for determining the cyclicality of markups would thus be to determine the importance of employment adjustment costs from estimation of (2.17), and compute the implied markup variations using (2.15) – (2.16).

Insofar as the specification (2.17) is consistent with the data, both approaches should yield the same implied markup series. It follows that Bils* results using his second method give an indication of the size of the correction that would result from taking account of adjustment costs for employment, if these are of the size that he estimated. His estimate of these adjustment costs imply an elasticity of even greater than the value of 1.4 discussed above.