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.
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PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 7

The intuition for this result is that high lagged levels of hours imply that the current cost of producing an additional unit is relatively low (because adjustment costs are low) so that current markups must be relatively high. Since, as we showed earlier, the labor share is more positively correlated with lags of hours (and more negatively correlated with leads of hours) this correction tends to make computed markup fluctuations more nearly coincident with fluctuations in hours.

To put this differently, consider the peak of the business cycle where hours are still rising but expected future hours are low. This correction suggests that marginal cost are particular high at this time because there is little future benefit from the hours that are currently being added.
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PRICES AND COSTS: Corrections to the Labor-Share Measure of Real Marginal Cost 6

An additional reason why marginal hours may be more expensive in booms is the presence of adjustment costs. It is simplest to illustrate this point if we assume, as, for example, in Pindyck and Rotemberg (1983), that there are convex costs of changing the labor input #.

Suppose that, in addition to the direct wage costs wtHt of hiring Ht hours in period t, there is an adjustment cost of KtHt(l){Ht/Here nt represents a price index in period t for the inputs that must be purchased as part of the adjustment process; we shall assume that the (logarithms of the) factor prices к and w are со-integrated, even if each is only difference-stationary. (More specifically, we shall assume that к/w is stationary.) The factor Ht4>(Ht/Ht-i) represents the physical quantity of inputs that must be expended in order to adjust the labor input; note that adjustment costs increase in proportion to the quantity of labor used by a given firm. This specification implies that adjustment costs remain of the same magnitude relative to direct labor costs, even if both H and w exhibit (deterministic or stochastic) trend growth.

The exposition is simplest if we treat the adjustment costs as “external”, in the sense that the additional inputs that must be purchased are something other than additional labor, so that both the production function (2.1) and the formula for the labor share can still be written as before in terms of a single state variable “H” . Finally, we assume that ф is a convex function, with 0(1) = ф({ 1) = 0; thus adjustment costs are non-negative, and minimized (equal to zero) in the case of no change in the labor input.
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