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.

We can then compute the marginal cost associated with an increase in output at date t. assuming that production is increased solely through an increase in the labor input at date t. with no change in the inputs used in production at other dates, except for the necessary changes in the inputs used in the adjustment process at both dates t and t + 1. In this case, (2.5) becomes
in which in turn 7nt = 1, 7^ = ^/^г_ь29 and Rt,t+i is the stochastic discount factor by which firms discount random income at date t + 1 back to date t. (Here we have written (2.13) solely in terms of variables that we expect to be stationary, even if there are unit roots in both H and w, to indicate that we expect ft to be a stationary random variable.
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If ф is strictly convex (i.e., if there are non-zero adjustment costs), the cyclical variation in the factor ft changes the nature of implied markup fluctuations. Because фг is positive when the labor input is rising and negative when it is falling. ft should be a procyclical factor, though with a less exact coincidence with standard business cycle indicators than the cyclical correction factors discussed thus far. If we take a log-linear approximation to (2.13), near a steady-state in which the variables #, k/w, 7*, and R are constant over time, we obtain
to obtain a formula to be used in computing markup variations. Equation (2.14) makes it clear that the cyclical variations in the labor input are the main determinant of the cyclical variations in ft. The factor ft will tend to be high when hours are temporarily high (both because they have risen relative to the past and because they are expected to fall in the future), and correspondingly low when they are temporarily low. Thus, it tends to increase the degree to which implied markups are countercyclical.

More precisely, the factor ft tends to introduce a greater negative correlation between measured markups and future hours. Consider, as a simple example, the case in which hours follow a stationary AR(1) process given by