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.

The last two columns of Table 2 show the effect of this correction for с equal to 4 and 8 while 13 is equal to .99. To carry out this analysis, we need an estimate of Erfm+i- We obtained this estimate by using one of the regressions used to compute expected output growth in Rotemberg and Woodford (1996a). In particular, the expectation at t of Ht+i is the fitted value of a regression of Ht+1 on the values at t and t — 1 of #, the rate of growth of private value added and the ratio of consumption of nondurables and services to GDP.

Subtracting the actual value of Ht from this fitted value, we obtain Etjm+i- This correction makes the markup strongly countercyclical and ensures that the correlation of the markup with the contemporaneous value of the cyclical indicator is larger in absolute value than the correlation with lagged values of this indicator. On the other hand, the correlation with leads of the indicator is both negative and larger still in absolute value, particularly when с is equal to 8.
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The same calculations apply, to a log-linear approximation, in the case that the adjustment costs take the form of less output from a given quantity of labor inputs. Suppose that in the above description of production costs, H refers to the hours that are used for production purposes in a given period, while Нф indicates the number of hours that employees must work on tasks that are created by a firm’s variation of its labor input over time. (In this case, к = w.)

Equations (2.12) – (2.13) continue to apply, as long as one recalls that H and Sh now refer solely to hours used directly in production. Total hours worked equal AH instead, and the total labor share equals Ash, where Л = 1 + 0(7#). But in the log-linear approximation, we obtain A = 0, and so equations (2.14) – (2.15) still apply, even if 7я and s’h refer to fluctuations in the total labor inputs hired by firms.

A more realistic specification of adjustment costs would assume costs of adjusting employment, rather than costs of adjusting the total labor input as above.31 Indeed, theoretical discussions that assume convex costs of adjusting the labor input, as above, generally motivate such a model by assuming that the hours worked per employee cannot be varied, so that the adjustment costs are in fact costs of varying employment.