Manufacturing Earnings and Cycles: New Evidence

Lanczos^,s σ factors (Bloomfield 1976, p. 129-137). We replace the truncated
weights of the optimal filter b_j in equation (A4) by the modified weights δ*,
which are obtained from

sin((2πj)∕(2∕< + 1))
^{j 3} f2πj)∕f2K ₊ l)

After this step, the modified filter weights of the Baxter-King filter a* can be
calculated as demonstrated above (Woitek, 1998).

A.3 Derviation of earnings expression (7) from BLS
data

The BLS calculate the earnings rate²¹ (i.e. average hourly earnings), by divid-
ing gross payrolls, GF, by total hours, thus

We can decompose equation (A6) by differentiating between overtime workers
and workers working only standard hours. Under the FLSA, overtime is com-
pensated at a premium rate for hours in excess of 40 per week. We assume
that overtime workers are compensated for 40 weekly hours at the standard
rate and then at the mandated premium rate for additional weekly hours.²²

Accordingly, we may re-express equation (A6) in the form

N_tW_tH_t + AflUt(40 - H_t^s) + N_t⁰W_t⁰V_t

where, N⁰ ≤ N are the number of employees working overtime, W and W⁰ are
the average standard and average overtime hourly wage rates, H^s is average
weekly standard hours of non-overtime workers, V = (H⁰ — 40) is average
overtime hours of overtime workers, and H⁰ is average total weekly hours
worked by overtime workers. The numerator of (A7) comprises three parts.

²¹See, BLS Handbook of Methods, 1997, Ch. 2 Employment, Hours, and Earnings from
the Establishment Survey.

²²Actually, there is evidence (Trejo, 1993) that some overtime workers receive the premium
before the 40 hour limit. Unfortunately, our data are such that we cannot accommodate
this possibility.

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