Three Strikes and You.re Out: Reply to Cooper and Willis



1 Summary of the case

Cooper and Willis (2003), henceforth CW, is the third version of the authors’ “The Economics of
Labor Adjustment: Mind the Gap.” In this comment CW argue that our finding (in Caballero and
Engel (1993), henceforth CE, and Caballero, Engel and Haltiwanger (1997), henceforth CEH) that
lumpy microeconomic adjustment matters for aggregate employment dynamics, is not warranted.
They base their case on “reproducing” our main findings using artificial data generated by a model
where microeconomic agents face quadratic adjustment costs. That is, they supposedly find our
results where they should not be found.

In this reply we show that the three claims on which they base their case are incorrect. Their
mistakes range from misinterpreting their own simulation results to failing to understand the con-
text in which our procedures should be applied. They also claim that our approach
assumes that
employment decisions depend on the gap between the target and current level of unemployment.
This is incorrect as well, since the ‘gap approach’ has been derived formally from at least as so-
phisticated microeconomic models as the one they present. On a more positive note, the correct
interpretation of CW’s results shows that our procedures are surprisingly robust to significant de-
partures from the assumptions made in our original derivations.

Throughout, CE and CEH take as an assumption validated in many other studies that at the
microeconomic level adjustments are lumpy,1 and examine whether the implied features of the
distribution of microeconomic gaps are useful in explaining aggregate employment fluctuations.
Specifically, the basic regression in CE and CEH is:

Et = λMt(1) +γMt(3),                                  (1)

where E represents the rate of growth of aggregate employment, and M(i) is the i-th moment of
the cross section distribution of gaps between actual and desired employment at the firm level.

When λ > 0 and γ = 0, the equation above simplifies to a linear model where the left hand side
variable depends only on aggregates (first moments of the cross section distribution of gaps). This
case can be obtained either from a microeconomic model where agents adjust infrequently but with
a probability that is independent of their gap (the constant hazard model of Calvo, 1983) or from a
model where agents face quadratic adjustment costs and adjust all the time (Sargent, 1978).2

When γ > 0, on the other hand, higher moments of the cross-section distribution of gaps matter

1CW (2003) seem to agree with this assumption, in particular, in their conclusion they refer to “overwhelming
evidence” in favor of it.

2See Rotemberg (1987) for a formal proof of the aggregate equivalence of Calvo’s lumpy adjustment model and
the quadratic adjustment cost model.



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