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Interactions Between the Seasonal
and Business Cycles in Production
and Inventories
Stephen G. Cecchetti, Anil K. Kashyap and
David W. Wilcox
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
September 1997 (W P-97-6)
FEDERAL RESERVE BANK
O F CHICAGO
Interactions B e tween the Seasonal and Business Cycles in Production
and Inventories
Stephen G. Cecchetti, Anil K. Kashyap and David W. W ilcox1
Current Draft: September 1997
1Federal Reserve Bank of New York, on leave from Ohio State University and NBER;
Graduate School of Business, University of Chicago, Federal Reserve Bank of Chicago, and
NBER; and Federal Reserve Board, respectively. We thank Andrew Abel, William Bell, Mark
Bils, William Cleveland, Stephen Cosslett, Spencer Krane, Pok-sang Lam, Nelson Mark, Jeffrey
Miron, Alan Viard, Mark Watson, participants at numerous seminars and especially Michael
Woodford for comments. Cecchetti and Kashyap acknowledge the financial support of the
National Science Foundation through a grant to the National Bureau of Economic Research.
Cecchetti also thanks the Federal Reserve Bank of Cleveland for research support. The opinions
expressed here are those of the authors only and are not necessarily shared by the Federal
Reserve Board, or the Federal Reserve Banks of Chicago, Cleveland or New York or the other
members of the staff at any of these institutions.
Interactions Between the Seasonal and Business Cycles in Production
and Inventories
Abstract
This paper shows that in several U.S. manufacturing industries, the seasonal variability of
production and inventories varies with the state of the business cycle. We present a simple
model which implies that if firms reduce the seasonal variability of their production as the
economy strengthens, and they either hold constant or increase the stock of inventories they
bring into the high-production seasons of the year, then they must face upward-sloping and
convex marginal cost curves. We conclude that firms in a number of industries face upwardsloping and convex marginal-production-cost curves. (JEL E32, C49)
Stephen G. Cecchetti
Director of Research
Federal Reserve Bank
of New York
New York, NY 10045
(212) 720-8629
and Ohio State University,
and NBER.
Stephen.Cecchetti@ny.frb.org
Anil K Kashyap
Graduate School of Business
University of Chicago
1101 E. 58th Street
Chicago, IL 60637
(773) 702-7260
and Federal Reserve Bank of
Chicago, and NBER.
Anil.Kashyap@gsb.uchicago.edu
David W. Wilcox
Division of Monetary
Affairs
Board of Governors
of the Federal Reserve
System
Washington, DC 20551
(202) 452-2441
dwilcox@frb.gov
A growing literature examines the shape of the aggregate production function.
Recently, the orthodox view that marginal cost curves are upward-sloping and convex
has been attacked by Robert E. Hall (1991) and Valerie A. Ramey (1991), who argue
that a number of important macroeconomic phenomena are consistent with declining
marginal costs, i.e. increasing returns to scale or agglomeration economies. This
paper develops new evidence on the shape of marginal-production-cost curves based
on changes in the seasonal patterns of production and inventory holdings over the
business cycle.
The intuition for our analysis is that capacity constraints are most likely to bind
when both the business cycle is at its peak and production is seasonally high. During
a boom, the presence of a capacity constraint might cause firms to reorganize the
pattern of their production within the year in order to produce a larger fraction of
annual output in off-peak seasons, thereby avoiding the high marginal cost (in the
extreme case, the
in fin ite ly
high marginal cost) associated with additional production
during the normally busy periods of the year.
This intuition is incomplete because the change in the seasonal pattern of pro
duction over the business cycle generally will not be sufficient to reveal the shape
of firms’ cost functions. However, in the next section of the paper we show how in
formation on the interaction between seasonal and business cycles can be combined
with data on inventories to identify the shape of firms costs functions. If, as the
economy strengthens, firms both reduce the seasonal variability of production and
carry more inventories into the high-production season, we can conclude that firms
face an upward-sloping and convex marginal-production-cost curve.
We conduct the empirical aspect of our investigation using data for each of the
20 two-digit manufacturing industries in the United States. For all but one industry,
we find overwhelming evidence that the seasonal patterns of both production and
inventories change over the business cycle. These are the “interactions” referred to
in the title. In a number of these industries, these interactions are of such a nature
as to allow us to determine the shape of the marginal-production-cost curve faced
1
by the representative firm in the industry. In five industries, booms are associated
with a reduction in the seasonal amplitude of production and either no change or an
increase in inventory holdings coming into the high-production season; on the basis of
this information, we conclude that firms in these industries face upward-sloping and
convex marginal-production-cost curves. In one other industry we find that booms are
associated with an increase in the seasonal variability of production and a reduction
in the level of inventories brought into the high production seasons; on the basis of
this information, we conclude that firms in that industry face marginal-productioncost curves that flatten out, and hence have an incentive to bunch their production.
Unfortunately, in the other 14 industries, the nature of the interactions we detect
does not allow us to identify the shapes of the marginal-production-cost curves.
This work builds on that of Robert B. Barsky and Jeffrey A. Miron (1989), J.
Joseph Beaulieu and Miron (1991 and 1992), Spencer D. Krane (1993), and Miron
and Stephen P. Zeldes (1988 and 1989), all of whom use information on seasonal
cycles to provide insights into economic behavior; Olivier J. Blanchard (1983), Ken
neth D. West (1986), Krane and Steven N. Braun (1991), Ray C. Fair (1989), and
Anil K Kashyap and David W. Wilcox (1993) who analyze the cost structure of pro
duction; Eric Ghysels (1991), who documents the statistical asymmetries in seasonal
fluctuations; and Alan S. Blinder (1986) and Blinder and Louis J. Maccini (1991),
who study inventories and production smoothing.1 Our work is closest to that of
Beaulieu, Jeffrey K. Mackie-Mason and Miron (1992), who show that the amplitude
of seasonal cycles is positively correlated with the amplitude of business cycles, both
across industries and across countries. We view their finding as complementary to
ours. An important distinguishing feature of our effort is that by jointly analyzing
production and inventory data we are able to establish the conditions under which
any interactions between cyclical and seasonal variation can be used to learn about
the shape of industry cost curves.
1West’s (1990) work using inventory fluctuations to distinguish supply from demand shocks is
also related.
2
The remainder of this paper is organized as follows: Section I outlines the circum
stances under which we will be able to deliver evidence on the shape of the marginal
production cost function. Section II presents our empirical results, and Section III
contains our conclusions.
I. A
S im p le M o d e l
This section outlines the circumstances under which a change over the business
cycle in the seasonal amplitude of production reveals information about the shape of
the marginal-production-cost function. Marginal-production-cost schedules can take
on any of four generic shapes. The first shape is upward-sloping and convex. Firms
facing this type of curve have an incentive to smooth production. We refer to these
firms as facing capacity constraints. The second generic shape is either upward-sloping
and concave, or downward-sloping and convex. In either case, the first derivative of
the cost curve is a decreasing function of the level of production (the curve “flattens
out”). Firms facing this type of curve have an incentive to bunch production. The
third shape is linear. This type of curve gives no incentive either to smooth production
or to bunch it, regardless of whether the curve is upward-sloping, flat, or downwardsloping. Finally, there are marginal curves that are downward-sloping and concave.
These curves encourage bunching, but we dismiss them from further consideration
because they generally will not give rise to interior solutions to the cost minimization
problem unless the inventory holding cost function is sufficiently convex.2 Thus, our
task is to develop a technique for distinguishing among three marginal-productioncost curves: (1) capacity constrained, (2) flattening out, and (3) linear.
We illustrate our method using a simple two-period model. Together, the two
periods in the model span one seasonal cycle. The representative firm chooses its
productive capacity prior to the start of the first period. Once this choice has been
2In cases w here th e holding cost function is sufficiently convex so as to guarantee an interior
op tim u m , th e cu rv atu re of th e holding cost function will force th e firm to behave as if it is capacity
constrained.
3
made, the state of the business cycle is revealed; both capacity and the state of the
business cycle remain fixed for the rest of time. As a harmless norm alization, we
assume that production is higher in the second period than in the first. We ignore
discounting.
There are two key building blocks for our analysis. One is the requirement that
the firm allocate its production between the first and second periods so that the ex
pected marginal cost of producing an extra unit of output in the first period and
storing it until the second period equals the expected marginal cost of producing an
extra unit in the second period. Stated in slightly different terms, optimal produc
tion scheduling requires that the difference between marginal production costs in the
seasons must equal the marginal cost of holding inventories across the two seasons.3
We emphasize that th is r e q u ir e m e n t m u s t h o ld ir r e s p e c tiv e o f w h e th e r th e s h o c k s in
th e m ,o d el o r ig in a te fro m , th e c o s t s id e o f th e m .odel o r th e d em .a n d s id e .
The second
building block is the assumption that the holding-cost function is convex in the level
of inventories.4
In many circumstances we w ill be able to describe the marginal-production-cost
curve if we are allowed to observe two pieces of information: the change over the busi
ness cycle in the seasonal amplitude of production and the change over the business
cycle in the level of inventories that firms carry into the high-production (second)
season. For example, suppose the volume of inventories brought into the second pe
riod is an increasing function of the strength of the economy, and the amplitude of
the seasonal variation in production is either a decreasing function of, or invariant
w ith respect to, the same variable. Then we can conclude that the firm must be
facing a capacity-constraint-type marginal-production-cost function. How so? Given
the assumed convexity of the holding-cost function, the positive correlation between
3 A first-order condition of this type falls out of all standard production scheduling problems. See,
inter alia, Charles F. Holt, Franco Modigliani, John Muth and Herbert Simon (1960), West (1986),
Ramey (1991), and Kashyap and Wilcox (1993).
4 Our assumption in this regard is consistent with the long line of models descended from Holt et
al. (1960). In such models, the quadratic term in the level of inventories causes inventories to be
cointegrated with sales, provided a certain cost shock is stationary.
4
Table 1: Given the Change in the Seasonal Amplitude of Production
and the Change in Inventory Holdings over the Business Cycle,
Is the Marginal-Production-Cost Function Best Described as Linear,
Flattening Out or Exhibiting Capacity Constraints?
As the economy strengthens, do firms carry less,
the same amount, or more in v e n to r ie s
into the high-production season?
During a boom,
does the seasonal
amplitude of
p r o d u c tio n increase,
stay the same
or decrease?
Increase
Stay
the
same
Decrease
less
the same
amount
more
flattening
out
flattening
out
could be any
of the three
flattening
out
linear
capacity
constrained
could be any
of the three
capacity
constrained
capacity
constrained
the state of the business cycle and the level of inventories carried into the second
period implies that the difference between second- and first-period marginal produc
tion costs must increase as the economy strengthens. A greater difference between
marginal production costs in the two periods can be consistent with a diminished or
unchanged difference in the quantity produced in the two periods only if the marginalproduction-cost function is of the capacity constraint type. We catalogue this result
in the middle and lower blocks of the right-hand column in Table 1. Sim ilar reasoning
can be used to derive the other entries shown in the table.
Unfortunately, in two cases—when the level of inventories carried into the busy
season and the seasonal amplitude of production move in the same direction over
the business cycle— we cannot make any inference about the shape of the marginal-
5
production-cost curve: The marginal-production-cost function could be any of the
three shapes.
Thus, in the context of a two-period model, the results derived in this section
constitute a (nearly) complete guide to the identification of the curvature of the
marginal production cost function based on two pieces of information: the change
in the seasonal amplitude of production over the business cycle, and the change in
the seasonal pattern of inventory holdings over the business cycle. Unfortunately,
there is no guarantee that this guide w ill be as exhaustive once adapted for use w ith
12 seasons rather than just 2. For example, the seasonal pattern of production may
change over the business cycle, but not in a way that we can easily characterize as
smoothing or bunching. O r, the seasonal pattern of inventory holdings may change
over the business cycle but not in a way that is correlated with the seasonal pattern
of production. As a result, there is the possibility (which turns out to be realized)
that the apparent clarity of the two-period results are muddied a bit once applied to
monthly data.
II. E m p i r i c a l R e s u l t s
The objectives of this section are (1) to quantify the interactions between seasonal
and cyclical influences on production at the two-digit level in the manufacturing
sector, and (2) to examine simultaneously data on production and inventories for
clues as to the shape of the marginal-production-cost function.
A. Evidence on Seasonal and Cyclical Interactions in Produc
tion
Consider the following reduced-form expression for monthly production:
12
__
In Q t
-
In Q t
=
(1)
1=1
6
where the s f a s are conventional seasonal dummy variables ( s lt = 1 if month t is the
?th month of the year, 0 otherwise), Xt is a stationary variable indicating the stage
of the business cycle, /,;(.) is differentiable, and l n Q t is the level of production that
would prevail in the average season if the cycle were at a neutral position.
Substituting a linear expansion of the functions / f, /,(A t) « <rt + <pt Xt , into (1 ), we
have
__
ln Q t
12
- ln Q t = Y
12
+Y
i=l
i=
(2)
1
The coefficients fa, determine the interaction between the seasonal and cyclical influ
ences on production.
Following Bell and Hillm er (1984), we rewrite (2) as
__
ln Q t ~ l n Q t —
_
li
<7+
n
—cr) ( s»t —s i2t) + Y , ^ * ~~
1=1
_
—si2t)At
(3)
i —1
where a and 0 are the means of the cr,’s and </>j’s, respectively. The conventional
assumption is that fa = <f), in which case the deviation of production from its normal
value is a function only of the stage of the business cycle and seasonal dummies.
One possible interpretation of ln Q t is as combination of a linear trend and a
(presumably nonstationary) variable ut . Th is leads us to difference (3), so that
_
11
_
A l n Q t = a + 0AA t + Y j A {[ (° i — cr) + (fa — 0)At](sj( —s i 2t)} + A ut ,
i=
(4)
1
where a is the slope of the linear trend in l n Q t .
We estimate equation (4) using monthly data on production at the two-digit
level, constructed from Commerce Department estimates of shipments and inventories
following the procedures outlined in Miron and Zeldes (1989), Patricia Reagan and
Dennis P. Sheehan (1985), West (1983), and Douglas Holtz-Eakin and Blinder (1983).
We updated the data used by these other authors in two respects: F irst, of course,
we included additional observations not previously available. Second, we recomputed
the (separate) markup factors required to convert inventories at the finished-goods
7
and work-in-process levels from a “cost” basis to a “market” basis. Previous authors
(West (1983) and Holtz-Eakin and Blinder (1983)) computed markup factors for 1972,
which was the base year as of their writing; we computed (and used in constructing
our updated measures of output) factors for 1987, which is the base year as of our
writing.
For each industry except electronic equipment and instruments, the sample period
runs from March 1967 through March 1995. (Using data that begin in January 1967,
we computed output as shipments plus the change in inventories, accounting for one
lost observation at the front of the sample period, and then computed the log change
in production, accounting for the other lost observation.) For electronic equipment
and instruments, we ended the sample period in December 1986 in order to avoid a
discontinuity in the data resulting from the reclassification of certain four-digit in
dustries from electronic equipment to instruments. The regression for transportation
equipment includes dummy variables for September through December of 1970 to
control for the influence of the auto strike. We estimate the covariance m atrix of the
coefficient estimates using the W hitney K . Newey and West (1987) procedure w ith
24 lags, and we define At to be the one-month lag of capacity utilization in total
manufacturing.5
We begin by testing whether the interaction coefficients { f a — fa} are jo in tly sig
nificant. To this end, we define the variable <f> = ( f a — fa . . . , f a i ~ fa), and test the
hypothesis H q : (j> = 0 against H a : (f) fa 0 . Columns (2) and (7) of Table 2 present
our results. For every industry except tobacco (S IC 21), we reject the null hypothesis
overwhelmingly. Thus, interactions of some type between the business cycle and the
seasonal pattern of production appear to be nearly ubiquitous.6
We next investigate the nature of the changes in the seasonal pattern of produc5Prior to estimating equation (4), we removed the seasonal means from capacity utilization in
order to guarantee that our estimated coefficients reflect information solely about production and
not also about capacity utilization.
6We also note that these series are very seasonal. In sixteen of the twenty cases, seasonal dummy
variables explain over 50 percent of the variation in the data, implying that the interactions could
lead to important shifts in the overall variability of production.
8
tion over the business cycle, focusing specifically on whether the seasonal variability
of production generally increases or decreases as the economy strengthens. The mag
nitude of the overall seasonal for month i is related to D i ( \ t ) = [(cq —<r) + ((/>,- —0)At]2.
We summarize the behavior of all 12 ZVs over the business cycle by constructing the
following ratio 1Z =
— where A/,, and \ ( are, respectively, the means of all
recorded values of At above the 85th percentile and below the 15th percentile of A( .
If the seasonal variability of production tends to shrink as the economy strengthens,
1Z
w ill be less than 1. Columns (3) and (8) of Table 2 show our estimates of TZ and,
in brackets, the p-values for the tests that TZ equals 1. (We executed these tests
using the delta method.) For 15 of the 20 two-digit manufacturing industries, the
point estimate of 7Z is less than 1. In six of these cases, the discrepancy from 1 is
statistically significant at the 7 percent level or better.7 In no case is 7Z significantly
greater than 1 at anything better than the 20 percent level.
B. Evidence on the Curvature of the Marginal Production
Cost Function
In line w ith our objective of identifying the shape of the marginal-productioncost function, we now consider the joint behavior of production and inventories. To
that end, we introduce the analogue for inventories to the specification we examined
earlier for production:
n
A ln lt = 7
4- u A X t +
_
A {[(/?j —fS) + (u>i —u7)At](sit — Si2t)} + Aet
(5)
i —1
The coefficients
measure the extent to which inventory seasonals are influenced
by the business cycle. We develop evidence on the changes over the business cycle
in the seasonal pattern of inventory holdings—and the alignment of those changes
7We also re-estimated equation (4) using lagged own-industry capacity utilization as the proxy for
At, and instrumenting for own-industry capacity using aggregate manufacturing capacity utilization.
This procedure yielded very similar results to those shown in Table 2.
9
with respect to the seasonal in production—by stacking equations (4) and (5), and
calculating the correlation between the inventory interaction coefficient ( a —u7) and
the production seasonal in the following month, (cr,:+1 — <f). A positive value of this
correlation indicates that, as the overall economy strengthens, firms tend to increase
the stock of inventories they bring into the high-production seasons of the year.
The estimated values of this correlation are reported in columns (4) and (9) of
Table 2 under the heading “/?•” One °f the twenty correlations (the one for chemicals)
is significantly positive, while two correlations (tobacco and miscellaneous durable
goods) are significantly negative. For the remaining 17 industries, the correlation
is not statistically different from zero at anything better than the 10 percent level.
(Separately, we tested the hypothesis that the
are all zero, and rejected this
hypothesis at better than the 1 percent level for all 20 industries.)
Finally, we use the information reported in Table 2 to classify industries according
to the framework laid out in Table 1. W e summarize this classification in columns (5)
and (10) of Table 2. For five industries, namely lumber, chemicals, petroleum, primary
metals, and fabricated metals, the evidence isconsistent with capacity-constraint-type
marginal-production-cost curves. In these industries, either the seasonal amplitude
of production declines as the economy strengthens, or firms bring a larger stock of
inventories into the high-production seasons of the year the stronger is the overall
economy, or both. The only industry for which we have solid evidence of marginalproduction-cost curves that flatten out is the miscellaneous durable goods category.
In this industry, the seasonal amplitude of production does not vary significantly over
the business cycle, and the level of inventories brought into the busy seasons of the
year is a decreasing function of the strength of the economy.
In one industry— tobacco— the evidence is inconclusive because the seasonal am
plitude of production declines and inventories brought into the busy seasons vary
countercyclically. This case is further complicated by the fact that we cannot reject
the null hypothesis that 72.=1.
Unfortunately, we are unable to classify any of the remaining 13 industries. The
10
ambiguity come because we cannot reject either the null of constant seasonal variabil
ity over the business cycle of production (1Z
=
1) or the null of no correlation between
the inventory interaction and the production seasonal
(p =
0). Nevertheless, we are
quite confident (given our rejection of the null that the production interactions are
all zero) that the marginal cost curves are not linear. Evidently, the nature of this
nonlinearity is not classifiable using our framework, and we leave the relevant entries
in columns (5) and (10) blank.8
III. C o n c l u s i o n
This paper examines recent data for the 20 two-digit manufacturing industries
in the United States, and documents the following facts. First, there is a pervasive
tendency for the seasonal pattern of production to vary with the state of the business
cycle. Second, in five manufacturing industries, the seasonal amplitude of production
is a decreasing function of the strength of the economy; in one of these industries,
the level of inventories brought into the normally high-production season is an in
creasing function of the strength of the economy. Following the typology presented
in Section I, we conclude that the representative firm in all five of these industries
faces a marginal-production-cost curve that is upward-sloping and convex— an opera
tional definition, in our view, of a capacity constraint. In one industry, (the so-called
“miscellaneous durable goods” industry) the level of inventories brought into the
high-production season of the year is a decreasing function of the strength of the
economy. Such behavior may reflect that marginal-production-cost curves are either
upward-sloping and concave, or downward-sloping and convex. In either case, firms
8We investigated the robustness of our classifications with respect to various splits of the sample
period. We re-estimated our results over the following sub-periods: 67:3-80:12, 81:1-95:2, and 71:185:12. We found no instances in which an industry classification based on data for the full sample
period was contradicted by results for one of the sub-periods. There were several instances in which
industries that had defied classification over the full sample were classifiable over one or more of
the sub-periods. For example, the textile industry was not classifiable over the full period, but
showed evidence of upward-sloping and convex marginal cost curves over the 81:1-95:2 and 71:185:12 periods.
11
would have the incentive to bunch production rather than smooth it. The remaining
14 industries defy easy classification: In all cases but one, we reject the hypothe
sis of no interaction between seasonal and cyclical influences on production; that is,
the marginal-production-cost schedule appears to be nonlinear in those two factors.
However, the nonlinearity does not give rise to either a marked change in the overall
seasonal variability of production, or a change in the pattern of inventory holdings
that is systematically related to the pattern of production. As a result, we are un
able to classify these industries within the framework laid out in a simple two-period
model.
Aside from their implications for the shape of the marginal cost curve, interactions
between seasonal cycles and business cycles raise serious questions about standard
methods of seasonal adjustment. Krane and William L. Wascher (1995), building on
work of James H. Stock and Mark W. Watson (1989, 1991), develop a multivariate
framework that addresses some of the statistical difficulties involved in dealing with
such interactions. But there still remains the basic issue of whether the interaction
term should be treated as ‘seasonal’ or ‘cyclical,’ and, at a more fundamental level,
whether seasonal adjustment makes sense at all when seasonals and cycles do not
neatly decompose.
Another area for future exploration involves the implications of our capacity con
straint explanation for interactions between seasonal and cyclical variation. Ifthe key
to the interactions is the degree to which capacity can be adjusted in different indus
tries, then the amplitude of seasonal and cyclical interactions should be correlated
with capacity adjustment costs. A test of this correlation would be of considerable
interest.
12
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13
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