18
of the matrix of second derivatives. These statistics for the production and price series are
reported in Table IV. Superscripts on the means denote the series: production (q) and price (p).
The growth rates of production for the two states are significantly different for eight industries at

the a=.05 level of significance and ten industries at a=.10.

Figure 4 reports the estimated smoothed probability, DS)|¥ 1559739), that the industry

is in the recessionary regime (state 2) at date t.!5 A reasonable criteria for inferring that the

recessionary regime is more likely is p(s, = ly, »-¥730) > .5. The smoothed inferences on
state 2 indicate that the dates of the business cycle at the industry level are similar to the
traditional NBER dates of business cycles. This first month of the recession in the mid 1970’s is
nearly uniform among these 17 industries, indicating that the filter is capturing the dynamics of
the business cycle. For the chemical (SIC 28) industry, the filter dates the beginning of the
1973-75 recession as December, 1972, nearly a whole year earlier than the NBER date of
1973:IV. The turning point in April, 1975, however, is identical to the conventional NBER date,
1975:II. The chemical industry also experiences contractions conforming to the traditional
recessions of 1980, 1981-82, and 1990-91. These, too, are present in the other industries but
with more variation in the starting and ending dates. At the monthly and industry level the
algorithm also indicates the common feature across industries of contractions in the early months
of 1986 and in the latter half of 1994, but these rarely last more than three months.

Note that the filter typically permits a clear classification of months into regimes. The
smoothed probabilities are usually very close to 0% or to 100%. There are only 6 inferences

between 40% and 60% for the chemical industry. The lumber (SIC 24) industry is an exception

' The evolving probabilities are similar to the smoothed inferences. Using all the information in the sample reduces
the “noise” in the some of probabilities within a recession. For example, six months into the 1973-75 recession, the
probability of being in the recessionary state falls from 99% to 70%, but using all the information in the sample
revises that estimate back to 99%.


Table VI
Conditional Moment Normality Tests (Skewness and Kurtosis)*

Test: Joint Test of
Normality Skewness Skewness  Kurtosis Kurtosis

Variable: yr at Pt ot Pt
Industry
Food (SIC 20) 2.459 0.271 1,228 0.113 0.123
(0.652) (0.603) (0.268) (0.736) (0.726)
Lumber (SIC 24) 56.538 1.844 23.516 4.696 4.023
(0.000) (0.175) (0.000) (0.030) (0.045)
Furniture (SIC 25) 18.922 0.312 7.439 2.550 1.962
(0.001) (0.577) (0.006) (0.110) (0.161)
Paper (SIC 26) 8.984 3.222 4.427 4.085 3.397
(0.062) (0.073) (0.035) (0.043) (0.065)
Newspaper (SIC 271) 15.823 2.181 10.625 3.697 2.993
(0.003) (0.140) (0.001) (0.055) (0.084)
Chemicals (SIC 28) 10.670 4.349 2.088 0.309 4.218
(0.031) (0.037) (0.149) (0.578) (0.040)
Petroleum Refining (SIC 29) 10.007 0.003 1.656 2.084 8.345
(0.040) (0.959) (0.198) (0.149) (0.004)
Rubber (SIC 30) 23.788 12.757 3.774 12.229 5.255
(0.000) (0.000) (0.052) (0.001) (0.022)
Leather (SIC 31) 9.241 3.682 0.768 2.003 4.556
(0.055) (0.055) (0.381) (0.157) (0.033)
Stone, Clay, and Glass (SIC 32) 54.066 2.474 18.137 7.921 5.938
(0.000) (0.116) (0.000) (0.005) (0.015)
Nonferrous Metals (SIC 33:3-6, 9) 24.333 0.279 0.242 0.168 23.606
(0.000) (0.598) (0.623) (0.682) (0.000)
Fabricated Metals (SIC 34) 10.023 0.107 0.107 6.800 3.993
(0.040) (0.744) (0.744) (0.009) (0.046)
Agricultural Machinery (SIC 352) 22.579 1.473 10.208 0.478 4.401
(0.000) (0.225) (0.001) (0.489) (0.036)
Construction Machinery (SIC 353) 22.526 5.128 0.042 3.501 14.826
(0.000) (0.024) (0.838) (0.061) (0.000)
Metalworking Machinery (SIC 354) 22.620 1.076 9.348 0.055 2.626
(0.000) (0.300) (0.002) (0.814) (0.105)
Electrical Machinery (SIC 36) 44.848 3.755 15.090 1.693 3.317
(0.000) (0.053) (0.000) (0.193) (0.069)
Transportation Equipment (SIC 37) 36.802 1.288 11.152 9.713 3.963

(0.000) (0.256) (0.001) (0.002) (0.047)

Note: Column 2 statistics are asymptotically 7? . Statistics in columns 3-6 are asymptotically 77.
Asymptotic p-values less than .1 are in bold.

* The maximum likelihood estimates used in the calculations of the score were estimated without
the Bayesian correction.


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Figure 4b
Inferred Probability of Being in State 2 (Recessionary State)

Paper
(SIC 26)

E 10's6
s0'€6
B cols
F 10°06
50°88
© co'98
 io'ss
I sores
e 6018
# 10°08
@ soe
609L
F ose
# SO'EL
60'LL
10'0L
50°89
60'99
10's9
J soo
E 60°19
10°09

100%
80%
60%
40%
20%

0%

Newspaper
(SIC 271)

gs es xexgegeg
8828238 °

I 0's6

10°¥6
1%

LOT6
70°06
10°68
V8

40°98
P0'S8
108
14}

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008

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VLL
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TOL
VOL
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| rO'OL

10°69

Chemicals

(SIC 28)

100%

80%

60%

40%

20%

0%

10°S6
S06
60°16
10°06
$0°88
60°98
10°S8
SOEs
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10°08
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60°9L
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TOOL
$0'89
60°99
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60°19
10°09



12
different in recessions than in booms. For example, suppose that 5, is normally distributed.
Then prices in booms are normally distributed, while prices in recessions consist of a normally
distributed marginal cost component and a non-normally distributed price markup component.

This generates price changes in recessions that have a distribution that is not Gaussian.

Table I .
Model Predictions
St
1 2
b +m,
E(p,)=0 FC) =h Fi) <0
Eqy)= ine E(q,)=Inty
_ 2
1 | Yartpdd=2¢00% Var(p,) = (0 + ddo# + #103,
Yar(qe)=0 Var(q,) =0
Spl
E =I bev 0
(Pr) = In bm > E(p,)=0
2 | Eq)=inry ee aa
Vartp,)=(Bo+9n)08+4aom, | PPD = Mateo som)
Var(q,) = 0 alan) =
: 1 1
% == =z 19) == 19, > $99 > 4
obey?! (+m) 2 @ +i) pero 0

Il. Estimation Approach, Data and Empirical Analysis

The business cycle formulation of the previous section generates a specific time series
model of price and quantity changes. However, some features of the model are too restrictive.
For example, the predicted variance of quantity changes is zero in each state; only two different
quantity-change amounts are predicted to appear. In what follows we specify an empirical
model that is more general than what is laid out in Table I. We allow a nonzero covariance

between price and quantity changes. As we will discuss later, this permits a test of a prediction


Histogram of Price Changes by State

Figure 6e

Transportation Equipment (SIC 37)

4

3

32

z

a

0

bee MN
SU TAE Rate Ss raat eu so

Transportation Equipment (SIC 37)

40 State 1

B30

3 2»

3

: lh

= 10 |

° alll.
gxaRgesaegergseRgase gas
SU POG aT ESSA AAI FH



Frequency Frequency Frequency Frequency
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6.00 6.00 -6.00 -10.00
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S s
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gz g ~
1.50 a 1.50 ray 1.50 2.50
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5.25 5.25 5.25 8.75
6.00 6.00 6.00 10.00
Frequency Frequency Frequency Frequency
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eS88sSsa ous as
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5.25 °5.25 oe 5.25 -5.25
4.50 4.50 4.50 4.50
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sy = —
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3.00 3.00 3.00 3.00
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450 450 450 4.50
5.25 5.25 5.25 5.25
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6.00 6.0 6.00

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2.25 2.25 2.25

3.00 3.00 3.00

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4.50 4.50 4.50 4.50

5.25 5.25 5.25 5.25

6.00 6.00 6.00

aje7g Aq SaduvyD ao11g Jo WeAdOISIT

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22

Distribution of Price Draws
We have assumed that y, has bivariate normal distributions in all states. Assuming that
expansionary price changes come from a normal distribution, the model predicts that
recessionary price changes come from a non-normal distribution because firms use non-normal
mixed strategy distributions for prices during recessions.'’ Following the work of Newey (1985)
and Hamilton (1996), we make use of the score statistics in tests that certain moment restrictions
- of the normal distribution hold for the data for both regimes. In particular, we will take the
standard approach and examine the third and fourth moments, jointly and individually for each

series. The null hypothesis then implies the following expectations hold for the joint test:

2
#306 = SVs Yrs OMY, -n,)"}-0 and

s=1

sel oP

2 oy
E ee ee | =0

Hamilton (1996) provides the scores for the univariate case, which can be naturally extended to

the bivariate model. The sample counterparts to the above moment restrictions are

2
Ys, = Sly. ys Oy, -B,)°

5al

tel 2 A ~ Gig ,
PCS, = Sil¥15---5¥758)-] (Y.-H) = 3) 4
os,

sal ap

Let M be the 7x 4 matrix whose r" row is given by

'

2 ‘ 2 oF,
[Spe = Slyyoos¥ 0, -2,)"] [En Sanco) |, -a 4 ‘ | ,

sal sal 5.58

"Even though the recessionary price changes may not be Gaussian, the estimates in Table III are consistent quasi-
maximum likelihood estimates.


Figure 1
Capacity Regions for the Pricing Subgame

"

{@,y)ix 20,y20,x +y Sa}
{(@,y):x 2a,y 2a}
={(x,y):x 20,y 20, (2,y) € A, @&y) € B}

W

A
B
Cc


3
duopoly model of investment and pricing in which product demand may be either high or low.
This model can be viewed as an extension of Kreps and Scheinkman (1983) to allow for demand
uncertainty. As long as the difference in demand between high and low levels is limited,
equilibrium prices conditional on low demand involve a markup over the short run competitive
price and positive variance. Later in Section I, we embed the two-stage model in a dynamic
business cycle setting in which demand alternates stochastically between fast-growth (boom)
phases and slow-growth (recession) phases. This formulation provides the structure for our
empirical tests. This specification also corresponds to Bagwell and Staiger’s (1997)
specification of demand changes in their analysis of optimal collusive mechanisms.

The empirical analysis in section II does not attempt to estimate price markups directly.”
Instead, the analysis investigates an implication of our non-collusive oligopoly model for
observed prices. The dynamic version of the model predicts that price changes are more variable
during recessions than during booms, because firms employ mixed strategies during recessions.
Another prediction resulting from the mixed strategy equilibrium is that the distribution of price
changes differs in recessions. These implications are tested using a type of time series switching
regime filter employed by Hamilton (1989). We examine seventeen manufacturing industries at
the two- and three-digit Standard Industrial Classification (SIC) level. A separate time series
model is estimated for each industry. Fourteen of these industries have substantially higher price
change variances and distinctly different forms of the price change distribution when production
is in a recessionary state.

Before beginning our analysis, we briefly review the relevant empirical evidence on price
markups. Domowitz, Hubbard, and Petersen (1987) examine the empirical evidence on cyclical

responses of prices and price-cost margins. With a panel data set of industries at the four-digit

? The estimation of price-cost margins or of markups over a competitive price involves potential measurement


Table II
Industries and Associated Price Indices

Industry SIC Code Price Index (BLS series)

Foods 20 PPI, Processed foods and feeds (wpu02)

Lumber and products 24 PPI, Lumber and wood products (wpu08)

Furniture and fixtures 25 PPI, Furniture and household durables (wpu12)

Paper and products 26 PPI, Pulp, paper, and allied products (wpu09)
Newspapers 271 CPI, Newspapers (cuur0000se5901)

Chemicals and products 28 PPI, Chemicals and allied products (wpu06)
Petroleum products 29 PPI, Petroleum products refined (wpu057)

Rubber and plastic products 30 PPI, Rubber and plastic products (wpu07)

Leather and products 31 PPI, Leather (wpu042)

Clay, glass, and stone products 32 PPI, Nonmetallic mineral products (wpu13)
Nonferrous metals 33:3-6,9 PPI, Nonferrous metals (wpu102)

Fabricated metal products 34 PPI, Fabricated structural metal products (wpu107)
Agricultural machinery and equipment 352 PPI, Agricultural machinery and equipment (wpul 11)
Construction machinery and equipment 353 PPI, Construction machinery and equipment (wpu1 12)
Metalworking machinery and equipment 354 PPI, Metalworking machinery and equipment (wpul 13)
Electrical machinery and equipment 36 PPI, Electrical machinery and equipment (wpul 17)
Transportation equipment 37 PPI, Transportation equipment (wpul4)



19
with frequent regime switches and more months with less conclusive inferences on the state.

The estimated switching probabilities can be used to calculate the expected duration of a
regime. A regime i is expected to last (1 - pi! months. The postwar historical average
recession according to NBER dating is 4.7 quarters which would imply that the probability of
remaining in the production contraction regime is 92.01%. Many of the industries have smaller
probabilities suggesting shorter recessionary periods at the industry level.

Columns 3 and 4 in Table IV report the results of Wald tests of the null hypothesis that
the growth rate of real prices is identical for both regimes. The chemical (SIC 28) industry is the
only industry with countercyclical pricing trends, and the nonferrous metals industry (SIC 33:3-
6, 9) is the only industry with strong procyclical trends in real prices.

Estimates of the Price Variance
The primary testable implication of our non-collusive oligopoly model is that the

2

variance of the price changes, o¢ o
ir

, is greater in the recessionary regime than during times of
production growth. Using a likelihood ratio test, the null hypothesis that we now test is

Ao: Sip =o Fi pe
The results from these tests are reported in Table V.

For every industry, save one, we can reject the null hypotheses of equal price change
variance draws across regimes. As the estimates in Table III show, the recessionary variance for
price changes is anywhere from 6 to 17 times the expansionary variance. For these sixteen
industries, the average ratio of the state 2 price variance to the state 1 price variance is 11.69.
Because the inferred probabilities of being in either state are usually close to 0% or to 100%, we
are confident that the prices are much more variable in a recession than in an expansion.

The lone exception is the nonferrous metals industry (SIC 33:3-6, 9) for which the point


25
a fairly high level of aggregation (two- and three-digit SIC). The optimal collusion theory may
work very well for some selected industries in which collusion is likely to be successful.

However, it does not appear to be operative for a large number of U.S. manufacturing industries.

Til. Concluding Remarks

Our analysis of pricing over the business cycle focused on a non-collusive oligopoly
formulation. The theoretical analysis predicts that output price changes exhibit higher variance
during recessions than during boom periods and that the form of the distribution of price changes
differs across recession and boom periods. The analysis also predicts that the markup of output
price over the short run competitive price (conditional on demand and capacity levels) is higher
in recessions than in booms. Evidence from U.S. manufacturing industries is consistent with the
hypothesis that price changes have higher variance in recessions than in booms. In fact, price
changes are much more variable in recessions than in booms for almost all industries studied. In
addition, the frequency distributions of these price changes in recessions are distinctly unlike
their expansionary counterparts. These empirical results complement the direct support for the
non-collusive oligopoly model from laboratory experiments reported in Wilson (1997).

Our empirical analysis does not attempt to measure directly price-cost margins or price
markups over the business cycle. Instead, we take an indirect approach that evaluates a key
implication of our pricing model. The evidence is consistent with the view that oligopoly firms
charge prices involving a higher markup over the short run competitive price during recessions
than during boom periods. This pattern of markups (as we define markups) may amplify the
effects of demand shocks on output. This in turn may provide a mechanism through which
oligopoly market power exacerbates the effects of aggregate demand shocks on the

macroeconomy.


21
point estimates for the expansionary price change variance are identical (.168) for the
agricultural machinery (SIC 352) and metalworking machinery (SIC 354) industries, but they
differ in recessions, 1.620 and 2.341, respectively. This suggests that these two industries with
similar inputs have different oligopoly pricing in recessions.

The food (SIC 20) industry may also provide some insight into how much of the increase
in the price variance may be due to economy-wide reasons other than oligopoly pricing
incentives. Our theory of oligopoly pricing may not be applicable to this industry because
production in the food industry is growing at a constant rate throughout the sample (see Table
III). Nevertheless, the filter estimates two empirically distinct regimes. Figure 4a illustrates that
food industry is in regime 2 from 1973-1981 and from 1990-91. The former time period was a
period of high inflation and general economic stagnation and the latter was the 1990-91
recession. As Table V reports, the two regimes for the food industry have statistically different
price change variances, apparently for reasons not due to oligopoly pricing incentives. If this is a
macroeconomic phenomenon, it may be a component of the state 2 price variances for the other
industries which do experience expansions and contractions. However, the food industry’s price
change variance only increases by a factor of six, the smallest of all the significantly different
variances (see Table V). For the fourteen industries which have distinct expansionary and
recessionary states, the average ratio of the state 2 price change variance to the state 1 price
change variance is 12.20. This suggests that if there is a macroeconomic source for a higher
price change variance in recessions, it is not responsible for all of the increased variance.
Moreover, as the next subsection discusses, for the fourteen industries with expansions and
contractions we find support for the additional prediction on the change in the form of the price

change distribution.


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£95°0 850°0- tZ'0 L970- €£0°0 891°0- a
(1970) (o€'0) (092'0) (86€°0) (spv'0) (680°0)
6Eb'0- Lve0- 60'0- LIT £80°0- 8610 a
(€70'0) (v€0'0) (€z0'0) (0z0'0) (090°0) (o€0'0)
$20°0- rer'0 1€0°0 $90'0- 1S0°0 1v0'0 a
(€v0'0) (s90'0) (9L0'0) (060°0) (811'0) + (870°0)
r6r'0 7r0'0 920 OLE0 0970 6170
(gzoIs) (147 OIs) (97 OIS) (sz Is) (pz OIS) (02 Is)
syeormoyD Jodedsmon, sodeg eanyuwin yt soquiny poor Jojoureseg

(d*'b] = ’4 107 soqeunsy pooyyoyry wnaxeypy

TH 148.


16
deseasonalized using Harvey’s (1993) basic structural model (pp. 142-44). In terms of the
notation in the model described above, gq; is 100 times the difference in natural logs of the index
of industrial production, and p; is 100 times the difference in natural logs of the corresponding

price index.

Estimates and Business Cycle Specification Testing

The model was estimated for twelve industries at the two-digit SIC level and five
industries at the three-digit level.'*"? The estimated persistence of both boom and recessionary
states is high for all industries, so there are only a small number of months representing
transitions from one state to another. Because of this, the standard errors associated with
estimates of 44), 4;,Q,,,andQ,, are large. Therefore, in Table III we report parameter
estimates for the persistent states and the estimated state transition probabilities for each of the
seventeen industries.

It is clear from the estimates that the filter is capturing the dynamics of business cycles as
opposed to structural changes in positive growth rates. For fourteen of the seventeen industries,
the production mean growth rate is statistically different from zero in state 1. The mean change
in production for state 2 is also negative for these fourteen industries, but not always statistically

different from zero.’

" We thank James Hamilton for kindly distributing his Gauss code with which we replicated his results. Additional
code was added for the likelihood ratio tests and bivariate conditional moment tests.

' The following industries are not included: Tobacco (SIC 21), Textiles (SIC 22), Apparel SIC (23), and
Instruments (SIC 38). The real price of tobacco was fairly constant from January, 1960, to August, 1982, after
which it began a nearly deterministic climb from 102 in September, 1983, to 234 in July, 1993. Then the real price
fell 24% in August, 1993, and does not change much for the rest of the sample. Clearly a more complicated model
explaining these events is necessary before the switching regime model is applied. The BLS does not publish a
price index solely for textiles, but one for Textiles and Apparel. It did publish a separate series for Apparel, but it
ends in 1977, There is also no price index which adequately captures the general industry, Instruments (SIC 38).
The five industries at the three-digit level represent three industries for which good measures of the price could not
be found at the two-digit level.

4 Furthermore, following Engel and Hamilton (1990) we use a general null hypothesis to test whether a series
follows a random walk against the alternative of the two-state regime model. The Markov switching probabilities


equilibria, which includes a symmetric capacity pair. If @>a(1—c/v)/(3@-2a), then
reaction functions are as in Figure 2B. Again, there is a continuum of equilibria. However, the
equilibrium set does not include a symmetric capacity pair because reaction functions have a
discontinuity involving a “jump” across the 45 degree line.

The main results may be summarized as follows.’ There is a continuum of equilibria.
Equilibrium capacities sum to @, the level of high demand. If demand is high, then in stage two
both firms set price v, which is the upper bound of the set of short run market clearing prices.
There is no excess capacity. If demand is low, then in stage two both firms adopt mixed
strategies for prices, with prices above the short run market clearing price of zero. Prices have
positive variance and excess capacity emerges. An illustration of equilibrium price distributions
for a low demand realization is provided in Figure 3. The figure illustrates the skewness of
pricing distributions that is a feature of equilibrium mixed strategies in Bertrand-Edgeworth
models.

We use the term price markup to indicate the difference between a firm’s price and the
(maximum) short run competitive price.® Equilibrium prices are procyclical while equilibrium
price markups are countercyclical in the simple two-stage game.

The predicted variation of prices when demand is low may seem to be at odds with the
notion of price rigidity that was noted at the beginning of the paper. After all, New Keynesian’s
explain that the difficulties posed by imperfect competition are due to inflexible prices rather

than to excessive price variation. In spite of this apparent contradiction, there is in fact a form of

* The two-stage model is formulated as a duopoly. However, the basic results on markups and price variability
continue to hold for markets with more than two firms, as long as the market does not become too competitive. For

example, suppose that there are n 22 equal sized firms, with total capacity equal to 7. If n+1<@/(a@ —a), then
when demand is low, equilibrium prices are draws from a mixed strategy distribution; prices involve a positive
markup over short run marginal cost and prices have positive variance.

® Our usage of the term markup may be somewhat unconventional. The markup term is often used to indicate the
difference between price and short run marginal cost. In our setting, short run marginal cost is not defined for


31

This capacity choice is optimal for firm one when ¥(y) <a -— y, which is possible only when

(A.2) does not hold. If (A.2) does not hold then xg a@)< 4a and ¥(y) <@-—y for an interval

of y-values exceeding 4a . The best response to y € G @, al is,

(A.4) B(y) = min{(y), max{@ - y,5(E(a) — cy/ vy}.
If (A.2) does not hold there is also an interval of y-values below 4a for which the best response

is ¥(y)<@—-y. In such cases, there is a local maximum that involves firm one being the large
firm and choosing capacity, @— y; however, the global maximum is for firm one to respond as
a small firm, choosing ¥(y)<y. Once y becomes sufficiently small, firm one’s best response
“jumps” up to @— y. The reaction correspondence for firm one is depicted as R, in Figure 2B.

The set of equilibria is the set of capacity pairs that sum to @ such that, (1) neither x nor y
exceed a critical value defined in the previous paragraph and, (2) an interval of capacity pairs
around x=y is excluded (when firm two is the large firm, the excluded set involves y-values

such that ¥(y)<@-y).


28

Hamilton, J. D. “Specification Testing in Markov-Switching Time-Series Models, Journal of
Econometrics, 1996, 70, 127-157.

Harvey, A. C. Time Series Models. Cambridge, MA: MIT Press, 1993.

Klemperer, P. “Competition When Consumers Have Switching Costs: An Overview with
Applications to Industrial Organization, Macroeconomics, and International Trade,” Review of
Economic Studies, 1995, 62, 515-39.

Kreps, D. M., and J. A. Scheinkman, “Quantity Precommitment and Bertrand Competition
Yield Cournot Outcomes,” Bell Journal of Economics, 1983, 14, 326-337.

Newey, W. “Maximum Likelihood Specification Testing and Conditional Moment Tests,”
Econometrica, 1985, 53, 1047-70.

Reynolds, S. and B.J. Wilson, “Market Power, Price Markups, and Capacity Investment
under Uncertain Demand,” working paper, University of Arizona, 1997; available as PDF file at
http://www.u.arizona.edu/~sreynold/

Rotemberg, J. J., “Sticky Prices in the United States,” Journal of Political Economy, 1982, 90,
1187-1211,

Rotemberg, J. J., and G. Saloner, “A Supergame-Theoretic Model of Price Wars during
Booms,” American Economic Review, 1986, 76, 390-407.

Rotemberg, J. J., and M. Woodford, “Oligopolistic Pricing and the Effects of Aggregate
Demand on Economic Activity,” Journal of Political Economy, 1992, 100(6), 1153-207.

Rotemberg, J. J., and M. Woodford, “Markups and the Business Cycle,” Macroeconomics
Annual, 1991, 6, 63-129.

Stiglitz, J. “Price Rigidities and Market Structure,” American Economic Review (Papers
and Proceedings, 1984, 74(2), 350-55.

Wilson, B. J., “What Collusion? Unilateral Market Power as a Catalyst for Countercyclical
Pricing,” working paper, University of Arizona, 1997.

Wilson, B.J., Market Power and Cyclical Pricing: The Macroeconomic Implications of
Industrial Organization, Ph.D. Dissertation, University of Arizona, 1997.

Yellen, J. “Efficiency Wage Models of Unemployment,” American Economic Review, 1984, 74,
200-205.


by linear average variable costs. The aluminum industry is part of one of the industry groups
considered in our empirical analysis. Our empirical results for this industry are largely
consistent with theirs; we find no evidence of price markups above the competitive price in
recessions for this industry. However, this industry is the exception rather than the rule among
the industries that we examine.

Wilson (1997) reports evidence from laboratory experiments on oligopoly pricing. The
experiments are similar to Davis and Holt’s (1994) posted offer pricing experiments except that
Wilson considers the effects of a demand shift rather than the effects of a supply/capacity
change. The results are broadly consistent with the model’s predictions. When demand is high,
prices are near the short run competitive level. When demand is low, prices remain above the
short run competitive level and prices are more variable than when demand is high. When
demand is low prices fail to conform precisely to the equilibrium mixed strategy predictions.

Prices appear to follow a disequilibrium process similar to an Edgeworth cycle process.

I. The Theoretical Model

We develop a duopoly model of capacity investment and pricing. The level of demand is
uncertain when firms make investment decisions but is known before firms make pricing
decisions. The initial formulation involves a simple two-stage game with a step demand

function?

Later in this section we embed the two-stage game in a dynamic, business cycle
setting that provides the basis for the econometric analysis.
Two-Stage Model

We assume there are two levels of demand, “low” and “high”. Let a €{a,q@} represent

the total mass or number of customers, with 0<a<a. The probability that demand is low is

* Reynolds and Wilson (1997) analyze more general models of this type, including formulations with a downward


4
SIC level spanning 1958-1981, they find that more concentrated industries have more procyclical
margins. As they note, these estimates may be biased upward (downward) if marginal cost is
greater (less) than measured average variable cost. Consistent with the Rotemberg and Saloner
predictions, Domowitz et al. further find that industries with high price-cost margins have more
countercyclical price movements. However, Domowitz et al. use industry-level changes in
capacity utilization as a proxy for business cycle movements. Low capacity utilization at the
industry level may simply be a result of high prices, rather than the result of a downward shift in
demand. Bresnahan (1989) also points out the limitations of cross-industry comparisons of
competition when assessing cyclical variations of margins and prices.

To avoid the problems of using accounting data for estimating the price-cost margin,
Domowitz (1992) takes an approach that examines total factor productivity. He adjusts the
Solow residual to allow firms to price above marginal cost and then permits the price-cost
margin to vary with the level of aggregate demand as measured by capacity utilization in
manufacturing. Domowitz’s point estimates indicate that there is a negative correlation between
the margin and aggregate demand movements; however, the standard errors are large enough so
that the null hypothesis of acyclicality cannot be rejected.

Bresnahan and Suslow’s (1989) study of the aluminum industry does not reveal any
evidence of oligopoly market power. They develop an econometric model of short run supply,
capacity constraints, and long-lived capital. Employing a switching regression model, they find
evidence of two regimes in their reduced form quantity-produced and quantity-shipped
equations. The implication is that in the high demand regime, prices are determined by the
vertical portion of a supply curve when production is constrained at capacity. Output is

unconstrained in the second regime, output falls well short of capacity, and prices are determined

problems, as explained below.


17

The production growth rates for the newspaper (SIC 271) and metalworking machinery

(SIC 354) industries are not statistically different from zero in state 1 or 2. Leather (SIC 31)

production is trending down for the entire sample, and the mean growth rate in production in the
food (SIC 20) industry is constant.

An alternative empirical approach is to estimate a model in which price and quantity

changes depend on the current state (s, € {1,2} ) but not on the previous state. Such a model has

two bivariate distributions for p and g, rather than four. The point estimates and the standard
errors from estimating such a two-state model differ only slightly from those presented in Table
III. The statistical tests (below) that compare means and variances in different states are based

on estimates from this two state model that includes the effect of the transition observations
(s,, =i,s, = /) on the state j estimates (see footnote 9).
A Wald statistic for testing whether the means for one of the component series, yu, , are

different across states is given by

(4 -BY
van( ji) + var( ji) — 200vih, fi)’

which is asymptotically distributed y?. The asymptotic variance and covariance of the

parameters, denoted by var(. i) and cév(/i,, /,), are estimated from the inverse of the negative

are not identified in the model when Hy = Hy and Q, = Q,, but asymptotically valid tests of the following null

hypothesis do exist- Ho: Py, =!- P2> Hy #HQ> and Q, #Q,. As in Engel and Hamilton, we test the null
hypothesis of a random walk with both a Wald and likelihood ratio statistic. The Wald statistic is given by

- 5 ye
(Py, - U- Ba)I
var( By 1) + Var(Pyy ) + 2€6W(By 1. P 29)

rR. The likelihood ratio test compares the unconstrained log

likelihood to the largest log likelihood when p,, = 1 - p22 and is asymptotically distributed as u- These test
statistics are not reported here. The Wald test statistics are all larger (in some cases much larger) than the likelihood
ratio test statistics, but the even the latter soundly rejects the null hypothesis of a random walk in favor of the two
regime model. (The smallest LR statistic is greater than 68.) In sum, the switching regime model is capturing the
dynamics of the business cycle for these industries.


function (see Reynolds and Wilson (1997)):

vx (x,y) eA

v(a—x)x/y (x,y)EC,xsy<a
r(x,y,a)=4v(a—x)x/a (x,y)EC, x<asy

v(a- y) (iy) EC, x>y

0 (x,y)eB

Expected revenue for firm two is given by r(y,x,a). The function r(-) is continuous in each of its
arguments.

In stage one firms choose capacities. The level of demand is uncertain in stage one.

Expected profits as a function of capacities are
m(x,y) = &(x,y,a) + (1- O)r(x,y,@) - cx
for firm one and z(y,x) for firm two.
The following two assumptions are utilized.

Assumption Al: (1—@)v>c

Assumption A2:

nile

a>a

Al is a condition that makes it attractive for firms to invest so that total capacity is sufficient to
serve all consumers when demand is high. Without A/, firms would not (collectively) invest in
capacity beyond a, and excess capacity would never emerge. A2 limits the difference in
demand levels; high demand is no more than 50 percent above low demand. In the absence of a

condition like A2, an equilibrium may involve a Bertrand-type outcome with prices equal to
short run marginal cost when demand is low.

Under A/ and A2 a pure strategy equilibrium in capacity choices exists. The derivations
appear in Appendix A. There are two principal types of reaction function configurations. If

0<a@(1-c/y)/ (3a — 2a), then reaction functions are as in Figure 2A. There is a continuum of

bound on the interval of competitive prices.


23
1
and let the matrix D be the 7x 12 matrix of scores. For § = 7TM 'M -M'D(D'D)'D’M), the

Wald statistic, 7r’S“'r, is asymptotically distributed as a y; and tests whether g, and p, are
jointly normally distributed. For only testing one restriction (column) of M, the test statistic is
asymptotically 7? .

The results of the joint and individual tests for skewness and kurtosis are reported in
Table VI. Column 2 reports the statistic for the joint test of normality for both regimes. There
are clear rejections for all industries except food (SIC 20). Most of the individual series reject
the null hypotheses of no skewness and/or kurtosis equal to 3. Recall that the moments being
tested include both states.

Histograms of the price draws by state are shown in Figure 6. A price change is

considered to be in state j (j= 1,2) if p(s, = jly, ys ¥ 3) >.5. These histograms for the price
draws in state 1, the expansionary state, appear to be consistent with a normal distribution
whereas the state 2 draws clearly are not. The variances of these draws are larger as our tests
above indicate, and the skewness and non-normal kurtosis detected in the conditional moment
tests is apparently coming from the recessionary price draws. Many of the industries have state 2
price draws with means shifted to the left, but with longer tails to the right.'® Other industries
like electrical machinery (SIC 36) and metalworking machinery (SIC 353) appear to have bi-

modal price distributions during recessions.

Comparing Predictions of Collusive and Non-Collusive Models

Rotemberg and Saloner (1986) analyze a repeated game model of oligopoly pricing in

which demand fluctuates randomly and independently over time. Firms are informed about the

'® These conditional moments tests are influenced, but not overwhelmingly so, by a common outlier in August,
1973. Food prices jumped 9% that month causing the overall PPI to rise abruptly. For many industries, this real


Price Movements over the Business Cycle

in U.S. Manufacturing Industries’

by

Bart J. Wilson®
and

Stanley S. Reynolds?

Abstract: This paper develops and tests implications of an oligopoly pricing model. The model
involves capacity investments that are made before demand is revealed and pricing decisions that
are made after demand is known. The model predicts that during a demand expansion the short
run competitive price is a pure strategy Nash equilibrium, but in a recession firms set prices
above the competitive price. Thus, price markups over the competitive price are countercyclical.
Prices set during a recession are more variable than prices set during expansionary periods,
because firms use mixed strategies for prices in recessions. This model is confronted with data
from U.S. manufacturing industries. The empirical analysis utilizes a time series switching
regime filter to test the unique predictions of the model, namely that (1) price changes are more
variable in recessions than in expansions and (2) the form of the distribution of price changes
differs between recessionary and expansionary regimes. Fourteen out of fifteen industries have
fluctuations consistent with this oligopoly pricing model. The data is also analyzed to compare
the predictions of this model with those of an optimal collusion model.

Bureau of Economics, Federal Trade Commission, Washington D.C.
Department of Economics, University of Arizona, Tucson, Arizona.

* Wilson acknowledges research support from the John M. Olin Foundation and the National Science Foundation
(Grant No, SBR-9507526). This paper is an extension of Chapter 4 of Wilson’s Ph.D. dissertation. Doc Ghose,
Mark Walker, and John Wooders provided helpful suggestions. This paper has also benefited from comments by
seminar participants at the University of Arizona, the University of Houston, Johns Hopkins University, the
University of Kansas, the University of Toronto, and Virginia Polytechnic Institute and State University. This work
represents the opinions of the authors and is not intended to represent the position of the Federal Trade Commission
or any individual Commissioner,


Figure 4d
Inferred Probability of Being in State 2 (Recessionary State)

Stone, Clay, and Glass

(SIC 32)

80%
60%
40%
20%

0%

100%

10°S6
SO0'€6
60°16
10°06
S0'88
60°98
10°S8
SO'es
60°18
10°08
SO'8L
60°9L
Tose
SO'EL
6OTL
10'0L
$089
60°99
10°s9
so'e9
6019
10°09

Nonferrous Metals

3-6, 9)

(SIC 33:

Ba x0's6

16
90°76

| 2016
is

90°88

F COLES

vss

E o0'r8
R oes
| ris

£ 90:08

CO'6L

a ULL
B 90°9L
© wOSL
Bel
E902
oo
£169
F 90°89
+ 209

Fabricated Metals

(SIC 34)

80%
60%
40%
20%

0%

100%

10's6
S0'€6
6016
10°06
$0°88
60°98
10°S8
So'es
6018
10°08
so'sL
60°9L
TO'SL
SO’eL
6OTL
TOOL
$0'89
60°99
10'S9
so'e9
60°19
10°09



10
learn the actual level of demand and choose prices. For industries with large irreversible
capacity investments (e.g., chemicals), this must be the interpretation.

We add a feature to the model to allow for price variations due to variations in production
costs. Let 5; be the real short run marginal cost of production in ¢, and let r,; be consumers’ real

reservation price in ft. Assume that }, is an iid. random variable with E(b,)=5>0 and

Var(b,) = o;. Furthermore, we assume that r; = b, + v; i.e., reservation prices and short run
marginal costs are perfectly correlated. Firms observe r, and b, after capacities are set in ¢ and
before prices are set in ¢. Variations in r; and 5, do not influence firms’ expected revenue
associated with capacity investments, since v is constant.

The marginal cost of capacity investment is also permitted to be time dependent through
the state variable, s, Let c(s;1) be the marginal cost of capacity in f Assume that
0<c(1) <c(2) <v so that the marginal cost of capacity following slow growth periods is less
than or equal to marginal capacity cost following high growth periods. The following two
assumptions are counterparts to A/ and A2.

Assumption B1: (1— pj2)v > c(i), 1=1,2
Assumption B2: 31 >t

Consider first the two-stage game for a single period ¢, given any a, and s;;. Under
assumptions B/ and B2 an equilibrium pair of capacities exists for this two-stage game.
Equilibrium capacities in period ¢ sum to t,a,_,. The set of equilibrium capacities may differ
depending on the value of state s,.;, since the probability of low demand and the marginal cost of
capacity are both permitted to depend on s;.). If demand is high then both firms set a price equal
to r;, the real reservation price. If demand is low, then firms adopt mixed strategies for prices

over a support contained in (4, ,7,].


9
price rigidity in the our model. The rigidity can be understood by noting that when demand
changes from high to low, equilibrium prices decrease by less than short run competitive prices.

A Dynamic, Business Cycle Formulation

In an analysis of business cycle fluctuations, Hamilton (1989) formulates a model of the
growth rate of aggregate U.S. output. This formulation involves an unobserved state variable
that indicates whether the economy is in a low-growth or a high-growth state, and a Markov
process governing transitions between states. Hamilton develops an estimation procedure for
this model and finds that it provides a good characterization of aggregate business cycle
fluctuations.’ This type of two-state growth process is also utilized in Bagwell and Staiger’s
(1997) theoretical analysis of collusive pricing over the business cycle.

We adapt Hamilton’s business cycle model to formulate a model of demand fluctuations
in a single market. Let a; denote the market size in period f. 5; is the “state” at date ¢ and takes
on the value of one or two. The evolution of the state variable is governed by a Markov chain

process with, py =Prob[s, = j|s,;_, =i], i,j € {1,2}. The size of the market follows a state
dependent trend a,,;=7,,,a, where t;>t2>0. In percentage terms, demand growth

between ¢ and 1 +1 is, In(a,,,) — In(a,) = In(ts,)-

There are two stages within each time period. Capacities are chosen in the first stage of
each period. In stage one of period ¢, a, and s,.; are known, but a, is unknown. So, if s.1 = i,
then pj is the probability of low growth (or contraction if m <1) from ¢1 to ¢. Firms choose

prices in stage two of period ¢ after observing the level of demand, a;. Another interpretation is

that firms are producing to a stock each period, but it is after production occurs that the firms

output equal to capacity. We use the short run competitive price as a benchmark instead of marginal cost.

” The business cycle dates from Hamilton's filter closely correspond to the traditional NBER dates, It is also worth
noting that the unobservable state is only one of many influences on the growth rate so that output could be falling
even though the economy is in the “high” growth rate state.


Figure 3
Equilibrium Mixing Distributions
6= 1/2, clv=1/3, ala =5/4

Small Firm

TM

TMs—

Large Firm

13v/18


30
any best response must be in [a — y, y]. Firm one’s expected profit function is not differentiable
at x=@-—y; expected profit is strictly concave in x for x e{a—y,y]. This implies that there
is a unique best response to y €[}4, 4] that lies in [a-y,y].

If the condition,

io
(A2) o<——

~ 3a-2a?

holds then the best response to y € G @, al is,
(A.3) B(y) = max{@ - y,5(E(a) - ey! v)}.

From (A.3) it follows that b(4a)=4a and b(y)=@-y for some interval of y-values

extending above yoda. Inequality (A.2) also implies that @~y is a best response to
y e[0,4a); ie., the local best response for firm one as a large firm is also a global best

response to y €[0,4@). So, inequality (A.2) yields a reaction function for firm one that is as
2

depicted by R in Figure 2A. The reaction function for firm two is defined symmetrically. The
set of equilibria is the set of capacity pairs that sum to @ such that neither x nor y exceed some

critical value. This critical value is the lowest value of y such that b(y) in (A.1) or (A.3) exceeds
a-y.
If (A.2) does not hold then a best response to y el} a,a] may be less than @-y. For

some values of y € G G, a], expected profit for firm one is maximized by choosing capacity,

¥(y) = fla + (1-0 -c/ v)y/ 8].


Figure 4e
Inferred Probability of Being in State 2 (Recessionary State)

Agricultural Machinery and Equipment

(SIC 352)

100%

80%

60%

40%

20%

0%
Sssesesgpsgssegagraxesess sss sgesgssggaa
Ferrer ree* FEBSSERSERERBSRRKVSAB

Construction Machinery and Equipment
(SIC 353)

aena -~ a
ssgssas
a en eS ad
oo an a

Metalworking Machinery and Equipment

(SIC 354)

100%

80%

60%

40% +

20%

0%
Ssegsesesesrcssrseegsesgs sags
sef®eraef&ee Fest gst ee Fag hs



29

Appendix A
Analysis of Two-Stage Game
In stage one, firm one’s expected profit is,
A(x,y) = (x,y,a) + I O)r(x,y,4) - x,

where r(-) is the expected subgame revenue, defined on p. 7. 1 is continuous in (x,y) since
r(-)is continuous in these arguments. However, x is not differentiable in x for all capacity
levels, so one cannot rely exclusively on first-order conditions to characterize best responses.

Consider the investment incentives for firm one, conditional on firm one being the larger

firm (x > y). In this case assumption A/ implies that expected profit is strictly increasing in x for
x+y<@; n(x,y) =v(E(a)- y)-cx for x+y >a, so that 7 is strictly decreasing when total
capacity exceeds the high demand level; and 7 is continuous in x at x =@-y. Therefore, if
y< 4a then a reaches a local maximum at x =@-y. This may not be a global maximum

since ~ may attain a higher value for some x < y.

The payoffs for firm one are more complex when it is the smaller firm (x < y). Consider
first the best response to y €[a,@]. Any best response will be less than a, since A2 implies that
m is strictly decreasing in x for x > a. Given y €[a,@], assumption A2 implies that 7 is
strictly inx for O<x<@-y. mn is strictly concave inx for @~ y<x<a. So, there is a unique

best response to each y €[a,@], which lies in @- y<x<a. This best response is,

_ = _ ,, (Ov-c)ay+ (1- 9)vaa
(A.1) by) = mesa Ow +U—-Ova) (—@yvay }.

Suppose that yel4a,q). If x<a-y then n(x,y)=vx-cx, which is strictly

increasing inx. If x2y then =(x,y) = v(E(a) — y) — cx, which is strictly decreasing in x. so


6
6 €(0,1). All consumers are assumed to have a common value, v, for one unit of the product;
the market demand is then a step function with height v and length a.

The marginal cost of capacity in the first stage is c. Firm one chooses capacity x and firm
two, capacity y. The marginal cost of production is constant in stage 2, and is normalized to
zero. The parameter v can be interpreted as the value of the good minus the short run marginal
production cost. We assume that both firms know v and c, with y>c> 0.

The level of demand is observed before firms set prices in stage two. A subgame in stage
two is defined by the triple, (x,y,a). There are three cases to consider. First, suppose that x+y < a.
This is region A in the graph of capacity pairs in Figure 1. If the capacities are in region A, then
a price equal to v is a dominant strategy for each firm. Note that v is also the short run
competitive price when capacities are in 4.4 A firm’s subgame payoff (revenue) is simply v times
its capacity. Second, suppose that x2aand y>a. Each firm has enough capacity to serve the
market by itself and the pricing subgame corresponds to a situation of Bertrand price
competition. This is region B in Figure 1. The unique subgame Nash equilibrium (NE) has both
firms setting price equal to zero (the short run competitive price) and each firm earns zero in the
subgame. The final case is represented by region C in Figure 1. Total capacity exceeds the size
of the market, but at least one firm is too small to serve all consumers by itself. In this case there
is no pure strategy equilibrium for the pricing subgame (except for the trivial case in which one
firm’s capacity is zero). A subgame Nash equilibrium involves mixed strategies for prices.
Reynolds and Wilson (1997) derive equilibrium price distributions and expected revenues for
this case. Equilibrium prices exceed the short run competitive price of zero.

The expected revenue for firm one in a subgame equilibrium is given by the following

sloping demand. However, the simple formulation of the present paper yields the principal empirical implications
that we search for in the data.
* If total capacity is less than a then v is the unique competitive price. If total capacity equals a then v is the upper


20
estimates of the price variance are nearly identical. This classification includes the aluminum
industry. Notice from Tables Il] and IV that the nonferrous metals industry is the only industry

which has different mean price growth rates across regimes and which has negative price

changes in recessions and positive price changes in expansions. Using p(s, = 2ly 15-5738) >
.5 as the criteria for dating a recession, the levels of the real price by state are shown in Figure 5
for the nonferrous metals industry. Prices almost uniformly rise during expansions and fall
during recessions, and as mentioned earlier, these price changes have equal variances. The sharp
rise in the level of real prices as the industry reaches the peak of the business cycle is consistent
with the Bresnahan and Suslow (1989) model in which firms are producing at the vertical portion
of the short run supply curve. Bresnahan and Suslow document how competition has increased
in the aluminum market by observing that the Herfindahl index and concentration ratios fall for
the U.S. aluminum industry from 1957-1982. It appears that there is too much competition in the
Aluminum Industry for firms to set a price markup above the short run competitive price when
demand is low.’®

In addition to our theory, another source of increased price variation during contractions
may be more variable input prices. This explanation, however, is not consistent with current
New Keynesian research on labor markets, which contends that real wages are rigid because
firms pay efficiency wages (see Yellen (1984)). Furthermore, the estimates from the machinery
industry (SIC 35) seem to indicate that input costs are not likely responsible for the increase in
the price change variance. All three-digit machinery industries plausibly have similar mixes of
the same inputs. If input cost variation is responsible for the price variation in both recessions
and expansions, then a testable prediction is that the price change variance is the same in

recessions and expansions for the three-digit machinery industries. As Table III indicates, the

'® cf ft. 5. If n+1>@/(@~a) then Nash equilibrium prices correspond to the short run competitive price.


Table IV
Tests of Different Means across Regimes

Industry Ho: f= Hy: w? =P
Hy ul # Hy wy? # ee

Wald Test p-value Wald Test p-value
Food (SIC 20) 0.046 (0.830) 2.832 (0.092)
Lumber (SIC 24) 0.930 (0.335) 0.002 (0.969)
Furniture (SIC 25) 14.039 (0.000) 1.165 (0.280)
Paper (SIC 26) 2.006 (0.157) 0.951 (0.329)
Newspaper (SIC 271) 2.822 (0.093) 0.530 (0.467)
Chemicals (SIC 28) 19.296 (0.000) 7.836 (0.005)
Petroleum Refining (SIC 29) 3.674 (0.055) 0.408 (0.523)
Rubber (SIC 30) 2.638 (0.104) 0.445 (0.504)
Leather (SIC 31) 0.007 (0.933) 0.796 (0.372)
Stone, Clay, and Glass (SIC 32) 7.619 (0.006) 0.261 (0.610)
Nonferrous Metals (SIC 33:3-6, 9) 11.179 (0.001) 77.682 (0.000)
Fabricated Metals (SIC 34) 15.873 (0.000) 0.008 (0.930)
Agricultural Machinery (SIC 352) 8.881 (0.003) 0.047 (0.828)
Construction Machinery (SIC 353) 23.928 (0.000) 0.195 (0.659)
Metalworking Machinery (SIC 354) 2.495 (0.114) 0.016 (0.899)
Electrical Machinery (SIC 36) 7.764 (0.005) 0.085 (0.771)
Transportation Equipment (SIC 37) 1.982 (0.159) 1.197 (0.274)

Note: Statistics are asymptotically y?. Asymptotic p-values less than .1 are in bold.


Table V
Test of Different Price Variances Across Regimes

Industry H: o, = oc, .
Hy oy # iy Sin
Oty
“Likelihood p-value
Ratio Test
Food (SIC 20) 68.274 (0.000) 5.86
Lumber (SIC 24) 109.508 (0.000) 8.14
Furniture (SIC 25) 142.410 (0.000) 14.38
Paper (SIC 26) 144.794 (0.000) 9.84
Newspaper (SIC 271) 99.566 (0.000) 9.91
Chemicals (SIC 28) 421.102 (0.000) 13.32
Petroleum Refining (SIC 29) 256.162 (0.000) 15.15
Rubber (SIC 30) 149.436 (0.000) 8.88
Leather (SIC 31) 219.260 (0.000) 10.36
Stone, Clay, and Glass (SIC 32) 169.534 (0.000) 13.68
Nonferrous Metals (SIC 33:3-6, 9) 0.002 (0.964) 1.01
Fabricated Metals (SIC 34) 292.092 (0.000) 17.65
Agricultural Machinery (SIC 352) 103.262 (0.000) 9.23
Construction Machinery (SIC 353) 121.596 (0.000) 11.61
Metalworking Machinery (SIC 354) 119.200 (0.000) 13.54
Electrical Machinery (SIC 36) 201.972 (0.000) 15.20
Transportation Equipment (SIC 37) 144.254 (0.000) 10.29

Note: Statistics are asymptotically u . Asymptotic p-values less than .1 are in bold.

* Both the restricted and the unrestricted likelihood functions were estimated without the
Bayesian correction.


Frequency

seeeey
eos 888s

Frequency
onNneauan

Frequency

eusaBnse

Frequency

OrNuauan

6.00 6.00 -6.00 6.00

15.25 5.25 5.25 5.25

4.50 6 450 6 4.50 > “4.50 >
“3.15 g 3.75 z “3.75 S 3.15 g
3.00 5 3.00 g 3.00 e -3.00 2
2.25 & 2.25 £ 225 g 2.25 5
1,50 Le z -1.50 Lp Zz “1.50 LL, z +1.50 L, z
0.75 E s 0.75 EBs. 0.75 E é 0.78 E é
0.00 rc z 0.00 a Ee 0.00 rs 0.00 iat
075 ad 0.75 a 075 2 0.75 Pa
1.50 g 1.50 g 1.50 a 1.50 g
2.25 B 2.25 & 2.25 8 2.25 g
3.00 ~ 3.00 ~ 3.00 3.00

3.15 3.75 3.75 3.15

450 4.50 450 4.50

5.25 5.25 5.25 5.25

6.00 6.00 6.00 6.00

Frequency Frequency Frequency Frequency
oS888833 opeeeye es8eas ere es

6.00 6.00 6.00 6.00

15.25 “5.25 525 5.25 mem

450 450 -4.50 z 450 z
315 2 “3.75 a 315 E “3.75 g
-3.00 a 3.00 A 3.00 z 3.00 Zz
2.25 g -2.25 z 225 = 2235 — z
-1.50 Le -1.50 Ls “1.50 LD 1.50 Ly 8
075 Ee E & 0.75 E & 075 Ee 0.15 E =
ce ————— 0.00 fy 2 0.00 be 0.00 > z
7s | ae 4 0.75 4 075 a 0.75 &
1.50 i] 1.50 a 150 g 1.50 —— g
2.25 $8 2.25 8 2.25 e 2.25 i e
3.00 3.00 3.00 <= 3.00 =
3.15 3.15 3.15 3.15

4,50 4.50 4.50 450

5.25 5.25 5.25 5.25

6.00 6.00 6.00 6.00

ayeIg Aq SasueyD so11g Jo uleASOISTL

pg aansry


26

It should be emphasized that our formulation does not rely on any form of collusion
among firms, in contrast to much of the recent Industrial Organization literature on pricing over
the business cycle. At the two- and three-digit SIC level of aggregation for manufacturing
industries, we find less support for a model of optimal collusion over the business cycle than for
our non-collusive oligopoly model.

Future research may focus on how industry concentration is related to the variance of
price changes at a more disaggregated industry level and how nature of costs affects oligopoly
pricing incentives. Forming larger systems by grouping industries together may also improve
estimation efficiency and increase the number of significant parameters. Since the empirical
results suggest that the form of the distribution of price changes differs across recessionary and
boom regimes, it would also seem worthwhile to develop a more sophisticated econometric

model of the recessionary regime.


2
in prices and industry efficiency over the business cycle. We refer to this as a non-collusive
oligopoly model to distinguish it from collusive theories of oligopoly behavior over the business
cycle. Our model emphasizes the role of long run production capacity investments that must be
made before demand conditions are known. After capacity investments are made, firms learn
about the level of product demand and choose prices. The pricing incentives for firms differ
depending on the level of demand. If demand is high, then the short run competitive (market
clearing) price is a pure strategy Nash equilibrium. However, if demand is low, then capacity
constrained firms have an incentive to deviate from the short run competitive price. The typical
result is that firms adopt mixed strategies for prices that involve a markup over the short run
competitive price and that generate excess production capacity.

Our non-collusive oligopoly model predicts that output prices are procyclical, as do many
other theories. The key prediction that distinguishes the model from other theories is that output
prices are predicted to have greater variance during low demand periods than during high
demand periods. Two other implications of our non-collusive oligopoly model are noteworthy.
First, price adjustments are sluggish in the downward direction, relative to perfectly competitive
prices. If demand changes from high to low then oligopoly firms reduce prices by an amount
less than the change in the competitive price, since oligopoly firms charge a markup over the
short run competitive price when demand is low. A second (and related) implication is that
existing capacity is utilized efficiently when demand is high but may be utilized inefficiently
when demand is low. Oligopoly price markups above the short run competitive price can lead to
less output and employment than is efficient when demand is low.

In Section I we formalize the idea that ex ante capacity investments coupled with ex post

price setting lead to price variation over the business cycle. We begin with a formal two-stage

shocks can yield procyclical markups over marginal cost and procyclical price levels.


Figure 4c
Inferred Probability of Being in State 2 (Recessionary State)

Petroleum Refining

(SIC 29)

E 10's6
A so'e6
E 60°16
# 10°06
E so'es
iE 60°98
B lo'se
# sores
E 60'18
£ 10°08
So's
60'9L
R ose
E so'ee
B cole
100L
| so's9
60°99
10's9
so'e9
$ 60°19
10109

100%
80%
60%
40%
20%

0%

Rubber

(SIC 30)

Leather

10°S6
SO'E6
60°16

10°06
E So's
# 60°98
A io'ss
B coce
A Gols
f 1008
So's
60°92
E 10'SL
E SO'EL
601

TOOL
0°89
60°99

4 10's9

so'eo

E6019

1009



Figure 4a
Inferred Probability of Being in State 2 (Recessionary State)

Food
(SIC 20)

100%
80%
60%
40%
20%

0%

10°S6
S0'€6
60°16
10°06
S088
60°98
10°S8
SO'E8

E 6018

10°08
SO'8L
60°9L
TO'SL
SO'EL
601L
10°0L
$0'89
60°99
10°s9
soeo
60°19
10°09

Lumber

(SIC 24)

E 10°56
if sovce
"co's
E 10°06
J co's
60°98
Bio's
f so’es
60°18
# 10°08
ye soe
i 60°9L
IE Io'se

SO'EL

601L

10'0L

5 S0'89
E 60°99
E 10'S9
b S0'E9
F 6019
1009

Furniture

(SIC 25)

100%
80%
60%
40%
20%

0%

10'S6
S0'€6
60°16
10°06
$0'88
60°98
10'S8
SOEs
60°18
10°08
SO'8L
60'9L
1O'SL
SO'EL
6O1L
TOOL
$0'89
60°99
10°s9
so'e9
6019
10°09



13
in an optimal collusive model. More importantly, a positive variance for quantity change is
permitted in each state. The state variable is not observed, so that the observed path of price and
quantity changes must be used to make inferences about the values of the state variable (as with
the Kalman filter). The unobserved state is assumed to be one of two growth rates, “high” or
“low,” and the probabilistic switch from one state to another is assumed to follow a Markov
process. This approach to the issue of cyclical pricing is appealing because the data will
determine when an industry switches regimes and whether relative price changes are more

variable or not in recessions.

Time Series Model

The model we use is of the type formulated by Engel and Hamilton (1990) to test the
hypothesis of uncovered interest parity. Let y, =[g,,p,]' be a two-dimensional vector at date ¢
where q, is the percentage change in production and p, is the percentage change in price for a
particular industry. There is an unobservable variable, s,, which characterizes the “state” at date 1
and takes on the value of one or two. In Engel and Hamilton (1990), y, is drawn from one
bivariate distribution if the state in ¢ is equal to one and a second bivariate distribution if the state
in ¢ is equal to two. Our setting is somewhat more complex, in that we have predictions on the
distribution governing y, draws conditional on both the current and the prior state (recall Table
1). Depending on the current and prior states, the contemporaneous changes in the growth rates

of production and real prices are drawn from a N(p 5.5, > @s,_,,s,) distribution. Thus, there are

four different bivariate distributions governing y,draws, since the pair (s,_,,5,) can take on
four different values.
The state variable follows the Markov chain process specified in Section I. Only through

51 do past realizations of y and s affect the unobservable state variable. Note that the draws of


15

The Data

The price and output measures considered for this study are monthly time series. The
Federal Reserve publishes indices of industrial production associated with the Bureau of Labor
Statistics’ Producer Price Index commodity series at the monthly level.'° For this study we limit
ourselves to considering industries first at the two-digit Standard Industrial Classification level
and then at the three-digit level if there are industry definition problems at the two-digit level.

The PPI’s at various levels of aggregation are the natural choice for the price series.'! To
form a relative industry price, p;, the aggregate PPI is used to deflate an individual industry price
index. Ifthe corresponding PPI for an industry at the two-digit SIC level is too broad, we then
look at industries at the three-digit level where we can get a closer match of the production and
producer price series. The industries and associated price indices included in this study are listed
in Table II.

The first observation for y, is either January, 1960, or the earliest date following January,
1960, when both price and production series data first become available; the final observation is

December, 1995. The originally unseasonalized price and production series are also individually

Laird, and Rubin (1977). With the EM algorithm, analytic derivatives are easily calculated from smoothed
inferences. Furthermore, systems with a larger number of parameters, as in the bivariate model employed here, do
not require additional iterations or computing time because the analytic derivatives are calculated from the smoothed
probabilities. In addition to choosing the method for maximizing the likelihood function, another problem arises
when the mean of the first state is set exactly equal to the first observation and the variance of the first state is
allowed to vanish. In such a case the likelihood function blows up to infinity. Hamilton (1991) offers Bayesian
priors as a solution to this problem. These positive Bayesian priors imply that one is now maximizing a generalized
objective function that is not subject to these singularities. The first order conditions from maximizing the log
likelihood function with and without the Bayesian priors are given in Hamilton (1990) and Engel and Hamilton
(1990). In practice we find that this potential singularity in the likelihood function is not a problem with our data.

° The smoothed probability, PS} =is,=J| Ypon¥ 738), is calculated as part of the filter for the two-state
model employed by Engel and Hamilton. In a two-state model, the transition observations from state i in period #1

to state j in period ¢ are incorporated into the estimate of Hs, =j and OQ = In practice we estimate a two-state

j
model and use p(s)_) = 4,5 = j1Y)+-»¥ 758) to estimate the parameters from a four-state version of the EM
equations in Hamilton (1990) (p.54).

° The results are similar if capacity utilization rates are used instead of industrial production.

"' The PPI data were downloaded from the BLS World Wide Web site http://stats.bls.gov/.


The adjustment of product market prices to changes in demand plays an important role in
business cycle fluctuations. A greater degree of price inflexibility in response to market demand
changes may lead to larger swings in real output and employment when an economy is subjected
to aggregate demand shocks. Indeed, Rotemberg (1982) finds evidence of sluggish price
adjustment in aggregate U.S. price data.

Recent research on product market price adjustment has focused on firms’ market power
and how this power may be exercised differentially over the business cycle. For example,
studies by Greenwald et al. (1984), Gottfries (1991), Klemperer (1995), and Chevalier and
Scharfstein (1996) have emphasized the role of capital-market imperfections. When capital-
market imperfections exist, the incentives for firms to make investments may be reduced because
firms may not reap the profits associated with the investment. One form of investment is a low
price that builds a firm’s market share by attracting more customers in the future. During a
recession, firms may raise prices, forgoing any attempt to raise future market share, because the
probability of default is high. Chevalier and Scharfstein (1996) find support for this hypothesis in
data drawn from the supermarket industry. Another strand of the literature has emphasized the
role of collusion. Rotemberg and Saloner (1986), Rotemberg and Woodford (1991, 1992) and
Bagwell and Staiger (1997) show that a firm participating in a collusive group may have more
incentive to defect during a boom period, because the short term gains from defection are
relatively large. Thus, an optimal collusive mechanism may involve lower prices (or markups
over marginal cost) during booms than during recessions, in order to eliminate the incentive to
defect.’

We propose an alternative model of oligopoly pricing that has implications for variations

If marginal cost is increasing with output then optimal collusion may involve countercyclical markups over
marginal cost coupled with procyclical price levels. Bagwell and Staiger (1997) show that the structure of an
optimal collusive mechanism is sensitive to the autocorrelation of demand shocks. Positively correlated demand


“sasoyjuored Ut ase SIOLIO prepurIsy

(gse'0) (yz0'e)

(6sb'0) (ez)

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uonepodsuesy yeorno3|q BulyOMe}oV uorjonysuOF yeinyjnousy Joyoureseg

ponuyguos YT gel


24
current level of demand when they set prices, but are uncertain of future demand levels.
Rotemberg and Saloner show that a firm participating in a collusive group may have more
incentive to defect during a boom period than during a low demand period, because the short
term gains from defection are relatively large. Thus, an optimal collusive mechanism may
involve lower prices (or markups over marginal cost) during booms than during recessions, in
order to eliminate the incentive to defect.

A recent paper by Bagwell and Staiger (1997) presents a theory of collusive pricing with
demand alternating between expansionary and recessionary phases. Their formulation allows for
autocorrelation in demand shocks. They prove that the prices for an optimal collusive
mechanism are weakly procyclical when growth rates are positively correlated over time. Our
empirical analysis shows that growth rates are positively correlated over time for most U.S.
manufacturing industries. However, a variety of approaches are consistent with procyclical
pricing, including our non-collusive oligopoly model, a theory of competitive pricing with rising
marginal costs, and the Cournot oligopoly theory. Another prediction in Bagwell and Staiger
(1997) is that a higher transitory demand shock (within a recessionary regime or a boom regime)
is associated with a lower most-collusive price. This would generate a negative covariance
between price change and quantity change within recessions and within booms. Table III reports
our estimates of these covariances (see the results for Q,,; and Q37). There are about as many
positive as negative covariances; for most industries the covariance is not significantly different
from zero. Thus, the evidence does not support the unique prediction of an optimal collusion
regarding transitory demand shocks. In contrast, the evidence supports the unique predictions of
our non-collusive oligopoly model regarding price change variances and distributions. We

should point out that these conclusions are based on an examination of a number of industries at

price draw around -5% is the /one outlying negative real price change.




Figure 4f
Inferred Probability of Being in State 2 (Recessionary State)

Electrical Machinery

(SIC 36)

100%
80%
60%

40%

20%

10°S6
S0'€6
6016
10°06
S0'88
60°98
10°s8
SOEs

5 6018

10°08
SO'8L
60°9L
1O'SL
SO'EL
6OTL

E 10'0L

0%

$0°89
60°99
10°S9
sored
60°19
10°09

Transportation Equipment

(SIC 37)

50'S6
20'r6
11°26
B s0'16
S006
20°68
IV'L8
P sos
50's
z0'r8
117
$3018
B soos
4 20'6L
BTL
0°92
© corse
| cove
rz
# soe
4 SO'OL
20°69

100%
80%
60%
40%
20%

0%



32

Appendix B
This appendix illustrates the derivations of the results presented in Table I for the
predicted price and quantity changes. We define p, =InP —InP_,and g,=InQ,—-InQ,,
where P, is the average price and Q, is production in period t. The approximations of p, use a

first order Taylor expansion:

x= E(x) + y~ EQ)

In(x + y) © In(E(x) + E(y)) + E(x) + E(y)

A derivation for the case, s;.; = 1 and s, = 2, is provided here. Derivations for the other
three cases in Table I are similar to this. The period ¢ price is the sum of the short run marginal

cost, b, , and a markup term, mj, , where m, €(0,v], E(m,) =m <v, and Var(m,) = or

The price in f—1 is, b,_; +v. Thus we have,

py = In(b, + my) —In(by-y +9) © W(2TM) opty — 8 i By -5
v

b+m by
which is linear in };, mys, and 5,., and

q, = Ina, —Ina,_, = In(t2).

The first and second moments are:

E(p,) = rf 2) <0,

+
b+m)) b b-i, b,4-5)°
Var(p,) = E [>,-of +) =E| ( 1 Me em Prat }
bt+y b+m, b+yv
2,02
oO, +07, 2
= 4 Ob 77 = (0 + O1)05 +0195, »

(G+m)y (b+)?

E(q,) = In(t2), and Var(q,) = 0.


Figure 2A
Equilibrium Capacity Choices

Se

Ig

Ma/l4

lla/14 a a
Parameter Values:@=1/2,c/v=1/4,a=5/4@

Figure 2B

Ie

Ig
a

145°

Parameter Values: 0 =3/4,c/v=1/8,a=5/4a


27

References

Bagwell, K. and R. Staiger, “Collusion over the Business Cycle,” Rand Journal of Economics,
Spring 1997, 28, 82-106.

Bresnahan, T. F., “Empirical Studies of Industries with Market Power,” in R. Schmalensee and
R. Willig, eds., Handbook of Industrial Organization, 1989, 2, New York: North-Holland, 1011-
57.

Bresnahan, T. F., and V. Y. Suslow, “Short run Supply with Capacity Constraints,” Journal of
Law and Economics, 1989, 32, $11-S41.

Chevalier, J., and D. Scharfstein, “Capital-Market Imperfections and Countercyclical Markups:
Theory and Evidence,” American Economic Review, September 1996, 86, 703-725.

Davis, D. and C. Holt, “Market Power and Mergers in Laboratory Markets with Posted Prices,”
Rand Journal of Economics, Autumn 1994, 25, 467-487.

Dempster, A., N. Laird, and D. Rubin, “Maximum Likelihood from Incomplete Data via the EM
Algorithm, Journal of the Royal Statistical Society, 1977, B 39, 1-38.

Domowitz, I., “Oligopoly Pricing: Time-Varying Conduct and the Influence of Product
Durability as an Element of Market Structure,” in The New Industrial Economics, eds. G.
Norman and M. La Manna, Brookfield, VT: Elgar Publishing, 1992, 214-235.

Domowitz, I., R. G. Hubbard, and B. Petersen, “Oligopoly Supergames: Some Empirical
Evidence on Prices and Margins,” in T. Bresnahan and R. Schmalensee, eds., The Empirical
Renaissance in Industrial Organization: An Overview, Oxford: Blackwell, 1987, p. 9-28.

Engel, C., and J. D. Hamilton, “Long Swings in the Dollar: Are They in the Data and Do
Markets Know It?” American Economic Review, 1990, 80, 689-713.

Gottfries, N. “Customer Markets, Credit Market Imperfections, and Real Price Rigidity,”
Economica, 1991, 58(3), 317-23.

Greenwald, B., J. Stiglitz, and A. Weiss, “Informational Imperfections in the Capital
Market and Macroeconomic Fluctuations,” American Economic Review (Papers and
Proceedings, 1984, 74(2), 194-99.

Hamilton, J.D. “A New Approach to the Economic Analysis of Nonstationary Time Series and
the Business Cycle,” Econometrica, 1989, 57(2), 357-384.

Hamilton, J. D. “Analysis of Time Series Subject to Changes in Regime,” Journal of
Econometrics, 1990 45, 39-70.

Hamilton, J.D. “A Quasi-Bayesian Approach to Estimating Parameters for Mixtures of
Normal Distributions,” Journal of Business and Economic Statistics, 1991, 9, 27-39.


11

Now consider the infinite horizon game beginning in period one, given initial conditions,
(49,59). Let Sbe a common discount factor, where we assume that 76 <1, so that payoffs are
bounded. We focus on equilibria that have two features: (1) firms use strategies that constitute
an equilibrium for the two-stage game in each period 1, conditional on (a,_,,5,_,), and (2) the
market shares of capacities are the same whenever the previous period state is the same.
Equilibria with these features yield a stationary Markov process for price and quantity changes.
The next step of the analysis is to characterize such a process.

Suppose that (s,_,5,) = (1,2); i-e., there is high growth in f-1 and low growth in r. Then
in period ¢ firms utilize mixed strategy distributions of prices. As long as s,) = 1, this
distribution is the same, regardless of the value of a,.;, given the features of the equilibrium we
consider. Let m, be a random variable representing the average of the two firms’ prices less

marginal production cost, based on their mixed strategies, with m,, €(0,v], E(m,) =m <v,
_ 2
and Var(m,) = Om, -

Suppose instead that (s,_),s,)=(2,2); ie., there is low growth in periods f-1 and ¢.
Firms utilize mixed strategy distributions of prices and, as long as s,., = 2, this distribution is the
same, regardless of the value of a;.;. Let m2, be a random variable representing the average of the
two firms’ prices less marginal production cost, based on their mixed strategies, with
m,, €(0,v], E(m,,) =m, <v, and Var(m,,) = on :

With two states the model makes four predictions for price and quantity changes. Let p,
and q; represent the percentage change in prices and quantity, respectively. The results are
derived in Appendix B and summarized in Table I. The testable implications of the model are
twofold. First, during recessions (s, =2) the model predicts that changes in price will have a

larger variance than during booms. Second, the form of the distribution of price changes will be


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Frequency

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2.28 2.25 225 225
3.00 3.00 3.00 3.00
3.75 3.75 3.75 3.75
4.50 450 4.50 4.50
5.25 5.25 5.25 5.25
6.00 6.00 6.00 6.00
Frequency Frequency Frequency Frequency
eos8sséss Stee haan ous BB onaansikz
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3.00 3.00 3.00 95
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14
y, are not independent draws from a mixture of normal distributions. The inferred probability
that y, is drawn from one distribution depends on all the realizations of y prior to time ft. This
approach differs from the Bresnahan and Suslow (1989) switching regime regression in that all
prior information is used to infer the current probability that y, is drawn from a particular
distribution.

The bivariate model of stochastic segmented trends permits a wide variety of behavior for
the series. First, the industry mean production growth rates may pick up slow or fast growth
rates of production for an industry. The production growth rates could also be the same in both
regimes, or one state may reflect recessionary periods and the other expansionary periods.
Likewise, these combinations are possible for the contemporaneous real price regimes.

The model described above is the basis for estimation of a parameter vector,
8 = (Aye Mir Hays Hs 2), Qy Q4,,Q,), P\y. Py)’. This vector has 22 parameters. The sample
log likelihood function, /n p(y7,y7_},---,¥}38), which is to be maximized with respect to the
unknown parameters of @, can be constructed from the conditional likelihood of y,; The
conditional likelihood of y,, p(y,ly,,-..¥,;8), is a byproduct of Hamilton’s (1989) filtering
algorithm. We can infer the probability that y, was drawn from a particular state s, based on all
information available at time #: p(s, |y,,...,.y,;9). Furthermore, a full sample smoother can be

used to draw inference on the regime at date f using all the information available ex post:

PS, LY 1 ¥739) 2?

5 Two issues arise when attempting to find the maximum likelihood estimate 8. Both involve the process of
finding the global maximum of the sample log likelihood function. From the filter we can determine the sample log
likelihood function as the sum of the conditional log likelihoods:

T
In PY TY74 voy Zn POY, 1¥,_1+-5¥,)- As in Hamilton (1989) the sample log likelihood can be

maximized using numeric hill-climbing methods, but systems with a large number of parameters (e.g., twenty-two)
often have many local maxima and require lengthy computing time. Hamilton (1990) shows how the switching
regime filter can be estimated using the EM (expected maximum likelihood) algorithm developed by Dempster,