Measurement of Conjectural Variations in Oligopoly
Author(s): Gyoichi Iwata
Reviewed work(s):
Source: Econometrica, Vol. 42, No. 5 (Sep., 1974), pp. 947-966
Published by: The Econometric Society
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Econometrica, Vol. 42, No. 5 (September, 1974)
MEASUREMENT OF CONJECTURAL VARIATIONS IN OLIGOPOLY
BY GYOICHIIWATA1
This paper proposes a method for measuring the numerical value of the conjectural
variation which has been a key concept in oligopoly theories. The statistical property of
the estimator for the conjectural variation is examined and two statistical tests are presented.
One is a test for the hypothesis that the conjectural variation is a specified value. The other
is designed for the hypothesis that there is a certain type of collusion among oligopoly
firms. These are applied to the Japanese flat glass industry.
1. INTRODUCTION
THEPURPOSEof this paper is to propose an econometric approach to the problem
of price determination in oligopoly. This will be accomplished by estimating the
values of conjectural variation for firms supplying a homogeneous product in an
oligopolistic market.
The concept of conjectural variation has been at the central core in the controversy over oligopoly theories since it was proposed by A. L. Bowley [1] and
R. Frisch [6]. Various values have been assumed for the value of conjectural
variation, but few attempts to measure it empirically have been made. In this
paper, we shall try to build up the theory and method for the statistical estimation
of the conjectural variation. Moreover, two statistical tests will be presented. One
is a test for the hypothesis that the conjectural variation is equal to a specified
value, and the other is a test for the hypothesis that there is a certain type of collusion among oligopoly firms. These will then be applied to the Japanese flat
glass industry during the period from 1956 to 1965.
2. A THEORETICALFRAMEWORKOF ANALYSIS
Suppose there is an oligopolistic market of a few firms with one homogeneous
product sold to a large number of purchasers. Let the total demand for this-product
be D, the price p, and the market demand function
(2.1)
p = f(D),
the derivative of which is assumed to be negative for any positive D. Let the
number of firms be n and the supply of the jth firm be qj. Then the total supply
S = q, + q2 +
(2.2)
D
...
= q1 +
+ qnmustbeequaltoD:
q2
+
+ qn-
' This paper is a revision of an earlier paper [8], given at the Second World Congress of the Econometric Society at Cambridge University in September, 1970. I am indebted to many persons, especially
to Professors William J. Fellner, Merton J. Peck, Kotaro Tsujimura, and Robert Evans, Jr. for their
useful comments and illuminating discussions. I would also like to thank the referees and editor for
helpful comments on earlier drafts of this paper.
947
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948
GYOICHI IWATA
Let us assume profit maximizing behavior for each firm. Defining profit as =j
Cj, where Rj and Cj are total revenue and cost respectively, the marginal
revenue may be expressed as
Rj-
(2.3)
+ dp
dR3
djdqq=+7jJiP+dD
+ ddpdD
dqj=p
+ dp( +
dD
)
where
(2.4)
di
qq(>Jk)
yj=dqj
k* i
We call yj the conjectural variation (cf. Hicks [7]). This is the ratio of the variation
of the other firms' supply which firm j assumes will result if it increases its own
supply.As is well known,Cournot[3] assumedyi
= ...
=
=
0.
The first and second order conditions for profit maximization are
(2.5)
p + dp(l + yj)qj-cj
=O
(j = 1,. .., n)
and
(2.6)
++
J +
(2
d
j
dqj dD
dD2'l
cJ1 < 0?
dq1
where cj is the marginal cost dCjldqj. If the price elasticity of demand is denoted
by oc(<0), (2.5) can be expressed as
(2.7)
p + --(1
+ yj)qj-c
=O
(j= 1,. ..,n).
Then the market share of firmj, qj/D, may be expressed as
P
yj
Summing up this relation over j produces
(2.8)
q=
D
_
pI+
n
Z
q1/D _= (X n
_=
C
C
Pj= I1 1+
(j=
,...,n).
p
Vi
Here the left-hand side is equal to unity. It can be rewritten as
(2.9)
c (,zj
a
'1
)
+1)
which implies that in this market the price level is expressed as a function of three
factors: oc,cj, and yj.
In this paper no a priori assumptions concerning the value of the conjectural
variations are made; rather their numerical values are estimated indirectly.2
2 A measurement which bears some resemblance
to this one was the estimation of price elasticity for
individual demand curves developed by Wassily Leontief in 1940 [10].
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CONJECTURALVARIATIONS
949
From (2.7), the conjectural variation yj is expressed as
(2.10)
jjP:-
p
pD_
qj
1.
The value of the right-hand side could be calculated if one knew the values of the
marginal cost cj and the price elasticity a. These may be obtained from the estimation of the cost function for each firm and the market demand function respectively.
There is a problem in the estimation of the market demand function in an
oligopoly market, since price may not be taken as an independent variable in the
ordinary least square regression, because the price p and the demand D are determined simultaneously.3 However, as seen in (2.9), if three factors oc,c;, and y. are
constant or exogeneously determined, price can be taken as predetermined.
Moreover the demand function is identifiable in this case because the reduced
relation between total supply and price will be the demand function itself.
We now introduce the following three assumptions:
1: The price elasticity of market demanda is constant regardless of
ASSUMPTION
the level of demand.
2: The marginal cost cj of each firm is constant with respect to its
ASSUMPTION
short-runvariation in output.
3: The conjectural variation yj is,a constant parameterfor eachfirm
ASSUMPTION
in each period.
Although these assumptions are introduced as a first approximation, they are
not unrealistic. The first assumption of constant price elasticity is frequently used
in empirical demand analyses because the log-linear type demand functions have
produced good results. The second assumption of constant marginal cost has a
great deal of evidence to support it. J. Johnston, after his detailed survey on
statistical cost functions, concluded: "Two major impressions, however, stand out
clearly. The first is that the various short-run studies more often than not indicate
constant marginal cost and declining average cost as the pattern that best seems
to describe the data that have been analyzed" [9, p. 168]. The third assumption
means that each oligopolist ex ante has some definite conjecture about his rivals'
attitudes. This conjecture may have been formed on the basis of his past experience.
By these assumptions, we have d2p/dD2 =
-
(1/ox)(1 - (11/))(p/D2), dcj/dqj = 0,
and dyj/dqj = 0. Using these three equations, the second order condition (2.6)
becomes
(2.11)
1 p (q + y.) [2-(I
ocID
l qj <
0.
+ yj)
oIDI
3 Franklin M. Fisher [5] in his estimation of the demand function for aluminum ingots in the United
States took price as an independent variable. He assumed that under conditions of monopoly or tightlyknit oligopoly, a monopolist or an oligopolist would adopt a policy of inertia and keep his prices
constant over a year or longer. This assumption contradicts our model of short-run profit maximization.
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950
GYOICHI IWATA
In this inequality, if we assume yj - 1, then the inside of the bracket will be
positive, so that the left-hand side will be nonnegative and (2.11) will not hold.
Therefore, at least
(2.12)
y > -1
must hold to satisfy the second order condition.
3.
MEASUREMENTOF COST FUNCTIONS
This paper uses the above model to analyze the Japanese flat glass industry.
This industry was chosen because it was a typical oligopolistic industry consisting
of three producers whose products had little differentiation. The three producers
were Asahi Glass Co. Ltd., Nippon Sheet Glass Co. Ltd., and Central Glass Co. Ltd.
Flat glass may be divided into two categories: window glass, which includes
common sheet and plate glass, figured glass, and wire glass; and polished plate
glass. The period of analysis was confined to the years 1956 to 1965, because price
data were unavailable before 1956, and after 1966 the "float method," the biggest
technological innovation in flat glass production in recent years, was introduced
into Japan. The shares of production of Asahi, Nippon, and Central in 1965 were
52.7 per cent, 33.5 per cent, and 13.8 per cent for window glass, and 46.2 per cent,
30.2 per cent, and 23.6 per cent for polished plate glass. Since Central entered
these markets in 1959 and 1964, the cojectural variations of only Asahi and Nippon
were measured.
The cost functions were estimated using the time series of the firms' half-yearly
accounting data.4 In order to make' more reliable estimates of the cost function,
they were measured indirectly. First the input functions for all inputs for which
data were available in the accounting reports were measured. Then these input
functions were substituted into the cost equation in order to obtain the cost function.
The subscript j denotes the firm number, with Asahi being j = 1 and Nippon
being j = 2. The subscript i refers to the type of product, denoting window glass
by i = 1, polished plate glass by i = 2, and products other than flat glass5 by i = 3.
The output of product i by the firm] is denoted by X . The unit of Xlj is a converted
box ;6 X2;, box; and X3j, 1,000 yen. We define X3j as the real value evaluated at
1962-11price.
The total cost is defined by the following cost equation:
(3.1)
Cj =
5
o
E SkjmkJ + PO pOj
+
wjLj S rjKj,
k~=1
4 Unfortunately the two firms have different accounting periods. The accounting periods for Asahi
end in June and December, while those of Nippon end in September and March.
5 In addition to flat glass, Asahi produced soda-ash, caustic soda, fire brick, the glass for Braun
tubes of television sets, etc., but Nippon produced only flat glass in this period.
6 One box is equivalent to sheets of flat glass with a total area of 100 square feet, regardless of thickness. One converted box means sheets of flat glass which have the same volume as one box of flat glass
of 2 millimeters thickness.
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CONJECTURALVARIATIONS
951
where m1j is input of silica at firmj (ton/half-year); m2j is input of soda-ash at firm
j (ton/half-year); M3i is input of dolomite at firmj (ton/half-year); M41 is input of
coal and heavy oil at firmj (million Cal./half-year); min1 is input of material salt at
firm 1 (ton/half-year); M52 is input of electricity at firm 2 (KWH/half-year); Ski iS
respectivepriceoftheabovematerialsatfirmj(1,OOOyenpereach unit)(k = 1,..., 5);
Li is number of workers of firm j at the end of each period (men); Wjis average
wage of workers of firm j (1,000 yen/man half-year); Kj is real value of capital
equipment of firm j, defined as the tangible assets except land and construction
account. The value of K1 at the end of the tth period is obtained by the formula
where Ijt represents the gross investment at t,
Kjt= (1 - bjt)Kj,t - + Ijt/(PIt/PIO),
bjt is the rate of depreciation at t, PI, is the wholesale price index of investment goods
(1960 = 1.0; source: The Bank of Japan), Kjo (the initial value of Kj,) is the bookkeeping value of capital equipment defined above (denoted by K' ) at the end of
1955-II; rj is the unit price of Kj and is defined as rj = PKj(P + bj), where PKj is
the ratio of KJ to K , and p is the market interest rate; Coj is the other cost of firm
j which is equal to material cost other than E SkjMkJplus labor cost other than
wjLj plus manufacturing overhead cost plus general management and selling
expenses minus incidental profit and loss minus rjKj; Poj is the deflator for the
other cost Coj, defined as the simple arithmetic mean of the wholesale price index
of producer goods (source: The Bank of Japan) and the wage index of temporary
and day workers in the manufacturing industry (source: Ministry of Labor),
1960 = 1.0.
The last two terms of (3.1), wjLj and rjKj, can be regarded as constant with respect to short-run output variation. Since the object was to obtain the short-run
marginal cost, it was only necessary to measure the input functions of m1j,. . .,m
and CoJ/POJ.
We chose a linear type with respect to output variation. This implies that the
derived marginal cost is constant with respect to output variation. It, however,
does not mean that it is constant with respect to the variation of production
capacity. Thus the possibility that the marginal input coefficients may change with
the capacity level had to be examined.
It was found that the relative price of each input had a statistically insignificant
effect on the input demand in every case. The following input functions were selected, taking into account the statistical significance of the estimated regression
coefficient, the theoretical sign of the coefficient and the fit of the regression.
The parenthesized figure is the standard error, s is the standard error of estimate,
R2 is the coefficient of determination, and d is the Durbin-Watson statistic. The
notations ** and * mean that the corresponding statistic was significant at the
1 per cent and 5 per cent level respectively. The sample size N was 20 in all cases.
Asahi
(3.2)
Mi1l =
-13,169 + 0.03334**XG1 + 0.001205*X31,
(0.0005559)
(11,280) (0.004253)
s = 8.366, R2 = 0.9732**, d = 1.356;
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GYOICHI IWATA
952
(3.3)
M21 =
-8,563* + 0.01017**XGl + 0.0005068**X31,
(0.0001686)
(3,421) (0.001290)
s = 2,537, R2 = 0.9419**, d = 1.187;
(3.4)
M31
= 634.2 + 0.005589**XGl + 0.0002948*X31,
(0.0001087)
(2,206) (0.0008318)
s = 1,636, R2 = 0.9235**, d = 1.374;
(3.5)
M41
= 235,340 + 0.2548**XG1 + 0.006162X31,
(0.006382)
(129,480) (0.04883)
s = 96,037, R2 = 0.8435**, d = 1.395;
(3.6)
Mi51 =
li8,180** + 0.02952**XG1 + 0.001478*X31,
(0.0006803)
(0.005205)
(13,804)
s = 10,238, R2 = 0.8936**, d = 1.705;
(3.7)
C01/PO1 =
3,960,200** + 0.3084X1l + 7.413X21 + 0.3169**X3,
(0.04294)
(5.490)
(0.4054)
(937,400)
s
- 5.506**Q21,-1,
462,840, R2 = 0.9333**, d = 2.387.
=
(1.765)
Nippon
(3.8)
M12 =
-4,128 + (0.03226** + 0.6701 x 10 9QG2,
(3,992) (0.002888) (0.3362 x 10-9)
s
(3.9)
M22 =
=
1)XG2,
1,952, R2 = 0.9926**, d = 3.032*;
-555.0 + (0.007448** + 0.4801 x 109**QGZ,-1)XG2,
(0.1412 x 10-9)
(1,677) (0.001213)
s = 819.9, R2 = 0.9860**, d = 1.617;
(3.10)
M32 =
-4,545 + 0.01 100**XG2,
(2,701) (0.001131)
s = 2,806, R2 = 0.8401**,
d = 0.437**;
(3.11)
M42 =
286,000** + 0.1679**XG2,
(0.02231)
(53,270)
s = 55,324, R2 = 0.7588**,
d = 1.055*;
(3.12)
M52
= 2,261 + 0.01045**XG2,
s = 2,851, R2 = 0.8212**,d = 1.020*;
(2,745) (0.001150)
(3.13)
C02/P02
=
-459,830 + 0.7737X12 + 14.94X22 + 12.67Q22,
(9.662)
(8.399)
(1,102,190) (0.6319)
+ 2,440,810 ** V,
(478,450)
1
s = 536,870, R2 = 0.7983**, d = 1.877.
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953
CONJECTURALVARIATIONS
Here, XGj stands for the output of flat glass defined as Xlj + 2.5X2j, where the
figure 2.5 means the rate of conversion of a box into a converted box since the
average thickness of polished plate glass was about 5 millimeters in the sample
period (see footnote 6). In each material input function above, XGj was adopted as
the independent variable, because inputs of these materials mkj are related to the
weight or volume of flat glass produced. Since the required amount of the inputs
of other costs COJ/Pojmay be different for the two types of product (window glass
X1j and polished plate glass X2j), both of these have been used as independent
variables. Unfortunately the statistical significance of each coefficient was not very
high, possibly because of collinearity between the two variables.
Equations (3.8) and (3.9) mean that at Nippon the marginal input coefficients
depended on QG2, -, the production capacity of total flat glass at the beginning
of the period (=Q12-1
+ 2.5Q22 1). The same type of input function for the
other mkJwas tested, but the significance was too low to adopt it.
In the input functions of the other costs COJ/POJ(3.7) and (3.13), Q2j-1 were
related to the fixed part of the other costs. In (3.13), V is a dummy variable taking
the value V = 1 for 1956-1-1962-II and 0 for 1963-1-1965-II. This variable was
introduced because Nippon changed significantly its scheme of cost accounting
at the end of 1963-I.
Finally, it should be noted that at least four cases are indicative of serial correlation among the disturbances. In order to obtain better estimates, the so-called
iterative procedure was used (see Christ [2 pp. 483-4]). Since there.was no assurance
that this procedure would improve the estimate, it was applied only to equation
(3.10) which showed one per cent significance. The first-order sample serial correlation coefficient of residuals p was 0.7752. The estimated regression of the first
step was as follows:
(3.10')
M32 - 0.7752m32,1
=
-22.89 + 0.009287**(XG2 (876.4) (0.001322)
0.7752XG2,_l),
N = 19, s = 1,725, R2 = 0.7347**, d = 1.777.
Since d was not significant even at the 5 per cent level, it was possible to stop at this
step. This equation instead of (3.10) was used as an input function.
4.
MEASUREMENTOF THE MARKETDEMAND FUNCTION
Thle next step is to estimate the market demand functions of the two types of
flat glass. According to the model stated in Section 2, a log-linear type demand
function was assumed. Quarterly data from 1956 to 1965 were used. The notations
and data sources were as follows:
D1 is the total domestic demand of window glass (converted box), defined as the
sum of domestic shipments from three producers and the imports from foreign
countries (source: Ministry of International Trade and Industry); D2 is the total
domestic demand of polished plate glass (box), defined in the same way as D1
(source: MITI); P, is the price index of the first product, defined as the arithmetic
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954
GYOICHI IWATA
mean of the wholesale price indices of common sheet glass of 2 millimeters thickness and that of 3 millimeters thickness, 1960 = 1 (source: The Bank of Japan);
P2 is the price index of the second product, defined as the arithmetic mean of the
wholesale price indices of polished plate glass of 5 millimeters thickness with 17
sheets contained in a box and that with 4 sheets in a box (source: The Bank of
Japan); Tis the floor area of total building construction started (1,000 square meters)
(source: Ministry of Construction); cois the ratio of wooden building construction
in the above T; YG is the real gross national product, evaluated at 1960 prices
(billion yen) (source: Economic Planning Agency); and Yc is the real private
consumption expenditures evaluated at 1960 prices (billion yen) (source EPA).
In building construction, the demand for flat glass may be substitutable for
other investment goods. Consequently a relative price defined as the ratio of Pi to
the price index of investment goods PI (see Section 3) was adopted. We also took
into consideration the substitutability between D1 and D2, introducing both
P1/PI and P2/PI into each demand function but theoretically and statistically
meaningful results were not obtained. Thus the two demands may be regarded as
independent.
Since most window glass is used for building, T was considered as one factor to
explain D1, but T is highly correlated with YG.YG was preferred to T because not
only can YGbe a proxy for Tbut it also has some meaning in explaining the demand
derived from uses other than in buildings. Consideration was also given to the
fact that window glass consumption per unit floor area may be different depending
on whether the building is wooden or ferro-concrete. Thus w was adopted as the
independent variable.
These considerations led to the following equation:
(4.1)
log D1 = 3.143** - 0.9757* log (P1/PI) + 0.1925 log o
(0.1275)
(0.2917) (0.4144)
=
N
40, s = 0.02973, R2 = 0.9471**,
+ 0.8730** log YG,
(0.08546)
d = 1.681,
where log means common logarithm.
In the case of polished plate glass, the demand from two other major sources,
automobiles and mirrors, must be considered. The demand for mirror use could
be represented by private consumption Yc.A regression of D2 on the automobile
production A (passenger cars, four-wheeled trucks, and buses) was tried, but a
significant coefficient was not obtained, probably because of multicollinearity
between A and Yc. So A was dropped and Ycwas assumed to represent automobile
demand. The estimated regression equation was as follows:
(4.2)
log D2
= -
1.817 - 0.5355 log (P2/P,) + 0.7160** log T
(0.2635)
(0.9947) (0.5500)
+ 1.0767** log Yc
(0.3598)
N = 40, s = 0.08228, R2 = 0.9174**,
d = 0.484**.
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955
CONJECTURALVARIATIONS
Since d was significant even at the one per cent level, an attempt to improve the
estimate was made, using the iterative procedure (see Section 3). Using p = 0.7415,
the following was estimated:
(4.2')
log D2t
-
-
0.7415 log D2,t-
1
1.832 - 0.9197 log
(1.659) (0.4601)
P2t
- 0.7415 log
P2,t-1
Pit
PI,t-1
+ 0.4445**(og T, - 0.7415 log 7T )
(0.1571)
+ 0.6695**(log Yct- 0.7415 log Yc,,- 1),
(0.1686)
N = 39, s = 0.04352, R2 = 0.7230**, d = 1.682.
Since d was not significant even at the 5 per cent level, it was not necessary to
continue. The degree of significance of price elasticity has increased considerably
in (4.2') in spite of the possible underestimation of the standard error of the regression coefficient in (4.2).
5. A MODEL OF THE FLAT GLASS INDUSTRY AND ITS STATISTICALPROPERTIES
Taking into account the results of the previous sections, it is possible to specify
a model for the Japanese flat glass industry. Firms selling several products will be
treated, but the theory which relates to a firm selling only one product need not
be varied essentially because, as shown in Sections 3 and 4, the demand for each
product can be assumed to be independent of the other, and the marginal cost
of each product is not affected by the output of the other products.
The following additional notations are used:
xij is the total supply of product i by firm j (converted box/half-year for i = 1;
box/half-year for i = 2); qij is the supply to Japan (converted box/half-year or
box/half-year); eii is the supply to countries other than Japan (converted box/halfyear or box/half-year); pij is the price (1,000 yen per converted box or per box);
and ci, is the marginal cost (1,000 yen per converted box or per box).
These definitions yield Di = El=, qij and xij = qij + eij. It was assumed that
the export (or in the case of the foreign firm, the supply to countries other than
Japan) eij and the supply of products other than flat glass X3j are exogenous
variables.
On the basis of the analysis in the previous section, the market demand functions
of each product may be assumed to be:
(5.1)
D1 = oxj0(Pj/P )a11woa12y
(5.2)
D2
=
where u1 and
13u,
cT2o(P2/PiL21Tc22 Y23u2
u2
are random disturbances, logarithms of which obey normal
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GYOICHI IWATA
956
distributions with zero means and constant variances. The variables PI, a, T, YG,
and Ycwere assumed to be exogenous.
There is a problem in obtaining equations like (2.1) from the above. P1 and P2
are price indices and are not the prices themselves. Actually, each product is
composed of various items the prices of which are different from each other. The
price is affected by the change of composition of the items as well as by the price
changes of individual items. Therefore the average prices were not used to measure
the market demand functions. After they have been measured, however, one might
be able to use the average price as the price pij because the composition of the
items will be nearly constant in the short-run. Thus it was assumed that:
(5.3)
(i = 1, 2; j = 1, 2, ... , n),
Pij = HijPi
where Jiij is constant in the short-run. We call Mijthe "composition coefficient."
If (5.3) is substituted into (5.1) and (5.2), and they are solved with respect to Pij,
we have
(5.4)
Plj
(5.5)
P22j =
=
DlIO-/
2
IDihT
D'XOt2
- O22/O21
Y1
PU-
YCa23/a2lj
Pju-
1/a21
The following input functions of material inputs mkJand real other cost Co0/Poj
for two firms were assumed:
(5.6)
mkj
=
COkj + (Clkj + C2kJQGJ,-1)(A1X1j
+ )2X2j)
+ C3kjX3j
+ Vkj
(k = 19,29,...,95),9
(5.7)
CJolpOj =Oj
+ XliXlj
+ 42jX2i
+ 43jX3j
+
'4jQ2j,-1
+ V6j,
where Vkj (k = 1, 2,. .. , 6) are the mutually independent random disturbances
which distribute according to normal distributions with zero and constant
variances. Some of C2kj' C3kj' and 43j are zero. We use Aito denote the conversion
coefficient of unit, meaning Al = 1.0 and A2 = 2.5. Xij and Qij are exogenous
variables. In the case of j = 2, it was assumed that 4O2 = '02 + 452 V, where V is
an exogenous variable.
Then the marginal cost cij was expressed as
5
(5.8)
cij =
E
k= 1
Skjii)lkj
+ C2kjQGj,- 1) + PojAij,
where the input prices Skj and Poj were assumed to be exogenous variables.
If we assume that each firm has a constant conjectural variation yij about the
rivals' supply of product i and follows profit maximizing behavior, we must
have the relation
(5.9)
Pii +
?aD(1 + yij)qij-cij
= 0
(i = 1, 2),
as the first order condition. The second order condition for maximization of
profit itj is that the Hessian matrix composed of 02 tJ/aqijaqkj (i, k = 1, 2) be
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957
CONJECTURALVARIATIONS
negativedefinite.But since 02nj/a0q1 jq2j is equalto zero in this model,we require
only
to be negative, namely
02nj/aqi2
(5.10)
1
aii(l
Lxi
+ yij)(1 -
+ yij)F2 -(1
Di'
LLil
1 q,J
cz1DJ
< 0
(i
=
1,2),
(see (2.11)).
From (5.9) we have
(5.11)
?ij = ail
j-
PiD i
1
This is the relation by which the value of yij was estimated.
From (5.9) and (5.3), the market share is expressed as
(5.12)
qij - ail
Di
-
Pi
Pj1i + Yiij
Following the same procedure to get (2.9),
n
(5.13)
Pi = ailE_
,c/,
n
Ij
1
+
Y
+ 1
(i =1, 2),
is obtained. From this equation it will be clear that Pi is determined casually prior
to the determination of demand level Di. Therefore the least squares estimator
(i1 will be unbiased, namely
(5.14)
(i = 1, 2).
E(Qi 1) = ai
It was assumed that Xij and Qij, - 1 were exogenous and statistically independent
of Vkj, for in the usual cost minimization behavior through which the input functions are derived the values of Xij and Qij,- 1 are given parameters. Therefore, the
least squares estimators of Cikj and tij in (5.6) and (5.7) were unbiased ones. If
these are substituted into (5.8), the unbiased estimators of marginal cost cij are
obtained as follows:
5
(5.15)
cij =
Z Skji)(
lkj
+
42kjQGJ,-
1) +
Pojiii,
k= 1
(5.16)
cc. =
E(cij)
The estimator of conjectural variation yij is given from (5.11) as follows:
(5.17)
yij =-ai 1i_
i
Pij
Here pij and the market share qij/Di are strict functions of exogenous variables
only. It is also safe to assume the statistical independence between oeil and %,j.
Thus if we take (conditional) mathematical expectation of -ij, we get
__
= E(il) E(_ij)_
(5.18)
E(?ij)
1a e
Ds-n
Namely,
j Di_
t of
qm
j
1
Iij
-
p j Di-1
Namely, -, is an unbiased estimator of yij
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=
958
GYOICHI IWATA
The variance of -ij is given as follows :7
(5.19)
O3.j =
i +
2
(p_
x2h)
+ (Cij
-
pij)2oU6fl.
Here the variance of marginal cost estimator cij is given as8
(5.20)
Z5kjL1kj
)il
okiU =
2 coV (glkj'
+
+ U2kjQGj,-
42kj)QGj,-1
1] + Poj,
k
The estimators of these variances are given as follows:
(5.21)
tcj=(h
(5.22)
= I
82ij
X.2kjE&c,
a2 )
[ A 2 (&4.1 +
[~(~+A
A2
i
2 coy
kj +
+
(U
(Aij_
(tlkjk
Pij)2af
-
1]
I + U;2kjQGj,-
g2kj)QGj,-
1] +
Oj&tij-
k
I-2
6 i and cov (,lkj' p2k) are the estimated unbiased variances and
where A2 1 9 0ikj'
covariance of the estimated regression coefficients.
6. TWO STATISTICALTESTS
(i) A Statistical Test of Conjectural Variation
We would like here to propose a statistical test of the hypothesis about yij.
Let us consider first the distribution of the estimator -ij. Since it was assumed that
the disturbances log ui and Vkj distribute normally, the least squares estimators
?(i 9 Cikj9and tij also distribute normally. Then the estimator cij will also obey a
normal distribution. Moreover in (5.17) pij and qj./Di are strict functions of
7
yij)2 = (DJ1(pijqij))'JE[R&2
-
cdj- =E(2--
+ 2(cij - pij)E[-il(
(cij _Cij)2]
+ (Ci -pij)2
il_i)(tij_Cij)]
E(-il-il)2
Here from the statistical independence between Oil and cij,
= (Cr&2i+ X2a2),2
&2 (a-C)2]
and
E[2il((il
ail)(ij
0.
=
-cij)]
8
Elkj)
=ci
=
E[ ZZ
k
-
+ (C2kj -
-
k
skishj?i
h
(2hj)QGj,-
-
{(4lkj
kj)
+
C2kj)QGJ,-1}
(42kj
1} + 2 E SkjAi({(Qlkj -
+
C2kj)QGj,-1}
lkj)
+ ((2kj
I
-
{(lhj
-lhj)
-2kJ)QGJ,-l1}P(8(4i
+
((2hj
-
i)
k
+ P
-
ij)2j
Since we were assuming the statistical independence among vj, V2j,
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V6,, we obta}ied (5.20).
CONJECTURAL
959
VARIATIONS
exogenous variables only. Thus, if we put
(6.1)
,
XO=
x
= p
(j
(ij_-Pij)
=
1,...
then XO, X1,... , Xn are mutually independent normal variates obeying N(11o, o0),
.. ., N(yun,
a 2) respectively, where
(6.2)
Po = ail,
Pi
= pDq (Cij- Pij)
(j = ,
n),
2
(6.3)
arv
2?
Di
5
ULr
j
1..
)
Thus -ij + 1 obeys the distribution of a product of two independent normal
variates, XOand Xi . This distribution has too complicated a form to be used directly
for the test of yij.
This problem may be attacked in another way. From (5.11) we have
(6.4)
yij = PopIj- 1.
Suppose we want to test the hypothesis that the conjectural variation is equal to a
certain specified value y*, namely popj - 1 = y*. If u,u# 0 is assumed, this is
transformed as
(6.5)
+ l)/Pj =0 .
Po - (;
It is possible to test the hypothesis in this form as follows. Define a statistic:
(6.6)
gj = XO - (y;, + l)/Xj.
Neither mean nor variance of gj exists since l/Xj, the inverse of a normal variate,
has no finite mean and variance. The exact distribution of gj cannot easily be
derived. But we can derive its asymptotic distribution. Let us first consider the
distribution of l/Xj. If Xj obeys normal distribution N(pp, ao), the characteristic
function of
(6.7)
y
ffj Xj
lIj
is defined as follows:
(6.8)
p(t) = E(eetY)=
-
a
f
exp
r~~z2
1 rZ
PL
2-H
/iJ
%+expdz,
--
+ dtz(
-1)1dz
{tz
where i is an imaginary unit. If sample sizes in the regression analyses of the
input functions (Section 3) become large, the variances and covariances of
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960
GYOICHI IWATA
regression coefficients in (5.20) approach zero. So,
(6.9)
1
lim y(t)-=
or3
i2
also approaches zero. Thus
itz/l4) dz
exp (_Z2/2
-
exp [-
_t/j2)2
12rrJ-
=~
=
J
z-
-
t2/(2ttj4)]dz
exp [- t2/(24)] .
The last expression exp [- t2/(2Q4)]is identical with the characteristic function of
the normal distribution N(0, 1/y4). Therefore Y converges in distribution to a
normal variate having mean zero and variance 1/y4. This means I/Xj itself obeys
asymptoticallya normaldistributionN(/ii;, 2/l4).
Consequently, gj = XO- (yi*+ l)/Xj obeys asymptotically a normal distribution N[Lo - (y! + 1)/Itj, U + (y?i + 1)2U4/jtf].
Making use of this result, it is possible to carry out a rough statistical test of the
hypothesis (6.5). Namely, if sample sizes are sufficiently large, it will allow us to
?
/Xa , where
approximate the asymptotic variance of gj by r = co + (yW,1)2+
(6.10)
(TO=
=
vI
(Di)82
and to regard the distribution of tj
(6.5) is true.
(j=
=
l,...,n)
aj/gj as N(0, 1), provided the hypothesis
(ii) A Statistical Test on a Certain Type of Collusion
As pointed out by Fellner [4], cooperative agreement without side payment
may be more realistic in the actual world than that with side payment, since
there is unwillingness to pool resources and earnings and to agree on interfirm
compensation, etc. In the cooperative oligopoly case without side payment, firms
will bargain and will attain agreement at some point on the contract curve.
If we denote the partial derivative of profit OjlOqik by rJk, the contract curve
equation in the product market i is expressed as
11
712
(6.11)
721
...
7n1
7E22
...
7tn2
.
.
7ln
72n
.
...
=0.
7nn
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961
CONJECTURAL VARIATIONS
It can be easily shown9 that (6.11) is equivalent to
(6.12)
gO - 1/p l -
1/P = 0,
-
.
are defined in (6.2).
where o0,..., Mu,
If the price level Pi (so that pij through (5.3)) and the market shares qij/Di are
determined exogenously by the collusion amongfirms, then XOand Xj (j = 1,... , n)
in (6.1) will distribute in the same way as stated in (i). Then it can be shown in the
same way as in (i) that a statistic
(6.13)
g = XO - 1/xl -
.-.
- 1/Xn
distributes asymptotically according -to a normal distribution
n
~
n
N go - E llpj, U2+ E (af/4ly .
j=1
j=1
Therefore if sample size is sufficiently large, the asymptotic variance of g is
approximated by 6g = CT + X>.=1 i/X4. If the hypothesis (6.12) is true, the
distributionof t = gl/fg will be approximatedby N(O,1).
It is to be noted that although oligopoly firms maximize their profits independently without cooperation, the equilibrium point happens to be on the contract
curve. The necessary and sufficient condition for this situation to arise is that
they have the conjectural variations satisfying
-yjl
(6.14)
1
1
...
1
Yi2
.
.
1
.
.
*
=
.1
-Yzini
Using (6.4), it can be shown that (6.14) is equivalent to (6.12). Thus even if the
hypothesis (6.12) is not rejected, it does not exclude the possibility of this incidental
j =
If we substitute lrj = p. + (1/a1i)(pij/Di)qij- ci (j = 1.n)
1.
tk
n; k 1j) into (6.11), and divide both sides by (ai IDi)
and
l(Pikqik),
(1/ai1)(pi1/Di)qij(k,
we get
7Ejk =
=0.
*-to
1
,,n-
The left-hand'side is equal to (-0)i4V
1
Y8.
.
(YO-
1/t1-*
*.
*-
10
d__ _
dq= lrjj +
dqi
Z dq.
dql
dqij
n
k-
+ Y1j7ty.=
k =- 7t1j
jj+ij
ik
0,
k*j
where the subscript * denotes any firm number other than j (since, in the case of the homogeneous
product, all ltjk (k = 1,...,n, k : j) are equal). Therefore 7rjj= - 7tj*yij. Substituting this into (6.11),
(6.14) is obtained. Adversely if (6.14) is satisfied, (6.11) is derived by multiplying both sides of (6.14)
7r..z1*.
by
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962
GYOICHI IWATA
equilibrium on the contract curve. But if the hypothesis is not rejected in every
period, it will be natural to suspect that there may have existed some collusive
behavior.
7. ESTIMATESOF CONJECTURALVARIATIONS
The values of conjectural variations for Asahi and Nippon from 1956 to 1965
were estimated. The results are shown in column (4) of Tables I and II. The conjectural variations of Asahi for window glass 1 take fairly stable values of around
0.2 during the whole period. On the other hand, at Nippon those of y12 fluctuate
between 0.3 and 0.7. The conjectural variations of Asahi for polished plate glass
Y21 show a slow decreasing pattern, taking negative values of - 0.1 to - 0.3 except
one positive value at 1957-I. On the other hand the estimates of Nippon Y22
vary in a marked degree, starting from negative values, taking positive values
during 1957-II-1960-II, and again becoming negative.
The difference between the estimates for two types of products may be explained
as follows. The domestic price of window glass in Japan was substantially lower
than the international price in these periods. Exports accounted for nearly twenty
per cent of total production, while imports accounted for at the most 0.2 per cent
of total demand. Thus in this market, no foreign firm could probably be a rival of
a Japanese producer. On the other hand, the domestic price of polished plate
glass was much higher than the international price, and in spite of the high tariff
and ocean freight, imports were not negligible (2 to 20 per cent of total demand).
Therefore, foreign firms were competing with Japanese firms in this market. In
such a competitive situation, the absolute value of the price elasticity of individual
demand qij for Japanese producers will be very high, because if the domestic
price falls, the supply to Japan from the foreign firms will go to other countries,
leaving a much greater share of the market to the Japanese firms. The higher ijl
means the smaller conjectural variation, since we have the identity:
rij =
ilqij Di
If
(Xii
)lij
qij
Di
yij would have even a negative value. This may be the reason why most of the estimates of both firms for polished plate glass are negative.
In this paper, however, we have no intention of presenting a new pure theory in
which conjectural variation itself is explained. The approach taken in this paper
Since
taPii
aDi
Pij-_ dD Dqij pij qij|
I
Di
q.j Di
dqijopij
(Xii
qij
_ pij jj dDi jj _qij
apij qij dqij Di
this identity holds.
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=
I
j
i)il
i
963
CONJECTURALVARIATIONS
1958-I
1962-I
1961-I
1957-I
1956-I
1960-I
1963-I
1964-I
1959-I
1965-I
1957-II
1956-II
1960-II
1959-II
1958-II
1965-II
1964-II
1963-II
1962-II
1961-II
1,000
2.605
2.555
2.588
2.585
2.628
2.578
2.668
2.653
2.667
2.513
2.515
2.583
2.538
2.577
2.638
2.619
2.527
2.540
2.586
2.640
yen
.891
.881
.861
.991
.981
.8631,000
1.008
.935
.838
.925
.909
.919
.917
.896
.912
.860
.926
.914
.870
.984
yen
.395
.475
.378
.388
.406
.557
.553
.505
.413
.462
.438
.466
.404
.386
.404
.484
.564
.544
.394
.3801,000
yen
.325
.128
.237
.203
.249
.117
.218
.148
.267
.197
.292
.297
.159
.179
.184
.139
.146
.224
.244
.180
.551
.741
.645
.598
.658
.583
.557
.698
.727
.602
.613
.706
.622
.572
.587
.577
.556
.684
.664
.590
- - - - - - - - - - - - - - - - - - - -
Pi
(1)
Cii
(2)
a211
(i)
(3)
Asahi
9,1
(4)
a911
(5)
THE
ESTIMATES
OF
x
aq2,
(6)
.561
.551
.573
.363
.432
.468
.467
.594
.80710-6
.382
.459
.499
.642
.746
.674
.476
.404
.424
.566
.434
CONJECTURAL
TABLE
I
1958-I
1957-I
1964-I
1956-I
1963-I
1962-I
1961-I
1960-I
1959-I
1965-I
1956-II
1957-II
1963-II
1962-II
1961-II
1958-II
1964-II
1960-II
1959-II
VARIATIONS
1,000
2.721
2.791
2.791
2.733
2.802
2.733
2.728
2.812
2.668
2.987
2.663
2.738
2.764
2.686
2.670
2.684
2.680
2.690
2.673yen
P12
(1)
FOR
C12
(2)
WINDOW
1,000
.731
.625
.878
.743
.695
.863
.849
.759
.622
.599
.726
.630
.594
.594
.600
.660
.620
.632
.820
yen
OC2
(ii)
(3)
.471
.413
.733
.397
.298
.277
.532
.458
.395
.402
.318
.382
.392
.376
.450
.504
.454
.390
.460
912
(4)
.861
.901
.875
1.088
1.097
1.038
.948
.978
.837
.818
.922
.922
.872
.906
.906
1.099
1.022
1.009
.979
aV12
(5)
1,000
1.187
1.145
1.265
1.167
1.268
1.167
1.139
1.139
1.433
1.299
1.283
1.237
1.223
1.404
1.369
1.202
1.142
1.200
1.444
yen
- - - - - - - - - - - - - -
-
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Nippon
X
.771
.835
.908
-1.005
-1.145
.608
.725
.742
.777
.967
.956
-1.030
.666
.726
.754
-1.016
.626
.584
.970
10-6
GLASS
aql2
(6)
964
GYOICHI IWATA
1965-I
1964-I
1963-I
1962-I
1961-I
1959-I
1958-I
1960-I
1957-I
1956-1
1963-lI
1959-lI
1958-lI
1965-11
1964-II
1962-II
1961-II
1960-II
1957-II
1956-II
P21
20.865
19.351
22.645
25.453
21.998
20.711
17.771
19.791
19.334
20.089
19.833
19.804
19.435
20.332
20.160
18.210
20.038
20.158
20.365
20.592
1,000yen
11.552
11.545
11.347
11.692
10.548
9.848
10.229
10.049
9.413
8.797
9.884
8.996
8.599
8.289
8.444
8.404
8.849
8.836
8.580
8.327
1,000yen
a
21
7.522
7.616
7.462
7.347
6.809
6.523
6.408
5.431
6.227
6.276
5.892
5.552
5.288
5.217
5.464
5.426
5.316
5.184
5.074
5.102
1,000yen
-.241
-.188
-.196
-.146
-.191
-.334
-.187
-.362
-.276
-.078
-.234
-.244
-.254
-.046
-.215
-.065
-.227
.197
-.006
-.046
221
(1)
(2)
(i)
(3)
Asahi
THE
(4)
ESTIMATES
.784
.750
.757
.655
.638
.628
.593
.635
.681
.644
.642
.592
.623
.577
.883
.710
.617
.660
.712
.652
- - - - - - -
- - -
692
12
(5)
21
(6)
OF
x
.717
.858
.678
.799
.714
1.065
-1.743
.890
-1.009
-2.323
.864
.979
-1.484
-1.235
.892
-4.968
-2.114
-4.123
-2.742
-4.909
10-4
CONJECTURAL
TABLE
II
1965-I
1964-I
1963-I
1962-I
1961-I
1959-I
1960-I
1958-I
1957-I
1956-1
1963-lI
1957-lI
1960-Il
1959-Il
1964-II
1962-II
1961-II
1958-II
1956-II
VARLATIONS
FOR
P22
(1)
t22
(2)
kC22
(ii)
(3)
22.405
22.413
23.925
26.412
27.3,03
22.579
26.404
20.914
26.652
26.648
23.925
26.947
26.520
26.364
26.937
25.143
26.882
27.210
22.940
1,000yen
POLISHED
18.861
20.971
20.298
17.381
21.329
15.711
21.670
18.496
15.868
15.741
15.201
18.110
15.055
18.192
15.159
16.564
15.062
16.046
15.813
1,000yen
10.087
9.651
9.231
11.666
11.279
11.464
9.877
9.709
8.391
8.231
10.902
8.265
7.887
7.971
8.760
8.366
7.887
8.299
7.954
1,000yen
-.918
-.918
-.876
.131
-.306
-.083
-.662
-.036
-.552
.658
.093
-.218
-.114
.300
.189
-.139
.440
.264
-.307
- -
(4)
aY22
X
.141
.225
-1.931
-6.611
-1.507
.596
1.963
.140
-2.225
.982
-2.995
-5.605
-3.009
-2.632
-5.667
-5.358
-5.688
-5.310
-5.860
10-4
aq22
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GLASS
Nippon
(5)
1.477
1.093
1.446
1.617
1.343
1.501
1.131
.861
1.452
.871
1.132
1.176
1.328
1.236
1.234
1.346
1.125
1.262
.870
- - - -
722
PLATE
(6)
CONJECTURALVARIATIONS
965
has been an empirical one, but the estimated standard deviations of -ij in column
(5) were too large to use -ij for empirical analysis. As seen in (5.21), 6,ij depends
upon Uai1and cij . The latter, as shown in column (3), is very large compared with
cij in column (2) nor are 6al 1 and U&21 small (0.4144 and 0.4601).
As seen in column (6), the second order condition (5.10) for profit maximization
was satisfied in all cases.12
Now let us apply the methods of statistical test stated in Section 6. As the estimates
are not reliable we shall do it only as a demonstration. First let us test the Cournot
hypothesis, namely yij = y*= 0, for the window glass market in 1956-I. The related
figures are as follows. X0 = l = -0.9757, q11/D1 = 0.571, q12/D1 = 0.427,
c 1 = 0.863, C12 = 1.145, Plu = 2.619, P12 = 2.673, o = a, = 0.4144, 17l1 =
0.380, and c12 = 0.600. Therefore, in case of Asahi, t1 = g1/6g1 =-0.1246/0.4534
=-0.4509.
0.2748; and in case of Nippon, t2 = g2/6g2 =-0.2289/0.5076
Both figures are much smaller than 1.96 (5 per cent level of normal distribution),
and we cannot reject the hypothesis that ylj = 0 with the 5 per cent significance
level in either case.
Secondly let us make a statistical test of the existence of the type of collusion
suggested in the previous section in the window glass market in 1956-I. Since this
market then can be regarded as a duopoly one, the hypothesis to be tested is
PO _- (1/u1) - (l/,U2) = 0. Using the same figures as above, we have t = g/6g =
0.6222/0.5400 = 1.152. Therefore the hypothesis is not rejected at the 5 per cent
significance level.
8. CONCLUSION
In this paper, we tried to search a way for the empirical analysis of oligopoly.
Our findings and conclusions may be summarized as follows:
(i) Theoretically, it was shown that the price level in a homogeneous product
oligopoly market is determined as a function of three factors, i.e., the price elasticity
of demand, the marginal cost, and the conjectural variation of each firm. If these
three factors are constant, the price will not change. This can be one explanation
for the phenomena of price rigidity in oligopoly. It was also found that the -conjectural variation in our approach must be larger than -1.
(ii) Although the statistical reliability of the estimated results were not too
satisfactory, at least we know that it should be possible to obtain better results,
if the estimates of the price elasticity and the marginal costs could be improved.
(iii) Two statistical tests were proposed. One was a test for the hypothesis that
the conjectural variation of any firm is a specified value. The second was designed
to judge the possibility of collusion which designates price and market shares for
each firm. The asymptotic distributions of test statistics for the two tests were
shown.
12
The values of 02
j./q3
were calculated using 6i, and ij as oi1 and yij in (5.10).
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GYOICHI IWATA
966
These are our main conclusions. It seems to me that the approach we have
established here can generally be used as an effective method for an empirical
analysis of an oligopoly with a homogeneous product.
Keio University
Manuscript received October, 1970, last revision received February, 1973.
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[10] LEONTIEF,
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This content downloaded on Fri, 1 Feb 2013 12:02:09 PM
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