PRICE DETERMINATION IN AN OLIGOPOLISTIC MARKET
40
Transcript of PRICE DETERMINATION IN AN OLIGOPOLISTIC MARKET
PRICE DETERMINATION IN AN OLIGOPOLISTIC MARKET
A STUDY OF THE JAPANESE PLATE GLASS INDUSTRY
By Gyoichi Iwata
April, 1969
4
1.
PRICE =TT .T_T_3IT I~. JT OLIGO''OT I ST I C M J?TrET
-'~- ST J ~- J i r -y JJ r A~. CE PLATE GLASS I Cm V ST RY --~'
By Gyoichi Iwata
Introduction
The purpose of this paper is to present one possible solution to the
problem of price determination in an oligopoly market, by analyzing
empirically the Japanese plate glass industry.
In this paper; we are going to estimate the value of conjectural
variation namely the quantity which a.firm conjectures about the behavior
of its rival firms. Then we will explain conjectural variation and. the
method of measuring it.
Suppose there is en oligopoly market of one homogeneous product.
This market c'nsists of a few firms that supply the product and a great
number of purchP.sers. Let the total demand of this product be D,, the
price p, and the market demand function p=f (D) . Let us consider the
1) This study was carried out at the Keio Economic Observatory,
Keio University. I am indebted to Kotaro Tsujimura and the other members
of the KE0 for many helpful comments and discussions. The assistance of
Kanzo Otani (~sahi Glass Company), Tohru Karachi and Ninoru uarada
(Ministry of International Trade and Commerce) is atefully acknowledged.
The original version of this paper was given at the Zushi Conference on
Jan. 8, 1968, and the present version was presented to this October, 1968,
meeting of the Japanese conometric Society at Osaka University. I Wlsh
to thank T. Miyashita, T. Sawa and H. Niida for their comments at these
meetings.
-1-
behavior of one firm. Let the supply quantity of this firm be q and the
supply of the other firm q. And, let us denote the revenue of that firm
by R. Then R=pq. If we assume the profit maximizing behavior of this
firm and defining the profit as Z ==c-C, where C is the total cost of this
firm, its supply q must be determined at the point where the marginal revenue
is equal to marginal cost. The marginal revenue may be expressed as
dF. dp dr dD dp d a+q (1.1) dq = p + d~ = P + dh da - P + dD dq
_ d dN
4
P + dD ̀ + da
as in the third. term of the last expression is the quantity called the q conjectural variation (cf. lJ and [3J'). This quantity is the ratio
~f the increase of the other firms' supply conjectured by the firm in
consideratiornt corresponding to its sunrly increase, As is well kno'm,
Augustin Cournot (lJ made his oligopoly theory by assuming dq/dq = 0.
After that, various oligopoly theories have been built on the various
assumptions about this conjectul variation But until now few attempts
to empirically estimate this value have been made.
If we can assume the equality between the expected marginal revenue and
the marginal cost the relation
r (1.2) p + q + dq a = as
holds. Therefore the value of the conjectural variation dq/da_ is expressed as
dC
(1.3); dq = ~ - 1
dD
The value of the righthand-side expression could be calculated if we could
measure the cost function C(q) and_ the market demand function f(D)..
-2-
Using this equation (1, 3), we try to estimate the actual values of the
equation (1.3) we try to estimate the actual values of the conjectural
variations of some firms in the Jap~riese plete glass industry., 2)
2) For a measurement which bears some resemblance to ours we must refer
to the estimation of the price elasticity of the individual demand curve
done by Wassily Le ontief in 1940 CS). The price elasticity of the
individual demand curve can be written as
('") 7 dp q dD dp dB dp dD d q dD dq } dD dq
using (1.1). The denominator of the last expression is equal to d - p, if the relation (1.2) holds. Therefore. we have
dq p
By calculating the righthand-side expression using the price and marginal
cost data, he estimated the values of the price elasticity of the United
States Steel Corporation' s individual demand curve for each gear during
the periods from 1927 to 1938 end. obtained the estimates -3,25 -- -4.06
as . He says in his conclusion as follows:
"Under oligopolistic conditions such as prevail on the American steel
market, the opinion of the particular producer about the possible reaction
of his actual or potential competitors (the so-called conjectural elements)
plays an important part in determination of the shape of his individual
demand curve. Barring the uncertain device of personal questionnaires,
the indirect method of demand analysis described above represents the
only possible way of measuring these highly volatile demand relationships."
(p.817 in (8))
-3-
There are three firms prod oing plate glass in Japan. They are Asahi
Glass Company, Wip?on Plate Glass Company and Central Glass Company, We call
them briefly Asahi, ITippon and central in this paper. Their production shares
of the sheet and plate glass excluding polished plate glass in 1965 are 52.7,
33.5 and 13.8% respectively (cf. Table 4.3-a). In this paper we have analyzed
only the behavior of Asahi and Nippon, excluding Central, for the reason that
the Central started its production from 1959 and its sample size is not yet
large enough to concern us.
In what follows, we start :;y measuring the cost function of Asahi and
Nippon (s 2), and then estimate the market demand function of ordinary and
figured plate glass and polished plate glass ( 3). After that, we estimate
the conjectural variations of A sahi and Nipnon for each half year period 1 L
(~4)•
-4-
§ 2. The "leasuremet of Cost Function
In this section we are going to estimate the short-run cost functions of
Asahi and Nippon, by using their accounting data. There are many difficulties
in the use of time series accounting data to estimate the cost function as
stated e, g, in C5), (6) and (7). The main problems are: 1) Factor price
change, 2) Scale change in productive capacity, 3) Multi-products,
4) Depreciation cost is usually determined by taxation authorities rather
than by economic criteria, 5) Technological change.
We try to solve the first problem of factor price change by splitting
the accounting cost data as much as possible into the independent items,
which may be separated into price and physical Quantity, or can be deflated
by the appropriate deflator. The second problem of scale charge may be
solved by introducing productive capacities as independent variables into
the input functions (demand functions for inputs) of the above physical or
deflated inputs. The multi-products problem may be solved, if we use these
products as independent variables of input functions. About the depreciation
problem, we try to settle this by subtracting the depreciation costs from the
short-run variable cost, since it is our object to measure the short-run
marginal cost. We cannot avoid the technological change problem as far as
we have to use the time series data of about ten years, during which various
technological changes have occurred (The production of plate glass by means
of the so-called float method, the recent gratest technological change,
started from 1966).
-5-
1. Cost Function of Asa_ i
Asahi Class Company is the aigges.t plate glass maker in Japan. It settles
accounts in June and Leconber, In what follows we try to estimate the cost
function using mainly the data from half-yearly report of Asahi in the period
from 1955.1 ( the first period of 1955, i.e, from January to June ) to 1967.1.
In addition to plate glass, Asahi produces soda-ash, caustic soda, fire brick'
the glass for the Brown tubes of television sets etc.
For convenience of analysis, we divide these products into three
categories as follows:
X1 : Ordinary sheet and plate glass, figured glass and wire glass,
X2 : Polished plate glass,
X3 : The other products.
The unit of Xl is converted cases, X2 cases, and X3 1,000 yen in terms of the
price. X3 is defined as follows.
5 P3 (2.1) X3 =E z~; /(O1), ial j/ P3i
where zl:the money-term amounts of production of soda-ash, z32: that of
caustic soda, z33o that of fire brick, z34: that of tube glass, z35: that
of the other products. And .p3i (i~1,...,5) are their prices, p3i is the value of p3. at 1962.2.
3) One case is equivalent to the sheets of plate glass the total area of
which is 100 square feet (n9.29 m2). In this definition the thickness of
the plate glass is not considered. One converted case, on the other hand,
denotes the sheets of plate glass that have the same volume as one case of
plate glass whose thickness is 2 millimeters. For example, one case of
plate glass of 5 millimeters thick corresponds to 2.5 converted cases,
-6-
We divide the total cost of this firm into following categories.
"aim material cost CTZ
Variable cost C~
The other cost CO r
Total cost C 4 ;Main labor cost CL
Capital cost CK !Fixed cost CF
;Incidental profit and loss CL
The mai.n material cost Cis defined as the sum of costs corresponding to the
materials of which inputs and prices data are available.
Let us denote input of each material by its price by si and the number of
kinds of materials by n. Then
n
(2.2) 1 = ' simi . i=1
The main labor cost is defined as
(2.3) CT = wL
where L is the number of workers at the end of each period, and w is average
wage (1,000 yen / man. half-year) .
The capital cost is defined as follows. The quantity of capital equipment
K is defined as the real value of the capital equipment evaluated with the
price of the base period, 1956.1. Its value at the end of each period is
obtained by the formula
(2.4) Kt + ( 1 - t ) Kt-1 + It /( PIt ) "IO
where It represents the gross investment at the period t and is estimated as
follows. Let us denote the bookkeeping value of the capital equipment
(defined as the sum of tangible fixed assets after depreciation reserve
excluding lands and construction account) at tth period by Kt and the
depreciation cost by dt. Then
(2.5) It = Kt - Kt-1 + dt.
-7-
As the initial value of K ~ of (2.4), we use the value of Kt at the end of
1950'.2. PIt denotes wholesale price index of investment goods of which base
year is 1960 (:'ounce: the Bank of Japan). 3 t is the rate of depreciation
defined as
(2.6) t dt T~ t t-1,
V,je define the unit price of the real capital equipment f (namely, the cost
from holding one unit of K during one period) as
(2.7) r=pK.( r. + b),
where R, is tile ratio of K' to T, and r. is the market interest rate r_ 1
(average interest rate on loans of all banks; Source: the Bank of Japan).
Then the capital cost is defined as
(2.8) CK = r }r .
The incidental profit and loss is defined as
(2.10) CB Q (nonoperating cost) - (nonoperating
income) - ri Kc
Finally the other cost is defined as
(2.9) C0 (the material costs other than C~?) + (the manufacturing
labor costs other than CT } + (manufacturing overhead cost)
- ( depreciation cost) + (general management and selling
expense)
A s the next step, we are going to measure the input functions as to
the physical or real quantities, which constitute these cost items. We tried
to test various tees of regression equations in order to explain these
inputs, we will show only their final result.
-8-
As the main material inputs, we adopted the following five inputs:
ml : silica input (ton), m2 : soda-ash input (ton), m3 : dolomite input (ton),
m4 : material salt input (ton), m5 : coal and heavy oil input (million Cal.).
The regression equations finally adopted are as follows. )
(2.10) log ml = -2.480 + 1.058 log XG + 0.08807 log X3 (0.3549) (0.07837) (0.03355)
d.f.=22, s=0.02829, R=0.9841, R=0.9827, d=1.415
(2.11) log m2 = -3.599 + 1.058 log XG + 0.1690 log X3 (0.4111) (0.09077) (0.03885)
d.f. = 22, s=0.03277, R=0.9835, R=0.9819, d=1.415
(2.19) log m = -2.492 + 0.9186 log X + 0.1215 log X3 3 (0.8103) (0.07663) G (0.03280)
d.f.=22, s=0.02767, R=0.9329, R=0.9814, d=1.509
(2.13) log m = 2.249 + 04144 log X~ + 0.05907 log X3 (0.2515) (0.05553) (0.02373)
d.f.=22, s=0.02005, R=0.9590, R=0.9551, d=1.353
(2.14) log m = 0.4184 + 0.8631 log X
G
5 (0 .4529) (0.04693)
d.f.=23, s=0.04158, R=0.9330, R=0.9300, d=1.4,63.
XG means total output of plate glass defined as
(2.15) XX1+ 2.5 X2
where the figure 2.5 means the rate of conversion of case to converted
case since the actual average thickness of the polished plate glass was
about 5 millimeters,
4) The logarithm of each variable is its natural logarithm
Parenthesized figizres under the estimated regression coefficients are
their standard errors. D.f, is the degree of freedom, s the standard
error of estimate, R multiple correlation coefficient, R adjusted multiple
correlation coefficient, and d Durbin-Watson statistic.
-9-
The quantity of coal and heavy oil is measured in terms of calories. One ton
of coal represents a heat value of ?.0 million Cal. and one ton of heavy oil
represents 9.639 _million Cal..
Then we procede to the estimation of the input function of the other
cost. The items that constitute the other cost CO are various and it is very
difficult to find a single deflator for CO. Therefore we divided CO into
the next four items and found the appropriate deflators for each item.
(2.16) CO+CM' +CLt +CH' +CS ,
where
CMt = the other material costs = the material costs
in the manufacturing cost -
CL' = the other labor ccst = the labor costs in the manufacturing
cost + the labor costs in the general management and
selling exmense - CL,
Cxt = the manufacturing overhead cost other than depreciation cost,
CSC = the general management and selling expense other than labor cost
and depreciation cost.
The deflators for these items are
the wholesale price index of raw materials and fuels, 1960 = 1
( Source o the Bank of Japan)
PL a the cash earning index of regalar wor.ers in manufacturing
industry, 1960 = 1 (Source a Ministry of Labor),
PH : the arithmetic men of Pand PT ,
PS a the wholesale price index of producer goods,
1960 = 1 (Source a the Bank of Japan)
The real value of the other cost c0/P0 is defined as
- 10 -
(2 .17 ) CO CTS t CL t Cx, C U r + + p D P P
o M PL 1R 4
where PO is the implicit deflator of CO. The final form of regression
equation obtained is
(2.18) CO p- 39684.333 + 0.1404 x1 + 8.850 X2 + 0.3899 x3 0 (
1,206,302) (0.5249) (8.256) (0.05178)
d.f. = 19, s = 729,928, R = 0.9776, = 0.9740, d = 1.394,
Finally, we explain the input functions of labor and capital equipment.
If we assume a production function of the substitution type such as
(2.19) Q = F(L, K)
exists among the productive capacity Q, labor inputs L, and capital ecuip-
ments K, then the minimum cost inputs of L and K for the given Q will be
determined as
(2,20) L = g(Q, w, r)
(2.21) K = h(Q, w, r).
Actually, however, an instantaneous adjustment can not be made because labor
as well as capital equipment cannot be easily changed. Therefore there my
be some time-lag between the time at which a decisicm is made to alter them
and that of execution. The regression equations finally obtained are
(2.22) log L = 2.182 + 0.3024 log a - 0.1474 log (W) (0.1678) (0.01061) (0.03055) r
d.f. = 19, $ = 0.006215, R = 0.9959, R = 0.9955, d a 1.598
w 2
-
2.23) log K = -0.03893 + 0.8465 log Q + 0.3279 log (r ) (1.446) (0.09365) (0.2494) -2
d.f. = 17, s= 0.04,402, R = 0.9578, R = 0.9527, d = 0.908
where Q denotes the total productive capacity of Asahi, which is defined as
(2.24) Q = z_ 5-). t i =1 i0 0~.0
- 11 -
denotes the productive capacity of the ith product at the end of period t.5)
C
Now we have eight input functions about ml, ..., m5, ~~, L, and K. 0
They are (2.10), ..., (2.14), (2.18), (2.22) and (2.23). If we put them
into the cost equation
5 G (2.25) C = slmi + P0(-) 0 + wL + rK + CB,
i=1 0
we then obtain the total cost function
(2.26) C = C(X1, X2, X3 ;1, Q2, Q3).
In order to find the shape of this cost function (2.26), we calculated the 6)
values of C corresponding to various output levels using data at 1965.2,
and drew the curves of the short-run cost functions and average cost functions.
The composition of outputs in the above calculation is as:3umed to keep the same
ratios as that of Ql, Q2, Q3 at 1965.2. In figure 2.1, there are five short-
run cost curves which correspond to the capacity levels (2 =) 0.25, 0.5, 0.?5,
1.0 and 1.5, respectively. The unity of 2 corresponds to the capacity level
at 1965.2.
It is seen that the shape of the total cost curves is almost linear
in the range of output not near to the origin. In figure 2.1, the cost
curves are drawn until the output varies to 1.2 times to the capacity,
5) The data about the productive capacity Q5t are not available. We used
the money term anount of production z5t deflated by producerts goods
price index as QSt.
6) The actual values at 1965.2 which were used for the cost curves are as
follows: B1=1.850, s2=27.000, s3=2.295, s4=3.750, s5=0.579, P0=1.19038,
w=273.882, r=0.13148, w/r=2,083.069, w-2/r-2=1,693.632, C3=_922,840,
Qi=6,810,000, Q2=219,536, Q3=17,5 -,884, Q=32,193,679.
- 12 -
=1.5c2
/
of ASah1CurvesCostShort-roan Total lie2.1i i ' reb C CD N
N
O
O
cl- (D
w
i
I 40 .~
=1.002
/=0.75
2=0.50
~/~ j
output1.5
C
1.00.5
30
20
10
0
0
13
C
x
4.0
3.
2.0
1.
0
,r
Fi gt ire 2.2 ; he hor--run average Cost Curves of Asahi
i
E`
'
}I
\\\ \\
d =0.25
50
2 =0.752 =1.00 2= 1.50
0 0.5 1.0 1.'5 output
- 14 -
2.
but in actuality the cost carves will rise sharply at the point where the out-
put exceeds the capacity level.
The average cost curves are drawn in Fii.re 2.2. It will be seen that
the average cost at the point of full utilization becomes smaller as the scale
of capacity 2 increases. So, it can be concluded that there exists the
economy of large scale in the plate glass production of Asahi.
Cost Function of Nippon
Nippon Glass Company is in second position in the market share of plate
glass production in Japan. This company settles accounts in November and
March. Nippon produces plate glass only. Therefore the analysis is simpler
than Asahi. Except for these points the formulation of the analysis is
almost the sane as that of Asahi.
We adopted the following five input functions about material inputs:
(2.27) log ml = -2.442 + 1.1504 log XG
(0.1690) (0.02663)
d.f.=19, s=0.01364, R=0.9949, R=0.9947, d=3.018
(2.28) log m2 = -3.647 + 1.251 log XG
(0.2429) (0.03828)
d.f.=19, s=0.01961, R=0.9912, R=0.9908, d=2.076
(2.29) log m3 = -4.515 + 1.387 log X (0.6925) (0.1091) G
d.f.=19, s=0.05590, R=0.9459, R=0.9430, d=0.659
(2.30) log m5 = 1.278 + 0.6970 log X1 + 0.02875 log X2
(1.029) (0.2045) (0.06037)
d.f.=19, s=0.03693, R=0.9254, R=0.9172, d=0.821
(2.31) log m6 = 3.936 + 0.3901 log X1 + 0.2214 log X2
(1..039) (0.2066) (0.1047)
d.f.=19, s=0.03732, R=0.9579, R=0.,9534, d=0.696
where m6 is the input of electric power (KTH).
- 15 -
zs to the other cost Cog -re obtained the following:
(2.32) C log 5.397 + .02381 :2, + 0.2214 log X2 0 (1 .788) (:.3533) (0.1047)
d.f.=17, s=0:06051. 0.7946, R=0.7668, d=0,630.
Finally, the input functions of labor and capital equipment are
estimated as follows
(2.33) log L 2.549 + 0.3089 log Q - 0.02096 log () (0.1598) (0.03095) (0.07198)
d.f.=18, s=0.01397, 3=0.9707, R=0.9674, d=1.715
(2.34) log K = 1.204 + 0.6933 log c + 0.2043 log Q2
(0.8707) (0.2963) (0.1207)
w
-2
+ 0.03358 log (r ) (0.3721) t-2
d.f.=l5, s=0.05525, ==0.9518, R=0.9419, d=1.245
In the same way as the previous section, the total cost function is
obtained if we substitute these eight equations into the cost equation
6 C0
(2.35) C =~' simi + PO() + wL + rK + CB
1
i-4
It can be written as
(2.36) C = C(X1, X2 ; Q1' y2)
The Figure 2.3 shows the short-run cost curves,7)the abscissa being the output
of which composition is the same as that of capacities at 1965.1. This figure
shows the same characteristics as those of Asahi. We will omitt the figure of
the average cost curves.
7) The data at 1965.1 used for the cost curves are as follows:
s1=2.65, s2=21.0, s3=2.5, s5=0.66397, s6=0.0024176, P0=1.15177, w=311.67,
r=0.12492, w/r=2494.9, w-2/r-2=2475.2, C3=64,059, Q1=4,860,000, Q2=120,000,
Q=5,160,000.
- 16 -
Figure 2,3 The Short-run Total Cost Curves of Niu on
C
d
F-'
0
tT
0
ci-
u 2 =L50
200
2 =1.00
/100 7
0
0 0.5 1.0 1.5 output
- 17 --
S3. The Measurement of Market eand Function
Before starting to estirnte the demand functions, let us look into the
mechanism of plate glass distribution in Japan. The following parenthesized
figures show the percentages of transaction of ordinary sheet and plate glass
in 1965. j~Thole plate glass, except that for export (15%) and direct sale to
camera markers, is sent to about 380 agencies in Japan. The number of
agencies of Asahi and Nippon is about 180 respectively, but some of them
belong to both Asahi and Nippon. Nearly 130 other agencies belong to Central
Glass Company. The plate glass then is shipped for building use (660) and
general industrial purposes (17%), directly or through retail shops. There
are about 18,000 retail shops in Japan 5,000 shops dealing in plate•glass
only, and 13,000 shops taking it up as a side business. Plate glass for
repair (1%) is sold by retail shops.
Let us start to measure the demand functions of plate glass. Let Dl be
the total domestic demand for the ordinary sheet and plate glass, figured and
wire glass (converted case) and D2 be that for the polished plate glass (case).
The domestic demand of each product is the sun of the domestic shipment from
three makers (Asahi, Nippon and Central) and the import. , The data of the
domestic shipments are obtained from the "Annual Statistics of Ceramic
Industry" edited by the Ministry of International Trade & Commerce.
The time series monthly variations of the domestic demands D1 and D2 are
shown in Figure 4.1, by real line and dotted line respectively. In this
figure the monthly variation of p11, the price of the ordinary sheet glass of
which thickness is 2 millimeters, and p21, the price of the polished plate
glass of 5 millimeters thick and 17 sheets per case, are also shown by a
bold line and a chain line respectively. This price data is taken from the
wholesale price survey by The Bank of Japan.
=18-=
C tom:
t.0
r-1
c~
--.__,
.~~.~
C, - -- .
CT r-!
M
CR r-f
----- ~--
r
N
`0
r 4
a
r4
1"ns
I 1
1i
i
t
r
\ r
I-- - _~
- V
1
_ _ 4
t
~ . c ~ `~
Ir
* !
^~ fi
\I
I
rr"
I
i + }
r
~.
Q.
b QQ
1n Ch
19
U2 02 cd
r4
O
r-4
O
rn N G}
rd O
02
O d
02
4.4 O
01
O .r.4
ri
n
r~ O z
r4
a)
n orc.er to use :e r. a ; e x ian_a ~3Y"T ar r s . we .ade the following price
ind.icies at month
~ pt Ut p= 2( o + y o ) j=1, 2
j
Pjl Pj2
where p12 is the price of the ordinary sheet glass of 3 millimeters and p22
is the price of the polished plate glass of 5 millimeters thick and 4 sheets
contained in a case. We use pl as the price index for D1 and P2 for D2.
Polished plate glass was consumed in the following way in 1965: 27% for
building, 45,E for the windows of automobiles and. 28% for mirrors.
We tested several types of demand function and finally obtained the
following equations
(3.1) log D, = 3.928 - 0.8704 log (1) + 0.5034 log T - 0.2448 log - (0.2313) (0.3002) ~I (0.06796) (0.1153)
d.f.=104, s=0.05339, R=0.8929, =0.8896, d=1.085
(3.2) log D2 = 0.6117 - 0.1498 log (p2) + 0.2739 log T + 0.1309 log A (0.3870) (0.1071) I (0.1264) (0.07846)
+ 0.6450 log Y (0.1659)
d.f.=67, s=0.04,500, R=0.9230, R=0.9182, d=1.757
In the above ecuations, PI denotes the wholesale price index of investment
goods (cf. § 2), T denotes the floor area of total building construction
started (1,000 square meters per month ; Sources Ministry of Construction),
w is the ratio of wooden building construction to that of total building
construction, A denotes the monthly production of automobiles (including
passenger cars, four-wheeled trucks and buses, Sdusde: Mcsrithly Reb t do
Automobile Industry s Data, Automobile Industrial Association), and Y
denotes the real private consumption expenditure (1960 price, one ~ia1idri
yen, quarterly figure ; Source: Annual Report on National Income Statistics,
Economic Planning Agency).
-20-
S 4. he Istimation of Co_ijecturai -ar_a:i:rs
In this section, we are =oin to estimate the conjectural variations of
Asahi and Nippon by using the estimated cost functions and market demand
functions.
In the first place,, let us formulate a model. In the following formu-
lation, when we refer to one firms that firm can be either Asahi or Nippon.
But in the case of Nippon, the output and supply of the third product is zero.
The total market demand of the j th product (if j=1, it is ordinary sheet
and plate glass and figured glass, arid. if j=2, it is polished plate glass) Dj
is equal to the sum of the following three supply quantities
qDj : domestic supply of the j th product by a firm in concern,
q~ J domestic 5:,..Y%tr,j OI the V Vh _+.Vd.uV J y,' tua.. other firms,
import of the j th product.
Then
(4.1) Dj =qDj + qDj + M. j = 1,2
If we denote the firm's supply of the jth product by q., and export by qJj,
the relation
(4.2) qj = a,v + q~j j = 1,2
holds. In the same way, if we denote the other firms' supply and export
by q. and q . respectively, we have -J
(4.3) q. =qDj qJj j = 1,2
Let the domestic price of the jth product be pj and the market demand
function of the jth product be
(4.4) pj = P3(D3) j = 1,2
y.ci ty. where express nirL we do not exp~ss the shift va~.~.ia:, ~ies egplici t1~ for si._*^pli.._t
we denote the export price of the jth product by the total revenue of
this firm is expressed as
2 (4.5) R =(P33 + pHjcEj) + P3g3
J
-21-
In this model we assume that the exhort qE., the export price py,. of
first and second product, and the supply of the third product q3 and
price p3 are all exogenous variables.
Let us write the short-run cost function of this firm as
(4.6) C = C(X1, x2, x),
where X. is the output of the jth product. In this equation, we did
write the capacities Q1, Q2, Q3 as independent variables, because in
short-run the capacities can be considered as constant.
We assume that at the sales decision level this firm determines
supplies q1 so as to maximize the anticipated profit
(4.7) 7r = R - C
2
= (PjgDj + jgmj) + P3g3 - C(ql, q2, q3). j=1
In this case the first order condition to maximize the profit is
(4.8) a ~c = aR, _ aC = 0 j = 1,2 1 a q~ a q~ 8q
On the other hand. the anticipated revenue of the jth product is
a R dp. dp. dD. dqj, . = p . + -~- q = p + ---~- -~ q as 0. From (4 .1), D _ = aqj J dqj Dj j ~j dqj Dj dqj J
+ Mj and if we suppose import M. to be given exogenously, we obtain
(4.9) aD dqD dq
q =pj + dq =1+q j =l, 2, J a j
where we assume dgEa =0. Therefore, the anticipated revenue is dqj
dp. dq. dp. (4,.10) aq = pj + qDj + q dD qDj j = 1, 2. dq ---~- in the right -hand expression is the conjectural variation which d
qj
are going to estimate.
- 22 -
the
its
not
the
its
we
.:le secora oraer corn: :_or mize prozi is
(4.11) a22 C 0 j = 1,2 aq.
and
' a2
~z a2r
8q~ aq1 aq2 a2~ a27t / a2TC ~2 (4,.12) = aq2 aq2 - aq1 aq2, > 0
a 2~ a 2
ag2ag1 aq2
From (4,.10) we obtain the relations
a2`y = (2+2i± d~J ) dpJ + (1+ dgJ)2 d2P g - _ (4.13) a q 2 dq1 d q 2 ll! dD d dD2 DJ a q 2
J J J gl J J
a2r 020
°`q1 vq2 aq1 aq2
As market demand functions, we have (3.1) and (3.2) in the previous
section. But these equations are the relations based on the monthly data.
If we adopt a half year as the tine unit, we have to adjust the constant
terms in the following way: Let the constant terms in (3.1) and (3.2) be
b10t and b201. Then the constant terms of the half year base b10 and b20
are expressed as
b10 = 61-b12 b10'
b -b24 r 20 s L1-b12-b24 2 b 20
After adjusting the constant terms in this way, the market demand
functions are
(4.15) ( p1 b11 Tb1 2 b13 D1 = b1 u P I
(4.16) D _ b (P2 } b21 T b22 A b_3 Y b24
-
2 20 pI
There is another problem. P1 and P2 in these equations are price
indicies and are not the prices r_ and o., stated in (4.4). 1 ~G
- 23 -
etually, each product is oomposed of various kinds of items the prices of
which are different from each other. The reason why we used the price
indicies p1 and a2 is that we considered that these price indicies could
be proxy variables for these prices of various items. Even if each price
of the various items fluctuated pro~aortionately, the effective price, i.e.,
average sales amount per case, would not change in that same proportion
necessarily4 because the composition of sales of these items might be changed.
Namely, the effective price is affected by the charge of item composition
as well as the price changes of various items. Thus we did not use the
effective prices at the measurement of the market demand functions.
A.i per we have measured the demand functions, however, we might be
able to use the effective prices as the prices p1 and p2, because the
items composition will not change in the short-run.
Let the sales money amount of the jth product be Sj and its price be
defined as
(4,.17) p = s./q. j = 1,2 j 3 '; And we assume the relation
(4.18) pj = j = 1,2
holds, where iLj is the proportional constant which is supposed to be
constant in the short-run.
If we substitute (4.18) into (4.15) and (4.16), and solve them with
respect to pj, we have
J_ _ _ (4.19) p1 = b10 b11 D1 b11 T b11 bit
_ 1. b 22 b 22 b23 b23
(4, 2o) p2 = b20 b21 D2 b21 Tb21 A b21 y b~2 u1 PI
-2Y_
Then we are _ _- o ~s Lira:.. values of he ccnjec ~ural -rariatio r_s
•j of Asani and _ : D c_''_. ~"ro (a.3) and (4.9) the con jeca? var^ atlors da.
of3each fir^1 is calculated by
ac
(4.21) da a p 1
aD J j
The results of estimation are shown at Table 4.1 and Table 4.2.
Column (1) and (11) show the actual values of the effective prices pl and
p2 respectively. The nrginai cost is calculated at (2) or (2') column.
7 It is seen that the marginal costs of the ordinary sheet and plate glass
are about 600 900 yen a t A sahi and 500 -- 800 yen at :;ippon.
The marginal the pelis=e pi costs of d nte _ ' g?as..; ~ ~ are about 9, 8C0 12,000
yen at :?sahi, but t _ose of :Iippon are declining from 28,331 yen at 1956.1
to 5,522 yen at 1965.1. `
(3) and (3') columns of Table 4.1 and Table 4.2 are the partial
derivatives of the r:{et demand functions, a .
aD. J They all have negative signs
. dq
The values of the conjectural variations da are shown at column (4) -J
and (4'). We show these time series variations a t Figure 4.1, where the
real lines stand for 3sahi and the dotted lines stand for Nipp_ on. dq= +
It will be seer that the conjectural variations about the ordinary =1
sheet and plate glass are 0 - 0.3 at Asahi and 0.7 - 1.0 at Nippon in the
periods from 1956.1 to 1965.2. If we use the estimated value a t 1965.1,
Asari conjectures that i it increases its supply by one unit, the other
~., tims will incr .... ea...se their s ...~rplies by 0.24 unit .n retaliation. 1y.~ On ~L ly'e
other hand, Nippon_ anticipates a supply increase of 0.99 unit from the
other firms in retaliation if Nippon_ increases its supply by one unit.
- 2 5 ...
Tab
le
4.1
The
Res
ults
on
A
sahi
I N
rn
Yea
r
1956
56 57
57
58 5e
59 59
60
60
61
61 62
62
63
63
64 64
65
65
peri
od 1 2 1 2 1 2 1.
2 1 2 1 2 1 2 1 2 1 2 1 2
n1
2.61
9
2.66
7
2,65
3
2.66
8
2.57
8
2,62
8
2.55
5
2.64
0
2.58
5
2.63
8
2.56
6
2.60
5
2.52
7
2.57
7
?.54
0
2.53
8
2.58
8
2.50
3
2.51
5
2.51
3
an
a4~
3
a >
> 1
an.
(4)
`~i1
~I
q1
0.03
0
0.72
6
o.92
'/
0.84
7
0.'7
6
0.70
1
0.74
7
0.73
5
0.90
3
0.74
7
0.78
8
0.72
9
0.77
0.65
7
0.75
7
0.75
2
0.82
0
0.63
9
0.84
2
0,80
7
-1,1
69
xlo-
6
-0.6
0
-0.9
54
-0.7
34
-1.0
25
-0,0
30
-o.e
73
-0.6
93
-0.7
72
-0.6
44
-0.7
02
-0.5
64
-0.6
33
-0.5
32
-0.5
74
_0,6
9
-0.5
13
-0.4
50
-0.4
7
-0.4
25
0.04
1
0.27
0
-0.0
03
0.14
1
0.1
3
0.1
08
0.13
3
0.13
9
0.45
4
0,12
2
0.16
8
0.15
6
0.1
01
0.2
90
0.22
7
0.2
42
0.2
71
0.1
82
0,2¢
1
0.1
95
5 a2
2-aq
1
-0.8
11
xlo-
6
-o.5
ao
-0.6
89
-0.5
74
-0.7
59
-0.5
51
-0.6
39
-0,4
91
-o.5
R9
-0.4
55
-0.5
50
-0.4
09
-0.4
57
-0.3
99
-o.4
E2
-0.3
94
-0.4
49
-0.3
75
_0,Q
29
-0.3
56
6)
a2
an1
aq-2
1.•7
15
xl0-
7
0.91
3
1.3e
5
0.60
6
0.99
o,C
'J1
0.40
0.57
5
0.83
1
0.52
7
0.59
9
0.41
5
0.47
7
0.16
0.53
0
0.32
0.53
7
0.42
1
a.q(
3c3
0.37
* T
heun
itof
th
efi
gure
s in
co
lum
n 1
and
(2)
is1,
000
yen
per
conv
erte
dca
se,
i
ti} 02
U
0
X d- ~- ~- LL\ r-4 rn d- N- ~- N- a\ d- cT N- CO C\i O C ~z-E ~ D O CT \C K1 ( r--! ~' N- 0 CD N N d' N- Chi CG U C H S`z H KZ C N- H eO N- d' `O UZ
.
rr~ N N r--4 r4 d r-1 0 C O 0 0 O C C O C` 0
a
d H X 'O CS z CO O 0 CG H N l0 K1 KZ l3 CO N K \ N r-I K\ t13 CT
CO e \ `O N CO N tf1 Cr N\ d' H M H 0 CO K1 CT H CO ~O L ~O KZ K1 N N H H H H H H H H 0 H O H O
0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 ! 1 t t t S t i Y I ! S f f 1 ! 1 ! ! !
NCV
LV1.C1
U
if
U\ N d' U \ 0 tD K'1 N- Ci\ tt\ C Gti Cf\ r-3 \O tX3 a ~' ~f" d" `O \ 0 ' N N- m CT. C~ \O Cx7 N a) N- CTS c 0 W\ N '-O CQ Q) a) t) a) C) CO CO CQ a CO d a) a b a) C31 a) a W CC . . . . .
p G O C1 o d G O o o ddC C7 O d c C c C 1 1 1 3 1 3 1 1 ! t f i i i i i 1 1 1 1
a)
a)
0 b
0
r-4
41 •ri
CV
b
M U
,r{
a)
a) s..
.ia
4»~ Q
-F~
I
a)
E
C~1
U
n b Cl)
4.3
Q U
u
IVi
mO
tf1
O
K1H
CVH
t1"\
Ld
CVCM
O
.
Ot
CV
HH
C".7
HCV
NOU'
O#
H
I-'H
C'-rCO
M
CD'C
Ot
NHH
HH
LL \CV
N-t1'\
C?
HH
tr
C'H
N
O#
OOLo
OH
HU'rt1
C1H
LC\C\O
O
N -N-C-
0H
ChH
N--CTC--
O#
Q'r
H
OH
CV
rn
O
CVH1-
O#
'OC-
OH
O
H
O
CTO
O#
N-'C
OH
'
O
OCV
CT
O#
C^-
L1f
O H
HH
C-
C
toCVO
Hs
K1H
OH
rn
CrH
CTaC~-'
0
L1"1C-
OH
OHCV
tC~H
U'\Q\K1
H#
C-
'
H
N-C-
N-H
tt1
C-
H#
'r4
.'
M
CTH
CVC-
H#
CDMcr
s'
HC3`tC-
H
OC-C-'
H#
K'\
'
OH
U'\
GCV
m
C--
CV
mU'O
HH
cK1O
aCV
'
O
CUs
c
Or...i
cf'l
H
ON
M
O#
X C--
0
N#
t
CV
OI-4
CV
I C'
CV CV
CL ~to C
CM
CVC.
P<'1
CM
**
H
ad m
O O
OQ -ice
01
O r»
r-f
(D r-~
ad
,-{ N r-! N r 4 N r{ N r 4 N r-4 N r! N r^4 N r-4 N r t N
* C- C-- 0 0 r4 r4 C'J C4
r-1
d O
4)
tll
- 27 -
N m
Tab
le
4,2
The
Res
ults
on
Nip
pon
Yea
r 56
56
57
57
58
58
59
59
60
lg
60
JZ
61
62
62
63
63
64
64
65
peri
od 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
cf
.
pl
2.67
3
2.60
2.73
3
2.66
3
2.69
0
2.']2
1
2.68
0
2.97
2.68
6
2.E
68
2.72
8
2.'7
33
2.79
1
2.91
2
2.80
2
2.68
4
2.79
1
2.76
4
2,'7
38
foot
note
of
T
able
4.1.
ao
aa
1
0.57
9
0.58
8
0.69
3
0.77
7
0.58
3
0.61
5
0.62
0
0.04
0,58
6
0.62
9
0.57
4
0.60
4
0.62
1
0.62
7
0.55
8
0.41
2
0.56
7
0.63
9
x.53
1
(3)
ap ant
x la
-6
-0.9
00
-0,8
19
-0.6
95
-0,8
38
-0.9
75
-0.8
70
-0.8
43
-0.7
88
-x.7
51
-0.6
65
-0.6
73
-0.6
26
-0,6
49
-0.5
87
-0.5
95
-0.4
97
-0.5
27
-0.4
97
-0.5
03
(4
d
1 t
0.83
9
0,81
7
0.68
9
0.69
].
0.66
c>.7
37
0.66
6
0.93
9
0.74
3
0.72
6
0.79
8
0.79
2
0.84
9
0.95
2
0.93
6
0.96
8
0.98
9
0.86
5
0.94
9
i
(5)
a2~c
a4~
x
10-
6 -o
.9i4
-0.7
73
-o.a
63
-0.9
14
-o.a
33
-o.e
oi
-0.7
65
-o.e
42
-u.6
77
-0.6
35
-0.6
11
-0.5
91
-0.6
36
-0.6
05
-0.5
65
-0.4
13
-0.5
24
-0,5
15
-0.4
86
(6)
a2~c
an~a
a2
x io
-6
-0.9
67
-0./6
2
-0.1
4
-o.s
o3
-0.8
69
-0.3
5
-0.5
39
-0.4
72
-0.3
55
-0.3
7.0
-o.2
a4
-0.2
60
-0.2
64
-0,2
53
-0.1
92
-0.1
34
-0,1
61
-O.lE
2
-0.1
35
I IV
I
Tab
le
4.2
The
R
esul
tson
N
ippo
nco
ntin
ued
(3')
(4')
(5')
Yea
r pe
riod
as aq2
op2
8D2
d`q2
dq2
2a
rt
aq2
e
x10-
3xl
o-3
x10-
10
1956
1
22.9
42x
•331
-3.7
25-1
.079
1.51
9-1
3.0
94
56
225
.143
27.1
80-3
.573
-1.0
280.
890
-6,8
82
57
126
.003
24.3
'19
-3.1
17-o
.yfo
0.34
6-2
, i)9
0
57
227
.211
19.7
5-2
.457
-o.a
74-0
.086
0,78
1
58
127
.304
26.0
24-2
.475
-0.9
750.
665
-5.5
52
58
226
.93e
19.1
37-2
.236
-0,x
75-0
.070
0,55
4
59
126
.947
17.7
85-1
.7x9
-0.8
35-0
.7.6
21.
. 23
7
59
223
.925
15.2
36-1
.260
-0.8
37-0
.134
1.
13D
60
126
.364
13.
142
-1.1
45-0
.774
-0.2
411.
62)
60
226
.648
12.4
10-1
.121
-o,a
o7-o
.2oz
1.21
9
61
126
.652
11.8
86-1
.064
-0.7
70-0
.237
1.45
61
226
.404
11.1
H6
-x.9
50-0
.613
-0.1
71],
.010
62
122
.580
11.3
01-0
.826
-o.A
52-0
.111
0.70
4
62
226
.520
10.5
43-0
.926
-0.8
42-0
.137
0.82
7
63
126
.412
7.50
7-0
.919
-0.7
9-0
.195
1.10
2
63
220
.914
6.84
1-0
.548
-0.7
97-0
.111
0.60
64
123
.926
6.70
4-0
.615
-0.7
54-0
.156
0.81
6
64
222
.4].
6.53
-0.5
37-0
,780
-0.1
160.
59"/
65
122
.x}0
65.
5?3
-0.5
84-o
.77z
-0.1
350.
655
*)
cf
,fo
otno
te
of
Tab
le
4,2.
Figu
re
4.1
The
Est
imat
ed
Val
ues
of
the
Con
ject
ural
Var
iatio
ns
1
dq J
dqj 1
0
I
W O
1
-1
11
IPPO
Ndq
1
d q~
1
i/
ASA
HI
cl q
1
dq 1
rr7r
Porr
dq 2
aq 2
dq
2
Yea
r
Peri
od
1956
1
iTT
'
57
2
1 2
5e
59
50
1 2
1 2
1 2
2
61 1
Fit 1
263
1
2
64
12
65 1
2
A s seer at Table 4. ; the market share of Nipcon is smaller than that of AsaPi
an_d therefore i ca: be ..hou`'t that _;ippon is more vulnerable to retaliation
by the other fir=-:s than '-sari is. In this respect, the result that the con-
jectural variations of Tippon are arrays larger that those of Asahi in the
whole observation ~e=iods is reasonable. The conjectural variations of both
firms increased a little since 1962 when Central's market share became
appreciable.
The conjectural variations d~2 for the polished plate glass are about dq2
-0 .9 at Asahi and -l.1 - -0.7 at Nippon, as shown at column (4'). From
these figures it can be stated that each of these firms conjectures that
the other firms reach in such a direction as to stop the price fall which
was caused by that firm' s supply increase, even if their market shares
should decrease. Conversely speaking, each firm thinks that the supply
reduction by one u~.~t will cause at once the other firms' supply increase
of almost one unit.
Let us see whether the second order conditions (4,.11) and (4.12), dal
are satisfied at each firm. ':e have to know the value dq? which is the
I first derivative o f the conjectural variation in order to calculate the a2~ a2:~ value aq2 in (4.13). But we cannot estimate a q 2- in our scheme.2 So, we
i d q. a L calculate this on the assumption that_ = 0. The values of 2 2 a
a2 dq q and 2 are shown at (5) and (5') columns respectively. aq2
At (6') column the :clue of
i
a2~ a2T
Sq aq1 aq2
d =
+ a 2 ~ a 2JZ'
a q2 a q1
aq2
- 31 -
Table 4.
Year
1955
56
57
58
59
60
61
62
63
64.
65
3-a
Asahi
(%)
57.1
59.4
60.8
58.2
57.8
58.6
57.6
55.0
53.1
52.5
52 7
(Footnote)
The oroduc ;,ion
Figured Glass
Until 19
polished
r~ippon
(;?)
42.9
40.6
39.2
41.8
41.7
39.0
38.5
38.5
38.1
37.2
58 the plate
Share of
and '/ire
0rdina j Sheet and
Glass
Central
(%)
0.5
2.4
3.9
6.5
8.8
10.2
production includes glass.
the
Plate Glass
Total Production
(1,000 converted cases)
6,650
7,724
9,102 8,509
9,396
10,765
11,493
12,261
13,68
15,186 'Ic -, ,0
raw plate glass for
Year
1955
56
57
58
5a
60
61
62
63
J
64
65
Source
Table
A aahi
(%)
65.6
62.2
57.6
71.5
70.6
67.2
65.9
61.1
63.1
55.0
46,2
"
he
4.3-b The
Preser T Monthly- S Aug. 1967,
Production
Nippon
34.4
37.8
42.4
28.5
29.4
32.8
34.1
38.9
36.9
31.1
30.2
.t State urvet of
pp. 3
and the
66.
Share of Polished Plate Glass
Central Total Production
(o) (1.000 cases) //
- vo
- 91
- 102
- lab
- 193
- 283
- 367
- 377
- 431
13.9 620
23.6 690
Problems in the Plate Glass Industry", Japanese Developsent Bank,
- 32 -
§5.
y L 2 ?.lld is shown It i s seen that :_^_e second order conditions, i.e.,
0, are satisfied in 1?:e all ceriods at ~~sahi, but in the case of _.i peon these
conditions are not satisfied in the four periods 1965.1, 1956.2, 1957.1, and
d 2q 82 ' 1958.1. It can be said that the fact that q2 < 0 when 2J =0 means
d4 aq
0 when d 2q >O, if we look at the equation (4.13). dq,
Concluding Remarks
1. The conjectural variation so far estimated is nothing but the
gradient of the iso-profit curve o== .e to firm a t the market equilibrium
point in (a, oj) plane. For simplicity we will explain it by the single
product modes in section 1. The profit is expressed as
(5.l) -- f(Q+c).Q - C(a)
If we make the total derivative of this ec uation and let i t be zero,
we obtain
Therefore, the relation
dC
dQ da _ _ l (5'3) ddD-
is obtained. This is the same expression that we have taken as the value
of conjectural variation in (1.3).
Let us draw the iso-profit curves fixing the profit 7t at various
levels in (5.1). They are shown in Figure 5.1. If we assume the market
equilibrium point is E in the figure, the gradient of the iso-profit curve
(%r -3) at E is assumed to be ecu • to the conjectural variation . de
This conjectural variation is the differential coefficient of the so-called
"conjectural function" at E . The conjectural function is the relation that
expresses the locus of q which the firm conjectures corresponding to the
various levels of q. This function may be written as
(5.4) q = (q) - 33 -
Fig 5,1 iso-profit
Con jectu.al
Curves and
Function
C
t
,% !
r r `
w
ci
L 1
'p()
I 7r=3
=4
7r=5
0 ~7
- 34
2. O'flode_ is cased on t.:e assumption of a ulomogeneous prod. ct.
_ ~ D_r If ~ ae assume a differentiated oduct, can we estimate =:e conjectural
variation?
Let us first consider the case of duopoly. Let two firms be denoted as
I arid. II, and each supply as each price as pi and let each individual
demand function be
(5.5) Pi = fi(g1'gIT) i = 1,11.
Then the ith firm's profit is expressed as
(5.5) - c(q) i = 1,11,
where c.(ai) is each firmts short-run cost function. At the profit
9-; mi zing point ,
(5•i') d?i _ afi afi dgk - dC i - dQi - n i oqi qi aqk dqi Qi dqi 0
holds. Therefore the conjectural variation is estimated by
dCi of i
(5.8) dqk _ dqi r P _ i aqi qi
i k = I II i k dqi of i q i a
qi Therefore the :measurement of the individual demand function
is necessary in this case.
Next, we are going to consider the case of triopoly.
Let the individual demand function be
(5.9) = i = 1,1?,111
Then at the profit meg dmizing point
d?t'i = afi i ~~, afi dqk i - dCi = 0 (x.10 dqi Pi aqi q k=i aqk dqi q dqi
holds in each firm. Therefore the conjectural v.ation of the firm I, say,
about the other fi:s II and III are given as
(5•.11) of I dgIl of I dgIII - (do1 _ of I agII dqI T agIlI dqI `dqI PI dqI q I q I .
of I So in this case we have only the weighted sum, the weights of which are ag
11
-35-
a fI ar d a q ~
In the latter case, if a homogeneous product is assumed, the relation
pI pII = p1T1 p holds, and the indi-ridual demand function (5.?) i s
changed as
(5.12) p = f(a-T+qII+QIII).
As a f - a f = a f = d f (where D = qI±o +qIIS) , (5.11) will be changed as aq I agII aq~ dD -II
dot
(5.13) dgli dg1II dqT p da I + da 1 - ~~ - 1.
In this case we can estimate the sum of the conjectural variations about
the firm II and iii, dgIT da-IlI dgI where q-= + a - ) do dc_ dc L..
model the separation of d g II and d q is im cssibz e . dqI dqI -
3. Let as look into the possibility of building an econometric model of
an oligopoly market, in which the price determination mechanism should be
explained. We assume a duopoly market of a homogeneous product. A triopoly
or general oligopoly case can be explained in the same way. If we express
to i+h firm+t s conjectural function as
(5.14) _ c i(qi) i, k = I, II; i k,
then the following relation must hold
dTCI _ f (q I+gII)+f '(q I+gII) C1±9' (q1)) q r0 I (q1) = 0 dqI
(5.15) d?zII _ f (q I+q~)-E-f''(q I+qII) C 1-~-yII (gII)1 qII-CII C qII) = 0
We can determine the supply q and a, and the price p = f(g1+q==) by ls.
~ the above egT,zatiors. Therefore errefore i~ is necessary to measure the form of
the derivative function of the conjectural function, i.e., co
-36-
4. t the present stage, we have no means to test directly whether the he
estimated conjectural variations are true values or not. The result obtained
is completely dependent on the degree of precision in the estimation of the
market demand functions and. the firm cost functions. Of coerce more fun-
damental questions are left. Are the firms actually determining their supply
quantities so as to maximize their profits? Do the conjectural functions
have a differentiable form at the market equilibrium point? If they have
kinks at the equilibrium points kicked individual demand curves such as
P, H. Sweezy pointed in (4 will be derived.
One way to answer these questions would be the extension of the above
measurements to other industries which have homogeneous products and
oligopolistic markets.
-3~_
RE E CES
(1J
12)
(4
~5.
(6J
C7J
CsJ
Cournot, A., Hecherches sur les principes mathematiaues de la theorie
des richesses, Paris, 1838.
Frisch, H., "Monopole--Polypole--La notion de force Bans
l'economie", Festschrift til 3arald Westergaard.", sup-
plement to National konomisk Tidsokrift, April, 1933
Hicks, J. H., "Annual Survey of Economic Theory : The Theory of Monopoly,"
Econometrica, Vol.3, Jan. 1936, pp.1--20.
Sweezy, P. M., "Demand under Conditions of Oligopoly,"
The Journal of Political Econc~y, Vol.CVII, 1939, pp.36^--73.
Staehle, H., "The Measurement of Statistical Cost Functions.
An Appraisal of Hecent Contributions," American Economic Review,
Vol.32,1942, pp.321--333.
Johnston, J., Statistical Cost Funnctions, McGraw-Hill, 1960.
Walters, A. A., "Production and Cost Functions: An Econometric
Survey," Ecanometrica, 1101.31. No.l-2, Jan.-Apr., 1963,
pp.1--66.
Leontief, W., "Elasticity of Demand Computed from Cost Data,"
The American Economic Review, Vol. 30, Dec., 1940, pp.814--817.
-38-
