LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
OPTIMAL COMPETITIVE fJARKETING BEHAVIOR
IN OLIGOPOLY
6A3-73
Jean-Jacques Lambin*, Philippe A. Naert**
and Alain V. Bultez***
January, 1973
MASSACHUSETECHNOLOG
50 MEMORIAL DRIVE
INSTITUTE 01
f..1ASS.
.
Z_..
RECEIVED
APR 11 1973
M.
I.
T.
LIBRARIES
INTRODUCTION
The problem of marketing mix optimization has received
considerable attention in the literature. For the profit maximizing
firm such an optimization is the extension of the "marginal revenue
equals marginal cost" rule to determine optimal levels for all marketing instruments rather than just price. Dorfman and Steiner derived an optimization rule for
a
sequently applied by Palda [27l
A
monopolistic firm [9], which was suband Lambin
[20l
in empirical
studies.
large number of normative models of competitive mar-
keting behavior in oligopolistic markets have also been formulated.
Various assumptions related to industry demand (stable versus expansible) and to the type of competitive reaction
der)
have lead to
a
large variety of models.
(follower versus lea-
Many of these are theo-
retical in nature and have not been directly applied, such as the
competitive models, by Mills [231, Gupta and Krishnan [14.
15,
19],
Shakun [29l. Baligh and Richartz [2], Kotler [161. Naert [241. and
others. Examples of empirical studies dealing with stable industry
sales are Lambin
typp reaction)
(follower)
[231
(
fo
1
[sol,
lower- type reaction)
taken into account
and Telser
and with expansible industry sales
and Bass
(follower)
In those studies where
ly
[221
(leader),
it
(leader-
are Schultz
[3].
competitive reaction is explicitis
implicitly assumed that compe-
titors react with the same marketing instrument as the one which
C37234
causes their reactions, that is, they react to
by
change in price, to
a
a
change in prices
change in advertising by
a
a
change in ad-
vertising. We will identify this kind of reaction with the simple
competitive reaction case. It is more realistic, however,
in the
spirit of the very concept of marketing mix
we will call multiple competitive reaction
competitor may react to
price,
a
,
1
,
to
and more
consider what
that is for example,
a
change in price not just by changing his
but also by changing his advertising and,
keting instruments as well.
To our knowledge
no
possibly other mar-
optimality condi-
tions have been derived yet for the multiple competitive reaction
case
.
In this
analysis.
the multiple
through
a
developed,
paper we will remain within the realm of static
First, we will derive profit maximization conditions for
competitive reaction case. We will then demonstrate
series of corollaries how optimization rules, previously
are special cases of our more general rule.
The second
part of this article will be devoted to the estimation problems arising in
a
multiple competitive reaction framework. Through the ana-
lysis of the data collected on
a
stable industry demand market, evi-
dence of the importance of considering multiple competitive reactions
will be presented.
.
I.
PROFIT rAXiniZATION
A
:
GENERALIZATION
First, we will derive optimality conditions in terms of
the vector of total sales elasticities
to decompose
les
E
its various
in
q.u
E
We will then show how
.
q .u
components related to industry sa-
effects and marKet share effects. The notation adopted is defi-
ned in Tab le
I
INSfRT TABLE
430UT HERE
I
3
Consider the following company profit function
(1)
Tr
q.[p-c.(q,x}]-s
=
For the profit maximizing producer,
ned by setting the derivative of
sion variables equal to zero
u
be
of vector
a
u.
-
(p
3u
X
-
n)
1 1
here
ip ly
i
(p,s,x].
-q
.-1]
9q
With marginal cost MC
=
.
9u
q
9u
•
9q
=
)
9u
9u
matrix of the elements
diagonal
^
-
by D /q,
9q
.—
3s
gq
^
-
9tt/9u
D
-i^
9c
9C
(3x3]
ng
3c^
c
u'
9u
Premu
(p.
respect to each of the deci-
q-(
c)
9u
(n
v/ith
9P
=
Let D
Let
.
8q
Sir
it
optimality conditions are obtai-
9c.
9p
.
"
9u
9s
i
D
D
"^
9u
-ii
q
._
9u
=
-—
3q
•
-r—
q. u
12)
where
condition obtc. ins
the following& optimality
j
^
E
•=
3u
E
-
9c.
no
-s/q.
H
I* ^H
Ip.
+
-X
g^
identity matrix. From which
the
is
I
(P
•
q.U
can deduct
(3)
Equation
rule].
generally known as the Dorf man -S t e i ner theorem (or
is
(3)
In fact,
merely states that at optimality. marginal reve-
it
5
cost for each marketing instrument
nue must equal marginal
.
The
Dorf man-St ei ner rule is generally written in the following form
(4)
n
°ij
q.p
8q
—
ds
.
ihere
p =
p
•
•
Equation (4) holds,
I
BR
.
s
£
X
Equations
.
*
t3)
n
q.s
q> X
9c/3x
and
=
ri
q.x
._PX
1
,.
9q/3x
•
_,
and
,
—
=
n
q,x
=
v
'
3c/ ^9x
,
•
-c
q
sine
=
pq
^^-^
•
_
"
(4)
p
X
—s
•
q
s
^
*
3q/ 9x
3c/3x
—
9q
3s
.
*
9q
~
=
p
'^9s
x^
_
q
•
'
£
x
.
*
are thus equivalent.
=
V
•
"^q.x
,
and
:
l-rcm
equation (2) one can easily dei'ivethe optimal values for the
decision variables.
(5J
u
In
\t-^
=
vector notation, we obtain
•
1
D^
h.
.
-q
:
.
q.u
where D^
E
is
a
diagonal matrix of the elements of vector
q.u
We now want to show how the vector
E
E
q.u
^
of total sales
q. u
elasticities can be decomposed into various elements in the case of
an
oligopoly. First,
\-je
can distinguish between
and an industry sales effect.
effects
a
market share effect
And secondly, we can separate direct
(assuming no competitive reaction) from indirect effect
(com-
petitive reaction effects). The theorem proposed below traces the
linKs between tctal sales elasticities,
industry demand, market share,
and competitive reaction elasticities.
THEOREM
The vector of elasticities of company sales with respect
to its decision variables is equal to a matrix partitioned into an
identity matrix and the matrix of competitive reaction elasticities
postmultiplied by the sum of the vectors of industry sales and market share elasticities,
Proof
:
i.e.
E
=
[lyR]
•
[F„
+
E
]
Company sales can be written as industry sales times market
share
:
—
q
=
q
=
m
•
(u,
Q^
The derivative of
U.
2)
'
(u,U)
m^
with respect to the decision vector
q
3Q,
9U
8(
3u
9U
•
9u
9U,
9U.
9U
9u
9u
du
r
9u
Premu
1
1
is
u
3U
—
3m
9u
9U
ip
1
-
y
i
by
ng
D
/q.
we obtain
D
9q
"q"
Tu
D^
~
*
Q
9Q^
D^
9u
"q"
3U
*
Tu
30^
*
'9
U
"'
9m^
^
m
*
u
'
9m^
9U
^
m
3u
*
u
*
"Fu
'
9lj
whi ch reduces to
(6)
E
=
q
(7)
E
.1
E_
0-
u
=
q.u
R
.
•
u
[I.R]
E„
0-p
•
[E^
Ot
*
E
,
U
,
.
m
E
m.,u
+
R
•
E
ni.,U
]
'
The optimality conditions are then obtained by substituting the ele-
ments on the right hand side of equation (7) for the elempnts of
in equation
(2
)
.
E
q.u
COROLLARIES
The optimization rules used in the empirical studies re-
ferred to in the introduction can be derived very easily from corollaries to the theorem.
In deriving
these corollaries, we will use
both forms of the theorem as obtained in equations
(6)
and
(7).
When
we use the phrase "no competitive reaction" we mean that the company
in deciding on values
for its decision variables
follower or
a
assumes no change in
decision variables. That is the case of
the values of competitors'
Cournot-type oligopolist.
a
"Competitive reaction" will
refer to the case where competitive reactions are explicitly taken
into account in determining values for the decision variables. This
7
is
the
case of
a
leader,
as
defined by Stackelberg
.
MON_OPOLY_AN_D_MONOPOLlST_IC_ COMPETITION^
Coro 1 lary
that
E
a
:
In monopoly
q
=
0,
and
U
and
R
do not
exist.
It
follows
:
(6)
In
1
=
q, u
E„
0^, u
monopolistic competition situation, each firm faces its own de-
mand curve and no industry demand is defined. This is the case derived by Dorfman and Steiner [9],
stein [13]
industry,
[20].
to
and applied by Palda \27],
and Feld"
study cases of monopolistic competition in the drug
and by Lambin for
a
frequently purchased consumer product
OLiGOPOLi
a;:p S_TAB_LE
INDUSTR1'_D,MAND (E
= 0)
Qrp
elasticities
(9)
Wit.
R
E
a
=
=
In which
0.
case
:
F
.ultlpllctlvs „..Ket sH.re function
of t.o -roUowln,
n
*
s
i
.
It
is
(ID)
t
ype
PI
i
.1211
easil y seen that
E
=
-
-
E
F
Equation (7) then reduces
to
Mn
E
=
q. u
E
m
.
u
This has been used by
Lambin in
a
consumer durable good [22].
study of
a
low price
Sometimes one use'JSB. market
r^^r<-c=^ share
~k
functions with the
cision variables in share
form.
'm^.pO
P
P
e...
m.
,
s'
de-
9.
For this type of marKet share function it is ea-ily
shown that
(12)
"
^m^.U
where
D
q
is
''^'
^mj.u°
*
diagonal matrix of the elements of vector
a
-^^
Equation
°U°
^
^
^2
'
^
^2
^n
^
^
can then be written as
(9)
%o
=
^.u
'
E^^^^.
Equation (13) is appropriate for Cowling and Cubbin's study of the
car market in the United Kingdom
med
E
q
.
m
u
Corollary
3
:
,
[8],
u'
Simple Competitive Reaction
R
*
R^, where R. has the same diagonal elements as R
d
d
but has zero off-diagonal entries.
(14)
E
q, u
altough they implicitly assu-
=
[I'R^l
•
d
and in the special case of
Equation
(7)
becomes
:
E
I
a
market share function in relative form,
u
10,
(15)
E
=
[I
R^]
-
•
since according to equation
E
=
E
(10'
[I.R,]
[I.R^l
d
q, u
an expression equivalent
to equation
.
Telser,
(15).
advertising in the cigarette industry [30], derived
ween absolute advertising elasticity
n
*
,
r\
[221,
and Telser's findings,
We may also observe that for
dustry demand
all
:
E
=
q, u
E
m
,
u
Using this in equation (15) gives
E^
=
m,
,
[I
-
R^l
d
•
E^
m
,
.
,
u
which is the form actually used by Telser.
Coro 1 lary
a
4
:
Multiple Competitive Reaction
Only En
=
0,
end hence
study of
relation bet-
and relative elasticity
which is the advertising equation in (15). For
of Lambin's results
flS^
in his
a
see Naert
comparison
[251.
cases with stable in-
u
11.
(17)
E
q.u
=
[I.R]
•
E
m^
or with the market share function in relative form
(IB)
E
q.u
=
[I
R]
-
•
E
I
The special case of equation
(IB)
has been derived by Bultez
[6]
We refer to section II for an empirical example.
OLI_GOPqLY_ AND EXPAllSI_BLE_INDUST_RY_ DEMAND
Corollary
mes
5
No Competitive Reaction. R
:
=
0,
(E
^ 0)
and equation
(6)
beco-
:
(19)
E
q.u
=
E„
0-
+
.
u
E
m
.
,
With the market share function in relative form this becomes
(20)
E
q.u
=
E„
*
0-p
u
.
E
m
,
u
The study by Bass on advertising and
Given
In share
(21)
f
E
q.u
=
market share function with the decision variables
a
orm, eoua t
i
on
E^
0^
cigarettes is of this kind [3].
.
u
(19)
D,,o
U
can be written as
•
E^
m
_n
,
u
:
12
Equation (21) is thp relevant form of equation
dy of competition between airlines in
Corollary
6
(22)
E^
Simple Competitive Reaction.
:
q
=
,,
»
u
n.R.l
d
•
[E
E^
*
(6)
for Schultz'
stu-
two-city market [28].
a
R
R
=
,
and equation
(5)
]
m
Telser's study of advertising and cigarettes contains
a
relation equi-
valent to (22) for advertising [3D]. This equivalence is demonstrated
intheappendix.
With the market share function in relative form one gets
(23)
Core 1 lary
E_
q
7
=
,
[I.R^]
•
d
u
petitive Reaction
-
*
[I
-
R^]
E^
•
.*
'i'
Oligopoly
:
E„
-
Expansible Industry Demand
-
Multiple Com-
Specific Functional Form for Market Share.
The theorem holds for any kind of market share functionIn the special
case of
a
market share function with the decision va-
riables in relative form, we find
(24)
E
=
q
a
I
u
[I.R]
•
E„
Q_
+
[I
:
-
result also derived by Bultez [6].
R]
•
E
m
i
.
,
u
13
The theorem and corollaries
rized in Tab
le
II
2
to
6
are succintly summa-
.
INSERT TABLE
Corollaries
2
to
7
I
I
ABOUT HERE
demonstrate how total sales elasti-
cities relate to industry sales and market share elasticities for
various kinds of competitive reaction in different types of oligopolies.
In each
case, we have presented this relationship in general,
and also for specific forms of the market share function.
For each
corollary wo have referred to at least one application, except for
the multiple competitive reaction cases.
Empirical evidence demons-
trating the importance of considering multiple competitive reaction
will be presented in the following section.
Figure
1
illustrates the explicit decomposition of
n
q
for the cases considered in this section.
INSERT FIGURE
1
ABOUT HERE
,
s
14.
II.
THE EMPIRICAL STUDY
The objective of this section is to present an empiri-
cal application of the multiple competitive reaction concept intro-
duced in the first part of this paper. The product studied is
low
a
price consumer durable good having reached the saturation phasis on
its
life cycle curve.
The market is dominated by three brands which
represent altogether approximately 90
Firms
A,
B
and
C
%
of total
industry sales.
manufacture different brands of the same general
product category. They do not offer,
however,
identical quality nor
do they
charge identical prices. Advertising expenditures represent
in each
case
a
significant but variable proportion of total sales
revenue. Since primary demand is stable,
a
all the increased sales of
given firm come necessarily at the expenses of its rivals and
con-
sequently the degree of recognition of mutual interdependence is very
high.
Thus,
the market
has
the structural
characteristics of
diffe-
a
rentiated oligopoly.
The data available for this study are fairly reliable.
Market shares,
retail prices and distribution rates come from
a
dealer panel. These panel data, which are purchased by the three
firms are available on
a
quarterly basis from 1960 to 1966 and over
four geographic regions. Company sales data come from company sources
and total
industry sales can also be estimated through statistics
published by governmental agencies. Media advertising expenditures
radio and television) per brand, month and region are provi-
(press,
ded by
a
commercial service from which monitors the media and values
15
list price.
time and space at the rate card
Point of sales promo-
tion expenditures arc not included in the sample. Nedia expenditures
advertising outlays. Each brand quali-
represent about 85
\
estimated by
a
synthetic index, derived from
;
empirically selected product attributes where
ty is
rential scale [26]
of total
a
semantic diffe-
period by period, by the marKeting staff
then subjectively rated,
of one of the firms.
Bain [1] suggests that product differentiation inhibits
changes in the percent of the market held by the oligopolistic firms
In contradiction with this view,
a
high degree of market share ins-
tability is observed in this market during the period under study.
10
The estimated indices of market share instability
for each region in Table III.
among the three firms
30.2
%,
37.9
%
and
In
1960,
the partition of the market
the others not included
-
31.9
for firms A,
\
are presented
B
and
C
-
was as follows
:
respectively. The
dominant fact of the 1962-66 period is the aggressive and successful
marketing strategy of firm
32.8
in
\
policy,
1962 to 66.1
in
whose market share increased from
1966.
As
a
result of firm
B
aggressive
drastic changes occurred in the degree of market concentra1
tion.
%
B,
The Herfindahl
on figure 2,
1
index of Table III,
graphically represented
illustrates this fact
INSERT TABLE
III
AND FIGURE 2 ABOUT HERE
16,
In acsuming the
leadership of the market, firm
3
pla-
ced heavy emphasis on advertising a"d its share of total industry
advertising expenditures increased from 18.9
in
1965.
%
1962 to 61.7
in
%
Because of the ratchet nature of advertising and of reac-
tions from the competing firms,
(deflated)
total
industry adverti-
sing expenditures have dramatically escalated and were four times
higher by the end of the period studied. The advert i si ng- sa Ics ratios of the firms are presented in Table IV.
They show that adver-
tising has been particularly instrumental in the growth of firm's
B
marKet position and present evidence of strong competition by
means of advertising.
INSERT TABLE
Selected ordinary
IV ABOUT HERE
1
eas
t -
square s estimates of market
share elasticities with respect to price,
quality and advertising
are also presented in Table V.
These results suggest that price
and quality have also played
significant role since
a
a
significant
part of the firms market share variance can be associated,
not only
with changes in advertising intensities as expected in oligopoly,
but also with price and quality adjustments.
tegy has been that of tne marketing mix.
ve situation it
is
of
In
Thus,
the firms stra-
view of this comptiliti-
interest to see how the competing firms have
reacted to the aggressive policy of firm
B
and,
successfully or not,
adjusted their own marketing variables to the new competitive situation created by the leader. The multiple reaction functions concept
17
should be
a„
for that purpose,
instructive analytical tool
INSERT TABLE V ABOUT HERE
RJAJTJOJ JLASTICIJ IJS,
ESTIMAlIOJ^_Ol THE C_OrFETlTlVl
e.poctod rival re^pons. Is
pointed out Oy Bre.s [Si.
dimension.
dl.snslon as .ell as an e.t.ot
„„oertaln ..d Has o tl.o
soon will
rival --espood and How
o.estlons, .ow will t.o
AS
0. th. two
.e respond,
for.er.
less relevant tHan the
the latter Is not
T.e
cnan.es Is .ad>caoo.petitlve reaction to price
tl„ln, o. expected
rival reactions to cnan.es
the tl.ln. of expected
,, different fro™
and puaUtv Price
variables sucH as advertising
in tne non-price
variations in
action immediately, wt,hile
variations can be put into
technological research. Similarly,
attributes often require
q .ality
required for the
limitations and of the time
because of media spac
reaction Pv
ca»pal.n, competitive
preparation of an advertising
reactions.
generally slower than price
„eans of advertising are
Thus,
follows
the functions are as
Pt
V
"P.P
Pt
'P.=
=t
.
:
x"'"-"
"t
The douPle
for advertising and duality.
with the inter
here to cope explicitly
.o.arlth.lc for™ is adopted
varlaOles. Thus, the
existing amon, the .arKetin,
and sl^llarlv.
Put
la..ed.
,,,,„, pattern
elasticities [R].
matr IX of reaction
is
constant
The periodicity of the data used to estjma'.^
t h
r-
Key-
porameters is annual because sales being very concentrated in one period of the year,
act within
a
it
is
practically impossible for competition to re-
shorter period of time. By pooling the data over time
and regions we obtain
a
total of 28 yearly observations. However,
ce the quality of products
sin-
charged to the consumers
and the prices
are identical across regions, we only have
7
degrees of freedom left
1 1
ower reaction theory we
for these variables.
On the basis
of
the
1
eade r-
fo
designed various simultaneous equation models and estimated them usinj
12
the well-known two-stage
le as t
major assumption,
:
namely
-
squares method
in
interdependence between
order to test one
P,
S
and
X,
which
would result from the followers full marketing mix orientation. Because of the postulated fundamental
three leader's decision variables
detected dependence between
P.
S
dependence of P,
S
and
X
on the same
p,
s
and x, we wondered whether the
and
X
might not merely be due to
a
1
3
spurious correlation effect. The test we devised was unconclusive
Furthermore, the contemporaneous covariances of the residuals across
equations were not sufficiently high to justify the application of
the joint generalized
As
a
least- squares technique [32,
pp.
295-311].
result of this preliminary and rather unsuccessful
econometric investigation, we turned back
to
a
much simpler set of
independent equations to estimate [R]. We used then the CochraneOrcutt [CDRO
iterative procedure [32, pp. 253-256]. The results are
presented in Table VI.
f
19.
INSERT TABLE VI A30UT HERE
The direct reaction elanticities cre all three positive
and statistically significant at the 97.5
per cent
level or higher. Their
order of magnitude imply fairly strong response from rivals to the leader's moves in price and quality. As expected in the phase of innovative maturity the high quality reaction elasticity suggests an almost
explosive reaction pattern in terms of product quality adjustments.
Significant support is also given to the multiple reaction concept,
(
of -diagona
1
since three out of six indirect reaction elasticities
cellsJ
are significant.
significant indirect
The most
reaction elasticities are observed in the quality reaction function
(despite the number of degrees of freedom left}. The signs of the price
and advertising variables are as
B's competitors
expected. They indicate that brand
not only have reacted in terms of product quality ad-
justments to the leader's move In quality, but also to the leader's
move in price and advertising.
In the
advertising reaction function,
the positive sign of the price variable is surprizing at first sight.
We may have expected,
indeed,
a
reverse counteraction,
i.e.
an
In-
crease in competitor's advertising to annihilate the leader's aggres-
siveness in price.
In
fact,
this non -
i n t
ui
t i
ve result
is
simply the
manifestation of brand B's competitors failure to counteract
ciently. During the period under stuay, brand
B
effi-
has consistently do-
minated his rivals not cnly in terms of his pr oduct - qua 1 1 ty ratio,
but also in terms of its advertising intensity.
Thus,
this positive
20,
sign may
sirr.ply
reflect the passivity of brand B's competitors who
succeeded to fill up their gap vis-^-vis the leader. By drama-
r.over
tically Increasing itz share of total Industry advertising expenditure
(up
to
65
%)
brand
.
B
has prevented its rivals from effective
conteraction by means of advertising.
In
other words,
competition
was unable to follow the advertising escalation in order to compen-
sate for this price-quality gap.
pretation
is
A
further confirmation of the inter-
given by the modest order of magnitude of the direct ad-
vertising reaction elasticity.
Thus,
those results seem to be consistent with the com-
peting firms actual marketing behavior. They also support the proposition that the firm competitive behavior must be analyzed within
the
context of the whole marketing mix.
EST I HAT I ON _0F
T^HE
LE ADER^S _S ALE_S J^ LA ST I_C IT I_E S_
We now report results obtained when we assume
a
simul-
taneous dependence between the level of advertising expenditures and
the
ble.
level of sales or market share since the industry demand is staA
static and
a
dynamic version of the model were estimated using
the instrumental variables method
(equivalent to the two-stage least-
squares in this case). Brand B's market share elasticities are reported in Table VII.
INSERT TABLE VII ABOUT HERE
21.
static and dynamic versions
In both
solute values of
fr.s
tlie
signs and ab-
oaramcters are as expected and are comparable
with the ordinary leas t - squares estimates of Table V. They are all
significant at the 95 per cent level or higher
del,
m
the static mo-
In
.
the test for serial correlation is performed by calculating the
Durbin statistic
sion of
[
m
T)
m
based on the residuals from the multiple regres-
d
.•
,
s
«
s
]
on all
t
the predetermined variables
of
the
system, whether or not they occur in the fitted equation with non-
zero coefficients [10, p.
ble [32,
pp.
correlation.
377].
Referring to Durbin and Watson ta-
724-725], we realize that
In
the dynamic version,
indicates
d
a
positive auto-
correlation
the test for serial
is
carried out by testing the significance of the lagged residuals
in
the regression of the residuals on the lagged residuals and the
explanatory variables [33, pp. 420-421]. The
namic market share model is in fact the
the lagged residuals coefficient.
We
can thus
t
d
statistic of the dy-
statistic corresponding
1
the dynamic market share model are uncorrelated
cal specification point of view,
to
conclude that the residuals
•
From
the dynamic version
seems preferable to the static version. However,
of
5
of
a
statisti-
the model
given the charac-
teristics of the market and the periodicity of the data used, the
static version is subjectively perfectly acceptable. As both versions
have their merits we will continue to take both of them into account
for the rest of the analysis.
Having ascertained the validity of the empirical results we may now return to the theorem we proposed in the first sec-
22
tlon of this paper. The leader's sales elasticities are to
ved.
In
E
=
from the application of corollary 4,
this case,
[I
-
R]
of table VI
suiting
E
•
^.
and the
E
m
deri-
i.e.
Table VIII shows how from the [R] matrix
^,
.
h"^
u*
vectors extracted from table VII, the re-
vectors arc obtained. Equilibrium long-run market share
E
elasticities, defining the full impact over time of changes in price
16
advertising and quality, wers also computed
quality,
.
Except
for product-
important disparity appears between these long-run elas-
no
ticities derived from the dynamic model and the elasticities deriVRd from the static model.
INSERT TABLE VIII ABOUT HERE
ECONOMIC INTERPRETATION OF THE RESULTS
The
vector contains the total sales elasticities
E
taking into account the reactions of competitors while the elements
of vector E^
^*
do not reoct
not react
.
[R]
are the total
This can easily be proven since if competitors do
[0]
=
sales elasticities when competitors
and ^n
^
"^
*
^m
*
^^^^ important feature ena-
bles us to visualize the demand for the leader's products as
ked resrcnsc curve,
ticities
(E
elasticities
»
5
ABOUT HERE
worth noting that these estimates of sales elas-
are significantly' different from the market share
^
1
q
is
kin-
drawn on figure 3.
INSERT FIGURE
It
a
u
(E
rn
,, u
*).
Thus,
anyJ judgment on the firms' behavior ba~i
t^
23.
sed on the
pie,
the
If we
E
m
vector would have been roriously' bifjsed. Fcr exam-
*
,
u
leader's monopoly power would have been underestimaL--^:
asc'jme
prof i t -maximi
z i
potential degree of monopoly power, 100
marK-up on marginal cost,
then our estimate of the
ng behavior,
is
—
•
equal to [-100/(1
+
the percentage
Our estima-
)].
n
tes of price elasticities are -1.673 in the case of the static
version and -1.734 in the case of the dynamic version. This gives
a
percentage mark-up of price above marginal cost of 114.54
136.29
%
or
respectively and the corresponding percentages of gross
%,
margin are
5
3.39
%
and 57.68
%
.
To
appreciate the welfare loss
attribuable to this monopoly power the ratio of competitive price
(nC]
to moriopoly
price (p)
is
also very instructive
creasing function of the price elasticity
:
(1
:
1/n
it
).
is
a
de-
The ratio
estimates are 0.46G1 and 0.4232 in this case. All these results conhigh degree of market concentration observed by the end
firm the
of 1966.
As far as
the advertising policy of the
leader's is
concerned, we can compute the value of the adve r t i s i ng- sa 1 es ratio
at
optimum derived from the Dorf man- S t ei ner rule, according to equa-
tion
[3]
:
thus at optimum
24.
The static model give^
estimate of
in
equal to
a
0.063 and the dynamic form an estimate equal to 0.002.
of
r>
cc^n,-? rl
these values v/ith the observed advertising-sales ratios,
mean of which is 0.0B8,
shows that although firm
B
son
the
had been over-
spending from 1363 to 1965, on the average, its advertising behavior had been rational. An adriitionnal confirmation of this fact
is
obtained when the firm's marginal return on advertising is com-
puted:
MRA
(p
-
no
(p
-
no
.
-^
on the ave raj
MRA
Assuming
we get
a
53
^
=
(q/s)
q.s
gross margin and using the static version estimatrs,
_.
:
MRA =(10.71)
•
(0.118)
(983,000/1,407.257)
•
_=
$
0.6B3
Thus the last dollar spent in advertising produces approximately
1
a
8
net return of 88 cents
If we rely on the dynamic version estimates we should
19
take the long-term impact of advertising into account
.
The
long-
term marginal cost marginal revenue equality rule in actualized
formthen becomes:
3q
(25)
z
1
=
(p
-
nc
Z
1
=
(1
+
r)
8
25
20
where
the cut-off rate.
is
r
price and quality fixea
Assumiriji
q
=
^t*B
Y
=
"^
•
1/(1
n
^*,
-
t +
.
,
m
s
,
s
^
s
T=0
t+B-T-1
A)
Thus,
for
(26)
^
B
=
'i'
^^t^B
[n
Substituting the value of
—
-
q
,
(1
-X)
s
n
m
.
.
s
*]
.when
/
B
extracted from (26) into equation (25)
z
(1-A) -n,
'^m^.s*'^t
1
At equilibrium.
(28)
li-n
t-coo
ts
]
=
(p
-
MC:
(1
*
r
-
A)
r
"t*6
26
and as
a
consequenci
n
(29)
(p
-
MO
•
Q
q.s
1
where
r
•
n m^
r
-
X
,
s
k
•
6
1
-
Y
.
n,
The long-run optimal equilibrium value of the
advertising expenditures
s
=
I
(11 .57)
•
is
thus equal to
(2.100,0003
•
(0. 149
1
:
.524
,
0.093
1
(1.524)
•
+
+
r
r
0. 147
X
-
0.344
(0.093)
To discount the flov; of returns produced by the advertising invest-
ment various interest rates may be considered. Yield on shares in
the same industry would give an idea of profit opportunities offernd
to the firm by alternative investment decisions.
However over the
period studied, dividends were not high enough to compensate for
the going down stocks so that yield on shares is not indicative cf
the external investment opportunities. As
a
result we turn to tha
interest borne by commercial bills over the same period.
A
6.5
'<.
average rate was observed and the corresponding long-run optimal
27,
levGl of
should be
tht;
advertising expenditures amounts to
cor,.p::rBd
with firm B's real expenditures
though the average amount spent is ^clov;
ced to conclude that firm B has
1966 period.
s
e
A
s
s
we are,
=$1,424,079
"s
= $ 1
,
407 257
,
wM.h
.
once again,
Alindu-
b een overspend i ng over the 1964-
sensivity analysis of the long-r'jn equilibrium level
to the discount
rate
rather insensitive to
r,
INSERT TABLF
IX
ABOUT HERE
28
CONCLUSIOi\.
In this
paper we first generalized the well known
Dorf man-St ei ner theorem in considering explicitly the multiple compe-
titive reaction case. We then reviewed the literature through
ries of corollaries.
On the basis
meters we turned to
a
a
se-
the theorem we developed, we
of
attempted to shed light on an oligopolistic market over
racterized by the rapid raise of
a
a
period cha-
leader. To estimate the key-para-
very simple model. The shortage of degrees of
freedom prevented us from trying any more complex form. Hov/ever weak
the statistical properties of some of our estimates might be we are
nevertheless confident in their descriptive as well as in their normative value. Indead they seem to perfectly describe the firms' com-
petitive behavior and reflect our subjective
the market.
From
a
a
priori feelings about
predictive point of view, the competitive reac-
tion elasticities should probably be Judgmental ly guessed and
2
rathEir
than econometrically aerived
a Justed
1
.
econometric estimates were converging.
In
our case.
Judgmental and
29,
.S
^"-..S
30.
EVOLUTION OF MARKET CONCENTRATION
0, 33
)
FIGIJRF
5 X
-
4
ESTITATEn KINrFO RESPONSE CURVEJ
3
X -
3
«
- ?
q
NO COWETITIVE RtAC
=
f
(s/p,
32
c
*
3
3--
3-
33,
TABLE
II
DECnnPOSITION OF TOTAL SALES ELASTICITIES
IN OLIGOPOLY
No
competitive
reaction
E
SfLECTEO INDICATOR'S OF MARKET STR'JCTURE
IN'OICATGRS
TABLE
I
V
ADVERTISING SALES RATIOS'
PERIODS
36
TABLE
SELECTED MARKET SHAP^
V
ELASTICITIES
O
O
CvJ
lo
39
TABLE
VIII
TOTAL SALES ELASTICITIES
0.023'
0.273
-0.774
0.901
Dynamic riodel
Static nodcl
Long-run
Short -run
m
0. 147'
.
u
-1
.673
116
0. 188
i
,
0.223
u
0.667
0.583*
0.503
0.
rL-R
1
.
1
1
5.676
-3.726'
5.543
,L-R
q.u
.
138
0.093
0. 171
1
L-R
.734
0. 142
0.261
TABLE
I
X
SENSITIVITY ANALYSIS OF THE LONG
-
RUN OPTIMAL
EQUILIBRIUn LEVEL OF ADVERTISING EXPENDITURES
r
41
APPENDIX
Tslser derived the following relation (translated in
•
between the various elasticities with respect to
our notation)
ad-
I
vertising
(A.I]
n
m,
=
m)
(1
n
,
s
Hn
5
,
s
=
n
m
,
,
s
"
-
»S.s'
this should have been written as two separa-
fact,
In
(A. 2)
{
q
they result from different assumptions.
te equalities because
let us
•
s
First,
take
n„
m,
=
s
n
m
(1
.
,
'S.s
s
This holds in the case of an oligopoly with stable in-
dustry sales,
simple competitive reaction and the marKet share func-
tion in relative form (see equation
On the
(A. 3)
^^
other hand
-
"^^
•
(15]).
,
^\.s
-
Is
the advertising equivalent
ly
with expansible industry demand,
-^q
,s
of equation
that is an oligopo-
simple competitive reaction and
the marKet share function in general form.
now be demonstrated.
(22),
This equivalence will
12.
Equation
1-m
(A. 4}
q
with
=
q
m,s
=
n
+
.
(A. 3}
+77
^
^r'
Q
30
(A. 5]
n
O.s
q,s
Ti„
s
can be written as
q,s
CO
3s
Q
3rq/ (q+0
n
s
3rn
s
3m
m
9s
m
33
3q
30
3q
q
c
3q
q
30
I
(A. 6]
3s
3s
3s
Substituting (A.B) in the second term of
s
30
_
q.s
S
[1
-
Hi
=
_s
'q.s
q.s
^
_^
B^
3^
_
'
_
q
»
3s
C
^
,
r,,
£-1
_
30
s
^
_s
m
c
y
1
.
Is.
*
3s
+
*
1
m
(A. 5)
-
c
3s
I
3s
*
nee
c
3s
3s
i
*
^
_3_m
3s
In
3s
^
_3_0
£
3s
m
3s
_3n^
,
3s
3S
3s
m
3s
m
3S
3s
^^•^^
%,s
°
^0^,s
*
^S,s
•
^Q^.S
*
"^m^.s
*
Ps.s
*
^m^.S
which is the advertising equation from vector equation (22
FOOT
N'
DIES
presentation of the marKeting mix concept, see Borden
^
For
^
Some reactions may be virtually instantaneous, v/hile others may
a
occur after
considerable time lag. For example, price reactions
a
Advertising reactions, however, usual-
are often almost immediate.
involve substantial time lags. So,
ly
'\.t
3u
^
A
.
k.t-1
**
"K.t-1
,
,
Whereas
^^^^^^^
^k.t
p
%,.u.
^l'"l
could be defined as
p
'"^2ht
~ l.t
1
be
,t
1
.t
l.t
more general profit function could be considered including dis-
tribution,
a
[
markup,
etc.
This simple form is used here to establish
link with Dorfman and Steiner's work.
Throughout we will assume that the second order conditions are
satisfied.
^
For
a
comparison of alternative models of determinants of marke-
ting expenditures in the firm,
6
30t/3U is an
(n
x
1)
see Elliott
[121.
vector, and 3U/3u is an
(n
x
n)
To be conformable for multiplication we should write
^
For
a
matrix.
30t
basic discussion of the Cournot and Stackelberg competitivf
reaction models,
see for example
Intriligator [17, pp. 205-19].
45.
The direct elasticity
not taking
(i.e.
into account possibli
competitive reaction^ of market share with respect to price
3m
.
3pm—
has been defined as
•
i
.5*
m
m
.x
^
i
>•
_
^
m
p*
m
.p*
m
£
i.e.
.
3p
m, .p'
a
i'
*^
.(p*)
(X*)
(s*)
which simplifies to.
'i'
Similarly one obtains
n
r,
m^.P
=
"
n
m^.p''1
Analogous results obtain for the other decision variables and in
vector notation we can write
E
^
m
,
u
=-E m
,
M
U
=
E
m
,
u
*
The direct elasticity of market share with respect to price is
then
P
m
.^
'
P-
P
a'n„
i.P
p*P
I
m
m. ,p
i'
(p
+
P)
3p
m
p*P
.s
^
(s°)
(x°)
.
,
X
o
n
m
,
.(x°]
1
_,
^
.
X
I
which reduces to.
"
"^m^.p
m
,
'^m^.p°
'
_o
•
P
o
•
n
p
m
,
tP^P
p
Similarly one obtains
n
=
m^.P„
n
More generally,
m
1°
,
m^.p
we can writi
m
u
An index devised
P
.
,
m
U
U
,
u
by Hymer and Pashigian
[161.
I^
and
I^
are res-
pectively the unv/eighted and weighted sum of absolute changes
in market
shares
1=7;
•2
=
J^
:
W
'^.T
I^.T
-
-
'"i.ii
^.ll
•
%.1
where m
is
the marKet share of brand i.
n
is
the number of competing brands.
at
time
t
47
an average instability
is
I
the extent of the observation period.
is
t
index computed over the whole period
T
of Observation,
i.e.
I ^
=
1
^^
.
-1
j
Hymer and Pashigian's indices. However,
^^
For
see Theil
^^
jm^^^
-
m^^^_J
they were computed as
pp.
[31,
-
also computed and reported in Table III,
316-18].
These simultaneous equation models are discussed in Bultez
X
,
P
S
,
-»
^
t-1
riables
and m
s
,
x^
-
t-1
»
s^ ^,
t-1
\
were the jointly dependent variables,
P^ ^,
t-1
S^ ^
t-1
and m^ ^
t-1
7
]
x
left in the estimation that the
of freedom were
results are not worth reporting
is
are willing to accept that the
distributed according to
a
t^
statistic computed here
Student's density function, an
assumption, which may be asymptotically valid in the case of
simultaneous equation model.
,
,
the predetermined va-
.
So few degrees
If we
.
^
comparison of the merits of the Herfindahl index with those
a
the entropy measure
of
n
^E^
.
'
48
'^
Note
the new Durbin statistic tests
thrit
against the alternative hypothesis of
pendence scheme.
the case of
In
a
a
the non-autocorrelation
first-order Markov de-
KoycK model where the original
disturbances are independent, when we apply Koyck's transiformation the resulting error terms get correlated according to
a
first-order moving average scheme and thus the Durbin statistic
is
ir re levant
.
this asymptotically valid test has been desi-
Moreover,
gned for single equation regression model.
Its
performance is
un-
known when we apply it to residuals from one equation which is
part of
^^
system.
a
The long term market share elasticities are the elements of
1
the vector
(log
rn
)
•
variable,
E
m
*
,
u
where
X
is
the coefficient of the
0.344.
i.e.
The significance of the
,
estimator is unknown and depends
E
critically on the statistical properties of the [R] and E^
^'
i
estimators. However the distribution of such an estimator may
be empirically approximated by Monte Carlo experiments with
various sample sizes.
^^
Assuming
a
58
%
gross margin and using the dynamic ver<=inn short-
run estimates, we obtain
MRA
^'^
=
(11 .57)
:
(0.093]
•
(963,000/1 ,407,2571
=
$
0.752
49
*^
A
detailed demonstration of the long-run result may be found
in BULTEZ
20
[7].
Since we expect
<_
X
<
1
.
I
X*^
=
1/(1
-
A)
T =
2^
Since we need
estimate them.
a
long period of stable competitive behavior to
n
.
REFERENCES
[l]
[2]
BAIN,
J.S., Industrial Organizations
York,
1959.
and L.E. RICHARTZ.
BALIGH, H.H.
.
John Wiley and Sons. New
"Varl ah 1 e-Sum Game flodels of
Marketing Problems". Journal of Marketing Research
No
[3]
2
(May,
P.M.,
BASS,
1967), pp.
"A
Vol.
.
IV,
173-1B3.
Simultaneous Equation Regression Study of Adver-
tising and Sales of Cigarettes", Journal of Marketing Research
Vol.
[4]
VI.
No
3
BORDEN, N.H.,
(August.
BREMS,
HANSEN,
H.,
M.
pp.
291-300.
"The Concept of the Marketing Mix", Journal of Ad-
vertising Research
[5]
1969),
"Price,
.
Vol.
4,
No
2
(June,
BULTEZ, A.,
in
2-7.
:
KJAER
(ed.). Readings in Danish Thcorij of Marketing
1966,
pp.
.
-
North-
149-159.
"Leader's Marketing Behavior in Oligopoly", C.I.M.
Working Paper 6-71, College
I
cicr.ccs du Msnagcment
[7]
pp.
Duality and Rival Response",
Holland Publishing Company, Amsterdam,
[B]
1964),
teruni vers i t aire d'§tudes doctora(C.I.M.),
Eru;
ine
197
BULTEZ, A., Competitive Strategy for Interrelated Markets,
published Ph.
D.
1
Un'
dissertation, Louvain University, in process
,
51.
[ol
COWLING,
CUBBIN.
and J.
K.
Competition
:
[9]
pp.
DORFriAN,
Quality and Advertising
An Econometric Investigation of the United King-
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1971),
"Pr::ce,
,
Vol.
(November,
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376-394.
STEINER,
and P.O.
R.
"Optimal Advertising and Optimal
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