Jean-Jacques Lambin.

Optimal competitive marketing behavior in oligopoly online

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system, whether or not they occur in the fitted equation with non-
zero coefficients [10, p. 377]. Referring to Durbin and Watson ta-
ble [32, pp. 724-725], we realize that d indicates a positive auto-
correlation. In the dynamic version, the test for serial correlation
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 d statistic of the dy-
namic market share model is in fact the t statistic corresponding to

the lagged residuals coefficient. We can thus conclude that the residuals of

1 5
the dynamic market share model are uncorrelated • From a statisti-
cal specification point of view, the dynamic version of the model
seems preferable to the static version. However, 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 re-
sults 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 h"^ deri-
ved. In this case, from the application of corollary 4, i.e.
E = [I - R] • E ^. Table VIII shows how from the [R] matrix

of table VI and the E ^, vectors extracted from table VII, the re-
m . u*

suiting E vectors arc obtained. Equilibrium long-run market share

elasticities, defining the full impact over time of changes in price

16
advertising and quality, wers also computed . Except for product-
quality, no important disparity appears between these long-run elas-
ticities derived from the dynamic model and the elasticities deri-
VRd from the static model.



INSERT TABLE VIII ABOUT HERE



ECONOMIC INTERPRETATION OF THE RESULTS



The E vector contains the total sales elasticities
taking into account the reactions of competitors while the elements
of vector E^ ^* are the total sales elasticities when competitors
do not reoct . This can easily be proven since if competitors do
not react [R] = [0] and ^n ^ "^ ^m * * ^^^^ important feature ena-
bles us to visualize the demand for the leader's products as a kin-
ked resrcnsc curve, drawn on figure 3.



INSERT FIGURE 5 ABOUT HERE



It is worth noting that these estimates of sales elas-
ticities (E 1 are significantly different from the market share
q » u ^ '

elasticities (E *). Thus, any judgment on the firms' behavior ba-
rn ,, u J ~i t^



23.



sed on the E * vector would have been roriously bifjsed. Fcr exam-
m , u '

pie, the leader's monopoly power would have been underestimaL - ^:
If we asc'jme prof i t -maximi z i ng behavior, then our estimate of the
potential degree of monopoly power, 100 • —



the percentage



marK-up on marginal cost, is equal to [-100/(1 + n )]. Our estima-
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 % or
136.29 %, 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 : it is a de-
creasing function of the price elasticity : (1 ♦ 1/n ). The ratio
estimates are 0.46G1 and 0.4232 in this case. All these results con-
firm the high degree of market concentration observed by the end
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^ in estimate of a equal to
0.063 and the dynamic form an estimate equal to 0.002. r> cc^n,-? rl son
of these values v/ith the observed advertising-sales ratios, the
mean of which is 0.0B8, shows that although firm B had been over-
spending from 1363 to 1965, on the average, its advertising beha-
vior 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 . -^



on the ave raj



MRA = (p - no



q.s



(q/s)



Assuming a 53 ^ gross margin and using the static version estimatrs,
we get : _. _-

MRA =(10.71) • (0.118) • (983,000/1,407.257) = $ 0.6B3



Thus the last dollar spent in advertising produces approximately

1 8
a 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:



(25)



z (p - nc

1 =



3q



Z (1 + r)

1 =



25



20

where r is the cut-off rate. Assumiriji price and quality fixea ,



^t*B = "^ • ^*,



t + 8



q . s m , s ^



n s

T=0 t+B-T-1



Y = 1/(1 - A)



Thus,



(26)



^ ^^t^B



'i'



for B =



[n - (1 -X) ♦ n *] .when B /
q , s m . . s



Substituting the value of — z extracted from (26) into equation (25)



'^■m^.s*'^t



(1-A) -n,



1 ♦ r "t*6



At equilibrium.



(28)



li-n ts ] = (p - MC:

t-coo



(1 * r - A)



26



and as a consequenci



(29)



(p - MO • Q • k



n ♦ r • n
q.s



m^ , s



1 ♦ r - X



where 6



1 - Y . n,



The long-run optimal equilibrium value of the
advertising expenditures is thus equal to :



s = I (11 .57) • (2.100,0003 • (0. 149



1 .524 , 0.093 + r X 0. 147



1 + r - 0.344



(1.524) • (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 'j
















^






















-













o








d


d


pJ


d


d lo


^


^


















Z


i'


§


°


in


1






,




o


d


d


T-*
































^




















a.


-








00





XI




— Si


o




1






CT>


CM




en


i? s


2


en








CO


00


*^




o r-




o










o




d


d CM






d


1
d


Si


'






1










?




(M


to


CTi


o


uD in












(N


CM


o-


o


r - in




■r-


c^


t*


CD








in








d


d


t-'


?■


Kl


?


'"


d CD
























C t)








































-D 2








*•












C «


0-






(O












D ._




















O. 1.


cn












o>






t> n


o












o






o >








-■











37,



3 C



C rM ♦J



O O CvJ



lo


T-


A
t^


s


iy


to


o


o


CJ








r-






't








o


in


?


o


c
















s^


?>



T- CM CO



TABLE VIII



TOTAL SALES ELASTICITIES



39



0.273

-0.774



0.023'



0.901



Static nodcl



5.543



0.503



Short -run



m . 1
1 . u



Dynamic riodel



-3.726'
0. 147'
0.583*



Long-run



rL-R
i , u



■5.676
0.223
0.667



1 .673
0. 116
0. 188



,L-R
q.u



-1 . 138
0.093
0. 171



L-R



1 .734
0. 142
0.261



TABLE



I X



SENSITIVITY ANALYSIS OF THE LONG - RUN OPTIMAL
EQUILIBRIUn LEVEL OF ADVERTISING EXPENDITURES



r


%


3 %


$ 1 .421 .246


4 \


$ 1.422.063


5 %


$ 1.422.899


6 %


$ 1,423.689


7 \


$ 1.424.461


8 %


$ 1.425.209


9 %


$ 1.425.940


10 %


$ 1.426.650



41



APPENDIX



Tslser derived the following relation (translated in
I • our notation) between the various elasticities with respect to ad-
vertising



(A.I] n = (1
m, s



m) • { n - Hn 5 = n ,
q , s , s m , s



" - »S.s'



In fact, this should have been written as two separa-
te equalities because they result from different assumptions. First,
let us take



(A. 2) n„ = n

m, s m . , s



(1



'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 (15]).

On the other hand ,



(A. 3)



^^ - "^^ • ^\.s - -^q ,s



Is the advertising equivalent of equation (22), that is an oligopo-
ly with expansible industry demand, simple competitive reaction and
the marKet share function in general form. This equivalence will
now be demonstrated.



Equation (A. 3} can be written as



1-m m,s q,s O.s



(A. 4} n = Ti„ +77

q . s ^r' ^



with = q + Q



12.



30



(A. 5] n



CO s 3rn s 3m



q,s Q 3s m 9s m 33



(A. 6]



3rq/ (q+0 n 3q q



3s



3s



3q 30



3q q 30

c I



3s 3s



Substituting (A.B) in the second term of (A. 5)



30 ^ 30 -

s _ c _^ B^ _ 3^ _ q » s ^ c _s ^ _3_m

I 3s m 3s

c



q.s 3s 3s



S ^ C , r,, _ £-1 y 1 . Is. + 1 i In

* 3s * 3s * m * 3s



nee [1 - Hi = ^



_s _3_0 £ , _3n^
'q.s 3s m 3s



q.s ' 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



^ For a presentation of the marKeting mix concept, see Borden [



^ Some reactions may be virtually instantaneous, v/hile others may
occur after a considerable time lag. For example, price reactions
are often almost immediate. Advertising reactions, however, usual-



ly involve substantial time lags. So, p

'\.t "K.t-1 ,

. , Whereas p

k.t-1 ^k.t ^l'"l



could be defined as



3u



^^^^^^^ %,.u. '"^2ht be ~



1 ,t
l.t



1 .t
l.t



^ A more general profit function could be considered including dis-
tribution, markup, etc. This simple form is used here to establish
a 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, see Elliott [121.



6



30t/3U is an (n x 1) vector, and 3U/3u is an (n x n) matrix.



To be conformable for multiplication we should write



30t



^ For a basic discussion of the Cournot and Stackelberg competitivf
reaction models, see for example Intriligator [17, pp. 205-19].



45.



The direct elasticity (i.e. not taking into account possibli

competitive reaction^ of market share with respect to price

3m .

has been defined as • — . i.e.

3pm



m .p* m p* m .5* m .x

£ ^ i _ >• i ^

m 3p



a .(p*)



*^ i'



m, .p'



(s*)



(X*)



which simplifies to.



'i'



Similarly one obtains



n r, = " n 1
m^.P m^.p''



Analogous results obtain for the other decision variables and in
vector notation we can write



E =-E M = E *
m , u m , U m , u



^ The direct elasticity of market share with respect to price is
then

P- a'n„



P .^
m ' 3p



i.P



m. ,p



P I i'
p*P



(p + P)



o n 1

m _, . X I



m .s ,
(s°) ^ .(x°] ^



p*P



m . , X



(x°)



which reduces to.



"^m^.p " '^m^.p° ' tP^P



n _o • P



m , p m , p



Similarly one obtains



n „ = ■ n o • P
m^.P m^.p



More generally, we can writi



m , u m . , U U m , u



1° An index devised 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; W '^.T - '"i.ii



•2 = J^ I^.T - ^.ll • %.1



where m is the marKet share of brand i. at time t
n is the number of competing brands.



47



t is the extent of the observation period.



I is an average instability index computed over the whole period

1 1 T n
of Observation, i.e. I ^ = ^^ . - . j ^ ^E^ jm^^^ - m^^^_J .



Hymer and Pashigian's indices. However, they were computed as

^^ For a comparison of the merits of the Herfindahl index with those
of the entropy measure - also computed and reported in Table III,
see Theil [31, pp. 316-18].

^^ These simultaneous equation models are discussed in Bultez \ 7 ] ,
X , P , S , s and m were the jointly dependent variables, x ,



^ -» x^ - » s^ ^, P^ ^, S^ ^ and m^ ^ the predetermined va-
t-1 t-1 t-1 t-1 t-1 t-1



riables .



So few degrees of freedom were left in the estimation that the



results are not worth reporting



If we are willing to accept that the t^ statistic computed here

is distributed according to a Student's density function, an

assumption, which may be asymptotically valid in the case of
simultaneous equation model.



48



'^ Note thrit the new Durbin statistic tests the non-autocorrelation
against the alternative hypothesis of a first-order Markov de-
pendence scheme. In the case of a KoycK model where the original
disturbances are independent, when we apply Koyck's transiforma-
tion the resulting error terms get correlated according to a
first-order moving average scheme and thus the Durbin statistic
is ir re levant .

Moreover, this asymptotically valid test has been desi-
gned for single equation regression model. Its performance is un-
known when we apply it to residuals from one equation which is
part of a system.

^^ The long term market share elasticities are the elements of
1



the vector



• E * , where X is the coefficient of the
m , u



(log rn ) variable, i.e. 0.344.



The significance of the E estimator is unknown and depends

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


2

Online LibraryJean-Jacques LambinOptimal competitive marketing behavior in oligopoly → online text (page 2 of 3)