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THOMAS S. GRUCA and D. SUDHARSHAN*
Previous research suggests that a multinomial logit model of market share can
be used in optimization or equilibrium analyses of marketing decision making. How-
ever, there
are
significant problems with the solutions obtained from such an anal-
ysis. The authors show that for any firm with less than a
50%
market share, the
profit maximizing solution is to increase advertising as much as possible. The same
holds true for any other positively signed marketing mix variable. The exponential
property of the MNl model precludes reasonable descriptions of firm behavior in
this case. Thus, the MNl market share model is inappropriate for equilibrium anal-
ysis of marketing competition.
Equilibrium
Characteristics
of
Multinomial
Logit
Market
Share
Models
MODEL
AND
ASSUMPTIONS
Consider an MNL model with two variables, price
(P)
and marketing effort
(m).
We assume that attrj =
aJeo,jP
j
e~jmj.
Market share for firm jwould be given by
Unfortunately, the use of MNL market share models
in those proposed analyses is not appropriate. Under rea-
sonable assumptions, the structure
of
the MNL model
precludes sensible descriptions of competitive marketing
behavior. Specifically, for any firm with less than a 50%
market share, the exponential properties
of
the model do
not allow the modeling
of
decreasing returns to scale for
advertising or promotion expenditures. Thus, the profit-
maximizing equilibrium marketing mix for those firms
is: spend as much money as possible on marketing ac-
tivities.
In the next section, we specify a simple MNL to il-
lustrate this behavior. We follow with the equilibrium
conditions and proof of the preceding assertions, and
conclude with a discussion of the results.
The multinomial logit (MNL) has a long history of
modeling marketing phenomena.
It
has been used to in-
vestigate the determinants of grocery store choice (Gensch
and Recker 1979), graduate school choice (Punj and
Staelin 1978), and brand choice in a wide range of mar-
kets (see, for example, Guadagni and Little 1983).
In their widely known book, Cooper and Nakanishi
(1988) compare various market share models. They con-
clude that the MNL and multiplicative competitive in-
teraction (MCI) models are the feasible alternatives for
marketing decision making. The choice between the MNL
and MCI models, they suggest, depends on the market-
ing variable under consideration (p. 35). We submit that
the choice between these alternatives is also determined
by the ultimate application
of
the model.
Some authors suggest that an MNL model be used for
optimization or equilibrium analyses
of
marketing de-
cision making. For example, Carpenter and Lehmann
(1985) suggest that their MNL model can be used to de-
rive optimal marketing mix policies in a game-theoretic
framework. Both Lilien and Kotler (1983, p. 680) and
Lilien and Ruzdic (1982) discuss how an MNL market
share model can be used to derive the optimal level of
marketing expenditure. (1)
attr)
MSHR =
-----'---
J
attr)
+2.j",)a
tt
r j
*Thomas S. Gruca is Assistant Professor of Marketing, College of
Business Administration, University of Iowa. D. Sudharshan is As-
sociate Professor of Business Administration, University of Illinois,
Urbana-Champaign.
480
(2)
(3)
1Tj =(P, -
MC}MSHR/SP
-
m)
-FC).
where P, is price,
MC
jis marginal cost,
MSHR
jis market
Journal
of
Marketing Research
Vol. XXVIII (November 1991),
480-2
MULTINOMIAL LOGIT MARKET SHARE MODELS 481
(7)
(8)
(10)
(9)
P3:A twice differentiable
functionf
does not attain a lo-
cal minimum at x* if
a'i(x)/axfevaluated
at x* <O.
Proof:
The elements of the Hessian matrix
Hf(x)
are the
cross-partial derivatives
off(x).
The diagonal ele-
ments thus are the second partial derivatives of
f(x)
or
a'i(x)/axf
for i=1, n. From PI' we know
that a necessary condition for a local minimum of
fat
x* is that the Hessian
Hf(x)
is positive sem-
idefinite at x*. From Pz,we know that if any di-
agonal element is negative, the matrix is not pos-
itive semidefinite. Therefore, if
a'i(x)/axf
evaluated at x* <0, then
Hf(x)
is not positive
semidefinite and, consequently, x* is not a local
minimizer of
f(x).
Q.E.D.
From these three propositions, we can construct the fol-
lowing theorem:
Theorem: For any firm that is
not
the cumulatively dom-
inant firm (having more than a 50% market
share), then any point satisfying
a(-7Tj)/am
j=
ois not a minimum point.
Proof" the second derivative
of
(-7T)
is given by
a
Z(
-7T)
a(mi
(pi attr,
L,j.,<.
attr,
= _ (Pj_
MC)
xJ J J
tattr, +
L,j.,<j
attr;)
_ 2
(pi
(attri
L,j.,<j
attr
j)
tattr, +
L,j.,<j
attrl
(P)Z
attr
L,j.,<.
attr,
=
_(po
_MC.) x}J J
J J
(attr,
+
L,j.,<j
attrl
x (
~
attr,
-
attrj
)<0
if
(L,j.,<j
aur,
-
attr)
>O. Thus, by P3,the point
(P*,
m*)
is not a minimizing point of
-7Tj
and
therefore it is not a maximizing point
of
7Tj.
Q.E.D.
We have shown that any interior point
(m)
is not an
equilibrium level of marketing expenditure. Because a
continuous function over a compact and convex space
must have a maximum point, it must occur at the bound-
ary because it is not the interior.
For firms that are not cumulatively dominant, the op-
timal marketing mix is to spend as much as possible on
marketing efforts. Note that this result holds for any pos-
itively signed marketing variable. If all firms follow the
equilibrium strategy, the one with the deepest pockets
WIllS.
For any firm with less than a 50% market share (which
will always be at least N-1 of Nfirms in a market),
the MNL cannot model diminishing returns to scale for
changes in advertising, promotion, or any other posi-
tively signed control variable. However, most empirical
research shows that the advertising response curve is
EQUILIBRIUM CONDITIONS
We assume that the strategy space (Pj,
m)
for every
firm jis a closed, bounded, convex set. All firms are
assumed to choose their marketing mix (Pj'
m)
to max-
imize profits
('IT)
given that all other firms are also max-
imizing profits through their choice of marketing mix
(P
-j'
m_).
The problem faced by every firm is
(4) MAX:
7r/P
j, mj) I(P _j,
m_).
In the next section, we discuss the characteristics of any
Nash equilibrium.
First-Order Conditions
Any interior (nonboundary) Nash equilibrium (P*, m*)
would have to satisfy the first-order conditions for a
maximum. Namely, the first partial derivatives of the
profit function would have to be zero at (P*, m*):
(5)
a7riaPj
=0
and
(6) a7rj/amj =
O.
Looking more closely at equation 6, we have
a7T
l3·attr xL,.,<attr
_J
=(Pj_
MC
j)xJ J IJ
~
-1=
o.
Bm, tattr, +L,j.,<jattr
j)
share, SP is the market sales potential, mjis marketing
effort, 'ITj is profit, and FCjis fixed costs. We assume
that all
u's
<0 and all B's >0, and that
SP-the
sales
potential for the
market-is
fixed.
For any point (P*, m*) that satisfies equation 7 to be an
equilibrium, 'ITj must be maximized at that point.
Second-Order Conditions
To simplify
our
determination of whether there is an
interior maximization point of
'IT,
we examine the char-
acteristics of -'IT. Any point where
iJ(-'IT)/iJm
j=0 that
is a minimizer of (-'IT) is also the point at which
'IT
is
maximized because MIN: (-'IT)
==
MAX:
'IT.
Before the main theorem, we state the following three
propositions.
PI (Gill, Murray, and Wright 1981, p.
63-64):
The nec-
essary conditions that a point x* must satisfy in order
to be a local minimizer
off(x)
=
f(xl,
x2,
...
,
xn)
are (a) the gradient of
f(x)
evaluated at the point x*
must be zero and (b) The Hessian of
f(x)
evaluated
at the point x* must be positive semidefinite.
Pz(Noble and Daniel 1977, p. 427): A necessary con-
dition for a Hermitian matrix A to be positive semi-
definite is that the diagonal elements of A are all non-
negative.
Proof" In the quadratic form (x, Ax), choose all Xk to be
zero except x.. Then (x, Ax) =
aiilxjlZ
and, be-
cause x, is not zero, we must have a., be non-neg-
ative for A to be positive semidefinite.
Q.E.D.
482
concave (diminishing marginal returns) at higher levels
of
spending (Little 1979). The MNL captures the di-
minishing marginal returns effect only if there is a cu-
mulatively dominant brand and only for that one brand.
DISCUSSION
The exponential nature of the MNL model is well suited
to modeling the volatile short-term changes in house-
hold-level purchase probabilities that are observed in
scanner panel data (Carpenter and Lehmann 1985; Guad-
agni and Little 1983; Gupta 1988). The response of choice
probability to changes in the marketing mix is S-shaped,
with larger marginal increases in probabilities coming at
lower levels of expenditure. This convex-concave re-
sponse pattern allows the MNL to capture the large
changes in choice probabilities that are observed even
when marketing mixes are not greatly altered.
However, the MNL cannot be generalized to equilib-
rium analyses of marketing competition. In such an ap-
plication, the alternative is to use the multiplicative com-
petitive interaction (or attraction) model. Like the MNL
market share model, the MCI model directly incorpo-
rates competition, is logically consistent (Bell, Keeney,
and Little 1975) and can be easily estimated by using
standard statistical packages (Nakanishi and Cooper 1974,
1982). Furthermore, through proper choice of elasticity
values (0 <lal's, I3's <1), it exhibits decreasing mar-
ginal market share response to changes in the marketing
mix. For that reason, it has been used often in equilib-
rium analyses of marketing competition (Kamani 1983,
1984, 1985; Monahan 1987; Steckel 1984).
CONCLUSIONS
The MNL model is often used for modeling individ-
ual-level choice behavior. However, its exponential
structure renders it of little use in equilibrium modeling
of market share competition. Because the
MNL
is not
generalizable to modeling competitive behavior, the MCI
would be a superior choice for that type
of
application.
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Reprint Nn. JMR284109