LIBRARY

OF THE

MASSACHUSETTS INSTITUTE

OF TECHNOLOGY

i

Dewey

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

OPTIMAL ALLOCATION OF COMPETITIVE

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MARKETING EFFORTS REVISITED

Philippe A. Naert*

557-71

September 1971

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

I MASS. INST. TECH.

OCT 5 1971'

DEWEY LIBRARY

OPTIMAL ALLOCATION OF COMPETITIVE

MARKETING EFFORTS REVISITED

Philippe A. Jaert* "^

551-11

September 1971

*Assista.it Professor of Management, Sloan School of Management,

M.I.T.

HOP?

no. 7

RECEIVED

OCT 6 1971

M. I. T. LIBkakilS

ABSTRACT

In a recent article in this Journal , Lambin presented an

extension of the Dorfman-Steiner theorem to the case of an

oligopolistic market. It is demonstrated that the market share

optimization rule derived by Lambin is incorrect. A correct

formulation is presented, A comparison of the absolute sales

and market share optimization rules yields a relationship between

absolute and relative advertising elasticity, previously obtained

by Telser and used in the Lambin article. Analogoxis results for

price and quality elasticity are also reported. Finally, some

problems associated with how Lambin interprets his results are

discussed.

5,'-so;i

- 2 -

Introduction

In a recent article in this Journal, Jean-Jacques Lambin

presents an interesting theoretical, but empirically verified,

extension of the Dorfman-Stelner theorem to the case of an

oligopolistic market . The objective was to derive a market share

optimization rule, that is, a rule where competitive effects are

explicitly taken into account. In the first part of this paper

we will rederive the market share optimization rule and we will

2

thus demonstrate that the result obtained by Lambin is incorrect .

Comparing the absolute sales and the market share optimization

rules yields a relationship between absolute and relative adver-

tising elasticity, preriously obtained by Telser and used in the

3

Lambin article , Analogous results for price and for quality

elasticity are also derived. In the second part of this paper we

will caraaents on the economic interpretation of the results .

First w€'. will point out a case of suboptimization, and finally we

will prove that what Lambin calls a long-term optimization rule

is really something else. The discussion in the second part is of

a more general nature in that most of it would remain valid even

If Lambin 's market share optimization rule were correct.

- 3 -

Notation

Let q = q(p,s,x) = Unit sales of brand i per time period .

p = i's sales price

s = i's advertising outlays

X = i's index of quality

c = c(q(p,s,x) ,x) = c(q,x) = unit average cost function

m = i's market share

Q = total industry sales, i.e. m = q/Q

P = average market price

S. ■ competitiors ' advertising outlays excluding brand i's

X = average product - quality index

p* = i's relative price, i.e. p*"p/P

s* = i's relative advertising, i.e. s* » s/S.

X* = i's relative quality, i.e. x* = x/X

n = -(9q/9p) (p/q) = i's absolute price elasticity

y p9q/3s " i's absolute marginal revenue product

of advertising

n = i's absolute advertising - sales elasticity

a = r„" .. V — = i's absolute product quality elasticity

x v.oC/dxy q

n * = -(9m/3p*) (p*/m) = i's market share elasticity with

respect to i's relative price

n ^ = (9m/9s*) (s*/m) = i's market share elasticity with

respect to i's relative advertising outlay.

n^ •= (9m/9x*) (x*/m) ■= i's market share elasticity with

respect to i's relative quality

Other symbols will be defined when needed.

- 4 -

The market share optimization rule

Brand i's profit function can be written as

IT = pq - qc - s (1)

or

IT = q(p,s,x) fp - c(q(p,s,x), x)j - s

(2)

The Dorfman-Steiner theorem Is simply the optimization of equation (2)

with respect to absolute price, advertising and quality. The following

5

well known result obtains'

2. = 1

p X c w (3)

where, with MC equal to marginal cost,

w = (p - MC) /p = the percentage of gross margin

Using the relative values defined in the previous section, equation

(2) can be rewritten as

t; = Qm (p*,s*, x*) [^*P - c(m (p*,s*,x*), k*X^ - s*S. (4)

Necessary conditions for optimality are

"9^* ~ 8s* " ax* ~ ^ ''

1^* - 4f. 'p*^ - =' - "» ff - = ° '"

- 5 -

37* = ^*

1