Hermann Simon.

Marketing multiplier and marketing strategy simplified dynamic decision rules online

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Hermann Simon**

Working Paper 1050-79

March 1979






Hermann Simon**
Working Paper 1050-79 March 1979

* The author is indebted to Manu Kalwani and Alvin J,
Silk for helpful comments.

** Assistant Professor of Management Science,
University of Bonn, and

Visiting Fellow, Sloan School of Management,
Massachusetts Institute of Technology


For a wide class of empirically tested dynamic marketing response
models a heuristic method for the determination of dynamically optimal
price and promotion levels is being developed. The future effects of
present marketing actions are measured by a marketing multiplier which
is partially based on managerial estimates. Very simple dynamic opti-
mality conditions for both single marketing variables and the marketing
mix are formulated. The application of the method to a number of em-
pirical models yields interesting insights into realistic magnitudes of
the dynamic impact.


- 1 -


Within the last few years a large body of empirical work on dynamic marketing
response models has emerged. In these models, the dependent variable sales or
market share is usually considered as a function of the lagged dependent variab-
le and of one or several marketing instruments. A wery genereral form of these
dynamic response functions can be written as

^t = ^^^ Vi ^ ^(^i,t" - "^,t^ (^)

where q^ sales in period t (units or market share)
f.(.) marketing response function in period t
X. . value of marketing variable j in period t (units or share)
a. absolute term (parameter)
A. carryover-coefficient (parameter)

The variables in (1) can be either in natural or in logarithmic dimension
so that the function comprehends both the linear and the multiplicative sales
model. Moreover, the parameters and/ or the marketing responses can be either
constant or time-varying.

Reviews of a great number of empirically tested models of type (1) can be
found in [10, 11, 28, 39]. Table 1 provides a synopsis of the different
versions encountered in the marketing literature. These studies comprehend
more than 200 products or product-market-combinations.


Due to the wide coverage in the literature it doesn't seem necessary to repeat
the rationale which underlies the different versions of model (1). We shall
focus on deriving a simple dynamic optimization heuristic which can be applied
to almost all of these versions.

2 -

The numerous optimization approaches in the literature are almost exclusively
limited to advertising and the linear carryover-function. Optimal ity conditions
of this type are, for instance, given in [3, 15, 17, 19, 35, 48]; numerical
solutions can be found in [8, 27, 47]. Another group of approaches use modern
control theory in order to derive dynamic optimality conditions for advertising.
Schmalensee [32] gives very general conditions of this type but does not pro-
vide any numerical solution. Most of the control theoretic approaches are
limited to specific advertising models (in particular to the models of Vidale-
Wolfe [46] and Nerlove-Arrow [25]) which will not be investigated in this
paper, [4, 13, 14, 33, 43, 44] are of this type). An exception is the pricing
model of Spremann [40, 41] a version of which can be compared with function (1).
To date, the practical relevance of the control theoretic models has remained
very limited.

We are not aware of any work in which a unified optimization approach for all
versions of model (1) and different marketing instruments is provided. The
present paper is structured as follows. In the next section we define a simple
measure of the cumulative effects of a marketing action. This measure is sub-
sequently used to formulate optimality conditions. Finally a simplified pro-
cedure which takes advantage of these conditions in order to obtain numerical
solutions is proposed and a number of applications is discussed.


The applicability of the following derivations is limited to models in which
the carryover-effect and the sales response to the marketing variable on which
a decision is to be made are separable . This separability condition holds for
all models in table 1 with the exception of [24, 45, 50].
Measuring the short-run sales effect of a certain marketing activity x- ^ by

3 -

means of the partial derivative 9q^./9x. ^., we obtain the total cumulative
sales effect (over time) attributable to this short-run response as

°° ^q^ ^q^.,_ 3q^. " 9q^.,^
^ T=0 ^^j,t ^^t ^^j,t 1=0 ^"^t

The sum term on the right hand side of (2) gives the total cumulative sales
effect of a marketing action as a multiple of the action's short-run effect .
Therefore, it seems reasonable to denote this sum term as marketing multiplier .

In the case of a linear carryover-function the multiplier for which we write
m is simply obtained as

^ = 1 ^h^hh^] ^\h^^ h^2^ ^3)

and if the carryover-coefficient A is constant over time and < A < 1

m^ = 1/(1 - A) (4)

which is the expression first derived by Palda [26]. Kotler [16] denoted (4)
as "long run marketing expenditure multiplier" and more recently - obviously
unaware of Kotler's denomination - Dhalla [11] used the label "long-term
marketing multiplier" for (4).

In the linear case, m. is independent from the future marketing activities,
whereas it depends on those activities in the multiplicative form of model (1).
The marketing multipliers of all models under consideration are given in table



By means of the marketing multiplier m. and the short-run elasticity a long-run
marketing elasticity E. ^ can be defined in the following way .

^j,t = ^-^j,t

where e. . = 9q^./9x- ^ • x. ./q. is the usual short-run elasticity.

E. . gives the cumulative sales effect which is induced by a 1%-change in

marketing variable j as a percentage of current sales. Though, to date, little

observed in the literature, this elasticity seems to have highly interesting

empirical properties. Comparing the respective price elasticities of 12

detergents and 21 pharmaceuticals the author obtained the following amazing

results [39]

short-run marketing long-run

elasticity multiplier elasticity
(mean) (mean)

Detergents 2.37 1.75 4.15

Pharmaceuticals .76 3.64 2.77

Though the short-run price elasticities are highly different, the long-run
elasticities of both product groups are rather close together (not significant-
ly different at the 5%-level) due to the differences in the carryover-patterns.

It should be observed that both m. and E. . are sales ( quantity ) related

t J > t

measures of long-run marketing effects. In order to make optimal marketing
decisions, however, a value -related measure of the long-run effects is re-
quired. We obtain this measure by weighing each period (t+T)'s term in m
by the contribution margin dl^ and the discount factor (1 + r)"T The

resulting value-adjusted marketing multiplier is


f:, "t+T 8q.
T=l ^t

"I'^l* I 'In Ji^

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