Marketing Spending Models - Innovation and Marketing in the Pharmaceutical Industry - page 570

The star indicates that variables are in their optimum. This result holds under the
assumption that both competitors play a Nash game in all marketing instruments or
competitor i plays a Stackelberg follower in one or all of the instruments. The solu-
tions ( 19.11 ) and ( 19.12 ) have to be extended by cross-effects if firm i acts as a
Stackelberg leader in at least one of the instruments. It may be striking that the
optimal solutions for the marketing budgets under competition do not depend on
competitor activity levels. This is a feature specific to the double-log response
model. It should, however, be noted that the optimal advertising and detailing bud-
gets depend implicitly on the equilibrium levels of competitor budgets via the equi-
librium brand sales level S *.
The optimality conditions also allow identifying the ratio of optimal distribution
of a limited marketing budget on advertising and sales force. Let R denote the fixed
marketing budget that is to be allocated across advertising and sales force. Then,
consistent with ( 19.11 ) and ( 19.12 ), the optimal budgets are (Dorfman and Steiner
1954 )

b
bd R

* =

A

i

(19.13)

+

i i

d
bd R

* =

D

i

.

(19.14)

+

i i

19.4.2 Dynamic Optimization

Optimizing expenditures across physicians and time . The evolution of a new drug is
a dynamic process of growth and decline by definition, i.e., drugs follow a life cycle
(e.g., Grabowski and Vernon 1990 ; Fischer et al. 2010 ). The responsiveness towards
marketing activities may change over the life cycle (e.g., Osinga et al. 2010 ). In
addition, a marketing impulse today usually has an effect on future sales for various
reasons (see Sect. 19.3.3 again). Normative models that do not take into account
dynamic marketing and life cycle effects are likely to yield suboptimal results.
Narayanan and Manchanda ( 2009 ) develop a structural brand choice model that
allows physicians to learn about the quality of a new drug in a Bayesian manner (for
a related model see Crawford and Shum 2005 ). Specifically, they assume that physi-
cians are risk-averse and uncertain about the quality of a new drug. They have some
prior beliefs about the quality and are assumed to be Bayesian updaters, i.e., physi-
cians update their prior beliefs with new information they obtain through a detailing
call, for example, by using Bayes rule to generate posterior beliefs. Hence, com-
munication activities may have an informative effect because they contribute to
reduce the uncertainty in physician's decision-making. A specific feature of
Narayanan and Manchanda's model is that it allows for heterogeneous learning
rates across physicians.