Biomedical Engineering Reference
In-Depth Information
Assuming an international firm that operates across the world and sells several
products, Fischer et al. (
2011b
) attempt to solve the dynamic portfolio-optimization
problem. To promote the drugs management can use various marketing activities
that have short-term and long-term effects on sales. These effects are captured by a
marketing stock variable (Nerlove and Arrow
1962
). In addition, the authors assume
that drug sales follow a life cycle whose shape can be influenced by (early) market-
ing investments. Top management sets the total marketing budget
R
that is fixed
over the planning horizon
T
. But it can be adjusted during next year's planning
cycle. Consistent with the budgeting practice in many firms, budget allocation deci-
sions are made at the end of each year for the next year.
If the firm wishes to maximize the net present value of its product portfolio, Π,
the following constrained dynamic profit maximization problem needs to be solved:
(
)
∑
∑
pc
−
⋅
qETt
ki
+
,
sz
kK iI
∈
∈
ki
ki
ki
ki
ki
T
k
∫
−
rt
Max∏
=
e
Discounting
dt
Profit contribu
tion
unitsales
s
ki
∑
kK iI
∑
∑
x
t
=
0
−
kin
∈
∈
nN
∈
k
i
Marketingexpenditures
(19.15)
Discounted netvalue of product portfolic
dR
dt
∑
kK iI
∑
∑
(
)
subject toR
=
x
,
with
=
0
Budgetconstraint
∈
∈
nN kin
∈
(19.16)
k
i
dS
dt
(
)
kin
(19.17)
=−
d
Sx
+
,
with
x
≥
0
,
Statevariableequation
kin in
kin
kin
S
³
0
,
S
( )
0
=
S
,
and
S TS
()
=
. (
Boundaryconditions
)
(19.18)
kin
kin
kin
0
kin
kinT
Here,
k
denotes the country with the index set
K
and
i
denotes the product, whereas
the set of products offered in country
k
may vary and is given by
I
k
. The index
n
denotes the spending category and
N
i
is the associated index set that may vary across
products.
S
ki
is an
N
i
-dimensional row vector summarizing the activity-speciic mar-
keting stocks for product
i
.
ET
measures the elapsed time since launch of a product
in
t
= 0,
r
is a discount rate, 0 <
r
<∞,
p
denotes price,
c
is marginal cost,
q
measures
sales,
Z
is a vector of covariates, and
x
n
denotes activity-specific marketing expen-
ditures. Using the calculus of variations together with the Lagrange approach, the
following solution for the optimal budget can be derived:
{
}
( )
Max
wt
*
,
0
( )
≅
kin
*
xt
R
,
{
}
kin
∑
∑∑
( )
*
Max
wt
,
0
lk jh mN
∈
∈
∈
ljm
j
∈∈∈
[
]
∀∈
kK
iInNt T
k
,
,
0
,
,
(19.19)
i
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