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Optimal Control Theory and its Applications pp 301–337Cite as

Optimal Control Problems in Advertising

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 106))

Abstract

My assignment for this lecture is to discuss applications of optimal control theory to Management Science problems. Since the field of Management Science encompasses production, finance, and marketing as its main functional areas which themselves are rather vast, it will not be possible to review all the optimal control problems arising in the literature of Management Science in the next hour. For this reason, I am taking liberty to narrow the scope of my lecture to problems dealing with optimal advertising policy, an area of marketing which has received quite a bit of attention for applications of optimal control theory.1

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Abbreviations

*:

denotes the values on the optimal path

-:

denotes the instantaneous optimal levels denotes the desired levels

‘:

denotes differentiation with respect to the argument

.:

denotes time derivative

A:

stock of advertising capital (a state variable)

B:

denotes a particular brand

E:

consumer’s attitude

F(x(T),T):

salvage value function; F e-rtx(T) if linear

G(·):

gain operator

J:

value of the objective function to be maximized

L(·):

loss operator

M:

consumer’s motivation

P:

market power

Q:

upperbound on the rate of advertising effort

R:

rate of profit margin gross of advertising

S:

sales rate

T:

denotes horizon

U:

total advertising expenditure in dollars by all other firms in the industry, i.e., excluding the firm under consideration

Y:

exogenous variable denoting the total market demand

Z:

the exogenous variables a, a0, a1, b, d, e, are constants

c(S):

cost of producing at rate S; cS if linear

g(·):

rate of change of sales rate or captured market potential

h(x,ẋ):

cost of advertising expressed as a function of x and ẋ

k:

decay constant

m:

superscript m denotes maximum level

p:

price per unit (a control variable or an exogenous variable)

r:

rate of discounting

s:

denotes singular levels

s:

denotes singular levels

t:

denotes time

u:

rate of advertising effort by the firm (a control variable)

w(u):

cost of advertising at rate u; wu if linear

x:

fraction of the market potential captured

y:

fraction of the market potential captured for the second product

z:

a dummy variable

α, μ, ψ:

are constants

β:

elasticity of demand with respect to price

β(Q):

a parameter function

δ:

rate of depreciation for the stock of advertising capital

λ:

the adjoint variable

η:

elasticity of demand with respect to advertising capital

ϰ(·):

rate of profit margin gross of advertising; ϰX if linear

ρ:

response constant

ς:

elasticity of demand with respect to the exogenous variable Z

Π:

present value of total profits

References

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Author information

Authors and Affiliations

  1. Faculty of Management Studies, University of Toronto, Canada

    Suresh P. Sethi

Authors
  1. Suresh P. Sethi
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Editor information

Editors and Affiliations

  1. Mathematics Department, Queen’s University, Kingston, Ontario, Canada

    Bruce J. Kirby

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© 1974 Springer-Verlag Berlin · Heidelberg

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Sethi, S.P. (1974). Optimal Control Problems in Advertising. In: Kirby, B.J. (eds) Optimal Control Theory and its Applications. Lecture Notes in Economics and Mathematical Systems, vol 106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48290-8_14

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  • DOI: https://doi.org/10.1007/978-3-642-48290-8_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07026-9

  • Online ISBN: 978-3-642-48290-8

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