Goodwill

Math Meth Oper Res (2002) 55 : 55–68

aaaaa 2002

New product introduction: goodwill, time and advertising cost* Alessandra Buratto1, Bruno Viscolani2 1 Universita` di Padova, Dipartimento di Matematica Pura e Applicata, Via Belzoni, n. 7, I-35131 Padova, Italy (e-mail: [email protected]) 2 Universita` di Padova, Dipartimento di Matematica Pura e Applicata, Via Belzoni, n. 7, I-35131 Padova, Italy (e-mail: [email protected]) Manuscript received: July 2001/Final version received: October 2001

Abstract. The advertising activity has an important role in the introduction of a new product. We tackle the problem of determining the advertising plan for preparing the product introduction, having the objectives of maximizing the product image at the launch time, minimizing the campaign length and minimizing the total advertising expenditure, within a fixed time interval. After formulating the new product introduction optimal control problem, we study and solve two special cases of it, namely the minimum time and the advertising cost problems. From the former we derive the existence of an optimal solution of the original problem, whereas we obtain its structure from the latter. Finally we present two alternative formulations of the original problem in the non-linear programming framework and suggest using them to solve the bounded time problem. Key words: Marketing, goodwill, new product introduction, optimal control, non-linear programming Mathematics Subject Classification 2000: Primary: 49N90; Secondary: 90B60. 1 Introduction The firm must plan carefully the di¤erent marketing actions which it has to take to introduce a new product in the market ([7], p. 307). Such coordinated actions must be performed before the sales start as well as after it. The firm communication has an important role to play, as one can observe for instance in the car industry, where an awareness advertising campaign usually precedes the distribution of a new car model. * Supported by MURST, CNR and University of Padua.

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Here we formulate and discuss the problem of determining the advertising policy to prepare the introduction of a new product into a market. We assume that the firm has three objectives: to maximize the product image (goodwill) at the launch time T, to minimize the launch time T, to minimize the total discounted advertising cost. The firm must determine the launch (or market entry) time T, at which the introduction of the product will begin, and plan the launch advertising campaign on the interval ½0; T (see [7], p. 333). We formulate an optimal control problem in which the goodwill evolves depending on the firm advertising policy, as in the so-called ‘‘capital stocks generated by advertising’’ models (see [5]). On the other hand, we do not take into account directly the sales, as they only begin after the launch time T, when the launch advertising policy stops. In fact, we assume that the launch advertising policy is a first part of a more complex advertising process: a second part of it starts at time T and pursues a di¤erent sales dependent objective. We focus only on the first part of the process, that is restricted to the pre-launch interval up to time T. We aim at obtaining some qualitative information on the optimal advertising policies and refer to a goodwill concept which is analogous to the Nerlove-Arrow’s one (see [9]). The first objective of maximizing the goodwill at the launch time is motivated by the firm taking into account the ‘‘communication e¤ects of advertising’’ on the consumer behavior, (see [3], p. 579). On the other hand, the e¤ects of the advertising policy until the time T on the sales, which will occur later, depend on the goodwil value at T: the greater the goodwill value at T, the higher the following sale rate, no matter which advertising policy will be chosen after T. The second objective of minimizing the launch time T reflects the importance of reducing the product time to market (see [7], p. 308). Time plays a crucial role in the product introduction problem, expecially for the ‘‘fast cycle’’ industries products [4] or for those products which are new both to the market and to the company, namely the ‘‘new product innovation’’ type ([8], p. 458), or the ‘‘new-to-the-world’’ type (see [7], p. 307), as some competitor could launch a similar product in advance. Yet, also for those types of products which are new only for the company, delaying the introduction entails the risk of losing revenues, as customers may prefer to buy a competitor’s product instead of waiting too much the announced one. Furthermore, the risk of a long-term loss in market share has to be considered, in particular when either the switching costs are high, or the quality of the competitors products is good, or the products di¤erentiation is low [6]. The third objective of minimizing the total discounted advertising expenditure is a quite natural e‰ciency requirement. In Section 2 we state the optimal control problem of the introduction of a new product, under the assumption that a single function of the three objective components, i.e. the final goodwill, the total discounted advertising cost and the final (launch) time, can represent the firm global objective. In Sections 3 and 4 we analyse two special problems: the objective of the first one is that of minimizing the launch time, whereas the objective of the second one is that of minimizing the discounted advertising cost for the introduction of the product at a given time. In Section 5 we present two alternative formulations of the original problem in the non-linear programming framework and suggest one of them as the way to solve the bounded time problem.

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2 Introduction of a new product: an optimal control problem Let kðtÞ be the goodwill level at time t, i.e. the product image by the consumers, let wðtÞ be the advertising expenditure intensity and let yðtÞ be the total discounted advertising expenditure up to time t, at the discount rate r > 0. After assuming that a) the goodwill kðtÞ depends on the expenditure rate through a special linear di¤erential equation, b) the firm image level at time 0 contributes to determine the initial product image, and hence the product may have a positive goodwill level kð0Þ ¼ k 0 b 0, c) a known level kT of goodwill must be reached in order to introduce profitably a given product in the market, d) the success of the product introduction depends on the time of the launch T and on the final goodwill kðTÞ, e) the time of the launch T is constrained to belong to a closed interval T J ½0; þy½, either bounded or unbounded, we are led to formulate the problem of determining the advertising policy for the product introduction and the launch time as the following optimal control problem I ðkT ; TÞ: maximize

ð1Þ

SðkðTÞ; T; yðTÞÞ;

subject to k_ ðtÞ ¼ bw a ðtÞ  ckðtÞ;

ð2Þ

y_ ðtÞ ¼ ert wðtÞ;

ð3Þ

kð0Þ ¼ k 0 ;

ð4Þ

kðTÞ b kT ;

yð0Þ ¼ 0;

ð5Þ

wðtÞ A ½0; w;

ð6Þ

T A T:

ð7Þ

We assume that the objective function S : ½0; þy½ 3 ! R represents the utility obtained from the introduction of the product and is a continuously di¤erentiable function, such that qSðk; t; yÞ b 0; qk

qSðk; t; yÞ a 0; qt

qSðk; t; yÞ a 0: qy

ð8Þ

These conditions require the utility to be increasing in the goodwill and to be decreasing in the final time and in the total communication cost. As for the admissibility conditions (2)–(7), we assume that

. k is the initial goodwill level, 0 a k < k ; . ckðtÞ is the goodwill spontaneous decay term, c > 0; 0

0

T

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A. Buratto, B. Viscolani

. bw ðtÞ is the advertising production function in terms of goodwill, b > 0, 1; . 0w 0. a

We observe that the goodwill evolution is described by a di¤erential equation which is a generalization of the linear motion equation proposed by Nerlove and Arrow [9]. Here we have a non-linear advertising productivity in terms of goodwill and the parameter a allows to represent a variety of goodwill production possibilities. In the limit, as a approaches 1, we find the Nerlove-Arrow goodwill motion equation. The constraint wðtÞ A ½0; w represents a general technological costraint, e.g. due to the communication channels capacity. In the following sections we discuss the problem by using an indirect method, which requires our studying first two special cases of the problem, to obtain some relevant information, and then reformulate the original problem as a non-linear programming problem. The special problems we study first are those of introducing a new product in minimum time and at minimum discounted cost respectively. The non-linear programming problem we obtain has two decision variables, namely the goodwill level at the introduction (launch) time and the launch time. This indirect approach is useful in order to avoid the complexity of the system of the Pontryagim Maximum Principle conditions and of the naturally resulting representation of the optimal solution. An idea of the optimal solutions representation complexity is given by the discussion of the optimal solution for a similar, but much simpler problem in [1] and [2]. 3 Introducing a new product in minimum time The minimum time problem It ðk Þ is defined, for k > k 0 , as minimize

T;

subject to

ð2Þ; ð6Þ;

and to

kð0Þ ¼ k 0 ; T b 0:

ð9Þ

kðTÞ b k;

ð10Þ ð11Þ

The minimum time problem It ðk Þ is equivalent to the problem I ðk; ½0; þy½Þ with the special objective function Sðk; t; yÞ ¼ t. Let us call sustainable goodwill the parameter ks ¼

bw a ; c

ð12Þ

which is in fact the least upper bound of the reachable goodwill values, under the assumptions of the problem I ðkT ; ½0; þy½Þ and the additional condition k 0 < ks . Then we can prove that

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i) the problem It ðk Þ has an optimal solution if and only if k < ks ; ii) the optimal value function of problem It ðk Þ, i.e. the minimum time, is Tmin ðk Þ ¼


 1 ks  k 0 ln ; c ks  k

k 0 < k < ks ;

ð13Þ

iii) the (unique continuous) optimal control of problem It ðk Þ is wðtÞ 1 w, t A ½0; Tmin . Hence we obtain a characterization of the feasibility of the original problem I ðkT ; TÞ. Theorem 1. A feasible solution for the problem I ðkT ; TÞ exists if and only if ks > kT

and

sup T b Tmin ðkT Þ:

Proof. Let the conditions hold and let us define T1 ¼ maxfmin T; Tmin ðkT Þg and 8 > w; > > <  ^ ðtÞ ¼
w ckT 1=a > > > ; : b

t A ½0; Tmin ðkT Þ; t A ½Tmin ðkT Þ; T1 ;

^ðtÞ, t A ½0; T1  is a feasible control function which determines a state then w ^ ðtÞ; T1 Þ is function ^ k ðtÞ such that ^ k ðT1 Þ ¼ kT . Therefore the triplet ð^k ðtÞ; w feasible. On the other hand, if the condition did not hold, then every feasible control would lead to a non feasible state function, with kðtÞ < kT for all t. r We observe that the minimum time Tmin ðkÞ is a 1-1 function, and its inverse, k max ðtÞ ¼ ks  ðks  k 0 Þect ;

ð14Þ

is the maximum reachable value of the goodwill at the time t. Corollary 1. A feasible solution for the problem I ðkT ; TÞ exists if and only if either kT a sup kmax ðTÞ ¼ kmax ðmax TÞ;

if T is bounded;

kT < sup kmax ðTÞ ¼ lim kmax ðtÞ;

if T is unbounded:

or t!þy

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Symmetrically, if k 0 > 0, solving the minimum time problem for k A 0; k 0  gives the decay time to the goodwill k,
 1 k0 Td ðk Þ ¼ ln ð15Þ ; 0 < k a k 0; c k and its inverse, the minimum reachable value of the goodwill at the time t, is k min ðtÞ ¼ k 0 ect :

ð16Þ

4 Introducing a new product at minimum discounted cost Let us denote by Iy ðk; TÞ the problem of minimizing the discounted advertising cost to reach at least the goodwill k at the fixed time T, which we call the advertising cost problem, ðT minimize

ert wðtÞ dt;

ð17Þ

0

subject to and to

ð2Þ; ð6Þ kð0Þ ¼ k 0 ;

kðTÞ b k;

ð18Þ

where T > 0 is fixed. The problem Iy ðk; TÞ just introduced is equivalent to the problem I ðk; TÞ with the special objective function Sðk; t; yÞ ¼  y, in the sense that a quadruple ðk  ðtÞ; y  ðtÞ; w  ðtÞ; TÞ is an optimal solution of the latter, where the optimal value of the objective functional (1) is y  ðTÞ, if and only if ðk  ðtÞ; w  ðtÞÞ is an optimal solution of the former and the optimal value of the objective functional (17) equals y  ðTÞ. We will denote by Y ðk; TÞ the optimal value function of problem Iy ðk; TÞ and call it the minimum discounted cost function. The minimum discounted cost function Y ðk; tÞ is defined for all ðk; tÞ such that k a k max ðtÞ, because of Corollary 1; moreover, Y ðk; tÞ ¼ 0 for all k a k min ðtÞ, because the zero control wðtÞ 1 0 is an optimal solution of the problem Iy ðk; TÞ if and only if k a k min ðTÞ. Lemma 1. Let T > 0 be fixed; the continuously di¤erentiable control function wðtÞ ¼ weððcþrÞ=ð1aÞÞðtTÞ ;

t A ½0; T;

ð19Þ

determines an admissible solution ðkðtÞ; wðtÞÞ for the problem Iy ðk; TÞ, if and only if k a ^ k ðTÞ, where ^ k ðTÞ ¼ k 0 ecT þ ^ ky ð1  eððcþarÞ=ð1aÞÞT Þ:

ð20Þ

and ð1  aÞcks ^ : ky ¼ c þ ar

ð21Þ

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Proof. In fact, the function (19) is an admissible control for problem Iy ðk; TÞ and ^ k ðTÞ is the value, at time T, of the goodwill function which is associated to the control (19) through the motion equation (2) and the initial condition given in (18). r The advertising policy (19) is an increasing exponential function with the parameter ðc þ rÞ=ð1  aÞ and reaches the maximum feasible expenditure rate w at the final time T. We further observe that ^k ð0Þ ¼ k 0 and ^k ðtÞ ! ^ky as t ! y. Moreover, if 0 < k 0 < ks , then ^k 0 ð0Þ > 0 and ^k ðtÞ has a maximum at 1a ks ln . t¼ aðc þ rÞ k 0 Theorem 2. For all T > 0 and k A kmin ðTÞ; kmax ðTÞ, there exists t ¼ tðk; TÞ b 0, such that the optimal control of Iy ðk; TÞ is wt ðtÞ; t A ½0; T, where wt ðtÞ ¼ w minf1; eððcþrÞ=ð1aÞÞðttÞ g;

ð22Þ

the function t ¼ tðk; TÞ is continuous, strictly decreasing in k and tðk; TÞ ! þy

as k ! kmin ðTÞ

ð23Þ

tð^ k ðTÞ; TÞ ¼ T;

ð24Þ

tðkmax ðTÞ; TÞ ¼ 0;

ð25Þ

furthermore, t ¼ tðk; TÞ is continuously di¤erentiable at all points ðk; TÞ 0 ð^ k ðTÞ; TÞ. Proof. In the special case that k ¼ k max ðTÞ, we notice that there exists a unique (continuous) feasible solution, which is determined by the control funcion wðtÞ 1 w. Of course it is the optimal solution and has the form (22) with t ¼ 0, so that tðk max ðTÞ; TÞ ¼ 0. Then, let us consider the problem Iy ðk; TÞ with the assumption that k min ðTÞ < k < k max ðTÞ. The ‘‘current value’’ Hamiltonian function is H c ðk; w; pÞ ¼ p0 w þ p½bw a  ck and if ðk  ðtÞ; w  ðtÞÞ is an optimal solution, then it satisfies the Pontryagin’s Maximum Principle necessary conditions i) ii) iii) iv) v)

ð p0 ; pðtÞÞ 0 ð0; 0Þ, for all t, w  ðtÞ A arg maxfp0 w þ bpðtÞw a j w A ½0; wg, for all t, p_ ðtÞ ¼ ðc þ rÞ pðtÞ, a.e., p0 A f0; 1g, pðTÞ b 0, pðTÞðk  ðTÞ  k Þ ¼ 0.

From (iii) we obtain that the adjoint function is pðtÞ ¼ pðTÞeðcþrÞðtTÞ and hence, from (iv), (v) and (i), that a feasible solution to the necessary conditions only exists with p0 ¼ 1, pðTÞ > 0 and k  ðTÞ ¼ k. From the condition (ii) we have that

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A. Buratto, B. Viscolani

w  ðtÞ ¼ minfw; ðabpðTÞeðcþrÞðtTÞ Þ 1=ð1aÞ g; and we notice that w  ðtÞ ¼ wt ðtÞ provided that t¼T

1 abpðTÞ ln 1a : cþr w

Let kt ðtÞ, t b 0, be the solution of equation (2) using the control wt ðtÞ and the initial condition kt ð0Þ ¼ k 0 , then we obtain that ( kt ðtÞ ¼

ky ect eððaÞ=ð1aÞÞðcþrÞt ðeððcþarÞ=ð1aÞÞt  1Þ; k 0 ect þ ^ ð^ k ðtÞ  ks ÞecðttÞ þ ks ;

t a t, t > t.

ð26Þ

We observe that ðwt ðtÞ; kt ðtÞÞ, t A ½0; T, is an admissible solution of the advertising cost problem which satisfies the transversality condition (v) if and only if kt ðTÞ ¼ k, i.e. if and only if fðk; t; TÞ ¼ kt ðTÞ  k ¼ 0. The function fðk; t; tÞ is continuously di¤erentiable at all ðk; t; tÞ such that t 0 t. Moreover, for all ðk; tÞ, the function fðk; t; tÞ is strictly decreasing in t and has range k min ðtÞ  k; k max ðtÞ  k; therefore, there exists a function tðk; tÞ, which is the unique solution to the equation fðk; t; tÞ ¼ 0. From the definition (20) and the meaning of ^k ðtÞ given in the proof of Lemma 1, we observe that tð^k ðtÞ; tÞ ¼ t. Moreover, we observe that the implicit function theorem assumptions hold at all solutions ðk 0 ; t 0 ; t 0 Þ of fðk; t; tÞ ¼ 0 such that t 0 0 t 0 , i.e. such that k 0 0 ^k ðtÞ: hence tðk; tÞ is continuous at all ðk; tÞ and continuously di¤erentiable for k 0 ^k ðtÞ. Furthermore, tðk; tÞ is strictly decreasing in k and satisfies the conditions (23)–(25). The unique solution we have obtained satisfies the assumptions of the Mangasarian su‰ciency theorem ([10], Th. 4, p. 105). r We call phase the parameter t of the advertising policy wt , defined in (22). Therefore, we will call optimal phase the function tðk; tÞ which is defined implicitly by the equation fðk; t; tÞ ¼ 0. The function ^ k ðtÞ has the meaning of maximum goodwill value in ½0; t, when using an increasing exponential control function with ‘‘optimal’’ parameter ðc þ rÞ=ð1  aÞ. Remark. If k min ðtÞ < k a ^k ðtÞ, then we can solve explicitly the equation fðk; t; tÞ ¼ 0 for the optimal phase function and obtain
 1a eððcþarÞ=ð1aÞÞt  1 ^ ln ky tðk; tÞ ¼ : aðc þ rÞ ke ct  k 0

ð27Þ

In this case the optimal control is in fact an increasing exponential function with ‘‘optimal’’ parameter ðc þ rÞ=ð1  aÞ. Lemma 2. If k 0 ^ k ðtÞ, then qtðk; tÞ < 0; qk

for all t;

ð28Þ

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63

and qtðk; tÞ > 0; qt

for all k > ^k ðtÞ;

ð29Þ

moreover, for all k < ks , lim tðk; tÞ ¼ þy:

t!þy

ð30Þ

Proof. From (26) we obtain that qfðk; t; tÞ < 0; qt

for all k; t; t;

and qfðk; t; tÞ > 0; qt

for all t; t and for k > ^k ðtÞ:

Hence (28) and (29) are consequences of the implicit function theorem as applied to the equation fðk; t; tÞ ¼ 0. Now, if k min ðtÞ < k a ^ k ðtÞ, then (30) follows from (27). On the other hand, if ^ k ðtÞ < k < k max ðtÞ, then the limit limt!þy tðk; tÞ exists, because tðk; tÞ is a monotonically increasing function of t for t > t. If (30) did not hold, i.e. if the limit were finite, then the equation fðk; t; tÞ ¼ 0 would not hold in the limit as t ! þy. r We observe that the discounted advertising cost over the interval ½0; t, which is associated with the control function wt ðtÞ, defined in the statement of Theorem 2, is 8 1a > > weððcþrÞ=ð1aÞÞt ðeððcþarÞ=ð1aÞÞt  1Þ; t a t, > < c þ ar ð31Þ zðt; tÞ ¼ > 1a w rt > rt ððcþrÞ=ð1aÞÞt rt > : wðe  e Þ þ ðe  e Þ; t > t. c þ ar r The discounted interval advertising cost function zðt; tÞ is a continuous function, strictly monotonically increasing in t for all t and strictly monotonically decreasing in t for all t. Moreover, zðt; tÞ is di¤erentiable at all points such that t 0 t. Therefore, the optimal value of the advertising cost problem Iy ðk; tÞ, i.e. the minimum discounted cost function Y ðk; tÞ, may be obtained from (31), for k > k min ðtÞ, by substituting the variable t with the function tðk; tÞ, provided by Theorem 2. To this end it has to be noted that t a tðk; tÞ , k a ^ k ðtÞ:

ð32Þ

In the following corollaries we state some general results on the minimum discounted cost function.

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Corollary 2. The minimum discounted cost function 0; k ¼ kmin ðtÞ, Y ðk; tÞ ¼ zðtðk; tÞ; tÞ; kmin ðtÞ < k a kmax ðtÞ, i) ii) iii) iv)

ð33Þ

is continuous, is strictly monotonically increasing in k, is strictly monotonically decreasing in t, for k > ^k ðtÞ, has the limit lim Y ðk; tÞ ¼ 0;

t!þy

for all k < ks ;

ð34Þ

v) is di¤erentiable at all points such that k 0 ^k ðtÞ. Proof. The continuity of Y ðk; tÞ follows from the continuity of zðt; tÞ and tðk; tÞ. The monotonicity properties (ii) and (iii) hold because the discounted interval advertising cost function zðt; tÞ is strictly decreasing in t, for which the previous Lemma holds. The limit of Y ðk; tÞ follows from the limit of tðk; tÞ provided by the previous Lemma. The di¤erentiability statement (v) follows from the di¤erentiability of zðt; tÞ, for t 0 t, and of tðk; tÞ, for k 0 ^k ðtÞ. r We recall, in particular, that tðk max ðtÞ; tÞ ¼ 0 and hence Y ðk max ðtÞ; tÞ ¼ zð0; tÞ ¼

w ð1  ert Þ: r

ð35Þ

A remarkable consequence of the above Corollary is that the problem of minimizing the discounted advertising cost with free final time has not any optimal solution. Corollary 3. The minimum discounted cost function Y ðk; tÞ is strictly convex in k for all t > 0. Proof. From Corollary 2 and Theorem 1 it follows that the function Y ðk; tÞ is continuously di¤erentiable in k, as k 0 ^k ðtÞ. Then the sensitivity results for the optimal value function and the adjoint function give qY ðk; tÞ=qk ¼ pðtÞ ([10], p. 213). We observe that pðtÞ can be written as pðtÞ ¼

w 1a ðcþrÞðttðk; tÞÞ e ; ab

it is a function of ðk; tÞ (some function compound with tðk; tÞ) which is strictly increasing in k as t is strictly decreasing in k (Lemma 2). r 5 Solution of the bounded time problem Let us consider the new product introduction problem I ðkT ; TÞ with a bounded time interval: T ¼ ½T1 ; T2 , 0 a T1 a T2 . A natural and correct way to solve the problem I ðkT ; ½T1 ; T2 Þ is to use the Pontryagin Maximum Prin-

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65

ciple necessary conditions, as we have done for the minimum time problem It ðk Þ and the advertising cost problem Iy ðk; TÞ. Nevertheless, most of the relevant information on the optimal solution of problem I ðkT ; ½T1 ; T2 Þ, which is obtainable from the Pontryagin conditions, has in fact been used to determine the solutions to the problems It ðk Þ and Iy ðk; TÞ. Therefore we can exploit the results of the analysis of Sections 3 and 4 in order to simplify the discussion of the bounded time problem I ðkT ; ½T1 ; T2 Þ. In fact we argue in the present Section that the original optimal control problem can be restated as a non-linear programming problem and then solved by using the appropriate necessary conditions and, possibly, some suitable numerical algorithms. A first equivalence result is provided by the following theorem which exploits the knowledge of the minimum discounted cost function Y ðk; tÞ. Theorem 3. The bounded time optimal control problem I ðkT ; ½T1 ; T2 Þ is equivalent to the non-linear programming problem I  ðkT ; ½T1 ; T2 Þ: maximize

f ðk; TÞ ¼ Sðk; T; Y ðk; TÞÞ;

subject to k A ½kT ; kmax ðTÞ; T A ½T1 ; T2 ;

ð36Þ ð37Þ ð38Þ

in the sense that a) if the problem I ðkT ; ½T1 ; T2 Þ has an optimal solution ðw  ðtÞ; k  ðtÞ; y  ðtÞÞ with final time T  , then ðk  ðT  Þ; T  Þ is an optimal solution to the problem I  ðkT ; ½T1 ; T2 Þ and f ðk  ðT  Þ; T  Þ ¼ Sðk  ðT  Þ; T  ; y  ðT  ÞÞ; b) if the problem I  ðkT ; ½T1 ; T2 Þ has an optimal solution ðk  ; T  Þ, then the optimal solution ðw  ðtÞ; k  ðtÞÞ to the minimum cost problem Iy ðk  ; T  Þ determines, through the equation (3) and the initial condition (5), the optimal solution ðw  ðtÞ; k  ðtÞ; y  ðtÞÞ to the problem I ðkT ; ½T1 ; T2 Þ. Proof. a) Let ðw  ðtÞ; k  ðtÞ; y  ðtÞÞ be an optimal solution of I ðkT ; ½T1 ; T2 Þ with final time T  , then y  ðT  Þ ¼ Y ðk  ðT  Þ; T  Þ, because of the definition of Y ðk; tÞ and of the fact that S is decreasing in y. Moreover, Sðk  ðT  Þ; T  , y  ðT  ÞÞ b Sðk; t; yÞ, for all ðk; t; yÞ which is the final goodwill, time and discounted advertising cost triplet associated with an admissible solution of I ðkT ; ½T1 ; T2 Þ; in particular, Sðk  ðT  Þ; T  ; y  ðT  ÞÞ b Sðk; t; Y ðk; tÞÞ, for all ðk; tÞ which are feasible solutions to I  ðkT ; ½T1 ; T2 Þ. b) Let ðk  ; t  Þ, be an optimal solution of I  ðkT ; ½T1 ; T2 Þ and let  ðw ðtÞ; k  ðtÞÞ be an optimal solution of Iy ðk  ; t  Þ. The optimal value of the objective functional of the latter problem is Y ðk  ; t  Þ ¼ y  ðt  Þ, where y  ðtÞ is the function which is obtained from the integration of the equation (3) with the initial condition (5). We have that Sðk  ðt  Þ; t  ; y  ðt  ÞÞ ¼ f ðk  ; t  Þ ¼ Sðk  ; t  ; Y ðk  ; t  ÞÞ b Sðk; t; Y ðk; tÞÞ, for all ðk; tÞ which are feasible solutions of I  ðkT ; ½T1 ; T2 Þ. Hence, in view of the definition of Y ðk; tÞ, it follows that Sðk  ðt  Þ; t  ; y  ðt  ÞÞ b Sðk; t; yÞ, for all ðk; t; yÞ ¼ ðkð^t Þ; ^t; yð^t ÞÞ which is the final goodwill, time and discounted advertising cost triplet associated with some admissible solution of I ðkT ; ½T1 ; T2 Þ. r

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An immediate consequence is the following existence result. Corollary 4. If the admissibility condition kT a kmax ðT2 Þ is satisfied, then the bounded time problem has an optimal solution. Proof. The problem I  ðkT ; ½T1 ; T2 Þ has an optimal solution, because it has a continuous objective function f ðk; TÞ and a nonempty and compact feasible set. Hence, the thesis follows from the equivalence between problems stated by the Theorem 3. r A second equivalence result is provided by the following theorem which exploits the knowledge of the discounted interval cost function zðt; tÞ and of the goodwill function kt ðtÞ, that are associated with an advertising expenditure rate function wt ðtÞ of the form (22). Theorem 4. Let I~ðkT ; ½T1 ; T2 Þ denote the non-linear programming problem maximize

gðt; TÞ ¼ Sðkt ðTÞ; T; zðt; TÞÞ;

subject to kt ðTÞ b kT ;

ð39Þ ð40Þ

t b 0;

ð41Þ

T A ½T1 ; T2 ;

ð42Þ

then the following implications hold: a) if wðtÞ 1 0 is an optimal control for the problem I ðkT ; ½T1 ; T2 Þ, then the problem I~ðkT ; ½T1 ; T2 Þ has not any optimal solution; b) if the problem I ðkT ; ½T1 ; T2 Þ has an optimal solution ðw  ðtÞ; k  ðtÞ; y  ðtÞÞ with final time T  and phase t  , then ðt  ; T  Þ is an optimal solution to the problem I~ðkT ; ½T1 ; T2 Þ and kt  ðT  Þ ¼ k  ðT  Þ, zðt  ; T  Þ ¼ y  ðT  Þ; 0 a ) if the problem I~ðkT ; ½T1 ; T2 Þ has no optimal solution, then the problem I ðkT ; ½T1 ; T2 Þ has the optimal control wðtÞ 1 0; b 0 ) if the problem I~ðkT ; ½T1 ; T2 Þ has an optimal solution ðt  ; T  Þ, then w  ðtÞ ¼ wt  ðtÞ is an optimal control for the problem I ðkT ; ½T1 ; T2 Þ. Proof. The components of a feasible solution ðt; TÞ of the problem I~ðkT ; ½T1 ; T2 Þ are the final time T and the phase t of a control wt ðtÞ of the form (22), which determine the feasible goodwill function kt ðtÞ for the problem I ðkT ; ½T1 ; T2 Þ. Moreover, the value of the I~ðkT ; ½T1 ; T2 Þ objective function at ðt; TÞ is the value of the I ðkT ; ½T1 ; T2 Þ objective functional at the solution ðwt ðtÞ; kt ðtÞÞ with final time T. If I~ðkT ; ½T1 ; T2 Þ has an optimal solution ðt; TÞ, it is associated with an optimal solution of I ðkT ; ½T1 ; T2 Þ under the requirement that the control has the form (22). On the other hand, if I ðkT ; ½T1 ; T2 Þ has an optimal solution

New product introduction

67

ðw  ðtÞ; k  ðtÞ; y  ðtÞÞ with final time T  and final goodwill k  ðT  Þ, then it must be an optimal solution of the advertising cost problem Iy ðk  ðT  Þ; T  Þ, so that it must have the form (22), because of Theorem 2. The control wðtÞ 1 0 may be optimal for the problem I ðkT ; ½T1 ; T2 Þ, if kT a k min ðT1 Þ: in such a case the optimal control function has not the form (22) for any real t and the problem I~ðkT ; ½T1 ; T2 Þ has not any optimal solution. If kT > k min ðT1 Þ, then the problem I ðkT ; ½T1 ; T2 Þ has an optimal solution with control w  ðtÞ ¼ wt  ðtÞ, for some t  b 0: then the problem I~ðkT ; ½T1 ; T2 Þ has a bounded feasible set and has the optimal solution ðt  ; T  Þ. If kT a k min ðT1 Þ, then either the problem I ðkT ; ½T1 ; T2 Þ has the optimal control w  ðtÞ 1 0, and then T  ¼ T1 and k  ðT1 Þ ¼ k min ðT1 Þ, or I ðkT ; ½T1 ; T2 Þ has the optimal control is w  ðtÞ ¼ wt  ðtÞ, for some t  b 0: in the first case the problem I~ðkT ; ½T1 ; T2 Þ has not any optimal solution, whereas in the second case it has an unbounded feasible set, but has the optimal solution ðt  ; T  Þ. r We obtain in particular that the fixed time new product introduction problem I ðkT ; fTgÞ has an optimal solution, as far as kT a k max ðTÞ. Such a problem is characterized by an exogenously given launch time and represents e.g. those cases in which the product has to be introduced in the market on the occasion of an exhibition or other special event. From the Theorems 3 and 4 we can derive immediately two special representations of the problem I ðkT ; fTgÞ as a 1-dimensional non-linear programming problem. Corollary 5. The fixed time new product introduction problem I ðkT ; fTgÞ is the problem of maximizing the function F ðkÞ ¼ f ðk; TÞ on the bounded interval ½kT ; kmax ðTÞ; it has an optimal control w  ðtÞ and i) either w  ðtÞ 1 0, ii) or w  ðtÞ ¼ wt  ðtÞ, for some t  b 0. In the latter case t  is a maximum point of the function GðtÞ ¼ gðt; TÞ, ii.1) either on the bounded interval ½0; tðkT ; TÞ, if kmin ðTÞ < kT a kmax ðTÞ, ii.2) or on the unbounded interval ½0; þy½, if kT a kmin ðTÞ. We recall that the objective functions f ðk; TÞ and gðt; TÞ of the nonlinear programming problems I  ðkT ; ½T1 ; T2 Þ and I~ðkT ; ½T1 ; T2 Þ are continuous and so are F ðkÞ and GðtÞ from Corollary 5. The functions f ðk; TÞ and gðt; TÞ are also continuously di¤erentiable except at the points ð^k ðTÞ; TÞ, the first one, and ðt; tÞ, the second one. No general monotonicity or concavity properties of the objective functions hold: therefore we cannot state a priori any su‰ciency or uniqueness result for the optimal solutions. The feasible set of probem I  ðkT ; ½T1 ; T2 Þ is compact and convex, whereas the one of problem I~ðkT ; ½T1 ; T2 Þ may be compact and convex as well as not. As for the I ðkT ; fTgÞ-equivalent 1-dimensional problems, we observe that the objective functions F ðkÞ and GðtÞ are continuously di¤erentiable except at the points ^ k ðTÞ, the first one, and T, the second one. In general we cannot state whether F ðkÞ and GðtÞ are concave or not.

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6 Conclusion We have considered the control problem of determining an optimal advertising policy for preparing the launch of a new product. The objective functional depends on the goodwill level at the launch time, the launch time and the total discounted advertising cost. The solutions of two special instances, namely the minimum time problem and the minimum discounted advertising cost problem, allow us to represent the original problem as a non-linear programming one. In fact, we have obtained two di¤erent non-linear programming representations of the original control problem. From them, on one hand, we have proved the existence of an optimal solution of the bounded time control problem. On the other hand, the non-linear programming problems are the natural way to find an optimal solution, by using some suitable numerical method. A numerical analysis of the optimal solutions for special instances of the utility function Sðk; t; yÞ is the natural prosecution of the present research. Here, two real parameters allow the model to represent a variety of goodwill dynamics, but di¤erent goodwill motion equations might be considered inside the same general framework. On the other hand, certain real life advertising practices might be better represented by a discrete time model. A relevant extension of the work should concern the advertising campaign for the entire launch time period and not only for the one which precedes immediately the sales. This would lead to a two phase problem in which, at the launch time T, the further advertising policy should be planned taking into account both the foreseeable sales and the goodwill obtained in the previous period.

Acknowledgements. We thank Francesco Casarin, for some useful management comments, and Paolo Pellizzari, for an illuminating observation concerning the non-linear programming representation of the new product introduction problem.

References 1. Buratto A, Viscolani B (1994) An optimal control student problem and a marketing counterpart. Mathematical and Computer Modelling 20:19–33 2. Buratto A, Viscolani B (1996) Errata to: An optimal control student problem and a marketing counterpart. Mathematical and Computer Modelling 23:123 3. Collesei U (1994) Marketing. CEDAM, Padova 4. Datar S, Jordan C, Kekre S, Rajiv S, Srinivasan K (1997) New product development structures and time to market. Management Science 43:452–464 5. Feichtinger G, Hartl RF, Sethi SP (1994) Dynamic optimal control models in advertising: recent developments. Management Science 40:195–226 6. Hendricks KB, Singhal VR (1997) Delays in new product introductions and the market value of the firm: the consequences of being late to the market. Management Science 43:129–142 7. Kotler P (1997) Marketing Management: Analysis, Planning, Implementation, and Control. Prentice-Hall, Upper Saddle River 8. Lilien GL, Kotler P, Moorthy KS (1992) Marketing models. Prentice-Hall, Englewood Cli¤s 9. Nerlove M, Arrow KJ (1962) Optimal advertising policy under dynamic conditions. Economica 29:129–142 10. Seierstad A, Sydsaeter K (1987) Optimal control theory with economic applications. NorthHolland, Amsterdam