ation channels that can be divided into two classes: the first includes all means of communication before described and which are directly controlled by the organizers while the second includes those communication activities that are not directly controlled by the organizers. In this latter category there is the verbal communication (the word-of-mouth) that can definitely be considered as an excellent communication tool as it is able to emotionally involve the supposed beneficiaries of the event. According Godes & Mayzlin  , the word-of-mouth communication “is often an important driver of consumer buying behavior such as the adoption of a new technology, the decision to watch a TV show, or the choice of which laptop to purchase”.
The problem is solved formulating a model that describes the situation and using the theory of stochastic control, in particular, a cost function and a quadratic penalty function. The results obtained allow to compare the effectiveness of advertising channels and to establish how the risk connected with the channel stochastic influences its use; besides, even if the cost of the channel stochastic become null, the problem continues to be well placed because the risk associated with the action of the decision can be interpreted as a additional cost.
The presence of the stochastic control makes the problem mathematically interesting and different from others. For the generality of the problem it is necessary to study some sub-problems which are both simple from a mathematical point of view and relevant for their economic features.
3. An Optimal Advertising
In this section, the authors adopt a diffusion model where the organizers control their advertising expenditures and wish to maximize profit or better they consider a typical optimal control problem where an advertising intensity is looked for to optimize a suitable firm objective. The main objective is, then, to determine, using optimal control techniques, an optimal price or advertising rate over time.
The optimal advertising policy, besides being dependent on the cost and revenue, is also affected by the length of the planning period and by the relation between the number of seats and the total number of potential attendees. To this end, the authors consider an organization that has planned a social event (a concert, a workshop, a football), at a fixed time T, in a place with a limited number of seats and they suppose that, in order to stimulate as wide a participation as possible, has been organized an advertising campaign, that the customers pay the tickets only at time T and that the demand depends on the event goodwill at T. Furthermore, they take into consideration a word-of-mouth effect (this means that who buys a ticket talks about it to its friends), and therefore is unnecessary to inform the people with other advertising: newspapers, radio and television. Such advertising is, furthermore, costly. Instead, the word-of-mouth advertising, with which individuals who are aware of the event forward information about the event to individuals who are unaware, is a type of advertising free.
In the model that the authors present, the time t is a continuous variable and the planning period is equal to interval
where T represent the instant of time in which the event will take place; T is, therefore, fixed and unmovable  .
be the fraction of the number of potential attendees that have bought a ticket by time t. The time-derivate of
, henceforth denoted by
, represents the sales rate at time t and satisfies the following equation:
is the rate of advertising effort used by the organizers
represents the efficiency of the advertising of the organizers
represents the efficiency of the word-of-mouth communication.
) are time-independent which is a plausible assumption if the time horizon, as here, is short.
Putting in evidence
, the (1) becomes:
is a nondecreasing function of time and is upper delimited by one, which represents the normalized number of potential participants in the event. As
tends to one, the effect of the advertising decreases, because the number of participants in the concert is predetermined, as established by a careful market survey, performed before every organizational decision, from which it emerges the forecast of the number of potential attendees.
Since the number of seats is fixed and the number of potential attendees is normalized to 1, it is need to consider the state constraint
where r is the fraction of the number of potential attendees that can sit down and
. This is a plausible assumption, unless the stadium or concert hall or theater is very large and the event is utterly uninteresting.
, that is, the number of seats does not exceed the number of potential attendees.
Therefore, the objective organizers are to minimize:
1) The advertising costs
2) The number of unsold seats on the day the event takes place
In the particular case that is presented here, the advertising efforts are costly; consequently, the advertising costs are represented by a quadratic and convex function
and this is mainly a matter of mathematical convenience and is consistent with the choice done.
represents the total costs supported for advertising and it is equal to:
are constants. For
The implication is that, for a sufficiently large value of v, it can not pay to advertise at all, but there is a need to make informed choices. In fact, if this is true, no tickets will be sold at all without distinction because
is equal to zero. The value of w determines the magnitude of the advertising rate whenever it is positive. Therefore, higher will be the value of w, smaller will be the advertising rate.
Consequently, the function that must be minimized is:
is the total advertising cost incurred while
Therefore, the function to minimize becomes:
The problem that comes out is a problem of optimal control and the number of seats sold and the advertising effort of the organizers are, respectively, the state and control variables; this problem can be solved by using the stochastic control theory; more precisely, the optimal control problem can be solved using either the Pontryagin Maximum Principle or the Hamilton Jacobi Bellman Equation approach (the first leads to open-loop strategies which only depend on time while the second leads to closed-loop strategies which depend also on the state variables). To this end must be defined the Hamiltonian function (omitting for simplicity the time argument t):
Assuming the existence of an optimal solution, the Maximum Principle provides the necessary conditions of optimality (Seierrstad & Sydsaeter  ).
Exists, thus, a constant
and a differential function
such that for each
it holds that
. Consequently, the Lagrangian:
provides the following conditions of optimality (omitting for simplicity the time argument t):
It is readily shown that
. Hence the Hamiltonian is strictly concave in a and the maximization yields a unique control a.
Differentiating in (10) with respect to time, and using (1), (11), and (12) it is possible get
which is negative.
Thus, the advertising efforts are decreasing over time. In fact, from (14) readily follows
which means that advertising efforts decrease over time with an increasing rate.
This result is due to the word-of-mouth effect; the advertising is used to generate initial awareness of the event, but as soon as the word-of-mouth process gains momentum, there is less need for advertising.
In the case in which there is not a word-of -mouth effect (therefore
) the number of tickets sold,
, can be increased only by advertising. Using the (14) it is possible show that the advertising effort is constant in time. Therefore, in the absence of a word-of-mouth effect, it becomes optimal to achieve this objective a policy of advertising spending more expensive because it must fill, to ensure the success of event, the absence and the benefits brought about by word of mouth. In other words, in the absence of a word-of-mouth effect, the advertising efforts should actually be spread evenly over time.
An interesting variant of the problem concerns the introduction of a quadratic final penalty
describes the payoff obtained by the organizers of an event like a concert or a theatre performance. For such events the number of available seats is a crucial parameter. Moreover, it is possible to hypothesize a goodwill threshold
- if the final goodwill exceeds it, i.e. if
, then the demand is greater than the available seats, and there are some unsatisfied consumers;
- if the final goodwill is less than it, i.e. if
then the demand is less than the available seats, and some tickets remain unsold.
In the first case the organization suffers a loss of reputation, whereas in the latter a loss of revenue. The quadratic penalty function
is a symmetric representation of both these kinds of loss: it is a compromise between the analytical tractability of the problem and a precise description of the economic consequences of observing a demand level above or below the available seats.
4. Basic Model Advertising for the Launch of a Product
In this section, the authors study the planning of a pre-launch publicity campaign using the stochastic control theory and some recent results of the stochastic linear quadratic control theory. First of all it must be said that the firm must plan carefully the different marketing actions which it has to take to introduce a new product in the market; in other words, the firm must determine the launch time T (or market entry time), at which the introduction of the product will begin, and must plan the launch advertising campaign on the interval; the sales do not are take into account directly, as they only begin after the launch time T, when the launch advertising policy stops. On the other hand, the effects 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.
Consequently, the aim of this section is to determine an optimal advertising policy for preparing the launch of a new product, assuming that a firm can control, in the programming interval
, the goodwill evolution of a product through the advertising flow or through some other communication channel and that it wants to maximize the expected utility given by the product goodwill at the (fixed) launch time T and to minimize the total advertising cost.
The firm, namely, wants to drive the goodwill in order to obtain a final demand as close as possible to the congestion threshold minimizing the total advertising costs.
The most natural approach to the problem is the description of the goodwill evolution using the stochastic processes theory and this is not new in advertising models. To use the definition of the goodwill is a more promising analytical approach and has considerable intuitive appeal because summarizes the effects of current and past advertising outlays on demand.
The authors present, namely, a linear model of the goodwill evolution under advertising investment. They refer to the concept of goodwill, introduced by Nerlove and Arrow in  , which resumes the effects of advertising on the demand and whose evolution is described by a linear differential equation which is a generalization of the linear motion equation proposed by Nerlove and Arrow; this last one is a starting point for some practical and theoretical studies in marketing.
Nerlove and Arrow consider the advertising as an investment in a stock (the goodwill
), which summarizes the effects of current and past advertising flow
and assume that the evolution of the goodwill satisfies the first order linear differential equation:
is current advertising expense,
are understood to be functions of time, and the dot denotes differentiation with respect to time. Equation (15) states that the net investment in goodwill is the difference between the gross investment (current advertising outlay) and the depreciation of the stock of goodwill  .
is the goodwill level at time
, the organization image, as well as the event features, contribute to affect the initial goodwill of the event,
, which is therefore positive; namely:
In other words, if it is assumed further that current advertising expenditure cannot be negative and that depreciation occurs at a constant proportional rate,
The goodwill decreases spontaneously with decay coefficient
is always positive and is sustained by the advertising investment
Assuming, then, that the publicity campaign is short enough, it is possible suppose that the consumers do not forget the advertising messages, therefore the goodwill decay is negligible. Hence, the goodwill decay term in the Nerlove and Arrow’s model is assumed to be zero.
Consequently, the goodwill evolution, without any advertising flow, can be described by the following linear stochastic differential equation (SDE):
where the parameter
represents the advertising volatility,
is the marginal productivities in terms of goodwill while
is a standard Brownian motion.
In other words, the goodwill is the solution of a controlled linear stochastic differential equation.
The above stochastic differential equation shows that the weight of the word-of-mouth communication is proportional to the actual goodwill and that it affects the goodwill randomly. Actual consumers communicate their product experience randomly, either favorably or unfavorably. The advertising message introduces also an uncertainty source in the system, namely, it has an unforeseeable effect on the goodwill, because either it may not be either completely understood by the consumers, or it may completely meet the taste of the public.
In other words, the potential consumers react randomly to advertising because they can be attracted or repelled by the advertising message. This can be described assuming that the goodwill evolution is represented by the following linear stochastic differential equation:
are independent Wiener processes, while
represents the advertising volatility and describes the rate of uncertainty introduced by advertising. Under these assumptions, also the advertising flow becomes a stochastic process and therefore the problem of determining the advertising policy for the product introduction and the launch time can be described using the stochastic control theory; in other words, the problem of determining an advertising policy in order to have an optimal goodwill level at the final time T, when the goodwill is subject to a random evolution, is the following optimal control problem:
• the function
represents the cost of advertising that the firm has to sustain in order to obtain an advertising flow
and, according to literature, it is non-linear, increasing, and convex
• the function
represents the utility considered as a prospective profit given by a final level of goodwill
from the final goodwill and is homogeneous with the cost, it is twice continuously differentiable and increasing
; in particular, the function
describes an estimate of the expected revenue since all the utility obtained by the firm is concentrated at the end of the programming interval
is the decay coefficient
) is a constant which represents the goodwill level at the time 0, namely the initial value of the goodwill