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Chapter 7 Applications to marketing

State Equation:

Sale expressed in terms of advertising (which is a

control variable)

Objective:

Profit maximization

Constraints:

Advertising levels to be nonnegative

The Nerlove-Arrow Advertising Model:

Let G(t) 0 denote the stock of goodwill at time t .

where is the advertising effort at time t

measured in dollars per unit time. Sale S is given by

Assuming the rate of total production costs is c(S), we

can write the total revenue net of production costs as

the revenue net of advertising expenditure is therefore

.

The firm wants to maximize the present value of net

revenue streams discounted at a fixed rate , i.e.,

subject to (7.1).

Since the only place that p occurs is in the integrand,

we can maximize J by first maximizing R with to price

p holding G fixed, and then maximize the result with

respect to u. Thus,

which implicitly gives the optimal price

Defining as the elasticity of demand

with respect to price, we can rewrite condition (7.5) as

which is the usual price formula for a monopolist,

known sometimes as the Amoroso-Robinson relation.

In words, the formula means that the marginal revenue

must equal the marginal cost . See,

e.g., Cohen and Cyert (1965, p.189).

Defining , the objective function in

(7.4) can be rewritten as

For convenience, we assume Z to be a given constant

and restate the optimal problem which we have just

formulated:

Solution by the Maximum Principle

The adjoint variable (t) is the shadow price

associated with the goodwill at time t . Thus, the

Hamiltonian in (7.8) can be interpreted as the dynamic

profit rate which consist of two terms: (i) the current

net profit rate and (ii) the value

of the new goodwill created by advertising at rate u.

Equation (7.9) corresponding to the usual

equilibrium relation for investment in capital goods:

see Arrow and Kurz (1970) and Jacquemin (1973). It

states that the marginal opportunity cost of

investment in goodwill should equal the sum of the

marginal profit from increased goodwill and

the capital gain:

Defining as the elasticity of demand

with respect to goodwill and using (7.3), (7.5), and

(7.9), we can derive ( see exercise 7.3)

We use (3.74) to obtain the optimal long-run stationary

equilibrium or turnpike . That is, we obtain

from (7.8) by using . We then set

and in (7.9). Finally, from (7.11) and

(7.9), or also the singular level can be obtained

as

The property of is that the optimal policy is to go to

as fast as possible. If , it is optimal to jump

instantaneously to by applying an appropriate

impulse at t =0 and then set for t >0. If

, the optimal control u*(t)=0 until the stock of

goodwill depreciates to the level , at which time

the control switches to and stays at this level

to maintain the level of goodwill. See Figure 7.1.

For a time-dependent Z, however, will be a

function of time. To maintain this level of , the

required control is . If is decreasing

sufficiently fast, then may become negative and

thus infeasible.If for all t, then the optimal policy

is as before. However, suppose is infeasible in the

interval [t1,t2] shown in Figure 7.2. In such a case, it is

feasible to set for t < t1 ; at t = t1 ( which is

point A in figure 7.2) we can no longer stay on the

turnpike and must set u(t)=0 until we hit the turnpike

again (at point B in figure 7.2). However, such a policy

is not necessarily optimal.

Figure 7.2: A Case of a Time-Dependent Turnpike and the Nature of Optimal Control

For instance, suppose we leave the turnpike at point C

anticipating the infeasibility at point A. The new path

CDEB may be better than the old path CAB. Roughly

the reason this may happen is that path CDEB is

“closer” to the turnpike than CAB. The picture in Figure

7.2 illustrates such a case. The optimal policy is the

one that is “closer” to the turnpike. This discussion will

become clearer in Section 7.2.2, when a similar

situation arises in connection with the Vidale-Wolfe

model. For further details, see Sethi (1977b) and

Breakwell (1968).

which implies

Because of these conditions it is clear that for a given

G0 , a choice of 0 such that (0 ,G0 ) is in Regions II

andIII, will not lead to a path converging to the

turnpike point . On the other hand, the choice of

(0 ,G0 ) in Region I when or (0 ,G0 ) in Region

IV when , can give a path that converges to

From a result in Coddington and Levinson(1955), it

can be shown that at least in the neighborhood of ,

there exists a locus of optimum starting points .

Given , we choose 0 on the saddle point path in

Region I of figure 7.3. Clearly, the initial control

u*(0)=f1(0). Furthermore, (t) is increasing and by

(7.17), u(t) is increasing, so that in this case the

optimal policy is to advertise at a low rate initially and

and gradually increase advertising to the turnpike level

. If , it can be shown similarly that

the optimal policy is to advertise most heavily in the

beginning and gradually decrease it to the turnpike

level as G approaches .

Note that the approach to the equilibrium is no

longer via the bang-bang control as in the Nerlove-

Arrow model. This, of course, is what one would

expect when a model is made nonlinear with respect

to the control variable u .

The Vidale-Wolfe Advertising Model

Now we can rewrite (7.19) as

Solution Using Green’s Theorem when Q is Large

To make use of Green’s theorem, it is convenient to

consider times and , where , and the

problem:

subject to

To change the objective function in (7.24) into a line

integral along any feasible arc from to in

(t,x)-space as shown in figure 7.4, we multiply by dt

and obtain the formal relation:

which we substitute into the objective function (7.24).

Thus,

Consider another feasible arc from to lying

above as shown in figure 7.4. Let ,where

is a simple closed curve traversed in the counter-

clockwise direction. That is, goes along in the

direction of its arrow and along in the direction

opposite its arrow. We now have

Since is a simple closed curve, we can use Green’s

theorem to express as an area integral over the

region R enclosed by . Thus, treating x and t as

independent variables, we can write

Denote the term in brackets of the integrand of (7.28)

by

Note that the sign of the integrand is the same as the

sign of I(x).

Lemma 7.1(Comparison Lemma). Let and be

the lower and upper feasible arcs as shown in figure

7.4. If I(x) 0 for all (x,t)R, then the lower arc is at

least as profitable as the upper arc . Analogously, if

I(x) 0 for all (x,t)R, then is at least as profitable

as .

Proof. If I(x) 0 for all (x,t)R, then 0 from (7.28)

and (7.29). Hence from (7.27), . The proof of

the other statement is similar.

To make use of this lemma to find the optimal control

for the problem stated in (7.23), we need to find

regions where I(x) is positive and where it is negative.

For this, note first that I(x) is an increasing function of x

in [0,1]. Solving I(x)=0 will give that value of x, above

which I(x) is positive and below which I(x) is negative.

Since I(x) is quadratic in 1/(1-x), we can use the

quadratic formula ( See Exercise 7.15) to get

To keep x in the interval [0,1], we must choose the

positive sign before the radical.

The optimal x must be nonnegative so we have

where the superscript s is used because this will turn

out to be a singular trajectory.Since is nonnegative,

the control

Note that and if, and only if,

Furthermore, the firm is better of with larger and r ,

and smaller and . Thus r/( +) represents a

measure of favorable circumstances.

Theorem 7.1 Let T be large and let xT be reachable

From x0. For the Cases 1-4 of inequalities relating x0

and xT to xs , theoptimal trajectories are given in

figures 7.5-7.8, respectively.

Proof. We give details for Case 1 only. The proofs for

the other cases are similar. Figure 7.9 shows the

optimal trajectory for figure 7.5 together with an

arbitrarily chosen feasible trajectory, shown dotted. It

should be clear that the dotted trajectory cannot cross

the arc x0 to C ,since u=Q on that arc. Similarly the

dotted trajectory cannot cross the arc G to xT,because

u=0 on that arc.

We subdivide the interval [0,T ] into subintervals over

which the dotted arc is either above, below, or identical

To the solid arc. In figure 7.9 these sub-intervals are

[0,d],[d,e],[e,f], and [f,T]. Because I(x) is positive for

and I(x) is negative for , the regions enclosed

by the two trajectories have been marked with + or –

sign depending on whether I(x) is positive or negative

on the regions, respectively. By Lemma 7.1, the solid

arc is better than the dotted arc in the subintervals

[0,d],[d,e],and [f,T]; in interval [e,f], they have identical

values. Hence the dotted trajectory is inferior to the

solid trajectory. This proof can be extended to any

(countable) number of crossing of the trajectories; see

Sethi(1977b).

Theorem 7.2 Let T be small, i.e., T < t1+t2 , and let xT

be reachable from x0 . For the two possible Case1 and

2 of inequalities relating x0 to xT and xs , the optimal

trajectories are given in figure 7.10 and 7.11,

respectively.

Proof. Therequirement of feasibility when T is small

rules out cases where x0and xT are on opposite

sides of or equal to xs. The proofs of optimality of the

trajectories shown in figures 7.10 and 7.11 are similar

to proofs of the parts of theorem 7.1, and are left as

exercise 7.23. In figures 7.10 and 7.11, it is possible to

have either t1 T or t2 T. Try sketching some of

these special cases.

Figure 7.10: Optimal Trajectory When T is Small inCase 1: x0< xs and xT > xs

Figure 7.11: Optimal Trajectory When T is Small inCase 2: x0> xs and xT < xs

Solution When Q is Small

where (T) is a constant, as in Row 2 of Table 3.1,

that must be determined. Furthermore, the Lagrange

multiplier u in (7.34) must satisfy

Solution When T is Infinite

We now formulate the infinite horizon version of (7.23)

When Q is small, i.e., Q< us , it is not possible to follow

the turnpike x = xs, because that would require u = us ,

which is not a feasible control. Intuitively, it seems

clear that the “closest” stationary path which we can

follow is the path obtained by setting and u=Q,

the largest possible control, in the state equation of

(7.39) .

This gives

by setting u = Q, and in (7.35) and solving for .

More specifically, we state the following theorems

which give the turnpike and optimal control when Q is

small. To prove these theorems we need to define two

more quantities, namely,

Theorem 7.3when Q is small, the following quantities

form a turnpike.

Proof. We show that the conditions in (3.73) hold for

(7.44). The first two are obvious. By exercise 7.28 we

know , which, from definitions (7.42) and (7.43),

implies . Furthermore , so (7.36) holds

and the third condition of (3.73) also holds. Finally

because from (7.38) and (7.43), it follows that

W 0 , so the Hamiltonian maximizing condition of

(3.73) holds with , and the proof is complete.

Theorem 7.4When Q is small, the optimal control at

time is given by:

Proof. (a) we set for all and note that

satisfies the adjoint equation (7.35) and the

transversality condition (3.70).

By Exercise 7.28 and the assumption that , we

know that for all t . the proof that (7.36) and

(7.37) hold for all relies on the fact that

and on an argument similar to the proof of the previous

theorem.

Figure 7.13 shows the optimal trajectories for

and two different starting values x(0), one above and

the other below . Note that in this figure we are

always in Case (a) since for all .

(b) Assume . For this case we will show that

the optimal trajectory is as shown in figure 7.14,

which is obtained by applying u=0 until and

thereafter. Using this policy we can find the

time t1 at which , by solving the state equation

in (7.39) with u = 0. This gives

Clearly for t t1, the policy u=Q is optimal because

Case (a) applies. We now consider the interval [0, t1).

Let be any time in this interval as shown in figure

7.14, and let be the corresponding value of the

sate variable. Consider the following two-point

boundary value problem in the interval

In Exercise 7.31 you are asked to show that the

switching function W(t) defined in (7.38) is negative in

the interval and W(t1)=0.

Therefore by (7.37), the policy u=0 used in deriving

(7.46) satisfies the maximum principle. This policy

“joins” the optimal policy after t1because

In this book the sufficiency of the transversality

condition (3.70) was stated under the hypothesis that

the derived Hamiltonian was concave; see Theorem

2.1. In the present example, this hypothesis does not

hold. However, as mentioned in Section 7.2.3, for this

simple bilinear problem, it can be shown that (3.70) is

sufficient for optimality. Because of the technical

nature of this issue we omit the details.

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