Optimal advertising policy with the contagion model - Springer Link

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 29, No. 4, DECEMBER 1979

Optimal Advertising Policy with the Contagion Model 1 S. P.

SETHI 2

Communicated by J. V. Breakwell

Abstract. This paper considers an optimal control problem for the dynamics of a contagion model, the optimal control being the rate of advertising expenditure that maximizes the present value of net profit streams over an infinite horizon. By using a Green's theorem approach, it is shown that there are multiple optimal stationary equilibria and that the optimal path from any given initial condition is a nearest feasible path to one of these equilibria. Key Words. Optimal control, advertising, Green's theorem approach, nearest feasible path, infinite horizon, optimal stationary equilibria, economic application.

1. Introduction A n u m b e r of empirical studies [beginning with Palda (Ref. 1), see Ref. 2] in the marketing literature demonstrate that the effect of an advertising expenditure persist for some time after the expenditure occurs. The existence of these carryover effects of advertising is now generally accepted (Ref. 3). Because of these effects, economists have treated advertising as an investment. The capital so formed is described by various names. Nerlove and A r r o w (Ref. 4) t e r m it a stock of goodwill, Ireland and Jones (Ref. 5) call it market power, and Jacquemin (Ref. 6) calls it the level of product differentiation. All of these names refer to the cumulative effect of advertising without making it operational. In contrast, the introduction of a diffusion process in the d e v e l o p m e n t of a sales-advertising response m o d e l allows us to define the cumulative 1This work was partially supported by the National Research Council of Canada, Grant No. A46.19. 2 Professor of Management Science, Faculty of Management Studies, University of Toronto, Toronto, Canada. 615 0022-3239/79/1200-0615503.00/0 © 1979 Plenum Publishing Corporation

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effect of advertising as the number of individuals who are aware of the given piece of information (Ref. 7). The fundamental concept of this approach is that individuals comprising the market learn of a particular piece of information by coming in contact with an advertising medium or by word of mouth. The first case is treated by Vidale and Wolfe (Ref. 8) and, in a different context, by Stigler (Ref. 9). The second case, in which information spreads by word of mouth, is treated by Ozga (Ref. 10) in a related context.

2. Contagion Model of Advertising The contagion model of advertising assumes that information spreads by word of mouth, rather than by an impersonal advertising medium. In the absence of forgetting, therefore, the likelihood of a given individual becoming informed of the given piece of information increases over time. T o develop the model, we define these terms: x = fraction of the population who know of the firm (awareness fraction, for short); size of the population is assumed constant and it is set at 1; u = rate of advertising; obviously, u -> O; p = contact coefficient (or, response constant), measured at u = 1; k = decay constant, implying forgetfulness on the part of the individuals who know of the firm. With these definitions, it is easy to see that the rate of change of the awareness fraction x is given by the differential equation

~c= p u x ( 1 - x ) - k x ,

x(0) = x 0 > 0.

(1)

Note that this model holds also for a growing population, provided p and k are made appropriately time dependent (Ref. 11). Note also that similar models have been extensively used in epidemic control and other related problems (Refs. 12-14). 3

3. Previous Analysis Gould (Ref. 7) has partially analyzed the implications of the contagion model for optimal advertising policy. H e assumes zr(x) to denote the rate of 3 Strictlyspeaking, a diffusionprocess approachwould yielda stochasticdifferentialequation of the form dX =[puX(1 - X ) - k X ] dt +cr(X, u) d/~, where X denotes the stochastic awareness fraction process and d~: is a standard Wiener process. Note that x = E(X) is the expected awareness fraction and obeys (1). The reader is referred to Tapiero (Ref. 15) for this type of development in a related context.


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profit (gross of advertising), with ~-'(x) > 0 H e also assumes

and

~'"(x) --- O.

w(u) to be the cost of advertising, with w'(u)>0

and

w"(u) > 0.

We note that, in Gould's model, u denotes the effective rate of advertising. The assumption on the form of the function w(u) reflects the diminishing marginal effectiveness of actual advertising expenditures. With r as the discount rate, the optimal control problem of Gould can be stated as follows: oo

max f [ z r ( x ) u-->0

w(u) e x p ( - r t ) dt,

(2)

d 0

subject to (1). Gould applies the maximum principle to characterize the optimal control. H e shows that there exists a stable optimal stationary equilibrium (g, ti). With respect to this, the optimal paths show two different modes of behavior, depending on the parameters of the problem. In one case, the optimal path for large enough initial state x0 < g begins with a low level of advertising, builds up to a level higher than the equilibrium level tT, and then cuts back toward t7 as x approaches ~. Of course, if Xo is not much smaller than ~, it is possible to begin with a higher-than-equilibrium level of advertising and continuously decrease to t7 as x approaches ~. This is simply the later part of the optimal path, when Xo is much smaller than ~. It is also noted that, for Xo > ~, the optimal advertising starts at a lower level and monotonically increases to tT. In the other case, the optimal path for large enough x0 < g begins with a low level of advertising and continuously builds up to tT. For Xo > ~, the optimal advertising starts low, increases to a level above tT, and then decreases gradually to ~7 as x approaches ~. It is to be noted that, in both of the above cases, the change from the initial state x0 to the equilibrium level ~ is monotonic. In connection with Gould's analysis, several remarks are in order. First, his analysis does not exhaust all the possibilities which might arise in the course of solving the problem. Second, Gould does not prove the existence and sufficiency of the optimal solution. Finally, the case of linear w(u) is not a special case of (2). Furthermore, a solution of the linear case cannot be traced from the steps in the strictly nonlinear solution, on account of the fact that we can no longer equate the derivative of the Hamiltonian to zero in its optimal control solution.

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Moreover, the linear case is an extremely interesting case, especially from a theoretical standpoint. It allows multiple optimal stationary equilibria. With the given values of the problem parameters, the convergence of the optimal path to a particular equilibrium depends on the initial conditions. This must certainly have practical implications for the firm interested in knowing its optimal equilibrium position. For all of these reasons, the analysis of the contagion model of advertising will not be complete without the linear case. 4

4. Optimal Control Problem In this paper, therefore, we choose to solve the following problem: oo

max{J= I

(zrx-u)exp(-rt)dt},

(3)

0

subject to (1) and u ~ [0, U].

(4)

Note, that, in formulating the objective function in (3), we have assumed that dollar sales are proportional to the awareness fraction x and that the profit margin (gross of advertising) per unit sales is constant. Furthermore, the control variable u denotes the rate of advertising expenditure in dollars per unit time. Finally, for simplicity in exposition, we assume U to be sufficiently large (see Ref. 17, for possible complications which arise when it is not so, and how to deal with them). Because of the existence of multiple optimal stationary equilibria, the analysis of this problem would be quite difficult. Probably, this is why this particular problem has been neglected for so long. Recently, however, we have developed a general methodology (Ref. 19) to deal with a class of problems exhibiting multiple optimal equilibria. This theory is based on a Green's theorem approach, which has been used to analyze problems in this class with single optimal equilibrium (Refs. 20, 17, and 13). We also note that the existence and sufficiency of the optimal solution is automatic in this approach. 4 Note that a more realistic problem would incorporate in (1) a term tzu(1 - x ) for effects of direct contact with the impersonal advertising medium (note that tz denotes the contact coefficient). See Sethi (Ref. 11) and Zufryden (Ref. 16) for partial analyses of this problem. However, our main interest is in examining the optimal advertising policy implications of the underlying contagion mechanism of information spread. The policy implication of the information spread solely by contact with the inanimate advertising medium has been fully analyzed in Sethi (Ref. 17) and Gould (Ref. 7). See Sethi (Ref. 18) for a survey of dynamic optimal control models in advertising.

JOTA: VOL. 29, NO. 4, D E C E M B E R 1979

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5. Green's Theorem Approach and Extensions First, we substitute u dt = dx + kx dt/flx(1 - x ) , derived from (1), into the objective function (3) to obtain the line integral along any curve F in (t, x) space: Jr = I {[¢rx- k / p ( 1 - x ) ] exp(-rt) d t - [ 1 / p x ( 1 - x ) ] exp(-rt) dx}. F

For a simple closed curve F, we can use Green's theorem to express this line integral over the area R bounded by F: J = f I {(O/Ot)[-exp(-rt)/px(1-x)] R

-(O/Ox)[~rx - k/p(1 - x)] exp(-rt)} dt dx = I I [ r / p x ( 1 - x ) - z r + k / p ( 1 - x ) 2 ] e x p ( - r t ) dtdx R

=I I {[-¢rOx3+2~x2+(k-r-~'O)x+r]exp(-rt)/Ox(1-x)2}dtdx R

R

To specify the optimal controls, we need to partition the (t, x)-space into the regions where the integrand I(x) takes positive and negative values. For this, we equate the integrand to zero and solve the resulting cubic equation E(x) = x 3 - 2x z - [(k - r - zrp)/lrp]x - r/rrp = 0. (6) From the theory of cubic equations, the solution of (6) can be specified as follows (see Spiegel, Ref. 21). Let (2 = [3(r + zrp - k )/ lrp - 4 ] / 9 , R = [-18(r + ¢rp - k ) / z r p + 27r/~rp + 16]/54, D=Q3+R 2

s = (R +4b) "~,

T = (R _,,/-~) i/2;

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then, the roots of (6) are

Xl=S+T+2/3, x2 = - ( 1 / 2 ) ( S + T) + 2 / 3 + (1/2)~/---3(S - T),

(7)

x3 = - ( 1 / 2 ) ( S + T) + 2 / 3 - (1/2)~/~-3(S - r ) . Furthermore, depending on the sign of the discriminant D, we have the following true situations: (a) one root is real and two are complex conjugate if D > 0; (b) all roots are real and unequal if D < 0; (c) all roots are real and at least two are equal if D = 0. Note that the situation D = 0 is a fortuitous one and can be ignored. W e have, therefore, only the first two situations. Because of the multiple roots of (6), there will in general be multiple optimal equilibria. If this were not so, the solution of the problem could have been easily obtained as in Ref. 17. In this simpler case, if there is a unique root 2 of (6) such that I(x)0

for x > £ ,

then £ is an optimal equilibrium. As in Ref. 17, the optimal path can easily be shown to be the nearest feasible path from the initial state Xo to £. This is shown in Fig. 1. That is, if Xo > £, use u* = 0 as long as x > £, and then switch to u* = a when x = ~. When Xo < £, then the optimal policy would be to use u* = U until x = $, and then switch to u* = 6. Note that t~ is the control which will maintain the state at ~. Note that, in the economic literature, the equilibrium level ~ is known as the turnpike level (Ref. 22). This solution approach is very appealing. One cannot help but wish that the approach be adapted to multiple equilibria cases. Fortunately, this has been done. In ref. 19, we have shown that the optimal path in the multiple x=!

xo~ l ( x ) > O uW=

Xo~ x=O

Fig. 1.

I(x) £ and Xo < £.


621

equilibria case is the nearest feasible path to one of the equilibria. A s applied to the contagion model, we can restate T h e o r e m 7.2 of Ref. 19 as follows.

Theorem 5.1.

Let

Y ={xlI(x) = O, I(x + E ) > 0, I(x - E ) < 0

for e > 0}w {0 if I ( 0 ) > 0} w {1 if I(1) < 0}.

Then, the optimal control f r o m a given Xo is the nearest feasible path f r o m Xo to s o m e £ ~ Y. Of course, the choice of £ will d e p e n d on the initial state Xo. This is a constructive t h e o r e m . 5 That is, to find the optimal solution, we first obtain the set Y of multiple optimal equilibria. This is d o n e by examining the b e h a v i o r of I(x). H a v i n g o b t a i n e d Y, we derive the nearest feasible paths f r o m x0 to each £ ~ Y. W e then select the best of these nearest feasible paths (i.e., the o n e which gives the highest value of the obiective function) as the optimal path. This is a straightforward procedure, as we shall show in the following section for the contagion model.

6. Optimal Solution To obtain the set of optimal equilibria, we n e e d only to check the sign of

I(x) for x ~ [0, 1]. F o r this we note that I(x) = -zrE(x)/x(1 - x ) 2 ; therefore, sign I(x) = - s i g n E(x),

0 < x < 1.

(8)

Since the sign of I (x) or that of - E (x) d e p e n d s critically on the discriminant D, we will carrry out this examination, first for D > 0 and then for D < 0. First Situation: D > 0. In this situation, there is only one real solution of the cubic e q u a t i o n (6). Let ~ be this solution. F u r t h e r m o r e , the x 3 term in the cubic E(x) has a positive coefficient; we have E(oo) = oo

and

E(-oo) = -oo.

(9)

5 Note that Y is the set of local maxima of the function x

V(x) = - I l(s) ds. I

0

The interior maxima are the roots of V'(x) = -l(x) = 0, satisfying the second-order condition V"(x) =-I'(x) < 0. Furthermore, x = 0 and x = 1 are constrained local maxima if V'(0)= -I(0) < 0 and V'(1) = -I(1) > 0, respectively (Ref. 19).

622


of-------

® E(x) =0

E ( x ) i"

Fig. 2. Possibleplots of E(x) with D > 0. We can now plot E ( x ) in Fig. 2 for the first situation. From Fig. 2, it is easy to see that we can have three subcases: (i)

2--1,

(ii)

0 0 is shown in Table 1 for each of the three subcases. Having obtained the results in Table 1, the specification of the optimal paths in the various subcases is very easy. An additional notation would make this task even easier. Let NFP (£, x0) denote the nearest feasible path to2 E Y from the initial state x0. For D > 0, we need only be concerned with two values of 2; these are 0 and 1. It is obvious that NFP (0, x0) is obtained by applying u(t) = 0, Vt, NFP(1, Xo) is obtained by applying u(t) = U, Vt. We note that the controls along the nearest feasible paths for 2 = 0 and 2 = 1 are independent of the initial values. In subcase (i), there is only one optimal equilibrium, namely, 2 = 0. The optimal path, therefore, in this subcase is NFP(0, Xo). Similarly, the optimal Table 1. Optimal stationary equilibria for D > 0. Subcase

Y

(i) (ii) (iii)

{o} {0, 1} {1}


623

path in subcase (iii) is NFP (1, x0). We note that NFP (1, Xo) will never reach the state where everyone is aware of the firm's advertising message. In fact, NFP (1, Xo) is asymptotic to the awareness fraction level 1 - k p / U . Finally, in subcase (ii), we have two optimal equilibria, i.e., ~ = 0 or 1. Given an initial state x0, either NFP (0, x0) is optimal or NFP (1, x0) is optimal. The selection can be made by evaluating the objective function value associated with each of the two paths. The path giving the higher value is the optimal path. It should be obvious that there exists an a such that x0 < a ~ NFP(0, x0) is optimal,

Xo>a ~ NFP(1, Xo) is optimal. This completes the analysis of the first situation. Second Situation: D < 0. In this situation, there are three unequal real solutions of the cubic equation (6). Let these solutions be £1,22, 23, and let £i < £2 < 23. Since it is the same cubic E ( x ) which satisfies (9), we can plot E ( x ) for D < 0 as in Fig. 3. From this figure, it is easy to see that we can have ten subcases depending on the location of 0 and 1 with respect to the three roots. For this consideration, we ignore the fortuitous cases when one or more of the roots are exactly 0 or 1. In Table 2, we provide the sets of optimal equilibria in each of these subcases. Note that empty boxes represent infeasible subcases. Further, note that the subcase [0 < £1, 1 > £3] has the maximum number of equilibria, i.e., three. Also, note that four subcases contain an interior equilibrium £2. The optimal solutions in subcases with Y = {0}, Y = {1}, Y = {0, 1} are the same as those in subcases (i), (iii), (ii) of D > 0, respectively. The subcase [0 ~ (xl,Ax2),^ 1 ~ (~2, ~3)] has a umque . . interior . . . eqmhbrlum. . . The optimal path is NFP (22, x0),'which can be obtained as follows: For x0 ;3

1 c (22, 23)

1 e (21, 22)

1 < ;1

0>;3 0 s (;2, ;3) 0 s (;1, ;2) 0 < £1

{1} {0, 1} {;2, 1} {0, ;2, 1}

{0} {¢2} {0, h}

11} (0, 1}

(ol

For Xo > ~2, use u (t) = 0 until x(t) = £2, and then switch to u =/~2. Note that the sketch of NFP (£2, x0) is available in Fig. 1 with ~ = 22 and a = a2. Also, note that a2 is the control which will maintain the state at 3~2 forever following the switching instant. Furthermore, /~2 £2 (or 0 for the problem to be meaningful, in view of the form of (1). The results obtained are also theoretically significant. The economic literature is replete with problems with a single unique optimal equilibrium, called a turnpike. There are, however, not many papers dealing with multiple equilibria, even though such problems arise quite frequently. This is because a general theory to deal with such problems was unavailable in the past. Now that the theory has been made available, we expect quite a few management science and economic applications of this theory in the future. 6 We conclude by noting that an important advantage of applying this theory is that one need not be concerned with special issues, such as the nonconcavity of the problem and the existence and sufficiency of the optimal solution. 6 See Sethi (Ref. 23) for the application of this theory to obtain the optimal quarantine program for an epidemic model.

626


References 1. PALDA, K. S., The Measurement of Cumulative Advertising Effects, PrenticeHall, Englewood Cliffs, New Jersey, 1964. 2. CLARKE. D. G., Econometric Measurement of the Duration of the Advertising Effect on Sales, Harvard University, Business School, Division of Research, Working Paper, 1975. 3. TULL, D. S., The Carry-overEffect of Advertising, Journal of Marketing, Vol. 20, pp. 46-53, 1965. 4. NERLOVE, M., and ARROW, K. J., Optimal Advertising Policy under Dynamic Conditions, Economica, Vol. 39, pp. 129-142, 1962. 5. IRELAND, N. J., and JONES, H. G., Optimality in Advertising: A Control Theory Approach, Proceedings of the IFORS/IFAC International Conference, Coventry, England, 1973 (IEE Conference Publication No. 101, pp. 186-199). 6. JACQUEMIN, A. P., Market Structure and the Firm's Market Power, Journal of Industrial Economics, Vol. 20, No. 2, 1972. 7. GOULD, J. P., Diffusion Processes and Optimal Advertising Policy, Microeconomic Foundation of Employment and Inflation Theory, Edited by E. S. Phelps, et al., W. W. Norton, New York, New York, 1970. 8. VIDALE, M. L., and WOLFE, H. B., An Operations Research Study of Sales Response to Advertising, Operations Research, Vol. 5, pp. 370-381, 1957. 9. STIGLER,G., The Economics of Information, Journal of Political Economy, Vol. 69, pp. 213-225, 1961. 10. OZGA, S., ImperfectMarkets Through Lack of Knowledge, Quarterly Journal of Economics, Vol. 74, pp. 29-52, 1960. 11. SETHI, S. P., Some Explanatory Remarks on the Optimal Control for the Vidale-Wolfe Advertising Model Operations Research, Vol. 22, No. 4, 1974. 12. BAILEY, N. T. J., The Mathematical Theory of Epidemics, Hafner, New York, New York, 1957. 13. SETHI, S. P., Quantitative Guidelines for Communicable Disease Control Program: A Complete Synthesis, Biometrics, Vol. 30, pp. 681-691, 1974. 14. WICKWIRE, K., Mathematical Models for the Control of Pests and Infectious Diseases: A Survey, Theoretical Population Biology, Vol. 11, pp. 182-232, 1977. 15. TAPIERO, C. S., On-Line and Adaptive Optimum Advertising Control by a Diffusion Approximation, Operations Research, Vol. 23, No. 5, 1975. 16. ZUFRYDEN,F. S., Optimal Multi-PeriodAdvertising Budget Allocation within a Competitive Environment, Operational Research Quarterly, Vol. 26, No. 4, 1975. 17. SETHI, S. P., Optimal Control of the Vidale-Wolfe Advertising Model, Operations Research, Vol. 21, No. 4, 1973. 18. SETHI, S. P., Dynamic Optimal ControlModels in Advertising: A Survey, SIAM Review, Vol. 19, No. 4, 1977. 19. SETHI, S. P., Nearest Feasible Paths in Optimal Control Problems: Theory, Examples, and Counterexamples, Journal of Optimization Theory and Applications, Vol. 23, No. 4, 1977.


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20. MIELE, A., Extremization o[ Linear lntegrals by Green ' s Theorem, Optimization Techniques, Edited by G. Leitmann, Academic Press, New York, New York, 1962. 21. SPIEGEL, M., Mathematical Handbooks o[ Formulas and Tables, McGraw-Hill Book Co., New York, New York, 1968. 22. ARROW, K. J., and KURZ, M., Public Investment, the Rate of Return, and Optimal Fiscal Policy, The Johns Hopkins Press, Baltimore, Maryland, pp. 26-57, 1971. 23. SETHI, S. P., Optimal Quarantine Programs[or Controlling an Epidemic Spread, Journal of Operational Research Society, Vol. 29, No. 3, 1978.