JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 117, No. 2, pp. 349–375, May 2003 ( 2003)

Optimal Pricing of a Product Diffusingin Rich and Poor Populations

R. F. HARTL,1 A. J. NOVAK,2 A. G. RAO,3 AND S. P. SETHI4

Communicated by G. Leitmann

Abstract. We consider a market consisting of two populations, termedrich and poor for convenience. If a product is priced such that it is veryexpensive for the poor, but affordable to the rich, then it becomes astatus symbol for the poor and this makes it more desirable for thepoor. At a lower price, the product is affordable by both populations.However, as more of the poor buy the product, it ceases to be a statussymbol and becomes less appealing to the rich. We present a two-statenonlinear optimal control problem that aims to obtain profit-maximiz-ing prices over time in this environment. We find that there are threecategories of optimal price paths. One is status-symbol pricing with highinitial price, declining over time. The other two are mass-market pricing,with price declining in one, and price increasing and then decreasing inthe other.

Key Words. Optimal control, marketing, pricing, market diffusion,aspirational group.

1. Introduction

Sociologists have argued that status is evaluated based primarily onmaterial items (e.g., Ref. 1). The ability of a product to confer status restson the extent to which its ownership and consumption is confined to adesirable group which is aspirational to an outside group (Ref. 2). A highprice assures potential buyers that the total sales of the product will belimited, and by thus restricting ownership to a small, higher-income group,

1Professor, Department of Business Studies, University of Vienna, Vienna, Austria.2Associate Professor, Department of Statistics and Decision Support Systems, University ofVienna, Vienna, Austria.

3Professor, John M. Olin School of Business, Washington University, St. Louis, Missouri.4Professor, School of Management, University of Texas at Dallas, Richardson, Texas.

3490022-3239�03�0500-0349�0 2003 Plenum Publishing Corporation

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creates exclusivity that turns the product into a status symbol. This groupfunctions as aspirational for an outside and relatively larger lower-incomegroup for which the product becomes exclusive due to its high price. Thus,the product differentiates status between the target group and the outsidegroup. However, since price can rarely guarantee absolute exclusivity, somemembers of the outside group may be able to acquire the product. Owner-ship of the exclusive product elevates the status of these outside group mem-bers, because of its association with membership in the aspirational group.However, this exclusivity is based upon a price high enough to make owner-ship by members of the outside group rare. Otherwise, it is in danger ofbecoming a fraudulent symbol (Ref. 1), i.e., a symbol that has become dif-fused across levels of the class hierarchy so that potential owners are notassured that consumption and ownership of these products guarantees acertain level of status. Thus, appropriately priced, the brand differentiatesstatus within the outside group. Because members of the high-income groupcan easily acquire the product, it is not exclusive for them. However, asownership of the product by the outside group becomes more widespread,the product becomes less desirable to the high-income group. Reference 3contains an extensive discussion of exclusivity phenomena and experimentalevidence supporting them. The paragraph above is based on that discussion.

Rao and Sethi (Ref. 4) have modeled the situation described above asfollows. They consider a monopolist who markets a durable good that ispurchased only once. The market consists of two populations, called forsimplicity rich and poor and subscripted by 1 and 2, respectively. LetFi (t), 0⁄Fi (t)⁄1, iG1, 2, denote the proportion of each population whichhas adopted the brand by time t. An alternative interpretation of Fi (t) isthat it is the probability that a member of population i has purchased theproduct by time t. The corresponding probability density is dFi (t)�dt. There-fore, for population i, [1�(1AFi (t))] dFi (t)�dt represents the hazard rate orthe conditional probability density that a member of population i makes apurchase at time t given that he has not done so yet.

Let ai be the price at and beyond which demand from population i iszero, a2Fa1 . The demand for the product in population i is modeled asaiAp, for 0⁄p⁄ai . As adoption in the poor population increases (i.e., F2increases), the product becomes less desirable for the rich population. How-ever, as adoption in the rich population increases (i.e., F1 increases), withF2 held constant, the product establishes itself as a status symbol for thepoor population. Using these arguments, Rao and Sethi propose the follow-ing model for the diffusion of the product in the two populations:

[1�(1AF1(t))] dF1(t)�dtG(1AF2(t))[a1Ap (t)], (1)

[1�(1AF2(t))] dF2(t)�dtGF1(t) max{0, a2Ap(t)}. (2)

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In (1) and (2), the l.h.s. expressions represent the hazard rates in therich and poor populations, respectively. In (1), the term 1AF2 models thereduction of demand for the product in the rich population, as adoption F2in the poor population increases. Similarly, in (2), the term F1 models theincrease in demand for the brand in the poor population as penetrationin the rich population increases, serving to make it even more of a statussymbol.

In this paper, we formulate an optimal control problem faced by aprofit maximizing monopolist in a market described by (1) and (2). We useoptimal control theory along with numerical computations to develop pricepaths under different parameter constellations. It would appear that thereare three possible categories of policies. First, the monopolist can sell toboth rich and poor populations throughout, by maintaining a2¤p, thusavoiding any effort at creating exclusivity. Second, he could begin withpHa2 , selling to the rich population and establishing the product as a statussymbol, and then drop the price below a2 and sell primarily to the poorpopulation. Third, he could maintain pHa2 throughout, and address onlythe rich population.

From the perspective of optimal control theory, the model presented hereis a two-state nonlinear optimal control problem. Moreover, unlike in theprey-predator models (see Ref. 5), the rich-poor dynamics here is asymmetric.This makes the problem quite interesting as well as complex. Furthermore,the model is quite different from the existing optimal control problems inthe marketing literature, which have been surveyed in Refs. 6–10.

The plan of the paper is as follows. In Section 2, we formulate theoptimization problem and obtain some preliminary results. Section 3 pro-vides the necessary conditions for optimality using the maximum principle.These conditions give rise to various possible scenarios. Section 4 analyzesthe movements between these scenarios depending on the values of the prob-lem parameters. Explicit solutions are provided in some scenarios. InSection 5, we use the results to obtain computational solutions and providetheir interpretation. Section 6 concludes the paper along with suggestionsfor future research. Section 7 (Appendix) contains the proofs of the prop-ositions and lemmas.

2. Model Formulation and Preliminaries

We specify the notation and develop the optimal control problem underconsideration:

F1(t)Gfraction of the rich population that has bought the product bytime t, t¤0;

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F2(t)Gfraction of the poor population that has bought the product bytime t, t¤0;

a1Gprohibitive price for the rich market;a2Gprohibitive price for the poor market, a2Fa1 ;cGconstant unit cost, 0⁄cFa2 ;SGsalvage value parameter, S¤0;MGsize of the poor population compared to the size of the rich

population;p(t)Gprice of the product at time t, t¤0.

The aim of the monopolist is to optimally set the price p(t) over timeso that the expected present value of profits from sales to both populationsplus the end-of-the-horizon salvage value based on the captured share of therich population is maximized. Thus, the objective function can be written as

JG�T

0

[dF1�dtCM(dF2�dt)]( pAc) dtCSF1(T ). (3)

The last term of (3) merits further discussion. In practice, at some timeT, the monopolist will introduce a new product to replace the existing oneand solve the pricing problem again and this process will be repeated. Toaccount for these future product introductions, we have introduced a sal-vage-value term proportional to the fraction of the rich population whichhas purchased the current product by time T. If this proportion is high, thenew product may be able to position itself as an exclusive product. However,if the proportion is low, it will be impossible for it to draw on its heritageto position itself as exclusive. As S increases, these impacts are magnified.

Thus, the optimal control problem faced by the monopolist is to maxi-mize (3) subject to the differential equations

dF1�dtG(1AF1)(1AF2)[a1Ap], F1(0)G0, (4)

dF2�dtGF1(1AF2) max{0, a2Ap}, F2(0)G0, (5)

the control inequality constraint

0⁄p(t)⁄a1 , t¤0, (6)

and the state inequality constraints

0⁄F1(t)⁄1, 0⁄F2(t)⁄1. (7)

Before we apply the maximum principle to solve this problem, we willmake a few important observations.

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Proposition 2.1. Along any optimal solution, p⁄a1 and the state con-straints (7) hold for every admissible price trajectory.

Proof. These follow immediately from (4) and (5). �

In view of Proposition 2.1, we can solve the optimal control problemwith the control constraint p(t)¤0 in place of (6) and without the stateconstraints (7).

Proposition 2.2. An upper limit for the profit is a1CMa2CS.

Proof. The maximum possible value of F1 and F2 is 1. Thus, the maxi-mum revenue that can be obtained from the rich population is a1 , the maxi-mum revenue that can be obtained from the poor population is Ma2 , andthe maximum salvage value is S. Note that this observation is true for anyvalue of T including TGS. Thus, a1 and Ma2 could be termed the sizes ofthe rich market and the poor market, respectively. As stated earlier, thevalue S represents the maximum value of the goodwill associated with thesales in the rich market. �

In the next proposition, we show that, even when TGS and cG0, nooptimal solution can obtain the objective function value equal to the upperbound a1CMa2CS. However, in that case, it is possible to have a profitthat is arbitrarily close to the upper bound.

Proposition 2.3. For cG0 and TGS, there exists no optimal policy.Nevertheless, one can obtain a policy which gives a profit arbitrarily closeto the upper bound a1CMa2CS.

Proof. It is easy to see that no pricing strategy will give a profit equalto the upper bound a1CMa2CS. To obtain a profit close to the upperbound, consider a small (H0 such that a1A(Ha2 and a2A(H0. Use theprice

p(t)Ga1A(, 0⁄ t⁄τ ,

where τ is arbitrarily large. Set

p(t)Ga2A(, for t¤τ .

The profit associated with this policy in the interval [0, t], tHτ , has the lowerbound

(a1A(CS )F1(τ )CM(a2A()[F2(t)AF2(τ )],

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where

F1(τ )G1Aexp(−(τ )and

F2(t)AF2(τ )→1, as t→S.

Note that the profit occurring in the rich market after time τ is not includedin the lower bound expression. For ( sufficiently small and τ sufficientlylarge, this lower bound expression can be made arbitrarily close to the upperbound a1CMa2CS. Clearly, this means that there does not exist an optimalsolution to the infinite horizon problem under consideration. �

Note that, when TGS, the most reasonable value for S is zero, sincethere is only one product to be considered. So, for TGS, the upper boundwhen SG0 is a1CMa2 . This provides a reason for having a finite productlife cycle for the product. With SH0, it is then possible to obtain a profitgreater than a1CMa2 even when TFS. Therefore, from now on, we willonly consider the case TFS.

3. Necessary Optimality Conditions

In order to obtain necessary conditions for optimal solution of ourcontrol problem, we make use of the Pontryagin maximum principle; seeRef. 11 or Ref. 12. The Hamiltonian is given by

HG( pAcCλ1)(1AF1)(1AF2)[a1Ap]

C[M( pAc)Cλ2 ]F1(1AF2) max{0, a2Ap}, (8)

where λ1 and λ2 denote the costates or adjoint variables associated with F1and F2 , respectively. They follow the adjoint equations

dλ1�dtG( pAcCλ1)(1AF2)[a1Ap]

A[M( pAc)Cλ2 ](1AF2) max{0, a2Ap}, λ1(T )GS, (9)

dλ2�dtG( pAcCλ1)(1AF1)[a1Ap]

C[M( pAc)Cλ2 ]F1max{0, a2Ap}, λ2(T )G0. (10)

The interpretation of the adjoint variables λ1(t) and λ2(t) is as follows.λ i (t) represents the marginal value of an increase in Fi (t), iG1, 2, expressedin current value or time t dollars. Thus, λ i (t)( represents the increase in theobjective function value if Fi (t) increases by a small amount (H0. Since anincrease in F2(t) at any time 0⁄ tFT hurts the future sales of the productin both rich and poor markets, we should have λ2(t)F0 for all t, except at

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the terminal time T where λ2(T )G0 as required by (10). Indeed, we shallprove this intuition in Proposition 4.6. On the other hand, an increase inthe value of F1(t) decreases future sales in the rich market and increasesfuture sales in the poor market. Thus, the value of λ1(t) can be positive ornegative depending on the parameters of the problem.

Economic interpretations to the adjoint Eqs. (9) and (10) and Hamil-tonian in (8) are standard; see Ref. 11 [Section 2.2.4].

The Hamiltonian maximization conditions on the feasible price range[0, a1 ] is

∂H�∂p�pG0G(1AF1)(1AF2)[a1 ]C(cAλ1)(1AF1)(1AF2)

CMF1(1AF2)a2C[cMAλ2 ]F1(1AF2)⁄0, for pG0, (11)

∂H�∂pG(1AF1)(1AF2)[a1Ap]A( pAcCλ1)(1AF1)(1AF2)

CMF1(1AF2)[a2Ap]

−[M( pAc)Cλ2 ]F1(1AF2)G0, for 0FpFa2 , (12)

∂H�∂pG(1AF1)(1AF2)[a1Ap]

−( pAcCλ1)(1AF1)(1AF2)G0, for a2FpFa1 . (13)

The problem with cH0 is not significantly different from the one withcG0, since the former can be reduced to the latter by defining the pricecontrol qGpAc and by redefining a1Ac and a2Ac as new a1 and a2. Thecontrol constraint 0⁄p reduces to −c⁄q. Henceforth, we deal only with thecase cG0.

3.1. Analysis in the Case cG0. For cG0, Eqs. (9) and (10) reduce to

dλ1�dtG( pCλ1)(1AF2)[a1Ap]

A[MpCλ2 ](1AF2) max{0, a2Ap}, λ1(T )GS, (14)

dλ2�dtG( pCλ1)(1AF1)[a1Ap]

C[MpCλ2 ]F1 max{0, a2Ap}, λ2(T )G0. (15)

The Hamiltonian maximization conditions (11)–(13) reduce to

∂H�∂p�pG0G(1AF2){(1AF1)[a1Aλ1 ]

CF1 [Ma2Aλ2 ]}⁄0, for pG0, (16a)

∂H�∂pG(1AF2){(1AF1)[a1A2pAλ1 ]

CF1 [Ma2A2MpAλ2 ]}G0, for 0FpFa2 , (16b)

∂H�∂pG(1AF1)(1AF2)[a1A2pAλ1 ]G0, for pHa2 . (16c)

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Before we derive the locally maximizing price, we define the following termsfor notational convenience:

p1G(a1Aλ1)�2, (17a)

p2G{(1AF1)[a1Aλ1 ]CF1 [Ma2Aλ2 ]}�[2(1AF1)C2MF1 ], (17b)

λLGa1A2a2A(Ma2Cλ2)F1�(1AF1), (18a)

λHGa1C(Ma2Aλ2)F1�(1AF1), (18b)

λ*Ga1A2a2A(a2Cλ2�M )[11CMF1�(1AF1)A1], (18c)

S*Gλ*�λ2G0Ga1Aa2 [1C11CMF1�(1AF1)]. (18d)

From (16), we can derive the locally maximizing price, respectively, asfollows:

pG0, if λ1¤λH, (19)

pGp2 , if λHHλ1HλL. (20)

Note that the upper limit λH in (20) is always bigger than the lower limitλL (r.h.s.) so that the specified interval for λ1 is not empty. Note also that

pGp1 , ifAa1⁄λ1Fa1A2a2 . (21)

Lemma 3.1. For pHa2 , it holds that λ1¤Aa1 .

Proof. By Proposition 2.1, pFa1 which implies the result by (21). �

Note that the conditions on λ1 in (20) and (21) are only necessaryconditions because there may be an overlap of the corresponding regions.This will be investigated further in the next section.

3.2. Further Analysis of the Hamiltonian Maximization Condition.First, we note that the Hamiltonian has a kink w.r.t. p at point a2. Thejump in the derivative is

∂H�∂p�pGa+2A∂H�∂p�pGa−2G[Ma2Cλ2 ]F1(1AF2). (22)

Clearly, this expression can be positive or negative. Thus, we have to con-sider two different regions depending on the sign of Ma2Cλ2 .

Region 1: Ma2Cλ2H0. In this region, by (22) the Hamiltonian has aconvex kink w.r.t. p and there may be two local maxima. Using the notationdefined in (17) and (18), the result can be summarized as follows.

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Proposition 3.1. Let Ma2Cλ2H0.

(A) If λ1¤λH, then pG0 is optimal.(B) If λHHλ1Ha1A2a2 , then pGp2 is a local maximum.(C) If a1A2a2Hλ1HλL, then H has two local maxima. Evaluating

H at both of these maxima and identifying the global maximumyields the following three subcases:

(C1) If a1A2a2¤λ1Hλ*, then pGp2Fa2 is optimal.(C2) If λ1Gλ*, then pGp2 and pGp1 are both optimal.(C3) If λ*Hλ1¤λL, then pGp1Ha2 is optimal.(D) If −a1⁄λ1FλL, then pGp1 is optimal.

Proof. See Appendix, Section 7.

Figure 1 shows the value of the expression H�(1AF2) as a function ofp in each of the above cases. The price at which the maximum of thisexpression occurs is indicated by solid-line circles. In Case (A), the maxi-mum occurs at pG0. In Cases (B) and (C1) it occurs for p between 0 anda2. In Case (C2), there are two maxima, one for p between 0 and a2 and onefor p between a2 and a1. Finally in Cases (C3) and (D), the maximum occursfor p between a2 and a1. Note that H(a1)G0 from (8) and that we havedepicted an instance of H(0)H0 in Fig. 1. In the hairline Case (C2), the twolocal maxima in the two regions are both globally optimal. In Fig. 2, themaximizing price in each of the cases is shown as a function of λ1. The linepGp1 in Cases (D) and (C3) is steeper than the line pGp2 in Cases (B) and(C1). Note that the Case (C2), where there are two maximizing prices,occurs at λ1Gλ*. This implies a downward jump in price at any time twhen the λ1 trajectory crosses the value λ*; see the example in Fig. 8.

Fig. 1. Different cases in Region 1.

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Fig. 2. Optimal price p(t) expressed in terms of λ1 for a fixed λ2.

Region 2: Ma2Cλ2⁄0. In this region, by (22) the Hamiltonian has aconcave kink (if Ma2Cλ2F0) w.r.t. p; therefore, the Hamiltonian is con-cave in the control variable p. The result in this case is summarized in thefollowing proposition.

Proposition 3.2. Let Ma2Cλ2⁄0. In every case, there is only one localmax that is also optimal:

(A) If λ1¤λH, then pG0 is optimal.(B) If λHHλ1HλL, then pGp2 is optimal.(D) If λ1Fa1A2a2 , then pGp1 is optimal.(E) If λLHλ1Ha1A2a2 , then pGa2 is optimal.

Proof. See Appendix, Section 7.

These cases are depicted in Fig. 3. Once again, we have depicted aninstance of H(0)H0 in Fig. 3. In Fig. 4, the price is shown as a function ofλ1. Note that the line pGp1 in Case (D) is steeper than the line pGp2 inCase (B). It is interesting to contrast Figs. 3 and 4 with Figs. 1 and 2. InCases (C1), (C2), (C3) in Figs. 1 and 2, we have two locally maximizingprices, one above and one below a2. In Case (E) in Figs. 3 and 4, we havea single maximizing price which occurs at a2. This results in a price equalto a2 over an interval of time during which λ1 takes values in the intervalfrom λL to a1A2a2 ; see the example in Fig. 11.

For any given (F1 , F2), Fig. 5 shows the regions in the (λ1 , λ2) spacewhere the various cases occur. It may be noted that the boundaries separat-ing these regions are straight lines. Note that a1A2a2 can be negative, whichin turn implies that S* can be negative. This is reasonable, since if the poor

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Fig. 3. Different cases in Region 2.

Fig. 4. Optimal price p(t) expressed in terms of λ1 for a fixed λ2.

people are willing to pay more than half of what the rich people are willingto pay (i.e., a1F2a2), it does not make much sense to sell only to the richpeople. Thus, for a1A2 a2F0, the regions where Cases (C) and (D) areoptimal occur only if λ1F0.

In the rest of the paper, we will investigate possible movements betweenthese regions and provide some typical time paths of price and marketshares.

4. Analysis of the Movement between and within the Regions

In the previous section, we have investigated in detail which price isoptimal for which combination of the shadow prices (λ1 , λ2) and have

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Fig. 5. Boundary lines separating the cases in the (λ1 , λ2) space for any given (F1 , F2).

identified various cases shown in Fig. 5. For the purposes of the followinganalysis, it is convenient to group these cases into four possible scenarioson the basis of the price ranges that apply in these cases.

Scenario 1: pG0, i.e., Case (A).Scenario 2: 0FpFa2 , i.e., Cases (B) and (C1).Scenario 3: pGa2 , i.e., Case (E).Scenario 4: a2FpFa1 , i.e., Cases (C3) and (D).

Note that we have not included (C2) in any of the scenarios. This isbecause Case (C2) has only a fleeting existence at an instant of time whenthe (λ1 , λ2) trajectory crosses the boundary marked λ*.

In what follows, we state first the results concerning the movementbetween and within the scenarios�cases. Proofs of these results are inAppendix. Then, we discuss the implications of these results.

Proposition 4.1. If the optimal solution stays in Scenario 1 from anytime t1 on, then the penetration of the rich market has the following upperbound:

F1(t)⁄2AB (F2(t))C1B (F2(t))2A2B (F2(t))A1, ∀t∈[t1 , T], (23)

where

B (F2(t))Gexp[1AF1(t1)C(a1�a2)(F2(t)AF2(t1))]�(1AF1(t1)). (24)

Proposition 4.2.

(i) A transition from Case (E) to Case (D) is not optimal.(ii) A transition from Case (C1) to (C3) is not optimal.

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Fig. 6. Boundary lines separating the scenarios in the (λ1 , λ2) space for any given (F1 , F2).

Proposition 4.3. A necessary condition for Scenario 4 to hold at theterminal time T is that a1¤2a2 .

Proposition 4.4. If Scenario 4 holds throughout, then the solution isgiven by

λ1(t)G−a1A4�(tAk), where kGTC4�(SCa1),

p(t)G(a1Aλ1(t))�2Ga1C2�(tAk), see Fig. 7 ,

F1(t)G1A{[(SCa1)(TAt)C4]�[(SCa1)TC4]}2,

λ2(t)G{4(SCa1)2�[(SCa1)TC4]2}(tAT ).

Proposition 4.5. If Scenario 4 holds at the end of the horizon, then itholds throughout.

Proposition 4.6. The adjoint variable λ2(t)F0, ∀t∈[0,T ) andλ2(T )G0.

Discussion of Results. Proposition 4.1 provides an upper bound onthe penetration of the rich market in terms of the penetration of the poormarket when pG0 throughout. This suggests that setting pG0 throughoutis not optimal and the optimal solution will leave Scenario 1 at some pointin time. The resulting increase in price from pG0 to a strictly positive valueresults in a reduced growth of F2 , but the effect on the growth of F1 isambiguous. The reasons for the ambiguity are two opposing effects. The

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Fig. 7. Price trajectory in Scenario 4.

direct effect reduces the growth of F1. Indirectly, the slower growing F2yields an increasing growth rate of F1. If this indirect effect surpasses thedirect effect, F1(t) may then exceed the upper bound given in Proposition4.1. Numerical computations confirm this observation.

Proposition 4.3 says that, if a1 is not sufficiently high compared to a2 ,then we will never have an exclusive policy throughout the horizon. Thismeans that we can go from Scenario 4 to Scenario 2, but not vice versa.

Proposition 4.4 gives the price trajectory when the product is sold onlyto the rich market. This price is decreasing as shown in Fig. 7. It may switchto Case (C1), where price is below a2. Depending on the value of S, thismay be the last regime, or it may switch to Case (C1).

5. Numerical Exploration of Policies

We have so far explored the canonical system numerically to try tocharacterize the types of marketing policies obtained and their related pricepaths. We also examined a large number of cases varying each parameterover a wide range, as shown in Table 1.

Table 1. Parameter values investigatedin the numerical analysis.

Parameter Values

a1 2.5, 5, 10S 0, 10, 20, 30, 83, 100M 5, 10, 20t 0.7, 0.5

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The price to the poor population, parameter a2 , was held at 1 through-out; thus, we explored cases where the ratio a1�a2 between the reservationprices for the rich and poor populations was 2.5, 5, 10. Since a1 and Ma2

are the sales potentials of the rich and poor markets respectively, our par-ameter choices enabled us to examine poor markets that are from 0.5 to 8times the size of the rich market.

Three broad categories of policies emerge. The first category consistsof price paths which we will refer to as exclusivity strategies. This categoryof policies emerges when the poor market is not more than twice the size ofthe rich market and S is small. Here, the price path is a declining one, butwith pHa2 for at least the initial period. Then, the price jumps down to alevel below a2 and declines for the rest of the time horizon. Included in thiscategory are price paths with pHa2 for the entire time horizon. Sales aretherefore limited to the rich population for at least the first part of the timehorizon, and this makes the product more exclusive for the poor populationwhen it finally becomes affordable to this group toward the end of the timehorizon. And, once the price drops, the remaining potential of the rich mar-ket is still accessible to the product. This type of strategy may be followedby a monopolist who periodically replaces an existing model of the productwith a new one. At the end of the life cycle of each model, the price isdropped, to obtain some additional sales from the poor population withoutdamaging much the exclusive image of the product and still enjoy close tothe maximum salvage value. Included in this category of policies is thespecial case where the price declines to a2 , remains at a2 for some time (stilldenying access to the poor population), and then declines below a2 .

Now, we relate these price paths to the scenarios and cases discussedearlier. The solutions start in Scenario 4, Cases (D) and (C3), with pHa2 ;the price then declines until it encounters the border between Cases (C3)and (C1). There the price jumps and enters Scenario 2, Cases (C3) and (B).In the special case mentioned, the price path again starts in Case (D), entersCase (E), and eventually enters Case (B). These price paths are illustratedin Figs. 8 and 9.

The impact of S on the price path is quite significant. When S is verylarge, the monopolist might do well to essentially give away the product sothat F1(T ) is very close to 1 and SF1(T ) is as large as possible. However,when S is small, the monopolist makes money by selling the product at ahigh price, rather than relying on a high salvage value. Thus, for the samemarket potential, the length of time t* for which pHa2 depends on S: thehigher the salvage value S, the lower is the value of t*.

The second category of policies is observed when the potential of thepoor market is large compared to the rich market. Here, p⁄a2 and is mono-tonically declining throughout. Thus, no effort is made to position the prod-uct as exclusive. Of course, some members of the rich population purchase

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Fig. 8. Price path with a jump. Cases (D)–(C3)–(C1)–(B). The parameters chosen area1G5, a2G1, SG30, MG10, TG0.7.

the product, but they are doing so because of its price and in spite of thefact that many members of the poor population have purchased the productas well. This price path corresponds to Case (B) or (C1). A typical exampleis illustrated in Fig. 10.

The third category of policies is also observed for large potential poormarkets, but here, we do not have a monotonically decreasing price path,but rather the price increases and then declines. Here, p⁄a2 always, soagain there is no effort to position the product as exclusive. A special caseof this category of price paths occurs when pG0 for a small initial period.In practice, this implies that the product would be given away (or sold atcost) to create a market buzz, to stimulate sales to both rich and poorpopulations. One possibility here would be to restrict these giveaways tomembers of the rich population with attendant publicity so that some exclu-sivity effect can be obtained, and used to stimulate sales further to the poorpopulation. A second special case ends the time horizon with a brief periodwith pG0, driving up the salvage value for cases where S is large. This casecorresponds to the situation where inventories of an old model of the prod-uct are being disposed off in order to clear space for a new model. A thirdspecial case arises when the initial increasing price hits pGa2 and stays there

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Fig. 9. Price path without a jump with constant segment pGa2 . Cases (D)–(E)–(B). Theparameters chosen are a1G5, a2G1, SG50, MG5, TG0.7.

a while before it starts declining. The declining phase could terminate withpFa2 or go all the way to pG0. In terms of the scenarios analyzed earlier,we have Case (B), with special cases of the price path transiting from Case(A) to Case (B), from Case (A) to Case (B) back to Case (A), and fromCase (B) to Case (E) and then back to Case (B) and then possibly to Case(A). Figures 11–14 illustrate these price paths.

6. Concluding Remarks

We have formulated an optimal control problem to obtain the priceover time in order to maximize profit from a product that diffuses in richand poor populations. Depending on the relative sales to each of thesepopulations, the product could be characterized as a status symbol or not.We find that, depending on the parameters of the problem, price can beclassified into one of three categories. In the first category, the product is astatus symbol, whose initial price is prohibitive for the poor. Price declinesover time to levels affordable to the poor. In the second category, the prod-uct is not a status symbol. The initial price is affordable to the poor and

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Fig. 10. Decreasing price path with pFa2 . Cases (B)–(C1). The parameters chosen area1G2.5, a2G1, SG0, MG20, TG0.7.

price declines over time. In the third category, the product again is not astatus symbol, and price always remains affordable to the poor. However,here price increases initially and eventually declines.

The optimal control problem solved here is nonlinear with two statevariables. The dynamics of the rich and the poor markets are asymmetric,which makes the problem interesting as well as complicated. A substantialpart of the problem is addressed analytically. The results are obtained usingboth analytical and numerical means.

The model presented here considers the undiscounted case. An obviousextension is to consider a positive discount rate. This will substantially com-plicate the problem, and may result in price paths that may have more thanone change in the sign of the derivative of price. However, we do not expectmajor changes to the results presented here.

A more interesting extension would consider future generations ofproducts, with one generation being replaced by the next. In our model, wehave recognized this issue through the salvage value S. A more realistictreatment would involve not only the optimal pricing of the successive prod-ucts but also the timing of their introductions. Some work along these lines

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Fig. 11. One-humped price path with 0FpFa2 . Case (B). The parameters chosen area1G2.5, a2G1, SG10, MG5, TG0.7.

has been done in the context of machine maintenance and replacement (seeRef. 11, Chapter 9).

7. Appendix: Proofs

Proof of Proposition 3.1. Part (A) follows immediately from (19).Note that λ1¤λH can also be expressed as

λ1(1AF1)Cλ2F1¤a1(1AF1)CMa2F1 .

Part (B) follows immediately from (20) and (21), since in this region thereis no ambiguity. Part (C) is a result of the overlap of the conditions in (20)and (21). Solving this ambiguity by evaluating H for both values of p andafter some tedious calculations identifying the global maximum yields λ* asthe hairline case. Note that λ* is clearly a value of λ1 that satisfies

a1A2a2¤λ1¤λL,

since

11CMF1�(1AF1)F1CMF1�(1AF1).

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Fig. 12. One-humped price path starting with pG0. Cases (A)–(B). The parameters chosenare a1G2.5, a2G1, SG10, MG20, TG0.7.

Part (D) follows again from (20), (21), and Lemma 3.1, since in this regionthere is no ambiguity. �

Proof of Proposition 3.2. Part (A) follows immediately from (19). Part(B) follows immediately from (20). Part (D) follows again from (21).

Unlike in Region 2, now there is no overlap of the conditions in (20)and (21). Thus, Case (C) cannot occur. However, now there is a regionwhen λLHλ1Ha1A2a2 , where neither the conditions in (20) nor in (21) aresatisfied. In this case, there are no interior local optima. Rather, the optimalp is at the kink a2. �

The following lemma is needed for the proof of the next proposition.

Lemma 7.1. The following result holds:

−exp(1Ax)⁄ [−5�2C2xAx2�2], ∀x⁄1.

Proof. This follows from a Taylor series expansion of Aexp(1Ax) upto order 2 around xG1. �

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Fig. 13. One-humped price path starting and ending with pG0. Cases (A)–(B)–(A). Theparameters chosen are a1G2.5, a2G1, SG20, MG5, TG0.7.

Proof of Proposition 4.1. For pG0, the state equations reduce to

dF1�dtG(1AF1)(1AF2)a1 , (25)

dF2�dtGF1(1AF2)a2 . (26)

It is possible to give an upper bound for F1(t) in terms of F2(t) for solutionswhich stay in Case (A). Along optimal solutions within Case (A), thedifferential equation

dF1�dF2Ga1(1AF1)�a2F1

has to hold, which implies

dF2G[a2F1�a1(1AF1)]dF1 .

Integrating yields

F2(t)GF2(t1)A(a2�a1){F1(t)AF1(t1)Clog[(1AF1(t))�(1AF1(t1))]},

which after some algebraic manipulations gives

F1(t)G1Aexp(1AF1(t))�B(F2(t)).

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Fig. 14. One-humped price path spending some time at pGa2. Cases (B)–(E)–(B). The par-ameters chosen are a1G5, a2G1, SG63, MG5, TG0.65.

From Lemma 7.1, it then follows that

F1(t)⁄1C[−5�2C2F1(t)AF1(t)2�2]�B(F2(t)),

which is equivalent to

0⁄B(F2(t))A5�2CF1(t)[2AB(F2(t))]AF1(t)2�2. (27)

As

B(F2(t))¤exp(1AF1(t1))�(1AF1(t1))¤e,

(27) is equivalent to (23). �

Lemma 7.2. Parts of solutions which follow Scenario 4 within the timeinterval (t1 , t2) are given by

F1(t)G1A[1AF1(t1)][(kAt)�(kAt1)]2, (28)

F2(t) ≡ F2(t1), (29)

λ1(t)G−a1A4�[(1AF2(t1))(tAk)], (30)

λ2(t)Gλ2(t1)C4(1AF1(t1))(tAt1)�[(1AF2(t1))2(kAt1)

2], (31)

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where

kGt1C4�[(λ1(t1)Ca1)(1AF2(t1))],

for all t∈(t1 ,t2). Note that kHt2 .

Proof. Under Scenario 4, i.e., for a2FpFa1 , the price is given by

pG(a1Aλ1)�2.

The state and the adjoint equations are

dF1�dtG(1AF1)(1AF2)[(a1Cλ1)�2], (32)

dF2�dtG0, (33)

dλ1�dtG[(a1Aλ1)�2Cλ1 ](1AF2)[a1A(a1Aλ1)�2]

G[(a1Cλ1)�2]2(1AF2), (34)

dλ2�dtG[(a1Aλ1)�2Cλ1 ](1AF1)[a1A(a1Aλ1)�2]

G[(a1Cλ1)�2]2(1AF1). (35)

Equation (29) is an immediate consequence of (33). The solution of (34) isthen given by (30), where the constant k follows from the initial conditionat tGt1 . From (32), we obtain also

F1(t)G1A[1AF1(t1)] exp�−�t

t1

[1AF2(t1)][(a1Cλ1(τ ))�2] dτ�G1A[1AF1(t1)][(kAt)�(kAt1)]

2,

where we have used

−�t

t1

[1AF2(t1)][(a1Cλ1(τ ))�2] dτ

G−�t

t1

[2�(kAτ )] dτG2 log[(kAt)�(kAt1)].

Furthermore, (35) reduces to

dλ2�dtG{−2�[1AF2(t1)](tAk)}2[1AF1(t1)][(kAt)�(kAt1)]2

G4[1AF1(t1)]�{[1AF2(t1)]2(kAt1)

2}, (36)

with its solution given by (31). �

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Proof of Proposition 4.2.

(i) The time derivative of λ1 along the border line λ1Ga1A2a2 isgiven by

λ̇1 �λ1Ga1A2 a2G(1AF2)(a1Aa2)2H0.

(ii) A transition from Scenario 2 to Scenario 4 can only occur inRegion 1. In Region 1, Scenario 4 holds if and only if

λ1(t)Fλ*(t)

Ga1Aa2Cλ2(t)�MA[a2Cλ2(t)�M]11CMF1(t)�[1AF1(t)],

which is equivalent to

g(t)Fa1A2a2 ,

with

g(t)Gλ1(t)C[a2Cλ2(t)�M]{11CMF1(t)�[1AF1(t)]A1}.

Note that g(t) is monotonically increasing under Scenario 4 in Region 1, asthe solutions given by (28), (30), (31) are monotonically increasing.

Let us assume that the transition from Scenario 2 to Scenario 4 occursat time t1. However, this is not possible as it would imply

g(t1)Ga1A2a2 , g(t)Fa1A2a2 , for all t∈(t1 , t1C(),

for some (H0, which is a contradiction to the monotonicity of g(t). �

Proof of Proposition 4.3. Scenario 4 holds at time T ifSFS*Ga1Aa2 [1C11CMF1�(1AF1)]. Obviously, S*Fa1A2a2 ; therefore,0⁄S together with SFS* require a1¤2a2 . �

Proof of Proposition 4.4. Apply the result of Lemma 7.2 to the caset1G0, t2GT together with the boundary conditions

F1(0)GF2(0)G0, λ1(T )GS, λ2(T )G0.

Equation (30) together with F2(0)G0 then yields

λ1(t)G−a1A4�(tAk),

where

kGTC4�(SCa1)

follows from

λ1(T )GS and p(t)G[a1Aλ1(t)]�2Ga1C2�(tAk).

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Inserting k into (28) gives

F1(t)G1A{[(SCa1)(TAt)C4]�[(SCa1) TC4]}2.

The solution for λ2(t) follows from the differential equation (36) togetherwith the transversality condition λ2(T )G0 �

Proof of Proposition 4.5. Proposition 4.5 is an immediate consequenceof Proposition 4.2. Solutions which end in Scenario 4 have to completely liein Scenario 4 as a transition from Scenario 3 as well as from Scenario 2 toScenario 4 is not possible. �

Proof of Proposition 4.6. Since we have λ2(T )G0 from the trans-versality condition, it is sufficient to show that λ2(t)G0 implies λ̇2(t)H0.Assume λ2(t)G0 at some t. Corresponding to the different scenarios, weconsider the following cases:

Scenario 1. Here,

λ̇2 �λ2G0G(1AF1)λ1a1H0

as in Scenario 1 and

λ1¤a1CMa2F1�(1AF1)H0.

Scenario 2. At λ2G0, this scenario occurs for

S*Ga1Aa2 [1C11CMF1�(1AF1)]Fλ1Fa1Ca2MF1�(1AF1).

The price is given by

p�λ2G0G[(1AF1)(a1Aλ1)CF1M a2 ]�[2(1AF1)C2MF1 ].

Defining

αGF1M�(1AF1),

we get

λ̇2 �λ2G0G(1AF1)[( pCλ1)(a1Ap)Cαp(a2Ap)],

with

pG(a1Aλ1Cαa2)�2(1Cα ).

Inserting for p yields

λ̇2 �λ2G0G[(1AF1)�4(1Cα )] f (λ1),

with

f (λ1)Gλ21C2λ1(a1C2αa1Aαa2)C(a1Cαa2)

2.

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The quadratic equation f (λ1)G0 has real roots

−(a1C2αa1Aαa2)J21α (1Cα )a1(a1Aa2).

Since f (λ1) is convex, it is sufficient to show that

S*¤A(a1C2αa1Aαa2)C21α (1Cα )a1(a1Aa2), (37)

as this implies

f (λ1)H0, ∀λ1HS*.

Inequality (37) is equivalent to

(2a1Aa2)(1Cα )Aa211CαA21α (1Cα )a1(a1Aa2)¤0. (38)

Inequality (38) holds, since the LHS of (38) is equal to

1(1Cα )[1C11Cα ][1(a1Aa2)A1αa1�(1C11Cα )]2.

Scenario 4. Here,

λ̇2 �λ2G0G(1AF1)[(a1Cλ1)�2]2H0. �

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