J Optim Theory Appl (2008) 137: 363–379DOI 10.1007/s10957-007-9330-x

Dynamic Slotting and Pricing Decisions in a DurableProduct Supply Chain

X. He · S.P. Sethi

Published online: 13 December 2007© Springer Science+Business Media, LLC 2007

Abstract We consider a supply chain in which a manufacturer sells an innovativedurable product to an independent retailer over its life cycle. We assume that theproduct demand follows a Bass-type diffusion process and that it is determined bythe market influences, retail price of the product, and shelf space allocated to it. Weconsider the following retailer profit optimization strategies: (i) the myopic strategyof maximizing the current-period profit and (ii) the far-sighted strategy of maximiz-ing the life-cycle profit. We characterize the optimal dynamic shelf-space allocationand retail pricing policies for the retailer and wholesale pricing policies for the man-ufacturer. We compute also these policies numerically. Surprisingly, we find that themanufacturer, and sometimes even the retailer, is better off with a myopic retailerstrategy in some cases.

Keywords Stackelberg differential games · Bass model · Pricing · Slotting · Supplychain management

1 Introduction

Consider a supply chain in which a manufacturer (he) sells an innovative durableproduct (IDP) to an independent retailer (she) over its life cycle of a fixed time hori-zon. During this period, the retailer makes decisions to influence the retail demandfor the IDP in order to maximize her profit objective. We focus on four importantfactors that affect the retail demand of the IDP. (a) Diffusion effect (network effect)by which we mean that the customers who have purchased the IDP inform thosewho have not. (b) Saturation effect: Because the product is durable, the consumers

Communicated by G. Leitmann.

X. He · S.P. Sethi (�)School of Management, University of Texas at Dallas, Richardson, TX, USAe-mail: sethi@utdallas.edu

364 J Optim Theory Appl (2008) 137: 363–379

purchase only once during the selling horizon of the IDP. Therefore, the more the cu-mulative adopters, the smaller is the remaining potential market. (c) Retail price: Theconsumer demand for the product is inversely related to the unit retail price charged.(d) Shelf space: While the shelf space has a positive impact on the demand for theIDP, it is a scare resource for the retailer.

We formulate the problem in a Stackelberg differential game framework. We as-sume that the manufacturer takes the role of the leader in his relationship with theretailer. Thus, the manufacturer announces his wholesale price to the retailer, and theretailer decides on the retail price and the shelf-space allocation over time in order tomaximize her profit objective, taking the wholesale price as given. When setting thewholesale price, the manufacturer takes the retailer’s best response into considerationin order to maximize his life-cycle profit. We consider the following two retailer’sprofit strategies: (i) far-sighted strategy and (ii) myopic strategy. By far-sighted wemean that the retailer maximizes her life-cycle profit, whereas by myopic we meanthat the retailer maximizes her instantaneous profit rates at each time instant in theselling horizon.

We address the following research questions: (Q1) For a given retailer’s profitstrategy, what are the optimal pricing and slotting policies for the retailer and the op-timal wholesale pricing policy for the manufacturer? (Q2) Should the retailer be far-sighted or myopic? (Q3) Does the manufacturer prefer the retailer to be far-sightedor myopic? (Q4) Is there a conflict of preference between the manufacturer and theretailer?

The solution concept for the Stackelberg differential game that we use is the open-loop equilibrium. This means that the manufacturer and the retailer decide on theirrespective policies at the start of the game. It is known that an open-loop equilibriumin the far-sighted case is time inconsistent if the leader cannot credibly commit tohis policy. In other words, if the leader is given the opportunity to revise his policy,he would, at sometime during the selling horizon, switch to another policy differentfrom the one he chose at the beginning of the game. Hence, open-loop policies makesense only when the leader can pre-commit to his policy. In this study, we assumethat the manufacturer commits to his wholesale pricing policy.

There is a vast literature on supply chain management dealing with stochastic de-mand and modeled as newsvendor problems (Ref. [1]). A serious limitation of suchanalyses is that they address only single period models. Yet the real world prob-lems are dynamic. Our work is an early attempt to formulate supply chain man-agement problems as dynamic games of Stackelberg type. In order for us to have atractable model, we focus on a deterministic demand and look for open-loop equi-libria. However, this is just a beginning, and we expect this work to inspire re-searchers to analyze many dynamic and stochastic problems in supply chain man-agement.

Our contributions include characterization of optimal solutions and their numeri-cal computations. Furthermore, based on these, we conclude that the following possi-bilities may arise in terms of the preferred profit strategies of the game players. First,the manufacturer may not always prefer the retailer to have a far-sighted strategy,i.e. the manufacturer sometimes is better off if the retailer is myopic. On the otherhand, the retailer’s preference (myopic/far-sighted) changes with market character-istics, and they do not always agree with the manufacturer’s preference. Our results

J Optim Theory Appl (2008) 137: 363–379 365

show that both the manufacturer and the retailer are better off if the retailer is far-sighted when the final market saturation level is low. However, if the level is high atthe end of the horizon, the manufacturer is better off with a myopic retailer, while theretailer prefers that the manufacturer sets his wholesale prices assuming that she isfar-sighted. The conflict between the manufacturer and the retailer over the preferredretail profit focus raises some contract implementation issues that are worth exploringas a future research topic.

Our research is closely related to the new product diffusion and shelf-space al-location models in the marketing literature and the differential game models in theoptimal control literature. The Bass diffusion model and its variants have been widelyused to forecast demand of a durable new product. The variants incorporate the priceimpact on the retail demand of an IDP in monopoly and oligopoly settings. We referthe readers to Ref. [2] for a comprehensive review of this literature.

In the context of optimal dynamic pricing, an important consideration is whether afirm maximizes its short-term profit or its long-term profit. Bass and Bultez (Ref. [3])assume that the firm maximizes the instantaneous profit, and the pricing strategies thatthey obtain are myopic in nature, as compared to (far-sighted) optimal pricing policieswhich maximize the firm’s aggregated profit over the product’s life cycle. Robinsonand Lakhani (Ref. [4]) compared the total profits resulting from far-sighted optimalpricing and myopic optimal pricing. Their numerical results show that the differencesin profits are significant, whereas Bass and Bultez report only small differences. Asnoted in Ref. [5], it is very critical to properly incorporate the impact of pricinginto the demand model. Several papers including Ref. [4] assume demand to be anexponential function of price. In contrast, as in Ref. [6], we assume that the demandis a linearly decreasing function of price.

Differential game models have been applied to analyze the strategic dynamic in-teractions between the players in supply chains, as reported in a review by He et al.(Ref. [7]). Our work is closest in spirit and structure to Ref. [8], who analyze dynamicpricing decisions in a Stackelberg differential game framework in the context of anIDP. We extend Ref. [8] by considering the impact of shelf-space allocation on theretail demand. As in Ref. [9], we assume the retail demand to be an increasing andconcave function of the merchandise displayed on the shelf. This enables us to studythe dynamic slotting decision of the retailer in addition to her pricing decision. Toour knowledge, this is the first paper that determines optimal pricing and shelf-spaceallocation simultaneously in a dynamic game framework.

The rest of the paper is organized as follows. In Sect. 2, we introduce the demandmodel. In Sect. 3 we study the case of a myopic retailer. In Section 4 we study thecase of a far-sighted retailer. In Sect. 5, we present a numerical study that comparesthe cases of far-sighted and myopic foci. In Sect. 6, we conclude by summarizing theresults and pointing out future research directions.

2 Demand Model

A manufacturer produces an IDP whose retail demand follows a Bass-type diffusionprocess. Let X(t) and X(t) = x(t) be the cumulative sales and the instantaneous

366 J Optim Theory Appl (2008) 137: 363–379

demand (rate of sales) at time t , 0 ≤ t ≤ T , where T denotes the selling horizon ofthe IDP. The demand dynamics are described by the differential equation

X(t) = √p(t)(M − X(t))(α + βX(t))(1 − γ r(t)), X(0) = X0, (1)

where p(t) is the shelf space allocated to the product at time t ∈ [0, T ], M denotes thepotential market size, (M −X(t)) is the unsaturated market size, α and β are positivecoefficients of external and internal market influences, respectively, and the parame-ter γ > 0 measures the customer’s sensitivity to the retail price r(t). According toour formulation, the sales rate x(t) is determined by four factors: the external marketinfluence, the internal market influence, the slotting decision, and the retail price. Weuse a multiplicatively separable function in (1) to model the impact of price, shelfspace, and cumulative sales on the instantaneous demand x(t). The instantaneousdemand is a linearly decreasing function of the retail price. The

√p(t) term in (1)

signifies that the shelf space has marginal diminishing returns with respect to the in-stantaneous demand. The effect of the cumulative sales on the instantaneous demandis as follows. Initially, the market is not saturated and the diffusion effect outweighsthe saturation effect. Over time, the market gets saturated, which makes additionalsales more difficult; thus as time progresses, the saturation effect starts dominatingthe diffusion effect.

We use superscripts M , R, and C to denote the manufacturer, the retailer, andthe channel variables, respectively. We let πR(t), ΠR(T ), and ΠM(T ) denote the re-tailer’s profit rate at time t, her total profit over the horizon T , and the manufacturer’stotal profit over the horizon T , respectively.

3 Myopic Retailer

We consider the case of a retailer with short-term profit focus. The manufacturer andthe retailer play a Stackelberg differential game. The sequence of plays is as follows.The manufacturer announces the wholesale price trajectory w(·). Then the retailersimultaneously decides the retail price trajectory r(·) and the shelf-space trajectoryp(·). Rewriting shelf-space allocation p(t) as (c(t))2 ≡ c2(t), and calling c(t) as theslotting decision, and with s0 denoting the retailer’s unit selling cost and s denotingthe unit cost of the shelf space, we formulate the retailer’s optimization problemat instant t for any given wholesale price trajectory w(·) = (w(t),0 ≤ t ≤ T ). Theretailer’s myopic profit is

πR(t) ≡ πR(X(t), r(t), c(t);w(·)) = [r(t) − w(t) − s0]X(t) − sc2(t). (2)

The problem is to choose r(t) and c(t) to maximize πR(t),0 ≤ t ≤ T , subject to

X(t) = c(t)(M − X(t))(α + βX(t))[1 − γ r(t)], X(0) = X0. (3)

It should be obvious in the myopic case that the retailer’s best response at timet will depend only on the past of the announced wholesale price trajectory, i.e. on{w(τ), 0 ≤ τ ≤ t}. Moreover, we can characterize the structure of these decisions

J Optim Theory Appl (2008) 137: 363–379 367

by using the first-order conditions for the maximum of πR(t). Specifically, solving∂πR(t)/∂r(t) = 0 and ∂πR(t)/∂c(t) = 0 simultaneously gives the best response

r(t) = r∗(X(t);w(·)) ≡ [1 + γ (w(t) + s0)]/2γ, (4)

c(t) = c∗(X(t);w(·)) ≡ F(X(t))[1 − γ (w(t) + s0)]2/8γ s, (5)

where F(X) = (M − X)(α + βX). Note that the best retail price response r(t) de-pends only on w(t) and the best slotting response c(t) depend on the cumulative salesX(t) and the wholesale price w(t) at time t. The manufacturer takes the retailer’s bestresponse (4) and (5) into consideration when solving his profit maximization prob-lem over the selling horizon T . That is, he uses (4) and (5) in (3) to obtain his stateequation, and his problem is

maxw(·)

∫ T

0[w(t) − c0]X(t)dt, (6)

X(t) = F 2(X(t))[1 − γ (w(t) + s0)]3/16γ s,X(0) = X0, (7)

where c0 is the manufacturer’s unit production cost. We use the Maximum principle tosolve this optimal control problem (see Ref. [10]). The manufacturer’s Hamiltonian is

HM(t) ≡ HM(X(t),w(t), λM(t)) = (w(t) − c0 + λM(t))X(t)

= F 2(X(t))(w(t) − c0 + λM(t))[1 − γ (w(t) + s0)]3/16γ s, (8)

where λM(t), the shadow price associated with the state variable X(t), satisfies theadjoint equation λM(t) = ∂HM(t)/∂X(t), i.e.,

λM(t) = −F ′(X(t))F (X(t))(w(t) − c0 + λM(t))[1 − γ (w(t) + s0)]3

8γ s,

λM(T ) = 0.

(9)

The transversality condition λM(T ) = 0 arises from the fact that X(T ) is free. Usingthe first-order condition ∂HM(t)/∂w(t) = 0 gives the optimal wholesale price

w(t) = {1 + γ [3c0 − 3λM(t) − s0]}/4γ. (10)

We note that the higher the λM(t), the lower is the wholesale price w(t). This resultis the demonstration of the economic interpretation of the shadow price: λM(t) is thefuture value of an additional unit of sales at time t. When λM(t) > 0, the higher theλM(t), the larger is the future value of the additional sales at time t. Thus, the man-ufacturer has an incentive to lower the wholesale price w(t) to stimulate immediatesales. Substituting the optimal wholesale price w(t) from (10) into r(t), c(t), x(t),

and λM(t), we have their values in equilibrium:

r(t) = {5 + 3γ [c0 + s0 − λM(t)]}/8γ, (11)

c(t) = F(X(t))[1 − γ (c0 + s0 − λM(t))]2/128γ s, (12)

x(t) = 27F(X(t))2[1 − γ (c0 + s0 − λM(t))]3/1024γ s, (13)

λM(t) = −27F ′(X(t))F (X(t))[1 − γ (c0 + s0 − λM(t))]4/2048sγ 2. (14)

368 J Optim Theory Appl (2008) 137: 363–379

Note that the equilibrium values of w(t), r(t), c(t), x(t), and λM(t) are functionsof the state variable X(t) and the shadow price λM(t). Also, since the equilibriumvalues of r(t) and x(t) depend on λM(t), the retailer’s optimal decisions depend onthe entire wholesale price trajectory w(·). Furthermore, one can easily see that thereis no reason for either player to change their policy in the middle of the game, andtherefore, the equilibrium in the myopic case is time consistent.

In the following lemma, we express λM(t) in terms of X(t) and X(T ).

Lemma 3.1 The shadow price trajectory λM(t) is given by

λM(t) = 1 − γ (c0 + s0)

γ

[√(M − X(T ))(α + βX(T ))

(M − X(t))(α + βX(t))− 1

], t ∈ [0, T ]. (15)

Proof From (13) and (14), we have

X(t)

λM(t)= − 2γF(X(t))

F ′(X(t))[1 − γ (c0 + s0 − λM(t))] ,

which can be integrated from t to T to obtain (15). �

We observe that the sign of λM(t) depends on the ratio of F(X(T )) to F(X(t)).

Thus, when F(X(T )) ≥ F(X(t)), λM(t) > 0 and when F(X(T )) ≤ F(X(t)),λM(t) < 0.

Lemma 3.2 In an optimal solution for the myopic retailer, the wholesale price, theretail price, the slotting decision, and the instantaneous sales rate are, respectively,

w(t) = {4(1 − γ s0) − 3√

F(X(T ))/F (X(t))[1 − γ (c0 + s0)]}/4γ, (16)

r(t) = {8 − 3√

F(X(T ))/F (X(t))[1 − γ (c0 + s0)]}/8γ, (17)

c(t) = 9F(X(T ))[1 − γ (c0 + s0)]2/128γ s, (18)

x(t) = 27F12 (X(t))F

32 (X(T ))[1 − γ (c0 + s0)]3/1024γ s. (19)

Furthermore, the slotting decision is constant over time, and the retail price, thewholesale price, and the instantaneous sales rate all peak at the same time.

Proof Substitute for λM(t) from (15) to (11), (10), (13) and (12) to obtain (17), (16),(19), and (18), respectively. The last part of the lemma is obvious from a comparisonof (17), (16) and (19). �

Lemma 3.3 The optimal cumulative sales X(t) is the unique solution to the equation

tan−1[α − βM + 2βX(t)

2√

βF(X(t))

]

= tan−1[α − βM + 2βX(0)

2√

β(X(0))

]+ 27

√βtF 3/2(X(T ))[1 − γ (c0 + s0)]3

1024γ s. (20)

J Optim Theory Appl (2008) 137: 363–379 369

Proof Integrate (19) from 0 to t and rearrange terms to obtain (20). In order to showthe uniqueness, one can show that for t = T , (20) has a unique solution for X(T ). Inview of the fact that the left-hand side is increasing in X(t), it follows that X(t) isunique. �

Lemma 3.4 In the optimal solution for the myopic retailer,

πR(t) = 81F 2(X(T ))[1 − γ (c0 + s0)]4/s(128γ )2 (21)

is constant over time, and her total profit over the horizon T is

ΠR(T ) =∫ T

0πR(τ)dτ = 81T F 2(X(T ))[1 − γ (c0 + s0)]4/s(128γ )2. (22)

The manufacturer’s total profit over the horizon T is

ΠM(T ) = [X(T ) − X(0)][1 − γ (c0 + s0)]γ

− 81T tF 2(X(T ))[1 − γ (c0 + s0)]4

s(64γ )2. (23)

Proof Substituting w(t), r(t), c(t), and x(t) obtained in Lemma 3.2 into (2) gives(21). Integrating (21) from 0 to T gives (22). Substituting w(t) and x(t) obtained inLemma 3.2 in (6) gives (23). �

Based on these results, we conduct a numerical study with the following parametervalues M = 1 × 107,X0 = 0, α = 0.016, β = 8 × 10−9, c0 = 100, s0 = 20, s =5 × 107, and γ = 5 × 10−4. We compute the optimal decisions for various valuesof the horizon T . Figure 1 shows the wholesale price trajectories for different valuesof T . We observe two patterns of wholesale price trajectories: increasing over time(for T = 25,40,45,60) and initially increasing then decreasing (for T = 75,80).Also, a comparison of (17), (16), and (19), or of Figs. 1–3, reveals that the trajectoriesof w(·), r(·), and x(·) mimic one another. Furthermore, we observe that for lowervalues of T , the wholesale price curves move downward as T increases, and beyonda certain value of T , the wholesale price curves move upward as T increases.

Using the same parameter values, we continue our numerical study to get furtherinsights. Table 1 reports the manufacturer’s profit ΠM(T ), the retailer’s profit ΠR(T ),and the profit ratios ΠM(T )/ΠC(T ) and ΠR(T )/ΠC(T ), where ΠC(T ) = ΠM(T )+ΠR(T ) denotes the channel’s profit, and the slotting decision c, which is constantthroughout the horizon, and the market saturation level X(T )/M for different valuesof T . Interestingly, we find that the impact of T on the manufacturer’s (retailer’s)share of the channel profit is not uniform in T . That is, his share of the channelprofit initially decreases (increases) as T increases, and beyond a certain value of T ,his share of profit increases (decreases) as T increases. As for the constant slottingdecision c, it initially increases as T increases, and beyond a certain value of T , itdecreases in T .

370 J Optim Theory Appl (2008) 137: 363–379

Fig. 1 Wholesale price trajectories for different horizons: Myopic retailer

4 Far-Sighted Retailer

In this section, we study the model with a retailer focused on the long-term profit. Weuse a bar over the variables to signify the retailer’s far-sighted focus. The retailer’sproblem for a given manufacturer’s wholesale price trajectory w(·) is

maxr(·),c(·)

∫ T

0{[r(t) − w(t) − s0]x(t) − sc2(t)}dt, (24)

x(t) = ˙X(t) = c(t)(M − X(t))(α + βX(t))[1 − γ r(t)], X(0) = X0. (25)

The retailer’s Hamiltonian HR(t) at time t , given the wholesale price trajectory w(·)is

HR(t) = HR(X(t), r(t), c(t), λR(t); w(·))= cF (X(t))(1 − γ r(t))(r(t) − w(t) − s0 + λR(t)) − sc2(t), (26)

where λR(t), the shadow price associated with X(t), satisfies the adjoint equation

˙λR(t) = −[∂HR(t)/∂X(t)], λR(T ) = 0. (27)

J Optim Theory Appl (2008) 137: 363–379 371

Fig. 2 Retail price trajectories with different horizons: Myopic retailer

The first-order conditions for maximization of HR(t) are given by

∂HR(t)/∂r(t) = −γF(X(t))[r − w(t) − s0 + λR(t)]+ F(X(t))(1 − γ r(t)) = 0, (28)

∂HR(t)/∂c(t) = F(X(t))(1 − γ r(t))[r(t) − w(t) − s0 + λR(t)]− 2sc(t) = 0. (29)

Their solution yields the best response retail price r(t) and slotting decision c(t) asfollows:

r(t) = r∗(X(t), λR(t); w(·)) ≡ [1 + γ (w(t) + s0 − λR(t))]/2γ, (30)

c(t) = c∗(X(t), λR(t); w(·)) ≡ F(X(t))[1 + γ (λR(t) − w(t) − s0)]2/8γ s. (31)

We make a number of observations from (30) and (31). First, for a given w(·),the retail price r(t) is a decreasing function of λR(t). Intuitively, when λR(t) > 0,meaning that there is a positive future value of additional sales at time t, the retaileris willing to lower the retail price below the myopic level (λR(t) = 0). On the otherhand, for a given w(·), the shelf-space allocation c2(t) is an increasing function ofλR(t). Thus, when there is positive future value of additional sales, the retailer iswilling to allocate more shelf space to the product to increase its sales when comparedto the myopic case.

372 J Optim Theory Appl (2008) 137: 363–379

Fig. 3 Instantaneous sales rate with T =75: Myopic retailer

Finally, since the retailer’s best response also depends on λR(t), which is af-fected by the future portion of the announced wholesale trajectory, i.e. on {w(τ ), t <

τ ≤ T }, we see that the best response depends indeed on the entire wholesale pricetrajectory w(·). This provides an important contrast to the case of the myopic retailer.As we will see, this dependence on λR(t) requires us to treat λR(t) as a state variablein the formulation of the manufacturer’s optimization problem. It is this requirementthat causes time inconsistency in the far-sighted case. For further details on the theoryof the Stackelberg differential games, see Ref. [11].

Substitution of (30) and (31) into (25) gives the instantaneous sales rate

x(t) = ˙X(t) = F 2(X(t))[1 + γ (λR(t) − w(t) − s0)]3/16γ s,

X(0) = X0.(32)

By substituting (30), (31), (32) into (26), we obtain the maximized Hamiltonian

H ∗R(t) ≡ H ∗

R(X(t), λR(t); w(·))= F 2(X(t))[1 + γ (λR(t) − w(t) − s0)]4/64sγ 2. (33)

J Optim Theory Appl (2008) 137: 363–379 373

Tabl

e1

Profi

ts,p

rofit

ratio

s,sl

ottin

gde

cisi

ons,

and

mar

kets

atur

atio

nle

vels

for

diff

eren

tT:M

yopi

cre

taile

r

T5

1025

4045

6070

7580

150

250

ΠM

(T)(

×108

)0.

5521

1.16

103.

4700

6.93

688.

4111

13.3

7716

.766

18.4

1920

.031

37.9

3053

.689

ΠR

(T)(

×108

)0.

431

0.96

183.

4819

8.40

2510

.407

15.3

6717

.543

18.3

7519

.076

22.8

2123

.177

ΠM

(T)

ΠC

(T)

×10

0%55

.69

54.0

547

.73

40.9

840

.05

42.0

444

.95

46.4

547

.90

61.6

170

.15

ΠR

(T)

ΠC

(T)×

100%

44.3

145

.95

52.2

759

.02

59.9

557

.96

55.0

553

.55

52.1

038

.39

29.8

5

c=

√ p0.

4167

0.43

860.

5278

0.64

820.

6801

0.71

570.

708

0.70

0.69

0.56

0.43

06

X(T

)M

×10

0%1.

222.

669.

2521

.57

26.6

239

.81

46.2

448

.89

51.2

468

.73

77.8

7

374 J Optim Theory Appl (2008) 137: 363–379

The adjoint equation (27) for the shadow price λR(t) = −[∂H ∗R(t)/∂X(t)] is

˙λR(t) = −F ′(X(t))F (X(t))[1 + γ (λR(t) − w(t) − s0)]4/32sγ 2,

λR(T ) = 0.(34)

The manufacturer takes the retailer’s best response into consideration. Thus, hisproblem is to obtain w(·) in order to maximize his profit

∫ T

0 [w(t)−c0]x(t)dt, subjectto (32) and (34). The manufacturer’s Hamiltonian is

HM(t) = HM(X(t), λR(t), w(t), λM(t), μ(t))

≡ [w(t) − c0 + λM(t)]X(t) + μ(t) ˙λR(t)

= F 2(X(t))(w(t) − c0 + λM(t))[1 + γ (λR(t) − w(t) − s0)]3/16γ s

− μ(t)F ′(X(t))F (X(t))[1 + γ (λR(t) − w(t) − s0)]4/32γ 2s, (35)

where λM(t) = −∂HM(t)/∂X(t) and μ(t) = −∂HM(t)/∂λR(t) are the shadowprices associated with X(t) and λR(t), and they satisfy the adjoint equations

˙λM(t) = − F ′(X(t))F (X(t))(w(t) − c0 + λM(t))[1 + γ (λR(t) − w(t) − s0)]3

8γ s

+ μ((F ′(X(t)))2 − 2βF(X(t)))[1 + γ (λR(t) − w(t) − s0)]4

32γ 2s, (36)

λM(T ) = 0,

˙μ(t) = −3F 2(X(t))(w(t) − c0 + λM(t))[1 + γ (λR(t) − w(t) − s0)]2

16s

+ μ(t)F ′(X(t))F (X(t))[1 + γ (λR(t) − w(t) − s0)]3

8γ s, (37)

μ(0) = 0.

The boundary conditions λ(T ) = 0 and μ(0) = 0 arise from the fact that x(T ) andλR(0) are free. Note that λR(t) > 0 (resp. < 0) when F ′(X(t)) > 0 (resp. < 0).

To derive the optimal wholesale price w(t), we use the first-order condition

∂HM(t)

∂w(t)= F 2(X(t))[1 + γ (λR(t) − w(t) − s0)]3

16γ s

− 3γF 2(X(t))(w(t) − c0 + λM(t))[1 + γ (λR(t) − w(t) − s0)]2

16γ s

+ μF ′(X(t))F (X(t))[1 + γ (λR(t) − w(t) − s0)]3

8γ s= 0. (38)

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Note that 1+γ (λR(t)−w(t)−s0) = 0 is ruled out as it would lead to x = 0 accordingto (32). Then, the other factor in (38) gives the equilibrium wholesale price

w(t) = {F(X(t))[1 + γ (3c0 − 3λM(t) + λR(t) − s0)]+ 2F ′(X(t))μ(t)[1 + γ (λR(t) − s0)]}× [2γ (2F(X(t)) + F ′(X(t))μ(t))]−1. (39)

By substituting (39) into (32), (34), (36), and (37), we obtain a two-boundary valueproblem consisting of four differential equations. We solve this problem numericallyfor the same set of the parameters values used in Sect. 3 for the myopic retailer.

The open-loop equilibrium we obtain in this case is time inconsistent. This isbecause μ(t) does not stay at its initial value of zero. So if μ(τ ) = 0 at some time τ >

0, then it is in the manufacturer’s interest to re-solve the problem at τ and choose anew wholesale price trajectory from τ on that satisfies μ(τ ) = 0. The intuition behindthis behavior is that the manufacturer announces a wholesale price trajectory at timezero that leads to the retailer’s decisions that are favorable to him. But by time τ , theretailer has executed his decisions in the interval [0,τ ], and the manufacturer has noincentive to keep his promise. It is for this reason, we have assumed at the outset ofthis paper that the manufacturer commits to his announced wholesale price policy.

We make a number of observations. The retail and wholesale price trajectoriesno longer mimic the instantaneous demand trajectory (Figs. 4–6), as they did in themyopic case. Instead, we observe two patterns for the wholesale price over time t :

Fig. 4 Wholesale and retail price trajectories: Far-sighted retailer

376 J Optim Theory Appl (2008) 137: 363–379

Fig. 5 Instantaneous sales rate: Far-sighted retailer

decreasing (for T = 5,15,20,25) and initially decreasing then increasing (for T =45,75,100). As for the retail price, it increases over time t for all values of T in thisstudy. We observe that for lower values of T , the retail price trajectory r(·) movesdownward as T increases, and beyond a certain value of T , it moves upward as T

increases. A similar observation holds for the wholesale price trajectories. We observethat for all values of T , the instantaneous sales rate trajectory x(·) rises upward in T

(Fig. 5). On the other hand, the behavior of the slotting decision c(t) is not constantover time, and is not uniform in T (see Fig. 6). We see that c(·) initially rises in T , andbeyond a certain value of T , it moves downward as T increases. Unlike the case of themyopic retailer, the manufacturer’s (retailer’s) share of the channel profit in the far-sighted case is not uniform in T (Table 2). Instead, his/her share of the entire channelprofit initially decreases(increases) as T increases, and beyond a certain value of T ,his/her share of the channel profit increases/decreases as T increases.

5 Myopic Focus versus Far-Sighted Focus

So far, we have derived the pricing and slotting decisions separately for the myopicand the far-sighted retailers. We now address the following interesting questions: Willthe retailer be better off with a far-sighted or myopic focus? If so, when? What is

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Fig. 6 Shelf space allocation: Far-sighted retailer

Table 2 Profits, profit ratios, and market saturation levels for different T : Far-sighted retailer

T 5 10 15 25 40 75 100

ΠM(T )(×108) 0.6002 1.3343 2.2860 5.4804 12.085 22.827 30.481

ΠR(T )(×108) 0.3801 1.0138 1.9584 5.1236 8.3747 11.563 13.316

ΠM(T )

ΠC(T )×100% 64.18 59.00 56.47 56.50 63.47 69.06 71.40

ΠR(T )

ΠC(T )×100% 35.82 41.00 43.53 43.50 36.53 30.94 28.60

X(T )M

× 100% 1.35 3.30 6.45 19.28 33.26 44.93 50.75

the manufacturer’s preference for the retailer’s focus? Will the manufacturer and theretailer have conflict over the retailer’s focus? If so, when? For a fixed wholesale pricecharged by the manufacturer, the retailer is certainly better off having a far-sightedfocus. However, the manufacturer adjusts his wholesale price according to the bestresponse of the retailer. Therefore, it is not obvious that the retailer will always bebetter off with a far-sighted focus.

In Table 3, we report the results based on our numerical computations, and com-pare the player’s life-cycle profits in the far-sighted retailer case to their profits inthe myopic retailer case. Also, the higher of the two profit numbers is boldfaced. Weobserve that the manufacturer and the retailer may have four different combinations

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Table 3 Players’ profits and saturation levels for different T in far-sighted (first number) and myopicretailer cases (second number)

T M’s profit (×108) R’s profit (×108) C’s profit (×108) % Saturation level

30 7.664,4.467 6.476,4.818 14.140,9.286 25.29,12.63

50 16.107,10.006 9.713,12.295 25.822,22.301 38.39,31.48

250 1.021,53.689 16.791,23.177 67.812,76.866 62.23,77.87

1200 84.314,98.274 19.457,18.707 103.777,116.98 75.44,92.08

of preferred retailer’s profit focus, i.e. both prefer a far-sighted retailer, both prefera myopic retailer, one prefers a far-sighted retailer and the other prefers a myopicretailer, and vice verse. Specifically, the manufacturer as well as the supply chainprefers a far-sighted retailer when the market saturation level is low (correspondingto the horizons of T = 30 and T = 50), whereas they both prefer a myopic retailerwhen the market saturation level is high (corresponding to the horizons of T = 250and T = 1200). When the market saturation level is low, the retailer’s preference isaligned with the manufacturer’s. That is, the retailer prefers the manufacturer to offerwholesale prices assuming a far-sighted profit focus on her part. However, when asthe market saturation level increases, the retailer switches her preference to a my-opic profit focus. If the market saturation level is extremely high (for example whenT = 1200), the retailer again prefers to be far-sighted.

6 Conclusions

In this paper, we study the dynamic optimal wholesale and retail pricing and shelf-space allocation in a decentralized durable product supply chain consisting of a man-ufacturer and a retailer. We formulate the problems as open-loop Stackelberg differ-ential games with the manufacturer as the leader and the retailer as the follower. Ourdemand model extends the Bass-type diffusion model by incorporating the impactsof retail price and shelf-space allocation on the retail demand. We study two retailerfoci: myopic and far-sighted. We provide analytical results and numerical analysisto obtain the Stackelberg equilibria for different life-cycle lengths. Furthermore, wedevelop insights into conditions under which the both players prefer a far-sighted re-tailer, both prefer a myopic retailer, and one prefers a far-sighted retailer and the otherprefers a myopic retailer and vice verse.

Our analysis opens up several opportunities for future research. First, the open-loop equilibrium that we use is time inconsistent in the far-sighted retailer case. Itwould be interesting to look into feedback Stackelberg equilibria and related timeconsistency issues. Second, further analysis of our model can be carried out to studythe issue of channel coordination. Third, our model can be extended to allow formultiple competing retailers. This would combine our demand model with the earlierresearch by Refs. [6, 8]. Finally, our model can be extended to allow for multipleproducts competing for a limited shelf space.

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