Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain

J Optim Theory Appl (2012) 154:615–643DOI 10.1007/s10957-012-0034-5

Optimal Advertising and Pricing in a Dynamic DurableGoods Supply Chain

Anshuman Chutani · Suresh P. Sethi

Received: 18 October 2011 / Accepted: 14 March 2012 / Published online: 5 April 2012© Springer Science+Business Media, LLC 2012

Abstract Cooperative advertising is an incentive offered by a manufacturer to influ-ence retailers’ promotional decisions. We study a dynamic durable goods duopolywith a manufacturer and two independent and competing retailers. The manufacturer,as a Stackelberg leader, announces his wholesale prices and his shares of retailers’ ad-vertising costs, and the retailers in response play a Nash differential game in choosingtheir optimal retail prices and advertising efforts over time. We obtain the feedbackequilibrium policies for the manufacturer and the retailers in explicit form for a lin-ear demand formulation. We investigate issues, like channel coordination and antidis-criminatory legislation, and also study a case, when the manufacturer sells throughonly one retailer and the second retailer sells a competing brand.

Keywords Cooperative advertising · Stackelberg differential game · Nashdifferential game · Sales–advertising dynamics · Feedback Stackelberg equilibrium ·Durable goods

1 Introduction

For any supply chain, pricing and advertising decisions are essentially dynamic innature. More often than not, these decisions are taken in conjunction with each

Communicated by George Leitmann.

A. ChutaniSchool of Management, Binghamton University, State University of New York, PO Box 6000,Binghamton, NY, 13902, USAe-mail: [email protected]

S.P. Sethi (�)School of Management, The University of Texas at Dallas, Mail Station SM30,800 W. Campbell Rd., Richardson, TX, 75080-3021, USAe-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

616 J Optim Theory Appl (2012) 154:615–643

other as they interact continuously over time, having an impact on sales and ul-timately on the profits of the supply chain members. While many researchers inthe past have studied dynamic advertising strategies or dynamic pricing policies,far fewer models consider both pricing and advertising decisions together, mainlydue to the analytical complexity of such models. The tractability of such prob-lems reduces further in the models, which consider a competitive setting involv-ing multiple brands within a product category, or within brand retail level compe-tition.

Some of the recent models incorporating optimal pricing and advertising deci-sions include, Teng and Thompson [1], which studied price and advertising in a newproduct oligopoly model. Their analysis was largely numerical in nature. Sethi et al.[2] proposed a new product adoption model and examined dynamic advertising andpricing decisions for a monopolist firm. Krishnamoorthy et al. [3] extended the Sethiet al. [2] model to study optimal pricing and advertising policies in a durable goodsduopoly and solved the resulting differential game explicitly.

Within the group of researchers and practitioners studying dynamic advertisingmodels and strategies, an increasing attention is being devoted to the practice of co-operative advertising. Cooperative advertising is an important incentive offered by amanufacturer to influence retailers’ promotional decisions. In a typical arrangement,a manufacturer agrees to reimburse a fraction of each retailer’s advertising expendi-ture in selling his product (Bergen and John [4]). This fraction is typically known asthe “subsidy rate” offered by the manufacturer to a retailer. Cooperative advertising isa fast increasing activity amounting to billions of dollars a year. Nagler [5] found thatthe total expenditure on cooperative advertising in 2000 was estimated at $15 billion,compared with $900 million in 1970 and according to some recent estimates, it wasmore than $25 billion in 2007. Cooperative advertising can be a significant part of themanufacturer’s expense according to Dant and Berger [6], and as many as 25–40 %of local advertisements and promotions are cooperatively funded. In addition, Duttaet al. [7] report that the subsidy rates differ from industry to industry: it is 88.38 % forconsumer convenience products, 69.85 % for other consumer products, and 69.29 %for industrial products.

Many researchers in the past have used static models to study cooperative ad-vertising. Berger [8] modeled cooperative advertising in the form of a wholesaleprice discount offered by the manufacturer to its retailer as an advertising allowance.He concluded that both the manufacturer and the retailer can do better with co-operative advertising. Dant and Berger [6] extended the Berger model to incor-porate demand uncertainty. Kali [9] studied cooperative advertising from the per-spective of coordinating a manufacturer-retailer channel. Huang et al. [10] allowedfor advertising by the manufacturer in addition to cooperative advertising. Theyalso justified their static model by making a case for short-term effects of promo-tion.

Jørgensen et al. [11] formulated a dynamic model with cooperative advertisingas a Stackelberg differential game between a manufacturer and a retailer with themanufacturer as the leader. They considered short term as well as long term formsof advertising efforts made by the retailer as well as the manufacturer. They showedthat manufacturer’s support of both types of retailer advertising benefits both channel

J Optim Theory Appl (2012) 154:615–643 617

members more than the support of only one type; moreover, support of one type isbetter than no support at all. Jørgensen et al. [12] modified the above model by in-troducing decreasing marginal returns to goodwill and studied two scenarios: a Nashgame without advertising support and a Stackelberg game with support from the man-ufacturer as the leader. They characterized stationary feedback policies in both cases.Jørgensen et al. [13] explored the possibility of advertising cooperation even whenthe retailer’s promotional efforts may erode the brand image. Karray and Zaccour[14] extended the above model to consider both the manufacturer’s national advertis-ing and the retailer’s local promotional effort. He et al. [15] solved a manufacturer-retailer Stackelberg game with cooperative advertising using the stochastic version ofthe Sethi [16] model. He et al. [17] considered a cooperative advertising channel con-sisting of a manufacturer selling its product through two independent and competingretailers.

Although there have been several studies on cooperative advertising decisions,very few models consider a competitive setting, and even fewer incorporate pric-ing decisions as well. In this paper, we study a cooperative advertising model for aretail market duopoly with one manufacturer as the Stackelberg leader and two re-tailers as followers, in the case of a market for durable goods. A durable good canbe defined as a commodity which, once purchased by the consumer, does not needto be repurchased for a lengthy period of time. Examples of durable goods includecars, TV’s, microwave ovens, washing machines, etc. The market potential of suchitems depletes with time as cumulative sales increase and, eventually, saturation isreached.

A celebrated sales dynamics model of durable goods is the model of innova-tion diffusion by Bass [18]. Many researchers extended the Bass model to includeprice and advertising effects. Mahajan et al. [19] provided a review of such mod-els. Robinson and Lakhani [20] extended the Bass model to include pricing deci-sions. A more recent study of optimal pricing policies for a monopolist has beenby Krishnan et al. [21], who have found that either a monotonically declining oran increasing-decreasing pricing pattern is optimal. Krishnan et al. [22] proposed abrand-level diffusion model to analyze the impact of a late entrant on the diffusionof different brands of a new consumer durable and that of the product category aswhole. Teng and Thompson [1] incorporated price and advertising in a new productoligopoly model. They showed that the optimal price and advertising patterns arehigh initially and then decrease over time. Their analysis though was limited to nu-merical in nature. More recently, Sethi et al. [2] proposed a sales dynamics modelfor durable goods and examined advertising and price decisions by a monopolistfirm. Krishnamoorthy et al. [3] proposed a competitive extension of the Sethi et al.model and solved the resulting differential game explicitly to obtain the optimal pric-ing and advertising policies. Contrary to Teng and Thompson [1], they found thatoptimal price should be constant and the optimal advertising should decrease overtime. We extend the competitive dynamics introduced in Krishnamoorthy et al. [3] tostudy pricing policies and cooperative advertising decisions for durable goods in thepresence of retail-level competition and obtain explicit results for a linear demandformulation.

With this study, we intend to bridge the gap in the cooperative (also referred toas “coop” in this paper) advertising literature by including the pricing decisions for


the manufacturer and the two retailers in the case of durable goods. To the best ofour knowledge, ours is the first study to incorporate pricing and advertising decisionswith cooperative advertising in a dynamic, competitive durable goods retail market.We aim at answering some key research questions as follows:

• What is the optimal wholesale price and subsidy rate policy of the manufacturer,and the optimal retail price policies and advertising responses by the retailers infeedback form?

• What is the impact of a coop advertising program on the profits of all the membersin the channel? How do channel profits with coop program compare to those with-out coop advertising, and to the integrated channel profit? Can coop advertisinglead to better channel coordination?

• What are the effects of an antidiscriminatory legislation which would restrict themanufacturer to offer equal subsidy rates to his retailers. How does it impact theoptimal subsidy rates, profits of all the members in the channel, and the total chan-nel profits?

• What is the impact of competition from another brand on the optimal policies andprofits of channel members?

The rest of the paper is organized as follows. We describe the model in Sect. 2,followed by analysis and some results in Sect. 3. In Sect. 4, we investigate the is-sue of channel coordination and compute the effect of cooperative advertising onthe profit functions of the manufacturer, both the retailers, and that of the overallchannel. In Sect. 5, we study a model in which only one retailer (say retailer 1)buys from the manufacturer and the second retailer (say retailer 2) buys from an-other manufacturer, and thus acts as a competitor to the manufacturer and his re-tailer (retailer 1) by selling a competing brand. We look into the impact of cooper-ative advertising on the value functions of the different parties in this case as well.In Sect. 6, we discuss an extension where an antidiscriminatory legislation forcesthe manufacturer to offer equal wholesale prices and subsidy rates to the two retail-ers. We also study the impact of such a legislation on the profits of all the membersin the channel and that of the channel as whole. We finally conclude our work inSect. 7.

2 The Model

We consider a dynamic retail duopoly, where a manufacturer sells its product throughone or both of two independent and competing retailers, labeled 1 and 2. The man-ufacturer decides wholesale price for the two retailers (w1,w2), and may choose tosubsidize the advertising expenditures of the retailers. The subsidy, expressed as afraction of a retailer’s total advertising expenditure, is referred to as the manufac-turer’s subsidy rate for that retailer. We use the following notation:

t Time t ∈ [0,∞[.i Indicates retailer i, i = 1,2, when used as a subscript.Xi(t) ∈ [0,1] Cumulative normalized sales of retailer i.ui(t) Retailer i’s advertising effort rate at time t .


wi(t) Wholesale price for Retailer i at time t .pi(t) Retail price of Retailer i at time t .θi(t) ≥ 0 Manufacturer’s subsidy rate for retailer i at time t .Di(pi) Demand of goods sold by retailer i as a function of his own retail price

0 ≤ Di(pi) ≤ 1, ∂Di(pi)/∂pi < 0.

ρi > 0 Advertising effectiveness parameter of retailer i.r > 0 Discount rate of the manufacturer and the retailers.Vi,Vm Value functions of retailer i and of the manufacturer, respectively.V I Value function of the integrated channel.

Without loss of generality, we assume that the unit manufacturing cost of theproduct be zero. Thus, the margin for the manufacturer from retailer i is equalto the wholesale price wi(t) charged from retailer i. The margin for retailer i

can be written as mi(t) := pi(t) − wi(t). Furthermore, we use the standard no-tations ViXj

:= ∂Vi/∂Xj , i = 1,2, j = 1,2, and VmXi:= ∂Vm/∂Xi and V I

Xi:=

∂V I /∂Xi, i = 1,2.We consider a total market potential of one with the cumulative normalized sales

of the firm i at time t denoted as Xi(t), i = 1,2. The rate of change of cumulativeunits sold, which is the instantaneous sales, is denoted by Xi(t), and is given by

Xi(t) = dXi(t)

dt= ρiui(t)Di

(pi(t)

)√1 − X1(t) − X2(t),

Xi(0) = xi ∈ [0,1], i = 1,2,

(1)

where X1(t)+X2(t) is the cumulative sales at time t , ui(t) is the retailer i’s advertis-ing effort at time t , ρi is the effectiveness of firm i’s advertising, and Di(pi(t)) is thedemand of retailer i as a function of own price, pi(t) at time t . This sales dynamics isa differential-game extension of a model by Sethi et al. [2] proposed by Krishnamoor-thy et al. [3]. We employ the idea of a feedback Stackelberg solution in our analysis.The sequence of events is shown in Fig. 1. The manufacturer, who is the Stackel-berg leader of the game, announces the wholesale price policy wi(X1(t),X2(t)) andthe subsidy rate policy θi(X1(t),X2(t)) for retailer i, i = 1,2 at time t . The retail-ers, acting as followers, choose their respective retail prices and advertising effortsin response, and thereby play a Nash differential game to increase their sales. Thus,the wholesale prices and subsidy rates at time t ≥ 0 are wi(X1,X2), i = 1,2, and

Fig. 1 Sequence of events


θi(X1,X2), i = 1,2, respectively. The retailers in response choose their optimal re-tail price and advertising effort by solving their respective optimization problems.The retailer i’s optimal control problem to maximize the present value of his profitstream over the infinite horizon, given the manufacturer’s wholesale prices and sub-sidy rates policies, is given by

Vi(X1,X2)

:= maxpi(t), ui (t)≥0, i=1,2, t≥0

∫ ∞

0e−rt

{(pi(t) − wi

(X1(t),X2(t)

))Xi(t)

− (1 − θi

(X1(t),X2(t)

))u2

i (t)}dt, i = 1,2, (2)

subject to (1), where pi(t) − wi(t) equals the margin of retailer i and Vi(X1,X2)

can be defined as the value function of retailer i. The solution to the Nash dif-ferential game defined by (1)–(2) would give retailer i’s feedback retail pricepi(X1(t),X2(t)), and advertising effort ui(X1(t),X2(t)), i = 1,2, which with aslight abuse of notation, can be written as pi(X1,X2 | w1(X1,X2),w2(X1,X2),

θ1(X1,X2), θ2(X1,X2)), and ui(X1,X2 | w1(X1,X2),w2(X1,X2), θ1(X1,X2),

θ2(X1,X2)), i = 1,2, respectively.The manufacturer anticipates the retailers’ optimal responses and incorporates

them into his optimization problem, which is a stationary infinite horizon optimalcontrol problem. The manufacturer’s problem is given by

Vm(X1,X2)

:= maxwi(t)≥0, 0≤θi (t)≤1, i=1,2, t≥0

∫ ∞

0e−rt

2∑

i=1

{wi(t)Xi(t)

− θi(t)[ui

(X1(t),X2(t) | w1(t),w2(t), θ1(t), θ2(t)

)]2}dt, (3)

subject to

Xi(t) = ρi �uiDi(�pi)√

1 − X1(t) − X2(t), Xi(0) = Xi ∈ [0,1], i = 1,2, (4)

where �ui = ui(X1(t),X2(t) | w1(t),w2(t), θ1(t), θ2(t)) and �pi = (X1(t),X2(t) |w1(t),w2(t), θ1(t), θ2(t)), i = 1,2, are the feedback advertising level and retail price,respectively, of retailer i, given the wholesale price and subsidy rate policy declaredby the manufacturer.

The solution to the optimal control problem (3)–(4) gives the optimal whole-sale price and subsidy rate in feedback form, which, with a slight abuse ofnotation, can be expressed as w∗

i (X1,X2) and θ∗i (X1,X2), i = 1,2, respec-

tively. Similarly, we can express retailer i’s retail price and advertising policy asp∗

i (X1,X2) = p∗i (X1,X2 | w∗

1(X1,X2),w∗2(X1,X2), θ

∗1 (X1,X2), θ

∗2 (X1,X2)) and

u∗i (X1,X2) = u∗

i (X1,X2 | w∗1(X1,X2),w

∗2(X1,X2), θ

∗1 (X1,X2), θ

∗2 (X1,X2)), i =

1,2, respectively. The optimal feedback policies of the manufacturer and the retail-ers, i.e., [w∗

i (X1,X2), θ∗i (X1,X2)] and [p∗

i (X1,X2), u∗i (X1,X2)], i = 1,2, respec-

tively, constitute a time-consistent feedback Stackelberg equilibrium of the prob-lems (1)–(4). Substituting these policies into the state equation (1), we get the


cumulative sales vector (X∗1(t),X∗

2(t)), t ≥ 0, and the decisions of the manufac-turer and the retailers as [w∗

i (t) = w∗i (X

∗1(t),X∗

2(t)); θ∗i (t) = θ∗

i (X∗1(t),X∗

2(t))], and[p∗

i (t) = p∗i (X

∗1(t),X∗

2(t));u∗i (t) = u∗

i (x∗1 (t), x∗

2 (t))], t ≥ 0, i = 1,2, respectively.

3 Preliminary Results

We first solve retailer i’s problem to find the optimal pricing and advertising policy,i.e., p∗

i (X1,X2 | w1(X1,X2),w2(X1,X2), θ1(X1,X2), θ2(X1,X2)), and u∗i (X1,X2 |

w1(X1,X2),w2(X1,X2), θ1(X1,X2), θ2(X1,X2)), respectively, given the wholesaleprice and subsidy rate policies announced by the manufacturer. We write Hamilton–Jacobi–Bellman (HJB) equations for the value functions of the two retailers, i.e.,V1(X1,X2) and V2(X1,X2) as follows:

rVi(X1,X2) = maxpi,ui≥0

[(pi − wi(X1,X2) + ViXi

)ρiuiDi(pi)

√1 − X1 − X2

− (1 − θi(X1,X2)

)u2

i + ViXjρjujDj (pj )

√1 − X1 − X2

],

i = 1,2, j = 1,2, i �= j, (5)

where ViXjrepresents a marginal increase in the total discounted profit of retailer

i, i = 1,2, with respect to increase in the cumulative sale of retailer j, j = 1,2.Writing the first-order conditions for pi and ui, i = 1,2, from the HJB equation

in (5), we get the following set of equations in pi and ui . For i = 1,2,

ρiui

√1 − X1 − X2

(Di(pi) + (pi − wi + ViXi

)∂Di(pi)/∂pi

) = 0, (6)

ρiDi(pi)√

1 − X1 − X2(pi − wi + ViXi) − 2ui(1 − θi) = 0. (7)

Solution of the first-order conditions in (6)–(7) gives the optimal retail price andadvertising policies for retailer i in feedback form, i.e., p∗

i (X1,X2) and u∗i (X1,X2),

i = 1,2, respectively.The manufacturer takes into account each retailer’s optimal response to his whole-

sale price and subsidy rate policy, and solves his problem to determine the optimalwholesale prices and subsidy rates for the two retailers. The HJB equation for themanufacturer’s value function Vm(X1,X2) is

rVm(X1,X2)

= maxw1,w2,θ1,θ2≥0

[(w1 + VmX1)ρ1u

∗1(X1,X2)D1

(p∗

1(X1,X2))√

1 − X1 − X2

+ (w2 + VmX2)ρ2u∗2(X1,X2)D2

(p∗

2(X1,X2))√

1 − X1 − X2

− θ1u∗2

1(X1,X2) − θ2u∗2

2(X1,X2)]. (8)

The solution to the manufacturer’s optimization problem (8) gives the optimal whole-sale price and subsidy rate in feedback form, i.e., w∗

i (X1,X2) and θ∗i (X1,X2), i =


1,2, respectively. To explore further, we need to specify the demand function for thetwo retailers. We consider the linear demand specification

Di(pi) = 1 − ηipi, i = 1,2, (9)

where ηi represents the price sensitivity of demand. The linear demand function isvery common in the literature (e.g., [2, 3], and [23]).

Using the demand specification (9) in the first-order conditions (6)–(7) and solvingfor pi and ui , we obtain the following result for the retailers’ problems.

Proposition 3.1 For a given subsidy wholesale price and rate policy, wi(X1,X2)

and θi(X1,X2), i = 1,2, respectively, the optimal feedback pricing and advertisingdecision of retailer i, i = 1,2, for the linear demand specification (9) is

p∗i = p∗

i (X1,X2 | w1,w2, θ1, θ2) = 1 − ηiViXi+ ηiwi(X1,X2)

2ηi

, (10)

u∗i = u∗

i (X1,X2 | w1,w2, θ1, θ2)

= (1 + ηiViXi− ηiwi(X1,X2))

2ρ2i

√1 − X1 − X2

8ηi(1 − θi(X1,X2)), (11)

and the value function Vi(X1,X2) satisfies

64rVi(X1,X2) = (1 − X1 − X2)

[(1 + ηiViXi

− ηiwi(X1,X2))4ρ2

i

η2i (1 − θi(X1,X2))

+ 4Vix3−i(1 + η3−iV(3−i)X3−i

− η3−iw3−i (X1,X2))3ρ2

3−i

η3−i (1 − θ3−i (X1,X2))

]. (12)

Proof We obtain (10) and (11) by using (9) in the first-order conditions (6)–(7)and solving them. To verify the second-order conditions, we consider Hessian ma-trix for the value function of retailer i, i = 1,2, by computing ∂2Vi/∂p

2i , ∂

2Vi/∂u2i ,

and ∂2Vi/∂pi∂ui , and find that the Hessian matrix is negative definite for pi = p∗i ,

ui = u∗i obtained in (10)–(11), thus indicating a maximum. We then use (10)–(11)

in (5) to obtain (12). �

We can see from (10) that the optimal retail price for retailer i increases with hiswholesale price and decreases with his marginal benefit with respect to his cumulativesales. The advertising effort by retailer i increases with his marginal benefit withrespect to his cumulative sales and decreases with his wholesale price. Moreover, theadvertising effort is greater for a higher uncaptured market, i.e., (1 − X1 − X2).

Taking into account the retailers’ optimal responses in (10)–(12), we can rewritethe HJB equation for the manufacturer’s value function V (X1,X2), given by (8), inthe following way:

64rVm(X1,X2)

(1 − X1 − X2)


= maxw1,w2,θ1,θ2≥0

[(−1 − η1V1X1 + η1w1)

3

η21(−1 + θ1)2

× (−4(VmX1 + w1)η1 + θ1 + (V1X1 + 4VmX1 + 3w1)η1θ1)ρ2

1

+ (−1 − η2V2X2 + η2w2)3

η22(−1 + θ2)2


2

]. (13)

We can now obtain the manufacturer’s optimal wholesale prices and subsidy ratespolicy as shown in the following result.

Proposition 3.2 The manufacturer’s optimal wholesale prices are

w∗i = w∗

i (X1,X2) = 1 + ηi(ViXi− 2VmXi

)

3ηi

, i = 1,2, (14)

and the optimal subsidy rates for the two retailers are

θ∗i = θ∗

i (X1,X2) = 1/3, i = 1,2. (15)

The manufacturer’s value function Vm(X1,X2) satisfies

144rVm(X1,X2) = (1 − X1 − X2)

[(1 + (VmX1 + V1X1)η1)

4ρ21

η21

+ (1 + (VmX2 + V2X2)η2)4ρ2

2

η22

]. (16)

Proof Solving the first-order conditions with respect to wi and θi, i = 1,2, in (13)gives the following:

wi = ViXi+ 1

ηi

, or wi = 1 + ηi(ViXi+ 3VmXi

(−1 + θi))

ηi(4 − 3θi), (17)

θi = 4ηi(wi + VmXi) − (1 + ηi(ViXi

− wi))

4ηi(wi + VmXi) + (1 + ηi(ViXi

− wi)). (18)

We can see from (13) that the manufacturer’s value function can be written as a sumof two terms, one containing manufacturer’s policy for retailer 1 only (i.e., w1, θ1),and the other containing the same for retailer 2 only (i.e., w2, θ2). Thus, the manu-facturer’s value function is separable in decisions for the two retailers. Therefore, toverify the second order conditions, we consider each of these two separable terms(concerning decisions for each retailer) one by one and show that their respectiveHessian matrix is negative definite. Thus, for retailer i, i = 1,2, the Hessian includes∂2Vm/∂w2

i , ∂2Vm/∂θ2

i , and ∂2Vm/∂wi∂θi . For retailer i, i = 1,2, we find that the


first-order conditions give three solutions, out of which two are identical. Thus solv-ing first-order conditions w.r.t. retailer i gives two distinct solutions. We can showthat one of them, which gives wi = ViXi

+ 1ηi

, yields p∗i = 1/ηi from (10) with

the corresponding demand Di(p∗i ) = 0 in (9), and can therefore be ruled out from

further consideration. The remaining solution with wi = 1+ηi (ViXi+3VmXi

(−1+θi ))

ηi (4−3θi )

gives a Hessian which is negative definite, and is therefore a maximizer. Conse-quently, we solve (17)–(18) to get the optimal values of w∗

i and θ∗i , i = 1,2, as

shown in (14) and (15), respectively. We then use (14) and (15) in (13) to ob-tain (16). �

Equation (14) shows that the optimal wholesale price for retailer i decreases withthe manufacturer’s marginal benefit with respect to cumulative sales from retailer i

and increases with the marginal benefit of retailer i with respect to his cumulativesales. Thus, if VmXi

is high, the manufacturer incentivizes retailer i to increase salesby reducing the wholesale price for retailer i. We can see from (10) and (11) thata decrease in wi reduces p∗

i and increases u∗i , which together act in increasing the

sales of retailer i. On the other hand, if ViXiis high, the manufacturer increases

the wholesale prices, since he knows that retailer i has his own incentive to in-crease his sales by reducing p∗

i and increasing u∗i , which is again evident from (10)

and (11).In fact, using (14) and (15) in (10) and (11), we get, for i = 1,2,

p∗i = 2 − ηi(ViXi

+ VmXi)

3ηi

, (19)

u∗i = (1 + ηi(ViXi

+ VmXi))2ρi

√1 − X1 − X2

12ηi

, (20)

and

216rVi(X1,X2) = (1 − X1 − X2)

[(1 + ηi(ViXi

+ VmXi)4ρ2

i

η2i

+ 6Vix3−i(1 + η3−i (V(3−i)X3−i

+ VmX3−i))3ρ2

3−i

η3−i

]. (21)

Equations (19) and (20) show that the retailer i’s retail price decreases and his adver-tising effort increases with increase in ViXi

as well as VmXi, thereby increasing the

cumulative sales rate of retailer i.In the dynamic programming equations (16) and (21), we see that with θ∗

1 and θ∗2

being constants, the value functions Vi(X1,X2) and Vm(X1,X2) are linear in X1 andX2 and are a multiple of (1 − X1 − X2). We therefore, conjecture the following formof the value functions

Vi(X1,X2) = βi(1 − X1 − X2), i = 1,2, (22)

Vm(X1,X2) = α(1 − X1 − X2), (23)


and try to solve for the coefficients α,β1 and β2 to obtain the optimal strategiesin feedback form. With this form of value functions, we see that ViXi

= ViX3−i=

−βi , and VmXi= −α, i = 1,2. We compare the coefficients of X1 and X2 and the

constant term of the value functions V1(X1,X2),V2(X1,X2), and Vm(X1,X2) in (16)and (21), with those in (22)–(23), and obtain the following system of equations to besolved for the coefficients α,β1 and β2. For i = 1,2,

216rβi = (−1 + ηi(βi + αi))4ρ2

i

η2i

+ 6βi(−1 + η3−i (β3−i + α+))3ρ23−i

η3−i

, (24)

144rαi = (−1 + (α + β1)η1)4ρ2

1

η21

+ (−1 + (α + β2)η2)4ρ2

2

η22

. (25)

In general, it is difficult to obtain an explicit solution of the system of (24)–(25).Nevertheless, it is easy to solve these equations numerically and study the dependenceof p∗

i ,w∗i and u∗

i , i = 1,2, on model parameters, i.e., ηi and ρi .

General Case: Numerical Analysis We perform numerical analysis using Mathe-matica to study the dependence of the optimal wholesale prices, retail prices andadvertising efforts on the model parameters.

(a) Effect of price sensitivity of demand (Figs. 2, 3): as η1 increases, the advertisingeffort by retailer 1 decreases and that by retailer 2 increases. Figure 2 showsthe variation of u∗

i /√

1 − X1 − X2, i = 1,2, with respect to η1. Figure 3 showsthe variation of optimal wholesale and retail prices brought about by changesin η1. We find that as η1 increases, the retail and the wholesale prices of both theretailers decrease. The decrease in p∗

1 and w∗1 is much faster than the decrease in

p∗2 and w∗

2 . As η1 increases, retailer 1’s demand decreases, and thus the retailerand the manufacturer compensate by reducing the wholesale and retail prices to

Fig. 2 Impact of η1 onadvertising efforts of retailers


Fig. 3 Optimal wholesale andretail prices vs. η1

Fig. 4 Impact of ρ1 onadvertising efforts of retailers

stimulate the demand. Since retailer 2 competes with retailer 1, his retail price(p∗

2) also decreases, but at a much slower rate.(b) Effect of the advertising effectiveness parameter (Figs. 4, 5): as the advertising

effectiveness of retailer 1 increases, we find that the advertising effort by retailers1 increases whereas the same by retailer 2 decreases. Also, the retailer with ahigher advertising effectiveness parameter puts a greater advertising effort. Wefind that as ρ1 increases, the wholesale prices for both the retailers as well as theretail prices of both the retailers increase. Moreover, the retailer with a higheradvertising effectiveness parameter has to pay a lower wholesale price to themanufacturer, and has a higher retail price compared to the other retailer.


Fig. 5 Optimal wholesale andretail prices vs. ρ1

4 Channel Coordination

In this segment, we compare the profits of all the channel members and the total chan-nel profit in our model with the corresponding values in the following two settings.

(i) An integrated channel in which the retail price and advertising decisions are takenbased on the maximization of the combined profit of the manufacturer and thetwo retailers. The wholesale price and subsidy rates do not play a part in thissetting.

(ii) A decentralized channel without cooperative advertising, with pi,wi and ui asdecision variables, i = 1,2 (i.e., θ1 = θ2 = 0).

Our objective from these comparisons is to study the effect of cooperative advertisingon the profits of all the channel members, and thereby, to find out if cooperativeadvertising can act as a tool to increase channel profits and thus, improve channelcoordination.

We first define the optimization problem of an integrated channel to decide theoptimal retail prices (p1,p2) and optimal levels of advertising (u1, u2):

V I (X1,X2)

:= maxpi(t),ui (t)≥0,i=1,2,t≥0

∫ ∞

0e−rt

(p1X1(t) + p2X2(t) − u2

1(t) − u22(t)

)dt, (26)

subject to

Xi(t) = dXi(t)

dt= ρiui(t)Di

(pi(t)

)√1 − X1(t) − X2(t),

Xi(0) = Xi ∈ [0,1], i = 1,2.

(27)

The HJB equation for the integrated channel function V I is


rV I (X1,X2) = maxpi(t), ui (t)≥0, i=1,2, t≥0

[p1X1 +p2X2 −u2

1 −u22 +V I

X1X1 +V I

X2X2

],

(28)where V I

Xi= ∂V I (X1,X2)/∂Xi, i = 1,2, and X1 and X2 are given by (27). To com-

pute the value function of the integrated channel in explicit form, and to compare itwith the total channel value functions of a decentralized channel, with and withoutcooperative advertising, we use the linear demand specification (9). We then obtainthe following result.

Proposition 4.1 The optimal feedback retail price and advertising policies for theintegrated channel with linear demand are

p∗i = p∗

i (X1,X2) = 1

2

(1

ηi

− V IXi

), i = 1,2, (29)

u∗i = u∗

i (X1,X2) = ρi(1 + ηiVIXi

)2√1 − X1 − X2

8ηi

, i = 1,2, (30)

and the value function of the integrated channel satisfies the following equation:

64rV I (X1,X2) = (1 − X1 − X2)

[ρ2

1(1 + η1VIX1)

4

η21

+ ρ22(1 + η2V

IX2)4

η22

]. (31)

Proof The proof is similar to that of Proposition 3.1. Using (27) in the HJB equa-tion (28), we can obtain first order conditions for maximization with respect to pi

and ui, i = 1,2, similar to (6)–(7). By solving these first-order conditions with theuse of (9), we obtain (29) and (30), and then use (29)–(30) in (28) to get (31). �

Here again, we conjecture a linear value function of the form V I (X1,X2) =αI (1 − X1 − X2), where αI = −V I

X1= −V I

X2, is constant and solves the equation

64rαI = (1−αI η1)4ρ2

1η2

1+ (1−αI η2)

4ρ22

η22

.

In the second case, we consider a decentralized channel with cooperative adver-tising and optimal values of wholesale prices, subsidy rates, retail prices, and adver-tising levels given by (14), (15), (19), and (20), respectively. We define the channelvalue function in this case as V c(X1,X2) := V c

m(X1,X2) + V cr (X1,X2), where V c

m

is the manufacturer’s value function (given by (23)) and V cr is the sum of the value

functions of both retailers obtained by (22).The third case is of a decentralized channel with no cooperation. The decision

variables for the manufacturer in this scenario are the wholesale prices and those forretailers are their respective retail prices and advertising levels. This case is treated byusing θi = 0, i = 1,2, in the first-order conditions (6), (7), and (17), and then solving


for optimal values of p∗i , u

∗i , and w∗

i . The value functions of the manufacturer andthe retailers in this case are defined as V n

m and V ni , i = 1,2, respectively, and are

expressed using the following:

V n1 (X1,X2) = βn

1 (1−X1 −X2), i = 1,2, and V nm(X1,X2) = αn(1−X1 −X2).

The channel value function is defined as V n(X1,X2) := V nm(X1,X2) + V n

r (X1,X2),where V n

r is the sum of the two retailers’ value functions in the noncooperative set-ting.

Since the manufacturer is the leader and decides his subsidy rates by maximizinghis total discounted profit, it is obvious that V c

m(X1,X2) ≥ V nm(X1,X2). Thus, it re-

mains to study the effect of cooperative advertising on the value functions of the tworetailers and the total value function of the channel.

Before we proceed further, we recall that the value functions are linear in X1and X2, and can be written as a constant coefficient times (1 − X1 − X2). It is there-fore sufficient to compare the values of their respective coefficients of (1−X1 −X2).Thus, to compare V,V c , and V n, we compare the values of αI , (α + β1 + β2) and(αn + βn

1 + βn2 ), respectively. Similarly, a comparison between α and αn, and be-

tween βi and βni , is equivalent to a comparison between the value functions V c

m andV n

m, and V ci and V n

i , respectively, for i = 1,2.In general, it is difficult to compute the value functions in explicit form. We, there-

fore, resort to numerical analysis using Mathematica to compute the values of thecoefficients and thus compare the value functions. Although we performed numer-ical analysis for several sets of values of model parameters, we report our findingsfor the setting described as follows. We use the linear demand formulation (9) and abase case for numerical computations using η1 = η2 = 1, ρ1 = ρ2 = 1, and r = 0.05.We then vary the parameters η1 and ρ1, one by one and compute the value of thecoefficients. Figures 6 and 7 compare (α + β1 + β2) and αI , with changes in η1

Fig. 6 (α + β1 + β2) and αI

vs. η1


Fig. 7 (α + β1 + β2) and αI

vs. ρ1

Fig. 8 Impact of cooperativeadvertising on value functioncoefficients with changing η1

and ρ1, respectively. Figures 8 and 9 show the impact of cooperative advertising onthe value function coefficients of the manufacturer, retailers 1 and 2, and channelas a whole, by changing η1 and ρ1, respectively. The data sets for the calculationsshown are the same as those used for the results in Figs. 2–5. Thus, for any pointin Figs. 6–9, the values of w∗

i , p∗i , and u∗

i are the same as the corresponding valuesin Figs. 2–5. Figures 6 and 7 show that the channel value function with cooperativeadvertising and the integrated channel value function decrease with η1 and increasewith ρ1. Figures 8–9 plot the difference between the value function coefficients for


Fig. 9 Impact of cooperativeadvertising on value functioncoefficients with changing ρ1

a channel with cooperative advertising and those for a channel without any, for allthe members of the channel and that for the channel as whole. Thus, Figs. 8–9 showthe impact of cooperative advertising on the profit functions of the manufacturer, theretailers and the overall channel. As anticipated, we find that the manufacturer al-ways gains from cooperative advertising. The gain for the manufacturer is greater forlower values of η1 and higher values of ρ1. However, we find that no retailer ben-efits from cooperative advertising. A retailer’s value functions in the two scenarioscould be approximately equal at best, for e.g., retailer 2 for low values of η1 andretailer 1 for low values of ρ1. Interestingly, we find that retailer 1’s losses show apattern similar to the manufacturer’s gains, i.e., decreasing with η1 and increasingwith ρ1. Finally, in most of the instances, we find that the total channel value func-tion with cooperative advertising is marginally lower than that without any cooper-ative advertising. These observations indicate that cooperative advertising does notseem to increase channel profits and improve channel coordination. Roughly speak-ing, cooperative advertising transfers some of the profits from the retailers’ side to themanufacturer’s, thereby keeping the total channel profits at approximately the samelevel, decreasing the channel profit in most cases and marginally improving it in afew.

5 The Case of Brand Level Competition

The numerical analysis of value functions in the previous section indicates that theretailers might not benefit from cooperative advertising when both of them are sell-ing the product of the same manufacturer. Thus, there might be no incentive for re-tailer(s) to participate in a cooperative advertising program when only one brandis being sold. It will be interesting however, to look into the benefit for a retailer


Fig. 10 Model when retailer 2buys from a competingmanufacturer

(say retailer 1) by participating in a cooperative advertising program with the man-ufacturer when the second retailer (say retailer 2) buys from another manufacturer(of a competing brand), and thus acts as a competitor to retailer 1 and the man-ufacturer by selling an alternate brand. In other words, we would like to see theprofits of all the parties in presence of brand level competition when the two re-tailers sell different brands of the same product category. We consider a channelas shown in Fig. 10. The manufacturer sells his product to retailer 1 who sells itin the market. Retailer 2, sells a brand which competes in the market with that ofretailer 1. We assume that the products sold by the two retailers are perfectly sub-stitutable. Here again, we assume that X1(t) and X2(t) denote the cumulative salesby retailer 1 and 2, respectively. Thus, the manufacturer’s cumulative sales is onlyX1(t). All the other notations remain the same as in the general model. The man-ufacturer announces his wholesale price w1(X1,X2) and subsidy rate θ1(X1,X2)

policy for retailer 1. Meanwhile, retailer 2 purchases from competing manufac-turer at a wholesale price w2, which is not a decision variable of the manufac-turer in our model, and is thus exogenous in nature. Clearly, θ2 = 0 in this set-ting as retailer 2 does not buy from the main manufacturer. The two retailers playa Nash game to compete in the consumers market and determine their optimal re-tail prices and advertising efforts, i.e., pi and ui, i = 1,2, respectively. The re-tailer i’s optimization problem is to maximize the present value of his profit streamover infinite horizon, given their wholesale prices, and given the subsidy rate pol-icy of the manufacturer for retailer 1. This problem can be written by using θ2 = 0in (2), subject to (1). The manufacturer’s optimal control problem is now givenby

Vm(X1,X2) := maxw1(t)≥0,0≤θ1(t)≤1,t≥0

∫ ∞

0e−rt

{w1(t)X1(t)

− θ1(t)[u1

(X1(t),X2(t) | w1(t),w2(t), θ1(t), θ2(t)

)]2}dt, (32)

subject to

X1(t) = ρ1�u1D1(�p1)√

1 − X1(t) − X2(t), X1(0) = X1 ∈ [0,1], (33)

where �u1 and �p1 are the feedback advertising level and retail price, respectively, ofretailer 1, given the wholesale price and subsidy rate policy declared by the manu-facturer.


We write the HJB equations for the value functions of the two retailers, i.e.,V1(X1,X2) and V2(X1,X2), as follows:

rVi(X1,X2)

= maxpi,ui≥0

[(pi − wi(X1,X2) + ViXi

)ρiuiDi(pi)

√1 − X1 − X2

− (1 − θi(X1,X2)

)u2

i + ViXjρjujDj (pj )

√1 − X1 − X2

], i = 1,2. (34)

The first-order conditions for pi and ui, i = 1,2, from the HJB equation in (34) can beobtained by simply using θ2 = 0 in (6)–(7). Solving these first-order conditions givesthe optimal retail price and advertising efforts for the two retailers in feedback form,i.e., p∗

i (X1,X2) and u∗i (X1,X2), i = 1,2, respectively. Similarly, the manufacturer’s

HJB equation, taking into account retailer 1’s optimal response to his wholesale priceand subsidy rate policy can be obtained from (7) by simply using θ2 = 0 and ignoringthe terms of w2, as retailer 2 does not contribute to the manufacturer’s sales. We thushave the following:

rVm(X1,X2)

= maxw1, θ1, ≥0

[(w1 + VmX1)ρ1u

∗1(X1,X2)D1

(p∗

1(X1,X2))√

1 − X1 − X2

+ VmX2ρ2u∗2(X1,X2)D2

(p∗

2(X1,X2))√

1 − X1 − X2

− θ1u∗2

1(X1,X2) − θ2u∗2

2(X1,X2)]. (35)

The solution to the manufacturer’s optimization problem gives the optimal wholesaleprice and subsidy rate policy for retailer 1 in feedback form, i.e., w∗

1(X1,X2) andθ∗

1 (X1,X2).Once again, to explore in further detail, we use the linear demand functions (9).

Using this demand specification, we can easily obtain the optimal feedback pricingand advertising decisions of the two retailers by simply using θ2 = 0 in (10) and(11) in Proposition 3.1. Furthermore, the value functions of the two retailers can beobtained by using θ2 = 0 in (12). We thus have the following results:

p∗1 = p∗

1(X1,X2 | w1,w2, θ1) = 1 − η1V1X1 + η1w1(X1,X2)

2η1, (36)

p∗2 = p∗

2(X1,X2 | w1,w2, θ1) = 1 − η2V2X2 + η2w2

2η2, (37)

u∗1 = u∗

1(X1,X2 | w1,w2, θ1)

= (1 + η1V1X1 − η1w1(X1,X2))2ρ2

1

√1 − X1 − X2

8η1(1 − θ1(X1,X2)), (38)

u∗2 = u∗

2(X1,X2 | w1,w2, θ1) = (1 + η2V2X2 − η2w2)2ρ2

2

√1 − X1 − X2

8η2, (39)


and the value function Vi(X1,X2), i = 1,2, satisfies

64rV1(X1,X2)

(1 − X1 − X2)

=[(1 + η1V1X1 − η1w1(X1,X2))

4ρ21

η21(1 − θ1(X1,X2))

+ 4V1x2(1 + η2V2X1 − η2w2)3ρ2

2

η2

],

64rV2(X1,X2)

(1 − X1 − X2)

=[(1 + η2V2X2 − η2w2)

4ρ22

η22

+ 4V2x1(1 + η1VX1 − η1w1(X1,X2))3ρ2

1

η1(1 − θ1(X1,X2))

].

Taking into account the retailers’ optimal pricing and advertising strategies,we canrewrite the HJB equation for the manufacturer’s value function Vm(X1,X2) givenby (35) in the following way:

64rVm(X1,X2)

(1 − X1 − X2)

= maxw1, θ1, ≥0

[(−1 − η1V1X1 + η1w1)

3

η21(−1 + θ1)2


1

× (−1 − η2V2X2 + η2w2)3(−4VmX2)ρ

22

η2

]. (40)

We now consider two cases. First, a cooperative equilibrium in which the manufac-turer offers a positive optimal subsidy rate to the retailer 1 and second, a noncooper-ative equilibrium in which θ1 = 0. Our objective is to study and compare the valuefunctions of all the parties in these two cases and investigate the impact of a cooper-ative advertising program in the presence of brand level competition.

Case 1: A Cooperative Solution We first consider a cooperative solution in whichthe manufacturer chooses his optimal subsidy rate for retailer 1. Using the first-orderconditions w.r.t. w1 and θ1 in (40), we find the manufacturer’s optimal wholesaleprice and optimal subsidy rate policy, given by the following:

w∗1 = w∗

1(X1,X2) = 1 + ηi(V1X1 − 2VmX1)

3η1, (41)

θ∗1 = θ∗

1 (X1,X2) = 1/3. (42)

The manufacturer’s value function satisfies

64rVm(X1,X2)

(1 − X1 − X2)

=[

4(1 + (VmX1 + V1X1)η1)4ρ2

1

9η21

+ 4(VmX2(1 + (V2X2 − w2)η2))3ρ2

2

η2

]. (43)


Now, using (41) and (42) in (36)–(39), we get the optimal retail prices and advertisingefforts of the two retailers as

p∗1 = 2 − η1(V1X1 + VmX1)

3η1, p∗

2 = 1 − η2(V2X2 − w2)

2η2,

u∗1 = (1 + η1(V1X1 + VmX1))

2ρ1√

1 − X1 − X2

12η1,

u∗2 = (1 + η2(V2X2 − w2))

2ρ2√

1 − X1 − X2

8η2.

Furthermore, the value functions for the two retailers solve

64rV1(X1,X2)

(1 − X1 − X2)

=[

8(1 + η1(V1X1 + VmX1))4ρ2

1

27η21

+ 4V1x2(1 + η2(V2X2 − w2))3ρ2

2

η2

], (44)

and

64rV2(X1,X2)

(1 − X1 − X2)

=[(1 + η2(V2X2 − w2))

4ρ22

η22

+ 16V2x1(1 + η1(V1X1 + VmX1))3ρ2

1

9η1

]. (45)

Once again, we propose the following linear form of the value functions.

Vi(X1,X2) = βi (1 − X1 − X2), i = 1,2, (46)

Vm(X1,X2) = α(1 − X1 − X2), (47)

and try to solve for the coefficients α, β1 and β2.We compare the coefficients of X1 and X2 and the constant term of the value func-

tions V1(X1,X2),V2(X1,X2), and Vm(X1,X2) in (43), (44)–(45) with those in (46)–(47), and obtain the following system of equations: to be solved for the coefficientsα, β1, and β2.

64rβ1 = 8(1 − η1(β1 + α))4ρ21

27η21

− 4β1(1 − η2(β2 + w2))3ρ2

2

η2, (48)

64rβ2 = (1 − η2(β2 + w2))4ρ2

2

η22

− 16β2(1 − η1(β1 + α))3ρ21

9η1, (49)

64rα = 4(1 − (α + β1)η1)4ρ2

1

9η21

− 4(α(1 − (β2 + w2)η2))3ρ2

2

η2. (50)


Case 2: A Noncooperative Solution (θ1 = 0) Next, we consider the case of no-cooperation, i.e., when the manufacturer does not offer any subsidy to retailer 1,and thus θ1 = 0 as well. Using θ1 = 0 in (36)–(39) gives the optimal retail pricesand advertising levels of the two retailers. Using θ1 = 0 in the manufacturer’s HJBequation, and solving the first-order conditions w.r.t. w1 gives the optimal wholesaleprice of the manufacturer as follows:

w∗1 = w∗

1(X1,X2) = 1 + ηi(V1X1 − 3VmX1)

4η1. (51)

Now using (51), along with θ1 = 0 in (36)–(39), we get the optimal retail prices andadvertising efforts of the two retailers as follows:

p∗1 = 5 − 3η1(V1X1 + VmX1)

8η1, p∗

2 = 1 − η2(V2X2 − w2)

2η2,

u∗1 = 9(1 + η1(V1X1 + VmX1))

2ρ1√

1 − X1 − X2

128η1,

u∗2 = (1 + η2(V2X2 − w2))

2ρ2√

1 − X1 − X2

8η2.

Furthermore, the value functions of retailer 1, retailer 2, and the manufacturer nowsolve the following equations, respectively:

64rV1(X1,X2)

(1 − X1 − X2)

=[

81(1 + η1(V1X1 + VmX1))4ρ2

1

256η21

+ 4V1x2(1 + η2(V2X2 − w2))3ρ2

2

η2

],

64rV2(X1,X2)

(1 − X1 − X2)

=[(1 + η2(V2X2 − w2))

4ρ22

η22

+ 27V2x1(1 + η1(V1X1 + VmX1))3ρ2

1

16η1

],

64rVm(X1,X2)

(1 − X1 − X2)

=[

27(1 + (VmX1 + V1X1)η1)4ρ2

1

64η21

+ 4(VmX2(1 + (V2X2 − w2)η2))3ρ2

2

η2

].

We use linear value functions of the form

Vi(X1,X2) = βni (1 − X1 − X2), i = 1,2, (52)

Vm(X1,X2) = αn(1 − X1 − X2), (53)


and solve for the coefficients αn, βn1 , and βn

2 using the following set of equations:

64rβn1 = 81(1 − η1(β

n1 + αn))4ρ2

1

256η21

− 4βn1 (1 − η2(β

n2 + w2))

3ρ22

η2, (54)

64rβn2 = (1 − η2(β

n2 + w2))

4ρ22

η22

− 27βn2 (1 − η1(β

n1 + αn))3ρ2

1

16η1, (55)

64rαn = 27(1 − (αn + βn1 )η1)

4ρ21

64η21

− 4(αn(1 − (βn2 + w2)η2))

3ρ22

η2, (56)

where the superscript n denotes a non-cooperative solution.

Comparison of Value Functions in Cooperative and Noncooperative Solution Wenow compare the value functions of all the parties (retailer 1, retailer 2, and the man-ufacturer) in a cooperative solution given by (46)–(47) with the corresponding valuesin a noncooperative solution, given by (52)–(53). Since it is quite obvious that themanufacturer will always gain from cooperative advertising, our aim is to investigateits impact on the profits of the two retailers. In particular, we want to find if retailer 1has any incentive to join a coop advertising program, when he faces a competingretailer who is selling another brand, and hence not getting any advertising subsidyfrom retailer 1’s supplier.

Looking at the linear formulation of the value functions given by (46)–(47)

and (52)–(53), we see that a comparison of coefficients α, β1, and β2 with αn, βn1 ,

and βn2 , is sufficient to compare the value functions of the manufacturer, retailer 1,

and retailer 2, respectively, in the two cases. In general, it is difficult to solve thesystem of (48)–(50) and (54)–(56) explicitly and, therefore, we resort to numeri-cal analysis to get some insights. We solved the system of (48)–(50) and (54)–(56)for various values of η1, η2, ρ1, and ρ2, and report some of our key findings inFigs. 11–12.

We solved the equations for several values of problem parameters. Figure 11shows the difference in value function coefficients of all the parties for various val-ues of η1, with ρ1 = ρ2 = 0.1, r = 0.05, and η2 = 1. Since w2 is also an exogenousparameter for this model, in the analysis shown here, we chose w2 equal to w∗

1 , i.e.,optimal wholesale price for retailer 1, given any set of parameters ηi, ρi, i = 1,2,and r . Thus, the figures shown here use the assumption that both the retailers getthe same wholesale price from their respective manufacturers. We find that whilethe manufacturer always gains from cooperative advertising, retailer 1 might gain orloose depending on the values of the parameters. We find that retailer 1 benefits atlow values of η1. Figure 12 shows the difference in the value functions with vary-ing values of ρ1 when η1 = η2 = 0.1, r = 0.05. and ρ2 = 1. Here again, we find thatretailer 1 can benefit from cooperative advertising, in this case as long as ρ1 is nottoo high. These results indicate that a cooperative advertising program can benefitretailer 1 as well, depending on the model parameters. This is in contrast to the sce-nario in which both the retailers buy from the same manufacturer, where numerical


Fig. 11 Benefit fromcooperative advertising vs. η1when retailer 2 buys from acompeting manufacturer

Fig. 12 Benefit fromcooperative advertising vs. ρ1when retailer 2 buys from acompeting manufacturer

analysis showed little indication of the retailers benefiting from cooperative adver-tising. A general trend observed in several numerical calculations was that retailer 1is more likely to benefit at lower values of ρ1 and η1. It would be interesting if onecould obtain an explicit condition based on the model parameters which would indi-cate the scenarios under which retailer 1 would benefit. However, given the difficultyin solving (46)–(47) and (54)–(56) explicitly, it is hard to obtain such a condition.


6 Equal Subsidy Rate for Both Retailers

In this section, we consider the case of an anti discriminatory act in effect, such asthe Robinson–Patman Act of 1936. The Robinson–Patman Act, along with other suchlegislations, was designed to prevent creation of monopolies in the market and to en-hance competition. Such acts prevent price discrimination between two or more com-peting buyers of a product. To study the impact of such a legislation in our model, weconsider a case where the manufacturer is restricted to offer equal wholesale pricesand equal subsidy rates to the two retailers.

We define V RPm (X1,X2),V

RP1 (X1,X2),V

RP2 (X1,X2), and V RP as the value func-

tions of the manufacturer, retailer 1, retailer 2, and the total channel, respectively,with the superscript RP standing for Robinson and Patman. These value func-tions solve the optimal control problems defined by (1)–(3), with w1 = w2 = w

and θ1 = θ2 = θ . Here again, we look for linear value functions of the following

form: V RPm (X1,X2) = αRP(1 − X1 − X2),V

RPi = βRP

i (1 − X1 − X2), i = 1,2, andV RP = (αRP + βRP

1 + βRP2 )(1 − X1 − X2). In the following analysis, we consider the

linear demand function (9), and use numerical analysis to compute the value of thecoefficients αRP, βRP

i , optimal retail prices p∗i , optimal advertising levels u∗

i , i = 1,2,optimal common wholesale price for the two retailers w∗, and the optimal commonsubsidy rate for the two retailers θ∗. Figure 13 compares the common optimal whole-sale price for the two retailers in the case of an anti-discriminatory act (denoted as w∗)with the optimal wholesale prices for the two retailers without such legislation (i.e.,w∗

1,w∗2). We find that as η1 increases, the common optimal wholesale price for the

two retailers decreases. Moreover, we see that a retailer with a higher price sensitiv-ity of demand (ηi) pays a higher wholesale price in the case of antidiscriminatoryact in effect than in the absence of such an act. Figure 14 shows the variation of θ∗

Fig. 13 Optimal wholesaleprices with and withoutanti-discriminatory act vs. η1


Fig. 14 Optimal subsidy rateswith and withoutanti-discriminatory act vs. η1

with respect to η1. Recall that in the general model with linear demand, the optimalsubsidy rates for the two retailers are independent of the values of η1, ρ1, and r , andequal to 1/3. In the case when the manufacturer has to offer equal wholesale pricesand subsidy rates to the two retailers, we find that the common optimal subsidy rateis no longer a fixed value and varies with η1. We find that the maximum value ofθ∗ achieved is equal to the subsidy rates without any legislation (θ∗

1 = θ∗2 = 1/3).

Thus, the introduction of additional constraint on the manufacturer (i.e., to offer equalsubsidy rates to the two retailers) makes the manufacturer to offer lower subsidyrates.

Next, we study the impact of an anti-discriminatory act on the profit functions ofall the channel members, and that of the channel as whole. Figure 15 shows the differ-ence in the value function coefficients with and without an antidiscriminatory act forthe manufacturer, retailers and the total channel, i.e., α −αRP, βi −βRP

i , i = 1,2, and(α+β1 +β2)− (αRP +βRP

1 +βRP2 ), respectively. As expected, the manufacturer does

not benefit from an antidiscriminatory act because of an additional constraint on hisoptimization problem. We also find that only the retailer with a lower value of pricesensitivity of demand (ηi) benefits from such legislation and the other retailer loses.Furthermore, the total channel seems to benefit from such an act as the gain for oneretailer offsets the losses of the other two parties. Figure 16 shows the ratios of thechannel value function to the integrated channel value function in three cases. Thesethree cases are: a channel with cooperative advertising and no legislation, a chan-nel with cooperative advertising and an antidiscriminatory legislation in effect, anda channel with no cooperative advertising. Thus, Fig. 16 indicates the extent of sup-ply chain coordination achievable in these three scenarios. We find that in all theinstances, a channel with no cooperative advertising performs better than a channelwith cooperative advertising and no legislation. Moreover, the value function for achannel with cooperative advertising and an antidiscriminatory act is highest of the


Fig. 15 Value functioncoefficients with anantidiscriminatory act minusthose without any act, withchanging η1

Fig. 16 Ratio of channel valuefunction to integrated channelvalue function in different cases,with changing η1

three, except when the difference between η1 and η2 is small. These results indicatethat in most cases, highest degree of channel coordination can be achieved when wehave cooperative advertising along with an anti-discriminatory legislation in effect.However, in some cases, when roughly speaking, the retailers are nearly identical, achannel with no cooperative advertising is able to achieve the highest total profits andthus the highest level of coordination.


7 Concluding Remarks

We obtain the feedback Stackelberg equilibrium and compute the optimal values ofthe advertising levels and retail prices by the two retailers, and wholesale prices andsubsidy rates offered by the manufacturer to the two retailers, in the case of a linearprice-dependent demand for durable goods. We find that the optimal subsidy ratesare independent of the model parameters and are equal to 1/3. We also provide thesensitivities of the optimal advertising levels, retail prices, and wholesale prices withrespect to η1 and ρ1. We study the effect of cooperative advertising on the profits ofall the members of the channel and that of the total channel. We find that the coop-erative advertising benefits only the manufacturer whereas the two retailers earn lessprofit, and except in a few cases, the total channel does not benefit either. An interest-ing conclusion that appears from our analysis is that with wholesale and retail pricesas decision variables as well, cooperative advertising is ineffective and a redundantmechanism. In fact, since there seems to be little evidence of retailers benefiting fromcooperative advertising, such an arrangement might not hold. We also analyze a sce-nario when retailer 2 buys from another manufacturer and find that under such brandlevel competition, cooperative advertising program can be beneficial for retailer 1as well. Finally, we consider the case of an antidiscriminatory legislation where themanufacturer has to offer equal subsidy rates to the two retailers and find that higherchannel profits can be achieved with such a legislation in most cases, except whenthe two retailers appear to have nearly identical model parameters.

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Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain

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