Optimal Advertising and Pricing in a Dynamic ... - Semantic Scholar
Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain Anshuman Chutani ∗, Suresh P. Sethi
†
June 7, 2011
Abstract Cooperative advertising is an incentive offered by a manufacturer to influence retailers’ promotional decisions. We study a dynamic durable goods duopoly with a manufacturer and two independent and competing retailers. The manufacturer as a Stackelberg leader announces his wholesale prices and his shares of retailers’ advertising costs, and the retailers in response play a Nash differential game in choosing their optimal retail prices and advertising efforts over time. We obtain the feedback equilibrium policies in explicit form for a linear demand formulation. We investigate issues like channel coordination and anti-discriminatory legislation and also study a case when the manufacturer sells through only one retailer and the second retailer sells a competing brand.
Keywords: Cooperative advertising, Stackelberg differential game, Nash differential game, sales-advertising dynamics, feedback Stackelberg equilibrium, Durable goods.
∗ Visiting Assistant Professor, School of Management, Binghamton University, State University of New York, PO Box 6000, Binghamton, NY, 13902 e-mail: [email protected] † Charles & Nancy Davidson Distinguished Professor of Operations Management, School of Management, Mail Station SM30, The University of Texas at Dallas, 800 W. Campbell Rd. Richardson, Texas 75080-3021 e-mail: [email protected]
Electronic copy available at: http://ssrn.com/abstract=1898309
1
Introduction
For any supply chain, pricing and advertising decisions are essentially dynamic in nature. More often than not, these decisions are taken in conjunction with each other as they interact continuously over time, having an impact on sales and ultimately on the profits of the supply chain members. While many researchers in the past have studied dynamic advertising strategies or dynamic pricing policies, far fewer models consider both pricing and advertising decisions together, mainly due to the analytical complexity of such models. The tractability of such problems reduces further in the models which consider a competitive setting involving multiple brands within a product category, or within brand retail level competition. Some of the recent models which do incorporate optimal pricing and advertising decisions include Teng and Thompson [1], which studied price and advertising in a new product oligopoly model. Their analysis was largely numerical in nature. Sethi et al. [2] proposed a new product adoption model and examined dynamic advertising and pricing decisions for a monopolist firm. Krishnamoorthy et al. [3] extended the Sethi et al. [2] model to study optimal pricing and advertising policies in a durable goods duopoly and solved the resulting differential game explicitly. Within the group of researchers and practitioners studying dynamic advertising models and strategies, an increasing attention is being devoted to the practice of cooperative advertising. Cooperative advertising is an important incentive offered by a manufacturer to influence retailers’ promotional decisions. In a typical arrangement, a manufacturer agrees to reimburse a fraction of each retailers advertising expenditures in selling his product (Bergen and John [4]). This fraction is typically known as the ’subsidy rate’ offered by the manufacturer to a retailer. Cooperative advertising is a fast increasing activity amounting to billions of dollars a year. Nagler [5] found that the 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 was more than $25 billion in 2007. Cooperative advertising can be a significant part of the manufacturer’s expense according to Dant and Berger [6], and as many as 25-40% of local advertisements and promotions are cooperatively funded. In addition, Dutta et al. [7] report that the subsidy rates differ from industry to industry: it is 88.38% for consumer 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 advertising. Berger [8] modeled cooperative advertising in the form of a wholesale price 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 cooperative advertising. Dant and Berger [6] extended the Berger model to incorporate demand uncertainty. 1
Electronic copy available at: http://ssrn.com/abstract=1898309
Kali [9] studied cooperative advertising from the perspective of coordinating a manufacturer-retailer channel. Huang et al. [10] allowed for advertising by the manufacturer in addition to cooperative advertising. They also justified their static model by making a case for short-term effects of promotion. Jørgensen et al. [11] formulate a dynamic model with cooperative advertising as a Stackelberg differential game between a manufacturer and a retailer with the manufacturer as the leader. They consider short term as well as long term forms of advertising efforts made by the retailer as well as the manufacturer. They show that manufacturer’s support of both types of retailer advertising benefits both channel members more than the support of only one type; moreover, support of one type is better than no support at all. Jørgensen et al. [12] modified the above model by introducing decreasing marginal returns to goodwill and studied two scenarios: a Nash game without advertising support and a Stackelberg game with support from the manufacturer as the leader. They characterized stationary feedback policies in both cases. Jørgensen et al. [13] explored the possibility of advertising cooperation even when the retailer’s promotional efforts may erode the brand image. Karray and Zaccour [14] extended the above model to consider both the manufacturer’s national advertising and the retailer’s local promotional effort. Recently, He et al. [15] solved a manufacturerretailer Stackelberg game with cooperative advertising using the stochastic version of the Sethi [16] model. He et al. [17] consider a cooperative advertising channel consisting of a manufacturer selling its product through two independent and competing retailers. Although there have been a number of studies on cooperative advertising decisions, very few models consider a competitive setting, and even fewer incorporate pricing decisions as well. In this paper, we study a cooperative advertising model for a retail market duopoly with one manufacturer as the Stackelberg leader and two retailers as followers, in the case of a market for durable goods. A durable good can be defined as a commodity which once purchased by the consumer, does not need to be repurchased for a lengthy period of time. Examples of durable goods include cars, TV’s, microwave ovens, washing machines etc. The market potential of such items depletes with time as cumulative sales increase, and eventually, saturation is reached. A celebrated sales dynamics model of durable goods is the Bass [18] model of innovation diffusion. Many researchers extended the Bass model to include price and advertising effects. Mahajan et al. [19] provide a review of such models. Robinson and Lakhani [20] extended the Bass model to include pricing decisions. A more recent study of optimal pricing policies for a monopolist is by Krishnan et al. [21], who found that either a monotonically declining or an increasing-decreasing pricing pattern is optimal. Krishnan et al. [22] proposed a brand-level diffusion model to analyze the impact of a late entrant on the diffusion of brands of a new consumer durable and that of a category as whole. Teng and Thompson [1] incorporated price and
2
advertising in a new product oligopoly model. They showed that the optimal price and advertising patterns are high initially and then decrease over time. Their analysis though was limited to numerical in nature. More recently, Sethi et al. [2] proposed a sales dynamics model for durable goods and examined advertising and price decisions by a monopolist firm. 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 pricing and advertising policies. Contrary to Teng and Thompson [1], they found that optimal price should be constant and the optimal advertising should decrease over time. We extend the competitive dynamics introduced in Krishnamoorthy et al. [3] to study pricing policies and cooperative advertising for durable goods in the presence of retail-level competition and obtain explicit results for a linear demand formulation. With this study, we intend to bridge the gap in the cooperative advertising literature by including the pricing decisions for the manufacturer and the two retailers in the model for durable goods. To the best of our knowledge, ours is the first study to incorporate pricing and advertising decisions with cooperative advertising in a dynamic, competitive durable goods retail market. We therefore look to answer some key research questions as follows. • What is the optimal wholesale price and subsidy rate policy of the manufacturer, and the optimal retail prices policies and advertising responses by the retailers in feedback form? • What is the impact of a coop advertising program on the profits of all the members in the channel? How do channel profits with coop program compare to those without coop advertising, and to the integrated channel profit? Can coop advertising lead to better channel coordination? • What are the effects of an anti-discriminatory legislation which would restrict the manufacturer to offer equal subsidy rates to his retailers. How does it impact the optimal subsidy rates, profits of all the members in the channel, and the total channel profits? • What is the impact of competition from another brand on the optimal policies and profits of channel members? The rest of the paper is organized as follows. We describe the model in section 2, followed by analysis and some results in section 3. In section 4 we investigate the issue of channel coordination and compute the effect of cooperative advertising on the profit functions of the manufacturer, of the two retailers, and of the overall channel. In section 5, we study a model in which only one retailer buys from the manufacturer and the second retailer acts as a competitor to the manufacturer and his retailer and look into the impact of
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cooperative advertising on the value functions of the different parties. In section 6 we discuss an extension where an anti-discriminatory legislation forces the manufacturer to offer equal wholesale prices and subsidy rates to the two retailers. We also study the impact of such a legislation on the profits of all the members in the channel and that of the channel as whole. We finally conclude our work in section 7.
2
The model
We consider a dynamic retail duopoly where a manufacturer sells its product through one or both of two independent and competing retailers, labeled 1 and 2. The manufacturer decides wholesale price for the two retailers (w1 , w2 ), and may choose to subsidize the advertising expenditures of the retailers. The subsidy, expressed as a fraction of a retailer’s total advertising expenditure, is referred to as the manufacturer’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 Di (pi )
Manufacturer’s subsidy rate for retailer i at time t, 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 , V m
Value functions of retailer i and of the manufacturer, respectively,
VI
Value function of the integrated channel.
Without loss of generality, we assume that the manufacturing cost of the the product is zero. Thus, the margin for the manufacturer from retailer i is equal to the wholesale price wi (t) charged from retailer i. The margin for retailer i can be defined as mi (t) = pi (t) − wi (t). Furthermore, we use the standard notations
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ViXj = ∂Vi /∂Xj , i = 1, 2, j = 1, 2, and VmXi = ∂Vm /∂Xi and VXI i = ∂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 cumulative units sold, which is the instantaneous sales, is denoted by X˙ i (t), and is given by X˙ i (t)
=
p dXi (t) = ρi ui (t)Di (pi (t)) 1 − X1 (t) − X2 (t), dt
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 advertising effort at time t, ρi is the effectiveness of firm i’s advertising, and Di (pi (t)) is the demand of retailer i as a function of own price, pi (t) at time t. We employ the idea of a feedback Stackelberg solution in our analysis. The sequence
Figure 1: Sequence of Events of events is shown in Fig. 1. The manufacturer, who is the Stackelberg leader of the game, announces the wholesale price policy wi (X1 (t), X2 (t)) and the subsidy rate policy θi (X1 (t), X2 (t)) for retailer i, i = 1, 2 at time t. The retailers, acting as followers choose their respective retail prices and advertising efforts in 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 θi (X1 , X2 ), i = 1, 2, respectively. The retailers in response choose their optimal retail price and advertising effort by solving their respective optimization problems. The retailer i’s optimal control problem to maximize the present value of his profit stream over the infinite horizon, given the manufacturer’s wholesale prices and subsidy rates policies, is given by Z Vi (X1 , X2 ) =
max pi (t),ui (t)≥0, i=1,2, t≥0
0
∞
e−rt (pi (t) − wi (X1 (t), X2 (t)))X˙ i (t) − (1 − θi (X1 (t), X2 (t)))u2i (t) dt,
5
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 differential game defined by (1)-(2) would give retailer i’s feedback retail price pi (X1 (t), X2 (t)) and advertising effort ui (X1 (t), X2 (t)), i = 1, 2, which, with a slight 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 optimal control problem. The manufacturer’s problem is given by Z Vm (X1 , X2 ) =
∞
e
max wi (t)≥0, 0≤θi (t)≤1, i=1,2, t≥0
0
−rt
2 X
wi (t)X˙ i (t)
i=1 2
− θi (t) [ui (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t))]
dt
(3)
1 = 1, 2,
(4)
subject to
X˙ i (t)
p = ρi u¯i Di (p¯i ) 1 − X1 (t) − X2 (t),
Xi (0) = Xi ∈ [0, 1],
where u¯i = ui (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t)) and p¯i = (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 declared by the manufacturer. The solution to the optimal control problem (3)-(4) gives the optimal wholesale price and subsidy rate in feedback form, which, with a slight abuse of notation, can be expressed as wi∗ (X1 , X2 ) and θi∗ (X1 , X2 ), i = 1, 2, respectively. Similarly, we can express retailer i’s retail price and advertising policy as p∗i (X1 , X2 ) = p∗i (X1 , X2 | w1∗ (X1 , X2 ), w2∗ (X1 , X2 ), θ1∗ (X1 , X2 ), θ2∗ (X1 , X2 )) and u∗i (X1 , X2 ) = u∗i (X1 , X2 | w1∗ (X1 , X2 ), w2∗ (X1 , X2 ), θ1∗ (X1 , X2 ), θ2∗ (X1 , X2 )), i = 1, 2, respectively. The optimal feedback policies of the manufacturer and the retailers, i.e., [wi∗ (X1 , X2 ), θi∗ (X1 , X2 )] and [p∗i (X1 , X2 ), u∗i (X1 , X2 )] , i = 1, 2, respectively, constitute a time-consistent feedback Stackelberg equilibrium of the problem (1)-(4). Substituting these policies into the state equation (1), we get the cumulative sales vector (X1∗ (t), X2∗ (t)), t ≥ 0, and the decisions of the manufacturer and the retailers as [wi∗ (t) = wi∗ (X1∗ (t), X2∗ (t)); θi∗ (t) = θi∗ (X1∗ (t), X2∗ (t))] and [p∗i (t) = p∗i (X1∗ (t), X2∗ (t)); u∗i (t) = u∗i (x∗1 (t), x∗2 (t))] , t ≥ 0, i = 1, 2, respectively.
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3
Preliminary results
We first solve retailer i’s problem to find the optimal pricing and advertising policy 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 wholesale price 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 ) =
p max [(pi − wi (X1 , X2 ) + ViXi )ρi ui Di (pi ) 1 − X1 − X2 pi ,ui ≥0 p −(1 − θi (X1 , X2 ))u2i + ViXj ρj uj Dj (pj ) 1 − X1 − X2 ],
i = 1, 2, j = 1, 2, i 6= j,(5)
where ViXj represents 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, p 1 − X1 − X2 (Di (pi ) + (pi − wi + ViXi )∂Di (pi )/∂pi ) = 0, p ρi Di (pi ) 1 − X1 − X2 (pi − wi + ViXi ) − 2ui (1 − θi ) = 0.
ρi ui
(6) (7)
Solution of the first order conditions in (6)-(7) gives the optimal retail price and advertising 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 wholesale price and subsidy rate policy, and solves his problem to determine the optimal wholesale prices and subsidy rates for the two retailers. The HJB equation for the manufacturer’s value function Vm (X1 , X2 ) is
rVm (X1 , X2 ) =
max
p (w1 + VmX1 )ρ1 u∗1 (X1 , X2 )D1 (p∗1 (X1 , X2 )) 1 − X1 − X2
w1 ,w2 ,θ1 ,θ2 ≥0
p + (w2 + VmX2 )ρ2 u∗2 (X1 , X2 )D2 (p∗2 (X1 , X2 )) 1 − X1 − X2 − θ1 u∗ 21 (X1 , X2 ) − θ2 u∗ 22 (X1 , X2 ) .
(8)
The solution to the manufacturer’s optimization problem (8) gives the optimal wholesale price and subsidy rate in feedback form, i.e., wi∗ (X1 , X2 ) and θi∗ (X1 , X2 ), i = 1, 2, respectively. To explore further, we need to
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specify the demand function for the two retailers. We consider the linear demand specification
Di (pi ) = 1 − ηi pi ,
i = 1, 2,
(9)
where ηi represents the price sensitivity of demand. The linear demand function is very common in the literature (e.g., [2], [3], and [23]). Using the demand specification (9) in the first-order conditions (6)-(7) and solving for 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 advertising decision of retailer i, i = 1, 2, for the linear demand specification (9) is 1 − ηi ViXi + ηi wi (X1 , X2 ) 2ηi √ (1 + ηi ViXi − ηi wi (X1 , X2 ))2 ρ2i 1 − X1 − X2 , u∗i = u∗i (X1 , X2 | w1 , w2 , θ1 , θ2 ) = 8ηi (1 − θi (X1 , X2 )) p∗i = p∗i (X1 , X2 | w1 , w2 , θ1 , θ2 ) =
(10) (11)
and the value function Vi (X1 , X2 ) satisfies (1 + ηi ViXi − ηi wi (X1 , X2 ))4 ρ2i 64rVi (X1 , X2 ) =(1 − X1 − X2 ) ηi2 (1 − θi (X1 , X2 )) 4Vix3−i (1 + η3−i V(3−i)X3−i − η3−i w3−i (X1 , X2 ))3 ρ23−i . + η3−i (1 − θ3−i (X1 , X2 ))
(12)
Proof: Using (9) in the first-order conditions (6)-(7), we obtain (10) and (11), and then use (10)-(11) in (5) to obtain (12). We can see from (10) that the optimal retail price for retailer i increases with his wholesale price and decreases with his marginal benefit with respect to his cumulative sales. The advertising effort by retailer i increases with his marginal benefit with respect to his cumulative sales and decreases with his wholesale price. Moreover, the advertising 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 rewrite the HJB equation for the manufacturer’s value function V (X1 , X2 ), given by (8), in the following way. 64rVm (X1 , X2 ) = (1 − X1 − X2 ) (−1 − η1 V1X1 + η1 w1 )3 (−4(VmX1 + w1 )η1 + θ1 + (V1X1 + 4VmX1 + 3w1 )η1 θ1 )ρ21 max w1 ,w2 ,θ1 ,θ2 ≥0 η12 (−1 + θ1 )2 8
(−1 − η2 V2X2 + η2 w2 )3 (−4(VmX2 + w2 )η2 + θ2 + (V2X2 + 4VmX1 + 3w2 )η2 θ2 )ρ22 . η22 (−1 + θ2 )2
(13)
We can now obtain the manufacturer’s optimal wholesale prices and subsidy rates policy as shown in the following result. Proposition 3.2: The manufacturer’s optimal wholesale prices are
wi∗ = wi∗ (X1 , X2 ) =
1 + ηi (ViXi − 2VmXi ) 3ηi
i = 1, 2,
(14)
and the optimal subsidy rates for the two retailers are
θi∗ = θi∗ (X1 , X2 ) = 1/3
i = 1, 2.
(15)
The manufacturer’s value function Vm (X1 , X2 ) satisfies (1 + (VmX1 + V1X1 )η1 )4 ρ21 (1 + (VmX2 + V2X2 )η2 )4 ρ22 144rVm (X1 , X2 ) = (1 − X1 − X2 ) + . η12 η22
(16)
Proof: Solving the first-order conditions with respect to wi and θi , i = 1, 2, in (13) gives the following: 1 1 + ηi (ViXi + 3VmXi (−1 + θi )) , or wi = , ηi ηi (4 − 3θi ) 4ηi (wi + VmXi ) − (1 + ηi (ViXi − wi )) θi = . 4ηi (wi + VmXi ) + (1 + ηi (ViXi − wi )) wi = ViXi +
We ignore the possibility that wi = ViXi +
1 ηi ,
(17) (18)
as it yields p∗i = 1/ηi from (10), which means Di (p∗i ) = 0 in
(9). With this observation, we solve equations (17)-(18) to get the optimal values of wi∗ and θi∗ , i = 1, 2, as shown in (14) and (15), respectively. We then use (14) and (15) in (13) to obtain (16). Equation (17) shows that the optimal wholesale price for retailer i decreases with the 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 cumulative sales. Thus, if VmXi is high, the manufacturer incentivizes retailer i to increase sales by reducing the wholesale price for retailer i. We can see from (10) and (11) that a 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 ViXi is high, the manufacturer increases the wholesale prices, since he knows that retailer i has his own incentive to increase his sales by reducing p∗i and increasing u∗i , which is again evident from (10) and (11).
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In fact, using (14) and (15) in (10) and (11), we get, for i = 1, 2, 2 − ηi (ViXi + VmXi ) , 3ηi √ (1 + ηi (ViXi + VmXi ))2 ρi 1 − X1 − X2 , u∗i = 12ηi p∗i =
(19) (20)
and 216rVi (X1 , X2 ) = (1 − X1 − X2 )
(1 + ηi (ViXi + VmXi )4 ρ2i ηi2 6Vix3−i (1 + η3−i (V(3−i)X3−i + VmX3−i ))3 ρ23−i + . η3−i
(21)
Equations (19) and (20) show that the retailer i’s retail price decreases and his advertising effort increases with increase in ViXi as well as VmXi , thereby increasing the sales rate of retailer i. In the dynamic programming equations (16) and (21), we see that with θ1∗ and θ2∗ being constants, the value functions Vi (X1 , X2 ) and Vm (X1 , X2 ) are linear in X1 and X2 and are a multiple of (1 − X1 − X2 ). We therefore, propose the following form of the value functions
Vi (X1 , X2 )
=
βi (1 − X1 − X2 ),
Vm (X1 , X2 )
=
α(1 − X1 − X2 ),
i = 1, 2,
(22) (23)
and try to solve for the coefficients α, β1 and β2 to obtain the optimal strategies in feedback form. With this proposed form, 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 equations (16), (21) and (22)-(23), and obtain the following system of equations to be solved for the coefficients α, β1 and β2 . For i = 1, 2,
216rβi
=
144rαi
=
6βi (−1 + η3−i (β3−i + α+))3 ρ23−i (−1 + ηi (βi + αi ))4 ρ2i + , ηi2 η3−i (−1 + (α + β1 )η1 )4 ρ21 (−1 + (α + β2 )η2 )4 ρ22 + . η12 η22
(24) (25)
In general, it is difficult to obtain an explicit solution of the system of equations (24)-(25). Nevertheless, it is easy to solve these equations numerically and study the dependence of p∗i , wi∗ and u∗i , i = 1, 2, on model parameters, i.e., ηi and ρi .
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General Case: numerical analysis We perform numerical analysis using Mathematica to study the dependence of the optimal wholesale price, retail price and advertising effectiveness on the model parameters. (a) Effect of price sensitivity of demand (Figures 2, 3): As η1 increases, the advertising effort by retailer √ 1 decreases and that by retailer 2 increases. Fig. 2 shows the variation of u∗i / 1 − X1 − X2 , i = 1, 2, with respect to η1 . Fig. 3 shows the variation of optimal wholesale and retail price brought about by changes in η1 . We find that as η1 increases, the retail and the wholesale prices of both the retailers decrease. The decrease in p∗1 and w1∗ is much faster than the decrease in p∗2 and w2∗ . As η1 increases, retailer 1’s demand decreases and thus, the retailer and the manufacturer compensate by reducing the wholesale and retail prices to 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 (Figures 4, 5): As the advertising effectiveness of retailer 1 increases, the subsidy rates for both retailers decrease. The rate of decrease is higher for retailer 2 than for retailer 1. All other parameters being the same, the retailer with a more effective advertising gets a higher subsidy rate.
Figure 2: Impact of η1 on advertising efforts of retailers
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Figure 3: Optimal wholesale and retail prices vs η1
Figure 4: Impact of ρ1 on advertising efforts of retailers
4
Figure 5: Optimal wholesale and retail prices vs ρ1
Channel coordination
In this segment we compare the profits of all the channel members and the total channel 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 taken based on the maximization of the combined profit of the manufacturer and the two retailers. The wholesale price and subsidy rates do not play a part in this setting. ii.) A decentralized channel without cooperative advertising, with pi , wi and ui as decision variables, i = 1, 2 (i.e., θ1 = θ2 = 0). Our objective from these comparisons is to study the effect of cooperative advertising on the profits of all the channel members, and thereby, to find out if cooperative advertising can act as a tool to increase channel profits and thus, improve channel coordination. We first define the optimization problem of an integrated channel to decide the optimal retail prices (p1 , p2 ) and optimal levels of advertising (u1 , u2 ) : I
V (X1 , X2 ) =
Z max pi (t),ui (t)≥0, i=1,2, t≥0
∞
e−rt (p1 X˙ 1 (t) + p2 X˙ 2 (t) − u21 (t) − u22 (t))dt,
0
12
(26)
subject to p dXi (t) X˙ i (t) = = ρi ui (t)Di (pi (t)) 1 − X1 (t) − X2 (t), dt
Xi (0) = Xi ∈ [0, 1], i = 1, 2.
(27)
The HJB equation for the integrated channel function V I is
rV I (X1 , X2 ) =
max pi (t),ui (t)≥0, i=1,2, t≥0
h i p1 X˙ 1 + p2 X˙ 2 − u21 − u22 + VXI 1 X˙ 1 + VXI 2 X˙ 2 ,
(28)
where VXI i = ∂V I (X1 , X2 )/∂Xi , i = 1, 2, and X˙ 1 and X˙ 2 are given by (27). To compute the value function of the integrated channel in explicit form and compare it with the total channel value functions with and without cooperative advertising, we shall use the linear demand specification (9). We then obtain the following result. Proposition 4.1: The optimal feedback retail price and advertising policies for the integrated channel with linear demand are 1 1 ( − VXI i ) i = 1, 2, 2 ηi √ ρi (1 + ηi VXI i )2 1 − X1 − X2 ∗ ∗ , i = 1, 2, ui = ui (X1 , X2 ) = 8ηi p∗i = p∗i (X1 , X2 ) =
(29) (30)
and the value function of the integrated channel satisfies the following equation
64rV I (X1 , X2 ) = (1 − X1 − X2 )
ρ21 (1 + η1 V I X1 )4 ρ22 (1 + η2 V I X2 )4 + . η12 η22
(31)
Proof: The proof is similar to that of Proposition 3.1. Using (27) in the HJB equation (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 the use 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 = −VXI 1 = −VXI 2 , is constant and solves the equation 64rαI =
(1−αI η1 )4 ρ21 η12
+
(1−αI η2 )4 ρ22 . η22
In the second case, we consider a decentralized channel with cooperative advertising and optimal values of wholesale prices, subsidy rates, retail prices and advertising levels given by (14), (15), (19) and (20), respectively. We define the value function in this case as V c (X1 , X2 ) = Vmc (X1 , X2 ) + Vrc (X1 , X2 ), where Vmc is the manufacturer’s value function (given by (23)) and Vrc is the sum of the value functions of both retailers obtained by (22).
13
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 for retailers are their respective retail prices and advertising levels. This case is treated by using 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 wi∗ . The value functions of the manufacturer and the retailers in this case are defined as Vmn and Vin , i = 1, 2, respectively, and are computed using the following. V1n (X1 , X2 ) = β1n (1 − X1 − X2 ),
i = 1, 2,
and
Vmn (X1 , X2 ) = αn (1 − X1 − X2 ).
The channel value function is defined as V n (X1 , X2 ) = Vmn (X1 , X2 ) + Vrn (X1 , X2 ), where Vrn is the sum of the two retailers’ value functions in the non-cooperative setting. Since the manufacturer is the leader and decides his subsidy rates by maximizing his total discounted profit, it is obvious that Vmc (X1 , X2 ) ≥ Vmn (X1 , X2 ). Thus it remains to study the effect of cooperative advertising on the value functions of the two retailers and the total value function of the channel. Before we proceed further, we recall that the value functions are linear in X1 and X2 , and can be written as a constant coefficient times (1−X1 −X2 ). It is therefore 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 +β1n +β2n ), respectively. Similarly, a comparison between α and αn , and between βi and βin , is equivalent to a comparison between the value functions Vmc and Vmn , and Vic and Vin , respectively, for i = 1, 2. In general, it is difficult to compute the value functions in explicit form. We, therefore, resort to numerical analysis using Mathematica to compute the values of the coefficients and thus compare the value functions. Although we perform numerical analysis for several sets of values of model parameters, we report our findings for the setting described as follows. We use the linear demand formulation (9) and a base 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 the coefficients. Figures 6 and 7 compare (α + β1 + β2 ) and αI , with changes in η1 and ρ1 , respectively. Figures 8 and 9 show the impact of cooperative advertising on the value function coefficients of the manufacturer, retailers 1 and 2, and channel as a whole, by changing η1 and ρ1 , respectively. The data sets for the calculations shown are the same as those used for the results in Figures 2-5. Thus, for any point in Figures 6-9, the values of wi∗ , p∗i and u∗i are the same as the corresponding values in Figures 2-5. Figures 6 and 7 show that the channel value function with cooperative advertising and the integrated channel value function decrease with η1 and increase with ρ1 . Figures 8-9 plot the difference between the value function coefficients for a channel with cooperative advertising and a channel without any, for all the
14
Figure 6: (α + β1 + β2 ) and αI vs η1
Figure 7: (α + β1 + β2 ) and αI vs ρ1
Figure 8: Impact of cooperative advertising on value Figure 9: Impact of cooperative advertising on value function coefficients with changing η1 function coefficients with changing ρ1 members of the channel and for the channel as whole. Thus, Figures 8-9 show the impact of cooperative advertising on the profit functions of the manufacturer, the retailers and the overall channel. As anticipated, we find that the manufacturer always gains from cooperative advertising. The gain for the manufacturer
15
is greater for lower values of η1 and higher values of ρ1 . However, we find that no retailer benefits from cooperative advertising. A retailer’s value functions in the two scenarios could be approximately equal at best, for e.g., retailer 2 for low values of η1 and retailer 1 for low values of ρ1 . Interestingly, we find that retailer 1’s losses show a pattern similar to the manufacturer’s gains, i.e., decreasing with η1 and increasing with ρ1 . Finally, in most of the instances, we find that the total channel value function with cooperative advertising is marginally lower than that without any cooperative advertising. These observations indicate that cooperative advertising does not seem to increase channel profits and improve channel coordination. Roughly speaking, cooperative advertising transfers some of the profits from the retailers’ side to the manufacturer’s, thereby keeping the total channel profits at approximately the same level, decreasing the channel profit in most cases and marginally improving it in a few.
5
The case of brand level competition
The numerical analysis of value functions indicates that the retailers might not benefit from cooperative advertising when the two retailers are selling the product of the same manufacturer. Thus, there might be no incentive for retailers to participate in a cooperative advertising program. It will be interesting to look into this aspect when the second retailer, say, for example, retailer 2, buys from a manufacturer of a competing brand and thus acts as a competitor to retailer 1. In other words, we would like to see the profits of all the parties in presence of brand level competition when the two retailers sell different brands of the same product category. We consider a channel as shown in Figure 10. The manufacturer sells his product to retailer 1 who sells it in the market. Retailer 2, sells a brand which competes in the market with that of retailer 1. We assume that the products sold by the two retailers are perfectly substitutable. Here again, we assume that X1 (t) and X2 (t) denote the cumulative sales by retailer 1 and 2, respectively. Thus the manufacturer’s cumulative sales is only X1 (t). All the other notations remain the same as in the general model. The manufacturer announces his wholesale price w1 (X1 , X2 ) and subsidy rate θ1 (X1 , X2 ) policy for retailer 1. Meanwhile, retailer 2 purchases from competing manufacturer at a wholesale price w2 , which is exogenous in this model. Clearly, θ2 = 0 in this setting. The two retailers play a Nash game to compete in the consumers market and determine their optimal retail prices and advertising efforts, i.e., pi and ui , i = 1, 2, respectively. The retailer i’s optimization problem to maximize the present value of his profit stream over infinite horizon, given their wholesale prices and the subsidy rate policy of the manufacturer for retailer 1, can be obtained by using θ2 = 0 in (2), subject to (1). The manufacturer’s optimal control problem is now
16
Figure 10: Model when retailer 2 buys from a competing manufacturer. given by Z Vm (X1 , X2 ) =
max w1 (t)≥0, 0≤θ1 (t)≤1, t≥0
∞
e−rt w1 (t)X˙ 1 (t)
0 2
− θ1 (t) [u1 (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t))]
dt
(32)
subject to
p X˙ 1 (t) = ρ1 u ¯1 D1 (¯ p1 ) 1 − X1 (t) − X2 (t),
X1 (0) = X1 ∈ [0, 1],
(33)
where u ¯1 and p¯1 are the feedback advertising level and retail price, respectively, of retailer 1, given the wholesale price and subsidy rate policy declared by the manufacturer. 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 ) =
p max [(pi − wi (X1 , X2 ) + ViXi )ρi ui Di (pi ) 1 − X1 − X2 pi ,ui ≥0 p −(1 − θi (X1 , X2 ))u2i + ViXj ρj uj Dj (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 be obtained by simply using θ2 = 0 in (6)-(7). Solving these first-order conditions gives the 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 price and subsidy rate policy can be obtained from (7) by simply using θ2 = 0 and ignoring the terms of w2 , as
17
retailer 2 does not contribute to the manufacturer’s sales. We thus have the following.
rVm (X1 , X2 ) = max
w1 ,θ1 ,≥0
p (w1 + VmX1 )ρ1 u∗1 (X1 , X2 )D1 (p∗1 (X1 , X2 )) 1 − X1 − X2
+ VmX2 ρ2 u∗2 (X1 , X2 )D2 (p∗2 (X1 , X2 ))
p
1 − X1 − X2 − θ1 u∗ 21 (X1 , X2 ) − θ2 u∗ 22 (X1 , X2 ) .
(35)
The solution to the manufacturer’s optimization problem gives the optimal wholesale price and subsidy rate policy for retailer 1 in feedback form, i.e., w1∗ (X1 , X2 ) and θ1∗ (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 pricing and advertising decisions of the two retailers by simply using θ2 = 0 in equations (10) and (11) in Proposition 3.1. Furthermore, the value functions of the two retailers can be obtained by using θ2 = 0 in (12). We thus have the following results. 1 − η1 V1X1 + η1 w1 (X1 , X2 ) , 2η1 1 − η2 V2X2 + η2 w2 p∗2 = p∗2 (X1 , X2 | w1 , w2 , θ1 ) = , 2η2 √ (1 + η1 V1X1 − η1 w1 (X1 , X2 ))2 ρ21 1 − X1 − X2 ∗ ∗ , u1 = u1 (X1 , X2 | w1 , w2 , θ1 ) = 8η1 (1 − θ1 (X1 , X2 )) √ (1 + η2 V2X2 − η2 w2 )2 ρ22 1 − X1 − X2 u∗2 = u∗2 (X1 , X2 | w1 , w2 , θ1 ) = , 8η2 ) p∗1 = p∗1 (X1 , X2 | w1 , w2 , θ1 ) =
(36) (37) (38) (39)
and the value function Vi (X1 , X2 ), i = 1, 2, satisfies (1 + η1 V1X1 − η1 w1 (X1 , X2 ))4 ρ21 64rV1 (X1 , X2 ) 4V1x2 (1 + η2 V2X1 − η2 w2 )3 ρ22 = + , (1 − X1 − X2 ) η12 (1 − θ1 (X1 , X2 )) η2 64rV2 (X1 , X2 ) (1 + η2 V2X2 − η2 w2 )4 ρ22 4V2x1 (1 + η1 VX1 − η1 w1 (X1 , X2 ))3 ρ21 = + . (1 − X1 − X2 ) η22 η1 (1 − θ1 (X1 , X2 )) Taking into account the retailers’ optimal pricing and advertising strategies,we can rewrite the HJB equation for the manufacturer’s value function Vm (X1 , X2 ) given by (35) in the following way. 64rVm (X1 , X2 ) = (1 − X1 − X2 ) (−1 − η1 V1X1 + η1 w1 )3 (−4(VmX1 + w1 )η1 + θ1 + (V1X1 + 4VmX1 + 3w1 )η1 θ1 )ρ21 max w1 ,θ1 ,≥0 η12 (−1 + θ1 )2 3 2 (−1 − η2 V2X2 + η2 w2 ) (−4VmX2 )ρ2 . η2
(40)
We now consider two cases. First, a cooperative equilibrium in which the manufacturer offers a positive op18
timal subsidy rate to the retailer 1 and second, a non-cooperative equilibrium in which θ1 = 0. Our objective is to study and compare the value functions of all the parties in these two cases and investigate the impact of a cooperative advertising program in the presence of brand level competition.
Case 1: A Cooperative Solution We first consider a cooperative solution in which the manufacturer chooses his optimal subsidy rate for retailer 1. Using the first-order conditions w.r.t. w1 and θ1 in (40), we find the manufacturer’s optimal wholesale price and optimal subsidy rate policy, given by the following:
w1∗ = w1∗ (X1 , X2 ) =
1 + ηi (V1X1 − 2VmX1 ) , 3η1
θ1∗ = θ1∗ (X1 , X2 ) = 1/3.
(41)
(42)
The manufacturer’s value function satisfies 4(1 + (VmX1 + V1X1 )η1 )4 ρ21 4(VmX2 (1 + (V2X2 − w2 )η2 ))3 ρ22 64rVm (X1 , X2 ) = + . (1 − X1 − X2 ) 9η12 η2
(43)
Now, using (41) and (42) in (36)-(39), we get the optimal retail prices and advertising efforts of the two retailers as, 2 − η1 (V1X1 + VmX1 ) 1 − η2 (V2X2 − w2 ) , p∗2 = , 3η1 2η2 √ √ (1 + η1 (V1X1 + VmX1 ))2 ρ1 1 − X1 − X2 (1 + η2 (V2X2 − w2 ))2 ρ2 1 − X1 − X2 ∗ ∗ u1 = , u2 = . 12η1 8η2 p∗1 =
Furthermore, the value functions for the two retailers solve 4V1x2 (1 + η2 (V2X2 − w2 ))3 ρ22 64rV1 (X1 , X2 ) 8(1 + η1 (V1X1 + VmX1 ))4 ρ21 + = (1 − X1 − X2 ) 27η12 η2
(44)
64rV2 (X1 , X2 ) (1 + η2 (V2X2 − w2 ))4 ρ22 16V2x1 (1 + η1 (V1X1 + VmX1 ))3 ρ21 = + . (1 − X1 − X2 ) η22 9η1
(45)
and
19
Once again, we propose the following linear form of the value functions.
Vi (X1 , X2 )
=
β˜i (1 − X1 − X2 ),
Vm (X1 , X2 )
=
α ˜ (1 − X1 − X2 ),
i = 1, 2,
(46) (47)
and try to solve for the coefficients α ˜ , β˜1 and β˜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 equations (43), (44)-(45) and (46)-(47), and obtain the following system of equations to be solved for the coefficients α ˜ , β˜1 and β˜2 . 64rβ˜1
=
64rβ˜2
=
64rα ˜
=
8(1 − η1 (β˜1 + α ˜ ))4 ρ21 4β˜1 (1 − η2 (β˜2 + w2 ))3 ρ22 − , 27η12 η2 (1 − η2 (β˜2 + w2 ))4 ρ22 16β˜2 (1 − η1 (β˜1 + α ˜ ))3 ρ21 , − 2 η2 9η1 4(1 − (˜ α + β˜1 )η1 )4 ρ21 4(˜ α(1 − (β˜2 + w2 )η2 ))3 ρ22 . − 9η12 η2
(48) (49) (50)
Case 2: A Non-Cooperative 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 equations (36)-(39) gives the optimal retail prices and advertising levels of the two retailers. Using θ1 = 0 in the manufacturer’s HJB equation, and solving the first-order conditions w.r.t. w1 gives the optimal wholesale price of the manufacturer as follows: w1∗ = w1∗ (X1 , X2 ) =
1 + ηi (V1X1 − 3VmX1 ) . 4η1
(51)
Now using (51), along with θ1 = 0 in (36)-(39), we get the optimal retail prices and advertising efforts of the two retailers as follows: 1 − η2 (V2X2 − w2 ) 5 − 3η1 (V1X1 + VmX1 ) , p∗2 = , 8η1 2η2 √ √ 9(1 + η1 (V1X1 + VmX1 ))2 ρ1 1 − X1 − X2 (1 + η2 (V2X2 − w2 ))2 ρ2 1 − X1 − X2 u∗1 = , u∗2 = . 128η1 8η2 p∗1 =
Furthermore, the value functions of retailer 1, retailer 2 and the manufacturer now solve the following
20
equations, respectively: 4V1x2 (1 + η2 (V2X2 − w2 ))3 ρ22 64rV1 (X1 , X2 ) 81(1 + η1 (V1X1 + VmX1 ))4 ρ21 + = , (1 − X1 − X2 ) 256η12 η2 27V2x1 (1 + η1 (V1X1 + VmX1 ))3 ρ21 64rV2 (X1 , X2 ) (1 + η2 (V2X2 − w2 ))4 ρ22 + = , (1 − X1 − X2 ) η22 16η1 4(VmX2 (1 + (V2X2 − w2 )η2 ))3 ρ22 27(1 + (VmX1 + V1X1 )η1 )4 ρ21 64rVm (X1 , X2 ) + = . (1 − X1 − X2 ) 64η12 η2 We use linear value functions of the form
Vi (X1 , X2 )
= β˜n i (1 − X1 − X2 ),
Vm (X1 , X2 )
= α˜n (1 − X1 − X2 ),
i = 1, 2,
(52) (53)
and solve for the coefficients α˜n , β˜n 1 and β˜2n using the following set of equations: 64rβ˜1n
=
64rβ˜2n
=
64rα˜n
=
4β˜n (1 − η2 (β˜2n + w2 ))3 ρ22 81(1 − η1 (β˜1n + α˜n ))4 ρ21 − 1 , 2 256η1 η2 (1 − η2 (β˜2n + w2 ))4 ρ22 27β˜2n (1 − η1 (β˜1n + α˜n ))3 ρ21 − , η22 16η1 27(1 − (α˜n + β˜1n )η1 )4 ρ21 4(α˜n (1 − (β˜2n + w2 )η2 ))3 ρ22 − , 2 64η1 η2
(54) (55) (56)
where the superscript n denotes a non-cooperative solution.
Comparison of Value Functions in Cooperative and Non-Cooperative Solution We now compare the value functions of all the parties (retailer 1, retailer 2 and the manufacturer) in a cooperative solution given by (46)-(47) with the corresponding values in a non-cooperative solution, given by (52)-(53). Since it is quite obvious that the manufacturer will always gain from cooperative advertising, our aim is to investigate its impact on the profits of the two retailers. In particular, we want to find if retailer 1 has any incentive to join a coop advertising program, when he faces a competing retailer who is selling another brand and hence not getting any advertising subsidy from 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 ˜ , β˜1 , and β˜2 with α˜n , β˜n 1 and β˜2n , 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 the system of equations (48)-(50) and (54)-(56) explicitly, and therefore, we resort to numerical analysis to get some insights. We solved the system of equations (48)-(50) and (54)-(56) for various values of η1 , η2 , ρ1
21
and ρ2 , and report some of our key findings in figures 11-12. We solved the equations for several values of problem parameters. Figure 11 shows the difference in value function coefficients of all the parties for various values of η1 , with ρ1 = ρ2 = 0.1, r = 0.05, and η2 = 1. Since w2 is also an exogenous parameter for this model, in the analysis shown here we chose w2 equal to w1∗ , 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 get the same wholesale price from their respective manufacturers. We find that while the manufacturer always gains from cooperative advertising, retailer 1 might gain or loose depending on the values of the parameters. We also find that retailer 1 benefits at low values of η1 . Figure 12 shows the difference in the value functions with varying values of ρ1 when η1 = η2 = 0.1, r = 0.05. and ρ2 = 1. Here again, we find that retailer 1 can benefit from cooperative advertising, in this case as long as ρ1 is not too high. These results indicate that a cooperative
Figure 11: Benefit from cooperative advertising vs η1 when retailer 2 buys from a competing manufacturer
Figure 12: Benefit from cooperative advertising vs ρ1 when retailer 2 buys from a competing manufacturer
advertising program can benefit retailer 1 as well, depending on the model parameters. This is in contrast to the scenario in which both the retailers buy from the same manufacturer, where numerical analysis showed little indication of retailers benefiting from cooperative advertising. A general trend observed in several numerical calculations was that retailer 1 is more likely to benefit at lower values of ρ1 and η1 . It would be interesting if one could obtain an explicit condition based on the model parameters which would indicate the scenarios under which retailer 1 would benefit. However, given the difficulty in solving the equations 22
(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 as the Robinson-Patman Act of 1936. The Robinson-Patman act, along with other such legislations, was designed to prevent creation of monopolies in the market and to enhance competition. Such acts prevent price discrimination between two or more competing buyers of a product. To study the impact of such a legislation in our model, we consider a case when the manufacturer is restricted to offer equal wholesale prices and equal subsidy rates to the two retailers. We define VmRP (X1 , X2 ), V1RP (X1 , X2 ), V2RP (X1 , X2 ) and V RP as the value functions of the manufacturer, retailer 1, retailer 2, and the total channel, respectively, with the superscript RP standing for Robinson and Patman. These value functions solve the optimal control problems defined by (1)-(3), with w1 = w2 = w and θ1 = θ2 = θ. We look for linear value functions of the following form: VmRP (X1 , X2 ) = αRP (1 − X1 − X2 ), ViRP = βiRP (1 − X1 − X2 ), i = 1, 2, and V RP = (αRP + β1RP + β2RP )(1 − X1 − X2 ). In the following analysis, we consider the linear demand function (9), and use numerical analysis to compute the value of the coefficients αRP , βiRP , 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 common subsidy rate for the two retailers θ∗ . Figure 13 compares the common optimal wholesale price for the two retailers in the case of an anti-discriminatory act (w∗ ) with the optimal wholesale prices for the two retailers without such legislation (w1∗ , w2∗ ). 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 sensitivity of demand (ηi ) pays a higher wholesale price in the case of anti-discriminatory act in effect than in the absence of such an act. Figure 14 shows the variation of θ∗ with respect to η1 . Recall that in the general model with linear demand, the optimal subsidy rates for the two retailers are independent of the values of η1 , ρ1 and r, and equal to 1/3. In the case when the manufacturer has to offer equal wholesale prices and subsidy rates to the two retailers, we find that the common optimal subsidy rate is no longer a fixed value and varies with η1 . We find that the maximum value of θ∗ achieved is equal to the subsidy rates without any legislation (θ1∗ = θ2∗ = 1/3). Thus, the introduction of additional constraint on the manufacturer (i.e., to offer equal subsidy rates to the two retailers) makes the manufacturer to offer lower subsidy rates. Next, we study the impact of an anti-discriminatory act on the profit functions of all the channel members,
23
and of the channel as whole. Figure 15 shows the difference in the value function coefficients with and without an anti-discriminatory act for the manufacturer, retailers and the total channel, i.e., α − αRP , βi = βiRP , i = 1, 2, and (α+β1 −β2 )−(αRP −β1RP −β2RP ), respectively. As expected, the manufacturer does not benefit from an anti-discriminatory act because of an additional constraint on his optimization problem. We also find that only the retailer with a lower value of price sensitivity 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 one retailer offsets the losses of the other two parties. Figure 16 shows the ratios of the channel value function in three cases, namely, the ratio of a channel with cooperative advertising and no legislation, the ratio of a channel with cooperative advertising and an anti-discriminatory legislation in effect, and the ratio of a channel with no cooperative advertising and the integrated channel value function. Thus Figure 16 shows the level of coordination possible in these three scenarios. We find that in all the instances, a channel with no cooperative advertising performs better than a channel with cooperative advertising and no legislation. Moreover the value function for a channel with cooperative advertising and an anti-discriminatory act is highest of the three, except when the difference between η1 and η2 is small. These results indicate that in most cases, highest degree of channel coordination can be achieved when we have cooperative advertising along with an anti-discriminatory legislation in effect. However, in some cases when, roughly speaking, the retailers are nearly identical, a channel with no cooperative advertising is able to achieve the highest total profits and thus the highest level of coordination.
7
Concluding Remarks
We obtain the feedback Stackelberg equilibrium and compute the optimal values of the advertising levels by the two retailers, retail prices, wholesale prices and subsidy rates in the case of a linear demand. We find that the optimal subsidy rates are independent of the model parameters and are equal to 1/3. We also provide the sensitivities of the optimal advertising levels, retail prices and wholesale prices with respect to η1 and ρ1 . We study the effect of cooperative advertising on the profits of all the members of the channel and that of the total channel. We find that the cooperative advertising benefits only the manufacturer whereas the two retailers earn less profit, and except in a few cases, the total channel does not benefit either. An interesting conclusion that appears from our analysis is that with wholesale and retail prices as decision variables as well, cooperative advertising is ineffective and a redundant mechanism. In fact, since there seems to be no evidence of retailers benefiting from cooperative advertising, such an arrangement might not hold. We also
24
Figure 13: Optimal wholesale prices with and without anti-discriminatory act vs η1
Figure 14: Optimal subsidy rates with and without anti-discriminatory act vs η1
Figure 15: Value function coefficients with an anti-discriminatory act minus those without any act, with changing η1
Figure 16: Ratio of channel value function to integrated channel value function in different cases, with changing η1
analyze a scenario when retailer 2 buys from another manufacturer and find that under such brand level competition, cooperative advertising program can be beneficial for retailer 1 as well. Finally, we consider
25
the case of an anti discriminatory legislation where the manufacturer has to offer equal subsidy rates to the two retailers and find that higher channel profits can be achieved with such a legislation.
References [1] Teng, J., Thompson, G.L.: Optimal pricing and advertising policies for new product oligopoly models. Market. Sci. 3(2), 148-168 (1984) [2] Sethi, S.P., Prasad, A., He, X.: Optimal Advertising and Pricing in a New-Product Adoption Model. J. Optim. Theory Appl. 139(2), 351-360 (2008) [3] Krishnamoorthy, A., Prasad, A., Sethi, S. P.: Optimal Pricing and Advertising in a Durable-Good Duopoly. Eur. J. Operat. Res. 200(2), 486-497 (2010) [4] Bergen, M., John, G.: Understanding cooperative advertising participation rates in conventional channels. J. of Market. Res. 34, 357-369 (1997) [5] Nagler, M.G.: An exploratory analysis of the determinants of cooperative advertising participation rates. Market. Lett. 17(2) 91-102 (2006) [6] Dant, R.P., Berger, P.D.: Modeling cooperative advertising decisions in franchising. J. Operat. Res. Soc. 47(9) 1120-1136 (1996) [7] Dutta, S., Bergen, M., John, G., Rao, A.: Variations in the Contractual Terms of Cooperative Advertising Contracts: An Empirical Investigation. Market. Lett. 6(1) 15-22 (1995) [8] Berger, P.D.: Vertical cooperative advertising ventures. J. Market. Res. 9(3) 309-312 (1972) [9] Kali, R.: Minimum advertised price. J. Econ. Manag. strategy 7(4) 647-668 (1998) [10] Huang, Z., Li, S.X., Mahajan, V.: An analysis of manufacturer-retailer supply chain coordination in cooperative advertising. Dec. Sci. 33(3) 469-494 (2002) [11] Jørgensen, S., Sigu´e, S.P., Zaccour, G.: Dynamic cooperative advertising in a channel. J. retail. 76(1) 71-92 (2000) [12] Jørgensen, S., Taboubi, S., Zaccour, G.: Cooperative advertising in a marketing channel. J. Optim. Theory. Appl. 110(1): 145-158 (2001) [13] Jørgensen, S., Taboubi, S., Zaccour, G.: Retail promotions with negative brand image effects: Is cooperation possible? Eur. J. Oper. Res. 150(2) 395-405 (2003) [14] Karray, S., Zaccour, G.: A differential game of advertising for national brand and store brands. A. Haurie, G. Zaccour eds. Dynamic Games: Theory and Applications. Springer, Berlin, 213-229 (2005)
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[15] He, X., Prasad, A., Sethi, S.P.: Cooperative advertising and pricing in a dynamic stochastic supply chain: feedback Stackelberg strategies. Prod. and. Oper. Mgmt. 18(1) 78-94 (2009) [16] Sethi, S.P.: Deterministic and stochastic optimization of a dynamic advertising model. Optim. Control Appl. Meth. 4(2) 179-184 (1983) [17] He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P.: Retail Competition and Cooperative Advertising. Oper. Res. Letters. 39, 11-16 (2011) [18] Bass, F.M.: A new product growth model for consumer durables. Mgmt. Sci. 15, 5, 215-227 (1969) [19] Mahajan, N., Muller, E., Bass, F. M.: New product diffusion models in marketing: A review and directions for research. J. Market. Res. 54 (1), 1-26 (1990) [20] Robinson, B., Lakhani, C.: Dynamic price models for new-product planning. Mgmt. Sci. 21(10), 11131122 (1975) [21] Krishnan, T. V., Bass, F. M., Jain, D. C.: Optimal pricing strategy for new products. Mgmt. Sc. 45(12), 1650-1663 (1999) [22] Krishnan, T. V., Kumar, V.: Impact of a late entrant on the diffusion of a new product/service. J. of Mkt. Res. 37(2), 269-278 (2000) [23] Petruzzi, N., Dada, M.: Pricing and the newsvendor problem: A review with extensions. Oper. Res, 47, 183194 (1999)
27
†
June 7, 2011
Abstract Cooperative advertising is an incentive offered by a manufacturer to influence retailers’ promotional decisions. We study a dynamic durable goods duopoly with a manufacturer and two independent and competing retailers. The manufacturer as a Stackelberg leader announces his wholesale prices and his shares of retailers’ advertising costs, and the retailers in response play a Nash differential game in choosing their optimal retail prices and advertising efforts over time. We obtain the feedback equilibrium policies in explicit form for a linear demand formulation. We investigate issues like channel coordination and anti-discriminatory legislation and also study a case when the manufacturer sells through only one retailer and the second retailer sells a competing brand.
Keywords: Cooperative advertising, Stackelberg differential game, Nash differential game, sales-advertising dynamics, feedback Stackelberg equilibrium, Durable goods.
∗ Visiting Assistant Professor, School of Management, Binghamton University, State University of New York, PO Box 6000, Binghamton, NY, 13902 e-mail: [email protected] † Charles & Nancy Davidson Distinguished Professor of Operations Management, School of Management, Mail Station SM30, The University of Texas at Dallas, 800 W. Campbell Rd. Richardson, Texas 75080-3021 e-mail: [email protected]
Electronic copy available at: http://ssrn.com/abstract=1898309
1
Introduction
For any supply chain, pricing and advertising decisions are essentially dynamic in nature. More often than not, these decisions are taken in conjunction with each other as they interact continuously over time, having an impact on sales and ultimately on the profits of the supply chain members. While many researchers in the past have studied dynamic advertising strategies or dynamic pricing policies, far fewer models consider both pricing and advertising decisions together, mainly due to the analytical complexity of such models. The tractability of such problems reduces further in the models which consider a competitive setting involving multiple brands within a product category, or within brand retail level competition. Some of the recent models which do incorporate optimal pricing and advertising decisions include Teng and Thompson [1], which studied price and advertising in a new product oligopoly model. Their analysis was largely numerical in nature. Sethi et al. [2] proposed a new product adoption model and examined dynamic advertising and pricing decisions for a monopolist firm. Krishnamoorthy et al. [3] extended the Sethi et al. [2] model to study optimal pricing and advertising policies in a durable goods duopoly and solved the resulting differential game explicitly. Within the group of researchers and practitioners studying dynamic advertising models and strategies, an increasing attention is being devoted to the practice of cooperative advertising. Cooperative advertising is an important incentive offered by a manufacturer to influence retailers’ promotional decisions. In a typical arrangement, a manufacturer agrees to reimburse a fraction of each retailers advertising expenditures in selling his product (Bergen and John [4]). This fraction is typically known as the ’subsidy rate’ offered by the manufacturer to a retailer. Cooperative advertising is a fast increasing activity amounting to billions of dollars a year. Nagler [5] found that the 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 was more than $25 billion in 2007. Cooperative advertising can be a significant part of the manufacturer’s expense according to Dant and Berger [6], and as many as 25-40% of local advertisements and promotions are cooperatively funded. In addition, Dutta et al. [7] report that the subsidy rates differ from industry to industry: it is 88.38% for consumer 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 advertising. Berger [8] modeled cooperative advertising in the form of a wholesale price 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 cooperative advertising. Dant and Berger [6] extended the Berger model to incorporate demand uncertainty. 1
Electronic copy available at: http://ssrn.com/abstract=1898309
Kali [9] studied cooperative advertising from the perspective of coordinating a manufacturer-retailer channel. Huang et al. [10] allowed for advertising by the manufacturer in addition to cooperative advertising. They also justified their static model by making a case for short-term effects of promotion. Jørgensen et al. [11] formulate a dynamic model with cooperative advertising as a Stackelberg differential game between a manufacturer and a retailer with the manufacturer as the leader. They consider short term as well as long term forms of advertising efforts made by the retailer as well as the manufacturer. They show that manufacturer’s support of both types of retailer advertising benefits both channel members more than the support of only one type; moreover, support of one type is better than no support at all. Jørgensen et al. [12] modified the above model by introducing decreasing marginal returns to goodwill and studied two scenarios: a Nash game without advertising support and a Stackelberg game with support from the manufacturer as the leader. They characterized stationary feedback policies in both cases. Jørgensen et al. [13] explored the possibility of advertising cooperation even when the retailer’s promotional efforts may erode the brand image. Karray and Zaccour [14] extended the above model to consider both the manufacturer’s national advertising and the retailer’s local promotional effort. Recently, He et al. [15] solved a manufacturerretailer Stackelberg game with cooperative advertising using the stochastic version of the Sethi [16] model. He et al. [17] consider a cooperative advertising channel consisting of a manufacturer selling its product through two independent and competing retailers. Although there have been a number of studies on cooperative advertising decisions, very few models consider a competitive setting, and even fewer incorporate pricing decisions as well. In this paper, we study a cooperative advertising model for a retail market duopoly with one manufacturer as the Stackelberg leader and two retailers as followers, in the case of a market for durable goods. A durable good can be defined as a commodity which once purchased by the consumer, does not need to be repurchased for a lengthy period of time. Examples of durable goods include cars, TV’s, microwave ovens, washing machines etc. The market potential of such items depletes with time as cumulative sales increase, and eventually, saturation is reached. A celebrated sales dynamics model of durable goods is the Bass [18] model of innovation diffusion. Many researchers extended the Bass model to include price and advertising effects. Mahajan et al. [19] provide a review of such models. Robinson and Lakhani [20] extended the Bass model to include pricing decisions. A more recent study of optimal pricing policies for a monopolist is by Krishnan et al. [21], who found that either a monotonically declining or an increasing-decreasing pricing pattern is optimal. Krishnan et al. [22] proposed a brand-level diffusion model to analyze the impact of a late entrant on the diffusion of brands of a new consumer durable and that of a category as whole. Teng and Thompson [1] incorporated price and
2
advertising in a new product oligopoly model. They showed that the optimal price and advertising patterns are high initially and then decrease over time. Their analysis though was limited to numerical in nature. More recently, Sethi et al. [2] proposed a sales dynamics model for durable goods and examined advertising and price decisions by a monopolist firm. 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 pricing and advertising policies. Contrary to Teng and Thompson [1], they found that optimal price should be constant and the optimal advertising should decrease over time. We extend the competitive dynamics introduced in Krishnamoorthy et al. [3] to study pricing policies and cooperative advertising for durable goods in the presence of retail-level competition and obtain explicit results for a linear demand formulation. With this study, we intend to bridge the gap in the cooperative advertising literature by including the pricing decisions for the manufacturer and the two retailers in the model for durable goods. To the best of our knowledge, ours is the first study to incorporate pricing and advertising decisions with cooperative advertising in a dynamic, competitive durable goods retail market. We therefore look to answer some key research questions as follows. • What is the optimal wholesale price and subsidy rate policy of the manufacturer, and the optimal retail prices policies and advertising responses by the retailers in feedback form? • What is the impact of a coop advertising program on the profits of all the members in the channel? How do channel profits with coop program compare to those without coop advertising, and to the integrated channel profit? Can coop advertising lead to better channel coordination? • What are the effects of an anti-discriminatory legislation which would restrict the manufacturer to offer equal subsidy rates to his retailers. How does it impact the optimal subsidy rates, profits of all the members in the channel, and the total channel profits? • What is the impact of competition from another brand on the optimal policies and profits of channel members? The rest of the paper is organized as follows. We describe the model in section 2, followed by analysis and some results in section 3. In section 4 we investigate the issue of channel coordination and compute the effect of cooperative advertising on the profit functions of the manufacturer, of the two retailers, and of the overall channel. In section 5, we study a model in which only one retailer buys from the manufacturer and the second retailer acts as a competitor to the manufacturer and his retailer and look into the impact of
3
cooperative advertising on the value functions of the different parties. In section 6 we discuss an extension where an anti-discriminatory legislation forces the manufacturer to offer equal wholesale prices and subsidy rates to the two retailers. We also study the impact of such a legislation on the profits of all the members in the channel and that of the channel as whole. We finally conclude our work in section 7.
2
The model
We consider a dynamic retail duopoly where a manufacturer sells its product through one or both of two independent and competing retailers, labeled 1 and 2. The manufacturer decides wholesale price for the two retailers (w1 , w2 ), and may choose to subsidize the advertising expenditures of the retailers. The subsidy, expressed as a fraction of a retailer’s total advertising expenditure, is referred to as the manufacturer’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 Di (pi )
Manufacturer’s subsidy rate for retailer i at time t, 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 , V m
Value functions of retailer i and of the manufacturer, respectively,
VI
Value function of the integrated channel.
Without loss of generality, we assume that the manufacturing cost of the the product is zero. Thus, the margin for the manufacturer from retailer i is equal to the wholesale price wi (t) charged from retailer i. The margin for retailer i can be defined as mi (t) = pi (t) − wi (t). Furthermore, we use the standard notations
4
ViXj = ∂Vi /∂Xj , i = 1, 2, j = 1, 2, and VmXi = ∂Vm /∂Xi and VXI i = ∂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 cumulative units sold, which is the instantaneous sales, is denoted by X˙ i (t), and is given by X˙ i (t)
=
p dXi (t) = ρi ui (t)Di (pi (t)) 1 − X1 (t) − X2 (t), dt
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 advertising effort at time t, ρi is the effectiveness of firm i’s advertising, and Di (pi (t)) is the demand of retailer i as a function of own price, pi (t) at time t. We employ the idea of a feedback Stackelberg solution in our analysis. The sequence
Figure 1: Sequence of Events of events is shown in Fig. 1. The manufacturer, who is the Stackelberg leader of the game, announces the wholesale price policy wi (X1 (t), X2 (t)) and the subsidy rate policy θi (X1 (t), X2 (t)) for retailer i, i = 1, 2 at time t. The retailers, acting as followers choose their respective retail prices and advertising efforts in 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 θi (X1 , X2 ), i = 1, 2, respectively. The retailers in response choose their optimal retail price and advertising effort by solving their respective optimization problems. The retailer i’s optimal control problem to maximize the present value of his profit stream over the infinite horizon, given the manufacturer’s wholesale prices and subsidy rates policies, is given by Z Vi (X1 , X2 ) =
max pi (t),ui (t)≥0, i=1,2, t≥0
0
∞
e−rt (pi (t) − wi (X1 (t), X2 (t)))X˙ i (t) − (1 − θi (X1 (t), X2 (t)))u2i (t) dt,
5
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 differential game defined by (1)-(2) would give retailer i’s feedback retail price pi (X1 (t), X2 (t)) and advertising effort ui (X1 (t), X2 (t)), i = 1, 2, which, with a slight 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 optimal control problem. The manufacturer’s problem is given by Z Vm (X1 , X2 ) =
∞
e
max wi (t)≥0, 0≤θi (t)≤1, i=1,2, t≥0
0
−rt
2 X
wi (t)X˙ i (t)
i=1 2
− θi (t) [ui (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t))]
dt
(3)
1 = 1, 2,
(4)
subject to
X˙ i (t)
p = ρi u¯i Di (p¯i ) 1 − X1 (t) − X2 (t),
Xi (0) = Xi ∈ [0, 1],
where u¯i = ui (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t)) and p¯i = (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 declared by the manufacturer. The solution to the optimal control problem (3)-(4) gives the optimal wholesale price and subsidy rate in feedback form, which, with a slight abuse of notation, can be expressed as wi∗ (X1 , X2 ) and θi∗ (X1 , X2 ), i = 1, 2, respectively. Similarly, we can express retailer i’s retail price and advertising policy as p∗i (X1 , X2 ) = p∗i (X1 , X2 | w1∗ (X1 , X2 ), w2∗ (X1 , X2 ), θ1∗ (X1 , X2 ), θ2∗ (X1 , X2 )) and u∗i (X1 , X2 ) = u∗i (X1 , X2 | w1∗ (X1 , X2 ), w2∗ (X1 , X2 ), θ1∗ (X1 , X2 ), θ2∗ (X1 , X2 )), i = 1, 2, respectively. The optimal feedback policies of the manufacturer and the retailers, i.e., [wi∗ (X1 , X2 ), θi∗ (X1 , X2 )] and [p∗i (X1 , X2 ), u∗i (X1 , X2 )] , i = 1, 2, respectively, constitute a time-consistent feedback Stackelberg equilibrium of the problem (1)-(4). Substituting these policies into the state equation (1), we get the cumulative sales vector (X1∗ (t), X2∗ (t)), t ≥ 0, and the decisions of the manufacturer and the retailers as [wi∗ (t) = wi∗ (X1∗ (t), X2∗ (t)); θi∗ (t) = θi∗ (X1∗ (t), X2∗ (t))] and [p∗i (t) = p∗i (X1∗ (t), X2∗ (t)); u∗i (t) = u∗i (x∗1 (t), x∗2 (t))] , t ≥ 0, i = 1, 2, respectively.
6
3
Preliminary results
We first solve retailer i’s problem to find the optimal pricing and advertising policy 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 wholesale price 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 ) =
p max [(pi − wi (X1 , X2 ) + ViXi )ρi ui Di (pi ) 1 − X1 − X2 pi ,ui ≥0 p −(1 − θi (X1 , X2 ))u2i + ViXj ρj uj Dj (pj ) 1 − X1 − X2 ],
i = 1, 2, j = 1, 2, i 6= j,(5)
where ViXj represents 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, p 1 − X1 − X2 (Di (pi ) + (pi − wi + ViXi )∂Di (pi )/∂pi ) = 0, p ρi Di (pi ) 1 − X1 − X2 (pi − wi + ViXi ) − 2ui (1 − θi ) = 0.
ρi ui
(6) (7)
Solution of the first order conditions in (6)-(7) gives the optimal retail price and advertising 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 wholesale price and subsidy rate policy, and solves his problem to determine the optimal wholesale prices and subsidy rates for the two retailers. The HJB equation for the manufacturer’s value function Vm (X1 , X2 ) is
rVm (X1 , X2 ) =
max
p (w1 + VmX1 )ρ1 u∗1 (X1 , X2 )D1 (p∗1 (X1 , X2 )) 1 − X1 − X2
w1 ,w2 ,θ1 ,θ2 ≥0
p + (w2 + VmX2 )ρ2 u∗2 (X1 , X2 )D2 (p∗2 (X1 , X2 )) 1 − X1 − X2 − θ1 u∗ 21 (X1 , X2 ) − θ2 u∗ 22 (X1 , X2 ) .
(8)
The solution to the manufacturer’s optimization problem (8) gives the optimal wholesale price and subsidy rate in feedback form, i.e., wi∗ (X1 , X2 ) and θi∗ (X1 , X2 ), i = 1, 2, respectively. To explore further, we need to
7
specify the demand function for the two retailers. We consider the linear demand specification
Di (pi ) = 1 − ηi pi ,
i = 1, 2,
(9)
where ηi represents the price sensitivity of demand. The linear demand function is very common in the literature (e.g., [2], [3], and [23]). Using the demand specification (9) in the first-order conditions (6)-(7) and solving for 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 advertising decision of retailer i, i = 1, 2, for the linear demand specification (9) is 1 − ηi ViXi + ηi wi (X1 , X2 ) 2ηi √ (1 + ηi ViXi − ηi wi (X1 , X2 ))2 ρ2i 1 − X1 − X2 , u∗i = u∗i (X1 , X2 | w1 , w2 , θ1 , θ2 ) = 8ηi (1 − θi (X1 , X2 )) p∗i = p∗i (X1 , X2 | w1 , w2 , θ1 , θ2 ) =
(10) (11)
and the value function Vi (X1 , X2 ) satisfies (1 + ηi ViXi − ηi wi (X1 , X2 ))4 ρ2i 64rVi (X1 , X2 ) =(1 − X1 − X2 ) ηi2 (1 − θi (X1 , X2 )) 4Vix3−i (1 + η3−i V(3−i)X3−i − η3−i w3−i (X1 , X2 ))3 ρ23−i . + η3−i (1 − θ3−i (X1 , X2 ))
(12)
Proof: Using (9) in the first-order conditions (6)-(7), we obtain (10) and (11), and then use (10)-(11) in (5) to obtain (12). We can see from (10) that the optimal retail price for retailer i increases with his wholesale price and decreases with his marginal benefit with respect to his cumulative sales. The advertising effort by retailer i increases with his marginal benefit with respect to his cumulative sales and decreases with his wholesale price. Moreover, the advertising 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 rewrite the HJB equation for the manufacturer’s value function V (X1 , X2 ), given by (8), in the following way. 64rVm (X1 , X2 ) = (1 − X1 − X2 ) (−1 − η1 V1X1 + η1 w1 )3 (−4(VmX1 + w1 )η1 + θ1 + (V1X1 + 4VmX1 + 3w1 )η1 θ1 )ρ21 max w1 ,w2 ,θ1 ,θ2 ≥0 η12 (−1 + θ1 )2 8
(−1 − η2 V2X2 + η2 w2 )3 (−4(VmX2 + w2 )η2 + θ2 + (V2X2 + 4VmX1 + 3w2 )η2 θ2 )ρ22 . η22 (−1 + θ2 )2
(13)
We can now obtain the manufacturer’s optimal wholesale prices and subsidy rates policy as shown in the following result. Proposition 3.2: The manufacturer’s optimal wholesale prices are
wi∗ = wi∗ (X1 , X2 ) =
1 + ηi (ViXi − 2VmXi ) 3ηi
i = 1, 2,
(14)
and the optimal subsidy rates for the two retailers are
θi∗ = θi∗ (X1 , X2 ) = 1/3
i = 1, 2.
(15)
The manufacturer’s value function Vm (X1 , X2 ) satisfies (1 + (VmX1 + V1X1 )η1 )4 ρ21 (1 + (VmX2 + V2X2 )η2 )4 ρ22 144rVm (X1 , X2 ) = (1 − X1 − X2 ) + . η12 η22
(16)
Proof: Solving the first-order conditions with respect to wi and θi , i = 1, 2, in (13) gives the following: 1 1 + ηi (ViXi + 3VmXi (−1 + θi )) , or wi = , ηi ηi (4 − 3θi ) 4ηi (wi + VmXi ) − (1 + ηi (ViXi − wi )) θi = . 4ηi (wi + VmXi ) + (1 + ηi (ViXi − wi )) wi = ViXi +
We ignore the possibility that wi = ViXi +
1 ηi ,
(17) (18)
as it yields p∗i = 1/ηi from (10), which means Di (p∗i ) = 0 in
(9). With this observation, we solve equations (17)-(18) to get the optimal values of wi∗ and θi∗ , i = 1, 2, as shown in (14) and (15), respectively. We then use (14) and (15) in (13) to obtain (16). Equation (17) shows that the optimal wholesale price for retailer i decreases with the 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 cumulative sales. Thus, if VmXi is high, the manufacturer incentivizes retailer i to increase sales by reducing the wholesale price for retailer i. We can see from (10) and (11) that a 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 ViXi is high, the manufacturer increases the wholesale prices, since he knows that retailer i has his own incentive to increase his sales by reducing p∗i and increasing u∗i , which is again evident from (10) and (11).
9
In fact, using (14) and (15) in (10) and (11), we get, for i = 1, 2, 2 − ηi (ViXi + VmXi ) , 3ηi √ (1 + ηi (ViXi + VmXi ))2 ρi 1 − X1 − X2 , u∗i = 12ηi p∗i =
(19) (20)
and 216rVi (X1 , X2 ) = (1 − X1 − X2 )
(1 + ηi (ViXi + VmXi )4 ρ2i ηi2 6Vix3−i (1 + η3−i (V(3−i)X3−i + VmX3−i ))3 ρ23−i + . η3−i
(21)
Equations (19) and (20) show that the retailer i’s retail price decreases and his advertising effort increases with increase in ViXi as well as VmXi , thereby increasing the sales rate of retailer i. In the dynamic programming equations (16) and (21), we see that with θ1∗ and θ2∗ being constants, the value functions Vi (X1 , X2 ) and Vm (X1 , X2 ) are linear in X1 and X2 and are a multiple of (1 − X1 − X2 ). We therefore, propose the following form of the value functions
Vi (X1 , X2 )
=
βi (1 − X1 − X2 ),
Vm (X1 , X2 )
=
α(1 − X1 − X2 ),
i = 1, 2,
(22) (23)
and try to solve for the coefficients α, β1 and β2 to obtain the optimal strategies in feedback form. With this proposed form, 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 equations (16), (21) and (22)-(23), and obtain the following system of equations to be solved for the coefficients α, β1 and β2 . For i = 1, 2,
216rβi
=
144rαi
=
6βi (−1 + η3−i (β3−i + α+))3 ρ23−i (−1 + ηi (βi + αi ))4 ρ2i + , ηi2 η3−i (−1 + (α + β1 )η1 )4 ρ21 (−1 + (α + β2 )η2 )4 ρ22 + . η12 η22
(24) (25)
In general, it is difficult to obtain an explicit solution of the system of equations (24)-(25). Nevertheless, it is easy to solve these equations numerically and study the dependence of p∗i , wi∗ and u∗i , i = 1, 2, on model parameters, i.e., ηi and ρi .
10
General Case: numerical analysis We perform numerical analysis using Mathematica to study the dependence of the optimal wholesale price, retail price and advertising effectiveness on the model parameters. (a) Effect of price sensitivity of demand (Figures 2, 3): As η1 increases, the advertising effort by retailer √ 1 decreases and that by retailer 2 increases. Fig. 2 shows the variation of u∗i / 1 − X1 − X2 , i = 1, 2, with respect to η1 . Fig. 3 shows the variation of optimal wholesale and retail price brought about by changes in η1 . We find that as η1 increases, the retail and the wholesale prices of both the retailers decrease. The decrease in p∗1 and w1∗ is much faster than the decrease in p∗2 and w2∗ . As η1 increases, retailer 1’s demand decreases and thus, the retailer and the manufacturer compensate by reducing the wholesale and retail prices to 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 (Figures 4, 5): As the advertising effectiveness of retailer 1 increases, the subsidy rates for both retailers decrease. The rate of decrease is higher for retailer 2 than for retailer 1. All other parameters being the same, the retailer with a more effective advertising gets a higher subsidy rate.
Figure 2: Impact of η1 on advertising efforts of retailers
11
Figure 3: Optimal wholesale and retail prices vs η1
Figure 4: Impact of ρ1 on advertising efforts of retailers
4
Figure 5: Optimal wholesale and retail prices vs ρ1
Channel coordination
In this segment we compare the profits of all the channel members and the total channel 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 taken based on the maximization of the combined profit of the manufacturer and the two retailers. The wholesale price and subsidy rates do not play a part in this setting. ii.) A decentralized channel without cooperative advertising, with pi , wi and ui as decision variables, i = 1, 2 (i.e., θ1 = θ2 = 0). Our objective from these comparisons is to study the effect of cooperative advertising on the profits of all the channel members, and thereby, to find out if cooperative advertising can act as a tool to increase channel profits and thus, improve channel coordination. We first define the optimization problem of an integrated channel to decide the optimal retail prices (p1 , p2 ) and optimal levels of advertising (u1 , u2 ) : I
V (X1 , X2 ) =
Z max pi (t),ui (t)≥0, i=1,2, t≥0
∞
e−rt (p1 X˙ 1 (t) + p2 X˙ 2 (t) − u21 (t) − u22 (t))dt,
0
12
(26)
subject to p dXi (t) X˙ i (t) = = ρi ui (t)Di (pi (t)) 1 − X1 (t) − X2 (t), dt
Xi (0) = Xi ∈ [0, 1], i = 1, 2.
(27)
The HJB equation for the integrated channel function V I is
rV I (X1 , X2 ) =
max pi (t),ui (t)≥0, i=1,2, t≥0
h i p1 X˙ 1 + p2 X˙ 2 − u21 − u22 + VXI 1 X˙ 1 + VXI 2 X˙ 2 ,
(28)
where VXI i = ∂V I (X1 , X2 )/∂Xi , i = 1, 2, and X˙ 1 and X˙ 2 are given by (27). To compute the value function of the integrated channel in explicit form and compare it with the total channel value functions with and without cooperative advertising, we shall use the linear demand specification (9). We then obtain the following result. Proposition 4.1: The optimal feedback retail price and advertising policies for the integrated channel with linear demand are 1 1 ( − VXI i ) i = 1, 2, 2 ηi √ ρi (1 + ηi VXI i )2 1 − X1 − X2 ∗ ∗ , i = 1, 2, ui = ui (X1 , X2 ) = 8ηi p∗i = p∗i (X1 , X2 ) =
(29) (30)
and the value function of the integrated channel satisfies the following equation
64rV I (X1 , X2 ) = (1 − X1 − X2 )
ρ21 (1 + η1 V I X1 )4 ρ22 (1 + η2 V I X2 )4 + . η12 η22
(31)
Proof: The proof is similar to that of Proposition 3.1. Using (27) in the HJB equation (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 the use 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 = −VXI 1 = −VXI 2 , is constant and solves the equation 64rαI =
(1−αI η1 )4 ρ21 η12
+
(1−αI η2 )4 ρ22 . η22
In the second case, we consider a decentralized channel with cooperative advertising and optimal values of wholesale prices, subsidy rates, retail prices and advertising levels given by (14), (15), (19) and (20), respectively. We define the value function in this case as V c (X1 , X2 ) = Vmc (X1 , X2 ) + Vrc (X1 , X2 ), where Vmc is the manufacturer’s value function (given by (23)) and Vrc is the sum of the value functions of both retailers obtained by (22).
13
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 for retailers are their respective retail prices and advertising levels. This case is treated by using 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 wi∗ . The value functions of the manufacturer and the retailers in this case are defined as Vmn and Vin , i = 1, 2, respectively, and are computed using the following. V1n (X1 , X2 ) = β1n (1 − X1 − X2 ),
i = 1, 2,
and
Vmn (X1 , X2 ) = αn (1 − X1 − X2 ).
The channel value function is defined as V n (X1 , X2 ) = Vmn (X1 , X2 ) + Vrn (X1 , X2 ), where Vrn is the sum of the two retailers’ value functions in the non-cooperative setting. Since the manufacturer is the leader and decides his subsidy rates by maximizing his total discounted profit, it is obvious that Vmc (X1 , X2 ) ≥ Vmn (X1 , X2 ). Thus it remains to study the effect of cooperative advertising on the value functions of the two retailers and the total value function of the channel. Before we proceed further, we recall that the value functions are linear in X1 and X2 , and can be written as a constant coefficient times (1−X1 −X2 ). It is therefore 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 +β1n +β2n ), respectively. Similarly, a comparison between α and αn , and between βi and βin , is equivalent to a comparison between the value functions Vmc and Vmn , and Vic and Vin , respectively, for i = 1, 2. In general, it is difficult to compute the value functions in explicit form. We, therefore, resort to numerical analysis using Mathematica to compute the values of the coefficients and thus compare the value functions. Although we perform numerical analysis for several sets of values of model parameters, we report our findings for the setting described as follows. We use the linear demand formulation (9) and a base 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 the coefficients. Figures 6 and 7 compare (α + β1 + β2 ) and αI , with changes in η1 and ρ1 , respectively. Figures 8 and 9 show the impact of cooperative advertising on the value function coefficients of the manufacturer, retailers 1 and 2, and channel as a whole, by changing η1 and ρ1 , respectively. The data sets for the calculations shown are the same as those used for the results in Figures 2-5. Thus, for any point in Figures 6-9, the values of wi∗ , p∗i and u∗i are the same as the corresponding values in Figures 2-5. Figures 6 and 7 show that the channel value function with cooperative advertising and the integrated channel value function decrease with η1 and increase with ρ1 . Figures 8-9 plot the difference between the value function coefficients for a channel with cooperative advertising and a channel without any, for all the
14
Figure 6: (α + β1 + β2 ) and αI vs η1
Figure 7: (α + β1 + β2 ) and αI vs ρ1
Figure 8: Impact of cooperative advertising on value Figure 9: Impact of cooperative advertising on value function coefficients with changing η1 function coefficients with changing ρ1 members of the channel and for the channel as whole. Thus, Figures 8-9 show the impact of cooperative advertising on the profit functions of the manufacturer, the retailers and the overall channel. As anticipated, we find that the manufacturer always gains from cooperative advertising. The gain for the manufacturer
15
is greater for lower values of η1 and higher values of ρ1 . However, we find that no retailer benefits from cooperative advertising. A retailer’s value functions in the two scenarios could be approximately equal at best, for e.g., retailer 2 for low values of η1 and retailer 1 for low values of ρ1 . Interestingly, we find that retailer 1’s losses show a pattern similar to the manufacturer’s gains, i.e., decreasing with η1 and increasing with ρ1 . Finally, in most of the instances, we find that the total channel value function with cooperative advertising is marginally lower than that without any cooperative advertising. These observations indicate that cooperative advertising does not seem to increase channel profits and improve channel coordination. Roughly speaking, cooperative advertising transfers some of the profits from the retailers’ side to the manufacturer’s, thereby keeping the total channel profits at approximately the same level, decreasing the channel profit in most cases and marginally improving it in a few.
5
The case of brand level competition
The numerical analysis of value functions indicates that the retailers might not benefit from cooperative advertising when the two retailers are selling the product of the same manufacturer. Thus, there might be no incentive for retailers to participate in a cooperative advertising program. It will be interesting to look into this aspect when the second retailer, say, for example, retailer 2, buys from a manufacturer of a competing brand and thus acts as a competitor to retailer 1. In other words, we would like to see the profits of all the parties in presence of brand level competition when the two retailers sell different brands of the same product category. We consider a channel as shown in Figure 10. The manufacturer sells his product to retailer 1 who sells it in the market. Retailer 2, sells a brand which competes in the market with that of retailer 1. We assume that the products sold by the two retailers are perfectly substitutable. Here again, we assume that X1 (t) and X2 (t) denote the cumulative sales by retailer 1 and 2, respectively. Thus the manufacturer’s cumulative sales is only X1 (t). All the other notations remain the same as in the general model. The manufacturer announces his wholesale price w1 (X1 , X2 ) and subsidy rate θ1 (X1 , X2 ) policy for retailer 1. Meanwhile, retailer 2 purchases from competing manufacturer at a wholesale price w2 , which is exogenous in this model. Clearly, θ2 = 0 in this setting. The two retailers play a Nash game to compete in the consumers market and determine their optimal retail prices and advertising efforts, i.e., pi and ui , i = 1, 2, respectively. The retailer i’s optimization problem to maximize the present value of his profit stream over infinite horizon, given their wholesale prices and the subsidy rate policy of the manufacturer for retailer 1, can be obtained by using θ2 = 0 in (2), subject to (1). The manufacturer’s optimal control problem is now
16
Figure 10: Model when retailer 2 buys from a competing manufacturer. given by Z Vm (X1 , X2 ) =
max w1 (t)≥0, 0≤θ1 (t)≤1, t≥0
∞
e−rt w1 (t)X˙ 1 (t)
0 2
− θ1 (t) [u1 (X1 (t), X2 (t) | w1 (t), w2 (t), θ1 (t), θ2 (t))]
dt
(32)
subject to
p X˙ 1 (t) = ρ1 u ¯1 D1 (¯ p1 ) 1 − X1 (t) − X2 (t),
X1 (0) = X1 ∈ [0, 1],
(33)
where u ¯1 and p¯1 are the feedback advertising level and retail price, respectively, of retailer 1, given the wholesale price and subsidy rate policy declared by the manufacturer. 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 ) =
p max [(pi − wi (X1 , X2 ) + ViXi )ρi ui Di (pi ) 1 − X1 − X2 pi ,ui ≥0 p −(1 − θi (X1 , X2 ))u2i + ViXj ρj uj Dj (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 be obtained by simply using θ2 = 0 in (6)-(7). Solving these first-order conditions gives the 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 price and subsidy rate policy can be obtained from (7) by simply using θ2 = 0 and ignoring the terms of w2 , as
17
retailer 2 does not contribute to the manufacturer’s sales. We thus have the following.
rVm (X1 , X2 ) = max
w1 ,θ1 ,≥0
p (w1 + VmX1 )ρ1 u∗1 (X1 , X2 )D1 (p∗1 (X1 , X2 )) 1 − X1 − X2
+ VmX2 ρ2 u∗2 (X1 , X2 )D2 (p∗2 (X1 , X2 ))
p
1 − X1 − X2 − θ1 u∗ 21 (X1 , X2 ) − θ2 u∗ 22 (X1 , X2 ) .
(35)
The solution to the manufacturer’s optimization problem gives the optimal wholesale price and subsidy rate policy for retailer 1 in feedback form, i.e., w1∗ (X1 , X2 ) and θ1∗ (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 pricing and advertising decisions of the two retailers by simply using θ2 = 0 in equations (10) and (11) in Proposition 3.1. Furthermore, the value functions of the two retailers can be obtained by using θ2 = 0 in (12). We thus have the following results. 1 − η1 V1X1 + η1 w1 (X1 , X2 ) , 2η1 1 − η2 V2X2 + η2 w2 p∗2 = p∗2 (X1 , X2 | w1 , w2 , θ1 ) = , 2η2 √ (1 + η1 V1X1 − η1 w1 (X1 , X2 ))2 ρ21 1 − X1 − X2 ∗ ∗ , u1 = u1 (X1 , X2 | w1 , w2 , θ1 ) = 8η1 (1 − θ1 (X1 , X2 )) √ (1 + η2 V2X2 − η2 w2 )2 ρ22 1 − X1 − X2 u∗2 = u∗2 (X1 , X2 | w1 , w2 , θ1 ) = , 8η2 ) p∗1 = p∗1 (X1 , X2 | w1 , w2 , θ1 ) =
(36) (37) (38) (39)
and the value function Vi (X1 , X2 ), i = 1, 2, satisfies (1 + η1 V1X1 − η1 w1 (X1 , X2 ))4 ρ21 64rV1 (X1 , X2 ) 4V1x2 (1 + η2 V2X1 − η2 w2 )3 ρ22 = + , (1 − X1 − X2 ) η12 (1 − θ1 (X1 , X2 )) η2 64rV2 (X1 , X2 ) (1 + η2 V2X2 − η2 w2 )4 ρ22 4V2x1 (1 + η1 VX1 − η1 w1 (X1 , X2 ))3 ρ21 = + . (1 − X1 − X2 ) η22 η1 (1 − θ1 (X1 , X2 )) Taking into account the retailers’ optimal pricing and advertising strategies,we can rewrite the HJB equation for the manufacturer’s value function Vm (X1 , X2 ) given by (35) in the following way. 64rVm (X1 , X2 ) = (1 − X1 − X2 ) (−1 − η1 V1X1 + η1 w1 )3 (−4(VmX1 + w1 )η1 + θ1 + (V1X1 + 4VmX1 + 3w1 )η1 θ1 )ρ21 max w1 ,θ1 ,≥0 η12 (−1 + θ1 )2 3 2 (−1 − η2 V2X2 + η2 w2 ) (−4VmX2 )ρ2 . η2
(40)
We now consider two cases. First, a cooperative equilibrium in which the manufacturer offers a positive op18
timal subsidy rate to the retailer 1 and second, a non-cooperative equilibrium in which θ1 = 0. Our objective is to study and compare the value functions of all the parties in these two cases and investigate the impact of a cooperative advertising program in the presence of brand level competition.
Case 1: A Cooperative Solution We first consider a cooperative solution in which the manufacturer chooses his optimal subsidy rate for retailer 1. Using the first-order conditions w.r.t. w1 and θ1 in (40), we find the manufacturer’s optimal wholesale price and optimal subsidy rate policy, given by the following:
w1∗ = w1∗ (X1 , X2 ) =
1 + ηi (V1X1 − 2VmX1 ) , 3η1
θ1∗ = θ1∗ (X1 , X2 ) = 1/3.
(41)
(42)
The manufacturer’s value function satisfies 4(1 + (VmX1 + V1X1 )η1 )4 ρ21 4(VmX2 (1 + (V2X2 − w2 )η2 ))3 ρ22 64rVm (X1 , X2 ) = + . (1 − X1 − X2 ) 9η12 η2
(43)
Now, using (41) and (42) in (36)-(39), we get the optimal retail prices and advertising efforts of the two retailers as, 2 − η1 (V1X1 + VmX1 ) 1 − η2 (V2X2 − w2 ) , p∗2 = , 3η1 2η2 √ √ (1 + η1 (V1X1 + VmX1 ))2 ρ1 1 − X1 − X2 (1 + η2 (V2X2 − w2 ))2 ρ2 1 − X1 − X2 ∗ ∗ u1 = , u2 = . 12η1 8η2 p∗1 =
Furthermore, the value functions for the two retailers solve 4V1x2 (1 + η2 (V2X2 − w2 ))3 ρ22 64rV1 (X1 , X2 ) 8(1 + η1 (V1X1 + VmX1 ))4 ρ21 + = (1 − X1 − X2 ) 27η12 η2
(44)
64rV2 (X1 , X2 ) (1 + η2 (V2X2 − w2 ))4 ρ22 16V2x1 (1 + η1 (V1X1 + VmX1 ))3 ρ21 = + . (1 − X1 − X2 ) η22 9η1
(45)
and
19
Once again, we propose the following linear form of the value functions.
Vi (X1 , X2 )
=
β˜i (1 − X1 − X2 ),
Vm (X1 , X2 )
=
α ˜ (1 − X1 − X2 ),
i = 1, 2,
(46) (47)
and try to solve for the coefficients α ˜ , β˜1 and β˜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 equations (43), (44)-(45) and (46)-(47), and obtain the following system of equations to be solved for the coefficients α ˜ , β˜1 and β˜2 . 64rβ˜1
=
64rβ˜2
=
64rα ˜
=
8(1 − η1 (β˜1 + α ˜ ))4 ρ21 4β˜1 (1 − η2 (β˜2 + w2 ))3 ρ22 − , 27η12 η2 (1 − η2 (β˜2 + w2 ))4 ρ22 16β˜2 (1 − η1 (β˜1 + α ˜ ))3 ρ21 , − 2 η2 9η1 4(1 − (˜ α + β˜1 )η1 )4 ρ21 4(˜ α(1 − (β˜2 + w2 )η2 ))3 ρ22 . − 9η12 η2
(48) (49) (50)
Case 2: A Non-Cooperative 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 equations (36)-(39) gives the optimal retail prices and advertising levels of the two retailers. Using θ1 = 0 in the manufacturer’s HJB equation, and solving the first-order conditions w.r.t. w1 gives the optimal wholesale price of the manufacturer as follows: w1∗ = w1∗ (X1 , X2 ) =
1 + ηi (V1X1 − 3VmX1 ) . 4η1
(51)
Now using (51), along with θ1 = 0 in (36)-(39), we get the optimal retail prices and advertising efforts of the two retailers as follows: 1 − η2 (V2X2 − w2 ) 5 − 3η1 (V1X1 + VmX1 ) , p∗2 = , 8η1 2η2 √ √ 9(1 + η1 (V1X1 + VmX1 ))2 ρ1 1 − X1 − X2 (1 + η2 (V2X2 − w2 ))2 ρ2 1 − X1 − X2 u∗1 = , u∗2 = . 128η1 8η2 p∗1 =
Furthermore, the value functions of retailer 1, retailer 2 and the manufacturer now solve the following
20
equations, respectively: 4V1x2 (1 + η2 (V2X2 − w2 ))3 ρ22 64rV1 (X1 , X2 ) 81(1 + η1 (V1X1 + VmX1 ))4 ρ21 + = , (1 − X1 − X2 ) 256η12 η2 27V2x1 (1 + η1 (V1X1 + VmX1 ))3 ρ21 64rV2 (X1 , X2 ) (1 + η2 (V2X2 − w2 ))4 ρ22 + = , (1 − X1 − X2 ) η22 16η1 4(VmX2 (1 + (V2X2 − w2 )η2 ))3 ρ22 27(1 + (VmX1 + V1X1 )η1 )4 ρ21 64rVm (X1 , X2 ) + = . (1 − X1 − X2 ) 64η12 η2 We use linear value functions of the form
Vi (X1 , X2 )
= β˜n i (1 − X1 − X2 ),
Vm (X1 , X2 )
= α˜n (1 − X1 − X2 ),
i = 1, 2,
(52) (53)
and solve for the coefficients α˜n , β˜n 1 and β˜2n using the following set of equations: 64rβ˜1n
=
64rβ˜2n
=
64rα˜n
=
4β˜n (1 − η2 (β˜2n + w2 ))3 ρ22 81(1 − η1 (β˜1n + α˜n ))4 ρ21 − 1 , 2 256η1 η2 (1 − η2 (β˜2n + w2 ))4 ρ22 27β˜2n (1 − η1 (β˜1n + α˜n ))3 ρ21 − , η22 16η1 27(1 − (α˜n + β˜1n )η1 )4 ρ21 4(α˜n (1 − (β˜2n + w2 )η2 ))3 ρ22 − , 2 64η1 η2
(54) (55) (56)
where the superscript n denotes a non-cooperative solution.
Comparison of Value Functions in Cooperative and Non-Cooperative Solution We now compare the value functions of all the parties (retailer 1, retailer 2 and the manufacturer) in a cooperative solution given by (46)-(47) with the corresponding values in a non-cooperative solution, given by (52)-(53). Since it is quite obvious that the manufacturer will always gain from cooperative advertising, our aim is to investigate its impact on the profits of the two retailers. In particular, we want to find if retailer 1 has any incentive to join a coop advertising program, when he faces a competing retailer who is selling another brand and hence not getting any advertising subsidy from 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 ˜ , β˜1 , and β˜2 with α˜n , β˜n 1 and β˜2n , 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 the system of equations (48)-(50) and (54)-(56) explicitly, and therefore, we resort to numerical analysis to get some insights. We solved the system of equations (48)-(50) and (54)-(56) for various values of η1 , η2 , ρ1
21
and ρ2 , and report some of our key findings in figures 11-12. We solved the equations for several values of problem parameters. Figure 11 shows the difference in value function coefficients of all the parties for various values of η1 , with ρ1 = ρ2 = 0.1, r = 0.05, and η2 = 1. Since w2 is also an exogenous parameter for this model, in the analysis shown here we chose w2 equal to w1∗ , 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 get the same wholesale price from their respective manufacturers. We find that while the manufacturer always gains from cooperative advertising, retailer 1 might gain or loose depending on the values of the parameters. We also find that retailer 1 benefits at low values of η1 . Figure 12 shows the difference in the value functions with varying values of ρ1 when η1 = η2 = 0.1, r = 0.05. and ρ2 = 1. Here again, we find that retailer 1 can benefit from cooperative advertising, in this case as long as ρ1 is not too high. These results indicate that a cooperative
Figure 11: Benefit from cooperative advertising vs η1 when retailer 2 buys from a competing manufacturer
Figure 12: Benefit from cooperative advertising vs ρ1 when retailer 2 buys from a competing manufacturer
advertising program can benefit retailer 1 as well, depending on the model parameters. This is in contrast to the scenario in which both the retailers buy from the same manufacturer, where numerical analysis showed little indication of retailers benefiting from cooperative advertising. A general trend observed in several numerical calculations was that retailer 1 is more likely to benefit at lower values of ρ1 and η1 . It would be interesting if one could obtain an explicit condition based on the model parameters which would indicate the scenarios under which retailer 1 would benefit. However, given the difficulty in solving the equations 22
(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 as the Robinson-Patman Act of 1936. The Robinson-Patman act, along with other such legislations, was designed to prevent creation of monopolies in the market and to enhance competition. Such acts prevent price discrimination between two or more competing buyers of a product. To study the impact of such a legislation in our model, we consider a case when the manufacturer is restricted to offer equal wholesale prices and equal subsidy rates to the two retailers. We define VmRP (X1 , X2 ), V1RP (X1 , X2 ), V2RP (X1 , X2 ) and V RP as the value functions of the manufacturer, retailer 1, retailer 2, and the total channel, respectively, with the superscript RP standing for Robinson and Patman. These value functions solve the optimal control problems defined by (1)-(3), with w1 = w2 = w and θ1 = θ2 = θ. We look for linear value functions of the following form: VmRP (X1 , X2 ) = αRP (1 − X1 − X2 ), ViRP = βiRP (1 − X1 − X2 ), i = 1, 2, and V RP = (αRP + β1RP + β2RP )(1 − X1 − X2 ). In the following analysis, we consider the linear demand function (9), and use numerical analysis to compute the value of the coefficients αRP , βiRP , 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 common subsidy rate for the two retailers θ∗ . Figure 13 compares the common optimal wholesale price for the two retailers in the case of an anti-discriminatory act (w∗ ) with the optimal wholesale prices for the two retailers without such legislation (w1∗ , w2∗ ). 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 sensitivity of demand (ηi ) pays a higher wholesale price in the case of anti-discriminatory act in effect than in the absence of such an act. Figure 14 shows the variation of θ∗ with respect to η1 . Recall that in the general model with linear demand, the optimal subsidy rates for the two retailers are independent of the values of η1 , ρ1 and r, and equal to 1/3. In the case when the manufacturer has to offer equal wholesale prices and subsidy rates to the two retailers, we find that the common optimal subsidy rate is no longer a fixed value and varies with η1 . We find that the maximum value of θ∗ achieved is equal to the subsidy rates without any legislation (θ1∗ = θ2∗ = 1/3). Thus, the introduction of additional constraint on the manufacturer (i.e., to offer equal subsidy rates to the two retailers) makes the manufacturer to offer lower subsidy rates. Next, we study the impact of an anti-discriminatory act on the profit functions of all the channel members,
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and of the channel as whole. Figure 15 shows the difference in the value function coefficients with and without an anti-discriminatory act for the manufacturer, retailers and the total channel, i.e., α − αRP , βi = βiRP , i = 1, 2, and (α+β1 −β2 )−(αRP −β1RP −β2RP ), respectively. As expected, the manufacturer does not benefit from an anti-discriminatory act because of an additional constraint on his optimization problem. We also find that only the retailer with a lower value of price sensitivity 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 one retailer offsets the losses of the other two parties. Figure 16 shows the ratios of the channel value function in three cases, namely, the ratio of a channel with cooperative advertising and no legislation, the ratio of a channel with cooperative advertising and an anti-discriminatory legislation in effect, and the ratio of a channel with no cooperative advertising and the integrated channel value function. Thus Figure 16 shows the level of coordination possible in these three scenarios. We find that in all the instances, a channel with no cooperative advertising performs better than a channel with cooperative advertising and no legislation. Moreover the value function for a channel with cooperative advertising and an anti-discriminatory act is highest of the three, except when the difference between η1 and η2 is small. These results indicate that in most cases, highest degree of channel coordination can be achieved when we have cooperative advertising along with an anti-discriminatory legislation in effect. However, in some cases when, roughly speaking, the retailers are nearly identical, a channel with no cooperative advertising is able to achieve the highest total profits and thus the highest level of coordination.
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Concluding Remarks
We obtain the feedback Stackelberg equilibrium and compute the optimal values of the advertising levels by the two retailers, retail prices, wholesale prices and subsidy rates in the case of a linear demand. We find that the optimal subsidy rates are independent of the model parameters and are equal to 1/3. We also provide the sensitivities of the optimal advertising levels, retail prices and wholesale prices with respect to η1 and ρ1 . We study the effect of cooperative advertising on the profits of all the members of the channel and that of the total channel. We find that the cooperative advertising benefits only the manufacturer whereas the two retailers earn less profit, and except in a few cases, the total channel does not benefit either. An interesting conclusion that appears from our analysis is that with wholesale and retail prices as decision variables as well, cooperative advertising is ineffective and a redundant mechanism. In fact, since there seems to be no evidence of retailers benefiting from cooperative advertising, such an arrangement might not hold. We also
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Figure 13: Optimal wholesale prices with and without anti-discriminatory act vs η1
Figure 14: Optimal subsidy rates with and without anti-discriminatory act vs η1
Figure 15: Value function coefficients with an anti-discriminatory act minus those without any act, with changing η1
Figure 16: Ratio of channel value function to integrated channel value function in different cases, with changing η1
analyze a scenario when retailer 2 buys from another manufacturer and find that under such brand level competition, cooperative advertising program can be beneficial for retailer 1 as well. Finally, we consider
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the case of an anti discriminatory legislation where the manufacturer has to offer equal subsidy rates to the two retailers and find that higher channel profits can be achieved with such a legislation.
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