Cooperative Advertising and Pricing in a Dynamic Stochastic Supply Chain: Feedback Stackelberg Strategies

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1 PRODUCTION AND OPERATIONS MANAGEMENT Vol. 18, No. 1, January February 009, ISSN EISSN POMS DOI /oms r 009 Production and Oerations Management Society Cooerative Advertising and Pricing in a Dynamic Stochastic Suly Chain: Feedback Stackelberg Strategies Xiuli He Belk College of Business, The University of North Carolina at Charlotte, Charlotte, North Carolina 83, xhe8@uncc.edu Ashutosh Prasad, Suresh P. Sethi School of Management, The University of Texas at Dallas, Richardson, Texas sethi@utdallas.edu, arasad@utdallas.edu ooerative (co-o) advertising is an imortant instrument for aligning manufacturer and retailer decisions in C suly chains. In this, the manufacturer announces a co-o advertising olicy, i.e., a articiation rate that secifies the ercentage of the retailer s advertising exenditure that it will rovide. In addition, it also announces the wholesale rice. In resonse, the retailer chooses its otimal advertising and ricing olicies. We model this suly chain roblem as a stochastic Stackelberg differential game whose dynamics follows Sethi s stochastic sales-advertising model. We obtain the condition when offering co-o advertising is otimal for the manufacturer. We rovide in feedback form the otimal advertising and ricing olicies for the manufacturer and the retailer. We contrast the results with the advertising and rice decisions of the vertically integrated channel, and suggest a method for coordinating the channel. Key words: Co-o advertising; sales-advertising dynamics; differential games; sethi model; distribution channel History: Received: October 007; Acceted: Aril 008, after 1 revision. 1. Introduction Cooerative (or co-o) advertising is commonly used in suly chains as an incentive by the manufacturer to influence retailer behavior. In co-o advertising rograms, the manufacturer contributes a ercentage of the retailer s advertising exenditure to increase the sales of the manufacturer s roduct (Dutta et al. 1995, Nagler 006). Co-o advertising rograms can be a significant exense for the manufacturer. As much as 5 40% of local advertisements are cooeratively funded (Dant and Berger 1996). Total exenditure on co-o advertising in 000 was estimated at $15 billion, comared with $900 million in 1970 (Nagler 006). The manufacturer s decision roblem, if offering a co-o advertising rogram to the retailer, is to determine what ercentage of advertising exense to contribute. This is called its articiation rate. The articiation rate need not be constant but can be a function of the state, which in our case is the roortion of the market aware of the roduct. If it is assumed that retail rice and retail advertising are the rimary marketing mix drivers of sales, then the effect of the articiation rate on these must be considered 78 when designing a co-o rogram. That is, it should be exected that the wholesale rice and the articiation rate of the manufacturer will influence the retail rice and the amount of retail advertising by the retailer, and, by backward induction, anticiation of the latter should influence the manufacturer s decisions. Furthermore, it should be ket in mind that advertising decisions have dynamic effects on awareness and sales, which are also subject to uncertainty. Nevertheless, generating actual decisions is difficult and managerial guidelines do not resently incororate all of these elements. For the manager, it is also imortant to understand how costs, advertising effectiveness, and other market and firm secific arameters will affect these decisions. In attemting to rovide managerial guidelines for co-o advertising, we study co-o advertising in the context of a suly chain consisting of a manufacturer selling through a retailer. Following a standard structure, the manufacturer is considered to be the Stackelberg leader and the retailer is the follower. Secifically, the manufacturer announces a wholesale rice and the ercentage of retail advertising that it will contribute. These can be state deendent, though

2 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society 79 we shall show that in equilibrium and under the assumtions of our model they are not. In turn, the retailer decides on its advertising exenditure and the retail rice. These can also be state deendent, but in our case only the otimal advertising is and the retail rice is not. These determine the sales for the channel, which also is subject to uncertainty. We determine the condition when offering co-o advertising is otimal. We rovide exlicit solutions and comarative statics. We then describe how the channel can be coordinated using co-o advertising. Furthermore, we show that, in the absence of coordinated co-o advertising and ricing strategies, the decentralized channel will always have higher than otimal rice and lower than otimal advertising. Prior literature on co-o advertising has often used static game models to cature the interaction between the manufacturer and retailer, thus ignoring the dynamic imact of advertising on sales. Dynamic models of co-o advertising have only recently been roosed (e.g., Jrgensen et al. 000, 001, 003, Karray and Zaccour 005), not least because the Stackelberg game structure with dynamics tends to make the modeling intractable. Often the solution rocedure must take recourse to oen-loo olicies, which are not time consistent, or exclude the ricing decisions, or omit demand uncertainty (see He et al. 007 for a survey). In contrast, this aer rooses a continuous-time dynamic model of co-o advertising and ricing decisions with uncertainty. In a dynamic stochastic model, we rovide, for the first time, exlicit feedback Stackelberg solutions of the otimal co-o olicy advertising and ricing decisions. These decisions are time consistent, i.e., neither the manufacturer not the retailer will have an incentive to revise their olicy at any future time. We discuss the literature and contribution in more detail in the next section. The remainder of the aer is organized as follows. In Section we survey the relevant background literature. Section 3 develos the roosed model, and Section 4 rovides the analysis and results. We comare the vertically integrated and decentralized channels analytically and with numerical examles and grahs. In Section 5, we study the use of co-o advertising and revenue sharing contracts. Section 6 rovides conclusions and directions for future research. Proofs of all results in the aer are given in Aendix A.. Background Literature This is an interdiscilinary aer related to the literature on suly chain coordination in oerations management and the literature on co-o advertising in marketing. In oerations management, Cachon (003) is a good reference on suly chain coordination. In Gerchak and Wang (004), the retailer sets the shares of the revenue and then the suliers decide delivery quantities. They show that revenue sharing alone cannot coordinate the assembly system, however, a revenue sharing scheme couled with a subsidy aid by the retailer to suliers can. Cachon and Lariviere (005), in a newsvendor setting, study the revenue sharing contracts between a retailer and a manufacturer who sets the wholesale rice. Gerchak et al. (006) study the revenue sharing contracts in a decentralized Stackelberg setting in which the video rental channel and the studio make indeendent decisions. Consistent with the coordination motif of this literature, we will examine the value of co-o advertising in coordinating the suly chain. The co-o advertising literature consists of relatively few quantitative studies. We can divide them into two categories: Static models (Berger 197, Bergen and John 1997, Dant and Berger 1996, Kim and Staelin 1999, Huang et al. 00) and dynamic models (Jrgensen and Zaccour, 003, Jrgensen et al. 000, 001, 003, Karray and Zaccour 005). We roceed to discuss these further. In the static setting, Berger (197) aears to be the first aer to analyze co-o advertising. It oerationalizes co-o advertising as a wholesale rice discount given by the manufacturer to the retailer as an advertising allowance. Sales are a concave function of the level of advertising chosen by the retailer. The conclusion is that both the manufacturer and retailer can be better off from co-o advertising. Dant and Berger (1996) extend the Berger model to study the coo advertising decisions in franchising systems with demand uncertainty and where there is disagreement between the manufacturer and retailer on anticiated sales. Bergen and John (1997) study the effects of advertising sillover, differentiation across cometing retailers, and differentiation across cometing manufacturers on the articiation rate. Kali (1998) finds that if the co-o advertising subsidy is made conditional on the advertised rice being above a secified rice threshold, then the channel can be coordinated on rice and advertising. Finally, Huang et al. (00) analyze both manufacturer-as-leader and artnershi advertising structures. They consider the imact of local advertising and the national brand name investment on the retailer s sales. Huang et al. (00) justify undertaking a static analysis of co-o advertising by arguing that romotion effects tend to be less long lasting than advertising, and are rimarily a driver of short-term sales. Nevertheless, the emirical evidence in studies of concurrent advertising and romotion dynamics by Naik et al. (005) and concurrent advertising and detailing by Chintagunta and Vilcassim (1994) and

3 80 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society Fruchter and Kalish (1998) finds significant suort for dynamic romotion effects. In our dynamic model, we use two arameters that can be adjusted to cature how imortant the resent is in relation to the future. One is the finite decay term in the dynamics originating from Vidale and Wolfe (1957) that catures loss of market share due to forgetting and background cometition. The other is the firm s ositive discount rate. If this rate is high, then the firm effectively behaves like a myoic agent and if it is low, like a foresighted firm. Setting these two arameters high should therefore resemble a static analysis, yet their resence gives the flexibility to exlain dynamic imlications. Another difference with the aforementioned aers is that we model the ricing decision in the channel exlicitly. But comared with Huang et al. (00), we have not included the ossibility that the manufacturer may searately advertise its roduct. Some justification for this comes from the study by Nagler (006, Table 1) who finds that of 86 brands in the survey, 1470 listed a co-o advertising rogram but only 599 did national advertising. In the dynamic setting, Jrgensen et al. (000) aly a Stackelberg differential game to model the interaction between the manufacturer and the retailer in a decentralized channel with the manufacturer as the Stackelberg leader in an extended Nerlove and Arrow (196) framework. They assume that advertising may have both long-term and short-term effects on retail sales and that the brand s goodwill is affected by the manufacturer s and retailer s long-term advertising only. They show that suorting both tyes of retailer advertising rovides more rofits to both channel members than any of the cases of artial suort, i.e., roviding suort to only one of the two tyes (either the retailer s long-term or short-term advertising), while the artial suort is better than no suort. Jrgensen and Zaccour (001) study dynamic co-o advertising and comare results from two games: Nash game without advertising suort and Stackelberg game with the manufacturer as the leader with suort. They conclude that co-o suort is Pareto imroving. Karray and Zaccour (005) extend Jrgensen et al. (001) to consider both the manufacturer s national advertising and the retailer s local romotional effort. In this model, the retailer sells a rivate label as well as the manufacturer s roduct and can choose the romotional effort for each roduct. The results show that a co-o advertising rogram can hel the manufacturer mitigate the cometitive imact of the rivate label. Jrgensen et al. (003) study the case where the retailer s romotions can damage the brand image and examine whether a co-o advertising rogram can still work in such a context. They find that it is feasible when the initial brand image is weak or if the initial brand image is at an intermediate level and retailer romotions are not too damaging to the brand image. To this existing dynamic co-o advertising literature, our roosed model and results make a contribution in the following resects. First, the difficulty of solving differential Stackelberg games means that one often either considers oen-loo decisions, or resorts to a class of roblems where Basar and Olsder (1999) and Rubio (006) have shown that the Stackelberg solution reduces to the solution of the Nash (simultaneous move) game, or imoses the linearquadratic structure in the roblem. However, our model does not have these simlifying features. Moreover, we obtain feedback Stackelberg solutions. The imortance of this solution is that a manufacturer would not feel temted to change its decisions deending on the state of the market. Second, we include the rice decision as an endogenous decision variable and show how co-o advertising can overcome the double marginalization roblem and coordinate the channel. Third, we include uncertainty into the demand dynamics and show that the results are robust to its inclusion. To achieve these goals, we emloy the Sethi (1983) advertising model as the dynamics of the otimal control and differential game roblems under consideration. 3. The Model We consider a channel consisting of a manufacturer selling a roduct to end users through a retailer. The roduct is in a mature category where sales, exressed as a fraction of the otential market, is influenced through advertising sending and retail rice. The manufacturer decides on the wholesale rice and imlements an advertising suort scheme via a articiation rate, i.e., for every dollar sent by the retailer, the manufacturer will contribute a certain ercentage. Secifically, the manufacturer decides on the wholesale rice w(t) and a co-o articiation rate y(t) at time t 0. The retailer decides on the channel s total advertising effort level u(t) and the retail rice (t), t 0. The costs of advertising, i.e., advertising exenditure, are quadratic in the advertising effort u(t) and the manufacturer s and retailer s advertising exenditures at time t are given by yu(t) and (1 y)u(t), resectively. The assumtion of a quadratic cost function is common in revious research (e.g., Chintagunta and Jain 199, Deal 1979, Jrgensen et al. 000, Prasad and Sethi 004) and imlies increasing marginal cost of advertising effort. Equivalently, the retailer decides on the channel s total advertising exenditure v(t), which results in a mar- ginally ffiffiffiffiffiffiffiffi diminishing advertising effect roortional to vðtþ ; t 0.

4 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society 81 We introduce the following additional notation. Notation t Time t, t 0 x(t)a[0, 1] xa[0, 1] u(t) 0 U(t) 0 w(t) 0 (t) 0 P(t) 0 y(t) 0 D() c 0 r40 d40 r40 m M,m R,m I V M, V R, V I a M ; a R ; a I b M ; b R ; b I Proortion of the market aware of roduct at time t Aware roortion, also denotes an arbitrary initial aware roortion Advertising effort rate at time t Advertising effort rate for the integrated channel at time t Wholesale rice at time t Retail rice at time t Retail rice for the integrated channel at time t Co-o advertising articiation rate offered by manufacturer at time t Demand function Constant unit roduction cost for manufacturer Advertising effectiveness arameter Awareness share decay arameter Discount rate Gross margins for manufacturer (M), retailer (R), and integrated channel (I) Value functions for M, R, and I, resectively Intercets and sloes of the value functions of M, R and I, resectively To model the dynamic effect of advertising on sales, we use the Sethi advertising model, which is related to the classical Vidale-Wolfe advertising model. Variants of the Sethi model have been used, for examle, by Bass et al. (005) and references therein, and emirically validated in studies such as Chintagunta and Jain (1995) and Naik et al. (008). We use the stochastic formulation of the Sethi model which is given by the Itô equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxðtþ ¼ ruðtþ 1 xðtþ dxðtþ dt þ sðxðtþþdzðtþ; xð0þ ¼x ½0; 1Š; t 0; ð1þ where x(t) in this aer reresents the awareness share, i.e., the number of aware (or informed) customers exressed as a fraction of the total market at time t, x denotes the initial condition, r is a resonse constant, and d determines the rate at which otential consumers are lost due to background cometition, roduct obsolescence, forgetting, etc. Because we aim to solve the roblem for any initial x, we can treat it as arbitrarily chosen. Thus, it will denote as well the variable reresenting the awareness share. In the second term on the right-hand side, s(x(t)) reresents a variance term and z(t), t 0, reresents a standard Wiener rocess on the underlying robability sace ðo; =; PÞ. We should stress that the admissible class of decisions u(t), t 0, are nonnegative stochastic rocesses, non-anticiative with resect to the Wiener rocess z(t), t 0. This requirement alies also to the other decisions that will be introduced later. Some elucidation of the roerties of Equation (1) is needed. Note that the secification of the dynamics has desirable roerties such as concave resonse with saturation. The awareness share is non-decreasing in advertising and subject to decay. The awareness share dynamicsareaffectedlinearlybytheadvertisingeffort u(t), which is the square root of the advertising exenditure u(t) and is thus a concave function for the advertising exenditure. The rate of awareness share increase is unaffected by the articiation rule, which is an internal transfer between the channel members. Advertising affects the roortion of unaware consumers as in Sethi (1983), but sales are generated only by the fraction of aware consumers who are willing to ay the retail rice. Price was similarly included in the objective function by Bass et al. (005) in a simultaneous-move Nash advertising game. REMARK 3.1. The awareness share remains bounded within [0, 1] desite the stochastic disturbances, as we assume that the function s : ð0; 1Þ!<is continuous and Lischitz on every closed subinterval of (0, 1), uðtþ 0; t 0; and sð0þ ¼ sð1þ ¼ 0. This gives a strictly ositive drift when the awareness share is 0 and a strictly negative drift when it is 1. Then, from Gihman and Skorohod (197,. 149, ), 0 and 1 are the natural boundaries for the solutions of (1) with xð0þ ¼x ½0; 1Š, i.e., xðtþ ð0; 1Þ almost surely for t40. & The channel members have a constant and ositive discount rate r and lay a Stackelberg differential game over an infinite horizon. We regard the manufacturer as the Stackelberg leader and the retailer as the follower. We restrict our attention to feedback Stackelberg solutions where the otimal olicy, in general, deends on the current state and time (see Basar and Olsder 1999). However, in an infinite horizon setting with time-indeendent arameters, we can with good reason choose to focus on stationary equilibria. Thus, the feedback olicies will not deend exlicitly on time t. There may be non-stationary equilibria as well, which we do not discover, however. The sequence of the events is as follows: First, the manufacturer announces the feedback wholesale rice olicy wðxþ 0 and the feedback articiation rate yðxþ ½0; 1Š. This means that at any time t 0, if the

5 8 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society state is x(t), then at time t, the wholesale rice, denoted as w(t), would be w(x(t)) and the articiation rate, denoted as y(t), would be y(x(t)). Note here a slight abuse of notation in using w(x) and y(x) as feedback olicies, and w(t) and y(t) as the decisions at time t. Second, the retailer sets the retail rice (t) and the advertising effort rate u(t) as its otimal resonse to the manufacturer s announced decisions. The retailer accomlishes this by solving an otimization roblem to maximize the resent value of its rofit stream over the infinite horizon. Given the manufacturer s announced olicies, the retailer s otimal control roblem is given by V R ðxþ ¼ Max ðtþ;uðtþ0;t0 E Z 1 e rt 0 ð1 yðxðtþþþuðtþ o dt fððtþ wðxðtþþþdððtþþxðtþ o ðþ subject to (1), where E denotes the exectation oerator. We will assume that the demand function is downwards-sloing and differentiable, and satisfies the usual conditions to ensure an interior solution for rice. Moreover, 0 DðÞ 1. We have denoted the otimal value of the retailer s discounted total rofit at time zero by V R (x), clearly indicating that it deends only on the initial value x. Because x is arbitrary, the function V R (x) is defined on the entire domain [0, 1]. Furthermore, because ours is an infinite horizon roblem with stationary arameters, the future at any time t looks the same as it does at time zero so long as x(t) 5 x. This means that V R (x) will also rovide us with the resent value of the rofit stream associated with an otimal olicy on the interval ½t; 1Þ; t 0, discounted to time t if x(t) 5 x at that time. Thus, V R (x) defines the value function for our roblem. The otimal solution of the roblem (1), () yields the retailer s feedback retail rice ðxjw; yþ and advertising effort uðxjw; yþ in resonse to the announced olicies w(x) and y(x). While these resonses are simly functions of x, the exlicit notation we are using emhasizes their deendence on the announced olicies. As an examle, if uðxjw; yþ ¼xw þ y; wðxþ ¼ ax ; and yðxþ ¼bx, then uðxjw; yþ means uðxjwðxþ; yðxþþ ¼ ax 3 þ bx. Furthermore, if x(t) denotes the awareness share at any time t 0, then the retailer s advertising effort u(t) at time t will be uðxðtþjw; yþ ¼ uðxðtþjwðxðtþþ; yðxðtþþþ ¼ axðtþ 3 þ bxðtþ. The manufacturer anticiates the retailer s reaction functions and incororates them into its otimal control roblem, and solves for its wholesale rice olicy w(x) and the articiation rate olicy y(x). Therefore, the manufacturer s roblem can be stated as Z 1 V M ðxþ ¼ Max E e rt fðwðtþ cþdðxðtþjwðtþ; ð yðtþþþxðtþ wðtþ0; 0 0yðtÞ1 yðtþðuðxðtþjwðtþ; yðtþþþ o o dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxðtþ ¼ ruðxðtþjwðtþ; yðtþþ 1 xðtþ dxðtþ dt þ sðxðtþþdzðtþ; xð0þ ¼x ½0; 1Š; t 0; ð3þ where V M ðxþ is the manufacturer s value function. The solution to roblem (3) yields the equilibrium feedback olicies w ðxþ and y ðxþ for the manufacturer. Once we have the solution w ðxþ and y ðxþ, we can then exress the retailer s feedback rice and advertising effort as ðxþ ¼ðxjw ; y Þ and u ðxþ ¼uðxjw ; y Þ. We would like to stress here that the olicies w ðxþ; y ðxþ; ðxþ and u ðxþ constitute a feedback Stackelberg equilibrium, which is time consistent. Using these olicies in the state Equation (1) results in the stochastic awareness rocess x ðtþ; t 0, and the resective decisions will be w ðx ðtþþ; y ðx ðtþþ; ðx ðtþþ and u ðx ðtþþ at time t 0. We will obtain these olicies exlicitly in this aer. Thus, this aer reresents a major advance over the bulk of the literature on Stackelberg differential game models in advertising and ricing, where oen-loo olicies are obtained, which are in general not time consistent. Before we roceed to analysis in the next section, we also formulate the roblem for the vertically integrated channel. Here the ricing and advertising decisions are made by a centralized decision maker in order to maximize the resent value of the total channel rofit. By denoting the retail rice as P(t) and the joint advertising effort by U(t) at time t, the roblem can be written as Z 1 h i V I ðxþ ¼ Max E e rt ðpðtþ cþdðpðtþþxðtþ UðtÞ dt ; PðtÞ;UðtÞ0;t0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxðtþ ¼ ruðtþ 1 xðtþ dxðtþ dt þ sðxðtþþdzðtþ; xð0þ ¼x ½0; 1Š; t 0: ð4þ We now roceed to the analysis of the decentralized and integrated channels. 4. Analysis We first consider the case of the integrated channel in which there is a central decision maker making the advertising and ricing decisions. This rovides the benchmark against which the decentralized channel results can be comared The Integrated Channel The analysis can be broken into two arts that rovide the otimal rice and otimal advertising, resectively. First, observe that P occurs only in the

6 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society 83 integrand and not in the dynamics of (4). Thus, we can maximize the channel rofit by first maximizing the integrand of (4) with resect to P, holding U fixed, and then solve the resulting otimal control roblem to obtain the otimal U (see, e.g., Sethi and Thomson 000, 7.1.1). The otimal rice P satisfies the firstorder condition (FOC) of maximizing ðp cþdðpþ with resect to P. Thus, the otimal ricing decision can be obtained indeendent of the advertising decisions. Secifically, the otimal rice P is constant over time, and it is the solution of the equation ðp cþd 0 ðp ÞþDðP Þ¼0. Let m I ðp cþdðp Þ denote the integrated channel s otimal gross margin er unit share of awareness. Note that this is different from the standard margin ðp cþ er unit sale. Using the imlicit function theorem, it can be shown =@c40 I =@co0. The former occurs because marginal revenue equals marginal cost at the otimal demand, and when marginal cost increases, the firm sells fewer units by charging a higher rice so that the marginal revenue is higher. But overall, due to cost increases, the gross margin is reduced. We next solve the otimal control roblem for the advertising decision. Here the objective function in (4) is rewritten with the maximized retail rice. PROPOSITION 1: For the integrated channel, we have the following results. (a) The integrated channel s otimal rofit is linear in x, i.e., V I ¼ a I þ b I x, where a I and b I are ositive constants given in terms of the system arameters as follows: a I ¼ b I r =4r; b I ¼ q m I ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr þ dþ þ r m I : þðrþdþ Comarative o0. (b) The otimal feedback advertising olicy is U ðxþ ¼ b Ir ffiffiffiffiffiffiffiffiffiffiffi 1 x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra I ð1 xþ: Comarative o0. & The solutions are exlicit, and the comarative statics on the decision variables and the value function aear to be intuitive. When the decay arameter is higher or the effectiveness of advertising lower, it is otimal to have less advertising. Advertising is higher when the firm is less myoic, i.e., it discounts the future less. This is due to the carryover effect of advertising. To obtain the effect of demand function arameters on the results, which will have an imact via the gross margins, we can emloy, for examle, the linear demand function DðPÞ ¼1 ZP, where Z catures rice sensitivity and 0 P 1=Z. In this case, P ¼ð1 þ ZcÞ= Z and m I ¼ð1 ZcÞ =4Z, I =@Zo0. When demand is more sensitive to rice, the otimal rice is lower and the otimal advertising should also be lower, because the returns from advertising have decreased. ffiffiffiffiffiffiffiffiffiffiffi The otimal advertising effort is roortional to 1 x, i.e., the otimal advertising budget is roortional to (1 x). In other words, when the awareness share is higher, there should be less advertising than when it is lower. Thus, the benefit of advertising is greatest at the beginning, over the steeest art of the sales-advertising resonse function when the awareness share is low, rather than when it is closer to saturation. 4.. The Decentralized Channel The results for the Stackelberg differential game are obtained as described in the revious section. We resent them in the following two roositions. Proosition will secify the retailer s best resonse and Proosition 3 will characterize the Stackelberg equilibrium for the suly chain. PROPOSITION : For any given olicies w(x) and y(x) of the manufacturer, the retailer s reaction rice is indeendent of y(x) and can be exressed as ðxjwþ ¼^ðwðxÞÞ, where the function ^ðwþ solves Dð^Þþð^ wþd 0 ð^þ ¼0; and its advertising reaction olicy is indeendent of w(x), and can be exressed as uðxjyþ ¼ VR x r 1 x & ð1 yðxþþ We insert these reaction functions into the maximization roblem for the manufacturer and roceed further. It turns out that, deending uon the arameter values, there can be two tyes of equilibria. In the first, which we term as the no co-o equilibrium, the manufacturer does not rovide any co-o advertising rogram to the retailer, whereas in the second, it does. We state these results in the following roosition. PROPOSITION 3 (Feedback Stackelberg Equilibrium): For any given set of arameters, there exists a unique feedback Stackelberg equilibrium ðw ðxþ; ðxþ; y ðxþ; u ðtþþ for the game, given by w ðxþ ¼w satisfying ðw cþ ddð^ðwþþ dw þ w Dð^ðw ÞÞ ¼ 0; ðxþ ¼ ¼ ^ðw Þ; with the function m ^ðwþ as defined in Proosition, and with Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m R ðrþdþ þr m R Dð Þ we have: & ðrþdþ þr m R þðrþdþ ; m M ðw cþdð Þ; m R ð w Þ First, we discuss the otimal rices obtained in Proosition 3. The retail rice is higher in a decen-

7 84 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society (a) if D 0 (b) if D40 No Co-o Equilibrium Co-o Equilibrium Retailer s rofit V R V R ðxþ ¼a R þ b R x V R ðxþ ¼a R þ b R x Manufacturer s rofit V M V M ðxþ ¼a M þ b M x V M ðxþ ¼a M þ b M x Coefficients of rofit functions, a R ; b R ; a M ; b M obtained from: m R b R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr þ dþ þ r m R þðrþdþ m M b M ¼ ðr þ dþþb R r a R ¼ b R r 4r a M ¼ b Rb M r r b R ¼ m R r þ d b Rðb R þ b M Þr 8ðr þ dþ b M ¼ m M r þ d ðb R þ b M Þ r 16ðr þ dþ a R ¼ b Rðb R þ b M Þr 8r a M ¼ ðb R þ b M Þ r 16r Particiation rate y ðxþ ¼ y ¼ 0 y ¼ b M b R b M þ b R ¼ 1 a R a M Advertising effort u ðxþ ¼ ffiffiffiffiffiffiffiffiffiffiffi rb R 1 x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra R ð1 xþ ffiffiffiffiffiffiffiffiffiffiffi rðb R þ b M Þ 1 x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra M ð1 xþ 4 tralized channel than in the integrated channel er the double marginalization effect. EXAMPLE: With the demand function DðPÞ ¼1 ZP; we see that ¼ð3 þ ZcÞ=4Z; w ¼ðZc þ 1Þ=Z; m R ¼ð1 ZcÞ =16Z; m M ¼ð1 ZcÞ =8Z; and m M þ m R ¼ 3ð1 ZcÞ =8Z: Thus, 4P ¼ð1 þ ZcÞ=Z, because ZcoZP 1: & Next, we discuss the no co-o solution aearing in the left column (a) of the table in Proosition 3, which haens when D 0 The salient oints of the no coo solution, which is comletely exlicit, are, first, ffiffiffiffiffi that advertising is roortional to a R, and a comarison with the integrated channel case shows that it is necessarily subotimal for the channel. Second, when the margin of the manufacturer is sufficiently smaller than that of the retailer, then the condition under which the no co-o equilibrium holds is satisfied. Thus, the manufacturer should not offer a co-o advertising incentive when the retailer has sufficient channel ower to swing the slit of channel margins in its direction. It is useful that the analysis rovides a result with no co-o advertising, because in ractice co-o advertising is not observed all the time. While noting that the condition for no co-o advertising secified in the roosition also alies when m M and m R are exogenously secified, we can verify that there are demand functions from which the condition or its negation emerge endogenously. EXAMPLES: The linear demand function and the isoelastic demand function rovide the verification being sought, since they generate oosite imlications for whether there should be co-o advertising or not in equilibrium. Note that these are the most commonly used demand functions (e.g., Petruzzi and Dada 1999). With the linear demand function DðÞ ¼1 Z; Z ð0; 1=Þ, it can be derived that ^ðwþ ¼ð1 þ ZwÞ=Z; ) ^ðwþ w ¼ð1 ZwÞ=Z, and further that w ¼ð1þZcÞ=Z; ) w c ¼ð1 ZcÞ=Z. Thus m R =m M ¼ð1 Zw Þ=ð1 ZcÞ. With the isoelastic demand function DðÞ ¼ W ; W41, and c 1, where W is the rice elasticity of demand, we get ^ðwþ ¼wW=ðW 1Þ; ) ^ðwþ w ¼ w=ðw 1Þ, and further that w ¼ cw=ðw 1Þ; ) w c ¼ c=ðw 1Þ. Thus m R =m M ¼ w =c. Noting further that w 4c in both cases, it is clear that in the linear demand case m R om M, which means that Proosition 3(a) never alies and co-o advertising is otimal. On the other hand, with isoelastic demand, we have m R 4m M, which imlies that Proosition 3(a) alies for low values of the elasticity W, and thereby rules out the

8 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society 85 Figure 1 Exlanation of Under-Advertising Using Marginal Analysis MC = u MR I = rα I (1 x) MR D = rα R (1 x) in No Co-o solution, rα M (1 x) in Co-o solution. u* U* Advertising Effort u otimality of co-o advertising. For high values of W, co-o advertising is otimal. & Finally, we consider the case where the condition (a) for no co-o advertising does not hold. Then, the right column condition (b) holds, i.e., D40. The arameters a R ; b R ; a M ; b M are obtained from the simultaneous solution of the four imlicit equations. Because it is the value of D that determines whether the manufacturer will share in advertising or not, we could term D as the differential ower of the manufacturer over the retailer. Note that m M m R is sufficient for D40, but not necessary. It is now ossible to see that the advertising in the decentralized channel is lower than in the vertically integrated channel. The ratio of theadvertising ffiffiffiffiffiffiffiffiffiffiffiffi exenditures ffiffiffiffiffiffiffiffiffiffiffiu ðxþ=u ðxþ is given by a M =a I if D40 and a R =a I if D 0. Because the integrated channel must have at least as high a value function as the decentralized channel for any x, including x 5 0, we have a I a M 1a R. And further, because the manufacturer cannot extract all the surlus from the decentralized channel, a I 4a M and a I 4a R. Therefore, the advertising in the decentralized channel is lower than the channel otimal rate. This extends the result obtained by Huang et al. (00) in the static case. We may also note that the ratio is constant, deending only on the roblem arameters and not on the current awareness share. Figure 1 shows how we can interret the otimal advertising olicy results such as U ðxþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra I ð1 xþ, obtained in Proosition 1 for the integrated case, in the usual marginal benefit equals marginal cost argument. For the integrated channel, we see by differentiating Equation (A1) with resect to U that the marginal cost of advertising is U and its marginal benefit is Vx I r 1 x. To see it more clearly, let us see what haens when we aly a constant control U in a small interval [t, t1dt], when x(t) 5 x. From the Itó differential Equation (1), we can see that Eðxðtþ dtþþ ¼ x þ ru ffiffiffiffiffiffiffiffiffiffiffi 1 x dt, because the exectation of the stochastic term is zero. But the marginal value (or the shadow rice) of the state is given by Vx I : Thus, the exected benefit is Vx I ½Eðxðt þ dtþþ xš ¼VI x ru 1 x dt: Moreover, the total advertising cost in the small interval is U dt: The difference of these two terms gives the exected rofit Vx I ru ffiffiffiffiffiffiffiffiffiffiffi 1 x U dt resulting from our action in the interval [t, t1dt]. We can maximize this rofit by taking its derivative with resect to U and set it equal to zero. But this gives recisely the FOC (A), which can now be seen as the condition of equating marginal revenue and marginal cost of advertising. Further analysis reveals that the marginal revenue in the integrated case can also be exressed as MR I ¼ Vx I r 1 x ¼ b I r ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x ¼ ra I ð1 xþ. For the decentralized channel as a whole, the instantaneous advertising exense rate is u, and so the marginal cost is u. We can disaggregate this for the retailer and manufacturer as follows: The marginal costs are (1 y)u and yu, resectively. Furthermore, the condition showing the equality between the marginal cost (1 y)u and the marginal benefit Vx Rr 1 x for the retailer is given by (A1). Although more comlicated to show, the equality between marginal benefit and marginal cost also holds for the manufacturer. To see this, let us solve the manufacturer s roblem (3) subject to the retailer s resonse (A13). This can be done by either using the Lagrange multilier method or by substituting for y from (A13) into (3). If we did the latter, then instead of the HJB Equation (A15), we would get the equation rv M ¼ max u0 m M ðxþ u þ VR x ru 1 x þ Vx M ru ffiffiffiffiffiffiffiffiffiffiffi 1 x : The first order condition for u is u þ VR x r 1 x þ Vx M r 1 x ¼ 0;

9 86 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society which, by using (A13) and adding yu on both sides, can be written as Vx M r 1 x uð1 yþ ¼yu: ð5þ Here we can see that without constraint (A13), the manufacturer s marginal benefit would be simly Vx Mr 1 x. The decrease in the marginal benefit by u(1 y) is because of constraint (A13). If we had used the Lagrange multilier aroach, then it can be shown that u(1 y) will be relaced by l(1 y) with l 5 u/ as the value of the Lagrange multilier, which we know reresents the shadow rice associated with the constraint (A13). Finally, by summing (5) and (A1), we obtain ðvx M þ ffiffiffiffiffiffiffiffiffiffiffi VR x Þr 1 x uð1 yþ ¼u: Figure Particiation Rate vs. Manufacturer s Margin The right-hand side is the total marginal cost of advertising for the decentralized channel and the lefthand side reresents the marginal revenue MR D of the decentralized channel, which can be written as ( MR D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra R ð1 xþ in the no co-o solution; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra M ð1 xþ in the co-o solution: We can now see that the otimal advertising effort derived in Proosition 3 for the decentralized channel can be obtained from MC D ¼ u. We have seen already that a I 4a M and a I 4a R, and hence Figure 1 shows that under-advertising, leading to lower sales and lower channel rofit, takes lace in the decentralized solution in both co-o and no co-o equilibria. Moreover, this under-advertising comes about from the alication of the marginal cost equals the margin cost argument two times once by the retailer and once by the manufacturer. This effectively generalizes the inefficiency caused by double marginalization for ricing in static suly chains to advertising decisions in dynamic stochastic suly chains. For the comarative statics on the articiation rate with resect to the gross margins of the firms, we emloy numerical analysis. For this we assume the margins to be given. This analysis is fairly straightforward because we have the formula for the otimal articiation rate in Proosition 3. In Figures and 3, we use arameters r , d 5 1 and r 5. We fix the retailer s margin to m R when lotting the relationshi between the manufacturer s margin m M and the articiation rate y in Figure, and the manufacturer s margin to m M when lotting m R against y in Figure 3. Figure shows that as the manufacturer s margin increases, the articiation rate increases. In contrast, Figure 3 shows that as the retailer s margin increases, the articiation decreases. The result aears to be consistent with revious literature (e.g., Berger 197) that has found that the articiation rate increases with the manufacturer s Figure 3 Particiation Rate vs. Retailer s Margin 4.3. Comarative Statics in the Decentralized Channel The comarative statics on the advertising decision in the no co-o solution follow from direct observations, and R o0: ð6þ These are as in the integrated channel case, excet for the clarification that advertising effort does not deend on the manufacturer s gross margin, but only on the retailer s. The comarative statics for advertising in the co-o case are likely identical, but the intractability of the algebraic exressions resents a hurdle to confirming this.

10 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society 87 margin, and this is suorted by the emirical studies as well, such as by Nagler (006). The argument is that the manufacturer has an incentive to motivate the retailer to increase advertising as it has a higher value for sales. However, a substantiation of this result may not hold in general. An investigation of the conditions under which itwouldholdisleftasatoicoffurtherresearch. When the margins are not given, they need to be derived in terms of the roblem arameters such as r, d, and r for any given demand functions. While we do not obtain comarative statics in terms of these arameters, these can be easily obtained numerically. The analysis in the next section roceeds with the case that there is co-o advertising in equilibrium Evolution Process of Awareness Share We next examine the awareness share rocesses analytically for both the integrated and the decentralized channels. Inserting the values of the advertising effort into the state equation, we obtain, dxðtþ ¼ðAð1 xðtþþ dxðtþþdt þ sðxðtþþdzðtþ; ð7þ xð0þ ¼x ½0; 1Š; t 0; where A ¼ r ffiffiffiffiffiffi ra I for the integrated channel, and A ¼ r ffiffiffiffiffiffiffiffi ra M for the decentralized channel. To characterize the evolution rocesses, a secification of the disturbance function is required. Following Prasad and Sethi (004), we use sðxþ ¼s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xð1 xþ, where s is a constant. This secification has the roerties discussed in Remark 3.1, and it ensures the awareness share to remain bounded within [0, 1] desite the stochastic disturbances. With this secification, we get the following result. PROPOSITION 4: The density of the stationary distribution of the awareness share is given by the Beta density, i.e., fðyþ ¼ G A þ d s s G A s y A s 1 ð1 yþ d G d s s 1 ; where GðsÞ ¼ R 1 0 q s 1 e q dq; s40 is the gamma function. Furthermore, the long-run equilibrium awareness shares, denoted by x I for the integrated channel and x D for the decentralized suly chain, are given by x I ¼ r ffiffiffiffiffiffi ra I r ffiffiffiffiffiffi ; x D ¼ r ffiffiffiffiffiffiffiffi ra M ra I þ d r ffiffiffiffiffiffiffiffi ra M þ d : & Clearly, as a consequence of a I 4a M, we get x I 4x D. What this means is that, due to higher advertising in the integrated channel, the average awareness about the roduct is higher in this case in the long run than when it is sold through a decentralized suly chain, ceteris aribus. Illustrative examles can be obtained for different arameter values. For the grahs in Figures 4 7, we Value Function (V) Figure 4 Value Function Comarison Vi Vr Vm Vm+Vr Market Awareness (x) used the linear demand function and arameter values c 5 0 and Z This generates m I 5 1, m M and m R Next, the values r , d 5 1, s 5 0.5, and advertising effectiveness r 5 are used. In ractice, a decision calculus aroach could be followed to obtain the arameter values. The solutions of the unknown coefficients are a I ¼ 7:3; a M ¼ 3:9; a R ¼ 1:39; b I ¼ 0:6; b M ¼ 0:3; b R ¼ 0:17. Therefore, y From this information, the value functions are lotted. In Figure 4, it can be seen that the value function for the integrated channel is over 1.5 times that of the decentralized channel. This is higher than m I =ðm M þ m R Þ¼1:33 times, which would be the result given by a static analysis. The additional inefficiency is, of course, due to lower than the centralized otimal level of advertising as shown in Figure 1 and as can be seen from Figure 5. Channel coordination is thus of greater value when co-o advertising is used. Another thing to note is that the manufacturer enjoys the first mover advantage, obtaining over twice as much rofit as the retailer. Next, we grah the samle aths of awareness shares and advertising efforts over time, to make the Advertising Effort Figure 5 Feedback Advertising Policies U* u* Market Awareness (x)

11 88 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society Figure 6 Awareness shares effect of the stochastic term much clearer. To do this, note that the SDE dxðtþ ¼aðxÞdt þ bðxþdwðtþ can be numerically aroximated ffiffiffi by xðt þ DÞ ¼xðtÞþ aðxðtþþd þ bðxðtþþ D sðtþ. The fsðtþg are i.i.d. Normal with mean 0 and variance 1, generated using Excel s random number generator, and using the time ste D Figure 6 shows a samle ath from an initial starting oint x Because the equilibrium awareness shares for both channel structures are higher than the initial value, there is a rise over time towards the long-run equilibrium awareness shares x I ¼ 0:55 and x D ¼ 0:45. A comarison of Figures 6 and 7 together will show that when the awareness share decreases below its mean value, then the advertising is higher, and vice versa. When the stochastic disturbances move the awareness share away from the mean, the otimal advertising changes in a way that tends to return the awareness share back to the mean. Thus the advertising acts to reduce the effects of stochastic disturbances. Figure 7 also shows that the advertising effort at each instant of time is Figure 7 Advertising Effort Samle Paths of Awareness Shares for Integrated and Decentralized Channels X E[X] x E[x] Time Samle Paths of Otimal Advertising Efforts for Integrated and Decentralized Channels U* u* Time also lower in the decentralized case than in the integrated case. 5. Revenue Sharing Contracts and Co-o Advertising We consider the case in which the co-o advertising rogram is combined with a revenue sharing contract. We want to examine whether the combination of these two schemes can coordinate the channel, i.e., achieve the same rofit level in the vertically integrated channel. We assume that revenue sharing is alied to all units. The firms rofit functions for any constant sharing rule y are V R ¼ E Z 1 e rt 0 ð1 yþu ðtþ dt V M ¼ E Z 1 e rt 0 yu ðtþ dt fðð1 yþðtþ wðtþþdððtþþxðtþ f½yðtþþwðtþ cšdððtþþxðtþ PROPOSITION 5: A revenue sharing and co-o advertising contract where the manufacturer receives a constant fraction y of the retail revenue and contributes the same fraction towards co-o advertising will coordinate the channel. In equilibrium, w ðxþ ¼cð1 yþ; ðxþ ¼P ; u ðxþ ¼U ðxþ; V R ¼ð1 yþv I and V M ¼ yv I : We make a few observations. First, the retailer shares the suly chain s rofit by sharing both its cost and revenue. Second, the manufacturer s wholesale rice is less than its roduction cost, which imlies that it loses money in selling the roduct to the retailer, but makes rofit by articiating in revenue sharing. Third, the revenue sharing contract can arbitrarily slit the rofit between the manufacturer and the retailer. These three observations carry over in the dynamics case, the results obtained in Cachon and Lariviere (005) in a one-eriod newsvendor model. There are other contracts by which the channel can be coordinated. For examle, a two-art tariff consisting of a wholesale rice er unit and a fixed fee is ossible. The wholesale rice is set to the marginal cost and the articiation rate is set to zero, which will make the retailer s roblem equivalent to the integrated channel roblem. The fixed fee extracts the maximum surlus and hence the manufacturer sets it equal to the integrated channel rofit leaving the retailer with no surlus. Instead of this, another method is to have a bargained slit of channel revenues so that it is ossible for both arties to have some surlus. &

12 Production and Oerations Management 18(1), , r 009 Production and Oerations Management Society Conclusions Co-o advertising is a widely used marketing tool that affects advertising and ricing olicies throughout the suly chain. A co-o advertising lan secifies a articiation rate, which is defined as the ercentage of the retailer s advertising exense on the manufacturer s roduct that is contributed by the manufacturer to the retailer. In this aer we rovide a theoretical analysis of co-o advertising lans in a dynamic stochastic suly chain. We first develo a model that has the following salient features. Unlike much of the work in the co-o literature, this is a dynamic model. Modeling dynamics is articularly imortant when studying advertising, because the carryover of advertising and romotion effects is well documented. We emloy the dynamics of the Sethi model, which is analytically tractable and has been validated in emirical studies. In further contrast to the literature, we incororate uncertainty in awareness share and derive otimal feedback olicies for rice and advertising by the manufacturer and retailer. The sequence of events is that the manufacturer announces a articiation rate olicy and wholesale rice olicy, and subsequently the retailer determines its selling rice and local advertising olicy. These decisions constitute a feedback strategy and are thus time consistent. There are two tyes of equilibria identified in the analysis that may occur deending on the arameter values. One corresonds to no co-o advertising (Proosition 3(a)) and the other to ositive co-o advertising (Proosition 3(b)). Thus, we note that it is not always otimal for the manufacturer to offer a co-o advertising rogram. In the case when the retailer s margin is lower than the manufacturer s margin, or close to it, then the manufacturer will offer a co-o rogram. Otherwise, when the retailer s margin is significantly higher than the manufacturer s, then the manufacturer will not offer a co-o rogram. We discussed how the shae of the demand curve is relevant to this conclusion, with co-o advertising being likely when there is linear demand and less likely with isoelastic demand. We find that the otimal articiation rate, the wholesale rice, and the retail rice are constants that deend only on the model arameters (Proosition 3). The otimal advertising exenditure on the other hand deends on the awareness share and follows an inverse law (Proosition 3). The results are robust against uncertainty. With a secification of the random noise, we are able to show that the awareness rocess has a Beta distribution (Proosition 4). We solve the model for a vertically integrated channel (Proosition 1). Comaring its results to those for the decentralized channel, we find that in the absence of co-o advertising, the decentralized channel has higher than otimal rices and lower than otimal advertising. The higher rices can be exlained by the standard double marginalization argument. Underadvertising, on the other hand, has a related but more involved exlanation. Whereas wholesale rice by itself cannot correct for these roblems, we demonstrate that a revenue and advertising sharing contract allows the channel to achieve the coordinated outcome (Proosition 5). Thus, we show that for the manufacturer, decision making that jointly otimizes co-o advertising and rice has an imortant benefit. A few limitations and future extensions of the model should be noted. We assume the manufacturer to be the Stackelberg leader, but there are ractical examles of large retailers such as Walmart that have the channel ower to dictate terms to the manufacturer. In that case, the retailer as Stackelberg leader should be studied. Another extension is suggested by the survey of Dutta et al. (1995) of over two thousand co-o advertising lans, which finds that these lans secify not only a articiation rate but also an accrual rate. For examle, a 50% articiation rate caed by a 3% accrual rate means that the manufacturer will comensate 50% of the retailer s local advertising exenses u to 3% of the urchases made by the retailer. A ossible reason for the use of accrual rates is to limit the maximum liability of the manufacturer in the absence of full information about the retailer and market conditions. Thus, incororation of the accrual rate in our model resents a toic of future research. It is also imortant to consider the imact of cometition between manufacturers, although tractability can become an issue there. Dutta et al. (1995) find that articiation rates are higher when there are fewer cometitors, which seems surrising; but they argue that manufacturers with monooly ower can use co-o advertising to resolve coordination roblems. Missing in their data is cometitive intensity among retailers. Both manufacturer and retailer level cometition deserve further attention in analytical and emirical models. Acknowledgments This article is dedicated to the memory of Lev Semenovich Pontryagin ( ) and resented as a lenary talk at the International Conference on Differential Equations and Toology in commemoration of the Centennial Anniversary of L.S. Pontryagin, Lomonosov Moscow State University, Moscow, Russia, June 17, 008. This aer won the PICMET (Portland International Center for Management of Engineering and Technology) 008 Outstanding Student Paer Award and was