Impact of demand price elasticity on advantages of cooperative advertising in a two-tier supply chain

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1 International Journal of Production Research ISSN: (Print) X (Online) Journal homeage: htt:// Imact of demand rice elasticity on advantages of cooerative advertising in a two-tier suly chain Lei Zhao, Jihua Zhang & Jinxing Xie To cite this article: Lei Zhao, Jihua Zhang & Jinxing Xie (06) Imact of demand rice elasticity on advantages of cooerative advertising in a two-tier suly chain, International Journal of Production Research, 54:9, 54-55, DOI: 0.080/ To link to this article: htt://dx.doi.org/0.080/ Published online: 5 Oct 05. Submit your article to this journal Article views: 4 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at htt:// Download by: [Tsinghua University] Date: 07 June 06, At: 8:08

2 International Journal of Production Research, 06 Vol. 54, No. 9, 54 55, htt://dx.doi.org/0.080/ Imact of demand rice elasticity on advantages of cooerative advertising in a two-tier suly chain Lei Zhao, Jihua Zhang and Jinxing Xie* Deartment of Mathematical Sciences, Tsinghua University, Beijing, China (Received January 05; acceted 6 Setember 05) This aer focuses on ricing and vertical cooerative advertising decisions in a two-tier suly chain. Using a Stackelberg game model where the manufacturer acts as the game leader and the retailer acts as the game follower, we obtain closed-form equilibrium solution and exlicitly show how ricing and advertising decisions are made. When market demand decreases exonentially with resect to the retail rice and increases with resect to national and local advertising exenditures in an additive way, the manufacturer benefits from roviding ercentage reimbursement for the retailer s local advertising exenditure when demand rice elasticity is large enough. Whether the manufacturer benefits from cooerative advertising is also closely related to suly chain member s relative advertising efficiency. In the decision for adoting coo advertising strategy, it is critical for the manufacturer to identify how market demand deends on national and local advertisements. The findings from this research can enhance our understanding of cooerative advertising decisions in a two-tier suly chain with rice-deendent demand. Keywords: cooerative advertising; ricing; game theory; suly chain management. Introduction When market demand of a certain roduct is advertising deendent, both national advertising and local advertising lay a role. National advertising is usually brand name oriented and aims at enlarging otential client base, whereas local advertising is end customer oriented and is basically used to stimulate short-term sales. In a tyical two-tier suly chain, national advertising is usually undertaken by the ustream member (the manufacturer) and local advertising is usually undertaken by the downstream member (the retailer). Vertical cooerative (coo) advertising is an arrangement in which the manufacturer shares a ortion of the retailer s local advertising cost. The fraction shared is referred to as the (manufacturer s) articiation rate. In the absence of coo advertising, the retailer would tyically advertise less than that desired by the manufacturer. Thus, articiation rates, as well as suly chain members advertising exenditures, are fundamental decisions in a suly chain. Many studies about coo advertising and various extensions have been resented in the literature (Berger 97; Jørgensen and Zaccour 999, 003; Huang and Li 00; Karray and Zaccour 007; Wang et al. 0; He et al. 0, 0; Ahmadi-Javid and Hoseinour 0; Zhang et al. 03; Aust and Buscher 04a; Gou et al. 04; Karray and Amin 05). Recently, Jørgensen and Zaccour (04) and Aust and Buscher (04b) rovide comrehensive reviews of researches on coo advertising. Pricing is another fundamental decision in a suly chain. It tyically includes decisions for the manufacturer s wholesale rice and the retailer s retail rice. Both ricing and coo advertising are significant determinants of market demand and hence rofits of both suly chain members. However, analytical models that simultaneously deal with coo advertising and ricing decisions are relatively sarse. Karray and Zaccour (006) allow rice cometition in their model and address the coo advertising as an efficient counterstrategy for the manufacturer in the resence of the retailer s rivate label. Yet, the manufacturer s national advertising decision is not included in their model. Yue et al. (006) incororate demand rice elasticity in the customer demand function, while the manufacturer, byassing the retailer, directly gives the consumer a rice deduction from the suggested retail rice. However, their model takes neither wholesale rice nor retail rice as suly chain member s decision variables. More recently, there is increasing research interest in analytical models that simultaneously deal with coo advertising and ricing decisions. Assuming that market demand decreases exonentially with resect to the retail rice and increases with resect to national and local advertising exenditures in a multilicative way, Szmerekovsky and Zhang *Corresonding author. jxie@math.tsinghua.edu.cn 05 Informa UK Limited, trading as Taylor & Francis Grou

3 54 L. Zhao et al. (009) show that it is otimal for the manufacturer not to rovide reimbursement for the retailer s local advertising exenditure. Xie and Neyret (009), Xie and Wei (009) and Yan(00) assume that the demand decreases linearly with resect to the retail rice and find that the manufacturer usually benefits from roviding ercentage reimbursement for the retailer s local advertising exenditure. SeyedEsfahani, Biazaran, and Gharakhani (0) introduce a demand function with a new arameter that can induce either a convex or a concave demand curve. Aust and Buscher (0) extend the work by relaxing restrictions on the ratio between the manufacturer s and the retailer s rofit margins. A common assumtion made in these studies is that the demand function is multilicatively searable in advertisement and rice. Besides, all these studies adot a deterministic and static game model, which imlicitly assumes that layers (firms) make decisions for a single-eriod roblem. There are also stochastic and dynamic models that deal with the advertising goodwill evolution. For examle, He, Prasad, and Sethi (009) roose a stochastic Stackelberg differential game and rovide in the feedback form the otimal advertising and ricing olicies for the manufacturer and the retailer. This aer focuses on the deterministic and static game model only. Noticing that both the linear demand function (Xie and Neyret 009; Xie and Wei 009; Yan 00) and the non-linear demand function (SeyedEsfahani, Biazaran, and Gharakhani 0; Aust and Buscher 0) in the abovementioned literature imly a non-constant rice elasticity, while the exonential demand function (Szmerekovsky and Zhang 009) reflects a constant rice elasticity, we are interested in the following question: What s the imact of demand rice elasticity on the manufacturer s decision on cooerative advertising? Secifically, assuming market demand decreases exonentially with resect to the retail rice and increases with resect to national and local advertising exenditures in an additive way, we find that the manufacturer benefits from roviding ercentage reimbursement for the retailer s local advertising when demand rice elasticity is larger than a certain value. Whether the manufacturer benefits from cost sharing is also related to suly chain members relative advertising efficiency. These results can enhance our understanding about the coo advertising and ricing decisions in a two-tier suly chain. The remainder of this aer is organised as follows: Section resents the model and assumtions. Section 3 solves the unique equilibrium for the Stackelberg game. Section 4 does sensitive analysis regarding imacts of different arameters on the equilibrium ricing and coo advertising decisions. Section 5 concludes the aer and rooses future research directions.. Model and assumtions Consider a two-tier suly chain consisting of a single manufacturer and a single retailer, in which they lay a sequential Stackelberg game. At the first stage of the game, the manufacturer acts as the game leader and decides the wholesale rice w (w > 0), the national advertising exenditure A (A 0) and the articiation rate t (0 t < ) simultaneously. At the second stage of the game, the retailer acts as the game follower and decides the local advertising exenditure a (a 0) and meanwhile sets the retail rice ( w). Assume market demand V deends jointly on a, A and as follows: V ð; a; AÞ ¼ e ffiffiffi ffiffiffi ðk r a þ km A Þ () where e is the demand rice elasticity and k r and k m are ositive arameters taking account of the different effectiveness of local and national advertising ffiffi exenditures. ffiffiffi Equation () imlies that market demand deends on ricing effect ( e ) and advertising effect (k r a þ km A ) in a multilicative attern, which is in accord with most of the existing studies (e.g. Xie and Wei 009; Aust and Buscher 0). As in Szmerekovsky and Zhang (009), we assume e >, so the demand is exonentially decreasing with resect to. In order to simlify exositions, we define the advertising ratio as k ¼ km =k r. Let the manufacturer s unit roduction cost be C (C > 0), then we can write the rofit functions for the manufacturer and the retailer as follows: P m ¼ðw CÞ e ffiffiffi ffiffiffi ðk r a þ km A Þ ta A; () P r ¼ð wþ e ffiffiffi ffiffiffi ðk r a þ km A Þ ð tþa: (3)

4 International Journal of Production Research Stackelberg equilibrium In this section, we solve the Stackelberg game and obtain the equilibrium solution by backward induction. At the second stage of the game, with the variables A, w and t being given, the retailer faces the following decision roblem: Max P r ¼ð wþ e ffiffi ffiffiffi ðk r a þ km A Þ ð tþa (4) s.t. w and a 0: Although the objective function Π r may not be jointly concave with resect to the decision variables a and, the articular function form allows to obtain a closed-form solution to the otimisation roblem. Secifically, it is straightforward to r =@ ¼ e ffiffi ffiffiffi ðk r a þ km A Þ½ew ðe ÞŠ: (5) with the assumtion e >, the retailer s rofit Π r will increase in when w ew/(e ) and decrease in when ew/(e ). This justifies that the otimal retail rice for the retailer should be ¼ ew=ðe Þ; (6) for all a 0, A 0, w C and 0 t <. Equation (6) indicates that the retail rice () should be set roortionally to the wholesale rice (w), with the roortional coefficient e/(e ) being a decreasing function of the demand rice elasticity (e). But it does not exlicitly deend on the variables A, t and a and the arameters k r and k m. It is easy to check whether Π r is a concave function with resect to a. By setting Π r / a = 0, we can obtain the retailer s otimal local advertising exenditure as a ¼ kr e ð wþ =4ð tþ : (7) Equation (7) holds for all w. Substituting (6) into (7), we have a ¼ kr ½ew=ðe ÞŠ e =4e ð tþ : (8) Equation (8) indicates that the local advertising exenditure (a) should be set as a decreasing function of the wholesale rice (w),and as an increasing function of the articiation rate (t). But the national advertising exenditure (A) does not exlicitly imact the decision for the local advertising exenditure (a). Therefore, at the first stage of the game, the manufacturer faces the following decision roblem: Max P m ¼ðw CÞ e ffiffi ffiffiffi ðk r a þ km A Þ ta A (9) s.t. 0 t\; w C; and A 0; where the variables and a satisfy (6) and (8), resectively. According to (9), the Stackelberg equilibrium (w *, A *, t *, *, a * ) can be characterised as in Proosition (see also Table, and the roof is rovided in Aendix). Proosition. The Stackelberg game has a unique equilibrium (w *,A *,t *, *,a * ) exressed as: When e >+/k, w * = w,t * = t, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðe Þ½e þ ðe ÞkŠþ ðe Þ þ 4ðe Þ k þ 4ðe Þ k w ¼ ðe Þ þ 4ðe Þ ; (0) k Table. Equilibriums under two different cases. e >+/k <e +/k w * = w w * = w t * = t = [(e 3)w (e )C]/[(e )w (e )C] t * = t =0 * = = ew /(e ) * = = ew /(e ) A ¼ A ¼ km ðw CÞ e=4 ew e A ¼ A ¼ km ðw CÞ e=4 ew e eððe a ¼ a ¼ kr Þw ðe ÞCÞ =6ðe Þ a ¼ a ¼ kr ðew =ðe ÞÞ e =4e ew e

5 544 L. Zhao et al. t ¼ ðe 3Þw ðe ÞC ðe Þw ðe ÞC : () When e +/k, w * =w,t * =t, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðe Þ½ þðe ÞkŠþ ðe Þ þ ðe Þk þðe Þ k w ¼ ; () ðe Þ½ þðe ÞkŠ t ¼ 0: (3) Under both cases, after the values for (w *,t * ) are determined, the values for (A *, *,a * ) can be calculated as: A ¼ km ðw CÞ ew e =4; (4) e ¼ ew =ðe Þ; (5) a ¼ kr ½ew =ðe ÞŠ e =4e ð t Þ : (6) Proosition reveals that under our assumtions for the demand function, the manufacturer has no incentive to share the retailer s local advertising exenditure when the rice elasticity is smaller than a certain value. However, when the rice elasticity is larger than a certain value, the manufacturer benefits from roviding ercentage reimbursement for the retailer s local advertising exenditure. This result is different from that in Szmerekovsky and Zhang (009). Assuming the demand function takes the form of V(, a, A) = e (α βa γ A δ )(e > and α, β, γ, δ are ositive constants), they find that for all values of rice elasticity greater than, the otimal strategy for the manufacturer is not to rovide any subsidy to the retailer s local advertisement. This difference is due to the different assumtions about the effect of advertisements on the demand. National and local advertisements are assumed to increase market demand in a multilicative way in their model, while they are assumed to work in an additive way in our model. These two different findings comlement each other and suggest that when the manufacturer considers whether coo advertising strategy should be adoted, it is critical to identify how the demand deends on national and local advertisements. Usually, roducts with many substitutes or whose urchase can be easily ostoned, or that are considered luxuries other than essential to everyday living, have higher rice elasticities. For examle, the demand for a secific brand soft drink would likely be highly elastic, and its rice elasticity can be more than 4 (Ayers and Collinge 005). Similarly, goods and services include secific-model automobiles, fresh tomatoes and long-run foreign travels. (Gwartney et al. 04). For these industries, if national advertising and local advertising increase market demand in an additive way as in our model, our result suggests that coo advertising strategy will be very attractive to the goods manufacturers (or service roviders). Now we focus on the case where the rice elasticity is large enough (i.e. e >+/k), so that the manufacturer s otimal strategy is to rovide a ositive reimbursement to the retailer s local advertising exenditure. If the manufacturer makes a wrong decision and rovides no reimbursement for the retailer s local advertising exenditure, his otimal strategy is to use t * = t = 0, and the associated otimal wholesale rice is obtained by w * = w, as exressed in (). However, when he rovides a ositive reimbursement to the retailer s local advertising exenditure, his otimal strategy is to use t * = t and w * = w, as exressed in (0) and (). Obviously, the reimbursement from the manufacturer will induce the retailer to invest more on local advertising, i.e. a > a. The following roosition reveals the imact of the decision mistake on other decision variables (the wholesale rice, the retail rice and the manufacturer s national advertising exenditure). Proosition. Suose e >+/k, then w > w, A > A, >, a > a. Proosition concludes that when e > + /k, if the manufacturer rovides no reimbursement for the retailer s local advertising exenditure, the retailer will invest less on national advertising and the manufacturer will also invest less on national advertising. Thus, the demand generated from both advertising will be lower. In order to maximise his rofit, the retailer has to announce a lower retail rice to induce more sales. Similarly, the manufacturer should also charge a lower wholesale rice. This leads to low rofit margins for both the manufacturer and the retailer, resulting in a loselose situation for both suly chain members. On the contrary, by adoting coo advertising, the demand generated from advertising will be higher and both the suly chain members can increase their rofit margins and imrove the erformance of the suly chain.

6 International Journal of Production Research Effects of demand arameters on the ricing and coo advertising decisions In this section, we concentrate our discussions on the case where the rice elasticity is large enough (i.e. e >+/k), so that the manufacturer s otimal strategy is to rovide a ositive reimbursement to the retailer s local advertising cost. Proosition 3 summaries how the manufacturer s and retailer s decision variables will be influenced by the arameters of the model. Proosition 3. Suose e >+/k, [ r m \0; [ m Parts (i) and (ii) of Proosition 3 mean that both the manufacturer s wholesale rice (w ) and the retailer s selling rice ( ) increase with the advertising ratio k and decrease with the rice elasticity e. They suggest that the suly chain sees a high level of wholesale rice (and retail rice) if either the local advertising is oorly effective comared with the national advertising or the market demand is highly sensitive with the unit retail rice changes. This observation is reasonable. We take w / e < 0 for examle. Because e reresents the sensitivity of the sales volume to the retail rice as defined in (4), we can exect that the more sensitive the sales volume to the retail rice is, the lower the wholesale rice will be. Part (iii) of Proosition 3 indicates that the manufacturer s cost sharing ercentage t will increase with both the advertising ratio k and the ricing elasticity e. This is intuitively understandable. On the one hand, when the local advertising is not very effective, the retailer has little incentive to imose massive local advertising. So the manufacturer will have to raise the articiation rate and induce a higher level of local advertising exenditure. On the other hand, when the market demand is less rice sensitive, the retailer s ricing instrument has limited ower in generating end customer demand. So the manufacturer can raise its articiation rate in the hoe that the retailer s non-ricing romotion (local advertising) induces more sales. Finally, Parts (iv) and (v) of Proosition 3 show that both members advertising exenditures will increase with resect to their own advertising efficacy and decrease with resect to their counterarty s advertising efficacy. This is also intuitively understandable, and it coincides with the findings in Xie and Wei (009). Please note that Proosition 3 does not say anything about the monotonicity of a and A with resect to the rice elasticity e. In fact, when e >+/k and e increase, a and A can either increase or decrease. In order to dislay how the demand rice elasticity will imact the advertising decisions for both arties, we rovide the following numerical examle. Examle. Fix k r = and k m = (thus k = 4). For two different levels of unit roduction costs C = 0.83 and C =.00, Figure lots how a and A change with e (e >+/k =.5) in a non-monotonic way. Figure reveals the following observations. () With different unit roduction costs, the monotonicity for the equilibrium advertising exenditures with resect to the rice elasticity does not travel well. Take a for examle. With a higher level of C =.00, a decreases in e for all e >.5; this decreasing trend, however, does not remain with a lower level C = When e is relatively high, a even goes in totally oosite directions with e. Secifically, when e is greater than a certain critical value, a will increase in e with C = 0.83 and decrease in e with C =.00. () Even with the same unit roduction cost, there can be no simle monotonic relations between the equilibrium advertising exenses and the rice elasticity. Taking A for examle, when C = 0.83, A will decrease in e when e is low, but will increase in e when e is relatively high.

7 546 L. Zhao et al. Figure. How a and A change with e. 5. Conclusions and future researches With the hel of a game theoretical model, this aer shows that the manufacturer benefits from roviding ercentage reimbursement for the retailer s local advertising exenditure when the rice elasticity is larger than a certain value. In case that the manufacturer makes a wrong decision by choosing a zero articiation rate, both national and local advertising exenditures and both wholesale and retail rices will systematically dro to lower levels. Sensitivity analyses show that (i) both the wholesale rice and the retail rice will increase with the advertising ratio and decrease with the rice elasticity; (ii) the manufacturer s otimal articiation rate increases with both the advertising ratio and the rice elasticity; and (iii) both members advertising exenditures will increase with resect to their own advertising efficacy and decrease with resect to their counterarty s advertising efficacy. Our contributions to the coo advertising literature are twofold. Qualitatively, we find that the manufacturer benefits from coo advertising when demand rice elasticity is high if national and local advertising exenditures influence the demand in an additive way. This result, along with existent studies (e.g. Szmerekovsky and Zhang 009; Xie and Wei 009), can enhance our understanding about the coo advertising and ricing decisions in a two-tier suly chain. In the decision for coo advertising, it is critical for the manufacturer to identify how market demand deends on the retail rice and national and local advertisements. Quantitatively, we managed to identify the unique Stackelberg equilibrium for suly chain members ricing and coo advertising decisions in all closed-form solutions. There are some directions that our model can be extended. First, considering cometition between manufacturers and retailers may lead to more interesting results. Second, our findings are made under a very secial and secific demand function. It is not yet clear for which tye of demand functions does the manufacturer benefit from roviding a ositive articiation rate, or, as imosed conversely, does there exist a broad class of demand functions under which the manufacturer s otimal articiation strategy is not to articiate? Finally, taking an emirical investigation is always a good idea. The associated managerial insights mentioned in this article may serve as the baseline for ossible hyothesis testing with industrial data. Disclosure statement No otential conflict of interest was reorted by the authors. Funding This work was suorted by the National Natural Science Foundation of China under [grant number 7000].

8 International Journal of Production Research 547 References Ahmadi-Javid, A., and P. Hoseinour. 0. On a Cooerative Advertising Model for a Suly Chain with One Manufacturer and One Retailer. Euroean Journal of Oerational Research 9: Aust, G., and U. Buscher. 0. Vertical Cooerative Advertising and Pricing Decisions in a Manufacturer Retailer Suly Chain: A Game-theoretical Aroach. Euroean Journal of Oerational Research 3: Aust, G., and U. Buscher. 04a. Vertical Cooerative Advertising in a Retailer Duooly. Comuters & Industrial Engineering 7: Aust, G., and U. Buscher. 04b. Cooerative Advertising Models in Suly Chain Management: A Review. Euroean Journal of Oerational Research 34: 4. Ayers, R., and R. Collinge Microeconomics: Exlore and Aly. Uer Saddle River, NJ: Pearson Prentice Hall. Berger, P. D. 97. Vertical Cooerative Advertising Ventures. Journal of Marketing Research 9: Gou, Q., J. Zhang, L. Liang, Z. Huang, and A. Ashley. 04. Horizontal Cooerative Programmes and Cooerative Advertising. International Journal of Production Research 5 (3): Gwartney, J., R. Strou, R. Sobel, and D. Macherson. 04. Economics: Private and Public Choice. Mason, OH: Cengage Learning. He, X., A. Prasad, and S. P. Sethi Co-o Advertising and Pricing in a Stochastic Suly Chain: Feedback Stackelberg Strategies. Production and Oerations Management 8 (): He, X., A. Krishnamoorthy, A. Prasad, and S. P. Sethi. 0. Retail Cometition and Cooerative Advertising. Oerations Research Letters 39 (): 6. He, X., A. Krishnamoorthy, A. Prasad, and S. P. Sethi. 0. Co-o Advertising in Dynamic Retail Oligoolies. Decision Sciences 43 (): Huang, Z., and S. X. Li. 00. Coo Advertising Models in a Manufacturer Retailer Suly Chain: A Game Theory Aroach. Euroean Journal of Oerational Research 35 (3): Jorgensen, S., and G. Zaccour Equilibrium Pricing and Advertising Strategies in a Marketing Channel. Journal of Otimization Theory and Alications 0: 5. Jorgensen, S., and G. Zaccour Channel Coordination Over Time: Incentive Equilibria and Credibility. Journal of Economic Dynamics and Control 7: Jorgensen, S., and G. Zaccour. 04. A Survey of Game-theoretical Models of Cooerative Advertising. Euroean Journal of Oerational Research 37: 4. Karray, S., and S. H. Amin. 05. Cooerative Advertising in a Suly Chain with Retail Cometition. International Journal of Production Research 53 (): Karray, S., and G. Zaccour Could Co-o Advertising be a Manufacturer s Counterstrategy to Store Brands? Journal of Business Research 59 (9): Karray, S., and G. Zaccour Effectiveness of Coo Advertising Programs in Cometitive Distribution Channels. International Game Theory Review 9 (): SeyedEsfahani, M. M., M. Biazaran, and M. Gharakhani. 0. A game Theoretic Aroach to Coordinate Pricing and Vertical Co-o Advertising in Manufacturer Retailer Suly Chains. Euroean Journal of Oerational Research (): Szmerekovsky, J. G., and J. Zhang Pricing and Two-tier Advertising with One Manufacturer and One Retailer. Euroean Journal of Oerational Research 9 (3): Wang, S., Y. Zhou, J. Min, and Y. Zhong. 0. Coordination of Cooerative Advertising Models in a One-manufacturer Tworetailer Suly Chain System. Comuters & Industrial Engineering 6 (4): Xie, J., and A. Neyret Co-o Advertising and Pricing Models in Manufacturer Retailer Suly Chains Co-o Advertising and Pricing Models in Manufacturer Retailer Suly Chains. Comuters & Industrial Engineering 56 (4): Xie, J., and J. C. Wei Coordinating Advertising and Pricing in a Manufacturer Retailer Channel. Euroean Journal of Oerational Research 97 (): Yan, R. 00. Cooerative Advertising, Pricing Strategy and Firm Performance in the E-Marketing Age. Journal of the Academy of Marketing Science 38 (4): Yue, J., J. Austin, M. Wang, and Z. Huang Coordination of Cooerative Advertising in a Two-level Suly Chain When Manufacturer Offers Discount. Euroean Journal of Oerational Research 68 (): Zhang, J., Q. Gou, L. Liang, and X. He. 03. Ingredient Branding Strategies in an Assembly Suly Chain: Models and Analysis. International Journal of Production Research 5 (3 4):

9 548 L. Zhao et al. Aendix A. Proof of Proosition Substituting Equations (6) and (8) into the exression of Π m, the manufacturer s decision roblem (9) becomes 0 ew k ew e e Max P m ¼ðw CÞ r e ffiffiffi tk B C ew e þ k m AA e e eð tþ 4e ð tþ A s.t. 0 t\; w C and A 0: In order to determine the unique otimal solution (w *, t *, A * ), we first examine the manufacturer s otimal decision on the national advertising exenditure. P m =@A ¼ 4 k ea mðw CÞ ew = 3= ðe Þ 0 and w C, we know Π m is a concave function with resect to A for all A 0. Thus, we can solve the first-order condition Π m / A = 0 and uniquely obtain the otimal national advertising exenditure by A ¼ km ew e=4: ðw CÞ (A.) e Substituting (A.) into (A.), we reduce the manufacturer s maximisation roblem to fmax P m ðw; tþ; s.t. 0 t\; w Cg; where P m ðw; tþ ¼ ew k ew e 3 e ðw CÞ r e þ k ew e mðw CÞ tk 6 e 7 ew e r k ew e mðw CÞ 4 5 e e eð tþ 4e ð tþ e (A.3) 4 Next we consider the manufacturer s otimal decisions on t and w. (i) Consider the case with e >+/k. The first-order derivative of P m ðw; tþ with resect to t is given m ðw; ¼ k r (A.) ½ð eþw þ Cðe ÞŠt þðe 3Þw Cðe Þ 4ð tþ 3 e e w e ðe Þ e : (A.4) With the constraints 0 t < and w C, the sign of the derivative (A.4) is determined by the formula f(t) = [( e)w +(e )C] t +(e 3)w (e )C, which is a decreasing function in t since the sloe ( e)w +(e )C [( e) +(e )] C = C <0. We define t 0 = [(e 3)w (e )C]/[(e )w (e )C] such that f(t 0 ) = 0, then f(t) will be ositive for t < t 0 and negative for t > t 0. Taking the constraint 0 t < into consideration, two subcases will follow as below. Subcase [I]. If f(0) > 0, or equivalently w > C(e )/(e 3), then f(t) will be ositive for 0 t < t 0 and negative for t 0 < t <, which suggests that P m ðw; tþ will first increase in t for 0 t < t 0 and then decrease in t for t 0 < t <. Thus, the manufacturer s otimal choice of t should be given by t ½IŠ ðe 3Þw ðe ÞC ðwþ ¼t 0 ¼ ðe Þw ðe ÞC : (A.5) Substituting (A.5) into (A.3), the manufacturer s otimisation roblem is now further reduced to fmax P ½IŠ m ðwþ; s.t. w [ Cðe Þ=ðe 3Þg, where Pm ½IŠ n o Þe ðwþ ¼ðe 6ðewÞ e kr ½ðw CÞðe ÞþwŠ þ 4km ðw CÞ ðe Þ (A.6) is a function of the decision variable w only. ½IŠ m ðwþ=@w ¼ 0, and making use of k ¼ k m =k r, after some algebraic simlifications, we have ½ðe Þ þ 4ðe Þ kšw þ ðe Þ½ðe Þþðe ÞkŠCw 4eðe Þðk þ ÞC ¼ 0: (A.7) The left-hand side of (A.7) is a quadratic and concave function that has two real roots given as follows: w ½IŠ C ðe Þ½ðe Þþðe ÞkŠþ ffiffiffiffiffi ¼ ðe Þ þ 4ðe Þ ; (A.8) k where n C ðe Þ½ðe Þþðe ÞkŠ ffiffiffi o D w ½IŠ ¼ ðe Þ þ 4ðe Þ ; (A.9) k ¼ðe Þ þ 4ðe Þ k þ 4ðe Þ k : (A.0)

10 It is easy to verify that w ½IŠ \C\Cðe Þ=ðe 3Þ. Furthermore, we have International Journal of Production Research 549 w ½IŠ Cðe Þ=ðe 3Þ ¼ 4Cðe Þ½ðe Þ þ 4ðe Þ kš½ke ðk þ ÞŠ ; (A.) ðe 3Þ½ðe Þ þ 4ðe Þ kš½kðe Þþðe Þ ffiffiffiffiffi þðe 3Þ Š which is greater than zero given our assumtion e >+/k. So, w ½IŠ < C < C(e )/(e 3) < w [I]. It follows ½IŠ m ðwþ=@w takes ositive values when w ðcðe Þ=ðe 3Þ; w ½IŠ Þ and takes negative values when w ðw ½IŠ ; þþ, indicating that P ½IŠ m ðwþ increases in w when C(e )/(e 3)<w < w [I] and decreases in w when w > w [I]. This argument concludes that the otimal wholesale rice under Subcase [I] should be w ½IŠ ¼ w ½IŠ and the associated otimal articiation rate should be t ½IŠ ðw ½IŠ Þ. Subcase [II]. If f(0) 0, or equivalently (C )w C(e )/(e 3), then f(t) 0 for all t (0 t < ), i.e. P m ðt; wþ is decreasing in t for all t. So, the otimal choice of t should be t [II] = 0. With t = t [II] = 0, the manufacturer s otimisation roblem is then reduced to: Max P ½IIŠ m ðwþ; s.t. ðc ÞwCðe Þ=ðe 3Þ, where P ½IIŠ ew e CÞ e m ðwþ ¼ðw kr 4ðe Þ w þ k mðe Þðw CÞ : (A.) ½IIŠ m ðwþ=@w ¼ 0, and making use of k ¼ k m =k r, after some algebraic simlifications, we have ðe Þ½kðe ÞþŠw ðe Þ½kðe ÞþŠwC þ eðe ÞkC ¼ 0: (A.3) The left-hand side of (A.3) is again a quadratic and concave function of w with two real roots as follows: w ½IIŠ C ðe Þ½ þðe ÞkŠþ ffiffiffiffiffi D ¼ ; (A.4) ðe Þ½ þðe ÞkŠ w ½IIŠ C ðe Þ½ þðe ÞkŠ ffiffiffiffiffi ¼ ðe Þ½ þðe ÞkŠ D ; (A.5) where D ¼ðe Þ þ ðe Þk þðe Þ k : (A.6) It is easy to check that w ½IIŠ < C < C(e )/(e 3) < w [II]. ½IIŠ m ðwþ=@w takes ositive values for all w in [C, C(e )/ (e 3)], which indicates that Pm ½IIŠ ðwþ is increasing in w for all C w C(e )/(e 3). This argument concludes that the otimal wholesale rice under Subcase [II] should be w ½IIŠ ¼ Cðe Þ=ðe 3Þ, with the associated otimal articiation rate t [II] =0. In order to find the global otimal solution (w *, t * ) for P m ðw; tþ with the condition e >+/k, we should comare the manufacturer s otimal rofits P ½IŠ m ðw½iš Þ¼P m ðw ½IŠ ; t ½IŠ ðw ½IŠ ÞÞ under Subcase [I] and P ½IIŠ m ðw½iiš Þ¼P m ðw ½IIŠ ; t ½IIŠ Þ¼P m ðw ½IIŠ ; 0Þ under Subcase [II]. By definition, we have Pm ½IŠ ðw½iš Þ¼P m ðw ½IŠ ; t ½IŠ ðw ½IŠ ÞÞ P m ðw; t ½IŠ ðwþþ for all w >C(e )/(e 3). Since Π m (w, t) is a continuous function in w, when w goes down to the lower bound C(e )/(e 3), we have P m ðw ½IŠ ; t ½IŠ ðw ½IŠ ÞÞ lim P mðw; t ½IŠ ðwþþ w!þcðe Þ=ðe 3Þ (A.7) ¼ P m ðw ½IIŠ ; t ½IŠ ðw ½IIŠ ÞÞ ¼ P m ðw ½IIŠ ; 0Þ ¼P m ðw ½IIŠ ; t ½IIŠ Þ; which concludes that P ½IŠ m ðw½iš ÞP ½IIŠ m ðw½iiš Þ. Therefore, when e >+/k, the otimal decision for the manufacturer should be w * = w [I] = w and t * = t [I] = t, which justifies Equations (0) and (). (ii) Now we consider the case with e +/k. By (A.4), the sign of the first-order m ðw; tþ=@t is determined by the formula f(t) = [( e)w +(e )C] t +(e 3)w (e )C, which is decreasing in t for all w C. Recalling that t 0 is defined such that f(t 0 ) = 0, we have t 0 > 0 if and only if f(0) = (e 3)w (e )C > 0. Taking the constraint 0 t < into consideration, we have three indeendent subcases as follows. Subcase [III]. If e 3/, then f(0) = (e 3)w (e )C 0. Consequently, f(t) 0 for all t (0 t < ), i.e. P m ðw; tþ is decreasing in t for all given values of w C. So the otimal choice of t in Subcase [III] should be t [III] =0. Subcase [IV]. If 3/ < e( +/k) and f(0) = (e 3)w (e )C 0, or equivalently, w C(e )/(e 3), then we still have f(t) 0 for 0 t <, i.e. P m ðw; tþ is decreasing in t for all given values of w such that C w C(e )/(e 3). For this situation, the otimal choice of t is t [IV] =0. Subcase [V]. If 3/ < e( +/k) and f(0) = (e 3)w (e )C > 0, or equivalently, w > C(e )/(e 3), then f(t) takes ositive values for 0 t < t 0 and negative values for t 0 < t <. Therefore, P m ðw; tþ will increase in t for 0 t < t 0 and decrease in t for t 0 < t <. So the manufacturer s otimal choice of t should be t ½VŠ ¼ t 0.

11 550 L. Zhao et al. Using similar arguments as in Part (i), by comaring all the three subcases listed above, we can show that when e +/k, the manufacturer s (globally) otimal articiation rate should be zero, i.e. t * =0=t. The otimal wholesale rice should be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðe Þ½ þðe ÞkŠþ ðe Þ þ ðe Þk þðe Þ k w ¼ ðe Þ½ þðe ÞkŠ ¼ w ; (A.8) which justifies Equations () and (3). This comletes the roof. A. Proof of Proosition By Equations (4) (6), A > A, >, a > a can be easily verified if w > w. Thus, it suffices to rove w > w. Using exressions of w and w in (0) and (), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ½ðe Þ þ kðe Þðe ÞŠ ðe Þ þ 4kðe Þ þ 4k ðe Þ ¼ ; w 4eðe Þð þ kþ (A.9) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ðe Þ½ þ kðe ÞŠ ðe Þ þ kðe Þþk ðe Þ ¼ : w keðe Þ (A.0) In order to show w > w, we only need to rove C w [ C w, which, after some algebraic simlifications, is equivalent to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (e )ð þ kþþk ðe Þ þ 4ðe Þ k þ 4ðe Þ k q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (A.) [ ð þ kþ ðe Þ þ ðe Þk þðe Þ k Squaring both sides, this inequality is equivalent to (e )ð þ kþk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe Þ þ 4ðe Þ k þ 4ðe Þ k (A.) [ ðe Þ ð þ kþk þ 4ðe Þkð þ kþ þ ðe Þ ð þ kþk : Squaring both sides again, this inequality is equivalent to ½3ðe Þk þ 8e 6Š½ðe Þk Š [ 0: (A.3) Since e >+/k, this is obviously true. This comletes the roof. A.3 Proof of Proosition 3 (i) According to (A.0), Δ =(e ) +4(e ) k +4(e ) k, thus C ðe Þ½e þ ðe ÞkŠþ ffiffiffiffiffi 4eðe Þð þ kþc w ¼ ðe Þ þ 4ðe Þ ¼ k ðe Þ½e þ ðe ÞkŠ ffiffiffiffiffi : (A.4) After tedious algebraic calculation, we obtain h ffiffiffiffiffi 4eðe ÞC ðe Þ e þ þ ðe Þ ¼ ffiffiffiffiffi ffiffiffiffiffi : (A.5) ðe Þ½e þ ðe ÞkŠ Since Δ =(e ) +4(e ) k +4(e ) k >(e ), [ 4eðe ÞC h i ffiffiffiffiffi ffiffiffiffiffi ðe Þ e þ þ ðe Þ k ðe Þ½e þ ðe ÞkŠ 4eðe ÞC h i ¼ ffiffiffiffiffi ffiffiffiffiffi ðe Þ ð þ kþ [ 0 ðe Þ½e þ ðe ÞkŠ Similarly, after tedious algebraic calculation, we ¼ 4ð þ kþc ffiffiffiffiffi ffiffiffiffiffi n ðe Þ½e þ ðe ÞkŠ ½e ðe Þ ffiffiffiffiffi o (A.7) kš ðe Þ½e þðe Þ Š 4ðe Þ 3 ðk þ k Þ

12 Since e >+/k, or equivalently, k > /(e ), we have International Journal of Production Research 55 e ðe Þ k\e 4ðe Þ =ðe Þ ¼ ½eþðe Þ Š=ðe Þ\0: (A.8) Therefore, w / e <0. (ii) Noticing that ¼ ew =ðe Þ is a linear function of w, and w is an increasing function of k according to Part (i), thus is also an increasing function of k, which means / k > 0. Besides, since e=ðe Þ [ 0 is a decreasing function of e, and w is a decreasing function of e according to Part (i), thus is also a decreasing function of e, which means / e < 0 is true. (iii) Since t = [(e 3)w (e )C]/[(e )w (e )C] is an increasing function of w, and w is an increasing function of k according to Part (i), thus t is also an increasing function of e, which means t / k >0. In order to rove t / e > 0, =½e ðe ÞC=w Šg =½e ðe C w w ¼ ðe Þw ðe ÞC C C þ e Thus, we only need to show that (w /C) (w /C) +(e )/C w / e >0. According to Equations (0) and (A.7), this is equivalent to 4eðe Þð þ kþ ðe Þ½e þ ðe ÞkŠ ffiffiffiffiffi 4eðe Þð þ kþ ðe Þ½e þ ðe ÞkŠ ffiffiffiffiffi 4ðe Þð þ kþ þ ffiffiffiffiffi ðe Þ½e þ ðe ÞkŠ ffiffiffiffiffi n ½e ðe Þ ffiffiffiffiffi o kš ðe Þ½e þðe Þ Š 4ðe Þ 3 ðk þ k Þ [ 0: After tedious algebraic calculation, we obtain this is equivalent to ffiffiffiffiffi ½ðe þðe Þð þ kþš þðe Þðe Þþ4ðe Þ ðk þ k Þ [ 0: (A.3) This is obviously true; thus, t / e > 0 is also true. (iv) From a ¼ kr ðew =ðe ÞÞ e ½ðe Þw ðe ÞCŠ =6ðe Þ, we ¼ k r ew e½ðe Þw ðe ÞCŠ ec ðe Þ : 8ðe Þ e w Since e >+/k is equivalent to C/w <(e 3)/(e ), we have ec ðe Þ\eðe 3Þ=ðe Þ ðe Þ ¼ =ðe Þ\0: (A.33) w Therefore, a / w < 0. When k r increases, k will decrease and w will decrease according to (i); thus, a will increase. Therefore, a / k r > 0. However, when k m increases, k will increase and w will increase according to Part (i); thus, a will decrease. Therefore, a / k m <0. (v) From A ¼ km ðw CÞ ðew =ðe ÞÞ e =4, we ¼ k m ew eðw CÞ ec ðe Þ : e w Making use of Part (i), w is increasing with k, thus w \ lim w ¼ ec=ðe Þ according to the exression of w in (0). Therefore, we have ec/w k! (e ) > 0 and thus A / w >0. When k r increases, k will decrease and w will decrease according to Part (i); thus, A will increase. Therefore, A / k r <0. However, when k m increases, k will increase and w will increase according to Part (i); thus, A will increase. Therefore, A / k m >0. This comletes the roof. (A.30)