Research Article Vertical Cooperative Advertising with Substitute Brands

Transcription

1 Appled Mathematcs Volume 013, Artcle ID 8001, 8 pages Research Artcle Vertcal Cooperatve Advertsng wth Substtute Brands You-Hua Chen and Xao-We Wen College of Economcs & Management, South Chna Agrcultural Unversty, Guangzhou 5106, Chna Correspondence should be addressed to You-Hua Chen; chenyhua1@163.com Receved 11 July 013; Accepted August 013 Academc Edtor: X. Henry Wang Copyrght 013 Y.-H. Chen and X.-W. Wen. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Cooperatve (co-op advertsng s attractng more and more attenton. Ths paper analyzes co-op advertsng behavor based on a dual-brand model wth a sngle manufacturer and a sngle retaler, and some nterestng conclusons are acheved. Frstly, the frm n the supply chan advertses both brands, and the dfference of advertsng expendture s not very large n equlbrum. Secondly, the retaler s advertsng and the manufacturer s partcpaton ratos depend on both the retaler s and the manufacturer s margnal profts. Thrdly, the stmulatng effect ncreases the advertsng nvestment whle the competton effect decreases t, but they have no effect on the manufacturer s partcpaton rato. Fourthly, co-op advertsng s more senstve to the manufacturer s margnal profts than those of the retaler. Lastly, total advertsng nvestment and proft are greater under cooperatve decson than under Stackelberg decson. 1. Introducton Takng cooperatve advertsng as an example, the ndustral frm becomes more dependent on ts cooperatve partners [1]. Co-op advertsng s a practce n whch the manufacturer pays hs retaler some proporton of the expendture of local advertsng [, 3]. Co-op advertsng s a type of cost-sharng contracts to promote sales. The orgnal justfcaton of co-op advertsng s the exstence of advertsng spllover between frms n the vertcal supply chan. More and more supply chans take part n co-op advertsng and, co-op advertsng s often used n consumer goods ndustres [, 5]. For example, Nagler [6] reported that the total co-op advertsng expendture doubled every two years from 900 mllon dollars n 1970 to 15 bllon dollars n 000. Based on Sen [7] and Dant and Berger [8], 5% 0% of the manufacturers, such as General Electrc, Apple, and IBM Corporaton, nvest n coop advertsng wth ther retalers. In practce, the manufacturer wth co-op advertsng contrbutes at least 50% of the advertsng expendture. Berger [9] clamed that generally the cost-sharng proporton between manufacturer and retaler s 50%-50%. Brennan [10] dentfed that IBM Corporaton bears 50% of the co-op advertsng cost wth ts retaler, whle Apple Corporaton s partcpaton rato s 75%. Bergen and John [11] showedthat household applances manufacturers pay 70% of the co-op advertsng expendture to ther retalers. Berger [9] ntally addressed co-op advertsng usng a mathematcal method. Subsequently, many people dscussed t from dfferent perspectves [, 3, 5, 1 15]. Yue et al. [16], Xe and We [3], He et al. [17], and Vscolan [18] employed the dynamc model whle Chntagunta and Jan [19], Jørgensen et al. [], and Jørgensen et al. [0] used the statc model to nvestgate co-op advertsng. (Dynamc model has an advantage n analyzng long-term effect of advertsng whle statc model makes t easy to study how the varaton of parameter affects the equlbrum soluton. BasedonNerloveandArrow s[1]goodwllmodel,jørgensen et al. [] and Jørgensen et al.[0] examned the phenomenon that consumer goods producers prefer to launch co-op advertsng wth ther retalers. They classfed advertsng nto short-term advertsng and long-term advertsng. Longterm advertsng affects the goodwll, whle short-term advertsng only nduces local sales. Huang and L [, 5] argued that manufacturers spend on both co-op advertsng and brand name nvestment, but co-op advertsng nvestment dmnshes brand name expendture. Both retaler s advertsng expendture and manufacturer s partcpaton ratos are

2 Appled Mathematcs affected by ther margnal profts. In co-op advertsng, a manufacturer always domnates a retaler, but Huang et al. [5] declared that the retaler s market power s becomng stronger and stronger. Karray and Zaccour [1] dscussed co-op advertsng wth multple products. (The retaler sells both the manufacturer s goods and ts own brands. For example, Wal-Mart sells selfowned property brands, such as Great Value, Manstays, and SmplyBascaswellasothermanufacturers products.in ther study, a retaler s a collaborator as well as a compettor to the manufacturer, and the retaler s prvate brands hurt the manufacturer s profts. Interestngly, Alston et al. [] addressed horzontal co-op advertsng between beef market and pork market and declared that the beggar-thyneghbor effect (beggar-thy-neghbor effect s mplct when a substantal part of the benefts to the producers authorzng the program comes at the expense of the producers of competng commodtes [] lowers advertsng expendture whle co-op advertsng ncreases profts. By employng the compettoneffect,(sncetherearetworetalersnthesupply chan, the competton between the two retalers affects the manufacturer s strategy Wang et al. [1] extended coop advertsng behavor to one-manufacturer two-retaler system. In ther model, the manufacturer nvests both n local advertsng and n natonal brand name advertsng, but natonal brand name advertsng reduces local advertsng, and the manufacturer does not always support the retaler s local advertsng. Zhang and Xe [15] clamed that natonal brand name advertsng ncreases as the number of retalers ncrease, whle Xe and We [3]declaredthatmostofthepror studes assumed sales to be only affected by advertsng and gnored the prce. But n ther research, they found that co-op advertsng decreases product prce and ncreases consumer welfare. Xe and Neyret [3]and He et al.[]also employed a prce model to nvestgate supply chan co-op advertsng actvty. In summary, most of exstent papers used sngle-manufacturer sngle-retaler sngle-brand or sngle-manufacturer dual-retalers sngle-brand model, and few of them consdered the stuaton that the supply chan supples dual/multple substtute brands. But t s common for one manufacturer to produce two or more substtute brands smultaneously. (Procter & Gamble produces several knds of shampoos, such as Rejoce, Head & Shoulders, Pantene, and Sassoon. Stmulated by ths phenomenon, ths paper employs a dualbrand model wth a sngle manufacturer and a sngle retaler to study co-op advertsng and reaches some nterestng conclusons. (It s easy to be expended to multple-brand case. Assumng that the manufacturer supples two substtute. (The substtute relatonshp of the two brands s reflected by the competton effect of co-op advertsng, whch means that the ncrease n advertsng of one brand decreases the sales of the other brand brands and launches co-op advertsng wth the retaler and co-op advertsng has both the stmulatng effect and the competton effect, (the stmulatng effect means that co-op advertsng ncreases the sale of the advertsed brand, whle the competton effect the mples advertsng decreases the sale of the substtute brand ths study obtans the followng conclusons. Frstly, co-op advertsng promotes both the proft of the manufacturer and the proft of the retaler, but the gap between the advertsng expendture of dfferent brands s smaller than a certan amount. Secondly, margnal profts of the manufacturer are more than half of the margnal profts of the retaler, or else the manufacturer has no ncentve to share any advertsng expendture wth ts retaler. Besdes, the stmulatng effect promotes advertsng whle the competton effect nhbts t. Co-op advertsng ncreases wth the margnal profts of the manufacturer and the retaler, but t s more senstve to the manufacturer s margnal proft than the retaler s. More nterestngly, the partcpaton rato of the manufacturer has nothng to do wth those effects of advertsng. Fnally, profts of both partcpants are hgher n the cooperatve stuaton than n the Stackelberg case. The rest of ths paper proceeds as follows. The basc model and assumptons are presented n the next secton. The model s analyzed n Secton 3, under both cooperatve stuaton and Stackelberg stuaton. Then some concludng remarks are gven n the last secton.. Model and Assumptons Assumethatthereareonemanufacturerandoneretaler n the supply chan, and they supply two substtute brands, denoted to brand ( 1, smultaneously. (The substtute of the two brands s represented by the competton effect of co-op advertsng. Advertsng has two effects: stmulatng effect and competton effect. The stmulatng effect s the postve effect of co-op advertsng, whch means that advertsng motvates the sales of the advertsed brand whle the competton effect s the negatve effect of co-op advertsng, whch ndcates advertsng for one brand nhbts the sales of the other. The retaler determnes that the co-op advertsng expendture whle the manufacturer decdes ts best partcpaton rato. (Ths s a general assumpton, and almost all other studes propose that assumpton. Then, a dual-brand model wth a sngle manufacturer and a sngle retaler s establshed as follows. The manufacturer solves the followng problem: Max t 1,t π M ρ M1 (z 1 +σ a 1 a +ρ M (z +σ a a 1 t 1 a 1 t a, whle the retaler solves the followng problem: Max π a 1,a R ρ R1 (z 1 +σ a 1 a +ρ R (z +σ a a 1 (1 t 1 a 1 (1 t a. z s a postve constant that represents the mnmum sales of brand ( 1, ; wthout loss of generalty, ths paper assumes z 1 z z. (Mnmum sale s the sale wthout advertsng. And z ( 1, has nothng to do wth the equlbrum solutons, so we assume z z j z. π M and π R are the total profts of the manufacturer and the retaler, respectvely; ρ M and ρ R ( 1,represent (1 (

3 Appled Mathematcs 3 the margnal profts of the manufacturer and the retaler; a,t ( 1, are co-op advertsng expendture of the retaler and partcpaton rato of the manufacturer. σ s the stmulatng effect parameter of advertsng, whle denotes the competton effect parameter. σ means that the advertsng mpacts on the two brands are exactly the same, and σ mples that the advertsng mpacts on the two brands are dfferent. Before analyzng the model, ths paper puts forward the followng assumptons..1. Assumptons. We have the followng assumptons: ( σ>, ( ρ M1 ρ M, ρ R1 ρ R, ( ρ R1 ρ R >0, ρ M1 ρ M >0. Assumpton ( means that the stmulatng effect s stronger than the competton effect and the advertsng mpacts on the two brands are dfference. Assumpton ( llustrates that the margnal profts dfferent of the manufacturer (retaler should not be too large. Combnng assumptons ( and ( yelds assumpton (. Furthermore, ρ M1 ρ M and ρ R1 ρ R are the net effect nvolved by co-op advertsng on the profts of the manufacturer and the retaler. 3. Model Analyss and Man Results Many frms supply products wthout advertsng, whch meansthattsnotessentalforfrmsnthesupplychanto launch co-op advertsng for both brands. Indeed, we reach Proposton 1. Proposton 1. If the sale of brand (or 1 s more than zero even wthout advertsng, advertsng expendture for brand 1 (or should be no more than z / or a z /. Proof. When a 0,theproftfrombrandsπ R ρ R (z σ a 1.Ifπ R 0,thenz σ a 1 0or a 1 z /.Forthe same reason we have a z /.Theproofscomplete. Proposton 1 mples that the dfference of advertsng expendture between dfferent brands should not be too large. If the supply chan only advertses one brand, the total advertsng expendture should be lmted to a certan range, [0, z / ]. Otherwse, the proft from the brand wthout advertsng s negatve and the supply chan wll not supply t. Ths successfully explans why advertsng tmes and frequences of Procter & Gamble for dfferent shampoos, such as Rejoce and Head & Shoulders, are almost the same. Proposton. The proft functons of the manufacturer and the retaler are concave. Proposton makes sure that functons (1 and( have unque optmal solutons. Usually, the manufacturer n the supply chan s domnant over the retaler [3, 1, 13, 15]. But as the power of the retaler ncreases, t acts as a cooperator to the manufacturer sometmes [5]. Accordngly, the model s analyzed henceforth under both the Stackelberg (leaderfollower stuaton and the cooperatve stuaton Stackelberg Stuaton. In the Stackelberg stuaton, the manufacturer acts as the leader and the retaler acts as the follower.thetwofrmsnthesupplychanplayatwo-stage Stackelberg game. The manufacturer decdes ts partcpaton rato n the frst stage. Then, gven the partcpaton rato, the retaler decdes ts best advertsng expendture n the second stage. To get the Stackelberg equlbrum soluton, backward nducton method s ntroduced, whch means that we should solve the reacton functon of the retaler n the second stage frst. So solvng functon (, we get a 1 (ρ R 1 ρ R (1 t 1, (3 a (ρ R. ( (1 t Substtutng (3-( nto(1 yelds the followng solutons, whch are the best partcpaton ratos of the manufacturer: t 1 1 t 1 (ρ R1 ρ R (ρ M1 ρ M + (ρ R1 ρ R, (5 (ρ R (ρ M ρ M1 + (ρ R. (6 From (5-(6, we acheve Propostons 3 and. Proposton 3. ( The theoretcal condton for the manufacturer to partcpate n co-op advertsng s (ρ M σ / > 1/, (, j 1,, j,and( the emprcal condton for the manufacturer to partcpate n coop advertsng s (ρ M / > 3/. We call (ρ M / > 1/ the theoretcal condton because t only exsts n theory. ((ρ M / > 1/ s equvalent to t >0. In fact, the manufacturer s partcpaton rato s always equal to or more than 1/ [9, 10] (BergenandJohn,1997.Ift > 1/,then (ρ M / > 3/.So(ρ M / > 3/ s called the emprcal condton. ((ρ M / > 3/ s equvalent to t > 1/. By ρ M1 ρ M, ρ R1 ρ R, the theoretcal condton s rewrtten as ρ M /ρ R > 1/,whchmeansthatthemargnalproft ofthemanufacturershouldbeatleast1/oftheretaler s. Otherwse, the manufacturer has no ncentve to take part n co-op advertsng. And the emprcal condton becomes ρ M /ρ R > 3/, whch llustrates that the manufacturer has more power than the retaler. (If ρ M >ρ R,themanufacturer can merge wth the retaler to maxmze ts combned proft.

4 Appled Mathematcs Proposton. ( t ncreases as ρ M ncreases and decreases as ncreases, ( t ncreases as ncreases and decreases as ρ R ncreases, and ( t s ndependent of σ and, (, j 1,, j. Proposton demonstrates that the partcpaton rato of the manufacturer s affected by the margnal profts of ts own as well as the retaler s. Interestngly, the partcpaton rato of the manufacturer has nothng to do wth co-op advertsng effects. The reason for ths s that, as the leader of the supply chan, the manufacturer s strategc acton s ndependent of the behavor of the retaler, whch means that the manufacturer s partcpaton rato does not depend on the retaler s advertsng expendture. So t s also ndependent of the effects of advertsng. Gven the best partcpaton rato t, ( 1,,tseasyto get the ratonal co-op advertsng expendture of the retaler at thesecondstage.sosubsttutngt n (3and(wth(5-(6, we reach the fnal expressons of a, ( 1, and acheve Propostons 5 and 6: a [ (ρ M 1 ρ M + (ρ R1 ρ R ], a 1 16 [ (ρ M ρ M1 + (ρ R ]. Proposton 5. ( a ncreases (decreases as ρ M ( ncreases, ( a ncreases (decreases as ρ R ( ncreases,( a ncreases (decreases as σ( ncreases, and (v a ncreases as t ncreases, (,j1,,jj. Comparngwth Proposton, the retaler s co-op advertsng expendture s also determned by the margnal profts oftheretaleraswellasthoseofthemanufacturer.buta decreases when ρ M and ρ R ncrease, whle t decreases as ρ M and ρ R1 ncrease. Furthermore, co-op advertsng expendture s also affected by the stmulatng effect and the competton effectofadvertsng. By Propostons and 5, wefndthattheeffectsof advertsng (ncludng the stmulatng effect and the competton effect only affect the co-op advertsng expendture of the retaler but have no effect on the manufacturer s equlbrum partcpaton rato decson. Advertsng has a spllover effect on the supply chan frms profts, and the retaler acts as a cooperator as well as a compettor to the manufacturer. Besdes, the dfference n margnal profts between the manufacturer and the retaler should not be too large. Proposton 6. Consder the followng a 1 / ρ M1 > a 1 / ρ R1, a 1 / ρ M > a 1 / ρ R. Proof. Followng the proof of Proposton 5, t s easy to acheve a 1 / ρ M1 a 1 / ρ R1 and a 1 / ρ M a 1 / ρ R, whch means a 1 / ρ M1 > a 1 / ρ R1, a 1 / ρ M > a 1 / ρ R. The proof of Proposton 6 s complete. (7 Proposton 6 ndcatesthat, comparedwththeretaler s, the manufacturer s margnal proft effects on advertsng expendture are stronger, whch means that co-op advertsng s more senstve to the margnal profts of the manufacturer. Consderng that σ>,boththenetmargnalprofteffect of the manufacturer and that of the retaler on advertsng are postve, and the manufacturer s net postve margnal proft effect on advertsng s larger than the retaler s. In other words, the manufacturer n the supply chan has more ncentve to expend on advertsng, whch provdes an explanaton for why manufacturers partcpaton ratos are always hgher than 50%. Total profts of the manufacturer and the retaler are denoted as π B M and πb R, respectvely, f the supply chan advertses for both brands, π S M and πs R f t advertses for only one brand, and π N M and πn R f t advertses for none. Then, we acqure Proposton 7. Proposton 7. The profts of the manufacturer and the retaler under dfferent condtons satsfy the followng relatonshps: π B M >πs M >πn M and πb R >πs R >πn R. Advertsng ncreases the profts of both the manufacturer and the retaler. So f the frms n the supply chan have no cash flow constrant, they always advertse for all ther brands. We can fnd evdence n the realty. For example, Procter & Gamble always advertses all of ts shampoo brands, such as Rejoce, Head & Shoulders, Pantene, and Sassoon, and all detergent brands, such as Tde, Lenor, and Arel, at the same tme. 3.. Cooperatve Stuaton. Many researches [, 5, 1] clam that market power of superretalers, such as Wal-Mart, Carrefour,Sunng,andGome(SunngandGomearethetwo bggest household applance retalers n Chna, becomes stronger and stronger and they become the strategc partner of manufacturer. So, n ths part, we analyze co-op advertsng under the cooperatve stuaton. And the maxmzaton problem for the supply chan under the cooperaton stuaton s Max Ππ a 1,a M +π R (ρ M1 +ρ R1 (z+σ a 1 a +(ρ M +ρ R (z+σ a a 1 a 1 a. The optmal equlbrum solutons are a 1 1 [(ρ M 1 +ρ R1 (ρ M +ρ R ], a 1 [(ρ M +ρ R (ρ M1 +ρ R1 ]. The total proft of the manufacturer and the retaler under the Stackelberg stuaton s presented as follows: (8 (9 Π π M +π R (ρ M 1 +ρ R1 (z+σ a 1 a +(ρ M +ρ R (z+σ a a 1 a 1 a. (10 Then, ths study acheves Proposton 8.

5 Appled Mathematcs 5 Proposton 8. a >a, (, j 1,, j,andπ>π. Proposton 8 llustrates that the co-op advertsng expendture and the total profts under the cooperatve stuaton are all more than those under the Stackelbreg stuaton. These conclusons are n accord wth other nvestgatons [5, 9, 16, 18]. More mportantly, although the condton analyzed and the base model employed n ths study are dfferent from n other papers, we acheve the same conclusons as others, whch meansthatthe conclusonsofproposton 8 are robust. Proposton 8 also shows us that f the retal prce of the retaler s fxed, cooperatve decson heghtens the socal welfare.. Concludng Remarks Advertsng has strong spllover effect, and frms n the supply chan cannot acqure all the profts brought by advertsng. So, more and more supply chan frms choose co-op advertsng to mprove sales. Stmulated by these phenomena, ths paper employs a sngle-manufacturer sngle-retaler two-brand model to address co-op advertsng. Dfferent from other studes, ths paper consders the phenomenon that the supply chan supples two substtute brands smultaneously. Besdes, ths paper separates advertsng effects nto stmulatng effect and competton effect. Ths paper acheves several nterestng conclusons. Frstly, f the supply chan frms supply two (or multple substtute brands, the dfference of total expendture between the two (or multple brands should not be too large. Secondly, cooperatve (co-op advertsng ncreases the total supply chan profts and advertsng expendture. More nterestngly, although both the stmulatng effect and the competton effect affect the retaler s advertsng decson, they have no effect on the manufacturer s partcpaton rato decson. The manufacturer margnal proft as well as the retaler s margnal proft affects both the manufacturer s and the retaler s behavor. Moreover, the manufacturer s margnal proft has more effect on the supply chan decson than the retaler s. The stmulatng effect ncreases co-op advertsng whle the competton effect decreases co-op advertsng. Next, the study can extend to multple brands or consders the effect of captal market, such as debt fnancng behavor, whchwllalsocomplcatetheanalyss.theseareourfurther studes. π M a 1 (ρ R (a 3/ <0, ρ R >0. (A.1 Besdes, we have π R / t 1 t 0,sofuncton(sconcave. Solvng π R / a 0,weobtan a 1 (ρ R 1 ρ R (1 t 1, a (ρ R (1 t. Substtutng (A.ntofuncton(1, we have So, ( π M ρ M (z + σ (1 t (1 t j t ( (1 t (A. ( + (z + σ (1 t j (1 t ( t j (. (1 t j π (ρ M M t (1 t ( (1 t (A.3 (A. Appendx Proof of Proposton. The frst- and second-order optmal condtons of (1and(yeld π R a 1 (ρ R (a 1/ (1 t, (, j 1,, j, π M t t (1 t 3, (ρ M [ ] (1 t 3 (1 t 3 t 3[ ] (1 t 3

6 6 Appled Mathematcs 1 {(ρ M (1 t { (1 t { 1 (1 t ( (1 t [ ] t (1 t 3 3 (ρ ( R (1 t 1 (ρ t R } (1 t 3 } } π [ 3 (ρ t ( R (1 t [ 1 t (1 t 3 ] ] ] 1 3 [ (1 t (ρ R (1 t [ + 1 t (1 t 3 ] ] ] <0. (A.5 t (ρ R [ (ρ M ] <0, t σ (ρ Mj σρ M ρ R [ (ρ M ] <0, t (σρ M [ (ρ M ] >0. (A.6 Consderng ρ M, ρ R, (, j, thenfrom(5 or (6, ths study acheves that t (ρ lm ρ M σ Mj ρ R ρ M ρ R [ (ρ M ] 0, t σ (ρ lm ρ M M ρ R ρ R [ (ρ M ] 0. Proof of Proposton s complete. Proof of Proposton 5. We have the followng: ρ R 1 8 σ[(ρ M ]> 0, (, j 1,, j, 1 8 [(ρ M ]< 0, (A.7 π R / a 1 a 0.Itfollowsthatfuncton(1 sconcaveand the proof s complete. Proof of Proposton 3. If t 1 ( /((ρ M + > 0, (, j 1,, j, then (ρ M > 0,and( ρ M /( > 1/.Smlarly,ft 1 ( /((ρ M + 1/,then(ρ M 3 > 0, and ( ρ M /( 3/. Proof of Proposton 3 s complete. Proof of Proposton. From (5, t ρ M σ [ (ρ M ] >0, (, j 1,, j, ρ M 1 σ[(ρ M ]> 0, 1 [(ρ M ]< 0, σ 1 8 [ (ρ M ] (ρ M +ρ R >0. By dfferentatng (3wthrespecttot,weacheve (A.8 a (ρ R t (1 t 3 >0. (A.9 Proof of Proposton 5 s complete.

7 Appled Mathematcs 7 Proof of Proposton 7. For, j 1, and j, And, by (A.10 (A.1, we acheve π B M πn M ρ M (σ a a j + (σ a j a π B M ρ M (z + σ a a j + (z + σ a j a t a t j a j t a t j a j, π S M ρ M (z + σ a + (z a t a, π N M ρ M z+ z. (A.10 (A.11 (A.1 1 ρ M (ρ M +ρ R ( ρ M ρ R 16ρ M +8ρ R [(ρ M +ρ R (] 1 8 (ρ M +ρ R ( (ρ M ρ R >0. (A.15 If ρ M ρ M, ρ R ρ R, ρ M /ρ R >1/, ρ M /ρ R > 1/, by(a.10-(a.11, we get So π B M >πs M >πn M.Forthesamereason,wehaveπB R >πs R > π N R. Proof of Proposton 7 s complete. π B M πs M ρ M a j + σ a j t j a j ρ M [ ( ρ M + ( ] ( (([( ρ M + ( ] ( (ρ M [( ρ M ( ] (16 1 By (A.11-(A.1, we have + ρ Mρ R (ρ R 16 ( >0. (A.13 Proof of Proposton 8. If ρ M, ρ R, (, j 1,, j σ>,thenwehave a a 1 [(ρ M +ρ R ( + ] 1 16 [ (ρ M ] 1 16 [(ρ M j +3 σ (ρ M +ρ R ] ( > 0, Π Π (ρ M +ρ R (z+σ a a j +( + (z+σ a j a a a j (ρ M +ρ R (z+σ a aj ( + π S M πn M σρ M a a t a (z+σ a j a +a +a j ρ M [ (ρ M ] ( (([(ρ M ] [(ρ M ] ρ R (ρ M +ρ R ( a a (a a ρ R (ρ M +ρ R ( 1 8 [(ρ M +3ρ R (ρ M +ρ R ] ρ R ( ( (ρ M ( ρ Mρ R (ρ R 16 ( >0. (A.1 >[(ρ M +ρ R 1 8 (ρ M +3ρ R ] ρ R ( >0. Proof of Proposton 8 s complete. (A.16

8 8 Appled Mathematcs Acknowledgments Ths work was supported by the Natonal Socal Scence Foundaton of Chna (11BGL03 and the Soft Scence Foundaton of Guangdong Provnce of Chna (01B Sncere thanks are offered to the anonymous revewers for ther helpful suggestons. References [1] J. V. Durme, R. J. Brode, and D. Redmore, Brand equty n cooperatve busness relatonshps: explorng the development of a conceptual model, Marketng Theory, vol. 3, no. 1, pp , 003. [] S.Jørgensen,S.P.Sgué, and G. Zaccour, Dynamc cooperatve advertsng n a channel, Retalng,vol.76,no.1,pp. 71 9, 000. [3] J. Xe and J. C. We, Coordnatng advertsng and prcng n a manufacturer-retaler channel, European Operatonal Research,vol.197,no.,pp ,009. [] Z. Huang and S. X. L, Co-op advertsng models n manufacturer-retaler supply chans: a game theory approach, European Operatonal Research, vol.135,no.3,pp. 57 5, 001. [5] Z. Huang, S. X. L, and V. Mahajan, An analyss of manufacturer-retaler supply chan coordnaton n cooperatve advertsng, Decson Scences, vol. 33, no. 3, pp. 69 9, 00. [6] M. G. Nagler, An exploratory analyss of the determnants of cooperatve advertsng partcpaton rates, Marketng Letters, vol. 17, no., pp , 006. [7] K. Sen, Advertsng fees n the franchsed channel, n Franchsng: Contemporary Issues and Research,P.J.KaufmannandR.P. Dant,Eds.,TheHaworthPress,NewYork,NY,USA,199. [8] R. P. Dant and P. D. Berger, Modellng cooperatve advertsng decsons n franchsng, the Operatonal Research Socety, vol. 7, no. 9, pp , [9] P. D. Berger, Vertcal cooperatve advertsng ventures, Journal of Marketng Research,vol.9,no.3,pp ,197. [10] L. Brennan, How retalers are puttng t all together, Sales and Marketng Management, vol. 5, pp. 6 65, [11] M. Bergen and G. John, Understandng cooperatve advertsng partcpaton rates n conventonal channels, Marketng Research,vol.3,no.3,pp ,1997. [1] S. Karray and G. Zaccour, Could co-op advertsng be a manufacturer s counterstrategy to store brands? Busness Research,vol.59,no.9,pp ,006. [13]R.Yan, Cooperatveadvertsng,prcngstrategyandfrm performance n the e-marketng age, the Academy of Marketng Scence,vol.38,no.,pp ,010. [1] S.-D. Wang, Y.-W. Zhou, J. Mn, and Y.-G. Zhong, Coordnaton of cooperatve advertsng models n a one-manufacturer two-retaler supply chan system, Computers & Industral Engneerng, vol. 61, no., pp , 011. [15] J. Zhang and J. Xe, A game theoretcal study of cooperatve advertsng wth multple retalers n a dstrbuton channel, Systems Scence and Systems Engneerng, vol.1,no. 1, pp , 01. [16] J. Yue, J. Austn, M.-C. Wang, and Z. Huang, Coordnaton of cooperatve advertsng n a two-level supply chan when manufacturer offers dscount, European Operatonal Research,vol.168,no.1,pp.65 85,006. [17]X.He,A.Krshnamoorthy,A.Prasad,andS.P.Seth, Retal competton and cooperatve advertsng, Operatons Research Letters, vol. 39, no. 1, pp , 011. [18] B. Vscolan, Pure-strategy Nash equlbra n an advertsng game wth nterference, European Operatonal Research,vol.16,no.3,pp ,01. [19] P. K. Chntagunta and D. Jan, A dynamc model of channel member strateges for marketng expendtures, Marketng Scence, vol. 11, no., pp , 199. [0] S. Jørgensen, S. Taboub, and G. Zaccour, Cooperatve advertsng n a marketng channel, Optmzaton Theory and Applcatons,vol.110,no.1,pp ,001. [1] M. Nerlove and K. J. Arrow, Optmal advertsng polcy under dynamc condtons, Economca, vol. 39, pp. 19 1, 196. [] J. M. Alston, J. W. Freebarn, and J. S. James, Beggar-thyneghbor advertsng: theory and applcaton to generc commodty promoton programs, Amercan Agrcultural Economcs, vol. 83, no., pp , 001. [3] J. Xe and A. Neyret, Co-op advertsng and prcng models n manufacturer-retaler supply chans, Computers & Industral Engneerng,vol.56,no.,pp ,009. []X.He,A.Prasad,andS.P.Seth, Cooperatveadvertsng and prcng n a dynamc stochastc supply chan: feedback stackelberg strateges, Producton and Operatons Management, vol.18,no.1,pp.78 9,009.

9 Advances n Operatons Research Advances n Decson Scences Appled Mathematcs Algebra Probablty and Statstcs The Scentfc World Journal Internatonal Dfferental Equatons Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Complex Analyss Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Dscrete Mathematcs Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Internatonal Stochastc Analyss Optmzaton