# Research Article Pricing Model for Dual Sales Channel with Promotion Effect Consideration

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6 6 Mathematical Problems in Engineering Table : Equilibrium strategies when price discounts are provided only by online channel ( + denotes the positive value) Pricing strategies Conditions Prices Sales Profits p R p D p D S R S D S D π R π D π D Pricing Strategy I C B c Null 4 θ 4 θ θ Null + + Null + + Pricing Strategy II C A c<c B c c+ (c + )θ Zero + + Zero + + (4θ Pricing Strategy III 0 c<c θ + θ )( +c) A (4 θ + θ ) ( θ +c) θ ( +c) 4θ θ +θ θ (4θ +θ θ θ θ )

10 0 Mathematical Problems in Engineering his/her online competitor he/she will adopt a price strategy that p R (p D ) p D /θ; otherwise he/she will use a strategy that p R (p D ) < p D /θ Specifically(i)when0 c < θ p R (p D ) p D ++c if θ c D < ( + c ) p D if (+c ) θ D < (6 8θ +θ +(8 4θ +θ )c) ( 4θ +θ ) 6 8θ +θ +(8 4θ +θ )c ( 4θ +θ ) if p D (6 8θ +θ +(8 4θ +θ )c) ( 4θ +θ ) (33) and (ii) when θ c p R (p D ) p D ++c if c D < (+c ) p D if (+c ) θ D < (+c) +c if p D (+c) (34) Proposition 9 illustrates effect of the online channel on traditional channel s first-period price If he/she anticipates a low first-period price of online channel he/she will charge arelativehighpricep R > p D / to get more marginal profits and if he/she anticipates a high first-period price of online channel he/she will charge a relative low price that p R < p D / tocapturemoremarketshare;otherwise if he/she anticipates a medium price from his/her online competitor he/she will give equal weight to the market share and marginal profits by charging a price that p R p D / 53 The Online Channel s Pricing Problem iven the firstperiod price of traditional seller the total profit of the online channel is π D (p D p R )π D (p D p R )+π D (p D p R ) ( p D )p θ D +π D (p D V p D ) c D <p R + ( p R p D )p D +π ( ) D (p D V p D ) p R + D R 0+π D (p D V p R ) p D >p R (35) where π D (p D V p D ) θ p D 4 (p D θ p D + c) (p D c) θ θ (p D p D θ + c) θ (4 θ ) ( θ ) θ p R 4 (p R θ p R +c)(p R c) π D (p D V p R ) θ θ (p R p R θ +c) (4 θ ) ( θ ) if c D < c θ θ if c θ D < ( θ ) c ( θ ) if p D ( θ ) c ( θ ) if c R < c θ c if θ R < ( θ )c ( θ ) if p R ( θ )c ( θ ) (36)

11 Mathematical Problems in Engineering Since π D (p D p R ) is nonquasiconcave we cannot ascertain the existence and uniqueness of a Nash equilibrium Our objective is to study the feasible regions for the existence of Nash equilibrium prices In order to obtain the feasible regions associated with a given value for the parameters c and θ in which a Nash price equilibrium exists we first derive the necessary and sufficient conditions on those variables in which an equilibrium exists and secondly we calculate all equilibria in prices for each firm and each critical interval Similar to the traditional channel the online channel also has three pricing strategy options when his/her rival s firstperiod price is given as follows: (i) high price strategy: p D > p R and V p R ; (ii) medium price strategy: p R + D R and V p D / ; (iii) low price strategy: p D R + and V p D / Notice that the total profit function in (35) is continuous and π D (p D p R )/ p D is negative at the left of the point p D p R and is zero at the right implying that π D (p D p R ) is nonincreasing at the point p D p R Thusfor the online seller it is never optimal for him/her to use a high price strategy that p D >p R This implies that anticipating a peak period for parcel deliveries the online seller will set a low first-period price to transfer more consumers to the normal sales period from the promotion sales period WecanalsoprovethatinaNashpriceequilibriump R < c+ InaNashpriceequilibrium(p R p D ) p R and p D should satisfy traditional seller s response function proposed in Proposition 9 Since p R (p D ++c )/ c+ ifp R c+ thenp R D / Recall that for the online seller it is never optimal to set such price that p R D / ThusinaNashpriceequilibriumthe first-period price should not be higher than c+ Now we consider the case p R < c+ andtheprofit function is changed to π D (p D p R )π D (p D p R )+π D (p D p R ) p D ( p R p D ) ( ) θ p D 4 (p D θ p D + c) (p D c) + θ θ θ (p D p D θ + c) θ (4 θ ) ( θ ) if c D < c θ θ if c θ D < ( θ ) c ( θ ) if p D ( θ ) c ( θ ) (37) The optimization problem of online channel is given by (37) subject to c D R Lemma 0 iven the traditional channel s first-period price p R and p R < c+ the optimal price for the online channel is c θ p R p 4 θ + θ D (p R ) θ θ p R +( θ )( ) c ( θ + θ ) θ (4 θ ) p R + θ ( )c (6 7 θ θ +θ ) if c p R < (4 θ + θ )c if (4 θ + θ )c if (4 θ + θ )c ( θ ) R < PR if PR p R R < (4 θ + θ )c ( θ ) (38) where PR (3 8 θ 4θ +3 θ +5θ θ 3 +θ θ (6 7 θ θ +θ )( θ + θ )) c (8 4θ +θ )( θ ) (39)

12 Mathematical Problems in Engineering 533 The Equilibrium Prices for the Two Channels Solving (33) (34) and (38) simultaneously we can get the following lemma Lemma There are Nash price equilibria for the traditional channel and online channel Specifically ( ) (6 θ +θ +7 θ )+ c ( )( θ + θ )+(+3 θ θ θ +θ θ )c p R θ θ +8 θ θ (4 θ + θ )( +c) (4 θ θ + θ ) +(+ )c θ ( )(4 θ ) + c ( θ p )θ θ +(4+3 θ 4 θ )c D θ θ +8 θ θ θ ( +c) 4 θ θ + θ c if 0 c < C E if C D c<c F if C F c< C if C c if 0 c < C E if C D c<c F if C F c< C if C c (40) where C D C E C D 4(8 4θ +θ ) ( )( θ ) θ + θ ( θ +8 θ θ )θ θ (6 7 θ θ +θ )+ 3 θ + θ C E 4(8 4θ +θ ) ( )( θ ) 6 7 θ θ +θ (θ θ ( θ + θ )) θ θ +θ C F ( )( θ ) 6 +3 θ θ θ C ( ) 3 θ θ + θ (4 θ )θ (8 8θ +4)θ 6θ + 64 >0 (4) 6 +θ 6 θ θ θ >0 6 θ +4θ + θ >0 3 ( ) θ 3 +4(5 )θ + 6 ( 4 )θ From Lemma we note that when C c theonline seller s price in the first period is citmaybecausedbythe assumption that V min(p R p D / ) csincep R c whether V cor not depends on the size relation between p D and c For the online channel it may be optimal for him/her to set a price that V cto capture the entire market If V c Scenario II will reduce to Scenario I where only the online channel provides the sales promotion By profits

14 4 Mathematical Problems in Engineering Table : Equilibrium strategies when price discounts are provided by both channels ( + denotes the positive value) Pricing strategies Conditions Pricing Strategy I C B c Null Prices Sales Profits p R p D p R p D S R S D S R S D π R π D π R π D 4 θ Null θ θ 4 θ Null + Null + Null + Null + Pricing Strategy II C A c<c B c c+ Null (c + )θ Zero + Null + Zero + Null + Pricing Strategy III C c<c (4 θ + θ )( +c) (4 θ + θ ) Pricing Strategy IV C D c<c F ( )( θ + θ ) θ +8 θ θ + ( + 3θ θ θ + θ )c θ +8 θ θ Pricing Strategy V 0 c<c E ( )(6 θ + c +θ +7 θ ) ( θ +c) Null 4 θ + θ ( ) θ θ +8 θ θ + (4 + 3θ 4 θ )c θ +8 θ θ ( θ )(4 θ ) + c c ( )( θ )(4 θ ) + (4 θ )(5 θ )c θ ( +c) (4θ +θ θ θ θ ) + + Null Null + ( )( θ ) θ θ +8 θ θ + ( + θ θ +3 θ )c θ +8 θ θ θ ( )( θ )(4 θ ) + θ (4 θ )(5 θ )c + + Zero Zero

17 Mathematical Problems in Engineering 7 (ii) iven p R wehaverewrittentheonlineseller sprofit function as p D ( p D )+ (p D) θ θ 4 π D (p R ) p D ( p R p D ) + (p D) θ ( ) 4 p R θ 4 0 D <p R + p R + D < p R p D p R (A4) The first-order conditions yield two interior solutions as follows (denoted by Sol i i ): Sol Sol (4 θ ) p R (4 + θ θ ) (A5) We can easily verify that the second-order conditions with respect to p D are nonpositive Comparing these two solutions with the boundary points p R + and p R we find Sol R + Sol R + when p R +( θ 4 ) Sol >p R + Sol <p R + (A6) when ( )(4 + θ θ ) <p R < θ +( θ 4 + θ θ 4 ) Sol >p R + p R + Sol p R when p R ( )(4 + θ θ ) 4 + θ θ From (A6) we then directly get the corresponding conditions where the interior solutions Sol and Sol andthe corner solutions Sol 3 p R + are optimal Proof of Proposition3 It can be easy to verify that p R / θ 0 p D / θ 0 p D / θ 0 π R / θ >0and π D / θ >0 ProofofLemma4 (i) Using the first-order conditions for the profit function in (0) we get two interior solutions as follows (denoted by PR i i ): PR V +c PR V Vθ +p D +c (A7) We can easily verify that the second-order conditions with respect to p R are nonpositive Similar to the proof of Lemma we get the corresponding conditions where these interior solutions and boundary points are the equilibrium prices by comparing the values of interior solutions with the boundary points By comparing we find that when 0 D < c V+Vθ π R (p R ) always increases in p R until it reaches the point (( θ )V+p D 0)and then remains unchanged at zero This implies that if the online channel sets a very low price the traditional channel cannot get any profit and will exit the market (ii) Using the first-order conditions for the profit function in () we get two interior solutions as follows (denoted by PD i i ): PD θ V/ PD θ p R / Wecaneasily verify that the second-order conditions with respect to p D are nonpositive Similar to the proof of Lemma we get the corresponding conditions where these interior solutions and boundary points are the equilibrium prices by comparing the values of interior solutions with the boundary points ProofofLemma7Using the first-order conditions for the profit function in (7) we get two interior solutions as follows (denoted by PT i i ): PT +c c PT 6 8θ +θ +(8 4θ +θ )c ( 4θ +θ ) (A8) We can easily verify that the second-order conditions with respect to p R are negative Comparing the values of the two interior solutions with the boundary points ( θ )c/(( θ )) wefindthatifc < θ thenpt > ( θ )c/(( θ )) and PT >( θ )c/(( θ ));otherwise PT ( θ )c/(( θ )) and PT ( θ )c/(( θ )) In the former case π R (p R p D ) reaches its maximum value at point p R (p D ) PT Considering the condition p R <p D / ifpt <p D / the traditional seller s optimal decision is p R (p D )PT and if PT p D / theoptimal decision for the traditional seller is p R (p D ) p D / In the latter case the traditional seller s optimal decision is p R (p D ) PT or p R (p D ) p D / depending on whether PT <p D / or not

18 8 Mathematical Problems in Engineering ProofofLemma8Using the first-order condition for the profit function in (3) we get the optimal price p R (p D ) (p D ++c )/ but only if this price is between p D / and p D + Otherwise the optimal price for the traditional seller is p R (p D )p D / Proof of Proposition 9 From Lemma 7 the profit for the traditional channel under low price strategy is as follows: (i) when c< θ π R (p D ) ( p D )(p D c ) ( 4θ +θ )p D θ (4 θ ) ( + (8 4θ +θ )c (4 θ ) )p D c+ (6 θ 3 ) ( c) + (5c 0c+9)θ (6c 40c + 4) θ 4( θ ) ( 4θ +θ ) if c D < ( θ )c ( θ ) ( θ ) c ( θ )(4 θ ) if ( θ )c ( θ ) D < [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) if p D [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) (A9) and (ii) when c θ π R (p D ) From Lemma 8 the profit for the traditional channel under medium pricing strategy is ( p D )(p D c ) ( c) 4 if c D < (c+) if p D (c+) (A0) where π R (p D )π R (p D )+π R ( p D ) (A) (p D + c ) π 4( θ R (p D ) ) if c D (+c ) ( p D )(p D c ) if p D > ( + c ) 0 if cθ π D < ( θ )c ( θ ) R (p D ) (p D p D θ c +c θ ) (4 θ ) if p D ( θ )c ( θ ) ( θ ) (A) Recall thatthemedium price strategy is always better than the high price strategy for the online channel in the first periodthusinthefollowingweonlycomparethelow price strategy with the medium price strategy Comparing the values of c (+c )/( ) [ ( θ )c]/[( θ )] [6 8θ +θ +(8 4θ +θ )c]/[( 4θ +θ )]and (c + )/wehave c ( θ )c ( θ ) (+c ) [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) if 0 c< ( )( θ ) + θ c ( + c ) ( θ )c ( θ ) [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) c ( + c ) if ( )( θ ) + θ c< θ (c+) ( θ )c ( θ ) if θ c (A3) We use superscripts l to represent the low price strategy and m torepresentthemediumpricestrategy

19 Mathematical Problems in Engineering 9 When 0 c<( )( θ )/( + θ ) 0 if c D < ( θ )c ( θ ) 0 if ( θ )c <p π m R (pd ) π l R (pd ) ( θ ) D ( + c ) 0 if ( + c ) θ D < [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) 0 if p D [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) (A4) When ( )( θ )/( + θ ) c< θ π m R (pd ) π l R 0 if c D < ( + c ) 0 if (+c ) (pd ) 0 if ( θ )c ( θ ) 0 D < ( θ )c ( θ ) D < [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) if p D [6 8θ +θ +(8 4θ +θ )c] ( 4θ +θ ) (A5) When θ c π m R (pd ) π l R (pd ) 0 if c D < (+c ) 0 if (+c ) θ D < (c+) 0 if (c+) D < ( θ )c ( θ ) 0 if p D ( θ )c ( θ ) (A6) From (A4) (A5) and (A6) we get the prices as follows: (i) If 0 c< θ the optimal first-period price for the traditional seller with a given p D is shown in (33) (ii) If θ c the optimal first-period price for the traditional seller with a given p D is shown in (34) Proof of Lemma 0 Using the first-order conditions for the profit function in (37) we get two interior solutions as follows (denoted by PO i i ): PO p R 4 θ + θ (A7) PO θ p R +( θ )( ) c (A8) ( θ + θ ) PO 3 (4 θ ) p R + θ ( )c (6 7 θ θ +θ (A9) ) We can easily verify that the second-order conditions with respect to p D are negative Comparing the values of the two interior solutions with the boundary points BP c BP c/( θ )andbp 3 [( θ ) c]/[( θ )] we find that (i) if c p R < [(4 θ + θ )c]/ then PO c PO < BP PO 3 < BP 3 implying that the profit function in (37) decreases in p D sothe optimal price for the online channel is p D (p R ) c (ii) If [(4 θ + θ )c]/ p R < [(4 θ + θ )c]/[( θ ) ]then c PO BP PO < BP and PO 3 < BP 3 andtheoptimalpricefortheonlinechannelis p D (p R )PO (iii) If [(4 θ + θ )c]/[( θ ) ]/ p R < [(8 +θ 5 θ θ )c]/[ ( θ )(4 θ )]

20 0 Mathematical Problems in Engineering then PO > BP BP PO BP 3 andpo 3 BP 3 and theoptimalpricefortheonlinechannelisp D (p R )PO (iv) If [(8 +θ 5 θ θ )c]/[ ( θ )(4 θ )] p R < [( θ )c/( θ )]thenpo > BP BP PO BP 3 and PO 3 BP 3 ;inthiscasetheonlineseller soptimaldecision is PO or PO 3 depending on whether π D (PO p R ) π D (PO 3 p R ) or not Some simple calculations show that π D (PO p R ) π D (PO 3 p R ) 0if [(8 +θ 5 θ θ )c]/[ ( θ )(4 θ )] p R < PR holds and π D (PO p R ) π D (PO 3 p R ) 0if PR p R <[( θ )c/( θ )] (v) If p R [( θ )c/( θ )]thenpo > BP PO BP 3 and PO 3 BP 3 The online seller s optimal decision in this case is p D (p R )PO 3 Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper Acknowledgments This paper was supported by National Social Science Foundation China no 4BL00 Youth Foundation of Humanities and Social Sciences of the Ministry China no 0YJC63045 and Anhui Philosophy and Social Science Foundation no AHSK-D359 References [] K L Webb and C J Lambe Internal multi-channel conflict: an exploratory investigation and conceptual framework Industrial Marketing Managementvol36nopp []SKMukhopadhyayD-QYaoandXYue Information sharing of value-adding retailer in a mixed channel hi-tech supply chain Journal of Business Research vol 6 no 9 pp [3] J Chen H Zhang and Y Sun Implementing coordination contracts in a manufacturer Stackelberg dual-channel supply chain Omega vol 40no 5 pp [4] YXiaTXiaoandPZhang Distributionchannelstrategies for a manufacturer with complementary products Decision Sciencesvol44nopp [5] J K Ryan D Sun and X Zhao Coordinating a supply chain with a manufacturer-owned online channel: a dual channel model under price competition IEEE Transactions on Engineering Managementvol60nopp [6] E Widodo K Takahashi K Morikawa I N Pujawan and B Santosa Managing sales return in dual sales channel: its product substitution and return channel analysis International Journal of Industrial and Systems Engineering vol9nopp 49 0 [7] W-YKChiangDChhajedandJDHess Directmarketing indirect profits: a strategic analysis of dual-channel supplychain design Management Sciencevol49nopp 0003 [8] A Arya B Mittendorf and D E M Sappington The bright side of supplier encroachment Marketing Science vol 6 no 5 pp [9] R Zhang B Liu and W Wang Pricing decisions in a dual channels system with different power structures Economic Modellingvol9nopp [0] S-H Chun B-D Rhee S Y Park and J-C Kim Emerging dual channel system and manufacturer s direct retail channel strategy International Review of Economics and Finance vol 0no4pp8 850 [] F Bernstein J-S Song and X Zheng Bricks-and-mortar vs clicks-and-mortar : an equilibrium analysis European Journal of Operational Researchvol87no3pp [] K-Y Chen M Kaya and Ö Özer Dual sales channel management with service competition Manufacturing & Service Operations Managementvol0no4pp [3] T Boyaci Competitive stocking and coordination in a multiple-channel distribution system IIE Transactionsvol37 no5pp [4] K Cattani W illand H S Heese and J Swaminathan Boiling frogs: pricing strategies for a manufacturer adding a direct channel that competes with the traditional channel Production and Operations Management vol5nopp [5] H Kurata D-Q Yao and J J Liu Pricing policies under direct vs indirect channel competition and national vs store brand competition European Journal of Operational Research vol 80nopp [6] S K Mukhopadhyay X Zhu and X Yue Optimal contract design for mixed channels under information asymmetry Production and Operations Managementvol7no6pp [7] Cai Z Zhang and M Zhang ame theoretical perspectives on dual-channel supply chain competition with price discounts and pricing schemes International Journal of Production Economicsvol7nopp [8]YXuBDanXMZhangandCLiu Coordinatinga dual-channel supply chain with risk-averse under a two-way revenue sharing contract International Journal of Production Economicsvol47pp [9] J K Ryan D Sun and X Zhao Coordinating a supply chain with a manufacturer-owned online channel: a dual channel model under price competition IEEE Transactions on Engineering Managementvol60nopp [0] R Yan and S hose Forecast information and traditional retailer performance in a dual-channel competitive market Journal of Business Researchvol63nopp [] S-H Chun and J-C Kim Pricing strategies in BC electronic commerce: analytical and empirical approaches Decision Support Systems vol 40 no pp [] A Dumrongsiri M Fan A Jain and K Moinzadeh A supply chain model with direct and retail channels European Journal of Operational Researchvol87no3pp [3] B Dan Xu and C Liu Pricing policies in a dual-channel supply chain with retail services International Journal of Production Economicsvol39nopp3 300

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