ON THE EQUIVALENCE OF SELECTED SUPPLY CHAIN CONTRACT MECHANISM

Transcription

1 NRG WORKING PAPER SERIES ON THE EQUIVALENCE OF SELECTED SUPPLY CHAIN CONTRACT MECHANISM Nyenrode Research Group Vijayender Reddy Nalla Jack van der Veen Venu Venugopal November 005 no

2 NRG WORKING PAPER SERIES On the Equivalence of Selected Supply Chain Contract Mechanism Vijayender Reddy Nalla Prof.dr. Jack van der Veen Prof.dr. Venu Venugopal November 005 NRG Working Paper no ISSN NRG The Nyenrode Research Group (NRG) is a research institute consisting of researchers from Nyenrode Business Universiteit and Hogeschool INHOLLAND, within the domain of Management and Business Administration. Straatweg 5, 36 BG Breukelen P.O. Box 30, 360 AC Breukelen The Netherlands Tel: +3 (0) Fax: +3 (0) nrg@nyenrode.nl NRG working papers can be downloaded at

3 Abstract This paper models a situation where a Supplier sells a fashion product to a Buyer who in turn sells the product to the consumers. Both the Supplier and the Buyer set their own selling price. For the above setting this paper designs different contract mechanisms such as Revenue sharing, Profit sharing, Quantity discounts, License fee and Mail-in-rebate contract mechanisms. The paper shows that the designed contract mechanism can provide both coordination and win-win. The paper also establishes the equivalence between the designed contract mechanisms and argues that industries can use one mechanism over the other in case of implementation problem. Keywords Coordination, Win-Win, Supply Chain Contracts, Mail-in-rebates Addresses for correspondence Drs. Vijayender Reddy Nalla r.vijayender@nyenrode.nl Phone: (+3) Prof.dr. Venu Venugopal v.venugopal@nyenrode.nl Phone: (+3) Prof.dr. Jack van der Veen j.vdveen@nyenrode.nl Phone: (+3) Nyenrode Business Universiteit Straatweg 5, 36 BG Breukelen, The Netherlands

4 . Introduction Companies in the past perceived themselves as a stand-alone entity in the business environment. Times have changed. Today companies begin to perceive themselves as a part of a chain network of entities. The reason is simple: Without deep co-operation and collaboration, no single company can survive and prosper on its own. This has lead to considerable interest among both practitioners and academics in the field of Supply Chain Management (SCM). Although the term SCM is widely used, there is no unique general definition in the literature. In this paper, we will define SCM as Management activities focused on the (voluntary) co-ordination of several entities in the supply chain in order to optimise the entire supply chain as it were one unit, rather than optimising each entity separately. Basically, the concept of SCM emphasises the need for co-ordination and collaboration within and between the firms to achieve a win-win situation for all the firms involved. The win-win situation in a co-ordinated supply chain results from various sources such as co-ordinated order process and delivery process [Clarck & Hammond, 997] and coordinated product introductions and promotional activities [Buzell et al., 990]. Frequently, in literature the win-win concept is discussed on a conceptual level only [Maloni & Benton, 997]. The purpose of this paper is to illustrate the win-win feature through a simple quantitative model. In this paper, we make use of the same model as (Veen & Venu (000 (a)) which is a simple supply chain consisting of two entities (firms) namely a Supplier and a Buyer. The Supplier sells a product with a short life cycle (e.g. a fashion product) to a Buyer who in turn sells the product to the consumers (see Figure ). Both the Supplier and the Buyer are free to set the price they charge to their customer(s). Both the players have symmetric and perfect information of the demand and the cost functions. Venugopal 3 of 47

5 supplier buyer C O N S U M E R S Figure : A simple supply chain. Firstly, will shall discuss two scenarios. In the first scenario, called Solitaire, the supplier and the buyer do not collaborate. They set their price to maximise their own profit. In the second scenario ( Partnership ), both firms collaborate and sit together to decide the prices with the objective to optimise the entire supply chain as well as their own profits. It will be shown that in the partnership-scenario there is a significant improvement (higher profit for both) over the solitaire-scenario. Also, it is shown how the prices can be set so that the additional profit is fairly split between the supplier and the buyer. Interestingly, coordination can also be achieved by modifying the terms of trade by using incentives or contracts over certain variables. This approach basically seeks to identify mechanisms or arrangements through which,. The performance of the whole chain (as if it were one unit) is maximized (in such a case, we say that the mechanism coordinates the chain), and. All the players improve their performance when compared to an uncoordinated situation (in such a case, we say that the mechanism leads to a win-win situation) or at least some of the members improve with other members not being worse off. The two conditions (coordination and win-win) are related but do not imply one another. For instance, maximum supply chain profit can be achieved when one company is forced to reduce its profit to enable the other company to make more than proportional additional profit. Likewise, the other way around, even in a win-win situation the maximum supply chain profit might not be achieved. From an implementation point of Venugopal 4 of 47

6 view, the win-win condition is a necessary (but not sufficient) condition to achieve supply chain coordination whereas coordination condition is desired but not required. Before implementing any incentive or contract mechanism, the obvious research questions are: Does the mechanism coordinate the chain? and Does a win-win situation exist under the mechanism? This paper analyses the effectiveness of the revenue sharing, profit sharing, quantity discount, license fee, mail-in rebate and wholesale price discount mechanisms as different scenarios. The profit sharing mechanism is governed by the supplier receiving a share of buyers profits. In the revenue sharing mechanism the transactions between the supplier and buyer are governed by the supplier receiving a share of buyers revenues. With the quantity discount mechanism, the supplier fixes a base price and charges an additional price which is a decreasing function of the ordered quantity. With the license fee mechanism the supplier gets a fixed amount along with the wholesale price. With the mail-in-rebate mechanisms the supplier is obliged to credit the end customer with an amount equal to the value of the rebate coupon. With the wholesale price discount mechanism the supplier provides a rebate to the retailer. A detailed discussion and explanation of each of the above contract mechanisms is provided in section 5. The remaining part of this paper is organised as follows. In Section the basic model and assumptions are introduced. The solitaire-scenario and partnership-scenario are discussed in Sections 3 and 4 respectively. Section 5 presents the analysis of all the contract mechanisms. Section 6 compares the performance of the mail-in-rebates and wholesale price discount mechanism. Finally in section 7 some conclusions are drawn.. The basic Model Consider the situation in which the supplier offers a quantity of products to the buyer at a price of W per unit. It is assumed that the product has a short life cycle, so that there is only a one-time order placed by the buyer (i.e. reordering is not possible and there is no stock from previous periods). The supplier incurs manufacturing and transportation cost of C per unit and is free to set the price W at which he will sell the product to the buyer. Venugopal 5 of 47

7 Furthermore, it is assumed that final customer demand D for the product depends linearly on the price P set by the Buyer: D( P) = α βp () with α, β > 0. Once the buyer has set the price P, he can determine the demand D and places an order of size Q = D(P) on the supplier. The buyer does not have any cost associated with his operation other than the purchasing cost WQ. It is also assumed that both the players have symmetric and perfect information. Throughout, we will make the realistic assumptions: 0 C < P α/β; (a) C W P. (b) In the above setting, the supplier has cost-information C and can set W such that his profit S : = ( W C)Q (3) is maximised. The buyer has information on the final customer demand (i.e. knows α and β), and is faced with price W. He can set the price P such that his profit B : = ( P W )Q (4) is maximised. The total supply chain profit T is determined by ( P C). T : = S + B = Q (5) Note that the expression for supply chain profit T does not include W. In fact, T only depends on the parameters α, β and C and the buyer s price P. In other words, the buyer s price P determines the supply chain profit and the supplier s price W determines how the supply chain profit is split between the supplier and the buyer. Venugopal 6 of 47

8 The above model has been discussed in many papers. Its origin can be traced back to [Sprengler, 950]. We will study the above model under two scenarios. In the so-called solitaire-scenario, firstly the supplier sets his price W. Then, the buyer facing price W and demand function D(P) determines his optimal price P, i.e. the price that maximises profit B. Then, an order of size Q = D(P) is placed to the supplier. In the partnership-scenario, the supplier and buyer jointly determine P and W. Note that the partnership-scenario differs from the solitaire-scenario in the following sense: In the Solitaire-scenario W is determined first and its value drives the value of P, whereas in the partnership environment, P is determined first and then W. In the Solitaire-scenario, the Supplier only knows C and the Buyer knows only α, β and W, whereas in the partnership environment both will have full information (α, β and C). Later, we introduce different contract mechanisms and analyse their parameters to understand their sensitivity for the above setting. With the contract mechanisms both the buyer and the seller can jointly agree on the terms of trade. By choosing the appropriate values for the parameters it is possible for the players to agree upon a coordinating solution and achieve win-win. Throughout this paper we will use the following example. Let D(P) = 00-P, i.e. α = 00 and β =. Furthermore, let C = 30. The profit function for the buyer is B = (P W) (00 P) = 00P 00W + WP P. The profit function for the supplier is S = (W 30)(00 P) = 00W P WP. The total supply chain profit is equal to T = 60P 3000 P. 3. The Solitaire-scenario (Scenario ) In this section we will discuss the model under the solitaire-scenario. Assume the supplier has set a price W. Obviously, the Buyer likes to maximise his profit, i.e. he chooses P such that Venugopal 7 of 47

9 B = ( P W ) α βp) = βp + ( α + βw ) P αw ( (6) is maximised. It is easy to see that the optimal price is given by α + βw P : =. (7) β It follows from () that the optimal order size is Q = ( α β ). (8) : W Assuming the optimal price P and order quantity Q are used, the following profits can be derived from ()-(4): for the buyer, B ( W ) ( α βw ) α : = αw βw = 4 + β (9) 4β for the supplier and S ( C W β ) : = W (0) ( W ) αc + ( α + β ) for the total supply chain. T α ( W ): = αc + βcw βw 4 β () Note that all the profits under the solitaire-scenario depend on the supplier s price W. Given this fact, the question is: What is value that the supplier should have chosen for W? Clearly, from (b), we know that C W P, (i.e. realistically, the supplier can set the value of W anywhere in between C and P). But, which specific value should the supplier choose? Obviously, a realistic assumption is that the supplier would choose the value of W so as to maximise his own profit. However, in general he is not able to do that since he has no information on the demand function (). However, if we do assume that the Venugopal 8 of 47

10 demand-function D(P) in () is known to the supplier, i.e. α, β and C are known, the supplier obviously likes to maximise his profit-function (0). It is easy to see that (0) is maximised by Defining α + βc W : =. () β ( α βc), K : = (3) 6β using price W and substituting this in (9)-(), we obtain B (W ) = K; S (W ) = K and T (W ) = 3K. At first glance it may seem somewhat surprising that in this situation the profit of the supplier is twice as large as the profit of the buyer. An intuitive explanation would be that the Supplier s advantage stems from the fact that here it is assumed that the supplier has full information (α, β and C). Consider the relation given in (b). Let us assume that the supplier had chosen the value of W as W = W T := C, i.e. the supplier decides to sell the products to the Buyer at cost. Substituting W T in (9)-() gives B (W T ) = 4K, S (W T ) = 0 and T (W T ) = 4K. Note that when compared to the previous situation the total supply chain profit is larger (4K instead of 3K). It can be easily shown that in fact W = C maximises the total supply chain profit function given in (). It is obvious that this situation is very profitable for the buyer but not for the supplier. In other words, there is absolutely no incentive for the supplier to optimise the supply chain profit: The more profit for the total supply chain, the less profit for the supplier! Consider the relations given in (a) and (b). Let us shortly review the case that, by coincidence, the supplier had chosen the value of W as α W = W R : =. (4) β Venugopal 9 of 47

11 It is easy to see from (8) that with price W R it is not profitable for the buyer to place an order (i.e. Q = 0), hence there is no profit for either the Buyer or the supplier: B (W R ) = 0; S (W R ) = 0 and T (W R ) = 0. The results in this section are summarised in Figure. Reviewing the example discussed at the end of Section, we obtain the following. The optimal price P for the Buyer is P = 5 + ½W, and the order size is Q = 50 W. The profits for the buyer, supplier and total supply chain are given by B (W) = 50 50W + ½W ; S (W) = W W and T (W) = W ½W respectively. If W = W = 40 then R (40) = 50; M (40) = 00 and T (40) = 50 (note that K = 50). 4 K Profit T 3 K B K K 0 S C W W B W Figure : Profits for Supplier (S), Buyer (B) and total supply chain (T) for various prices W under the Solitaire-scenario. Venugopal 0 of 47

12 4. The Partnership-scenario ( Scenario ) In this scenario we assume that the supplier and buyer jointly determine P first and then W. Since they are working together, all information (i.e. α, β and C) is available to them. In optimising the joint profit, P should be chosen to maximise T as given in (5). It is easy to see that the optimal price is given by α + βc P : =, (5) β hence the order size is given by Q = ( α β ). (6) : C Note that P P and Q Q, i.e. if the total supply chain is optimised then the price P is lower and the quantity ordered Q is higher. It may be concluded that the consumers are profiting from the collaboration between the Buyer and the Supplier in the sense that the price for the product is lower. Substituting (5) and (6) in (5) shows that the total supply chain profit in this scenario is T = 4K. It is important to note that the supply chain profit is in fact independent of W. When compared to the supply chain profit under the Solitaire-scenario, it can be observed that under the Partnership-scenario the supply chain profit value is larger (or equal in case of W = C). The feature that the supply chain profit is lower if companies do not collaborate was first observed in [Spengler, 950] and is known under the term double marginalisation (see e.g. [Cachon, 999]). The difference in supply chain profit between the two scenarios (i.e. T T ) is given by β Δ : = 4 ( W ), C (7) where W is the price set by the supplier under the solitaire-scenario. But what about the profits for the supplier and the buyer under the partnership-scenario? Are they also better (greater or equal) compared to the profits under the solitaire- Venugopal of 47

13 scenario? Let us analyse the same. Obviously, unlike the total supply chain profit, the profits of the buyer and the supplier do depend on the price W. The exact relation can be derived from (3) and (4) by substituting (5) and (6). This gives B ( W ) ( α β ) ( α βc)( α + βc) = C W + (8) 4β : for the Buyer and ( W ) ( α βc) W ( α βc)c S : (9) = for the supplier. Clearly, both R and M are linear functions of W. Furthermore, it is easy to see that R (W) = 0 if and only if α + βc W = = P = W, β (0) (where W is as defined in Section 3), and B (W) = S (W) if and only if W = W E, defined by α + 3βC W E : =. () 4β The profits of the buyer, supplier and the total supply chain as functions of W are depicted in Figure 3. Venugopal of 47

14 Profit 4 K T B 3 K K K 0 S C W - W + W W Figure 3: Profits for Supplier (S), Buyer (B) and total supply chain (T) for various prices W under the Partnership-scenario. So, what should the supplier s price W be? A possible answer can be W = W E because that is the only price at which the buyer and the supplier split the supply chain profit in two equal parts (B (W E ) = S (W E ) = K). However, it might also be the case that when forming the partnership the two companies decide that a fair price is a price W at which both the buyer and the supplier are better off (i.e. have higher profit) than in the solitaire-scenario. So, in other words, both companies are looking for a win-win situation. Theorem : For any W under the solitaire scenario there is a W under the partnership scenario such that both the buyer and the supplier have a higher profit in partnership scenario, i.e. B ( W ) B ( W ) and S ( W ) S( ) W Proof: Let, as before, W denote the price charged by the supplier in the solitairescenario. Note that when that price would remain under the partnership-scenario, the buyer will lose and the supplier will gain profit, which obviously is not acceptable for the buyer. Therefore, the price W should be lower than W. To a certain extent this is This statement holds if W > C. In the (unlikely) case that W = C, the Solitaire and the Partnershipscenario are the same. Venugopal 3 of 47

15 acceptable for the supplier because the order-size Q in the partnership-scenario is larger than Q in the solitaire-scenario, i.e. the supplier will earn less per unit but will sell more units. Using (9) and (8) we find that any price W is acceptable to the buyer as long as B (W) B (W ), which is equivalent to βc + αw βw W W + : =. () ( α βc) Similarly, by using (0) and (9), any price W is acceptable to the supplier as long as M (W) M (W ) or Note that ( α + β ) ( α βc) βc + C W βw W W : =. (3) ( W C) ( α βc). β W + W = (4) In fact, it is one of the key results in this paper that W W + (from the relation (4)) and, moreover, that W = W + if and only if W = C. This implies that there is always a price W such that both the supplier and the buyer have a strict higher profit in the partnershipscenario than in the solitaire-scenario (except for the unlikely case that W = C). In other words, win-win situations do actually exist! Let us come back to the issue of determining W. So far we have seen that a win-win situation occurs whenever W satisfies W < W < W +. If the price is set at W = W + then the supplier will get all the additional profit (i.e. B (W + ) = B (W ) and S (W + ) = B (W ) + Δ) and if the price is set at W then the buyer will get all the additional profit (B (W ) = B (W ) + Δ and S (W ) = S (W )). One approach to splitting the additional profit fairly is by setting W + W+ W =. (5) In that case the additional profits for the two companies (i.e. profit under the partnership-scenario minus the profit under the solitaire-scenario) are equally large and equal to ½Δ. Venugopal 4 of 47

16 Another realistic possibility would be that the buyer and supplier agree that when working together, the percentage of growth in profit compared to the solitaire-scenario should be equal for both companies. This is achieved if the price is set at At that price, we have ( W ) ( W ) ( W ) ( W ) B S W = W + W+. (6) T T B B ( W ) ( W ) = S S ( W ) Δ = ( ) + W T ( W ) 00%. (7) Again returning to the example in Sections and 3, we have the following results. The total supply chain profit is optimised by P = 40. Using this price, the supply chain profit is equal to T = 00 (= 4S). The profit of the buyer and supplier are given by B (W) = 800 0W and S (W) = 0W 600 respectively. If W = W E = 35 then both companies have a profit of 00. Let us assume that W = 40, i.e. B = 50 and S = 00. Note that the increase in profit due to the collaboration is equal to 50 (i.e. 50/50 = 33.33%). Furthermore, W + = 37.5 and W = 35. If the price is set at W = ( )/ = 36.5 then the profit of the buyer is 75 (an increase of 5) and the profit of the supplier is 5 (an increase of 5 also). If the price is set at W = (50W + 00W + )/50 = then the profit of the buyer is (an increase of 33.33%) and the profit of the supplier is (an increase of 33.33% also). The above analysis helps us to conclude that a partnership always will always provide win-win. However, in reality partnerships are difficult to create and sustain and pose several implementation issues. In the next section we discuss the applicability of the contract mechanisms (Tsay et al. (998)) as these mechanisms are powerful enough to create supply chain optimisation and win-win, and implementation is not too difficult. Venugopal 5 of 47

17 5. Contract mechanisms Supply chain (SC) contracts are coordination mechanisms that utilize incentives to make SC players decisions coherent among each other. In particular, the incentives let the risk and the revenue (which arise from different sources of uncertainty and from channel coordination, respectively) be shared by all players. Different models of SC contracts have been developed in the literature. They include the quantity flexibility contracts [Tsay, 999], the backup agreements [Eppen and Iyer, 997], the buy back or return policies [Pasternack, 985, Emmons and Gilbert, 998, Padmanabhan and Png, 997], the incentive mechanisms [Lee and Whang, 999], the revenue sharing contracts [Cachon and Lariviere, 005, Dana and Speir, 00] and the Quantity discounts [Weng, 995]. Contract Mechanisms have also been studied extensively in the economics and marketing science literature where the issue of designing contractual agreements arises when a supplier uses intermediate firms to reach final consumers. [Tirole, 988] and [Katz, 989] provide excellent reviews of contracts in economics literature. [Cachon 003] has discussed the supply chain coordination with the contract mechanisms. Profit sharing and the revenue sharing mechanism are discussed in the subsection 5. & 5.. The quantity discounts, and license fee mechanisms are dealt with in the subsection 5.3 & 5.4. The mail-in-rebate mechanism the wholesale price discount mechanisms are discussed in the subsection 5.5 &5.6. The performance of the mail-in rebate mechanism has been compared with the wholesale price discount in section Profit sharing mechanism (Scenario 3) In the profit sharing mechanism the transactions between the supplier and buyer are governed by the supplier receiving a share of the buyers profits. [Jeuland and Shugan, 983] have used profit sharing as a mechanism to achieve channel coordination. The profit sharing mechanism can be identified by two parameters, namely, wholesale price W and a percentage of the buyers profit δ (0<δ<), that goes to the supplier. The profit functions are, Venugopal 6 of 47

18 for the buyer ( α Q3 ) B 3( W3, Q3, δ ) = [[ ] Q3 W3Q3 ]( δ ) (8) β for the supplier ( α Q3 ) S3 ( W3, Q3, δ ) = W3Q3 + [[ ] Q3 CQ3 ] δ CQ3 (9) β for the total supply chain ( α Q3 ) T3 Q3 ) = [[ ] Q3 CQ β ( 3 ] (30) The quantity Q 3 which will optimize supply chain profit can be obtained from equation 30 as, Q 3 ( α βc) = (3) The optimal value of Q 3 for the buyer can be obtained from equation 8 as Q 3 ( α βw ) 3 = (3) The value of W 3 (wholesale price with profit sharing mechanism) at which optimal Q 3 will be ordered by the buyer can be obtained from equation 3 and equation 3 W 3 = C (33) The optimal quantity will be ordered by the buyer when the supplier s wholesale price is equal to his cost of manufacturing. A similar result has been obtained for the profit sharing by [Veen & Venu, 000(b)] in a setting in which demand is stochastic. This would be acceptable to the supplier as he is also receiving a share of profit from the buyer. The supplier only makes sure that the percentage of the buyer s profit that he is entitled to will yield a higher profit than the solitaire scenario. Venugopal 7 of 47

19 The profits for the players at this value of W 3 can be obtained as follows, for the buyer for the supplier ( α βc) B 3( δ ) = [ ]( δ ) 4β (34) ( α βc) S3 ( δ ) = [ ] δ 4β (35) for the total supply chain ( α βc) T3 = : = 4K 4β (36) It is clear from the above Equations that the profit for the supplier and the buyer is dependent on δ, hence δ can be used to divide the profits between them in the required proportions. As can be seen from equation 36 that the profit sharing mechanism will optimize the supply chain. Also here achieving win-win is possible, as can be seen from the following theorem. Theorem : For any price W3 W under the Solitaire scenario, there is a value for δ under the profit sharing scenario such that both the supplier and the buyer have a higher profit than the realized profit under the solitaire scenario. i.e. S δ ) S ( ) 3 ( W and B δ ) B ( ) 3( W Proof: Win-Win is said to be achieved if profit with profit sharing is higher than the profit in a solitaire scenario. The profit for the supplier with profit sharing should be greater than his profit in a solitaire scenario. S δ ) S ( ) if 3 ( W δ 4K S ) (37) ( W Venugopal 8 of 47

20 Lower bound for δ (defined as δ - ) can be obtained from equation 37 as S( W) δ =: δ (38) 4K The profit for the buyer with profit sharing should be greater than his profit in a solitaire scenario. B δ ) B ( ) if 3( W ( δ )4K B ) (39) ( W Upper bound for δ (defined as δ + ) can be obtained from equation 39 as δ 4 K B ( W ) 4K + =: δ (40) In order to complete the proof, it remains to be shown that the upper bound is always greater than the lower bound as follows, 4K [ B W S W δ + ( ) + ( )] δ = (4) 4K This expression 4, will always be greater than zero as the difference between the total supply chain profit with centralization will be greater than that of the solitaire. And when the profits under a centralized scenario is equal to the profits under the solitaire scenario the above equation will be equal to zero. When that happens both the upper and the lower bound of the profits sharing mechanisms will be equal. Coming back to our numerical example, the supply chain profit can be obtained as 00. the lower bound and the upper bounds for δ can be obtained as 0.5 and 0.75 respectively. The difference between the upper and the lower bound is Revenue sharing mechanism (Scenario 4) In the revenue sharing mechanism the transactions between the supplier and buyer are governed by the supplier receiving a share of the buyer s revenues. The revenue sharing Venugopal 9 of 47

21 mechanism can be identified by two parameters, namely, wholesale price W and a percentage of the buyers profit γ (0<γ<), that goes to the supplier. Many supply contracts in vertically-separated industries include revenue sharing. One recent example is from video-cassette rental industry, see [Cachon & Lariviere, 005, Dana & Spier, 00]. The profit functions are, ( α Q4 ) B4 ( W4, Q4, γ ) = ( γ )[ ] Q4 W4Q4 (4) β for the buyer ( α Q4 ) S 4 ( W4, Q4, γ ) = W4Q4 + γ [ ] Q4 CQ4 (43) β for the supplier for the total supply chain ( α Q4 ) T4 Q4 ) = [[ ] Q4 CQ β ( 4 ] (44) The quantity Q 4 which will optimize supply chain profit can be obtained from 44 as, Q 4 ( α βc) = (45) The value optimal value of Q 4 for the buyer can be obtained from 4 as α( γ ) βw4 Q 4 = (46) ( γ ) The value of W 4 at which optimal Q will be ordered by the buyer can be obtained from equation 45 and equation 46, as W = ( )C (47) 4 γ The optimal quantity will be ordered by the buyer when the supplier sets his wholesale price at a level lower than his cost of manufacturing. A similar result has been obtained Venugopal 0 of 47

22 for the revenue sharing mechanism by [Veen & Venu, 000(b)] in a setting in which demand is stochastic. This would be acceptable to the supplier as he is also receiving a share of buyer s revenues. The supplier only makes sure that the percentage of the buyer s revenues that he is entitled to will yield a higher profit than the solitaire scenario. The profits for the players at this value of W 4 can be obtained as follows, for the buyer for the supplier for the total supply chain ( α βc) B 4 ( γ ) = [ ]( γ ) 4β (48) ( α βc) S 4 ( γ ) = [ ] γ 4β (49) ( α βc) T3 ( γ ) = : = 4K 4β (50) It is clear from the above Equations that the profit for the supplier and the buyer is dependent on γ, hence γ can be used to divide the profits between them in the required proportions. As can be seen from equation 50, the revenue sharing mechanism will optimize the supply chain. Also here achieving win-win is possible, as can be seen from the following theorem. Theorem 3: For any price W4 W under the solitaire scenario, there is a value for γ under the revenue sharing scenario such that both the supplier and the buyer have a higher profit than the realized profit under the solitaire scenario. i.e. S γ ) S ( ) 4 ( W and B γ ) B ( ) 4 ( W Proof: Win-Win is said to be achieved if profit with revenue sharing is higher than the profit in a solitaire scenario. Venugopal of 47

23 The profit for the supplier with revenue sharing should be greater than his profit in a solitaire scenario. S γ ) S ( ) if 4 ( W γ 4K S ) (5) ( W Lower bound for γ (defined as γ - ) can be obtained from equation 5 as S ( W ) γ =: γ (5) 4K The profit for the buyer with revenue sharing should be greater than his profit in a solitaire scenario. B δ ) B ( ) if 4 ( W ( γ )4K B ) (53) ( W Upper bound for γ (defined as γ + ) can be obtained as γ ( 4K B ( W )) 4K + =: γ (54) In order to complete the proof, it remains to be shown that the upper bound is always greater than the lower bound as follows, + γ γ = 4 + K [ B ( W ) 4K S ( W ) (55) This expression 55, will always be greater than zero as the difference between the total supply chain profit with centralization will be greater than that the profit with the solitaire scenario. And when the profits under a centralized scenario is equal to the profits under the solitaire scenario the above equation will be equal to zero. When that happens both the upper and the lower bound of the profits sharing mechanisms will be equal. Venugopal of 47

24 Coming back to our numerical example, the supply chain profit can be obtained as 00. the lower bound and the upper bounds for γ can be obtained as 0.5 and 0.75 respectively. The difference between the upper and the lower bound is 0.5. Interestingly, it can be observed from equations 38 and 5 that the lower bound values for the profit sharing and the revenue sharing mechanism are the same. And from equation 40 and 54 the upper bound values for both the mechanisms are the same as well. The above analysis helps us to conclude that for the considered setting the performance of both the mechanism is equivalent as they provide the same range of parametric values for the win-win and will yield same profits for a considered revenue or profit sharing percentage. Figure 4 depicts the profit profiles for the players for different values of the parameters of the profit sharing and revenue sharing mechanisms. Profit Parameter values Buyers profit Suppliers profit Supply chain profit Figure 4: Profit profiles for the buyer, supplier and the supply chain for different parameter values of the profit and the revenue sharing mechanisms Venugopal 3 of 47

25 5.3 Quantity discount (Scenario 5) Quantity discount is one of the most commonly used contract mechanisms in practice. This mechanism has been studied extensively in literature. [Jeuland and Shugan, 983] were the first to show explicitly that, for a two member channel facing deterministic demand, the quantity discount scheme can coordinate the channel and also share the efficiency gains. [Lee and Rosenblatt, 985] study a linear discount schedule in which both the parties can benefit whereas in [Manohan, 984] and [Rosenblatt & Lee, 986] the supplier accrues all the benefits. [Weng, 995] extends Jeuland & Shugan s results by specifically incorporating mechanisms that determine the relationship between operating decisions (such as order quantities and selling prices) and profits (revenue less inventory and set-up costs) calculated using the typical EOQ framework for a buyer and supplier and deriving policies between them that can coordinate the channels activities. With the quantity discount mechanism, the supplier fixes a base price λ and charges an additional price μ which is a decreasing function in Q. At its simplest form the price at which the seller sells the product to the buyer is dependent upon the order quantity is given as, μ W5 ( Q) = λ + (56) Q The price function in Equation 56, yields the following profits, for the supplier, S ( λ, μ, Q ) ( λ C Q + μ (57) 5 5 = ) 5 for the buyer, and, B ( λ, μ, Q ) ( P λ Q μ (58) 5 5 = ) 5 for the supply chain. T ( = Q (59) 5 Q5 ) ( P C) 5 Venugopal 4 of 47

26 Again the supply chain profit is independent of the parameters of quantity discounts and is same as the profit in Equation 5. The quantity Q 5 which will optimize supply chain profit can be obtained from (59) as, Q 5 α βc = ( ) (60) The optimal value of Q 5 for the buyer can be obtained from equation 58 as α βλ Q 5 = ( ) (6) The value of λ which will optimize supply chain profit can be obtained from equations (60) and (6) as, λ = C (6) Substituting this value of λ in (57)-(59) yields the following profits, for the supply chain, ( α βc) T5 = [ ] = 4K 4β (63) for the supplier and, S μ ) = μ (64) 5 ( for the buyer. ( α βc) B 5 ( μ) = μ (65) 4β The numerical example yields the following profits for the supplier, buyer and supply chain μ, ( 00 μ) and 00. Venugopal 5 of 47

27 The conclusion is that for the chosen value of λ, the supply chain is optimized. The question then remains whether also win-win can be achieved when compared to the solitaire scenario. The next theorem shows that this is the case. Theorem 4: For any price W5 W under the solitaire scenario, there are values λ and μ under the quantity discount scenario, such that both the supplier and the buyer have a higher profit than the realized profit under the solitaire scenario, i.e. S μ ) S ( ) 5 ( W and B5 ( μ) B ( W ) Proof: Win-Win is said to be achieved if profit with quantity discount is higher than the profit in a solitaire scenario. The profit for the supplier with quantity discount should be greater than his profit in a solitaire scenario. S μ) S ( ) implies 5 ( W μ ( α βc) ( 8β ) = : μ (66) The profit for the buyer with quantity discount should be greater than his profit in a solitaire scenario. B 5 ( W μ ) B ( ) implies μ 3 ( α 6 β C β ) = : μ + (67) The theorem will be valid if we can prove that the difference between the upper bound and lower bound obtained in above (66) and (67) will always be positive. μ μ ( α βc) = 6 + β 0 (68) Venugopal 6 of 47

28 Equation 68 will always be positive. For our numerical example the value of μ + (the upper bound for μ) will be equal to 50, and that for μ - (the lower bound for μ) will be equal to 00. The difference between the bounds is 50. It is clear from the mathematical analysis and the numerical example that the quantity discount mechanism can coordinate the channel as well as distribute profits such that both the parties can benefit. We continue our discussion with the license fee mechanism. 5.4 License fee mechanism (Scenario 6) The Two part tariff or License fee contract is governed by the parameters (W,L) where W is the wholesale price and L is a license fee. This type of contract is common in franchising situations. For example, the transactions between McDonalds and its franchisers are governed through a franchise fee L and a wholesale price W that would be quite close to marginal cost. [Moorthy, 987] and [Jeuland & shugan, 983] have tested the ability of the License fee mechanisms to enable coordination. [Veen & Venu, 000(b)] have derived a range within which win-win can be obtained with the license fee mechanism. License fee with L and W has exactly the same effect as Quantity discounts with μ and γ where μ L and λ W. The range for L will be same as that for μ in the quantity discount model. Theorem 5: For any price W6 W under solitaire, there are values for L under the License fee mechanism such that both the supplier and the buyer have a higher profit than the realized profit under the solitaire scenario. The proof for theorem 5 follows the same lines as the one for the quantity discounts. Venugopal 7 of 47

29 Figure 5 depicts the profit profiles for the players with the parameters of the quantity 4K 3K K profit K Parameters discount and the license fee mechanism Buyers profit Suppliers profit Supply chain profit Figure 5: Profit profiles for the buyer, supplier and the supply chain for different parameter values of the quantity discount and license fee mechanism 5.5 Mail-in-rebate contract (Scenario 7) With the mail-in rebate mechanism, the supplier is obliged to credit the end customer with an amount equal to the value of the rebate coupon. All the customer has to do is to purchase the product at the buyer and mail the rebate coupon back to the supplier. In this case the supplier directly influences customer demand by providing an upside incentive to the buyer to increase its order quantity. By introducing the rebate, the effective price paid by the customers is reduced and hence the buyer faces a higher (μ L) demand level. As the buyer is also aware of the rebate provided to the end consumer, he would increase his order quantity. A brief description of the mail-in-rebate mechanism has been provided by Simchi-levi et.al ( nd edition). Gerstner and Hess(995) have applied a targeted pull pricing strategy, in which the manufacturer offers discounts directly to price-conscious consumers, expecting them to ask the retailer for the product. They apply this pricing strategy in a setting in which two different segments of customers Venugopal 8 of 47

30 make up the market: high willingness-to-pay consumers and low willingness-to-pay consumers. Their study lead to a conclusion that channel coordination is achieved when the manufacturer carefully designs the pull so the discount is accessible to the target price-conscious consumers but hard to get for the non-targeted segment. Mail-in-rebate mechanisms designed in this paper are similar to the targeted pull pricing strategy used by Gerstner and Hess. However, the setting in which we analyse the mail-in-rebates is different from that of Gerstner and Hess. With the mail in rebate contract mechanism the effective price that the customer pays will be equal to (P 7 -R), where P 7 is the price charged by the retailer and R is the rebate provided by the supplier. The price charged by the retailer in the solitaire scenario can be obtained from equation 7 as, 3α + βc P = 4β The effective price paid by the end consumer when the supplier provides a rebate R is 3α + βc P7 = R (69) 4β Supplier would be interested in providing a rebate value which would maximise his profit. The performance of the mail-in-rebate mechanism is brought out as the proof of theorem 6. Theorem 6: At the rebate value which is optimal for the supplier R M, though the supply chain profit is not optimal, both the buyer and the supplier has higher profit than the realized profit under the solitaire scenario. Venugopal 9 of 47

31 Proof: A rebate value which will optimise the suppliers profit can be obtained by optimising the suppliers profit function given in equation 76, S = W RM C)( α β ) (70) 7 ( 7 P7 The optimal rebate for the supplier obtained from the above equation 70 is, R M = [ α βc] (7) 8β When the supplier provides the rebate which is optimal for him, the profit for the supplier, buyer and the supply chain can be obtained as follows, 9 S 7 = ( α βc) (7) 64β 6 B7 = ( α βc) (73) 64β 5 T7 = ( α βc) (74) 64β The effect of the profits of the players and the supply chain when the supplier chooses the rebate value which maximises his profit can be analysed as, B 7 ( α βc) B = (75) 3β Venugopal 30 of 47

32 The positive value for the above equation suggests that the buyer obtains an additional profit. The difference in profits for the supplier can be obtained as, S 7 ( α βc) S = (76) 64β It is important to note that even when the supplier optimises his profit the buyers additional profit is twice that of the supplier. One intuitive explanation could be that the buyer is benefiting with the mail-in-rebate contract without contributing anything from his share. The additional total profit for the supply chain compared to the solitaire scenario, T 7 3( α βc) T = (77) 64β The value by which the supply chain profit is away from the optimal supply chain profit can be obtained as T ( α βc) T8 = (78) 64β In the next scenario we look at the optimal mail-in-rebate value from the supply chain perspective. 5.6 Optimal rebate value for the supply chain (Scenario 8) The profit function for the supply chain with price P 8 can be obtained as T 8 ( 8 P8 = P C)( α β ) (79) The optimal rebate for the supply chain can be obtained from equation 79 as Venugopal 3 of 47

33 R opt = [ α βc] (80) 4β The selling price and the order quantity at the optimal rebate value can be obtained from equation 69 and equation as, P 8 ( α + βc) = (8) β Q 8 ( α βc) = (8) β Theorem 7: At the optimal rebate value for the supply chain R opt the suppliers profit is same as that the profit he has realised during the solitaire scenario. The proof for the above theorem can be obtained by using the R opt values in the profit functions of the players. optimal rebate in equation 80 allocates the following profits for the supplier and the retailer, S 8 ( α βc) = = K (83) 8β B 8 ( α βc) = = K (84) 8β It is clear from equation 83 that at the optimal rebate for the supply chain the suppliers profit is same as that he has obtained during the solitaire scenario. All the additional profit goes to the retailer. Another interesting observation which can be made from the above analysis is that the suppliers profit is same as that of the buyer. Supplier doesn t have any monetary benefits by providing the rebate which is optimal for the supply chain, but what can make the optimal rebate attractive to the supplier is the possibility to realise a higher market share and a better brand image for his products. Venugopal 3 of 47

34 Reviewing the example discussed at the end of Section, the wholesale price set for the buyer can be obtained as 40. The buyer will fix the final selling price at 45. If the supplier provides a rebate R to the end consumer, the profit functions for the buyer can be obtained as B R = R, For the supplier it is = R R, For the supply chain it is T R = R R and the supply chain at different rebate values. S R. Figure 6 indicates the profit profiles for the players 4K 3K T K Profits S K B R M Rebate value Ropt Figure 6: Profits for Supplier (S), Buyer (B) and total supply chain (T) with mail-in- rebate mechanism The analysis with the mail-in-rebate mechanism gives us the following insights: ) At the rebate value which is optimal for the supply chain the suppliers profit is same as that he receives in a solitaire scenario. All the additional profit goes to the buyer. Interestingly, at the optimal rebate value the buyers profit is same as that of the suppliers. ) At the rebate value which is optimal for the supply chain the entire monetary benefits are enjoyed by the buyers and the end consumer. Venugopal 33 of 47

35 3) If the supplier decides to stick to the rebate value which is optimal for the supply chain, it helps him to achieve a greater market share and helps to build better brand image. It would automatically motivate the buyer to promote suppliers product more than that of his competitors. All the above reasons can help him to obtained profits which are sustainable for a greater period of time. 4) If the supplier decides to provide a rebate value which is optimal for him, both he and the buyer achieves win-win, but the supply chain profit is not optimised. Any rebate value between 0 and the optimal rebate provides win-win. 5) Any rebate value greater than the optimal rebate has detrimental effect on the profit for the supplier and the supply chain. In our next analysis we analyse the effectiveness of the wholesale price discount provided to the retailer. 5.7 Wholesale price discount mechanism (Scenario 9) Instead of providing the rebate to the end consumer as discussed in the previous section the supplier can choose to provide a wholesale price discount to the buyer. With this mechanism the discount value is independent of the number of units purchased, unlike the quantity discount mechanisms in which the discount value increases with buyers order quantity. The present section compares the effectiveness of the mail-in-rebate with an equivalent wholesale price discount. We analyse the case for which the supplier provides wholesale price discount equal to the rebate value which is optimal for him. This value can be obtained from equation 7 as d w = [ α βc] (85) 8β The effective wholesale price can be obtained as follows, Venugopal 34 of 47

36 W9 = W d w = [3α + 5βC] (86) 8β When the supplier sets the wholesale price as given in equation 86, the buyer sets the following selling price P = [α + 5β ] (87) 6β 9 C The total number of units demanded at the above selling price can be obtained as: Q 5 = [ α β ] (88) 6 9 C The profits for the supplier; buyer and the supply chain can be obtained as; 5 S9 = ( α βc) (89) 8β 5 B9 = ( α βc) (90) 56β 55 T9 = ( α βc) (9) 56β Theorem 8: For the wholesale price discount equivalent to the suppliers optimal rebate value, the buyer and the supply chain makes a greater profit and the supplier makes a lower profit compared to the solitaire scenario. Proof: Comparison of the profits made by the players with the wholesale price discount mechanism and the solitaire scenario gives the following result, Venugopal 35 of 47

37 B 9 9( α βc) B = (9) 56β The positive value for the above equation suggests that the retailer obtains an additional profit. The difference in profits for the supplier can be obtained as, S 9 ( α βc) S = (93) 8β It is important to note that supplier receives a lower profit than the solitaire scenario. The wholesale price discount is not effective from the suppliers point of view. The reason for this is that the additional contribution due to the increase in the demand is lower than the loss due to the reduction of the wholesale price. The additional total profit for the supply chain is, T 9 7( α βc) T = (94) 56β We can conclude from the above analysis that with the wholesale price discount mechanism the buyer and the supply chain benefits in comparison to the solitaire scenario, but the supplier looses. 6. Comparing the performance of the mail-in-rebate and the wholesale price discount Mechanism. Next we compare the profits for the players and the supply chain with the wholesale price discount and the mail-in-rebate mechanism. The difference in the profit for the buyer can be obtained as, Venugopal 36 of 47

38 B 9 ( α βc) B7 = (95) 56β The positive value for the above equation suggests that the retailer obtains an additional profit with the wholesale price discount when compared to the mail-in-rebate mechanism. The graph below indicates the profit profile for the buyer for the two different mechanisms (for our numerical example). 4K 3K K Profit K Mail-in/Linear(rebate) 0 Mail-in-Rebate wholesale price discount Figure 7: Comparing the performance of the buyer with mail-in-rebate and wholesale price discount mechanism It is very clear from Figure 7 that the buyer can make greater profit with the wholesale price discount than what he can make with the mail-in-rebate. The difference in profits for the supplier can be obtained as, Venugopal 37 of 47

39 S 9 3( α βc) S 7 = (96) 8β A negative value in the equation 96 indicates that the supplier profit is higher with the mail in rebate mechanism. The reason for this is that with the mail-in-rebate mechanism all the discount is passed on to the end consumer which enhances the end consumer demand. The increase in the consumer demand with the mail-in-rebate mechanism will be higher than that of the wholesale price discount mechanism. The graph below indicates the profit profile of the supplier with different mechanism(for our numerical example). K Profit Mail-in-rebate/Wholesale price discount 0 Mail-in-Rebate wholesale price discount Figure 8: Comparing the performance of the supplier with mail-in-rebate and wholesale price discount mechanism It is clear from the Figure 8 that the supplier gets a greater profit with the mail-in-rebate as compared to the linear rebate. The additional total profit for the supply chain is, Venugopal 38 of 47

40 T 9 5( α βc) T7 = (97) 56β The negative value in equation 97 indicates that the supply chain profit is higher with the mail-in-rebate mechanism. The reason is that the increase in profit due to additional demand is higher than the decrease due to reducing the effective final selling price. The graph below indicates the profit profile for the supply chain with different mechanism (for our numerical example). 4K 3K K Profit K Mail-in-Rebate/wholesale price discount 0 Mail-in-Rebate wholesale price discount Figure 9: Comparing the performance of the supply chain with mail-in and wholesale price discount mechanism It is clear from Figure 9 that from a supply chain perspective the mail-in-rebate performs better than the linear rebate up to a certain threshold value (Optimal mail-in rebate value from the supply chain perspective) beyond which the linear rebate performs better. The reason for the above behaviour can be explained as the gain in profit for the supply chain Venugopal 39 of 47